continuum mechanics Los Alamos

Finite Deformation of Materials with an Ensemble of DefectsLA-13994-MS Approved for public release;distribution is unlimited. This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the Regents of the University of California, the United States Government norany agency thereof, nor any of their employees make any warranty, express or implied, or assume any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represent that its use would not infringe privately ownedrights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the Regents of the University of California, the United StatesGovernment, or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the Regents of the University of California, the United States Government, or any agency thereof. Los Alamos National Laboratory strongly supports academicfreedom and a researcher's right to publish; as an institution, however, the Laboratory does not endorse the viewpoint of a publication or guarantee its technical correctness.Los Alamos National Laboratory, an affirmative action/equal opportunity employer, is operated by the University of California for the United States Department of Energy under contract W-7405-ENG-36.The previous report in this unclassified series is LA-11063-MS, Vol. I. Finite Deformation of Materials with an Ensemble of Defects John K. DienesLA-13994-MS Issued: January 2003 vCONTENTS ABSTRACT .......................................................................................................1 1. INTRODUCTION AND SUMMARY ....................................................................2 2. POLAR DECOMPOSITION ...............................................................................5 3. RELATIONS BETWEEN VARIOUS RATES .........................................................8 4. ENERGETICS AND EQUILIBRIUM ELASTICITY ...............................................14 5. SIMPLE SHEAR ............................................................................................18 6. RIGID-BODY MOTION ..................................................................................24 7. TORSION ....................................................................................................25 8. SIMPLE VORTEX .........................................................................................30 9. STRESS RATE IN CONSTITUTIVE RELATIONS ................................................32 10. SUPERPOSITION OF STRAIN RATES .............................................................35 11. SUPERPOSITION WITH TIME-DEPENDENT STATISTICS ..................................41 12. CONCLUSIONS AND CLOSURE ....................................................................42 APPENDIX A. MATRIX THEORY .......................................................................45 APPENDIX B. REMARKS ON THE MEASURE OF STRAIN .....................................57 APPENDIX C. REMARKS ON THE TENSOR CHARACTER OF STRAIN ....................61 APPENDIX D. REMARKS ON ELASTIC CONSTITUTIVE LAWS ..............................67 APPENDIX E. POLAR RATE OF ROTATION ........................................................69 REFERENCES ..................................................................................................73 1Finite Deformation of Materials with an Ensemble of Defects John K. Dienes ABSTRACT The theory of large deformations developed here is closely related to continuum mechanics but it differs in several major respects, especially in considering the deformation associated with various types of physical behavior, making it possible to synthesize a general approach to formulating constitutive laws. One goal is to derivegeneral concepts of strain, strain rate, stress, and stress rate that are somewhat more physics-based than in most standard works on continuum mechanics, and to demonstrate some new relations between these quantities. With these concepts it is possible to developa generalized principle of superposition of strain rates (GSSR) that accounts for damage as well as plastic flow. The traditional superposition of strain rates allows for addition of elastic and plastic strain rates and is commonly thought to be valid only for small strains.The GSSR allows us to compute deformations involving plastic flow and, in addition, brittle failure, fragmentation, high-pressure effects and other types of behavior as necessary, and the theory is valid for arbitrarily large deformations. In fact, GSSR isderived from more basic ideas and has broader application than the standard superposition of strain rates. The physical basis for calculations of complex material response is developed in a separate report. The implementation into the SCRAM computer program isdocumented separately. The polar decomposition theorem is taken as a starting point for the theory of large deformation, an approach somewhat different from that usually taken in continuummechanics. Two sets of orthogonal axes are distinguished, space axes that are fixed in ambient space, and polar axes that are related to material deformation. This clarifies several concepts; for example, it is shown that the Signorini and Green-St. Venant strainsare actually measures of the same physical entity, one in space axes and the other in polar axes. It follows that they are not competing measures, as is often implied in traditional continuum mechanics. It also follows that Piola stress is a measure in polar axes, while Cauchy stress is a measure in space axes. Another consequence of polar decomposition is a proof that vorticity is not a measure of the rate of material rotation (as is often stated in the hydrodynamics literature) but that they are related. This allows us to develop an exactapproach to computing rates of tensor quantities, called polar rates, that account for material rotation in an exact way. This leads to a simple relation between Signorini strain rate and stretching (the symmetric part of the velocity gradient). It also follows that thepolar stress rate is the appropriate measure for the rate of change of Cauchy stress, and that the more traditional stress rate of Zaremba, Jaumann, and Noll is only an approximation, valid at small strains. Examples are described for materials undergoing simple shear,vortex motion, and torsion. _______________________ 21. INTRODUCTION AND SUMMARY Though the mathematical theory of deformation has been studied extensively, the early work assumed continuity of deformation and, in this respect, failed to model real materials, which contain voids, cracks, grain boundaries, dislocations, disclinations, twins, shear bands, inclusions, and combinations of these and other defects that arise in innumerable ways. A completely satisfactoryapproach to understanding deformation in the presence of real defects is beyond our abilities, but progress toward analyzing the micromechanics of plastic flow, failure, and fragmentation is possible with our current knowledge, and is necessary if we are to take full advantage of the massivecomputing resources that are becoming available. This monograph is intended to bridge the gap between the classical theory of deformation and a statistical theory of imperfect materials. The computer program in which these ideas are being implemented is SCRAM. The theory in this volume draws heavily on ideas from continuum mechanics, including, especially, use of the polar decomposition theorem to determine the extent of rotation in a general deformation. (M athematical notation, definitions and theorems are described in Appendix A.) Various definitions of strain and stress, and the general theory of elastic deformation as a process that depends only on the final state of a material, are taken from continuum mechanics. The emphasis in this work is different from continuum mechanics, however, for the computed deformation is thoughtof as the average of an ensemble of deformations with different but statistically similar defects. This difference is not emphasized until Section 10, when the principle of Superposition of Strain Rates is introduced, but it is the main subject of Volume II. In addition to these topics, we discuss at lengthmaterial rotation and its rate, which have been imperfectly treated in the literature of continuum mechanics. Whereas many theorists take the view that many different rates that account for material rotation are valid, here we consider such a view to be physically unreasonable, and argue that goodmeasurements would measure only one rate, which is the polar rate. In addition, we do not allow various definitions of strain on the same grounds. The view taken here is that the physical strain is unique, and that Almansi, Green, and Signorini strains are merely different representations of thesame physical quantity. Other definitions of strain have no interest. This view is woefully undemocratic, but in writing a computer program to represent physical deformation a definite choice must be made. Furthermore, the physical basis for this view seems compelling. The relation to Hill’sgeneral classification is given in Appendix B. With regard to polar decomposition, the usual emphasis and terminology are somewhat modified. In standard works the deformation gradient is factored with a stretch matrix written either on the rightor left of the rotation matrix, and, hence, the stretches are called right and left stretches. Here, the approach is physical rather than typographical, and the stretches have reference axes either fixed in space (left stretch) or rotating with the material (right stretch). Other quantities such as stress andstrain can also be referred to either spatial or material axes by a suitable rotation. This distinction clarifies many concepts in the theory of deformation. For example, it becomes clear that Signorini strain and Green strain are related by a rotation from one set of axes to the other, and that Piola stressis defined in polar, not spatial, axes. Rate equations describing deformation can be formulated by differentiating the expression for polar decomposition. It is shown that the rate of polar rotation, ΩΩ, is different, in general, from 3vorticity, W, though they can be made to coincide at any particular instant by a suitable choice of reference time (i.e., choosing reference time as current time). It is also shown that vorticity is not an adequate measure of the rate of rotation for solids. In computing rates of matrix quantities such as stress, strain, and compliance, it is necessary to accountfor the rate of material rotation when the reference axes are fixed in space, but this is not necessary if polar axes are chosen. Both approaches and their relationship are discussed in detail. Rates of change of matrices referred to space axes are called polar rates here, since the analysis uses polardecomposition as its starting point. 1 If polar axes are used in computing the stress, conversions must be made from spatial axes. A natural definition of strain εε in the current context is that of Signorini, though the motivation is quite different from his. It turns out that its polar rate ˆεε is proportional (in a certain sense) to the stretching, D (the symmetric part of the velocity gradient, using Truesdell’s terminology). The stretching is not the rate of change of any path-independent strain, though it is often called strain rate in the older literature. When D is known, a unique strain can be found by integrating ˆεε=VDV, in which the carat accounts for the polar rate of rotation, as discussed in the body of this report. It is shown in Section 10 that D can be written as the sum of several parts, each one concerned with a separate physical process, such as elastic deformation of the material matrix, plastic flow, opening of voids, and shearing of closed cracks. In view of the proportionality of Signorini strain rate and stretching, the Signorini strain rate can also be written as the sum of parts in the same way. ThisSuperposition of Strain Rates goes back to Reuss, but the current treatment is motivated by microphysical considerations rather than by the macroscopic consideration that motivated Reuss, and the application is more general, accounting for crack growth and nonlinear elasticity. The tensor character of strain and stretching is considered at some length in Appendix C. The tensor concept is used here in a more restrictive sense than in continuum mechanics. For many authors (especially mathematicians), quantities that transform according to the appropriate mathematical laws in any space are considered tensors. Here, however, we follow the view common in physics that only quantities that transform like tensors in physical space are considered admissible. It is shown that the usual Green strain 2 is not a tensor in this restricted sense, though Signorini strain is. The selection of a suitable measure of strain is important for several reasons. First, in formulating constitutive laws it is important to adopt relations between appropriate physical quantities. An example of an inappropriate formulation is the use of the deformation gradient F as a measure of distortion, since it involves material rotation as well as strain. (This is discussed in Appendix D.) In the analysis of isotropic elastic bodies, it turns out that an analysis using F may not actually be erroneous, because the constitutive laws are formulated in such a way that the rotation disappearsfrom the constitutive laws, but this approach is misleading since it will not work for anisotropic or 1 The names of various tensor quantities vary widely. In a previous version of this report the variables in the frame obtained by rotating to polar axes were called “material quantities” since they tend to follow material rotation. On reflection, however, this is co nsidered a poor choice because axes fixed in a material do not normally remain orthogonal, especially under extreme shear, while polar axe s do. Hence, the rotated axes are called “polar axes” in this revised report. Thus, we deal with spatial and polar axes. These are called Eulerian (or Eulerean) by some authors, but this practice is considered confusing since it involves a rather poor analogy to th e choice of Eulerian and Lagrangean independent variables in fluid dynamics. These terms, in turn, do not reflect the contributions of Eule r and Lagrange, as discussed at length by Truesdell. (Truesdell attributes the Lagrangean description to Euler and the Eulerian descr iption to d’Alembert.) 2 This is frequently called the right Cauchy-Green tensor, but in the interest of brevity it is called the Green strain here. 4inelastic materials. It is preferable to select variables that measure physical behavior directly in the first place, and to adopt an approach that promotes physical thinking rather than abstract formalism. A second reason for narrowing down the choice of variables in constitutive laws concerns computational methods, which involve much of our time and resources nowadays. Computerprograms can be written involving functional relations between any measures of strain, stress, strain rate and stress rate, but it would be unreasonable to suppose that all measures are equally valid and useful. The premise in the current approach is that a combination of mathematical and physicalanalysis can lead us to a good selection of variables without resorting to experiment or comparisons between computational schemes. The plan of this monograph is to develop a self-contained analysis of finite deformation. The theory of polar decomposition is discussed in Section 2. Rates arising from polar decomposition are considered in Section 3, including an important explicit relation between the vorticity W and the polar rate of rotation, ΩΩ. In Section 4, the form of the most general constitutive law for an elastic deformation is derived. The next four sections give illustrations of the general theory, comparing especially the spatial and polar representations of deformation. Simple shear, discussed in Section 5, involves displacement in a fixed direction, and is a common example of deformation in continuummechanics. In practice it arises in boundary layers of fluids, in shear bands in solids, and in shear cracks under some conditions. It constitutes an interesting example of finite deformation because both the shear and rotation can be very large. Rigid body motion, Section 6, is trivial, but it is of some interest to see how the theory of deformation reduces to rigid-body motion as a limiting case. Torsion, characterized in Section 7, is quite complex. The stretch and rotation matrices are computed exactly (for the first time, I believe), as well as the strain and strain rate. The full characterization ofstress and strain given here is necessary to interpret experiments with large torsion exactly, but no such effort is currently in progress insofar as I know. Though the simple vortex is observed in fluids rather than solids, it illustrates nicely the differences between vorticity, which is zero for an idealvortex, and rate of polar rotation, which does not vanish. This is discussed in Section 8. The formulation of constitutive laws is considered next, and it is shown in Section 9 that the general form involves the polar stress rate. The argument is similar to Noll’s (1955), but by means ofa more general choice of reference axes it is shown that it is the polar stress rate that is appropriate in general, not the more traditional stress rate of Z aremba (1903), Jaumann (1911), and Noll. (Noll assumes that the reference time is the current time.) In this line of argument, the form of the stressrate is derived. This approach is considered stronger and more compelling physically than the usual approach of testing for invariance of the stress rate. The concept of Superposition of Strain Rates is discussed in Section 10 in general terms. An alternative, more general derivation is given in Section 11. This superposition principle is fundamental to the analysis of deformation. It is believed to provide a more rigorous and general starting point than the product decomposition sometimes used in finite-deformation plasticity. Details of the physical theory are outlined in Volume II, PhysicalTheory. 3 3 Comments on any aspect of this analysis are most welcome ([email protected]). 5I am indebted to many colleagues for useful discussions, including P. Blewett, R. Brannon, J. Middleditch, Q. Zuo, and J. Kershner. The original typing was done by A. Rosen and the current version was retyped by Ellie Habbersett using Microsoft Word and Mathtype. The financial support of the Munitions Directorate, and of J. V. Repa, S. E. Newfield, and the Joint DoD/DOE Munitions Technology Development Program is appreciated. This work was performed under the auspices of the U.S. Department of Energy. 2. POLAR DECOMPOSITION The deformation of a continuum at any point is characterized by F=()=∂ ∂   Fijx Xi j , 2.1 the deformation gradient, where xiXtj(,) denotes the cartesian coordinates at time t of a point with initial cartesian coordinates Xi. It includes both rotation and stretch, which can be characterized by means of the polar decomposition theorem.4 This theorem makes it possible to write the matrix F5 as the product of a positive definite matrix V (stretch) and an orthogonal matrix R (rotation), so that F=VR . 2.2 This equation allows for a mathematical interpretation of the deformation of an infinitesimal line element, ∆Xj, between two material points as if it occurred in two steps: a rotation R, followed by a stretch V. The notation and relevant aspects of the theory of matrices are summarized in Appendix A. The tensor character of the variables is not an issue in the matrix theory discussed here, and all indexed variables will be described with subscripts. This is not to suggest that the tensor character ofthe variables is not important; rather, it is considered vital, but it is just not relevant to the current discussion. The tensor character of the variables is addressed in Appendix C. The discussion in this section follows many of the ideas standard in continuum mechanics, but continuity of thedeformation is considered to be of a mathematical rather than physical character since the medium of interest may not be continuous. If the physical medium contains defects, we can consider the average of an ensemble of deformations in materials with defects that have a certain statistical character. Themean motion can then be treated as continuous, though individual members of the ensemble are discontinuous in the neighborhood of defects. The positive-definite character of V, which requires that Vxxij i j be positive for any choice of the xi, follows from the positive-definite character of FFT (Truesdell’s B). To show this, we may begin by noting that 4 Polar decomposition was first demonstrated by Autonne (1902) and was introduced into thermoelasticity by Signorini (1930). Noll (1945) exploited the theorem in his thesis work, which is a standard reference in continuum mechanics. 5 F can be said to transform as a mixed tensor involving both spatial and reference coordinate systems, but this mathematical description does not have any useful physical interpretation (invariance) in the way the contravariant stress tensor and covari ant strain tensor do. This writer finds it misleading to characterize F as a tensor. 6xiFijFkjxkjxF xB xii j i ik k =()=ΣΣ2 , is nonnegative since it can be expressed as a sum of squares. This defines B, as discussed further in Appendix C. Now, let T denote the matrix of eigenvectors of B, and let ΛΛ denote the (diagonal) matrix of its eigenvalues. Then, as indicated in Eq. A.36 (here and in what follows, the A denotes anequation in Appendix A), BT T==FFTTΛΛ . 2.3 The diagonal components of ΛΛ are all positive since B is positive definite. It can be readily verified that the stretch can be expressed in the form VT T=λλT , 2.4 where the components λi of the diagonal matrix λλ are the positive square roots of the corresponding elements Λi of ΛΛ. Since Vxx xTij i j k kiik =()∑λ22.5 is positive, V is positive definite. To show that R is orthogonal, write, using Eqs. 2.2 and 2.3, RR V FF V V BV ITT== =−− − −11 1 1 . 2.6 This completes the proof, for if the product of a matrix and its transpose is the identity matrix I, then it is orthogonal, as indicated by Eq. A.44. Another polar decomposition is also possible in which the stretch occurs second typographically. It can be written FQ V= . 2.7 To show the relation to Eq. 2.2, the expression for F given therein can be rewritten as F=RR V R =RV R VRTT12 12//() () . 2.8 Now, the product of the two quantities in parentheses is positive definite and, hence, equal to V by comparison with Eq. 2.7, provided polar decomposition is unique, for then Q = R and VR V R=T. To show that the decomposition is indeed unique, assume for the moment that there is another decomposition FQ V=˜˜ , 2.9 where ˜Q is orthogonal and ˜V is positive-definite. Then, FF V QQ V VTT==˜˜˜˜˜ 2 . 2.10 7But FFT has a unique, positive-definite square root, by the same argument as above, so ˜VV=, and it follows that ˜Q=Q=R . Using these results (Eqs. 2.2 and 2.4), F can be written as the triple product F=T Sλλ 2.11 in a unique way where S=TRT . 2.12 Thus, any deformation can be represented as a rotation S, followed by an orthogonal stretch λλ, followed by another rotation T. These operations can be grouped in two ways: F= TS S S RV()()=Tλλ 2.13 and F= T T TS VRλλT()()= . 2.14 It is appropriate to describe the grouping of Eq. 2.13 as a material polar decomposition, since V is associated with the stretch ellipsoid whose principal axes ( S) are measured from rotated axes (i.e., axes that rotate with the material), in view of Eq. 2.12. Then, it is also appropriate to describe thegrouping of Eq. 2.14 as a spatial polar decomposition, since V is associated with the stretch ellipsoid whose principal axes ( T) are specified with respect to axes fixed in space. The distinction is discussed in more detail in connection with the example of simple shear in Section 5, and is illustrated in Fig. 5.1, which can be examined without reading Section 5. It seems natural to call the orthogonal axes specified by R “polar axes,” since they are uniquely specified by polar decomposition. These axes are orthogonal and rotate with the deformation thoughthey are not fixed in the material. Another set of axes is specified by the principal axes of the stretch ellipsoid. When the stretch is specified with respect to polar axes, which rotate with the deformation as in Eq. 2.13, it is natural to call it V, either polar or material stretch. When the stretch is specified with respect to space axes, which are fixed as in Eq. 2.14, it is natural to call it the spatial stretch V. In either case, the stretch is specified by the same ellipsoid . The difference lies in the reference axes. As mentioned above, though the polar axes rotate with the deformation, they are not fixed in thematerial. In particular, they are orthogonal, whereas axes fixed in the material become skewed. The stretch V is generally called the right stretch in continuum mechanics because it is written to the right of R. The principal axes of the stretch ellipsoid are called a Lagrangean triad by Hill (1978) when measured from the polar axes. The spatial stretch V is often called the left stretch to recall its formal location to the left of R in Eq. 2.14. The principal axes when defined with respect to spatial axes T are called a Eulerian triad by Hill. The relation to Hill’s approach is discussed in Appendix B following Eq. B.5. The matrix R is determined algebraically from F. It has the properties of a rotation matrix since it is orthogonal, but its geometrical significance is not intuitively obvious. Of course, if a body rotateswithout stretching, then VI=, and it follows that F = R, and the significance is unambiguous. But if the stretching is very large, it is not intuitively obvious that the mapping from an initially orthogonal 8set of axes to a highly skewed set of axes involves an unambiguous rotation, though it is mathematically straightforward. An interpretation of R can be obtained by considering the triple product TSλλ of Eq. 2.11. Then, consider another mapping from the deformed state in which the orthogonal stretching does not take place, i.e., λλ=I. Then F = TS = R. Thus, R is the transformation that obtains if the lengths of the line elements were to remain unchanged in the second of the three stages of the transformation of Eq. 2.11. Since stress is a tensor, constitutive laws should relate stress to a tensor measure of strain, stretch, and/or related tensors. As shown in Appendix C, B is a tensor measure of strain, though B is not. The deformation F itself does not transform like a tensor, so it should not appear in formulating constitutive laws. (In some simple elastic constitutive laws, F appears in the form FFT, and this is admissible because the rotation R disappears, and this is simply a nonphysical way of writing B (Eq. 2.3). Since F involves the deformation of a fiber ( dXi), it might be supposed that R is useful only when the underlying physics involves stretching of fibers. However, R also appears when shear processes are to be accounted for. To see this, note that the rotation of an element of area in shear processes is characterized by rotation of its normal, n, as discussed in deriving Eq. A.66 (nF N=−ΓT). Now F−T has the polar decomposition ˜˜VR (any matrix has such a decomposition). Let us consider the relation of these matrices to the stretch V and rotation R. Since polar decomposition is unique, ˜,˜ VV RR==−1. 2.15 Thus, R also characterizes the rotation of a triad of normals (not necessarily orthogonal) that characterize 3 planes in a general deformation, and it thereby characterizes the rotation of sheardefects (such as shear cracks, shear bands, or dislocation loops) as well as lengthening (or shortening) of fibers. 3. RELATIONS BETWEEN VARIOUS RATES The velocity gradient G is related to the deformation gradient F through GF F=  =    =∂ ∂∂ ∂∂ ∂u xx XX xi ki jj k˙˙ ,-13.1 where F is given by Eq. 2.1, and the factorization expresses an application of the chain rule. In this expression ui denotes the velocity ux txXt txXtiij ij ,,˙, ()=∂() ∂=() , 3.2 that is, the rate of change of position of a point fixed in the material (having a specified Xj). Now, any matrix can be represented as the sum of a symmetric part and an antisymmetric part. Thus, the velocity gradient can be represented as the sum of a symmetric stretching matrix D and an antisymmetric vorticity matrix W, so that 9G=D+W . 3.3 By combining Eqs. 3.1, 3.3, and 2.2, it can be shown that D+W=VV +V V ˙−−11ΩΩ , 3.4 where ΩΩ=−˙RR13.5 is an antisymmetric matrix representing the rate of polar rotation. By premultiplying and postmultiplying Eq. 3.4 by V and adding the former to the transpose of the latter, it can be shown that ˙BB B VDV +−=ΩΩΩΩ2 , 3.6 where B, the square of V, is defined by Eq. 2.3. It will prove useful to define the Signorini (1931) strain εε= −()1 2BI . 3.7 This spatial strain (based on the spatial stretch V) satisfies the relation ˙εεεεΩΩΩΩεε+−= VDV 3.8 that follows readily from Eqs. 3.6 and 3.7. The relation of the Signorini strain to other strain measures commonly used in continuum mechanics is discussed in Appendices B and C. The quantity on the left side of Eq. 3.8 I call the polar strain rate. The stretching D is called the strain rate in some works on continuum mechanics, but this seems misleading since it is not the rate of change of any well-defined, history-independent quantity. On the other hand, the left side of Eq. 3.8 is the rate of change of a material state εε involving a rate (the polar rate) appropriate to spatial tensors. This rate will be discussed further in this section and examples given in subsequent sections. For small deformations, V is nearly equal to I, and the strain rate can be approximated as D. Matrices of the form ˙AA A+−ΩΩΩΩ appearing in Eq. 3.8 represent the rate of change of A in axes having fixed orientation, accounting for the effect of polar rotation on the components of A. These may be called polar rates, since ΩΩ arises from polar decomposition. Alternatively, it may be useful to work in polar axes x, which are related to the fixed axes by xR x=T . 3.9 (In an earlier paper I called these “material axes.” This is not a particularly good name, since axes specified in a material at some reference time may also be reasonably called material axes but theybecome skewed at later times. Such axes are different from the axes characterized by Eq. 3.9, which remain orthogonal as a material deforms. “Polar axes” seems a natural term for the axes of Eq. 3.9, since determination of these axes requires polar decomposition. It is consistent with the phrase “polarrate” introduced above.) The polar axes represent a set of coordinates from which the deformation can be considered a pure stretch (that is, without rotation). As in Section 2, quantities represented in 10polar axes are denoted with bars. In particular, matrix quantities in polar axes are related to those in fixed axes by AR A R=T , ARAR=T , 3.10 a,b RA RA A A Ad dtT  =+ − =˙ ˆ ΩΩΩΩ . 3.11 This rate, denoted by a carat and called (here) a polar rate, has some of the properties of an ordinary derivative. For, if Z = X + Y, then ˆˆˆZX + Y= . 3.12 Also, if Z = XY, then it can be readily shown that ˆˆ ˆZX YX Y=+ . 3.13 With this notation, Eqs. 3.6 and 3.8 become ˆBVDV=2 3.14 and ˆεε=VDV . 3.15 If we define the Green metric B as BR B RF F==TT , 3.16a,b and polar strain as εεεε=RRT , 3.17 then Eqs. 3.14 and 3.15 become ˙BV D V=2 3.18 and ˙εε=VDV , 3.19 with the dot denoting an ordinary time derivative. In using these relations, it should be kept in mind that the polar axes are constantly changing as deformation proceeds. Relation of Polar-Rotation Rate and Vorticity A relation between rate of polar rotation ΩΩ and vorticity W can be obtained by p ostmultiplying Eq. 3.4 by V and subtracting the transpose of the result, leading to VVΩΩΩΩ+−− =VW WV Z , 3.20 where Z denotes the antisymmetric commutator matrix 11ZD VV D=− . 3.21 The matrix equation 3.20 can be considered as a set of three scalar equations in three unknowns, the independent components of ΩΩ. Since Eq. 3.20 involves three antisymmetric matrices, ΩΩ, W, and Z, it is natural to consider its form when expressed in terms of the associated vectors ωω, w, and z. The necessary relations are Ωik ijk je=ω , 3.22 We wik ijk j= , 3.23 and Ze zik ijk j= , 3.24 where eijk is defined following Eq. A.15. To simplify the calculations it is convenient to define a relative rate HW=−ΩΩ 3.25 and the associated vector hj by He hik ijk j= . 3.26 Writing out Eq. 3.20 in this notation, we find hV e eV Zijjkijjk ik l l l+() = . 3.27 This set of equations can be put in the equivalent form using Eq. A.30, hV hV zii i ill l−= , 3.28 or in matrix notation, treating h and z as vectors (I tr V-V) h = z , 3.29 where tr V denotes the trace of V, that is, tr ViiV= 3.30 summed on i. When Eq. 3.29 is solved for h, and h is expressed in terms of ωω and w, an explicit expression for the rate of polar rotation is obtained: ωω= + −()−wI V Vztr1 . 3.31 It is sometimes convenient to separate the deformation into an isotropic part and a deviatoric part (denoted by primes) whose trace vanishes, so that DD I=+'˙1 3θ 3.32 12and VV I=+'1 3v , 3.33 where ˙θ is the rate of change of compression and v is the dilatation. Then, ZD VV DD ' V 'V ' D ' =−= − , 3.34 indicating that the isotropic part of the deformation does not influence Z. Z vanishes for rigid body motion D = 0, as well as for pure dilatation, V = a I. It also vanishes if the rate of deformation is isotropic, so that DI=˙/θ3. More generally, as pointed out by Zuo (2001), Z vanishes when D and V are parallel (in the sense that they have the same eigenvectors). The rate of polar rotation is expressed in Eq. 3.31 as the sum of two parts, the vorticity and a historical part, the former independent of previous motion and the latter involving V as well as D. It is shown in Appendix A that the rotation R can be expressed in a more physical manner by Eq. A.45 (RQ=eφ), where φ is the angle of rotation and Q represents the axis of rotation, through Eq. A.47 (Qe rik ijk j=) , with r=(,,)rrr123 a unit vector along the axis of rotation. This leads to a representation of Ω in terms of φ and Q given as Eq. A.59, which can be expressed in vector form as ωω= + + −˙sin˙cos˙ φφ φrr r r ()1 x . 3.35 The rate of rotation can be expressed explicitly in terms of ωω by solving Eq. 3.35 for ˙r. That is, Eq. 3.35 can be written as Ar p˙= , 3.36 where A=−− − −− − −− −    sin ( cos ) ( cos ) (cos ) sin ( cos ) (cos ) ( cos ) sinφφφ φφ φ φφ φ11 11 1132 31 21rr rr rr3.37 and pr=−ωω˙φ . 3.38 Then, ˙rp1=−A . 3.39 The angular rate can be expressed in terms of ωω by taking the dot product of Eq. 3.35 with r, so that ˙φ=•rωω . 3.40 Now, since ωω can be expressed as the sum of two parts, according to Eq. 3.31, so can ˙r. This is done by writing ωω11=+ = − ()−wh h IVV z , tr . 3.41a,b 13Then, pwh rp p=+ − + = + (˙˙)φφwhwh , 3.42,a,b where ˙φw=•rw , ˙φh=•rh . 3.43a,b Thus, ˙rww==− ()−−• Ap A wwr r113.44a,b ˙rhh==− ()−−• Ap A hhr r113.45a,b The stress rate that arises in computing the normal component of traction on a crack is slightly different. (For cracks with reasonably simple shapes, such as the plane two-dimensional crack or the penny-shaped crack, this quantity determines the crack opening). To compute the rate of change of crack opening, we require the rate of change of the normal component of traction, σn, which can be written ˙σσn ij i jd dtnn=() . 3.46 Now, as the material in which the crack is imbedded rotates, the crack normal rotates. To compute the rate of rotation, we can make use of Eq. A.66 written in index form as nL Niik k= , 3.47 where Nk represents the normal at t = 0. Then, ˙˜nniik k=Ω , 3.48 where, using Eq. A.72, ˜˙/ΩΩ= −ΓΓIGT3.49 is different from ΩΩ given by Eq. 3.5 and, in particular, is not antisymmetric. Thus, the expression for ˙σn becomes, after some algebra, ˙˜σσn ij i jnn= , 3.50 where the operation (~) defines a “planar rate” ˜˙˜˜ AAA A=+ +ΩΩΩΩT . 3.51 Thus, in the calculation of invariants, it may be necessary to account for polar-like rates. If σn represents a “constant” normal stress, this requires that ˜σij=0, and not that ˙σij=0, since the latter does not account for changing axes. 144. ENERGETICS AND EQUILIBRIUM ELASTICITY The rate at which quasistatic work is done on a mass M of material bounded by a surface S, as illustrated in Fig. 4.1, is given by ˙Wn u d Sij j i S=∫σ . 4.1 Here, σij denotes the Cauchy stress tensor, which can be written in matrix form as σσ=()σij , 4.2 the components of velocity are written as ui, and nj represents the jth component of the outward unit normal to S. By mean of Gauss’ theorem, ˙W can be put in either of the forms ˙Wd d V dv d Mij ij Vij ij M==∫∫σσ , 4.3 where the first integral is taken over the volume V and the second over the mass M, provided the mass is considered to be in static equilibrium so that σij j,=0. The stress tensor is assumed symmetric, and v denotes the specific volume. Thus, the stretching D=()dij introduced in Eq. 3.3 arises naturally in energy considerations. If only mechanical effects are considered, the internal energy I per unit mass is given by II d v d to oij ijt =+∫σσ , 4.4 where Io represents the initial specific internal energy. Then, Eq. 4.3 becomes ˙˙WI d V V=∫ρ , 4.5 where ρ=1/v. T Sn u~ ~ ~ Fig. 4.1 Illustrati on of a mass M bounded by the surface S. In index notation, the local normal is given by ni; the traction, Tnji j i=σ ; and the local velocity, ui. 15A constitutive law represents elastic behavior if the strain energy per unit mass depends only on the current strain εε, and is, consequently, independent of strain history, i.e., II v t r d tfot−= ()=() ∫σσε ε D 0 , 4.6 where tr denotes the trace of the matrix product. The function fεε() may be an anisotropic function of the strain εε. The integrand is written in terms of the polar quantities σσ and D, rather than in terms of the spatial quantities σσ and D, in order to account for material anisotropy in the subsequent analysis. It is straightforward to show that this is valid, since tr tr , 4.7 using Eq. 3.14 and the definition of trace given as Eq. 3.30. In order to determine a criterion for the internal energy I to be history independent, it is useful to define a quasi-stress ττσ σ =−−v voVV11 . 4.8 Then, using Eq. 3.19, vtr v troσστ τεε D=˙ , 4.9 and the integral of Eq. 4.6 becomes II v t r dooo−= ()∫εττεε . 4.10 This integral is history-independent if τρεij o ijf=∂ ∂ . 4.11 Consequently, from Eq. 4.8, the stress must be of the form σσΣ Σ=ρ ρoVV 4.12 to ensure elastic behavior, where Σij ijof=∂ ∂ρε4.13 and f denotes an elastic potential. The relation of this formula to similar formulas given by Rivlin and Ericson (1955) and discussed by Truesdell (1952) is discussed in Appendix D. Isotropic Material If a material is isotropic, the elastic potential f depends on the strain only through its invariants. An equivalent but more convenient formulation of isotropy is to let it depend on the invariants of stretch. It is straightforward to show that the invariants of B and B are the same by means of Eq. 3.10. The invariants can be selected in numerous ways, but it is particularly convenient to select 16It r t r BBii ii 1=== =BB , 4.14 ′=′′=′′ It r t r21 21 2BB BB , 4.15 I3==BB , 4.16 where B' B I=−1 31I 4.17 is the deviator of B. We can write the potential in terms of these invariants as fg I I Iεε()=′()123,, . 4.18 Now, it is straightforward to show that ∂ ∂=I Bijij1δ , 4.19 ∂′ ∂=′I BB ijij2 , 4.20 and ∂ ∂=I BB ijij3˜ , 4.21 where ˜Bij is the cofactor of Bij in the determinant of B and, consequently, ˜BB=I3I , 4.22 as indicated in Eq. A.21. Denoting the matrix of cofactors of V by ˜V, so that ˜VV I=I3 , 4.23 then Eqs. 4.12 and 4.13 reduce to σσ=∂ ∂+∂ ∂−  +∂ ∂   21 3122 1 33 ρg Ig IIg II BBB I . 4.24 By means of the rotation operation indicated in Eq. 3.10, this can be put into the spatial form σσ=∂ ∂+∂ ∂−  +∂ ∂   21 3122 1 33 ρg Ig IIg II BBB I . 4.25 The density is related to I3 through ρρoI/()=2 3 . 4.26 17The validity of Eq. 4.25 can be verified in two steps. The first involves Eq. 3.14, from which the relations 21 Bd t r Iij ij==ˆ˙B , 4.27 21 32 12 tr I IBB D−    =′˙ , 4.28 and 233Id d Iii ii==B˙ 4.29 follow readily. Proof of Eq. 4.29 requires the use of Euler’s identity, given in Appendix A as Eq. A.23. In the second step, the relations 4.25 and 4.27–4.29 are combined to show that vd v t r gij ijσ=()=σσD˙ . 4.30 This verifies that the rate of change of potential for an isotropic elastic material is equal to the rate at which work is done by the applied stresses. When the deformation vanishes so that B = I, the stress should vanish (unless the material is in a prestress condition). In order for this condition to hold, it is necessary that ∂ ∂+∂ ∂=g Ig I130 . 4.31 Since it follows from Eq. 4.14 that I13=, from Eq. 4.15 that ′=I20, and from Eq. 4.16 that I31=, then Eq. 4.31 is sufficient to make the stress vanish when B = I. Anisotropic Material We now return to the more general case in which the elastic material is not isotropic. If the expression for material stress given as Eqs. 4.12 and 4.13 is converted to spatial (Cauchy) stress by rotation, then σσσσΣ Σ ==RR F FT oTρ ρ . 4.32 This expression is given by Truesdell (1952) as Cosserat’s form for an elastic material. (The dependence on F is something of an accident which occurs because the polar stress is rotated into spatial axes. As discussed elsewhere, the constitutive equation for an isotropic material should notinvolve the rotation R, and, hence, should not involve F.) In linear elasticity, the elastic potential of Eq. 4.13 is quadratic in the strains, as discussed by Landau and Lifshitz (1986). This requires 36 elastic constants, which reduce to 21 as a result of energy considerations.It is interesting that τij, defined by Eq. 4.8, is the Piola-Kirchoff stress, usually written as ττσ σ=−−ρ ρo TFF1 . 4.33 18This expression can be obtained by combining Eqs. 4.32, 4.11, and 4.13. Thus, Eq. 4.11 shows that the Piola-Kirchoff stress is the gradient of an elastic potential with respect to the material strain. It is possible to define a spatial Piola-Kirchoff tensor by rotation of ττ, ττττ=RRT , 4.34 using the transformation of Eq. 3.10. This presumes that the Piola-Kirchoff stress is a polar stress (a point not emphasized in the continuum mechanics literature). This is natural since the polar strain used in obtaining Eq. 4.11 is regarded as a material strain, as is indicated by the use of bars inEq. 3.17. The rate of change of internal energy per unit volume can be written ρρ ρ˙It r t r oT=() = () σστ τ DF F D , 4.35 in view of Eq. 4.33. Then, using the polar decomposition expressed in Eqs. 2.2, 4.34, and 3.15, therate of change of energy per unit mass can be expressed in terms of ττ as ˙ ˆ Iv t ro= ()ττεε . 4.36 This pair of conjugate variables ττ and ˆεε is not standard in continuum mechanics, but it arises naturally when considering elastic behavior of anisotropic materials in spatial coordinates. The spatialPiola-Kirchoff stress for an isotropic elastic material has the form ττ=+ −  +  −21 3 121 331ρ∂ ∂∂ ∂∂ ∂g Ig IIg II IB I B , 4.37 as can be deduced from Eqs. 4.34, 4.33 with 2.2, and 4.25. 5. SIMPLE SHEAR In simple shear, material is displaced parallel to a fixed plane. The coordinates of displaced material can be written in the general case as xXU X tiii j =+ (), , 5.1 where Ui is the displacement. When the x1 axis is selected as the displacement direction, the only coordinate that varies is xX X11 22 =+ε , 5.2 where ε is a scalar measure of shear. The deformation gradient takes the form F=12 0 010 001ε      . 5.3 19In general, ε can depend on time in an arbitrary way. The velocity gradient, given by Eq. 3.1, reduces to GF F==    −˙˙ 102 0 000000ε , 5.4 and the stretching and vorticity are, from Eq. 3.3, D=    00 00 000˙ ˙ε ε , W=−    00 00 00 0˙ ˙ε ε . 5.5 It will prove convenient to define an angle of shear, β, by εβ=tan 5.6 and to abbreviate the form of various matrices that follow by writing s=sinβ , c=cosβ . 5.7 It is straightforward to calculate that the spatial deformation tensor is BF F==+    T14 2 0 21 0 00 12εε ε . 5.8 The matrix of eigenvalues of B can be readily found by solving the characteristic equation, A.42 of Appendix A, with the result ΛΛ=+() −()    10 0 01 000 122 22sc sc/ / . 5.9 The corresponding matrix of eigenvectors can be found by solving the appropriate simultaneous equations and normalizing, leading to T=+() −−() −() +()    1212 0 1212 0 00 1ss ss// // . 5.10 It follows that the spatial (left) stretch is, using Eq. 2.4, V=+    () /10 0 00 12sc s sc , 5.11 20and the Signorini strain of Eq. 3.7 is εε=    20 00 00 02εε ε . 5.12 Construction of Mohr’s circle, or direct calculation of the eigenvalues of εε, shows that the principal strains are εε ε221±+ and 0, and that the mean strain is 232ε/, which is tensile. Thus, shearing produces a second-order effect that tends to open up voids and cracks with certain orientations and, hence, this effect is termed dilatancy. The compressive principal strain εε ε221−+() decreases monotonically from −ε to –1/2 with increasing deformation, whereas the tensile principal strain increases without bound. Though the density does not change in simple shear, the tensile character of ε11 suggests that cracks whose normals lie in a shearing plane will open, while those that are perpendicular to the shearing plane will remain closed, as they are in compression. This is a nonlinear effect not accounted for in ordinary elasticity. If cracks are present, they will cause the elastic constants to depend on the sign of the stress. This subject is outside the scope of the currenttreatment, which is essentially kinematic, but is an essential part of Statistical Crack Mechanics. The rotation matrix RVF=−1 is, from Eq. 2.2, R=−    cs sc0 0 00 1 . 5.13 Differentiation of this result shows that the rate of polar rotation is ΩΩ= = −    −˙˙ ˙RR100 00 00 0β β . 5.14 For small deformations, the vorticity W, given by Eq. 5.5, and ΩΩ are nearly equal, but for large deformations, ΩΩ is smaller by a factor cos2β, in view of Eq. 5.6. This is consistent with the observation that the amount of polar rotation is limited to π/2, so its rate must go to zero as β approaches π/2. It is interesting to verify the general relation between the polar rate of rotation and vorticity given by Eq. 3.35. This can be done in two steps. The first is to compute the deformation commutator of Eq. 3.25, ZD VV D=−=−    201 0 100 0002˙sin cosεβ β , 5.15 21and its vector counterpart, defined by Eq. 3.28: z=    20 0 12˙sin cosεβ β . 5.16 Also, IVVtrc ccc c c−() =++− +() +()    −122 1 4142 22 0 22 1 10 00 2 1 1// / /ε ε . 5.17 By combining this result with Eqs. 5.5 and 5.15, we find wI V Vz+−() = −    −tr10 0 ˙β , 5.18 which is indeed equal to ωω, as computed directly from 5.14. In polar axes, the deformation is characterized by the polar stretching (using Eq. 3.10) D=−    ˙sin cos cos sinεββ ββ22 0 22 0 00 0 , 5.19 the stretch V=+()    cs ss c0 10 0012/ , 5.20 and the strain εε=    00 20 0012ε εε . 5.21 Then, the relation between polar strain rate ˙εε and polar stretching D given by Eq. 3.19 reduces, in this example, to ˙˙ ˙˙ εε= =    VDV00 40 000ε εε ε , 5.22 as can be verified by direct calculation. The corresponding relation in spatial axes, given by Eq. 3.8, becomes, for simple shear, 22ˆ˙ ˙εεεεεεΩΩΩΩεε =+ − = =++ +    VDVβεε ε εε24 1 3 0 13 2 0 00 032 2 , 5.23 which can also be verified by direct calculation. It is straightforward to show that the invariants of ˙εε and ˆεε are the same. The relation of material response in spatial and polar axes is illustrated in Fig. 5.1, which illustrates the stretch ellipsoid and the three important systems in which it can be represented—the spatial, polar, and principal axes. The stretch can be represented by the ellipsoid Bxx B xx xxij i j ij i j ij i j=== Λ˜˜1 5.24 in any of several ways which are equivalent. The xi are defined by Eq. 3.9. The ellipsoid is analogous to the strain quadric discussed by Sokolnikoff and Specht (1946), but is simpler to think about than the strain quadric because its character does not change (the strain quadric may be hyperbolic or anysecond-order surface). The stretch is always positive, but is less than one when compressive and exceeds one when tensile. The stretch quadric is always an ellipsoid. (We refer to either quadric, Vxxij i j or Bxxij i j , as a stretch ellipsoid—they are the same except for the magnitude of the principle stretches.) The ellipsoid is related to the Signorini strain by Eq. 3.7. The spatial axes are fixed, but as the shear increases, the principal and polar axes rotate. The principal axes, initially bisecting the spatial axes, rotate clockwise until those two systems coincide. The polar axes also rotate clockwisefrom the initial orientation, in which they coincide with the spatial axes, until they have rotated 90 degrees, when the shear becomes infinite. The stretch ellipsoid is initially a circle (actually a cylinder), but at small strains its principal axes essentially bisect the spatial axes, as mentioned above.A principal stretch is the square root of the inverse of the length of the principal axis of the stretch ellipsoid. The product of the principal stretches, given by Eq. 5.9, is unity, indicating that the specific volume does not change for simple shear. The Λ matrix, by definition, is diagonal, and the ˜xi denotes coordinates in principal axes. Thus, xT xiik k=˜ . 5.25 The matrix of eigenvectors given by Eq. 5.10 can be written in the alternative form T=−    cos sin sin cosυυ υυ00 00 1 5.26 involving the angle from the principal to the spatial axes υπ β=−//42 . 5.27 The relation of the xi and the xi is given by Eq. 3.9, which, in analogy with the notation of Eq. 5.25, can be written xR xij i j= . 5.28 23χνΛ1-1/2 x1 x1–x2x2x2 – x1~~ Λ2-1/2β Fig. 5.1. The stretch ellipsoid for simple shear. The ellipsoid is the same whether referred to spatial, polar, or principal axes. It is initially a circle of unit radius. At early times, itbecomes an ellipse with principal axes roughly bisecting the spatial axes. At late times, it becomes very thin and elongated, with the major axis approaching the x2 axis. Its size is not the stretch, but is related to it. For example, the length of the major axis is 112/cos / sinΛ= − ()ββ , and the length of the minor axis is 111/cos / sinΛ= + ()ββ , with tan β equal to the scalar strain ε. Both the polar and principal axes rotate clockwise with increasing deformation. The Signorini strain is related to the difference between the stretch ellipsoid and the reference circle in spatial axes,while the Green-St. Venant strain is related to the difference in polar axes. The transformation from polar to principle axes is given by xS xT=˜ , with S given by Eq. 2.12. For simple shear, this reduces to S=−    =+cos sin sin cos ,χχ χχ χπβ0 0 00 142 . 5.29 The strain measures εε and εε discussed above measure the deviation of the ellipsoid from a circle in the various systems. These strains should not, therefore, be considered as competitive, but as alternate representations of the same physical entity. A broader discussion of measures of strain is given in Appendix C. To compute the strain energy for isotropic elastic deformation, it is necessary to determine the invariants of stretch, which are, of course, the same in the various systems discussed: 24IBii1234== + ε , 5.30 IB Bij ij 224 1 2416 3=′′=+εε , 5.31 I31= . 5.32 In application, it is necessary to relate g to these invariants by means of experimental data or, possibly, by atomic or microstructural models. Still, it is possible to make Eq. 4.25 for the stress explicit, by use of Eq. 5.8 in the example of simple shear: σρ ε ε ζ11 1 22 2425 3432 3=+  +  +ogg g , 5.33 σζ ρ ε22 22 16 3=+og , 5.34 σζρ ε33 228 3=−og , 5.35 σµ ερ ε12 23232 3=+og , 5.36 where ρ has been set to ρo, since I31=, and the abbreviation gg In n=∂ ∂5.37 has been introduced. In addition, it is convenient to set 212ρ µogg+()= 5.38 and 213ρζogg+()= . 5.39 The term gg13+ in Eq. 5.39 vanishes for ε=0 in view of Eq. 4.31, but it may not vanish for large ε, so the general form is retained. The fact that σ12 is linear in ε up to a cubic term is consistent with a comment by Mooney (1940) that for rubber, linearity of the shear stress as a function of strain holds to very large strains. If it is linear to 200% strain, as he finds, then g2 must be very small in addition to the quadratic term being absent. 6. RIGID-BODY MOTION In the case of rigid-body rotation, the deformation is trivial, but it is interesting to see how the general results are specialized. Such a deformation can be represented by xr yr==cos , sinθθ , 6.1 25where the z-axis is taken as the axis of rotation and θω φ=+t 6.2 and r and φ depend on the initial values X and Y of x and y through rX Y Y X=+ =22,tan /φ . 6.3 Then, applying Eq. 2.1, F=−     cos sin sin cosωω ωωtt tt0 0 00 1 . 6.4 It can be readily shown that BVI , RF== = . 6.5 It follows that the velocity gradient is GF F==−    −˙101 0 100 000ω , 6.6 and D = 0, Z = 0. It is straightforward to verify that ΩΩ= =˙RR WT , 6.7 demonstrating that the vorticity and rate of material rotation are equal in this case. 7. TORSION When a uniform circular cylinder is twisted so that each section normal to its axis rotates without strain and there is no elongation, the deformation may be described as simple torsion. Let the material points be defined by their initial coordinates X and Y, Z, and take the Z axis as the axis of the cylinder. Then, the angular orientation θ can be written θθ φ=()+() oZt XY,, , 7.1 where φ denotes the initial orientation of a point with the initial coordinates X, Y, and θo the amount of twist. The Cartesian coordinates of material position at time t are given by xr yr zZ===cos , sin ,θθ , 7.2 with r remaining constant during the deformation. Then, substitution of these expressions into Eq. 2.1 for the deformation gradient leads to F=−     cos sin sin cosθθ θθoo ooa b 00 1 , 7.3 26where ar b rZoo oo=−′ =′′ =∂ ∂θθ θθ θθsin , cos , , 7.4 and ′θ0 is constant in view of the uniformity of deformation of the cylinder. The corresponding Cauchy-Green tensor is BF F==+ +    Taa b a ab b b ab1 1 12 2 , 7.5 which is independent of φ. It proves convenient to define tan /αθ=′2ro , 7.6 which is closely related to the helical angle of twist, as discussed below and illustrated in Fig. 7.1. Z Y Xφθo γ Fig. 7.1. Cylinder in torsion. The angle γθ=−tan ( / )1 02rZ proves to be a convenient measure of deformation. 27The eigenvalues of B are given by ΛΛ=      cot tan 2 22 200 00 00 1α α . 7.7 The determinant of ΛΛ is unity, indicating that the deformation proceeds without change in volume. Calculation of the normalized eigenvectors of B leads to the expression T=− −      cos sin sin sin cos cos cos sin cos sin sin cosαα αα ααθθ θ θθ θ22 22 220 . 7.8 In subsequent calculations, it proves convenient to define the complement of α γπ α=−/2 , 7.9 which has the convenient property that it vanishes when the twist vanishes, since 2222tan cotγθ α=′== = +rA a bo . 7.10 The angle of twist is tan−′1roθ. Thus, γ is closely related to the angle of twist, but is algebraically more convenient. This equation also defines A, a quantity useful in shortening some lengthyexpressions. Then, using Eq. 2.4, the stretch is given by V=+− − −+ −    1 12 2QQ QQsin sin cos sin sin sin cos cos cos sin sin sin cos sin cosθθ θ θ γ θθ θ θγ θγ θγ γ , 7.11 where Q=−−21coscosγγ . 7.12 It can be verified that V=1 by direct calculations, demonstrating that simple torsion causes no change in volume, even in the strongly nonlinear regime. When the twist is zero, γ=0, Q = 0, and V = I. To calculate V−1, one can either determine the matrix of cofactors of Eq. 7.11 or use the spectral relation V−=1 12ΤΤΤΤ−−Λ/T , 7.13 28with the result V−=+− () −() +− −−     122 221 1 2cos sin cos sin cos cos sin sin sin cos cos sin cos cos cos sin sin sin cos sincoscosγθ θ θ θ γ θ γ θθ γ θ θγ θγ θγ θγγγ . 7.14 Then, the rotation matrix is RVF==+− +− −+ −    −1sin sin cos cos cos sin cos cos cos sin sin sin sin cos cos sin cos sin sin cos cos cos cos sin sin sin cos sin cosθφγ θφ θφγ θφ θγ θφ θφγ θ φθ φ γ θ γ φγ φγ γ.7.15 Further calculations are greatly simplified by considering the case φ=0, which is sufficiently general because the deformation is the same for all generators of a cylinder in torsion. Let Ro denote the value of R on φ=0, given by Rooo o oo o=−− −    cos sin cos sin sin sin cos cos cos sin sin cosθθ γ θ γ θθ γθ γ γγ 0 . 7.16 When there is no twist, θo=0. Consequently, ′=θo0, and it follows from Eq. 7.10 that γ=0 and the rotation reduces to RIo=. The rate of polar rotation is given by ΩΩoo oo o ooo ==− −    +−    −˙sin cos sin cos˙ ˙ RR100 00 001 0 100 000θ θ θθγθ , 7.17 where ˙˙cos˙ sin γγθ θγ ==1 222 22Az ro o . 7.18 Though ˙γ and ˙θo are related, they are different, since γ is related to the helical twist angle, while θo is the angle of rotation in the plane normal to the Z axis, as shown in Fig. 7.1. The vorticity is found by application of Eqs. 3.1 and 3.3. As above, only the value Wo on the generator φ=0 is required: Woo o oooA=− −    +−    00 00 0201 0 100 000sin cos sin cos˙˙θ θ θθθ . 7.19 The first term is the same as the first term of ΩΩo, except for a factor of cos2γ, just as in simple shear. The second term is the same as the second term of ΩΩo. For θo small, the first term is a rotation essentially about the x1 axis, and is like simple shear except for the difference in axes and calling the 29shear angle γ instead of β. The second term represents rotation about the axis of symmetry, the Z axis. In view of Eqs. 7.10 and 7.18, ˙ cos˙,˙ cos˙ θ γγ γγoz rA ==22 22 . 7.20 It is shown in Appendix A that the rotation matrix can be expressed in the form RQ=eα , 7.21 where Q is an antisymmetric matrix associated with the unit vector q characterizing the axis of rotation, and α is the angle of rotation about q, not the α of Eq. 7.6. It is straightforward to show that qoo 1 1 =− −νγθ θsin sin / cos , 7.22 qo 2 1 =− −νγ θsin cos , 7.23 qo 311=+() − νγ θcos cos , 7.24 where νγ θθ γ=+() −− +() []{}−13 112cos cos cos cos/ oo . 7.25 When r is very small, so that γ is small, q3 is very near to 1, so that the rotation is clockwise about the x3 axis, and the deformation is essentially the same as in simple shear. However, as r increases without bound, γ approaches π/2, and the rotation vector becomes qq qo oo oo o11 2 21 2 31 2 1 31 31 3=−+ −  =−− −   =− −  cos cos,cos cos,cos cosθ θθ θθ θ . 7.26 This direction varies in a complex fashion, but for θo small, it approaches the x1 axis. The angle of rotation α of Eq. 7.21 is given by cos cos cos cos cosαθ θ γ γ=+ + −()1 21oo . 7.27 When r is zero, γ is zero, and this reduces to αθ=o. In general, however, α and θo are quite different. For example, if r and z are equal and θo=°45, then α=°49 6.. The stretching is given by D=− − −    ˙sin cos sin cosA 200 00 0θ θ θθ . 7.28 For θ small, this is the same as for simple shear, with γ playing the same role as β in simple shear. There is a difference in the labeling of axes, with X2 in the torsion analysis playing the role of X1 in the shear analysis when φ=0, and X3 in the torsion analysis playing the role of X2 in the shear analysis. 30As in the analysis of shear, it is possible to verify the relation Eq. 3.35 between ωω and w. On φ=0, ωωωωοο==− −    ˙cos ˙sin ˙γθ γθ θo o o , 7.29 ww==− −     oo o oA A˙ ˙cos sin ˙2 2θ θ θ , 7.30 and z=    ˙sin coscos sin Ao o2 0γ γθ θ . 7.31 It is straightforward, then, to verify that zI V V woo o o otr=−() −()ωω . 7.32 8. SIMPLE VORTEX To illustrate the difference between ΩΩ and W, it is useful to consider a simple vortex. For such an irrotational motion, W is zero everywhere (except the origin), whereas ΩΩ increases monotonically with time. (If it seems paradoxical that the motion in a simple vortex is irrotational , this is a consequence of some naïve statements in books on hydrodynamics—zero vorticity does not imply zero rotation, as shown by Eq. 3.31.) The components of velocity are given by the expressions uKrvKr=− =sin,cos αα , 8.1 where K is the strength of the vortex and αω φ=+t . 8.2 The position vector is given by xr yr==cos , sinαα , 8.3 so that the phase φ can be determined by the initial coordinates through tan /φ=YX . 8.4 31The angular velocity is ω=Kr/2 . 8.5 A straightforward calculation leads to the expression F=+− + −−  cos sin cos sin sin sin sin cos cos cos cos sinββαφ ββαφ ββαφ ββαφ22 228.6 for the deformation gradient, where βω=t . 8.7 The left Cauchy-Green tensor is B=++ −− −− −+  12 2 4 2 2 2 2 22 2 2 1 22 422 2 22 2βα β α βα β α βα β α βα β αsin sin cos sin cos sin sin cos . 8.8 In previous sections, V has been determined by computing its eigenvectors and eigenvalues, but here we use the relations VBIBB =+ +==J JJJJ t r2 12212 , , 8.9 that follow from the Cayley-Hamilton theorem (see Appendix A) for 2 x 2 matrices. Then, V=+()++ −− −− −+  −112 2 2 2 22 1 2 221 222 2 22 2ββα β α βα β α βα β α βα β αsin sin cos sin cos sin sin cos . 8.10 A fairly lengthy calculation leads to the simple result ˙sin cos cos sinFF−=− −−  122 22ωαα αα , 8.11 which, in view of Eq. 3.3, leads to the conclusion that W=0 and DF F=−˙1. Consider now the rate of polar rotation, ωω, which can be determined algebraically by use of Eq. 3.35. To begin, one computes DV= +++() −+() − −+() +− +()  ω βββα βα β βα β ββα 112 12 12 12222 2 22 2sin cos cos sin . 8.12 Then, Eqs. 3.21 and 3.24 lead to z=+ ()  −002 12212,,/βω β . 8.13 The computation of ωω can be completed using the relation hI V Vz=−() =+ ()()−tr z V V1 31 12 200,, / , 8.14 32which holds for plane flows. Completion of the calculation using Eq. 3.31 then leads to ωω= +()00 122,,/()ωβ β . 8.15 Thus, the polar rotation has an angular velocity that is initially zero, but approaches a constant value of ω at late times, when β is large, i.e., the fluid rotates though the vorticity is zero. 9. STRESS RATE IN CONSTITUTIVE RELATIONS Given an isotropic constitutive law having the rate form ˙,σσσσ=()ψD 9.1 in the absence of rotation, it is often necessary to construct a general form that allows for rotation, especially when formulating computer algorithms. In particular, this situation arises in connection with elastic-plastic behavior, kinematic hardening, and Maxwell solids. In this section, it will be shown that the appropriate generalization of Eq. 9.1 is ˆ,σσσσ=()ψD , 9.2 where the polar rate operator (∧) is defined by Eq. 3.11. When a material is rotating, Eq. 9.1 must apply in the rotating, or polar, axes. The stress in polar axes is σσσσ=RRT , 9.3 as indicated by Eq. 3.10. By differentiating, we find that ˙˙ ˙ ˙ σσσσσσσσ =++RRRRRRTTT . 9.4 Now, for isotropic materials, ψψσσσ σ ,,DR R R R D R R()=()TTT9.5 since, by the Cayley-Hamilton theorem (Appendix A), ψ can always be expressed as a polynomial in σσ and D, and the property indicated by Eq. 9.5 holds for polynomials. Then, since Eq. 9.1 must hold in rotating axes, neglecting dynamical effects so that ˙,σσσσ=()ψD , 9.6 premultiplication by R and postmultiplication by RT leads to ˆ˙˙˙ , σσσσσσσσσ σ =+ + = () RR RR DTTψ , 9.7 which is equivalent to Eq. 9.2, and had to be proved. This stress rate is different from the ZJN stress rate of Zaremba (1903), Jaumann (1911), and Noll (1955), σσσσσσσσ∨ =− +˙WW , 9.8 33though it is similar in form. At any time, one can choose the reference system so that F = I, forcing V = I and consequently, in view of Eqs. 3.21 and 3.31, ΩΩ=W. However, this cannot hold for all time, since the reference system can only be selected once. This point has led to great confusion in the literature. Note that here we have derived Eq. 9.7, not merely conjectured the form as was done byZaremba and Jaumann. This analysis is similar to Noll’s, but his final step of setting Ω = W is not generally valid. To illustrate the difference in using the exact (polar) stress rate of Eq. 9.7 and the approximate ZJN stress rate of Eq. 9.8, consider the hypo-elastic material defined by ˆ,σσσσ=()=+ ψλ µDI DDtr2 , 9.9 where λ and µ are the Lamé constants, as it undergoes simple shear. By virtue of 5.5, the trace of D is zero. Then, applying the stress rate of Eq. 9.7, we find ˆ˙σσ σ µ11 11 12 21 11 22 0=− = =Ω D , 9.10 ˆ˙σσ σσ µ12 12 12 22 11 12 12 2 =− + =ΩΩ D , 9.11 ˆ˙σσ σ µ22 22 12 12 22 22 0=+ = = Ω D . 9.12 These equations can be combined to obtain the ordinary differential equation d d2 11 2 11 244 σ βσµ β+= cos9.13 whose general solution is σµββ β β β β β11242 2 2 2=+ −() ++ cos lncos sin sin cos sin AB . 9.14 If we assume that all the stresses are initially zero, then σµββ β β β11242 2=+ −()cos lncos sin sin , 9.15 σµ β β β β β1222 2 2 2=− () ( ) − ()cos tan lncos tan , 9.16 σσ22 11=− . 9.17 Note that there was an error in Dienes (1979) in which tan 2 β was printed as tan2β. Also note that εβ=tan here (Eq. 5.6), where e==22εβtan. Thus, D12=˙ε here while De122=˙/ in Dienes. As the strain approaches infinity at late time, β approaches π/2, as indicated by Eq. 5.6, and the stress approaches infinity logarithmically. The stress-strain relation is given graphically in Fig. 9.1. If the ZJN stress rate is used for this process, the governing equations are ˙σσ11 21 0 −=a , 9.18 342.8 2.4 2.0 1.6 1.2 .8 .4 0 0. 2 . 4. 6 . 8 1.0 1.2 1.4 1.6 1.8 2.0 2.2Non-Dimensional Stress, σ12/µ Strain, ∋Current Theory Zaremba - Jaumann - Noll Theory Fig. 9.1 Comparis on of the stress-strain relation for simple shear, using the traditional method of Zaremba-Jaumann-Noll and the current finite-deformation theory based on polardecomposition. Note that a negative slope would lead to instability. ˙σσ σµ12 22 112−−() =aa , 9.19 ˙σσ22 21 0 +=a , 9.20 where ax X=∂ ∂=˙˙1 22ε 9.21 is considered constant in this example. It is straightforward to show that, if the stresses vanish initially, σµ12=sinat , 9.22 σµ111=−()cosat , 9.23 σµ221=− −()cosat . 9.24 This behavior is compared with the exact solution in Fig. 9.1. Clearly, the periodic behavior of this solution is not physically realistic, though the solution agrees with the exact solution for small strains.The difference in these behaviors can be traced to the difference between ΩΩ and W. The rate of polar rotation Ω in simple shear gradually drops from its initial value to zero as deformation proceeds, as 35given by Eqs. 5.14 and 5.6. The vorticity W, however, is constant, as shown by Eq. 5.5, when ˙ε is constant. Thus, vorticity does not measure the polar rotation rate, except at the initial (reference) time. 10. SUPERPOSITION OF STRAIN RATES Though superposition of strain rates dates back to Reuss (1931), who applied it in 1928 to characterize elastic-plastic behavior, no physical motivation was given, nor adumbrated, nor was it suggested that the principle could be extended to more complex situations in mechanics where processes other than plastic flow are involved. In this section, a motivation for the superpositionprinciple is described and the extension to fragmentation and the behavior of porous materials is also considered. Extension to cover polycrystalline and composite materials is probably straightforward. Though the title of the section contains the phrase “strain rates,” the details of the analysis involveprimarily the stretching D rather than the strain rate ˆε. However, in view of Eq. 3.15, if DD=∑α α , 10.1 with Dα denoting the stretching due to the αth process, then ˆˆεε=  =∑∑VVDα αα α, 10.2 where ˆεεα is the Signorini strain rate due to the αth process. Thus, superposition of stretchings and superposition of Signorini strain rates are equivalent. Of course, V denotes the total stretch, and cannot be separated into parts in a corresponding way. The premise of this section is that deformation is often the consequence of numerous independent physical processes which can be superimposed to obtain an overall deformation. This is illustrated inFig. 10.1. The difference in velocity between two points P1 and P2 can be written ∆∆ ∆uu uiicid=∑+∑ , 10.3 where the first sum is taken over the continuous elastic regions between defects, and the second sum is taken over the defects between elastic regions. The continuous regions can, in fact, be considered to have a more general character, such as thermoelastic behavior characterized by rather general thermodynamic laws. They can be anisotropic and may involve plastic flow or various phases. Theflaws are considered to be either open or closed. (In real materials, flaws may be partially open and partially closed, and dislocations may move out of their plane, but the treatment of such flaws is beyond the current treatment.) The rules for elastic behavior were discussed in Section 4. Morecomplex thermodynamic behavior is deferred to a subsequent volume. The success of the superposition method depends on the adequacy of the constitutive laws characterizing the various kinds of defects and their statistics. The independence of the behavior of defects is crucial to themethod. This, too, will be treated in a separate volume. 36u1 iP1P2 u2 i u – 2 iu = 1 i∆u + e i∆ud i Stable shear crackUnstable shear crackLocked shear crackPlastic slip planeStable open crackUnstable open crackShear band Fig. 10.1 Superposition of strain rates. An illustration of how the effects of various defects can combine to influence the velocity increment between two points P1 and P2. The number of defects per unit length is denoted by ˜/L=1λ, where λ is the average separation in the reference state for a particular kind of flaw. Use of ˜L makes it possible to replace the summation over individual defects in Eq. 10.3 by a sum over sets of defects involving ˜L, so that ∆∆vS L vidi =∑˜ααδ 10.4 represents the contribution to the velocity difference of the defects intersecting a segment between points P1 and P2 of length ∆S, as illustrated in Fig. 10.2. It is assumed that ˜L is not time dependent. The time-dependent case is addressed in Section 11. It is very convenient to take the flaws as circular, and quite possibly a good approximation in many situations (Zhurkov 1975). This may be adduced with an analogy. In the development of the kinetic theory of gases, molecules such as O2 were considered spherical for the purpose of computing collisions, though that is clearly far from precise. Nevertheless, the resulting theory was verysuccessful in explaining the behavior of gases; for example, diffusion, conduction, and viscosity coefficients were computed to an accuracy typically on the order of 1%, according to Jeans (1940). P2 ∆sP1u2 ~ u1 ~ i~N γ Fig. 10.2 The contribution of a defect to the difference in velocity between P1 and P2. 37In the theory of solid defects, approximating their shape as circular greatly simplifies the statistical arguments (which are much more complex than for gases) as well as the behavior of individual defects. (Recall, in this connection, that the theory of circular cracks was not developed until the 1950s when Sack, Segedin, Sneddon, and Keer were able to solve the governing boundary valueproblems.) A more explicit result is stated by Kachanov (1994): “Replacement of an elliptic crack by the equivalent circular crack is adequate with very good accuracy.” The number of defects of each type per unit length is equal to the projected area per unit volume, as illustrated in Fig. 10.3. A set of circular defects with radii in the range cc c,+()∆ and orientations roughly in the direction ΩΩ occupying a spherical cap of size ∆Ω on the unit sphere, as illustrated in Fig. 10.4, is said to be a defect set. (Cracks have symmetry such that a reversal of 180° leaves them unchanged. Thus, half the unit sphere is sufficient to characterize defect orientation.) A useful subdivision of half the unit sphere is illustrated in Fig. 10.5. With these restrictions, the number of defects per unit length in each crack set can be written ˜ cos Lc L cc=−πγφθ2∆∆ Ω , 10.5 where L is the number of cracks per unit volume exceeding c in radius, and the subscripts denote differentiation. The angles φ and θ are the usual polar coordinates (having nothing to do with polar axes), so that ∆Ω ∆ ∆=θφ θsin . 10.6 γ nr ~ Fig. 10.3 Illustration of a control volume of radius r and length l containing a random assortment of circular defects of area “a” used in estimating the number of defects per unit length ˜LL a=π, when the defects have a number density “ L” per 2π, per unit volume. 38Ω ∆Ω Fig. 10.4 Hemisphere d efining crack orientation. The cap of area ∆Ω and center Ω defines one crack set. 9 3 58 4 7 261 Fig. 10.5 Subdivision of a unit hemisphere into elements defining flaw orientation. The angle between PP12 and the crack normal N is denoted by γ. Now, observe that ∆∆Si Xkk= 10.7a and cosγ=Nikk , 10.7b where ∆Xk is the projection of the element of arc ∆S in a reference direction. Then, upon dividing Eq. 10.4 by ∆xj and considering the mean (in the sense of an ensemble average) deformation to be continuous, we may write uF c L u N cijd kj cik ,=−−∑12πδφθ ∆∆ Ω , 10.8 where Fkj−1 denotes an element of F−1. This relation can be put into spatial axes by means of Eq. A.66, so that un c L u cijd jci ,=−∑νπ δφθ2∆∆ Ω , 10.9 where ν==()−121 2 /ΓNV N . 10.10 The summation is taken over defect orientation, size, and type. 39The treatment of plastic deformation has a somewhat different character. We consider plastic flow to consist of the motion of dislocation loops in slip planes. This is, of course, a great idealization, since loops may vanish, move out of their own plane, lose their circular character, coalesce, become pinned, or be nucleated by thermal or athermal mechanisms. Equation 10.3 can be written ∆∆uuii=∑α α , 10.11 where the sum is taken over various types of defects and over the continuum regions between defects, which are assumed to have the same strain rate. Dividing by ∆xj, we find, on taking the limit, uuij ij,,=∑α α . 10.12 These gradients may be separated into symmetric and antisymmetric parts, as indicated by Eq. 3.3. Then, ddij ij=∑α α10.13 and wwij ij=∑α α . 10.14 Thus, the proposed superposition of velocity gradients implies the superposition of stretchings, which, in turn, implies the superposition of strain rates as indicated in Eq. 10.2. It also impliessuperposition of spins. This principle also implies superposition of energy rates. This conclusion is derived by operating on Eq. 10.13 with σij, and writing ˙edij ijαασ= . 10.15 The energy associated with each deformation mechanism is eα. This energy may go into elastic deformation, and thereby represents a reversible process; into heating, and thereby represents anirreversible process; into formation of new surfaces; and into other microscopic processes such as local kinetic energy. The energy balance is written ˙˙ee=∑α α . 10.16 The rate of energy deposition into each type of mechanism can be written ˙˙ea eαα= , 10.17 with aα a fraction less than one and aα α∑=1 . 10.18 40Each deformation mechanism involves a stretching which can be expressed with a constitutive law of the form DV , Dαα=()fσσσσ,ˆ, , 10.19 though it may possibly involve other variables. This determines the aα. However, this implies that WWαα=a , 10.20 for only if this relation holds can the constitutive laws embodied in Eq. 10.13 be made consistent with Eq. 10.12. This equation is not a necessary ingredient for calculating deformation, but once the deformation is known, it can be used to interpret the motion and determine the amount of spin associated with each type of deformation. If the stretching and spinning associated with each mode ofdeformation are applied to Eq. 3.31, we find ωωαα α=+ −()−wI V V ztr1 , 10.21 where zα is the vector associated with the matrix ZD V V Dαα α=− , 10.22 generalizing Eq. 3.21. Then, Eq. 3.5 can be resolved into ˙RRαα=ΩΩ . 10.23 In addition, if D is represented by a superposition, this suggests that Eq. 3.14 can be resolved into ˆBV D Vαα=2 , 10.24 and Eq. 3.18 can be resolved into ˙BV D Vαα=2 . 10.25 Thus, one can determine by integration Bα, Bα, BB=∑α10.26 and BB=∑α . 10.27 Similarly, the G reen and Signorini strain rates of Eqs. 3.15 and 3.19 can be integrated with the results εεεε=∑α10.28 and εεεε=∑α . 10.29 We can also define 41˙FG Fαα= 10.30 and, thereby, define Fα as the integral of ˙Fα. Then, a rotation can be defined by RF Vαααα=−1 . 10.31 These Rαα are not the same as those obtained by integrating Eq. 10.23. They are not orthogonal, nor are they the matrices that would be obtained by polar decomposition of Fα. This is not surprising, for rotations do not add. The superposition works only for rates. The separation of F and other quantities into parts, each associated with a particular kind of defect, leads to new insights. In particular, the energy may be separated into parts associated with each defect, and this lends some insight into the dominant features of the overall motion in specificcases. More research into types of deformation and the conditions that allow one to dominate, as in brittle-ductile transition, would be of great interest. 11. SUPERPOSITION WITH TIME-DEPENDENT STATISTICS If the statistics are time dependent as, for example, when cracks grow, the equations of Section 10 need to be generalized. One approach is to recognize that Eq. 10.3 for velocity increment has an analogue for displacement increment ∆∆ ∆vv viic id=+Σ , 11.1 where ∆vi denotes the displacement in the xi direction, ∆vic is the contribution from the continuum, and ∆vid denotes the contribution from defects. The summation is taken over all the individual defects. If now it is assumed that the cracks have a statistical distribution ˜Lα, then the contribution from defects can be expressed as ∆∆vS L vidi =∑˜ααδ , 11.2 where, as before, ˜Lα denotes the number of defects of type α per unit length, and the displacement across a defect of type α (α may characterize the radius) is written δαvi. The summation involves distributions of defects, which may change in the course of deformation. Thus, ˜Lα may be time dependent. As before, the defects are taken to be circular, not a bad assumption in view of the statement by Kachanov quoted in the paragraph following Fig. 10.2. The relation of Eq. 10.5continues to hold, but it should be recalled that L is time dependent here, and is considered to evolve according to an evolution law such as that of Eq. 3.2 of Vol. 2. Using Eq. 10.7a,b, we obtain ∆∆ ∆ ∆ Ω Σ vX c c L v Nid k cik =− πδφθ2 . 11.3 Here, it is noted that the summation is over crack orientation Ω as well as size, and could include type as well. We can find the displacement rate, the velocity due to defects, by differentiating Eq. 11.3 to obtain ∆∆ ∆ ∆ ∆ Ω˙ (˙ ˙) vu X c cL v L v Nid id k cicik == − + Σπδ δφθ φθ2 . 11.4 42Dividing by ∆xj , taking the ensemble average of deformations with the same statistics, and letting ∆S go to zero, we have GFc c L v Lv Nijd kj cicik =− +−12Σπδ δφθ φθ ∆∆ Ω(˙ ˙) . 11.5 Noting, as before, Eqs. A.66 and 10.10, we get the part of the velocity gradient due to defects Gc c L v L v nijd cicij =− +Σ∆∆ Ω Γπδ δφθ φθ2(˙ ˙)/ . 11.6 The first term in parentheses is a contribution from the unsteady statistics that does not occur in the previous derivation. The second term is the same as Eq. 10.9. However, we use 1/ Γ rather than υ of Eq.10.10, since it is difficult to distinguish between italic v ( v) and Greek υ. The deformation of foams provides a nice example of the GSSR, for the total deformation of a solid filled with spherical voids can be shown to involve two parts that can be added, one due to the matrix material and the other involving collapse of the voids. This is discussed by Dienes and Solem (1999). 12. CONCLUSIONS AND CLOSURE This report began as a collection of notes concerning the feasibility of a novel theory of brittle materials in which the defects in an ensemble grow and coalesce, increasing the permeability, reducing strength, and forming hot spots, while the material experiences large strain and rotation. Such a theory was needed to understand the anisotropic response of oil shale and other geologicmaterials to blasting, the dynamics of gross structural failure, and the sensitivity of explosives and propellants to impact. Many other applications are extant, such as collapse of buildings, formation of the moon, and stimulation of reservoirs. When this work began in the 1970s, continuum theories shed little light on the evolution of porosity and permeability or on the formation of hot spots in propellants. The work on NAGFRAG by Seaman and others at SRI (Seaman et al. 1972) appeared provocative but did not account for shearbands or shear cracks and made no reference to classical fracture theory. The work on associated flow laws and, later, nonassociated flow laws, 6 promised no insight or accounting for what happened on the mesoscopic or microscopic scales, and could not account for permeability or hot spots. For thesereasons a new direction was taken. The goal was to have an approach that accounted for both microstructural behavior and large deformations. This report contains the framework for such a kinematic theory, though not in the complete form that would be expected of a formal treatise—it isstill a collection of notes, but it should provide a useful reference for those interested in Statistical Crack Mechanics and its implementation in the SCRAM algorithm for material response. It is shown that polar decomposition provides a useful and unique basis for a theory of large deformation, allowing us to place many of the standard measures of stress, strain, stress rate, strain 6 The associated flow law was based on the idea that the strain rate is parallel to the normal to the yield surface (in six-dimen sional space), as discussed by Hill (1950). However, the use of this normality rule to account for dilatancy, as proposed by Drucker a nd Prager (1952), predicted far too much dilatancy (pore volume) due to shear, as discussed by Sandler and Baron (1985); so, vario us modifications were made to force the calculations to conform to experimental data. Experiments of Spitzig and Richmond (1984) demonstrated that there is no dilatancy at all in metals though the yield stress depends strongly on pressure, pretty much inva lidating the whole idea of normality (and related concepts) in the view of this writer. 43rate, rotation, and vorticity into a coherent framework. It allows us to show that the polar rate of rotation is different from the vorticity, that they are related by an algebraic formula, and that the polar rate is the appropriate choice for computing stress rate. It follows that vorticity does not provide a suitable measure of rotation for solids. In particular, it is shown that irrotational flows (zero vorticity)normally involve material rotation. Thus, the standard formulas for stress rate are not valid at large deformations, they exhibit unphysical periodic behavior under monotonic loading. The alternative, based on the polar stress rate, avoids this difficulty. It also turns out that polar decompositionprovides a useful relation between stretching and Signorini strain rate, which, surprisingly, improves the stability of finite-element calculations. It has not been previously emphasized that stretching (the symmetric part of the velocity gradient) is not associated with any unique strain, casting doubt on itsutility as a measure of strain rate. On the other hand, the Signorini strain rate is the polar rate of Signorini strain. Stretching and strain rate are related in a simple way (Eq. 3.6) useful in numerical calculations. The utility of the results from polar decomposition is shown in a variety of examples. The main goal of this effort was to generalize the concept of superposition of strain rates. The traditional use was in superimposing elastic and plastic strain rates, an idea due to Reuss, but it was commonly thought that this concept is useful only for small deformations. Its utility lay in smoothingthe transition between elastic and plastic states. In the current work, it is shown that the superposition principle is valid for large deformations as well, and that it applies when the Signorini strain rate is used as a measure of strain rate rather than stretching. It is considered especially useful to extend thesuperposition principle to a GSSR that allows us to combine the effects of various physical mechanisms by adding their strain rates. Thus, we can add the strain rates due to elasticity; plasticity; opening, shear, growth and coalescence of microcracks; and high-pressure equation of state. Thisallows for a very general formulation of constitutive laws, accounting for the failure of materials, the compaction of foams, and a myriad of other physical processes. This is an alternative to the popular concept of product decomposition, which takes the deformation gradient as a product of elastic andplastic contributions. 7 That concept has the drawback of leading to very complex algebra, even in the case of two contributions, elastic and plastic deformation. Were one to try to include fracture, high- pressure response, and void formation, the algebra would be quite intractable. GSSR is based on the observation that the effects of defects can be added so long as they are geometrically separate and statistically independent. Large separation is not necessary, as illustrated by Kachanov (1984) in calculations of crack intersections. One effect of this rule is to allow forgreatly reduced modulus of elasticity due to microcracking, and to allow for plasticity in the presence of this reduced elasticity. To summarize GSSR, it extends the standard use of superposition in three ways. First, it is derived from a more fundamental basis, rather than presented as an ad hoc concept (a sort of deus 7 Product decomposition takes the deformation gradient as the product of elastic and plastic parts, FFeFp=, claiming that the plastic part is physically superposed on the elastic part. It seems equally plausible, however, to argue that the plastic part dominate s, so one should write FFpFe=. This approach to finite deformation, which is often considered exact, leads to great algebraic complications. These are avoided by many researchers by arguing that for small strains, the rates of stretching due to elasticity and plastici ty can be added, i.e., DDe+Dp= . On the other hand, in GSSR, this latter relation is considered exact, a logical consequence of defect statistics and polar decomposition. 44ex machina ). Second, the derivation shows that GSSR is valid for arbitrarily large overall deformation. Third, superposition is valid for combining any number of physical processes. These generalizations may be taken with a grain of salt, since counterexamples can surely be found involving strong defect interactions or unusual materials. Still, GSSR is believed to be a usefulextension of the more traditional concept. Appendices discuss a number of issues that would detract from the flow of the main document. Matrix concepts are reviewed in Appendix A, emphasizing simplicity and relevance to kinematics. Itstarts with the simplest concepts and ends with some new results. Appendix B compares the results of the main text with more traditional views on strain. The difficult subject of allowable formulations of constitutive laws is addressed in Appendices C and D, in which we attempt to delimit the use oftensor concepts in mechanics. In Appendix E, a very specialized calculation is undertaken to reconcile two apparently different points of view relating polar rate of rotation and vorticity (Hill 1978, Dienes 1978a), and it is shown that they are actually consistent. [This is in contrast to theimplication by Nemat-Nasser (1983) that they are different.] In fact, this work can be considered an effort to reconcile various points of view on many aspects of material behavior. It began with the effort to reconcile the rotation R obtained by polar decomposition and the vorticity, ω. The most recent effort concerns the formation of macroscopic cracks from microscopic defects, and reconciliation of continuum mechanics with the behavior of an ensemble of defects that exhibit discontinuities (Dienes 1996). 45APPENDIX A MATRIX THEORY Summary of Matrix Notation and Concepts A matrix is an array of numbers. It can be expressed in a variety of ways, such as AAA AAA AAAAij11 12 13 21 22 23 31 32 33    =()=A . A.1 In general, the numbers Aij can be complex, and the arrays need not be square, but in the theory of deformation the matrices of interest are usually 3 x 3, as in Eq. A.1, and the numbers are real. The transpose of A is obtained by diagonal reflection, viz., AAA AAA AAAAjiT11 21 31 12 22 32 13 23 33    =()=A . A.2 Symmetric matrices have the property AAT ij jior A A== , A.3 whereas, for antisymmetric matrices, AAT ji ijor A A=− =− . A.4 Any matrix can be written as the sum of symmetric and antisymmetric parts, viz., AA A=+sa , A.5 where AA AsT=+()1 2A.6 and AA AaT=−()1 2 . A.7 In the case of a 3 x 3 matrix, the symmetric matrix contains 6 independent elements and the antisymmetric matrix contains 3 independent elements. The sum of two matrices is given by AB+=()+()=+() ABA Bij ij ij ij . A.8 A linear combination of matrices is given by ab a A bBij ij AB+= +() . A.9 46The product of two matrices can be written C=AB A.10 or CA B A Bij ik kj kik kj ==∑ . A.11 It is often convenient to drop the sum over repeated indices, as indicated in the second part of Eq. A.11. The summation convention indicated above for matrices is more rudimentary than theconvention in tensor analysis, which brings in a variety of physical and geometrical concepts. The transpose of a product is the product of the transposed matrices in reverse order, CB ATT T= A.12 or CB A A Bji ki jk kjk ki ==∑ . A.13 Note that AB BA≠ in general and that symmetry of A or B does not imply any symmetry properties for the product. Inverse Matrices The inverse of a matrix is written A−1. It has the property that when either premultiplied or postmultiplied by A, the product is the identity matrix I=()=    δij100 010001 , A.14 where δij is the Kronecker delta. Thus, AA A A I−−==11 . A.15 The transpose of the inverse is the inverse of the transpose, allowing us to define A−T unambiguously: AA A−−−()=()=11 TTT . A.16 The proof is given here to illustrate the power of matrix methods. Applying Eqs. A.12 to A.15 we have AAII−() ==1TTT . Then, postmultiplying by AT()−1, one finds A.16. It is very awkward to show this with index notation. 47Determinants In the preceding paragraph, the existence and properties of inverse matrices were treated in outline form. Consider now the calculation of inverse matrices. For this, it is necessary to make use of determinants, written as AAij==A . A.17 For a 2 x 2 matrix AAA AAAA AAij== −11 12 21 2211 22 12 21 . A.18 For a 3 x 3 matrix Ae A A Aijijkijk =123 , A.19 where eijk is the classic permutation symbol: if any 2 of ijk are the same, it is zero. Otherwise, it is 1 if ijk can be obtained from 123 by interchanging pairs of indices an even number of times, and –1 if obtained by an odd number of interchanges. A cofactor aij is formed by striking out the ith row and jth column of the matrix A and forming the determinant of the remaining terms, with appropriate sign ()−+1ij. Thus, for example, aAA AAAA AA232311 12 31 3211 32 12 31 1=−() =− −()+A AA A A13 21 22 23 33 , where the terms to be struck out are indicated in copperplate bold. Then, the inverse of a matrix is given by Aa Aij ji−=1/ . A.20 This is sometimes called the reduced cofactor. Also, aA Aijkj ik=δ or aaAITA= A.21a is called the Laplace expansion of A. If the Aij are considered functions of a parameter s, then the derivative of the determinant A is given by dA dsad dsAij ij= A.21b or 48d dstrd dsTAA=   aa , A.22 where aij is the cofactor of Aij. An important application of this formula is the proof of Euler’s identity concerning conservation of mass ˙,JJ uii= , A.23 where the superimposed dot denotes the time derivative, and Jv voo== =//ρρF A.24 denotes the relative volume of an element of mass of specific volume v and density ρ. The zero subscripts denote initial values. The second term on the right of Eq. A.23 denotes the divergence of the velocity field, and F denotes the deformation gradient. To prove Eq. A.23, write Eq. 3.1 as ˙FG F= . A.25 Differentiating Eq. A.24, applying Eq. A.21 and Eq. A.22, and denoting the cofactor of Fij by fij, ˙˙, Jd dtfF fGF G uij ij ij ik kj ii i i === = =FF F , verifying Eq. A.23. Alternatively, in matrix notation, ˙˙Jd dttr f tr f trTT T==()=() = FF F G F G . A useful formula given by Flugge (1972) is δδ δ δδ δ δδ δilim in jljm jn kl km knijk lmnee= . A.26 This formula is, in fact, only a special case of a formula that nicely characterizes the behavior of determinants, viz., AAA AAA AAAAe eor os ot pr ps pt qr qs qtopq rst= , A.27 where A is the value of the determinant AAA AAA AAAA11 12 13 21 22 23 31 32 33= . A.28 Expansion of the determinant in Eq. A.26 leads to eeijk lmn il jmkn il jnkminjl km in jmklim jnklimjl kn = −+−+− δδ δ δδ δ δδδ δδ δ δ δ δ δ δδ . A.29 49Contraction leads to the following useful relations: eeijkimn jm knjnkm =−δδ δδ , A.30 eeijkijn kn=2δ , A.31 eeijk ijk=6 . A.32 Eigenvectors and Eigenvalues Matrices appear in the solution of simultaneous equations, which can be written Ax yij j i= , or Ax y= . The solution can be written formally as xA y=−1 provided that the determinant of A is not zero. An important variation on this theme occurs when the vectors x and y are parallel, so that Ax x Ix==λλ . A.33 Here, λ is described as an eigenvalue of A. Since Eq. A.33 is homogeneous in x, it has a nonzero solution only if the determinant of the matrix AI−λ is zero. If A is an n x n matrix, then there are n values of λ, denoted by λα, which are eigenvalues; each has its own eigenvector, xα, for which Eq. A.33 is satisfied. Expanding the determinant results in a polynomial in λ. The solutions satisfy the n relations Ax xαα αλ= A.34 not summed on α. In some applications, there may be double roots, so that the number of distinct eigenvalues is less than n, but this does not present an important aspect of the theory of deformation.It is generally useful to deal with normalized eigenvectors having the property xxi iiαα∑=1 . This relation is subsumed in the more general relation xxiiαβαβδ= A.35 expressing orthogonality as well as normalization; the summation on i is now understood. When A is symmetric, the λα are real. This is often the case in the theory of deformation. However, when 50dealing with rotations, A is orthogonal and, consequently, not symmetric. Then, it can be shown that one of the λα is real (assuming 3 dimensions) and the others are complex. This subject is discussed in detail by Goldstein (1959). To show that the λα are real when A is symmetric, suppose for a moment that they are complex, and let an overbar denote the complex conjugate. Then, conjugation of Eq. A.34 leads to Ax xααααααλλ= . Switching to index notation, and then operating with xiα, we find xAx xxii j j iiααααααααααλλ= Similarly, operating on Eq. A.34 with xiα leads to xAx xxii j j iiααααααααααλλ= . But if AAij ji=, the left sides of these two equations are equal. Then, the λα are equal to their conjugates, implying that they are real, which was to be shown. Similar arguments lead to Eq. A.35. A symmetric matrix can be expanded into the form AX X=ΛΛT , A.36 where X denotes the matrix of normalized eigenvectors, X=()xiα . A.37 In this notation, A.35 can be written XX IT= . A.38 Matrices with this property are said to be orthogonal. If a matrix is interpreted as defining a quadric surface Ayyij i j=1 , this has a simple geometric interpretation, for quadrics of this form have principal axes about which they are symmetric. The transformation X rotates the coordinate axes to the principal axes of the quadric. Orthogonality is, thus, the algebraic equivalent of a rotation in geometry. The expansion of Eq. A.36 can be regarded as a reduction of A to diagonal form. It is useful in many contexts. For example, any integer power of a matrix can be written AX Xnn T=ΛΛ , A.39 in view of Eq. A.38. It readily follows that Eq. A.39 holds for any rational or negative n. The extension to irrational powers is beyond the scope of this material. The eigenvalues of Eq. A.33 can be found in practice in numerous ways, but one important way in analysis is to notice that Eq. A.33 can be written in the form AI x−() =λ0 , A.40 51which is homogeneous in x. This equation has nontrivial solutions only if the determinant of the coefficients is zero, AI−=λ0 . A.41 This determinant can be expanded as a polynomial of degree n in λ, λλ λn mm mn af−= ()=−− =∑ 11 10 , A.42 where the am are “invariants” of A. This term expresses the fact that if A is transformed to different axes, the am are unchanged. The n roots of Eq. A.42 are the eigenvalues of A. Equation A.42 is often called the characteristic equation, and the roots the characteristic values, a term equivalent toeigenvalues. These properties can be used to show that a matrix satisfies its own characteristic equation, the Cayley-Hamilton theorem. This is of considerable value in many aspects of the analysis ofdeformation. To show this, note that in view of Eq. A.39 ffTAX X()=()ΛΛ , where the polynomial function f is defined by Eq. A.42. Since ΛΛ is diagonal, we have, for n = 3, ff f fΛΛ()=() () ()    λ λ λ1 2 300 0000 , and each of the diagonal terms of this matrix is zero in view of Eq. A.42. The extension to general n is obvious. Thus, fA()=0 , A.43 and A satisfies its own characteristic equation. If the λα are all positive, then the matrix is said to be positive definite. It is often convenient to refer to the quadric in X, ffTAX X()=()ΛΛ. The quadric is an ellipsoid when the λα are all positive. There are many criteria concerning the conditions under which the λα are positive, but these are beyond the current scope. Rotation A matrix having the orthogonal property RRT=−1A.44 represents a rotation. To see that this algebraic property represents the geometric process that we call a rotation, consider the process in which an element of length ds is unchanged, though the solidexperiencing this process is deformed. (We use “deformation” to include stretch and rotation). Then, 52if dxi defines the element after rotation and dxi′ the element before rotation, we must have, using Eq. 2.1, ds dx dx F dx F dx dx dxii i j jik k jj2== ′′=′′ . This holds if FFijik jk=δ , that is, FF IT= . Thus, FFT=−1 , showing that the length of an element of material is unchanged if the transpose and inverse of the deformation matrix are equal, and this is what is meant by a rotation. An orthogonal matrix can be expressed as the exponential RQ=eφ , A.45 where Q is a skew-symmetric matrix representing the axis of rotation and φ is the angle of rotation. To show this, let r be a vector that is unchanged by the rotation R, so that rR r=λ λ=1, . A.46 That is, if one of the eigenvalues of R is unity, R represents a rotation about r. To prove this, first apply Eq. A.44 and then Eq. A.46 to the product given below on the left, with an overbar denoting the complex conjugate, rRrR rr r rii jkk j ii j j==λλ . Then, on division by rrii, one finds that λλ=1. Now, there are 3 eigenvalues for a rotation in three dimensions which are the roots of the cubic equation given as Eq. A.41. Since at least one root mustbe real, it must be ±1. But R = I, at the beginning of deformation, so λ=1. Since rotations are continuous, λ=1 must continue to be the real root as deformation proceeds. It is straightforward to show that there are no other real roots. This shows that Eq. 46 holds.To continue the proof of Eq. 45, we relate the antisymmetric matrix Q to the unit vector r by Qe r Qik ijk j ik == (),Q . A.47 Now, expand the exponential of A.45 in a power series, so that RI Q Q=+ + +φφ222/! . . . . A.48 But it can be shown by means of Eq. A.30 that Q2=−()rrili lδ , A.49 53QQ42=− , A.50 QQ22 2nn+=− , A.51 and QQ21 21nn+−=− . A.52 Care must be expressed in manipulations involving Q because its determinant is zero and, consequently, it has no inverse. Thus, n in Eqs. A.51 and A.52 is restricted to be a positive integer.Using these relations, Eq.A.48 can be rearranged so that RI Q Q=+ +− ()sin cosφφ12 . A.53 It readily follows from Eq. A.47 that Rr=r, proving that this representation of R generates a rotation about r. By taking the trace of both sides of Eq. A.53, it can be shown that cosφ=−()1 21trR . A.54 The angle φ is the amount of rotation represented by R about the axis of rotation, r. This can be verified by considering a rotation S of coordinate axes such that SR S P S R ST==−1 , A.55 where P=−    cos sin sin cosθθ θθ00 00 1 . A.56 The similarity transformation indicated in Eq. A.55 is discussed by Bellman (1960), Gantmacher (1959), and other treatises on matrices. Since SSijkj ik=δ , it can be immediately shown that tr trPR= , which, in view of Eqs. A.53 and A.56, is tantamount to 12 12+= +cos cosφθ , or θφ=. Thus, the rotation represented by R is equivalent to a rotation about an axis r, the real, normalized eigenvector of R, through an angle φ determined by the trace of R. If R1 and R2 represent successive rotations, their order matters, for RR12 is different from RR21. However, any two rotations are equivalent to a single rotation about a suitable axis, for it is readily shown that 54RR RR12 121() =()− T , so that the product has the orthogonal property. It can be shown that the angle of rotation is the same whatever the order of the rotations R1 and R2. To see this, let φ1 and φ2 denote the two rotation angles, and let cosα=rrii12A.57 so that α is the angle between the two axes of rotation. Then, it can be shown that trRR12 1 2 1 22 12 1 2 12 1=+ − + + −− () (cos cos sin sin cos ) sin cos cos cos cosφφ φφα αφ φ φφ A.58 Since φ1 and φ2 can be interchanged in this formula, and cosα is symmetric in r1 and r2, the order of rotations does not affect trRR12 and, hence, the angle of rotation. Of course, the axis depends on the order. If the two rotations have the same axis, so that α=0, it follows from Eq. A.58 that the rotation angle is the sum of φ1 and φ2. It can also be shown that when the rotation is time dependent, ΩΩ= = + + − () −()˙˙sin˙cos˙˙ RR Q Q QQ QQTφφ φ 1 A.59 and ˙˙ ˙˙ QQ QQ−=−()rr rrij ij . A.60 The three terms on the right of Eq. A.59 represent rates about perpendicular axes. This can be shown by noting that the angle between two unit vectors is simply related to the inner product of the corresponding antisymmetric matrices, for, using Eq. A.31, AB e aeb abij ij ikj k ilj l k k== 2 . Thus, if the inner product is zero, the corresponding vectors are perpendicular. This result can be exploited to show that the three terms contributing to ΩΩ in Eq. A.59 represent rotations about three perpendicular axes. Deformation of an Element of Area In the course of deformation, perpendicular material fibers do not remain perpendicular, nor do perpendicular planes. A material fiber characterized by dXi is transformed into Fd Xij j in the course of deformation. The change in element of area, however, requires a more elaborate calculation, as follows. The directed elements of area of a pair of fibers dxi1 and dxi2 are ds dx dx e F F dX dX ekijijk il jmlmijk ==12 12 . A.61 Now, it can be shown that FF e e Hiljmijk lmn nk=F , A.62 55where HF=−1 . A.63 (This can be obtained from the determinant expansion ae a a a elmn il jmkn ijk= , which is a slight generalization of Eq. A.19, by multiplying by the inverse Alp). Then dx dx e H e dX dXijijk nk lmn l m12 12=F . A.64 Thus, a differential vector area ed X d X d Slmn l mno 12= in reference axes is transformed into ds H dSkn k no=F in current spatial axes, or, in vector notation, dd doT osF S F F F S==−−1 . A.65 The last term on the right is the same as Eq. 1.12.3 of Eringen (1967) or Eq. 1.32 of Hill (1978). In the absence of strain, this reduces to the simple rotation indicated by Eq. 3.13. The unit normal to thedeformed element of area is given by normalizing d S and doS nFN N VN L N=() =−−−T 212/ , A.66 where N is the unit normal in reference axes and LF=−ΓT , A.67 with Γ=()−−NV N21 2 . A.68 But L is not a rotation matrix, as it depends on the normal N, nor does it satisfy Eq. A.44. The quantity Γ can be written in terms of spatial variables by rewriting Eq. A.66 as nF Niik k=Γ . A.69 Since NNkk=1, it readily follows that Γ2=nBnii jj . A.70 Thus, Γ2 can be regarded as the normal component of position . (Just as traction is a force per unit area related to stress by Tnii j j=σ, so can position be related to stretch by PB nii j j= .) The derivative of L is needed in some calculations, requiring the derivative of an inverse matrix. To find the needed relation, take the derivative of Eq. A.15. Then, ˙AA A A−−+()=110d dt , 56and it follows that d dtAA A A−− −()=−11 1 ˙ . A.71 The derivative of Γ can be found by using the result ˙ ˙/ nG n n=− +()TΓΓ , A.72 which follows from differentiating Eq. A.66 and using Eqs. A.71 and 3.1. Then, since nnii=1, it follows that nnii˙=0 . A.73 Applying this to Eq. A.72, we find, using Eq. 3.3, that ˙/ΓΓ =nDnii jj , A.74 since the antisymmetry of Wij results in nWnii jj=0 . A.75 Alternatively, Eq. A.74 can be found by differentiating the relation Γ−−=21NB N A.76 that follows from Eq. A.67, but the calculation is considerably longer. Using Eq. A.75, Eq. A.72 can be written as ˙nW n nDn I D n =+ −() . A.77 If n is chosen so that it lies parallel to a principal direction of D, then the second term on the right of Eq. A.77 vanishes, leaving ˙′=′nW n , A.78 where the prime denotes a vector that is aligned with one of the principal directions of D. This result is erroneously described by Truesdell and Toupin (1960) and reproduced by Eringen (1967) as “the spin is the angular velocity of the principle axes of extension.” The correct statement is, “A materialline element m of unit length that is aligned with one of the principle directions of D rotates with velocity ˙mW m=.” Note that the skew character of W leads to mm•=˙0 . It is no coincidence that m and n satisfy the same relations, since they represent dual triads. An n vector is the cross product of two material vectors, the pair defining the orientation of a surface element. 57APPENDIX B REMARKS ON THE MEASURE OF STRAIN For small deformations, the analysis of strain presents no uniqueness problem, but for large deformations, numerous competing definitions have been proposed, of which many have been discussed by Truesdell and Toupin (1960). The choice of strain measure is of more than academic interest, for mechanical data are usually presented in the form of stress-strain curves. Unfortunately,the choice of strain measure is rarely discussed by experimentalists. It is, however, crucial to nonlinear material science that theory and experiment should refer to the same thing, and desirable that the choice should be a reasonably good one. In Appendix C, the Signorini strain εε= −()1 2BI, as given by Eq. 3.7, is compared with the two most common strain measures, that of Green and St. Venant, EB I=−()1 2, and that of Almansi and Hamel, eI B=−()−1 21, from the point of view of metric tensors. In this appendix, some general remarks are made about the ambiguities in finite- deformation strain. In Truesdell and Toupin (1960), the problem of simple shear is discussed at length, and it is shown that second-order terms appear in the strain tensors described above with E2222=ε, e2222=−ε, and ε220= in current notation. The fact that these alternatives are entirely different is not discussed! In fact, the measures E, e, and εε are different representations of the same physical strain. The theory underlying this apparent discrepancy is resolved in Section 3 a nd an example is discussed in detail in Section 5. The relations between various theoretical measures of strain are discussed in Appendix C. In this section, we are more concerned with the actual measurement of strain. The approach taken by Hill is also discussed, and it is shown that his strain measures are fixedin the material. Thus, though his treatment is said to be general, the Signorini strain is not one of those included in Hill’s general theory. Strain gauges can be used to measure the Green-St. Venant strain, but do not provide any information about rotation. This may not matter for some purposes, but there is a risk that stress may be measured (or inferred) in one system and strain in another. This will not matter much in small deformations, but could lead to large errors when the deformation (stretch and rotation) is large.Strain gauge measurements are made in polar axes. They could be obtained in space axes by application of Eq. 3.10 if R were known, but R cannot be deduced without further measurements. Consequently, it appears that the spatial (Signorini) strain cannot be obtained with strain gauges,though the Green-St. Venant strain can be found in this way. Of course, strain gauges are not reliable at large strains anyway, so such approaches are not of interest for large deformations. A more general approach to measurement of strain is to observe the deformation of a square grid of lines inscribed on the test sample. The deformation gradient F for the surface can be obtained, approximately, by the finite difference Fx Xiji j~∆ ∆ , B.1 where ∆Xj denotes the initial grid length and ∆xi the changing grid length. The stretch and rotation can be obtained, in two dimensions, by computing B with Eq. 2.3 and the V from Eq. 8.9. The 58rotation follows from Eq. 2.2. The spatial (Signorini) strain can be obtained from Eq. 3.7, and material (Green) strain follows from Eq. 3.17. This approach to measurement of strain is not used much (if at all), though it would appear to lead to more complete results than strain gauges for large deformation. Extension to three dimensions is possible, in principle, by use of internal markers andprecise depth measurement, but is probably very difficult. Alternatively, thickness measurements of thin sheets might be used to obtain the full three-dimensional strain matrix. A different approach to the analysis of strain has been taken by Hill (1978), who defines a general measure of strain as ˜˜Eeij iL jL=∑αα ααµµ , B.2 where the µαiL are equivalent to S (Eq. 2.12), and ˜ / emm ααλ=−()212 . B.3 For m = 1, this is the Green strain. For m=1 2, it reduces to engineering strain. For m = 0, it is the logarithmic strain, and for m = -1, it is the Almansi strain. In the current notation this is equivalent to ˜˜ES e S=T , B.4 where ˜e is the diagonal matrix of strains. Though Hill makes use of different notation and emphasizes vector triads more than matrices, his results are identical to those given here. Hill’srelations and the corresponding matrix equations are listed in Table B-2. Unit vectors in the directions of the principal axes of the stretch ellipsoid measure in space axes (Hill’s “ Eulerian” frame) are denoted by µµαE. The µµαL are the same unit vectors in polar axes (Hill’s “ Lagrangean” frame). The Signorini strain given in Eq. 3.7 would be εε= −()∑ ×1 221λλαα ααααααmmEEB.5 in Hill’s notation. It lies outside the realm of strains included in Eq. B.5. It is common for authors and computer programmers to describe their work as “general” or “unified” when it combines several (or perhaps only two) ideas. Keynes’s general theory, Einstein’sgeneral and unified theories, and Hill’s (1978) “unified and definitive” article are sometimes considered by readers (or those who have not read, but have heard of, those works) to encompass everything that is relevant. This is a great danger indeed. Caveat lector . Though a magnificent contribution, Hill’s article does not discuss ˙,ˆRRTσσ, Signorini strain, or the topic of strain rate in a manner suitable for computation or measurement of large deformation. These subjects deserve more attention. 59Table B-2. Comparison of Hill’s and Current Notations Current Hill G ΓΓ D εε W ΩΩ FA DWG−=T ′ΓΓ F−T B SµµααµL iL[]=() see note TµµααµE iE[]=() see note VSS=Tλλ ΛΛµ µµµ =×∑λαα α αLL R=TSR=×∑µµµµαα αEL TR S=TµµµµααEL=R Note: The square bracket denotes a one-dimensional array. The one-dimensional array of vectors is a matrix. 60 61APPENDIX C REMARKS ON THE TENSOR CHARACTER OF STRAIN In much of the literature of continuum mechanics, the quantities CF F=T and BF F=T are said to be tensors on the grounds that they appear in quadratic forms representing the square of arc length, and they satisfy certain transformation laws. In this appendix, a more physical and restricted notation of tensor is considered which requires that the transformation involve coordinates that characterizecurrent physical space and not just some mathematical reference space. Thus, the quantity represented by the components Aij is considered a contravariant physical tensor only if it transforms according to ′=∂′ ∂∂′ ∂AAx xx xij kli kj l , C.1 and a covariant physical tensor only if it transforms according to ′=∂ ∂′∂ ∂′AAx xx xijklk il j , C.2 where the xi and ′xi are coordinates of the current physical space in which physical processes take place. Many authors call a tensor a quantity that transforms according to Eqs. C.1 or C.2 when the xi are any system of coordinates. With the more limited notion of physical tensor used here, some quantities that are called strains in the literature are not tensor invariant and, thus, are not physical tensors. According to this restricted definition, B (C in Truesdell’s 1952 notation), defined in Eq. 3.16, is not a physical tensor, for under a change of coordinates ′=∂′ ∂∑∂′ ∂=∂ ∂∂ ∂∂′ ∂∑∂′ ∂Bx Xx Xx Xx Xx xx xkli kii lm kn li mii n , C.3a,b which does not have the form of Eq. C.2. Though B represents a mathematical tensor in the reference space with coordinates Xi, in view of Eq. C.3a, this does not have direct physical significance. Though often called a tensor in the literature of continuum mechanics, it is not a tensor in the sensecommonly used in physics. The deformation matrix BF F=T, or Bx Xx Xiji k kj k=∂ ∂∂ ∂∑ , C.4 does transform as a physical tensor, for application of the chain rule shows that it transforms as indicated by Eq. C.1. It is no coincidence that B is said to be “frame indifferent” ( Truesdell and Toupin 1960, Truesdell 1966) while B (Truesdell’s C) is not, for the same sorts of transformations are involved in specifying frame indifference as for showing tensor invariance in the sense used inphysics. 62Let us consider now the Cauchy deformation cX xX xkmK k KK m=∂ ∂∂ ∂∑ . C.5 This metric appears when the euclidian element of arc in reference space dS dX dXkk K2=∑ C.6 is expressed in physical space as dS c dx dxkmkm 2= . C.7 Application of the chain rule shows that in the ′xm coordinate system ′=∂ ∂′∂ ∂′ccx xx xmliji mj l , C.8 which is a transformation of the form indicated by Eq. C.2. Thus, c is a covariant physical tensor. It appears to be useful in tensor analysis in many instances to adopt matrix notation. Let us introduce here the transformation matrix γγ=()=∂ ∂′   •γjii jx x . C.9 Then, the contravariant tensor property indicated in Eq. C.1 can be written ′=−−AAγγγγ1T , C.10 and the covariant tensor property indicated in Eq. C.2 can be written ′=AAγγγγT . C.11 Then, if A is a tensor (contravariant or covariant), it follows that its inverse, ′()=−−− −AA111γγγγT , C.12 is also a tensor (covariant or contravariant). Now, in matrix notion, Eq. C.5 is cFF B==−− −T11 . C.13 Thus, if B is a contravariant physical tensor, then c is a covariant physical tensor, as concluded above on the basis of a longer calculation. These results can be summarized in tabular form as indicated in Table C-1, illustrating three ways to characterize the element of arc in continuum mechanics and the corresponding three measures of stretch and strain. In the current view of the theory of deformation, quantities that transform like tensors only in the mathematical sense are called “metrics.” The term “tensor” is then reserved for quantities that transform in this way in physical space, those called “physical tensors” above. 63Table C-1. Comparison of Various Arc Meaures, Metrics, and Strains Rotated reference coordinates (polar axes)Euclidian physical space with reference coordinatesEuclidian space of reference coordinates dB d X dXklklσ2= ds dx dx B dX dXkk KMKL 2== dS dX dX c dx dxII kmkm 2== Signorini strain tensor Green-St. Venant metric Almansi strain tensor εε=−()1 2BI εε=−()=1 2BI E eI c=−()1 2 dXRdX= εεεε=RRT The first column of Table C-1 characterizes an element of arc in the material as it deforms, with dX selected as the differential element to eliminate the rotational part of the deformation. Note that the rotation indicated in the last row of the first column rotates the reference axes so as to eliminate the rotation from the deformation. This is the inverse of the rotation indicated by Eq. 3.9. The metric in reference axes is noneuclidian. The arcs in such a space interpolate between the atoms characterizing the material. Integrating d σ along the arc C0 in the rotated reference axes results in the length of the deformed arc C. To show this, put dX RdX V FdX V dx == =−−11 (using Eqs. 3.9 and 2.2), and note that B cancels the two occurrences of V−1in the expression for dσ2. Thus, the length of the arc is that of the map of C0 in the current Euclidean space. The metric B is a physical tensor related to the stretching D by Eq. 3.14. The Cauchy stress for an isotropic elastic material can be expressed in terms of B, as indicated by Eq. 4.25. In the second column, the element of arc of euclidean physical space is defined, as well as its characterization in the reference space involving the initial coordinates. Note that dx dx dx dxii ii=, as discussed following Eq. A44. The elements of arc ds and d σ are actually the same, but the representation is different. The metric B is not a physical tensor, as indicated in the discussion of Eq. C.5. The polar elastic stress for isotropic materials can be expressed in terms of B, as indicated by Eq. 4.24. For anisotropic materials, it is necessary to use the more basic formulation of Eq. 4.12. The Euclidian element of arc in reference space is given in the third column. The metric c transforms like a physical tensor, but its rate is not related to the stretching in any simple way like Eq. 3.14. For small deformation, all three strains are equivalent. The three strains in the table are closely related, so that any one determines the other two, though in the case of the Green-St. Venantstrain it is necessary to use the rotation R to make the connection, viz., ER==εεTεεR . C.14 There is a fourth measure of arc, as pointed out to me by W. Cook, which seems to have not aroused any previous interest. It arises from the lack of symmetry in the three strains of Table C-1, 64which can conceptually be considered as part of a 2 x 2 array whose elements are spatial, and rotated axes and spatial or reference axes. We can define by analogy with the third column of the table eR e R IB== −()− T1 21 , C.15 with BF Fb−− −== =∂ ∂∂ ∂  ∑11 T i jk j jX xX x . C.16 The associated element of arc is db d X dXikik Σ2= , C.17 representing a Riemannian geometry with coordinates Xi and metric bik. However, b does not transform like a physical tensor. Consider now the velocity gradient GF F==∂ ∂  ∂ ∂  −˙˙1x XX xi jj k . C.18 Analysis of the metric properties of G requires considerations from tensor analysis, rather than simply the tensor algebra used in the analysis of strain. The covariant derivative of the velocity vector is uu xi jkujii jk;=∂ ∂+ , C.19 where i jkgj k lil =[], C.20 is the Christoffel symbol of the second kind, and ik l g g gil k kl i ik l,,,, []=+ −()1 2C.21 is the Christoffel symbol of the first kind. Here, the gij are the components of the metric tensor of the space in which the velocity is imbedded, and commas denote partial differentiation. Now, the Xi and t are independent variables, so one can write ∂ ∂=∂ ∂∂ ∂  ∂ ∂u xtx XX xi ji kk j . C.22 Let us consider the effect of a coordinate transformation from coordinates ′xk, so that xf xjj k=′() but there is not explicit dependence on t. Then, from 65ux xuii kk=∂ ∂′′ , it follows that ∂ ∂=∂ ∂′∂′∂′ ∂′+∂ ∂′∂′ ∂∂′ ∂′u xx xxx xux xx xu xi ji mnm jni mn jm n2 . C.23 Now, it is shown in texts on tensor analysis, such as Eisenhart (1947), that ∂ ∂′∂′+∂ ∂′∂ ∂′=′∂ ∂′2x xxi lkx xx xk mnx xi mnl mk ni k , C.24 where k mn′ denotes the Christoffel symbol for the coordinate system ′xi. Then, direct calculation shows that un mluu xx xx xjiln mi nm j ; , =′ ′+′ ′    ′′ ∂ ∂∂ ∂∂ ∂ which can be written uux xx xji mni nm j ; . =′ ′′;∂ ∂∂ ∂C.25 This demonstrates that ujl ; is a mixed tensor. It is useful to know that it is a tensor, but its mixed character precludes a separation into symmetric and antisymmetric parts. However, a covariant tensor Gg umj mi ji=; C.26 can be formed from the mixed tensor. It can be readily shown that Gmj transforms like a covariant tensor, and it can be separated into symmetric and antisymmetric parts as indicated in Eq. 3.3. It follows that both D and W are covariant tensors. The tensor character of strain, compelling it to satisfy a transformation of the type ′=∂ ∂′∂ ∂′EEx xx xijklk il j , is also obligatory for stress. Tensor properties of geometric quantities such as strain can be examined by direct calculation, but the tensor character of stress must be stated as a physical principle. In special cases where the stress is given by a constitutive law involving strain, it should be possible to verify its tensor character by direct calculation. The tensor character of stress, however, arises fromphysical necessity rather than direct calculation. The underlying basis for this is that physical relations must hold in any coordinate system. For example, since the first law of thermodynamics, ˙ ˙ IVd Qij ij=+σ , 66is a scalar equation, σij and dij must both satisfy tensor transformations given above with σij contravariant and dij covariant. Here, I denotes internal energy, V; specific volume; and Q, heat supplied to the system. Formulating the constitutive laws with physical tensor quantities ensures physical validity and that the formulation is valid in various nonrotating coordinate systems. The laws would not be expected to hold in rotating systems since, then, centrifugal force would play a role. Note that energy conservation coupled with Galilean invariance ensures momentum conservation, as shown by Dienes(1978b). This illustrates the importance of invariance (which is not essentially different from Truesdell’s “indifference”). On the other hand, it is useful to have constitutive laws fixed in the material, insofar as possible, so polar axes may be appropriate for certain purposes. This discussionhas not included constitutive laws, but clearly they should be written with all quantities defined in the same coordinate system. This is assured if all quantities are physical tensors. It may be useful for computational purposes to express the constitutive relations in polar axes as an alternative, in order toeliminate rotation, but the constitutive laws should be valid in fixed axes. 67APPENDIX D REMARKS ON ELASTIC CONSTITUTIVE LAWS Truesdell (1952) gives various expressions obtained by Boussinesq, Kelvin, Cosserat, Neumann, and Kirchhoff relating stress and deformation for an elastic material. Equation 4.12 is equivalent to those results. I argue here that B (Truesdell’s C) is not a physical tensor, though it is an adequate measure of stretch. Nor is B frame indifferent. In numerical simulation, where the velocity and, hence, D and D, are given, Eq. 3.18 ˙BV D V=2 (which does not involve the rate of rotation) is useful to update B, but then calculations must be made in polar axes. Alternatively, B can be obtained by computing FFT and using Eq. 3.10. Another way of viewing an essential difficulty in some formulations is to consider the dangerous formulation “ σ = f(F) ” in the special (distortion free) case when V = I. Then “ σ = f(R) ,” which is clearly unphysical since the left side is the stress tensor and the right side is not a tensor. Of course, if R drops out because f involves the argument FFT, then the formulation works out physically, but this is an additional consideration. In the expression of Neumann and Kirchoff involving the potential f, and in Eqs. 39.6 and 39.7 of Truesdell (1952), the stress is given by Tf xi iα α=∂ ∂ , with the components xia of F taken as the independent variables; but, since there are nine and the elastic potential involves only six strains. This seems confusing and unnatural. It is as though the potential depended on rotation! It is formally possible if the B are written out in terms of the elements of F as indicated above, but this is not usually done. On th e other hand, in Section 4, the potential is considered to depend only on the 6 components of strain. If the stress of Eq. 4.12 is put into space axes by application of Eq. 3.14, then the spatial (Cauchy) stress, which can be expressed as σσ= =∂ ∂ρ ρ εοFFΣΣT ij ijf,Σ , D.1 is equivalent to the second expression given by Truesdell (1952) in his Eq. 39.2 as Boussinesq’s form of the stress-strain relation. Rivlin and Ericson (1955) write an expression equivalent to Eq. 4.24, except that they use C to denote what is called B here (and by Truesdell); also, their I2 (Eq. 1.2) is −′I2, as given in Eq. 4.15. A factor of 1/3 seems to be omitted in their expression for I2, when the expression for I2 is compared with Eq. 4.15. The definition of I2′ given here as Eq. 4.15 has the advantage that its gradient is the deviatoric metric ′Bij defined by Eq. 4.20, ∂′ ∂=′I BB ijij2 , D.2 which is convenient to remember and arises often in theoretical calculations. 68 69APPENDIX E POLAR RATE OF ROTATION An alternative expression for the difference H of the rate of polar rotation ΩΩ and the vorticity W, defined in Eq. 3.29, is HD T Tjktr trrtrt jr kt =− −∑λλ λλ,˜ , E.1 where ˜DT D Trt qr qs st= E.2 is the stretching in the principal axes of stretch. This expression for H was derived by Nemat-Nasser (1983) (who denotes the left side of Eq. E.1 by εjk) using the formalism developed by Hill (1978). Here, D is the stretching defined by Eq. 3.3, and T is the matrix of eigenvectors of V given by Eq. 2.4. The purpose of this appendix is to demonstrate that the result given above as Eq. E.1 is equivalent to that given as Eq. 3.35, and originally derived by Dienes (1978). This is considered to be of some interest since Nemat-Nasser and others have been critical of the use of polar stress rate, given here as Eq. 9.2. In essence, their argument seems to be that the ZJN stress rate (see Section 9) is adequate, and (argues Nemat-Nasser) that I introduced the polar stress rate as an ad hoc device to eliminate certain oscillations observed by Nagtegaal and de Jong (1981) in kinematic-hardening calculations. Nemat-Nasser takes the position that the difference in stress rates σσσσHH−, with H expressed above as E. 1, can be added to the right-hand side of an arbitrary constitutive law of the form ˆ,σσσσ=()fD , E.3 which, thereby, becomes σσσσσσσσσ σσ σσσ∨ =− + = ′()=+ − ˙ , WW D HH ff . E.4 Since the right side is said to be arbitrary, he maintains that it makes no difference whether one uses ˆσ or σ∨ . However, in the calculation of specific plastic and viscoelastic flows, the function fσσ,D() is presumed known. Changing the function changes the material. Thus, subtracting ˆσσσσσσσσ −= −∨ HH E.5 from fσσ,D() results in a different flow rule. Furthermore, the expression in E.5 has no physical meaning (in the sense of representing material response), though it influences the motion and,consequently, should not be arbitrarily added to E.3. It seems that the ZJN stress rate has become so entrenched in the literature that its validity cannot be questioned without vigorous reaction. One should recall, however, that it is usually introduced inan ad hoc manner, having the property that it is invariant under a change of frame. It is not unique, however, in this respect! It is preferable to some other stress rates, since it leaves the second invariant of stress unchanged, as discussed by Prager (1961). However, the combination of frame indifference, 70together with this invariance argument, forms an unwieldy and unnatural foundation for the selection of stress rate. The result is not unique and, as shown in Section 3, it is only approximate. Noll actually derives an expression for stress rate. A similar procedure is fo llowed in Section 9, except that we do not assume V = I, as does Noll. The choice V = I leads to the ZJN stress rate. It is not sensible, however, to assume that V= I for all time! Hence, it is not true in general that Ω = W, nor that the stress rate is given by σσσσσσσσ∨ =− +˙WW . Hill (1978) supports the use of ZJN stress rate (Jaumann flux). In deriving his Eq. 1.20, however, he assumes that A = B = Q and ˙QW Q=, with Q orthogonal. Thus, the ZJN stress rate is proven valid only when the axes are orthogonal. His discussion of objective rates of stress in his Section 3follows the same line. He states, “It is straightforward to calculate the representation of T on a fixed background with which the Lagrangian triad is momentarily coincident.” Whereas his statements and views are quite correct, they are not what is wanted in calculations in which the initial conditions aregiven and it is necessary to follow large deformation to late times with numerical calculation. In such calculations, the reference axes are fixed, and it would not be useful to keep changing them. The difference in these stress rates (ZJN and polar) is not important in flows with small deformation.However, when deformation is large, as in the analysis of shear bands and other m icrostructural instabilities, and in rubber, many solid rocket propellan ts, other viscoelastic materials, impact problems, geological flows, and a variety of other large deformation problems, the difference can beimportant. In plastic flow with kinematic hardening, the difference arising from different stress rates can be large, and in materials undergoing fragmentation or shear banding, the material shear and rotation can be enormous. To demonstrate that Eq. E.1 follows from Eq. 3.31, it is convenient to define a matrix YT S T=T , E.6 where SIV V=−()−tr1 . E.7 In view of Eq. 2.4, Y is diagonal, and can be written explicitly Y=+() +() +()    − − −λλ λλ λλ231 131 12100 00 00 . E.8 It is also convenient to define QS e ejkpq il jik plq= . E.9 Now, Eq. 3.29 can be written hS z= . E.10 71Then, using Eq. E.9, the difference h can be expressed in matrix form as HQ Zik ikpq pq =1 2 . E.11 It is also convenient to define BT eijk li jlk= . E.12 Then, on combining Eqs. E.6, E.7, E.9, and E.12, we find that QB B Yjkpq ijk lpq il il=∑ , . E.13 Now, using Eq. 2.4, Z can be written ZT D Tpq pr rt t r rtqt =− ()∑˜ ,λλ . E.14 When substituted into Eq. E.11, a lengthy expression for H is obtained containing a group of terms that simplifies to the permutation symbol, viz., BT T e TT T espq pr qt plqprlsqt rst == . E.15 This formula expresses the expansion of the determinant of T when r = 1, s = 2, t = 3, which is 1. (When the rows of the determinant are interchanged, its sign is changed, as indicated by thepermutation symbol erst. If any two of r, s, t are the same, then the determinant has two rows the same, and is zero). With this result, Eq. E.11 reduces to HB Y hjk ijk ii i=∑1 2 , E.16 where he Di irt r t rtrt =−()∑λλ ,˜ . E.17 In Eq. E.16, we express the diagonal terms of Y as Yi, so that Y=()=()YYiliilδ . E.18 The expression in E.16 can be brought into agreement with E.1 by noting that Te TT einijk jl km lmn= . E.19 With this relation, it can be shown that HDT T e Y ejk rt t r jl km lmn nnrt n rt=−() ∑ ∑1 2˜ ,λλ . E.20 The sum over n can be simplified by means of Eq. A.29, whence Ye e Y Y Ynlmnnrtii lrmtltmr t r ltmrlrmt ∑ =−() ++() −() δδ δδ δδ δδ . E.21 72Substitution into Eq. E.20 leads to HD T T T T Y Y Yjk tr r t rtjrktjtkrii t r =−() −() −−() ∑1 2λλ˜ , . E.22 It can be verified by detailed enumeration that, as a result of the definition of Y given as Eq. E.8, the last term on the right can be replaced by 1/λλrt+(), and that the two terms in the middle parentheses make equal contributions to the sum. This completes the demonstration that Nemat- Nasser’s result, given as Eq. E.1, is equivalent to Dienes’s Eq. 3.29, which is repeated in this appendix as Eq. E.10. 73REFERENCES Autonne, M.L., 1902, “Sur les groupes lineaires, reels, et orthogonaux,” 30, pp. 121–130, Bull. De la Soc. Math. de France. Bellman, R., 1960, Introduction to Matrix Analysis , McGraw-Hill. Cook, W., Personal communication, 1982.Dienes, J.K., 1978a, “On the Analysis of Rotation and Stress Rate in Deforming Bodies,” Acta Mechanica 32, 217–232. Dienes, J.K., 1978b, “The Method of Generalized Coordinates and an Application to Rayleigh-Taylor Instability,” Phys. Fluids 21, 5, pp. 736–744. Dienes, J.K., 1989, “Theory of Deformation,” Part II, Physical theory, LA-11063-MS, Vol. II, September 1989. Dienes, J.K., 1996, “A Unified Theory of Flow, Hot Spots, and Fragmentation with an Application to Explosive Sensitvity,” in High-Pressure Shock Compression of Solids II , Springer-Verlag New York, Inc., Lee Davison et al., eds. Dienes, J.K. and J.C. Solem, 1999, “Nonlinear behavior of some hydrostatically stressed isotropic elastomeric foams,” Acta Mechanica 138, 155–162. Drucker, D.C. and W. Prager, 1952, “Soil mechanics and plastic analysis or limit design,” Quarterly of Applied Mechanics, 10, 157–175. Eringen, A.D., 1967, Mechanics of Continua , Wiley, Theorem 2, p. 81. Eisenhart, L.P., 1947, An Introduction to Differential Geometry , Princeton University Press. Flugge, W., 1972, Tensor Analysis and Continuum Mechanics , Springer-Verlag. Gantmacher, F.R., 1959, The Theory of Matrices , Chelsea. Goldstein, H., 1959, Classical Mechanics , Addison-Wesley. Green, G., 1839, “On the Laws of Reflexion and Refraction of Light at the Common Surface of Two Non-Crystallized Media,” Trans. Cambr. Phil. Soc. 7, 245–269. Hill, R., 1950, The Mathematical Theory of Plasticity , Oxford. Hill, R., 1978, “Aspects of Invariance in Solid Mechanics,” in Advances in Solid Mechanics, 18 . Jeans, J., 1940, An Introduction to the Kinetic Theory of Gases , Cambridge. Jaumann, G., 1911, “Geschlossenes System physikalischer und chemischer Differentialgesetze,” Sitzber. Akad. Wiss. Wien (IIA), 120, 385–530. Kachanov, M., 1984, “Elastic Solids with Many Cracks and Related Problems,” in Advances in Applied Mechanics , Academic Press. Landau, L., and E. Lifschitz, 1986, Theory of Elasticity , Third Edition. Mooney, M., 1940, “A Theory of Large Elastic Deformation,” J. Appl. Phys. 2, 582. Nagtegaal, J.C. and J.E. de Jong, 1981, “Some Aspects of Non-Isotropic Work Hardening in Finite Strain Plasticity,” Proc. Workshop on Plasticity of Metals at Finite Strain, E.H. Lee, and R.L. Mallet, eds., Stanford Univ., July 29–Aug. 1. 74Nemat-Nasser, S., 1983, “On Finite Plastic Flow of Crystalline Solids and Geomaterials,” J. Appl. Mech. 50, 1114–1126. Noll, W., 1955, “On the Continuity of the Solid and Fluid States,” J. Rat. Mech. Anal. 4, 3–1. Prager, W., 1961, “An Elementary Discussion of Definitions of Stress Rate,” Quart. Appl. Math. 18, 403–407. Reuss, A., 1931, “Beruchsichtigung der elastischen Formanderung in der Plastizitatstheorie,” Zeits. Ang. Math. Mech., 10, pp. 266–269. Rivlin, R.S. and Ericson, J.L., 1955, “Stress-Deformation Relations for Isotropic Materials,” J. Rat. Mech. Anal., 4, 3, 323–425. Also reprinted in Continuum Mechanics II, The Rational Mechanics of Materials , C. Truesdell, ed., Gordon and Breach, 1965. Sandler, I. and M. Baron, 1985, “Numerical Models for Dynamic Loading,” in Mechanics of Geomaterials , Z.P. Bazant, ed., John Wiley and Sons. Seaman, L., D.A. Shockey, and D.R. Curran, 1972, “The growth law for crack propagation under shock conditions,” in Dynamic crack propagation, Proceedings of an international conference ,” G.C. Sih, ed., Lehigh University, Bethlehem, PA, July 10–12, 1972, pp. 629–647. Signorini, A., 1931, “Sulle deformazioi termoelastische finite,” Third International Congress for Applied Mechanics , Proceedings , Vol. 2, Pt. 2, pp. 80–89, August 24–29, 1930, Stockholm, Kungl, Boktryckeriet, P.A., Norstedt & Soener. Sokolnikoff, I.S. and R.D. Specht, 1946, Mathematical Theory of Elasticity , McGraw-Hill. Spitzig , W.A. and O. Richmond, 1984, “The Effect of Pressure on the Flow Stress of Metals,” Acta Metall., 32, 3, pp. 457–463. Truesdell, C., 1952, “The Mechanical Foundations of Elasticity and Fluid Dynamics,” J. Rat. Mech. Anal., 1, 125–300. Truesdell, C., 1954, The Kinematics of Vorticity , Indiana University Press. Truesdell, C. and Toupin, R.A., 1960, “The Classical Field Theories,” in Handbuch der Physik , Vol. III. Truesdell, C., 1966, The Elements of Continuum Mechanics , Springer-Verlag. Zaremba, S., 1903, “Sur une forme perfectionnée de la théorie de la relaxation,” Bull. Int. Acad. Sci. Cracovie No. 8, 594–616. Zhurkov, S.N. and V.S. Kuksenko, 1975, “The Micromechanics of Polymer Fracture,” Int. J. Fracture, 11, No. 4. Zuo, K., Personal communication, 2001. This report has been reproduced directly from the best available copy. It is available electronicallyon the Web ( http://www .doe.gov/bridge). Copies are available for sale to U.S. Department of Energy employees and contractors from: Office of Scientific and Technical Information P.O. Box 62 Oak Ridge, TN 37831 (865) 576-8401 Copies are available for sale to the public from: National Technical Information Service U.S. Department of Commerce5285 Port Royal Road Springfield, VA 22616 (800) 553-6847