Caveats concerning conjugate stress and strain measures for frame indifferent anisotropic elasticity

Transcription

1 This is a modified version of the paper by the same title that appears in Acta Mech, Vol. 129, No. 1 2, In this version, the tensor order of a variable equals the number of underlines beneath it. Also, several sections have been expanded to add more extended discussions than page limits would allow in the original publication. Caveats concerning conjugate stress and strain measures for frame indifferent anisotropic elasticity R. M. Brannon, Albuquerque, New Mexico Summary. Small-distortion constitutive laws are often extemporaneously generalized to large deformations by merely applying them in the unrotated material frame or, equivalently, by using polar rates. This approach does render the resulting constitutive model indifferent to large superimposed rigid rotations, but it may lead to incorrect predictions for both the magnitude and direction of stress whenever there is significant material distortion. To demonstrate this claim, an exact large-deformation solution is derived for the stress in an idealized fiber-reinforced composite. This example shows that the Cauchy tangent stiffness tensor (corresponding to the conjugate pair of Cauchy stress and the symmetric part of the velocity gradient) must evolve in both magnitude and direction whenever the material distorts. Volume changes necessarily lead to a loss of major-symmetry of this Cauchy tangent stiffness tensor, which can be rectified by instead using specific or Kirchhoff stress. A previous work that correctly pointed out the need for the Cauchy stiffness tensor to distort is shown to have overlooked an additional contribution from the rate of distortion. Some of the anomalous properties of the Cauchy stiffness tensor are eradicated by instead using the second Piola-Kirchhoff stress or, equivalently, convected coordinates. Such an approach, however, demands accurate measurements of large-distortion material response, not only to obtain physically realistic results, but also to avoid potential instabilities in numerical computations. 1 Introduction This paper explores some counterexamples that remind us why a sensible large-distortion (or even moderate-distortion) constitutive law cannot be constructed from a small-distortion law by merely applying the small-distortion law in the unrotated configuration. Likewise, it is well known that numerous choices exist for special rates in spatial formulation that will satisfy frame indifference. Because each choice generally gives a different answer, obtaining a satisfactory large-deformation constitutive law must require considerations beyond frame indifference. This fact is, of course, well-known and acknowledged within the academic research community (especially in the fluid mechanics literature), but is nevertheless occasionally ignored amidst pressures to solve complete problems. Alternative stress-strain measures or approaches such as convected coordinates can improve results, but can be difficult to implement and still do not liberate the researcher from actually measuring material response at large distortions. Within the solids field, popular conjugate stress and strain rate measures are the Cauchy stress and the symmetric part of the velocity gradient, (which is not actually the rate of σ anything). To satisfy the principle of material frame indifference, relations involving these two variables are often phrased in terms of the unrotated Cauchy stress, σ ij R T ip R T jq σ pq, (1) 1

2 and the unrotated symmetric part of the velocity gradient, D ij R T ip R T jq D pq. (2) Here, repeated indices are summed from 1 to 3, a superscript T denotes the transpose (i.e., R ip T R pi ), and is the proper rotation from the polar decomposition of the deformation gradient, (i.e., F ij V ik R kj R ik V kj ), (3) where is the symmetric positive-definite left stretch, and (more commonly denoted ) Ũ is the symmetric positive-definite right stretch. 2

3 zero distortion zero strain large displacement gradients large distortion large strain large displacement gradients small distortion small strain large displacement gradients zero distortion large strain large displacement gradients In this paper, we use various phrases to describe the severity or nature of a deformation, each having fundamentally different meanings as summarized in the following table: Table 1: phrase overall scalar measure value includes the effects of displacement gradients rotation, volume Ĩ changes, shape changes deformation dilatation Ĩ J 1 volume changes, shape changes volume changes distortion rotation angle J 1 3 Ĩ θ cos 1 [--( 1) ] 2 tr 1 shape changes bulk rotation angular rotation rate ω Ω :Ω changes in rotation axis or rotation angle vorticity w W :W changes in rotation and shape 3

4 In Table 1, J is the Jacobian of the deformation, defined, J det det (4) The spatial gradient of the velocity field, deformation gradient tensor according to L ij v i x j, is related to the rate of the 1 (5) and the vorticity tensor in Table 1 is W is defined T. Ω 1 -- ( T). The bulk (also called polar ) spin tensor 2 When we refer to the deformed configuration at a given instant of time, the phrase zero net rotation simply means that, which applies if and only if θ 0. A deformation Ĩ interval is arotational if θ remains constant and therefore ω 0. A deformation interval is irrotational if w 0. Ironically, a simple vortex is an irrotational field that is not arotational. Simple shear is a deformation for which vorticity is constant, but the polar rotation rate is varying. In fact, the polar rotation rate for simple shear initially equals the vorticity, but approaches zero as the deformation proceeds. This simple observation leads constitutive modelers to ask themselves which is a more useful measure of spin, the vorticity or polar rotation rate? Vorticity is of questionable value as a measure of rotation rate because, as indicated in Table 1, its value can be altered by distortion or bulk rotation. In his well-known paper, Dienes [2] advocated the polar rate because using vorticity results to oscillatory shear stress predictions. Subtle criticisms of Dienes s work will be discussed later in this paper. A large deformation problem is one for which is large. Large deformation can Ĩ include contributions from both size and shape changes. A deformation involves small distortions (i.e., negligible shape changes but, permissibly, large volume changes) if the stretch tensor is nearly isotropic (i.e., for some multiple λ ). Stated in the λĩ contrapositive, a necessary condition for large distortions is that the deviatoric part of must be large in magnitude. A key theme in this paper is that the principle of material frame indifference is nothing more than a consistency requirement that ensures that material response in arotational loading will be related in a specific way to the response under the same loading with superimposed rigid rotation. Material frame indifference is a consistency requirement because satisfying it has no bearing whatsoever on whether or not these two model predictions are any good in the first 4

5 place. Frame indifference simply ensures that model predictions good or bad transform in particular ways under superimposed rigid material motion. When Dienes and others favor one objective rate over another, they are not addressing the issue of frame indifference. They are instead addressing the question of whether the predictions are good or bad compared to measured data. These authors almost always take the structure of the constitutive equations to be unchanged when comparing rates. Hence, what is really being identified is which special rates are most appropriate to a specific constitutive model structure. A different model structure might have better results with a different objective rate, so a rate cannot per se be declared good or bad it is actually the marriage of model structure with certain rates that is good or bad. These statements will be later clarified in the context of Dienes s arguments that favored the polar rate over the Jaumann rate. Throughout this paper, an overbar denotes an operation that unrotates the basis of any spatial tensor back to the reference configuration, leaving the components unchanged. Thus, If is a vector, w w i R T ip w p. (6) If is a second-order tensor, à A ij R T ip R T jq A pq. (7) If is a fourth-order tensor, Ỹ Y ijkl R T ip R T jq R kr R T ls Y pqrs, (8) and so on. The polar rate o A of any second-order tensor A is defined by o à à + Ω Ã Ã, (9) Ω where T is the polar spin and a superposed dot denotes the usual material time rate. Ω The more general definition of the polar rate (applicable for tensors of any order) is a Lie derivative. Specifically, the polar rate is found by taking the true rate of the unrotated tensor, and then rotating the result back to the spatial configuration. Using our overbar notation, the polar rate is then defined simply by o à o à Ã, (10) This result shows that any linear constitutive relation phrased in terms of polar rates of spatial tensors is equivalent to the same relation phrased in terms of material rates of the associated unrotated tensors as long as material vectors such as fiber orientations, which are implicitly part of the constitutive model, are appropriately unrotated as well. 5

6 As will be illustrated in this paper via a simple fiber-composite example, the unrotated orientation of a material fiber is not generally coincident with the initial orientation of that fiber. Material distortion (shape changes even in the absence of rotation) makes these orientations differ. Specifically, if denotes a material fiber in the initial configuration, then, after deformation, it will become in the spatial configuration. Unrotation of this material fiber gives T ( ) T ( ). (11) The fact that does not discount the usefulness of casting all constitutive models in an unrotated reference configuration. To the contrary, doing so is very useful because in the reference configuration, the material model automatically satisfies frame indifference, which allows the modeler to focus attention on how to improve results under large distortion without concern for rotation or special objective time rates. In the unrotated configuration, all rates are genuine rates, which can be exploited to significantly improve accuracy and speed of numerical calculations. To reiterate, casting material models in the unrotated configuration is one of the most straightforward methods for satisfying frame indifference without any loss in generality of the solution upon re-rotating into the spatial configuration. Physical considerations in the unrotated configuration, especially for anisotropic media, are the focus of this paper. Frame indifference is a different issue. It is physical considerations associated with material distortion that often guide a constitutive modeler s choice of stress and strain measures. As explained later in the context of an exact fiber-composite solution, the choice of stress and strain measures affects the simplicity of the governing equations. Any other choices can be made without loss in generality, at the cost of simplicity in the constitutive structure. The second Piola-Kirchhoff (PK2) stress is a commonly-used alternative stress measure s defined by 1 T. In indicial notation, s J σ s ij JF 1 ip F 1 jq JV 1 ip σ pq V 1 jq σ pq, (12) where the Jacobian J is the ratio of the initial mass density to final density ρ: ρ o J det( ) ρ ---- o ρ. (13) The Lagrange strain, defined by ε 6

7 1 -- ( T ) ε 2 Ĩ 1 -- ( ) 2 Ĩ, (14) is conjugate to the PK2 stress. s The unrotated symmetric part of the velocity gradient (Eq. (2)) is related to the stretching rate V and the Lagrange strain rate by ε ( + ). (15) 2 ε Constitutive laws phrased in terms of the PK2 stress and the Lagrange strain s ε automatically satisfy the principle of material frame indifference, as do those that use the unrotated Cauchy stress and unrotated rate-of-deformation, though the two approaches σ do not give identical results. Importantly, merely satisfying the principle of material frame indifference is not sufficient to make a model sensible for high-distortion problems such as penetration. Many so-called large-deformation constitutive laws are in fact valid only for large rotations, not large distortions. When users or developers apply such a model beyond its applicability, the model itself might be wrongly blamed for the incorrect results. More seriously, material constants might be inappropriately tuned to match experimental results for deformations that lie beyond the model s capability, thereby marring the model s credibility for any other loading paths. To illustrate these points in an extreme and idealized example, an exact stress-strain relation will be derived for a microstructure consisting of thin idealized fibers embedded in a negligibly stiff matrix (air). For very thin fibers, the Cauchy stress must always be uniaxial in the fiber direction. However, even for pure stretch deformations, the fiber direction generally changes with time, so the rate of the unrotated Cauchy stress is not uniaxial. This directional rate effect (negligible for small distortions without residual stress) is rarely captured in modern large-deformation constitutive laws and can be naturally accommodated by phrasing the constitutive law in terms of the PK2 stress. However, the magnitude of the result must still be governed by nonlinear high-distortion experimental data. For our fibers-in-air example, will be shown to be linear in. In other words, there σ does exist a fourth-order tensor such that σ ij L ijpq D pq. By using the definition of the unrotation overbar, this result may be automatically converted to the spatial constitutive o equation σ ij L ijpq D pq. However, we show here that will not be major-symmetric (that is, ) if there are dilatation rates. Major symmetry may be recovered, if the stress L ijpq L pqij 7

8 measure is replaced by the thermodynamically consistent specific stress (stress divided by density), or equivalently by the Kirchhoff stress, (which may explain its increased use in Jσ modern constitutive models). Using the exact solution for fibers-in-air, we will demonstrate that there exists a fourth-order tensor E ijkl such that the exact solution may be expressed in terms of the second Piola-Kirchhoff stress and Lagrangean strain rate in the linear form s ij E ijkl ε kl, which likewise has an analogous spatial form in terms of polar rates, o o s ij E ijkl ε kl. Not surprisingly, L ijkl and E ijkl are not equal. The transformation relations that relate conjugate stress-strain pairs naturally influence the structure of the tangent stiffness tensor. It would be grossly inappropriate to retain the tangent stiffness structure from one conjugate pair to judge merits of other stress-strain definitions. Likewise, the exact solution of our fibers-in-air example is always expressible in linear rate form using polar rates and, because other objective rates are linearly related to polar objective rates, the exact solution is always linear with respect to objective rates no matter what rates are selected. The distinction is that each formulation of the unique exact solution has different tangent stiffness structures. It would be grossly inappropriate to retain the structure from one formulation to assess the merits of other objective rates. This is essentially what Dienes (and others) are doing when they touts one rate over another. Instead, we hope to clarify through our exact fibers-inair example that it is not special material rates that are good or bad. Instead, it is the simplicity of the associated constitutive structure that is good or bad. The preferred model is the one with the simplest structure. 8

9 2 Exact solution for an idealized fiber-reinforced material Consider a material consisting of stiff fibers uniformly distributed in a very weak matrix (air). Single fibers are presumed well-characterized. Specifically, if a single fiber is stretched so that its current length divided by its initial length is λ, then the force in that fiber is given by some known function F (λ) satisfying F (1)0. Initial Cross-sectional area A o Deformed area A J n A o NM nf - T N γ mf M Force in single fiber F(λ) area of force A f A cosγ A/( n m ) JA o /λ This is a color version of the figure in the original paper. Suppose all the fibers have an initial orientation parallel to a unit vector, and they are distributed uniformly with fibers per unit initial cross-sectional area. Then the ν o representative volume element sketched in Fig. 1 contains a total of ν o A o fibers. A homogeneous deformation will distort those fibers to a new orientation parallel to. (16) Note that is not a unit vector. The fiber stretch λ is just the magnitude of : λ T. (17) In Fig. 1, the initial cross-sectional area A o has a unit normal. By Nanson s relation [1], Ñ this initial area deforms to a new orientation parallel to ñ T Ñ. (18) 9

10 Also by Nanson s relation, the magnitude of the deformed area is A JA o ñ. Figure 1 depicts the deformed element as seen in the plane containing and ñ where it is clear that the new cross-sectional area A f (i.e., the area normal to the uniaxial fiber force) is simply Acosγ JA o ñ cosγ, where γ is the angle between and ñ. Hence, cosγ ( ñ) ( ñ ). Noting that ñ 1 and recalling that λ, we find that A f JA o λ. The Cauchy stress must be uniaxial in the fiber direction. The force per fiber is F (λ). σ The number of fibers is ν o A o. Thus, the uniaxial stress is the total force F (λ)ν o A o divided by the area currently normal to the fiber direction, JA o /λ. Recalling that the magnitude of is λ, the exact solution (valid for any deformation ) for the Cauchy stress tensor is σ σ ij G( λ) m, where. (19) J i m F λ j G( λ) ν ( ) o λ Hence, the exact solution for the unrotated Cauchy stress is σ ij G ( λ) m J i m j where m i R T ip m p V ij M j. (20) Substituting Eq. (20) into Eq. (12) shows that the exact solution for the PK2 stress is s ij G( λ)m i M j. (21) These exact solutions demonstrate that the stress is a nonlinear function of strain even if the fiber force function F (λ) is linear (affine). In other words, nonlinearity of large-distortion material response can arise as much from kinematics as from inherent material nonlinearities. An implicit goal of stress and strain measures is to (approximately) capture these kinematic contributions. 3 Exact tangent modulus tensors. We now verify that each exact stress rate can be written as a fourth-order tangent modulus operating on the respective conjugate strain rate, as is generally assumed a priori in the literature. The PK2 tangent modulus (i.e., the modulus associated with the PK2 stress) possesses many of the properties usually assumed about modulus tensors. However, we will 10

11 show that the Cauchy tangent modulus strongly depends on the deformation and is not even major-symmetric. Differentiating Eq. (17), the material time rate of the fiber stretch is M λ i M j ε ij λ m i m j D ij λ. (22) The rate of the Jacobian is J Jδ kl D kl. (23) Differentiating Eq. (21) using Eq. (22) shows that s ij E ijkl ε G' ( λ) kl, where E ijkl M. (24) λ i M j M k M l In the limiting case of small distortions and volume changes (where modulus tensor is ), the PK2 tangent Ĩ 0 E ijkl ν o F'1 ( )M i M j M k M l valid for λ 1. (25) Referring to Eq. (24), the exact large-distortion PK2 tangent modulus is not generally 0 Ẽ equal to the small-distortion PK2 modulus except for the highly nonphysical case that F (λ) Ẽ is proportional to λ(λ 2-1). Since this function has a zero slope at λ 1 3, cavalierly using a constant PK2 modulus could lead to compression instabilities in numerical calculations. On the other hand, the small distortion PK2 modulus does possess an appealing feature of having the same material direction vectors as the exact large-distortion modulus. This would be an attractive feature in numerical calculations because it would bypass any need to alter material directions in response to material distortion. Such is not true for the unrotated Cauchy stress, as shown below. When selecting strain and conjugate stress measures, we will explain that one is faced with two choices: (1) use the Cauchy stress to obtain qualitatively reasonable predictions from a linear elastic model for uniaxial deformations in the fiber direction, but face extraordinarily complicated non-linearity of the tangent modulus with respect to fiber orientation, or (2) use the PK2 stress and retain all natural material symmetries (i.e., no special treatment to handle fiber reorientation), but necessarily incorporate non-linear elastic constants to avoid compression instabilities. Which of one of these choices constitutes a simpler 11

12 constitutive framework is a matter of taste. Either one, properly implemented, will give the exact solution, so it boils down to asking: which one is easier to implement? Deriving an exact solution for the Cauchy stress rate is straightforward but moderately complicated and, as it happens, not necessary. The quantities λ, J and are all functions of. Consequently, their rates are all linear in. Equation (15) may be solved to give as a linear function of (invertibility follows by considering the equation in the principal basis of ). Thus the chain rule may be applied to Eq. (20) to show that the exact rate of the unrotated Cauchy stress is linear in, i.e., there does exist a (very complicated) fourth-order Cauchy tangent tensor which is a function of the stretch, but is independent of such that the exact solution for the unrotated Cauchy stress may be written σ ij L ijkl D kl. (26) The existence of a Cauchy tangent tensor is often presumed in the literature, and this result validates such an assumption for our fiber material, However, its complicated dependence on the state of deformation adulterates its usual interpretation as a material property. Furthermore, another common assumption about the nature of (namely, that it is major-symmetric) is generally inappropriate, as discussed in Section 4. L ijkl We now examine the structure of the tensor Eq. (20), for the unrotated Cauchy stress may be written in more detail. The exact solution, σ ij hp i p j, (27) where p λ and h. (28) λ 2 G( λ) J Here, p is simply a unit vector in the direction of and h is a scalar function of the stretch λ and the Jacobian J (also note that h0 if and only if λ1). Recalling Eqs. (22) and (23), h ( η p k p l h δ kl )D kl where η λ [ 2G( λ) + λg' ( λ) ]. (29) J Hence, the exact solution for the rate of the unrotated Cauchy stress is of the form σ ij p i p j ( η p k p l h δ kl )D kl + hp ( + p i pj ) i p j. (30) 12

13 Because is a unit vector, its rate is always perpendicular to. Consequently, the stress p p p rate is not uniaxial even though the stress itself is uniaxial. The rate of p is given by 1 pi A ij p j p i ( p j p k D jk ), where A ij V ikv kj D ij H ij. (31) Here, where is the unrotated spin ( T ) and is the unrotated vorticity. H Ω W Ω W Dienes [2] showed that can be expressed as a linear function of and, hence, so can, H à thereby again demonstrating that the entire right-hand side of Eq. (30) may be expressed in the form of Eq. (26). However, is equal to if and only if and share the same principal à axes, which describes a class of proportional stretching deformations not uncommon in laboratory experiments and useful in subsequent examples. Below, we confirm Zheng s assertion [3] that the tensor in Eq. (26) must be transversely isotropic about a privileged direction parallel to, not. We show, however, that has an unexpected structure. Since the fibers are all parallel to, and because the material has no resistance to shear along the fiber directions, it would seem natural to take L ijkl Ep i p j p k p l, where E is a material constant. If this conjecture were true, however, the unrotated stress rate would be proportional to p i p j that is, it would be uniaxial, in contradiction to the exact solution Eq. (30) except for small fiber stretches (so that h 0) or for deformations that do not distort the fibers from their initial orientation (so that ). p 0 Recall that in Eq. (26) exists for any deformation. We now examine the fundamental structure of for a class of deformations in which and share the same principal axes. Then, and Eqs. (30) and (31) combine to give à σ ij [ h( δ kl D kl ) + ( η 2h)P kl D kl ]P ij + hp ( ik D kj + D ik P kj ), (32) where P ij is a simple projection tensor defined by P ij p i p j. (33) Factoring out the rate of deformation gives σ ij L * ijkl Dkl h ( P ij δ )D kl kl, (34) In terms of Mandel components (ordered 11, 22, 33, 23, 31, 12) corresponding to a basis having the 1-direction aligned with, the components of L* ijkl are p 13

14 L* ijkl η h h. (35) The tensor * is indeed transversely isotropic about, but the presence of h makes the p stress rate have nonzero shear (23 and 31) components even though the stress itself is always uniaxial. These components arise because the fiber axis changes with material distortion even if the polar rotation tensor is the identity, a contribution overlooked by Zheng [3]. If the distortion is severe enough that, then generally h 0 (neglecting h would be tantamount to neglecting stress itself). Put differently, it would be inconsistent to assert that * must be transversely isotropic about while not allowing its components to include contributions from the rate of change of p p. It would be wrong to generalize, say, Spencer s small-distortion expression of stress as a transverse function of strain [4] to large deformations in the unrotated configuration. Doing so misses contributions like the last term in Eq. (30). Recalling Eqs. (7) and (10), Eq. (32) may be easily converted to spatial form for direct comparison with Zheng s generalization of a transversely isotropic constitutive equation from its arotational form [3]. Zheng s Eq. (4.5) fails to capture the last term in Eq. (34), unless h 0 (which is possible only for the uninteresting case of unstretched fibers) or D kk 0 (also a very restrictive condition). Furthermore, recall there is no comparison whatsoever without the assumption that and share the same principal axes, representing yet another restriction that would have to be placed on the deformation in order to obtain good results using Zheng s expression. It might appear at first glance that the last term in Zheng s Eq. (4.5) captures at least the qualitative character of the last term in Eq. (32), but Zheng s term arises not from the rate of rotation of the fibers, but simply from the shear resistance of the material, which must be zero for the idealized fiber material. As discussed in Section 5, Zheng s approach for generalizing arotational constitutive laws seems incapable of capturing certain distinctions in material response that arise from the microstructure. Of course, despite these defects, Zheng s model is far superior to any model that takes the reference transverse axis of symmetry to be unchanged by distortion. 14

15 4 Major-symmetry of tangent tensors The fourth-order tensor hp p in the second term of Eq. (34) is transversely isotropic i j δ kl about, but it is not major-symmetric. Hence, in the exact solution (Eq. 26), the tensor is p not major-symmetric (i.e., ). The existence of an elastic energy function is often L ijkl L klij used to justify an assumption of major-symmetry. The argument is subtle for large deformations where, in the absence of thermal power, 1 the first law of thermodynamics requires that ρe σ ij D ij σ ij D ij. (36) Here, ρ is the density and e is the internal energy per unit mass. Note that σ ij D ij is not the true rate of any path-independent quantity, while ( σ ij ρ)d ij is a true rate. Equivalently, ( Jσ ij )D ij is a true rate. The non-major-symmetric term in Eq. (34) would not be present if we had instead differentiated the exact solution for Jσ ij. This is, in fact, a general result as shown below. In terms of the PK2 stress (without thermal power) is s and Lagrange strain, the first law of thermodynamics ε ρ o e s ij ε ij, (37) For a closed elastic system, the PK2 stress is taken to be a (material) function of strain, and it follows from Eq. (37) that this function is derivable from a potential (the energy): s ij ---- ρ o e ε ij. (38) Hence, applying the chain rule noting that the material rate of ρ o is zero, sij E ijkl ε kl, where E ijkl ρ 2 e o (39) ε ij ε kl As long as the energy e is sufficiently differentiable, the fourth-order tensor Ẽ symmetry. possesses major A similar analysis cannot be performed directly on Eq. (36) because is not a true rate with respect to deformation. 2 Instead, Eq. (39) must be rephrased in terms of Cauchy stress. Recalling that J ρ o ρ, Eq. (12) may be written 1 Including thermal power would not change our ultimate conclusions. 2 There does not generally exist a tensor that depends only on the deformation such that. ξ ξ 15

16 σ ij s pq V ρ ip V jq ρ o. (40) Differentiating this expression using Eqs. (39), (15), and (13) shows that the material rate of the specific stress is D σ ij (41) Dt ρ V ip V jq V kr V E pqrs ls ρj Dkl V 1 ipv σ qj σ pq ip V 1 ρ ρ pq V qj Recalling that is linear in, this result proves that for any deformation path there does exist a fourth-order tensor such that C D σ ij C Dt ρ ρ ijkl D kl. (42) Noting that ρ ρδ ij D ij, this result also confirms the existence of a Cauchy tangent tensor such that σ ij L ijkl D kl. To demonstrate the claim that is not necessarily major symmetric, it is sufficient to consider proportional stretching paths where and share principal axes. Then recalling the discussion following Eq. (31), Eq. (41) becomes 1 1 C ijkl --V. (43) J ip V jq V kr V ls E pqrs + -- ( δ 2 ik σ jl + δ jl σ ik + δ il σ jk + δ jk σ il ) Since possesses major symmetry, so does. Noting that ρ ρδ ij D ij, Eq. (42) may be Ẽ C written σ ij L ijkl D kl, where L ijkl C ijkl σ ij δ kl, (44) which is not major symmetric. This example of proportional stretching proves that the rate of stress itself is not generally expressible as a major-symmetric tensor operating on because volumetric contributions [second term in Eq. (44), or second term in Eq. (34)] are not majorsymmetric. Bergander [5] mentions that Kirchhoff s stress tensor is routinely used in Jσ modern constitutive models, though sometimes without any justification other than convenience. The above discussion shows that using specific stress (or Kirchhoff s stress tensor, ) endows a major symmetry to the conjugate tangent modulus tensor, which is an Jσ appealing and useful property for many applications. However, the fiber example studied earlier shows that even if Kirchhoff s stress is used, the associated tangent modulus tensor cannot sensibly be regarded as a material property it must change as the material distorts. 16

17 5 Microstructure affects macrostructure We now mention that the Cauchy tangent tensor must distort with the material in a way that depends on the microstructure. In Fig. 2, a fiber composite and a laminate are each subjected to the same macroscopic pure stretch. Both composites are initially transversely isotropic with the same axis of symmetry ( Ñ ). Upon distortion, the symmetry axis moves with the material in the fiber case, but with the material planes in the laminate case. For the fiber case, Zheng [3] correctly pointed out that a large deformation law of the form f( ) must also depend on the distorted fiber direction, not the reference σ direction. We have already shown [Eq. (30)] that it must additionally depend on the rate. In the laminate case, the function f apparently must depend on the distorted laminate plane normal ñ (and its rate). To accommodate these microstructural considerations, a numerical model (e.g., [6]) might require the user to specify whether the material possesses a fiber or a laminate microstructure, though such an approach might be stymied by exotic laminates whose layers are fiber composites. Zheng did not discuss the alternative law of the form g( ), where the PK2 function g apparently depends on the undistorted reference direction s ε in both the fiber and laminate cases, again suggesting that the PK2 description might be better suited for large-distortion analyses of anisotropic materials. Fibers embedded in a weak matrix. FIBERS-IN-AIR Ñ ñ Membranes embedded in a weak matrix, like a deck of cards. DECK OF CARDS T Ñ Fiber Composite Layered Composite This is a revised version of the figure from the original paper that uses pure shear (a pure stretch is just this with a superimposed rigid rotation that removes the rotation from shearing). The key point is that the material direction that defines material symmetry will not necessarily flow with the material. 17

18 6 Conclusions This paper has reviewed several important caveats regarding large-distortion constitutive laws. Namely, a sensible large-distortion constitutive law generally cannot be constructed from a small-distortion law by simply applying the small-distortion law in the unrotated configuration, even though frame indifference is satisfied. Any anisotropic constitutive law phrased in terms of unrotated Cauchy stress and rate-of-deformation must account for σ distorted material directions. For our fiber example, this means that the Cauchy tangent stiffness tensor must depend not on the initial fiber orientation, but on the distorted orientation, and its rate. Furthermore, the Cauchy stress must be replaced by the σ specific stress ρ or Kirchhoff s stress if the associated tangent modulus tensor is to σ Jσ possess major symmetry for general deformations. Comparing a fiber composite with a laminate composite having the same transverse axis of symmetry demonstrates that the microstructural source of the anisotropy must be explicitly accommodated for any largedistortion problem modeled using unrotated Cauchy stress with the unrotated rate-ofdeformation. These complications associated with distortions of the material directions can be managed by using the second Piola-Kirchhoff stress of Eq. (12) together with the Lagrange strain of Eq. (14). For this conjugate pair, the tangent modulus possesses major symmetry and depends on the initial (not distorted) material directions; hence, there are no counterintuitive contributions from material direction rates. However, to avoid numerical instabilities, the PK2 modulus magnitude must still be determined by high-distortion experiments. Acknowledgments This work was supported by the United States Department of Energy under Contract DE-AC04-94AL Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy. References [1] Malvern, L. E.: Introduction to the Mechanics of a Continuous Medium, Englewood Cliffs: Prentice-Hall, [2] Dienes, J. K.: On the analysis of rotation and stress rate in deforming bodies. Acta Mech. 32, (1979). [3] Zheng, Q.-S.: On the generalization of constitutive laws from their arotational forms. Acta Mech., 91, (1992). [4] Spencer, A.J.M (ed.): Continuum theory of the mechanics of fibre-reinforced composites. CISM Courses and Lectures No Wien New York: Springer [5] Bergander, H.: Finite plastic constitutive laws for finite deformations. Acta Mech. 109, (1995). [6] Taylor, P. A.: The CTH transverse isotropy model. Sandia National Laboratories Report SAND (1995). Author s address: Department of Computational Physics and Mechanics, Sandia National Laboratories, Albuquerque, NM

19 Deformed area A J n A o ~ Initial Cross-sectional area A o T Ñ ñ Ñ γ area of force A f A cosγ ( A ñ ) ñ JA o /λ Fig. 1. An idealized array of fibers in a negligibly stiff matrix (e.g., air). The fibers distort to a new orientation parallel to (the magnitude of which is the fiber stretch λ). The crosssectional area A o originally normal to the fiber direction distorts to a new orientation parallel to ñ (not a unit vector). The right side of the figure shows the distorted shape as seen in the plane spanned by ñ and. 19

20 Ñ Fiber Composite Laminate Composite Fig. 2. Two initially transversely isotropic microstructures suffering pure shear with enough superimposed rotation to make. The macroscopic symmetry axes move in a way that depends on the microscopic source of Ĩ the anisotropy. ñ 1 Ñ 20