Interface waves in pre-stressed incompressible solids
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Interface Wavesin Pre-Stressed Incompressible Solids
Michel Destrade
Institut Jean Le Rond d’Alembert, CNRS/Universit´e Pierre et Marie Curie, Paris, France
Abstract
We study incremental wave propagation for what is seemingly the sim-plest boundary value problem, namely that constitued by the plane interface of asemi-infinite solid. With a view to model loaded elastomers and soft tissues, wefocus on incompressible solids, subjected to large homogeneous static deformations.The resulting strain-induced anisotropy complicates matters for the incrementalboundary value problem, but we transpose and take advantage of powerful tech-niques and results from the linear anisotropic elastodynamics theory. In particularwe cover several situations where fully explicit secular equations can be derived,including Rayleigh and Stoneley waves in principal directions, and Rayleigh wavespolarized in a principal plane or propagating in any direction in a principal plane.We also discuss the merits of polynomial secular equations with respect to morerobust, but less transparent, exact secular equations.
The term “acousto-elastic effect” describes the interplay between the static deformationof an elastic solid and the motion of an elastic wave. If both the deformation and themotion are of infinitesimal amplitude, then all the governing equations are linearized, seefor instance the Chapter by Norris for examples and applications or the experimentalresults of Pao et al. (1984). If both the deformation and the motion are of finite ampli-tude, then the resulting governing equations are highly nonlinear, and their resolution isthe subject of much research, see the Chapter by Fu for the weakly nonlinear theory andthe Chapter by Saccomandi for the fully nonlinear theory.In between those two situations lies the theory of “small-on-large”, also known as thetheory of “incremental” motions, where the wave is an infinitesimal perturbation super-imposed onto the large static homogeneous deformation of a generic hyperelastic solid.There, the homogeneous character of the static deformation and the linear character ofthe incremental equations of motion ensure that the calculations are valid for any strainenergy density (to be specified later for applications, if necessary). The next Sectionof this Chapter briefly recalls the governing equations of incremental motions (see theChapter by Ogden for their derivation).It turns out that many similarities can be drawn between the equations of incrementalmotions and those of linear anisotropic elasticity, with the main difference that in thelatter case, the anisotropy is set once and for all for a given crystal whereas in the former1ase, it is strain-induced and susceptible to great variations from one configuration toanother. Using the similarities, we may transpose the so-called Stroh formulation andexploit its many results; on the other hand, when focussing on the differences, we mayhighlight the influences of the pre-stress and of the choice of a strain-energy densityon the propagation of waves. In this Chapter, attention is restricted to waves at theinterface of pre-deformed, semi-infinite solids, in contact either with vacuum (Rayleighwaves) or with another solid (Stoneley waves). With a view to model elastomers andbiological soft tissues, the solids are considered to be incompressible (mathematically,this internal constraint lightens somewhat the expressions but does not prove essentialto the resolution).Several situations are treated: principal wave propagation in Section 3, principal po-larization in Section 4, and principal plane propagation in Section 5. The emphasis ison deriving explicit secular equations in polynomial form, using some simple “funda-mental equations” derived at the end of Section 2. Of course, as the setting gets moreand more involved, so does the search for a polynomial secular equation; eventually itsdegree becomes too high for comfort and other techniques are required. The concludingsection (Section 6) discusses the pros and cons of such equations, as opposed to exact,non-explicit, secular equations, free of spurious roots.
Consider an isotropic, incompressible, hyperelastic solid at rest, characterized by amass density ρ and a strain energy function W . Then subject it to a large, static, ho-mogeneous deformation (“the pre-strain”) carrying the particle at X in the undeformedconfiguration to the position x in the deformed configuration.Call F = ∂ x /∂ X the corresponding constant deformation gradient and B = F F t theassociated left Cauchy-Green strain tensor. This tensor being symmetric, the directionsof its eigenvectors are orthogonal; they are called the principal axes of pre-strain or inshort, the principal axes . Also, the eigenvalues of B are positive, λ , λ , λ , say, and λ , λ , λ are called the principal stretches . Figure 1 shows how a unit cube with edgesaligned with the principal axes, is transformed by the pre-strain. Note that because thesolid is incompressible , its volume is preserved through any deformation so that here, λ λ λ = 1 . (2.1)The first two principal invariants of strain are defined as I = tr B , I = [(tr B ) − tr ( B )] / . (2.2)In the Cartesian coordinate system aligned with the principal axes, B is diagonal. Calling e , e , e , the unit vectors in the x , x , x directions, respectively, we have B = λ e ⊗ e + λ e ⊗ e + λ e ⊗ e , (2.3)and the computation of I , I there gives I = λ + λ + λ , I = λ λ + λ λ + λ λ . (2.4)2 igure 1. Finite homogeneous deformation of a unit cube.For an isotropic solid, W may be given as a function of the invariants: W = W ( I , I )or equivalently, as a symmetric function of the principal stretches: W = W ( λ , λ , λ ),according to what is most convenient for the analysis or according to how W has beendetermined experimentally. With the first choice, the constant Cauchy stress (“the pre-stress”) necessary to maintain the solid in its state of finite homogeneous deformationis σ = − p I + 2( ∂W/∂I + I ∂W/∂I ) B − ∂W/∂I ) B , (2.5)where p is a Lagrange multiplier due to the constraint of incompressibility (a yet arbitraryconstant scalar to be determined from initial and boundary conditions.) With the secondchoice, the non-zero components of σ relative to the principal axes are written as σ i = − p + λ i ∂W/∂λ i , i = 1 , , . (2.6)The proof for the equivalence between (2.5) and (2.6) relies on the connections (2.4). Consider a half-space filled with an incompressible hyperelastic solid subject to a largehomogeneous deformation. We take the Cartesian coordinate system (ˆ x , ˆ x , ˆ x ) to beoriented so that the boundary is at ˆ x = 0, and we study the propagation in the ˆ x direction of an infinitesimal interface wave in the solid.This wave is inhomogeneous as it progresses in an harmonic manner in a directionlying in the interface, while its amplitude decays with distance from the boundary.We call u the mechanical displacement associated with the wave, and ˙ p the incrementin the Lagrange multiplier p due to incompressibility. The incremental nominal stress tensor s has components s ji = A jilk u k,l + pu j,i − ˙ pδ ij , (2.7)3here the comma denotes partial differentiation with respect to the coordinates ˆ x , ˆ x ,ˆ x . Here, A is the fourth-order tensor of instantaneous elastic moduli , with components A jilk = F jα F lβ ∂ WF iα F kβ = A lkji . (2.8)Note that due to the symmetry above, A has in general 45 independent components.In the principal axes coordinate system ( x x x ) however, there are only 15 independentnon-zero components; they are (Ogden, 2001): A iijj = λ i λ j W ij , A ijij = ( λ i W i − λ j W j ) λ i / ( λ i − λ j ) , i = j, λ i = λ j , A ijij = ( A iiii − A iijj + λ i W i ) / , i = j, λ i = λ j , A ijji = A jiij = A ijij − λ i W i , i = j, (2.9)(no sums on repeated indexes here), where W j = ∂W/∂λ j and W ij = ∂ W/ ( ∂λ i ∂λ j ).Finally, the governing equations are the incremental equations of motion and theincremental constraint of incompressibility; they read s ji,j = ρ∂ u i /∂t , u j,j = 0 , (2.10)respectively.Now everything is in place to solve an interface wave problem. We take u and ˙ p inthe form { u , ˙ p } = { U ( k ˆ x ) , i kP ( k ˆ x ) } e i k ( n · ˆ x − vt ) , (2.11)where k is the wave number, U and P are functions of the variable k ˆ x only, n is theunit vector in the direction of propagation, and v is the speed. Clearly by (2.7), s has asimilar form, say s = i k S ( k ˆ x )e i k ( n · ˆ x − vt ) , (2.12)where S is a function of k ˆ x only.After substitution, it turns out that the incremental governing equations (2.10) canbe cast as the following first-order differential system (Chadwick, 1997), ξ ′ = i N ξ , where ξ = [ U , t ] t , (2.13)the prime denotes differentiation with respect to the variable k ˆ x , and t are the tractions acting on planes parallel to the boundary, with components t j = S j . Here the matrix N has the following block structure N = (cid:20) N N N + ρv I N t (cid:21) , (2.14)where N , N = N t , and N = N t are square matrices. This is the so-called Stroh for-mulation . In effect, many of the results established thanks to the Stroh (1962) formalismin linear anisotropic elasticity can formally be carried over to the context of incrementaldynamics in nonlinear elasticity, as shown by Chadwick and Jarvis (1979a), Chadwick(1997), and Fu (2005a,b). 4 .3 Resolution
The solution to the first-order differential system (2.13) is an exponential function in k ˆ x , ξ ( k ˆ x ) = e i kq ˆ x ζ , (2.15)where ζ is a constant vector and q is a scalar. Then the following eigenvalue problememerges: N ζ = q ζ . Its resolution is in two steps.First, find the eigenvalues by solving the propagation condition ,det ( N − q I ) = 0 , (2.16)for q , and keep those q j ’s which satisfy the decay condition . For instance, when the solidfills up the ˆ x > ℑ ( q ) > , (2.17)ensuring that the solution (2.15) is localized near the interface and vanishes away fromit. The penetration depth of the interface wave is clearly related to the magnitude of ℑ ( q ): the smaller this quantity is, the deeper the wave penetrates into the solid.The propagation condition (2.16) is a polynomial in q with real coefficients and ithas only complex roots (Fu, 2005b), which come therefore in pairs of complex conjugatequantities. Hence, half of all the roots to the propagation condition qualify as satisfyingthe decay condition. Let ζ j be the eigenvector corresponding to the qualifying root q j .Now proceed to the second step, which is to construct the general localized solutionto the equations of motion, as ξ ( k ˆ x ) = P γ j e i kq j ˆ x ζ j , (2.18)for some arbitrary constants γ j . Then compute this vector at the interface ˆ x = 0 andapply the boundary conditions . The vector ξ (0) is often decomposed as follows, ξ (0) = (cid:20) U (0) t (0) (cid:21) = P γ j ζ j = (cid:20) AB (cid:21) γ , (2.19)where A and B are square matrices and γ is the vector with components γ j . For instance,the archetype of interface waves is the Rayleigh (1885) surface wave, which propagatesat the interface between a solid half-space and the vacuum, leaving the boundary free oftractions. Mathematically, the corresponding boundary condition is that t (0) = Bγ = ,leading to det B = 0 . (2.20)This (complex) form of the secular equation is however not the optimal form, and it mightlead to unsatisfactory answers to the questions of existence and uniqueness of the wave(see Barnett (2000) for an historical account of this point). From the Stroh formalism,and its application to the present context, we learn that it is much more efficient towork with the surface impedance matrix (Ingebrigsten and Tonning, 1969) than with thematrices A and B ; this matrix M is defined by M = − i BA − . (2.21)5t is Hermitian (Barnett and Lothe, 1985; Fu, 2005b) and so det M = − i(det B ) / (det A )is a real quantity and the secular equation for Rayleigh surface waves , written in the formdet M = 0 , (2.22)is a real equation, in contrast to (2.20). Moreover, if there is a root to this equation inthe subsonic regime (where v is less than the speed of any bulk wave), then it is unique;also, the existence of a root is equivalent to the existence of a surface wave.Similar results also exist for other types of interface waves as seen in the course of thisChapter. For instance, the boundary conditions for Stoneley (1924) interface waves arethat displacements and tractions are continuous across the boundary between two rigidlybonded semi-infinite solids. Then ξ (0) = ξ ∗ (0) where the asterisk refers to quantitiesfor the solid in ˆ x
0. Equivalently, Aγ = A ∗ γ ∗ , Bγ = B ∗ γ ∗ , from which comes[ BA − A ∗ − B ∗ ] γ ∗ = , leading todet( M + M ∗ ) = 0 , (2.23)the optimal form of the secular equation for Stoneley interface waves . Here M ∗ is thesurface impedance matrix for the solid in the ˆ x M ∗ = i B ∗ ( A ∗ ) − . (2.24) The derivation of a secular equation, preferably in the optimal form involving thesurface impedance matrix, is no sinecure in general. The problematic step lies in theresolution of the propagation condition (2.16).For principal wave propagation (ˆ x , ˆ x , ˆ x are aligned with the principal axes), thepropagation condition factorizes into the product of a term linear in q and a termquadratic in q . Here we can compute the roots explicitly, keep the qualifying ones (see(2.17)), and solve the boundary value problem in its entirety. Many problems falling inthis category have been solved over the years, and some are presented in Section 3.For a non-principal wave with propagation direction and attenuation direction bothin a principal plane (the saggital plane (ˆ x ˆ x ) is a principal plane but ˆ x is not a principalaxis), the propagation condition factorizes into the product of a term linear in q and aterm quartic in q . We treat this case in Section 4. Although it is possible to write downformally the qualifying roots of the quartic (Fu, 2005a; Destrade and Fu, 2006; Fu andBrookes, 2006), the formulas involved are cumbersome to interpret.For a wave propagating in a principal plane but not in a principal direction (ˆ x isaligned with a principal axis but neither ˆ x nor ˆ x are aligned with principal axes), thepropagation condition is a cubic in q . We treat this case in Section 5. Now it is adaunting task to find analytical expressions for the roots q satisfying the decay condition(2.17).Finally, for wave propagation in any other case, the propagation condition is a sexticin q , unsolvable analytically according to Galois theory.These observations suggest that, except in the case of principal waves, numericalprocedures are required in order to make progress. It is indeed the case that sophisticated6ools and efficient numerical recipes have been developed by Barnett and Lothe (1985),Fu and Mielke (2002), and several others, with most satisfying results. However it isalso the case that some interface wave problems can be solved analytically, up to thederivation of the secular equation in explicit polynomial form . The first steps in thatdirection were taken by Currie (1979), and his advances were later refined by Taylor andCurrie (1981) and Taziev (1989), revisited by Mozhaev (1995) and by Ting (2004), andextended by Destrade (2003).The equations that turn out to be fundamental in the derivation of explicit polynomialsecular equations are ξ (0) · ˆ IN n ξ (0) = 0 , (2.25)where ˆ I = (cid:20) II (cid:21) and n is an integer. Their derivation is most simple. First, it can beshown by induction (Ting, 2004) that N n has a block structure similar to that of N ,that is N n = " N ( n )1 N ( n )2 K ( n ) N ( n ) t , (2.26)with K ( n ) = K ( n ) t , N ( n )2 = N ( n ) t . It then follows that ˆ IN n = " K ( n ) N ( n ) t N ( n )1 N ( n )2 (2.27)is symmetric for all n . Now take the scalar product of both sides of the governingequation (2.13) by ˆ IN n ξ to get ξ ′ · ˆ IN n ξ = i ξ · ˆ IN n +1 ξ ; (2.28)finally add its complex conjugate to this equality to end up with ξ ′ · ˆ IN n ξ + ξ · ˆ IN n ξ ′ = 0 , (2.29)and, by integration between the interface (at ˆ x = 0) and infinity (where U and t , andthus ξ , vanish), arrive at (2.25).For instance, the boundary condition for Rayleigh surface waves is that there areno incremental tractions at the interface; thus ξ (0) = [ U (0) , ] t , and the fundamentalequations (2.25) reduce to U (0) · K ( n ) U (0) = 0 . (2.30) Here we take (ˆ x , ˆ x , ˆ x ) to coincide with the principal axes ( x , x , x ). The pre-deformation is thus ˆ x = λ X , ˆ x = λ X , ˆ x = λ X . (3.1)Figure 2 summarizes the situation with respect to the waves’ characteristics near theinterface. Bear in mind that the wave analysis is linear and gives no indication aboutthe amplitude; moreover, a half-space has no characteristic length so that the secularequation is non-dispersive and the wavelength remains undetermined.7 ropagationattenuation xxx Figure 2.
Incremental wave propagation localized near the surface of a semi-infinitedeformed solid. The analysis does not give the amplitude nor the wavelength.
For principal waves, the fields (2.11) and (2.12) are independent of ˆ x = x , becauseˆ x = x and n · ˆ x = ˆ x = x . Also, recall from (2.9) that the non-zero components of A in the ( x , x , x ) coordinate system of principal axes are A = λ W , A = λ λ W , A = λ W ,λ − A = λ − A = λ W − λ W λ − λ , A = λ W − λ W λ − λ λ λ , (3.2)and also A , A , A , A , A , A , A , A , and A ,whose expressions are not needed in this Section.From these observations follows that the third equation of motion (2.10) reduces to − S + i S ′ = − ρv U , (3.3)where by (2.7), i S = i A U , i S = A U ′ . (3.4)Hence the movement along the x principal axis is governed by an equation which dependsonly on U . For this equation, governing what is termed the anti-plane motion, wetake the trivial solution: U = 0, and we focus on the in-plane motion. According to(2.10) , , , it is governed by − S + i S ′ = − ρv U , − S + i S ′ = − ρv U , i U + U ′ = 0 , (3.5)8here by (2.7), i S = i( A + p ) U + A U ′ − P, i S = A U ′ + i( A + p ) U , i S = ( A + p ) U ′ + i A U , i S = i A U + ( A + p ) U ′ − P. (3.6)We eliminate p in favour of the pre-stress: by (2.6) at j = 2, we have p = λ W − σ andso by (3.2), A + p = A − σ . (3.7)It then follows from the second equation above that U ′ = i (cid:20) − A − σ A U + 1 A S (cid:21) , (3.8)and this constitutes the first line of the first-order system (2.13). The second line comesfrom the incremental incompressibility constraint (3.5) as U ′ = i [ − U ] . (3.9)Proceeding similarly for S ′ , S ′ , we find eventually that the governing equations areindeed in the form (2.13), where ξ = [ U , U , S , S ] t and − N , N , and − N aregiven by γ − σ γ , γ
00 0 , β + γ − σ ) 00 γ − ( γ − σ ) γ , (3.10)respectively, where we used the following short-hand notations (no sums), γ ij = A ijij = λ i λ − j γ ji , β ij = A iiii + A jjjj − A iijj − A ijji = 2 β ji . (3.11)or equivalently, γ ij = ( λ i W i − λ j W j ) λ i / ( λ i − λ j ) = λ i λ − j γ ji , β ij = λ i W ii − λ i λ j W ij + λ j W jj + 2( λ i W j − λ j W i ) λ i λ j / ( λ i − λ j ) = 2 β ji . (3.12) The propagation condition (2.16) reduces to a quadratic in q , γ q + (2 β − ρv ) q + γ − ρv = 0 . (3.13)Notice how σ , though present in N , does not appear explicitly in this equation.9alling q , q , the roots of the quadratic, we have q q = γ − ρv γ , q + q = − β − ρv γ . (3.14)The roots q , q of the biquadratic satisfying the decay condition (2.17) are in one ofthe two following forms; either: q = i β , q = i β , where β > β >
0, or: q = α + i β , q = − α + i β , where β >
0. Whatever the case, q q > q q <
0, and q + q isa purely imaginary quantity. From the first inequality we deduce that (Dowaikh andOgden, 1990) η = s γ − ρv γ (3.15)is a real quantity. From the second, and using the definitions of η and γ ij , we find q q = − η, ( q + q ) = γ − β γ − η − η = λ λ − − β γ − η − η . (3.16)We compute the eigenvectors ζ and ζ of N corresponding to q and q as anycolumn of the matrix adjoint to N − q I and to N − q I , respectively. Choosing thethird column, we find ζ = (cid:20) a b (cid:21) , ζ = (cid:20) a b (cid:21) , (3.17)where a j = " − q j γ , q j γ t , b j = (cid:2) − q j ( q j − σ ) , q j (1 − σ ) − η (cid:3) t , (3.18)and σ = σ /γ is a non-dimensional measure of the pre-stress.We can now construct the A = [ a | a ] and B = [ b | b ] matrices, and the surfaceimpedance matrix M = − i BA − . It turns out to be M = − i γ (cid:20) q + q − σ − η − (1 − σ − η ) ( q + q ) η (cid:21) , (3.19)which is indeed Hermitian because q + q is a purely imaginary quantity and η is real.For Rayleigh surface waves , the secular equation is (2.22), or here, using (3.16), η + η + (2 − λ λ − + 2 β γ − σ ) η − (1 − σ ) = 0 . (3.20)Dowaikh and Ogden (1990) established this form of the secular equation for principalsurface waves in pre-stressed incompressible solids, following other works by Hayes andRivlin (1961), Flavin (1963), Willson (1973a,b), Chadwick and Jarvis (1979a), Guz (2002,review with an extensive bibliography), and many others. It is of course consistent withLord Rayleigh’s own analysis of surface waves in linear isotropic incompressible solids.To check this, let the solid be un-stressed ( σ i = 0) and un-deformed ( λ i = 1); then η p − ρv /µ , where µ is the infinitesimal shear modulus; also, β = γ and η is the real root of η + η + 3 η − η ≃ . ρv /µ ≃ . Stoneley interface waves , the secular equation is (2.23) where M + M ∗ = − i (cid:20) γ ( q + q ) − γ ∗ ( q ∗ + q ∗ ) γ (1 − η ) − γ ∗ (1 − η ∗ ) − γ (1 − η ) + γ ∗ (1 − η ∗ ) γ ( q + q ) η − γ ∗ ( q ∗ + q ∗ ) η ∗ (cid:21) . (3.21)This equation was studied in great detail by Dowaikh and Ogden (1991) and by Chadwick(1995). It is consistent with the analysis of Stoneley (1924) of interface waves in linearisotropic incompressible solids. A remarkable feature of this secular equation for principalStoneley interface waves in deformed incompressible solids – first noted by Chadwick andJarvis (1979b) – is that the pre-stress σ does not appear explicitly in it, in contrast tothe equation for surface waves (3.20). This quantity, which is continuous across theinterface ( σ = σ ∗ ), disappears in the addition of the two surface impedance matrices.Of course it still plays an implicit role, in determining the pre-strain.Dowaikh and Ogden (1990, 1991), Chadwick (1995), and Guz (2002, review) havecovered almost every aspect of principal interface wave propagation and more informationcan be found in their respective articles. In the next Subsection we rapidly work out twoexamples of surface waves. First we present an example taken from the literature on elastomers, where theMooney-Rivlin strain energy function is often encountered. It is given by W = D ( λ + λ + λ − / D ( λ λ + λ λ + λ λ − / , (3.22)where D and D are positive constants with the dimensions of a stiffness. The Mooney-Rivlin material enjoys special properties with respect to wave propagation (the neo-Hookean material, which corresponds to the special case D = 0, enjoys even morespecial properties as is seen in Section 5.3). For instance, once subjected to a largehomogeneous pre-strain, it permits the propagation of bulk waves in every direction ;these waves can be infinitesimal, but also of arbitrary finite amplitude (Boulanger andHayes, 1992); they can be homogeneous plane waves but also inhomogeneous plane waves(Destrade, 2000, 2002). The quantities (3.12) are also quite special; they are γ ij = ( D + D λ k ) λ i , β ij = ( D + D λ k )( λ i + λ j ) , (3.23)where k = i, j , and thus they satisfy2 β ij = γ ij + γ ji . (3.24)These relationships mean that the biquadratic (3.13) factorizes to( q + 1)( q + η ) = 0 , (3.25)and that the secular equation (3.20) reduces to η + η + (3 − σ ) η − (1 − σ ) = 0 . (3.26)11ence one qualifying root is q = i, whatever the values of the material constants D and D . The other root is q = i η . When there is no pre-stress normal to the boundary( σ = 0), then η is the real root of η + η + 3 η − η ≃ . ρv = γ − γ η = ( D + D λ )( λ − . λ ) , (3.27)a result first established by Flavin (1963). Here q = i and q ≃ . completely independent of the pre-strain and of the material parameters D and D . We say that the penetration depth isuniversal relative to the class of Mooney-Rivlin materials.The second example is taken from the biomechanics literature. From a series ofuniaxial tests on human aortic aneurysms, Raghavan and Vorp (2000) deduced that thefollowing strain energy density gave a satisfying fit with the data plots, W = C ( λ + λ + λ −
3) + C ( λ + λ + λ − , (3.28)where, typically, C = 0 .
175 MPa, C = 1 . σ = 0, σ = σ = 0, leading through (2.6) to the following equi-biaxial pre-strain, λ = λ, λ = λ − / , λ = λ − / , (3.29)where λ is calculated from σ = 2( λ − λ − )[ C + 2 C (2 − λ − + λ − )] . (3.30)Finally, using the following expressions for the relevant moduli, γ = 2 C λ − + 4 C (2 λ − − λ − + λ − ) , γ = λ γ ,β = C ( λ + λ − ) + 2 C (4 λ − λ − λ − − λ − + 3 λ − ) , (3.31)it is a simple matter to solve the secular equation (3.20) numerically and plot the vari-ations of the squared wave speed, scaled with respect to the squared bulk wave speed γ /ρ , with the pre-stretch λ . Figure 3 b displays these variations; for comparison pur-poses, Figure 3 a shows the variations of the scaled squared wave speed in the case of aMooney-Rivlin material in uniaxial stress; in that later case the graph is independent ofthe material parameters D and D because by (3.27), ρv /γ = 1 − . λ − . Thedashed lines indicate the speed of Lord Rayleigh’s squared speed in the isotropic (nopre-strain) case where λ = 1, ρv /γ ≃ . λ ≃ . / ≃ .
444 for allMooney-Rivlin materials, as shown by Biot (1963), and at λ ≃ .
315 for the soft biolog-ical tissue model above. Beyond that critical compression stretch , v <
0, leading to apurely imaginary v , an amplitude which then grows exponentially with time accordingto (2.11), and a breakdown of the linearized analysis. The search for critical compressionstretches is an extremely active area of research, clearly linked to the geometric stabilityanalysis of solids. 12 λρ v / γ ρ v / γ λ Figure 3.
Variations of the scaled squared surface wave speed with the stretch in uniaxialpre-stress (a) for any Mooney-Rivlin material and (b) for the solid with strain energydensity (3.28) where C = 0 .
175 MPa, C = 1 . In this Section we study the case where ˆ x is aligned with the principal axis x but neitherˆ x (propagation direction) nor ˆ x (attenuation direction) are aligned with principal axes.The components ˆ A jilk in the (ˆ x ˆ x ˆ x ) coordinate system of the instantaneous elasticmoduli tensor A are related to the components A jilk , given by (3.2), in the ( x x x )coordinate system of principal axes through the tensor transformationsˆ A jilk = Ω jp Ω iq Ω lr Ω ks A pqrs , where Ω = cos Θ − sin Θ 0sin Θ cos Θ 00 0 1 , (4.1)and Θ is the angle between x and ˆ x .In particular we find that the non-zero components in the forms ˆ A lk and ˆ A lk are ˆ A , ˆ A , ˆ A , ˆ A , ˆ A , and ˆ A . The mechanical fields (2.11) and(2.12) are independent of ˆ x = x because n · ˆ x = ˆ x here. As a consequence, the thirdequation of motion (2.10) reduces to − S + i S ′ = − ρv U , (4.2)where by (2.7),i S = i ˆ A U + ˆ A U ′ , i S = i ˆ A U + ˆ A U ′ . (4.3)Hence the movement along the x principal axis is governed by an equation which dependsonly on U , and for this anti-plane motion we take the trivial solution: U = 0.The equations governing the in-plane motion have been derived in the case of ageneral plane pre-strain by Fu (2005a) and solved for surface waves in the case of a pre-strain consisting in a triaxial stretch followed by a simple shear by Destrade and Ogden(2005). Instead of treating these cases again, we revisit the case relative to one of the13ost important pre-strain fitting into the present context, that of finite simple shear ,presented originally by Connor and Ogden (1995) for surface waves.Figure 4 sketches what happens to a unit cube when a solid is subject to the simpleshear of amount K , ˆ x = X + KX , ˆ x = X , ˆ x = X . (4.4)Here the principal axes are x = X and x , x which make an angle ψ with X andwith X , respectively. That angle, and the corresponding principal stretches are (e.g.Chadwick (1976)), ψ = (1 /
2) tan − (2 /K ) , λ , = p K / ± K/ , λ = 1 . (4.5)These relations highlight a major difference between this homogeneous pre-strain andthe triaxial pre-stretch (3.1): here the orientation of the principal axes with respect tothe plane interface changes as the magnitude of the pre-strain changes. xx x xK < K > x x Figure 4.
Finite simple shear of amount K of a block near the interface. The ˆ x directionis the direction of shear ; the (ˆ x ˆ x ) plane is the plane of shear ; the (ˆ x ˆ x ) plane is the glide plane . The deformation (4.4) is an example of plane strain . As we focus on two-partialincremental waves in this Section, we may take advantage of formulas established byMerodio and Ogden (2002) in a similar context. The components of the deformationgradient tensor F and of the left Cauchy-Green strain tensor B for 2D pre-strain and2D incremental motions, in the (ˆ x , ˆ x ) coordinate system (aligned with the ( X , X )system), are F = (cid:20) K (cid:21) , B = (cid:20) K KK (cid:21) . (4.6)14or plane strain, λ = 1, so that in incompressible solids λ = λ − by (2.1). It followsby (2.4) that I = I . Accordingly, we define the single-variable function c W ( I ) by theidentity c W ( I ) = W ( I , I ) . (4.7)Then the 2D version of the constitutive equation (2.5) is σ = − ˆ p I + 2 c W B , (4.8)where ˆ p is the Lagrange multiplier due to the incompressibility constraint. Also, thecomponents of A are (Merodio and Ogden, 2002),ˆ A jilk = 2 c W δ ik B jl + 4 c W B ij B lk , (4.9)where c W = c W ′ ( I ), c W = c W ′′ ( I ). With the help of (4.6) , we find that in the (ˆ x , ˆ x )coordinate system, the non-zero components ˆ A jilk relevant to in-plane motion areˆ A = ˆ A = 4 c W K (1 + K ) , ˆ A = ˆ A = 2 c W K + 4 c W K (1 + K ) , ˆ A = ˆ A = 4 c W (1 + K ) , ˆ A = ˆ A = 4 c W K , ˆ A = 2 c W (1 + K ) + 4 c W K , ˆ A = 2 c W + 4 c W K , ˆ A = ˆ A = 2 c W K + 4 c W K, ˆ A = ˆ A = 4 c W K, ˆ A = 2 c W (1 + K ) + 4 c W (1 + K ) , ˆ A = 2 c W + 4 c W . (4.10)Now the remaining incremental governing equations (2.10) , , reduce to − S + i S ′ = − ρv U , − S + i S ′ = − ρv U , i U + U ′ = 0 , (4.11)where by (2.7),i S = i( ˆ A + ˆ p ) U + i ˆ A U + ˆ A U ′ + ˆ A U ′ − i ˆ P , i S = i ˆ A U + i ˆ A U + ( ˆ A + ˆ p ) U ′ + ˆ A U ′ . i S = i ˆ A U + i( ˆ A + ˆ p ) U + ˆ A U ′ + ˆ A U ′ , i S = i ˆ A U + i ˆ A U + ˆ A U ′ + ( ˆ A + ˆ p ) U ′ − i ˆ P , (4.12)where ˆ P is the increment of ˆ p . The pre-stress necessary to maintain the solid in thestatic state of large simple shear is (4.8). In particular, the ˆ σ component along the ˆ x axis is found using (4.6) and (4.7) asˆ σ = − ˆ p + 2 c W , (4.13)leading to the connection ˆ A + ˆ p = ˆ A − ˆ σ . (4.14)15ote that Connor and Ogden (1995) and Fu (2005a) keep σ rather than ˆ σ as a measureof the pre-stress. It is the component of the pre-stress along the principal axis x , whoseorientation changes with the pre-strain, in contrast to ˆ σ , the component of the Cauchypre-stress tensor along the unchanged normal to the interface. As pointed out by Hussainand Ogden (2000), it is ˆ σ which is continuous across the interface of two bonded shearedsolids.Following the same procedure as in Section 3.1, we find the Stroh formulation of thegoverning equations in the expected form (2.10), where ξ = [ U , U , S , S ] t and − N and N , are given by K − ˆ σ c W + 4 c W K , c W + 4 c W K
00 0 , (4.15)respectively, and − N by c W − ˆ σ ) − (4 c W − ˆ σ ) K − (4 c W − ˆ σ ) K c W K + 2ˆ σ − ˆ σ c W + 4 c W K . (4.16) The propagation condition (2.16) reduces to a quartic in q ,2( c W + 2 c W K ) q + 4( c W + 2 c W K ) Kq + [2( c W (2 + K ) − c W K (2 − K ) − ρv ] q + 4( c W − c W K ) Kq + 2 c W (1 + K ) + 4 c W K − ρv = 0 . (4.17)Notice how ˆ σ , though present in N , does not appear explicitly in this equation.Quite surprisingly, there are two instances – both pointed out by Connor and Ogden(1995, 1996) – where we can solve this quartic exactly and simply. The first instanceoccurs for incremental deformations , when v = 0; it is applicable whatever the strainenergy density W might be. The reason for the simplicity of this resolution is madeapparent by the change of unknown from q to e q = q − K/
2. Then the quartic becomes2( c W + 2 c W K ) e q + [ c W (4 − K ) − c W K (4 + K ) − ρv ] e q + ( ρv ) K e q + [( c W + 2 c W K )(4 + K ) − ρv ](4 + K ) / , (4.18)which is clearly a biquadratic at v = 0. The consequence is that any incremental static problem can be solved in its entirety for sheared solids because the roots q are accessibleexplicitly. This was to be expected though, because of an important theorem by Fu andMielke (2002) which states that “the buckling condition for a pre-stressed elastic half-space is independent of the orientation of the surface as long as the surface normal remains16n the ( x , x ) plane”; the buckling condition is what corresponds to the marginallystable static solution obtained at v = 0; it is also called the wrinkling condition , or the bifurcation criterion , or any other denomination associated with the onset of instabilityin the linearised (incremental) theory.The second instance where the quartic is easy to solve is when the solid is a Mooney-Rivlin material, see (3.22); this case is treated in Section 4.3.In general however, the quartic is difficult (but not impossible, see Fu (2005a,b),Destrade and Fu (2006), and the concluding Section) to solve analytically and othermethods, such as those relying on the fundamental equations (2.25), are required. Forthe time being, we complete the picture with formal calculations.Assuming the roots of the quartic have been computed, and calling them q , q , q , q , where q and q both satisfy the decay condition (2.17), we find that the eigenvectors ζ and ζ associated with q and q , respectively, are ζ = (cid:20) a b (cid:21) , ζ = (cid:20) a b (cid:21) , (4.19)where a j = [ q j , − q j ] t , b j = (cid:2) q j (cid:0) ˆ γ ( q j + Kq j −
1) + ˆ σ (cid:1) , − (ˆ γ − ˆ σ ) q j + ˆ ν q j + ˆ γ − ρv (cid:3) t . (4.20)Here, ˆ γ = 2 c W + 4 c W K , ˆ γ = 2 c W (1 + K ) + 4 c W K , ˆ β = c W (2 + K ) − c W K (2 − K ) , ˆ ν = 2 c W K − c W K . (4.21)Then we compute the A = [ a | a ] and B = [ b | b ] matrices, and eventually the surfaceimpedance matrix M = − i BA − , as M = − i ˆ γ ( K + q + q ) ˆ γ (1 + q q ) + ˆ σ ρv − ˆ γ q q − ˆ γ − ˆ σ ( ρv − ˆ γ ) q + q q q − ˆ ν . (4.22)Note that this matrix is indeed Hermitian; this is easy to show by using the quartic (4.17)and the definitions (4.21) to uncover the identities: q + q + q + q = − K,q q q q = (ˆ γ − ρv ) / ˆ γ ,q q ( q + q ) + q q ( q + q ) = − ν / ˆ γ . (4.23)These identities allow us to rewrite the surface impedance matrix as M = − i " ˆ γ ( q + q − q − q ) / γ (1 + q q ) − ˆ σ − ˆ γ (1 + q q ) + ˆ σ − ˆ γ [ q q ( q + q ) − q q ( q + q )] / . (4.24)17u (2005a) derived the same form of the surface impedance matrix for the more generalcase of any plane strain (any pre-strain where λ = 1), which includes the present caseof finite shear.Notice also how the pre-stress ˆ σ is going to be explicitly present in the secularequation for Rayleigh surface waves (2.22), but absent from the secular equation forStoneley interface waves (2.23). These secular equations remain implicit as long as theroots q and q are not known. In the general case where the quartic (4.17) is not solvablein a simple manner, we seek an explicit secular equation using the fundamental equations(2.25). Rayleigh surface waves.
With the explicit expressions (4.15) and (4.16) for the blocksof the matrix N , we can compute N − and N . The lower left corner of these givesin turn K (1) = N + ρv I and K ( − , K (2) , respectively. Then the equations (2.30)written at n = 1 , − ,
2, yield the linear homogeneous system K (1)11 K (1)12 K (1)22 K ( − K ( − K ( − K (2)11 K (2)12 K (2)22 U (0) U (0) U (0) U (0) + U (0) U (0) U (0) U (0) = . (4.25)The vanishing of the determinant of the 3 × ρv . It is too long to reproduce in general buteasy to obtain (and solve numerically) with a computer algebra system. Here we presentits expression in the case where ˆ σ = 0. Then, K (1) and K (2) are given by " ρv − c W c W c W ρv + 2 c W , " c W − ρv ) K c W (4 − K ) − ρv c W (4 − K ) − ρv − c W K , (4.26)respectively, the components of K ( − are K ( − = 2(2 c W − ρv ) ,K ( − = − c W (1 + K ) − ρv ] K,K ( − = 4 c W (2 + K ) + ρv ρv − c W (5 + K ) − c W K (1 + K ) c W + 2 c W K , (4.27)(up to an inessential common factor), and the secular equation is the quartic x − x + K K + K c W c W ! x − K K + K c W c W ! x + 8 c W + 2 c W K c W (4 + K ) = 0 , (4.28)where x is the following non-dimensional measure of the squared wave speed, x = ρv / [ c W (4 + K )]. 18s stated above, the secular equation is also a quartic in the squared wave speedwhen ˆ σ = 0. For a given material, a given pre-stress, and a given shear, its numericalresolution may yield more than one positive real root. If such is the case, then foreach corresponding speed, compute the roots to the quartic (4.17) and discard those(supersonic) speeds which do not give two complex conjugate pairs of roots. Finally, findwhich of the remaining speeds (if there is more than one) satisfies the secular equationwritten in optimal form (2.22). Stoneley shear-twin interface waves.
For Stoneley interface waves, the fundamen-tal equations (2.25) are not practical to derive secular equations in general, and we mustresort to other methods, such as those developed by Destrade and Fu (2006). The ex-ception is the special case when each half-space is filled with Mooney-Rivlin materials,because then the roots to the quartic (4.17) can be found easily, see Section 4.3.We now focus on the possibility of propagating incremental waves at a shear-twininterface . In this configuration, two solids, made of the same incompressible material, aresubject to equal and opposite shears, see Figure 5. The study of wave propagation at thistype of interface can have important repercussions in the non-destructive evaluation of atwinned interface because this bimaterial can “simulate the finite (plastic) deformationassociated with a crystal twin” (Hussain and Ogden, 2000). - KK xx
12 ^^ x Figure 5.
Stoneley wave propagation at a shear-twin interface. Both half-spaces areoccupied with the same incompressible solid, one subject to a simple shear of amount K , the other subject to a simple shear of amount − K . The analysis shows that Stoneleywaves cannot actually travel in the direction of shear.For one half-space, the propagation condition is the quartic (4.17). For the other half-space, the amount of shear is changed to its opposite, but c W and c W do not changesigns because they are functions of K . It follows that if q and q are qualifying rootsof the propagation condition in one half-space, then − q and − q are qualifying roots inthe other half-space. Also, we see from (4.21) that ˆ γ , ˆ γ , and ˆ β in one half-space19re equal to their counterparts in the other half-space, but that the counterpart to ˆ ν is − ˆ ν . Finally, ˆ σ is continuous across the interface. Then we conclude from (4.20) thatthe counterparts to the matrices A = (cid:20) A A A A (cid:21) , B = (cid:20) B B B B (cid:21) , (4.29)in one half-space are the matrices (Destrade, 2003; Ting, 2005) A ∗ = (cid:20) A A − A − A (cid:21) , B ∗ = (cid:20) − B − B B B (cid:21) , (4.30)in the other half-space. In turn, this conclusion leads to the following sum of surfaceimpedance matrices M + M ∗ = " M
00 2 M , (4.31)where M ∗ = i B ∗ ( A ∗ ) − . Consequently, the exact secular equation for Stoneley shear-twin interface waves is, according to (2.23), (4.24), and (4.31) that either ℑ ( q + q ) = 0 , or ℑ [ q q ( q + q )] = 0 . (4.32)However, as is easily proved, neither of these quantities can be zero when both q and q have positive imaginary parts. It follows that Stoneley waves cannot propagate inthe direction of sheared at a shear-twin interface , whatever the strain energy function is,and whatever what the shear is. Note however that Hussain and Ogden (2000) show, intheir study of reflection and transmission of plane waves at shear-twin interface, that anincident harmonic plane wave can give rise to an interfacial wave (but of a different typethan the Stoneley type).
For the Mooney-Rivlin strain energy density (3.22)we have c W = D / , c W = 0 , (4.33)where D = D + D is the shear modulus. The quartic propagation condition (4.17) thenfactorizes to ( q + 1)( q + 2 Kq + K + η ) = 0 , where η = p − ρv / D , (4.34)and its roots with positive imaginary parts are q = i , q = − K + i η. (4.35)We also haveˆ γ = D , ˆ γ = D (1 + K ) , ˆ β = D (2 + K ) , ˆ ν = D K, (4.36)20nd the surface impedance matrix of (4.24) reduces to M = D " η + 1 − K + i( η −
1) + iˆ σ / D− K − i( η − − iˆ σ / D η + η + K . (4.37)For Rayleigh surface waves in a sheared Mooney-Rivlin material, the secular equation(2.22) is a cubic in η , η + η + (3 + K − σ / D ) η − (1 − ˆ σ / D ) = 0 . (4.38)See Connor and Ogden (1995) for this equation with σ instead of ˆ σ , and Destrade andOgden (2005) for a generalization of this equation to a triaxial stretch followed by a shear.See also Figure 6 for the variations of the squared scaled wave speed ρv / D = 1 − η withthe amount of shear K , for several values of the pre-stress ˆ σ . In particular, the plotsat ˆ σ = ± D show that the Mooney-Rivlin material is unstable when subject only to ahydrostatic pressure of that amount (that is, ρv = 0 at K = 0) but regains stability assoon as it is sheared ( ρv > K = 0) . K r v / (cid:2) Figure 6.
Squared scaled surface wave speed ρv / D as a function of the amount of shear K for a Mooney-Rivlin material. For the solid plot, there is no pre-stress normal to theboundary (ˆ σ = 0); for the dashed plots, the displayed number indicates the value ofthe non-dimensional pre-stress ˆ σ / D . More plots are found in the article by Connor andOgden (1995).As is clear from the roots (4.35), from the secular equation (4.38), and from Figure 6,both the penetration depth and the speed are even functions of K . Also, once η is foundby solving the secular equation, we have ∂η/∂ ( K ) = − η / [2 η + η + (1 − ˆ σ / D ) ] < ∂ ℑ ( q ) /∂ ( K ) = ∂η/∂ ( K ) <
0, indicating that both the surface wave speed andthe penetration depth increase with the magnitude of the shear.21e note that having found an explicit expression (4.37) for the surface impedancematrix allows us to solve the problem of Stoneley interface waves in its entirety, for half-spaces made of different Mooney-Rivlin solids, subject to different amounts of shear; seeChadwick and Jarvis (1979b) for more discussion on this point.
Sheared Gent solids.
As an example of a strain energy density for which the quartic(4.17) is not easily solved, we now work with the following function, W = − C J m ln (cid:18) − I − J m (cid:19) , I < J m , (4.39)proposed initially by Gent (1996) to describe strain-stiffening elastomers and since thenextensively used by Horgan and Saccomandi (2003, and references therein) to modelstrain-stiffening soft biological tissues such as arteries. Here C > J m > limits the amount of shear by imposing − p J m < K < p J m . (4.40)Turning our attention to the propagation of surface waves, we compute the followingquantities, c W = C J m J m − K ) , c W = C J m J m − K ) . (4.41)To fix the ideas, we take two Gent materials, one with J m = 9 .
0, the other (stiffer) with J m = 1 .
0. Figure 7 shows the variations of the non-dimensional measure of the surfacewave speed p ρv / C with the amount of shear K . The bounds due to the inequalities(4.40) are clearly visible, indicating that the solids become more and more rigid as theirlimit of chain extensibility is approached. The speed is an even function of K and onlythe K > σ = 0), we solvenumerically the quartic secular equation (4.28) and find in general that there is two realpositive roots for ρv ; one gives a supersonic speed and is discarded; the other gives aspeed for which the exact secular equation (2.22) is satisfied and it is kept. The Figurealso shows (dotted plots) the effect of the compressive pre-stress ˆ σ = − . C , which is toslow the wave down in the small-to-moderate shear region; here again a quartic secularequation is solved numerically and only one speed is kept. In this Section we consider an interface wave propagating in a principal plane, x = 0 say,but not in a principal direction. We call θ the angle between the propagation directionˆ x and the principal axis x , see Figure 8. Although some results exist for non-principalStoneley waves (Destrade, 2005), the focus of this Section is on Rayleigh surface waves.22 [ ρ v /C ] K Figure 7.
Scaled surface wave speed p ρv / C as a function of the amount of shear K fortwo Gent materials. For the solid plots, there is no pre-stress normal to the boundary(ˆ σ = 0); for the dashed plots, the pre-stress is ˆ σ = − . C . The vertical asymptotescorrespond to the maximal amount of shear, that is K max = ± . J m = 9 .
0, and K max = ± . J m = 1 . Hence we model this motion as { u , ˙ p, s } = { U ( kx ) , i kP ( kx ) , i k S ( kx ) } e i k ( c θ x + s θ x − vt ) , (5.1)where c θ = cos θ , s θ = sin θ . By projecting the governing equations in the coordinate axisof the principal axes (where A has 15 independent non-zero components, see Section2.2), we can write them in the Stroh form (2.13) as a homogeneous linear system of sixfirst-order differential equations, where ξ = [ U , U , U , S , S , S ] t , see Destrade et al.(2005) for details. Here the matrices − N , N , − N are given by c θ ( γ − σ ) /γ c θ s θ s θ ( γ − σ ) /γ , /γ /γ , χ − κ ν − κ µ , (5.2)23 ll l x x q x x Figure 8.
Interface wave propagation when the boundary x = 0 is a principal plane ofpre-strain. The ˆ x direction is the direction of propagation, making an angle θ with theprincipal direction of pre-strain x .respectively, where χ = 2 c θ ( β + γ − σ ) + s θ γ ,ν = c θ [ γ − ( γ − σ ) /γ ] + s θ [ γ − ( γ − σ ) /γ ] ,µ = c θ γ + 2 s θ ( β + γ − σ ) ,κ = c θ s θ ( β − β − β − γ − γ + 2 σ ) , (5.3)and the γ ij , β ij are defined in (3.12).Rogerson and Sandiford (1999) show that the propagation condition (2.16) is a cubicin q , γ γ q − [( γ + γ ) X − c ] q + ( X − c X + c ) q + ( X − c )( X − c ) = 0 , (5.4)with X = ρv and c = ( γ γ + 2 β γ ) c θ + ( γ γ + 2 β γ ) s θ ,c = ( γ + γ + 2 β ) c θ + ( γ + γ + 2 β ) s θ ,c = ( γ γ + 2 β γ ) c θ + ( γ γ + 2 β γ ) s θ + [ γ γ + γ γ + γ γ − ( β − β − β ) + 4 β β ] c θ s θ ,c = γ c θ + γ s θ ,c = γ c θ + 2 β c θ s θ + γ s θ . (5.5)24 .2 Resolution for Rayleigh surface waves Finding the eigenvectors ζ , ζ , ζ of N corresponding to the qualifying roots q , q , q (say) is a lengthy task by hand, best left to a computer algebra system. In the end,we find that the ζ i are of the forms: ζ = (cid:20) a b (cid:21) , ζ = (cid:20) a b (cid:21) , ζ = (cid:20) a b (cid:21) , (5.6)where a j = a q i + a q i + a − q j + b q j + b q j d q j + d q j + d , b j = h q j + h q j ( ν − X )( q j + mq j + n ) g q j + g q j . (5.7)Here the quantities m and n are given by m = (cid:18) γ + ( γ − σ ) γ ( ν − X ) c θ (cid:19) [ η − X ] + (cid:18) γ + ( γ − σ ) γ ( ν − X ) s θ (cid:19) [ µ − X ] − κ ( γ − σ )( γ − σ ) γ γ ( ν − X ) c θ s θ , (5.8) n = (cid:26) (cid:20) ( γ − σ ) γ c θ + ( γ − σ ) γ s θ (cid:21) ( ν − X ) − (cid:27) × [( µ − X )( η − X ) − κ ] / ( γ γ ) . (5.9)The expressions for the quantities a i , b i , d i , h i , g i are too lengthy to reproduce but theyare easily obtained in a formal manner using a computer algebra system; as it turns out,these constants are not needed in the secular equation for Rayleigh surface waves. Indeed,the expression for the surface impedance matrix M is very lengthy, but its determinantfactorizes greatly. Hence we find that the exact secular equation for Rayleigh surfacewaves (2.22) reduces to (Taziev, 1987; Destrade et al., 2005) nω I − ω III ( m − ω II ) = 0 , (5.10)where ω I = − ( q + q + q ) , ω II = q q + q q + q q , ω III = − q q q . (5.11)This secular equation remains implicit as long as the roots q , q , q satisfying the decaycondition are not known. To compute them, we must find the wave speed, by using thefundamental equations (2.30).First, using the explicit expression (5.2) of the Stroh matrix N , we compute N − and N , and in particular we find explicit expressions for the lower left blocks K (1) = N + ρv I , K ( − , K (3) . Next, we use the result (Destrade, 2005) that U (0) is in theform U (0) = U (0)[1 , i α, β ] T , (5.12)25here α , β are real numbers, to write the fundamental equations (2.30) at n = − , , K ( − K ( − K ( − K (1)13 K (1)33 K (1)22 K (3)13 K (3)33 K (3)22 ββ α = − K ( − − K (1)11 − K (3)11 . (5.13)Then, we solve this non-homogeneous system by Cramer’s rule to find2 β = ∆ / ∆ , β = ∆ / ∆ , (5.14)where ∆, ∆ , and ∆ are the following determinants,∆ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K ( − K ( − K ( − K (1)13 K (1)33 K (1)22 K (3)13 K (3)33 K (3)22 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ∆ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − K ( − K ( − K ( − − K (1)11 K (1)33 K (1)22 − K (3)11 K (3)33 K (3)22 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ∆ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K ( − − K ( − K ( − K (1)13 − K (1)11 K (1)22 K (3)13 − K (3)11 K (3)22 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (5.15)Finally we write down the compatibility of equations (5.14) as∆ − = 0 , (5.16)which is the explicit secular equation for non-principal surface waves in deformed incom-pressible materials .In general this equation is a polynomial of degree 12 in ρv (Taziev, 1989; Destradeet al., 2005), easy to solve numerically. Of the 12 possible roots, we keep those which arereal, positive, and give a subsonic speed (that is a speed for which the bicubic (5.4) hasthree pairs of complex conjugate roots). We then test the remaining speeds against theexact secular equation (5.10). The neo-Hookean form of the strain energy density for an in-compressible isotropic solid is a sub-case of the Mooney-Rivlin form, namely D = 0 in(3.22) so that W = D ( λ + λ + λ ) / . (5.17)It leads to a stress-strain relationship (2.5) which is “linear” with respect to the leftCauchy-green strain tensor, σ = − p I + D B . (5.18)26he neo-Hookean model is unable to capture neither qualitatively nor quantitativelyexperimental data, over any range of deformations (Saccomandi, 2004). It is nonetheless ahighly popular model in the literature because it forms the basis of a statistical treatmentfor the molecular description of rubber elasticity (Treloar, 1949).With respect to wave propagation, it has more peculiar properties than the Mooney-Rivlin material, because here, γ ij = D λ i , β ij = D ( λ i + λ j ) , (5.19)which leads to great simplifications in the matrices N , N , N of (5.2).Now placing ourselves in the coordinates system (ˆ x , x , ˆ x ) attached to the wavepropagation, see Figure 8, we introduce the functions ˆ U i , ˆ S i ( i = 1 , , U i = Ω ij U j , ˆ S i = Ω ij S j , where Ω ij = c θ − s θ s θ c θ . (5.20)With these functions, the governing equations (2.13) decouple the anti-saggital motion[ ˆ U , ˆ S ] from its saggital counterpart (recall that the direction of propagation and thenormal to the interface define what is called the saggital plane .) For the latter motionwe find [ ˆ U ′ , ˆ U ′ , ˆ S ′ , ˆ S ′ ] t = i N [ ˆ U , ˆ U , ˆ S , ˆ S ] t , (5.21)with N = − σ / ( D λ ) 0 − ρv − ˆ χ − ρv − ˆ ν − σ , (5.22)whereˆ χ = D ( c θ λ + s θ λ + 3 λ − σ ) , ˆ ν = D [ c θ λ + s θ λ + λ (1 − σ ) ] , (5.23)and σ = σ / ( D λ ) is a non-dimensional measure of the pre-stress. The associatedpropagation condition is( q + 1)[ λ q + ( c θ λ + s θ λ ) − ρv / D ] = 0 . (5.24)In other words, the situation is formally the same as that for principal waves inMooney-Rivlin material, see Section 3.3. The conclusion is that the speed is given by ρv = D ( c θ λ + s θ λ − λ η ) , (5.25)where η is the real root of (3.26). Flavin (1963) established this result at σ = 0. As healso showed, the situation gets more complicated for Mooney-Rivlin solids, because thewave is no longer plane polarized for a triaxial pre-stretch.27 ooney-Rivlin solid. The Mooney-Rivlin strain energy density (3.22) at D = 0 doesnot lead to a decoupling of the saggital motion from the anti-saggital motion. However,as in Sections 3 and 4, we find that the propagation condition factorizes, here as( q + 1)( q − Sq + P ) = 0 , (5.26)where S = (cid:18) γ + 1 γ (cid:19) X − (cid:18) γ γ + γ γ (cid:19) c θ − (cid:18) γ γ + γ γ (cid:19) s θ ,P = ( X − γ c θ − γ s θ )( X − γ c θ − γ s θ ) / ( γ γ ) . (5.27)Flavin (1963) was the first to notice that q = i is an attenuation factor for non-principal waves in Mooney-Rivlin solids. Pichugin (2001) showed that a necessary andsufficient condition for the factorization (5.26) to occur is that the relations (3.24) hold;he also showed, completing earlier work by Willson (1973a), that another factorization ofthe general bicubic (5.4) also occurs, this time for any strain energy function , when two ofthe principal stretches of pre-strain are equal (equi-biaxial pre-strain). Finally note that( q + 1) always comes out as a factor in the propagation condition for inhomogeneouswaves in Mooney-Rivlin solids, whatever the direction of propagation is (not necessarilyin a principal plane as here), see Destrade (2002) for details.Thanks to the factorization (5.26), we can actually compute explicitly the quantities ω I , ω II , ω III of (5.11). Indeed, we now have q = i , q q = −√ P , q + q = i q √ P − S, (5.28)so that ω I = − i(1 + q √ P − S ) , − ω II = √ P + q √ P − S, ω
III = i √ P , (5.29)leading to an explicit and exact form of the secular equation (5.10), n (cid:16) q √ P − S (cid:17) + √ P (cid:16) m + √ P + q √ P − S (cid:17) = 0 . (5.30)This result allows us to investigate the influence of pre-stress on surface wave propaga-tion (see Destrade et al. (2005) for an example) and to address an important question inthe study of surface stability: how much can a Mooney-Rivlin half-space be compressedbefore it buckles? In Section 3 we found the critical stretch in a principal direction ( x ),indicating the appearance of wrinkles parallel to x , see Figure 2 for a visualization.However, could it be that wrinkles appeared earlier in the compression, in another di-rection? To answer this we take X (= ρv ) = 0 (onset of instability) and solve (5.30) for λ , for each value of θ .For instance, take the case of the following plane pre-strain, λ = λ, λ = λ − , λ = 1 , (5.31)28mposed on a half-space made of the Mooney-Rivlin solid with material parameters D = 2 . µ, D = 0 . µ, (5.32)where µ has the dimension of a stiffness (the shear modulus of this Mooney-Rivlin solid is( D + D ) / . µ ). Figure 9 displays the values of the critical stretch ratio, measuredin the principal direction x , for each angle θ and for several values of the pre-stress σ .The solid line corresponds to σ = 0. At θ = 0 we find the critical compression ratio λ (say) for wrinkles aligned along x ; hence the solid curve starts at λ = 0 .
544 asexpected from solving (3.27) with v = 0 and λ = λ = λ − . At θ = 0 we find that thecritical compression stretch is below λ ; it follows that λ is the absolute critical stretchof compression for our example (5.31), (5.32). Of course, for Mooney-Rivlin solids otherthan (5.32), or for pre-strains other than (5.31), or for solids other than Mooney-Rivlinsolids, we might end up with a different behaviour in compression. However the sameanalysis can be brought in each case to its conclusion, with no additional difficulty. l c q Figure 9.
Critical stretch ratio of compression for a Mooney-Rivlin material as a func-tion of the angle between the normal to the wrinkles and the principal axis of greatestcompression. The Mooney-Rivlin solid is subject to a finite plane strain compression( λ = 1); its material parameters are D = 2 . µ , D = 0 . µ ; the pre-stress normal to theboundary is given by σ /µ = -3.0, -1.0, 0.0, 1.0, 3.0 ( µ has the dimension of a stiffness). Gent solid.
We find that the elastic moduli (3.12) for Gent materials (4.39) are givenin general by γ ij = C J m J m + 3 − λ − λ − λ λ i , β ij = C J m J m + 3 − λ − λ − λ " λ i + λ j + 2( λ i − λ j ) J m + 3 − λ − λ − λ . (5.33)29ow consider that a half-space made of a Gent material with J m = 9 . x = 0 is free of tractions. Then the principalstretches are λ = p K / K/ , λ = 1 , λ = p K / − K/ , (5.34)and of course, λ + λ + λ − K .Figure 10 shows the variations of the surface wave speed in the plane of shear, withrespect to the angle between the direction of propagation (ˆ x ) and the direction of shear( X ). For a shear of amount K = 0 .
5, the principal axis of greatest stretch x makesan angle 37 . ◦ with the direction of shear, and the principal axis of smallest stretch x makes an angle 127 . ◦ with the direction of shear. For K = 1, those two angles are 31 . ◦ and 121 . ◦ , respectively; for K = 2, those two angles are 22 . ◦ and 112 . ◦ , respectively.Clearly, the surface wave reaches extremal values along those directions, indicating thatthe principal directions can be determined acoustically. K =
K =
K = [ r v /C ] q Figure 10.
Wave propagating in the plane of shear of a semi-infinite sheared Gent solidwith J m = 9 .
0: scaled speed p ρv /C as a function of the angle between the propagationdirection and the direction of shear, for several values of the amount of shear. This Chapter has presented several situations where it is possible to derive explicit secularequations in exact form and in polynomial form, for incremental waves propagatingalong the plane interface of one (or two) deformed semi-infinite hyperelastic solid (rigidly30onded solids). The subject of interface waves in general is broad and includes manyother situations and geometries such as, to name but a few: the addition of a layer of finitethickness (Love waves, Lamb waves, etc.), or of a fluid (Scholte waves for inviscid fluids,Stoneley waves for viscous fluids, etc.), the consideration of an anisotropy due to familiesof extensible fibres, of cylindrical or spherical coordinates, or of curved boundaries, andso on. Reading the works by Ogden (2003, 2004) and by Guz (2002) and the referencestherein gives a good overview of the vast panorama spanned by these other types ofinterface wave problems.This concluding Section expands on the reasons put forth to explain why it can beadvantageous at times to seek explicit secular equations rather than to turn to numericsoutright. Among other things, explicit secular equations in polynomial form (i) are easy to solve numerically, with the greatest precision required; (ii) are sometimes surprisingly simple and short, in spite of a complicated or evenunsolvable propagation condition; (iii) lend themselves to simple asymptotic treatments, leading to approximate analyticalexpressions for the wave speed; (iv) account for all the solutions satisfying the propagation condition and the boundarycondition at the interface.
Point (i) goes without saying because the numerical methods used to determine theroots of a polynomial are safe and robust. Of course, at most only one root corresponds tothe actual solution and all the other roots must be discarded as being spurious . So, oncea likely root is found numerically (that is a root ρv which is real and positive), it mustbe checked that at that speed the exact secular equation is satisfied. This routine checkis simple enough to perform (solve the propagation condition, find the correspondingimpedance matrix, check that (2.22) (for Rayleigh waves) or that (2.23) (for Stoneleywaves) is satisfied).Other numerical techniques to find the interface wave speed invoke the Barnett-Lothe-Stroh integral formalism (see Ting (1996) for an account), computational algorithms foreigenvalue problems Taylor (1981), or an algebraic Ricatti equation for the impedancematrix (see the Chapter by Fu in this book and the references therein) to determine M ( v ) for any numerical value of v ; then v is varied in order to satisfy the exact secularequation up to the desired precision. This however is not always an easy task, as seen inthe following example.Consider the Mooney-Rivlin solid with material parameters (5.32), maintained in astate of static pre-strain, with (Rogerson and Sandiford, 1999), λ = √ . , λ = √ . , λ = ( λ λ ) − , σ = 0 . µ. (6.1)Take the interface between the solid and the vacuum to be the principal plane x = 0and study the propagation of a Rayleigh surface wave in a direction close to x ( θ is closeto 90 ◦ in Figure 8.)Along x we have a principal wave, travelling with a speed v (90) say, found fromSection 3.2. Here (Destrade et al., 2005) we find that v (90) ≃ . p µ/ρ . On the other31and, a shear (homogeneous) bulk wave linearly polarized along x travels with speed v say, given by ρv = γ , and a shear (homogeneous) bulk wave linearly polarized along x travels with speed v say, given by ρv = γ . Here we find that v ≃ . p µ/ρ and that v ≃ . p µ/ρ , showing that the surface wave travels with a speed which is intermediate between those of the shear bulk waves. That principal wave is two-partial and polarized in the ( x x ) plane, which is why it can afford to be faster than the shearbulk wave polarized in the x direction. Also, it is isolated , in the sense that the transitiontoward a surface wave propagating in a direction θ = 90 ◦ is abrupt, because this latterwave is tri-partial and must therefore travel with a speed v ( θ ) say, which is less than thespeed of any homogeneous bulk shear wave, in particular less than p c /ρ and less than p c /ρ , where c and c are defined in (5.5). Figure 11 shows the variations of these 3speeds in the neighbourhood of the x axis and makes those features clear. . r v _ q bulk shear waves principal surface wavenon-principal surface wave_ m Figure 11.
Wave propagation in the x principal plane of semi-infinite Mooney-Rivlinsolid (5.32) deformed by (6.1): variations of scaled speed p ρv /µ as a function of theangle, near the x direction.Now if our numerical method for the surface wave speed relies on using the knownprincipal wave speed as an “initial guess”, then it might run into difficulties here becauseof the isolation of the principal wave. This special feature is characteristic of stronglyanisotropic crystals in linear elasticity. In incremental dynamics, it can occur at will,simply by deforming the solid sufficiently to create a strong strain-induced anisotropy.Also, as is clear from the Figure, the surface wave travels with a speed which isextremely close to that of the bulk shear wave p c /ρ . Hence at θ = 89 . ◦ , the speedof the former is given by p ρv /µ = 0 . . . ., (6.2)32hilst the speed of the latter is given by p c /µ = 0 . . . .. (6.3)Consequently, if a numerical scheme for the surface wave speed relies on increasing v insmall steps until det M ( v ) = 0 is satisfied to any desired precision, then it will have tobe pushed to the 14th significant digit to insure that the speed does not correspond to abulk wave! Here the explicit polynomial secular gives only two positive roots, and it is aroutine check to verify that only (6.2) satisfies the exact secular equation. For an interface wave propagating in any direction in a deformed solid , the propaga-tion condition is a sextic in general. Such is for instance the case for a wave propagatingin any direction in the glide plane of the sheared block in Figure 4. Although a sextic isunsolvable analytically, it is nonetheless possible to find a polynomial secular equationfor a Rayleigh surface wave, see Taziev (1989) or Ting (2004) for details. The polyno-mial turns out to be of degree 27 in ρv . For Stoneley interface waves it does not seempractical to look for a polynomial secular equation.For an interface wave propagating in any direction in a principal plane of pre-defor-mation, the propagation condition is the bicubic (5.4). Although formulas exist for theroots of a cubic, the analytical resolution of (5.4) proves to be complicated because itis not known a priori whether the roots are purely imaginary or have a non-zero realpart and accordingly, which formula should be used. However, the polynomial explicitsecular equation for a Rayleigh surface wave is of degree 12 in ρv as seen earlier. Thisis also the case in the symmetry plane of an orthorhombic or monoclinic crystal. Inthe symmetry plane of a cubic crystal, the degree of the polynomial comes down to 10,as noted by Taylor (1981). For a two-partial surface wave, coupled to electrical fieldsthrough piezoelectricity in 2mm crystals, the polynomial is also of degree 10 in ρv formetallized boundary conditions (there the propagation condition is also a bicubic, seeCollet and Destrade (2005)).For a two-partial (non-principal) interface wave polarized in a principal plane of pre-deformation, the propagation condition is a quartic, q + d q + d q + d q + d = 0 , (6.4)say (see (4.17) for the case of simple shear). In contrast to the case of the bicubic above,we know here what the form of the roots should be, because the roots of a quartic areeither two pairs of complex conjugate numbers, or one double real root with one pair ofcomplex conjugate numbers, or four real roots. Here only the first scenario is acceptablein order to satisfy the decay condition (2.17) and thus a single formula, always valid, isrequired; it can be found in the textbooks. First introduce in turn the quantities r , s ,and h , defined by r = d − d , s = d − d d + d , h = d − d d + d d − d , (6.5)the quantities λ and φ ∈ ]0 , π/ λ = (cid:0) h + r (cid:1) / , φ = arccos (cid:2) λ (cid:0) r + s − rh (cid:1)(cid:3) , (6.6)33nd the quantities z , z , and z , defined by z = 2 λ / cos( φ ) − r,z = 2 λ / cos( φ + 2 π/ − r,z = 2 λ / cos( φ + 4 π/ − r. (6.7)Then the qualifying roots are p = sign( s ) √ z − d + i( √− z + √− z ) ,p = − sign( s ) √ z − d + i( √− z − √− z ) , (6.8)where sign( s ) equals 1 if s is non-negative and − M , and ultimately the exactsecular equation, can be found. It then means that any interface wave problem can besolved exactly. Nonetheless it might still be rewarding to look for the polynomial secularequation, to check whether they turn out to be simple. For instance, the polynomialsecular equation for Rayleigh waves is a quartic in the squared speed when the solid issheared (Section 4.2), or tri-axially stretched and then sheared (Destrade and Ogden,2005), or when the wave is polarized in the symmetry plane of a crystal (Currie, 1979).Now consider a flexural wave travelling along the free edge of a thin (Love-Kirchhoff)orthotropic plate where the edge makes an arbitrary angle with a principal axis of sym-metry. Thompson et al. (2002) show that the corresponding propagation condition isa quartic, but turn to a numerical resolution. However Fu (2003) shows that the poly-nomial secular equation is just a cubic in the squared speed . Finally consider the caseof a piezoacoustic Bleustein-Gulyaev surface wave in a rotated Y -cut about the Z axisfor 4 crystals. There also the propagation condition is a quartic, but the fundamentalequations reveal that the polynomial secular equation for metallized boundary conditionsis simply a quadratic in the squared speed (Collet and Destrade, 2004). Clearly it is aworthy enterprise to unearth these polynomials rather than use numerics or the formulas(6.5)-(6.8). Once the polynomial secular equation is established, in the form P ( v ) = 0, say, it is astraightforward matter to derive an analytical approximation for v . Calling v an initialapproximation, we find in the first order that v ≃ v − P ( v ) / P ′ ( v ) . (6.9)Of course we must choose v judiciously for an optimal expression.Here we work out a simple example. We take a solid half-space subject to a hydrostaticpressure only, so that λ = λ = λ ≡ λ say, and σ = σ = σ ≡ σ say. Fromthe incompressibility constraint (2.1), λ = 1 follows and we have a pre-stressed , but unstrained , solid. It is thus isotropic and a surface wave propagates with the same speedin every direction. To derive the secular equation, we specialize for instance (3.20) to the34sotropic case. We find that γ = γ = β = µ , the infinitesimal shear modulus, andthe exact secular equation is (Dowaikh and Ogden, 1990) f ( η ) = η + η + (3 − σ ) η − (1 − σ ) = 0 , (6.10)where σ ≡ σ/µ and η = p − ρv /µ . Calling c ≡ p ρv /µ a dimensionless measure ofthe wave speed, and multiplying (6.10) by f ( − η ), gives the polynomial secular equation(Dowaikh and Ogden, 1990), P ( v ) ≡ c − − σ ) c + 6( σ − c + ( σ − σ − = 0 . (6.11)In the region where c is close to 1, the first order approximation (6.9) gives c ≃ −P (1) / P ′ (1), that is c ≃ − σ + 6 σ + 4 σ − σ − σ + 12 σ . (6.12)In particular we find that for an unstressed incompressible solid, c ≃ /
22 = 0 . . . . ,which is a good approximation to the value found from the exact equation 0.9553. . . (LordRayleigh, 1885). On Figure 12 we superpose the approximate and exact curves, and findgood agreement in the σ = − σ = 1 region. Beyond that range, c departs from theneighbourhood of 1, and a better choice must be made for the initial guess. This pointis not developed further here but it is clearly exposed in the paper by Mozhaev (1991). c σ Figure 12.
Surface wave speed in an incompressible solid subject to hydrostatic pressure only.Solid plot: exact calculation, dashed plot: approximate calculation.
Taylor (1981) and Taziev (1989) both remarked that many of the roots to an explicitsecular equation in polynomial form have a physical origin, because they correspond35o a motion satisfying both the equations of motion and the boundary conditions atthe interface. In particular for the solid/vacuum interface, the polynomial provides theparameters not only for the Rayleigh surface wave, but also for pseudo-surface waves,for Brewster reflection of bulk waves, for the reflection of inhomogeneous waves, for“organ-pipe” modes, etc.One of the most important root is that corresponding to a leaky surface wave (if itexists), whose energy is diffused slowly into the half-space, and for which the classicalmethods evoked in Section 6.1 run into major difficulties.Another application for this wealth of information is that many more wavefronts canbe found, in particular those observed in the neighbourhood of cusps, which correspondto the interference of inhomogeneous plane waves (Huet, 2006). Figure 13 shows theremarkable correspondence obtained between experiment and predictions, for wavefrontspropagating on the surface of a copper sample (cubic symmetry). Here the explicit secularequation proves useful because it is easy to differentiate, as required for the derivationof the curve. –1–0.500.51–1 –0.5 0.5 1
Figure 13.
Wavefronts on the surface of copper cut along a symmetry plane. Left: experi-mental results obtained by LASER impact (Huet, 2006). Right: predictions derived from thesecular equation in polynomial form, showing in black, the Rayleigh wave wavefront and in grey,the wavefronts due to leaky waves and interference of inhomogeneous waves.
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