Appendix K

Appendix J, Version 2 ready to install PhL 10.2.12 Appendix J: Deformation Tensors in Continuum Mechanics 1 (a) A Preliminary Deformation Flow Picture 1 (b) A More Complicated Deformation Flow Picture 6 (c) Form of a solid constitutive equation involving the deformation tensor 11 (d) Some fluid constitutive equations 12 (e) Corotational and other objective time derivatives of the stress tensor 13 // reviewed this 10.16.12, filled in all references, it is ready to publish. // reviewed and edited again 10.18.12, it is ready to install with no further changes. // installed this as Appendix K at 8:50 PM 10.19.12 (after doing J↔K name swap) Appendix K: Deformation Tensors in Continuum Mechanics Tensor-like objects appear everywhere in continuum mechanics. As was noted in Section 2 (k), and later in Appendix E (i), continuum mechanics texts generally refer to all objects having indices as being "tensors", whether or not these objects actually transform as tensors with respect to some underlying transformation. The most commonly appearing tensors have two indices and are just 3x3 matrices associated with 3D space. In this category, there are several kinds of stress tensors, and many kinds of strain and deformation tensors which describe how a tiny volume of continuous matter (perhaps a tiny cube near some point x) changes shape in response to some applied stress. For a fluid, a fixed stress pattern can cause a continuous ongoing change of shape which is measured then by a "rate of deformation tensor" often called D. An equation relating stress to strain/deformation is called a constitutive equation and describes the "response" of some physical system to "stimulus". The constitutive equation for a spring is F = -kΔx, for example, which is distinct from the equation of motion for a mass on a spring which is F = ma. For the spring, the stimulus is the force F ("stress"), and the response is the spring stretch Δx ("strain"). In this section we shall study the tensor aspects of several kinds of deformation tensors appearing in continuum mechanics. In the book of Lai et. al. this material is spread over several chapters, but here it will all be put in one place with Lai references provided. Section (c) below considers the form of a candidate constitutive equation for a continuous solid whose form is determined by the requirement that the equation be "covariant" as discussed in Section 7 (u). Similarly, sections (d) and (e) consider covariant constitutive equations for fluids (a) A Preliminary Deformation Flow Picture Recall the general nature of the Pictures appearing through this document, such as The arrow represents an underlying generally non-linear transformation between x-space and x'-space given by x' = F(x), while R(x) (S = R-1) is the linearized-at-point-x version of the transformation which defines the notion of a vector as in dx' = R dx (see Section 2). Since F will have another meaning below, we shall change the transformation name so that x' = F(x). Now consider the picture below in which appear two sequential transformations Fto and Ft. There are three spaces called X-space on the bottom, x-space in the middle, and x'-space on the top. The spaces are associated with frames of reference S0 , S and S'. Each frame has some set of basis vectors to be discussed below. The linearized R matrix objects associated with the two transformations are shown to the right and are given the names Rt0 = F on the bottom and Rt= Ft on the top. The transformation x' = Ft(x,τ) is spatial coordinate transformation only, the time coordinate τ is a parameter. In the picture below, time increases in the upward direction, so τ > t > t0. Fig 1 Consider for the moment just the bottom transformation. The transformation Fto ≡ F describes the deformation of a particle of continuous matter which starts at position X and time t0 and ends up at position x at time t. If we look at a large cube of continuous matter, we might find that it deforms in a very complicated manner as determined by the non-linear transformation F applied to all the particles within this large cube. The cube gets stirred up and is probably no longer recognizable. But if instead we consider a differentially small starting cube at X and t0, we shall find that at time t that cube is at location x but has been transformed into a tiny rotated parallelepiped whose axes are no longer orthogonal. It is assumed that the flow is reasonable and smooth, we are not considering some kind of "explosion" here. We use the words flow and fluid, but the deformation concept applies to elastic solids as well as fluids since these deform in some way when they are stressed (think jello or even steel). Rather than think of the flow in terms of the edges of this tiny cube, one can instead consider two very closely spaced points in the fluid close to X which are separated by spacing dX at time t0, which we think of as "a little dumbbell". At time t, if one carefully tracks the "pathlines" of the ends of the dumbbell, one finds that the dumbbell tumbles and stretches and ends up as dx at time t and location x, Fig 2 This differential dumbbell can be regarded as a mathematical "probe" embedded in the continuous medium. The relationship between dx and dX is given by dx = F(X,t) dX dxi = FijdXj // Lai p 105 (3.18.13) where the matrix Fij is called "the deformation gradient". It is also known as "the deformation gradient tensor" even though it is not a "tensorial tensor" with respect to any identifiable transformation. In Section 5 (o) the above equation was identified with dx' = R(x)dx with R(x) here being F(x,t). The picture above is then the second figure shown in Section 2 (left-right switched). Thus, the deformation gradient F is the linearized version (at point x) of some fancy non-linear (and unknown) "flow transformation" F. Since one can write dxi = (∂xi/∂Xj)dXj , one finds that Fij = (∂xi/∂Xj) ≡ ∂jXi or F = (x) // Lai p 105 (3.18.4) where the gradient is with respect to X, so it is really = (X). Thus the name "deformation gradient". [ Notice that (x) is a matrix. In Appendix G the form of (v) for arbitrary vector v is found in arbitrary curvilinear coordinates. The index order reversal Fij = ∂jXi is mentioned there as well. ] The deformation gradient F(X,t) depends implicitly on the time t0. At t = t0+ε (with a very small ε) no flow has yet taken place, so dxi = dXi and then F(X,t0) = 1. Time t0 is called the reference time and one could display it by writing F(X,t) = Ft0(X,t), but normally this t0 label is suppressed. Now consider the upper flow in Fig 1 above. It is entirely analogous to the lower flow, but names are changed. We get from the lower flow to the upper flow by making these replacements : t0 → t dX → dx Fto(X,t) → Ft(x,τ) S0 → S Fto(X,t0) = 1 → Ft(x,t) = 1 t → τ dx → dx' Fto(X,t) → Ft(x,τ) S → S' Fto(X,t0) = 1 → Ft(x,t) = 1 The equations corresponding to those shown above are dx' = Ft(x,τ) dx dx'i = (Ft)ijdxj // Lai p 457 (8.7.2) Since one can write dx'i = (∂x'i/∂xj)dxj , one finds that (Ft)ij = (∂x'i/∂xj) or Ft = (x') // Lai p 457 (8.7.3) where the gradient is with respect to x, so it is really = (x). In the lower flow, t0 is the reference time, and t is the "current time". In the upper flow, the current time t is the reference time, and τ is some time τ > t. The upper flow is relative to current time t as reference, and for that reason the word "relative" is pre-pended to the names of all related tensors. Thus, Ft is called the "relative deformation gradient ", whereas F is just the "deformation gradient ". Why are relative tensors useful? The main motivation for use of the relative tensors concerns differentiation with respect to time in the vicinity of the current time t. One can write dtFt(x,t) ≡ [∂τFt(x,τ)]τ=t // x fixed (for example, x = x1) where ∂τFt(x,τ) ≈ [ Ft(x,τ+dτ) - Ft(x,τ) ] / dτ . Here dt = Dt = d/dt = D/Dt = the total time derivative, and ∂t = ∂/∂t = the partial time derivative. Since x is fixed, dx = 0 so dt = ∂t. We want to know the rate of deformation at some fixed current time t, and it is the τ argument of the function Ft(x,τ) that lets this derivative be computed. One could in theory carry out this same differentiation using the "non-relative" tensors by doing d/dt0 with t0 near t: dtFt(X,t) ≡ [∂t0Ft0(X,t)]t0=t // X fixed where ∂t0Ft0(X,t) ≈ [ Ft0(X,t) - Ft0-dt0(X,t) ] / dt0 , but this goes against the grain of the idea that X is a material coordinate at initial time t0 which is earlier than t. And in the above, we end up with a statement about Lagrangian functions f(X,t) rather than Eulerian functions f(x,t), though one might argue that as t0→ t, one has X → x. The relative tensor approach makes the differentiation process clearer, and will be used below for that purpose. The frames of reference and the cameraman Each of the three frames of reference S0, S and S' in Fig 1 has its own set of orthonormal basis vectors which we are completely free to set in any manner. As a construct it is helpful to imagine that, as the flow proceeds, it is observed by a cameraman who flies around on a camera platform which translates and rotates in some arbitrary manner. Since our main concern will be with the dumbbells like dX, dx and dx', the translational part of the camera platform motion is irrelevant since dX is invariant under translations. We allow the cameraman's arbitrary orientation at times t0, t and τ to determine the axes of the three frames S0, S and S'. The cameraman is an "observer". The values of the deformation gradient matrix elements Fij depend on the choice of basis vectors in frames S0 and S, so they in fact are dependent on how the cameraman flies his platform. Consider, dx = dx11(S) + dx2 2(S) + dx3 3(S) dX = dX11(S0) + dX2 2(S0) + dX3 3(S0) Once the axes are chosen, the value of the Fij are determined, for example, F12 ≈ (dx1)/(dX2) . If we were to rotate the basis vectors in frame S, for example, dx1 would change, dX2 would stay the same, and F12 would change. Tensor expansions of the deformation gradients These two expansions won't be used below, they are just inserted here "for interest". The expansions use the basis vectors just defined above which are n(S0) for frame S0 and n(S) for frame S. Consider this candidate expansion for the deformation gradient F, F = Σij Fij i(S) j(S0) = Σij Fij [i(S)] [j(S0)]T where Fij = [F(S,S0)]ij = [i(S)]T F [j(S0)] = (∂xi/∂Xj) , where we write the expansion in both direct product and matrix form as in Appendix E. This is a "mixed basis expansion" as discussed in Appendix E (i). Consider the application of this expansion to dX : { Σij Fij [i(S)] [j(S0)]T } dX = Σij Fij [i(S)] [j(S0)]T { ΣkdXkk(S0)} = Σij Fij ΣkdXk [i(S)] [j(S0)]T [ k(S0)] = Σij Fij ΣkdXk [i(S)] δj,k = [ Σij Fij dXj] i(S) = [ dxi ] i(S) // using the fact that F dX = dx = dx . Since {expansion}dX = dx and since ((X)x) dX = dx by the chain rule, it seems reasonable to conclude that {expansion} = ((X)x) = F. If we agree to use the i(S) for both dx and dx', so that dx = dxii(S) and dx' = dx'ii(S), then the following is a viable expansion for the relative deformation gradient Ft: Ft = Σij (Ft)ij i(S) j(S) = Σij (Ft)ij [i(S)] [j(S)]T where (Ft)ij = [Ft(S,S)]ij = [i(S)]T Ft [j(S)] = (∂x'i/∂xj) The verification is similar to the above, { Σij (Ft)ij [i(S)] [j(S)]T } dx = Σij (Ft)ij [i(S)] [j(S)]T { Σkdxkk(S)} = Σij (Ft)ij Σkdxk [i(S)] [j(S)]T [ k(S)] = Σij (Ft)ij Σkdxk [i(S)] δj,k = [ Σij (Ft)ij dxj] i(S) = [ dx'i ] i(S) // using the fact that Ft dx = dx' = dx' . Since {expansion}dx = dx' and since ((x)x') dx = dx' by the chain rule, it seems reasonable to conclude that {expansion} = ((x)x') = Ft. (b) A More Complicated Deformation Flow Picture Consider now this flow picture, Fig 3 There is much to be said about this drawing. The left side is the same as in Fig 1 shown above. The picture is simplified in that it shows only the linearized R-type transformations like F and not the full transformations like F, and these R-type matrices are now labeled right on the transformation arrows. We only care about these F matrices because we only care about the dumbbells like dx. For example, on the lower left we have dx = F dX . The two sides of the picture represent observations of the same flow by two independent flying cameraman observers, call them C and C*. On each side the basis vectors of the various frames are set by the motions of these cameramen. The two cameramen have agreed to start off at time t0 with their camera platforms in exact alignment, so there is no need for a frame S0*. The two frames of reference S and S* are related by some Galilean transformation (rotation plus translation) which brings the two independent camera platforms into alignment at time t : x* = Q(t) (x-x0) + c(t) => dx* = Q(t) dx Here x0 is a randomly selected center for the rotation Q(t), and c(t) the corresponding translation. As above, we only care about the Q(t) part of this transformation, so in terms of dx objects, the frames S and S* are in effect related by the rotation Q(t). The arrows in the picture correctly describe the transformations of the dx type objects in moving between frames. In the lower part, for example, we have dx* = Q(t) dx dx = F dX dx* = F* dX Comparing the left and right equations one has Q(t) dx = F* dX and then the center equation can be used on the left side to get Q(t) F dX = F*dX . Since this has to be true for any dX, we have Q(t) F = F* as shown in the drawing. This result is trivially obtained just by looking at the alternate arrow paths from frame S0 to frame S*. So: F* = Q(t) F or F*(x*,t*) = Q(t) F(x,t) t* = t At time τ we have a similar situation, but there are four arrows instead of three. Comparing the arrow paths from frame S to frame S'* one finds Ft*Q(t) = Q(τ)Ft => Ft* = Q(τ)FtQ(t)T or Ft*(x*,τ*) = Q(τ) Ft(x,τ)Q(t)T t* = t The two Q's are rotations (reflections included) and are therefore orthogonal so Q-1 = QT. Does F transform as a tensor with respect to rotation Q(t) ? We can think of F(x,t) as a property of the continuous material at location x and current time t. F describes the "state of deformation". If F transformed as a tensor with respect to Q(t), one would need this to be true, F* = Q(t) F Q(t)T // not true! which is the matrix form for the transformation of a rank-2 tensor as shown, for example, in Section 5 (f). But we have just seen that F* = Q(t) F so the required Q(t)T on the right is missing. We conclude therefore that in fact F, although it is called a tensor, does not transform as a tensor under Q(t). One then says that F is a non-objective tensor with respect to Q(t). Equation F* = Q(t) F in fact says that the columns of matrix F transform as vectors under Q(t), which is very different frame saying F transforms as a rank-2 tensor under Q(t). If one is trying to construct a phenomenological equation modeling a continuous material at point x and time t, one must make sure that equation is "covariant" (frame-indifferent) with respect to rotation Q(t). The observers (cameramen) in frame S and frame S* must see equations which have exactly the same form, which means the elements in the equations must be objective with respect to Q(t). See Section 7 (u) for a general discussion of "covariance". Since F is non-objective, it is not directly useful in the construction of covariant model equations. Do any of the usual "derived tensors" transform as tensors with respect to Q(t) ? By "the usual derived tensors" we mean B, C, U, V and associated R all defined as follows: B = FFT = the left Cauchy-Green deformation tensor = the Piola deformation tensor C = FTF = the right Cauchy-Green deformation tensor = the Finger deformation tensor F = RU = VR R = rotation U,V = symmetric positive definite Tensors B and C are defined simply as shown, and both are therefore symmetric tensors. The last line is a statement of the polar decomposition theorem which says that any (real) non-singular matrix (det ≠ 0) can be uniquely written in these two ways (we apply this theorem to the deformation tensor F) F = RU = VR => U = RTVR and V = RURT // Lai p 114 (3.21.1,2,4) where R is a rotation matrix and V and U are symmetric positive definite matrices (meaning the eigenvalues are all positive) known as the left and right stretch tensors. Note that R is the same matrix in both the RU and VR forms. The idea is that the R matrix takes into account the rotational part of the deformation F, while U or V take into account the stretch component of the deformation. If the deformation is a pure rotation, U = V = 1, whereas if the deformation is a pure stretch then R = 1. A general deformation is a rotation/stretch/shear affair and one will find that none of R, U, V are unity. One can combine the three equations above to find that B = FFT = (VR)(VR)T = VRRTVT = VVT = VV = V2 // Lai p 125 (3.25.1) C = FTF = (RU)T(RU) = UTRTRU = UTU = U2 // Lai p 115 (3.22.1,2) So our task is to discover whether any of these derived tensors transform as a tensor relative to Q(t). If they do transform as tensors (if they are objective), then they are candidates for use in constructing model equations for the continuous material. We start with B and C: B* = F*F*T = (QF)(QF)T = QF FTQT = QBQT => B* = Q(t)BQ(t)T . C* = F*TF* = (QF)T(QF) = FTQTQF = FTF = C => C* = C Thus, the left Cauchy-Green deformation tensor B actually does transform as a rank-2 tensor with respect to Q(t), so it is a tensorial tensor, it is "objective". In contrast, since C* = C, the right Cauchy-Green deformation tensor does not transform as a rank-2 tensor. In fact each element of matrix C transforms as a tensorial scalar with respect to Q(t). What about V and U as defined above, the left and right stretch tensors? F = RU = VR F* = R*U* = V*R* Consider, F* = QF = Q(RU) = (QR) (U) = R*U* Since U is positive definite symmetric, and since QR is a rotation, and since the polar decomposition is unique, it must be that R* = QR and U* = U . Next write F* = QF = Q(VR) = (QVQT)(QR) = V* R* Since the eigenvalues of symmetric V are determined by det(V-λI) = 0, and since this is the same as the equation det(QVQT-λI) = 0, QVQT has the same eigenvalues as V and so (QVQT) is symmetric and positive definite. Due to this fact and the fact that QR is a rotation, and the fact that the polar decomposition is unique, it must be that V* = QVQT and QR = R* . So here is a summary for our tensors of interest. Only two of the five deformation tensors actually transform as tensors. The references are to Lai page 336-337 : F* = Q(t)F // Lai (5.56.21) B* = Q(t)BQ(t)T // rank-2 tensor with respect to Q(t) so objective // Lai (5.56.31) C* = C // Lai (5.56.28) U* = U V* = Q(t)VQ(t)T // rank-2 tensor with respect to Q(t) so objective R* = Q(t)R Comment: Recall that F = F(x,t) has a hidden parameter t0 so in fact F = Ft0(x,t) . Similarly, all derived tensors have this same hidden parameter. Thus, for example, one could write the transformation of B as Bt0*(x*,t*) = Q(t) Bt0(x,t)Q(t)T t* = t x* = Q(t) (x-x0) + c(t) dx* = Q(t) dx . The parameter t0 is treated as a fixed constant here and plays no role in the question of whether or not B transforms as a rank-2 tensor. The important time argument of B is the current time t, and the main idea is that B*(t) = Q(t) B(t)Q(t)T so that B(t) is objective with respect to the rotation Q(t). The transformation is valid for any value of t0. In the limit that t0 → t, the equation says 1 = Q(t) 1 Q(t)T which of course is true since rotation Q(t) is orthogonal. Do any of the usual relative derived tensors transform as tensors with respect to Q(t) ? Again we think of a relative tensor Wt as being a property of the continuous material at current time t, a measure of the state of deformation. Such a tensor is objective only if Wt* = Q(t)WtQ(t)T. With regard to the above Comment, in this new situation it is the t of Wt which is the time variable of interest (the current time), and time τ is regarded as a fixed parameter, as was t0 in the Comment. It just happens that the notational positions of the current time t and the parameter time τ are swapped in this case relative to the last, so now we have Wt*(x*,τ*) = Q(t)Wt(x,τ) Q(t)T τ* = τ x* = Q(t) (x-x0) + c(t) dx* = Q(t) dx . A tensor Wt which transforms as a rank-2 tensor (is objective) with respect to rotation Q(t) must satisfy the rule above, where the arguments of both Q rotations are t. As in the Comment above, this transformation is valid for any value of parameter τ, and as τ→t, the equation says 1 = Q(t) 1 Q(t)T. Our study of the transformation properties of the relative tensors proceeds in a manner similar to that used for the regular tensors above. We start with Bt ≡ FtFtT : Bt* = Ft*Ft*T = [Q(τ) Ft QT(t)] [Q(τ) Ft QT(t)]T = Q(τ) Ft QT(t) Q(t) FtT Q(τ)T = Q(τ) Ft FtT Q(τ)T = Q(τ) Bt Q(τ)T // not a rank-2 tensor since t ≠ τ Next comes Ct ≡ FtTFt : Ct* = Ft*TFt* = [Q(τ) Ft QT(t)]T [Q(τ) Ft QT(t)] = Q(t) FtT Q(τ)T Q(τ) Ft QT(t) = Q(t) FtT Ft QT(t) = Q(t) CtQT(t) // yes a rank-2 tensor with respect to Q(t) What about the left and right relative stretch tensors Vt and Ut? Ft= RtUt = VtRt Ft* = Rt*Ut* = Vt*Rt* Consider, Ft* = Q(τ) Ft QT(t) = Q(τ) RtUtQT(t) = [Q(τ) RtQT(t)] [Q(t)UtQT(t)] = Rt*Ut* . Since [Q(τ) RtQT(t)] is a rotation and since [Q(t)UtQT(t)] is a symmetric positive definite matrix by the argument given in the previous section, and since the polar decomposition is unique, it must be that Rt* = Q(τ) RtQT(t) and Ut* = Q(t)UtQT(t) // Ut is a rank-2 tensor Finally, write Ft* = Q(τ) FtQT(t) = Q(τ)VtRtQT(t) = [Q(τ)VtQT(τ)] [Q(τ) RtQT(t)] = Vt* Rt* By the same argument used several times above, we conclude that Rt* = Q(τ)RtQT(t) and Vt* = Q(τ)VtQT(τ) // Vt is not a rank-2 tensor, τ ≠ t The rule for transforming Rt is the same as found a few lines above. Here then are the conclusions, with references to Lai page 472: Ft* = Q(τ)FtQT(t) // Lai (8.13.6) Bt* = Q(τ)BtQ(τ)T // Lai (8.13.12) Ct* = Q(t)CtQT(t) // rank-2 tensor with respect to Q(t) so objective // Lai (8.13.10) Ut* = Q(t)UtQT(t) // rank-2 tensor with respect to Q(t) so objective // Lai (8.13.9) Vt* = Q(τ)VtQT(τ) // Lai (8.13.12) Rt* = Q(τ)RtQT(t) // Lai (8.13.8) Notice that among the "normal" tensors, B and V are objective, whereas among the "relative tensors" it is Ct and Ut that are objective. All the other tensors are "non-objective". (c) Form of a solid constitutive equation involving the deformation tensor For a solid continuous material in frame S one can consider a stress/deformation relationship of the form T = f(B), where T is the Cauchy stress tensor, B is the left Cauchy-Green deformation tensor mentioned in section (b) above, and f is "some function". In frame S*, there will be some covariant version of the equation T* = f*(B*). If the medium is isotropic (rotationally invariant in its properties), then f* = f and one will have T* = f(B*) in Frame S*. Two observers of the same system in frames related by a rotation cannot observe different functions f ≠ f* if the material is isotropic. Notice that there are two separate issues here: (1) equation must be covariant under rotations to be viable; (2) isotropic implies f = f*. If f is a polynomial, or a function which can be approximated by one (f is smooth), then T = f(B) with polynomial coefficients which are rotational scalars (with respect to Q) is a viable equation form for the following reason: since B is a rank-2 tensor, so is any power of B, B*2 = [QBQT][QBQT] = Q B2QT etc. and if the polynomial coefficients are scalars, then f(B) is a rank-2 tensor. Just as a particle force F transforms as a rank-1 tensor under rotations, the Cauchy stress tensor T transforms as a rank-2 tensor under rotations, and then both sides of T = f(B) transform in the same way -- as rank-2 tensors. Any candidate equation between T and a deformation tensor which did not have both sides transforming the same way would be invalid from the get-go (except perhaps as an approximation). The scalar coefficients must be functions of the Bij and there are three such scalars known as the principal scalar invariants of B (Lai p 40), one of which is det(B), so the scalar coefficients can be any functions of these three scalar invariants. Furthermore, one can use the fact that B = FFT is symmetric along with the Cayley-Hamilton theorem (symmetric matrix B satisfies its own secular equation, whose coefficients by the way are those scalar invariants) to show that any powers of B in polynomial f(B) larger than degree 2 can be expressed as a linear combination of I, B and B2. One ends up then with T = aI + bB + cB2 where a,b,c are functions of the three scalar invariants of tensor B. Since both sides of T = f(B) transform in the same way (rank-2 tensors), the equation T = f(B) is "covariant" as discussed in Section 7 (u), meaning it has the same form in frame S* as it has in S. The equation T = f(B) is a relation between stress and strain in the form of deformation, and as such is called a constitutive equation for the continuous material. One wants such equations to be covariant between frames of reference related by any Galilean transformation (rotation + translation), even if one or both of these frames are non-inertial. This is an extension of Hooke's Law for a spring, F = -k Δx , which is covariant under rotations and translations. In contrast, equations of motion are only covariant if both frame S and S* are inertial frames. Notice that this entire discussion falls apart completely if one tries T = f(F) or T = f(C) as a candidate constitutive relation, since then the two sides of the equation don't transform the same way. This subject is discussed in Lai pp 334-342 and p 40 for the scalar invariants. The requirement of covariance for an isotropic material and the fact that B is symmetric and transforms as a tensor puts a severe restriction on the form of the constitutive equation and we end up with T = aI + bB + cB2. Since one can replace B3 = αB2 + βB + γI, if B is invertible (detB ≠ 0) one has B2 = αB + βI + γB-1 and this allows the alternate form T = a'I + b'B + c'B-1 . This last equation is used to model large deformations of an isotropic elastic material. An example is the Mooney-Rivlin theory for rubber. (d) Some fluid constitutive equations It was noted in section (b) that the relative deformation tensors are appropriate when one is interested in time derivatives of the tensors. It was also noted that the relative deformation tensor Ct is objective. One can expand Ct(x,τ) in a Taylor series about current time t in this manner (∂τ ≡ ∂/∂τ) , Ct(x,τ) = Σn=0∞ [ ∂τnCt(x,τ)]τ=t (τ-t)n/n! = Σn=0∞An(x,t) (τ-t)n/n! // Lai p 463 (8.10.1) An(x,t) ≡ [ ∂τnCt(x,τ)]τ=t , where the coefficient derivatives are given the names An(x,t) called Rivlin-Ericksen tensors. Each of these coefficient tensors is in fact objective, just as is Ct, since (as usual, t = t*, τ = τ* ) Q(t) [ ∂τnCt(x,τ)]τ=t QT(t) = { ∂τn [Q(t) Ct(x,τ) QT(t)]}τ=t = { ∂τ*n Ct*(x*,τ*)}τ*=t => Q(t) An(x,t) QT(t) = A*n(x*,t) These An(x,t) tensors appear in various models of "non-Newtonian" fluid behavior, the general study of which is called rheology, based on the Greek word for a current flow (a rheostat controls electric current), // OED2 Here are a few covariant constitutive equations and the names assigned to them (Lai p 481). Note that for any normal fluid, there is always a -pI tensor term in the expression for stress T, where p is the fluid pressure and I is the identity matrix. The diagonal elements of matrix -pI are the equal normal stresses of the surroundings of a tiny cube of fluid pulling out on the cube faces, hence the -p (p > 0) since we know the fluid actually pushes in on the cube. T = -pI + functional of Ct(τ), τ ≤ t // "simple" fluid, since nFt not involved (Ct=FtT Ft) T = -pI +!Syntax Error, I dτ f1(τ) Ct(τ) // single-integral simple fluid. f1(τ) = a memory weight function T = -pI + f(A1, A2....AN) // Rivlin-Ericksen incompressible fluid of complexity N T = -pI + μ1A1 + μ2A12 + μ3A2 // second order fluid (paint, blood, polymers) T = -pI + μA1 // incompressible Newtonian fluid (fluids like water) It turns out that A1 = 2D where D = [(v) + (v)T] /2 ≡ (v)sym , so A1 is twice the rate of deformation tensor D. The other An can then be found from this recursion relation, An+1 = dtAn + An(v) + (v)TAn // Lai p 468 (8.11.2) Here v is the fluid velocity vector and dt = d/dt = D/Dt. Again, (v) is the subject of Appendix G. (e) Corotational and other objective time derivatives of the Cauchy stress tensor The Cauchy stress tensor T transforms as a tensor under Q(t); it is objective. One can write therefore, T*(x*,t*) = Q(t) T(x,t) Q(t)T t* = t x* = Q(t) (x-x0) + c(t) dx* = Q(t) dx Clarification of the above equation One can think of the above equation T* = QTQT as involving operators in Hilbert Space, as outlined in Appendix E (g). In the upper part of Fig 3 above we show four different frames of reference called S, S*, S' and S'* each of which has its own set of basis vectors we might call un, u*n, u'n and u*'n. It happens that the picture refers to S and S* at time t, and S' and S'* at time τ, but any basis vectors can be "used" at any time one wants. For example, here are four expansions of the operator T(x,t) T(x,t) = Σab Tab(x,t) ua ub = Σab T*ab(x*,t) u*a u*b = Σab T'ab(x',t) u'a u'b = Σab T'*ab(x'*,t) u'*a u'*b in which we see four different kinds of components Tab, T*ab, T'ab, T'*ab . The spatial arguments of each component are written as appropriate for that frame of reference and of course all "correspond" to each other (for example, x' = Ft(x,τ)). Recall from Section 2 (i) the notion of the transformation of a contravariant vector field in developmental notation V'(x') = R V(x) contravariant Rik(x) ≡ (∂x'i/∂xk) R = S-1 where the argument is appropriate to the space of interest. The time argument t in the above four expansions of T can be set to any arbitrary value. The stress tensor at a point x is in general a function of time t. One could for example set t = τ in all the expansions. Having said this, we now decide that only the frame S basis vectors un shall be used in our expansions and components. Then for example ( these un were called n(S) earlier) T(x,t) = Σab Tab(x,t) ua ub T*(x*,t) = Σab T*ab(x*,t) ua ub Q(t) = Σab Qab(t) ua ub . Our operator statement of objectivity then becomes the following when expressed in components, T*(x*,t*)ij = Q(t)ia T(x,t)ab Q(t)Tbj t* = t Thus, there should be no confusion about the following two equations which we express back in operator notation with the position arguments suppressed (but shown on the right) T*(t) = Q(t) T(t) Q(t)T // T*(x*, t) = Q(t) T(x,t) Q(t)T T*(τ) = Q(τ) T(τ) Q(τ)T // T*(x*, τ) = Q(τ) T(x,τ) Q(τ)T Problem: The tensor dT/dt fails to transform as a rank-2 tensor, even though T does so transform. If one tries to construct covariant constitutive equations involving dT/dt, a problem arises because dT/dt is non-objective, T*(t) = Q(t) T(t) Q(t)T (dT*/dt) = Q (dT/dt) QT + [ (dQ/dt) T QT + Q T (dQ/dt)T ] , so there are two extra unwanted terms. Just as B = FFT is constructed to provide an objective derived tensor from non-objective F, one can construct a derived version of (dT/dt) which is objective. In the next three sections, three different derived versions are described. The corotational/Jaumann derivatives The first step is to define an adjusted stress tensor Jt(τ) at time τ according to (see Lai p 483, (8.19.3); Lai does not have a t subscript on J). Jt(τ) ≡ RtT(τ) T(τ) Rt(τ) // Jt(x,τ) ≡ RtT(x,τ) T(x,τ) Rt(x,τ) where Rt(τ) is the rotation which appears above in section (b), where we had (showing τ arguments), Rt*(τ) = Q(τ) Rt(τ)QT(t) . The tensor Rt(τ) is non-objective due to appearance of Q(τ) instead of Q(t) on the left (see comments on Wt above). Recall that this rotation Rt(τ) is unique and is determined from the deformation tensor by the polar decomposition Ft(τ) = Rt(τ)Ut(τ) = Vt(τ) Rt(τ). Thus, in some sense Jt(τ) knows about the stress tensor T(τ), and it knows something about the deformation tensor through Rt(τ). [ The meaning of the term "corotational" is explained far below. ] The claim now is that the time derivative of this corotating stress tensor Jt is objective, meaning that tensor dtJt transforms as a rank-2 tensor under the rotation Q(t). Here is a proof : We first assemble the following facts, T*(τ) = Q(τ) T(τ) Q(τ)T // transformation of stress tensor T at time τ (see prev section) Rt*(τ) = Q(τ) Rt(τ)QT(t) // how Rt(τ) transforms, where Ft(τ) = Rt(τ)Ut(τ) Jt(τ) ≡ RtT(τ) T(τ) Rt(τ) // definition of Jt(τ) in frame S Jt*(τ) ≡ Rt*T(τ) T*(τ) R*t(τ) // corresponding Jt* in frame S* and then we combine these ingredients to obtain a transformation rule for Jt : Jt*(τ) ≡ Rt*T(τ) T*(τ) R*t(τ) = [Q(τ) Rt(τ)QT(t)]T [Q(τ) T(τ) Q(τ)T] [Q(τ) Rt(τ)QT(t)] = [Q(t) RtT(τ)QT(τ)] [Q(τ) T(τ) Q(τ)T] [Q(τ) Rt(τ)QT(t)] = Q(t) RtT(τ) [QT(τ)Q(τ)] T(τ) [Q(τ)TQ(τ)] Rt(τ)QT(t) = Q(t) [ RtT(τ)T(τ) Rt(τ)] QT(t) = Q(t) Jt(τ) QT(t) . (*) Since this equation Jt*(τ) = Q(t) Jt(τ) QT(t) fulfills the condition described earlier for Wt to be objective, we conclude that the corotating stress transforms as a rank-2 tensor, where τ is treated as a parameter. Consider now the limit of (*) as τ → t. One finds, Jt(τ) ≡ RtT(τ) T(τ) Rt(τ) Jt(t) ≡ RtT(t) T(t) Rt(t) = 1 T(t) 1 = T(t) and J*t(τ) ≡ R*tT(τ) T*(τ) R*t(τ) J*t(t) ≡ R*tT(t) T*(t) R*t(t) = 1 T*(t) 1 = T*(t) . In this limit, the corotation Rt-1(τ) has come to a halt, and (*) becomes a statement that T is objective. More interestingly, we can apply ∂τn = ∂n/∂τn to both sides of (*) to get dτn Jt*(τ) = Q(t)[ dτn Jt(τ) ] QT(t) . Taking the limit τ→t then gives [dtn Jt*](t) = Q(t) [dtn Jt](t) QT(t) which says that dtnJt are all objective tensors. And in particular, for n = 1, (dtJt)* = Q(t) (dtJt) QT(t) , and this concludes our proof that dJt/dt is objective, whereas dT/dt is not objective. The above objective tensor time derivatives are sometimes written using the following strange notation n ≡ [dnJt(t)/dtn], n = 1,2,3... ≡ 1 Jt(τ) ≡ RtT(τ) T(τ) Rt(τ) and these are called corotational or Jaumann derivatives (Lai p 484) [Jaumann-Zaremba]. It can be shown that = dtT + TW-WT where W = [(v) – (v)T]/2 = "the spin tensor" // Lai p 484 (8.19.10) The Oldroyd Lower convected derivatives An alternative solution to the same problem uses a different adjusted stress tensor, JL(τ) ≡ FtT(τ) T(τ) Ft(τ) // Lai p 484 (8.19.12) We suppress the t subscript on JL just to avoid having to write (JL)t(τ). In the table at the end of section (b) one sees that Ft transforms the same way Rt does, so one can repeat the above analysis to conclude that the derivatives [dnJL(t)/dtn] are all objective tensors (just replace Rt→ Ft everywhere), so n ≡ [dnJL(t)/dtn] , n = 1,2,3... ≡ 1, JL(τ) ≡ FtT(τ) T(τ) Ft(τ) // = n and these are the "Oldroyd lower convected derivatives" (Lai p 485, called n) . is sometimes called the Cotter-Rivlin stress rate. It can be shown that = = dtT + T(v) + (v)TT // Lai p 484 (8.19.21) The Oldroyd Upper convected derivatives Finally, consider again the non-objective way that Ft transforms (table end of section (b)) Ft*(τ) = Q(τ)Ft(τ)QT(t) => (Ft-1)*(τ) = Q(t) (Ft-1(τ)) QT(τ) // inverted => (Ft-1)T*(τ) = Q(τ) (Ft-1)T(τ) QT(t) // then transposed This object Ft-1,T therefore transforms the same way Rt and Ft transform, so we obtain a third set of objective time derivatives called the Oldroyd upper convected derivatives (Lai p 486 uses n) n ≡ [dnJU(t)/dtn] , n = 1,2,3... ≡ 1, JU(τ) ≡ Ft-1(τ) T(τ) Ft-1,T(τ) // = n The meaning of the term "convected" is explained below. It can be shown that = = dtT – (v)T – T(v)T // Lai p 486 (8.19.26) Covariant constitutive equations Constitutive equations involving an objective time derivative of the stress tensor are called "rate type constitutive equations". Here are some models for incompressible fluids : T = -pI + S where S + λ = 2μD // a convected Maxwell fluid T = -pI + S where S + λ(∂S/∂t) = 2μD // linear Maxwell fluid, see below (non-covariant) T = -pI + S where S = 2μD // Newtonian fluid T = -pI + S where S + λ1 = 2μ(D + λ2) // a corotational Jeffrey fluid T = -pI + S where S + λ1 = 2μ(D + λ2) // Oldroyd fluid A The objective derivatives can be written out in terms of other tensors, for example, = Oldroyd lower = (dT/dt) + T (v) + (v)T T . // Lai p 485 (8.19.21) Fluids with stress time derivatives in their constitutive equations exhibit both elastic and viscous behavior at the same time. Pull on a chunk of such a fluid and the pull is initially resisted by an elastic force, but after a while the internal stress field damps out (molasses, honey) and that elastic force goes away, as if the fluid were microscopically constructed of little springs and dragging dashpots. When a constitutive equation includes a time derivative of stress, the "response" (in this case D = [(v) + (v)T]/2 ) to the "stimulus" (T or S) includes factors of the form e-t/c where the c are decay time constants which are functions of the fluid parameters λi. In this case, the fluid has memory of its past over a time period less than these time constants, as with the honey example. For flow that is very slow relative to these time constants, the time derivative term may be neglected. In the moderately slow flow case, it can be shown that the distinction between the corotational time derivative and (dS/dt) can be neglected and then the convected Maxwell fluid shown above becomes the traditional linear Maxwell fluid which is modeled on those springs and dashpots with S + λ (∂S/∂t) = 2μD. (The time derivatives here are meant to act only on the second argument of S(x,τ) so may be regarded as partial derivatives. ) If λ = 0, the linear Maxwell fluid becomes an (incompressible) Newtonian fluid like water which has no memory. Comment: The linear Maxwell fluid equation S + λ (∂S/∂t) = 2μD can be solved for S using the standard Green's Function method and the solution is S(t) = 2 !Syntax Error, I dt' [ (μ/λ)e-(t-t')/λ] D(t') where the bracketed quantity (the Green's Function or kernel) is called the stress relaxation function φ(t-t'). One can see in this solution the notion of memory (history) with time constant λ: the stress of the present is a function of the rate of deformation D going on in the entire past history. This solution fits into the "simple fluid" form shown earlier, where recall that D = (1/2)A1 and A1 = [ ∂τCt(x,τ)]τ=t. Our main point is to demonstrate the construction of constitutive equations which are covariant with respect to rotations, and which therefore can contain only tensors which in fact transform as tensors under rotations. In continuum mechanics, such tensors are said to be objective tensors. Interpretation of the adjusted stress tensors discussed above. In Section 2 we discuss the notion of the transformation of a contravariant vector V' = RV in developmental notation. In x'-space, the vector components are V'i = RijVi where Vi are the components in x-space. If R is a rotation matrix, then the unit basis vectors in the two spaces can be taken as Cartesian, call them u'n in x'-space and un in x-space. We have these two expansions of V: V = Σn Vn un = Σn V'n u'n where Vn = V un V'n = V u'n In the "active view" of things, we can think of V' = RV as creating a new vector V' in x-space from the old vector V created by rotating the vector V by R. In the "passive view", we think of the V'i as the components of the original vector V projected onto the backwards-rotated basis vectors u'n = R-1 un. To verify this relation between the basis vectors, we can write V'n = V u'n = V R-1un = RV RR-1un = RV un = V' un = V'n . So one can think either of V being rotated forward in x-space into V' where V' has x-space components V'n , or one can think of the V'n as the components of V one measures in frame that is backwards rotated by R-1 , that is, u'n = R-1 un. Consider then a rank-2 tensor that transforms as in Section 5 (f) according to M' = R M RT. The passive interpretation is that the components M'ij are those one observes in a frame of reference whose basis vectors are rotated by R-1 relative to the basis vectors of the unprimed frame, just as in the vector case of the last paragraph. If we now set R = R-1, then M' = R-1 M (R-1)T tells us that the components M'ij of tensor M are those measured in a frame whose basis vectors are rotated forward by R relative to the unprimed frame. If it happens that R = R(t), we would say that M'ij are the components of M which are observed in a frame of reference which is rotating by R(t) relative to the frame of the unprimed components Mij. The basis vectors of the primed frame are then u'n = R un . With this long-winded introduction, we now consider the corotational stress tenser Jt(τ) from above, Jt(τ) ≡ RtT(τ) T(τ) Rt(τ) Since Rt(τ) is a rotation, RtT(τ) = Rt-1(τ), so we have, suppressing τ, Jt = Rt-1T Rt . Therefore, we can regard (Jt)ij as T'ij, the components of stress T measured in a frame which is rotating by Rt relative to the frame in which the Tij are measured. Since this primed frame rotates by Rt relative to the unprimed frame, it is called a corotating frame, and Jt is then called the corotational stress, and its time derivative is called the corotational stress rate. We next consider the upper Oldroyd stress given above a JU ≡ Ft-1 T (Ft-1)T In analogy with the above discussion, we can regard (JU)ij as T'ij, the components of stress T measured in a frame which is deforming by Ft relative to the unprimed frame. That is to say, the basis vectors of the primed frame are given by u'n = Ft un. In this case, since Ft is not a rotation, if the un start as unit vectors, then the u'n are not unit vectors. One can think of each basis vector un as being aligned with its own dumbbell dx(n) and then we have dx'(n)= Ft dx(n). What this says is that the basis vectors are embedded in the fluid which deforms as it flows according to Ft. The basis vectors "convect" with the fluid, so this upper Oldroyd stress is called the upper convective stress tensor. For the lower Oldroyd stress we have JL ≡ FtT T Ft and this cannot be written in the form JL = R-1 M (R-1)T so this does not fit into our interpretive template. But in the next section we show that JL is the covariant partner to the contravariant tensor JU so they are both the same animal and we are happy to have the interpretation above for JU. The Oldroyd convected stresses in developmental and standard notation In the previous section two adjusted stress tensors were introduced, JU ≡ Ft-1 T Ft-1T // upper JL ≡ FtT T Ft // lower In developmental notation, a covariant tensor gets an overbar while a contravariant one does not. If we assume that frame S is Cartesian, then T = as per Section 5 (h) for vectors. We can interpret the above two equations in this manner J ≡ Ft-1 T Ft-1T // upper ≡ FtT Ft // lower which we compare with the first equations in Section 5 (f) where we change generic matrix name M to T, T' = R T RT // contravariant rank-2 tensor transforms this way ' = ST S // covariant rank-2 tensor transforms this way . Setting R = Ft-1 and S = R-1 = Ft gives T' = Ft-1 T Ft-1T // contravariant rank-2 tensor ' = FtT Ft // covariant rank-2 tensor Therefore we identify JU = J = T' = contravariant stress tensor T viewed from a frame convecting at Ft-1 JL = = ' = covariant stress tensor T viewed from a frame convecting at Ft-1 In standard notation the two equations J ≡ Ft-1 T Ft-1T // upper ≡ FtT Ft // lower become Jij = (Ft-1)ia (Ft-1)jb Tab (JU)ij = Jij = contravariant Jij = (Ft-1)ia (Ft-1)jb Tab (JL)ij = Jij = covariant and this explains the meaning of the words "upper" and "lower" in respect to the Oldroyd objects. See footnote on Lai page 485.