GGeophys. J. Int. (0000) , 000–000
On the stress dependence of the elastic tensor
Matthew Maitra & David Al-Attar
Department of Earth Sciences, Bullard Laboratories, University of Cambridge,Madingley Road, Cambridge CB3 OEZ, United Kingdom.Email: [email protected]
29 July 2020
SUMMARY
The dependence of the elastic tensor on the equilibrium stress is investigated theoretically.Using ideas from finite-elasticity, it is first shown that both the equilibrium stress and elastictensor are given uniquely in terms of the equilibrium deformation gradient relative to a fixedchoice of reference body. Inversion of the relation between the deformation gradient and stressmight, therefore, be expected to lead neatly to the desired expression for the elastic tensor.Unfortunately, the deformation gradient can only be recovered from the stress up to a choiceof rotation matrix. Hence it is not possible in general to express the elastic tensor as a uniquefunction of the equilibrium stress. By considering material symmetries, though, it is shownthat the degree of non-uniqueness can sometimes be reduced, and in some cases even removedentirely. These results are illustrated through a range numerical calculations, and we also obtainlinearised relations applicable to small perturbations in equilibrium stress. Finally, we make acomparison with previous studies, before considering implications for geophysical forward- andinverse-modelling.
Key words:
Theoretical seismology; elasticity and anelasticity; seismic anisotropy.
Classical linear elasticity is based on an assumption of small deformations away from a stress-free equilibrium. Its applicability to seismologyis therefore unclear, given the presence of large equilibrium stresses within the Earth. Indeed, approaches to seismic wave propagation withina pre-stressed Earth have a long and complicated history (e.g. Dahlen & Tromp 1998, Chapter 1) and it was not until Dahlen (1972b), inconjunction with the work of Biot (1965), that a complete and correct treatment was given.Dahlen’s work, based on linearisation of the equations of finite-elasticity, led to an elastic tensor of the form Λ ijkl = Γ ijkl + 12 (cid:0) T ij δ kl + T kl δ ij + T ik δ jl − T jk δ il − T il δ jk − T jl δ ik (cid:1) , (1.1)where we have followed the notation of Dahlen & Tromp (1998). Note that, in fact, this is one of a range of equivalent elastic tensors that canbe used depending on how the equations of motion are formulated. Here T ij denotes the equilibrium stress tensor, while we have set Γ ijkl = (cid:18) κ − µ (cid:19) δ ij δ kl + µ ( δ ik δ jl + δ il δ jk ) + γ ijkl , (1.2)with κ the bulk modulus, µ the shear modulus, and γ ijkl representing what Dahlen interpreted as the inherent anisotropy of the medium, theanisotropy that would be present in the absence of pre-stress. Notably, the first term in eq.(1.1) has the classical symmetries Γ ijkl = Γ jikl = Γ ijlk = Γ klij , (1.3)and so involves at most 21 independent components. When the equilibrium stress vanishes, Dahlen’s expression is identical in form to that ofclassical linear elasticity. In general, however, his elastic tensor retains only the so-called hyperelastic symmetry Λ ijkl = Λ klij , (1.4)lacking the minor symmetries Λ ijkl = Λ jikl = Λ ijlk . This expression for the elastic tensor has a number of important implications. Firstly, theelastic wave speeds display no explicit dependence on equilibrium pressure. Secondly, Dahlen (1972a) showed that, in the absence of inherentanisotropy, deviatoric stresses have no first-order effect on P-wave velocities whilst S-waves are split to the same accuracy (Fig.1). It shouldalso be emphasised that although T ij only appears linearly in eq.(1.1), Dahlen’s theory is not linearised with respect to equilibrium stress. a r X i v : . [ phy s i c s . g e o - ph ] J u l M. A. Maitra & D. Al-Attar
Figure 1.
The effect of deviatoric stress on an isotropic body, according to Dahlen (1972b). The left panel shows the slowness surface of a stress-free isotropicmaterial, sliced through the x – y plane. The sheets are circular and those for the two shear-waves (outermost) are indistinguishable. The right panel shows theslowness surface of the same material under the application of a small deviatoric stress according to Dahlen’s theory. The isotropic result is shown faintly in thebackground for reference. Whilst the the S-waves are noticeably split, the P-wave surface has not perceptibly moved from its isotropic position, illustratingDahlen’s result that P-wave speeds are not changed to first-order by a small stress. The work of Dahlen (1972b) has underlain most of global seismology and related fields since its publication. Recently, however, Tromp& Trampert (2018) revisited seismic wave propagation in the presence of non-zero equilibrium stress. Their starting point was the expressionfor the elastic tensor (Dahlen & Tromp 1998, Section 3.6.2) Λ ijkl = Γ ijkl + a (cid:0) T ij δ kl + T kl δ ij (cid:1) + (1 + b ) T ik δ jl + b (cid:0) T jk δ il + T il δ jk + T jl δ ik (cid:1) . (1.5)Here Γ ijkl and T ij are as above, while a and b are dimensionless constants that are introduced through a symmetry argument, but whosevalues are not fixed by the derivation of eq.(1.5). The elastic tensor of Dahlen (1972b) arises from this expression by setting a = − b = 12 , (1.6)which is the unique combination of a and b that leads to seismic wave speeds with no explicit dependence on pressure. It is clear physicallythat pressure must affect seismic wave speeds, so the elastic moduli occurring in Dahlen’s tensor Γ ijkl must be implicit functions thereof. Animmediate limitation of Dahlen’s theory, then, is that it gives no way to determine how the elastic wave speeds vary with pressure. In returningto this topic Tromp & Trampert suggested that one should instead take a = 12 + 13 µ (cid:48) − κ (cid:48) b = − − µ (cid:48) , (1.7)with κ (cid:48) and µ (cid:48) denoting pressure-derivatives of the bulk and shear moduli, respectively. The motivation behind this choice is that, in theabsence of inherent anisotropy, it leads to the wave speeds ρc p = (cid:0) κ + κ (cid:48) p (cid:1) + 43 (cid:0) µ + µ (cid:48) p (cid:1) ρc s = (cid:0) µ + µ (cid:48) p (cid:1) (1.8)in a hydrostatically stressed medium with pressure p and density ρ . The authors argue that this is desirable on the grounds that these are theclassical expressions for isotropic wave speeds, but with elastic constants that are explicitly corrected for a non-zero pressure. Building on thiswork, Tromp et al. (2019, eq.25) then generalised eq.(1.5), using Tromp & Trampert’s (2018) values of a and b , to give the elastic tensor Ξ ijkl = Γ ijkl + p Γ (cid:48) ijkl − p ( δ ij δ kl − δ ik δ jl − δ jk δ il )+ 12 (cid:0) τ ij δ kl + τ kl δ ij (cid:1) − (cid:0) τ ik δ jl + τ jk δ il + τ il δ jk + τ jl δ ik (cid:1) − (cid:0) Γ (cid:48) imkl τ mj + Γ (cid:48) jmkl τ mi + Γ (cid:48) kmij τ ml + Γ (cid:48) lmij τ mk (cid:1) , (1.9)where Γ (cid:48) ijkl denotes the pressure-derivative of Γ ijkl , τ ij is the deviatoric part of T ij as given above, and the elastic tensor Λ ijkl can be n the stress dependence of the elastic tensor Figure 2.
Dahlen’s theory compared with that of Tromp et al. (2019). As above, we have taken an isotropic material (slowness surface shown faintly on eachplot) and applied a given, small deviatoric stress to it according to the theories of the respective authors. As stated in the main text, Tromp et al.’s theory (right)does indeed predict that P-wave speeds are perturbed to first-order in small stress, a distinct physical prediction from Dahlen’s theory (left). For the sake ofillustration we have taken µ (cid:48) = κ (cid:48) = 1 , consistent with Stacey (1992). obtained through (Tromp & Trampert 2018, eq.31) Λ ijkl = Ξ ijkl + T ik δ jl . (1.10)This theory displays quantitatively and qualitatively different behaviour from that of Dahlen (1972b). In particular, and in direct contraventionof Dahlen, Tromp et al. suggest that both P- and S-wave speeds change to first-order if a small deviatoric stress is added to an isotropic medium(Fig.2).We are left with two distinct theories for the effect of equilibrium stress on the elastic tensor. Both results can be obtained from eq.(1.5),but through different physical arguments, both of which are based on specific choices of a and b . Because the theories lead to distinctobservable behaviour, mineral physics experiments or ab initio calculations have the potential to distinguish between them (e.g. Tromp et al.2019). Nonetheless, it is not clear to the present authors that the resolution of this issue lies simply in finding a correct choice for theseconstants. For example, if this were the case then the elastic tensor would, as noted above, have an exact linear dependence on equilibriumstress. While it is not impossible that this holds, it would surely be unexpected. Indeed, Dahlen’s original work makes clear that eq.(1.5) is avalid representation of the elastic tensor, so the real question is whether eq.(1.5) makes explicit the full dependence on equilibrium stress. Inorder to make progress with this issue we take a new approach rooted firmly in the theory of finite-elasticity. We present that argument inSection 3, having first reviewed the necessary ideas in Section 2. After stating our main result we give some examples, both numerical andanalytical, in Section 4, and discuss the implications of our theory in Section 5. In this section we summarise the aspects of elasticity pertinent to this paper. For more details see, for example, Marsden & Hughes (1994),Dahlen & Tromp (1998) or Truesdell & Noll (2004). Appendix A contains some futher mathematical definitions along with a few non-standardnotations we have found helpful.
The deformation of an elastic body is described relative to a fixed reference configuration , with each particle labelled by its position withinthe associated reference body
M ⊆ R , which is assumed to be connected, bounded, and have an open interior. At a time t , the position inphysical space of the particle at x ∈ M is written ϕ ( x , t ) . In this manner, we define a mapping ϕ : M × R → R , (2.1)which is called the motion of the body relative to the reference configuration. For a fixed time, t , the image of the mapping ϕ ( · , t ) is written M t and represents the region of physical space the body instantaneously occupies. It will be assumed that for each fixed time the mapping M. A. Maitra & D. Al-Attar ϕ ( · , t ) : M → M t is smooth with a smooth inverse. A fundamental object derived from the motion is its deformation gradient , F = D x ϕ , (2.2)where D x denotes partial differentiation with respect to position as defined through ϕ ( x + δ x , t ) = ϕ ( x , t ) +( D x ϕ )( x , t ) · δ x + O (cid:0) (cid:107) δ x (cid:107) (cid:1) . (2.3)(We will generally neglect the subscript on D where it is unambiguous which variable we are differentiating with respect to.) Equivalently, theCartesian components of the deformation gradient are F ij = ∂ϕ i ∂x j . (2.4)The Jacobian is then defined as J = det F . (2.5)Due to our assumption that ϕ ( · , t ) : M → M t has a smooth inverse, it follows from the inverse function theorem (e.g. Marsden & Hughes1994) that the deformation gradient takes values in the general linear group GL (3) (Appendix A1). We assume without loss of generality that J is everywhere positive, meaning that the motion is orientation preserving.The density at time t at the point y ∈ M t in physical space is written (cid:37) ( y , t ) . From conservation of mass, we are led to define the referential density , ρ ( x ) = J ( x , t ) (cid:37) [ ϕ ( x , t ) , t ] , (2.6)a time- independent function within the reference body. Cauchy’s theorem implies that the traction t acting on a surface-element within M t is related linearly to the unit-normal ˆ n of the corresponding reference surface element within M (e.g. Marsden & Hughes 1994). We cantherefore define the first Piola-Kirchhoff stress tensor T through t = T · ˆ n . (2.7)This expression is equivalent to eq.(2.41) of Dahlen & Tromp (1998), but we place our indices according to a different convention; there areconsequent effects within expressions for the elastic tensor. From Newton’s second law we obtain the momentum equation ρ ¨ ϕ − Div T = f , (2.8)where dots are used to represent time differentiation, the divergence of a tensor field is given by (Div T ) i = ∂ j T ij , (2.9)and f denotes the body forces acting on M . To complete the equations of motion we need to relate T and ϕ through a suitable constitutive relation. We follow Dahlen (1972b) byrestricting attention to hyperelastic materials , in which case the first Piola-Kirchhoff stress depends on the motion through the expression T ( x , t ) = ( D F W )[ x , F ( x , t )] , (2.10)where W : M × GL (3) → R is the strain-energy function and D F W denotes its partial derivative with respect to F .The form of the strain-energy function is constrained by the principle of material-frame indifference . Discussed at length by Marsden &Hughes (1994) and Truesdell & Noll (2004), it requires that W ( x , RF ) = W ( x , F ) (2.11)for all rotation matrices R ∈ SO (3) and F ∈ GL (3) (see Appendix A1). It can be shown using the polar decomposition theorem (e.g.Marsden & Hughes 1994) that this condition holds if and only if for some auxiliary strain-energy function V we can write W ( x , F ) = V ( x , C ) , (2.12)with the symmetric right Cauchy–Green deformation tensor defined to be C = F T F . (2.13)Applying the chain rule to differentiate eq.(2.12), we arrive at an alternative expression for the first Piola-Kirchhoff stress in terms of V .Although this calculation can readily be carried out using index notation, it provides an opportunity to explain in a simple situation some of theinvariant notations that will play an important role later. First, by definition of the derivative of W with respect to F we have W ( F + δ F ) − W ( F ) = (cid:104) DW ( F ) , δ F (cid:105) + O ( (cid:107) δ F (cid:107) ) , (2.14)where to avoid clutter we have suppressed spatial arguments, while (cid:104)· , ·(cid:105) is the inner product for matrices defined in eq.(A.15) and (cid:107) · (cid:107) is thecorresponding norm (eq.A.17). Using eq.(2.12), the left hand side is, by definition, equal to V ( C + δ C ) − V ( C ) = (cid:104) DV ( C ) , δ C (cid:105) + O ( (cid:107) δ C (cid:107) ) , (2.15) n the stress dependence of the elastic tensor with δ C the perturbation to C associated with δ F . Using eq.(2.13) it follows that δ C = F T δ F + δ F T F + O ( (cid:107) δ F (cid:107) ) , (2.16)which allows us to write (cid:104) DW ( F ) , δ F (cid:105) = (cid:68) DV ( C ) , F T δ F + δ F T F (cid:69) . (2.17)Looking at the terms on the right hand side, the left multiplication operator defined in eq.(A.3) lets us write F T δ F = L T F · δ F , (2.18)where we have made use of eq.(A.13). Using this notation we find that (cid:68) DV ( C ) , F T δ F (cid:69) = (cid:68) DV ( C ) , L T F · δ F (cid:69) = (cid:104) L F · DV ( C ) , δ F (cid:105) . (2.19)To deal with the remaining term, we note that for any matrices A and B our inner product satisfies (cid:104) A , B (cid:105) = (cid:10) A T , B T (cid:11) . This identity, alongwith the fact that DV ( C ) is necessarily symmetric, implies that (cid:68) DV ( C ) , δ F T F (cid:69) = (cid:68) DV ( C ) , F T δ F (cid:69) , (2.20)and so we conclude that (cid:104) DW ( F ) , δ F (cid:105) = (cid:104) L F · DV ( C ) , δ F (cid:105) . (2.21)Because the perturbation δ F was arbitrary, we have established the identity DW ( F ) = 2 L F · DV ( C ) , (2.22)which, putting back in the spatial arguments, is equivalent to D F W ( x , F ) = 2 F D C V ( x , C ) . (2.23)From this expression we are led to define the symmetric second Piola-Kirchhoff stress tensor S ( x , t ) = 2 D C V [ x , C ( x , t )] . (2.24)Hence we obtain the identity T = FS . (2.25)A third useful stress tensor is the Cauchy stress , σ . It relates the traction on a surface element at y ∈ M t to the surface’s instantaneousunit-normal ˆ n t . From this definition, it can be shown (e.g. Marsden & Hughes 1994) that T = J ( σ ◦ ϕ ) F − T . (2.26)Although it is perhaps not obvious from this expression, the Cauchy stress is symmetric. Using eq.(2.25) we obtain σ ◦ ϕ = 1 J FSF T , (2.27)and the symmetry of σ follows from that of S . For seismological purposes it is generally sufficient to study linearised elastodynamics. We define an equilibrium configuration ϕ e : M → R to be a time-independent solution of the equations of motion subject to a time-independent body force f e and surface traction t e . The resultingequilibrium first Piola-Kirchhoff stress is given by T e ( x ) = D F W [ x , F e ( x )] , (2.28)where F e = D x ϕ e . Using eq.(2.25) and (2.27), the three equilibrium stress tensors are then related by T e = F e S e = J e ( σ e ◦ ϕ e ) F − Te . (2.29)If such a body is subject to a small disturbance from equilibrium, we can look for solutions of the form ϕ ( x , t ) = ϕ e ( x ) + (cid:15) u ( x , t ) + O (cid:0) (cid:15) (cid:1) , (2.30)where u is the displacement vector and (cid:15) a dimensionless perturbation parameter. Under this ansatz the deformation gradient becomes F ( x , t ) = F e ( x ) + (cid:15) D x u ( x , t ) + O (cid:0) (cid:15) (cid:1) , (2.31)while the first Piola-Kirchhoff stress expands to T ( x , t ) = T e ( x ) + (cid:15) Λ ( x ) · D x u ( x , t ) + O ( (cid:15) ) , (2.32)where we have defined the elastic tensor , Λ ( x ) = D F W [ x , F e ( x )] . (2.33) M. A. Maitra & D. Al-Attar
Note that, due to the equality of mixed partial derivatives, the elastic tensor possesses the so-called hyperelastic symmetry Λ T = Λ , (2.34)which, in index notation, takes the familiar form Λ ijkl = Λ klij .As shown by eq.(2.32), the elastic tensor completely describes the linearised constitutive behaviour of the body. In particular, at a point x ∈ M there will be three possible elastic wave speeds in the direction of the unit vector ˆ p . These wave speeds, c , are determined through theeigenvalue problem (e.g. Dahlen & Tromp 1998, Section 3.6.3) Γ ( x , ˆ p ) · a = c a , (2.35)where a is the corresponding polarisation vector, and the Christoffel matrix has components Γ ik ( x , ˆ p ) = 1 ρ ( x ) Λ ijkl ( x )ˆ p j ˆ p l . (2.36)This matrix is symmetric due to eq.(2.34), hence the c are real. Within an elastic solid it is conventionally assumed that these squared wavespeeds are positive, a necessary condition for well-posedness of the linearised equations of motion (e.g. Marsden & Hughes 1994). As thepropagation direction ˆ p varies over the unit two-sphere, the three positive wave-speeds define the so-called slowness surface at x . In general,this surface will be comprised of three distinct sheets, though they can sometimes touch due to degenerate eigenvalues within eq.(2.35).Finally, it is useful to express the elastic tensor in terms of the auxiliary strain-energy function V . The derivation of this result mirrors thatof eq.(2.22), and again we suppress all spatial arguments to avoid clutter. By definition of the elastic tensor, we take as a starting point DW ( F e + δ F ) − DW ( F e ) = Λ · δ F + O ( (cid:107) δ F (cid:107) ) , (2.37)for arbitrary δ F . Using eq.(2.12), the left hand side can be written DW ( F e + δ F ) − DW ( F e ) = 2 L F e + δ F · DV ( C e + δ C ) − L F e · DV ( C e ) , (2.38)where δ C is the perturbation to C associated with δ F . Expanding out the right hand side to first-order it follows that Λ · δ F = L δ F · S e + 12 L F e · [ A · ( F Te δ F + δ F T F e )] , (2.39)where we have used eqs. (2.16) and (2.24), and introduced the fourth-order tensor A = 4 D V ( C e ) . (2.40)To proceed, we first note that L δ F · S e = R S e · δ F by definition of the left and right multiplication operators. Secondly, A · ( F Te δ F + δ F T F e ) = 2 A · ( F Te δ F ) (2.41)because V is a function on symmetric matrices. Thus Λ · δ F = R S e · δ F + (cid:16) L F e A L T F e (cid:17) · δ F , (2.42)and, given that δ F is arbitrary, this establishes our desired expression for the elastic tensor, Λ = R S e + L F e A L T F e . (2.43)It is worth emphasising that, due to the symmetry of C , the tensor A has the full set of classical elastic symmetries A ijkl = A jikl = A ijlk = A klij , (2.44)and so possesses at most 21 independent components. The motion of an elastic body has been described relative to a fixed reference configuration involving material parameters ρ and W . The samebody can, of course, be described using a different choice of reference configuration, and it is natural to ask how the two points of view arerelated. This question was discussed by Al-Attar & Crawford (2016) and Al-Attar et al. (2018); here we simply recall the results relevant tothis work. Let ϕ : M × R → R denote the motion of an elastic body relative to a given reference configuration. The same motion relative to a differentreference configuration will be written ˜ ϕ : ˜ M × R → R , where ˜ M is the associated reference body that will not, in general, be equal to M .At a time t , the particle labelled by x ∈ M lies at the point ϕ ( x , t ) in physical space. Relative to the second description of the motion, theremust be a unique point ˜ x ∈ ˜ M such that ϕ ( x , t ) = ˜ ϕ (˜ x , t ) . (2.45)This correspondence between x and ˜ x holds for all times, defining a mapping ξ : ˜ M → M that relates the two motions through ϕ ( x , t ) = ˜ ϕ [ ξ (˜ x ) , t ] . (2.46) n the stress dependence of the elastic tensor It is assumed for simplicity that ξ is smooth and has a smooth inverse. Under such a particle relabelling transformation the form of theequations of motion is clearly left unchanged, while it was shown by Al-Attar & Crawford (2016) that the material parameters ˜ ρ and ˜ W relative to the second reference configuration can be obtained from those in the first by ˜ ρ (˜ x ) = J ξ (˜ x ) ρ [ ξ (˜ x )] , (2.47) ˜ W (˜ x , ˜ F ) = J ξ (˜ x ) W [ ξ (˜ x ) , ˜ F F ξ (˜ x ) − ] , (2.48)where F ξ = D ξ and J ξ = det F ξ . When considering linearised motions of an elastic body it is conventional to select the reference configuration so that the equilibriumconfiguration takes the simple form ϕ e ( x ) = x . (2.49)In this manner, the label for each particle is simply its position in physical space at equilibrium. In the terminology of Al-Attar & Crawford(2016), such a reference configuration is said to be natural . The equilibrium deformation gradient then satisfies F e ( x ) = , (2.50)while its Jacobian is everywhere equal to one. Given this choice, the equilibrium first Piola-Kirchhoff stress is obtained by evaluating the firstderivative of the strain-energy at the identity: T e = D F W ( · , ) . (2.51)An attractive feature of natural reference configurations is that the distinction between the different equilibrium stress-tensors vanishes.Indeed, it is trivial to verify that equation (2.29) now reads T e = S e = σ e . (2.52)In particular, it follows that σ e = D F W ( · , ) = 2 D C V ( · , ) , (2.53)an expression we will dissect in Section 3. In the same manner, the elastic tensor takes the simpler form Λ = D F W ( · , ) , (2.54)which, from eq.(2.43), can be written equivalently as Λ = A + R σ e . (2.55)As noted above, the tensor A = 4 D C V ( · , ) (2.56)possesses all the classical elastic symmetries. In contrast, the second term in eq.(2.55) has components [ R σ e ] ijkl = δ ik [ σ e ] lj (2.57)which, for general σ e (cid:54) = , are not invariant under the interchange of either i ↔ j or k ↔ l (although the symmetry of σ e ensures that R σ e possesses the hyperelastic symmetry). We therefore see that Λ inherits the full complement of classical symmetries only with respect to astress-free natural reference configuration . Moreover, it is only with respect to a natural reference configuration that the propagation directions ˆ p within eq.(2.35) can be equated with directions in physical space, allowing the slowness surface to be interpreted in a straightforwardmanner.Finally, it is within a natural reference configuration that we can see most clearly the motivation for Dahlen & Tromp’s (1998) elastictensor with its constants a and b (eq.1.5). Eq.(2.55) prescribes the form that any elastic tensor must take – in particular, it tells us that A mustpossess all the classical elastic symmetries – but it does not tell us is how A depends on σ e . For instance, one could write A = ( term possessing classical symmetries ) +( terms explicitly linear in stress also possessing those symmetries ) . (2.58)Dahlen (1972b) did precisely this, but his expression is not the only way that such a decomposition can be achieved. A more general version isthat of Dahlen & Tromp (1998), with different choices of a and b leading to different representations of the elastic tensor. These decompositionsare perfectly valid, but they do not make clear A ’s dependence on stress. Relative to a fixed reference configuration, the material symmetry group of a strain-energy function W at a point x ∈ M is Sym( W, x ) = { Q ∈ SL (3) | W ( x , FQ ) = W ( x , F ) , ∀ F ∈ GL (3) } , (2.59) M. A. Maitra & D. Al-Attar where SL (3) denotes the special linear group on R whose definition is recalled in Appendix A1. Physically, the symmetry group reflects theinvariance of the strain-energy with respect to orientation of stretching. Following Noll (1974), the body is said to be fluid at a point if thesymmetry group is equal to SL (3) , and solid if it is a proper subgroup thereof.Under a change of reference configuration, the symmetry group is not generally invariant. To see this, let ξ : ˜ M → M be a particlerelabelling transformation, and suppose that ξ (˜ x ) = x . If ˜ Q ∈ Sym( ˜
W , ˜ x ) then from eq.(2.48) we obtain W [ x , ˜ F ˜ QF ξ (˜ x ) − ] = W [ x , ˜ FF ξ (˜ x ) − ] (2.60)for all ˜ F ∈ GL (3) . Hence for a unique Q ∈ Sym( W, x ) we have ˜ QF ξ (˜ x ) − = F ξ (˜ x ) − Q . (2.61)This establishes a group isomorphism between Sym( W, x ) and Sym( ˜
W , ˜ x ) , which is given concretely through matrix conjugation: Sym( W, x ) (cid:51) Q (cid:55)→ F ξ (˜ x ) − QF ξ (˜ x ) ∈ Sym( ˜
W , ˜ x ) . (2.62)This mapping leaves SL (3) invariant, and hence our definitions of the material symmetry group itself and of an elastic fluid are sound.Noll (1965) showed that, up to this isomorphism, the largest proper subgroup of SL (3) is equal to the special orthogonal group SO (3) .We can, therefore, define an elastic solid to be isotropic at a point if its symmetry group relative to an arbitrary reference configuration isisomorphic under matrix conjugation to SO (3) . Equivalently, it is isotropic if for some reference configuration its symmetry group is equal to SO (3) . If the symmetry group is isomorphic to a proper subgroup of SO (3) we say the solid is anisotropic , with the extreme case being whenthis group consists of the identity matrix alone. In between these two end-members can be found, for example, transversely-isotropic materials,whose symmetry group is isomorphic under matrix conjugation to SO (2) . This corresponds physically to the strain-energy being invariantunder rotations about a certain fixed axis.Importantly, the preceding discussion is independent of any particular choice of reference configuration. It corrects a mistake of Al-Attar &Crawford (2016), who implied that material symmetries can be lost or gained through particle relabelling transformations. Such transformationssimply represent a change in our description of the body’s deformation. They cannot entail any physical consequences.As a final concept that we will need later, consider a natural reference configuration for a stress-free elastic body in equilibrium. Such aconfiguration is defined by DV ( C ∗ ) = with C ∗ = . From eq.(2.59), the material symmetry group acts on C according to C (cid:55)→ Q T CQ ,so by definition we have V (cid:16) Q T C ∗ Q (cid:17) = V ( C ∗ ) , (2.63)from which V (cid:16) Q T Q (cid:17) = V ( ) . (2.64)For the equilibrium to be stable, C ∗ = must lie at a strict local minimum of the strain-energy function. This allows us to equate thearguments of the left and right hand sides of eq.(2.64). The elements of the symmetry group then satisfy Q T Q = , (2.65)from which it is clear that Q ∈ SO (3) . The symmetry group of a stress-free elastic body, described with respect to a natural referenceconfiguration, is therefore equal to a subgroup of SO (3) , rather than just isomorphic thereto; this reflects the intuitive notion that it shouldbe impossible to stretch a stress-free solid without changing its energy. For example, in the stress-free case, an isotropic body in a naturalreference configuration has a material symmetry group equal to the whole of SO (3) , while that of a transversely-isotropic material is equal to SO (2) , having fixed the orientation of the symmetry-axis. Having recalled the necessary results and notations from the theory of elasticity, we now turn to our main question. Namely, we seek todetermine the functional dependence of the elastic tensor on the equilibrium Cauchy stress.
For an equilibrium body M , the Cauchy stress and elastic tensor take the form σ ( x ) = DW ( x , ) (3.1a) Λ ( x ) = D W ( x , ) , (3.1b)where W is the strain-energy function with respect to a natural reference configuration, and for notational simplicity we have dropped thesubscript from σ e . Our hope is to express the elastic tensor as a function of the equilibrium stress. Variations in σ arise, of course, thoughchanges to the equilibrium configuration, but the dependence of eqs.(3.1) thereon is masked by the use of a natural reference configuration.As a first step, we must reformulate eqs.(3.1) in a way that makes fully explicit the dependence of the two equations on the equilibriumconfiguration. n the stress dependence of the elastic tensor Figure 3.
The setup of our problem. The reference body ˜ M , with fixed elastic properties, is considered to be a reference configuration for the equilibrium body M . The two bodies are related by the equilibrium-mapping Φ , permitting us to use particle-relabelling transformations to write the strain-energy function W interms of ˜ W . We consider an arbitrary fixed reference configuration with the associated reference body denoted by ˜ M , and with strain-energy function ˜ W . The correspondence between this reference configuration and the natural reference configuration M is given by a mapping Φ : ˜ M → M . (3.2)This is just the equilibrium configuration of the body relative to our newly introduced reference configuration (see Fig. 3). Regarding theinverse mapping Φ − as a particle relabelling transformation, we can use eq.(2.48) to relate W to ˜ W : W (cid:2) Φ (˜ x ) , F (cid:48) (cid:3) = J F (˜ x ) − ˜ W (cid:2) ˜ x , F (cid:48) F (˜ x ) (cid:3) . (3.3)To avoid cluttered notations here and in what follows, we write F for the equilibrium deformation gradient D Φ and J F = det F , while F (cid:48) represents an arbitrary element of GL (3) . From this relation we see that W [ Φ (˜ x ) , · ] depends only on ˜ W at the fixed point ˜ x ∈ ˜ M .Furthermore, the relationship is parametrised by F evaluated at ˜ x . All in all, the two functions are related in a local manner; no generality islost by focusing on a single, arbitrary point in M and its pre-image in ˜ M . We do this from now on, dropping all spatial arguments to arrive atthe simpler relations σ = DW ( ) (3.4a) Λ = D W ( ) (3.4b)and W (cid:0) F (cid:48) (cid:1) = J − F ˜ W (cid:0) F (cid:48) F (cid:1) . (3.5)To complete the reformulation of eqs.(3.4), we apply the chain rule to eq.(3.5) so as to write σ and Λ explicitly in terms of ˜ W and F .It will be preferable to work not with W , but rather with the auxiliary strain-energy function V that encodes material-frame indifferenceautomatically. We see from eqs. (2.12) and (3.5) that the relationship between V in M and its counterpart ˜ V in ˜ M is given by V ( C (cid:48) ) = J − F ˜ V ( t T F · C (cid:48) ) , (3.6)where C (cid:48) = ( F (cid:48) ) T F (cid:48) and the operator t F is defined in eq.(A.19). From eqs.(2.53 – 2.56) it is clear that we require the first and secondderivatives of V evaluated at C (cid:48) = .Starting with the first derivative, we use eq.(3.6) to write J − F ˜ V [ t T F · ( C (cid:48) + δ C (cid:48) )] − J − F ˜ V [ t T F · C (cid:48) ] = (cid:10) DV ( C (cid:48) ) , δ C (cid:48) (cid:11) + O ( (cid:107) δ C (cid:48) (cid:107) ) , (3.7)while by expanding out the left hand side we obtain J − F (cid:68) D ˜ V ( t T F · C (cid:48) ) , t T F · δ C (cid:48) (cid:69) = (cid:10) DV ( C (cid:48) ) , δ C (cid:48) (cid:11) . (3.8)Using eqs. (A.16) and (A.21) the left hand side can be simplified and, as δ C (cid:48) is arbitrary, we have established DV ( C (cid:48) ) = J − F t F · D ˜ V ( t T F · C (cid:48) ) . (3.9)Evaluating this result at C (cid:48) = , noting that t T F · = C and using eq.(2.53), we obtain the first of our desired expressions, σ = 2 J − F t F · D ˜ V ( C ) . (3.10)Note that it is simply a restatement of eq.(2.27). M. A. Maitra & D. Al-Attar
From the definition of A in eq.(2.56), we have DV ( + δ C (cid:48) ) − DV ( ) = 14 A · δ C (cid:48) + O ( (cid:107) δ C (cid:48) (cid:107) ) (3.11)for arbitrary δ C (cid:48) , but using eq.(3.9) that is equivalent to J − F t F · (cid:104) D ˜ V ( C + t T F · δ C (cid:48) ) − D ˜ V ( C ) (cid:105) = 14 A · δ C (cid:48) + O ( (cid:107) δ C (cid:48) (cid:107) ) . (3.12)Expanding out the left hand side to first-order, we then obtain J − F t F · (cid:104) D ˜ V ( C ) · ( t T F · δ C (cid:48) ) (cid:105) = 14 A · δ C (cid:48) , (3.13)and as δ C (cid:48) was arbitrary we can read off the result A = 4 J − F t F D ˜ V ( C ) t T F . (3.14)Combining this with eq.(2.55), we arrive at the second of our desired expressions, Λ = 4 J − F t F D ˜ V ( C ) t T F + R σ . (3.15)Through eqs. (3.10) and (3.15) we have expressed both the equilibrium Cauchy stress and the elastic tensor as explicit functions of F , theequilibrium deformation gradient relative to the fixed reference configuration ˜ M . To emphasise this point we introduce the notation σ = ˆ σ ( F ) ≡ J − F t F · D ˜ V ( C ) (3.16a) Λ = ˆ Λ ( F ) ≡ J − F t F D ˜ V ( C ) t T F + R ˆ σ ( F ) , (3.16b)with ˆ σ (resp. ˆ Λ ) the function that takes an equilibrium deformation gradient to the corresponding equilibrium Cauchy stress (resp. elastictensor). It is worth observing that both of these relations are nonlinear for any non-trivial choice of strain-energy function ˜ V . Superficially,they suggest that we could write the elastic tensor as a function of equilibrium stress by inverting ˆ σ . However, there is rather a conspicuousproblem. The function ˆ σ maps elements in the nine-dimensional group GL (3) into the six-dimensional vector space of symmetric matriceson R . The equation ˆ σ ( F ) = σ for F is therefore underdetermined, so we cannot expect there to be a unique solution. To better understand,and indeed partially resolve, this issue we must examine how the principle of material-frame indifference and material symmetries manifestwithin eqs.(3.16). Material-frame indifference concerns the behaviour of the strain-energy function and related quantities under transformations of the deformationgradient of the form F (cid:55)→ RF with R ∈ SO (3) . Recalling that the value of the right Cauchy–Green deformation tensor is invariant undersuch a transformation, and using eq.(A.22), we see immediately from eqs.(3.16) that ˆ σ ( RF ) = t R · ˆ σ ( F ) (3.17a) ˆ Λ ( RF ) = t R ˆ Λ ( F ) t T R (3.17b)for all R ∈ SO (3) . Next, we exploit the polar decomposition theorem (e.g. Marsden & Hughes 1994), which shows that any F ∈ GL (3) canbe written as F = RU , (3.18)for unique R ∈ SO (3) and with U a unique, symmetric, positive-definite matrix. Substituting this decomposition into eqs.(3.16) and makinguse of eqs.(3.17) we then obtain ˆ σ ( F ) = t R · ˆ Σ ( U ) (3.19a) ˆ Λ ( F ) = t R ˆ λ ( U ) t T R , (3.19b)where we have defined the functions ˆ Σ ( U ) = 2 J − U t U · D ˜ V (cid:0) U (cid:1) (3.20a) ˆ λ ( U ) = 4 J − U t U D ˜ V (cid:0) U (cid:1) t U + R ˆ Σ ( U ) , (3.20b)whose arguments are required to be symmetric, positive-definite matrices. Importantly, the function ˆ Σ maps symmetric matrices into symmetricmatrices. As a result, there is no dimensional obstruction to its being invertible. We will assume that a unique inverse ˆ Σ − exists whereverrequired (a position that we argue for in Appendix B in order to avoid a tangent here).We now examine solutions of the equation ˆ σ ( F ) = σ for given σ . As above, we write F = RU , but now suppose that the value of R ∈ SO (3) has been fixed arbitrarily. Using eq.(3.19a) we then trivially obtain U = ˆ Σ − (cid:16) t T R · σ (cid:17) , (3.21)whence it follows that the equilibrium deformation gradient is given by F = ˆ F ( σ , R ) ≡ L R · ˆ Σ − (cid:16) t T R · σ (cid:17) . (3.22) n the stress dependence of the elastic tensor Here we see concretely where the missing three degrees of freedom enter into the inversion of ˆ σ . While eq.(3.22) constitutes one solutionof the given equation, it is clearly not unique: letting R vary over SO (3) generates a three-parameter family of solutions. Moreover, theuniqueness of the polar decomposition shows that one cannot vary R without changing F , and also that all solutions of the equation can beobtained in this manner. Crucially, this non-uniqueness carries over into the elastic tensor. Substitution of eq.(3.22) into eq.(3.16b) yields Λ = ˆ Λ (cid:104) ˆ F ( σ , R ) (cid:105) ≡ ¯ Λ ( σ , R ) , (3.23)where the function ¯ Λ can be written more expansively as ¯ Λ ( σ , R ) = t R (cid:16) ˆ λ ◦ ˆ Σ − (cid:17)(cid:16) t T R · σ (cid:17) t T R . (3.24)It follows that we cannot expect to write the elastic tensor as a function of the equilibrium Cauchy stress alone. A definite value for Λ dependson the arbitrary choice of an element of SO (3) .We can add some nuance to this possibly surprising result by considering material symmetries. Let Sym( ˜ W ) denote the materialsymmetry group (which could be trivial) at the point of interest, whose elements Q act on the deformation gradient on the right through F (cid:55)→ FQ . In terms of the right Cauchy–Green deformation tensor such a transformation takes the form C (cid:55)→ Q T CQ = t T Q · C , and bydefinition we have ˜ V ( t T Q · C ) = ˜ V ( C ) (3.25)for all Q ∈ Sym( ˜ W ) . Differentiating this relation in the now standard manner yields D ˜ V ( C ) = t Q · D ˜ V (cid:16) t T Q · C (cid:17) (3.26) D ˜ V ( C ) = t Q D ˜ V (cid:16) t T Q · C (cid:17) t T Q . (3.27)Using the properties of t F , we then find from eqs.(3.16) that ˆ σ ( FQ ) = 2 J − F t F · (cid:104) t Q · D ˜ V (cid:16) t T Q · C (cid:17)(cid:105) (3.28) ˆ Λ ( FQ ) = 4 J − F t F (cid:104) t Q D ˜ V (cid:16) t T Q · C (cid:17) t T Q (cid:105) t T F + R ˆ σ ( FQ ) , (3.29)from which it readily follows that ˆ σ ( FQ ) = ˆ σ ( F ) (3.30a) ˆ Λ ( FQ ) = ˆ Λ ( F ) (3.30b)for any Q ∈ Sym( ˜ W ) .To understand the implications of eqs.(3.30) it is simplest if we consider ˜ M to constitute a natural reference configuration for anequilibrium body that is stress-free. What follows is therefore based on the assumption that there exists some stress-free reference-configurationwith respect to which we can describe M . This way, we can take Sym( ˜ W ) to be equal , rather than just isomorphic, to a subgroup of SO (3) .Because all Q ∈ Sym( ˜ W ) are then rotations, the material symmetry group acts through ( R , U ) (cid:55)→ ( RQ , Q T UQ ) on the level of the polardecomposition. Therefore, using eq.(3.19a) we find that eq.(3.30a) requires ˆ Σ ( t Q · U ) = t Q · ˆ Σ ( U ) (3.31)for arbitrary Q ∈ Sym( ˜ W ) . Acting the inverse function ˆ Σ − on this expression we obtain the analogous result ˆ Σ − ( t Q · Σ ) = t Q · ˆ Σ − ( Σ ) (3.32)for arbitrary symmetric Σ , which we may substitute into eq.(3.22) to conclude that ˆ F ( σ , RQ ) = L RQ · ˆ Σ − ( t T RQ · σ )= ( L R L Q ) · ˆ Σ − [ t T Q · ( t T R · σ )]= ( L R L Q t T Q ) · ˆ Σ − ( t T R · σ )= ˆ F ( σ , R ) Q . (3.33)This relationship does not allow us to determine F any more precisely – it will always be known only up to an element of SO (3) – but wemay substitute it into eq.(3.23) to yield ˆ Λ (cid:104) ˆ F ( σ , RQ ) (cid:105) = ˆ Λ (cid:104) ˆ F ( σ , R ) Q (cid:105) . (3.34)It follows immediately from eq.(3.30b) that ˆ Λ (cid:104) ˆ F ( σ , RQ ) (cid:105) = ˆ Λ (cid:104) ˆ F ( σ , R ) (cid:105) . (3.35)Hence, using the notation of eq.(3.23), we have obtained the key identity ¯ Λ ( σ , RQ ) = ¯ Λ ( σ , R ) ∀ Q ∈ Sym( ˜ W ) . (3.36)Eq.(3.36) implies that two distinct rotation matrices R , R (cid:48) ∈ SO (3) will lead to the same elastic tensor via eq.(3.23) if R T R (cid:48) ∈ M. A. Maitra & D. Al-Attar
Sym( ˜ W ) . It is readily verified that this defines an equivalence relation, meaning that SO (3) can be partitioned into distinct equivalenceclasses, with the resulting quotient space denoted by SO (3) / Sym( ˜ W ) . The function ¯ Λ ( σ , · ) therefore depends not on the rotation matrix R directly, but only on the equivalence class in SO (3) / Sym( ˜ W ) to which it belongs. In this manner, the number of additional parametersrequired to fix a definite value of the elastic tensor can be reduced.In summary, we have shown that it is possible to express the elastic tensor as a function of equilibrium stress, but only at the costof introducing extra arbitrary parameters. Given our initial comments about the form of ˆ σ , the presence of these parameters is perhapsnot surprising. After all, we were essentially trying to fix nine numbers knowing only six. What is pleasing is that we have been able toexploit material-frame indifference to ‘package’ these extra degrees of freedom into a rotation matrix and write down a solution that is stillwell-defined. In addition, although our argument appeared at first to suggest that the rotation matrix was arbitrary, we have shown that thepresence of material symmetries can reduce the number of arbitrary parameters to be specified. In a geophysical context it will often be convenient to regard the total equilibrium Cauchy stress in the body M as some small perturbation tothe equilibrium Cauchy stress of the reference-body ˜ M . It is therefore useful to linearise expression (3.23), the calculations for which are laidout in Appendix C. We assume that there exists a zeroth-order equilibrium configuration ˜ M possessing Cauchy stress σ and elastic tensor Λ = A + R σ . Because this background state is known, we can set R to the identity without loss of generality.We linearise the system about a small perturbing stress. With (cid:15) a small parameter, the stress is set to σ = σ + (cid:15) σ . (3.37)We must also linearise the rotation matrix of eq.(3.23). It is set to the identity at zeroth-order, so its perturbation satisfies R = + (cid:15) ω + O (cid:0) (cid:15) (cid:1) , (3.38)with ω an antisymmetric matrix. Substituting these into eq.(3.23) and Taylor expanding, we may write Λ = Λ + (cid:15) Λ + O (cid:0) (cid:15) (cid:1) . (3.39)The perturbed elastic tensor Λ is decomposed as Λ = A + R σ , (3.40)consistent with eq.(2.55). Under these definitions, we show in Appendix C1 that A is given by A = X u + ω A + A X u − ω − A tr (cid:0) u (cid:1) + 8 D ˜ V ( ) · u , (3.41)where u is a symmetric matrix which satisfies a generalised linear stress-strain relationship σ − X ω · σ = (cid:0) A + X σ − σ ⊗ (cid:1) · u . (3.42)In these expressions we have introduced the tensor product on matrices (eq.A.18) and the notation X (eq.A.24). Thus, in order to fully specifythe perturbation to the elastic tensor we must provide:(i) the perturbation to the stress, σ ;(ii) σ and A , which encode information about the zeroth-order equilibrium body;(iii) further information about the zeroth-order equilibrium body, via the third derivatives of its strain-energy function at equilibrium;(iv) an arbitrary antisymmetric matrix, ω .Eq.(3.41) is a general result whose derivation makes no particular demands on the form of the zeroth-order equilibrium configuration, butwe can already make several observations. Firstly, no matter its functional form, A can be shown to possess all the classical elastic symmetries,as required by eq.(2.55). Secondly, given that it is explicitly linear in the stress-perturbation, this theory is superficially rather close to thoseof Dahlen (1972b), Dahlen & Tromp (1998), Tromp & Trampert (2018) and Tromp et al. (2019), discussed in Section 1. There are someimportant differences, though. For instance, our linearised theory makes explicit reference to third derivatives of the strain-energy function atequilibrium. In fact, those third derivatives can be seen as parametrising the theory. Moreover, eq.(3.41) is parametrised further by the arbitrarydegrees of freedom associated with ω . One might imagine that terms in ω would drop out when we compute, say, the Christoffel operator,so that it would have no effect on observable phenomena, but we show later, for the particular case of a transversely-isotropic material, thatthis does not happen. Eq.(3.41) also contains interactions between applied stress and material symmetries that are not captured by the theoriesof Dahlen (1972b), Dahlen & Tromp (1998), Tromp & Trampert (2018). Given that those authors interpret the tensor γ ijkl as representingthe ‘inherent anisotropy’ of the medium, it is clear that applied stress and ‘inherent anisotropy’ cannot couple in eqs. (1.1) and (1.5). Thegeneralised elastic tensor of Tromp et al. (2019) can represent such interactions, but the issues remain of ω and the strain-energy function’sthird derivatives. The relationship between these existing theories and our linearised theory will become clearer as we consider some moreconcrete examples. n the stress dependence of the elastic tensor We now illustrate how the preceding results apply to a few different physical situations. We will consider both large and small stresses, makinguse of eq.(3.41) for the latter. Henceforward, we will refer to the body M of the previous section simply as ‘the equilibrium body’. We willdescribe the fixed reference body ˜ M as the background body , and similarly for all its associated quantities, such as strain-energy function andmaterial symmetry group. All calculations for the following examples were carried out in Mathematica (with the relevant notebook included inthe supplementary material). The scenarios involving large stress required numerical inversion of the nonlinear function ˆ Σ , for which we usedMathematica’s inbuilt ‘FindRoot’ function. We begin by considering isotropic, stress-free background bodies. In the isotropic special case σ alone provides a unique specification of Λ .To see this, it is sufficient to return to the identity ¯ Λ ( σ , R ) = ¯ Λ ( σ , RQ ) , (4.1)for some R and Q both in SO (3) . By the standard group axioms we may write Q = R T R (cid:48) , (4.2)where R (cid:48) is another arbitrary element of SO (3) , whence ¯ Λ ( σ , R ) = ¯ Λ (cid:0) σ , R (cid:48) (cid:1) . (4.3)The matrices R and R (cid:48) are both arbitrary, so ¯ Λ ( σ , · ) is independent of the choice of rotation matrix. When evaluating ¯ Λ we may therefore set R = without loss of generality. The elastic tensor is then given conveniently by Λ = ¯ Λ ( σ , ) . (4.4)This argument shows that all rotations are equivalent up to right multiplication by SO (3) ; in other words, the quotient space SO (3) / Sym( ˜ W ) = SO (3) / SO (3) (4.5)is a set which contains just one element. With these preliminaries we are in position to investigate the behaviour of isotropic materials underapplied stress. The nonlinearity of expression (3.23) implies that a general material’s response to applied stress is influenced by derivatives of the strain-energyfunction higher than second-order. We demonstrate this in Fig. 4, contrasting the slowness surfaces of two different isotropic materialsunder the same applied stress. The strain-energy functions describing the background bodies are (e.g. Holzapfel 2000) modified Saint-VenantKirchhoff , ˜ W MSVK ( F ) = λ J + µ (cid:0) [ C − ] (cid:1) , (4.6)and neo-Hookean , ˜ W NH ( F ) = µ (cid:20) tr( C ) − µλ ( J − λµ − (cid:21) , (4.7)where λ and µ are constants. If no stress is applied – that is, if the background strain-energy functions and their derivatives are evaluated at F = – these materials are indistinguishable, each possessing a classical, stress-free, isotropic elastic tensor, Λ ijkl = λδ ij δ kl + µ ( δ ik δ jl + δ il δ jk ) . (4.8)It is also clear from this that λ and µ should be interpreted as the standard Lam´e parameters. Under large applied stress, though, the strain-energyfunctions are evaluated away from the identity, meaning that third- and higher-order derivatives become relevant. As shown in Fig.4, where wehave applied a deviatoric stress of magnitude (cid:107) σ (cid:107) ∼ . µ , the materials then display distinct behaviour. When a hydrostatic pressure is applied to a stress-free, isotropic solid, its only effect on the elastic tensor is to change the elastic moduli. Herewe derive exact expressions for λ and µ as functions of pressure. We return to eqs.(3.20) and write U = φ , (4.9)for some positive scalar φ . Given that the system is isotropic and the applied stress hydrostatic, it is clear from symmetry considerations andeq.(3.31) that U cannot take any other form. The deformation gradient itself can only ever be known up to an arbitrary rotation matrix, so it isgiven by F = RU = φ R , (4.10) M. A. Maitra & D. Al-Attar
Figure 4.
The behaviour of two different isotropic materials subject to the same applied stress. The left panel shows how a material governed by a modifiedSaint-Venant Kirchhoff strain-energy function reacts to the application of a certain deviatoric stress of magnitude (cid:107) σ (cid:107) ∼ . µ . We plot the x − y slownesssurface, with the zero-stress slowness surface shown faintly for reference. P-waves and S-waves are both affected by the stress. We show the same on the right,but for a material governed by a neo-Hookean constitutive relation. Shear-waves are not noticeably split here and the P-wave response is muted. Thus, the twomaterials behave differently under stress, despite being indistinguishable in its absence. with R ∈ SO (3) , while the right Cauchy–Green deformation tensor is C = φ . (4.11)This deformation gradient corresponds physically to a local compression or dilation of the background-body with a rotation superimposed; φ < effects a compression and vice versa. If the resulting equilibrium pressure in M is p , the Cauchy stress is σ = − p , so fromeqs.(3.16,3.19,3.20) − p = 2 φ D ˜ V (cid:0) φ (cid:1) (4.12a) Λ = 4 φD ˜ V (cid:0) φ (cid:1) + R σ . (4.12b)We can set R to the identity in these expressions because the background material is isotropic (see eq.4.4).Now, for the stress-free isotropic medium represented by ˜ M , the strain-energy function ˜ V is a function of the three scalar invariants of C (Holzapfel 2000). Defining the scalar invariants as I i ( C ) ≡ tr (cid:16) C i (cid:17) , i = 1 , , , (4.13)we can write the strain-energy function as ˜ V ( C ) ≡ ˆ V [ I ( C ) , I ( C ) , I ( C )] . (4.14)From the chain rule, its first and second derivatives are D ˜ V = (cid:88) i ˆ V i DI i (4.15) D ˜ V = (cid:88) ij ˆ V ij DI i ⊗ DI j + (cid:88) i ˆ V i D I i , (4.16)where we have written the derivatives of ˆ V with respect to its arguments in an obvious way. When the derivatives are evaluated at C = φ ,we will write e.g. v ( φ ) = ˆ V (cid:2) I (cid:0) φ (cid:1) , I (cid:0) φ (cid:1) , I (cid:0) φ (cid:1)(cid:3) , (4.17)and similarly for the other derivatives. With this, one finds after a little algebra that − p = 2 φ (cid:2) v ( φ ) + 2 v ( φ ) φ + 3 v ( φ ) φ (cid:3) (4.18a) Λ = 4 φ (cid:0)(cid:8) v ( φ ) + 4 v ( φ ) φ +[4 v ( φ ) + 6 v ( φ )] φ + 12 v ( φ ) φ + 9 v ( φ ) φ (cid:9) ⊗ + (cid:2) v ( φ ) + 6 v ( φ ) φ (cid:3) ˆid (cid:17) + R σ . (4.18b) n the stress dependence of the elastic tensor where the operator ˆid has been defined in eq.(A.34) and has components (cid:16) ˆid (cid:17) ijkl = 12 ( δ ik δ jl + δ il δ jk ) . (4.19)The coefficients of the first two operators in the expression for Λ are clearly the Lam´e parameters, given as functions of φ , so we reach theequations λ = 4 φ (cid:8) v ( φ ) + 4 v ( φ ) φ +[4 v ( φ ) + 6 v ( φ )] φ + 12 v ( φ ) φ + 9 v ( φ ) φ (cid:9) (4.20) µ = 4 φ (cid:2) v ( φ ) + 3 v ( φ ) φ (cid:3) . (4.21) φ is obtained as a function of p by inverting the relation − p = 2 φ (cid:2) v ( φ ) + 2 v ( φ ) φ + 3 v ( φ ) φ (cid:3) , (4.22)which would be performed numerically for a general strain-energy function. The Lam´e parameters are then given as explicit functions of theequilibrium pressure, parametrised by the derivatives of ˆ V . For comparison, if we substitute such a hydrostatic stress with pressure p into theisotropic versions of eqs. (1.5) and (1.9) respectively, we find: Λ ijkl = ( λ − ap ) δ ij δ kl + ( µ − bp )( δ ik δ jl + δ il δ jk ) − pδ ik δ jl (4.23) Λ ijkl = (cid:2) λ + p (cid:0) λ (cid:48) − (cid:1)(cid:3) δ ij δ kl + (cid:2) µ + p (cid:0) µ (cid:48) + 1 (cid:1)(cid:3) ( δ ik δ jl + δ il δ jk ) − pδ ik δ jl . (4.24)These expressions each contain only two explicit free parameters. As such, they are inconsistent with eqs. (4.20) and (4.21) unless their elasticmoduli are considered to have implicit pressure-dependence.It is also useful to note that the form of U considered here, despite producing a stressed configuration from an unstressed one, does notalter the material symmetry group, Sym( ˜ W ) = SO (3) . From eq.(2.62) and with F = φ R , Sym( W ) = (cid:110) F − QF , Q ∈ Sym( ˜ W ) (cid:111) = (cid:110) R T QR , Q ∈ Sym( ˜ W ) (cid:111) = Sym( ˜ W ) . (4.25)As expected on physical grounds, applying a pressure to an isotropic body does not break the isotropy. All our conclusions from Section 3therefore apply to an isotropic body even under hydrostatic stress. In particular, we can linearise about a hydrostatically stressed equilibriumgiven by U = (Appendix C1) without having to refer the system back to some unstressed state. Eq.(3.41) simplifies dramatically when we consider a small stress applied to a hydrostatically pre-stressed equilibrium. The total stress iswritten as σ = − p − p + τ , (4.26)with p (cid:28) p (cid:107) τ (cid:107) (cid:28) p tr (cid:0) τ (cid:1) = 0 , (4.27)and the complete elastic tensor is (Appendix C2) Λ ijkl = (cid:20)(cid:0) κ + κ (cid:48) p (cid:1) − (cid:0) µ + µ (cid:48) p (cid:1)(cid:21) δ ij δ kl + (cid:0) µ + µ (cid:48) p (cid:1) ( δ ik δ jl + δ il δ jk ) − (cid:0) p + p (cid:1) δ ik δ jl + a (cid:0) δ ij τ kl + δ kl τ ij (cid:1) + b (cid:0) δ ik τ jl + δ jl τ ik + δ il τ jk + δ jk τ il (cid:1) + δ ik τ jl . (4.28)The constants µ (cid:48) , κ (cid:48) , a and b are defined to be κ (cid:48) = − κ + 3 ζ + 3 ζ + ζ κ + p (4.29a) µ (cid:48) = − µ + ζ + ζ κ + p (4.29b) a = κ − µ + ζ µ − p (4.29c) b = µ + ζ µ − p , (4.29d)with ζ , ζ and ζ the Murnaghan constants , which offer a complete characterisation of the third derivatives of an isotropic strain-energyfunction about equilibrium (Murnaghan 1937). Up to third-order accuracy in the deformation gradient, specification of p , κ , µ , ζ , ζ and ζ M. A. Maitra & D. Al-Attar is sufficient to fix all the elastic properties of the background body. In light of Section 4.1.2, it should be emphasised that κ , µ , ζ , ζ and ζ are constants defined relative to the hydrostatically stressed background.Eq.(4.28) may be compared with eq.(1.5). Although the expressions look similar, our approach has introduced different features. Eq.(4.28)contains more numerical constants. Not only do we have a and b , but we have separately been led to define κ (cid:48) and µ (cid:48) , constants which we toointerpret as pressure-derivatives of the elastic moduli. Notably, under our approach κ (cid:48) and µ (cid:48) cannot be written uniquely in terms of a and b . Infact, all four of these constants are explicitly determined by the constitutive relation used to describe the background body, emerging from thetheory as dimensionless combinations of the equilibrium pressure, shear and bulk moduli, and the three Murnaghan constants (eqs.4.29). Both a = − b = 12 (4.30)(see eq.(1.1)) and Tromp & Trampert’s choice of a = + 12 + 13 µ (cid:48) − κ (cid:48) b = − − µ (cid:48) (4.31)amount to assuming an appropriate form for that constitutive relation. Crucially, the elastic tensor of eq.(4.28) depends on three material-dependent parameters besides p , κ and µ . This runs contrary to the conclusions of Tromp & Trampert (2018), and even the more generalelastic tensor (eq.1.9) given by Tromp et al. (2019) cannot be consistent with our linearised theory because it is parametrised, in the isotropiccase, just by κ (cid:48) and µ (cid:48) . Whilst an isotropic material’s response to a given applied stress is determined solely by its background strain-energy function, we also need toaccount for eq.(3.23)’s non-unique stress-dependence when considering materials with smaller symmetry groups. For example, we statedabove that the symmetry group of a stress-free, transversely-isotropic material is SO (2) . A definite value of Λ therefore depends upon thechoice of an arbitrary element of the quotient space SO (3) / SO (2) . This reflects the fact that ¯ Λ ( σ , · ) cannot distinguish between matricesthat only differ in how much rotation they cause about the symmetry axis. Therefore to evaluate ¯ Λ ( σ , R ) we should only choose arbitrarilybetween rotation matrices whose own axes of rotation lie in the plane perpendicular to the symmetry axis. In order to pick such a matrix wemust choose a direction for its rotation-axis – a direction in R described by an angle φ – and then specify an angle of rotation θ about thataxis. Careful consideration of the possibility of double-counting shows that θ ∈ [0 , π ) (4.32) φ ∈ [0 , π ) (4.33)(or vice versa). It is clear that we have effectively specified a point on S , the unit 2-sphere, or equivalently a direction in R . Indeed, it may beestablished by rigorous methods that SO (3) / SO (2) ∼ = S . The effect of this non-uniqueness is illustrated by Fig. 5, which shows slowness surfaces of a material described by the transversely-isotropicstrain-energy function ˜ W TI ( F ) = ˜ W MSVK ( F ) +[ α + 2 β log J + γ ( I − I − − α I − . (4.34)In this equation, α , β and γ are extra material constants required to describe a transversely-isotropic material, while I and I are the twofurther scalar invariants in terms of which a transversely-isotropic strain energy function is parametrised (Holzapfel 2000). They are defined as I = (cid:104) ν , C · ν (cid:105) (4.35) I = (cid:10) ν , C · ν (cid:11) , (4.36)with the unit-vector ν pointing along the material’s symmetry-axis. This strain-energy function is adapted from Bonet & Burton (1998),although our definition of β differs from theirs by a factor of two and we have defined the ‘isotropic part’ of the function differently. Theequilibrium configuration is unstressed, with elastic tensor Λ ijkl = λδ ij δ kl + µ ( δ ik δ jl + δ il δ jk )+ 8 γν i ν j ν k ν l + 4 β ( ν i ν j δ kl + δ ij ν k ν l ) − α ( ν i ν k δ jl + ν j ν k δ il + ν j ν l δ ik + ν i ν l δ jk ) . (4.37)We have applied the same stress to the material in both panels of the figure, but selected different ( θ, φ ) pairs, producing slowness surfaces ofdifferent shapes . It should be emphasised that the material is described by the same strain-energy function in both panels; that the two slownesssurfaces demonstrate distinct physical behaviour is due solely to eq.(3.23)’s non-unique dependence on the equilibrium stress. n the stress dependence of the elastic tensor Figure 5.
Two x – y slowness surfaces of a transversely-isotropic material subjected to the same large deviatoric stress, but with different choices of the arbitraryparameters θ and φ . The slow- and fast-directions of P-waves are different, as are the shear-wave splitting patterns. The distinct physical behaviour implied inthe two panels is a result of changing the arbitrary parameters in ¯ Λ ( σ , R ) . Finally, we consider how a transversely-isotropic material responds to a small applied stress. This is the simplest nontrivial example inwhich one can show analytically how the arbitrary rotation matrix of eq.(3.23) manifests in the linearised elastic tensor. Moreover, we candemonstrate concretely that the material’s symmetries interact with the applied stress.A transversely-isotropic material cannot support a purely hydrostatic equilibrium pre-stress, unlike an isotropic material. We thereforetake σ = − (cid:18) p + q (cid:19) + q ν ⊗ ν (4.38)as the zeroth-order stress. An argument analogous to that of Section 4.1.2 justifies this step, and demonstrates that the material symmetrygroup of such a pre-stressed transversely-isotropic material remains equal to SO (2) . Defining the shorthand N ≡ ν ⊗ ν , we then apply asmall stress σ , and a tedious calculation laid out in Appendix C3 leads to the linearised elastic tensor A = η ⊗ + η ˆid + η N ⊗ N + η ( ⊗ N + N ⊗ ) + η ˆ X N + η ˆ X σ + η (cid:0) ⊗ σ + σ ⊗ (cid:1) + η (cid:0) N ⊗ σ + σ ⊗ N (cid:1) + η (cid:2) ⊗ (cid:0) X N · σ (cid:1) + (cid:0) X N · σ (cid:1) ⊗ (cid:3) + η (cid:2) N ⊗ (cid:0) X N · σ (cid:1) + (cid:0) X N · σ (cid:1) ⊗ N (cid:3) + η ˆ X ( X N · σ ) + η (cid:16) ˆ X σ ˆ X N + ˆ X N ˆ X σ (cid:17) + η ˆ X ω + η [ ⊗ ( X ω · N ) +( X ω · N ) ⊗ ] + η [ N ⊗ ( X ω · N ) +( X ω · N ) ⊗ N ]+ η (cid:16) ˆ X ω ˆ X N − ˆ X N ˆ X ω (cid:17) , (4.39)with the constants { η i } defined in eqs.(C.90). The antisymmetric matrix ω defines a vector pointing in an arbitrary direction in the planeperpendicular to the unperturbed material symmetry axis. Its components are ω i = − (cid:15) ijk ω jk , (4.40)where (cid:15) ijk are the components of the Levi-Civita tensor, and its direction and magnitude are precisely the two arbitrary constants that must befixed. On the other hand the { η i } are unique, scalar-valued functions of:(i) the constants p and q which parametrise the zeroth-order equilibrium stress;(ii) the transversely-isotropic constants λ , µ , α , β and γ (which are implicitly functions of p , q and their respective stress-free values);(iii) tr (cid:0) σ (cid:1) and (cid:10) ν , σ · ν (cid:11) ;(iv) the seven constants { ζ i } defined in eq.(C.80) which, analogously to the Murnaghan constants, parametrise the third derivatives of atransversely-isotropic strain-energy function.The precise functional forms of the { η i } are not nearly as important as the fact that A contains ‘cross-terms’ between the symmetry-direction and the applied stress. The anisotropy displayed by A does not divide neatly into the sum of ‘anisotropy due to applied stress’ and M. A. Maitra & D. Al-Attar ‘anisotropy due to crystal structure’, as we noted when discussing the general linearised elastic tensor eq.(3.41). Of those discussed previously,only the elastic tensor of Tromp et al. (2019) contains cross terms between ‘inherent anisotropy’ and applied stress, but it does not containenough explicit free parameters to be consistent with eq.(4.39). The free parameters of eq.(1.9) are the pressure-derivatives of Γ ijkl , thetensor that Tromp et al. interpret as representing the ‘inherent’ properties of the material. In the transversely-isotropic case, Γ (cid:48) ijkl dependson the pressure-derivatives of λ , µ , α , β and γ , corresponding to five free parameters. By contrast, we have shown that the third derivativesof a transversely-isotropic strain-energy function are generally specified by seven constants. Furthermore, eq.(4.39) contains two arbitraryparameters, the components of ω .As a last point, let us consider the perturbation to the Christoffel operator associated with terms in ω . Continuing to ignore spatialarguments, one can show that definition (2.36) is equivalent to (cid:104) a , ρ Γ ( p ) · a (cid:105) = (cid:104) a ⊗ p , Λ · ( a ⊗ p ) (cid:105) (4.41)for arbitrary a . From this it is a matter of algebra to show that the contribution to ρ Γ of the terms in ω is ρ Γ = ( η + η ) (cid:104) ν , p (cid:105) (cid:2) p ⊗ (cid:0) ω × ν (cid:1) + (cid:0) ω × ν (cid:1) ⊗ p (cid:3) + (cid:18) η + 12 η (cid:19) (cid:10) ω × ν , p (cid:11) ( p ⊗ ν + ν ⊗ p )+ (cid:0) η (cid:104) ν , p (cid:105) + η (cid:107) p (cid:107) (cid:1)(cid:2) ν ⊗ (cid:0) ω × ν (cid:1) + (cid:0) ω × ν (cid:1) ⊗ ν (cid:3) + (cid:104) ν , p (cid:105) (cid:10) ω × ν , p (cid:11) ( η ν ⊗ ν + η ) . (4.42)Each of the coefficients η , η and η depends on a different combination of the { ζ i } . But the { ζ i } parametrise the third-derivatives of thestrain energy and are considered independent of the other elastic constants. It is therefore clear that ω will generally contribute nonzero termsto the Christoffel operator. Working under the theory of finite elasticity, we have derived an expression for the elastic tensor as an explicit function of equilibrium Cauchystress. Our results differ from previous treatments in two main ways: they suggest that the elastic tensor’s dependence on equilibrium stress isgenerally both nonlinear and non-unique. On account of the nonlinearity alone, knowledge of a material’s background elastic-tensor is notsufficient to determine the material’s response to an applied stress; we require the information contained within higher-order derivatives of thebackground strain-energy function. Furthermore, the elastic tensor is a function not only of the equilibrium stress, but also of an arbitraryrotation matrix. As such, even with a definite strain-energy function in hand, the change in the elastic tensor due to an applied stress dependson the non-unique choice of this matrix. However, we have also shown that the degree of non-uniqueness is reduced if the material under studyhas a nontrivial material symmetry group when no stress is applied. In the linearised case, our approach shows that the characterisation of amaterial’s response to a small applied stress depends on more parameters than have been made explicit in previous studies. It also gives clearmeaning to those parameters, distinguishing between those that are determined by the material itself and those that are arbitrary.It is important to note that previous expressions for the elastic tensor in the literature are correct as long as certain terms within them areinterpreted as depending implicitly on equilibrium stress. In Dahlen’s (1972b) expression, for example, the author regarded the anisotropic term γ ijkl as an ‘inherent’ property of the body, independent of stress. In so doing he deprived his expression of the ability to describe interactionsbetween applied stress and inherent anisotropy. Indeed, it was argued by Nikitin & Chesnokov (1981, 1984) that eq.(1.1) only applies toisotropic bodies. Moreover, even in the isotropic case, Dahlen’s theory does not make the pressure-dependence of the elastic moduli explicit.The same observations apply to eq.(1.5) for any constants a and b . On the other hand, Tromp et al.’s (2019) generalised expression for theelastic tensor does feature explicit interactions between inherent anisotropy and stress, but it does so only in a linearised sense and uses too fewindependent parameters to fully account for terms associated with the strain-energy function’s third derivatives.There are a number of potential geophysical applications of this work, but we must first mention a caveat. The Earth is not elastic overgeological time-scales, hence it is not reasonable to regard its equilibrium as having arisen through a finite deformation of an elastic materialaway from some hypothetical stress-free state. Within future work it would therefore be interesting to extend our methods to account forviscoelastic effects. Nevertheless, our framework should give valid descriptions of phenomena that occur over time-scales sufficiently short forthe Earth to respond in an elastic – or only slightly anelastic – manner. For example, one might consider the effect on elastic wave speeds ofprocesses that are fast relative to viscoelastic relaxation times but slow compared to those of seismic wave propagation, such as body tides,seasonal loading in the hydrosphere, or anthropogenic activity. In fact, it is precisely this kind of application that motivated the work of Tromp& Trampert (2018). Moreover, in these cases the applied stresses will be small relative to the background, so the linearised theory we havedeveloped should be relevant.In addition to simply modelling the seismological effect of such stresses, this theory might be useful within studies that seek to invertseismological observations for changes in the equilibrium stress. This inverse problem is already rendered challenging by the fact that theequilibrium stress is not an entirely free parameter, but is constrained to satisfy the equilibrium equations (Backus 1967; Al-Attar & Woodhouse2010). Our results make clear that it is also necessary to either provide or simultaneously invert for additional parameters related to thirdderivatives of the strain-energy function and, in some cases, infinitesimal rotations. n the stress dependence of the elastic tensor REFERENCES
Al-Attar, D. & Crawford, O., 2016. Particle relabelling transformations in elastodynamics,
Geophysical Journal International , (1), 575–593.Al-Attar, D. & Woodhouse, J. H., 2010. On the parametrization of equilibrium stress fields in the earth, Geophysical Journal International , (1), 567–576.Al-Attar, D., Crawford, O., Valentine, A. P., & Trampert, J., 2018. Hamilton’s principle and normal mode coupling in an aspherical planet with a fluid core, Geophysical Journal International .Backus, G. E., 1967. Converting vector and tensor equations to scalar equations in spherical coordinates,
Geophysical Journal International , (1-3), 71–101.Biot, M. A., 1965. Mechanics of incremental deformations , Wiley, New York.Bonet, J. & Burton, A., 1998. A simple orthotropic, transversely isotropic hyperelastic constitutive equation for large strain computations,
Computer Methodsin Applied Mechanics and Engineering , (1-4), 151–164.Dahlen, F., 1972a. Elastic velocity anisotropy in the presence of an anisotropic initial stress, Bulletin of the Seismological Society of America , (5), 1183–1193.Dahlen, F. & Tromp, J., 1998. Theoretical global seismology , Princeton university press.Dahlen, F. A., 1972b. Elastic dislocation theory for a self-gravitating elastic configuration with an initial static stress field,
Geophysical Journal International , (4), 357–383.Holzapfel, G., 2000. Nonlinear Solid Mechanics: A Continuum Approach for Engineering , John Wiley and Sons, Chichester, UK.Marsden, J. E. & Hughes, T. J., 1994.
Mathematical foundations of elasticity , Courier Corporation.Murnaghan, F. D., 1937. Finite deformations of an elastic solid,
American Journal of Mathematics , (2), 235.Nikitin, L. & Chesnokov, E., 1981. Influence of a stressed condition on the anisotropy of elastic properties in a medium, Izvestiya Earth Physics , , 174–183.Nikitin, L. V. & Chesnokov, E. M., 1984. Wave propagation in elastic media with stress-induced anisotropy, Geophysical Journal International , (1),129–133.Noll, W., 1965. Proof of the maximality of the orthogonal group in the unimodular group, Archive for Rational Mechanics and Analysis , (2), 100–102.Noll, W., 1974. A new mathematical theory of simple materials , Springer.Stacey, F. D., 1992.
Physics of the Earth , Brookfield Press.Tromp, J. & Trampert, J., 2018. Effects of induced stress on seismic forward modelling and inversion,
Geophysical Journal International , (2), 851–867.Tromp, J., Marcondes, M. L., Wentzcovitch, R. M. M., & Trampert, J., 2019. Effects of induced stress on seismic waves: Validation based on ab initiocalculations, Journal of Geophysical Research: Solid Earth , (1), 729–741.Truesdell, C. & Noll, W., 2004. The Non-Linear Field Theories of Mechanics , Springer Berlin Heidelberg, Berlin, Heidelberg.
APPENDIX A: NOTATIONS AND DEFINITIONSA1 Groups
We define GL ( n ) , the general linear group of dimension n , to be the set of invertible n × n matrices under the operation of matrix multiplication.For a general group G , a subgroup H of G is a subset of the elements of G that is itself a group; H is described as a proper subgroup of G if H (cid:54) = G . With this, we can define SL ( n ) , the special linear group of n × n matrices with unit determinant, which is a proper subgroupof GL ( n ) . A particularly important proper subgroup of SL ( n ) is SO ( n ) , the n-dimensional special orthogonal group whose elements arerotation matrices in n dimensions. For any R ∈ SO ( n ) , det R = 1 (A.1) R − = R T . (A.2) A2 Some nonstandard linear operators
We have found it useful to introduce the left- and right-multiplication operators L A and R A which act according to L A · B = AB (A.3) R A · B = BA (A.4)(A.5)for arbitrary matrices A , B ∈ GL (3) . The operators are expressed in index-notation as ( L A ) ijkl = A ik δ lj (A.6) ( R A ) ijkl = δ ik A lj , (A.7)and, as an example, ( L A · B ) ij = ( L A ) ijkl B kl = A ik δ lj B kl = A ik B kj = ( AB ) ij . (A.8)It is clear that L A and R B commute for any choice of A and B , and that they satisfy R AB = R B R A (A.9) L AB = L A L B , (A.10) M. A. Maitra & D. Al-Attar while their inverses have the property that L − A = L A − (A.11) R − A = R A − . (A.12)We also define the operators L T A = L A T (A.13) R T A = R A T . (A.14)Then, defining the inner-product on matrices by (cid:104) A , B (cid:105) = tr (cid:16) AB T (cid:17) (A.15)and introducing C ∈ GL (3) , it follows quickly that (cid:104) A , L C · B (cid:105) = (cid:68) L T C · A , B (cid:69) , (A.16)and similarly for R C . This is the origin of our suggestive notation for L T A and R T A : with the inner product as defined, they behave superficiallyas though they were the respective transpose operators of L A and R A . We also define a norm on matrices, (cid:107) A (cid:107) = (cid:112) (cid:104) A , A (cid:105) , (A.17)and tensor-product , ( A ⊗ B ) · C = (cid:104) B , C (cid:105) A . (A.18)In order to avoid clutter we have – just as for the inner product – simply used the standard notation for a tensor-product and norm on R ,trusting that context will make our meaning unambiguous. Finally, we define the particularly useful operator t A = L A R T A . (A.19)From the properties of L A and R A discussed above, we clearly have t − A = t A − (A.20) t T A = t A T , (A.21)as well as the useful functoral property t AB = t A t B . (A.22)For R ∈ SO (3) , t R · A = RAR T (A.23)evidently represents a rotation of A by R . The related operator X A = L A + R T A (A.24)is the term in t + A linear in A , and is particularly useful when we consider linearisation. Furthermore, if A and B are both symmetric theoperator satisfies X A · B = X B · A . (A.25)We will also need to symmetrise these operators at times. The reasons for this – and their solutions – are best illustrated by example.Consider the expression for the quadratic part of an isotropic strain-energy function (e.g. Holzapfel 2000): V ( C ) = λ C ) + µ (cid:0) C (cid:1) . (A.26)It is easy to show that the stress vanishes at C = . In such a stress-free equilibrium we have (eq.2.55) Λ = 4 D V ( ) . (A.27)We can evaluate the derivative by using the definitions above to rewrite V as V ( C ) = λ (cid:104) , C (cid:105) + µ (cid:104) C , id · C (cid:105) = (cid:28) C , (cid:18) λ ⊗ + µ (cid:19) · C (cid:29) , (A.28)where we further define the identity operator id , which has components (id) ijkl = δ ik δ jl . (A.29)Thereupon, we might be tempted to write the second derivative of V with respect to C about the identity as D V ( ) ? = λ ⊗ + 2 µ id . (A.30) n the stress dependence of the elastic tensor However, that gives the components of the elastic tensor as Λ ijkl = λδ ij δ kl + 2 µδ ik δ jl , (A.31)an expression that is clearly wrong except for when Λ operates on symmetric matrices. In general, though, Λ will operate on non-symmetricmatrices, when defining the Christoffel operator for instance. Therefore, when differentiating strain-energy functions that depend on thesymmetric matrix C we must explicitly symmetrise all the operators that arise. The process of symmetrisation is defined most convenientlyusing index notation. The operator M is symmetrised on its p ’th pair of indices by M i j i j ...i p j p ...i n j n → (cid:0) M i j i j ...i p j p ...i n j n + M i j i j ...j p i p ...i n j n (cid:1) . (A.32)Taking M to be a fourth-rank tensor, it is symmetrised on both pairs of indices (which we denote by a hat) by writing ( ˆ M ) ijkl = 14 ( M ijkl + M ijlk + M jikl + M jikl ) . (A.33)The symmetrised identity operator is thus written as ( ˆid) ijkl = 12 ( δ ik δ jl + δ il δ jk ) . (A.34)Substituting this into eq.(A.30) gives the correct expression Λ ijkl = λδ ij δ kl + µ ( δ ik δ jl + δ il δ jk ) . (A.35)Note that this process confers the minor elastic symmetries on a fourth-rank tensor, but not necessarily the hyperelastic symmetry. APPENDIX B: THE INVERTIBILITY OF ˆ Σ The existence of the inverse function ˆ Σ − depends on the specific choice of constitutive relation; without fixing ˜ V concretely it is difficultto obtain definite results. However, we can show that the inverse mapping exists at least locally in cases of some physical importance. Toproceed, we appeal to the inverse function theorem (e.g. Marsden & Hughes 1994). The theorem tells us that the nonlinear mapping ˆ Σ hasa well-defined inverse function in some open neighbourhood of ˆ Σ ( U ) if D ˆ Σ ( U ) , a linear mapping from the vector-space of symmetricmatrices into itself, is invertible.Let us work in the vicinity of U = . We will set R = for clarity, but that does not affect the validity of our argument. In that case wehave ˆ Σ ( ) = 2 DV ( ) . (B.1)Making use of eq.(A.25) of Appendix A, we can expand ˆ Σ about the identity as ˆ Σ ( + u ) = 2[1 − tr( u )][id + X u ] · (cid:2) DV ( ) + 2 D V ( ) · u (cid:3) = ˆ Σ ( ) − tr( u ) ˆ Σ ( ) + X u · ˆ Σ ( ) + 4 D V ( ) · u = ˆ Σ ( ) + (cid:16) D V ( ) + X ˆ Σ ( ) − ˆ Σ ( ) ⊗ (cid:17) · u , (B.2)with u a small symmetric matrix. We have shown that D ˆ Σ ( ) = 4 D V ( ) + X ˆ Σ ( ) − ˆ Σ ( ) ⊗ . (B.3)Inspection of this expression indicates that the invertibility of D V ( ) is a sufficient condition for D ˆ Σ ( ) to possess a unique inverselocally, in the absence of equilibrium stress. As long as ˆ Σ ( ) is not too large, invertibility of D V ( ) should remain sufficient for the localexistence of a unique ˆ Σ − . One can show that D V ( ) being invertible is equivalent to the condition of linearised stability of the equilibrium(e.g. Marsden & Hughes 1994). It is essential to enforce linearised stability, for otherwise unphysical motions are permitted in which thestrain-energy decreases upon deformation.In summary, we have argued that linearised stability of the equilibrium is a sufficient condition for ˆ Σ to possess a unique inverse in theneighbourhood of ˆ Σ ( ) , as long as the equilibrium stress is not too large. By no means is this a proof of the global invertibility of ˆ Σ , but webelieve that it makes it plausible that we should be able to write down a well-defined inverse function ˆ Σ − when required. APPENDIX C: CALCULATION OF THE LINEARISED ELASTIC TENSORC1 General expressions
We assume that the body is initially in a state described by σ = σ (C.1) Λ = Λ . (C.2) M. A. Maitra & D. Al-Attar
Both quantities are considered to be the derivatives of some background strain-energy evaluated at the identity, so that Λ is given uniquely by Λ = ¯ Λ (cid:0) σ , (cid:1) . (C.3)Once again, the function ¯ Λ has the form ¯ Λ ( σ , R ) = t R (cid:104)(cid:16) ˆ λ ◦ ˆ Σ − (cid:17)(cid:16) t T R · σ (cid:17)(cid:105) t T R . (C.4)We then introduce a small change in the stress: σ = σ + σ (C.5)for small σ . The system is assumed to be perturbative, so that we may write Λ = Λ + Λ (C.6)for small Λ . For the rest of this section we will neglect all terms higher than first order.Under the framework of Section 3, the change in stress is considered to be induced by some deformation gradient F , so it follows that F should take the form F = + f . (C.7)with f small. Now, the arbitrary rotation matrix R in eq.(C.4) is a relic of F , having been introduced by the identity F = RU . (C.8)Therefore it too must be considered to be a small perturbation to its background value: R = + ω , (C.9)with ω some small antisymmetric matrix. The same is true for U , that is U = + u , (C.10)but with the small matrix u of course symmetric. It is clear that the elastic tensor satisfies Λ + Λ = ¯ Λ (cid:0) σ + σ , + ω (cid:1) , (C.11)which we may expand in a Taylor series to find Λ = (cid:0) D σ ¯ Λ (cid:1)(cid:0) σ , (cid:1) · σ + (cid:0) D R ¯ Λ (cid:1)(cid:0) σ , (cid:1) · ω . (C.12)The R -derivative can conveniently be written in terms of the σ -derivative. Using the full form of ¯ Λ from eq.(C.4), as well as the operator id defined in eq.(A.29), ¯ Λ ( σ , + ω ) = (id + X ω ) (cid:104)(cid:16) ˆ λ ◦ ˆ Σ − (cid:17) ( σ − X ω · σ ) (cid:105) (id − X ω )= (cid:16) ˆ λ ◦ ˆ Σ − (cid:17) ( σ − X ω · σ ) + X ω (cid:104)(cid:16) ˆ λ ◦ ˆ Σ − (cid:17) ( σ ) (cid:105) − (cid:104)(cid:16) ˆ λ ◦ ˆ Σ − (cid:17) ( σ ) (cid:105) X ω = ¯ Λ ( σ − X ω · σ , ) + X ω ¯ Λ ( σ , ) − ¯ Λ ( σ , ) X ω = Λ − (cid:0) D σ ¯ Λ (cid:1) ( σ , ) · ( X ω · σ ) + X ω Λ − Λ X ω , (C.13)from which it is apparent that (cid:0) D R ¯ Λ (cid:1)(cid:0) σ , (cid:1) · ω = X ω Λ − Λ X ω − (cid:0) D σ ¯ Λ (cid:1)(cid:0) σ , (cid:1) · (cid:0) X ω · σ (cid:1) . (C.14)Substituting this into eq.(C.12), the perturbation to the elastic tensor is Λ = X ω Λ − Λ X ω + (cid:0) D σ ¯ Λ (cid:1)(cid:0) σ , (cid:1) · (cid:0) σ − X ω · σ (cid:1) . (C.15)Calculation of D σ ¯ Λ (cid:0) σ , (cid:1) requires us to find D ˆ λ ( ) and D ˆ Σ ( ) . They are evaluated most conveniently by forming explicit binomialexpansions of ˆ Σ and ˆ λ about U = + u . At zeroth order ˆ Σ ( ) = σ = 2 DV ( ) (C.16) ˆ λ ( ) = Λ = 4 D V ( ) + R ˆ Σ ( ) , (C.17)and it is useful to define A = 4 D V ( ) = ˆ λ ( ) − R ˆ Σ ( ) . (C.18)We showed in Appendix B that D ˆ Σ ( ) = A + X σ − σ ⊗ . (C.19)In the same manner, ˆ λ expands as ˆ λ ( + u ) = 4(1 − tr( u ))(id + X u ) (cid:2) D V ( ) + 2 D V ( ) · u (cid:3) (id + X u ) + R ˆ Σ ( + u ) = A + 8 D V ( ) · u + X u A + A X u − A tr( u ) + R σ + R D ˆ Σ ( ) · u , (C.20) n the stress dependence of the elastic tensor so its derivative at the identity is given by D ˆ λ ( ) · u = R D ˆ Σ ( ) · u + 8 D V ( ) · u + X u A + A X u − A tr( u ) (C.21)for any u .Now we can compute D σ ¯ Λ (cid:0) σ , (cid:1) by writing ¯ Λ (cid:0) σ + S , (cid:1) = ˆ λ (cid:104) ˆ Σ − (cid:0) σ + S (cid:1)(cid:105) = ˆ λ (cid:104) ˆ Σ − (cid:0) σ (cid:1) + (cid:104) D (cid:16) ˆ Σ − (cid:17)(cid:105)(cid:0) σ (cid:1) · S (cid:105) = ˆ λ (cid:26) + (cid:104) D ˆ Σ ( ) (cid:105) − · S (cid:27) = ˆ λ ( ) + D ˆ λ ( ) · (cid:26)(cid:104) D ˆ Σ ( ) (cid:105) − · S (cid:27) , (C.22)valid for small symmetric S , where we have used the identity D (cid:16) ˆ Σ − (cid:17)(cid:0) σ (cid:1) = (cid:110) D ˆ Σ (cid:104) ˆ Σ − (cid:0) σ (cid:1)(cid:105)(cid:111) − = (cid:104) D ˆ Σ ( ) (cid:105) − . (C.23)Thus, (cid:0) D σ ¯ Λ (cid:1)(cid:0) σ , (cid:1) · (cid:0) σ − X ω · σ (cid:1) = D ˆ λ ( ) · (cid:26)(cid:104) D ˆ Σ ( ) (cid:105) − · (cid:0) σ − X ω · σ (cid:1)(cid:27) , (C.24)and we can combine this with eqs. (C.21), (C.19) and (C.15) to write the perturbation to the elastic tensor as Λ = X ω Λ − Λ X ω + X u A + A X u − A tr( u ) + 8 D ˜ V ( ) · u + R ( σ − X ω · σ ) , (C.25)with u = (cid:0) A + X σ − σ ⊗ (cid:1) − · (cid:0) σ − X ω · σ (cid:1) . (C.26)A final simplification is obtained by noting that X ω Λ − Λ X ω = X ω A − A X ω + X ω R σ − R σ X ω . (C.27)It is then readily shown that X ω R σ − R σ X ω = R X ω · σ , (C.28)which partially cancels with the final term in eq.(C.25). Thus, our final expression for the perturbation to the elastic tensor is Λ = A + R σ A = X u + ω A + A X u − ω − A tr( u ) + 8 D ˜ V ( ) · uu = (cid:0) A + X σ − σ ⊗ (cid:1) − · (cid:0) σ − X ω · σ (cid:1) . (C.29)Note that the arbitrary antisymmetric matrix ω insinuates itself implicitly via the definition of u , as well as appearing explicitly in theexpression for A . When the background material is isotropic ω can be set to zero without loss of generality. C2 Isotropic, hydrostatically pre-stressed background
For a zeroth-order hydrostatic stress and elastic tensor given by σ = − p = λ ⊗ + 2 µ ˆid , (C.30)the perturbation to the stress satisfies σ = (cid:0) A + X σ − σ ⊗ (cid:1) · u = (cid:16) λ ⊗ + 2 µ ˆid − p id + p ⊗ (cid:17) · u = ˜ λ tr( u ) + 2˜ µ u , (C.31)with ˜ λ = λ + p (C.32a) ˜ µ = µ − p . (C.32b)It follows easily that tr (cid:0) σ (cid:1) = (cid:16) λ + 2˜ µ (cid:17) tr( u ) , (C.33) M. A. Maitra & D. Al-Attar from which u = 12˜ µ (cid:20) σ − ˜ λ λ + 2˜ µ tr (cid:0) σ (cid:1) (cid:21) . (C.34)Next, we split the stress into its hydrostatic and deviatoric components. Writing σ = − p + τ , (C.35)with tr (cid:0) τ (cid:1) = 0 , (C.36)we find that u = − x + 12˜ µ τ , (C.37)where we define the shorthand x ≡ p λ + 2˜ µ . (C.38)Let us now turn to the expression for A , which is given by A = X u A + A X u − tr( u ) A + 8 D ˜ V ( ) · u (C.39)in the isotropic case. For the third-derivative term, we follow the same philosophy as Murnaghan (1937), considering a Taylor-expansion of thestrain-energy function ˜ V up to third-order in the scalar-invariants of C . The most general such expression is, in index-notation, [8 D ˜ V ( )] ijklmn C ij C kl C mn = ζ tr( C ) + 3 ζ tr( C ) tr (cid:0) C (cid:1) + 2 ζ tr (cid:0) C (cid:1) , (C.40)where we have defined the constants ζ , ζ and ζ . This expression should be compared with Murnaghan (1937, p.250, 2nd equation frombottom), wherein the Murnaghan constants l , m and n are introduced, as well as the bottom equation of p.240 of that paper, where thescalar-invariants of a tensor are defined. Our constants ζ , ζ and ζ , which we have found convenient to use when performing calculations,are equivalent to Murnaghan’s, and we shall simply refer to them as ‘the Murnaghan constants’. Using our operator notation, the expressionfor the third-derivative term in A is D ˜ V ( ) · u = ζ tr( u ) ⊗ + ζ tr( u ) ˆid + ζ ( ⊗ u + u ⊗ ) + ζ ˆ X u , (C.41)where we have used hats to mark symmetrisation of certain operators (see eq.A.33). With A given above, and bearing in mind that X u ( ⊗ ) = 2 u ⊗ , (C.42) X u ˆid = ˆ X u , (C.43)we obtain A = ( ζ − λ ) tr( u ) ⊗ +( ζ − µ ) tr( u ) ˆid +(2 λ + ζ )( ⊗ u + u ⊗ ) +(4 µ + ζ ) ˆ X u . (C.44)We can now substitute in our expression for u in terms of p and τ , yielding A = − λ + 3 ζ + 2 ζ λ + 2 µ + p p ⊗ − µ + 3 ζ + 2 ζ λ + 2 µ + p p ˆid + 2 λ + ζ µ − p ) (cid:0) ⊗ τ + τ ⊗ (cid:1) + 4 µ + ζ µ − p ) ˆ X τ . (C.45)With this, we may write the perturbation to the elastic tensor itself as Λ = ( κ (cid:48) − µ (cid:48) ) p ⊗ + 2 µ (cid:48) p ˆid + a (cid:0) ⊗ τ + τ ⊗ (cid:1) + 2 b ˆ X τ − p id + R τ , (C.46)where µ (cid:48) , κ (cid:48) , a and b are given in terms of the Murnaghan constants as µ (cid:48) ≡ − µ + ζ + ζ λ + 2 µ + p (C.47) κ (cid:48) ≡ − κ + 3 ζ + 3 ζ + ζ λ + 2 µ + p (C.48) a ≡ κ − µ + ζ µ − p (C.49) b ≡ µ + ζ µ − p . (C.50)The full elastic tensor is written in index notation as Λ ijkl = (cid:20)(cid:0) κ + κ (cid:48) p (cid:1) − (cid:0) µ + µ (cid:48) p (cid:1)(cid:21) δ ij δ kl + (cid:0) µ + µ (cid:48) p (cid:1) ( δ ik δ jl + δ il δ jk ) − (cid:0) p + p (cid:1) δ ik δ jl + a (cid:0) δ ij τ kl + δ kl τ ij (cid:1) + b (cid:0) δ ik τ jl + δ jl τ ik + δ il τ jk + δ jk τ il (cid:1) + δ ik τ jl . (C.51) n the stress dependence of the elastic tensor C3 Transversely-isotropic, ‘quasi-hydrostatically’ pre-stressed background
The transversely-isotropic calculation is more intricate than the isotropic one for a number of reasons. Firstly, and perhaps most simplyremedied, we can no longer ignore the arbitrary antisymmetric matrix ω . Secondly, the stress-strain relationship u = (cid:0) A + X σ − σ ⊗ (cid:1) − · (cid:0) σ − X ω · σ (cid:1) (C.52)is not as easily inverted for a transversely-isotropic elastic tensor. Thirdly, there are many more terms to keep track of.Let us begin by recalling the stress-free transversely-isotropic elastic-tensor given in eq.(4.37): Λ ijkl = λδ ij δ kl + µ ( δ ik δ jl + δ il δ jk )+ 8 γν i ν j ν k ν l + 4 β ( ν i ν j δ kl + δ ij ν k ν l ) − α ( ν i ν k δ jl + ν j ν k δ il + ν j ν l δ ik + ν i ν l δ jk ) . (C.53)For the remainder of the calculation we will switch to our ‘operator-based’ notation. We also introduce a shorthand that is standard in manyareas of physics, whereby a symmetric expression is written as ( symmetric object ) = ( not-necessarily-symmetric-object ) + h.c. . (C.54)It will allow us to avoid significant clutter later on. (Here, h.c. technically stands for ‘hermitian conjugate’.) Defining the tensor N = ν ⊗ ν , (C.55)we can now rewrite the zeroth-order elastic tensor as A = λ ⊗ + 2 µ ˆid + 8 γ N ⊗ N + 4 β ( ⊗ N + h.c. ) − α ˆ X N , (C.56)and the pre-stress (eq.4.38) as σ = − (cid:18) p + q (cid:19) + q N . (C.57) N satisfies the relations tr( N ) = 1 , (C.58)and N n = N (C.59)for positive integer n , from which it follows by induction that X n N = X N +(2 n − N ⊗ N . (C.60)It is useful to note that X σ − σ ⊗ = − (cid:18) p + q (cid:19) id + q X N + (cid:18) p + q (cid:19) ⊗ − q N ⊗ , (C.61)for now we can rewrite the stress-strain relationship eq.(C.52) as σ − X ω · σ = (cid:20) A − (cid:18) p + q (cid:19) id + q X N + (cid:18) p + q (cid:19) ⊗ − q N ⊗ (cid:21) · u = ˜ λ tr( u ) + 2˜ µ u + 8˜ γ (cid:104) N , u (cid:105) N − α X N · u + 4 ˜ β (cid:104) N , u (cid:105) + 4 ˜ β (cid:48) tr( u ) N , (C.62)with ˜ λ = λ + p + q (C.63a) ˜ µ = µ − p − q (C.63b) ˜ γ = γ (C.63c) ˜ α = α − q (C.63d) ˜ β = β (C.63e) ˜ β (cid:48) = β − q . (C.63f)We are now in position to solve for u in terms of σ and ω . In the isotropic case we could invert for u simply by writing tr( u ) in termsof tr (cid:0) σ (cid:1) . Here, by analogy, we take not only the trace of both sides, but also the inner product with N , to find that (cid:32) tr (cid:0) σ (cid:1)(cid:10) N , σ (cid:11)(cid:33) = (cid:32) λ + 2˜ µ + 4 ˜ β (cid:48) γ − α + 12 ˜ β ˜ λ + 4 ˜ β (cid:48) µ + 8˜ γ − α + 4 ˜ β (cid:33) (cid:32) tr( u ) (cid:104) N , u (cid:105) (cid:33) . (C.64)Note that ω is not present here because tr (cid:0) X ω · σ (cid:1) = (cid:10) , X ω · σ (cid:11) = (cid:68) X T ω · , σ (cid:69) , (C.65) M. A. Maitra & D. Al-Attar which vanishes due to the antisymmetry of ω . By the same token, (cid:10) N , X ω · σ (cid:11) = − (cid:18) p + q (cid:19) (cid:104) N , X ω · (cid:105) + q (cid:104) N , X ω · N (cid:105) = 0 , (C.66)with the second term vanishing due to the antisymmetry of the operator X ω (which obviously follows from that of ω ). Thus, both tr( u ) and (cid:104) N , u (cid:105) can be written in terms of known quantities wherever they appear.Now, in a further step reminiscent of the isotropic case, we move most of the terms in eq.(C.62) over to the left hand side, giving µ u − α X N · u = 2˜ µ Σ (cid:48) , (C.67)where Σ (cid:48) is a shorthand defined as Σ (cid:48) = 12˜ µ (cid:104) σ − X ω · σ − ˜ η − ˜ θ N (cid:105) , (C.68)with (cid:32) ˜ η ˜ θ (cid:33) = (cid:32) ˜ λ β β (cid:48) γ (cid:33) (cid:32) tr( u ) (cid:104) N , u (cid:105) (cid:33) . (C.69)We are left to solve the equation (cid:18) id − ˜ α ˜ µ X N (cid:19) · u = Σ (cid:48) . (C.70)It is readily verified that the necessary inverse operator is (cid:18) id − ˜ α ˜ µ X N (cid:19) − = id + ˜ P X N + ˜ Q N ⊗ N , (C.71)with ˜ P = ˜ α ˜ µ − ˜ α ˜ µ (C.72a) ˜ Q = 2 ˜ α ˜ µ − ˜ α ˜ µ ˜ P , (C.72b)from which it follows that u = Σ (cid:48) + ˜ P X N · Σ (cid:48) + ˜ Q (cid:10) N , Σ (cid:48) (cid:11) N . (C.73)This procedure seems to break down when ˜ µ equals ˜ α or α , but the physical significance of these cases is not yet clear to the present authors.From here, with the help of the identity X N X ω · N = X ω · N , (C.74)it is a matter of relatively straightforward algebra to find that µ u = σ − ˜ η + ˜ P X N · σ + ˜ R N − q (1 + ˜ P ) X ω · N , (C.75)with ˜ R = ˜ Q (cid:16)(cid:10) N , σ (cid:11) − ˜ η − ˜ θ (cid:17) − ˜ θ − P (cid:16) ˜ η + ˜ θ (cid:17) . (C.76)We now have an expression for u in terms of:(i) the transversely-isotropic constants λ , µ , α , β and γ , as well as ν ;(ii) the constants p and q which define the equilibrium stress-state;(iii) the perturbation to the stress σ ;(iv) an antisymmetric matrix ω ‘pointing’ in an arbitrary direction in the plane perpendicular to the unperturbed symmetry-axis.This is to be substituted into the expression A = X u + ω A + A X u − ω − A tr( u ) + 8 D ˜ V ( ) · u (C.77)for the perturbed elastic tensor.In order to make progress we must parametrise the third derivatives of the transversely-isotropic strain-energy function. As stated in themain text, such a strain-energy function will generally depend on C not only through the invariants I , I and I defined in eq.(4.13), but alsothrough the terms (e.g. Holzapfel 2000) (cid:104) ν , C · ν (cid:105) = (cid:104) N , C (cid:105) (C.78) (cid:10) ν , C · ν (cid:11) = (cid:10) N , C (cid:11) = 12 (cid:104) C , X N · C (cid:105) . (C.79) n the stress dependence of the elastic tensor The third order terms of ˜ V ’s Taylor-series about the identity therefore satisfy [8 D ˜ V ( )] ijklmn C ij C kl C mn = ζ tr( C ) + 3 ζ tr( C ) tr (cid:0) C (cid:1) + 2 ζ tr (cid:0) C (cid:1) + 3 ζ tr( C ) (cid:104) C , N (cid:105) + 3 ζ tr( C ) (cid:104) C , X N · C (cid:105) + 3 ζ (cid:104) C , id · C (cid:105) (cid:104) C , N (cid:105) + 3 ζ (cid:104) C , N (cid:105) (cid:104) C , X N · C (cid:105) , (C.80)for some material-dependent constants { ζ i } , so that the corresponding term in the elastic tensor is D ˜ V ( ) · u = ζ tr( u ) ⊗ + ζ (cid:104) tr( u ) ˆid +( ⊗ u + h.c. ) (cid:105) + ζ ˆ X u + ζ [ (cid:104) N , u (cid:105) ( ⊗ N + h.c. ) + tr( u ) N ⊗ N ]+ ζ [( ⊗ ( X N · u ) + h.c. ) + tr( u ) X N ]+ ζ [( N ⊗ u + h.c. ) + (cid:104) N , u (cid:105) id]+ ζ [( N ⊗ ( X N · u ) + h.c. ) + (cid:104) N , u (cid:105) X N ]= ζ tr( u ) ⊗ +( ζ tr( u ) + ζ (cid:104) N , u (cid:105) ) ˆid + ζ ( ⊗ u + h.c. ) + ζ ˆ X u + ζ [ (cid:104) N , u (cid:105) ( ⊗ N + h.c. ) + tr( u ) N ⊗ N ] + ζ [ ⊗ ( X N · u ) + h.c. ]+ ζ ( N ⊗ u + h.c. ) + ζ [ N ⊗ ( X N · u ) + h.c. ] +( ζ tr( u ) + ζ (cid:104) N , u (cid:105) ) ˆ X N . (C.81)Whilst for an isotropic solid we needed three extra material-dependent constants to parametrise the third derivatives, here we need seven.With this, we are ready to write down an expression for the elastic tensor’s perturbation. Observe that X u A = (cid:0) A X u (cid:1) T (C.82) X ω A = (cid:0) A X ( − ω ) (cid:1) T . (C.83)Thus, A = tr( u ) (cid:104) ( h − λ ) ⊗ +( h − µ ) ˆid +( h − γ ) N ⊗ N +( h + 2 α ) ˆ X N +( h − β )( ⊗ N + h.c. ) (cid:105) +( ζ + 4 µ ) ˆ X u +( ζ + 2 λ )( ⊗ u + h.c. ) +( ζ + 8 β )( N ⊗ u + h.c. )+( ζ + 4 β )[ ⊗ ( X N · u ) + h.c. ] +( ζ + 8 γ )[ N ⊗ ( X N · u ) + h.c. ] − α (cid:16) ˆ X u ˆ X N + h.c. (cid:17) + 4 µ ˆ X ω + 8 γ [ N ⊗ ( X ω · N ) + h.c. ] + 4 β [ ⊗ ( X ω · N ) + h.c. ] − α (cid:16) ˆ X ω ˆ X N + h.c. (cid:17) . (C.84)It is now a matter of tedious algebra to complete the calculation. The nontrivial identities that we need are: X N = X N + 2 N ⊗ N (C.85) X N X ω · N = X ω · N (C.86) X ( X ω · N ) = X ω X N + h.c. (C.87) ˆ X X N · σ ˆ X N + h.c. = ˆ X X N · σ + 2 (cid:2) N ⊗ (cid:0) X N · σ (cid:1) + h.c. (cid:3) + 2 (cid:10) N , σ (cid:11) ˆ X N . (C.88)Finally, the perturbation to the elastic tensor is given by A = η ⊗ + η ˆid + η N ⊗ N + η ( ⊗ N + h.c. ) + η ˆ X N + η ˆ X σ + η (cid:0) ⊗ σ + h.c. (cid:1) + η (cid:0) N ⊗ σ + h.c. (cid:1) + η (cid:2) ⊗ (cid:0) X N · σ (cid:1) + h.c. (cid:3) + η (cid:2) N ⊗ (cid:0) X N · σ (cid:1) + h.c. (cid:3) + η ˆ X X N · σ + η (cid:16) ˆ X σ ˆ X N + h.c. (cid:17) + η ˆ X ω + η [ ⊗ ( X ω · N ) + h.c. ] + η [ N ⊗ ( X ω · N ) + h.c. ] + η (cid:16) ˆ X ω ˆ X N + h.c. (cid:17) . (C.89) M. A. Maitra & D. Al-Attar
The { η i } are defined as µη = 2˜ µ tr( u )( ζ − λ ) − [2˜ η ( ζ + 2 λ )] (C.90a) µη = 2˜ µ [( ζ − µ ) tr( u ) + ζ (cid:104) N , u (cid:105) ] − [2˜ η ( ζ + 4 µ )] (C.90b) µη = 2˜ µ tr( u )( ζ − γ ) + (cid:104) R ( ζ + 8 β ) − α ˜ R + (cid:16) R − η + 4 ˜ P (cid:10) N , σ (cid:11)(cid:17) ( ζ + 8 γ ) (cid:105) (C.90c) µη = 2˜ µ [ ζ (cid:104) N , u (cid:105) − β tr( u )] + (cid:104) ˜ R ( ζ + 2 λ ) − ˜ η ( ζ + 8 β ) + (cid:16) P (cid:10) N , σ (cid:11) + 2 ˜ R − η (cid:17) ( ζ + 4 β ) (cid:105) (C.90d) µη = 2˜ µ [( ζ + 2 α ) tr( u ) + ζ (cid:104) N , u (cid:105) ] + (cid:104) ˜ R ( ζ + 4 µ ) − α (cid:16) R − η + 2 ˜ P (cid:10) N , σ (cid:11)(cid:17)(cid:105) (C.90e) µη = ζ + 4 µ (C.90f) µη = ζ + 2 λ (C.90g) µη = ζ + 8 β (C.90h) µη = ˜ P ( ζ + 2 λ ) + (cid:16) P (cid:17) ( ζ + 4 β ) (C.90i) µη = ˜ P ( ζ + 8 β − α ) + (cid:16) P (cid:17) ( ζ + 8 γ ) (C.90j) µη = ˜ P ( ζ + 4 µ − α ) (C.90k) µη = − α (C.90l) µη = 8 µ ˜ µ (C.90m) µη = 8 β ˜ µ − q (cid:16) P (cid:17) ( ζ + ζ + 2 λ + 4 β ) (C.90n) µη = 16 γ ˜ µ − q (cid:16) P (cid:17) ( ζ + ζ − α + 8 β + 8 γ ) (C.90o) µη = − (cid:104) α ˜ µ + q (cid:16) P (cid:17) ( ζ + 4 µ − α ) (cid:105) , (C.90p)definitions which should in turn be combined with the earlier definitions eqs.(C.63,C.64,C.69,C.72,C.76,C.80). The perturbation to the elastictensor is given in terms of(i) the transversely-isotropic constants λ , µ , α , β and γ , as well as ν ;(ii) the constants p and q which parametrise the zeroth-order equilibrium stress;(iii) the seven constants { ζ i } defined in eq.(C.80), which parametrise the third derivatives of a transversely-isotropic strain-energy functionabout equilibrium;(iv) the small applied stress σ ;(v) the arbitrary 2-parameter antisymmetric matrix ωω