Propagation of waves in an incompressible transversely isotropic elastic solid with initial stress: Biot revisited

Journal of Mechanics of Materials and Structures PROPAGATION OF WAVES IN AN INCOMPRESSIBLE TRANSVERSELY ISOTROPIC ELASTIC SOLID WITH INITIAL STRESS: BIOT REVISITED Ray W. Ogden and Baljeet Singh Volume 6, No. 1-4 January–June 2011 mathematical sciences publishers JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES Vol. 6, No. 1-4, 2011 msp PROPAGATION OF WAVES IN AN INCOMPRESSIBLE TRANSVERSELY ISOTROPIC ELASTIC SOLID WITH INITIAL STRESS: BIOT REVISITED R AY W. O GDEN AND BALJEET S INGH In this paper, the general constitutive equation for a transversely isotropic hyperelastic solid in the presence of initial stress is derived, based on the theory of invariants. In the general finite deformation case for a compressible material this requires 18 invariants (17 for an incompressible material). The equations governing infinitesimal motions superimposed on a finite deformation are then used in conjunction with the constitutive law to examine the propagation of both homogeneous plane waves and, with the restriction to two dimensions, Rayleigh surface waves. For this purpose we consider incompressible materials and a restricted set of invariants that is sufficient to capture both the effects of initial stress and transverse isotropy. Moreover, the equations are specialized to the undeformed configuration in order to compare with the classical formulation of Biot. One feature of the general theory is that the speeds of homogeneous plane waves and surface waves depend nonlinearly on the initial stress, in contrast to the situation of the more specialized isotropic and orthotropic theories of Biot. The speeds of (homogeneous plane) shear waves and Rayleigh waves in an incompressible material are obtained and the significant differences from Biot’s results for both isotropic and transversely isotropic materials are highlighted with calculations based on a specific form of strain-energy function. 1. Introduction Initial stresses in solids have an important influence on the mechanical response of the material from an initially stressed configuration. Applications range from geophysics to the components of engineering structures and the behavior of soft biological tissues. The term initial stresses embraces situations in which the stress is accompanied by finite deformation from an unstressed configuration, in which case the term pre-stresses is commonly used, as, for example, in [Ieşan 1989], 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. In this latter case the initial stress is referred to as residual stress according to the definition of [Hoger 1985]. In the present paper we are concerned with the effect of initial stress on the propagation of small amplitude (linearized) elastic waves. A static theory of initial stress was developed long ago by Biot [1939]. He then extended it to wave propagation problems [1940]; this work is summarized in [Biot 1965]. In [Biot 1940] he states: “No assumption is made on how the initial state of stress is produced” — he requires only that it satisfy the equilibrium equations. Biot’s theory has since formed the basis of many contributions to the literature, particularly in the geophysical context; see, for example, [Tolstoy Singh’s work was supported by a Royal Society International Travel Grant. This is gratefully acknowledged. Keywords: hyperelasticity, initial stress, residual stress, transverse isotropy, invariants, plane waves, surface waves, Biot’s theory. 453 454 RAY W. OGDEN AND BALJEET SINGH 1982] and the more recent [Dey and De 1999; Sharma and Garg 2006]. The latter was concerned with an initially stressed anisotropic material and further references can be found therein. In the context of modern continuum theory, however, Biot’s formulation of the equations is not straightforward, and part of the purpose of the present work is to show how Biot’s formulation fits into a more general and more transparent framework. Surface waves in elastic solids were first studied by Lord Rayleigh [1885] for an isotropic elastic solid. The extension of surface wave analysis and other wave propagation problems to anisotropic elastic materials has been the subject of many studies; see, for example, [Musgrave 1959; Anderson 1961; Stoneley 1963; Chadwick and Smith 1977; Royer and Dieulesaint 1984; Barnett and Lothe 1985; Mozhaev 1995; Nair and Sotiropoulos 1997; 1999; Destrade 2001a; 2001b; Destrade et al. 2002; Ting 2002a; 2002c; 2002b; Destrade 2003; Ogden and Vinh 2004]. For problems involving surface waves in a finitely deformed pre-stressed elastic solid (strain-induced anisotropy) we refer to [Hayes and Rivlin 1961; Flavin 1963; Chadwick and Jarvis 1979; Dowaikh and Ogden 1990; 1991; Norris and Sinha 1995 (concerning a solid/fluid interface); Chadwick 1997; Prikazchikov and Rogerson 2004 (concerning prestressed transversely isotropic solids); Destrade et al. 2005; Edmondson and Fu 2009]; see also [Song and Fu 2007]. As representatives of other works concerning waves in initially stressed elastic solids we cite [Norris 1983] on plane waves, the review [Guz 2002] and the analysis [Akbarov and Guz 2004] of waves in circular cylinders. Here we shall study the effect of initial stress on the propagation of surface waves based on a general formulation of the constitutive law of an elastic material that would be transversely isotropic in the absence of initial stress. In Section 2, the equations governing small amplitude waves in a deformed transversely isotropic elastic solid with initial stress are derived, both for compressible and incompressible materials, the transverse isotropy being associated with a preferred direction in the initially stressed reference configuration. The constitutive law of the material is based on a strain-energy function (defined per unit reference volume) which depends on the combined invariants of the right Cauchy–Green deformation tensor, the initial stress tensor and the preferred direction. For a compressible material there are 18 such independent invariants in the general three-dimensional case, a number which reduces to 17 for an incompressible material. Expressions for the Cauchy stress and nominal stress tensors and the elasticity tensor are given in general forms but, because the large number of invariants makes the theory unwieldy in general, their forms are made explicit only for a restricted number of invariants, and attention is then confined largely to incompressible materials with seven invariants. In Section 3, the equations of motion are specialized in order to study the effect of initial stress on the wave speed of homogeneous plane waves. It is noted, in particular, that the wave speed depends in a nonlinear fashion on the initial stress. In order to make contact with Biot’s theory and to see how it sits within the general framework considered here, we give, in Appendix B, a derivation of Biot’s equations, their connection with the equations herein, and a formula for the relation between the components of the elasticity tensor used here (which depend nonlinearly on the initial stress) and the components of Biot’s elasticity tensor. In Section 4, the theory is specialized to two-dimensional motions (for incompressible materials) and then applied, in Section 5, to the study of Rayleigh waves in a half-space subject to initial stress parallel to its boundary with the preferred direction of transverse isotropy either parallel or normal to the boundary. The secular equation is derived and then specialized to give corresponding results for Biot’s isotropic and orthotropic theories. The final section, Section 6, provides numerical results that PROPAGATION OF WAVES IN AN INCOMPRESSIBLE TRANSVERSELY ISOTROPIC ELASTIC SOLID 455 show the significant differences between the predictions of the general theory and of Biot’s theory, and some concluding discussion is contained therein. 2. Equations of motion We consider an elastic body whose initial geometry defines a reference configuration, which we denote by Br . In this configuration the body is in equilibrium and may in general be subject to a stress distribution, and we denote the Cauchy stress in Br by T . If there is a body force br per unit mass acting then the equation of equilibrium is Div T C r br D 0; (1) where r is the mass density of the material in Br and Div is the divergence operator in Br , i.e., with respect to position vector X in Br . If the traction on the boundary @Br of Br vanishes pointwise then T is referred to as a residual stress, and it is necessarily non-uniform [Hoger 1985; Ogden 2003]. If the traction is not zero then we refer to T as an initial stress or pre-stress, and in general this may be accompanied by some prior deformation required to reach the configuration Br from an unstressed state. Here we shall not be concerned with how the initial stress is produced. A motion of the body from Br may be described by a function  so that the current position x of the material point initially at X is given by x D .X ; t /, where t is time. The deformation gradient tensor, denoted F , is given by F D Grad .X ; t /, where Grad is the gradient operator in Br . Let B be the configuration occupied by the body at time t . The constitutive law of an elastic material may be described in terms of a strain-energy function, which is a function of F and defined per unit volume in Br . We denote this by W .F /, but note that in general W depends also on the initial stress T and on implicit material symmetries, which are suppressed for the present. Let S denote the nominal stress tensor in the configuration B. Then, for a material not subject to any internal mechanical constraints S is given by @W : @F (2) det F D 1 (3) SD For an incompressible material the constraint is enforced and (2) is modified to @W pF 1 ; @F where p is a Lagrange multiplier associated with the constraint. The motion  is governed by the equation SD Div S C r b D r x;t t ; (4) (5) where b is the body force acting in the current configuration, which may in general be different from 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 is the particle velocity and x;t t the acceleration. 456 RAY W. OGDEN AND BALJEET SINGH Although this is not strictly necessary, we now assume, for simplicity, that the body force is uniform and independent of the deformation. Then, b D br and we get from (1) and (5) Div.S T / D r x;t t : (6) Now consider a finitely deformed equilibrium configuration B0 defined by 0 .X / and let F0 be the deformation gradient in this configuration and S0 the corresponding nominal stress. Then, Div.S0 T / D 0. Next, we consider an incremental motion from this latter configuration with displacement xP D .X ; t / 0 .X /, and corresponding increment FP D Grad xP D F F0 in the deformation gradient. Let SP D S S0 denote the increment in the nominal stress. Then, Div SP D r xP ;t t : (7) For a compressible material the linearized form of SP is SP D AFP ; SP ˛i D A˛iˇj FPjˇ ; (8) while for an incompressible material SP D AFP C pF 1 FP F 1 pF P 1 ; (9) where pP is the increment in p and A is the elasticity tensor, which, for either a compressible or an incompressible material, is defined by AD @2 W ; @F @F A˛iˇj D @2 W D Aˇj˛i ; @Fi˛ @Fjˇ (10) evaluated in the configuration B0 . The convention that Greek indices refer to the configuration Br and Roman indices to B0 is adopted here. The linearized incompressibility condition is tr.FP F0 1 / D 0: (11) For details of the background on the theory of incremental deformations superimposed on a finite deformation we refer to [Ogden 1984; 2007], for example. It is convenient to work with B0 as the reference configuration, which requires that all quantities are updated, i.e., pushed forward, from Br to B0 and incremental quantities are treated as functions of x and t. In particular, we define the Eulerian form of the displacement vector by u.x; t / D x.X P ; t / via the connection x D 0 .X /. The updated forms of the incremental constitutive laws (8) and (9) are, respectively, SP 0 D A0 L; SP 0pi D A0piqj Lj q ; (12) and SP 0 D A0 L C pL pI; P (13) where L D grad u is the displacement gradient, I is the identity tensor and a subscript 0 indicates a pushed forward quantity. In particular, J0 A0piqj D F0p˛ F0qˇ A˛iˇj ; (14) PROPAGATION OF WAVES IN AN INCOMPRESSIBLE TRANSVERSELY ISOTROPIC ELASTIC SOLID 457 where J0 D det F0 D r =0 and 0 is the density in B0 . For an incompressible material J0 D 1 and 0 D r . The (linearized) incremental form of the incompressibility constraint (11) is then expressed as tr L  div u D 0: (15) For a compressible material, the (Cartesian) component form of the equation of motion is .A0piqj uj ;q /;p D 0 ui;t t ; (16) and for an incompressible material .A0piqj uj ;q /;p pP;i C p;j uj ;i D r ui;t t ; with ui;i D 0: (17) Note that as well as possessing the major symmetry A0piqj D A0qjpi induced by (10), A has the property A0piqj C ıjp 0iq D A0ipqj C ıij 0pq (18) for a compressible material, and A0piqj C ıjp .0iq C pıiq / D A0ipqj C ıij .0pq C pıpq / (19) for an incompressible material, where 0ij are the components of the Cauchy stress tensor 0 in B0 . These are easily established by considering the symmetry of Cauchy stress expressed in the form FS D S T F T , taking the increment of this and then updating the reference configuration to B0 to obtain SP 0 C L0 D SP 0T C 0 LT : (20) Suppose now that the material is homogeneous. This requires, in particular, that the initial stress T is uniform. We recall, however, that a residual stress cannot be uniform [Hoger 1985; Ogden 2003], so the following analysis does not apply if the initial stress is a residual stress. Suppose further that 0 .X / is a homogeneous deformation. Then, the configuration B0 is uniform and hence A, A0 and p are constant, and the equations of motion (16) and (17) reduce to A0piqj uj ;pq D 0 ui;t t (21) and A0piqj uj ;pq pP;i D r ui;t t ; ui;i D 0; (22) respectively. In general A, and hence A0 , depends on the deformation through F , on the initial stress T and on any material symmetry present in the configuration Br . Here we consider a transversely isotropic material with preferred direction M in Br , where M is a unit vector. To make the dependence of A (and A0 ) on these quantities explicit we consider the scalar invariants of the tensors involved. 2.1. Invariant formulation. By objectivity, the dependence of the strain-energy function W on F is through the right Cauchy–Green tensor C , which is defined by C D F T F , and we therefore consider W to depend on the invariants of the three tensors C , T and M ˝ M since the material properties are assumed to be independent of the sense of M . 458 RAY W. OGDEN AND BALJEET SINGH The invariants of C most commonly used are the principal invariants, defined by I1 D tr C ; I2 D 21 Œ.tr C /2 tr.C 2 /; I3 D det C : (23) The (anisotropic) invariants associated with M and C are usually taken as I5 D M
.C 2 M /: I4 D M
.CM /; (24) The notation I1 ; : : : ; I5 is fairly standard for these invariants; see, for example, [Merodio and Ogden 2002; 2003]. In the reference configuration Br these reduce to I1 D I2 D 3, I3 D I4 D I5 D 1. A set of independent invariants of T that do not involve M may be taken as tr T ; tr.T 2 /; tr.T 3 /; tr.T C 2 /; tr.T C /; tr.T 2 C /; tr.T 2 C 2 /; (25) invariants of T independent of C as M
.TM /; M
.T 2 M /; (26) and invariants depending on C ; M and T as M
.T CM /; M
.T C 2 M /; M
.T 2 CM /; M
.T 2 C 2 M /: (27) These are the only independent invariants, 18 in total. For an incompressible material we have I3 D 1 and hence there are 17 independent invariants in this case. In the reference configuration Br the fourth, fifth, sixth and seventh invariants in (25) and the invariants (27) reduce to the first two in (25) and the two in (26). For full discussion of the relevant background on invariants of tensors we refer to [Adkins 1960; Spencer 1971; Zheng 1994]. A set of invariants equivalent to the above has been used by [Hoger 1993a; 1996]. For related work concerned with the constitutive equations and material symmetry for a residually stressed elastic material we refer to [Coleman and Noll 1964; Hoger 1986; 1993b; Man and Lu 1987; Johnson and Hoger 1993; Man 1998; Saravanan 2008; Tanuma and Man 2008]. We have not for the moment defined particular notation for the invariants (25)–(27). In the general case suppose there are N invariants, which we denote by Ii ; i D 1; 2; : : : ; N . Then, the expressions for the stress and elasticity tensors require the calculation of N X @W @Ii D Wi @F @F (28) iD1 and N N N X XX @Ij @2 Ii @2 W @Ii D C ˝ ; Wi Wij @F @F @F @F @F @F iD1 (29) iD1 j D1 where we have used the shorthand notations Wi D @W =@Ii ; Wij D @2 W =@Ii @Ij ; i; j D 1; 2; : : : ; N . Such a large number of invariants is impractical for applications, so for simplicity we restrict attention to incompressible materials and to the following invariants, which capture the main features of the combined anisotropy and initial stress. In particular, we omit invariants that are nonlinear in T . Thus, PROPAGATION OF WAVES IN AN INCOMPRESSIBLE TRANSVERSELY ISOTROPIC ELASTIC SOLID 459 we consider I1 D tr C ; I4 D M
.CM /; I5 D M
.C 2 M /; I6 D tr.T C /; I7 D tr.T C 2 /; I8 D M
.T CM /; (30) I9 D M
.T C 2 M /, (31) which identify the invariants I6 ; I7 ; I8 ; I9 , while the standard notation I1 ; I4 ; I5 is retained and, since we are considering incompressible materials, I3  1. We may also include tr T and M
.TM / in the functional dependence of W since they do not depend on C and hence their derivatives with respect to F do not contribute to the stress or elasticity tensors. In the following we give explicit expressions for the stress tensors and the elasticity tensor based on this restricted set of invariants. 2.2. Stress tensors. The strain-energy function W is now taken to depend on the seven deformationdependent invariants I1 ; I4 ; : : : ; I9 together (possibly) with tr T and M
.TM /. For the considered incompressible material we have SD @W @F pF 1 D X 1i9 i¤2;3 Wi @Ii @F pF 1 ; (32) and the corresponding Cauchy stress is  DF @W @F pI D X Wi F 1i9 i¤2;3 @Ii @F pI; (33) where Wi D @W =@Ii ; i D 1; 4; : : : ; 9. The required expressions for @Ii =@F are listed for convenience in Appendix A in component form. These enable the Cauchy stress to be expanded as  D pI C 2W1 B C 2W4 m ˝ m C 2W5 .m ˝ Bm C Bm ˝ m/ C 2W6 † C 2W7 .B† C †B / C W8 .F TM ˝ m C m ˝ F TM / C W9 .F TM ˝ Bm C Bm ˝ F TM C FC TM ˝ m C m ˝ FC TM /; (34) where m D FM , † D F TF T and B D FF T is the left Cauchy–Green deformation tensor. In the reference configuration Br the Cauchy stress must be equal to the initial stress. Thus, when (34) is evaluated in Br , we obtain T D .2W1 p/I C 2.W4 C 2W5 /M ˝ M C 2.W6 C 2W7 /T C .W8 C 2W9 /.TM ˝ M C M ˝ TM /; (35) where W1 ; W4 ; : : : ; W9 are evaluated in the reference configuration, where F D I, and may in general depend on tr T and M
.TM /. For consistency it is therefore appropriate to set p D 2W1 ; W4 C 2W5 D 0; W6 C 2W7 D 21 ; W8 C 2W9 D 0; (36) in the reference configuration. Indeed, if (35) holds for all possible T then these conditions necessarily follow. 460 RAY W. OGDEN AND BALJEET SINGH 2.3. The elasticity tensor. Next, we note that the elasticity tensor A is given by AD X X @2 Ii @2 W D C Wi @F @F @F @F 1i9 i¤2;3 X 1i9 1j 9 i¤2;3 j ¤2;3 Wij @Ij @Ii ˝ : @F @F (37) This requires expressions for the second derivatives of the invariants. In component form these are given in Appendix A. Since the resulting formula for the components of A is quite long we do not give it here. Instead, we give its specialization to the situation in which there is no finite deformation and B0 coincides with Br . The subscript 0 on A0 may now be omitted, and taking account of the conditions (36) prevailing in the reference configuration the components of A in the reference configuration can be arranged in the (still fairly lengthy) form Apiqj D 2W1 ıij ıpq C 2W5 .ıij Mp Mq C ıpq Mi Mj C ıiq Mj Mp C ıjp Mi Mq / C ıij Tpq C 2W7 .ıij Tpq C ıpq Tij C ıiq Tjp C ıjp Tiq / C W9 Œıij .Mp Tqr C Mq Tpr / C ıpq .Mi Tj r C Mj Ti r / C ıiq .Mj Tpr C Mp Tj r / C ıjp .Mi Tqr C Mq Ti r /Mr C 4W11 ıip ıj q C 4.W44 C 4W45 C 4W55 /Mi Mj Mp Mq C 4.W14 C 2W15 /.ıip Mj Mq C ıj q Mi Mp / C 4.W16 C 2W17 /.ıip Tj q C ıj q Tip / C 2.W18 C 2W19 /Œıip .Mj Tqr C Mq Tj r / C ıj q .Mi Tpr C Mp Ti r /Mr C 4.W46 C 2W47 C 2W56 C 4W57 /.Mi Mp Tj q C Mj Mq Tip / C 2.W48 C 2W49 C 2W58 C 4W59 /.Ti r Mj Mp Mq C Tj r Mi Mp Mq C Tpr Mi Mj Mq C Tqr Mi Mj Mp /Mr C 4.W66 C 4W67 C 4W77 /Tip Tj q C 2.W68 C 2W69 C 2W78 C 4W79 /ŒTip .Tqr Mj C Tj r Mq / C Tj q .Tpr Mi C Ti r Mp /Mr C .W88 C 4W89 C 4W99 /.Mi Tpr C Mp Ti r /.Mj Tqs C Mq Tj s /Mr Ms ; (38) all the derivatives of W being evaluated in Br . With the restricted set of invariants adopted these depend in general on tr T and M
.TM /. Note that when (38) is substituted into the equation of motion all the terms involving ıjp or ıj q disappear by virtue of the incompressibility condition. We now consider three special cases that will be used subsequently: (1) the underlying material is isotropic; (2) TM D 0; (3) T D T M ˝ M . Case 1: Isotropy. If there is no preferred direction in Br and in the absence of initial stress the material is isotropic then in the presence of initial stress (38) reduces simply to Apiqj D 2W1 ıij ıpq C ıij Tpq C 2W7 .ıij Tpq C ıpq Tij C ıiq Tjp C ıjp Tiq / C 4W11 ıip ıj q C 4.W16 C 2W17 /.ıip Tj q C ıj q Tip / C 4.W66 C 4W67 C 4W77 /Tip Tj q I (39) that is, all terms in which W has a subscript 4, 5, 8 or 9 are omitted. Case 2: TM D 0. In this case all the terms in (38) in which W has a subscript 8 or 9 vanish and (38) reduces to PROPAGATION OF WAVES IN AN INCOMPRESSIBLE TRANSVERSELY ISOTROPIC ELASTIC SOLID 461 Apiqj D 2W1 ıij ıpq C 2W5 .ıij Mp Mq C ıpq Mi Mj C ıiq Mj Mp C ıjp Mi Mq / C ıij Tpq C 2W7 .ıij Tpq C ıpq Tij C ıiq Tjp C ıjp Tiq / C 4W11 ıip ıj q C 4.W44 C 4W45 C 4W55 /Mi Mj Mp Mq C 4.W14 C 2W15 /.ıip Mj Mq C ıj q Mi Mp / C 4.W16 C 2W17 /.ıip Tj q C ıj q Tip / C 4.W46 C 2W47 C 2W56 C 4W57 /.Mi Mp Tj q C Mj Mq Tip / C 4.W66 C 4W67 C 4W77 /Tip Tj q : We note, in particular, the connections Aijj i D Aj i ij D Aij ij Aj ijj D Aijjj .2W1 C Ti i / D Aj ij i Tij ; i ¤ j; .2W1 C Tjj /; Aijj k D Aij kj i ¤ j; Tik ; (40) (41) where i; j ; k are distinct. Case 3: T D T M ˝ M . In this case the components of A have a relatively simple structure and can be written compactly as Apiqj D 2W1 ıij ıpq C 4W11 ıip ıj q C A.ıij Mp Mq C ıpq Mi Mj C ıiq Mj Mp C ıjp Mi Mq / CB.ıip Mj Mq C ıj q Mi Mp / C CMi Mj Mp Mq C T ıij Mp Mq ; (42) where A D 2ŒW5 CT .W7 CW9 /; (43) B D 4ŒW14 C2W15 CT .W16 C2W17 /CT 2 .W18 C2W19 /; (44) C D 4ŒW44 C4W45 C4W55 CT .W46 C2W47 CW48 C2W49 C2W56 C 4W57 C2W58 C4W59 / CT 2 .W66 C4W67 C4W77 CW68 C2W69 C2W78 C4W79 CW88 C4W89 C4W99 /: (45) It is noteworthy that in each of the three cases the components Apiqj are quadratic in the components of the initial stress even though we have not included in the model invariants that are nonlinear in the initial stress. 3. Homogeneous plane waves With the focus on incompressible materials we now apply the equation of motion and the incompressibility condition in (22) to the analysis of homogeneous plane waves. In particular, we consider the incremental displacement u and Lagrange multiplier pP to have the forms u D f .n
x vt/d; pP D g.n
x vt /; (46) where d is a constant unit (polarization) vector, the unit vector n is the direction of propagation of the plane wave, v is the wave speed, f is a function that need not be made explicit, but is subject to the restriction f 00 ¤ 0, and g is a function related to f . A prime on f or g indicates differentiation with respect to its argument. Substitution of (46) into (22) then yields ŒQ.n/d v 2 df 00 g 0 n D 0; d
n D 0; (47) 462 RAY W. OGDEN AND BALJEET SINGH where the (symmetric) acoustic tensor Q.n/ is defined by Qij .n/ D Apiqj np nq : (48) On taking the dot product of (47)1 with n we obtain g 0 D ŒQ.n/d
nf 00 , and on substituting back into (47)1 and eliminating the factor f 00 we may arrange the resulting propagation condition in the form n ˝ Q.n/nd D v 2 d; ŒQ.n/ (49) or equivalently as N Q.n/d D v 2 d; (50) N where Q.n/ is the projection of Q.n/ on to the plane normal to n defined by N N N Q.n/ D IQ.n/ I; (51) where IN D I n ˝ n. The symmetrization (51) was originally derived in [Scott and Hayes 1985]. Thus, N is symmetric and the two-dimensional eigenvalue problem (50) (in the plane normal to n) therefore Q.n/ has two real solutions for v 2 . The wave speeds are real if the two eigenvalues are positive, and this is guaranteed if the strong ellipticity condition holds. This is expressed as N ŒQ.n/d
d > 0 for all nonzero vectors d and n such that d
n D 0: (52) For given n and d the wave speed v is obtained from N v 2 D ŒQ.n/d
d D ŒQ.n/d
d: (53) As a first application we consider Case 1 in Section 2.3. Then we obtain N N n ˝ IT N n; Q.n/ D Œ2W1 C .2W7 C 1/.T n/
nIN C 2W7 TN C 2.W66 C 4W67 C 4W77 /IT (54) N IN is the projection of T on to the plane normal to n. The wave speed is then easily where TN D IT calculated from (53). For illustration we consider the initial stress to be uniaxial and along the x1 axis with T11 D T . Then, if we consider motion in the .x1 ; x2 / plane with n1 D cos , n2 D sin , d1 D sin , d2 D cos , we obtain v 2 D 2W1 C 2W7 T C T cos2  C .W66 C 4W67 C 4W77 /T 2 sin2 2: (55) Clearly, except for the angles  D 0 and  D =2 the squared wave speed depends quadratically (at least) on the initial stress T , bearing in mind that in general the coefficients W1 ; W7 ; W66 ; : : : could depend on T . If all the invariants had been included in W then the nonlinearity would be quartic (at least). This is in contrast to the predictions of Biot’s theory, and we now make contact with that for comparison. For this purpose we have provided in Appendix B a derivation of the connection between the general theory used here and Biot’s theory. If we denote by Bij kl the material constants used in Biot’s theory then, from Appendix B, we have Aij kl D Bij kl 1 2 ıil Tj k 1 2 ıik Tj l 1 2 ıj k Til C 21 ıj l Tik C ıkl Tij : (56) For the present special problem the relevant Bij kl components are given by B1111 D  T; B2222 D ; B1122 D  T; B2211 D ; B1212 D ; (57) PROPAGATION OF WAVES IN AN INCOMPRESSIBLE TRANSVERSELY ISOTROPIC ELASTIC SOLID 463 where  > 0 is the shear modulus. It follows that A1111 D  T; A2222 D  D A1122 ; A1212 D  C 21 T; A2121 D  1 2T D A1221 : (58) The wave speed in this case is then given by v 2 D  1 2T C T cos2 : (59) Clearly, this result can be recovered from (55) if we take 2W1 D , W7 D 41 (and hence, by (36)3 , W6 D 1), and W66 C 4W67 C 4W77 D 0. The individual components in (58) are recovered from the general expression (39) by also setting W11 D 0 and W16 C 2W17 D 0 in (39), except that we obtain A1122 D 0 instead of  and A1221 D T =2 instead of  T =2. However, this difference is of no consequence since the sum of these two terms is the same in each case and it is only their sum, as the coefficient of 2d1 d2 n1 n2 , that contributes to the expression for v 2 in (53). In fact, because of the incompressibility constraint, there is an intrinsic non-uniqueness in the components Aij kl (and hence in Bij kl ) since a term of the form p  ıil ıj k C q  ıij ıkl may be added to Aij kl , where p  and q  are arbitrary scalars, possibly dependent on tr T and M
.TM / in the general case. The term in q  disappears from the incremental constitutive relation by incompressibility and the term in p  vanishes identically in the incremental equation of motion, again by incompressibility. As a second example we consider Case 3 in Section 2.3, for which we obtain N N ˝M N; Q.n/ D Œ2W1 C .A C T /.M
n/2 IN C ŒA C C.M
n/2 M (60) N D IM N , and hence from (53) we obtain where M v 2 D 2W1 C .A C T /.M
n/2 C ŒA C C.M
n/2 .M
d/2 : (61) Let us take, for illustration, M to be along the x1 axis with the same d and n as in the previous example, so the motion is confined to the .x1 ; x2 / plane. Then v 2 D 2W1 C A C T cos2  C C sin2  cos2 ; (62) where A and C are given by (43) and (45), respectively. We note, in particular, that C is quadratic in T . Again we compare with the corresponding formula from Biot’s theory, which involves two material constants N and Q for this (two-dimensional orthotropic) situation, for which we obtain A1111 D N T; A2222 D N D A1122 ; A1212 D Q C 12 T; A2121 D Q 1 2T D A1221 ; (63) and hence v 2 D Q 1 2T C T cos2  C 4.N Q/ sin2  cos2 : (64) This can be obtained from (62) by setting 2W1 D N; 2W1 C 2W5 D Q; W7 D C  4.W44 C 4W45 C 4W55 / D 4.N 1 4; (65) Q/; (66) and all the other derivatives of W to zero. The case of isotropy is recovered by taking N D Q D . Case 2 produces very similar results when the motion is again restricted to the .x1 ; x2 / plane. If we take M now to be aligned along the x2 axis and T as before then we obtain v 2 D 2W1 C A0 C T cos2  C C 0 sin2  cos2 : (67) 464 RAY W. OGDEN AND BALJEET SINGH Compared with (62) A0 is obtained from A in (43) by omitting the term in W9 , while C 0 is given by C 0 D 4ŒW44 C 4W45 C 4W55 2T .W46 C 2W47 C 2W56 C 4W57 / C T 2 .W66 C 4W67 C 4W77 /; (68) which differs from C in (45). For each of the cases considered above, the appropriate specialization of the strong ellipticity condition (52) requires v 2 > 0, and conditions on the material parameters and the initial stress for this to be satisfied for all angles  may be inferred. For example, from (67), if T D 0 then necessary and sufficient conditions for v 2 > 0 are W1 C W5 > 0 and 2W1 C 2W5 C W44 C 4W45 C 4W55 > 0. On the other hand, if T ¤ 0 then restrictions are imposed on the permissible range of values of T . For example, by considering  D =2 and  D 0, we deduce that 2W1 C A0 > 0 and 2W1 C A0 C T > 0, respectively, are necessary conditions for v 2 > 0, but to obtain necessary and sufficient conditions for v 2 > 0 for all  requires consideration of several possible cases, depending on the signs of the material coefficients W7 ; W46 ; W66 ; : : : , and since this is algebraically lengthy the details are omitted. 4. Equations governing two-dimensional motions In this section we restrict attention to two-dimensional motions of incompressible materials. Specifically, we consider motions in the .x1 ; x2 / plane with displacement components u D .u1 ; u2 ; 0/, where u1 and u2 depend only on x1 ; x2 and t . The two in-plane components of the equations of motion (14) are Ap1qj uj ;pq pP;1 D u1;t t ; Ap2qj uj ;pq pP;2 D u2;t t ; (69) where the summation is over indices 1 and 2. In general, the third component of the equations of motion should also be considered; here this has the form Ap3qj uj ;pq pP;3 D 0. However, henceforth we specialize the form of T and the orientation of M so that this reduces to pP;3 D 0, i.e., pP depends only on x1 ; x2 and t. In particular, we take the .x1 ; x2 / plane to be a principal plane of T , with M either lying parallel to the .x1 ; x2 / plane or aligned with the x3 direction. Then, it is easy to show that Ap3qj uj ;pq vanishes identically. Elimination of pP by cross differentiation in (69) yields Ap1qj uj ;pq2 Ap2qj uj ;pq1 D .u1;2t t By incompressibility we can introduce a scalar function u1 D ;2 ; A1212 ;1111 C 2.A1222 A1211 / A2221 / ;1 ; (71) , namely ;1112 C .A1111 C A2222 C 2.A1121 (70) such that u2 D and substitution into (70) then yields an equation for u2;1t t /: 2A1122 ;1222 C A2121 2A1221 / ;2222 D . ;1122 ;11t t C ;22t t /: (72) We now apply these equations to Cases 2 and 3 specialized to the .x1 ; x2 / plane for T D T e1 ˝ e1 , with M D e2 (Case 2) and M D e1 (Case 3). Case 2: T D T e1 ˝ e1 and M D e2 . In this case (72) reduces to .a C T / ;1111 C .2a C T C b/ ;1122 C a ;2222 D . ;11t t C ;22t t /; (73) PROPAGATION OF WAVES IN AN INCOMPRESSIBLE TRANSVERSELY ISOTROPIC ELASTIC SOLID 465 where T D A2121 D 2W1 C A0 D 2W1 C 2W5 C 2T W7 ; a D A1212 b D A1111 C A2222 2A1122 2A1221 A2121 D C A1212 (74) 0 D 4.W44 C4W45 C4W55 / 8T .W46 C2W47 C2W56 C4W57 /C4T 2 .W66 C4W67 C4W77 /: (75) Case 3: T D T e1 ˝ e1 and M D e1 . Similarly to the previous case, (72) reduces to .˛ C T / ;1111 C .2˛ C T C ˇ/ ;1122 C ˛ ;2222 D . ;11t t C ;22t t /; (76) where ˛ D A1212 T D A2121 D 2W1 C A D 2W1 C 2W5 C 2T W7 C 2T W9 ; ˇ D A1111 C A2222 2A1122 2A1221 A1212 (77) A2121 D C D 4ŒW44 C 4W45 C 4W55 C T .W46 C 2W47 C 2W56 C 4W57 C W48 C 4W49 C 2W58 C 4W59 / C T 2 .W66 C 4W67 C 4W77 C W68 C 2W69 C 2W78 C 4W79 C W88 C 4W89 C 4W99 /: (78) If we compare the coefficients of Case 2 and Case 3, it may be remarked that ˛ D a C 2T W9 , while ˇ and b differ significantly, although their structures and the equations are very similar. If there is no initial stress then T D 0 and the two sets of equations are the same except that I4 and I5 will be different for the two cases because the directions of M are different. If there is initial stress but no preferred direction (underlying isotropy) then all the derivatives of W with a subscript 4, 5, 8 or 9 vanish and the two equations are identical, with coefficients reducing to a D ˛ D 2W1 C 2T W7 and b D ˇ D 4T 2 .W66 C 4W67 C 4W77 /. 5. Rayleigh surface waves We now consider a half-space occupying the region x2 < 0 in the reference configuration with boundary x2 D 0 and surface waves propagating along the direction x1 . The initial stress and preferred direction are confined to the .x1 ; x2 / plane so that the equations of motion (69) are applicable and the third equation is again satisfied trivially. Furthermore, we take the initial stress to have the form T D T e1 ˝ e1 . Then the components A1112 , A1121 , A2212 , A2221 all vanish, as for Cases 2 and 3 in Section 4, and the equations simplify accordingly. On the surface we take the incremental surface traction to vanish, so that SP 021 D 0; SP 022 D 0 on x2 D 0. For the considered incompressible material this yields A2121 u1;2 C .A2112 C p/u2;1 D 0; and with u1 D A2121 ;22 ;2 and u2 D .A2112 C p/ ;1 ;11 A2211 u1;1 C .A2222 C p/u2;2 pP D 0 on x2 D 0; (79) pP D 0 on x2 D 0: (80) these boundary conditions become D 0; A2211 ;12 .A2222 C p/ ;12 The second of these equations may be expressed in terms of by differentiating with respect to x1 and then using (69)1 , appropriately specialized, to eliminate pP;1 . The result is .A1111 C A2222 2A1122 2A2112 C A2121 / ;112 C A2121 ;222 D ;2t t : (81) 466 RAY W. OGDEN AND BALJEET SINGH The two boundary conditions can now be written compactly as a. ;22 ;11 / D 0; .3a C b C T / ;112 C a ;222 D ;2t t on x2 D 0: (82) These apply for Case 2. The corresponding equations for Case 3 are obtained by replacing a and b by ˛ and ˇ, respectively. Thus, in the following we work with the parameters a and b. We consider harmonic waves propagating in the x1 direction and we write in the form .x1 ; x2 ; t / D .z/ expŒik.x1 vt /; (83) where k is the wave number, v is the wave speed, z D kx2 , and the function  is to be determined. Substituting this into the equation of motion (73) we obtain a 0000 .2a C b C T v 2 / 00 C .a C T v 2 / D 0; (84) wherein and in the following equations a prime on  denotes differentiation with respect to z. In terms of  the boundary conditions (82) become  00 .0/ C .0/ D 0; a 000 .0/ v 2 / 0 .0/ D 0: .3a C b C T (85) The factor a has been omitted from the first of these equations on the assumption that a ¤ 0. For the solution of (84) we require the decay condition .x2 / ! 0 as x2 ! 1 to hold, and the general solution satisfying this condition is .z/ D c1 exp.s1 z/ C c2 exp.s2 z/; (86) where c1 and c2 are constants and s1 ; s2 are the solutions of as 4 .2a C b C T v 2 /s 2 C .a C T v 2 / D 0 (87) a C T v 2 : a (88) with positive real parts. From the latter it follows that s12 C s22 D 2a C b C T a v 2 ; s12 s22 D If s12 and s22 are real, then they must be positive to ensure that s1 and s2 have positive real part. If they are complex then they are conjugate. In either case the product s12 s22 must be positive. Assuming that a > 0, a real (surface) wave speed v satisfies the inequality 0 < v 2 < a C T: (89) Note that if T  0 then this cannot be satisfied unless a > 0, which requires W1 C W5 C T W7 > 0. In Biot’s theory, for example, this, in conjunction with (89), yields 2Q < T < 2Q. Substitution of the solution (86) into the boundary conditions (85) yields the equations .1 C s12 /c1 C .1 C s22 /c2 D 0; Œas13 .3a C b C T v 2 /s1 c1 C Œas23 .3a C b C T (90) v 2 /s2 c2 D 0: (91) In order to construct a nontrivial solution of this system, corresponding to vanishing of the determinant of coefficients, it is convenient to introduce the notations p T v 2 b TN D ;  D ; bN D ;  D 1 C TN ; (92) a a a PROPAGATION OF WAVES IN AN INCOMPRESSIBLE TRANSVERSELY ISOTROPIC ELASTIC SOLID 467 and then we obtain the cubic equation N 3 C 2 C .3 C b/ 1D0 (93) for . Once this is solved for (positive)  the wave speed is obtained from the formula v 2 D a D a C T a2 : The inequalities satisfied by  corresponding to (89) are 0 <  < Note that (93) can be written as .4 C bN C TN / C TN (94) p 1 C TN .  D 0: (95) N and hence we must have 4 C bN > 0 in this If T D 0 then this can only be satisfied for  > 0 if  < 4 C b, case. If T ¤ 0 then since, by definition,   0, (95) can only be satisfied if either TN <  < 4 C bN C TN ; (96) 4 C bN C TN <  < TN ; (97) which requires 4 C bN > 0, or which requires 4 C bN < 0, or the transitional case  D TN ; (98) which, since then  D 1, corresponds to 4 C bN D 0. Note that (96) can hold for either positive or negative T provided 4 C bN C TN > 0, but (97) and (98) are only possible if T > 0. 00 If we denote the function on the left-hand side of (93) by f ./ and p note that f .0/ p D 1 and f ./ > 0 for   0 we see that (93) has a unique positive solution for  2 .0; 1 C TN  if f . 1 C TN /  0. If T D 0 then this is guaranteed if 4 C bN > 0, and if this inequality holds a unique positive solution also exists for T > 0 (subject to a > 0). On the other hand, if T < 0 then there is a value TN0 of TN with 1 < TN0 < 0 such that a unique positive solution exists only for TN > TN0 . The value of TN0 is determined as the value p N N of T satisfying f . 1 C T / D 0, bearing in mind that in general bN depends on T . The exact details depend in a fairly complicated way on the form of the strain-energy function W and are not discussed here. Instead we illustrate the results using a specific form of W in the following section. 6. Numerical results and discussion We remain focused on Case 2 in order to compare the results of the general theory with those of the Biot theory for both anisotropic and isotropic models. In particular, we compare the wave speeds for both plane (shear) waves and Rayleigh waves. In each case this involves the initial stress T and material parameters a and b, which also depend on T . For the general constitutive law considered in Case 2 we record here the expressions for a and b from Section 4 for ease of reference: a D 2W1 C 2W5 C 2T W7 ; b D 4.W44 C4W45 C4W55 / (99) 8T .W46 C2W47 C2W56 C4W57 /C4T 2 .W66 C4W67 C4W77 /: (100) If the second derivative terms in W involving a subscript 6 or 7 are identically zero then these reduce to a D 2W1 C 2W5 C 2T W7 ; b D 4.W44 C 4W45 C 4W55 /; (101) 468 RAY W. OGDEN AND BALJEET SINGH which can be specialized to Biot’s anisotropic case by setting W7 D 14 , 2W1 C 2W5 D Q and W44 C 4W45 C 4W55 D N Q in terms of Biot’s constants N and Q. Finally, the case of isotropy is then recovered by also dropping terms with a subscript 4 or 5, yielding a D 2W1 C 2T W7 ; b D 0; (102) and Biot’s case is obtained by setting 2W1 D  and W7 D 14 , where  is the shear modulus. For plane waves in the .x1 ; x2 / plane the wave speed v is given by v 2 D a C T cos2  C b sin2  cos2 ; (103) while the corresponding formula for the Rayleigh wave speed is v 2 D a C T a2 ; (104) where  is the unique positive solution of the equation 3 C 2 C .3 C b=a/ 1 D 0; 0<< p 1 C T =a: (105) In order to capture the effect of the T 2 term in b, which provides the main distinction from the classical theory, it suffices for purposes of illustration to consider a specific form of the strain-energy function W . In the absence of initial stress an incompressible transversely isotropic linear elastic solid is characterized in terms of three elastic constants. Thus, for the transversely isotropic part of W three material constants are needed. The connections between the derivatives of W with respect to the invariants I1 ; I2 ; I4 and I5 evaluated in the undeformed configuration and the three classical constants were provided in [Merodio and Ogden 2005] and can be specialized to the present situation in which I2 is omitted. We denote the transversely isotropic constants here by ,  and  and consider a transversely isotropic strain-energy function of the form W D 12 .I1 3/ C 21 .I4 I5 /2 2.I4 1/ C .I5 1/; (106) which consists of an isotropic neo-Hookean term with constant  and two anisotropic terms associated with the preferred direction and involving two anisotropic constants  and . The condition W4 C2W5 D 0 identified in (36) is then satisfied in the reference configuration Br , W45 C W55 D 0 and W44 D . The initial stress is next incorporated in the model by introducing two additional material constants  and and the invariants I6 and I7 , and ensuring that the condition W6 C 2W7 D 12 in (36) is satisfied in the configuration Br . A simple example of this inclusion, which we adopt here, extends (106) to the form W D 12 .I1 3/ C 21 .I4 I5 /2 2.I4 1/ C .I5 C. 12 1/ C 12 I62 .tr T /I6 2/I6 C I7 C 1 2 .tr T /2 C . 1 2 / tr T ; (107) the final two terms being included merely to ensure that W vanishes in Br . They do not contribute to the stress. The only nonzero derivatives of W when evaluated in Br are the first derivatives W1 D 21 ; W4 D 2; W5 D ; W6 D 1 2 2; W7 D ; (108) PROPAGATION OF WAVES IN AN INCOMPRESSIBLE TRANSVERSELY ISOTROPIC ELASTIC SOLID 469 and the second derivatives W44 D ; W45 D ; W55 D ; W66 D : (109) The expressions for a and b become a D  C 2 C 2T; b D 4. C T 2 /: (110) Clearly, if  D 14 then the term in T within a has the same form as in the Biot theory. Inclusion of the dimensionless constant  allows more flexibility in the model. In what follows we present results in dimensionless form by defining the dimensionless quantities  D v 2 =; a D a=; b  D b=;   D =;   D =;  D ; T  D T =; (111) and for consistency of notation for the constants we set  D . Thus, for plane waves,  D 1 C 2  C 2 T  C T  cos2  C 4.  C  2 T  / sin2  cos2 : (112) In Figure 1 we plot  as a function of T  for the values 0; =6; =4; =3; =2 of  and specifically we take   D 0:4,   D 2:2,  D 0:5. The results are qualitatively very similar for other values of these parameters. The parameter  has been set at 41 . By considering the values  D 0 and  D =2 it can be deduced from (112) that T  must be restricted to the range of values 2.1 C 2  / < T  < 2.1 C 2  /. Also in Figure 1 we show, for comparison, the results corresponding to   D 0:4,   D 2:2,  D 0 (the thick dashed lines — essentially the specialization to Biot’s anisotropic theory) and to   D 0,   D 0, 6 ξ 5 c b d 4 b d 3 e d a e a 2 b 1 -2 -1 0 1 2 T∗ Figure 1. Plot of dimensionless squared wave speed  D v 2 = against dimensionless initial stress T  for  D 0 (a), =6 (b), =4 (c), =3 (d), =2 (e), based on (112): continuous curves for parameter values   D 0:4,   D 2:2,  D 0:5; thick dashed lines for   D 0:4,   D 2:2,  D 0; thin dashed lines for   D 0,   D 0,  D 0 (isotropy). In each case  D 41 . The horizontal dashed lines correspond to  D =4. 470 RAY W. OGDEN AND BALJEET SINGH 6 T (a) D 2 6 5 5 4 4 3 3 2 2 1 1 0.0 ξ 6 0.5 1.0 1.5 0.0 (c) T D1 6 5 5 4 4 3 3 2 2 1 1 0.0 0.5 1.0 T (b) D 1 1.5 0.0 0.5 1.0 1.5 0.5 1.0 1.5 (d) T D2 θ Figure 2. Dimensionless squared wave speed  D v 2 = versus the angle  2 Œ0; =2, based on (112), for  D 14 and various values of the dimensionless initial stress T  . Remaining parameter values are   D 0:4,   D 2:2,  D 0:5 (solid curves);   D 0:4,   D 2:2,  D 0 (thick dashed curves);   D 0,   D 0,  D 0 (thin dashed curves).  D 0 (the thin dashed lines — Biot’s isotropic theory). For  D 0 and  D =2 there is no difference between the two anisotropic models. For intermediate values of  there is a very significant difference between the results for the general model and Biot’s anisotropic model due to the term in T  2 . Note that in the isotropic case the wave speed vanishes for T  D ˙2, which correspond to the extreme values of T  identified previously. An alternative view of the results is shown in Figure 2, in which  is plotted against the angle  separately for four values of T  . For each value of T  the curves corresponding to the three models are shown. Again the significant difference between the general model and Biot’s anisotropic model should be noted, which is particularly strong for the larger values of T  . The parameter  also has a significant effect and we illustrate this in Figure 3 in which  is plotted against T  for three different values of  and for   D 0:4;   D 2:2;  D 0:2 and  D =3. Even though the (nonlinear) strain-energy function (107) is a considerable specialization of the most general such model it nevertheless demonstrates that the effect of initial stress on the speed of plane waves can be much stronger than is the case with the classical Biot theory. Next we illustrate the effect of the model (107) on the Rayleigh wave speed based on the solution of (105), with  D v 2 = then given by (104). In Figure 4,  is plotted against T  for representative values of the parameters (  D 0:4,   D 2:2,  D 0:5) and two values of  : 0:15 and 0:3. In each case the dependence on T  is effectively linear, and this is found also to be the case even when the coefficient  of T  2 is quite large. Also shown for comparison are the specializations corresponding to Biot’s PROPAGATION OF WAVES IN AN INCOMPRESSIBLE TRANSVERSELY ISOTROPIC ELASTIC SOLID 471 7 6 ξ c  D 0:05 5 b  D 0:25 a  D 0:45 4 3 2 1 -3 -2 0 -1 1 2 3 T∗ Figure 3. Dimensionless squared wave speed  D v 2 = versus T  , based on (112), for  D =3,   D 0:4,   D 2:2,  D 0:2 and different values of  . 4  D 0:3 b ξ  D 0:25 3 a D 0:15 2  D 0:25 1 0 -3 -2 -1 0 1 2 3 T∗ Figure 4. Dimensionless squared Rayleigh wave speed  D v 2 = versus T  , based on (104) and (105), for different values of  . Remaining parameter values are   D 0:4,   D 2:2,  D 0:5 (solid lines);   D 0:4,   D 2:2,  D 0 (dashed line);   D 0,   D 0,  D 0 (dotted line). anisotropic and isotropic results, for which, respectively, the parameter values are taken as   D 0:4,   D 2:2,  D 0,  D 0:25 and   D 0,   D 0,  D 0,  D 0:25. In the isotropic case we have  D 1 C 12 T  .1 1  2 2 T /0 ; (113) where 0 is the unique positive root of (105) when b D 0 and is given approximately as 0 D 0:2956. As noted earlier, a positive real upper limit for  requires that 1 C T  =a > 0, and for isotropy this gives 2 < T  < 2. The upper limit corresponds to  D 2, as can be seen in the figure, and this is a cut-off value beyond which a Rayleigh surface wave cannot propagate. At the lower limit  < 0 and there is also a cut-off value of T  , approximately 1:679. These limiting values of T  are associated with the onset of instability of the underlying initially stressed configuration. For a related discussion of 472 RAY W. OGDEN AND BALJEET SINGH 10 c T D2 8 ξ 6 c T D2 b D0 T T D 2 a T D 2 a 4 2 0 0 1 2 3 4 κ∗ Figure 5. Dimensionless squared Rayleigh wave speed  D v 2 = versus the parameter   for various values of T  , with   D 2:2,  D 0, and  D 0:15 (solid lines) or  D 0:25 (dashed lines). such instability in the case of a pre-stressed (and deformed) half-space we refer to [Dowaikh and Ogden 1990]. Analogous cut-off values of T  are evident also for the anisotropic models. As a further illustration of the influence of the parameter  , Figure 5 shows  plotted as a function of the parameter   for each of the initial stress values T  D 2; 0; 2 and for the set of parameters   D 2:2,  D 0,  D 0:15 compared with results for the set   D 2:2,  D 0,  D 0:25. Clearly,  has a significant effect on the value of the wave speed (except, of course, for T  D 0). Thus, as is the case with plane waves, the initial stress dependence of the nonlinear model (107) can have a strong influence on the value of the Rayleigh wave speed compared with its classical linear specialization. Moreover, other specific choices of the nonlinear model within the general framework outlined here can equally and even more substantially affect the wave speed, when, for example, invariants that are nonlinear in the initial stress are included in the form of W . In this paper we have developed a general theory of transversely isotropic hyperelasticity incorporating initial stress and used a particular specialization of the theory to calculate the elasticity tensor for an undeformed initially stressed configuration. This was then used to examine the propagation of homogeneous plane waves and Rayleigh surface waves with particular reference to the effect of the initial stress. The results, which involve nonlinear terms in the initial stress, are significantly different from those based on the classical theory of Biot, which can be recovered as a special case of the present formulation. Appendix A. Derivatives of the invariants First derivatives. The first derivatives of the invariants I1 ; I4 ; : : : ; I9 with respect to F are obtained, in component form, as @I1 D 2Fi˛ ; @Fi˛ @I4 D 2M˛ Fiˇ Mˇ ; @Fi˛ @I6 D 2T˛ˇ Fiˇ ; @Fi˛ @I5 D 2.M˛ Fiˇ Cˇ C C˛ˇ Mˇ Fi /M ; @Fi˛ @I7 D 2.T˛ˇ Cˇ C C˛ˇ Tˇ /Fi ; @Fi˛ PROPAGATION OF WAVES IN AN INCOMPRESSIBLE TRANSVERSELY ISOTROPIC ELASTIC SOLID 473 @I8 D .T˛ˇ Mˇ Fi C M˛ Fiˇ Tˇ /M ; @Fi˛ @I9 D .T˛ˇ Mˇ Fi C @Fi˛ ı C C˛ˇ Mˇ Fi T ı C M˛ Fiˇ Cˇ T ı C Fiˇ Mˇ C˛ T ı /Mı : When evaluated in the reference configuration these reduce to @I1 D 2ıi˛ ; @Fi˛ 2 2 @I4 @I5 D 4Mi M˛ D ; @Fi˛ @Fi˛ 2 @I6 @I7 D 4Ti˛ D ; @Fi˛ @Fi˛ @I8 @I9 D 2.Mi T˛ˇ C M˛ Tiˇ /Mˇ D : @Fi˛ @Fi˛ Since the latter apply in the reference configuration there is strictly no distinction between Greek and Roman indices in this case. Second derivatives. The second derivatives of the invariants I1 ; I4 ; : : : ; I9 with respect to F are, in component form, @2 I 1 D 2ıij ı˛ˇ ; @Fi˛ @Fjˇ @2 I4 D 2ıij M˛ Mˇ ; @Fi˛ @Fjˇ @2 I6 D 2ıij T˛ˇ ; @Fi˛ @Fjˇ @2 I 5 D 2ıij .M˛ Cˇ C Mˇ C˛ /M @Fi˛ @Fjˇ C 2Bij M˛ Mˇ C 2ı˛ˇ Fi M Fj ı Mı C 2.Fiˇ Fj M˛ C Fj˛ Fi Mˇ /M ; @2 I 7 D 2Œıij .T˛ Cˇ C Tˇ C˛ / C Bij T˛ˇ C ı˛ˇ †ij C Fiˇ Fj T˛ C Fj˛ Fi Tˇ ; @Fi˛ @Fjˇ @2 I 8 D ıij .M˛ Tˇ C Mˇ T˛ /M ; @Fi˛ @Fjˇ @2 I 9 D ıij .T˛ M Cˇı C Tˇ M C˛ı C C˛ T ı Mˇ C Cˇ T ı M˛ /Mı @Fi˛ @Fjˇ C ı˛ˇ .Fi T ı Fj " C Fj T ı Fi" /Mı M" C Bij .M˛ Tˇ C T˛ Mˇ /M C .M˛ Fiˇ Fj C Mˇ Fj˛ Fi /T ı Mı C .T˛ Fiˇ Fj ı C Tˇ Fj˛ Fiı /M Mı : When evaluated in the reference configuration these specialize to @2 I1 D 2ıij ı˛ˇ ; @Fi˛ @Fjˇ @2 I4 D 2ıij M˛ Mˇ ; @Fi˛ @Fjˇ @2 I 6 D 2ıij T˛ˇ ; @Fi˛ @Fjˇ @2 I5 D 6ıij M˛ Mˇ C 2.ı˛ˇ Mi Mj C ıiˇ Mj M˛ C ıj˛ Mi Mˇ /; @Fi˛ @Fjˇ @2 I 7 D 6ıij T˛ˇ C 2.ı˛ˇ Tij C ıiˇ Tj˛ C ıj˛ Tiˇ /; @Fi˛ @Fjˇ @2 I8 D ıij .M˛ Tˇ C Mˇ T˛ /M ; @Fi˛ @Fjˇ 474 RAY W. OGDEN AND BALJEET SINGH @2 I 9 D 3ıij .M˛ Tˇ C Mˇ T˛ /M C ı˛ˇ .Mi Tj C Mj Ti /M @Fi˛ @Fjˇ C ıiˇ .M˛ Tj C Mj T˛ /M C ıj˛ .Mi Tˇ C Mˇ Ti /M : Appendix B. Connections with Biot’s equations It is instructive to relate the formulation of the incremental equations adopted here to that of Biot (see, for example, [Biot 1965] for details). For a compressible material, which we deal with here, the nominal stress tensor S and Cauchy stress tensor  are related by S D J F 1  , where F is deformation gradient and J D det F . On taking the increment of this equation we obtain SP D J F 1 P C JP F 1  CJF P 1 D JF 1 ŒP C .tr L/ L ; where L D grad u and u is the displacement vector. On updating the reference configuration from Br to B0 we obtain SP 0 D P C .tr L/ L : Taking the divergence of both sides leads to div SP 0 D div P C div Œ.tr L/ L : If the configuration B0 is uniform then we have divŒ.tr L/ L  D Lkk;j j i Lj k;j ki D Lkk;j j i Ljj ;k ki D 0; and the incremental equation of motion becomes div SP 0  div P D u;t t : Now, Biot works in terms of components referred to different sets of axes, in particular a set of Cartesian axes and a set of axes obtained by rotation therefrom, with the rotation associated with the incremental deformation. Let ei ; i D 1; 2; 3, be a Cartesian coordinate basis in B0 , and let ei0 ; i D 1; 2; 3, be the axes obtained by the rotation. Then, ei0  ei C Wei , where the tensor W is the antisymmetric part of the displacement gradient L. After the increment the total Cauchy stress is  C P , which has components on the rotated axes given by, to the first order in incremental quantities, Œ. C P /ei0 
ej0 D Œ. C P /.ei C Wei /
.ej C Wej / D ij C P ij C ik Wkj Wik kj ; where the symmetry of  and the antisymmetry of W have been used, and ij and Wij are their components referred to the original axes (note that Wij here are not the same as the second derivatives of the strain-energy function with respect to the invariants introduced in (29)). The component form of the incremental equation of motion with uniform  therefore becomes Pj i;j C ik Wkj ;j j k Wik;j D ui;t t : This is translated into Biot’s notation [1965] by setting ij D Sij , P ij D sij and Wij D !ij , and then yields Biot’s equation sij ;j C Sj k !ik;j C Sik !j k;j D ui;t t PROPAGATION OF WAVES IN AN INCOMPRESSIBLE TRANSVERSELY ISOTROPIC ELASTIC SOLID 475 appropriate for homogeneous Sij in the absence of body forces [Biot 1965, p. 264]. 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