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One-semester course notes on solid mechanics by Krzysztof Wilmanski of the University of Zielona Gora, Poland, for the ROSE School in Pavia. They cover vectors and tensors, kinematics, balance of mass and momentum, thermodynamics, linear elasticity (Green functions, waves, plane strain), thermo- and poroelasticity, viscoelasticity, plasticity and dislocations. It is a downloaded book by another author, not Phil's own work.

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Fundamentals of Solid Mechanics Course at the European School for Advanced Studies in Earthq uake Risk Reduction (ROSE School), Pavia, Italy Krzysztof Wilmanski University of Zielona Gora, Poland http://www.mech-wilmanski.de Contents Introduction, historical sketch 5 1 Modicum of vectors and tensors 9 1.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Geometry and kinematics of continua 21 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 2.2 Reference conﬁguration and Lagrangian description . . . . . . . . . . . . . 22 2.3 Displacement, velocity, Eulerian description . . . . . . . . . . . . . . . . . 30 2.4 Inﬁnitesimal strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.5 Compatibility conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3 Balance of mass and momentum 43 3.1 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Conservation of momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.1 Lagrangian description . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.2 Eulerian description . . . . . . . . . . . . . . . . . . . . . . . . . . 4 9 3.2.3 Moment of momentum . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.4 Stress analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4 Thermodynamics of solids 61 4.1 Energy conservation law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 Second law of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 64 5 Elastic materials 69 5.1 Non-linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.2 Linear elasticity, isotropic and anisotropic material s . . . . . . . . . . . . 71 5.2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2.2 Navier-Cauchy equations, Green functions, displace ment potentials 77 5.2.3 Beltrami-Michell equations . . . . . . . . . . . . . . . . . . . . . . 89 5.2.4 Plane strain and plane stress . . . . . . . . . . . . . . . . . . . . . 90 5.2.5 Waves in linear elastic materials . . . . . . . . . . . . . . . . . . . 95 5.2.6 Principle of virtual work . . . . . . . . . . . . . . . . . . . . . . . . 104 3 4 CONTENTS 5.3 Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.4 Poroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6 Viscoelastic materials 115 6.1 Viscoelastic ﬂuids and solids . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 Rheological models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.3 Three-dimensional viscoelastic model . . . . . . . . . . . . . . . . . . . . . 123 6.4 Diﬀerential constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 128 6.5 Steady state processes and elastic-viscoelastic corre spondence principle . . 129 7 Plasticity 135 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 35 7.2 Plasticity of ductile materials . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.3 Plasticity of soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.4 Viscoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 8 Dislocations 157 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 57 8.2 Continuum with dislocations . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.3 On plasticity of metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 8.4 Dislocations in geophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 9 Appendix: Green functions for isotropic elastic material s 169 9.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 9.2 Statics of isotropic elastic materials . . . . . . . . . . . . . . . . . . . . . . 170 9.3 Dynamic Green function for isotropic elastic materials . . . . . . . . . . . 173 Introduction, historical sketch Remnants of old civil engineering constructions prove that already in ancient times the human being was able to build very complex structures. Th e old masters had both experience and intuition with which they were able to create the most daring structures. However, they did not use in their work any theoretical model s as we understand them today. Most likely, it was ﬁrst in XVI century that notions ne cessary for such a modeling were invented. Leonardo da Vinci (1452-1519) sketched in his notebooks a po ssible test of the tensile strength of a wire. Arc of Ctesiphon (the winter palace of Sassanids, Iraq) cons tructed in 129 B.C. (left panel) and the dome of the Florence cathedral desi gned and built by Filippo Brunelleschi in 1425 (right panel) — two examples of the early ingenious constructions. ’In his book on mechanics1Galileo (Galileo Galilei, 1564-1642) also dealt with the strength of materials, founding that branch of science as we ll. He was the ﬁrst to show 1Dialogo di Galileo Galilei Linceo Matematico Sopraordinar io dello Studio di Pisa, in Florenza, Per Gio: Batista Landini MDCXXXII 5 6 Introduction that if a structure increased in all dimensions equally it wo uld grow weaker — at least he was the ﬁrst to explain the theoretical basis for this. This i s what is known as the square- cube law. The volume increases as the cube of linear dimensio ns but the strength only as the square. For that reason larger animals require propor tionally sturdier supports than small ones. A deer expanded to the size of an elephant and kept in exact proportion would collapse, its legs would have to be thickened out of pro portion for proper support’2. Below we mention only a few scientists whose contributions a re particularly important for the development of continuum mechanics at its early stag e. Many further details can be found in the article of J. R. Rice3. Robert Hooke is considered to be the founder of linear elasti city. He discovered in 1660 (published in 1678), the observation that the displace ment under a load was for many materials proportional to force. However, he was not aw are yet of the necessity of terms of stress and strain. A similar discovery was made by E. Mariotte (France, 1680). He described as well the explanation for the resistance of be ams to transverse loadings. He considered the existence of bending moments caused by tra nsverse loadings and devel- oping extensional and compressional deformations, respec tively, in material ﬁbers along upper and lower portions of beams. The ﬁrst to introduce the r elation between stresses and strains was the Swiss mathematician and mechanician Jac ob Bernoulli (1654-1705). In his last paper of 1705 he indicated that the proper way of de scribing deformation was to give force per unit area, i.e. stress, as a function of the e longation per unit length, i.e. strain, of a material ﬁber under tension. Numerous most important contributions were made by the Swiss mathematician and mechanician Leonha rd Euler (1707-1783), who was taught mathematics by Jacob’ brother Johann Bernoul li (1667-1748). Among many other ideas he proposed a linear relation between stres sσand strainεin the form σ=Eε(1727). The coeﬃcient Eis now usually called Young’s modulus after English naturalist Thomas Young who developed a similar idea in 1807 . Since the proposition of Jacob Bernoulli the notion that the re is an internal tension acting across surfaces in a deformed solid has been commonly accepted. It was used, for example, by German mathematician and physicist Gottfried W ilhelm Leibniz in 1684. Euler introduced the idea that at a given section along the le ngth of a beam there were internal tensions amounting to a net force and a net bending m oment. Euler introduced the idea of compressive normal stress as the pressure in a ﬂui d in 1752. The French engineer and physicist Charles-Augustin Coulom b (1736-1806) was ap- parently the ﬁrst to relate the theory of a beam as a bent elast ic line to stress and strain in an actual beam. He discovered the famous expression σ=My/I for the stress due to the pure bending of a homogeneous linear elastic beam; here Mis the bending moment, yis the distance of a point from an axis that passes through the section centroid, parallel to the axis of bending, and Iis the integral of y2over the section area. The notion of shear stress was introduced by French mathematician Parent in 1713. It was the work of Coulomb in 1773 to develop extensively this idea in connec tion with beams and with the stressing and failure of soil. He also studied frictiona l slip in 1779. The most important and extensive contributions to the devel opment of continuum 2quotation from I. Asimov’s Biographical Encyclopedia of Sc ience and Technology, Doubleday, 1964. 3J. R. RIiscIcscIesc; Mechanics of Solids , published as a section of the article on Mechanics in the 199 3 printing of the 15th edition of Encyclopaedia Britannica (v olume 23, pages 734 - 747 and 773), 1993. Introduction 7 mechanics stem from the great French mathematician Augusti n Louis Cauchy (1789- 1857), originally educated as an engineer. ’In 1822 he forma lized the stress concept in the context of a general three-dimensional theory, showed i ts properties as consisting of a 3 by 3 symmetric array of numbers that transform as a tensor, derived the equations of motion for a continuum in terms of the components of stress , and gave the speciﬁc development of the theory of linear elastic response for iso tropic solids. As part of this work, Cauchy also introduced the equations which express th e six components of strain, three extensional and three shear, in terms of derivatives o f displacements for the case when all those derivatives are much smaller than unity; simi lar expressions had been given earlier by Euler in expressing rates of straining in te rms of the derivatives of the velocity ﬁeld in a ﬂuid’4. In this book we present a modern version of those models in whi ch not only linear elastic but also viscoelastic and plastic materials are inc luded. The full nonlinear theory is not included but some of its basic notions such as Lagrangi an and Eulerian descriptions are indicated. In the presentation we avoid many mathematical details in or der to be understandable for less mathematically skillful engineers and natural sci entists. Those who would like to clean up some mathematical points we refer to the work of M. Gurtin [4]. Nonlinear problems are presented in many modern monographs. We quote h ere only four examples [2], [11], [21], [22]. Linear models of elasticity, viscoelasticity, plasticity , viscoplasticity and dislocations are presented. To keep the volume of the notes related to the e xtent of the one-semester course we have not included such important subjects as coupl ing to thermal eﬀects, a theory of brittle materials (damage), some topics, such as critical states of soils or earthquake mechanics, are only indicated. The book almost d oes not contain exercises which we oﬀer to students separately. Some examples, auxiliary remarks and reminders which are no t necessary for the systematic presentation of the material are conﬁned by the s igns⋆...♣. References are made in two ways. I have selected a number of books and monogra phs — 24 to be exact — which were used extensively in preparing these notes and wh ich may serve as a help in homework of students. Many references to particular issu es and, especially, historical notes, are made in the form of footnotes. I would like to apolo gy to these readers who do not speak Polish for some references in this language. I di d it only occasionally when the English version is not available and when I wanted to pay a tribute to my collegues and masters for teaching me many years ago the subject of cont inuum mechanics. 4J. R. RIiscIcscIesc; Mechanics of Solids , published as a section of the article on Mechanics in the 199 3 printing of the 15th edition of Encyclopaedia Britannica (v olume 23, pages 734 - 747 and 773), 1993. 8 Introduction Chapter 1 Modicum of vectors and tensors 1.1 Algebra The most important notions of mechanics such as positions of points, velocities, ac- celerations, forces are vectors and many other important ob jects such as deformations, stresses, elasticity parameters form tensors. Therefore w e begin our presentation with a brief overview of a vector calculus in Euclidean spaces. We limit the presentation to three-dimensional spaces as continuum mechanics does no t require any more general approach. Vectors are objects characterized by the length and the dire ction. A vector space V is deﬁned by a set of axioms describing three basic operation s on vectors belonging to this space: a multiplication by a real number, an addition, a nd a scalar product. The ﬁrst operation ∀a∈V∀α∈ℜb=αa∈V (1.1) deﬁnes, for any vector a, a new vector b=αawhose direction is the same (it has the opposite orientation for α <0) as this of the vector aand the length is αtimes larger (or smaller for|α|<1) than this ofa. The second operation ∀a,b∈Va+b=c∈V (1.2) deﬁnes for two vectors a,ba new vectorcwhich is constructed according to the rule of 9 10 Modicum of vectors and tensors triangle shown in Fig.1.1. Fig. 1.1: Addition of vectors The third operation ∀a,bα=a·b, α∈ℜ, (1.3) is a scalar product which for any two vectors a,bdeﬁnes a real number. For any vector a a·a=|a|2, (1.4) it deﬁnes its length |a|, while for two vectors a,bit speciﬁes the angle ϕ=(a,b)between them a·b=|a||b|cosϕ. (1.5) These three operations satisfy a set of axioms, such as assoc iativity, commutativity, etc. which we do not specify here as in the analytical description these are replaced by similar axioms for operations on real numbers. One should also mention that, by means of the above operation s one can introduce the vector0whose length is 0and direction is arbitrary as well as a number of linearly independent vectors which deﬁnes the dimension of the vecto r spaceV. We say that the spaceVis three-dimensional if for any three diﬀerent non-zero vec torsa1,a2,a3of diﬀerent direction the relation α1a1+α2a2+α3a3=0, (1.6) is satisﬁed only if the numbers α1,α2,α3are all equal to zero. This property allows to replace the above presented geometrical approach to vector calculus by an analytical ap- proach. This was an ingenious idea of René Descartes (1596-1 650). For Euclidean spaces which we use in these notes we select in the vector space Vthree linearly independent vectorsei,i=1,2,3,which satisfy the following condition ei·ej=δij, i,j=1,2,3, (1.7) whereδijis the so-called Kronecker delta. It is equal to one for i=jand zero otherwise. Fori=jwe have ei·ei=1, (1.8) which means that each vector eihas the unit length. Simultaneously, for i/negationslash=j, ei·ej=0⇒ϕ=π 2, (1.9) 1.1 Algebra 11 whereϕis the angle between eiandej. It means that these vectors are perpendicular. We call such a set of three vectors {e1,e2,e3}the base (or basis) vectors of the space V. Obviously, any linear combination a=a1e1+a2e2+a3e3∈V, (1.10) is a non-zero vector for ainot simultaneously equal to zero. It is easy to see that any vector from the space Vcan be written in this form. If we choose such a vector athen a·ei=3summationdisplay j=1(ajej)·ei=3summationdisplay j=1ajδij=ai. (1.11) The numbers aiare called coordinates of the vector a. Clearly, they may be written in the form ai=|a||ei|cos((a,ei))=|a|cos((a,ei)), (1.12) i.e. geometrically it is the length (with an appropriate sig n depending on the choice of ei!) of projection of the vector aon the direction of the unit vector ei.Certainly,(a,ei) denotes the angle between the vectors aandei. The above rule of representation of an arbitrary vector allo ws to write the operations in the vector space in the following manner b=αa=(αai)ei,b=biei, bi=αai, c=a+b=(ai+bi)ei,c=ciei, ci=ai+bi, (1.13) α=a·b=aibi, α=aibi, where we have introduced the so-called Einstein convention that a repetition of an index in the product means the summation over all values of this ind ex. For instance, aibi≡3summationdisplay i=1aibi. (1.14) Relations (1.13) allow for the replacement of geometrical r ules of vector calculus by analytical rules for numbers denoting coordinates of the ve ctor. A vector ain the three- dimensional space is in this sense equivalent to the matrix (a1,a2,a3)Tprovided we choose a speciﬁc set of base vectors {e1,e2,e3}. However, in contrast to matrices which are collections of numbers, vectors satisfy certain rules o f transformation between ma- trices speciﬁed by the rotations of base vectors. It means th at two diﬀerent matrices (a1,a2,a3)Tand(a1′,a2′,a3′)Tmay deﬁne the same vector if the coordinates aiand ai′are connected by a certain rule of transformation. We can spe cify this rule immedi- ately if we consider a rotation of the base vectors {e1,e2,e3}to the new base vectors {e1′,e2′,e3′}. We have ei′=Ai′iei, Ai′i=ei′·ei=cos((ei′,ei)), ei=Aii′ei′, Aii′=ei·ei′=cos((ei,ei′)), (1.15) ei′=Ai′iAij′ej′⇒Ai′iAij′=δi′j′. 12 Modicum of vectors and tensors Hence the matrix (Aii′)is inverse to the matrix (Ai′i)Obviously, these matrices of transformation Ai′iandAii′are square and formed of sine and cosine functions of angles between base vectors {e1,e2,e3}and{e1′,e2′,e3′}. ⋆Example: Further we refer frequently to the rotation in the case of a tw o-dimensional space. In a three-dimensional space a rotation is deﬁned by t hree angles (e.g. Euler an- gles of crystallography). In the two-dimensional case we ne ed one angle, say φ, which we assume to be positive in the anticlockwise direction (see: F ig. 1.2). Fig. 1.2: Rotation of the basis on the plane perpendicular to e3 Then we have A11′=e1·e1′=cosφ, A12′=e1·e2′=cosparenleftBigπ 2+φparenrightBig =−sinφ, A21′=e2·e1′=cosparenleftBigπ 2−φparenrightBig =sinφ, A22′=e2·e2′=cosφ, (1.16) A33′=e3e3′=1, and the remaining components are zero. Similarly, we can ﬁnd the components of the matrixAi′i. Then we have (Aii′)= cosφ−sinφ0 sinφcosφ0 0 0 1,(A i′i)=cosφsinφ0 −sinφcosφ0 0 0 1,(A ii′)T=(Ai′i). (1.17) The last property, which holds also in a three-dimensional c ase, together with (1.15) means that the matrix (Ai′i)is orthogonal. This is the general property of all matrices of rotation. We shall return to this property in the discussi on of the deformation of continua. As an example let us consider the two-dimensional transform ation of the vector a=aiei,(ai)=(1,2,3), (1.18) 1.1 Algebra 13 withφ=π/6. We have a=ai′ei′, ai′=Aii′ai⇒ (1.19) ⇒(ai′)=(cosφ+2sinφ,−sinφ+2cosφ,3)= =parenleftBig 1+√ 3/2,√ 3−1/2,3parenrightBig . ♣ Bearing the above considerations in mind we can write for an a rbitrary vector a a=aiei=ai′ei′⇒ai′=(aiei)·ei′=Aii′ai, (1.20) In mathematics the rule of transformation (1.20) is conside red to be the formal deﬁ- nition of the vector. We return now to objects deﬁned in a general case on three-dim ensional vector spaces. One of the most important objects deﬁned on these spaces is a t ensor of the second rank which transforms an arbitrary vector into another vector. I n addition this transformation should be linear and homogeneous. Formally, we can write b=t(a)=Ta, (1.21) whereTis independent of a. The ﬁrst part of this relation means that the vector bis the value of the function tcalculated for a chosen vector a. The second part means that the functiontis linear. It should be invertible, i.e. the tensor T−1should exist and be unique a=T−1b,TT−1=1, (1.22) where1is the unit tensor. These properties indicate that a represe ntation of the tensor Tin any set of base vectors is a square matrix. For a chosen set of base vectors {e1,e2,e3}we can introduce the operation of the tensor product⊗which deﬁnes the unit tensor 1as the matrix (δij)and we write 1=δijei⊗ej. (1.23) Clearly, in order to be a unit tensor it must possess the follo wing property a=1a⇒aiei=(δijei⊗ej)(akek), (1.24) which means that the tensor product operates in this way that we take the scalar product of the second unit vector in 1(i.e.ejin our case) with the vector aappearing after the unit tensor. Then (1.24) becomes aiei=δijeiakδjk, (1.25) which is, of course, an identity. Making use of the tensor product we can write the following re presentation for an arbitrary tensor of the second rank T=Tijei⊗ej. (1.26) 14 Modicum of vectors and tensors Matrix(Tij)is the representation of the tensor Tin the basis{e1,e2,e3}. The numbers Tijare called components of the tensor T. Now the relation (1.21) can be written in the form biei=(Tijei⊗ej)(akek)=Tijakeiej·ek=Tijajei⇒bi=Tijaj. (1.27) The tensorTchanges both the direction and the length of the vector a. For the length we have the relation |b|2=b·b=(Ta)·(Ta)= (1.28) =TijajTilal=ajTijTilal=a·TTTa. Usually|b|/negationslash=|a|. However, if the tensor Tpossesses the property TTT=1the length of the vectorbremains the same as this of the vector a. Such tensors are called orthogonal and we denote them usually by O. Obviously, they have the property OT=O−1, (1.29) which may also be used as the deﬁnition of orthogonality of th e tensor. Orthogonal tensors yield only rotations of vectors. Once we have the length of the vector bwe can easily ﬁnd the angle of rotation caused by the tensorT. We have for the angle ϕ=(a,b) a·b=|a||b|cosϕ⇒cosϕ=Tijaiaj√aiairadicalbig (TijajTikak). (1.30) The representation (1.26) allows to specify rules of transf ormation of components of an arbitrary tensor of the second rank T. Performing the transformation of base vectors {e1,e2,e3}→{e1′,e2′,e3′}we obtain T=Tijei⊗ej=Tij(Aii′ei′)⊗(Ajj′ej′)= =Aii′Ajj′Tijei′⊗ej′=Ti′j′ei′⊗ej′. Hence Ti′j′=Aii′Ajj′Tij. (1.31) Similarly to vectors, this relation is used in tensor calcul us as a deﬁnition of the tensor of the second rank. Majority of second rank tensors in mechanics are symmetric. They possess six in- dependent components instead of nine components of a genera l case. However, some particular problems, such as Cosserat media and couple stre sses, interactions with elec- tromagnetic ﬁelds require full nonsymmetric tensors. An ar bitrary tensor can be always split into a symmetric and antisymmetric parts T=Ta+Ts,Ta=1 2parenleftBig T−TTparenrightBig ,Ts=1 2parenleftBig T+TTparenrightBig , (1.32) 1.1 Algebra 15 and, obviously, Tapossesses only three oﬀ-diagonal non-zero components whil eTspos- sesses six components. It is often convenient to replace the antisymmetric part by a vector. Usually, one uses the following deﬁnition V=Vkek, Vk=−1 2ǫijkTa ij⇒Ta ij=−ǫijkVk. (1.33) whereVis called the axial vector and ǫijkis the permutation symbol (Levi-Civita sym- bol). It is equal to one for the even permutation of indices {1,2,3},{2,3,1},{3,1,2}, minus one for the odd permutation of indices {2,1,3},{1,3,2},{3,2,1}and zero oth- erwise, i.e. ǫijk=1 2(j−i)(k−i)(k−j). (1.34) This symbol appears in the deﬁnition of the so-called exteri or or vector product of two vectors. The deﬁnition of this operation b=a1×a2, (1.35) is such that the vector bis perpendicular to both a1anda2,its length is given by the relation |b|=|a1||a2|sin((a1,a2)), (1.36) and the direction is determined by the anticlockwise screw r ule. For instance e3=e1×e2, (1.37) for the base vectors used in this work. In general, we have for these vectors ei×ej=ǫijkek. (1.38) Consequently, for the vector product of two arbitrary vecto rsa1=a1 iei,a2=a2iei,we obtain b=bkek=a1×a2=parenleftbiga1 ieiparenrightbig×parenleftbiga2 jejparenrightbig=a1ia2jǫijkek⇒bk=ǫijka1ia2j.(1.39) The vector product is well deﬁned within a theory of 3-dimens ional vector spaces. How- ever, for example for two-dimensional spaces of vectors tan gent to a surface at a given point, the result of this operation is a vector which does not belong any more to the vector space. It is rather a vector locally perpendicular to the surface. The permutation symbol can be also used in the evaluation of d eterminants. For a tensorT=Tijei⊗ej, we can easily prove the relation detT=ǫijkT1iT2jT3k. (1.40) In mechanics of isotropic materials we use also the followin g identity ("contracted epsilon identity") ǫijkǫimn=δjmδkn−δjnδkm. (1.41) 16 Modicum of vectors and tensors A certain choice of base vectors plays a very important role i n the description of properties of tensors of the second rank. This is the content s of the so-called eigenvalue problem. We proceed to present its details. First we shall ﬁnd a direction ndeﬁned by unit vector, |n|=1, whose transformation by the tensorTinto the vector Tnis extremal in the sense that the length of its projection onngiven byn·Tnis largest or smallest with respect to all changes of n. This is the variational problem δ(n·Tn−λ(n·n−1))=0, (1.42) for an arbitrary small change of the direction, δn. Hereλis the Lagrange multiplier which eliminates the constraint on the length of the vector n:n·n=1. Obviously, the problem can be written in the form δn·[(Ts−λ1)n]=0, (1.43) for arbitrary variations δn. Let us notice that n·Tn=n·(Ta+Ts)n=n·Tsn. (1.44) Consequently, the antisymmetric part Tahas no inﬂuence on the solution of the problem. Bearing the above remark in mind, we obtain from (1.43) (Ts−λ1)n=0, (1.45) or, in components,parenleftbigTs ij−λδijparenrightbignj=0. (1.46) It means that λare the eigenvalues of the symmetric tensor Tsandnare the corre- sponding eigenvectors. The existence of nontrivial solutions of the set of three equ ations (1.46) requires that its determinant is zero det(Ts−λ1)=0, (1.47) i.e. vextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsingleTs 11−λ Ts 12Ts 13 Ts 12Ts 22−λ Ts 23 Ts 13Ts 23Ts 33−λvextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsingle=0. (1.48) This can be written in the explicit form λ 3−Iλ2+IIλ−III=0, (1.49) where I=trTs=Ts ii, II=1 2parenleftbigI2−trTs2parenrightbig=1 2parenleftBig (Ts ii)2−Ts ijTs ijparenrightBig , III=detTs,(1.50) are the so-called principal invariants of the tensor Ts. Obviously, the cubic equation (1.49) possesses three roots. For symmetric real matrices t hey are all real. It is customary in mechanics to order them in the following manner λ(1)≥λ(2)≥λ(3). (1.51) 1.2 Analysis 17 However, in more general mathematical problems, in particu lar when the spectrum of eigenvalues is inﬁnite, the smallest eigenvalue is chosen a s the ﬁrst and then the eigenvalue sequence is growing. Clearly I=λ(1)+λ(2)+λ(3), II=λ(1)λ(2)+λ(1)λ(3)+λ(2)λ(3), (1.52) III=λ(1)λ(2)λ(3). Once we have these values we can ﬁnd the corresponding eigenv ectorsn(1),n(2),n(3)from the set of equations (1.45). Only two of these equations are i ndependent which means that we can normalize the eigenvectors by requiringvextendsinglevextendsinglen(α)vextendsinglevextendsingle=1, α=1,2,3.It is easy to show that these vectors are perpendicular to each other. Nam ely we have for α,β=1,2,3 n(β)·bracketleftBigparenleftBig Ts−λ(α)1parenrightBig n(α)bracketrightBig =0⇒parenleftBig λ(α)−λ(β)parenrightBig n(α)·n(β)=0. (1.53) If the eigenvalues are distinct, i.e. λ(α)/negationslash=λ(β)forα/negationslash=βwe obtain n(α)·n(β)=0, (1.54) and, consequently, the vectors n(α),n(β)are perpendicular (orthogonal). The proof can be easily extended on the case of twofold and threefold eigen values. The above property of eigenvectors allows to use them as a spe cial set of base vectorsbraceleftbig n(1),n(2),n(3)bracerightbig . Then the tensor Tscan be written in the form Ts=3summationdisplay α=1λ(α)n(α)⊗n(α). (1.55) We call this form the spectral representation of the tensor Ts. Obviously, the tensor in this representation has the form of diagonal matrix parenleftbigTs αβparenrightbig= λ(1)0 0 0λ(2)0 0 0λ(3) . (1.56) 1.2 Analysis Apart from algebraic properties of vectors and tensors whic h we presented above, prob- lems of continuum mechanics require diﬀerentiation of thes e objects with respect to spatial variables and with respect to time. Obviously, such problems appear when vec- tors and tensors are functions of coordinates in space and ti me. Then we speak about vector or tensor ﬁelds. One of the most important properties of the vector ﬁelds deﬁn ed on three-dimensional domains is the Helmholtz (1821-1894) decomposition1. For every square-integrable vector 1C. AImscIrscIoscIuscIcscIhscIesc, C. BIescIrscInscIascIrscIdscIisc, M. DIascIuscIgscIesc, V. GIiscIrscIascIuscIlscItsc ;. Vector potentials in three dimensional non-smooth domains, Mathematical Methods in the Applied Sciences, 21, 823—864, 1998. 18 Modicum of vectors and tensors ﬁeldvthe following orthogonal decomposition holds v(x)≡v(x1,x2,x3)=gradϕ+rotψ,x=xiei, (1.57) wherexiare Cartesian coordinates of the point xof the three-dimensional Euclidean spaceE3, gradϕ=∂ϕ ∂xiei,rotψ=ǫijk∂ψk ∂xjei, (1.58) andϕ,ψare called scalar and vector potential, respectively. The operator rot(it is identical with curlwhich is used in some texts to denote the same operation) can be easily related to the integration alo ng a closed curve. Namely G. Stokes (1819-1903) proved the Theorem that for all ﬁelds vdiﬀerentiable on an oriented surfaceSwith the normal vector nthe following relation holds integraldisplay S(rotv)·ndS=contintegraldisplay ∂Sv·dx, (1.59) i.e.integraldisplay Sǫijk∂vk ∂xjnidS=contintegraldisplay ∂Svidxi, where∂Sis the boundary curve of the surface S. On the other hand, integration over a closed surface is relat ed to the volume integra- tion. This is the subject of the Gauss (1777-1855) Theorem di scovered by J. L. Lagrange in 1762. For a compact domain V⊂E3with a piecewise smooth boundary ∂Vthe following relation for a continuously diﬀerentiable vector ﬁeld vholds integraldisplay VdivvdV=contintegraldisplay ∂Vv·ndS, (1.60) where∂Vdenotes the boundary of the domain of volume integration Vandnis a unit vector orthogonal to the boundary and oriented outwards. G. Leibniz (1646-1716) Theorem for volume integrals which w e use in analysis of bal- ance equations of mechanics describes the time diﬀerentiat ion of integrals whose domain is time-dependent. It has the following form d dtintegraldisplay V(t)f(t,x)dV=integraldisplay V∂f ∂t(t,x)dV+contintegraldisplay ∂Vf(t,x)v·ndS, (1.61) wheretdenotes time,xis the point within V,vis the velocity of boundary points of the domainV. Instead of a rigorous proof it is useful to observe the way in which the above structure of the time derivative arises. The right-hand sid e consists of two contributions: the ﬁrst one arises in the case of time independent domain of i ntegration, V, and due to time diﬀerentiation there is only a contribution of the inte grandfwhile the second one arises when the function fis time independent (i.e. time tis kept constant in f) and the 1.2 Analysis 19 volumeVchanges. They add due to the linearity of the operator of diﬀe rentiation as in the case of diﬀerentiation of the product d(fg)/dt=gdf/dt+fdg/dt . The structure of the second contribution is explained in Fig.1.3. Fig. 1.3: Interpretation of Leibniz Theorem Locally the change of the volume can be written in the form dV=dSn·vdt, where vis the velocity of the point of the boundary. Consequently, t he change of the domain of integration for the increment of time dthas the form integraldisplay V(t+dt)f(t)dV−integraldisplay V(t)f(t)dV= contintegraldisplay ∂V(t)f(t)v·ndS dt. (1.62) Now the second contribution of (1.61) easily follows. As a particular case of the last Theorem we have for f=1 d dtintegraldisplay VdV=contintegraldisplay ∂Vv·ndS=integraldisplay VdivvdV⇒dV dt=integraldisplay VdivvdV. (1.63) This relation indicates that divvdescribes the local time changes of the volume. This interpretation is useful in the analysis of the mass balance equation of continuum me- chanics. More details of the vector calculus and its applications in m echanics can be found in numerous books on continuum mechanics and thermodynamic s (e.g. [22]). In some exercises and examples we use curvilinear coordinates rath er than Cartesian coordinates applied in the above presentation. We demonstrate their pro perties and appropriate rules of transformations in these examples. 20 Modicum of vectors and tensors Chapter 2 Geometry and kinematics of continua 2.1 Preliminaries As mentioned in Chapter 1 a theoretical description of mecha nical behaviour of structures requires continuous models. It means that a collection of po ints of structures form a three- dimensional continuum of a certain mathematical construct (the so-called diﬀerentiable manifold). The points Xof the setB0of such a continuum move in a three-dimensional space of motionE3. The main purpose of continuum mechanics is to determine thi s motion for any given set of external agents (forces, or given displacements of boundaries of structure, or a mixture of both). It means that one has to so lve the set of governing equations in order to ﬁnd a current position of an arbitrary p ointX∈B0of the material bodyB0. A collectionBtof positions of all points from the material body B0at a given instant of time tis called the current conﬁguration of the body. Once a partic ular model is selected (elasticity, viscoelasticity, plastici ty, viscoplasticity, etc.) the knowledge of these current conﬁgurations allows to calculate deforma tions, stresses, dissipation of energy, work done on the system or any other quantity which ma y be of practical interest. In some cases one is interested only, for instance, in the dis tribution of stresses in the system. As we know from the classical linear elasticity such problems may be solved by a transformation of the governing set of equations of motion i nto equations for stresses. In the linear elasticity we call them Beltrami-Michell equati ons. However, in general such a transformation is diﬃcult if possible at all and, therefore we limit our attention in these notes to the formulation of governing equations of motion on ly marginally referring to other approaches. Engineering problems arise usually in connection with a par ticular geometry of struc- tures which may lead to considerable simpliﬁcations of mech anical models. This is related to the fact that one or two spacial dimensions of a structure a re much smaller than the remaining dimension. This yields models of shells, plates, rods, bars and their combi- nations such as ﬁbrous media, nets and so on. We refer to model s of such structures 21 22 Geometry and kinematics of continua in some examples but, due to a limited volume of these notes we shall not go into any details of their modeling. A conﬁgurationBtas a collection of points in E3occupied by points X∈B0of the material body is in some cases of practical interest not suﬃc ient to describe a geometry of the system. For instance, in the description of suspensio ns or liquid crystals one may need additional local degrees of freedom related to rotatio ns of microparticles. In some other cases one may even need additional tensors as a natural space in which the motion appears is not Euclidean. This appears in the description of a continuous distribution of dislocations important in the theory of plasticity. Someti mes an appropriate extension of the classical continuum necessary to describe such systems can be done by the so-called microstructural variables. We mention some of them further in this book. However, it is not always the case (e.g. a nonlinear theory of dislocatio ns, or diﬀusion processes of mixtures). Such problems shall not be considered in these no tes. 2.2 Reference conﬁguration and Lagrangian descrip- tion The choice of the reference conﬁguration B0with respect to which the motion of the body is measured is arbitrary. If possible we choose a natura l stress-free conﬁguration. This is not always convenient for solids (e.g. for prestress ed structures) and it is never possible for ﬂuids. These cases will be discussed separatel y but, in principle, the analysis presented below can be taken over also for such cases. In thos e many systems in which we can choose a natural conﬁguration we assume its deformation to be zero. The motion of the body, i.e. the function which describes the geometry of a ll subsequent conﬁgurations is described by the function of two variables: time, t, and the point, X, of the reference conﬁgurationB0. The latter speciﬁes the particle which is described. In cho sen Cartesian coordinate systems we have x=f(X,t),i.e.xk=fk(X1,X2,X3,t),X∈B0, k=1,2,3,(2.1) where x=xkek,X=XKeK, (2.2) are position vectors in the current conﬁguration and refere nce conﬁguration, respectively. As the reference conﬁguration is one of the conﬁgurations ap pearing in the real motion of the body, for instance, the one for t=0, we can, certainly, choose the same coordinate system for all conﬁgurations, i.e. xk=δkKXK,ek=δkKeK. (2.3) However, in the analysis of certain invariance properties ( e.g. isotropy of the material, material objectivity, etc.), it is convenient to distingui sh between these two systems as we did in the relation (2.1). The coordinate system with the b ase vectorseKis called 2.2 Reference conﬁguration and Lagrangian description 23 Lagrangian and the coordinate system with the base vectors ekis called Eulerian. Fig. 2.1: Transformation from the reference conﬁguration B0to the current conﬁgurationBt.The function of motion fdescribes time changes of position ofmaterialpoints X∈B0intox∈Btwhilethedeformationgradient Fdescribes the transformation of material vectors (e.g. a tangent vect ordXof the curve C0into a tangent vector dxof the curveCt) The vector function f(X,t), the function of motion, is assumed to be twice continu- ously diﬀerentiable with respect to all variables. Consequ ently, it must be also continuous. This property of the theories of continuum eliminates many i mportant processes from continuous models. Some examples are shown in Fig. 2.2. In pr actical applications, we overcome this diﬃculty by some additional sophisticated co nstructions (e.g. for cracks in solids or vorticities in ﬂuids). We shall point out some of th em further in these notes. Fig. 2.2: Some motions (nontopological) which cannot be des cribed by con- tinuous functions of motion. 24 Geometry and kinematics of continua The function of motion f(X,t)speciﬁes the local rule of transformation of the so- called material vectors. If we select a smooth curve C0in the reference conﬁguration B0, given by a parametric equation X=X(S), whereSis the parameter along this curve then in each point of this curve the tangent vector is given by the derivative dX/dS.A corresponding inﬁnitesimal vector dX=(dX/dS)dSchanges during the motion in the following way dx=(Gradf)dX≡(Gradf)dX dSdS=dx dSdS, (2.4) or, in coordinates, dxk=∂fk ∂XKdXK=∂fk ∂XKdXK dSdS=dxk dSdS. (2.5) Hence, in the current conﬁguration, the tangent vector dX/dSof the curveC0changes into the tangent vector dx/dSof the curveCtgiven by the relation x=f(X(S),t)in the conﬁgurationBtwhich is the image of the curve C0of the initial conﬁguration B0. This new vector has, in general, a diﬀerent length and a diﬀerent d irection than the vector dX/dS. All such vectors, V, which fulﬁl the above indicated rule of transformation whe n changing the conﬁguration B0intoBt v=FV,F=Gradf, (2.6) i.e.vk=FkKVK, FkK=∂fk ∂XK, are called material vectors. This transformation by means o f the objectFwhich is called the deformation gradient is the most important notion descr ibing geometrical changes of a continuum during the deformation. Before we present the ful l analysis of the deformation gradientFlet us consider two simple examples. ⋆We begin with the simplest case of a uniform extension of a cub e in three perpen- dicular directions (see: Fig. 2.3.) Fig. 2.3: Reference conﬁguration of a cube with the vector Vof the examples 2.2 Reference conﬁguration and Lagrangian description 25 described by the following function of motion in Cartesian c oordinates x1=X1(1+ε1), x2=X2(1+ε2), x3=X3(1+ε3), (2.7) whereε1,ε2,ε3are three constants. Then the deformation gradient is given by the relation F=FkKek⊗eK,(FkK)= 1+ε10 0 0 1+ε20 0 0 1+ ε3. (2.8) Obviously, it is independent of coordinates, i.e. the defor mation is homogeneous. We check the action of the deformation gradient on a chosen vect orV. An example of this vector is shown in Fig. 2.3. In coordinates indicated in this Figure the vector Vhas the following components V=−e 1+e2+e3. Its image after the deformation is as follows v=FV= (2.9) =((1+ε1)e1⊗e1+(1+ε2)e2⊗e2+(1+ε3)e3⊗e3)(−e1+e2+e3)= =−(1+ε1)e1+(1+ε2)e2+(1+ε3)e3. Hence the current image vof the vectorVhas a diﬀerent length and a diﬀerent direction |v|2=v·v=3summationdisplay i=1(1+εi)2,|V|2=V·V=3, (2.10) cos((v,V))=summationtext3 i=1(1+εi) √ 3radicalBigsummationtext3 i=1(1+εi)2. Obviously, for ε1=ε2=ε3the angle between vandVis equal to zero.♣ ⋆The second example describes the so-called simple shearing in the plane perpendic- ular toe1. Then x1=X1, x2=X2+X3tanϕ, x3=X3, (2.11) and the corresponding deformation gradient is as follows (L agrangian and Eulerian base vectors are identical ek=δkKeK) F=FkKek⊗eK,(FkK)= 1 0 0 0 1 tanϕ 0 0 1 . (2.12) The current image of the vector Vis now given by the relation v=FV= (2.13) =(e1⊗e1+e2⊗e2+e3⊗e3+tanϕe2⊗e3)(−e1+e2+e3)= =−e1+(1+tanϕ)e2+e3. 26 Geometry and kinematics of continua Also in this case the vector vhas the diﬀerent length and direction from the vector V |v|2=v·v=2+(1+tan ϕ)2,|V|2=V·V=3, (2.14) cos((v,V))=3+tanϕ √ 3radicalBig 2+(1+tan ϕ)2. ♣ The deformation gradient Fis, obviously, represented by a square matrix. However, it is not a tensor of the second rank. Clearly, if we change the La grangian basis eK→eK′= AK′KeK(compare (1.15)) but keep unchanged the Eulerian basis ekthe deformation gradient transforms as follows FkK′=AKK′FkK, (2.15) i.e. it transforms as three vectors for k=1,2,3rather than a tensor. The same property possesses the ﬁrst index under the transformation ek→ek′=Ak′kek,andeKis kept unchanged Fk′K=Akk′FkK. (2.16) This is one of the reasons why the notation for Lagrangian and Eulerian coordinates is diﬀerent. The deformation gradient can be written in a diﬀerent form in which these trans- formation rules possess an obvious interpretation. This is the subject of the so-called polar decomposition Theorem. For a nonsingular deformatio n gradientF,detF>0(we return to the justiﬁcation of this property in the Chapter on the conservation of mass) there exist an orthogonal matrix Rand a symmetric tensor Usuch that F=RU,RT=R−1,UT=U, (2.17) FkK=RkLUKL,(RkL)T=(RkL)−1, UKL=ULK, and this decomposition is unique. Rdescribes the local rotation and the right stretch tensorUthe local deformation. We show the construction of these obj ects. Let us note that in the following product C=FTF=UTRTRU=U2, (2.18) CKL=FkKFkL=UKMUML, the orthogonal part Rdoes not appear. Hence, in order to ﬁnd Uwe have to take "the square root" of C. Obviously, it is not the same operation which we perform wit h numbers. We deﬁne it through the eigenvalues. Namely, we cal culate ﬁrst the eigenvalues and eigenvectors of the tensor C (C−λC1)KC=0. (2.19) Obviously det(CKL−λCδKL)=0, (2.20) 2.2 Reference conﬁguration and Lagrangian description 27 and the solution of this cubic equation gives three eigenval uesλ(1) C,λ(2) C,λ(3) Cof the tensor C. They are all real and positive ( detC=(detF)2>0!). From (2.19) one can ﬁnd then unit eigenvectors K(1) C,K(2)C,K(3)C. They are orthogonal (compare (1.54)). Hence, we can write the tensor Cin the spectral representation C=3summationdisplay α=1λ(α) CK(α) C⊗K(α) C. (2.21) Simultaneously, the eigenvalue problem for Uhas the following form (U−λU1)KU=0. (2.22) If we multiply this equation by Ufrom the left we have (UU−λUU)KU=parenleftbig C−λ2 U1parenrightbig KU=0. (2.23) Consequently λU=radicalbig λC,KU=KC. (2.24) Hence, we obtain the following spectral representation for the right stretch U U=3summationdisplay α=1radicalBig λ(α) CK(α) C⊗K(α) C. (2.25) This is what was meant by taking a square root of C. OnceUis given we can calculate Rfrom (2.17) R=FU−1. (2.26) The above presented procedure is simultaneously the proof o f the polar decomposition Theorem. ⋆Before we proceed let us consider a simple example for the app lication of the above procedure. We ﬁnd the polar decomposition of the deformatio n gradient in the simple shearing given by the relation (2.12). The Lagrangian and Eu lerian base vectors are assumed to be identical ek=δkKeK. We have C=CKLeK⊗eL,(CKL)= 1 0 0 0 1 tan ϕ 0 tanϕ1+tan2ϕ . (2.27) The eigenvalue problem yields the following equation for ei genvaluesλC (1−λC)bracketleftbig(1−λC)parenleftbig(1−λC)+tan2ϕparenrightbig−tan2ϕbracketrightbig=0. (2.28) The solution has the form λ(1) C=1, λ(2)C=parenleftbigg1−sinα cosαparenrightbigg2 , λ(3) C=parenleftbigg1+sinα cosαparenrightbigg2 , (2.29) 28 Geometry and kinematics of continua where tanα=1 2tanϕ. (2.30) The corresponding eigenvectors follow from (2.19). We obta in K(1) C=e1, K(2) C=−parenleftbigg1√ 2cosα√1−sinαparenrightbigg e2+parenleftbigg1√ 2√ 1−sinαparenrightbigg e3, (2.31) K(3) C=parenleftbigg1√ 2cosα√1+sinαparenrightbigg e2+parenleftbigg1√ 2√ 1+sinαparenrightbigg e3. According to the relation (2.25) we obtain for the tensor U U=e1⊗e1+cosαe2⊗e2+sinα(e2⊗e3+e3⊗e2)+parenleftbigg2 cosα−cosαparenrightbigg e3⊗e3.(2.32) Its inverse has the form U−1=e1⊗e1+parenleftbigg2 cosα−cosαparenrightbigg e2⊗e2−sinα(e2⊗e3+e3⊗e2)+cosαe3⊗e3.(2.33) Consequently, according to the relation (2.26), we have for the orthogonal part R=FU−1=e1⊗e1+cosαe2⊗e2+sinα(e2⊗e3−e3⊗e2)+cosαe3⊗e3.(2.34) For the linear theory which we consider further in this book i t is useful to collect the above results in the case of the small angle ϕ. Thenα≈ϕ/2and we obtain U=e1⊗e1+e2⊗e2+ϕ 2(e2⊗e3+e3⊗e2)+e3⊗e3, U−1=e1⊗e1+e2⊗e2−ϕ 2(e2⊗e3+e3⊗e2)+e3⊗e3, (2.35) R=e1⊗e1+e2⊗e2+ϕ 2(e2⊗e3−e3⊗e2)+e3⊗e3. On the plane perpendicular to e1these objects yield the following transformations of the edges of unit length of the rectangular prism Ue2=e2+ϕ 2e3,Fe2=RUe2=e2, (2.36) Ue3=ϕ 2e2+e3,Fe3=RUe3=ϕe2+e3. Obviously, the vectors e2ande3along the edges are material. 2.2 Reference conﬁguration and Lagrangian description 29 We demonstrate these transformations in Fig. 2.4. Fig. 2.4: Geometry of the linear simple shearing on the plane perpendicular toe1. Obviously, the symmetric stretch tensor Uyields the change of shape from the rectan- gle to the parallelogram with the symmetry axis of the declin ationπ/4. The orthogonal tensorRrotates back the deformed parallelogram in such a way that th e horizontal edge before the deformation becomes horizontal after the deform ation as well.♣ The above presented example indicates that U=UKLeK⊗eLdescribes the true local deformation. This is the reason for calling it the right stre tch tensor while its square C=U2=CKLeK⊗eLis called the right Cauchy-Green deformation tensor. AsR=RkKek⊗eKdescribes local rotations — due to orthogonality — it does no t change the length of material vectors. The polar decomposition Theorem can be also written in the du al form F=VR,V=Vklek⊗el=VT,R=RkKek⊗eK,RT=R−1, (2.37) B=FFT=V2=BT. ThenVis called the left stretch tensor and Bthe left Cauchy-Green deformation tensor. For the latter we can easily prove λB=λC,kB=FKC |FKC|,B=1 |FKC|23summationdisplay α=1λ(α) CparenleftBig FK(α) CparenrightBig ⊗parenleftBig FK(α) CparenrightBig ,(2.38) i.e. Cauchy-Green tensors CandBhave the same eigenvalues. In Fig. 2.5 [5] we demonstrate schematically the interpreta tion of the polar decom- 30 Geometry and kinematics of continua position. Fig. 2.5: Polar decomposition of the deformation gradient Fas the composition of stretch Vfollowed by rotation R(i.e.VR) or vice versa ( RU) 2.3 Displacement, velocity, Eulerian description The motion of the body can be described not only by the functio n of motionfbut, as customary in the linear elasticity, by the displacement v ectoru.Similarly tof, it is deﬁned with respect to a chosen reference conﬁguration, say B0, x=f(X,t)=X+u(X,t). (2.39) Then the gradient of deformation has the form F=1+Gradu. (2.40) Provided we identify the Lagrangian and Eulerian coordinat e systems the Cauchy-Green deformation tensors have then the following form C=FTF=1+Gradu+(Gradu)T+(Gradu)TGradu,(2.41) i.e.CKL=δKL+δKk∂uk ∂XL+δLk∂uk ∂XK+∂uk ∂XK∂uk ∂XL, B=FFT=1+Gradu+(Gradu)T+Gradu(Gradu)T,(2.42) i.e.Bkl=δkl+δkK∂ul ∂XK+δlK∂uk ∂XK+∂uk ∂XK∂ul ∂XK. 2.3 Displacement, velocity, Eulerian description 31 These relations show that the displacement vector is not ver y convenient in nonlinear models. There are also other reasons for not using it in such t heories (see: [22]). However, it is a very useful notion in linear models and it shall be exte nsively used in this book. For our further considerations in these notes it is importan t to specify the above geometrical description under the assumption of small defo rmations. We proceed to do so. We base our considerations on the analysis of the right Cauch y-Green deformation tensorC. It is clear that rotations cannot be assumed to be small as ev en in the case of lack of deformation the system may rotate as a rigid body and t his rotation, of course, cannot be small. For this reason, we cannot constrain the def ormation gradient Fand rather measures of deformations are appropriate tools. The undeformed conﬁguration is characterized by the deformation gradient F=1. Consequently, this conﬁguration is described by the deformation tensors C=1andB=1. The spectral representation of the tensorC(2.21) indicates that this tensor diﬀers a little from the un it tensor if its eigenvalues λ(α) Cdeviate a little from unity. These eigenvalues are called pr incipal stretches. It is convenient to introduce a norm for the tenso rC−1rather than for C. It is done by the following relation /bardblC−1/bardbl= max α=1,2,3vextendsinglevextendsinglevextendsingleλ(α) C−1vextendsinglevextendsinglevextendsingle. (2.43) Then, we say that the body undergoes small deformations if th e norm ofCsatisﬁes the condition /bardblC−1/bardbl≪1. (2.44) Under this condition one does not have to distinguish betwee n Lagrangian and Eulerian coordinates. For an arbitrary function h(x,t)we have ∂h ∂XKeK=FkK∂h ∂xkeK≈∂h ∂xkek. (2.45) It means that Eulerian coordinates can be treated for small d eformations as Lagrangian coordinates. For small deformations, it is convenient to introduced diﬀe rent measures of deforma- tion. One deﬁnes for arbitrary deformations the following m easures — Green-St. Venant measure E=1 2(C−1), λE=λC−1 2, (2.46) — Almansi-Hamel measure e=1 2parenleftbig1−B−1parenrightbig, λe=1−1/λC 2. (2.47) For small deformations λe=λC−1 2λC≈λC−1 2=λE. (2.48) Hence, both these measures, Eande, are not distinguishable. 32 Geometry and kinematics of continua In terms of the displacement vector the Almansi-Hamel measu re of small deformations can be written in the following form e≈1 2parenleftBig gradu+(gradu)TparenrightBig +Oparenleftbigε2parenrightbig,i.e.ekl≈1 2parenleftbigg∂uk ∂xl+∂ul ∂xkparenrightbigg +Oparenleftbigε2parenrightbig,(2.49) where ε≡/bardblgradu/bardbl≪1,/bardblgradu/bardbl=radicalbig (gradu)·(gradu)=radicalbigg ∂uk ∂xl∂uk ∂xl. (2.50) andOparenleftbigε2parenrightbigare contributions of the order ε2=(gradu)·(gradu)and higher. This measure will be used in linear models discussed further . In the description of ﬂuids it is convenient to change the ref erence conﬁguration. Obviously, in contrast to solids one cannot expect an existe nce of conﬁgurations which are stress-free. Consequently, any choice of the reference con ﬁguration will be not natural for ﬂuids. The most suitable seems to be the current conﬁguratio n and, indeed, it is chosen as the reference conﬁguration in most works on ﬂuid mechanic s. Such a description does not identify particles. In any point xof the space of motion, occupied by the material one can specify the velocity v(x,t)as a function of time. By means of this ﬁeld one can ﬁnd trajectories of particles and, consequently, identify them. In a particular case of the reference conﬁguration B0these trajectories are labeled by points X∈B0. We proceed to describe the details of such a description. The velocity of particles is deﬁned in the Lagrangian descri ption by the time derivative of the function of motion ˙ x≡v=∂f ∂t(X,t),i.e.vk=∂fk ∂t(X1,X2,X3,t). (2.51) Consequently, if we deﬁne the trajectory of motion of the par ticleXas the curvef(X,t) parametrized by the time t, the velocity is the vector tangent to the trajectory. In the Eulerian description we change the variables X→xby the function X=f−1(x,t) inverse tofwith respect to X. Such a function exists because we assume the determinant detF/negationslash=0. Then the velocity in the Eulerian description is the follow ing vector function v=v(x,t),x∈Bt, (2.52) and it points in the direction tangent to the trajectory of th e particleXwhich is instan- taneously located in the point xof the conﬁguration space. Consequently, the equation of this trajectory is given by the set of three ordinary diﬀer ential equations dx dt=v(x,t),x(t=0)=X. (2.53) For a given velocity ﬁeld this is a highly nonlinear set which can be only seldom solved analytically. In order to appreciate the details of the Eulerian descripti on, we consider three con- ﬁgurations:B0which has been discussed before and two current conﬁguratio nsBt,Bτ 2.3 Displacement, velocity, Eulerian description 33 for instances of time tandτ, respectively. Then the function of motion mapping these conﬁgurations on each other forms the diagram shown in Fig. 2 .6. In these mappings we have x=f(X,t),ξ=f(X,τ)⇒ξ=ft(x,τ)=fparenleftbigf−1(x,t),τparenrightbig. (2.54) Hence, the function of the relative motion ft(.,τ)describes positions of points xof the conﬁguration in the instant of time tat the new instant of time τ. Obviously, these functions specify the corresponding gradients of deformat ion dx=F(X,t)dX, dξ=F(X,τ)dX,⇒ (2.55) ⇒dξ=Ft(x,τ)dx,Ft(x,τ)=F(X,τ)F−1(X,t)vextendsinglevextendsingle X=f−1(x,t). Fig. 2.6: Three conﬁgurations yielding Eulerian descripti on Consequently, Ft(x,τ=t)=1, (2.56) and, for this reason we say that Ft(.,τ)is the relative deformation gradient with respect to the conﬁguration at the instant of time t. The above notion allows to introduce time derivatives of arb itrary quantities in the Eulerian description. As an example let us consider a materi al vector function Q(X,t). Its current image is as follows q(X,t)=F(X,t)Q(X,t). (2.57) Hence ∂q ∂t(X,t) =F(X,t)∂Q ∂t(X,t)+(Gradv)Q(X,t),Gradv≡∂F ∂t(X,t),(2.58) i.e.∂qk ∂t=FkK∂QK ∂t+∂vk ∂XKQK,∂vk ∂XK=∂FkK ∂t≡∂2fk ∂t∂XK. 34 Geometry and kinematics of continua These rules of diﬀerentiation are straightforward. It is no t so in the Eulerian description. We have rather q(x,t) =Fparenleftbigf−1(x,t),tparenrightbigQparenleftbigf−1(x,t),tparenrightbig, q(ξ,τ) =Fparenleftbigf−1(ξ,τ),τparenrightbigQparenleftbigf−1(ξ,τ),τparenrightbig⇒ (2.59) ⇒q(x,t)=F−1 t(ξ,τ)q(ξ,τ)vextendsinglevextendsingle ξ=f−1 t(x,τ). The time derivative of q(x,t)is now deﬁned in the same way as in the Lagrangian description in the limit τ→t. We obtain Lvq(x,t) =dbracketleftbigF−1 t(ξ,τ)q(ξ,τ)bracketrightbig dτvextendsinglevextendsinglevextendsinglevextendsinglevextendsingle τ=t= =∂q ∂t(x,t)+(v·grad)q(x,t)+dbracketleftbigF−1 t(ξ,τ)bracketrightbig dτvextendsinglevextendsinglevextendsinglevextendsinglevextendsingle τ=tq(x,t).(2.60) The last contribution can be transformed in the following wa y dbracketleftbigF−1 tFtbracketrightbig dτ=0=dF−1 t dτFt+F−1 tdFt dτ⇒dF−1 t dτ=−F−1 tdFt dτF−1 t. (2.61) Hence dbracketleftbigF−1 t(ξ,τ)bracketrightbig dτvextendsinglevextendsinglevextendsinglevextendsinglevextendsingle τ=t=−dFparenleftbigf−1(x,t),tparenrightbig dtF−1parenleftbigf−1(x,t),tparenrightbig= =−gradv(x,t), (2.62) i.e. dF−1 tkl dτvextendsinglevextendsinglevextendsinglevextendsingle τ=t=−dFkK dtF−1 Kl=∂vk ∂XK∂XK ∂xl=∂vk ∂xl. (2.63) The quantity L=gradv, (2.64) is called the velocity gradient and it plays an important rol e in nonlinear ﬂuid mechanics. Obviously it can be split into symmetric and antisymmetric p arts L=D+W,D=1 2parenleftBig L+LTparenrightBig ,W=1 2parenleftBig L−LTparenrightBig , (2.65) i.e.∂vk ∂xl=Dkl+Wkl, Dkl=1 2parenleftbigg∂vk ∂xl+∂vl ∂xkparenrightbigg , Wkl=1 2parenleftbigg∂vk ∂xl−∂vl ∂xkparenrightbigg . The tensorDis called the stretching and the tensor Wis called the spin. Substitution of (2.64) in (2.60) yields Lvq(x,t) =∂q ∂t(x,t)+(v·grad)q(x,t)−Lq(x,t), (2.66) i.e.Lvqk=∂qk ∂t+vl∂qk ∂xl−∂vk ∂xlql. 2.3 Displacement, velocity, Eulerian description 35 This is the so-called Lie derivative of qrelated to the velocity ﬁeld v. Obviously, additional contributions to the standard partial time deri vative are nonlinear and they play an important role in nonlinear theories. This derivati ve of a vector ﬁeld in Eulerian description as well as analogous derivatives of tensor ﬁeld s have an important property that they are objective, i.e. invariant with respect to a cha nge of observer. We shall not discuss this subject in these notes. Let us complete this juxtaposition of Lagrangian and Euleri an description of geometry with the proof of the Euler-Piola-Jacobi identities which a re frequently used by the transformation of balance equations. Namely DivparenleftbigJF−Tparenrightbig(X,t)=0,divparenleftbigJ−1FTparenrightbig(x,t)=0. (2.67) We prove the ﬁrst one. The dual identity follows in the simila r manner. We write it in Cartesian coordinates ∂parenleftbigJF−1 Kkparenrightbig ∂XK=∂J ∂XKF−1 Kk+J∂F−1 Kk ∂XK =JF−1 Ll∂FlL ∂XKF−1 Kk−JF−1 KlF−1 Lk∂FlL ∂XK, and (2.67) follows when we use the symmetry ∂FlL/∂XK=∂FlK/∂XL. In the derivation we have used the identity ∂parenleftbig F−1 KkFkLparenrightbig ∂XK=0=∂F−1 Kk ∂XKFkL+F−1 Kk∂FkL ∂XK(2.68) =⇒∂F−1 Kk ∂XK=−F−1 KlF−1 Lk∂FlL ∂XK. The Lagrangian description yields as well an identity which is very useful in wave analysis. Usually, it is proved as a part of the so-called Had amard Theorem. We show this identity in the diﬀerent way. For a given set of two ﬁelds : deformation gradient F(X,t)and the velocity ﬁeld v=(X,t)one has to require the so-called integrability conditions∂FkK ∂t=∂vk ∂XK,∂FkK ∂XL=∂FkL ∂XK, (2.69) for the function of motion f(X,t)to exist. IfFandvare not given apriori but derived fromfthen the integrability conditions (2.69) are identically f ulﬁlled. On the other hand, ifFandvare not suﬃciently smooth, for example they possess a singul arity on a moving surfaceSthen we can require the integrability conditions to be fulﬁl led only in a weaker form. We investigate the ﬁrst one and write it as d dtintegraldisplay PFkKdV−contintegraldisplay ∂PvkNKdS=0, (2.70) for all subbodies P. Obviously, if there are no singularities this condition is equivalent to (2.69). We present the local form of this relation in Chapt er 4 on balance equations. 36 Geometry and kinematics of continua 2.4 Inﬁnitesimal strains Now we return to the analysis of the deformation under the ass umption (2.44), i.e. to the case of small deformations which is of the main concern in these notes. The most important property of small deformation is the iden tity of the Lagrangian and Eulerian description, i.e. we can identify the systems o f coordinates xk=δkKXK, ek=δkKeKand the dependence on xkandXKis the same. As a consequence, we can write the right Cauchy-Green deformation tensor in the form F=1+gradu⇒C=FTF≈gradu+(gradu)T+1≈FFT=B,(2.71) i.e.Ckl=∂uk ∂xl+∂ul ∂xk+δkl, asε=/bardblgradu/bardbl≪1. Simultaneously, the Almansi-Hamel deformation tensor (2. 48) has the following form e=1 2parenleftbig1−B−1parenrightbig=1 2parenleftBigg 1−3summationdisplay α=11 λ(α) Bk(α) B⊗k(α) BparenrightBigg = =1 23summationdisplay α=1parenleftBigg 1−1 λ(α) BparenrightBigg k(α) B⊗k(α) B=1 23summationdisplay α=1parenleftBigg 1−1 λ(α) CparenrightBigg k(α) B⊗k(α) B≈(2.72) ≈3summationdisplay α=1λ(α) C−1 2K(α) C⊗K(α) C=3summationdisplay α=1λ(α) eK(α) C⊗K(α) C,vextendsinglevextendsinglevextendsingleFK(α) Cvextendsinglevextendsinglevextendsingle≈1. Hence, for small deformations, as already indicated (compa re (2.49)), e≈1 2parenleftBig gradu+(gradu)TparenrightBig , (2.73) i.e.ekl≈∂uk ∂xl+∂ul ∂xk. This is the most commonly used measure of deformation in line ar theories. If relation (2.73) is used as the deﬁnition of the deformation measure eand not as an approximation of Almansi-Hamel tensor, eis called the strain ﬁeld. We proceed to investigate some of its properties. We begin with a certain invariance problem which plays an imp ortant role in all branches of continuum mechanics. One should expect that the description of deforma- tion should not change if we rotate very slowly the body as a wh ole. This rigid body rotation should be slow in this sense that we should not evoke inertial body forces such as centrifugal forces. These would, certainly, produce def ormations. Therefore we inves- tigate only a static problem. The rigid body rotation is then deﬁned by the following relation in the Lagrangian description x=OX,OT=O−1,F|rigid=1+Gradu|rigid=O, (2.74) 2.4 Inﬁnitesimal strains 37 whereOis the constant orthogonal matrix. Hence, in the case of the A lmansi-Hamel deformation tensor deﬁned by (2.73) C|rigid=FTF=OTO=1,e|rigid=1 2parenleftBig O+OTparenrightBig −1, (2.75) which means that the rigid rotation yields undeformed body w hose measure of defor- mation is the Cauchy-Green tensor Cbut it yields a deformation if it is deﬁned by the strain ﬁelde, i.e. by the simpliﬁed Almansi-Hamel tensor (2.73). Hence, it is not a proper measure for large deformations. The original deﬁnition (3. 46) yields, of course, for the rigid rotation the vanishing deformation e=1 2parenleftbig1−B−1parenrightbig=1 2parenleftBig 1−OOTparenrightBig =0. Displacements in the linear theory for the rigid rotation (r igid displacement) are given by the relation w(x)=u0+(O0−1)(x−x0), (2.76) whereu0is an arbitrary vector, x0is the reference point and xis the position of the point of the body. O0denotes a constant orthogonal matrix and, therefore, it des cribes an arbitrary time-independent rotation. As already mentio ned, we do not introduce the time dependence in order to eliminate inertial eﬀects. In th e above relation, the diﬀerence O0(x−x0)−(x−x0)is the displacement of point xdue to the rotation around the point x0. On this displacement we superpose the displacement u0=u(x0)of the pointx0. Consequently gradw=O0−1. (2.77) In order to see the consequences of this relation in the case o f the linear model, we consider the spectral representation of the matrix of rotat ionO0.We have (O0−λo1)ko=0⇒det(O0−λo1)=0, (2.78) from which it follows parenleftbig 1−λoOT 0parenrightbig ko=0⇒detparenleftbigg O0−1 λo1parenrightbigg =0, (2.79) Hence λ2 o=1⇒λo=±1. (2.80) It means that the matrix O0possesses two real eigenvalues. We skip the negative value which describes the mirror picture. Let us choose the refere nce system in such a way that one of the axes coincide with the eigenvector correspon ding toλo= 1. Then the matrixO0has the form (compare (1.17)) parenleftbigO0 klparenrightbig= 1 0 0 0 cosϕsinϕ 0−sinϕcosϕ . (2.81) 38 Geometry and kinematics of continua Clearly, this matrix possesses, in addition to real eigenva lues, the complex eigenvalues as well (1−λo)bracketleftBig (cosϕ−λo)2+(sinϕ)2bracketrightBig =0⇒ (2.82) ⇒λ(1) o=1orλ2o−2cosϕλo+1=0, i.e.λ(2,3) o=cosϕ±isinϕ=e±iϕ. These eigenvalues determine the angle of rotation ϕaround the single real eigenvector corresponding to λ0=1. For small angles of rotation, the orthogonal matrix contain s the nontrivial antisym- metric part O0=1+Ω0,parenleftbig Ω0 klparenrightbig = 0 0 0 0 0ϕ 0−ϕ0 ,ΩT 0=−Ω0. (2.83) The rigid displacement is characterized by the following Gu rtin Theorem: The following three statements are equivalent: 1.wis a rigid displacement ﬁeld, i.e. w=u0+Ω0(x−x0),ΩT0=−Ω0. (2.84) 2. The strain ﬁeld eis vanishing on the domain Bt. 3.whas the projection property on Bt, i.e. for any pair of points x,y∈Bt (w(x)−w(y))·(x−y)=0. (2.85) Namely, we have [u0+Ω0(x−x0)−u0−Ω0(y−x0)]·(x−y)=0, due to the antisymmetry of Ω0. Hence 1.⇒3. Now we take the derivative of (2.85) with respect toxand then with respect to y. Evaluating it at x=ywe obtain [(grad)xw(x)]T·(x−y)−(w(x)−w(y))=0 ⇒[gradw(x)]T+gradw(x)=2e(x)=0, i.e. 3.⇒2. Finally, for e=0we havegrad(gradw)=0. Hencewis a linear function of x:w=u0+A(x−x0)and, ase=0⇒A+AT=0, it must have the form of the rigid displacement which means 2. ⇒1. This completes the proof. The above Theorem implies immediately the Kirchhoﬀ Theorem : if two displacement ﬁeldsuandu′produce the same strain ﬁeld ethen u=u′+w, (2.86) wherewis a rigid displacement. We have, obviously, grad(u−u′)+[grad(u−u′)]T=0 which yields, according to Gurtin’s Theorem, that wis the rigid displacement. 2.4 Inﬁnitesimal strains 39 An arbitrary displacement u(x)can be split into the displacement caused by the deformation and this caused by the rigid rotation. Then for s mall deformations gradu=e+Ω, (2.87) where Ω=1 2parenleftBig gradu−(gradu)TparenrightBig , (2.88) and the vector ω=1 2rotu, ωk=−1 2εkijΩij, (2.89) is the rotation vector. Bearing for subbodies Pof small volumes Vthe relation (1.63) in mind as well as the relation (2.51) for the velocity under small deformatio ns (i.e. identical Lagrange and Euler reference systems) v(x,t)=∂u ∂t(x,t) (2.90) we have δV δt=integraldisplay Pdiv∂u ∂tdV=d dtintegraldisplay PdivudV≈Vtre δt⇒δV V=tre, (2.91) i.e.tredescribes the dilatation. A special important class of deformations is deﬁned by the ho mogeneous displacement ﬁeld u(x)=u0+A(x−x0), (2.92) where the point x0, the vectoru0and the tensor Aare independent of x. Ifu0=0and Ais symmetric then uis called a pure strain from x0. ⋆The following deformations for homogeneous displacement ﬁ elds should be men- tioned: 1) simple extension of amount e in the direction n,|n|=1, u=e((x−x0)·n)n,gradu=en⊗n, (2.93) e=en⊗ni.e.(eij)= e0 0 0 0 0 0 0 0 , Ω=0, for the base vectors {e1=n,e2,e3}; 2) uniform dilatation of amount e u=e(x−x0),gradu=e1, (2.94) e=e1i.e.(eij)= e0 0 0e0 0 0e, Ω=0; 40 Geometry and kinematics of continua 3) simple shear of amount κwith respect to the directions {m,n},m·n=0(compare (2.35)) and both these vectors are unit u=κm·(x−x0)n,gradu=κm⊗n (2.95) e=1 2κ(m⊗n+n⊗m)i.e.(eij)= 0κ/2 0 κ/2 0 0 0 0 0, Ω=1 2κ(m⊗n−n⊗m), for the base vectors {e1=m,e2=n,e3}.♣ 2.5 Compatibility conditions Providedeis given, the relation for the strain ﬁeld e=1 2parenleftBig gradu+(gradu)TparenrightBig , (2.96) can be considered to constitute the set of the ﬁrst order part ial diﬀerential equations for the displacement ﬁeld u. The uniqueness of solution is answered by the Kirchhoﬀ Theorem — two solutions diﬀer at most by a rigid displacement ﬁeldw. We answer now the question of existence of such solutions. Let us deﬁne the rotoperator for the tensor ein the following way ∀a=const[rote]a=rotbracketleftbigeTabracketrightbigi.e.rote=ǫijk∂emk ∂xjei⊗em. (2.97) Then, for the strain tensor rote=gradω, (2.98) where the rotation vector ωis deﬁned by the relation (2.89). It follows from the simple calculation rote=1 2rotparenleftBig gradu+(gradu)TparenrightBig =1 2gradrotu=gradω, (2.99) or, in coordinates ǫijk∂emk ∂xj=1 2ǫijk∂ ∂xjparenleftbigg∂um ∂xk+∂uk ∂xmparenrightbigg =1 2ǫijk∂2uk ∂xj∂xm=∂ωi ∂xm. (2.100) Hence rotrote=0, (2.101) where we have used the relation (2.99) rotrote=ǫpnm∂2ωi ∂xm∂xnep⊗ei≡0. 2.5 Compatibility conditions 41 The relation (2.101) is the equation of compatibility. It can be proved that the equation of compatibility is suﬃcie nt for the existence of solutions of the equation (2.96) provided the domain Btis simply connected. We leave out a rather simple proof (e.g. [4]). A similar but much more complicated relation can be shown for the nonlinear theory (see: Sect. 2.5 of [22]). It follows from the assumption that the space of motion is Euclidean which means that its tensor of curvature is zero. W e shall not present this problem in these notes. However, we discuss further the case of the body with continuous distribution of dislocations for which this condition is vi olated. The compatibility condition in the explicit form is as follo ws ǫijkǫlmn∂2ejm ∂xk∂xn=0. (2.102) It can be easily shown that for a particular case of plane disp lacements in which u=u1e1+u2e2the set of six independent equations (2.102) reduces to the s ingle relation 2∂2e12 ∂x1∂x2=∂2e11 ∂x2∂x2+∂2e22 ∂x1∂x1. (2.103) This equation is used in the construction of the so-called Be ltrami-Michell equations of linear elasticity which we have already mentioned at the beg inning of this Chapter and which we present in Chapter 5. 42 Geometry and kinematics of continua Chapter 3 Balance of mass and momentum Classical continuum mechanics is primarily concerned with the search for solutions for the function of motion. In the Lagrangian description curre nt values of mass density are then given by a simple kinematical relation. Field equat ions for these ﬁelds follow from two fundamental equations of physics — conservation of mass and momentum. Some additional ﬁelds such as density of dislocations or plastic deformations require additional evolution equations which we discuss further in these notes . In this Chapter we present these two conservation laws of mechanics in both Lagrangian and Eulerian description. We mention as well the consequences of the third mechanical c onservation law — moment of momentum conservation. It is a basis for the ﬁeld equation s in models of systems in which an additional local degree of freedom, the rotation (s pin), plays a rule. This is, for instance, the case for liquid crystals. For classical co ntinua considered in these notes it restricts the form of the stress tensor, an object appeari ng in the conservation law for momentum. 3.1 Conservation of mass As in the mechanics of points the mass in continuum mechanics is a measure of inertia of subbodies of the body B0. SubbodiesP⊂B0are subsets of a certain mathematical structure which we do not need to present in details. Each sub body has a prescribed massM(P)>0and we make the assumption (continuity) that this quantity c an be described by the density ρ0 M(P)=integraldisplay Pρ0(X)dV. (3.1) In contrast to material points of classical mechanics of poi nts and rigid bodies the ma- terial pointX∈B0possesses no mass, the mass density ρ0serves only the purpose of determining the mass of ﬁnite subbodies through the relatio n (3.1). In this sense, the 43 44 Balance of mass and momentum mass density is not measurable. We measure in laboratories t he mass of bodies and assuming the homogeneity we deﬁne the mass density as the fra ction ρ0≈M(P) V(P). (3.2) According to the law of mass conservation the mass M(P)does not change in any process dM(P) dt=0. (3.3) We may have mass supplies from other components due to chemic al reactions or phase changes but this requires the construction of continuum mec hanics of mixtures of many components which we shall not present in these notes. This as sumption on conservation does not mean that mass density remains constant in processe s. Its changes are connected with changes of the volume occupied by the subbody Pat any instant of time. On the other hand this volume is given by the function of motion. Namely, in a current conﬁguration the points X∈Poccupy the following domain in the space of motion E3 Pt={x|x=f(X,t),X∈P}. (3.4) Such domains we call material. Hence the mass of Pcan be written in the form M(P)=integraldisplay PtρdVt, (3.5) whereρis the current mass density and dVtis the volume element in the current conﬁg- uration. The element dVtis the transformation of the element dV=dX1dX2dX3to the current conﬁguration. In Cartesian coordinates we can writ e it in the following way dVt= [(Fe1dX1)×(Fe2dX2)]·(Fe3dX3)= = [(Fk1ek)×(Fl2el)]·(Fm3em)dV= (3.6) =εklmFk1Fl2Fm3dV=JdV, J=detF, where the relation (1.40) has been used and we have used the sa me base vectors in Lagrangian and Eulerian coordinates ek=δkKeK.Consequently ρ=ρ0J−1. (3.7) Hence for a given deformation gradient Fchanges of the mass density are determined. For this reason, the ﬁeld of mass density does not appear amon g unknown ﬁelds in the Lagrangian description. Obviously, in the reference conﬁg urationJ=1. Both the initial mass density ρ0and the current mass density must be positive. Consequently , we have the condition J >0. (3.8) This condition yields the local invertibility of the functi on of motion x=f(X,t): X=f−1(x,t). 3.1 Conservation of mass 45 The relation (3.5) can be also written in the local form in tim e d dtintegraldisplay PtρdVt=0, (3.9) for all material domains Pt. Making use of the Leibniz Theorem (1.56) we can write it in the following form integraldisplay Pt∂ρ ∂tdVt+contintegraldisplay ∂Ptρv·ndSt=0. (3.10) Bearing the Gauss Theorem (1.60) in mind, we obtain integraldisplay Ptparenleftbigg∂ρ ∂t+div(ρv)parenrightbigg dVt=0. (3.11) Hence, for almost all points of the body (except of points whe reρis discontinuous) the following local form of the mass conservation in Eulerian de scription follows ∂ρ ∂t+div(ρv)=0, (3.12) i.e.∂ρ ∂t+∂ ∂xk(ρvk)=0. Obviously, (3.7) fulﬁlls this equation identically. In ord er to see it, we rewrite the above equation in the form parenleftbigg∂ρ ∂t+∂ρ ∂xkvkparenrightbigg +ρ∂vk ∂xk=0. (3.13) The expression in the parenthesis is called the material tim e derivative. It is identical with the partial time derivative in the Lagrangian descript ion. Namely we have ∂ρ(fk(X,t),t) ∂t=∂ρ ∂t+∂ρ ∂xk∂fk ∂t=∂ρ ∂t+∂ρ ∂xkvk. (3.14) Hence, in Lagrangian coordinates, (3.13) yields ∂J−1 ∂t+J−1F−T·∂F ∂t=0,F−T·∂F ∂t=∂FkK ∂tF−1 Kk. (3.15) On the other hand, the rule of diﬀerentiation of determinant s yields ∂J−1 ∂t=−1 J2∂J ∂F·∂F ∂t=−1 J2JF−T·∂F ∂t, (3.16) and (3.15) becomes the identity. Apart from the continuity relation (3.12) the relation (3.1 0) yields also a very useful relation on moving surfaces on which the mass density posses ses a ﬁnite discontinuity. 46 Balance of mass and momentum In order to ﬁnd this relation, we consider ﬁrst a descending f amily of subbodies {Pi} intersecting the surface Sin the reference conﬁguration. We assume that this surface moves in the direction of its unit normal Nwith the speed U, i.e. the velocity of points of the singular surface is UN. The family of subbodies is descending in this sense that at each instant of time they have the same common part with the singular surface and their volume is diminishing to zero with the growing index i ∀i/negationslash=jPi∩S=Pj∩S,lim i→∞(volPi)=0,volPi=integraldisplay PidV. (3.17) This construction with an appropriate orientation of norma l vectors is shown in Fig. 3.1. Fig. 3.1: A singular surface Sintersecting the family of subbodies {Pi} Now the relation (3.3) can be written in the form d dtintegraldisplay Piρ0dV=d dtintegraldisplay P+ iρ0dV+d dtintegraldisplay P− iρ0dV,P+ i∪P− i=Pi, (3.18) whereP+ i,P− iare the parts ofPicreated by the intersection with S. Obviously, ρ0is constant within each of these two subbodies but it does not ha ve to possess the same value if the surface Sis the surface of discontinuity for the mass density. Simult aneously, due to the motion of Sthe subbodiesP+ i,P− ichange in time. Hence, according to the Leibniz Theorem, we obtain d dtintegraldisplay Piρ0dV=−integraldisplay ∂P+ i∩Sρ+ 0UdS+integraldisplay ∂P− i∩Sρ−0UdS=0, (3.19) where the opposite sign in the integrals follows from the opp osite orientation of the outward oriented normal vector of P+ iand the normal vector of the surface S. The 3.1 Conservation of mass 47 quantitiesρ+ 0andρ−0are, of course, the values of the mass density on two sides of t he surfaceS. Hence, this surface is carrying the discontinuity of the ma ss density [[ρ0]]=ρ+ 0−ρ−0. (3.20) Taking the limit i→∞in (3.19) we obtain the following local form of this relation U[[ρ0]]=0. (3.21) In a particular case of the surface Swhich is not moving — we say then that the surface is material — we have U=0and the jump of mass density [[ρ0]]may be arbitrary. If it is diﬀerent from zero the surface Sis the surface of contact of two diﬀerent materials. Otherwise the initial mass density ρ0must be continuous [[ρ0]]=0. The same considerations can be repeated for the current conﬁ guration. Instead of the surfaceSwe have to consider its current image Stmoving with the speed c, i.e. its velocity iscn. The subbodies in the reference conﬁguration have to be tran sformed to the current conﬁguration Pi(t)={x|x=f(X,t),X∈Pi}.Then (3.9) yields integraldisplay Pi(t)∂ρ ∂tdV+contintegraldisplay ∂P+ i(t)ρw·ndSt+contintegraldisplay ∂P− i(t)ρw·ndSt=0, (3.22) wherew=vfor points of the surfaces beyond Sandw=cnfor points onS. Again taking the limit, we obtain [[ρ(v·n−c)]]≡ρ+parenleftbig v+·n−cparenrightbig −ρ−parenleftbig v−·n−cparenrightbig =0 (3.23) where, obviously, v±are the limits of the particle velocity on the surface St. The relation (3.23) is Eulerian counterpart of Lagrangian relation (3.2 1). It is clear that for the material surfaceStwe havev·n=cand the particle velocity is continuous. The above presented analysis can be repeated for the equatio n (2.70) which has the formal structure of balance equation. We have d dtintegraldisplay PiFkKdV=d dtintegraldisplay P+ iFkKdV+d dtintegraldisplay P− iFkKdV= =integraldisplay Pi∂FkK ∂tdV−integraldisplay ∂P+ i∩SF+ kKUdS+integraldisplay ∂P− i∩SF− kKUdS= (3.24) =contintegraldisplay ∂PivkNKdS, whereF± kKare the limits of the deformation gradient on both sides of th e surfaceS. Taking the limit described above we obtain [[FkK]]U+[[vk]]NK=0,[[FkK]]=F+ kk−F− kK,[[vk]]=v+ k−v− k. (3.25) This is the kinematical compatibility condition which is th e part of the Hadamard The- orem and which we have mentioned in Section 2.3. It says that t he discontinuity of velocity on a singular surface — it may be, for instance, the f ront of the so-called shock wave — yields necessarily the discontinuity of deformation andviceversa . 48 Balance of mass and momentum 3.2 Conservation of momentum 3.2.1 Lagrangian description The momentum for continuum is again deﬁned for subbodies rat her than points as we know it from the classical mechanics of mass points. Namely, it refers to the motion of the center of gravity of small material portions and then for an arbitrary subbody Pit has the form M(P)=integraldisplay Pρ0vdV. (3.26) Its time changes depend on interactions with the external wo rld of the subbody. If these interactions are zero the momentum is conserved dM dt=0. (3.27) However, if we cut Pfrom the bodyB0which itself interacts with the external world it cannot be expected that the interactions of subbody vanis h. They are transmitted through the surface ∂Pas well as directly into the interior of P. The latter may be, for instance, gravitational or they may be created by the rot ational motion of the body in the form of centrifugal or Coriolis forces. Consequently , instead of (3.27) we have to writedM dt=contintegraldisplay ∂PtNdS+integraldisplay Pρ0bdV, (3.28) wheretNare the so-called tractions, i.e. forces acting on a unit sur face of∂Pin the reference conﬁguration, and bare called body forces. They may result from the action of the gravity but they may be as well a consequence of the moti on of the body as a whole which results, for instance, in centrifugal forces. A . L. Cauchy (1789-1857) has shown that the tractions are linear functions of the unit nor mal vectorNof the surface ∂P, i.e. [22] tN=PN,i.e.t(N) k=PkKNK,tN=t(N) kek,P=PkKek⊗eK, (3.29) wherePis independent of N. This object is called Piola-Kirchhof stress tensor even though it is not a tensor. t(N) kare components of the stress vector for a particular choice of the subbodyP, i.e. for a particular choice of the surface ∂P. Making use of the Gauss Theorem we can write (3.28) in the form integraldisplay Pparenleftbigg ρ0∂v ∂t−DivP−ρ0bparenrightbigg dV=0, (3.30) which must hold for all subbodies of B0. Consequently, for almost all points of the body B0 ρ0∂v ∂t=DivP+ρ0b, (3.31) i.e.ρ0∂vk ∂t=∂PkK ∂XK+ρ0bk. 3.2 Conservation of momentum 49 This is the local form of the momentum conservation law in Lag rangian description. Obviously∂vk/∂t≡∂2fk/∂t2is the acceleration in this description. The most important consequence of the above analysis is the e xistence of the stress tensorP. However, the Lagrangian description has the disadvantage of referring to a nonphysical reference surface. Practical applications ar e mostly based on the Eulerian description. We return later to the comparison of both appro aches. 3.2.2 Eulerian description Let us make the transformation from the Lagrangian to Euleri an description in the momentum balance (3.30). We have integraldisplay Ptparenleftbigg ρ0∂v ∂t−DivP−ρ0bparenrightbigg J−1dVt=0, (3.32) where (3.6) has been used and the function in parenthesis sti ll depends onxthrough the function of motion f−1(x,t). However DivP=∂PkK ∂XKek=∂PkK ∂xl∂xl ∂XKek=F·(gradP), (3.33) Using the identity (2.67) 2, the relation (3.7) and making the transformation of variab les X→xwe have integraldisplay Ptparenleftbigg ρparenleftbigg∂v ∂t+(v·gradv)parenrightbigg −divT−ρbparenrightbigg dVt=0,T=J−1PFT, (3.34) i.e.σkl=J−1PkKFlK,T=σklek⊗el. The new stress tensor Tis called the Cauchy stress tensor. Components of the Cauchy stress tensor are often deﬁned in Cartesian coordinates x1=x, x2=y, x3=zand base vectors{ex,ey,ez}. Then the components are denoted in the following way (σkl)= σxτxyτxz τxyσyτyz τxzτyzσz. (3.35) The components σ x,σy,σzare called normal stresses and the components τxy,τxz,τyz are called shear stresses. Normal components are also descr ibed by a double index σx=σxx,σy=σyy,σz=σzz. Now transforming the stress contribution from the volume to surface integral we obtain integraldisplay Ptρparenleftbigg∂v ∂t+(v·gradv)parenrightbigg dVt−contintegraldisplay ∂PtTndSt=integraldisplay PtρbdVt. (3.36) Consequently, the stress vector (traction) in the current c onﬁguration is given by the relation tn=Tni.e.t(n) k=σklnl. (3.37) 50 Balance of mass and momentum In contrast to the stress vector (traction) tNappearing in the relation (3.28) the above stress vector describes the force of surface interaction pe r unit area of the current surface ∂Pt. The relation (3.37) is called the Cauchy relation. The local form of the momentum balance (3.34) ρparenleftbigg∂v ∂t+(v·gradv)parenrightbigg =divT+ρb, (3.38) i.e.ρparenleftbigg∂vk ∂t+vl∂vk ∂xlparenrightbigg =∂σkl ∂xl+ρbk. is called the Cauchy equation. In ﬂuid mechanics the local momentum equation is also used in a diﬀerent form ρparenleftbigg∂vk ∂t+2vlWklparenrightbigg =parenleftbigg∂σkl ∂xl−1 2ρ∂v2 ∂xkparenrightbigg +ρbk, v2=v·v, (3.39) whereWklis the spin (compare (2.65)) and which follows from the ident ity vl∂vk ∂xl=2vlWkl+1 2∂v2 ∂xk. (3.40) Then for potential (irrotational) ﬂows v=gradϕ,i.e.vk=∂ϕ ∂xk⇒W=0, (3.41) and the momentum balance becomes ∂vk ∂t=1 ρ∂σkl ∂xl−∂ ∂xkparenleftbigg1 2v2+Φparenrightbigg ,b=−gradΦ, (3.42) which immediately yields the Bernoulli Theorems for ideal ﬂ uids (e.g. [21]). Namely, for ideal ﬂuids which do not carry the shear stresses the stre ss tensor is reduced to the diagonal form T=−p1i.e.σkl=−pδkl, (3.43) wherep=−1 3σkkis the pressure. Then an arbitrary direction is principal fo r this stress tensor. Equation (3.42) has now the following form (it is ass umed thatρ=ρ(p)!) ∂vk ∂t=−∂ ∂xkparenleftbigg P+1 2v2+Φparenrightbigg , P=integraldisplaydp ρ+const., (3.44) wherePis the so-called pressure function. Obviously, for slow mot ions in which one can neglect the acceleration this relation indicates that P+1 2v2+Φis only a function of time, and this, in turn, yields the Bernoulli Theorem. This f orm of momentum balance yields the theory of water waves. As in the case of mass balance equation, the conservation of m omentum yields condi- tions on a singular surface which may carry the discontinuit ies of mass and momentum. 3.2 Conservation of momentum 51 In the Lagrangian description, we have to introduce again a d escending family of sub- bodies{Pi}deﬁned by (3.18) and then (3.28) can be written in the form integraldisplay Piρ0∂v ∂tdV−integraldisplay ∂P+ i∩Sρ+ 0v+UdS+integraldisplay ∂P− i∩Sρ−0v−UdS=contintegraldisplay ∂PiPNdS+integraldisplay Piρ0bdV,(3.45) for each member Piof this family. Again the minus sign follows from the opposit e orientation of normal vectors. In the limit i→∞we obtain ρ0U[[v]]+[[PN]]=0, (3.46) whereNis the unit normal of the surface S. Consequently, a discontinuity of the stress vector (traction) on a singular surface yields the disconti nuity of the velocity and this, in turn, according to Hadamard Theorem (compare (3.25)), yi elds a discontinuity of the deformation. For the Eulerian description we obtain the following dual re lation integraldisplay Pi(t)ρ∂v ∂tdV+integraldisplay ∂P+ i(t)ρvw·ndS+integraldisplay ∂P− i(t)ρvw·ndS=contintegraldisplay ∂Pi(t)TndSt+integraldisplay Pi(t)ρbdV,(3.47) Again, in the limit we obtain (compare (3.22), (3.23)) [[ρ(v·n−c)v]]−[[Tn]]=0. (3.48) Obviously, the expression (v·n−c)describes the motion of the particles with respect to the moving singular surface in the direction normal to thi s surface. If we account for the mass conservation (3.23) then ρ(v·n−c)[[v]]−[[Tn]]=0. (3.49) where the value ρ(v·n−c)is taken on any side of the surface as they are equal. For material surfaces we obtain, of course, [[Tn]]=0, (3.50) i.e. the continuity of tractions. This relation is fundamen tal for the formulation of bound- ary conditions in terms of stresses. Otherwise, if the tract ion is discontinuous on the nonmaterial surface it yields necessarily the discontinui ty of velocity. The relations for mass and momentum conservations on singul ar surfaces (3.23), (3.49) are called the dynamic compatibility conditions. 3.2.3 Moment of momentum It can be easily shown that the moment of momentum conservati on yields the symmetry of the Cauchy stress tensor. The classical deﬁnition of the m oment of momentum for the subbodyP⊂B0has the form K(t)=integraldisplay Pρ0x×vdV, (3.51) 52 Balance of mass and momentum wherex=f(X,t)is the position of the point X∈Pat the instant of time t. Its time changes are described by the relation d dtintegraldisplay Pρ0x×vdV=contintegraldisplay ∂Px×(PN)dS+integraldisplay Pρ0x×bdV. (3.52) In Cartesian coordinates this relation can be written in the form integraldisplay Pρ0εijkxj∂vk ∂tdV=contintegraldisplay ∂PεijkxjPkKNKdS+integraldisplay Pρ0εijkxjbkdV. (3.53) In the ﬁrst integral, the observation that the contribution with the derivative ∂xj/∂t=vj is identically zero has been used. The surface integral can b e transformed in the following waycontintegraldisplay ∂PεijkxjPkKNKdS=integraldisplay Pεijkparenleftbigg FjKPkK+xj∂PkK ∂XKparenrightbigg dS. (3.54) Bearing (3.31) in mind, we obtain εijkFjKPkK=0, (3.55) in almost all points of B0. Consequently, the deﬁnition of the Cauchy stress tensor (3 .34) yields εijkσjk=0,i.e.T=TT. (3.56) Hence, the conservation of moment of momentum yields the sym metry of the Cauchy stress tensor. 3.2.4 Stress analysis Due to the symmetry of the Cauchy stress tensor we can easily s olve the problem of the biggest and smallest local stresses. We begin this analy sis with the maximum and minimum of normal stresses, i.e. these stress components wh ich are perpendicular to the surface of the cross-section of the system. If ndenotes the unit vector perpendicular to the surface at the point xin the current conﬁguration then we have to ﬁnd this vector for which the stress vector projected on n, i.e.n·tn=n·(Tn)is maximum or minimum. We have solved this problem already for tensors of deformati on (compare (2.19)). The directionnis given by the solution of the eigenvalue problem (T−σ1)n=0, (3.57) whereσare eigenvalues of T. These are given by the equation det(T−σ1)=0. (3.58) In the explicit form it reads σ3−Iσσ2+IIσσ−IIIσ=0, (3.59) 3.2 Conservation of momentum 53 where Iσ= trT=σ11+σ22+σ33=σ(1)+σ(2)+σ(3), IIσ=1 2parenleftbigI2 σ−trT2parenrightbig=vextendsinglevextendsinglevextendsinglevextendsingleσ11σ12 σ12σ22vextendsinglevextendsinglevextendsinglevextendsingle+vextendsinglevextendsinglevextendsinglevextendsingleσ 11σ13 σ13σ33vextendsinglevextendsinglevextendsinglevextendsingle+vextendsinglevextendsinglevextendsinglevextendsingleσ 22σ23 σ23σ22vextendsinglevextendsinglevextendsinglevextendsingle= =σ (1)σ(2)+σ(1)σ(3)+σ(2)σ(3), (3.60) IIIσ= detT=vextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsingleσ 11σ12σ13 σ12σ22σ23 σ13σ23σ33vextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsingle=σ (1)σ(2)σ(3), are the principal invariants of the stress tensor T. The name ’invariant’ refers to the fact that these quantities remain the same in all frames of refere nce, i.e. they are independent of the choice of the base vectors. Once we have the three solut ionsσ(1),σ(2),σ(3)of the equation (3.59), which are called principal values of the st ress tensorT, we can ﬁnd from the set (3.57) three corresponding normalized eigenve ctorsn(1),n(2),n(3). They are called principal directions of the stress tensor T. Obviously, they are orthogonal, i.e.n(α)·n(β)=δαβ. According to the relation (3.57) we have n(α)·Tn(β)=0forα/negationslash=β. (3.61) Hence, if we choosebraceleftbig n(1),n(2),n(3)bracerightbig as the basis vectors we obtain the following spectral representation of stress tensor T=3summationdisplay α=1σ(α)n(α)⊗n(α). (3.62) The graphical representations of the stress tensor Tin an arbitrary Cartesian basis {ex,ey,ez}and in the principal directionsbraceleftbign(1),n(2),n(3)bracerightbigare shown in Fig. 3.2. Fig. 3.2: Stress components in Cartesian coordinates — an ar bitrary coordinate system (left panel) and the system of principal d irections (right panel) 54 Balance of mass and momentum In order to appreciate the notions of stress tensors we prese nt a few simple examples. ⋆Let us begin with an example of the Cauchy stress tensor for wh ich we want to ﬁnd principal stresses and the principal directions. In a chose n frame of Cartesian coordinates it has the form of the following matrix (σij)= 5 2−3 2 4 1 −3 1−3,T=σ ijei⊗ej, (3.63) where the inputs are in MPa. The corresponding eigenvalue pr oblem has the form  5−σ2−3 2 4−σ1 −3 1−3−σn 1 n2 n3=0 0 0 , (3.64) and the eigenvector nshould be of the unit length n=n iei,n·n=(n1)2+(n2)2+(n3)2=1. (3.65) Obviously, the determinant of the set of equations (3.64) mu st be zero. Hence −det(T−σ1)=σ3−Iσσ2+IIσσ−IIIσ=0. (3.66) Coeﬃcients of this equation, the principal invariants of th e stress tensor T, are given by the relations Iσ= trT=6MPa, IIσ=1 2parenleftbigIσ−trT2parenrightbig=−21(MPa)2, (3.67) IIIσ= detT=−101(MPa)3. The solution of (3.66) (it has ben obtained by Maple7 )has the form σ(3)=−4.3099MPa, σ(2)=3.3832MPa, σ(1)=6.9267MPa. (3.68) It is easy to check that the following identities are satisﬁe d Iσ=σ(1)+σ(2)+σ(3), IIσ=σ(1)σ(2)+σ(1)σ(3)+σ(2)σ(3),(3.69) IIIσ=σ(1)σ(2)σ(3). Now, by means of the equations (3.64) and the normalization c ondition (3.65) we ﬁnd the corresponding eigenvectors. They are as follows n(1)= 0.8394e1+0.5043e2−0.2029e3, n(2)= 0.4256e1−0.8419e2−0.3319e3, (3.70) n(3)= 0.3382e1−0.1922e2+0.9212e3. 3.2 Conservation of momentum 55 Obviously, due to the relation n(α)·ei=cosparenleftbigparenleftbign(α),eiparenrightbigparenrightbig, the components of eigenvectors are cosines of the angles between eigenvectors and correspo nding base vectors. We have parenleftBig n(1),e1parenrightBig = 70.230,parenleftBig n(1),e2parenrightBig =101.080,parenleftBig n(1),e3parenrightBig =22.900, parenleftBig n(2),e1parenrightBig = 64.810,parenleftBig n(2),e2parenrightBig =147.340,parenleftBig n(2),e3parenrightBig =109.380,(3.71) parenleftBig n(3),e1parenrightBig = 32.920,parenleftBig n(3),e2parenrightBig =59.720,parenleftBig n(3),e3parenrightBig =101.710. The spectral representation of the stress tensor has the for m T=6.9267n(1)⊗n(1)+3.3832n(2)⊗n(2)−4.3099n(3)⊗n(3). (3.72) Hence, the biggest value of the stress is n(1)·Tn(1)= 6.9267MPa, it is tension (positive value!) in the direction n(1). The smallest value is n(3)·Tn(3)=−4.3099MPa, it is compression (negative value!). On surfaces perpendic ular to the principal directions n(1),n(2),n(3)shear stresses are equal to zero: n(α)·Tn(β)=0forα/negationslash=β.♣ ⋆In the second example we calculate the stress vector tnon the plane intersecting the cube which is loaded on the faces by the stress Tgiven by the formula (3.63). The plane crosses the points (1,0,0.5),(0,1,1),(1,1,0)as shown in Fig. 3.3. Fig. 3.3: Intersection of the cube discussed in the example On the same Figure we indicate the components of stresses act ing on the back faces of the cube. The plane intersecting the cube is, obviously, g iven by the equation p(x1,x2,x3)=x1+0.5x2+x3−1.5=0. (3.73) 56 Balance of mass and momentum The vector perpendicular to this plane is parallel to gradp=e1+0.5e2+e3and, after the normalization, it yields the following normal vector n n=2 3e1+1 3e2+2 3e3. (3.74) Now the stress vector on the plane is speciﬁed by the relation tn=Tn=σijnjei=2e1+10 3e2−3e3. (3.75) The normal component of the stress acting in the intersectio n by the plane follows as σn=n·tn=4 9=0.4444MPa. (3.76) Consequently, the shear stress on this plane is given by the v ector τn=tn−σnn=1 27(46e1+86e2−89e3),|τn|=4.8902MPa. (3.77) ♣ ⋆In this example we show the diﬀerence between the stress tens ors of Piola-Kirchhoﬀ and Cauchy. Let us consider again the deformation of the pris m described in the example (2.7). We make the simplifying assumption that the material is incompressible, i.e. its volume remains constant during the deformation. This is pra ctically fulﬁlled for many materials, e.g. for rubber-like materials. Then J=1⇒(1+ε1)(1+ε2)(1+ε3)=1. (3.78) Assuming that the prism is loaded by the force Pin the direction e3we conclude from symmetryε1=ε2. Hence 1+ε1=1+ε2=1√1+ε3. (3.79) The reference area on which the external force Pis distributed, say A0=(ae1)·(ae2)= a2, changes due to the deformation to A=(aFe1)·(aFe2)=a2(1+ε1)2=a2/(1+ε3). Consequently, the normal components of the Piola-Kirchhoﬀ and Cauchy stress in the e3-direction, respectively, are as follows P33=P a2, σ33=P a2(1+ε3). (3.80) Hence, even a moderate elongation of, say, ε3= 0.1yields the 10% diﬀerence in these components. The component σ33which is indeed measured in laboratory is bigger than the component of the Piola-Kirchhoﬀ stress. This diﬀerence may even yield a change in the behaviour of stresses in function of deformation. There are cases where the growing Cauchy stress corresponds to a decaying Piola-Kirchhoﬀ str ess andviceversa . This may lead to erroneous conclusions concerning the so-called mat erial stability.♣ 3.2 Conservation of momentum 57 There are two two-dimensional cases in which the above stres s analysis possesses a simple geometrical interpretation. The ﬁrst case — plane st resses — appears when the stress tensor consists of four non-zero components (σij)= σxτxy0 τxyσy0 0 0 0, (3.81) whereσ x≡σxx,σy=σyy. The second case — plane strains — appears when the stress tens or consists of ﬁve non-zero components (σij)= σxτxy0 τxyσy0 0 0σz, (3.82) withσ x≡σxx,σy=σyy,σz≡σzz. Then the eigenvalue problem (T−σ1)n=0, (3.83) Fig. 3.4: Transformation of the stress tensor for 2D-case yieldsσ(3)=0andn(3)=ezin the ﬁrst case and σ(3)=σzandn(3)=ezin the second case. For this reason it is suﬃcient to consider the problem o n the plane perpendicular toez. We consider the components of the stress on the plane perpen dicular to the unit vectorn= cosϕex+sinϕey(see Fig. 3.4.). It is convenient to introduce the vector perpendicular to n:n+=−sinϕex+cosϕey. The stress vector tnis obviously given by the relation tn=Tn=(σxcosϕ+τxysinϕ)ex+(τxycosϕ+σysinϕ)ey. (3.84) 58 Balance of mass and momentum Then the normal component σnand the tangential component τnare as follows σn=tn·n=σx+σy 2+σx−σy 2cos2ϕ+τxysin2ϕ, (3.85) τn=tn·n+=−σx−σy 2sin2ϕ+τxycos2ϕ. If we eliminate the angle ϕfrom these relations we obtain the relation which should hold for an arbitrary intersection of the square on the plane perpendicular to ez. This equation has the form τ2 n+parenleftbigg σn−σx+σy 2parenrightbigg2 =parenleftbiggσx−σy 2parenrightbigg2 +τ2 xy. (3.86) This is the equation of the circle on the plane of (σn,τn)-variables whose center lies on the axisτn=0at the point1 2(σx+σy). It is called Mohr’s circle. The radius of this circle is equal toradicalBig 1 4(σx−σy)2+τ2xy. For a given tensor (3.81) or (3.82) the angle ϕ yields the values of the components of the stress vector on th e plane of intersection as indicated in Fig. 3.5. As a particular case of relations (3.85) we can locate the pos ition of the principal direction and the principal values of stresses for this two- dimensional case. We use the property of the principal direction that the shear stress on the plane perpendicular to this direction is vanishing. Then (3.85) 2yields Fig. 3.5: Mohr’s circle tan2ϕ0=2τxy σx−σy. (3.87) 3.2 Conservation of momentum 59 Certainly, this corresponds to the point of intersection of the circle with the line τn=0. The values of normal stresses for this angle are σ(1),(2)=σx+σy 2±radicalBiggparenleftbiggσx−σy 2parenrightbigg2 +τ2xy, (3.88) σ(1)= maxσn, σ(2)=minσn, It follows as well from the Mohr circle that the maximum shear stresses are equal to the radius of the circle maxτn=radicalBiggparenleftbiggσx−σy 2parenrightbigg2 +τ2xy≡σ(1)−σ(2) 2, (3.89) and they appear on the plane which forms angles π/4and3π/4with principal directions. Let us mention that a similar construction can be also made fo r the general three- dimensional case. Below we present an example of such a const ruction. It is performed for three principal directions separately. Then for each di rection one can perform a two-dimensional construction described above. This follo ws from the fact that in the principal direction shear stress is equal to zero which mean s that the stress distribution in the remaining directions is two-dimensional. This const ruction is useful, for instance, in motivation of various yield criteria. Applications in ro ck mechanics are discussed by Jaeger, Cook and Zimerman [6]. ⋆We construct Mohr’s circles for the stress tensor (σij)= 1−9 2 −9 1 2 2 2 16. (3.90) Principal stresses and corresponding principal direction s are as follows σ (1)= 16.3288,n(1)=−0.7024e1−0.7024e2+0.1155e3. σ(2)= 10.00n(2)=0.7071e1−0.7071e2, (3.91) σ(3)=−8.3288,n(3)=0.08166e1+0.08166e2+0.9933e3, The construction of Mohr’s circles is shown in Fig. 3.6. Each circle is constructed for two-dimensional coordinates on planes perpendicular t o the corresponding principal direction. For instance, the left circle corresponds to str ess distribution on the plane perpendicular to n(3). Any state of stresses in an arbitrary cross-section is a poi nt of the 60 Balance of mass and momentum dashed area. Fig. 3.6: An example of Mohr’s circles for three-dimensiona l stress distribution.♣ However, in contrast to the two-dimensional case presented above, this three-dimen- sional construction does not seem to have any practical bear ing. It shows only that the most important principal stresses σ(1),σ(2),σ(3)determine the maximum shear stresses asparenleftbigσ(1)−σ(3)parenrightbig/2,parenleftbigσ(2)−σ(3)parenrightbig/2,parenleftbigσ(1)−σ(2)parenrightbig/2and the orientations of planes on which they act are determined by the angle π/4between the normals to those planes and the corresponding principal directions. Chapter 4 Thermodynamics of solids Continuum mechanics describes not only processes of deform ations caused by the me- chanical forces but it must account for various nonmechanic al eﬀects which are necessarily coupled to mechanical processes. For instance, any process in which a dissipation of en- ergy appears, and such are processes in viscoelastic or plas tic materials, must account for the nonmechanical transport of energy, for example by co nduction. It is also natural to include stresses which appear due to changes of temperatu re (the so-called thermal stresses) or stresses caused by chemical reactions in struc tural elements. These phe- nomena require some thermodynamical considerations. For t his reason, we present in this Chapter fundamental elements of continuum thermodyna mics, in particular the ﬁrst law, called the principle of conservation of energy and the s econd law, called the entropy inequality. Even in the case of purely mechanical processes in which the d issipation does not appear the energy conservation law may be quite useful as an e xample of some variational principles of classical elasticity clearly shows. We retur n to these problems later. 4.1 Energy conservation law We begin from the formulation of the law of the energy conserv ation in the Lagrangian description. Even though the origin of the subject goes back to the works of Fourier at the beginning of XIX century the formulation of this law can b e found ﬁrst in the work of J. R. Mayer in 1842. However the real modern development in this ﬁeld began at the end of XIX century with works of Maxwell, Boltzmann and many o thers. The classical systems without the so-called internal degre es of freedom are charac- terized by the total energy consisting of two contributions : the internal energy and the kinetic energy. The ﬁrst one describes the accumulation of e nergy in the system due to interactions of particles which change their microstates a s a consequence of deformation and temperature. If a subsystem is completely isolated from the external world its total energy remains constant. This is, of course, not the case if f orces acting on the sys- tem perform working or the contact with other subbodies yiel ds the exchange of energy 61 62 Thermodynamics of solids caused by the diﬀerence of temperature. Bearing all these ag ents in mind, we can write the energy conservation law for any subbody P⊂B0in the form d dtintegraldisplay Pρ0parenleftbigg ε+1 2v2parenrightbigg dV=contintegraldisplay ∂P(−Q·N+tN·v)dS+integraldisplay Pρ0(b·v+r)dV, (4.1) whereεdenotes the internal energy density per unit mass of the refe rence conﬁguration and1 2ρ0v2≡1 2ρ0v·vis the density of the kinetic energy per unit volume of the ini tial conﬁguration. The surface terms consist of the working of st ressestN·vand of the nonmechanical transfer of energy per unit reference surfac e and unit time, Q·N, in which Qis called the heat ﬂux vector. The vectorial form of the heat ﬂ ux (i.e. the linearity with respect to the normal vector N) is the consequence of the Cauchy Theorem analogous to this which we have used in the construction of the stress tens ors. The volume integral on the right-hand side describes the supply of energy by the wor king of body forces b·vand the radiation r. The last contribution plays no role for solids as the supply of energy in the form of radiation is essential only on boundaries of soli ds. Therefore we shall neglect it in further considerations. The transformation to the local form requires the same steps as in the case of mo- mentum conservation law. Bering the relation (3.29) in mind , we obtain for almost all points of the body B0 ρ0∂ ∂tparenleftbigg ε+1 2v2parenrightbigg +DivparenleftBig Q−PTvparenrightBig =ρ0b·v. (4.2) Diﬀerentiation of the kinetic energy and power of stresses y ields ρ0∂ε ∂t+DivQ+v·parenleftbigg ρ0∂v ∂t−DivPparenrightbigg −P·Gradv=v·(ρ0b). Bearing the momentum conservation (3.31) in mind we obtain ρ0∂ε ∂t+DivQ=P·Gradv, (4.3) i.e.ρ0∂ε ∂t+∂QK ∂XK=PkK∂vk ∂XK≡PkK∂FkK ∂t. This is the so-called equation of balance of internal energy . Due to the working of stresses on the right-hand side it is not a conservation law. On a singular surface the conservation of energy yields a con dition important for contact problems — boundary conditions, phase transformat ions, etc. The same procedure which we have used for the conservation of mass and momentum l eads to the following local relation ρ0Ubracketleftbiggbracketleftbigg ε+1 2v2bracketrightbiggbracketrightbigg −bracketleftBigbracketleftBig Q−PTvbracketrightBigbracketrightBig ·N=0, (4.4) at each point of the singular surface. In a particular case of the continuous velocity, the relation (3.46) yields immediately ρ0U[[ε]]−[[Q]]·N=0, (4.5) 4.1 Energy conservation law 63 i.e. on material surfaces on which U=0the heat ﬂuxQ·Nis continuous and this yields an important boundary condition commonly used, for instanc e, in physics of structures. Otherwise the relation (4.5) plays an important role in the t heory of phase transforma- tions. Performing straightforward transformation we can write th e above equations in the Eulerian form. The equation (4.1) becomes d dtintegraldisplay Ptρparenleftbigg ε+1 2v2parenrightbigg dVt=integraldisplay PtdivparenleftBig −J−1FQ+parenleftBig J−1FPTparenrightBig vparenrightBig dVt+integraldisplay Ptρ(b·v+r)dVt, (4.6) where we have used the Gauss Theorem and the Euler-Piola-Jac obi identities (2.67). Obviously q=J−1FQ, (4.7) i.e.qk=J−1FkKQK, is the current value of the heat ﬂux vector which describes th e nonmechanical transfer of energy per unit current surface and unit time. Bearing the relation (3.34) for Cauchy stresses in mind, we o btain d dtintegraldisplay Ptρparenleftbigg ε+1 2v2parenrightbigg dVt=integraldisplay Ptdiv(−q+T·v)dVt+integraldisplay Ptρ(b·v+r)dVt. (4.8) Now, as in the previous cases we perform the diﬀerentiation w ith respect to time d dtintegraldisplay Ptρparenleftbigg ε+1 2v2parenrightbigg dVt=integraldisplay Pt∂ ∂tbracketleftbigg ρparenleftbigg ε+1 2v2parenrightbiggbracketrightbigg dVt+contintegraldisplay ∂Ptρparenleftbigg ε+1 2v2parenrightbigg v·ndSt.(4.9) Accounting for the conservation of mass (3.12) we can transf orm this relation to the following local form ρ∂ ∂tparenleftbigg ε+1 2v2parenrightbigg +ρ(v·grad)parenleftbigg ε+1 2v2parenrightbigg = =div(−q+T·v)+ρ(b·v+r), i.e.ρ∂ ∂tparenleftbigg ε+1 2v2parenrightbigg +ρvk∂ ∂xkparenleftbigg ε+1 2v2parenrightbigg = (4.10) =∂ ∂xk(−qk+σklvl)+ρ(bkvk+r). This is the local form of the ﬁrst law of thermodynamics, i.e. the law of conservation of energy. As in the case of the Lagrangian description we can apply the l aw of conservation of 64 Thermodynamics of solids momentum (3.38). Then it follows ρparenleftbigg∂ε ∂t+v·gradεparenrightbigg +divq=T·(gradv)+ρr, (4.11) i.e.ρparenleftbigg∂ε ∂t+vk∂ε ∂xkparenrightbigg +∂qk ∂xk=σkl∂vk ∂xl+ρr. This is the Eulerian form of the balance equation of internal energyε. As before, it is not the conservation law due to the presence of the stress pow er∂σkl(∂vk/∂xl)which, as it is said in physics, does not have a divergence form. Similarly to the previous considerations we can construct t he energy conservation law on singular surfaces in the Eulerian description. We obtain the following counterpart of the Lagrangian relation (4.4) ρ(v·n−c)bracketleftbiggbracketleftbigg ε−1 2v2bracketrightbiggbracketrightbigg +[[q−Tv]]·n=0, (4.12) where the mass balance (3.23) has been used. In a particular c ase of ideal ﬂuids one can transform the last contribution in the following way [[Tv]]·n=−[[pv·n]]=ρ(v·n−c)bracketleftbiggbracketleftbiggp ρbracketrightbiggbracketrightbigg , where we have assumed [[v]]=0i.e.[[p]]=0. Consequently r=[[h]]=−[[q]]·n ρ(v·n−c), h=ε+p ρ, (4.13) wherehis called the speciﬁc enthalpy and rdenotes the so-called latent heat. This notion plays an important role in the theory of phase transfo rmations (e.g. evaporation, condensation, melting, etc.) and chemical reactions (e.g. combustion). 4.2 Second law of thermodynamics The ﬁrst traces of the second law of thermodynamics can be fou nd even in the works of Fourier whose relation for the heat ﬂux was constructed at the end of XVIII century (published in 1808 and republished in part in his book: Théorieanalytiquedelachaleur in 1822) in such a way that the heat transfer was possible only from hotter to colder areas of the body otherwise not loaded. S. Carnot in 1824 cons tructed a procedure for the determination of the eﬃciency of heat engines which was d irectly related to the second law of thermodynamics. The most essential step was done by L. Boltzmann whose works in years 1868-1872 yielded the famous H-Theorem. This theor em which corresponds to the modern entropy inequality shows that the macroscopic ir reversibility can be reﬂected by a single scalar function. The proof of Boltzmann was done f or ideal gases and it was based on the hypothesis that macroscopic modelling alwa ys enhances an element of 4.2 Second law of thermodynamics 65 probability. For the details of the motivation of the modern form of the second law of thermodynamics we refer to the book of K. Wilmanski [22]. In these notes we do not need to go into any details of thermody namics. The reference to the second law will be made occasionally but we shall not pr esent any details of the strategy of constructing thermodynamic models. In this Cha pter we demonstrate this strategy on the example of an ideal ﬂuid in order to explain so me basic notions. The construction of nonequilibrium thermodynamics is base d on the assumption of the existence of an entropy function which is a constitutive sca lar satisfying in the Lagrangian description the balance law of the following form d dtintegraldisplay Pρ0ηdV+contintegraldisplay ∂PH·NdS=integraldisplay PˆηdV, (4.14) for any subbodyP⊂B0whereηis the speciﬁc entropy, His the entropy ﬂux and ˆηis the entropy production density. It is assumed that for every process in the system the entropy production is nonnegative for any subbody P⊂B0 integraldisplay PˆηdV≥0. (4.15) This formulation of the second law of thermodynamics has two features. First of all the eﬀect of radiation is neglected. We have already mention ed that the energy radiation has no practical bearing and the same assumption is made for t he entropy. Secondly, the structure of balance law (4.14) is such that surfaces have no contributions to the entropy production. This may not be the case and it is assumed here onl y for simplicity. For many models of classical continuum thermodynamics one c an prove that the entropy ﬂuxHand the heat ﬂux Qare proportional H=Q T, (4.16) whereTis the absolute temperature. In many systems of practical im portance, such as mixtures, this relation does not hold (e.g. [22]). However, it is suﬃcient for our purposes. Then the second law can be written in the form d dtintegraldisplay Pρ0ηdV+contintegraldisplay ∂PQ·N TdS≥0, (4.17) which is called Clausius-Duhem inequality. We can easily construct its local counterparts ρ0∂η ∂t+DivparenleftbiggQ Tparenrightbigg ≥0in regular points of B0, (4.18) and this is called the entropy inequality as well as ρ0U[[η]]−bracketleftbiggbracketleftbiggQ·N Tbracketrightbiggbracketrightbigg =0 in points of singular surfaces. (4.19) 66 Thermodynamics of solids As before, we can transform these laws to the Eulerian descri ption. Then they have the form ρparenleftbigg∂η ∂t+(v·grad)ηparenrightbigg +divparenleftBigq TparenrightBig ≥0in regular points of Bt, (4.20) and ρ(v·n−c)[[η]]+bracketleftBigbracketleftBigq·n TbracketrightBigbracketrightBig =0 in points of singular surfaces. (4.21) The last relation shows that indeed singular surfaces descr ibed by the second law following from the balance law (4.14) do not produce entropy . An extension of thermo- dynamics on processes in which it is not the case is complicat ed and not much has been done in this direction. However, a particular case of (4.21) plays an important role in the theory of phase transformations in spite of their irreve rsibility. If we assume that the velocity is continuous [[v]]=0, and, additionally, that the temperature does not suﬀer a jump either, [[T]]=0, then we have ρ(v·n−c)bracketleftbiggbracketleftbigg ψ+p ρbracketrightbiggbracketrightbigg =0, ψ=ε−Tη, (4.22) whereψis the so-called Helmholtz free energy function. This relat ion for surfaces sepa- rating two phases yields the equation for phase equilibrium line. In a particular case of evaporation and condensation it leads to the so-called Maxw ell construction. As we indicated above the entropy inequality must hold only f or real processes in systems. We explain this limitation on a simple example of an ideal ﬂuid. ⋆In order to see the consequences of the second law of thermody namics, we have to construct ﬁeld equations for a chosen set of ﬁelds. In the c ase of ideal ﬂuids these ﬁelds are the mass density ρ(x,t),velocityv(x,t)and temperature T(x,t). They are described by the mass, momentum and energy conservation equ ations ∂ρ ∂t+div(ρv)=0, ρparenleftbigg∂v ∂t+(v·grad)vparenrightbigg =−gradp, (4.23) ρparenleftbigg∂ε ∂t+(v·grad)εparenrightbigg +divq=−pdivv, where the Cauchy stress tensor is reduced to the pressure p=−1 3trTasT=−p1. The above set becomes the set of ﬁeld equations if we deﬁne additi onallyp,ε,qin terms of ρ,v,Tand their derivatives. For ideal ﬂuids these constitutive r elations have the form p=p(ρ,T), ε=ε(ρ,T),q=−KTgradT, (4.24) whereKTis the so-called heat conductivity. The relation for the hea t ﬂux is called the Fourier relation of heat conduction. The entropy inequality (4.20), i.e. ρparenleftbigg∂η ∂t+(v·grad)ηparenrightbigg +divparenleftBigq TparenrightBig ≥0, η=η(ρ,T), (4.25) 4.2 Second law of thermodynamics 67 must hold for all solutions of the set (4.23) with constituti ve relations (4.24). It means that on the class of solutions of the entropy inequality we im pose constraints in the form of ﬁeld equations. Such constraints can be eliminated i n the same way as we do for problems of mechanics with constraints, i.e. by means of Lagrange multipliers. This technique is now commonly used in continuum thermodynamics (e.g. [11], [21], [22]). In our simple example we can eliminate constraints directly . As the derivatives of the velocity∂v/∂tandgradvdo not enter the entropy inequality, the laws of mass and momentum conservation do not restrict the class of solution of this inequality. Hence, we have to account only for the energy conservation. We do so b y eliminating the heat ﬂux. It follows ρparenleftbigg∂ψ ∂t+(v·grad)ψparenrightbigg +ηparenleftbigg∂T ∂t+(v·grad)Tparenrightbigg − (4.26) −p ρparenleftbigg∂ρ ∂t+(v·grad)ρparenrightbigg +1 Tq·gradT≤0, where, as already indicated in (4.22), ψ=ε−Tη=ψ(ρ,T), (4.27) denotes the Helmholtz free energy function and we have used t he mass balance equation to eliminate divv=(∂ρ/∂t+v·gradρ)/ρ. The chain rule of diﬀerentiation yields ∂ψ ∂t=∂ψ ∂ρ∂ρ ∂t+∂ψ ∂T∂T ∂t, (4.28) and similarly for gradψ. The contributions containing the derivatives ∂ρ/∂t,∂T/∂t are linear with respect to these derivatives. As they are not constraint anymore and, consequently, can be chosen arbitrarily, their coeﬃcients must be zero in order to fulﬁl the inequality. This yields the following results p=ρ2∂ψ ∂ρ, η=−∂ψ ∂T⇒ε=ψ−T∂ψ ∂T, (4.29) D=−q·gradT≥0⇒KT≥0. The identities for p,η,εshow that we need only a constitutive relation for the Helmho ltz free energy ψin order to reproduce the remaining relations. For this reas on, such func- tions are called thermodynamical potentials. Simultaneou sly, the entropy inequality is reduced to the so-called residual inequality which restric ts the functionDcalled the dis- sipation function. In our simple case it yields the conditio n for the heat conduction which is equivalent to Fourier’s assumption on the ﬂow of energy in the heat conductor from hotter to colder regions. In irreversible mechanical proce sses the dissipation is related to the viscosity (viscoelastic materials) and to the plastic w orking (viscoplastic materials). Let us complete this example with a relation which follows fr om the thermodynamic identities. We have dψ=dε−ηdT−Tdη=∂ψ ∂TdT+∂ψ ∂ρdρ=−ηdT+p ρ2dρ⇒ ⇒dη=1 Tparenleftbigg dε−p ρ2dρparenrightbigg . (4.30) 68 Thermodynamics of solids This is the so-called Gibbs equation for ideal ﬂuids. ♣ Chapter 5 Elastic materials Relations which we presented in previous Chapters describe properties of any continuous system. In this sense they are universal. However, in order t o ﬁnd the behaviour of a particular material system they are not suﬃcient. We have t o perform the so-called closure which means that we have to add certain relations whi ch yield ﬁeld equations for a chosen set of ﬁelds. Together with initial and boundary con ditions these ﬁeld equations form a mathematical problem of partial diﬀerential equatio ns. Solutions of the latter may be analytical and we show some of them in these notes or they ma y be approximations following, for example, from numerical procedures. This pr oblem shall not be considered in this booklet. We begin the demonstration of the closure procedure with a cl ass of materials called elastic. In the next Subsection we present brieﬂy equations of nonlinear elasticity and then we discuss extensively linear problems for isotropic m aterials. 5.1 Non-linear elasticity The fundamental ﬁeld which we want to ﬁnd in continuum mechan ics is the function of motionf. For the so-called elastic materials it is the only unknown ﬁ eld as thermal problems are ignored. This means that the temperature is ass umed to be constant, processes are isothermal. The governing equation for the function of motion follows fr om the momentum con- servation (3.31), i.e. ρ0∂v ∂t=DivP+ρ0b,v=∂f ∂t,x=f(X,t),P=P(X,t),X∈B0,(5.1) where the body force b(X,t)is assumed to be given. In order to transform (5.1) into the ﬁeld equation for fwe have to specify the Piola- Kirchhoﬀ stress tensor P. We do so with the help of the second law of thermodynamics. Elimination of the heat ﬂux contribution DivQby means of the energy balance (4.3) and the assumption of the constant temperature yields the condi tion that for all isothermal 69 70 Elastic materials processes the following inequality must be satisﬁed ρ0∂ψ ∂t≤P·Gradv, ψ=ε−Tη. (5.2) We now make the basic constitutive assumption which deﬁnes t he class of nonlinear elastic materials. Namely, we assume that the Helmholtz fre e energy function ψdepends on the motion of continuum solely through the deformation gr adient ψ=ψ(F). (5.3) It means that, for instance, rates of deformation do not have any inﬂuence on the free energy. Substitution of this assumption in the inequality ( 5.2) yields parenleftbigg ρ0∂ψ ∂F−Pparenrightbigg ·∂F ∂t≤0, (5.4) and this condition should hold for all derivatives ∂F/∂t. Consequently, we have to require P=ρ0∂ψ ∂F, (5.5) and the inequality (5.4) is identically satisﬁed as equalit y. This means that the dissipation in such processes is zero, i.e. all processes in elastic mate rials are reversible. Many applications of such a model are based on the isotropy as sumption. It means that reactions of the material on a given external loading ar e independent of the ori- entation of a sample. Simultaneously, we require that local rotations do not inﬂuence the reaction of the material. It means that the Helmholtz fre e energy depends on the deformation gradient F=RUin such a way that Rdoes not appear in the constitu- tive relation. Hence, we can replace the dependence on Fby the dependence on the right Cauchy-Green tensor C=FTF=U2. Simultaneously, the isotropy of the material means that the constitutive relation should not change by an arbitrary change of the reference coordinates. This is possible if the constitutiv e dependence of a scalar function onCis reduced to the dependence on the invariants of C. Finally, we have ψ=ψ(I,II,III), I=trC, II=1 2parenleftbigI2−trC2parenrightbig, III=detC.(5.6) Substitution in (5.5) and the chain rule of diﬀerentiation y ield P= 2ρ0Fparenleftbigg∂ψ ∂I∂I ∂C+∂ψ ∂II∂II ∂C+∂ψ ∂III∂III ∂Cparenrightbigg = = 2ρ0Fparenleftbigg∂ψ ∂I1+∂ψ ∂II(I1−C)+∂ψ ∂IIIIIIC−1parenrightbigg = = 2ρ0parenleftbigg∂ψ ∂IB+∂ψ ∂IIparenleftbigIB−B2parenrightbig+∂ψ ∂IIIIII1parenrightbigg F−T, (5.7) where, for convenience, we have used the left Cauchy-Green t ensor, deﬁned by the relation (2.37):B=FFT.This is equivalent to Cbecause both deformation measures CandB 5.2 Linear elasticity, isotropic and anisotropic material s 71 have the same invariants (compare (2.38)). Now, using the de ﬁnition (3.34) of the Cauchy stressT=J−1PFTwe obtain T=/beth01+/beth1B+/beth−1B−1, (5.8) where /beth0= 2ρparenleftbigg II∂ψ ∂II+III∂ψ ∂IIIparenrightbigg , (5.9) /beth1= 2ρ∂ψ ∂I,/beth−1=−2ρIII∂ψ ∂II, ρ=ρ0J−1≡ρ0√ III, (compare (3.7)). The coeﬃcients /beth0,/beth1,/beth−1are called elasticities or response coeﬃcients and they are functions of invariants I,II,III . The relation (5.8) deﬁnes the so-called compressible Mooney-Rivlin material. Many other constitu tive relations for non-linear elastic materials are presented, for instance, in [2], [14] or [22]. Their main applications serve the purpose of description of such materials as rubber , many polymeric materials, biological tissues, etc. We shall not discuss them any furth er in these notes. 5.2 Linear elasticity, isotropic and anisotropic mate- rials 5.2.1 Governing equations Let us summarize geometrical and dynamic relations for a lin ear model which we have presented in Chapters 2 and 3. As indicated the convenient way to describe the motion of the linear material is through the displacement ﬁeld uwhich is the function of Lagrangian coordinates and time. This is the so-called displacement approach. As in a li near model we do not dis- tinguish between Lagrangian and Eulerian reference system s, the displacement function which is the main ﬁeld of the linear elasticity is the followi ng suﬃciently smooth function u=u(x,t)i.e.uk=uk(x1,x2,x3,t),u=ukek,x=xkek. (5.10) This function deﬁnes the velocity vand the strain ﬁeld eof the linear model v=∂u ∂t,e=1 2parenleftBig gradu+(gradu)TparenrightBig , (5.11) i.e.vk=∂uk ∂t, ekl=1 2parenleftbigg∂uk ∂xl+∂ul ∂xkparenrightbigg . In some applications (e.g. theory of acoustic waves) it is co nvenient to introduce the velocityvand the strain ﬁeld eas unknown ﬁelds instead of the displacement u. Then these ﬁelds must satisfy the following integrability condi tion ∂e ∂t=1 2parenleftBig gradv+(gradv)TparenrightBig i.e.∂ekl ∂t=1 2parenleftbigg∂vk ∂xl+∂vl ∂xkparenrightbigg , (5.12) 72 Elastic materials which is, obviously, satisﬁed if the displacement ﬁeld uis given. Additionally, the strain ﬁeld must fulﬁl the compatibility condition (2.102) which follows from the Euclidean character of the space of motion, i.e. ǫijkǫlmn∂2ejm ∂xk∂xn=0. (5.13) As already indicated there are six independent relations wh ich follow from (5.13). Namely 2∂2e12 ∂x1∂x2=∂2e11 ∂x2 2+∂2e22 ∂x2 1, 2∂2e23 ∂x2∂x3=∂2e22 ∂x2 3+∂2e33 ∂x2 2, 2∂2e13 ∂x1∂x3=∂2e11 ∂x2 3+∂2e33 ∂x2 1, ∂2e11 ∂x2∂x3=∂2e12 ∂x3∂x1−∂2e23 ∂x2 1+∂2e13 ∂x1∂x2, (5.14) ∂2e22 ∂x1∂x3=∂2e23 ∂x1∂x2−∂2e13 ∂x2 2+∂2e12 ∂x2∂x3, ∂2e33 ∂x1∂x2=∂2e13 ∂x2∂x3−∂2e12 ∂x2 3+∂2e23 ∂x1∂x3. If we substitute the strain-stress relations in these compa tibility conditions we obtain equations for stresses. If the boundary value problem is als o formulated in stresses then this set can be solved. This approach is called the stress app roach. We show further some examples of such a formulation. In the displacement approach, purely mechanical problems ( isothermal processes) require the ﬁeld equation for the displacement u. This follows from the linear form of the momentum conservation ρ∂2u ∂t2=divT+ρb, (5.15) whereρ≈const, provided we specify the Cauchy stress tensor Tin terms of the dis- placement and its derivatives. As in the nonlinear case, the constitutive dependence of T is given by a function of the strain ﬁeld e. Assuming that initial stresses (i.e. stresses in the conﬁguration in which e=0) are zero the most general linear form of such a relation is as follows σkl=cklijeij,T=σijei⊗ej,e=eijei⊗ej, (5.16) wherecijklare 81 constants ( = 34). However, the symmetry of the Cauchy stresses σij=σjiand of the strain ﬁeld eij=ejireduces the number of independent constants to 21. This can be easily seen in a Voigt notation which is comm only used in the crystallography. It replaces the components of the second r ank tensors by six-dimensional vectors (e11,e22,e33,e23,e13,e12) = (e1,e2,e3,e4,e5,e6), (5.17) (σ11,σ22,σ33,σ23,σ13,σ12) = (σ1,σ2,σ3,σ4,σ5,σ6). 5.2 Linear elasticity, isotropic and anisotropic material s 73 Then the relation (5.16) can be replaced by the following mat rix relation σα=Cαβeβ, α,β=1,...,6. (5.18) One can easily ﬁnd the correspondence between cijklandCαβwhich we do not quote here. Obviously, the 6×6matrix(Cαβ)is symmetric which means that it possesses 6+30/2=21 independent components. This is the maximum number of indep endent material parameters which may appear in the relation (5.16) . Materials which require this number of constants are called anisotropic. They do not possess any particular symmetry properties which means that samples cut from such a material in diﬀerent directions yield diﬀerent response to the same external loa ding. Some particular cases of material symmetry are of practical importance. We call the material orthotropic if it possesses three orthogonal plan es of symmetry deﬁned by the base vectors{e1,e2,e3}. Then in this reference system the stress-strain relation r educes to the following relation in Voigt’s notation  σ1 σ2 σ3 σ4 σ5 σ6 = C11C12C130 0 0 C12C22C230 0 0 C13C23C330 0 0 0 0 0 C440 0 0 0 0 0 C550 0 0 0 0 0 C66  e1 e2 e3 e4 e5 e6 , (5.19) i.e. it is described by 9material parameters. The material is transversely isotrop ic if it is symmetric with respect to a rotation about an axis of symme try. Ife3is such an axis then in Vogt’s notation the stress-strain relation is as fol lows  σ1 σ2 σ3 σ4 σ5 σ6 = C11C12C130 0 0 C12C11C130 0 0 C13C13C330 0 0 0 0 0 C440 0 0 0 0 0 C440 0 0 0 0 01 2(C11−C12)  e1 e2 e3 e4 e5 e6 , (5.20) i.e. it is described by 5material parameters. In spite of many anisotropic materials appearing in practic al applications (e.g. wood, composites, many rock structures) in these notes we concent rate on isotropic materials which were already investigated in the nonlinear case. This limitation is connected with technical diﬃculties. Linear isotropic elastic material is characterized by two m aterial parameters. For instance, for such materials, we can write the relation (5.1 6) in the form T=λ(tre)1+2µei.e.σij=λekkδij+2µeij, (5.21) i.e.cijkl=λδijδkl+µ(δikδjl+δilδjk), whereλ,µare the so-called Lamé constants and the relation (5.21) is c alled Hooke’s law. 74 Elastic materials This law corresponds to the constitutive relation for the He lmholtz free energy ρψ=1 2T·e=1 2σijeij=1 2parenleftBig λ(tre)2+2µe·eparenrightBig =1 2(λ(ekk)(ell)+2µeklekl),(5.22) Namely, the nonlinear relation (5.5) can be easily transfor med to the thermodynamic relation of the linear model T=ρ∂ψ ∂e. (5.23) ⋆We have T=J−1PFT=J−1ρ0∂ψ ∂FFT= =ρparenleftbigg∂ψ ∂CKL∂CKL ∂FmMparenrightbigg FlMem⊗el= = 2ρ∂ψ ∂CKLFkKδmkδLMFlMem⊗el= = 2ρFkK∂ψ ∂CKLFlLem⊗el= (5.24) = 2ρFkK∂ψ ∂EMN∂EMN ∂CKLFlLem⊗el= =ρFkK∂ψ ∂EMNδMKδNLFlLem⊗el≈ρ∂ψ ∂eklek⊗el. ♣ The Hooke law yields in particular the following relation fo r the volume changes given in the linear theory by the trace of the strain ﬁeld, tre, p=−1 3trT=−Ktre, K=λ+2 3µ, (5.25) whereKis the so-called compressibility (or bulk) modulus. It is of ten used in soil mechanics together with G=µand the latter is called the shear (Kirchhoﬀ) modulus. Substitution of (5.25) in (5.21) yields immediately the fol lowing inverse relations e=1 2µparenleftbigg T−λ 3λ+2µ(trT)1parenrightbigg , (5.26) i.e.eij=1 2µparenleftbigg σij−λ 3λ+2µσkkδijparenrightbigg . Coeﬃcients1/2µandλ/[2µ(3λ+2µ)]are called compliances. They form the following isotropic compliance matrix c′ ijkl=1 2µδikδjl−λ 2µ(3λ+2µ)δijδkl⇒eij=c′ ijklσkl, (5.27) which is the inverse to the isotropic elasticity matrix cijkl. 5.2 Linear elasticity, isotropic and anisotropic material s 75 The relation (5.26) is usually written in the following expl icit form e11=1 E(σ11−ν(σ22+σ33)), e22=1 E(σ22−ν(σ11+σ33)), e33=1 E(σ33−ν(σ11+σ22)), (5.28) e12=1 2µσ12, e13=1 2µσ13, e23=1 2µσ23, and the comparison with (5.26) yields E=µ(3λ+2µ) λ+µλ=Eν (1+ν)(1−2ν) ⇒ ν=λ 2(λ+µ)µ=E 2(1+ν), (5.29) whereEis called Young (elasticity) modulus and νis the Poisson number. Then it follows for the compressibility modulus K=E 3(1−2ν). (5.30) Values of material parameters are not quite arbitrary and we shall demonstrate various limits which they have to fulﬁl as we proceed with the present ation of particular problems. We shall see, for example that the compressibility modulus Kand the shear modulus µare nonnegative. It means that, due to (5.29) 4, Poisson’s number is: ν >−1and, according to (5.29) 3,ν≤0.5. As Poisson’s number ν, by means of (5.28), assigns the expansion of the material under the loading in the perpen dicular (lateral) direction (e.g.σ22=σ33=0,σ11<0⇒e22=e33=−νσ11/E;e22,e33would be expected to be positive for positive E) it is often speculated that ν >0. It has been shown that it must not be the case1. 1e.g.R.LIascIkscIescIssc; Foam structures with a negative Poisson’s ratio, Science 235:1038—1040, 1987, G.W.MIiscIlscItscIoscInsc; Composite materials with Poisson’s ratios close to -1. Journal of the Mechanics and Physics of Solids, 40(5):1105—1137, 1992. 76 Elastic materials In Fig. 5.1. we show a two-dimensional structure2which possesses the property ν <0. Table:Elasticconstantsforchosenmaterialsattemperature 200C. λbracketleftbig 1010Pabracketrightbig µbracketleftbig 1010Pabracketrightbig Ebracketleftbig 1010Pabracketrightbig νKbracketleftbig 1010Pabracketrightbig aluminium 5.63 2.60 6.98 0.34 7.36 brass 8.90 3.60 9.76 0.36 11.30 copper 10.63 4.55 12.29 0.35 13.66 duralumin 5.78 2.70 7.24 0.34 7.58 ice (−40C)0.70 0.36 0.96 0.33 0.94 iron 10.49 8.20 21.00 0.28 15.96 lead 4.07 0.57 1.64 0.44 4.45 marble 4.15 2.70 7.04 0.30 5.95 plexiglass 0.28 0.12 0.32 0.35 0.36 polystyrene 0.28 0.12 0.32 0.35 0.36 steel 11.78 8.0 20.76 0.30 17.11 Fig. 5.1: A structure modelled by a linear elastic continuum with the negative Poisson number ν In the other limit ν=0.5there are no volume changes tre=1−2ν EtrT⇒ν=0.5tre=0andK=∞, (5.31) i.e.J=1andρ=ρ0. Such materials are called incompressible. 2U.D.LIascIrscIsscIescInsc,O.SIiscIgscImscIuscInscIdsc,S.BIoscIuscIwscIsscItscIrscIasc; Design and fabrication of compliant micro-mechanisms and structures with negative Poisson’s ratio, JournalofMicroelectromechanicalSystems, 6:99—106, 1997. 5.2 Linear elasticity, isotropic and anisotropic material s 77 5.2.2 Navier-Cauchy equations, Green functions, displace ment potentials Substitution of the Hooke law (5.21) in the momentum balance (5.15) yields immediately the governing equations for the displacement u ρ∂2u ∂t2= (λ+µ)graddivu+µdivgradu+ρb≡ (5.32) ≡(λ+µ)gradtre+µ∇2u+ρb, or in Cartesian coordinates ρ∂2uk ∂t2=(λ+µ)∂2ul ∂xk∂xl+µ∂2uk ∂xl∂xl+ρbk. (5.33) These are Navier-Cauchy equations. They are also called Lam é equations. ⋆In many practical problems it is convenient to use curviline ar coordinates. For the purpose of these notes we need only cylindrical and spherica l coordinates. We shall not go into any details and present below the basic relations for these two systems and for the so-called physical components of the quantities. Let us only brieﬂy explain the latter notion. We assume that curvilinear coordinates are introduced in th e three-dimensional Euclid- ean space, i.e. yα=yα(x1,x2,x3), α=1,2,3,⇒xk=xkparenleftbigy1,y2,y3parenrightbig, k=1,2,3,(5.34) are equations of parametric lines. Then the covariant and co ntravariant base vectors are given by the following relations gα=∂r ∂yα,r=xkek,gα·gβ=δα β, (5.35a) i.e.gαis tangent to the yα-parametric line, and gαis perpendicular to the parametric surfaceyα=const. Both cylindrical and spherical coordinates are orthogonal , i.e. the metric tensor gαβ=gα·gβ, (5.36) is diagonal. One can introduce unit base vectors e(α)=gα1√gαα(do not add!) ,e(α)·e(β)=δαβ. (5.37) Then the displacement vector uand the deformation tensor ehave the following compo- nents u=u(α)e(α), uα=u·gα, u(α)=uα√gαα(do not add!) , (5.38) e=e(α)(β)e(α)⊗e(β), eαβ=gα·egβ, e(α)(β)=eαβ√gαα√gββ(do not add!) , 78 Elastic materials which are called physical components. Diﬀerentiation of an arbitrary vector, say — the displaceme nt vectoru, yields ∂u ∂yα=∂ ∂yα(uµgµ)=∂uµ ∂yαgµ+uµ∂gµ ∂yα=parenleftbigg∂uµ ∂yα+Γµ αβuβparenrightbigg gµ, (5.39) whereΓµ αβ=gµ·∂gβ ∂yα≡gµ·∂2r ∂yα∂yβ,∂uµ ∂yα+Γµ αβuβ=∇αuµ, whereΓµ αβare called Christoﬀel symbols and ∇αis the covariant derivative of the vector. Similar relations follow for tensors of the second order. In the relations quoted below for cylindrical and spherical coordinates we have re- placed partial derivatives of the Cartesian coordinates by covariant derivatives of the corresponding curvilinear coordinates and components of t he vectors and tensors are physical components in the corresponding coordinate syste ms.♣ ⋆1) Cylindrical coordinates . These coordinates are deﬁned by the following transformati on of Cartesian coordinates x1=rcosθ, x2=rsinθ, x3=z,r=eixi, r=y1=radicalBig (x1)2+(x2)2, θ=y2=arctanparenleftbiggx2 x1parenrightbigg , z=y3=x3. (5.40) The corresponding unit base vectors are er=cosθe1+sinθe2,eθ=−sinθe1+cosθe2,ez=e3. (5.41) Then the strain function eis as follows err=∂ur ∂r, eθθ=ur r+1 r∂uθ ∂θ, ezz=∂uz ∂z, erθ=1 2parenleftbigg∂uθ ∂r−uθ r+1 r∂ur ∂θparenrightbigg , eθz=1 2parenleftbigg1 r∂uz ∂θ+∂uθ ∂zparenrightbigg , (5.42) ezr=1 2parenleftbigg∂ur ∂z+∂uz ∂rparenrightbigg . The momentum balance equations become ρ∂2ur ∂t2=∂σrr ∂r+1 r∂σrθ ∂θ+∂σrz ∂z+σrr−σθθ r+ρbr, ρ∂2uθ ∂t2=∂σrθ ∂r+1 r∂σθθ ∂θ+∂σθz ∂z+ρbθ, (5.43) ρ∂2uz ∂t2=∂σrz ∂r+1 r∂σθz ∂θ+∂σzz ∂z+ρbz. 5.2 Linear elasticity, isotropic and anisotropic material s 79 The displacement equations are in this case ρ∂2ur ∂t2=µparenleftbigg ∇2ur−ur r2−2 r2∂uθ ∂θparenrightbigg + +(λ+µ)∂ ∂rbracketleftbigg1 r∂ ∂r(rur)+1 r∂uθ ∂θ+∂uz ∂zbracketrightbigg +ρbr, ρ∂2uθ ∂t2=µparenleftbigg ∇2uθ−uθ r2+2 r2∂ur ∂θparenrightbigg + (5.44) +(λ+µ)1 r∂ ∂θbracketleftbigg1 r∂ ∂r(rur)+1 r∂uθ ∂θ+∂uz ∂zbracketrightbigg +ρbθ, ρ∂2uz ∂t2= (λ+µ)∂ ∂zbracketleftbigg1 r∂ ∂r(rur)+1 r∂uθ ∂θ+∂uz ∂zbracketrightbigg +ρbz. In addition, the Laplace operator in cylindrical coordinat es has the following form ∇2=∂2 ∂r2+1 r∂ ∂r+1 r2∂2 ∂θ2+∂2 ∂z2, (5.45) while the volume changes are tre=1 r∂ ∂r(rur)+1 r∂uϕ ∂ϕ+∂uz ∂z. (5.46) ♣ ⋆2) Spherical coordinates. We have here x1=rsinθcosϕ, x2=rsinθsinϕ, x3=rcosθ,r=xiei, r=y1=√xkxk, ϕ=y2=arctanparenleftbiggx2 x1parenrightbigg , θ=y3=arccosx3√xkxk,(5.47) and the corresponding base vectors er= sinθcosϕe1+sinθsinϕe2+cosθe3, eϕ=−sinϕe1+cosϕe2, (5.48) eθ= cosθcosϕe1+cosθsinϕe2−sinθe3. 80 Elastic materials The strain function ehas the form err=∂ur ∂r, eϕϕ=1 rsinθ∂uϕ ∂ϕ+ur r+uθ rcotθ, eθθ=1 r∂uθ ∂θ+ur r, erϕ=1 2parenleftbigg1 rsinθ∂ur ∂ϕ−uϕ r+∂uθ ∂rparenrightbigg , (5.49) erθ=1 2parenleftbigg1 r∂ur ∂θ−uθ r+∂uθ ∂rparenrightbigg , eϕθ=1 2parenleftbigg1 r∂uϕ ∂θ−uϕ rcotθ+1 rsinθ∂uθ ∂ϕparenrightbigg . The momentum balance equations are as follows ρ∂2ur ∂t2=∂σrr ∂r+1 rsinθ∂σrϕ ∂ϕ+1 r∂σrθ ∂θ+2σrr−σϕϕ+σrθcotθ r+ρbr, ρ∂2uϕ ∂t2=∂σrϕ ∂r+1 rsinθ∂σϕϕ ∂ϕ+1 r∂σϕθ ∂θ+3σrϕ+2σϕθcotθ r+ρbϕ,(5.50) ρ∂2uθ ∂t2=∂σrθ ∂r+1 rsinθ∂σϕθ ∂ϕ+1 r∂σθθ ∂θ+3σrθ+(σθθ−σϕϕ) r+ρbz. The displacement equations are in this case ρ∂2ur ∂t2=µbraceleftbigg ∇2ur−2 r2bracketleftbigg ur+1 sinθ∂ ∂θ(uθsinθ)+1 sinθ∂uϕ ∂ϕbracketrightbiggbracerightbigg + +(λ+µ)∂tre ∂r+ρbr, ρ∂2uϕ ∂t2=µbraceleftbigg ∇2uϕ+2 r2sinθbracketleftbigg∂ur ∂ϕ+∂uθ ∂θcotθ−uϕ 2sinθbracketrightbiggbracerightbigg + (5.51) +(λ+µ)1 rsinθ∂tre ∂ϕ+ρbϕ, ρ∂2uθ ∂t2=µbraceleftbigg ∇2uθ−2 r2bracketleftbigg∂uθ ∂θ−uθ 2sin2θ−∂uϕ ∂ϕcosθ sin2θbracketrightbiggbracerightbigg + +(λ+µ)1 r∂tre ∂ϕ+ρbθ, where the Laplace operator has the form ∇2=1 r2∂ ∂rparenleftbigg r2∂ ∂rparenrightbigg +1 r2sinθ∂ ∂θparenleftbigg sinθ∂ ∂θparenrightbigg +1 r2sinθ∂2 ∂ϕ2, (5.52) and the volume changes are tre=1 r2sinθbracketleftbigg∂ ∂rparenleftbigurr2sinθparenrightbig+∂ ∂θ(uθrsinθ)+∂ ∂ϕ(uϕr)bracketrightbigg . (5.53) ♣ The set of displacement equations can be solved provided we s pecify initial and bound- ary conditions. The latter may be given in terms of displacem ents or their derivatives. 5.2 Linear elasticity, isotropic and anisotropic material s 81 In Fig. 5.2. we show a domain Bton which the equation (5.33) is deﬁned and whose part of the boundary ∂Bσ tis loaded by a given traction tnand on the remaining part of the boundary ∂Bu t, ∂Bσ t∪∂Bu t=∂Bt, the displacement ubis given. Hence u(x,t)=ub(x,t)forx∈∂Bu t, (5.54) λ(divu)n+µparenleftBig gradu+(gradu)TparenrightBig n(x,t)=tn(x,t)forx∈∂Bσ t, i.e. uk(x,t)=ub k(x,t)forx∈∂Bu t, (5.55) λ∂ul ∂xlnk+µparenleftbigg∂uk ∂xl+∂ul ∂xkparenrightbigg nl(x,t)=tn k(x,t)forx∈∂Bσ t. In the sequel we show a few typical examples of this boundary v alue problem. Fig. 5.2: Boundary conditions of linear elasticity One of the important general features of the set (5.32) is its hyperbolicity. We shall not go into a detailed mathematical deﬁnition of this notion . We rather use its physical interpretation that hyperbolic systems describe the propa gation of waves of weak discon- tinuity (acoustic waves). We shall show this property farth er in a few diﬀerent ways but it can also be immediately seen when we use Helmholtz decompo sition (1.57) u=gradϕ+rotψ, (5.56) i.e.uk=∂ϕ ∂xk+ǫklm∂ψm ∂xl, 82 Elastic materials whereϕ,ψare scalar and vector displacement potentials. Substituti on in (5.33) yields ∂ ∂xkbracketleftbigg ρ∂2ϕ ∂t2−(λ+2µ)∇2ϕbracketrightbigg +ǫkpq∂ ∂xpbracketleftBigg ρ∂2ψq ∂t2−µ∇2ψqbracketrightBigg =0, (5.57) ∇2=∂2 ∂xm∂xm, if we neglect for simplicity the body forces. The contributi ons in square brackets should be zero independently as a diﬀerentiation of (5.57) shows. C onsequently ∂2ϕ ∂t2=c2 L∇2ϕ, c2 L=λ+2µ ρ=K+4 3µ ρ, (5.58) ∂2ψ ∂t2=c2 T∇2ψ, c2T=µ ρ<c2 L. The inequality follows for K >0,µ >0which we justify further. These are two linear wave equations of the second order. In the one-dimensional s pecial case they have the following form parenleftbigg∂ ∂t−cL∂ ∂x1parenrightbiggparenleftbigg∂ϕ ∂t+cL∂ϕ ∂x1parenrightbigg = 0, (5.59) parenleftbigg∂ ∂t−cT∂ ∂x1parenrightbiggparenleftbigg∂ψ ∂t+cT∂ψ ∂x1parenrightbigg = 0, with the following d’Alambert solutions (e.g. [21]) ϕ=ϕ−(x1+cLt)+ϕ+(x1−cLt), (5.60) ψ=ψ−(x1+cTt)+ψ+(x1−cTt), whereϕ−,ϕ+,ψ−,ψ+are arbitrary twice diﬀerentiable functions, i.e. they ind eed de- scribe waves with speeds of propagation cLandcT. As we show further the scalar potentialϕdescribes the so-called longitudinal wave, i.e. a wave in wh ich the motion of particles is in the same direction as the propagation of the w ave. Its speed of propagation iscL=radicalbig (λ+2µ)/ρ.The vector potential ψdescribes the so-called transversal wave with the motion of particles perpendicular to the direction of propagation and its speed of propagation is cT=radicalbig µ/ρ.We present properties of these dynamic solutions later in many details. Some solutions of the set of displacement equations for the i nﬁnite medium can be constructed by means of Green functions. These are function s specifying the displacement in an arbitrary point in the case of a single force f=fkekacting at the point x=0. This is the method well-known in many branches of physics as well as s tructural mechanics where it is called the inﬂuence function method [12]. In the static case (i.e. for the acceleration ∂2u/∂t2identically zero) the displacement u(x)can be written in the following form ui=Gikfk, (5.61) 5.2 Linear elasticity, isotropic and anisotropic material s 83 where Gik(x)=λ+3µ 8πµ(λ+2µ)parenleftbiggδik r+λ+µ λ+3µxixk r3parenrightbigg , r=√xpxp. (5.62) This solution has been found by W. Thomson (lord Kelvin) in 18 48. Once we have this solution we can ﬁnd the displacement in an inﬁnite mediu m for an arbitrary set of loading forces using the superposition. For instance, for t he forcePδ(r)δ(z)acting in the direction of the z-axis this function yields the following form of the displac ement in cylindrical coordinates {r,θ,z} u=P 4πµ√ r2+z2bracketleftbigg1 4(1−ν)rz (r2+z2)er+parenleftbigg 1−1 4(1−ν)r2 r2+z2parenrightbigg ezbracketrightbigg .(5.63) Hence, the component of the displacement uzin the direction of the force diminishes as√ r2+z2. There is also an additional component urin the radial direction er. Obviously, the solution is singular in the point r=0,z=0, i.e. in the point of action of the force. The similar procedure can be applied in the full dynamic case . We obtain Gij(x,t)=1 4πρbraceleftbigg δparenleftbigg t−r cTparenrightbiggparenleftbiggδij c2 Tr−xixj c2 Tr3parenrightbigg + +δparenleftbigg t−r cLparenrightbiggxixj c2 Lr3+ (5.64) +bracketleftbigg Hparenleftbigg t−r cLparenrightbigg −Hparenleftbigg t−r cTparenrightbiggbracketrightbiggt r3parenleftBig 3xixj r2−δijparenrightBigbracerightbigg , where the Dirac- δof the argument t−r/cTactivates the ﬁrst contribution when the shear wave of the speed cTarrives to a chosen point, the Dirac- δof the argument t−r/cLac- tivates the second contribution, when the longitudinal wav e of the speed cLarrives to a chosen point, and the Heaviside function contributions HparenleftBig t−r cLparenrightBig =braceleftbigg1fort>r cL 0fort<r cL andHparenleftBig t−r cTparenrightBig =braceleftbigg1fort>r cT 0fort<r cTof the third contribution activate this term be- tween both waves. Derivation of the above static Green function as well as the d ynamic Green function can be found in the Appendix. In their derivation we use integ ral Fourier transform method. Inspection of the relation (5.64) shows that it consists of c ontributions which possess diﬀerent properties when r→0and whenr→∞. In the ﬁrst case we can neglect the ﬁrst two contributions because they are proportional to 1/rin contrast to the last term which is proportional to 1/r3. Consequently, this term dominates for small r. We call it the near ﬁeld approximation. On the other hand, these are t he ﬁrst two terms which dominate for large rand we can neglect the last contribution. We call this case th e far ﬁeld approximation. We shall use these approximations i n estimates of the action of dislocations in the last Chapter of this book. 84 Elastic materials The method of Green function can be also extended on ﬁnite dom ains and this method is also indicated in the Appendix. However, construction of solutions of boundary value problems for ﬁnite domains by means of Green functions requi res a rather involved in- tegrations. These can be performed numerically but analyti cal solutions can be found easier using other methods. An extensive class of methods has been proposed for static pr oblems (compare the presentation of many examples in the classical books of Timo shenko and Goodier [18] and Landau and Lifshitz [8]). They lead to solutions of simpler L aplace or Poisson equations for functions called potentials. We present some of them. Some elements of the solution by means of one scalar and one ve ctor function can be found already in the early paper of J. Boussinesq (1878)3,4. It has been fully developed by P. F. Papkowich (1932) and H. Neuber (1934). Namely, they h ave shown that the following relation u=Φ−1 4(1−ν)grad[Φ0+x·Φ], (5.65) satisﬁes identically the displacement equations provided the scalar and vector Papkovich- Neuber potentials, Φ0,Φfulﬁl Poisson’s equations ∇2Φ0−ρx·b µ=0,∇2Φ+ρb µ=0, (5.66) i.e. forb=0they are harmonic functions. The proof is by substitution. The procedure of solution is as follows. We solve the equatio ns (5.66) with boundary conditions formulated in terms of functions Φ0,Φ. These follow either from relations (5.65) if they are given for displacements on the boundary or from the relations for the stress vector on the boundary. Then we have to use the followi ng relation for stresses σij=2µbracketleftbigg∂2Ψ ∂xi∂xj−2(1−ν)parenleftbigg∂Φi ∂xj+∂Φj ∂xiparenrightbigg −δijν∇2Ψbracketrightbigg , (5.67) where Ψ=Φ0+xkΦk,∇2Ψ=2∂Φk ∂xk, (5.68) which is the consequence of the Hooke law (5.21). In many problems it is suﬃcient to introduce three functions . For instance, for the half-space we choose Φ0andΦ=(Φ1,Φ2,0). Displacements in the space caused by the body forces are determined by means of Φ0=0,Φ=(Φ1,Φ2,Φ3). For axial symmetric cases one applies Boussinesq represent ation. Forb=0, we have in cylindrical coordinates ur=∂Ψ ∂r, uz=∂Ψ ∂z−4(1−ν)Φr,Ψ=Φ0+zΦz. (5.69) 3Valentin Joseph Boussinesq, 1842-1929. 4J.BIoscIuscIsscIsscIiscInscIescIsscIqsc; Équilibre d’élasticité d’un solide isotrope sans pesanteu r, supportant diﬀérents poids, C. R. Acad. Sci., Paris, 86, 1260-1263, 1878. P.F.PIascIpscIkscIoscIvscIiscIcscIhsc; Solution Générale des équations diﬀerentielles fondament ales d’élasticité exprimée par trois fonctions harmoniques, Compt. Rend. Acad. Sci. Paris 195: 513—515 , 1932. H. NIescIuscIbscIescIrsc; Ein neuer Ansatz zur Lösung räumlicher Probleme der Elastiz itätstheorie, Z. Angew. Math. Mech. 14: 203—212, 1934. 5.2 Linear elasticity, isotropic and anisotropic material s 85 Another class of potentials was introduced also by J. Boussi nesq and later rediscov- ered by Somigliana and Galerkin5. This Boussinesq-Somigliana-Galerkin solution has the form u=∇2g−1 2(1−ν)graddivg, (5.70) where ∇2∇2g=−1 µρb, (5.71) i.e. the vector potential gis biharmonic for b=0. All these classes give rise to complete solutions of the disp lacement equations. A diﬀerent harmonic potential, particularly useful for hal f-space problems, was intro- duced by E. Treﬀtz6. Namely u1=∂Φ ∂x1+λ+µ µx3∂2Φ ∂x1∂x3, u2=∂Φ ∂x2+λ+µ µx3∂2Φ ∂x2∂x3,∇2Φ=0. (5.72) u3=−λ+2µ µ∂Φ ∂x3+λ+µ µx3∂2Φ ∂x2 3. Fig. 5.3: Boussinesq problem 5J. BIoscIuscIsscIsscIiscInscIescIsscIqsc; Application des potentiels à l’étude de l’équilibre et des m ouvements des solides élasiques, Paris: Gauthier-Villars, 1885. C.SIoscImscIiscIgscIlscIiscIascInscIasc; Sulle equazioni della elasticità, Ann. Math. , (2)17, 37.64, 1889. B.GIascIlscIescIrscIkscIiscInsc; On an investigation of stresses and deformations in elastic isotropic solids (in Russian), Dokl. Akad. Nauk SSSR , 353-358, 1930. 6E.TIrscIescIfscIfscItscIzsc; Mathematische Elastizitätstheorie , in:Handbuch der Physik, Bd. VI, Berlin, Springer, 1928. 86 Elastic materials Substitution of these relations in (5.32) for the static cas e and without body forces yields the identity. By means of this potential we can constr uct the solution of the Boussinesq problem shown in Fig. 5.3. The stress tensor in terms of potential Φis given in the form σ11 2µ=∂2Φ ∂x2 1+λ+µ µx3∂3Φ ∂x2 1∂x3−λ µ∂2Φ ∂x2 3, σ22 2µ=∂2Φ ∂x2 2+λ+µ µx3∂3Φ ∂x2 2∂x3−λ µ∂2Φ ∂x2 3, σ33 2µ=−λ+µ µ∂2Φ ∂x2 3+λ+µ µx3∂3Φ ∂x3 3, (5.73) σ12 2µ=∂2Φ ∂x1∂x2+λ+µ µx3∂3Φ ∂x1∂x2∂x3, σ23 2µ=λ+µ µx3∂3Φ ∂x2∂x2 3, σ13 2µ=λ+µ µx3∂3Φ ∂x1∂x2 3. Consequently, the shear stresses σ23andσ13are zero on the plane x3= 0.Hence, the potential deﬁned by relations (5.72) can be used only for half-spaces loaded in the direction perpendicular to this surface. ⋆Now we specify the potential Φdescribing the load in the form of the force Pe3as indicated in Fig. 5.3. This solution is known as the Boussine sq problem. It is easy to check that the following form of the potential Φ=−P 4π(λ+µ)ln(x3+r), r=√xkxk, (5.74) is a harmonic function, i.e. it satisﬁes the Laplace equatio n (5.72). Diﬀerentiation with respect tox3yields ∂Φ ∂x3=−P 4π(λ+µ)1 r,∂2Φ ∂x2 3=P 4π(λ+µ)x3 r3, (5.75) ∂3Φ ∂x3 3=P 4π(λ+µ)parenleftbigg1 r3−3x2 3 r5parenrightbigg . Substitution in (5.73) 3yields the following relation for the normal stress in x3-direction σ33=−3P 2πx3 3 r5, r=√xkxk. (5.76) This component of stresses is zero on the plane x3=0except of the point r=0where it is singular. However, if we transform stresses to spherical coordinates and integrate over an arbitrary half-sphere of the radius rit becomes equal to P. In this sense, we satisfy the boundary conditions. 5.2 Linear elasticity, isotropic and anisotropic material s 87 It is a straightforward calculation to ﬁnd the displacement . For the vertical component u3we obtain u3=−P 4π(λ+µ)1 rparenleftbigg 1−λ+µ µparenleftbigg 1+x2 3 r2parenrightbiggparenrightbigg . (5.77) This function of x3andR=radicalbig x2 1+x22(i.e.r2=R2+x23) is shown in Fig. 5.4. Units are arbitrary and we have chosen λ=4.15×1010Pa andµ=2.7×1010Pa. Obviously, there is a singularity at the point of action of the force r=radicalbig R2+x2 3=0. Fig. 5.4: Vertical displacement in the Boussinesq problem a s a function of vertical distance z=x3from the surface and horizontal distance Rfrom the axisx3♣ The Boussinesq problem can be also solved by means of another Treﬀtz potentials which are deﬁned in the following way ui=ϕi+x3χi. (5.78) It is a convenient method for problems of the half-space beca use the function ϕisatisﬁes forx3=0the same boundary condition as the displacement ui. Both functions, ϕiand χi, are harmonic ∇2ϕi=0,∇2χi=0. (5.79) The function χiis connected to ϕithrough the compatibility condition with the static displacement equations. Substitution in (5.33) with ∂2ui/∂t2=0,bi=0yields 2∂χ3 ∂xi+λ+µ µ∂ ∂xiparenleftbigg χ3+∂ϕk ∂xk+x3∂χk ∂xkparenrightbigg =0. (5.80) In many cases it is suﬃcient to assume that χipossesses a potential χi=∂ψ ∂xi,∇2ψ=0. (5.81) 88 Elastic materials Then (5.80) becomes 2∂2ψ ∂xi∂x3+λ+µ µ∂ ∂xiparenleftbigg∂ψ ∂x3+∂ϕk ∂xkparenrightbigg =0. (5.82) We can integrate once this equation with respect to xi. Assuming the constant to be zero we obtain∂ψ ∂x3=−1 3−4ν∂ϕk ∂xk,1 3−4ν≡λ+µ λ+3µ. (5.83) Now, solving the boundary value problem for ϕiwe can ﬁnd from the above relation the functionψand then the displacement from the deﬁnition of the Treﬀtz po tential which has now the form ui=ϕi+x3∂ψ ∂xi. (5.84) This representation yields not only the solution of the Bous sinesq problem but also the solution of the so-called Cerrutti problem7in which the boundary x3is loaded by the tangential force Pin thex1-direction. We present the displacement for both problems in juxtaposition in the Table below. Both solutions were com bined by R. D. Mindlin8. Numerous solutions of similar problems can be found in the bo ok of K. L. Johnson9. Table:SolutionsofBoussinesqandCerruttiproblemsforhalf-spa ce J. Boussinesq V. Cerrutti u1=Px1 4πµparenleftBig x3 r3−1−2ν r(r+x3)parenrightBig Px1 4πµparenleftBig 1 r+x2 1 r3+(1−2ν)parenleftBig 1 r+x3−x2 1 r(r+x3)2parenrightBigparenrightBig u2=Px2 4πµparenleftBig x3 r3−1−2ν r(r+x3)parenrightBig Px1x2 4πµparenleftBig 1 r3−(1−2ν)1 r(r+x3)2parenrightBig u3=P 4πµparenleftBig x2 3 r3+1−2ν rparenrightBig P 4πµparenleftBig x1x3 r3+(1−2ν)x1 r(r+x3)parenrightBig The above presented displacement formulation shall be also used further in the wave analysis. 7Valentino Cerrutti, 1850-1909 8R. D. MIiscInscIdscIlscIiscInsc; Force at a point in the interior of a semi-inﬁnite solid, Oﬃce of Naval Research Project NR-064-388 Contract Nonr-266(09), Technical Repo rt No. HCU-9-s:K, >NR-266(09)-CE, May 1953. see also:I.A.OIkscIuscImscIuscIrscIasc; On the generalization of Cerrutti’s problem in an elastic ha lf-space,Struc- tural Engn./Earthquake Eng., 12, 2, 17-26, 1995, D.A.PIoscIzscIhscIascIrscIsscIkscIiscIisc; Generalization of the Cerruti Problem, Doklady Physics ,53, No. 5, pp. 283—286, Pleiades Publishing, Ltd, 2008. 9K.L.JIoscIhscInscIsscIoscInsc; Contact Mechanics, (ninth printing) Cambridge University Press , 2003. 5.2 Linear elasticity, isotropic and anisotropic material s 89 5.2.3 Beltrami-Michell equations Static problems of linear elasticity can be also solved in a d iﬀerent way. Namely, if the boundary conditions prescribe tractions then we can direct ly ﬁnd distributions of stresses. This can be done by use of the compatibility conditions (5.13 ) or rather (5.14) as there are only six independent compatibility conditions. These i ndependent conditions may be also obtained by the contraction in (5.13) and then written i n the form ∂2eij ∂xk∂xk+∂2ekk ∂xi∂xj−∂2eik ∂xj∂xk−∂2ejk ∂xj∂xk=0. (5.85) Substitution of the inverted Hooke law (5.26) yields ∇2σij+2(λ+µ) 3λ+2µ∂2σkk ∂xi∂xj−λ 3λ+2µδij∇2σkk− (5.86) −parenleftbigg∂2σkj ∂xi∂xk+∂2σki ∂xj∂xkparenrightbigg =0. Bearing the momentum balance equation (i.e. the equilibriu m condition in the static case!) in mind ∂σij ∂xj+ρbi=0, (5.87) we reduce the system of equations (5.86) to the following for m ∇2σij+2(λ+µ) 3λ+2µ∂2σkk ∂xi∂xj−λ 3λ+2µδij∇2σkk+ (5.88) +ρ∂bi ∂xj+ρ∂bj ∂xi=0. The trace of this relation yields ∇2σkk=−3λ+2µ λ+2µρ∂bk ∂xk. (5.89) Hence, we can eliminate this Laplace operator contribution in (5.88). We obtain ∇2σij+1 1+ν∂2σkk ∂xi∂xj=−ρparenleftbigg∂bi ∂xj+∂bj ∂xiparenrightbigg −ν 1−νδijρ∂bk ∂xk. (5.90) These are Beltrami-Michell stress equations. Together wit h boundary conditions for tractions they form the well-posed problem for the determin ation of stresses. Many examples of applications of these equations extended by the contribution of pore pressure can be found in geomechanics (e.g. [20]). In a particular case of potential body forces ρbi=−∂Γ ∂xi,∇2Γ=0, (5.91) 90 Elastic materials the set of equations (5.90) has the form ∇2σij+1 1+ν∂2σkk ∂xi∂xj=−2∂2Γ ∂xi∂xj. (5.92) Therefore for such external forces the pressure is a harmoni c function ∇2p=0, p=−1 3σkk. (5.93) Simultaneously, the application of Laplace operator to (5. 92) yields ∇2∇2σij=0, (5.94) i.e. all components of stresses are biharmonic functions. 5.2.4 Plane strain and plane stress We complete these considerations with a brief presentation of two special cases of statics: plane strain and plane stress systems (compare (3.81), (3.8 2). Then all functions depend only on two variables, say, xα,α=1,2. For a system extended to inﬁnity in the x3-direction we have the plane strains — u3-component of displacement is identically zero and, conseq uently e3k=0. (5.95) Constitutive relations reduce to the form σαβ=λ∂uγ ∂xγδαβ+µparenleftbigg∂uα ∂xβ+∂uβ ∂xαparenrightbigg , σ33=λ∂uγ ∂xγα,β,γ=1,2. (5.96) The displacement equations have then the following form (λ+µ)∂2uβ ∂xα∂xβ+µ∂2uα ∂xβ∂xβ+ρbα=0. (5.97) They can be solved, as in a general case, by means of various po tentials which satisfy either Laplace or Poisson equation. The most important of th em are: 1) Galerkin function F=(F1,F2,0) uα=λ+2µ µ∂2Fα ∂xβ∂xβ−λ+µ µ∂2Fβ ∂xα∂xβ⇒ (5.98) ⇒(λ+2µ)∂4Fα ∂xβ∂xβ∂xγ∂xγ+ρbα=0i.e.∇2∇2Fα+ρbα λ+2µ=0. In many cases it is suﬃcient to introduce only one component o f this function. 5.2 Linear elasticity, isotropic and anisotropic material s 91 The stress in the direction of x3-axis is as follows σ33=2µν 1−2ν∂2 ∂xα∂xα∂Fβ ∂xβ=2µν 1−2ν∂ ∂xβ∇2Fβ. (5.99) 2) Papkovich-Neuber potentials uα=∂(Φ0+xβΦβ) ∂xα−4(1−ν)Φα, (5.100) with the following equations for potentials 4µ(1−ν)∂2Φ0 ∂xβ∂xβ+ρxβbβ=0, 4µ(1−ν)∂2Φα ∂xβ∂xβ−ρbα=0, α=1,2. (5.101) 3) Airy function. Compatibility conditions (5.85) reduce i n the plane case to the single equation ∂2e11 ∂x2∂x2+∂2e22 ∂x1∂x1=2∂2e12 ∂x1∂x2. (5.102) In terms of stresses this condition and the equilibrium cond itions without body forces have the form ∂2σ11 ∂x2∂x2+∂2σ22 ∂x1∂x1−λ 2(λ+µ)∂2 ∂xβ∂xβ(σ11+σ22)=2∂2σ12 ∂x1∂x2, ∂σ11 ∂x1+∂σ12 ∂x2=0,∂σ12 ∂x1+∂σ22 ∂x2=0. (5.103) The following Airy function F σαβ=−∂2F ∂xα∂xβ+δαβ∂2F ∂xγ∂xγ, (5.104) satisﬁes identically the equilibrium conditions and yield s the following form of the com- patibility condition ∇2∇2F=∂4F ∂xα∂xα∂xβ∂xβ≡∂4F ∂x4 1+2∂4F ∂x2 1∂x2 2+∂4F ∂x4 2=0, (5.105) i.e. it is a biharmonic function. Now we consider the case of plane stresses. It appears in membranes, i.e. systems whose one dimension, say in the x3-direction, is much smaller than in the remaining two directions. Then we can use an approximation σ3k≈0. (5.106) 92 Elastic materials Simultaneously σαβ=2µeαβ+2µλ λ+2µδαβeγγ, α,β,γ=1,2, (5.107) which implies the following set of displacement equations µparenleftbigg∂2uα ∂xβ∂xβ+3λ+2µ λ+2µ∂2uβ ∂xα∂xβparenrightbigg +ρbα=0. (5.108) If we invert the constitutive relations (5.107) eαβ=1 2µparenleftbigg σαβ−λ 3λ+2µδαβσγγparenrightbigg , (5.109) then the compatibility relation (5.102) yields the followi ng equation ∂2σ11 ∂x2∂x2+∂2σ22 ∂x1∂x1−λ 3λ+2µ∂2 ∂xβ∂xβ(σ11+σ22)=2∂2σ12 ∂x1∂x2. (5.110) Hence, we can again introduce the Airy function Fby the relation (5.104) which satisﬁes identically equilibrium conditions and it is again a biharm onic function satisfying the equation (5.105). By means of this Airy function one solves in the linear elasti city the Flamant (1892) problem (stresses and displacements in a linear elastic wed ge loaded by point forces at its sharp end; in particular the solution for the half-plane ), punch problems and many others. We present here a simple example of a solution for a me mbrane with a hole. This solution indicates an important property of mechanical sys tems with imperfections that they yield stress concentration. ⋆In order to appreciate an inﬂuence of structure discontinui ties on the distribution of stresses we consider a simple example of an inﬁnite membra ne with hole (cavity) of radiusa(see: Fig. 5.5.)10. Fig. 5.5: Extension of a membrane with the circular cavity 10CIhscIisc-TIescIhscWIascInscIgsc ;Applied Elasticity , McGraw-Hill, N. Y., 1953 5.2 Linear elasticity, isotropic and anisotropic material s 93 The membrane is loaded uniformly in the x-direction by the lo ad of intensity S. Clearly, if the hole is not there the stress in the membrane ha s the following components σx=S, σy=τxy=0. (5.111) This solution corresponds to the Airy function F0=1 2Sy2=1 4Sr2(1−cos2θ). (5.112) This relation implies the following components of stresses in the polar coordinates σ0 rr=1 r∂F0 ∂r+1 r2∂2F0 ∂θ2=1 2S(1+cos2θ), σ0 θθ=∂2F0 ∂r2=1 2S(1−cos2θ), (5.113) τ0 rθ=−∂ ∂rparenleftbigg1 r∂F0 ∂θparenrightbigg =−1 2Ssin2θ. We have made this transformation of coordinates as the probl em with the cavity is easier in polar coordinates. For the problem with the cavity the fol lowing boundary conditions must be fulﬁlled σrr=τrθ=0forr=a, (5.114) σrr=σ0 rr, τrθ=τ0 rθ, σθθ=σ0 θθforr→∞. The structure of the function F0suggests that we can try to ﬁnd the Airy function for the more general case in the following form F=f1(r)+f2(r)cos2θ. (5.115) This function must be biharmonic. Substitution of (5.115) i n the equation for Airy function in polar coordinates (compare (5.45)) parenleftbigg∂2 ∂r2+1 r∂ ∂r+1 r2∂2 ∂θ2parenrightbiggparenleftbigg∂2F ∂r2+1 r∂F ∂r+1 r2∂2F ∂θ2parenrightbigg =0, (5.116) yields two equations as the general equation should hold for arbitrary angles θ. They have the following form parenleftbiggd2 dr2+1 rd drparenrightbiggparenleftbiggd2f1 dr2+1 rdf1 drparenrightbigg =0, (5.117) parenleftbiggd2 dr2+1 rd dr−4 r2parenrightbiggparenleftbiggd2f2 dr2+1 rdf2 dr−4f2 r2parenrightbigg =0. The solutions of these simple ordinary diﬀerential equatio ns have the form f1(r) =c1r2lnr+c2r2+c3lnr+c4, (5.118) f2(r) =c5r2+c6r4+c7 r2+c8. 94 Elastic materials This yields the Airy function in the form F=parenleftbigc1r2lnr+c2r2+c3lnr+c4parenrightbig+parenleftBig c5r2+c6r4+c7 r2+c8parenrightBig cos2θ. (5.119) Hence relations for stress components are as follows σrr=1 r∂F ∂r+1 r2∂2F ∂θ2=c1(1+2lnr)+2c2+c3 r2−parenleftbigg 2c5+6c7 r4+4c8 r2parenrightbigg cos2θ, σθθ=∂2F ∂r2=c1(3+2lnr)+2c2−c3 r2+parenleftbigg 2c5+12c6r2+6c7 r4parenrightbigg cos2θ,(5.120) τrθ=−∂ ∂rparenleftbigg1 r∂F ∂θparenrightbigg =parenleftbigg 2c5+6c6r2−5c7 r4−2c8 r2parenrightbigg sin2θ. Stresses should be ﬁnite in inﬁnity which means that constan tsc1andc6must be identically zero. The remaining boundary conditions lead t o the following solution σr=S 2parenleftbigg 1−a2 r2parenrightbigg +S 2parenleftbigg 1+3a4 r4−4a2 r2parenrightbigg cos2θ, σθ=S 2parenleftbigg 1+a2 r2parenrightbigg −S 2parenleftbigg 1+3a4 r4parenrightbigg cos2θ, (5.121) τrθ=−S 2parenleftbigg 1−3a4 r4+2a2 r2parenrightbigg sin2θ. It is seen that for r=aandθ=π/2andθ=3π/2the circumferential stress σθθhas the maximum value equal to 3S. This is three times more than in the case without the hole. In Fig. 5.6. we show the behaviour of σθθfor these two values of the angle. Fig. 5.6: Circumferential stresses σθθ/Sin function of the distance from the holer/a.♣ The problem of the concentration of stresses in the vicinity of various holes has a very extensive literature due to its practical bearing. 5.2 Linear elasticity, isotropic and anisotropic material s 95 5.2.5 Waves in linear elastic materials We return to the analysis of the displacement equations (5.3 2). We begin with a proof of existence of two waves described by these equations. This is based on Hadamard Theorem which we sketch for the purpose of the linear theory. We consider a point xA on a singular surface on which both the strain eand the velocity vare continuous but their higher derivatives such as the acceleration ∂v/∂tor the gradient of strain grademay suﬀer a ﬁnite discontinuity. The point xAchanges its position with the moving singular surface and, say, after a small time increment δtis located in a point xB=xA+δx. We assume that this change happens in the direction nperpendicular to the surface and with the speedc, i.e.δx=cnδt.Then the change of the value of the gradient of displacement between these two points can be calculated on the path ahead o f the singular surface and behind this surface and, consequently, along the paths o n which the gradient of displacement possesses continuous derivatives. We have ∂ui ∂xjvextendsinglevextendsinglevextendsinglevextendsingle B=∂ui ∂xjvextendsinglevextendsinglevextendsinglevextendsingle A+∂2ui ∂xj∂xkvextendsinglevextendsinglevextendsinglevextendsingle+ Aδxk+∂2ui ∂xj∂tvextendsinglevextendsinglevextendsinglevextendsingle+ Aδt= (5.122) =∂ui ∂xjvextendsinglevextendsinglevextendsinglevextendsingle A+∂2ui ∂xj∂xkvextendsinglevextendsinglevextendsinglevextendsingle− Aδxk+∂2ui ∂xj∂tvextendsinglevextendsinglevextendsinglevextendsingle− Aδt, where the signature " +" and "−" indicates the limits on both sides of the surface. Subtracting these relations, we easily arrive at the follow ing compatibility condition bracketleftbiggbracketleftbigg∂2ui ∂xj∂xkbracketrightbiggbracketrightbigg =−1 cbracketleftbiggbracketleftbigg∂2ui ∂xj∂tbracketrightbiggbracketrightbigg nk, (5.123) where[[...]]=(...)+−(...)−is the diﬀerence of limits on both sides of the surface. In the same way we prove the identity for the acceleration bracketleftbiggbracketleftbigg∂2ui ∂xj∂tbracketrightbiggbracketrightbigg =−1 cbracketleftbiggbracketleftbigg∂2ui ∂t2bracketrightbiggbracketrightbigg nj. (5.124) These identities form the contents of Hadamard Theorem. The y can be combined to give the following relationbracketleftbiggbracketleftbigg∂2ui ∂xj∂xkbracketrightbiggbracketrightbigg =1 c2bracketleftbiggbracketleftbigg∂2ui ∂t2bracketrightbiggbracketrightbigg njnk. (5.125) Now we form the jump of the displacement equations on the sing ular surface described above. We obtain ρbracketleftbiggbracketleftbigg∂2ui ∂t2bracketrightbiggbracketrightbigg =(λ+µ)bracketleftbiggbracketleftbigg∂2uk ∂xi∂xkbracketrightbiggbracketrightbigg +µbracketleftbiggbracketleftbigg∂2ui ∂xk∂xkbracketrightbiggbracketrightbigg . (5.126) Substitution of (5.125) yields parenleftbigg c2δik−λ+µ ρnink−µ ρδikparenrightbiggbracketleftbiggbracketleftbigg∂2uk ∂t2bracketrightbiggbracketrightbigg =0. (5.127) 96 Elastic materials Consequently, we obtain the eigenvalue problem in which the discontinuity of accelera- tion is the eigenvector. The eigenvalues can be found by the s eparation of longitudinal and transversal contributions. If we multiply the relation (5.127) by a unit vector t perpendicular to n(i.e.tini=0) then we obtain parenleftbigg c2−µ ρparenrightbiggparenleftbiggbracketleftbiggbracketleftbigg∂2uk ∂t2bracketrightbiggbracketrightbigg tkparenrightbigg =0. (5.128) Hence, either the projection of discontinuity on the direct ion perpendicular to nis zero and then we obtain the identity, or it is diﬀerent from zero an d then c2=c2 T=µ ρ. (5.129) This is the square of the speed of propagation of the front of t he wave on which the acceleration suﬀers the transversal discontinuity. We cal l such waves transversal (shear waves). Obviously, in order to be real the speed of propagati on yields the condition µ>0, (5.130) which is one of the limitations of material parameters menti oned at the beginning of this Chapter. Now we multiply the equation (5.127) by the vector n. It follows parenleftbigg c2δik−λ+2µ ρparenrightbiggparenleftbiggbracketleftbiggbracketleftbigg∂2uk ∂t2bracketrightbiggbracketrightbigg nkparenrightbigg =0. (5.131) Hence, either the projection of discontinuity on the direct ionnis zero and then we obtain the identity, or it is diﬀerent from zero and then c2=c2 L=λ+2µ ρ. (5.132) This is the square of the speed of propagation of the front of t he wave on which the acceleration suﬀers the longitudinal discontinuity. We ca ll such waves longitudinal. As before, for existence of these waves we have to require λ+2µ>0, (5.133) which is the second limitation of material parameters. When both conditions (5.130) and 5.2 Linear elasticity, isotropic and anisotropic material s 97 (5.133) are satisﬁed we say that the set of displacement equa tions (5.32) is hyperbolic. Fig. 5.7: Schematic picture of longitudinal and transversa l waves In Fig. 5.7. we demonstrate schematically the motion of part icles by the transition of longitudinal and transversal waves. Any dynamic solution of the displacement equations (5.32) d escribes the propagation of the wave front which divides the domain Btat the instant of time tinto a part which is not yet disturbed by the loading and the part behind t he front where the dynamic displacement evolves. Exact solutions of this art c an be constructed by the use of dynamic Green function for ﬁnite domains. However, te chnical diﬃculties in construction of such solutions are so extensive that it pays oﬀ to consider a local structure of dynamic disturbances. This is usually done by means of the Fourier analysis of plane waves. The latter assumption means that we replace the three -dimensional propagation by a one-dimensional local approximation. The solution is a ssumed to have the form u=RebraceleftBig Aei(k·x−ωt)bracerightBig , (5.134) whereAis a complex constant amplitude, ωis the so-called frequency of the wave. We assume that it is given. Such waves are called monochromatic . The vectorkhas the structurek=kn,n·n=1and the unit vector nis the direction of propagation of the wave. It is assumed to be constant which means that the wave is plane.kis the so-called wave number and it may be complex. Consequently the function (5.134) can be written in the form u=e−Imk(n·x)ReparenleftBig AeiRek(n·x−cpht)parenrightBig , cph=ω Rek, (5.135) andcphis called the phase speed. Imkdescribes the damping of the wave. The above relation can be also written in the form of a real function u=e−Imk(n·x)A0cos(Rek(n·x−cpht)+φ),A=A0eiφ, (5.136) andA0is the real amplitude. [Rek(n·x−cpht)+φ]is called the phase and φis called the phase shift. 98 Elastic materials It is convenient to change phase in the following manner Rek(n·x−cpht)+φ=2πparenleftbiggRek 2πn·x−ω 2πtparenrightbigg +φ= =2πparenleftBign·x l−ftparenrightBig +φ, l=2π Rek, f=ω 2π. (5.137) The quantity fis called the technical frequency and lis the wave length of the mono- chromatic wave of frequency ω. The full solution of the displacement equations (5.32) can b e constructed by means of the combination of monochromatic waves which form then co ntributions to a Fourier series. This representation of waves is called spectral. We do not need to go into all details of the spectral analysis a nd present only solutions of the form (5.134). Substitution of this relation in the dis placement equations yields ρ(−iω)2Ai=(λ+µ)Ak(ikk)(iki)+µAi(ikk)(ikk), i.e.bracketleftbigρω2δik−(λ+µ)kikk−µk2δikbracketrightbigAk=0, k=radicalbig kkkk. (5.138) This is again the eigenvalue problem. As in the case of the wav e front which we have discussed above we separate the tangential and longitudina l components. Scalar multi- plication by the unit vector tperpendicular to kyields bracketleftbigρω2−µk2bracketrightbigAktk=0. (5.139) Hence, the projection of the amplitude on the direction perp endicular to the direction of propagationn=k/kdiﬀerent from zero yields k2=ω2 c2 T. (5.140) This is the so-called dispersion relation for transversal m onochromatic waves. It shows that the wave number is real in this case and that the phase spe ed is equal to the speed of propagation of transversal waves cph=cT, (5.141) i.e. the phase speeds are independent of the frequency of the monochromatic wave. Such waves are called non-dispersive. As Imk=0they are not attenuated (damping is zero). Now scalar multiplication of the equation (5.138) by kyields bracketleftbigρω2−(λ+2µ)k2bracketrightbigAknk=0. (5.142) Again for the projection of the amplitude on the direction of propagation diﬀerent from zero we obtain k2=ω2 c2 L. (5.143) 5.2 Linear elasticity, isotropic and anisotropic material s 99 This dispersion relation for longitudinal monochromatic w aves yields the phase speed equal to the speed cL cph=cL, (5.144) and this is again independent of the frequency. As transvers al waves, also longitudinal monochromatic waves are non-dispersive and not attenuated . The two sorts of waves which we presented above are called bul k waves because they propagate inside of the body. In the case of a boundary the sit uation changes. We consider here a simple case of a half-space with the plane bou ndary which yields the so-called surface waves. In the case under consideration th ey were discovered by J. W. Rayleigh (1887)11. On the boundary perpendicular to the x3-axis we assume the boundary conditions Tn|x3=0i.e.σk3|x3=0=0,n=−e3, (5.145) i.e. the boundary is stress-free. It means that the wave has b een created far away from the origin of coordinates and its source will be ignored in th e analysis. Simultaneously, we have the following Sommerfeld condition u|x3→∞=0. (5.146) It is easier to seek the solution of the problem when we make th e following decompo- sition of the displacement vector u=uL+uT,rotuL=0,divuT=0, (5.147) whereuLis called the potential part and uTis the solenoidal part. Obviously, it is directly connected with the Helmholtz decomposition (1.57): uL= gradϕ,anduT= rotψ. These two parts must satisfy equations ∂2uL ∂t2=c2 L∇2uL,∂2uT ∂t2=c2 T∇2uT, (5.148) following directly from the displacement equations (5.32) (compare (5.58)). We seek the solution in the form of the following ansatz uL=ALe−γx3ei(kx1−ωt)e1+BLe−γx3ei(kx1−ωt)e3, (5.149) uT=ATe−βx3ei(kx1−ωt)e1+BTe−βx3ei(kx1−ωt)e3. Hence, we consider the plane problem. We anticipate a progre ssive wave solution in the x1-direction and the decay of the solution in the x3-direction provided the coeﬃcients γ,βare positive. If such a solution does not exist it means that t he surface wave does not appear. Substitution of the ansatz (5.149) in wave equations (5.148 ) leads to the compatibility conditions γ2 k2=1−c2 R c2 L,β2 k2=1−c2 R c2 T, cR=ω k, (5.150) 11J.W.(SItscIrscIuscItscItsc)RIascIyscIlscIescIiscIgscIhsc; On waves propagated along the plane surface of an elastic sol id,Proc. London Math. Soc., 17:4-11, 1887. 100 Elastic materials wherecRis the phase speed of the wave. Now we use the properties of the potential and solenoidal parts. We have ǫ231∂uL 1 ∂x3+ǫ213∂uL 3 ∂x1= 0⇒BL=iγ kAL, (5.151) ∂uT 1 ∂x1+∂uT 3 ∂x3= 0⇒BT=ik βAT. Consequently uL=parenleftBig e1+iγ ke3parenrightBig ALe−γx3ei(kx1−ωt), (5.152) uT=parenleftbigg e1+ik βe3parenrightbigg ATe−βx3ei(kx1−ωt). Obviously, the Sommerfeld condition is satisﬁed by these fu nctions ifγ,β >0. The stress components which we need in boundary conditions (5.1 45) can be written in the form 1 ρσ33=parenleftbigc2 L−2c2Tparenrightbig∂u1 ∂x1+c2 L∂u3 ∂x3, (5.153) 1 ρσ13=c2 Tparenleftbigg∂u1 ∂x3+∂u3 ∂x1parenrightbigg . Hence, forx3=0the substitution of (5.152) leads to the set of two equations parenleftbigg 2−c2 R c2 Tparenrightbigg AL+2AT= 0, (5.154) 2γβ k2AL+parenleftbigg 2−c2 R c2 Tparenrightbigg AT= 0. This is the homogeneous set of equations for amplitudes AL,AT. It possesses nontrivial solutions if the determinant is equal to zero parenleftbigg 2−c2R c2 Tparenrightbigg2 −4radicalBigg 1−c2 R c2 TradicalBigg 1−c2 R c2 L=0, (5.155) where relations (5.150) have been used. This equation for cRis called Rayleigh dispersion relation. It is clear that the speed of Rayleigh waves cRis independent of the frequency ω. Consequently, Rayleigh waves are non-dispersive. The solution of the equation (5.155) is shown in Fig. 5.8. As cT cL=radicalbiggµ λ+2µ=radicalbigg 1 21−2ν 1−ν<1, (5.156) the speed of Rayleigh waves is smaller than the speed of trans versal waves. It has a physical interpretation in terms of the so-called construc tive interference of longitudinal 5.2 Linear elasticity, isotropic and anisotropic material s 101 and transversal waves which happens after these waves are re ﬂected from the boundary and so create the surface wave. Fig. 5.8: Dimensionless speed of Rayleigh waves cR/cTas a function of the fraction of transversal and longitudinal speeds cT/cL. Let us inspect the relations for components of displacement s. According to (5.147) they have the form u1=parenleftbigALe−γx3+ATe−βx3parenrightbigei(kx1−ωt), (5.157) u3=iparenleftbiggγ kALe−γx3+k βATe−βx3parenrightbigg ei(kx1−ωt). We choose the real part of these relations with cos(kx1−ωt). Then by eliminating the time from these relations, we obtain (Reu1)2 α2 1+(Reu3)2 α2 3=1, (5.158) where α1=ALe−γx3+ATe−βx3, (5.159) α3=γ kALe−γx3+k βATe−βx3. Hence the orbits of particles are ellipses with semiaxes |α1|,|α3|. One can easily show that the motion is anticlockwise. This is opposite to the dir ection of motion of water particles in shallow water waves. The size of these ellipses diminishes exponentially with the depthx3. This is the reason for calling such a wave the surface wave. I t can be shown that the elliptic motion of particles in planes x2=constis the only motion possible for Rayleigh waves. Transversal Rayleigh waves do not exist. The lack of dispersion in the plane case presented above expl ains the disastrous action of surface waves in earthquakes. In contrast to bulk waves wh ich move from the point 102 Elastic materials source (hypocenter) in approximately spherical form and ar e the ﬁrst two arrivals, the surface wave moves from epicenter in approximately cylindr ical form and arrives as the last one (cR< cT< cL). Hence the energy which carries the wave is distributed on a much larger surfaces for bulk waves than for surface waves. Fig. 5.9: An application of surface waves: testing of rails The nondispersive character of three waves which we have con sidered above is rather exceptional. Whenever the problem comprises a characteris tic length waves become dispersive. This is the case for surface waves in boreholes, tunnels, layers, or media with microstructure such as porous materials (characteristic l ength — dimensions of channels). We consider here the simplest example of Love waves12. A. E. H. Love (1911) has solved the problem of propagation of w aves in a layer of thicknessHon an elastic half-space. The plane of contact of these two me dia (interface) is perpendicular to the upward oriented x3-axis and the origin of the coordinates lies on this plane. We distinguish the material properties by a ’p rime’, i.e.ρ′,c′ Tare the mass density and the speed of transversal waves in the layer w hileρ,cTare the mass density and the speed of transversal waves in the half-space . We assume that the motion of particles is perpendicular to the (x1,x3)-plane of propagation of waves. Then the problem is described by two wave equations ∂2u′ 2 ∂t2=c′2 T∇2u′ 2for0<x3<H, (5.160) ∂2u2 ∂t2=c2 T∇2u2forx3<0. One can show that longitudinal surface waves (i.e. waves wit h a displacement (u′ 1,0,u′3) in the plane x2=const) for this conﬁguration do not exist. 12C.G.LIascIisc, K.WIiscIlscImscIascInscIsscIkscIisc; (eds),Surface Waves in Geomechanics: Direct and Inverse Modellin g for Soils and Rocks , Springer, 2005. 5.2 Linear elasticity, isotropic and anisotropic material s 103 We seek the solution of the system (5.160) in the form of a mono chromatic wave of the frequency ω u′ 2=parenleftBig A′eiks′x3+B′e−iks′x3parenrightBig ei(kx1−ωt)≡ ≡2(ReA′cosks′x3−ImA′sinks′x3)ei(kx1−ωt), (5.161) u2=Beksx3ei(kx1−ωt). This solution should satisfy the boundary conditions 1. Shear stress on the plane x3=His equal to zero, i.e. ∂u′ 2 ∂yvextendsinglevextendsinglevextendsinglevextendsingle x3=H=0, (5.162) 2. shear stress and the displacement must be continuous on th e interfacex3=0 ρ′c′2 T∂u′ 2 ∂x3vextendsinglevextendsinglevextendsinglevextendsingle x3=0=ρc2 T∂u2 ∂x3vextendsinglevextendsinglevextendsinglevextendsingle x3=0, ρ′c′2 T=µ′, ρc2 T=µ, u′ 2|x3=0=u2|x3=0. (5.163) Substitution of (5.161) in (5.160) yields the compatibilit y conditions s′2=c2 c′2 T−1, s2=1−c2 c2 T, c=ω k. (5.164) The boundary condition (5.162) leads to the following displ acement in the layer u′ 2=2ReA′cos(ks′(H−x3)) cos(ks′H)ei(kx1−ωt). (5.165) Then the remaining two conditions (5.163) give rise to two eq uations for the constants ReA′,B. As this system of equations is homogeneous the determinant must be zero and we obtain the following Love dispersion relation ω=c Hs′bracketleftbigg arctanparenleftbiggρc2 Ts ρ′c′2 Ts′parenrightbigg +nπbracketrightbigg , n=1,2,3,..., (5.166) wheres,s′must be real and smust be positive for the amplitude of the wave to decay in the half-space. Consequently, according to (5.164), c′ T≤c≤cT. (5.167) Hence, the Love waves exist only in layers which are softer th an the foundation. Si- multaneously, the dispersion relation has inﬁnitely many s olutions, the so-called modes, and the corresponding speeds of propagation depend on the fr equencyω. Love waves are dispersive. This means that packages of waves of diﬀeren t frequency become broader during the propagation — some of their monochromatic contri butions are slower than the 104 Elastic materials others. For this reason, one introduces also an "average" sp eed of propagation which is called the group velocity cg=dω dk=dω dcphc2 ph dω dcphcph−ω, (5.168) where the second part of the relation follows immediately fr om the deﬁnitions. Conse- quently, we can ﬁnd the group velocity immediately substitu ting the dispersion relation (5.166). We show here only a simple numerical example for the following data (compare examples in [1]) cT=5km s, c′ T=3km s,ρ′ ρ=0.875, H=10km. (5.169) Results are plotted in Fig. 5.10. Fig. 5.10: Phase and group velocity of the ﬁrst mode of Love wa ve for the data(5.169) In contrast to the phase velocity the group velocity is not a m onotonous function of the frequency. This property yields the existence of the so- called Airy phase which has an important bearing in description of seismic waves13. 5.2.6 Principle of virtual work Motivated by the principle of conservation of energy we pres ent now one of the most important procedures in linear elasticity which forms both a method for mathematical proofs of existence and uniqueness of solutions as well as a f oundation for numerous approximate methods of solutions. 13for some details see: C.G.LIascIisc,K.WIiscIlscImscIascInscIsscIkscIisc; (eds),Surface Waves in Geomechanics: Direct and Inverse Modelling for Soils and Rocks , Springer, 2005. 5.2 Linear elasticity, isotropic and anisotropic material s 105 Let us consider ﬁrst the static problem. The body Btis loaded by external forces ρb and tractionstngiven on the surface ∂Bσ twith the prescribed displacement ubon the remaining part of the boundary ∂Bu t. This yields displacements u(x), stressesT(x)and straine(x)in the bodyBt. We impose on these displacements a virtual displacement δuwhich is small, continu- ously diﬀerentiable and admissible. It means that it has to c omply to conditions limiting the motion of the body. For instance, it must be zero on the bou ndary∂Bu t. The principle of virtual work says that the work done by virtu al displacements of external loadings is equal to the work of internal forces, i. e. integraldisplay Btρb·δudV+integraldisplay ∂Bttn·δudS=integraldisplay BtT·δedV, (5.170) where δe=1 2parenleftBig (gradδu)+(gradδu)TparenrightBig =1 2δparenleftBig (gradu)+(gradu)TparenrightBig . (5.171) We shall see that this statement is related to the energy cons ervation (4.1). In order to derive (5.170) we multiply the equilibrium condi tions byδuand integrate over the bodyBt. We have integraldisplay Bt(divT+ρb)·δudV=0. (5.172) This yields integraldisplay BtdivT·δudV=integraldisplay Bt[div(Tδu)−T·gradδu]dV= =integraldisplay ∂Bttn·δudS−integraldisplay BtT·δedV, (5.173) where we have used δu=0on∂Bu t. This relation indicates (5.170). Conversely the principle (5.170) yields local equilibrium conditions. For isotropic elastic solids we can also write T·δe=(λekkδij+2µeij)δeij=1 2λδ(eiiejj)+µδ(eijeij)=1 2δ(σijeij).(5.174) Consequently integraldisplay Btρb·δudV+integraldisplay ∂Bttn·δudS=δE, E=integraldisplay BtρεdV, ρε=1 2σijeij. (5.175) Obviously, the quantity ρεis the potential energy of the linear elastic material. The quantityEis called the work of deformations. 106 Elastic materials As the body forces band tractionstnare given their variations are zero. Hence, we can write δΠe=0,Πe=E−integraldisplay Btρb·udV−integraldisplay ∂Bttn·udS, (5.176) whereΠeis the potential energy of the displacement ﬁeld. The above c ondition shows that this potential possesses an extremum in real motion. It is easy to see that it is minimum. We have to compare the potentials Π′ eandΠefor displacements u+δuand u. We have ρε(eij+δeij)=ρε(eij)+ρ∂ε ∂eijδeij+1 2!∂2ε ∂eij∂eklδeijδekl+..., ⇒ρε(eij+δeij)−ρε(eij)=σijδeij+1 2∂σij ∂eklδeijδekl+... (5.177) Therefore Π′ e−Πe=integraldisplay BtσijδeijdV−integraldisplay Btρb·δudV−integraldisplay ∂Bttn·δudS+integraldisplay Bt1 2∂σij ∂eklδeijδekldV= =integraldisplay Bt1 2∂σij ∂eklδeijδekldV=integraldisplay Bt1 2(λδijδkl+2µδikδjl)δeijδekldV= =integraldisplay Bt1 2(λδeiiδejj+2µδeijδeij)dV= (5.178) =integraldisplay Bt1 2parenleftbiggparenleftbigg λ+2 3µparenrightbigg δeiiδejj+2µδeD ijδeD ijparenrightbigg dV, where δeD ij=δeij−1 3δekkδij, (5.179) is the deviatoric part of δeij. The quantity (5.178) is positive if each contribution to the sum is positive. This follows from the fact that δeiiandδeD ijare independent and arbitrary. Consequently, the potential Πepossesses the minimum in equilibrium if and only if the material parameters satisfy the conditions λ+2 3µ=K >0, µ>0. (5.180) We have mentioned these limitations before. They indicate a s wellλ+2µ>0, i.e. (5.130) and (5.133) which means that the minimum condition implies t he hyperbolicity of the displacement equations. We can easily extend the above principle on the full dynamic c ase. Then the principle of virtual work (5.176) must be modiﬁed in the following way. We consider the motion of the body between two instances of time, t1andt2. We consider the variation of real displacement δuwhich satisﬁes the conditions δu(x,t1)=δu(x,t2)=0. (5.181) 5.3 Thermoelasticity 107 Instead of (5.172) we have t2integraldisplay t1dtintegraldisplay Btparenleftbigg divT+ρb−ρ∂2u ∂t2parenrightbigg ·δudV=0. (5.182) We have to transform the contribution of acceleration. If we introduce the kinetic energy K=integraldisplay Bt1 2ρ∂u ∂t·∂u ∂tdV, (5.183) then t2integraldisplay t1δKdt=t2integraldisplay t1dtintegraldisplay Btρ∂ ∂tparenleftbigg∂u ∂t·δuparenrightbigg dV−t2integraldisplay t1dtintegraldisplay Btρ∂2u ∂t2·δudVdt. (5.184) Due to conditions (5.181) the ﬁrst integral on the right-han d side vanishes. Consequently, δt2integraldisplay t1(E−K)dt=t2integraldisplay t1dtintegraldisplay Btρb·udV+integraldisplay ∂Bttn·δudS, (5.185) or δt2integraldisplay t1Ldt=0,L=K−Πe. (5.186) This is the Hamilton principle. The functional Lis called Lagrangian of the system. The Hamilton principle says that Lagrangian has an extremum in the interval of time t1<t<t2, where in the endpoints of this interval the state of the body is known. Hamilton’s principle has a very extensive physical literat ure as it is the main tool in the derivation of model equations for numerous reversible proc esses of classical, relativistic and quantum mechanics. The method based on the construction of Lagrangian and various invariance principles yields equations of motion a nd conservation laws (see the classical reference of Landau and Lifschitz [7]). Attempts to extend the method on irreversible processes such as heat conduction or theory of dislocations in applications to plasticity were not successful because the so-called Lag range-Euler equations which follow as the equations of motion in this approach are invari ant with respect to time reversal, i.e. they must be reversible. 5.3 Thermoelasticity All processes in linear elastic materials which we have been discussing were assumed to be isothermal. This is almost never the case and an inﬂuence of t emperature diﬀerence may have a very substantial inﬂuence on the distribution of stre sses. We show the simplest 108 Elastic materials possible extension of the linear elasticity on processes in which the temperature is variable as well. In the thermodynamic construction of a model we have to deal w ith at least two ﬁelds in nonisothermal processes: displacement uand temperature T. For the displacement we expect as before that it follows from the ﬁeld equations cons tructed on the basis of the momentum conservation law. On the other hand, the temperatu reTshould satisfy the ﬁeld equation which we assume to follow from the energy conse rvation law. Consequently, we choose the equations (3.38) and (4.11) from which we const ruct the ﬁeld equations. In the linear problems they are as follows ρ∂vk ∂t=∂σkl ∂xl+ρbk, (5.187) ρ∂ε ∂t+∂qk ∂xk=σkl∂vk ∂xl, where we have neglected the radiation and the mass density ρis constant. For thermoelastic materials we assume that the stresses σkl, the internal energy εand the heat ﬂux qkare functions of the strain ekl, temperature Tand temperature gradient ∂T/∂xk. These constitutive functions should satisfy the second la w of thermodynamics (4.20) which in the linear form is as follows ρ∂η ∂t+divparenleftBigq TparenrightBig ≥0, (5.188) for all processes. The entropy density ηis also a function of the above listed constitutive variables. We simplify the considerations by assuming thes e constitutive laws in the form σkl=σkl(eij,T), ε=ε(eij,T), η=η(eij,T), qk=−KT∂T ∂xk, (5.189) whereKTis the thermal conductivity coeﬃcient.The last relation is called Fourier’s law. The detailed justiﬁcation of these assumptions can be found in books on continuum thermodynamics (e.g. [22]). Now we exploit the inequality (5.188). The procedure is the s ame as for ideal gases in Sec. 5.2. The momentum balance (5.187) does not impose any restrictions on the inequality (5.188) due to the presence of acceleration. The elimination of the heat ﬂux from the inequality eliminates as well the constraint impos ed by the energy conservation. We obtain ρ∂ψ ∂t+ρη∂T ∂t+1 Tqk∂T ∂xk−σkl∂ekl ∂t≤0, (5.190) ψ=ε−Tη=ψ(eij,T), whereψis the Helmholtz free energy function. This inequality shou ld hold for all ﬁelds of displacement and temperature. Due to the symmetry of the s tress tensor, we have made the following replacement σkl∂vk ∂xl=1 2σklparenleftbigg∂vk ∂xl+∂vl ∂xkparenrightbigg =σkl∂ekl ∂t. (5.191) 5.3 Thermoelasticity 109 Now, the chain rule of diﬀerentiation and the linearity of th e inequality with respect to the derivatives ∂T/∂t,∂e kl/∂tyield the following identities σkl=ρ∂ψ ∂ekl, η=−∂ψ ∂T, ε=ψ−T∂ψ ∂T, (5.192) and the residual inequality deﬁning the dissipation D=−qk∂T ∂xk=KTparenleftbigg∂T ∂xk∂T ∂xkparenrightbigg ≥0⇒K≥0. (5.193) These relations immediately imply the following Gibbs equa tion of linear thermoelas- ticity dη=1 Tparenleftbigg dε−1 ρσkldeklparenrightbigg . (5.194) Now, we are in the position to formulate the linear isotropic model. We assume that the current temperature deviates only a little from the homo geneous initial temperature T0, i.e. vextendsinglevextendsinglevextendsinglevextendsingleT−T0 T0vextendsinglevextendsinglevextendsinglevextendsingle≪1. (5.195) Assuming in addition that the undeformed state ( ekl= 0) is stress-free we write the Helmholtz free energy in the form of quadratic function with respect to the deviation from the initial natural conﬁguration ρψ=−1 2ρcv T0(T−T0)2−γ(T−T0)ekk+1 2cijkleijekl, (5.196) cijkl=λδijδkl+µ(δikδjl+δilδjk). The function must be quadratic due to relations (5.192) whic h imply that derivatives of ψ must be linear. The coeﬃcients in the above relation possess the following interpretation. The internal energy follows in the form ε=cv 2T0parenleftbigT2−T2 0parenrightbig+γ ρT0ekk+1 2ρcijkleijekl. (5.197) Consequently ∂ε ∂T=cvT T0≈cv. (5.198) Hence the coeﬃcient cvis the speciﬁc heat by constant volume. Incidentally, the sp eciﬁc heats by constant volume and constant pressure, respective ly, are practically identical for solids, in contrast to gases. We proceed to the coeﬃcient γ. For the stress tensor we obtain σij=λekkδij+2µeij−γ(T−T0)δij. (5.199) The ﬁrst part is, obviously, identical with Hooke’s law for i sothermal processes. The trace of this equation leads to ekk=σkk 3K−γ K(T−T0), K=λ+2 3µ. (5.200) 110 Elastic materials Hence the coeﬃcient γdescribes volume changes caused by the temperature diﬀeren ce. It is called volumetric thermal expansion coeﬃcient for sol ids. It is sometimes denoted by αV. In experiments usually the linear thermal expansion coeﬃc ient for solids α=γ/3K is measured. Some values of this coeﬃcient are shown in the Ta ble below. Table:Linearthermalexpansioncoeﬃcient α=γ/3Kbracketleftbig10−6/◦Kbracketrightbig aluminium 23.8 cast iron 11.8 asphalt 200 limescale 20 ice (0◦C) 0.502 marble 11 iron 12.1 polystyrene 60÷80 pyrex glass 3.2 porcelain 3÷4 crystal glass 0.45 sandstone 5 granite 3÷8ﬁrebrick 5 ⋆In order to appreciate the order of magnitude of stresses cre ated by the temperature diﬀerence we calculate the stress in a thin bar along x1-axis ﬁxed on both ends and heated uniformly from the temperature T0toT,T−T0=100◦. We choose steel as the material for which α= 11.8∗10−61/◦K, λ= 11.78∗1010Pa, (5.201) µ= 8.00∗1010Pa. Then we have e11= 0⇒σ22+σ33=2(λ+µ)ekk−2γ(T−T0)≈0, i.e.ekk=γ λ+µ(T−T0)=α3λ+2µ λ+µ(T−T0). (5.202) Hence, for our data, σ11=λekk−γ(T−T0)=−αµ3λ+2µ λ+µ(T−T0)=245 MPa. (5.203) As the yield limit (the limit of elastic behavior) for constr uction steel is app. 250MPa we see from the above example that relatively small temperat ure diﬀerence may create already an irreparable damage in the material. It may, of cou rse, yield the buckling as well. One should keep in mind that during a construction ﬁre t he temperature diﬀerence is app. 900◦and the temperature of magma is from 700◦C to 1300◦C, i.e. the rocks shortly before melting reach the temperature app. 600◦C and this temperature diﬀerence yields enormous thermal stresses. ⋆ By means of the above constitutive relations we can write the ﬁeld equations. They have the form ρ∂2u ∂t2=(λ+µ)graddivu+µ∇2u−γgradT+ρb, ∂T ∂t=KT ρcv∇2T−γT0 ρcv∂ ∂tdivu. (5.204) 5.4 Poroelasticity 111 Clearly, due to the thermal expansion the equations are coup led. In many cases, one can neglect the coupling in the equation for the temperature . Then the distribution of temperature is determined as in the so-called rigid heat con ductors, i.e. in an undeformed body and then this temperature ﬁeld can be introduced to the d isplacement equations as an external force. 5.4 Poroelasticity Appearance of porous materials in nature is so common that we will not list even ex- amples. Three of them are shown in Fig. 5.11. From the point of view of mechanics the main issue in description of such materials is the coupli ng between the motion of the solid skeleton and ﬂuids in pores and channels. A proper cont inuum thermodynamics of such systems requires a multicomponent modeling which is ca lled the theory of immis- cible mixtures. We shall not enter this ﬁeld of research in th ese notes and refer to many monographs on the subject14. Fig. 5.11: Examples of porous materials: sand, bronchus, to ilet paper In this Subsection we show the modeling initiated by K. von Te rzaghi (1883-1963) who proposed an extension of the classical linear elasticity by the diﬀusion equation for pore pressure15. The couplings between the pore pressure and stresses in the solid skeleton are similar to these in thermoelasticity. Consequently, th e model belongs to the class of one-component models. Its motivation and modern developme nts within geomechanics can be found in the book of H. F. Wang [20]. The main ﬁeld is again the displacement u(x,t)but, in addition, we have to con- sider volume changes of pore spaces. Macroscopically these changes are described by a quantityεwhich is analogous to the volume changes of the skeleton give n bye=tre, e=1 2parenleftBig gradu+(gradu)TparenrightBig . M. Biot has proposed in 194116a concept of the variation 14e.g.R.M.BIoscIwscIescInsc; Diﬀusion models implied by the theory of mixtures, in: C. Tru esdell,Rational Thermodynamics, Second Edition , Springer, 237-263, N. Y., 1984, J.BIescIascIrsc;Dynamics of Fluids in Porous Media , Dover, N.Y., 1972., as well as the monographs of Wilmanski [21], [22]. 15K.IvscIoscInscTIescIrscIzscIascIgscIhscIisc; Erdbaumechanik auf bodenphysikalischer Grundlage , Deuticke, Wien, 1925, K.IvscIoscInscTIescIrscIzscIascIgscIhscIisc; Theoretical Soil Mechanics , J. Wiley and Sons, New York, 1943. 16M.A.BIiscIoscItsc; General theory of three-dimensional consolidation, J. Appl. Physic ,12, 155-164, 1941. 112 Elastic materials in water content, ζ, which is deﬁned in terms of the volume changes by the relatio n ζ=n0(e−ε), (5.205) wheren0denotes the initial value of porosity. This is the fraction o f voids to the total volume of the porous material, provided we choose a small dom ain for this deﬁnition. These small domains are called Representative Elementary V olumes (REV). We shall not discuss them in these notes. It can be shown that in a therm odynamical equilibrium ζ=0which means that contributions of this variable to the model are irreversible. They are related to the relative motion of solid and ﬂuid componen ts, i.e. to the diﬀusion. We can write ∂ζ ∂t=n0∂ ∂t(divu−ε)=−n0divvseep,vseep=vF−∂u ∂t,∂ε ∂t=divvF,(5.206) wherevFis the velocity of the ﬂuid and vseepis the so-called seepage velocity. For this reason, one cannot principally construct a variati onal formalism for such models in spite of many publications in which it is though att empted. Once we have the additional ﬁeld ζwe can introduce a conjugate dynamic variable, p, which is called pore pressure. The fundamental relations de ﬁning the couplings between the skeleton and the ﬂuid can be written in principal coordin ates in the following form e(1)=1 EparenleftBig σ(1)−νparenleftBig σ(2)+σ(3)parenrightBigparenrightBig +p 3H, e(2)=1 EparenleftBig σ(2)−νparenleftBig σ(1)+σ(3)parenrightBigparenrightBig +p 3H, (5.207) e(3)=1 EparenleftBig σ(3)−νparenleftBig σ(1)+σ(2)parenrightBigparenrightBig +p 3H, ζ=1 3HparenleftBig σ(1)+σ(2)+σ(3)parenrightBig +p R, where1/H,1/Rare the so-called Biot moduli: 1/His the so-called poroelastic expansion coeﬃcient, while 1/Ris the unconstrained speciﬁc storage coeﬃcient. Their deta iled pre- sentation can be found in the book of H. Wang [20]. It should be mentioned that M. Biot, the founder of a systematic approach to the subject of porome chanics of saturated ma- terials, was changing his notation many times and, for this r eason, some care is required in reading his papers. Certainly, the transformation of (5.207) to arbitrary coor dinates yields the extended stress-strain relations. They have the form σij=λekkδij+2µeij−αpδij, α=K H, (5.208) ζ=αekk+α KuBp, B=αR K, whereαis called the Biot-Willis coeﬃcient, Bis the Skempton coeﬃcient, Ku=K 1−αB, K=λ+2 3µ, (5.209) 5.4 Poroelasticity 113 andKuis called the undrained bulk modulus, i.e. the bulk modulus w hich corresponds to ζ=0. Clearly, in addition to the Lamé constants λ,µ, the model contains two additional material parameters, for instance, αandB, orαandKu. In addition to the above described coupling properties betw een stresses, σijandp, and strainseijandζ, we have to describe the ﬂow of the ﬂuid through the porous mat erials. This is the subject of Darcy’s law. It relates the gradient of the pore pressure to the relative velocity of the ﬂuid and the skeleton, i.e. the seep age velocity,vseep=q/n0. It can be written in the following form q=−k ηgradp, (5.210) wherekis the so-called intrinsic permeability and ηdenotes the ﬂuid viscosity. The ratio k/ηis called the mobility. Some typical values of these paramet ers for water in pores (i.e. forη=10−3Pa·s,ρ=1000 kg/m3,g=10m/s2) are shown in the Table. Table:Permeabilityforafewrocktypes [20] Rock typepermeability kbracketleftbigm2bracketrightbigpermeability [Darcy]hydraulic con- ductivity[m/s] sand or sandstone10−121 10−5 sandstone or limestone10−1510−310−8 granite or shale 10−1810−610−11 Once we have the constitutive relations (5.207) we can const ruct ﬁeld equations using the classical momentum conservation law and an additional e quation for the pore pres- sure which follows from the deﬁnition of the variation in wat er content (5.205) and the kinematic relation (5.206) combined with Darcy’s law (5.21 0) ρ∂2u ∂t2= (λ+µ)graddivu+µ∇2u−αgradp+ρb, (5.211) ∂p ∂t=c∇2p−KuB∂ ∂tdivu, where c=KuB αk η, (5.212) is called the diﬀusivity. Comparison of equations of linear thermoelasticity (5.204 ) and linear poroelasticity (5.211) shows a full mathematical analogy of these models. T herefore many solution available in thermoelasticity such as Green’s functions ca n be taken over to poroelasticity. Let us mention that mechanics of poroelastic materials base d on the theory of mixtures reveals many additional features related to porous structu res which cannot be described 114 Elastic materials by the very simplistic model presented above. One of those fe atures is the existence of an additional bulk wave, the so-called P2-wave, discovered theoretically already by J. Frenkel (1944) and conﬁrmed experimentally during the last two decades. This mode of propagation cannot be described by equations (5.211) bec ause, like thermoelasticity equations (5.204), the contribution of the pore pressure is described by the parabolic equation (5.211) 2, corresponding to the heat conduction equation (5.204) 2which is par- abolic as well. Chapter 6 Viscoelastic materials 6.1 Viscoelastic ﬂuids and solids To the class of viscoelastic materials belong practically a ll elastic materials if they are observed in suﬃciently long times. The description of such m aterials contains a charac- teristic relaxation time (or many, maybe even inﬁnitely man y of them) and this may have values from milliseconds to millions of years. In the latter case structural materials are obviously considered to be elastic. Over the lengths of time required to build mountain ranges, however, rocks appear to deform as very viscous ﬂuid s via a process known as slow creeping ﬂow. Many materials such as polymers possess r elaxation times of some hours to some years and then, of course, their viscous proper ties must be incorporated in modelling as well. Models of viscoelastic materials deve loped from the combination of elastic materials and viscous ﬂuids. Development of models of viscous ﬂuids was initiated in time s of Isaac Newton. In Book II of his Principia (1687) Newton formulated laws of res istance to the motion of ﬂuids. The next essential step, deﬁnitions of various rheological materials including viscoelas- tic materials was made ﬁrst at the end of XIXth century. Maxwe ll, Boltzmann, Kelvin, Voigt and many others introduced ﬁrst simple and then more so phisticated models of materials which were combinations of springs and dash-pots . We show some of them in the next Subsection. At the end of 50th of XXth century the dev elopment of continu- ous models began. This was initiated by works of A. E. Green, J . Ericksen and R. S. Rivlin1and continued by B. D. Coleman, N. Gurtin, W. Noll, C. Truesde ll. Viscoelastic 1R. S. RIiscIvscIlscIiscInsc, J. L. EIrscIiscIcscIkscIsscIescInsc; Stress-deformation relations for isotropic materials, J. Rat. Mech. Anal.,4, 323-425, 1955. A.E.GIrscIescIescInsc,R.S.RIiscIvscIlscIiscInsc; The mechanics of non-linear materials with memory, I, Arch. Rat. Mech. Anal.,1, 1-21, 1957. A. E. GIrscIescIescInsc, R. S. RIiscIvscIlscIiscInsc; The mechanics of non-linear materials with memory, III, Arch. Rat. Mech. Anal. ,1, 387-404, 1960. A.E.GIrscIescIescInsc,R.S.RIiscIvscIlscIiscInsc,A.J.M.SIpscIescInscIcscIescIrsc; The mechanics of non-linear materials with memory, II,Arch. Rat. Mech. Anal. ,1, 82-90, 1959. B.D.CIoscIlscIescImscIascInsc,W.NIoscIlscIlsc ; Foundations of linear viscoelasticity, Rev. Mod. Phys. ,33, 239-249, 1961. 115 116 Viscoelastic materials materials belong to the broader class of materials with memo ry developed in these works. Mathematical Principles of Natural Philosophy, London, 1687 Fig. 6.1: The beginning of Section V, Book II of Isaac Newtons ’s Mathemat- ical Principles of Natural Philosophy (London, 1687), conta ining the model of viscous ﬂuids 6.1 Viscoelastic ﬂuids and solids 117 In this Chapter, we present the main features of the viscoela sticity of solids. We follow two classical books on the subject R. M. Christensen [ 3] and A. C. Pipkin [15]. In Fig. 6.2. we show schematically a distinction between vis coelastic solids and viscoelastic (non-Newtonian) ﬂuids. In the simple shear ex periment we apply to a rec- tangle the shear deformation κ(t)=κ0H(t), whereH(t)is the Heaviside function. The nonzero component of stretching (compare (2.65)) is then ∂κ/∂t=κ0δ(t). This means, as shown in Part B) of the Fig. 6.2., that the shear stress in th e linear elastic material remains constant and in the viscous ﬂuid becomes inﬁnite at t he initial instant of time and then it is zero. In the case of viscoelastic solid (see Part C) of Fig. 6.2.) th e stress would relax after a long time to a ﬁnite value smaller than the initial value but diﬀerent from zero (stress relaxation). For a viscoelastic ﬂuid the stress would begin with the same value as in the case of viscoelastic solid but it would relax to zero as in the case of viscous ﬂuid. Fig. 6.2: Viscoelastic solids vs. ﬂuids — stress relaxation in viscoelastic solids and ﬂuids These observations will be justiﬁed on simple rheological m odels in the next Subsec- tion and they will be incorporated in the construction of a st andard model of viscoelastic solids. The most important eﬀect appearing in viscoelastic materia ls is creep. We explain this notion on a simple example of a slab subjected to a one-step st ress history σ(t)=σ0H(t). We show the behaviour schematically in Fig. 6.3. The respons e of an elastic solid would beκ(t)=κ0H(t), i.e. constant shear for positive time t. In a viscous ﬂuid, the shear 118 Viscoelastic materials would increase at a constant rate, κ(t) =σt/η, whereηdenotes the viscosity. In a viscoelastic material, the shear at ﬁrst jumps so that the in stantaneous response is elastic. The shear then continues to increase with a decreasing rate a nd it approaches a ﬁnite limitκ(∞). This is the phenomenon of creep. Otherwise, if the shear inc reases linearly in long times it is characteristic for a viscoelastic ﬂuid. Fig. 6.3: Behaviour of various materials under a one-step st ress history (schematic) — creep in viscoelastic solids and ﬂuids The above mentioned creep eﬀects yield the notion of memory i n the material behav- iour. We quote here R. M. Christensen [3]: ’It is instructive to consider a situation which represents a generalization of the response to a single sudd enly applied change of surface traction. Suppose a material having instantaneous elastic ity and creep characteristics de- scribed above is subjected to two nonsimultaneously applie d sudden changes in uniform stress, superimposed upon each other. After the ﬁrst applic ation of stress, but before the second, the material responds in some time dependent man ner which depends upon the magnitude of the ﬁrst stress state. But now consider the s ituation that exists at an arbitrary small interval of time after the sudden applicati on of the second stress state. The material not only experiences the instantaneous respon se to the second change in surface traction but also it experiences a continuing time d ependent response to the ﬁrst applied level of stress. An elastic material would respond o nly to the total stress level at every instant of time. Thus, this more general type of materi al possesses a characteristic which can be descriptively referred to as a memory eﬀect. Tha t is, the material response is not only determined by the current state of stress, but is a lso determined by all past states of stress’. Let us mention in passing that the theory of non-newtonian (v iscoelastic) ﬂuids has a very extensive literature which we do not quote in these not es. 6.2 Rheological models We present here a few very simple models reﬂecting memory eﬀe cts arising in viscoelas- tic solids. Equations describing their behaviour follow as special cases of constitutive 6.2 Rheological models 119 relations for viscoelastic materials. R. M. Christensen in his book [3] presents ﬁrst these general constitutive relations and then simple rheologica l elements appear as illustration. It seems to be more appealing to proceed the other way around. Fig. 6.4.: Some rheological elements: spring and dashpot (u pper panel, on top), Maxwell element (upper panel, on bottom), standard el ement (lower panel). The one-dimensional models are constructed by the combinat ion of springs and dash- pot elements. The spring is an ideal elastic element obeying the linear force-extension relation. The force and extension are analogs for stress and strain. This element fulﬁls the relation σ=G0κ. (6.1) The dashpot is an ideal viscous element that extends at the ra te proportional to the applied force ˙κ=dκ dt=σ η, (6.2) whereηis the viscosity. Various combinations of these two elements yield simple rhe ological models of vis- coelastic materials. The simplest one is Maxwell’s model pr esented in Fig. 6.4. (upper panel, on bottom). It is described by the conditions σ=σel=σvis, κ=κel+κvis, (6.3) σel=G0κel, σvis=η˙κvis, 120 Viscoelastic materials with an obvious meaning of the notation. The elasticity cons tantG0,[Pa], is called in the theory of viscoelasticity the relaxation modulus, and η,[Pa·s]is the viscosity. Relations (6.3) yield the following constitutive equation for Maxwel l’s model ˙σ+1 τσ=G0˙κ, τ=η G0. (6.4) Obviously, it is an evolution equation for stresses σandτis the relaxation time. The formal solution of this equation has the following form σ=σ0e−t/τ−G0tintegraldisplay 0dκ ds(t−s)e−s/τds, σ0=σ(t=0). (6.5) For the constant stress σ=constthe equation (6.4) yields the constant rate of strain This corresponds to the ﬂuid-like behaviour in Fig. 6.3. The same conclusion follows in the case of a constant strain ˙κ=0which yields the decay of stresses in time to zero. In this case the solution of the equation (6.4) has the form σ=σ0e−t/τ. Another model which can be simply constructed from the sprin g and dashpot is the parallel connection. This is called the Kelvin model and it i s described by the conditions σ=σel+σvis, κ=κel=κvis, (6.6) σel=G0κel, σvis=η˙κvis. This corresponds to the evolution equation for the strain κ ˙κ+1 τκ=σ η. (6.7) Hence, for the constant strain we obtain the elastic behavio ur. Otherwise, the formal solution of the equation (6.7) has the form κ=κ0e−t/τ+1 ηtintegraldisplay 0σ(t−s)e−s/τds. (6.8) For the constant stresses σ0, this yields κ(t=0)=κ0andκ(t→∞)=σ0τ/η=σ0/G0. This is the behaviour of the simplest viscoelastic solid (co mpare Fig. 6.3). Another combination of springs and dashpots is the parallel combination of a spring of elasticityG1and a Maxwell model with parameters G2,η. This is the so-called standard rheological model. It is described by the relations σ=σ1+σ2, κ=κ1=κM, (6.9) σ1=G1κ1,˙σ2+η G2σ2=G2˙κM. These relations lead to the rate equation ˙σ+1 τ2σ=ητ1+τ2 τ1τ2parenleftbigg ˙κ+1 τ1+τ2κparenrightbigg , τ1=η G1, τ2=η G2. (6.10) 6.2 Rheological models 121 This equation contains two relaxation times τ1,τ2. In practical applications the above presented models are mu ch too simple. In order to incorporate more relaxation times parallel arrangement s ofNMaxwell models are constructed. This is called a generalized Maxwell model. Si milarly, by a series order ofNKelvin models one can construct a generalized Kelvin model. The analysis of properties of such models can be performed by the Laplace tra nsform. We shall not present any further details concerning this subject and ref er rather to the book of D. R. Bland2. ⋆It is instructive to investigate the form of the second law of thermodynamics for rheological materials. In contrast to the full viscoelasti c models consequences of the entropy inequality (4.20) can be easily found for those rheo logical cases. We consider Maxwell’s model as an example. In the one dimensional linear case under considerations the balance of internal energy (4.11) and the entropy inequa lity reduce to the following form ρ∂ε ∂t+∂q ∂x−σ∂κ ∂t=0, ρ∂η ∂t+∂ ∂xparenleftBigq TparenrightBig ≥0, (6.11) whereqis thex-component of the heat ﬂux and ηis the entropy density (not viscosity in this exercise!). Combination of these two relations yiel ds ρ∂ψ ∂t+ρη∂T ∂t+1 Tq∂T ∂x−σ∂κ ∂t≤0, ψ=ε−Tη. (6.12) We have to write additional constitutive relations for the H elmholtz free energy ψand the heat ﬂux q. As we are interested in isothermal processes, the contribu tions∂T/∂t and∂T/∂x vanish from the problem and we assume only that ψis of the following form ψ=ψ(κ,σ). (6.13) It is important to notice that both κandσare governed by diﬀerential equations and, for this reason, should be treated as two independent ﬁelds. Now the inequality (6.12) can be written in the form ρparenleftbigg∂ψ ∂κ−σparenrightbigg∂κ ∂t+ρ∂ψ ∂σparenleftbigg G0∂κ ∂t−σ τparenrightbigg ≤0. (6.14) where we have made use of (6.4). This inequality must hold for arbitrary derivatives ∂κ/∂t. Hence ∂ψ ∂κ+G0∂ψ ∂σ=σ ρ,D=ρ τ∂ψ ∂σσ≥0. (6.15) The ﬁrst relation can be considered to be the diﬀerential equ ation forψ. It is easy to ﬁnd its solution using the method of characteristics. It has the form ψ=ψel(σ−G0κ)+σ2 2ρG0, (6.16) 2D.F.BIlscIascInscIdsc; The Theory of Linear Viscoelasticity , Pergamon Press, Oxford, 1960. 122 Viscoelastic materials where independent variables are now ξ=σ−G0κandσ. The ﬁrst part remains constant along the straight lines on (σ,κ)-planes:ξ=σ−G0κ=const. This is an elastic part of the solution. The second part of (6.15), the dissipation ine quality, yields τG0≥0, i.e. the viscosity must be positive. The above calculations are typical for systems with an evolu tion equation describing additional ﬁelds in the model. In Maxwell’s model the stress σis such a ﬁeld. Let us mention in passing that the exploitation of the entrop y inequality (6.11) 2was so easy because the constitutive relation for stresses has i n the Maxwell model the form of diﬀerential equation. As we see further, general three-d imensional models may not have this form. Usually they appear as functional dependenc ies. Then the evaluation of thermodynamical restrictions requires special techniq ues, in particular the so-called linear extensions of functionals (Fréchet derivatives). T his problem has been extensively investigated in 60ties and it is sometimes called Coleman’s method from the name of its founder. We shall not enter this subject in these notes. ♣ ⋆Laplace transforms We recall here the basic deﬁnitions of the Laplace transform needed in the theory of viscoelasticity. It is well-known that many simple funct ions do not possess a Fourier transform as the deﬁning integrals fail to converge for inﬁn ite limits. Therefore it is convenient to consider not the transform of the function f(t)but rather of the function f(t)exp(−rt). Then the transform ¯f(z)=∞integraldisplay −∞f(t)exp(−zt)dt, z=r+iω, (6.17) is called two-sided Laplace transform of f(t). Obviously the inverse follows by multipli- cation of (6.17) by exp(rt)and integration f(t)=1 2π∞integraldisplay −∞¯f(r+iω)exp[(r+iω)t]dω. (6.18) If we are only interested in values of f(t)fortpositive, it is suﬃcient to transform the functionf(t)H(t)rather then f(t). In this manner we avoid problems with divergence of the above integral for the lower limit of exp(−rt). Then the Laplace transform is the cut oﬀ of the two-sided Laplace transform ¯f(z)=∞integraldisplay 0f(t)exp(−zt)dt. (6.19) Its inverse has the form f(t)H(t)=1 2π∞integraldisplay −∞¯f(r+iω)exp[(r+iω)t]dω=1 2πir+i∞integraldisplay r−i∞¯f(z)exp(zt)dz.(6.20) 6.3 Three-dimensional viscoelastic model 123 The ﬁrst integral clears the meaning of the second integral. If the integral (6.19) converges for some damping factor r=Rez, it also converges for all larger damping factors. Con- sequently, the integral converges in some half-plane Rez > r0in the complex z−plane. The liner=r0is called the abscissa of convergence. All values of rwhich lie to the right ofr0are big enough for convergence. This is the very important pr operty because the function¯f(z)has no singularity in the half-plane of convergence. Howeve r, it may have it on the abscissa of convergence. As a consequence the integra ting¯f(z)around any closed contour yields zero. It means that ¯f(z)is an analytic function of zin the half-plane of convergence. This property is the basis for the calculation of inverse integrals (6.20). Frequently used in application to viscoelasticity is the La place transform of the con- volution integral. It is as follows3 ∞integraldisplay 0 tintegraldisplay 0f(s)g(t−s)ds bracehtipupleftbracehtipdownrightbracehtipdownleftbracehtipupright convolution integral exp(−zt)dt=¯f(z)¯g(z). (6.21) Example : we consider the Maxwell model described by the evolution eq uation (6.4). The Laplace transform of this equation yields −σ0+z¯σ+1 τ¯σ=G0(−κ0+z¯κ). (6.22) At the instant of the time t= 0we have the elastic reaction σ0=G0κ0.This is the deformation of the spring in the Maxwell model before the das hpot had time to start moving. This yields the solution of the problem in the transf ormationz−plane ¯σ=G0z z+1/τ¯κ. (6.23) Hence, the abscissa is crossing the point r0=−1/τ,ω=0. The half-plane to the right of the vertical line r=r0contains no singular points of ¯σ. The inverse yields the solution (6.5)♣ 6.3 Three-dimensional viscoelastic model Motivated by the above rheological considerations we const ruct now a constitutive model for the three-dimensional viscoelastic continuum. We expe ct the stress tensor to depend on the history of strain. We can formally postulate the follo wing relation σij(t)=∞ Ψij s=0(ekl(t−s),ekl(t)), (6.24) 3R.BIrscIascIcscIescIwscIescIlscIlsc; The Fourier Transform and Its Applications, 2nd Edition, McGraw—Hill, 1986 . 124 Viscoelastic materials where∞ Ψij s=0denotes a linear tensor valued functional mapping the strai n historyeij(t), −∞ ≤t≤ ∞, into the stress history σij(t). In addition, the functional possesses a parametric dependence upon the current value of strain eij(t)which describes the instantaneous elastic response mentioned in the above pres ented properties of viscoelastic materials. We do not include a dependence on the spatial vari ablexas the material is assumed to be homogeneous. The above functional has an int egral representation for continuous histories of strain. It follows from the Ries z representation Theorem4. Namely, it has the form of the Stieltjes integral σij=∞integraldisplay 0ekl(t−s)dGijkl(s), (6.25) where each component of the fourth order tensor Gijklis of bounded variation. The components of this tensor are called relaxation functions. The above convolution of the constitutive law implies that it is invariant with respect t o arbitrary shifts in the time scale. This invariance is related to the energy conservatio n but we shall not discuss it any further in these notes. Integral constitutive relations of this type are called the Boltzmann integrals (the Boltzmann superposition principle). The tensor Gijklpossesses obvious symmetries following from the symmetry o f the stress and strain tensors Gijkl(t)=Gjikl(t)=Gijlk(t). (6.26) Assuming additionally the continuity of the ﬁrst derivativ e of the tensor Gijkland eij(t)=0fort<0we can write the relation (6.25) in the form σij=Gijkl(0)ekl(t)+tintegraldisplay 0ekl(t−s)dGijkl(s) dsds. (6.27) This form exposes the instantaneous elastic reaction of the material. Bearing the conti- nuity ofeij(t)in mind, we can integrate (6.27) by parts. It follows σij=tintegraldisplay 0Gijkl(t−s)dekl(s) dsds. (6.28) Under the weaker assumption of a step discontinuity at t= 0one can generalize the above relation5and obtain the following relation σij(t)=Gijkl(t)ekl(0)+tintegraldisplay 0Gijkl(t−s)dekl(s) dsds. (6.29) 4e.g.N.DIuscInscIfscIoscIrscIdsc,J.T.SIcscIhscIwscIascIrscItscIzsc; Linear Operators. Part I: General Theory , Interscience Publ., New York, 1958 5M.E.GIuscIrscItscIiscInsc,E.SItscIescIrscInscIbscIescIrscIgsc; On the linear theory of viscoelasticity, Arch. Rat. Mech. Anal. ,11, 291-356, 1962. 6.3 Three-dimensional viscoelastic model 125 In spite of the discontinuity at t=0the lower limit can be shifted from 0to−∞due to the above mentioned shift invariance provided eij(t→−∞)→0. Integration by parts yields then σij(t)=tintegraldisplay −∞Gijkl(t−s)dekl(s) dsds. (6.30) An alternative to the above constitutive relation is the inv erse eij(t)=tintegraldisplay −∞Jijkl(t−s)dσkl(s) dsds, (6.31) where Jijkl(t)=Jjikl(t)=Jijlk(t), Jijkl(t)=0 for−∞<t<0. (6.32) These functions are assumed to possess continuous ﬁrst deri vatives and they are called creep functions. We limit all further considerations to isotropic materials . The most general isotropic representation of the fourth order tensor contains two inde pendent parameters (compare (5.21)). It is convenient to write it in the form Gijkl(t)=1 3[G2(t)−G1(t)]δijδkl+1 2[G1(t)](δikδjl+δilδjk), (6.33) whereG1(t)andG2(t)are independent relaxation functions. If we separate spher ical and deviatoric parts of stress and strain tensors σij=1 3σkkδij+σD ij, σD kk=0, (6.34) eij=1 3ekkδij+eD ij, eD kk=0, then the relation (6.30) splits in the following way σD ij(t) =tintegraldisplay −∞G1(t−s)deD ij(s) dsds, (6.35) σkk=tintegraldisplay −∞G2(t−s)dekk(s) dsds. Similarly, the inverse relations obtain the form eD ij(t) =tintegraldisplay −∞J1(t−s)dσD ij(s) dsds, (6.36) ekk=tintegraldisplay −∞J2(t−s)dσkk(s) dsds. 126 Viscoelastic materials whereJ1(t),J2(t)are two independent isotropic creep functions. Obviously, the func- tionsG1,J1are appropriate for shear processes and G2,J2for dilatation processes. The relaxation and creep functions are, of course, related t o each other. The easiest way to ﬁnd this relation is to perform the Laplace transforma tion on relations (6.35) and (6.36)6. We obtain ¯σD ij=z¯G1¯eD ij,¯σkk=z¯G2¯ekk, (6.37) ¯eD ij=z¯J1¯σD ij,¯ekk=z¯J2¯σkk. These relations imply Jα=parenleftbig z2Gαparenrightbig−1, α=1,2. (6.38) The isotropic relations for elastic materials (5.21), (5.2 6) can be written in the form σkk= 3Kekk, σD ij=2µeD ij, (6.39) ekk=1 3Kσkk, eD ij=1 2µσD ij. They would suggest that Jα(t) = [Gα(t)]−1. Relations (6.38) show that this is not correct. However, it can be shown using properties of the Lap lace transform that limit values indeed satisfy such relations lim t→0Jα(t)=lim t→0[Gα(t)]−1,lim t→∞Jα(t)= lim t→∞[Gα(t)]−1. (6.40) Incidentally, to be consistent with the linear elasticity o f isotropic materials relations (6.39) suggest the following notation for relaxation and cr eep functions µ(t)=G1(t)/2, K(t)=G2(t)/3. (6.41) The above presented results suggest a useful short-hand not ation for Stieltjes convo- lution integrals which has been introduced by Gurtin and Ste rnberg in the earlier quoted paper. Namely, we write instead of (6.25) the following rela tion σij=ekl∗dGijkl, (6.42) i.e. we write for two arbitrary functions f,g f∗dg=tintegraldisplay −∞f(t−s)dg(s), g(t→−∞)=0, (6.43) 6It should be mentioned that, instead of the classical Laplac e transform (6.19) it may be more convenient to use a modiﬁcation which is called Laplace-Car son transform. It is deﬁned by the relation ¯f(z)=z∞/integraldisplay 0f(t)exp(−zt)dt. It is applied in the presentation of viscoelasticity by Lema itre and Chaboche [9]. We follow here rather the older approach of Christensen [3] and Pipkin [15]. 6.3 Three-dimensional viscoelastic model 127 wheref(t)is continuous for 0 ≤t≤∞. Iff(t) = 0 fort <0then one can show the commutativity relation f∗dg=g∗df. (6.44) Consequently, σij=Gijkl∗dekl, which is the counterpart of (6.30). The above notation leads to the following useful identities f∗d(g∗dh) = (f∗dg)∗dh=f∗dg∗dh, (6.45) f∗d(g+h) =f∗dg+f∗dh. We shall not enhance the subject of a general theory of materi als with memory. This can be found in classical monographs on the subject7. We proceed to investigate an example which helps to clear the distinction between the viscoelastic ﬂuid and the viscoelastic solid. We consid er the case of the simple shear deﬁned by the relation (2.95). The amount of shear κ=κ0H(t)is assumed to be the step function in time and it yields the stretching (2.65) in t he linear theory to be identical with the time derivative of strain which is given by the rate o f shearing˙κ=κ0δ(t) Dijei⊗ej=deij dtei⊗ej=˙κ 2(e1⊗e2+e2⊗e1). (6.46) Hence, σD 12=κ0 2tintegraldisplay 0G1(t−s)δ(s)ds=κ0 2G1(t), G1(t)=0fort<0. (6.47) It follows from the deﬁnition of a linear isotropic viscoela stic solid that the following condition must hold lim t→∞G1(t)→nonzero constant ⇒solids. (6.48) On the other hand, for viscoelastic (non-Newtonian) ﬂuid lim t→∞G1(t)=0⇒ﬂuids. (6.49) The latter condition is necessary but not suﬃcient. In addit ion, the relaxation function G1must fulﬁl a condition for the steady state ﬂow. Then the visc oelastic ﬂuid at large values of time, where the steady state will be achieved, must behave like a viscous ﬂuid of the viscosity η. Hence, the relaxation time must have the property η=1 2∞integraldisplay 0G1(s)ds(ﬂuids). (6.50) 7C.TIrscIuscIescIsscIdscIescIlscIlsc,W.NIoscIlscIlsc; The Non-Linear Field Theories of Mechanics, Encyclopedia of Physics, vol. III/3, S. Flügge (ed.), Springer, Berlin, 1965, C. TIrscIuscIescIsscIdscIescIlscIlsc; A First Course in Rational Continuum Mechanics, The Johns Hopkins University Press, Baltimore, 1972 (Chapter XIII). 128 Viscoelastic materials 6.4 Diﬀerential constitutive relations We have seen on examples the simple rheological models that c onstitutive relations of these models may have the form of evolution equations (rate- type constitutive relations). This can be taken over to the description of a three-dimensio nal continuum. Let us consider the following diﬀerential operator p0σD ij+p1dσD ij dt+p2d2σD ij dt2+...=q0eD ij+q1deD ij dt+q2d2eD ij dt2+..., (6.51) or, in the compact form, P(D)σD ij=Q(D)eD ij, P(D)=Nsummationdisplay k=0pkDk, Q(D)=Nsummationdisplay k=0qkDk, Dk=dk dtk.(6.52) This type of operators appear, for instance, for generalize d Maxwell and Kelvin models mentioned before. In order to see the signiﬁcance of such mod els for viscoelasticity we take the Laplace transform of (6.52) ¯P(z)¯σD ij−1 zNsummationdisplay k=1pkNsummationdisplay r=1zrbracketleftBigg dk−rσD ij dtk−r(t=0)bracketrightBigg = (6.53) =¯Q(z)¯eD ij−1 zNsummationdisplay k=1qkNsummationdisplay r=1zrbracketleftBigg dk−reD ij dtk−r(t=0)bracketrightBigg , where ¯P(z)=Nsummationdisplay k=0pkzk,¯Q(z)=Nsummationdisplay k=0qkzk. (6.54) These relations follow easily by integration by parts. If we compare these relations with (6.37) then they specify the relaxation function by the form ula z¯G1=¯Q(z)/¯P(z), (6.55) provided the initial conditions are constraint by the relat ions Nsummationdisplay r=kprbracketleftBigg dk−rσD ij dtk−r(t=0)bracketrightBigg =Nsummationdisplay r=kqrbracketleftBigg dk−reD ij dtk−r(t=0)bracketrightBigg , k=1,2,...,N. (6.56) Consequently the relaxation function G1is speciﬁed in terms of 2(N+1)parameters p0,...,pN,q0,...,qNwhich are related to a sequence of relaxation times. Similar relations can be introduced for the dilatational part of stress and str ain L(D)σkk(t)=M(D)ekk(t), (6.57) and these are again speciﬁed by a ﬁnite sequence of parameter s. 6.5 Steady state processes and elastic-viscoelastic corre spondence principle 129 The question if such models can be indeed physically plausib le is not simple. Some models of this art, introduced for non-Newtonian ﬂuids (the so-called Rivlin-Ericksen ﬂu- ids) show that there appear problems of stability8and convergence9for dynamic processes and these models seem to work well for steady state ﬂows. Ther e are claims that non- Newtonian ﬂuids require always the spectrum of inﬁnitely ma ny relaxation times and, consequently, such polynomial models as (6.52) are physica lly useless. For viscoelastic solids a simple diﬀerential model is often based on the simplest dif- ferential equation describing the evolution of stresses. T his is analogous to the Maxwell model (6.4) constructed within the classical rheology. It i s based on the equation for stresses τdσD ij dt+σD ij=2ηdeD ij dt, (6.58) whereτis the relaxation time and ηis the viscosity. It is the so-called standard linear viscoelastic solid [21] 6.5 Steadystateprocessesandelastic-viscoelasticcor- respondence principle Now we investigate a class of problems which appear in spectr al analysis of waves in which we seek solutions in the form of monochromatic waves. I t is then important to know the behaviour of the constitutive relation (6.35) if th e time dependence of the strain is harmonic. We denote representatives of deviatoric and sp herical strains and stresses by˜eand˜σ, respectively, and assume ˜e=˜e0eiωt, (6.59) whereωis the frequency and ˜e0an amplitude. We write the typical contribution to (6.35) in the following form ˜σ=tintegraldisplay −∞Gα(t−s)d˜e dsds, (6.60) whereα=1or2in dependence of the choice of ˜σ. It is convenient to split the relaxation modulus into two parts: G0 αandG1α, where the ﬁrst part is equal to the limit of the relaxation modulus for t→∞and, consequently, G1 α(t→∞)=0. Both parts are zero for the time smaller than 0. Hence ˜σ=G0 α˜e0eiωt+iω˜e0tintegraldisplay −∞G1α (t−s)eiωsds. (6.61) 8D. JIoscIsscIescIpscIhsc; Instability of the rest state of ﬂuids of arbitrary grade gre ater than one, Arch. Rat. Mech. Anal., 75,251-256, 1981 . 9R.S.RIiscIvscIlscIiscInsc,K.WIiscIlscImscIascInscIsscIkscIisc; The passage from memory functionals to Rivlin-Ericksen con stitutive equations, ZAMP, 38, 624-629, 1987 as well asK.WIiscIlscImscIascInscIsscIkscIisc [21] 130 Viscoelastic materials Changing the variables η=t−swe obtain ˜σ= G0 α+ω∞integraldisplay 0G1α(η)sinωηdη+iω∞integraldisplay 0G1α(η)cosωηdη ˜e0eiωt. (6.62) Consequently, the stress ˜σis given by the complex modulus G∗ α ˜σ=G∗ α˜e0eiωt, ReG∗ α=G0α+ω∞integraldisplay 0G1α(η)sinωηdη, (6.63) ImG∗ α=ω∞integraldisplay 0G1α(η)cosωηdη. The real part is called the storage modulus and the imaginary part the loss modulus. The integration by parts yields the frequency limit behavio ur of the above moduli. We have ReG∗α(ω) =G0α+G1α(0)+∞integraldisplay 0dG1 α(η) dηcosωηdη, (6.64) ImG∗ α(ω) =−∞integraldisplay 0dG1 α(η) dηsinωηdη. Hence ReG∗ α(ω=0)=G0α=Gα(t)|t→∞,ImG∗α(ω=0)=0. (6.65) For the other limit, changing the variables ωη=τwe easily obtain ReG∗α(ω→∞)=G0α+G1α(t=0)=Gα(t)|t→0,ImG∗α(ω→∞)=0.(6.66) These relations show that for very high frequency the viscoe lastic solid behaves as an elastic solid. The same concerns very low frequencies and th is diﬀers solids from viscous ﬂuids. The above considerations determine, obviously, the Fourie r transforms of the consti- tutive relations. With the deﬁnitions for an arbitrary func tionf ¯f(ω)=∞integraldisplay −∞f(t)e−iωtdt, f(t)=1 2π∞integraldisplay −∞¯f(ω)eiωtdω, (6.67) we have ¯σD ij(ω) =G∗1(iω)¯eD ij(ω), (6.68) ¯σkk=G∗2(iω)¯ekk(ω). 6.5 Steady state processes and elastic-viscoelastic corre spondence principle 131 The similarity of these relations to the elastic relations ( 6.39) is called the elastic- viscoelastic correspondence principle . It has been ﬁrst ob served by W. T. Read in 195010. With respect to the important practical aspects of this pri nciple we present it as well for the Laplace transform. Let us consider the full set of governing equations describi ng the boundary value problem of a linear isotropic viscoelastic material. We hav e eij=1 2parenleftbigg∂ui ∂xj+∂uj ∂xiparenrightbigg , (6.69) ∂σij ∂xj+ρbi=0,forx∈Bt (6.70) σD ij(t)=2tintegraldisplay −∞µ(t−s)∂eD ij ∂sds, σkk=3tintegraldisplay −∞K(t−s)∂ekk ∂sds, (6.71) σij=1 3σkkδij+σD ij, eij=1 3ekkδij+eD ij, (6.72) σij(t)nj=tn iforx∈∂Bσ t, ui(t)=uniforx∈∂Bu t, (6.73) ui(t)=eij(t)=σij(t)=0 for−∞<t<0, (6.74) where we have used the notation (6.41) for the relaxation mod uli. Laplace transform of these equations has the form ¯eij=1 2parenleftbigg∂¯ui ∂xj+∂¯uj ∂xiparenrightbigg , (6.75) ∂¯σij ∂xj+ρ¯bi=0,forx∈Bt (6.76) ¯σD ij(t)=2z¯µ(z)¯eD ij¯σkk=3z¯K(z)¯ekk, (6.77) ¯σij=1 3¯σkkδij+¯σD ij,¯eij=1 3¯ekkδij+¯eD ij, (6.78) ¯σij(z)nj=¯tniforx∈∂Bσ t,¯ui(z)=¯uniforx∈∂Bu t, (6.79) wherezis the transformation variable and bars denote Laplace tran sforms. Obviously the set (6.75)-(6.79) has a form identical with eq uations of linear elasticity except of complex moduli z¯K(z),z¯µ(z)which replace real moduli K,µof the elasticity theory. This correspondence reveals the possibility of con verting numerous static solu- tions of elasticity into quasi-static solutions of viscoel asticity. The main problem is now the inversion of the Laplace transform. One more general remark is appropriate for quasi-static pro blems of viscoelastic mate- rials. Before we formulate it, let us collect the material fu nctions corresponding to the ma- terial constants of elasticity. We have already the relatio ns (6.41), i.e. µ(t)=G1(t)/2, 10W. T. RIescIascIdsc; Stress analysis for compressible viscoelastic materials, J, Appl. Phys. , 21, 671-674, 1950. 132 Viscoelastic materials K(t)=G2(t)/3and they yield in the transformed form (compare (5.29)) ¯λ(z) =¯K(z)−2 3¯µ(z)=1 3parenleftbig¯G2(z)−¯G1(z)parenrightbig , ¯E(z) = =3¯µ(z)¯K(z) ¯K(z)+1 3¯µ(z)=3¯G1(z)¯G2(z) 2¯G2(z)+¯G1(z), (6.80) ¯ν(z) =¯λ(z) 2parenleftbig¯λ(z)+¯µ(z)parenrightbig=¯G2(z)−¯G1(z) 2¯G2(z)+¯G1(z). The question arises if we can apply the method of separation o f variables in a quasi- static problems of linear viscoelasticity. It means that, f or instance, the displacement should have the form ui(x,t)=ˇui(x)u(t), (6.81) whereu(t)is a common function for all components of the displacement. Hence, if we neglect the acceleration (quasi-static problem!) and body forces, the ﬁeld equations for displacements have the form ∂2ˇui ∂xk∂xktintegraldisplay −∞µ(t−s)du(s) dsds+∂2ˇuk ∂xi∂xktintegraldisplay −∞[λ(t−s)+µ(t−s)]du(s) dsds=0.(6.82) Obviously, the time contributions must be eliminated from t his equation and this yields λ(t)+µ(t)=βµ(t), β=const. (6.83) This yields immediately that Poisson’s ratio νmust be independent of time. Bearing the last relation (6.80) in mind, we obtain the restriction G2(t) G1(t)=1+ν 1−2ν=const. (6.84) In the similar manner we can prove that the ratio of creep func tions is a constant J2(t) J1(t)=1−2ν 1+ν=const. (6.85) These two conditions are necessary for the applicability of the method of separation of variables. ⋆We demonstrate the application of the correspondence princ iple on a simple ex- ample. We consider the axial symmetric problem of the cylind er under the given radial loading on both lateral surfaces. The outer surface is press urized by an elastic case11. The Laplace transform of the radial displacement ¯ur(r,z)must fulﬁl the equation (see: (5.44)) ∂2¯ur ∂r2+1 r∂¯ur ∂r−¯u r2=0, (6.86) 11R. M. CIhscIrscIiscIsscItscIescInscIsscIescInsc, R. N. SIcscIhscIrscIescIiscInscIescIrsc; Response to pressurization of a viscoelastic cylinder with an eroding internal boundary, AIAA J. ,3, 1451, 1965. 6.5 Steady state processes and elastic-viscoelastic corre spondence principle 133 with the solution ¯ur=¯C(z)r+¯D(z) r. (6.87) In order to apply the boundary conditions, we have to write st ress-strain relations in the transformed form. We have by the correspondence princip le ¯σrr= 2z¯µparenleftbigg∂¯ur ∂r+z¯ν 1−2z¯ν¯eparenrightbigg ,¯e=∂¯ur ∂r+¯ur r, ¯σθθ= 2z¯µparenleftbigg¯ur r+z¯ν 1−2z¯ν¯eparenrightbigg , (6.88) ¯σzz=2z2¯ν¯µ 1−2z¯ν. The boundary conditions for the pressurized cylinder have t he form σrr(r=a,t) =−p(t), σrr(r=b,t)=−q(t), (6.89) u(r=b,t) =q(t)bracketleftBigg b2parenleftbig 1−ν2 cparenrightbig EchbracketrightBigg , whereEc,νcare elastic properties of the case, his its thickness, bthe outer radius and athe inner radius of the viscoelastic cylinder. Easy calculations yield the following form of the transform ed stresses ¯σrr=C1−b2 r2C2,¯σθθ=C1+b2 r2C2,¯σzz=2z¯νC1, (6.90) where C1=−¯p(S−z¯µ) (S−z¯µ)+(b2/a2)[S(1−2z¯ν)+z¯µ], C2=¯p[S(1−2z¯ν)+z¯µ] (S−z¯µ)+(b2/a2)[S(1−2z¯ν)+z¯µ], (6.91) S=Ech 2b(1−ν2c). Inverse transformations performed in the quoted work of Chr istensen and Schreiner were made under the assumption that the modulus ¯µis a polynomial in zand the Poisson numberνis constant (see above). The polynomial form of ¯µfollows from the following considerations suggested by rheological models. It is assu med that the modulus µ(t)has the form µ(t)=G0+Nsummationdisplay n=1Gne−t/τn, (6.92) whereG0,Gn,τn,n= 1,...,N are constants. Obviously, τnhave the interpretation of relaxation times. The values of these parameters are obtain ed by ﬁtting to experimental 134 Viscoelastic materials data (e.g. for harmonic torsional loading experiments of cy lindrical samples). Laplace transform of (6.92) yields z¯µ(z)=A(z) Nproductdisplay n=1(z+1/τn), (6.93) whereA(z)is anNthgrade polynomial in zdetermined by coeﬃcients Gn. Now the inverse transformation can be made by the technique o f integration of the function¯f(z) =P(z)/Q(z)whose denominator yields simple pole singularities (i.e. zeros ofQ(z)) in the complex domain. We shall not quote rather complex ﬁna l results.♣ Chapter 7 Plasticity 7.1 Introduction Various plasticity models of mechanics are developed to des cribe a class of permanent deformations. These deformations are generated during loa ding processes and remain after the removal of the load. In this Chapter we present a few aspects of the classical linear plasticity. This model is based on the assumption on t he additive separation of elastic and plastic deformation increments. In nonlinear m odels it is the deformation gradientFin which these permanent deformations are separated F=FeFp, (7.1) where the plastic deformation is described by Fpand the elastic part is Fe. Only the product of these two objects is indeed the gradient of the fun ction of motion f. Neither FenorFpcan be written in such a form — they are not integrable. In spit e of this problem, material vectors transformed by Fpform a vector space for each material point X∈B0and these spaces are sometimes called intermediate conﬁgur ations. We shall not elaborate these issues of nonlinear models1. However, it should be mentioned that the assumption (7.1) indicates the additive separation of incr ements of deformation in the linear model. Namely, the time derivative of the deformatio n gradient has, obviously, the form ˙F=˙FeFp+Fe˙Fp, (7.2) which yields for small strains ˙e=˙ ee+˙ ep, (7.3) where˙eeis the elastic strain rate and ˙ epis the plastic strain rate. In some older models it is even assumed that this additive decomposition concerns s trains themselves: e=ee+ep which is obviously much stronger than (7.3) and yields certa in general doubts. 1e.g. see:AIlscIbscIrscIescIcscIhscItsc BIescIrscItscIrscIascImsc; Elasticity and Plasticity of Large Deformations , Springer Berlin, 2008. 135 136 Plasticity The aim of the elastoplastic models in the displacement form ulation is to ﬁnd the displacement vector uwhose gradient deﬁnes the strain e— as in the case of linear elasticity, and the plastic strain epwhich becomes an additional ﬁeld. Fig. 7.1: States of material in the stress space Σ.A— the elastic state, B — the plastic state. The most fundamental characteristic feature of classical p lasticity is the distinction of an elastic domain in the space of stresses Σ={T},T=σijei⊗ej. All paths of stresses which lie in the elastic domain produce solely elastic defor mations, i.e. after inverting the process of loading the material returns to its original s tate. This is schematically shown in Fig. 7.1. The elastic domain lies within the bounding yield surface al so called the yield limit or the yield locus. Stress states which lie beyond this limit ar e attainable only by moving the whole yield surface. Such processes are called hardenin g. In Fig. 7.1. we demonstrate the so-called isotropic hardening. We return to this notion in the sequel. Increments of plastic strains are described by stresses whose direction p oints in the outward direction of the yield surface. This is related to the so-called Drucke r2stability postulate which we present further. The above described way of construction of plasticity is som etimes called stress space formulation and it was motivated by properties of metals. Th ere is an alternative which has grown up from soil mechanics3. Such materials as rocks, soils and concrete reveal softening behaviour which violates Drucker’s postulate. I n order to avoid this problem, the so-called strain space formulation4was developed in which, instead of Drucker’s postulate one applies the Ilyushyn5postulate. The detailed discussion of these stability problems can be found, for instance, in the book of Wu [24]. 2D.C.DIrscIuscIcscIkscIescIrsc; A more fundamental approach to plastic stress-strain relat ions, in:Proc. 1st Nat. Congress Appl. Mech., ASME, 487, 1951. 3This formulation has been initiated by the work: Z.MIrscIoacutescIzsc ; Non-associated ﬂow laws in plasticity, Journ. de Mecanique ,2, 21-42, 1963. 4J.CIascIsscIescIysc,P.M.NIascIgscIhscIdscIisc; On the nonequivalence of the stress space and strain space fo rmulations of plasticity theory, J. Appl. Mech. ,50, 350, 1983. 5A.A.IIlscIyscIuscIsscIhscIiscInsc; On the postulate of plasticity, PMM, 25, 503, 1961. 7.1 Introduction 137 Fig. 7.2: Schematic plastic behaviour of various materials In Fig. 7.2. we show schematically strain-stress curves for diﬀerent types of materials. The steepest curve in both pictures correspond to the so-cal led brittle materials which practically do not reveal any plastic deformations prior to failure. Their deformations under high loading are small and they absorb only a little ene rgy before breaking. It should be underlined that many materials may behave this way in low temperatures whereas their properties are very diﬀerent in high temperat ures. This transition explains mysterious catastrophes of Liberty ships in 40th of the XXth century. The curves for ductile materials in Fig. 7.2. correspond to m aterials for which the classical plasticity was developed. They possess relative large irreversible deformations and by failure absorb a large amount of energy. Therefore the y are called tough. 138 Plasticity Many damages and accidents of cargo vessels were occurred, and especially for Liberty Ships. The vast majority of the sea accidents were related to brittle fracture. By 1st of April 1946, 1441 cases of damage had been reported for 970 cargo vessels, 1031 of which were to Liberty Ships. Total numbers of 4720 damages were reported. Seven ships were broken in two, e.g. "Schenectady". 7.2 Plasticity of ductile materials We proceed to specify the yield surface in the stress space. A s mentioned above this stress formulation was primarily motivated by plastic deformatio ns of metals. In such materials the pressure phas practically no inﬂuence on plastic strains which means t hat the yield surface should be described only by the stress deviator: σD ij=σij+pδij, p=−1 3σkk. The eigenvalues of the stress deviator follow from the eigen value problem parenleftbig σD ij−sδijparenrightbig nj=0, (7.4) and the solutions must satisfy the condition Is=s(1)+s(2)+s(3)=0. (7.5) The eigenvalues s(α)and the eigenvalues σ(α)of the full stress tensor σijare, of course, connected by the relation σ(α)=s(α)−p, α=1,2,3. (7.6) For the purpose of formulation of various hypotheses for the yield surface, it is con- venient to calculate invariants of the stress deviator and t he maximum shear stresses. As presented in Subsection 3.2.4 (compare the three-dimens ional Mohr circles), the ex- tremum values of shear stress are given by the diﬀerences of t hree principal values of the stress tensor (radii of Mohr’s circles) τ(1)=σ(2)−σ(3) 2, τ(2)=σ(1)−σ(3) 2, τ(3)=σ(1)−σ(2) 2. (7.7) Hence, we have as well τ(1)=s(2)−s(3) 2, τ(2)=s(1)−s(3) 2, τ(3)=s(1)−s(2) 2. (7.8) 7.2 Plasticity of ductile materials 139 Further we use the sum of squares of these quantities. In term s of invariants of the stress tensor and of the stress deviator it has the form 3summationdisplay α=1parenleftBig τ(α)parenrightBig2 =3 2bracketleftbigg1 3I2 σ−IIσbracketrightbigg =−3 2IIs=1 2σ2 eq, σeq=radicalbigg 3 2σD ijσD ij,(7.9) whereσeq=σ(1)in the uniaxial tension/compression for which σ(2)=σ(3)= 0. It is clear that the second invariant of the deviatoric stresse s must be negative. These relations follow from the deﬁnitions of the invariants Iσ=σkk=−3p, IIσ=1 2parenleftbigI2 σ−σijσijparenrightbig, IIIσ=det(σij), (7.10) Is= 0, IIs=−1 2σD ijσD ij=−J2=−σ2 eq 3, IIIs=J3=detparenleftbigσD ijparenrightbig. The quantity σeq=√3J2is called the equivalent (eﬀective) stress. Now, we are in the position to deﬁne the elastic domain in the s pace of stresses. It is convenient to represent it by a domain in the three-dimens ional space of principal stresses. In this space we choose the principal stresses σ(1),σ(2),σ(3)as coordinates. The assumption that the pressure does not inﬂuence plastic stra ins means that yield surfaces in this space must be cylindrical surfaces with generatrix p erpendicular to surfaces s(1)+ s(2)+s(3)=0, i.e.σ(1)+σ(2)+σ(3)+3p=0.The axis of those cylinders is, certainly, the straight line σ(1)=σ(2)=σ(3). This line is called the hydrostatic axis. In general, we can write the equation of the yield surface in the form fparenleftbigJ2,J3,ep ij,κ,Tparenrightbig=0, (7.11) with the parametric dependence on the plastic strain ep ij,temperature Tand the harden- ing parameter κ. We return to these parameters later. Two examples of yield s urfaces, discussed further in some details, are shown in Fig. 7.3. It is also convenient to introduce a normal (perpendicular) vector to the yield surface in the stress space given by its gradient in this space, i.e. N=∂f ∂Tvextendsinglevextendsinglevextendsingle∂f ∂Tvextendsinglevextendsinglevextendsingle,i.e.Nij=∂f ∂σijradicalBig ∂f ∂σkl∂f ∂σkl. (7.12) Then we can introduce local coordinates in which the yield fu nctionfidentiﬁes the elastic domain of the Σ-space assuming there negative values, i.e. for all elastic processesf <0. We skip here the presentation of the history of the deﬁnition of yield surfaces which goes back to Galileo Galilei. There are two fundamental form s of this surface which are still commonly used in the linear plasticity of solids. The o lder one was proposed by H. Tresca in 1864 and it is called Tresca-Guest surface. Its equ ation has the form max αvextendsinglevextendsinglevextendsingleτ(α)vextendsinglevextendsinglevextendsingle=σ0⇒σ(1)−σ(3)=2σ0>0, (7.13) whereσ0is the material parameter and we have ordered the principal s tressesσ(1)≥ σ(2)≥σ(3). It means that the beginning of the plastic deformation appe ars in the point of 140 Plasticity the maximum shear stress. In the space of principal stresses it is a prism of six sides and inﬁnite length (see: Fig. 7.3.). The parameter σ0may be dependent on all parameters listed in the general relation (7.11). Fig. 7.3: Yield surfaces in the space of principal stresses The second yield surface was proposed in 1904 by M. T. Huber6and then rediscovered in 1913 by R. von Mises and H. von Hencky. It says that the limit of elastic deformation is reached when the energy of shape changes (distortion ener gy) reaches the limit value ρεY. The distortion energy ρεDis deﬁned as a part of the full energy of deformation ρε reduced by the energy of volume changes ρεV(e.g. compare (5.175)). We have ρε=1 2σijeij=1 2parenleftBigg σijσkk 9Kδij+σijσD ij 2µparenrightBigg = =1 2KparenleftBigσkk 3parenrightBig2 +1 4µparenleftbigσD ijσD ijparenrightbig⇒ρεV=p2 2K, ρεD=1 4µparenleftbigσD ijσD ijparenrightbig, (7.14) i.e.ρεD=ρεY⇒ρεY=1 4µparenleftbig σD ijσD ijparenrightbig =σ2 eq 6µ. Making use of the identity, following from (7.5), 3parenleftBig s(1)s(2)+s(1)s(3)+s(2)s(3)parenrightBig =−1 2bracketleftbiggparenleftBig s(1)−s(2)parenrightBig2 +parenleftBig s(1)−s(2)parenrightBig2 +parenleftBig s(1)−s(2)parenrightBig2bracketrightbigg , (7.15) 6M. T. HIuscIbscIescIrsc; Przyczynek do podstaw wytrzymało ´sci,Czasop. Techn. , Lwów,22, 1904. Due to the publication of this work in Polish it remained unknown un til the hypothesis was rediscovered by von Mises and von Hencky. 7.2 Plasticity of ductile materials 141 we obtain σD ijσD ij=1 3bracketleftbiggparenleftBig s(1)−s(2)parenrightBig2 +parenleftBig s(1)−s(2)parenrightBig2 +parenleftBig s(1)−s(2)parenrightBig2bracketrightbigg . (7.16) Consequently, bearing (7.6) in mind, the yield limit is reac hed when the principal stresses fulﬁl the condition 3σD ijσD ij=parenleftBig σ(1)−σ(2)parenrightBig2 +parenleftBig σ(2)−σ(3)parenrightBig2 +parenleftBig σ(1)−σ(3)parenrightBig2 =2σ2Y, (7.17) i.e.σeq=σY, where σY=radicalbig 6µρεY, (7.18) is the yield limit ( σ(1)=σYin uniaxial tension, i.e. for σ(2)=σ(2)=0). It means that for processes in which σeq<σYall states are elastic ( f <0) and otherwise the system develops plastic deformations. Clearly, the relation (7.1 7) deﬁnes a circular cylinder in the space of principal stresses. Its axis is again identical with the line σ(1)=σ(2)=σ(3), it is extended to inﬁnity and it has common generatrix with th e prism of Tresca as shown in Fig. 7.3. ⋆In order to compare analytically both deﬁnitions of the yiel d surface we show that the yield stress σYcalculated by means of the distortion energy of the Huber-Mi ses- Hencky hypothesis (7.17) is not bigger than the material par ameter2σ0of the Tresca hypothesis (7.13). Let us write (7.17) in the following form σY=1√ 2radicalBigparenleftbigσ(1)−σ(2)parenrightbig2+parenleftbigσ(2)−σ(3)parenrightbig2+parenleftbigσ(1)−σ(3)parenrightbig2= =vextendsinglevextendsingleσ(1)−σ(3)vextendsinglevextendsingle √ 2radicalBiggparenleftbiggσ(1)−σ(2) σ(1)−σ(3)parenrightbigg2 +parenleftbiggσ(2)−σ(3) σ(1)−σ(3)parenrightbigg2 +1= =vextendsinglevextendsingleσ(1)−σ(3)vextendsinglevextendsingle √ 2radicalBiggparenleftbigg1−µσ 2parenrightbigg2 +parenleftbigg1+µσ 2parenrightbigg2 +1= =vextendsinglevextendsinglevextendsingleσ(1)−σ(3)vextendsinglevextendsinglevextendsingleradicalbigg 3+µ2σ 4, (7.19) where µσ=2σ(2)−parenleftbigσ(1)+σ(3)parenrightbig σ(1)−σ(3), (7.20) is the so-called Lode parameter which describes an inﬂuence of the middle principal stressσ(2). Obviously−1≤µσ≤1which corresponds to σ(2)=σ(3)for the lower bound, and σ(2)=σ(1)for the upper bound. It plays an important role in the theory o f civil engineering structures. Hence σY≤vextendsinglevextendsinglevextendsingleσ(1)−σ(3)vextendsinglevextendsinglevextendsingle=2σ 0.♣ (7.21) 142 Plasticity ⋆We demonstrate on a simple example an application of the noti on of the yield stress. We consider a circular ring of a constant thickness w ith external and internal radiiaandb, respectively, and an external loading by the pressures paandpbon these circumferences. We check when the material of the ring reach es in all points the yield stress according to the Huber-Mises-Hencky hypothesis. Th is is the so-called state of the load-carrying capacity of this structure. This is the axial symmetric problem of plane stresses. Conse quently, the principal stresses in cylindrical coordinates are given by σ(1)=σrr,σ(2)=σθθ,σ(3)=0. These components of stresses must fulﬁl the equilibrium conditio n (momentum balance (5.43)) dσrr dr+σrr−σθθ r=0, (7.22) and, according to (7.17), at each place of the ring (σrr−σθθ)2+(σrr)2+(σθθ)2=2σ2 Y. (7.23) By means of (7.22) we eliminate the component σθθof stresses and obtain the following equationparenleftbigg rds drparenrightbigg2 +sparenleftbigg rds drparenrightbigg +s2−1=0, s=σrr σY√ 2. (7.24) Solution of this quadratic equation with respect to the deri vativeds/dr yields ds dr=−s 2r±1 2rradicalbig 4−3s2. (7.25) Consequently ds −s±√ 4−3s2=dr 2r. (7.26) As we have to require |s|<2/√ 3we can change the variables s=2√ 3sinϕ, (7.27) and this yields −dϕ tanϕ∓√ 3=dr 2r. (7.28) Hence, we obtain two solutions but only one of them is real and it has the form r=Cradicaltpradicalvertexradicalvertexradicalvertexradicalvertexradicalvertexradicalvertexradicalbt1+3s2 4−3s2radicalBigg s√ 3 4−3s2+√ 3expbracketleftBigg −√ 3 2arctans√ 3√ 4−3s2bracketrightBigg , (7.29) whereCis the constant of integration. 7.2 Plasticity of ductile materials 143 The construction of solution is shown in Fig. 7.4. in arbitra ry units. For the radius b the value of the radial stress is given by σr=−pb. We adjust the curve described by the relation (7.29) in such a way that it intersects the point (−pb,b)indicated by the circle in Fig. 7.4. This yields the value of the constant Cin the solution. Then for the given value of the radius awe ﬁnd the value of the pressure pawhich yields the limit value for the load of this structure, i.e. its load-carrying capacity . Fig. 7.4: Construction of solution for the load-carrying ca pacity of the ring ♣ Yield surfaces impose conditions on elastic solutions unde r which the system possesses only elastic strains. In some design problems this is alread y suﬃcient. However, many engineering structures admit some plastic deformations — f or example, in the case of concrete it is the rule — and then we have to ﬁnd a way to describ e the evolution of plastic strains ep.We proceed to develop such models. First of all, we have to deﬁne not only the shape of the yield su rface in the stress space, as we did above, but also its dependence on parameters listed in (7.11). We indicate here a few important examples. It is convenient to write the yield function in the form fparenleftbig J2,J3,ep ij,κ,Tparenrightbig =F(J2,J3)−σYparenleftbig ep eq,κ,Tparenrightbig =0, (7.30) whereσYis the yield limit in the uniaxial tension/compression test for which σ22=σ33=σ(2)=σ(3)=0⇒σeq=σ(1)=σ11, (7.31) ˙ep 11=−2˙ep 22=−2˙ep 33,˙ep ij=0fori/negationslash=j⇒˙ep eq= ˙ep 11. whereσeqis given by (7.9), and it is assumed to be given in terms of argu ments listed in (7.30).ep eqis the equivalent plastic strain obtained from the integrat ion in time of the eﬀective rate of plastic deformation ˙ep eq=radicalbigg 2 3˙ep ij˙ep ij. (7.32) 144 Plasticity Obviously, for isotropic materials we expect ˙ep ijto be deviatoric. This yields relations (7.31). Let us begin with the simplest case. In a particular case of id eal plasticity we consider materials without hardening. Then the function (7.30) has t he form f=radicalbigg 3 2σD ijσD ij−σY=0, (7.33) with the constant yield limit σY. Of course, the stress tensor must be such that elastic processes remain within the elastic domain which is charact erized byf <0. Plastic deformations may develop for stresses which belong to the yi eld surface. Their changes yielding plastic deformation must remain on this surface wh ich means that the increments of stresses described by ˙σijmust be tangent to the yield surface. Hence, for such process es ˙f=∂f ∂σij˙σij=0, (7.34) as the gradient ∂f/∂σijis perpendicular to the yield surface. Now we make the fundamental constitutive assumption, speci fying the rate of plastic strain˙ep ijand require that this quantity follows from a potential G(σij)deﬁned on the stress space ˙ep ij=˙λ∂G ∂σij, (7.35) where˙λis a scalar function following from the so-called Prager con sistency condition. We demonstrate it further. Let us introduce the notion of the outward normal vector to th e surfacef=0(com- pare (7.12)). Clearly Nij=∂f ∂σijvextenddoublevextenddoublevextenddoublevextenddouble∂f ∂σklvextenddoublevextenddoublevextenddoublevextenddouble−1 ,vextenddoublevextenddoublevextenddoublevextenddouble∂f ∂σklvextenddoublevextenddoublevextenddoublevextenddouble=radicalbigg ∂f ∂σkl∂f ∂σkl, (7.36) is such a vector. For the yield surface (7.33) it becomes Nij=σD ijvextenddoublevextenddoubleσD klvextenddoublevextenddouble,vextenddoublevextenddoubleσD klvextenddoublevextenddouble=radicalBig σD klσD kl=radicalbigg 2 3σY. (7.37) Obviously, we have the following classiﬁcation ˙ep ij=braceleftBigg 0forf <0orf=0andNij˙σij<0. ˙λ∂G ∂σijforf=0andNij˙σij=0.(7.38) The condition Nij˙σij<0means that the process yields the unloading — as Nijis or- thogonal to the yield surface, ˙σijmust point in the direction of the elastic domain and, consequently, the process must be elastic. In a particular case when the potential Gand the yield function are identical we obtain 7.2 Plasticity of ductile materials 145 ˙ep ij=˙λ∂f ∂σij. (7.39) This is the so-called associated ﬂow rule. In this case, the r ate of plastic deformation ˙ep ij is perpendicular to the yield surface, i.e. it is parallel to the normal vector Nij. It is appropriate to make here the following remark concerni ng the geometry of the yield surface. If this surface were not convex then in points in which it is concave tangent changes of the stress tensor ˙σij, i.e.(∂f/∂σij)˙σij=0, would yield stresses in the interior of the yield surface, i.e. in the range f <0. This would be related to the development of pure elastic deformations in contrast to the assumption t hat tangent changes of stress yield plastic deformations. Therefore, in the classical pl asticity nonconvex yield surfaces are not admissible. This is the subject of the so-called Druc ker stability postulate. In the local form for the associated ﬂow rules (7.39) it can be wr itten as ˙ep ij˙σij>0. (7.40) It is also sometimes postulated in the global form integraldisplayparenleftbig σij−σ0 ijparenrightbig dep ij>0, (7.41) which shows that the postulate imposes a restriction on the w ork of plastic deformations between an arbitrary initial state of stress σ0 ijand an arbitrary ﬁnite state of stress σij. For the Huber-Mises-Hencky yield function (7.33) we obtain the associated ﬂow rule ˙ep ij=˙λradicalbigg 3 2σD ijvextenddoublevextenddoubleσD klvextenddoublevextenddouble=˙λradicalbigg 3 2Nij. (7.42) In the simple uniaxial tension/compression test we have the n ˙ep eq= ˙ep 11=˙λ⇒˙ep ijσij=˙λradicalbigg 3 2vextenddoublevextenddoubleσD klvextenddoublevextenddouble=˙λσ11= ˙ep eqσY. (7.43) Hence for the rate of work (working) we obtain ˙W= ˙eijσij=parenleftbig˙ee ij+ ˙ep ijparenrightbigσij⇒ ⇒˙Wp= ˙ep ijσij=˙λradicalbigg 3 2vextenddoublevextenddoubleσD klvextenddoublevextenddouble= ˙ep eqσeq= ˙ep eqσY. (7.44) The last expression — ˙ep eqσY— describes the plastic working in the one-dimensional test which is an amount of energy dissipated by the system per unit time due to the plastic deformation. Hence ˙λ≥0, (7.45) and the equality holds only for elastic deformations. This s tatement follows easily from the second law of thermodynamics. 146 Plasticity In order to construct an equation for plastic strains we acco unt for the additive de- composition (7.3). For f <0we have elastic processes and then this property indicates (compare (5.26)) the following Prandl-Reuss equation for t he rate of deformation ˙eij=parenleftBigg ˙σD ij 2µ+˙σkk 9KδijparenrightBigg +˙λradicalbigg 3 2Nij= (7.46) =parenleftBigg ˙σD ij 2µ+˙σkk 9KδijparenrightBigg +˙WpσD ijvextenddoublevextenddoubleσD klvextenddoublevextenddouble2. (7.47) This follows from the property of isotropic elastic materia ls for which the eigenvectors for the stress and for the strain are identical7. The spherical part is, obviously, purely elastic while the deviatoric part can be written in the form dσD ij dt+˙Wp4µ 3σ2 YσD ij=2µdeD ij dt. (7.48) This equation is very similar to the evolution equation for s tresses within the standard linear model of viscoelasticity (6.58) divided by the relax ation timeτ. However, there is a very essential diﬀerence between these two models. It is ea sy to check that the equation of viscoelasticity (6.58) is not invariant with respect to a change of time scale t→αt, whereαis an arbitrary constant. This indicates the rate dependenc e in the reaction of the material. It is not the case for the equation of plasticit y (7.48). Diﬀerentiation with respect to time appears in all terms of this equation and for t his reason the transformation parameterαcancels out. This is the reason for denoting the consistency parameter by ˙λ=dep eq/dt. It transforms: t→αt⇒˙λ→˙λ/α. Therefore, the classical plasticity is rate independent. The response of the system is the same for v ery fast and very slow time changes of the loading. In reality, metals do possess th is property when the rate of deformation ˙ep eqis approximately 10−6−10−41/s. For higher rates one has to incorporate the rate dependence (compare the book of Lemaitre, Chaboche [9] for further details). This yields viscoplastic models presented further in these notes. In the more general case of isotropic hardening and for isoth ermal processes σYbe- comes a function of ep eqalone. For many materials it is also important to include the temperature dependence. Then σYbecomes the function of these two quantities. The model is similar to this which we have considered above but on e has to correct the deﬁn- ition of the consistency parameter ˙λ. Finally, a dependence on the hardening parameter κmeans that we account for the accumulation of plastic deform ations in the material. The most common deﬁnitions of this parameter are as follows a) the parameter accounting for the accumulation of the plas tic energy κ=tintegraldisplay 0σij(ξ)˙ep ij(ξ)dξ, (7.49) 7In order to prove it, it is suﬃcient to compare the eigenvalue problems for deviatoric stress and strain tensors. 7.2 Plasticity of ductile materials 147 b) Odqvist parameter accounting for the accumulation of the plastic deformation (compare (7.30) and (7.32)) κ=tintegraldisplay 0radicalbigg 2 3˙ep ij˙ep ijdξ=tintegraldisplay 0dep eq dξdξ. (7.50) Then the relation (7.11) yields the consistency condition ˙f=0⇒∂f ∂σij˙σij+∂f ∂ep ij˙ep ij+∂f ∂T˙T+∂f ∂κ˙κ=0. (7.51) Simultaneously, ˙ep ij/negationslash=0only in processes of loading which are deﬁned by the relation ∂f ∂σij˙σij+∂f ∂T˙T >0. (7.52) In the limit case∂f ∂σij˙σij+∂f ∂T˙T=0, (7.53) we say that the process is neutral. Finally, for the process o f unloading, ∂f ∂σij˙σij+∂f ∂T˙T <0. (7.54) Consequently, for the evolution of plastic deformation we h ave the following relations ˙ep ij=  0for eitherf <0orf=0and∂f ∂σij˙σij+∂f ∂T˙T≤0 ˙λ∂f ∂σijforf=0and∂f ∂σij˙σij+∂f ∂T˙T >0.(7.55) In the case of the hardening parameter (7.49) the consistenc y condition (7.51) can be written in the form ˙f=∂f ∂σkl˙σkl+∂f ∂T˙T+bracketleftBigg ∂f ∂ep ij++∂f ∂κσijbracketrightBigg ˙ep ij=0. Hence, we obtain the following relation for the consistency parameter ˙λ=∂f ∂σkl˙σkl+∂f ∂T˙T D, D=−∂f ∂ep ij∂f ∂σij−∂f ∂κ∂f ∂σijσij. (7.56) The quantity Dis called the hardening function. We have ˙λ>0⇒D>0. (7.57) The ﬂow rule can be now written in the form ˙ep ij=1 D∂f ∂σijparenleftbigg∂f ∂σkl˙σkl+∂f ∂T˙Tparenrightbigg . (7.58) 148 Plasticity The right hand side is the homogeneous function of ˙σijand˙T. Consequently, this ﬂow rule possesses the same time invariance as the rule (7.39) fo r the model without hardening, i.e. the model is rate independent. Apart from the above presented isotropic hardening materia ls reveal changes of the yield limit which can be attributed to the shift of the origin in the space of stresses. A typical example is the growth of the yield stress in tensile l oading with the simultaneous decay of the yield stress for compression. In the uniaxial ca se it means that the whole stress-strain diagram will be shifted on a certain value of s tresses. This is the Bauschinger eﬀect. Fig. 7.5: An example of Bauschinger eﬀect in cyclic loading The corresponding hardening is called kinematical or aniso tropic. It is described by the so-called back-stress Z=Zijei⊗ejwhich speciﬁes the shift of the origin in the stress space. The yield function can be then written in the form f(T,Z,κ)=F(T,Z)−σYparenleftbigep eq,Tparenrightbig=radicalbigg 3 2¯σD ij¯σD ij−σY=0, (7.59) where ¯σD ij=σD ij−Zij. (7.60) One has to specify an equation for the back-stress. It is usua lly assumed to have the form of the evolution equation, e.g. ˙Zij=˙β(σij−Zij), (7.61) where˙βis a material parameter. We skip here the further details ref erring to numerous monographs on the subject8. 8e.g. see the book [9] or AIlscIbscIrscIescIcscIhscItscBIescIrscItscIrscIascImsc; Elasticity and Plasticity of Large Deformations , Springer Berlin, 2008. MIiscIcscIhscIascIlslashscKIlscIescIiscIbscIescIrsc; Handbook of Computational Solid Mechanics , Springer, Heidelberg, 1998. GIescIrscIascIrscIdscA.MIascIuscIgscIiscInsc; The Thermomechanics of Plasticity and Fracture, Cambridge University Press, 1992. 7.3 Plasticity of soils 149 7.3 Plasticity of soils Theories of irrecoverable, permanent deformations of soil s is very diﬀerent from the plasticity of metals presented above. Metals produce plast ic deformations primarily due to the redistribution and production of crystallograph ic defects called dislocations. Plastic behaviour of soils is mainly connected with the redi stribution of grains and it is strongly inﬂuenced by ﬂuids ﬁlling the voids (pores) of such a granular material. Strain due to the deformation of grains is often negligible in compa rison to the amount of shear and dilatation caused by relative motions of grains. The beh aviour is entirely diﬀerent in the case of dry granular materials (frictional materials ) than a material saturated by, for instance, water or oil where the cohesive forces play an i mportant role. A detailed modern presentation of the problem of permanent deformatio ns of soils can be found in the book of D. Muir Wood [23] (compare also a set of lectures vo n Verruijt [19]). Similar issues for rocks are presented in the classical book of Jaege r, Cook and Zimmerman[6]. We limit the attention only to few issues of this subject. Attempts to describe the plasticity of granular materials s tem from Coulomb, who formulated a simple relation between the normal stress σnon the surface with a normal vectornand the shear stress τnon this surface. It is a generalization of the law of friction between two bodies and has the form |τn|=c−σntanϕ, (7.62) whereϕis the so-called friction angle (angle of repose) and cdenotes the cohesion intercept. This relation is called Mohr-Coulomb law. For dr y granular materials the cohesion does not appear, c= 0, and then the angle of repose ϕis the only material parameter. It is, for instance, the slope of natural sand hil ls and pits (Fig. 7.6). Fig. 7.6: Sand pit trap of antlion in dry sand. Slope almost equal to ϕ The above relation leads immediately to the yield function i n terms of principal stressesσ(1)>σ(2)>σ(3).. Namely fparenleftBig σ(1),σ(2),σ(3)parenrightBig =parenleftBig σ(1)−σ(3)parenrightBig +parenleftBig σ(1)+σ(3)parenrightBig sinϕ−2ccosϕ=0. (7.63) 150 Plasticity The derivation from properties of Mohr’s circle is shown in F ig. 7.7. Namely |τn|=σ(1)−σ(3) 2cosϕ, σn=σ(1)+σ(3) 2−σ(1)−σ(3) 2sinϕ. (7.64) Substitution in (7.62) yields (7.63). Forϕ= 0andc=σ0the yield function (7.63) becomes the Tresca-Guest yield condition (7.13). Fig. 7.7: Construction of Mohr-Coulomb yield function Cohesive forces are inﬂuencing not only the relation betwee n normal and shear stresses. Due to the porosity of granular materials a ﬂuid in pores yiel ds cohesive interactions as well as it carries a part of external loading. This observati on was a main contribution of von Terzaghi to the theory of consolidation of soils9. He has made an assumption that the pore pressure pdoes not have an inﬂuence on the plastic deformation of soils . The meaning of pis here the same as in Subsection 6.4. and it should not be conf used with the trace of the bulk stress σij, i.e.p/negationslash=−1 3σkk. It means that the stress appearing in yield functions must be reduced by subtracting the contribu tion of this pressure. If we deﬁne the eﬀective stress σ′ ij=σij+pδij, (7.65) then the Mohr-Coulomb yield function becomes parenleftBig σ′(1)−σ′(3)parenrightBig +parenleftBig σ′(1)+σ′(3)parenrightBig sinϕ−2ccosϕ=0. (7.66) σ′(α)=σ(α)+p, α=1,2,3. This function is shown in the upper panel of Fig. 7.8. Further we distinguish by primes all quantities based on the eﬀective stress. 9K.IvscIoscInscTIescIrscIzscIascIgscIhscIisc; ErdbaumechanikaufbodenphysikalischerGrundlage, Franz Deuticke, Wien, 1925. 7.3 Plasticity of soils 151 Incidentally, a similar notion of eﬀective stresses appear s in the theory of damage — it is related to changes of reference surface due to the appeara nce of cracks. Such models shall be not presented in these notes. ⋆Remark. There exists some confusion within the soil mechanics conce rning the deﬁnition of positive stresses. Soils carry almost without exception only compressive loads (compare Fig. 7.7. and 7.8.) and, for this reason, in co ntrast to the classical continuum mechanics, a compressive stress is assumed to be p ositive. This is convenient in a ﬁxed system of coordinates related to experimental setu ps such as triaxial apparatus. Then pressure pin the deﬁnition of eﬀective stresses (7.65) would appear wi th the minus sign. Usually it is denoted in soil mechanics by u. In some textbooks10both conventions concerning the sign of stresses appear simultaneously. How ever, such a change of sign in a general stress tensor yields the lack of proper invarian ce with respect to changes of reference systems. It is also contradictory with the choice of the positive direction of vectors normal to material surfaces on which many mathemati cal problems of balance laws and the Cauchy Theorem rely. For these reasons, we work h ere with the same convention as in the rest of this book — tensile stress is posi tive. In addition, one should be careful in the case of relation of s uch one-component models to models following from the theory of immiscible mixtures, for instance to Biot’s model. Such models are based on partial quantities and then the pore pressurepis not the partial pressurepFof a multicomponent model but rather p=pF/n, wherenis the porosity.♣ In soil mechanics, where the deﬁnitions of elastic domains d escribed in the previous Subsection are not appropriate, it is convenient to introdu ce special systems of reference in the space of eﬀective principal stresses. One of them is di rectly related to the set of invariants (7.10) p′=1 3I′ 1=−1 3I′ σ, q=radicalbig 3J′ 2=σeq, r=33radicalbigg J′ 3 2. (7.67) Another one is a cylindrical system. One of the axes is the pre ssureparenleftbig−1 3σ′ kkparenrightbig, i.e. it is oriented along the line σ′(1)=σ′(2)=σ′(3). It is denoted by ξand scaledξ=I′ σ/√ 3. The other two coordinates are deﬁned by the relations ρ=radicalbig 2J′ 2≡radicalbigg 2 3σ′ eq≡radicalBig σ′D ijσ′D ij,cos(3θ)=parenleftbiggr σ′eqparenrightbigg3 . (7.68) These are the so-called Haigh—Westergaard coordinates. Th e(ξ,ρ)- planes are called Renduli ˇc planes and the angle θis called the Lode angle. The transformation from these coordinates back to principal stresses is given by the relat ion  σ′(1) σ′(2) σ′(3)=1 √ 3 ξ ξ ξ +radicalbigg 2 3ρ cosθ cosparenleftbigθ−2 3πparenrightbig cosparenleftbigθ+2 3πparenrightbig . (7.69) Mohr-Coulomb yield function in the Haigh—Westergaard coor dinates has the following formbracketleftBig√ 3sinparenleftBig θ+π 3parenrightBig −sinϕcosparenleftBig θ+π 3parenrightBigbracketrightBig ρ−√ 2ξsinϕ=√ 6ccosϕ. (7.70) 10e.g. [23] orR.LIascInscIcscIescIlscIlscIoscItscItscIasc; Geotechnical Engineering, Balkema, Rotterdam, 1995. 152 Plasticity Alternatively, in terms of the invariants (p′,q,r)we can write bracketleftbigg1√ 3cosϕsinparenleftBig θ+π 3parenrightBig −1 3tanϕcosparenleftBig θ+π 3parenrightBigbracketrightbigg q−p′tanϕ=c, (7.71) θ=1 3arccosparenleftbiggr qparenrightbigg3 . As in the classical theory of plasticity, modiﬁcations of Mo hr-Coulomb condition were introduced in order to eliminate corners in the yield surfac e. One of such modiﬁcations was introduced by D. C. Drucker and W. Prager. This condition for the limit state of soils has the following form radicalbig J′ 2−√ 3cosϕradicalbig 3+sin2ϕc−sinϕradicalBig 3parenleftbig 3+sin2ϕparenrightbigI′ σ=0, (7.72) where the invariants J′ 2andI′ σare deﬁned by relations for eﬀective stress analogous to (7.10). This function is shown in the lower panel of Fig. 7. 8. The dependence on the invariant I′ σfollows from the dependence of the yield in soils on volume ch anges, i.e. it describes an inﬂuence of dilatancy on the appearance of the critical limit state. Forϕ= 0andc=σY/√ 3this condition becomes identical with Huber-Mises-Hencky condition (7.17). Due to its simplicity the Mohr-Coulomb yield surface is ofte n used to model the plastic ﬂow of geomaterials (and other cohesive-frictiona l materials). However, many such materials show dilatational behavior under triaxial s tates of stress which the Mohr- Coulomb model does not include. Also, since the yield surfac e has corners, it may be inconvenient to use the original Mohr-Coulomb model to de termine the direction of plastic ﬂow. Therefore it is common to use a non-associated p lastic ﬂow potential that is smooth. For example, one is using the function g=radicalBig (αcYtanψ)2+G2(ϕ,θ)q2−p′tanϕ, (7.73) whereαis a parameter, cYis the value of cwhen the plastic strain is zero (also called the initial cohesion yield stress), ψis the angle made by the yield surface in the Renduli ˇc plane at high values of p′(this angle is also called the dilation angle), and G(ϕ,θ)is an appropriate function that is also smooth in the deviatoric s tress plane. 7.3 Plasticity of soils 153 Fig. 7.8: Yield surfaces (7.66)and(7.72)in the space of principal eﬀective stressesσ1=σ′(1),σ2=σ′(2), σ3=σ′(3). We shall not expand this subject anymore. Due to the vast ﬁeld of applications: soils, powders, avalanches, debris ﬂows and many others, the numbe r of models describing the critical behaviour of such materials is also very large. Cap plasticity models, Cam-Clay (CC) models, Modiﬁed-Cam-Clay (MCC) models, Mroz models, e tc. are based on similar ideas as the models presented above. There exists also a clas s of hypoplasticity models in which the notion of the yield surface does not appear at all and which seem to ﬁt well phenomena appearing in sands11. 11compare articles of E.BIascIuscIescIrsc : Analysis of Shear Banding with a Hypoplastic Constitutive Model for a Dry and Cohesionless Granular Material, 335-350, and D. KIoscIlscIyscImscIbscIascIssc : The Importance of Sand in Earth Sciences, both in: B.AIlscIbscIescIrscIssc (ed.);Continuous Media with Microstructure , Springer, Berlin, 2010. 154 Plasticity 7.4 Viscoplasticity There are many ways of extension of the classical plasticity to include rate dependence. Obviously, one of them would be to incorporate additionally some viscous properties as we did in Chapter 7. This kind of the model is developed sinc e early works of P. Perzyna12. The other way, less ambitous, is to incorporate a rate depen dence in the deﬁnition of the yield function. In principle, the classica l yield function cannot exist in such models but one gets results by direct extension of plast icity models presented in this Chapter. For such models it is advocated in the books of L emaitre, Chaboche [9] and Lemaitre, Desmorat [10]. We present here only a few hints to the model of the second kind . Namely, it is assumed that the yield criterion satisﬁes the relation f= 0,˙f=0— plasticity, f=σV>0— viscoplasticity, (7.74) withf <0satisﬁed in the elastic domain. σVis a viscous stress given by a viscosity law. In both cases fcan be chosen according to the rules discussed in previous Su bsections. For instance, in the case of Huber-Mises-Hencky model with i sotropic and kinematic hardening we have f= (T−Z)eq−κ−σY, (7.75) (T−Z)eq=radicalbigg 3 2parenleftbigσD ij−ZD ijparenrightbigparenleftbigσD ij−ZD ijparenrightbig, whereκdescribes the isotropic hardening related to the size growt h of the yield surface. It may be, for instance, assumed to have the exponential form κ=κ∞bracketleftbig1−expparenleftbig−bep eqparenrightbigbracketrightbig, (7.76) whereκ∞,bare material parameters depending on temperature. Sometim es a power law κ=Kpparenleftbigep eqparenrightbig1/Mis suﬃcient. Kinematic hardening described by the back-stresses Zijrequires an evolution equa- tion. It may have the form (7.61) or it may be the so-called Arm strong-Frederick law13 d dtparenleftbiggZij Cparenrightbigg =2 3˙ep ij−γ CZij˙ep eq, (7.77) for which the identiﬁcation of parameters is easier [10]. The viscous stress σVis also given by various empirical relations. Two of them hav e the form 1) Norton power law σV=KNparenleftbig˙ep eqparenrightbig1/N, (7.78) 12e.g.P.PIescIrscIzscIyscInscIasc; The constitutive equations for the rate sensitive plastic m aterials,Quart. Appl. Math.,20, 321-332, 1963. 13P.J.AIrscImscIsscItscIrscIoscInscIgsc,C.O.FIrscIescIdscIescIrscIiscIcscIksc; A mathematical representation of the multiaxial Bausching er eﬀect, CEGB Report, RD/B/N731, Berkeley Nuclear Laborator ies, 1966. 7.4 Viscoplasticity 155 2) exponential law leading to the saturation at large plasti c rates σV=K∞bracketleftbigg 1−expparenleftbigg −˙ep eq nparenrightbiggbracketrightbigg , (7.79) whereKN,K∞,Nandnare material parameters. In Fig. 7.9 we show a comparison of results for various viscou s models14. Fig. 7.9: Relaxation test for the identiﬁcation of viscosit y parameters — Inconel alloy at θ=6270C. For the alloy investigated by Lemaitre and Dufailly the foll owing parameters are appropriate E=160 GPa,KN=75GPa/s1/N, N=2.4, K∞=104GPa,n=1.4×10−2s−1. Rate-dependent viscoplastic models must be used in cases of high deformation rates. For metals, the rates up to app. 10−31/s do not inﬂuence substantially results in the plastic range of deformations. For higher rates the yield li mit may grow even three times by the rate1001/s15. 14J.LIescImscIascIiscItscIrscIesc,J.DIuscIfscIascIiscIlscIlscIysc; Damage measurements, Engn. Fracture Mech., 28, 1987 15P.PIescIrscIzscIyscInscIasc; Thermodynamics of Inelastic Materials (in Polish), PWN, Warsaw, 1978. 156 Plasticity Chapter 8 Dislocations 8.1 Introduction One of the diﬃcult questions of materials science some 100 ye ars ago was the elucidation of the mechanism of plastic deformation of crystalline bodi es. Plasticity of ductile mate- rials described in Subsection 8.2 is purely macroscopic and the range of its applicability can be explained only by means of microscopic properties of m aterials. In 30th of the XXth century it was shown by A. H. Cottrell, E. Orowan, M. Pola nyi, J. W. Taylor, that dislocations, line defects in crystalline bodies, are the source of plastic deformation. Since this discovery a new branch of plasticity has been deve loped — crystal plasticity. It began with works of Schmid, Boas, Taylor and yielded importa nt results in the ﬁeld of evolution of plastic anisotropy, textures, cold rolling an d forge techniques of metals. In this Chapter, we present some properties of discrete disl ocations as well as a contin- uum model of these defects. The theory of discrete dislocati ons found an application in modeling of rupture appearing by earthquakes. This applica tion shall be brieﬂy presented at the end of this Chapter. We begin with the formal deﬁnition of the dislocation. There are two possibilities. One of them was proposed by C. Somigliana (1914) and it is base d on the notion of the dislocation line. Another one was introduced by V. Volterra (1907) and it is using a notion of a singular surface on which the displacement vecto ruis discontinuous. Both deﬁnitions yield similar models but they are not equivalent1. In Fig. 8.1. and 8.2. we show a few schematic pictures of the dislocation in a crystal . In the left panel of Fig. 8.1. we show the result of a removal of half-planes of atoms fr om an inﬁnite ideal cubic crystal. In result the upper half-space and the lower half-s pace possess a misﬁt. In order to correct it, in the vicinity of the horizontal cut the lattice constants (distance between lines indicated in the Figure) must be diﬀerent. Thi s yields the existence of inﬁnite straight lines perpendicular to the page in which on e of the atomic half-planes terminates and in their neighborhood the lattice is distort ed. This line defect is called the 1see: introduction to the subject by T. Mura [13]. A detailed d iscussion of the problem can be found inZ. MIoscIsscIsscIascIkscIoscIwscIsscIkscIasc; Self-equilibrated stresses and dislocations (in Polish), in:Technical Mechanics. Vol. IV: Elasticity , M. Sokolowski (ed.), PWN, Warsaw, 1978, 157 158 Dislocations edge dislocation. Obviously, if we try to complete a closed c urve around such a line we have to make diﬀerent number of lattice steps in one directio n than we make backwards. This is seen even better in the cartoon of the screw dislocati on in the right panel of this Figure. The above described construction is demonstrated again in t he Föll cartoons of Fig. 8.2. The right panel shows the combination of the edge and scr ew dislocations in which the vectorbdescribing the misﬁt is neither perpendicular to the disloc ation line (edge dislocation) nor parallel to this line (screw dislocation) . It forms rather an angle of sixty degrees with this line. Obviously, depending on the combina tion, this angle may be arbitrary. Dislocation lines carry both an accumulated energy and self -equilibrated stresses in the reference state of the body. They try to minimize this ene rgy by minimizing the length. This means that in the inﬁnite ideal crystal they for m straight lines. However, in real ﬁnite crystals this is not possible. Consequently, t hey must either terminate on the boundaries or, which is mostly the case in reality, they m ust form closed circuits. This is also the reason for their motion. Under loading — shea ring in the plane of misﬁt, this lines change the curvature and, in attempt to minimize t he length, they are shifted along this plane. Fig. 8.1: Schematic picture of a two-dimensional misﬁt alon g the horizontal line yielding edge dislocation line every 20 steps (left pan el) and a schematic picture of the screw dislocation (right panel) 8.2 Continuum with dislocations 159 Fig. 8.2: Three characteristic types of dislocations: edge (left panel), screw (middle panel) and "sixty degree" (right panel)2 We proceed to the mathematical description of the dislocati on. 8.2 Continuum with dislocations The closed curve Dis said to be the dislocation line (dislocation loop) in a con tinuum if a line integral along any suﬃciently small closed circuit Bcircumventing once the curve Dpossesses the property contintegraldisplay Bdu=b, (8.1) for an arbitrary loading of the continuum. Obviously, uis the displacement vector and b/negationslash=0is called the Burgers vector of the dislocation line D. According to Stokes Theorem (1.59), for diﬀerentiable displacement ﬁeld u(x,t)we can write the integral in the form contintegraldisplay B∂ui ∂xjdxj=integraldisplay SBǫkij∂2ui ∂xi∂xjnkdS=0, (8.2) whereSBis a surface spanned on the curve B. Hence, the deﬁnition (8.1) is nontrivial only for displacement ﬁelds which are discontinuous on the s urfaceSB. This relates the Somigliana deﬁnition of the dislocation to the surface deﬁn ition introduced by Volterra. For energetic reasons bis usually the shortest translation vector of the lattice; e .g. |b|=a/2<110>for the fcc lattice. The assumption that the Burgers contour B is suﬃciently small means that it encloses only one dislocat ion lineDand that it is intersecting only once a surface span by the line D. This is schematically shown in Fig. 8.3.. The sign of the Burgers vector bis deﬁned by the right screw rule shown also in the Fig. 8.3.. 2afterHIescIlscImscIuscItscFIodieresisscIlscIlsc ;Defectsincrystals , http://www.tf.uni-kiel.de/matwis/amat/def_en/index .html 160 Dislocations Fig. 8.3: Dislocation loop D, Burgers contour B and the sign convention One of the surfaces related to the dislocation line is the cyl inder for which the curve Dis the directrix and whose generatrix are straight lines in t he direction of the Burg- ers vectorb. This is the so-called gliding surface along which dislocat ions move most frequently because the resistance of the crystal to such a mo tion is in this surface the smallest (the conservative motion). Another possibility w hich requires much more energy is the so-called climbing of dislocations (the nonconserva tive motion). The latter requires some atomic diﬀusion processes. For details we refer to nume rous books on the subject3. We proceed to describe the geometry of the dislocation line. It is described by the position vector x=ζ(l,t), (8.3) wherelis a parameter along the line. Then the tangent vector and the velocity of dislocation are deﬁned by the relations dζk=∂ζk ∂ldl⇒tk(x,t)=contintegraldisplay Dδ(x−ζ(l,t))dζk, (8.4) ˙ζk=∂ζk ∂t, where the vector ﬁeld t(x,t)is given in the whole continuum but it is diﬀerent from zero only on dislocation lines. In order to describe the continuous ﬁelds in the presence of d islocation we introduce the tensor of distortion βwhich smears out the Burgers condition (8.1). This tensor 3e.g.: J.WIescIescIrscItscImscIascInsc,J.R.WIescIescIrscItscImscIascInsc; Elementary Dislocation Theory, Macmillan, New York, 1967, D.HIuscIlscIlsc;Introduction to Dislocations, Pergamon, 1975, J.D.EIsscIhscIescIlscIbscIysc; Continuum Theory of Lattice Defects, Solid State Physics , vol.3, 79, Academic Press, N.Y., 1956, A.M.KIoscIsscIescIvscIiscIcscIhsc; Crystal Lattice: Photons, Solitons, Dislocations , John Wiley, 1999. 8.2 Continuum with dislocations 161 coincides with the gradient of displacement in simply conne cted domainsPwhich do not contain dislocation loops: P∩D=∅. Hence we deﬁne duj=βijdxi,withβij(x,t)=∂uj ∂xiforx∈P. (8.5) Then the condition (8.1) has the form bj=contintegraldisplay Bβijdxi=integraldisplay SBǫkli∂βij ∂xlnkdS, (8.6) where the second relation follows from the Stokes Theorem fo r an arbitrary surface SB spanned on the curve B. This relation allows to smear out the ﬁeld of distortion. Nam ely, for the dislocation loopDintersecting a surface SBin the pointx, as shown in Fig. 8.4: Fig. 8.4: Orientations of Burgers surfaceSBand dislocation loop D we use the following identity integraldisplay SB contintegraldisplay Dδ(x−ζ)dζkn kdS=  1 −1 0for the same orientation of nanddζ, for the opposite orientation of nanddζ, forDnot intersecting SB. (8.7) Then (8.6) can be written in the form bjintegraldisplay SB contintegraldisplay Dδ(x−ζ)dζkn kdS=integraldisplay SBǫkli∂βij ∂xlnkdS. (8.8) This relation must hold for an arbitrary surface SBspanned on the Burgers contour B. Consequently, ǫkli∂βij ∂xl=bjtk, (8.9) 162 Dislocations where we have used the deﬁnition (8.4) of the vector ttangent to the dislocation loop D. This is the diﬀerential form of the Burgers relation (8.1). As the vectortis diﬀerent from zero only on dislocation lines the above relation state s that the distortion βhas a vector potential beyond the dislocation line ( rotβ=0), i.e.β=graduas we have already mentioned before. The tensor α=b⊗ti.e.αij=bitj, (8.10) is called the tensor of dislocation density. Obviously, it s atisﬁes the relation ǫilk∂βkj ∂xl−αij=0. (8.11) The above considerations allow to write the full set of equat ions which determine the distortionβand other ﬁelds of a linear elastic continuum caused by a give n dislocation line. In the static case the problem is quite simple. We use th e equilibrium condition ∂σij ∂xj=0, (8.12) which follows from (3.38) and the Hooke law (5.21) σij=cijklβkl, cijkl=λδijδkl+µ(δikδjl+δilδjk). (8.13) We have to incorporate the relation (8.11) which describes t he loading by the dislocation. To do so we diﬀerentiate the equilibrium condition and subse quently we substitute (8.13) cijkl∂2βkl ∂xj∂xp=0. Now we multiply (8.11) by ǫiql, and use the identity ǫkplǫkqi∂βij ∂xq=(δpqδil−δqiδqj)∂βij ∂xq=∂βlj ∂xp−∂βpj ∂xl=ǫkplαkj, (8.14) which follows from the contracted epsilon identity (1.41). Finally, we have cijkl∂2βpl ∂xi∂xk=cijklǫqpk∂αql ∂xi. (8.15) Together with the condition ekl=1 2(βkl+βlk), (8.16) the equation (8.15) fully describes the problem. Once we ﬁnd βijwe can ﬁnd stresses from Hooke’s law (8.13). As this ﬁeld of stresses follows onl y from the presence of the dislocation without any external load we say that it is self- equilibrated. Solutions of this equilibrium problem are very important be cause they determine the stress concentration in the vicinity of the line defect. The y can be found by means of the 8.2 Continuum with dislocations 163 Green function of the linear elasticity. We quote here only t he result of W. G. Burgers for the displacement uin the case of a dislocation loop Dwith the constant Burgers vectorb. It has the form uk=−1 4πbkcontintegraldisplay Dǫijlrjkl r(r−riki)dζi+1 4πǫkijbicontintegraldisplay Ddζj r+1 4πλ+µ λ+2µǫijlbj∂ ∂xkcontintegraldisplay Drl rdζi, rk=xk−ζk(l), r=√rkrk, (8.17) wherekis the unit vector perpendicular to the plane of the loop D. Only in exceptional cases one can perform analytically the i ntegration in the above relation. It can be done for the straight line dislocations. In such a case, one obtains, for instance, the following components of stresses 1) screw dislocation given by l= (0,0,1)wherelpoints in the direction of the dislocation line D, andb=(0,0,b) σx=σy=σz=τxy=0, (8.18) τxz=µb 2πy x2+y2, τyz=−µb 2πx x2+y2, 2) edge dislocation given by l=(0,0,1),b=(b,0,0) τxz=τyz=0, σx=b 2π2µ(λ+µ) λ+2µyparenleftbig 3x2+y2parenrightbig (x2+y2)2, σy=−b 2π2µ(λ+µ) λ+2µyparenleftbig x2−y2parenrightbig (x2+y2)2, (8.19) σxz=b 2π2µλ λ+2µy x2+y2, σxy=−b 2π2µ(λ+µ) λ+2µxparenleftbig x2−y2parenrightbig (x2+y2)2. Solutions for dislocations in some anisotropic media can be found in the explicit form as well. We do not present details of the dynamic theory of dislocatio ns4. The Burgers con- dition holds true also in this general case but one has to cope with the problem of elimi- nation of the acceleration term in the equation of motion. On e can derive an additional equation for the evolution of the distortion β ∂βkl ∂t−∂vl ∂xk=Jkl, (8.20) 4some aspects of this theory can be found in the earlier quoted work of Z. Mossakowska as well as: H.ZIoscIrscIsscIkscIisc; Theory of discrete defects, Arch. Mech. Stos., 18, 3, 1966. 164 Dislocations whereJklis the dislocation ﬂux given by the relation Jkl=blǫkijcontintegraldisplay D(t)˙ζiδ(x−ζ(l,t))dζj. (8.21) Some universal solutions are known also in this case but we sh all not quote them in these notes. 8.3 On plasticity of metals As we have mentioned at the beginning of this Chapter, the veh ement research of the continua with dislocations was connected with the discover y that plastic deformations of metals are related to the redistribution and production of d islocations. In low tempera- tures, i.e. temperatures below app. 70% of the temperature o f melting point, dislocations are moving on characteristic crystallographic planes on wh ich they require the least en- ergy for the motion. During this motion they get stacked on bo undaries of grains of polycrystals and on other obstacles. One of them may be a poin t in which more than one dislocation appear simultaneously and their Burgers ve ctors annihilate each other, i.e.summationtext αb(α)= 0, whereαnumbers the dislocations in this point. Such knots play an important role in the production of dislocations. Namely, t he shear stress which acts in the slip plane of motion of a dislocation is bending a line of d islocation pinned to two such obstacles. The loop is trying to minimize the energy whi ch leads to overhanging shown in Fig. 8.5. When the two sides (green in Figure) meet th ey annihilate because their Burgers vectors are identical but of the opposite sign . The loop becomes free to move and the rest of the virginal line of dislocation begins t he process anew. This is the so-called Frank-Read source of dislocations. Fig. 8.5: Frank-Read source of dislocation 8.4 Dislocations in geophysics 165 This and similar mechanisms yield the production of disloca tions which is an irre- versible process related to the increment of plastic strain s. During plastic deformations the number density of dislocations may grow from some 1010to10201/cm2. Fig. 8.6: Electron microscope picture of Frank-Read source. Black traces are dislocations on the surface of the sample Modeling of such processes is based on certain microscopic o bservations transferred to the level of continuum. The fundamental role plays here the s o-called Orovan equation which relates the rate of shearing to the Burgers vector, spe ed of dislocations and dislo- cation density. Together with the evolution equation for th is density in which intensities of sources are incorporated one obtains a semistructural pl asticity model, the so-called crystal plasticity which successfully solved many problem s of mechanics of metals5. In high temperatures the process becomes more complicated b ecause the defects pinned to the grain boundaries begin to move as well. In this r ange the theory of dislocations as presented above cannot be applied anymore. 8.4 Dislocations in geophysics The origin of various models of dislocations goes back to the defective structure of crys- talline bodies such as metals. However, we have seen that the se models describe line defects in continua independently of a particular crystall ine lattice. Only some indica- tions concerning the Burgers vector bear on crystallograph y. Therefore one can apply such models in all cases in which a description of a discontin uous displacement ﬁeld is needed. This is indeed the case in modeling of earthquakes. M ost likely it was A. E. H. Love in 1945 who proposed to apply the Volterra dislocatio n model6in description of earthquakes. The problem of seismic sources was discusse d by Vvedenskaya (1956), Steketee (1958) and others7. The modern presentation of the subject can be found in the book of Aki and Richards [1]. 5for the introduction to crystal plasticity see the book of K. Wilmanski [22]. Many details can be found in the monograph R.W.K.HIoscInscIescIyscIcscIoscImscIbscIesc; The Plastic Deformation of Metals , E. Arnold, 1968, andU.F.KIoscIcscIkscIssc,A.AIrscIgscIoscInsc,M.AIsscIhscIbscIysc; Thermodynamics and Kinetics of Slip, vol.19, Chalmers, B., Christian, J. W. and Massalski, T. W. (eds.), Pergamon Press , Oxford, 1975. 6V.VIoscIlscItscIescIrscIrscIasc, Sur l’équilibre des corps élastiques multiplement connexe s, Ann. Sci. l’École Normale Supérieure, Paris, 24, 401-517, 1907. 7A.V.VIvscIescIdscIescInscIsscIkscIascIyscIasc; Determination of displacement ﬁelds for earthquakes by mea ns of the disloca- tion theory (in Russian), Izv. Akad. Nauk SSSR, Geoﬁz., 3, 277-284, 1956, 166 Dislocations The mechanism of earthquake rupture may be more complicated than this which can be described by the Volterra dislocation. It is related to cr ack formation and it is coupled to complicated tectonic processes which we do not discuss in this book. Reader interested in these problems is referred to an article of J. Rice [16]. We leave out the discussion of the structure of forces acting in the fault — according to Aki , Richard [1] the so-called double couple theory seems to be prevailing, and limit the at tention to modeling a slip in the fault and its action on the vicinity. A fault surface SDwith the boundary ∂SB=Dlies in a linear isotropic medium and it is assumed to be perpendicular to the x3-axis. A slip is presumed to take place in the direction of a unit vector l=l1e1+l2e2. Then the displacement vector possesses a discontinuity ∆u=u+−u−=b=|∆u|l, (8.22) wherebis the Burgers vector for the dislocation line D. Hence, Volterra dislocation allows to take over all results the theory of dislocations in the description of such a fault defect. Strictly speaking, the deﬁnition of Volterra dislocation requires as well that derivatives of uonSDare continuous which means, of course, also the continuity o f the stress. The ﬁeld of displacement created by the dislocat ion yields a system of self- equilibrated stresses and, for this reason, an accompanyin g distortion may be considered as a ﬁeld of initial deformations e0 ijgiven by the relation e0 ij=−bjintegraldisplay SDδ(x−ξ)nidS. (8.23) This relation yields immediately the notion of moment tenso r densitymijgiven by the Hooke’s law for the initial deformation mij=cijkle0 kl, (8.24) and, consequently, a deﬁnition of forces appearing in the eq uation for the real displace- mentu X0 k=−∂mkl ∂xl, ρ∂2ui ∂t2=cijkl∂2uk ∂xl∂xj+X0 i. (8.25) Obviously, the tensor of material parameters for isotropic materialscijklhas the form (5.21). Now, the dynamic Green function (5.64) yields solut ions for the displacement. For instance, for the source which is the Heaviside function bH(t)we obtain the displacement in the far ﬁeld approximation in the following form uL i=M0 4πρc3 Lr(nkll+nllk)xixkxl r3δparenleftbigg t−r cLparenrightbigg , nkek=e3,l=l1e1+l2e2, uS i=M0 4πρc3 Tr(nkll+nllk)parenleftBig δik−xixk r2parenrightBigxl rδparenleftbigg t−r cTparenrightbigg , (8.26) J .A. SItscIescIkscIescItscIescIesc; Some geophysical applications of the theory of dislocation s,Can. J. Phys. ,36, 1168-1198, 1958. Some details can be found in the book of A. Udias: PrinciplesofSeismology , Cambridge Univ. Press., 1999. 8.4 Dislocations in geophysics 167 whereM0=ηb(areaSD)is the seismic moment, ηdenotes the rigidity modulus. These are two arrivals, longitudinal and transversal, in a point w ith the distance rfrom the source. The model indicated above speciﬁes various notions of the ea rthquake such as seismic moment, its decomposition into various forces acting on the plane of the defect including the mentioned above double couples. However, in many respec ts it seems to be too simpliﬁed. For instance, it does not contain any criteria fo r the rupture. We shall not elaborate this subject any further. 168 Dislocations Chapter 9 Appendix: Green functions for isotropic elastic materials 9.1 Purpose Green’s functions known also as fundamental solutions serv e the purpose of construction of analytical solutions of linear diﬀerential equations. T hey also form the basis for at least two important procedures of approximation. The ﬁrst one yie lds the so-called boundary element methods. The second one evaluates average macrosco pic properties of materials with microstructure. The Green function allows to solve the linear equation Lu+f=0, (9.1) whereLis the linear operator, uan unknown vector function and fa given function. For the purpose of this Appendix we assume the operator Lto be of the second order and the domain to be inﬁnite. For ﬁnite domains one can obtain solutions by a simple transformation which we present further. The formal solution of the equation (9.1) in the static case c an be, for instance, written in the form of the following convolution integral1 u=G∗f=integraldisplay G(x−x1)f(x1)dV1, (9.2) whereGis the Green tensor for the static problem. In the following two Sections we present the construction of Green’s functions for static and dynamic problems of linear elasticity. We follow here the presentation of T. D. Shermergor [17]. 1e.g.R.IdscIescWIiscItsc; Continuum theory of stationary dislocations, Solid State Physics, 10,249, 1960. 169 170 Appendix: Green functions for isotropic elastic materi als 9.2 Statics of isotropic elastic materials In this Subsection we present the construction of Green’s fu nction for the linear elasticity in the static case. The operator Lhas in this case the following form L=Likei⊗ek, Lik=∂ ∂xjcijkl∂ ∂xl, (9.3) wherecijklis the tensor of elasticity. For heterogeneous materials it may be dependent on the pointx. We consider only homogeneous materials for which it consis ts of material constants. For isotropic materials it has the following exp licit form cijkl=λδikδjl+2µδijδkl, (9.4) whereλ,µare Lamé constants. Substitution of (9.2) in (9.1) yields L(x)integraldisplay G(x−x1)f(x1)dV1=−f(x). (9.5) Consequently, the Green function must satisfy the equation L(x)G(x−x1)=−1δ(x−x1),1=δijei⊗ej, (9.6) andδ(x−x1)is the Dirac function. Obviously, in the linear elasticity the vector udenotes the displacement. Then the components of the Green function Gij(x−x1)deﬁne the components of displacement ui(x)at the pointxof the inﬁnite medium caused by the unit force acting at the po intx1 in the direction ej. If the medium is ﬁnite the Green function speciﬁes the displ acement uby the relation2 um(x)=integraldisplay VGim(x−x1)fi(x1)dV1+ (9.7) +integraldisplay ∂Vbracketleftbigg ui(x1)cijkl∂Gkm ∂xl(x−x1)+Gim(x−x1)σij(x1)bracketrightbigg njdA1, where∂Vis the surface of the domain Vandniare components of the normal vector of this surface. The form of the Green tensor Gijfollows from the equation (9.6). It is usually found by means of the Fourier integral transform. We have ¯G(k)=integraldisplay G(x)e−ik·xdVx,G(x)=1 8π3integraldisplay ¯G(k)eik·xdVk. (9.8) Application of the Fourier transform to the equation (9.6) f orx1=0(this assumption is immaterial as for the inﬁnite domain we can always shift th e origin of the coordinates to the pointx1), i.e. to the equation cijkl∂2Gkn ∂xj∂xl(x)=−δinδ(x) 2R.IdscIescWIiscItsc; Continuum theory of stationary dislocations, Solid State Physics, 10,249, 1960. 9.2 Statics of isotropic elastic materials 171 yields ¯ζik¯Gkn(k)=δin,¯ζ ik=cijklkjkl. (9.9) This is the set of algebraic relations which can be solved by i nverting the matrix ¯ζ ik. In the case of isotropic materials given by (9.4) it is immediat e. It can be done as well for materials with the hexagonal symmetry3. In general some approximate methods such as the method of perturbation must be applied4,5. For isotropic materials we have ¯ ζik=µk2δik+(λ+µ)kikk, k2=kiki. (9.10) Consequently ¯Gik=¯ζ−1 ik=1 µparenleftbigg1 k2δik−κkikk k2parenrightbigg , κ=λ+µ λ+2µ. (9.11) It remains to invert the Fourier transform. Before we do so le t us quote an identity which follows from considerations of electrostatics. In su ch a case the Maxwell equations describing the electromagnetic ﬁeld reduce to the followin g two equations divE=4πρ,rotE=0, (9.12) whereEthe electric ﬁeld and ρis the electric charge density. The second relation implies the existence of the potential ϕsuch that E=−gradϕ. (9.13) The choice of signs is the matter of tradition. Consequently , the equation determining the potential ϕhas the form of the Poisson equation ∇2ϕ=−4πρ,∇2=∂2 ∂xk∂xk, (9.14) 3see:E. M. LIiscIfscIsscIhscIiscItscIzsc, L. N. RIoscIsscIescInscIzscIwscIescIiscIgsc; O postrojenii tensora Grina dla osnownogo urawnienia teorii uprugosti w sluczaje nieograniczenoj uprugo-aniso tropnoj sredy (in Russian), JETF, 17, 9, 783, 1947, E.KIrscIodieresisscInscIescIrsc; , Das Fundamentalintegral der anisotropen elastischen Diﬀ erentialgleichungen, Z. Phys. , 151, 4, 504, 1958, L. LIescIiscIcscIescIksc; The Green function of the theory of elasticity in an anisotro pic hexagonal medium, Czechosl. J. Phys. ,B19, 6, 799, 1969. 4e.g.T.MIuscIrscIasc;Micromechanics of Defects in Solids , 2nd ed., Martinus-Nijhoﬀ, Dordrecht, 1987. 5e.g. the quotation of the Abstract of the paper of L.J.GIrscIascIysc,D.GIhscIoscIsscIhsc,T.KIascIpscIlscIascInsc; Evaluation of the anisotropic Green’s function in three dimensional elas ticity, Computational Mechanics ,17, 4, 1996: A perturbation expansion technique for approximating the t hree dimensional anisotropic elastic Green’s function is presented. The method employs the usual series for the matrix (I—A)−1to ob- tain an expansion in which the zeroth order term is an isotrop ic fundamental solution. The higher order contributions are expressed as contour integrals of matrix products, and can be directly evaluated with a symbolic manipulation program. A convergence condition i s established for cubic crystals, and it is shown that convergence is enhanced by employing Voigt avera ged isotropic constants to deﬁne the expan- sion point. Example calculations demonstrate that, for mod erately anisotropic materials, employing the ﬁrst few terms in the series provides an accurate solution an d a fast computational algorithm. However, for strongly anisotropic solids, this approach will most li kely not be competitive with the Wilson-Cruse interpolation algorithm. 172 Appendix: Green functions for isotropic elastic materi als where the last relation for the Laplace operator holds for Ca rtesian coordinates. An easy argument based on the balance equation of charge6yields for the charge density ρ=eδ(x)the following solution of (9.14) E=ex r3⇒ϕ=e r, r2=x·x, (9.15) whereeis the electric charge. Obviously, it is the Coulomb law. Now the substitution of ϕin (9.12) yields a representation of the Dirac δ-function by a function regular beyond the pointx=0. We have δ(x)=−1 4π∇2parenleftbigg1 rparenrightbigg . (9.16) We use this identity in the derivation of the Green function. In addition to the above identity we have ∇2r=∂2r ∂xk∂xk=∂ ∂xkparenleftBigxk rparenrightBig =2 r. (9.17) Hence, bearing (9.16) in mind, −8πδ(x)=∂4r ∂xk∂xk∂xl∂xl. (9.18) We apply to this relation the Fourier transform. It follows integraldisplay∂4r ∂xk∂xk∂xl∂xle−ik·xdVx= (9.19) =integraldisplay∂ ∂xkparenleftbigg∂3r ∂xk∂xl∂xle−ik·xparenrightbigg dVx+ikkintegraldisplay∂3r ∂xk∂xl∂xle−ik·xdVx=... ...=k4integraldisplay re−ik·xdVx, where we have used the Gauss divergence theorem and accounte d for the fact that surface integrals must vanish for the inﬁnite domain. Application o f the inverse transform yields r=−1 π2integraldisplayeik·x k4dVk. (9.20) Diﬀerentiation of this relation leads to the following iden tities ∂2r ∂xk∂xl=1 π2integraldisplaykkkl k4eik·xdVk,∇2r=1 π2integraldisplayeik·x k2dVk. (9.21) Now we are in the position to ﬁnd the Fourier inverse of the rel ation (9.11). Bearing (9.8) in mind we obtain immediately Gik(x)=1 8πµparenleftbigg δik∇2r−κ∂2r ∂xi∂xkparenrightbigg = (9.22) =2−κ 8πµparenleftbiggδik r+κ 2−κxixk rparenrightbigg , κ=λ+µ λ+2µ≡1 2(1−ν). 6e.g.L.D.LIascInscIdscIascIusc,E.M.LIiscIfscIsscIhscIiscItscIzsc; Course of Theoretical Physic , vol. 2:The Classical Theory of Fields , 4th ed., Oxford, Butterworth-Heinemann, 1980. 9.3 Dynamic Green function for isotropic elastic materials 173 This is the Green function for static equations of the linear isotropic elasticity. 9.3 Dynamic Green function for isotropic elastic ma- terials In the dynamic case we have to include the inertial force in th e momentum balance equation. Hence for the linear elasticity the relation (9.3 ) for the operator Lmust be replaced by the following one L=Likei⊗ek, Lik=−δikρ∂2 ∂t2+∂ ∂xjcijkl∂ ∂xl, (9.23) whereρis the constant mass density. For isotropic materials deﬁned by (9.4), we have Lik=−−δikρ∂2 ∂t2+(λ+µ)∂2 ∂xi∂xk+µ∂2 ∂xl∂xlδik. (9.24) Green’s tensor for the above operator will be sought again by the Fourier transfor- mation. In the dynamic case we have to perform also the transf ormation with respect to time. To this aim we could use Laplace transform or, as we do below, we have to cut the Fourier transform to the range of nonnegative time. This reﬂects the principle of causality (determinism) of classical mechanics. Conseque ntly, the Green function has to satisfy the following equations LilGlj(x,t) =δijδ(x)δ(t)fort≥0, (9.25) Gij= 0 fort<0. Obviously, the Green function Gijdetermines the displacement ui(x,t)in the instant of timetand at the point xcaused by the unit force acting in the ej-direction in the instant of time t= 0at the pointx=0. Causality physically means that the displacement cannot be caused by incoming waves which shoul d not exist yet before the force was applied at the point x=0. As before, the displacement ui(x,t)for an arbitrary given external force fj(x,t)is then speciﬁed by the convolution integral ui=Gil∗fj. (9.26) Obviously the time integration in (9.25) yields immediatel y that the static Green functionGij(x), calculated in the previous Section should satisfy the rela tion Gij(x)=∞integraldisplay −∞Gij(x,t)dt. (9.27) 174 Appendix: Green functions for isotropic elastic materi als As already mentioned we ﬁnd the Green function for the inﬁnit e medium by the double Fourier transform ¯G(k,ω) =integraldisplay integraldisplay G(x,t)ei(k·x−ωt)dVxdt, (9.28) G(x,t) =1 16π4integraldisplay integraldisplay ¯G(k,ω)e−i(k·x−ωt)dVkdω. We callkthe wave vector and the scalar ωthe frequency. Then after the Fourier trans- form the operator Lij(x,t)for isotropic materials has the following form ¯Lij(k,ω)=ρω2δij−(λ+µ)kikj−µk2δij, (9.29) and the equation (9.25) becomes purely algebraic ¯Lik¯Gkj=−δij. (9.30) We have to invert the matrix (9.29). Hence, after easy calcul ations the Fourier trans- form of the dynamic Green function is as follows ¯Gij=1 µk2−ρω2bracketleftbigg δij−(λ+µ)kikj (λ+2µ)k2−ρω2bracketrightbigg . (9.31) Obviously, the static transform of the Green function (9.11 ) follows from (9.31) by the substitution ω=0. It is convenient to write the above relation by means of the sp eeds of propagation of longitudinal and transversal waves in a linear elastic mate rial c2 L=λ+2µ ρ, c2 T=µ ρ. (9.32) We obtain ρ¯Gij=1 c2 Lk2−ω2bracketleftBigg δij−parenleftbig c2 L−c2Tparenrightbig kikj c2 Lk2−ω2bracketrightBigg . (9.33) The inverse of the above relation is given by the double integ ration prescribed by (9.28)2. We perform ﬁrst the integration with respect to the wave vec tork. We have Gij(x,ω)=1 8π3integraldisplay ¯Gij(k,ω)e−ik·xdVk. (9.34) Hence Gij(x,ω)=I0δij−∂I ∂xi∂xj, (9.35) where I0=1 8π3integraldisplaye−ik·x c2 Tk2−ω2dVk=1 4πrc2 Texpparenleftbigg −iωr cTparenrightbigg , I=−c2 L−c2T 8π3integraldisplaye−ik·x (c2 Lk2−ω2)(c2Tk2−ω2)dVk (9.36) =1 4πrω2bracketleftbigg expparenleftbigg −iωr cLparenrightbigg −expparenleftbigg −iωr cTparenrightbiggbracketrightbigg . 9.3 Dynamic Green function for isotropic elastic materials 175 Calculations of integrals I0andIare made using the method of residua for complex func- tions. In order to use this method we assume formally that bot h speeds of propagation are complex. This would indeed be the case for viscoelastic m aterials. We demonstrate the calculations on the example of the integral I0. Then c2 T(ω)=Rec2T(ω)+iImc2T(ω)=[1+iβ(ω)]Rec2T(ω). (9.37) This extension yields the following form of the integral I0 I0=1 8π3integraldisplaye−ik·x c2 T(ω)k2−ω2dVk= =1 8π3integraldisplay∞ 0integraldisplayπ 0eikrcosθ2πsinθk2 c2 T(ω)k2−ω2dkdθ= (9.38) =1 4π2irintegraldisplay∞ 0kparenleftbigeikr−e−ikrparenrightbig c2 T(ω)k2−ω2dk=1 4π2irintegraldisplay∞ −∞keikr c2 T(ω)k2−ω2dk. This integral can be evaluated by the method of residua. Obvi ously, it possesses two poles k=±ω {[1+iβ(ω)]Rec2 T(ω)}0.5=±ωexpbracketleftbig−i 2arctanβ(ω)bracketrightbig braceleftbiggradicalBig 1+β2(ω)Rec2 T(ω)bracerightbigg0.5. (9.39) The pole with the minus sign lies in the second quadrant of the complex plane while the other pole lies in the fourth quadrant. Hence, we choose as th e path of integration the real axis and the semicircle of the inﬁnite radius in the uppe r part of the complex plane. We obtain I0=1 4πrc2 Texpbracketleftbigg −irω cT(ω)bracketrightbigg . (9.40) The transition cT(ω)→cTgives the desired result. In a similar manner one can calcula te the integral I. We have to use obvious identities when diﬀerentiating in (9 .35) ∂r ∂xi=ni,∂2r ∂xi∂xj=1 r(δij−ninj), nini=1. (9.41) It follows Gij(x,ω)=1 r[h(ωr)δij+g(ωr)ninj], (9.42) where h(ωr) =1 4πr2ρω2braceleftbiggbracketleftbiggparenleftbigg 1+irω cLparenrightbigg e−iωr/cL−parenleftbigg 1+irω cTparenrightbigg e−iωr/cTbracketrightbigg +r2ω2 c2 Te−iωr/cTbracerightbigg , g(ωr) =−1 4πr2ρω2braceleftbiggbracketleftbigg 3parenleftbigg 1+irω cLparenrightbigg −−r2ω2 c2 Lbracketrightbigg e−iωr/cL− (9.43) −bracketleftbigg 3parenleftbigg 1+irω cTparenrightbigg −r2ω2 c2 Tbracketrightbigg e−iωr/cTbracerightbigg 176 Appendix: Green functions for isotropic elastic materi als This is, obviously, the Green function for monochromatic wa ves of the given frequency ω. The time dependence of solution is then given by the factor exp(iωt). It remains to perform the second inverse transformation. We use here the following relations 1 2πintegraldisplay∞ −∞eiωtdω=δ(t), 1 2πintegraldisplay∞ −∞1 iωeiωtdω=H(t)=braceleftbigg1fort>0, 0fort<0, −1 2πintegraldisplay∞ −∞1 ω2eiωtdω= Ψ(t)=braceleftbiggtfort>0, 0fort<0,(9.44) ∂Ψ ∂t=H(t),∂H ∂t=δ(t). Bearing these relation in mind as well as ∂2 ∂xi∂xjparenleftbigg1 rparenrightbigg =3ninj−δij r3, (9.45) −1 2πintegraldisplay∞ −∞1 ω2parenleftbigg 1+irω cLparenrightbigg eiω(t−r/cL)dω= Ψparenleftbigg t−r cLparenrightbigg +r cLHparenleftbigg t−r cLparenrightbigg , and similarly for cT, we ﬁnally obtain the dynamic Green function 4πρGij(x,t)=δparenleftbigg t−r cTparenrightbiggparenleftbiggδij c2 Tr−xixj c2 Tr3parenrightbigg + +δparenleftbigg t−r cLparenrightbiggxixj c2 Lr3+ (9.46) +∂2 ∂xi∂xjparenleftbigg1 rparenrightbigg tparenleftbigg Hparenleftbigg t−r cLparenrightbigg −Hparenleftbigg t−r cTparenrightbiggparenrightbigg . The ﬁrst contribution describes the transversal part of the impulse which arrives with the speedcT, the second contribution is the longitudinal part of the imp ulse which arrives with the speed cLand the third contribution is the evolution of the impulse be tween the arrival of the longitudinal and transversal parts. Easy integration in the relation (9.27) yields the static Gr een function given by the relation (9.22). Bibliography [1]K. AIkscIisc, P. G. RIiscIcscIhscIascIrscIdscIssc; Quantitative Seismology , University Science Books, Sausalito, 2002. [2]RIoscImscIescIsscIhsc C. BIascItscIrscIasc; Elements of Continuum Mechanics , AIAA Education Series, Reston, 2006. [3]R.M.CIhscIrscIiscIsscItscIescInscIsscIescInsc; Theory of Viscoelasticity. An Introduction , Academic Press, N. Y., 1971. [4]M.E.GIuscIrscItscIiscInsc; TheLinearTheoryofElasticity , in: C. Truesdell (ed.), Encyclopedia of Physics, vol. VIa/2 (Mechanics of Solids), Springer, Ber lin, 1972. [5]K. 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[12]Y.A.MIescIlscInscIiscIkscIoscIvsc; InﬂuenceFunctionApproach: SelectedTopicsofStructural Me- chanics, WIT Press, 2008. [13]T.MIuscIrscIasc; The continuum theory of dislocations, Adv.Mater.Res., 3, 1, 1968. [14]R.W.OIgscIdscIescInsc; Non-LinearElasticDeformations , Dover, Mineola, N. Y., 1984. 177 178 BIBLIOGRAPHY [15]A.C.PIiscIpscIkscIiscInsc; LecturesonViscoelasticTheory , Springer, New York, 1972. [16]J. R. RIiscIcscIesc; The Mechanics of Earthquake Rupture, in: Physics of the Earth’s Interior , A. M. Dziewonski, E. Boschi (eds.), North Holland, 1980. [17]T. D. SIhscIescIrscImscIescIrscIgscIoscIrsc; Teoria uprugosti mikroneodnorodnych sred, (in Russian), "Nauka", Moscow, 1977. [18]S.TIiscImscIoscIsscIhscIescInscIkscIosc,J.N.GIoscIoscIdscIiscIescIrsc; TheoryofElasticity , McGraw-Hill, N. Y., 1951. [19]A.VIescIrscIrscIuscIiscIjscItsc; SoilMechanics, Delft University of Technology, 2006. [20]H. F. 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Index acoustic waves, 81 Airy function, 91 Airy phase, 104 Almansi-Hamel measure, 31 amplitude of wave, 97 angle of repose, 149 anisotropic hardening, seekinematical hard- ening anisotropic material, 73 Armstrong-Frederick law, 154 associated ﬂow rule, 145 back-stress, 148 balance of internal energy, 62 Bauschinger eﬀect, 148 Beltrami-Michell stress equations, 89 Bernoulli Theorems, 50 biharmonic function, 90 Biot moduli, 112 Biot-Willis coeﬃcient, 112 body force, 48 Boltzmann integral, 124 Boltzmann superposition principle, 124 boundary conditions, 80 Boussinesq problem, 86 Boussinesq representation, 84 Boussinesq-Somigliana-Galerkin solution, 85 brittle materials, 137 bulk modulus, seecompressibility modulus bulk waves, 99 Burgers vector, 159 Cauchy equation, 50 Cauchy relation, 50 Cauchy stress tensor, 49 Cerrutti problem, 88Christoﬀel symbols, 78 Clausius-Duhem inequality, 65 cohesion, 149 cohesion yield stress, 152 Coleman’s method, 122 compatibility condition, 41 compatibility equation, 41 compliance, 74 components of tensor, 13 compressibility modulus, 74 compressible Mooney-Rivlin material, 71 concentration of stresses, 94 consistency parameter, 146 consolidation of soils, 150 continuity of tractions, 51 continuity relation, 45 contracted epsilon identity, 15 contravariant base vector, 77 convolution integral, 123 coordinates of vector, 11 covariant base vector, 77 covariant derivative, 78 creep, 117 creep functions, 125 current conﬁguration, 21 curvilinear coordinates, 77 cylindrical coordinates, 78 d’Alambert solution, 82 damping, 97 damping factor, 123 Darcy’s law, 113 dashpot, 119 deformation gradient, 24 relative, 33 determinant, 15 179 180 INDEX diﬀusion, 112 diﬀusion equation, 111 diﬀusivity, 113 Dirac function, 83, 170 direction of propagation, 97 discontinuity of mass density, 47 dislocation density, 162 dislocation ﬂux, 164 dislocation line, 159 dislocation loop, seedislocation line dispersion relation, 98 displacement approach, 72 displacement vector, 30 dissipation, 109 distortion, 160 distortion energy, 140 Drucker stability postulate, 145 Drucker’s postulate, 136 Drucker-Prager yield surface, 152 dry granular materials, seefrictional mate- rials ductile materials, 137 dynamic compatibility conditions, 51 dynamic Green function, 174, 176 earthquake rupture, 166 edge dislocation, 158, 163 eﬀective stress, 150 eﬀective stress , seeequivalent stress eigenvalue, 16 eigenvalue problem, 15 eigenvector, 16 Einstein convention, 11 elastic domain, 136 elastic material, 69 elastic-viscoelastic correspondence principle, 131 elasticities,seeresponse coeﬃcient elasticity modulus, seeYoung modulus energy conservation, 61 entropy ﬂux, 65 entropy function, 65 entropy production, 65 equivalent plastic strain, 143 equivalent stress, 139Euler-Piola-Jacobi identities, 35 Eulerian coordinates, 23 evolution equations, 128 exterior product, seevector product far ﬁeld approximation, 83 ﬁrst law of thermodynamics, 63 Flamant problem, 92 Fourier integral transform, 122, 170 Fourier relation of heat conduction, 66 Fourier’s law, seeFourier relation of heat conduction Frank-Read source of dislocations, 164 frequency of wave, 97 friction angle, seeangle of respose frictional materials, 149 function of motion, 23 Galerkin function, 90 Gauss Theorem, 18 generalized Kelvin model, 121 generalized Maxwell model, 121 Gibbs equation for ideal ﬂuids, 68 Gibbs equation of linear thermoelasticity, 109 Green function, 82, 169 Green tensor for static problem, 169 Green-St. Venant measure, 31 group velocity, 104 Gurtin Theorem, 38 Hadamard Theorem, 47, 95 Haigh—Westergaard coordinates, 151 Hamilton principle, 107 hardening function, 147 hardening parameter, 146 harmonic function, 84 heat conductivity, 66 heat ﬂux vector, 62 Heaviside function, 83 Helmholtz decomposition, 17 Helmholtz free energy, 66, 67, 74, 108 Hooke law, 73 Huber-Mises-Hencky surface, 140 hyperbolicity, 81 INDEX 181 hypoplasticity models, 153 ideal ﬂuid, 66 Ilyushyn postulate, 136 immiscible mixture, 111 incompressible material, 76 inﬂuence function, 82 integrability conditions, 35 internal energy density, 62 intrinsic permeability, 113 irrotational ﬂow, seepotential ﬂow isotropic compliance matrix, 74 isotropy of the material, 70 Kelvin model, 120 kinematical hardening, 148 kinetic energy, 62 Kirchhoﬀ modulus, seeshear modulus Kirchhoﬀ Theorem, 38 knot, 164 Kronecker delta, 10 Lagrange-Euler equations, 107 Lagrangian, 107 Lagrangian coordinates, 23 Lamé constants, 73 Lamé equations, seeNavier-Cauchy equa- tions Laplace operator, 79 Laplace operator , 80 Laplace transform, 122 latent heat, 64 left Cauchy-Green deformation tensor, 29 left stretch tensor, 29 Leibniz Theorem, 18 Levi-Civita symbol, seepermutation sym- bol Lie derivative, 35 linear isotropic elastic material, 73 linear thermal expansion coeﬃcient, 110 load-carrying capacity, 142 local rotation, 26 Lode angle, 151 Lode parameter, 141 longitudinal wave, 82, 96loss modulus, 130 Love dispersion relation, 103 Love waves, 102 mass conservation, 44 mass density, 44 material body, 21 material domain, 44 material time derivative, 45 material vector, 24 maximum shear stresses, 59 Maxwell construction, 66 Maxwell’s model, 119 membrane with cavity, 92 memory eﬀect, 118 metric tensor, 77 mobility, 113 modes, 103 Mohr’s circle, 58 Mohr-Coulomb law, 149 moment of momentum, 51 moment tensor density, 166 momentum, 48 momentum conservation law, 49 monochromatic wave, 97 Navier-Cauchy equations, 77 near ﬁeld approximation, 83 non-associated plastic ﬂow potential, 152 non-Newtonian ﬂuid, seeviscoelastic ﬂuid normal stresses, 49 Norton power law, 154 Odqvist parameter, 147 Orovan equation, 165 orthogonal tensor, 14 orthotropic material, 73 Papkovich-Neuber potentials, 84, 91 permutation symbol, 15 phase, 97 phase equilibrium line, 66 phase shift, 97 phase speed, 97 physical components, 77, 78 Piola-Kirchhof stress tensor, 48 182 INDEX plane strains, 57, 90 plane stresses, 57, 91 plane wave, 97 plastic strain, 136 plastic working, 145 Poisson number, 75 polar decomposition, 26 pore pressure, 111, 112, 150 pore spaces, 111 poroelastic expansion coeﬃcient, 112 porosity, 112 porous material, 113 potential energy, 106 potential ﬂows, 50 Prager consistency condition, 144 Prandl-Reuss equation, 146 pressure, 50 pressure function, 50 principal direction of stress tensor, 53 principal invariants, 16, 53 principal stretch, 31 principal values of stress tensor, 53 principle of virtual work, 105 pure strain, 39 radiation, 62 rate-type constitutive relations, 128 Rayleigh waves, 100 reference conﬁguration, 22 relaxation functions, 124 relaxation time, 115 Rendulic planes, 151 Representative Elementary Volumes, 112 residual inequality, 67 response coeﬃcient, 71 rheological elements, 119 right Cauchy-Green deformation tensor, 29 right stretch tensor, 26 rigid body rotation, 36 rigid displacement, 37 rigid heat conductor, 111 Rivlin-Ericksen ﬂuids, 129 rotation vector, 39 rule of transformation, 11, 14scalar potential, 18 scalar product, 10 screw dislocation, 158, 163 second law of thermodynamics, 65 seepage velocity, 112 seismic waves, 104 self-equilibrated stresses, 158, 162 separation of variables, 132 shear modulus, 74 shear stresses, 49 shear wave,seetransversal wave simple extension, 39 simple shear, 40 simple shearing, 25 singular surface, 46 Skempton coeﬃcient, 112 small deformation, 31 solid skeleton, 111 Somigliana dislocation, 157 Sommerfeld condition, 99 speciﬁc enthalpy, 64 speciﬁc entropy, 65 speciﬁc heat, 109 spectral representation, 17, 98 spherical coordinates, 79 spin, 34 spring, 119 square-cube law, 6 standard linear viscoelastic solid, 129 standard rheological model, 120 Stokes Theorem, 18 storage modulus, 130 strain ﬁeld, 36, 40 strain space formulation, 136 stress approach, 72 stress relaxation, 117 stress space formulation, 136 stress vector, seetraction stretching, 34 supply of energy, 62 surface waves, 99 technical frequency, 98 tensor of the second rank, 13 tensor product, 13 INDEX 183 thermal conductivity, 108 thermodynamical potential, 67 thermoelastic materials, 108 traction, 48, 50 trajectory of motion, 32 transversal wave, 82, 96 transversely isotropic material, 73 Treﬀtz potential, 85 Tresca-Guest surface, 139 unconstrained speciﬁc storage coeﬃcient, 112 undrained bulk modulus, 113 uniform dilatation, 39 uniform extension, 24 variation in water content, 112 vector potential, 18 vector product, 15 vector space, 9 dimension, 10 velocity, 32 velocity gradient, 34 virtual displacement, 105 viscoelastic ﬂuid, 117 viscoelasticity of solids, 117 Voigt notation, 72 Volterra dislocation, 157 volumetric thermal expansion coeﬃcient, 110 wave length, 98 wave number, 97 waves of weak discontinuity, seeacoustic waves working of body forces, 62 yield function, 139 yield limit,seeyield surface yield locus,seeyield surface yield surface, 136, 139 Young modulus, 75