Wave Motion 48 (2011) 552–567 Contents lists available at ScienceDirect Wave Motion j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / wave m o t i Initial stresses in elastic solids: Constitutive laws and acoustoelasticity M. Shams a,e, M. Destrade b,c,⁎, R.W. Ogden a,c,d a b c d e School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow G12 8QW, Scotland, UK School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, University Road, Galway, Ireland School of Electrical, Electronic, and Mechanical Engineering, University College Dublin, Belfield, Dublin 4, Ireland School of Engineering, University of Aberdeen, King's College, Aberdeen AB24 3UE, Scotland, UK Centre for Advanced Mathematics and Physics, National University of Science and Technology (NUST), Sector H-12, Islamabad, Pakistan a r t i c l e i n f o Article history: Received 14 September 2010 Received in revised form 23 March 2011 Accepted 1 April 2011 Available online 13 April 2011 Keywords: Nonlinear elasticity Initial stress Residual stress Invariants Plane waves Biot's theory a b s t r a c t On the basis of the nonlinear theory of elasticity, the general constitutive equation for an isotropic hyperelastic solid in the presence of initial stress is derived. This derivation involves invariants that couple the deformation with the initial stress and in general, for a compressible material, it requires 10 invariants, reducing to 9 for an incompressible material. Expressions for the Cauchy and nominal stress tensors in a finitely deformed configuration are given along with the elasticity tensor and its specialization to the initially stressed undeformed configuration. The equations governing infinitesimal motions superimposed on a finite deformation are then used to study the combined effects of initial stress and finite deformation on the propagation of homogeneous plane waves in a homogeneously deformed and initially stressed solid of infinite extent. This general framework allows for various different specializations, which make contact with earlier works. In particular, connections with results derived within Biot's classical theory are highlighted. The general results are also specialized to the case of a small initial stress and a small pre-deformation, i.e. to the evaluation of the acoustoelastic effect. Here the formulas derived for the wave speeds cover the case of a second-order elastic solid without initial stress and subject to a uniaxial tension [Hughes and Kelly, Phys. Rev. 92 (1953) 1145] and are consistent with results for an undeformed solid subject to a residual stress [Man and Lu, J. Elasticity 17 (1987) 159]. These formulas provide a basis for acoustic evaluation of the secondand third-order elasticity constants and of the residual stresses. The results are further illustrated in respect of a prototype model of nonlinear elasticity with initial stress, allowing for both finite deformation and nonlinear dependence on the initial stress. © 2011 Elsevier B.V. All rights reserved. 1. Introduction The presence of initial stresses in solid materials can have a substantial effect on their subsequent response to applied loads that is very different from the corresponding response in the absence of initial stresses. In geophysics, for example, the high stress developed below the Earth's surface due to gravity has a strong influence on the propagation speed of elastic waves, while in soft biological tissues initial (or residual) stresses in artery walls ensure that the circumferential stress distribution through the thickness of the artery wall is close to uniform at typical physiological blood pressures. Initial stresses may arise, for example, from applied loads, as in the case of gravity, processes of growth and development in living tissue or, in the case of engineering components, from the manufacturing process, either by design to improve the performance of the component or unintentionally (which can lead to defective behaviour). Here we use the term initial stress in its broadest sense, irrespective of how the stress ⁎ Corresponding author at: School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, University Road, Galway, Ireland. E-mail address: michel.destrade@nuigalway.ie (M. Destrade). 0165-2125/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2011.04.004 M. Shams et al. / Wave Motion 48 (2011) 552–567 553 develops. This includes situations where the stress is due to an applied load leading to an accompanying finite deformation from an unstressed configuration, in which case the term prestress is commonly used, and situations in which the initial stress arises from some other process, such as manufacturing or growth, and is present in the absence of applied loads. If an initial stress is present in the absence of applied loads (body forces and surface tractions) it is referred to as residual stress, as in the definition adopted by Hoger [9]. In the context of finite deformation elasticity theory much motivation for the study of initial stresses, more especially residual stresses, comes from soft tissue biomechanics, and for a recent discussion of residual stresses in artery walls we refer to Holzapfel and Ogden [12] and references therein. The development of constitutive laws for residually stressed materials has been a focus of much of the work of Hoger, as exemplified by Hoger [9–11] and Johnson and Hoger [14]; see also the recent paper by Saravanan [22]. In the context of acoustoelasticity the work of Man and Lu [16] was based on the developments of Hoger, and it also relates to the much earlier work of Biot. Biot's work was developed within the geophysical context and he was particularly concerned with the effect of initial stress on the propagation of small amplitude elastic waves. He developed a static theory of small deformations influenced by initial stress [1], followed by a corresponding theory for wave propagation [2], all of this work being conveniently collected in his monograph [3]. Whatever the source of the initial stress and whether or not it is accompanied by a finite deformation, of particular interest in many applications is the effect that the initial stress has on small deformations (static or time-dependent), specifically deformations linearized relative to the initially-stressed state and often referred to as incremental deformations or motions. If there is an accompanying finite deformation then the theory is often referred to as the theory of small deformations superimposed on large deformations. This theory requires knowledge of the associated elasticity tensor, which in general depends on any finite deformation present and on the initial stress. Thus, part of the purpose of the present work is to obtain the general form of the elasticity tensor for an initially stressed material that is subject to a finite deformation where the initial stress is not itself associated with an initial finite deformation. We shall be concerned primarily with the case in which the material possesses no intrinsic anisotropy so that any anisotropy arises solely from the presence of initial stress, and we show how the components of the elasticity tensor relate to those of the elasticity coefficients in Biot's isotropic theory. In the present paper we use the elasticity tensor to examine the effect of initial stress on the propagation of small amplitude elastic waves for both finitely deformed and unstrained materials, with reference to the associated strong ellipticity condition. In Section 2, the basic equations for a (hyper)elastic material subject to initial stress are summarized together with the equations governing equilibrium of the initially stressed and finitely deformed initially stressed configurations. The equations of incremental motions superimposed on an initially stressed and finitely deformed equilibrium configuration are then given in Section 3, wherein the elasticity tensor is derived and its general symmetry properties recorded. In Section 4 we then specialize the general theory and focus on the development of the constitutive law for an initially stressed material that has no intrinsic anisotropy such as might be associated with preferred directions, for example, and any anisotropy in its response relative to the initially stressed configuration is entirely due to the initial stress. The constitutive law of the material is based on a strain-energy function (defined per unit reference volume) that depends on the combined invariants of the right Cauchy–Green deformation tensor and the initial stress tensor. For a compressible material there are 10 such independent invariants in the general three-dimensional case, a number which reduces to 9 for an incompressible material. Expressions for the Cauchy stress and nominal stress tensors and the elasticity tensor are given in general forms for both compressible and incompressible materials and then specialized for specific applications by reducing the number of invariants involved while ensuring that the effects of initial stress are adequately accounted for. Some details of the calculations are relegated to Appendix A for convenience of reference. In Section 5, the equations of motion are specialized in order to study the effect of initial stress on the speed of infinitesimal homogeneous plane waves. First, we give results for the situation in which there is initial stress but no finite deformation. It is noted, in particular, that the wave speed depends in a nonlinear fashion on the initial stress. Results are compared with those arising from Biot's isotropic theory based on connections between the components of the elasticity tensor used here and those used in Biot's theory. The general connection between these components, which was derived in a recent paper [21], is noted here for reference. Also, within the present framework, by considering a linear elastic material, we confirm a formula given by Man and Lu [16] in the general linear theory, which they attribute to Biot, concerning the effect of initial stress on the speed of homogeneous plane waves and the acoustoelastic effect. Next, for the Murnaghan form of strain-energy function [17], which is appropriate for second-order elastic deformations, and specializing to the case of a material without initial stress, we recover the results of Hughes and Kelly [13] concerned with the second-order correction for the speeds of longitudinal and transverse waves in an isotropic elastic material. The final illustration introduces a prototype nonlinear model of an elastic material with initial stress that allows for both finite deformation and nonlinear initial stress. 2. Basic equations for an elastic solid with initial stress Consider an elastic body in some reference configuration, which we denote by Br . Let X be the position vector of a material point in Br . Any subsequent deformation of the body is measured from Br and we assume that there is an initial (Cauchy) stress τ in this configuration. This initial stress is symmetric (rotational balance in the absence of intrinsic couple stresses) and satisfies the equilibrium equation Div τ = 0; ð2:1Þ 554 M. Shams et al. / Wave Motion 48 (2011) 552–567 in the absence of body forces, where Div denotes the divergence operator with respect to Br . If the traction on the boundary ∂Br of Br vanishes pointwise then τ is referred to as a residual stress, and it is necessarily non-uniform [9,19]. If the traction is not zero then the initial stress may or may not be accompanied by some prior deformation required to reach the configuration Br from a completely unstressed configuration. Here we shall not be concerned with how the initial stress is produced. Suppose now that the body is deformed quasi-statically into a new configuration B with boundary ∂B so that the material point X takes up the new position x given by x = χ(X), where the vector function χ defines the deformation for X ∈ Br . The so-called deformation χ is required to be a bijection and to possess appropriate regularity properties, which need not be made explicit here. The deformation gradient tensor, denoted F, is defined by F = Grad χ, where Grad is the gradient operator with respect to Br , and the left and right Cauchy–Green deformation tensors are defined by T B = FF ; T C = F F; ð2:2Þ respectively. Let σ and S denote the Cauchy stress tensor and the nominal stress tensor, respectively, in the configuration B. For equilibrium in the absence of body forces σ and S satisfy the equations div σ = 0; Div S = 0; ð2:3Þ and we note the standard connection σ = (det F)− 1FS. For a material without couple stresses, σ is symmetric and hence we have T T FS = S F : ð2:4Þ For an elastic material we consider a strain-energy function W defined per unit volume in Br . This function depends on the deformation gradient F and the initial stress τ. If τ depends on X, as would be the case for a residually-stressed material, then the material is necessarily inhomogeneous, but if τ is independent of X the material is homogeneous unless its properties depend separately on X. In either case we make the dependence on τ explicit and write W = W ðF; τÞ: ð2:5Þ Of course, by objectivity, W depends on F only through C = FTF, but otherwise this form of W is completely general and no material symmetry is invoked. Note, however, that in general, i.e. if τ is not a hydrostatic compression or tension, the presence of τ induces some anisotropy in the material, even if the material has no intrinsic anisotropy. Thus, τ has an effect on the constitutive law analogous to that of a structure tensor in anisotropic elasticity. For a material not subject to any internal constraints, the nominal and Cauchy stresses are given by S= ∂W ðF; τÞ; ∂F σ=J −1 FS = J −1 F ∂W ðF; τÞ; ∂F ð2:6Þ respectively, where J = det F N 0. When evaluated in Br , these give the connection τ= ∂W ðI; τÞ; ∂F ð2:7Þ where I is the identity tensor. For an incompressible material, the internal constraint J ≡ det F = 1 ð2:8Þ must hold for all deformations and the counterpart of Eq. (2.6) in this case is S= ∂W −1 ðF; τÞ−pF ; ∂F σ = FS = F ∂W ðF; τÞ−pI; ∂F ð2:9Þ where p is a Lagrange multiplier associated with the constraint. When evaluated in the residually stressed configuration, these both reduce to τ= ∂W ðr Þ ðI; τÞ−p I; ∂F where p(r) is the value of p in Br . ð2:10Þ M. Shams et al. / Wave Motion 48 (2011) 552–567 555 3. Incremental motion superimposed on an initially stressed configuration subject to finite deformation Superimposed on the equilibrium configuration B defined by x = χ(X), we now consider an incremental motion ẋðX; t Þ, where t is time. Here and in the following a superposed dot indicates an incremental quantity, increments are consider ‘small’, and the resulting incremental equations are linearized in the increments. Thus, ẋ represents the displacement from x, and we shall also express it in Eulerian form by writing the displacement vector as a function of x and t, namely u = u(x, t). The corresponding increment in the deformation gradient, Ḟ, is expressible as Ḟ = LF; ð3:1Þ where L = grad u is the displacement gradient. The (linearized) incremental nominal stress takes the forms Ṡ = ( A Ḟ ; ð3:2Þ A Ḟ + pF−1 ḞF−1 − ṗF−1 ; for unconstrained and incompressible materials, respectively, where A= ∂2 W ; ∂F∂F Aαiβj = ∂2 W ; ∂Fiα ∂Fjβ ð3:3Þ is the elasticity tensor and, in component form, A Ḟ ≡ Aαiβj Ḟ jβ defines the product used in the above. Here and henceforth the standard summation convention for repeated indices applies. We note in passing that Eq. (3.2) is applicable even if a strain-energy function is not assumed, in which case the major symmetry iα ↔ jβ from Eq. (3.3) is lost; moreover, the theory could be considered more general if the explicit dependence on structure tensors such as τ is omitted. The generality considered here, however, is sufficient for our purpose. For an incompressible material the incremental incompressibility constraint can be written in either of the forms ! " −1 tr ḞF = tr L = 0; div u = 0: ð3:4Þ In the absence of body forces the incremental motion is governed by the equation Div Ṡ = ρr x;tt ; ð3:5Þ where ρr is the mass density in Br and a subscript t following a comma signifies the material time derivative, i.e. the time derivative at fixed X, so that x,t = u,t is the particle velocity and x,tt = u,tt the acceleration. The incremental counterpart of the rotational balance equation (2.4) is F Ṡ + ḞS = ṠT FT + ST ḞT ; ð3:6Þ or, on use of Eqs. (2.6)2 and (3.1), F Ṡ + JLσ = ṠT FT + J σ LT : ð3:7Þ −1 On introducing the ‘push forward’ Ṡ0 of Ṡ, defined by Ṡ0 = J F Ṡ, and the corresponding push forward A 0 of the elasticity tensor such that Ṡ0 = A 0 L (or A 0 L + pL−ṗI in the case of an incompressible material) we may express Eq. (3.7) in the form T A 0 L + Lσ = ðA 0 LÞ + σL T ð3:8Þ for an unconstrained material, and as T T A 0 L + Lð σ + pIÞ = ðA 0 LÞ + ðσ + pIÞL ; ð3:9Þ for an incompressible material. In component form A 0 is related to A via JA0piqj = Fpα Fqβ Aαiβj ; ð3:10Þ with J = 1 in the incompressible case. For details of the background on the theory of incremental deformations superimposed on a finite deformation we refer to Ogden [18,20], for example. 556 M. Shams et al. / Wave Motion 48 (2011) 552–567 From Eq. (3.8) we deduce that tr½ðA 0 L + LσÞW$ = 0 for any skew symmetric second-order tensor W, and since this and the corresponding result from Eq. (3.9) holds for arbitrary L we deduce further that A 0 W = −σW; A 0 W = −ð σ + pIÞW; ð3:11Þ for an arbitrary skew symmetric W for unconstrained and incompressible materials, respectively. A formula equivalent to Eq. (3.11)1 was derived by Hoger [10] by a different method and stated explicitly as the formula (2.2.2) in the latter paper for a residually-stressed but undeformed configuration. Note that the derivation herein does not require the existence of a strainenergy function. Another result in Hoger [10] is recovered by choosing L to be symmetric, which leads to i 1h 1 T A 0 E−ðA 0 EÞ = ðσE−EσÞ; 2 2 ð3:12Þ for any symmetric E. This relation applies for both unconstrained and incompressible materials and corresponds to the formula (2.2.3) in Hoger [10], the right-hand side of which contains a sign error. Note that as well as possessing the major symmetry A0piqj = A0qjpi , which follows from Eqs. (3.3) and (3.10), A 0 has the property A0piqj + δjp σ iq = A0ipqj + δij σ pq ; ð3:13Þ for a compressible material, and ! " ! " A0piqj + δjp σ iq + pδiq = A0ipqj + δij σ pq + pδpq ; ð3:14Þ for an incompressible material. These formulas follow from Eqs. (3.8) and (3.9), respectively. Corresponding formulas for a prestressed material in the absence of residual stress were given by Chadwick and Ogden [6] and Chadwick [5]. In Eulerian form, i.e. in terms of Ṡ0 and u, the incremental equation of motion (3.5) becomes div Ṡ0 = ρu;tt ; ð3:15Þ where ρ = ρr J− 1 is the mass density in B. The (Cartesian) component form of this equation may be written, for an unconstrained material, as ! " A0piqj uj;q = ρui;tt ; ð3:16Þ ;p and for an incompressible material as ! " A0piqj uj;q −ṗ;i + p; j uj;i = ρr ui;tt ; ;p with ui;i = 0: ð3:17Þ Suppose now that the material properties are homogeneous, implying, in particular, that the initial stress τ is uniform, and that the deformation in B is homogeneous. Then, A, A 0 and p are constant, and the equations of motion (3.16) and (3.17) reduce to A0piqj uj;pq = ρui;tt ; ð3:18Þ and A0piqj uj;pq −ṗ;i = ρui;tt ; ui;i = 0; ð3:19Þ respectively, and we note that ρ = ρr in the latter case. As already indicated, we are considering A to depend on the deformation through F, on the initial stress τ, and on any material symmetry present in the configuration Br . In what follows we consider the intrinsic properties of the material to be isotropic relative to Br so that the only source of anisotropy is the initial stress. In order to account for the initial stress in the expressions for the stress and elasticity tensors we base the development in the following sections on scalar invariants involving the right Cauchy– Green deformation tensor C and the initial stress tensor τ. M. Shams et al. / Wave Motion 48 (2011) 552–567 557 4. Invariant-based formulation 4.1. Invariants of C and τ The strain-energy function W of an initially stressed material depends on τ as well as the deformation gradient F, and, by objectivity, it depends on F only through C = FTF. For a material that is isotropic relative to Br in the absence of initial stress, W can be treated as a function of three independent invariants of C, which are commonly taken to be the principal invariants defined by ! "i 1h 2 2 ðtr CÞ −tr C ; I3 = det C: I1 = tr C; I2 = ð4:1Þ 2 In the configuration Br these reduce to I1 = I2 = 3, I3 = 1. If there is an initial stress τ in Br then, in general, the material response relative to Br depends also on the invariants of τ and the combined invariants of τ and C. A possible set of independent invariants of τ, including those that depend on C, is ! " ! " 3 2 tr τ; tr τ ; tr τ ; trðτCÞ; ! " 2 tr τC ; ! " ! " 2 2 2 tr τ C ; tr τ C : ð4:2Þ Together, there are thus at most 10 independent invariants of C and τ in general, and for an incompressible material we have I3 = 1 and hence there are at most 9 independent invariants in this case. The number of independent invariants is reduced when the dimension of the considered problem is reduced from three to two, as for a plane strain setting, for example. In the reference configuration Br the fourth and fifth invariants listed in Eq. (4.2) reduce to the first, and the sixth and seventh to the second. For relevant background on invariants of tensors we refer to Spencer [25] and Zheng [28]. For subsequent reference we introduce the notation I6 = trðτCÞ; ! " ! " ! " 2 2 2 2 I7 = tr τC ; I8 = tr τ C ; I9 = tr τ C ; ð4:3Þ for the invariants that depend on both C and τ, noting that the notations I4 and I5 are not used since these tend to be associated with a preferred direction in a transversely isotropic material. Although not immediately obvious, it is worth noting that invariants such as tr(Cτ Cτ) and tr(Cτ C2τ) may be expressed in terms of the invariants listed above. This may be shown by first applying the Cayley–Hamilton theorem to C + λτ for an arbitrary scalar λ. The coefficient of λ yields the identity C2 τ + CτC + τC2 −I1 ðCτ + τCÞ−ðtr τÞC2 −½I1 ðtr τÞ−I6 $C + I2 τ−½I2 ðtr τÞ−I1 I6 + I7 $I = 0: ð4:4Þ Then, by multiplying by τ, for example, and then taking the trace of the result leads to h ! "i 2 2 2 ; trðCτCτÞ = 2I1 I8 −2I9 −2I1 I6 ðtr τÞ + 2I7 ðtr τÞ + I6 + I2 ðtr τÞ −tr τ ð4:5Þ which shows how tr(CτCτ) depends on the other invariants. The expressions for the stress and elasticity tensors given in Eqs. (2.6), (2.9) and (3.3) require the calculation of ∂W ∂I = ∑ Wi i ; ∂F ∂F i∈I ð4:6Þ 2 2 ∂Ij ∂ W ∂ Ii ∂I = ∑ Wi + ∑ Wij i ⊗ ; ∂F∂F ∂F∂F ∂F ∂F i∈I i; j ∈ I ð4:7Þ and where we have used the shorthand notations Wi = ∂W = ∂Ii ; Wij = ∂2 W = ∂Ii ∂Ij ; i; j∈ I, and I is the index set {1, 2, 3, 6, 7, 8, 9} (or {1, 2, 6, 7, 8, 9} in the case of an incompressible material). Although their derivatives with respect to F vanish and do not appear in the above expressions, the invariants tr τ, tr(τ2) and tr(τ3) may nevertheless be included in the functional dependence of W. In the following we give explicit expressions for the stress tensors and the elasticity tensor based on these summations. 4.2. Stress tensors For an unconstrained material the strain-energy function W depends on the seven deformation dependent invariants I1, I2, I3, I6,…, I9 together, in general, with tr τ, tr(τ2) and tr(τ3). From Eqs. (2.6) and (4.6) the nominal and Cauchy stresses are given by S= ∂W ∂I = ∑ Wi i ; ∂F ∂F i∈I σ=J −1 FS: ð4:8Þ 558 M. Shams et al. / Wave Motion 48 (2011) 552–567 The corresponding expressions for an incompressible material are S= ∂W ∂I −1 −1 −pF = ∑ Wi i −pF ; ∂F ∂F i∈I σ = FS: ð4:9Þ The required expressions for ∂ Ii/∂ F are listed for convenience in Appendix A. In particular, these enable the Cauchy stress to be expanded, so that ! " ! " 2 −1 −1 −1 Jσ = 2W1 B + 2W2 I1 B−B + 2I3 W3 I + 2W6 Σ + 2W7 ðΣB + BΣÞ + 2W8 ΣB Σ + 2W9 ΣB ΣB + BΣB Σ ; ð4:10Þ where the notation Σ = FτFT has been introduced and we recall that B = FFT is the left Cauchy–Green tensor. For an incompressible material Eq. (4.10) is replaced by ! " ! " 2 −1 −1 −1 σ = 2W1 B + 2W2 I1 B−B −pI + 2W6 Σ + 2W7 ðΣB + BΣÞ + 2W8 ΣB Σ + 2W9 ΣB ΣB + BΣB Σ ; ð4:11Þ with I3 ≡ 1. If we evaluate Eq. (4.10) in the reference configuration, it reduces to the appropriate specialization of Eq. (2.7), namely 2 τ = 2ðW1 + 2W2 + W3 ÞI + 2ðW6 + 2W7 Þτ + 2ðW8 + 2W9 Þτ ; ð4:12Þ where all Wi ; i ∈ I , are evaluated for I1 = I2 = 3, I3 = 1, I6 = I7 = tr τ, I8 = I9 = tr(τ2). This indicates that we should set W1 + 2W2 + W3 = 0; 2ðW6 + 2W7 Þ = 1; W8 + 2W9 = 0 ð4:13Þ there. In fact, if we require Eq. (4.12) to hold for all τ then the conditions in Eq. (4.13) necessarily follow. The corresponding reduction for an incompressible material yields ! " 2 ðr Þ τ = 2W1 + 4W2 −p I + 2ðW6 + 2W7 Þτ + 2ðW8 + 2W9 Þτ ; ð4:14Þ which specializes Eq. (2.10), and the counterpart of Eq. (4.13) is ðr Þ 2W1 + 4W2 −p = 0; 2ðW6 + 2W7 Þ = 1; W8 + 2W9 = 0; ð4:15Þ evaluated in the reference configuration. 4.3. The elasticity tensor Next, we note that the elasticity tensor A is given by A= ∂Ij ∂2 W ∂2 Ii ∂I + ∑ Wij i ⊗ = ∑ Wi : ∂F∂F ∂F ∂F∂F ∂F i; j ∈ I i∈I ð4:16Þ M. Shams et al. / Wave Motion 48 (2011) 552–567 559 This requires expressions for the second derivatives of the invariants, which are collected together for reference in Appendix A in component form. Here we give the component form of A 0 : h i JA0piqj = 2ðW1 + I1 W2 ÞBpq δij + 2W2 2Bpi Bqj −Biq Bjp −Bpr Brq δij −Bpq Bij ! " h i + 2I3 W3 2δip δjq −δiq δjp + 2W6 Σpq δij + 2W7 Σpq Bij + Σpr Brq δij + Bpr Σrq δij + Σij Bpq + Σpj Biq + Σiq Bpj ! "! " + 4W11 Bip Bjq + 4W22 I1 Bip −Bir Brp I1 Bjq −Bjs Bsq ! " + 4I32 W33 δip δjq + 4W12 2I1 Bip Bjq −Bip Bjr Brq −Bjq Bir Brp ! " " h ! i + 4I3 W13 Bip δjq + Bjq δip + 4I3 W23 I1 Bip δjq + Bjq δip −δip Bjr Brq −δjq Bir Brp ! " " h ! ! "i + 4W16 Bip Σjq + Bjq Σip + 4W17 Bip Σjr Brq + Bjr Σrq + Bjq Σir Brp + Bir Σrp " ! " i h! + 4W26 I1 Bip −Bir Brp Σjq + I1 Bjq −Bjr Brq Σip "! " ! "! h! "i + 4W27 I1 Bip −Bir Brp Σjs Bsq + Bjs Σsq + I1 Bjq −Bjr Brq Σis Bsp + Bis Σsp ! " " h ! ! "i + 4I3 W36 δip Σjq + δjq Σip + 4I3 W37 δip Σjr Brq + Bjr Σrq + δjq Σir Brp + Bir Σrp " h ! ! "i + 4W66 Σip Σjq + 4W67 Σip Σjr Brq + Bjr Σrq + Σjq Σir Brp + Bir Σrp ! "! " + 4W77 Σir Brp + Bir Σrp Σjs Bsq + Bjs Σsq ; ð4:17Þ and, in order to save space, terms involving derivatives with respect to I8 or I9 have not been included. Note, however, that it is straightforward to include these terms, just by repeating the terms involving derivatives with respect to I6 and I7, replacing the indices 6 and 7 by 8 and 9, respectively, and replacing Σ = FτFT by Fτ2FT. When there is no initial stress the above formula reduces to the result for a prestressed isotropic elastic solid, as given in, for example, the classical paper by Hayes and Rivlin [8]; see also Toupin and Bernstein [26] and Truesdell [27] for related basic theory. The corresponding formula for an incompressible material is obtained by setting J = 1 and I3 = 1 and omitting the terms involving derivatives of W with respect to I3, and yields h i A0piqj = 2ðW1 + I1 W2 ÞBpq δij + 2W2 2Bpi Bqj −Biq Bjp −Bpr Brq δij −Bpq Bij h i + 2W6 Σpq δij + 2W7 Σpq Bij + Σpr Brq δij + Bpr Σrq δij + Σij Bpq + Σpj Biq + Σiq Bpj ! "! " + 4W11 Bip Bjq + 4W22 I1 Bip −Bir Brp I1 Bjq −Bjs Bsq ! " + 4W12 2I1 Bip Bjq −Bip Bjr Brq −Bjq Bir Brp ! " " h ! ! "i + 4W16 Bip Σjq + Bjq Σip + 4W17 Bip Σjr Brq + Bjr Σrq + Bjq Σir Brp + Bir Σrp " ! " i h! + 4W26 I1 Bip −Bir Brp Σjq + I1 Bjq −Bjr Brq Σip h! "i ! "! " "! + 4W27 I1 Bip −Bir Brp Σjs Bsq + Bjs Σsq + I1 Bjq −Bjr Brq Σis Bsp + Bis Σsp " h ! ! "i + 4W66 Σip Σjq + 4W67 Σip Σjr Brq + Bjr Σrq + Σjq Σir Brp + Bir Σrp ! "! " + 4W77 Σir Brp + Bir Σrp Σjs Bsq + Bjs Σsq : ð4:18Þ When evaluated in the reference configuration, the moduli (4.17), with the missing terms included, reduce to ! " ! " A0piqj = α 1 δij δpq + δiq δjp + α2 δip δjq + δij τpq + β1 δij τpq + δpq τ ij + δiq τjp + δjp τ iq ! " ! " + β2 δip τ jq + δjq τ ip + β3 τ ip τ jq + γ1 δij τ pk τkq + δpq τik τkj + δiq τjk τkp + δjp τ ik τ kq ! " ! " + γ2 δip τjk τ kq + δjq τik τkp + γ3 τip τ jk τ kq + τ jq τ ik τkp + γ4 τik τ kp τjk τkq ; ð4:19Þ where the α's, β's, and γ's are defined by α 1 = 2ðW1 + W2 Þ; α2 = 4ðW11 + 4W12 + 4W22 + 2W13 + 4W23 + W33 Þ−2α1 ; β1 = 2W7 ; β2 = 4ðW16 + 2W17 + 2W26 + 4W27 + W36 + 2W37 Þ; β3 = 4ðW66 + 4W67 + 4W77 Þ; γ1 = 2W9 ; γ2 = 4ðW18 + 2W19 + 2W28 + 4W29 + W38 + 2W39 Þ; γ3 = 4ðW68 + 2W69 + 2W78 + 4W79 Þ; γ4 = 4ðW88 + 4W89 + 4W99 Þ; ð4:20Þ 560 M. Shams et al. / Wave Motion 48 (2011) 552–567 all derivatives being evaluated in the reference configuration and use having been made of the connections (4.13). Note that in general the expressions (4.20) may depend on the invariants tr τ, tr(τ2) and tr(τ3). It is worth noting here that when referred to axes that coincide with the principal axes of τ, the only non-zero components of Eq. (4.19) are given by A0iiii = 2α 1 + α 2 + ð1 + 4β1 + 2β2 Þτi + β3 τ 2i + 2ð2γ1 + γ2 Þτ2i + 2γ3 τ 3i + γ4 τ 4i ; ! " ! " ! " A0iijj = α 2 + β2 τi + τ j + β3 τi τj + γ2 τ2i + τ2j + γ3 τ i + τj τi τ j + γ4 τ2i τ 2j ; ! " ! " A0ijij = α 1 + τi + β1 τ i + τj + γ1 τ2i + τ2j = A0ijji + τi ; ð4:21Þ where there are no sums on repeated indices, i ≠ j, and τi (i = 1, 2, 3) are the principal values of τ (in general, 15 non-zero components of A 0 in total). It is interesting, but not surprising, to note that for a prestressed isotropic elastic material the only non-zero components of A 0 are also A0iiii , A0iijj , A0ijij and A0ijji , i ≠ j (see, for example, [18]). For an incompressible material there is a further reduction of Eq. (4.19), leading to ! " ! " A0piqj = α 1 δij δpq + δiq δjp + δij τpq + β1 δij τpq + δpq τ ij + δiq τjp + δjp τ iq ! " ! " + β2 δip τjq + δjq τip + β3 τip τjq + γ1 δij τpk τkq + δpq τ ik τ kj + δiq τ jk τ kp + δjp τik τkq ! " ! " + γ2 δip τjk τ kq + δjq τ ik τkp + γ3 τip τjk τ kq + τjq τ ik τ kp + γ4 τik τkp τjk τkq ; ð4:22Þ since the term involving α2 may be dropped because it vanishes from the expression for the incremental stress by virtue of the incompressibility condition. Here α1, β1, β3, γ1, γ3 and γ4 are unchanged, while β2 reduces to 4(W16 + 2W17 + 2W26 + 4W27) and γ2 reduces to 4(W18 + 2W19 + 2W28 + 4W29). The terms β2δjqτip and γ2δjqτikτkp also vanish from the incremental stress for the same reason but are retained here since otherwise the major symmetry would be lost. Note that the terms involving δjp in Eq. (4.22) do not contribute to the equation of motion, again by virtue of the incompressibility condition. 5. The effect of initial stress on infinitesimal wave propagation Now consider incremental motions in an infinite homogeneous medium subject to homogeneous deformation and/or homogeneous initial stress. From Section 3 we recall that the equation of incremental motion is A0piqj uj;pq = ρui;tt ; ð5:1Þ for a compressible material, and A0piqj uj;pq −ṗ;i = ρui;tt ; ui;i = 0; ð5:2Þ for an incompressible material. Consider a homogeneous plane wave of the form u = mf ðn⋅x−vt Þ; ð5:3Þ where m is a fixed unit vector (the polarization vector), f is an arbitrary function of the argument n ⋅ x − vt, n is a unit vector in the direction of propagation, and v is the wave speed. For an incompressible material we also set ṗ = g ðn⋅x−vt Þ; ð5:4Þ where g is a function related to f. Substitution into the equation of motion (Eq. (5.1)) (after dropping f″, which is assumed to be non-zero) leads to 2 A0piq j np nq mj = ρv mi ; ð5:5Þ for a compressible material, and into Eq. (5.2) 2 A0piq j np nq mj f ″ −g ′ ni = ρv mi f ″ ; mi ni = 0; ð5:6Þ for an incompressible material, where a prime on f or g indicates the derivative with respect to its argument. By multiplying the first equation in Eq. (5.6) by ni and using the second equation, we obtain g ′ = A0piq j np nq mj ni f ″ ; ð5:7Þ M. Shams et al. / Wave Motion 48 (2011) 552–567 and upon substitution back in Eq. (5.6) for g′ and again dropping f″ we arrive at ! " 2 A0piq j np nq −A0pkq j np nq nk ni mj = ρv mi : 561 ð5:8Þ It is convenient now to use the acoustic tensor Q(n), whose components are defined by Q ij ðnÞ = A0piq j np nq : ð5:9Þ Then Eqs. (5.5) and (5.8) can be written in the compact forms 2 Q ðnÞm = ρv m; 2 Q ðnÞm = ρv m; ð5:10Þ m⋅n = 0; ð5:11Þ for unconstrained and incompressible materials, respectively, where, noting that Im = m, with I = I−n ⊗ n defined as the projection operator on to the plane normal to n, the symmetric tensor Q ðnÞ is defined as Q ðnÞ = IQ ðnÞI: ð5:12Þ This is the projection of Q(n) on to the plane normal to n. The symmetrization of the acoustic tensor in the incompressible case is attributed to Hayes in the Ph.D. thesis of Scott [23]1; see also Scott and Hayes [24] and Boulanger and Hayes [4]. For any given direction of propagation n we now have a symmetric algebraic eigenvalue problem for determining ρv2 and m, which is three-dimensional in the compressible case and two-dimensional in the incompressible case. Because of the symmetry there are three mutually orthogonal eigenvectors m corresponding to the direction of displacement in the compressible case and two in the incompressible case, corresponding to transverse waves. The values of ρv2, three and two, respectively, are obtained from the characteristic equations h i 2 det Q ðnÞ−ρv I = 0; h i 2 det Q ðnÞ−ρv I = 0; ð5:13Þ for compressible and incompressible materials, respectively. If m is known then ρv2 is given by 2 ρv = ½Q ðnÞm$⋅m; ð5:14Þ which applies for either compressible or incompressible materials. To examine plane waves in detail we need to calculate Q(n) for the forms of A 0 given in Section 4.3. We now do this explicitly for the reference configuration Br . From Eqs. (4.19) and (5.9) we obtain h ! "i 2 2 Q ðnÞ = α 1 + ð1 + β1 Þðn⋅τnÞ + γ1 n⋅τ n I + β1 τ + γ1 τ + ðα 1 + α2 Þn⊗ n + ðβ1 + β2 Þðn⊗ τn + τn ⊗nÞ + β3 τn⊗τn ! " ! " + ðγ1 + γ2 Þ n⊗ τ2 n + τ2 n⊗n + γ3 τn ⊗τ2 n + τ2 n ⊗τn + γ4 τ2 n ⊗τ2 n ð5:15Þ for compressible materials, and from Eqs. (4.22), (5.9), and (5.12), we obtain h ! "i 2 2 Q ðnÞ = α 1 + ð1 + β1 Þðn⋅τnÞ + γ1 n⋅τ n I + β1 IτI + γ1 Iτ I + β3 Iτn ⊗Iτn h i 2 2 2 2 + γ3 Iτn ⊗Iτ n + Iτ n ⊗Iτn + γ4 Iτ n ⊗Iτ n ð5:16Þ for incompressible materials. Note, in particular, the explicit nonlinear dependence of the acoustic tensors on the initial stress τ (not forgetting that the α's, β's, and γ's may themselves be nonlinear in τ). Example 1. Longitudinal and transverse waves in compressible materials We consider the possibility of a pure longitudinal wave in the reference configuration (initial stress, no pre-strain), so that m = n. From Eqs. (5.10) and (5.15) we obtain ! " 2 2 a−ρv n + bτn + cτ n = 0; 1 We are grateful to a reviewer for bringing this reference to our attention. ð5:17Þ 562 M. Shams et al. / Wave Motion 48 (2011) 552–567 where the constants a, b, c are given by ! " a = 2α 1 + α 2 + ð1 + 2β1 + β2 Þðn⋅τnÞ + ðγ1 + γ2 Þ n⋅τ2 n ; ! " b = 2β1 + β2 + β3 ðn⋅τnÞ + γ3 n⋅τ2 n ; ! " 2 c = 2γ1 + γ2 + γ3 ðn⋅τnÞ + γ4 n⋅τ n : ð5:18Þ Several distinct cases arise for which a longitudinal wave can propagate provided the material parameters and the initial stresses are such that ρv2 N 0. First, for the degenerate case in which b = c = 0 we have ρv2 = a and a longitudinal wave can propagate in any direction. Second, if c = 0 and b ≠ 0 then n is an eigenvector of τ. Recalling that τi, i = 1, 2, 3, are the principal values of τ, a longitudinal wave can then propagate along the i-th principal direction, i = 1, 2, 3, with speed v given by ρv2 = a + bτi. Third, if c ≠ 0 then we refer Eq. (5.17) to the principal axes of τ, leading to ! " 2 2 a−ρv + bτi + cτi ni = 0; i = 1; 2; 3: ð5:19Þ There are now different possibilities depending on the multiplicity of the eigenvalues of τ. If τi = τ say, i = 1, 2, 3, then ρv2 = a + bτ + cτ2 and a longitudinal wave can propagate in any direction. If two of the eigenvalues are equal but distinct from the third, say τ1 = τ2 = τ ≠ τ3, then a longitudinal wave can propagate along the 3 direction with speed given by ρv2 = a + bτ3 + cτ23 or in the principal (1, 2) plane with speed given by ρv2 = a + bτ + cτ2. There is also a special case in which if b + c(τ + τ3) = 0 then a longitudinal wave can propagate in an arbitrary direction with speed ρv2 = a − cττ3. Finally, if the eigenvalues τi are distinct then, for propagation in the i-th principal direction, the wave speed, expanded in full, is given by 2 2 3 4 ρv = 2α 1 + α 2 + ð1 + 4β1 + 2β2 Þτ i + ðβ3 + 3γ1 + 2γ2 Þτi + 2γ3 τ i + γ4 τi : ð5:20Þ It is also easy to show that propagation is possible in any direction within an (i, j) principal plane if b + c(τi + τj) = 0, in which case the formula for the wave speed is ρv2 = a − cτiτj. Of course, a longitudinal wave is in general accompanied by a pair of transverse waves with mutually orthogonal polarizations. In the case where a pure longitudinal wave exists and n is aligned with the principal axis ei of τ, say, with i = 1, 2 or 3, it follows that the expression (5.15) for the acoustical tensor reduces to h i Q ðei Þ = α 1 + ð1 + β1 Þτ i + γ1 τ2i I + β1 τ + γ1 τ2 h i + α 1 + α 2 + 2ðβ1 + β2 Þτ i + ðβ3 + 2γ1 + 2γ2 Þτ2i + 2γ3 τ 3i + γ4 τ4i ei ⊗ei : ð5:21Þ Hence, for a corresponding transverse wave with polarization m and speed v, we have, on use of Eq. (5.10), h i 2 2 2 α 1 + ð1 + β1 Þτi + γ1 τi m + β1 τm + γ1 τ m = ρv m: ð5:22Þ In general it does not follow that m is also a principal axis of τ. However, in the special case in which m is also a principal axis of τ (in the plane normal to ei), let this correspond to principal initial stress τj, j ≠ i. Then, the wave speed, denoted vij, associated with m = ej is given by ! " ! " 2 2 2 ρvij = τi + α 1 + β1 τi + τ j + γ1 τ i + τ j : ð5:23Þ The difference ! " 2 2 ρ vij −vji = τi −τj ; ð5:24Þ recalls the universal relationship established by Man and Lu [16] for the general linear theory. Note, however, that in Man and Lu [16] the material was taken to be orthotropic with the principal axes of τ coinciding with the axes of orthotropy, whereas here the orthotropy is associated with τ itself and the underlying material is isotropic. The principal axes are eigenvectors of Q in each case although the values of the components Q ij differ. The situation is somewhat different if one starts by considering the existence of a pure transverse wave. There is then no general result showing that such a wave must be travelling and/or polarized along a principal axis of initial stress. We do not include here the details of this case, and refer instead to Man and Lu [16] for an example of a wave polarized along one principal direction of stress and propagating in any direction in the plane normal to the polarization. M. Shams et al. / Wave Motion 48 (2011) 552–567 563 In all the above examples, for a real wave to exist it is necessary that ρv2 N 0, which puts restrictions on the parameters and initial stress components involved. In general this requirement may be stated in terms of the strong ellipticity condition, i.e. m ⋅ ½Q ðnÞm$ N 0 for all non2zero vectors m; n: ð5:25Þ This also applies in the incompressible case, subject to the restriction m ⋅ n = 0. Example 2. Connections with the results of Man and Lu Suppose now that in Eq. (5.15) the coefficients α1, α2, β1, β2 are constants and only the terms that are linear in the initial stress τ are retained. In particular, the terms with coefficients β3, γi (i = 1, 2, 3, 4) are omitted. Then, referred to the principal axes of τ, the components of Q(n) can be written compactly as Q ii = α 1 + ðα 1 + α 2 Þn2i + ð1 + β1 Þðn⋅τnÞ + β1 τi + 2ðβ1 + β2 Þτ i n2i ; h ! "i Q ij = α 1 + α 2 + ðβ1 + β2 Þ τi + τj ni nj ; i≠j; i = 1; 2; 3; ð5:26Þ where τi (i = 1, 2, 3) are the principal values of τ. These formulas are consistent with corresponding expressions for the components of the acoustic tensor (or Christoffel tensor) obtained by Man and Lu [16] [their Eq. (18)] although this is not immediately clear since the notation differs. The correspondence can be established by noting that the tensor L used by Man and Lu [16] has components that are connected to those of A 0 by Lijkl = A0ijkl −δjl τik . Example 3. Connections with the classical theory of Biot In the classical theory of Biot [1–3] the incremental stress may be written (in the present notation) as BL or in index notation as Bpiqj uj;q , where the coefficients Bpiqj satisfy the relations Bpiqj −Bqjpi = δip τ jq −δjq τ ip ; Bpiqj = Bipqj = Bpijq ; ð5:27Þ the latter following from the existence of a strain-energy function. It was shown in [21] that the general connection between A0piqj and Bpiqj may be written in the form 1 1 1 1 A0piqj = Bpiqj − δpj τqi − δpq τ ij − δqi τpj + δij τpq + δqj τ pi : 2 2 2 2 ð5:28Þ It is worth noting that this connection does not depend on the existence of a strain-energy function and is independent of any material symmetry. For the general expression (4.19) to reduce to the Biot form for isotropic response the material parameters in Eq. (4.19) must be specialized to α 1 = μ; α 2 = λ; β1 = −1 = 2; β2 = 1; ð5:29Þ where μ and λ are the Lamé moduli, and the terms which are of order higher than 1 in τ must be neglected. This corresponds to Bpiqj having the form ! " Bpiqj = μ δij δpq + δqi δpj + λδpi δqj + δpi τ qj ; ð5:30Þ which satisfies the conditions (5.27). Clearly, the formula (5.30) relies on the special identifications (Eq.(5.29)), and is very much a specialization of the more general theory discussed here. In effect, if one was to linearize the general expression of the elastic moduli (4.19) with respect to the initial stress, then the expansions α 1 = μ + α̂ 1 τ; α 2 = λ + α̂ 2 τ; β1 = β̂1 ; β2 = β̂2 ; ð5:31Þ would be required, where “τ” is a term of first order in τ, and the scalars with a hat are constants, which cannot be determined a priori and must be measured experimentally. In conclusion, six material constants (λ, μ, α̂1 , α̂2 , β̂1 , β̂2 ) are required to describe the isotropic response of a compressible material with a small initial stress, not just the two Lamé coefficients as Biot implied. We refer to Man [15] for a discussion of this point in relation to Hartig's law and a proof that four additional constants are needed to describe the elastic response of currently known real materials. Example 4. Second-order elasticity Here we consider an isotropic elastic solid without initial stress and its specialization to second order in the components of the Green strain tensor E = 12 ðC−IÞ, where C is again the right Cauchy–Green deformation tensor. An appropriate form of the elastic 564 M. Shams et al. / Wave Motion 48 (2011) 552–567 strain-energy function, which is correct to second order, is that due to Murnaghan [17]. When expressed in terms of the invariants of C given by Eq. (4.1) the Murnaghan energy function has the form W= ! " " λ μ! 2 l m n 2 3 2 I1 −2I1 −2I2 + 3 + ðI1 −3Þ + ðI1 −3Þ + ðI1 −3Þ I1 −3I2 + ðI1 −I2 + I3 −1Þ; 8 4 24 12 8 ð5:32Þ where λ and μ are the classical Lamé constants of linear elasticity and l, m, n are the second-order constants of Murnaghan. For this energy function we have W22 = W13 = W23 = W33 = 0 and since we are not including initial stress the expression (4.17) reduces to h i ! " JA0piqj = 2ðW1 + I1 W2 ÞBpq δij + 2W2 2Bpi Bqj −Biq Bjp −Bpr Brq δij −Bpq Bij + 2I3 W3 2δip δjq −δiq δjp ! " + 4W11 Bip Bjq + 4W12 2I1 Bip Bjq −Bip Bjr Brq −Bjq Bir Brp ; ð5:33Þ and the remaining coefficients W1, W2, W3, W11 and W12 are simply obtained from Eq. (5.32). Then, with this specialization, the wave speed v is given by 2 ρv = A0piqj np nq mi mi : ð5:34Þ In working with second-order elasticity the corrections to the classical longitudinal and transverse wave speeds are obtained to first order in E. We have C = I + 2E and I1 = 3 + 2E, exactly, which we use together with the linear approximations I2 ≃ 3 + 4E, I3 ≃ 1 + 2E, B ≃ C = I + 2E, and we also note that ρ ≃ ρr(1 − E), where E = tr E. To the first order in E we then obtain 1 1 1 1 1 n + ðλ + 2μ + 2mÞE; W2 = − μ− n− mE; 8 2 2 8 2 1 1 1 = ðλ + 2μ + 4mÞ + ðl + 2mÞE; W12 = − m: 4 2 4 W1 = μ + W11 W3 = 1 n; 8 ð5:35Þ After some manipulations the above approximations enable us to obtain, to the first order in E, ! " ! " JA0piqj = μ δij δpq + δiq δjp + λδip δjq + 2μ 2δij Epq + δpq Eij + δiq Ejp + δjp Eiq ! " + λ Eδij δpq + 2δip Ejq + 2δjq Eip + 2lEδip δjq h ! ! "i " + m E δij δpq + δiq δjp −2δip δjq + 2 δip Ejq + δjq Eip + ð5:36Þ ! "i 1 h n δij Epq + δpq Eij + δiq Ejp + δjp Eiq −2δip Ejq −2δjq Eip −E δij δpq + δiq δjp −2δip δjq ; 2 and hence, for any m, n pair satisfying Eq. (5.10), the wave speed is obtained via 2 2 ρr v = μ + ðμ + λÞðm⋅nÞ + 2μ ½2ðn⋅EnÞ + ðm⋅EmÞ + 2ðm⋅nÞðm⋅EnÞ$ n h i o + λ½E + 4ðm⋅nÞðm⋅EnÞ$ + 2lEðm⋅nÞ2 + m E 1−ðm⋅nÞ2 + 4ðm⋅nÞðm⋅EnÞ h io 1 n 2 : + n ðn⋅EnÞ + ðm⋅EmÞ−2ðm⋅nÞðm⋅EnÞ−E 1−ðm⋅nÞ 2 ð5:37Þ For a longitudinal wave with m = n this reduces to 2 ρr v = λ + 2μ + ðλ + 2lÞE + ð10μ + 4λ + 4mÞðn⋅EnÞ; ð5:38Þ and for a transverse wave with m ⋅ n = 0 and m × n = l, 1 2 ρr v = μ + 2μ ðn⋅EnÞ + ðλ + 2μ + mÞE− ð4μ + nÞðl⋅ElÞ; 2 ð5:39Þ where we have made use of the connection l ⋅ El + m ⋅ Em + n ⋅ En = E. When n is specialized to the axis e1 and m is taken to be e2 and e3 in turn, where e1, e2 and e3 are principal axes of strain, then these results agree with those obtained by Hughes and Kelly [13] [their Eq. (11)]. It is straightforward to show that a longitudinal wave actually exists if either n is an eigenvector of E or the elastic constants satisfy λ + 2μ + m = 0. For the existence of a transverse wave there are more options. First, in the special case when the elastic constants are such that λ + 2μ + m = 0 and 4μ + n = 0, a transverse wave exists for any direction of propagation n. Second, if 4(λ + μ + m) − n = 0 and 4μ + n ≠ 0 then a transverse wave exists only if m is an eigenvector of E. Third, if 4μ + n = 0 and M. Shams et al. / Wave Motion 48 (2011) 552–567 565 λ + 2μ + m ≠ 0 then a transverse wave exists for any n for which Em lies in the plane normal to n. Finally, if none of these very special cases apply then the existence of a transverse wave requires that m be an eigenvector of E. Note that the case of incompressible materials is treated elsewhere; see Destrade et al. [7] and references therein. Example 5. A simple nonlinear model The above examples have involved different specializations that consider the strain and/or the initial stress to be small. In this final example we consider a finite deformation from a configuration that is subject to an initial stress that is nonlinear. For this purpose it suffices to adopt a simple prototype incompressible form of strain-energy function, which is taken to be W= 1 1 1 2 μ ðI −3Þ + μ ðI6 −tr τÞ + ðI6 −tr τÞ; 2 1 2 2 ð5:40Þ where the first term represents the classical (isotropic) strain-energy function of a neo-Hookean material with shear modulus μ, while the terms in I6 capture the effect of initial stress and satisfy Eq. (4.15)2. The constant μ has dimensions of [stress]− 1. We recall that I6 = tr(Cτ), which may also be written as tr Σ. From Eq. (4.11) the Cauchy stress is then obtained as σ = μB−pI + ½1 + 2μ ðI6 −tr τÞ$Σ; ð5:41Þ and from Eq. (4.18) the components of A 0 reduce to A0piqj = μBpq δij + ½1 + 2μ ðI6 −tr τÞ$Σpq δij + 4μΣpi Σqj : ð5:42Þ The corresponding components of the acoustic tensor are obtained from Eq. (5.9), and it is convenient to write it in tensor notation as Q ðnÞ = αI + βΣn⊗ Σn; ð5:43Þ where for compactness of representation we have introduced the notations α = μ ðn⋅BnÞ + ½1 + 2μ ðI6 −tr τÞ$ðn⋅ΣnÞ; β = 4μ: ð5:44Þ Since we are considering an incompressible material we form the projection of Q(n) on to the plane normal to n according to Eq. (5.12). This is Q ðnÞ = αI + βIΣn⊗ IΣn: ð5:45Þ Next we solve Eq. (5.13)2 and find that the two solutions for ρv2 are 2 ρv = α; 2 2 ρv = α + βðn⋅ΣmÞ ; ð5:46Þ and Eq. (5.11) becomes 2 αm + βðn⋅ΣmÞIΣn = ρv m; m⋅n = 0: ð5:47Þ Assuming that β ≠ 0, for the solution ρv2 = α this requires that either IΣn = 0 or n ⋅ Σm = 0. If IΣn = 0 then n is an eigenvector of Σ and both transverse waves have the same speed v, given by ρv2 = α. If, on the other hand, n ⋅ Σm = 0 and IΣn≠0 then the wave speed corresponding to polarization m is given by ρv2 = α, while for the second transverse wave, with polarization m′ say, the wave speed is given by ρv2 = α + β(n ⋅ Σm′)2. As a simple illustration of the above results we now consider the initial stress to be uniaxial, of the form τ = τ1e1 ⊗ e1, where e1 is a principal axis of B corresponding to principal stretch λ1. Then, Σ = τ1λ21e1 ⊗ e1. It follows that n ⋅ Σm = 0 and ! " 2 2 2 2 2 ρv = ðμ + τ 1 Þλ1 + 2μτ1 λ1 λ1 −1 : ð5:48Þ It now suffices to measure the wave speed for three different values of λ1 to evaluate the material constants μ and μ, and the initial stress τ1. If the deformation is achieved by applying a uniaxial stress along e1 then the associated principal Cauchy stress σ1 is obtained from Eq. (5.41) as ! " 2 2 2 2 σ 1 = ðμ + τ 1 Þλ1 −p + 2μτ1 λ1 λ1 −1 : ð5:49Þ 566 M. Shams et al. / Wave Motion 48 (2011) 552–567 1 By symmetry, incompressibility and vanishing of the lateral stress it also follows from Eq. (5.41) that p = μλ− 1 . If this is used in 1 . the above then the expression for the wave speed can be written simply as ρv2 = σ1 + µ λ− 1 Acknowledgments This work is supported by a Senior Marie Curie Fellowship awarded by the Seventh Framework Programme of the European Commission to the second author, and by an E.T.S. Walton Award given to the third author by the Science Foundation Ireland. This material is partly based upon works supported by the Science Foundation Ireland under Grant No. SFI 08/W.1/B2580. Appendix A A1. First derivatives of the invariants Here we list the first derivatives of the invariants I1, I2, I3, I6,…, I9 with respect to the deformation gradient. These are ∂I1 T = 2F ; ∂F ∂I6 T = 2τF ; ∂F ∂I8 2 T = 2τ F ; ∂F ! " ∂I2 T T T = 2 I1 F −F FF ; ∂F ∂I7 T T T T = 2τF FF + 2F FτF ; ∂F ∂I9 2 T T T 2 T = 2τ F FF + 2F Fτ F ; ∂F ∂I3 −1 = 2I3 F ; ∂F from which we obtain the expressions ∂I1 = 2B; ∂F ∂I6 = 2Σ; F ∂F ∂I −1 F 8 = 2ΣB Σ; ∂F F ! " ∂I2 2 = 2 I1 B−B ; ∂F ∂I F 7 = 2ΣB + 2BΣ; ∂F ∂I9 −1 −1 F = 2ΣB ΣB + 2BΣB Σ; ∂F F F ∂I3 = 2I3 I; ∂F T required in the expansion of Cauchy stress, where Σ = FτFT and B = FF . In the reference configuration Br these reduce to ∂I1 ∂I2 ∂I3 ∂I6 ∂I7 ∂I8 ∂I9 2 2 = 2I; = 4I; = 2I; = 2τ; = 4τ; = 2τ ; = 4τ : ∂F ∂F ∂F ∂F ∂F ∂F ∂F A2. Second derivatives of the invariants Here we present the second derivatives of the invariants in index notation, omitting the details of the calculations. In the form required for the calculation of the components of A 0 we have Fpα Fqβ ∂2 I1 = 2Bpq δij ; ∂Fiα ∂Fjβ Fpα Fqβ ∂ I2 = 2I1 Bpq δij + 4Bpi Bqj −2Biq Bjp −2Bpr Brq δij −2Bpq Bij ; ∂Fiα ∂Fjβ Fpα Fqβ ∂ I3 = 4I3 δip δjq −2I3 δiq δjp ; ∂Fiα ∂Fjβ Fpα Fqβ ∂ I6 = 2Σpq δij ; ∂Fiα ∂Fjβ Fpα Fqβ ∂ I7 = 2Σpq Bij + 2Σpr Brq δij + 2Bpr Σrq δij + 2Σij Bpq + 2Σpj Biq + 2Σqi Bjp : ∂Fiα ∂Fjβ 2 2 2 2 The second derivatives of I8 and I9 can be deduced from those of I6 and I7, respectively, by replacing τ by τ2 , or Σ by ΣB−1 Σ. In the reference configuration Br these reduce to ∂2 I 1 = 2δpq δij ; ∂Fip ∂Fjq ∂2 I2 = 2δpq δij + 4δip δjq −2δiq δjp ; ∂Fip ∂Fjq ∂2 I 3 = 4δip δjq −2δiq δjp ; ∂Fip ∂Fjq ∂2 I 6 = 2τ pq δij ; ∂Fip ∂Fjq ∂2 I 7 = 6τ pq δij + 2τ ij δpq + 2τ jp δiq + 2τ iq δjp : ∂Fip ∂Fjq M. 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