DSpace at VNU: On Rayleigh waves in incompressible orthotropic elastic solids

On Rayleigh waves in incompressible orthotropic elastic solids Ray W Ogdena) Department of Mathematics, University of Glasgow, Glasgow G12 8QW, United Kingdom Pham Chi Vinh Faculty of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnam ͑Received June 2003; accepted for publication November 2003͒ In this paper the secular equation for the Rayleigh wave speed in an incompressible orthotropic elastic solid is obtained in a form that does not admit spurious solutions It is then shown that inequalities on the material constants that ensure positive definiteness of the strain–energy function guarantee existence and uniqueness of the Rayleigh wave speed Finally, an explicit formula for the Rayleigh wave speed is obtained © 2004 Acoustical Society of America ͓DOI: 10.1121/1.1636464͔ PACS numbers: 43.20.Jr, 43.20.Bi, 43.20.Gp ͓JGH͔ I INTRODUCTION Rayleigh ͑surface͒ waves were first studied by Rayleigh ͑1885͒ for a compressible isotropic elastic solid The extension of surface wave analysis to anisotropic elastic materials has been the subject of many studies; see, for example, Stoneley ͑1963͒; Chadwick and Smith ͑1977͒; Royer and Dieulesaint ͑1984͒; Mozhaev ͑1995͒; Destrade ͑2001a͒; Ting ͑2002a, c͒, and references contained therein Some recent work has focused on incompressible anisotropic elastic solids ͑Nair and Sotiropoulos, 1997, 1999; Destrade, 2001b; Destrade et al., 2002͒, while application to materials, compressible or incompressible, subject to prestress has also attracted considerable attention ͑Dowaikh and Ogden, 1990, 1991; Chadwick, 1997, for example͒ Since Green’s functions for many elastodynamic problems for a half-space require the solution of the secular equation for Rayleigh waves, formulas for the Rayleigh wave speed in various elastic media are clearly of practical as well as theoretical interest A formula for the Rayleigh wave speed in compressible isotropic solids was first obtained by Rahman and Barber ͑1995͒ for a limited range of values of the parameter ⑀ ϵ␮/͑␭ϩ2␮͒, where ␭ and ␮ are the Lame´ constants For any range of values of ⑀ a formula was obtained by Nkemzi ͑1997͒; see also Malischewsky ͑2000͒ Recently, for some special cases of compressible monoclinic materials with symmetry plane x ϭ0, formulas for the Rayleigh wave speed were found by Ting ͑2002b͒ and Destrade ͑2003͒ as the roots of quadratic equations Rayleigh waves in incompressible orthotropic elastic materials were examined recently by Destrade ͑2001b͒ Destrade used the method of first integrals proposed by Mozhaev ͑1995͒ and found a form of the secular equation He used this to prove that Rayleigh waves exist and are unique in these materials for all values of the relevant material constants However, the form of the secular equation obtained by use of Mozhaev’s method necessarily admits spurious solutions Thus, the analysis of Destrade requires a͒ Electronic mail: rwo@maths.gla.ac.uk 530 J Acoust Soc Am 115 (2), February 2004 Pages: 530–533 some modification The secular equation for Rayleigh waves in incompressible orthotropic materials presented recently by Destrade et al ͑2002͒ also admits spurious solutions The aim of the present paper is to obtain a formula for the Rayleigh wave speed in an incompressible orthotropic elastic material For this purpose a form of the secular equation that does not admit spurious solutions is required In Sec II the basic equations and notation are presented for describing motion in an incompressible orthotropic elastic material We consider a half-space whose boundary is a symmetry plane of the material Since the equations for timeharmonic waves propagating parallel to the boundary of this half-space decouple into a plane motion, in the plane defined by the half-space normal and the direction of propagation, and a motion normal to that plane ͑see, for example, Destrade, 2001b͒, it suffices to consider the plane strain case In Sec III the secular equation for Rayleigh waves is derived in the desired form and it is shown how it relates to the form given by Destrade ͑2001b͒ Existence and uniqueness results are presented in Sec IV In particular, it is shown that Rayleigh waves exist and are unique in an incompressible orthotropic elastic solid, provided the inequalities ␥ ϵc 66Ͼ0, ␦ ϵc 11ϩc 22Ϫ2c 12Ͼ0 ͑1͒ are satisfied, where c 11 , c 12 , c 22 , and c 66 are material constants associated with the considered plane, which, in Cartesian coordinates, is taken to be the (x ,x ) plane These inequalities are necessary and sufficient for the strain energy ͑specialized to the considered plane͒ to be positive definite It is also easy to show that they are necessary and sufficient for strong ellipticity to hold for the considered motion An explicit formula for the Rayleigh speed is derived in Sec V by using the theory of cubic equations Corresponding results for the compressible theory will be discussed in a separate paper II BASIC EQUATIONS Let (x ,x ,x ) be Cartesian coordinates and consider an orthotropic elastic material occupying the half-space x Ͻ0, with traction-free boundary x ϭ0, which is a plane of sym- 0001-4966/2004/115(2)/530/4/$20.00 © 2004 Acoustical Society of America Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms Download to IP: 142.104.240.194 On: Tue, 16 Dec 2014 09:51:40 metry for the orthotropy We consider a plane motion in the (x ,x ) plane with displacement components (u ,u ,u ) such that u i ϭu i ͑ x ,x ,t ͒ , iϭ1,2, u ϵ0, ͑2͒ where t is time The nonzero components ⑀ i j of the infinitesimal strain tensor are given by ⑀ i j ϭ 12 ͑ u i, j ϩu j,i ͒ , ͑3͒ i, jϭ1,2, where a comma signifies partial differentiation with respect to spatial variables For an incompressible material, we have u 1,1ϩu 2,2ϭ0, ͑4͒ from which we deduce the existence of a scalar function, denoted ␺ (x ,x ,t), such that u ϭ ␺ ,2 , u ϭϪ ␺ ,1 ͑5͒ For the considered motion the relevant components of the stress are given by, for example, Nair and Sotiropoulos ͑1997͒ Thus, ␴ 11ϭϪpϩc 11⑀ 11ϩc 12⑀ 22 , ␴ 22ϭϪpϩc 12⑀ 11ϩc 22⑀ 22 , ␴ 12ϭ2c 66⑀ 12 , ͑6͒ where ␴ i j , iϭ1, 2, are components of the stress tensor, c i j are elastic constants of the material in standard notation, and pϭp(x ,x ,t) is the hydrostatic pressure associated with the incompressibility constraint Note that, in general, ␴ 33 0, but we shall not need to use this component of stress For the strain energy to be positive definite for the considered plane motion, the elastic constants c i j in Eq ͑6͒ must satisfy the inequalities given in Eq ͑1͒ In the absence of body forces the relevant components of the equation of motion are 11,1 12,2 uă , 12,1 22,2 uă , where dots denote partial differentiation with respect to t, and ␳ is the mass density of the material Use of Eqs ͑3͒, ͑5͒, and ͑6͒ in Eq ͑7͒ and elimination of p by cross differentiation leads to an equation for ␺, namely ,11112 ,1122 ,2222 ă ,11 ă ,22 , x ,x ,t ͒ →0 as x →Ϫϱ ͑12͒ III RAYLEIGH WAVES: SECULAR EQUATION We now consider harmonic waves propagating in the x direction, and we write ␺ in the form ␺ ͑ x ,x ,t ͒ ϭ ␾ ͑ y ͒ exp͓ ik ͑ x Ϫct ͔͒ , ͑13͒ where k is the wave number, c is the wave speed, yϭkx , and the function ␾ is to be determined Substitution of Eq ͑13͒ into Eq ͑8͒ yields ␥␾ ЉЉ Ϫ ͑ ␤ Ϫ ␳ c ͒ ␾ Љ ϩ ͑ ␥ Ϫ ␳ c ͒ ␾ ϭ0, ͑14͒ where, in Eq ͑14͒ and the following, a prime on ␾ indicates differentiation with respect to y In terms of ␾ the boundary conditions ͑11͒ become ␾ Љ ͑ ͒ ϩ ␾ ͑ ͒ ϭ0, ␥␾ ٞ ͑ ͒ ϩ ͑ ␥ Ϫ ␦ ϩ ␳ c ͒ ␾ Ј ͑ ͒ ϭ0, ͑15͒ in the first of which we have omitted the factor ␥ on the assumption that ␥ From Eq ͑12͒ we also require that ␾ ͑ x ͒ →0 as x →Ϫϱ ͑16͒ Thus, the problem is reduced to solving Eq ͑14͒ with the boundary conditions ͑15͒ and ͑16͒ The general solution for ␾ (y) that satisfies the condition ͑16͒ is ␾ ͑ y ͒ ϭA exp͑ s y ͒ ϩB exp͑ s y ͒ , ͑17͒ where A and B are constants, while s and s are the solutions of the equation ␥ s Ϫ ͑ ␤ Ϫ ␳ c ͒ s ϩ ͑ ␥ Ϫ ␳ c ͒ ϭ0, ͑18͒ with positive real parts From Eq ͑18͒ it follows that s 21 ϩs 22 ϭ ͑ ␤ Ϫ ␳ c ͒ / ␥ , s 21 s 22 ϭ ͑ ␥ Ϫ ␳ c ͒ / ␥ ␤ ϭ ␦ Ϫ2 ␥ , ͑9͒ and ␥ and ␦ are defined by Eq ͑1͒ In terms of the stress components the traction-free boundary conditions are written on x ϭ0 ͑10͒ Using Eqs ͑3͒, ͑5͒, ͑6͒, and the first of ͑7͒, Eq ͑10͒ can be expressed as conditions on ␺ This requires differentiation of ␴ 22 with respect to x so as to facilitate elimination of the term in p, as in Dowaikh and Ogden ͑1990͒ for Rayleigh waves on a prestrained half-space of incompressible isotropic elastic material The resulting boundary conditions are ␥ ͑ ␺ ,22Ϫ ␺ ,11͒ ϭ0, ␥ ͑ ␺ ,222Ϫ ␺ ,112 ,112 ă ,20 19 If the roots and of the quadratic Eq ͑18͒ are real, then they must be positive to ensure that s and s can have a positive real part If they are complex then they are conjugate In either case the product s 21 s 22 must be positive and hence a real ͑surface͒ wave speed c satisfies the inequalities s 21 s 22 0Ͻ ␳ c Ͻ ␥ where ␴ 12ϭ ␴ 22ϭ0 We shall also require that ͑20͒ Note that the limiting wave speed such that ␳ c ϭ ␥ is the speed of a shear body wave, not a surface wave Substituting Eq ͑17͒ into the boundary conditions ͑15͒, we obtain the equations ͑ 1ϩs 21 ͒ Aϩ ͑ 1ϩs 22 ͒ Bϭ0, ͓ ␥ ͑ s 21 ϩ1 ͒ ϩ ␳ c Ϫ ␦ ͔ s Aϩ ͓ ␥ ͑ s 22 ϩ1 ͒ ϩ ␳ c Ϫ ␦ ͔ s Bϭ0, ͑21͒ for A and B For nontrivial solution the determinant of coefficients of the system ͑21͒ must vanish After removal of the factor (s Ϫs ), this yields ␥ ͑ s 21 ϩs 22 ϩs 21 s 22 ͒ ϩ ͑ ␦ Ϫ ␳ c ͒ s s ϩ ␥ Ϫ ␦ ϩ ␳ c ϭ0 ͑22͒ Use of Eq ͑19͒ in Eq ͑22͒ then leads to on x ϭ0 J Acoust Soc Am., Vol 115, No 2, February 2004 ͑11͒ ͑ ␦ Ϫ ␳ c ͒ ͱ1Ϫ ␳ c / ␥ Ϫ ␳ c ϭ0, R W Ogden and P Chi Vinh: Rayleigh waves in orthotropic elastic solids ͑23͒ 531 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms Download to IP: 142.104.240.194 On: Tue, 16 Dec 2014 09:51:40 which is the required secular equation for the wave speed through ␳ c Note that for this equation to have a real solution for c it is necessary, in addition to Eq ͑20͒, that the inequality ␳ c 2Ͻ ␦ , ͑23͒ has no real nonzero solution for c whatever the sign of ␥, although it is not physically meaningful to admit nonpositive values of these constants ͑24͒ must hold Note that a solution of Eq ͑22͒ with s ϭs is admissible, but it can be shown that this corresponds to a real surface wave if and only if ␦ϭ32␥/9 and ␳ c ϭ ␦ /4, a solution that is given by ͑23͒ for this very restricted combination of material constants Of course, in this case, the solution ͑17͒ requires modification By rearranging Eq ͑23͒ and squaring to eliminate the square root, we obtain the secular equation derived by Destrade ͑2001b͒, which, in the present notation, can be written ͑ ␦ Ϫ ␳ c ͒ ͑ 1Ϫ ␳ c / ␥ ͒ ϭ ͑ ␳ c ͒ ͑25͒ Destrade ͑2001b͒ used the method of first integrals proposed by Mozhaev ͑1995͒ and needed to impose the restriction ␳ c ␦ , which in our derivation is automatically satisfied through Eq ͑24͒ Using this equation, Destrade concluded that a unique Rayleigh wave exists in an incompressible orthotropic elastic material for any values of ␥ and ␦ However, Eq ͑25͒ may have spurious solutions for ␳ c that are not solutions of Eq ͑23͒, and it is therefore advisable to avoid drawing conclusions on the basis of Eq ͑25͒ V A FORMULA FOR THE WAVE SPEED In this section we derive an explicit formula for the wave speed, given that ␥Ͼ0, ␦Ͼ0, by seeking the unique root, ␩ say, of Eq ͑26͒ in the interval ͑0,1͒ The wave speed c is then given by ␳ c ϭ ␥ ͑ 1Ϫ ␩ 20 ͒ ͑29͒ We now show that the cubic Eq ͑26͒ has only one real root, namely ␩ , the other two being complex According to the theory of cubic equation ͑see, for example, Cowles and Thompson, 1947 or Abramowitz and Stegun 1974͒, the nature of the three roots of the cubic ␩ ϩa ␩ ϩa ␩ ϩa ϭ0, is determined by the sign of the discriminant D defined by DϭR ϩQ , ͑31͒ where R and Q are given in terms of the coefficients a , a , a by Rϭ 541 ͑ 9a a Ϫ27a Ϫ2a 32 ͒ , IV EXISTENCE AND UNIQUENESS OF RAYLEIGH WAVES We now show that the inequalities ␥Ͼ0 and ␦Ͼ0 jointly ensure the existence and uniqueness of a Rayleigh wave For this purpose it is convenient to introduce the new variable ␩ ϭ ͱ1Ϫ ␳ c / ␥ so that the secular equation ͑23͒ may be rewritten as f ͑ ␩ ͒ ϵ ␩ ϩ ␩ ϩ ͑ ␦ / ␥ Ϫ1 ͒ ␩ Ϫ1ϭ0, 0Ͻ ␩ Ͻ1 ͑26͒ Then f ͑ ͒ ϭϪ1Ͻ0, f ͑ ͒ ϭ ␦ / ␥ Ͼ0, ͑27͒ which guarantees that Eq ͑26͒ has at least one solution in the interval ͑0,1͒ We also have f Ј ͑ ␩ ͒ ϭ3 ␩ ϩ2 ␩ ϩ ␦ / ␥ Ϫ1, f Љ ͑ ␩ ͒ Ͼ0 ͑ ␩ Ͼ0 ͒ ͑28͒ If ␦у␥ then it follows that f Ј ( ␩ )Ͼ0 for ␩Ͼ0 and hence f is monotonic increasing for ␩Ͼ0 In this case the solution for ␩ is unique If, on the other hand, 0Ͻ␦Ͻ␥ then f Ј (0)Ͻ0 Thus, f has a maximum for ␩Ͻ0 and a minimum for ␩Ͼ0 By the inequality in Eq ͑28͒ f therefore decreases to a minimum as ␩ increases from 0, and thereafter increases monotonically Hence, the solution is also unique in this case We therefore conclude that in an incompressible orthotropic elastic half-space there exists a unique Rayleigh wave provided the material constants satisfy the conditions ͑1͒, which ensure that the strain–energy function is positive definite for the considered plane strain restriction We note in passing that if ␦р0 then it can be seen immediately that Eq 532 J Acoust Soc Am., Vol 115, No 2, February 2004 ͑30͒ Qϭ 91 ͑ 3a Ϫa 22 ͒ ͑32͒ If DϾ0, Eq ͑30͒ has one real root and two complex conjugate roots If Dϭ0, the equation has three real roots, at least two of which are equal If DϽ0, Eq ͑30͒ has three distinct real roots In the first case (DϾ0) the single real root ␩ is given by Cardano’s formula ͑Cowles and Thompson, 1947; Abramowitz and Stegun, 1974͒ in the form ␩ ϭϪ 13 a ϩ ͑ Rϩ ͱD ͒ 1/3ϩ ͑ RϪ ͱD ͒ 1/3 ͑33͒ For the secular equation in the form Eq ͑26͒, we have a ϭϪ1, a ϭ⌬Ϫ1, a ϭ1, ͑34͒ and hence Rϭ ⌬ ϩ , 27 ⌬ Qϭ Ϫ , ͑35͒ where ⌬ϭ␦/␥ Using Eq ͑35͒ in Eq ͑31͒, it is easy to verify that Dϭ 108 ⌬ ͑ 4⌬ Ϫ13⌬ϩ32͒ ͑36͒ It is clear from Eq ͑36͒ that DϾ0 provided ⌬Ͼ0 Thus, Eq ͑30͒ has only one real root, necessarily within the required range of values Use of Eqs ͑34͒, ͑35͒, and ͑36͒ in Eq ͑33͒ leads to ␩ ϭ 13 ͓ Ϫ1ϩ ͱ͓ 9⌬ϩ16ϩ3 ͱ3 ͱ⌬ ͑ 4⌬ Ϫ13⌬ϩ32͔͒ /2 ϩ ͱ͓ 9⌬ϩ16Ϫ3 ͱ3 ͱ⌬ ͑ 4⌬ Ϫ13⌬ϩ32͔͒ /2͔ ͑37͒ From Eqs ͑29͒ and ͑37͒ the speed c of the Rayleigh wave is given by R W Ogden and P Chi Vinh: Rayleigh waves in orthotropic elastic solids Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms Download to IP: 142.104.240.194 On: Tue, 16 Dec 2014 09:51:40 FIG Plot of ␳ c / ␥ against ⌬͑Ͼ0͒ from Eq ͑38͒ ␳ c / ␥ ϭ1Ϫ 91 ͓ Ϫ1 ϩ ͱ͓ 9⌬ϩ16ϩ3 ͱ3 ͱ⌬ ͑ 4⌬ Ϫ13⌬ϩ32͔͒ /2 ϩ ͱ͓ 9⌬ϩ16Ϫ3 ͱ3 ͱ⌬ ͑ 4⌬ Ϫ13⌬ϩ32͔͒ /2͔ ͑38͒ For an ͑incompressible͒ isotropic material c 11ϭc 22 , c 11Ϫc 12ϭ2 ␮ , and c 66ϭ ␮ , where ␮ is the classical shear modulus, and hence, by Eq ͑1͒, ⌬ϭ4 In this case the formula ͑38͒ specializes to ␳ c / ␥ ϭ1Ϫ 91 ͓ 3ͱ6 ͱ33ϩ26Ϫ 3ͱ6 ͱ33Ϫ26Ϫ1 ͔ ͑39͒ This is approximately 0.9126, which is the classical value for an incompressible isotropic elastic solid ͑see, for example, Ewing et al., 1957͒ In Fig a plot of ␳ c / ␥ against ⌬͑Ͼ0͒ based on Eq ͑38͒ is shown in order to illustrate the dependence of the wave speed on the ratio of material constants The wave speed is very small for small ⌬ and increases rapidly as ⌬ increases, reaching its isotropic value for ⌬ϭ4 and then approaching an asymptotic value with ␳ c / ␥ →1 as ⌬ becomes very large The asymptotic limit corresponds to a wave speed equal to the shear wave speed Note that ␦ may be interpreted as a shear modulus of the material; indeed, in the isotropic case ␦ϭ2␮, where ␮ is the Lame´ shear modulus Thus, the limit ⌬→0 ͑which is not applicable for isotropic materials͒ corresponds to a material with one vanishingly small shear modulus Similarly, ␥ is a shear modulus and, if ␦ 0, in the limit ␥→0 we have ⌬→ϱ Thus, we have interpretations for the two extreme values of ⌬ ACKNOWLEDGMENTS The work is partly supported by the Ministry of Education and Training of Vietnam and completed during a visit of the second author to the Department of Mathematics, University of Glasgow, UK J Acoust Soc Am., Vol 115, No 2, February 2004 Abramowitz, M., and Stegun, I A ͑1974͒ Handbook of Mathematical Functions ͑Dover, New York͒ Chadwick, P ͑1997͒ ‘‘The application of the Stroh formulation to prestressed elastic media,’’ Math Mech Solids 2, 379– 403 Chadwick, P., and Smith, G D ͑1977͒ ‘‘Foundations of the theory of surface waves in anisotropic elastic materials,’’ Adv Appl Mech 17, 303– 376 Cowles, W H., and Thompson, J E ͑1947͒ Algebra ͑Van Nostrand, New York͒ Destrade, M ͑2001a͒ ‘‘The explicit secular equation for surface acoustic waves in monoclinic elastic crystals,’’ J Acoust Soc Am 109, 1398 – 1402 Destrade, M ͑2001b͒ ‘‘Surface waves in orthotropic incompressible materials,’’ J Acoust Soc Am 110, 837– 840 Destrade, M ͑2003͒ ‘‘Rayleigh waves in symmetry planes of crystals: explicit secular equations and some explicit wave speeds,’’ Mech Mater 35, 931–939 Destrade, M., Martin, P A., and Ting, T C T ͑2002͒ ‘‘The incompressible limit in linear anisotropic elasticity, with application to surface waves and elastostatics,’’ J Mech Phys Solids 50, 1453–1468 Dowaikh, M A., and Ogden, R W ͑1990͒ ‘‘On surface waves and deformations in a pre-stressed incompressible elastic solid,’’ IMA J Appl Math 44, 261–284 Dowaikh, M A., and Ogden, R W ͑1991͒ ‘‘On surface waves and deformations in a compressible elastic half-space,’’ Stab Appl Anal Cont Media 1, 27– 45 Ewing, W M., Jardetzky, W F., and Press, F ͑1957͒ Elastic Waves in Layered Media ͑McGraw-Hill, New York͒ Malischewsky, P G ͑2000͒ Comment on ‘‘A new formula for the velocity of Rayleigh waves by D Nkemzi,’’ Wave Motion 31, 93–96 Mozhaev, V G ͑1995͒ ‘‘Some new ideas in the theory of surface acoustic waves in anisotropic media,’’ in IUTAM Symposium on Anisotropy, Inhomogeneity and Nonlinearity in Solid Mechanics, edited by D F Parker and A H England ͑Kluwer, Dordrecht͒, pp 455– 462 Nair, S., and Sotiropoulos, D A ͑1997͒ ‘‘Elastic waves in orthotropic incompressible materials and reflection from an interface,’’ J Acoust Soc Am 102, 102–109 Nair, S., and Sotiropoulos, D A ͑1999͒ ‘‘Interfacial waves in incompressible monoclinic materials with an interlayer,’’ Mech Mater 31, 225–233 Nkemzi, D ͑1997͒ ‘‘A new formula for the velocity of Rayleigh waves,’’ Wave Motion 26, 199–205 Rahman, M., and Barber, J R ͑1995͒ ‘‘Exact expressions for the roots of the secular equation for Rayleigh waves,’’ J Appl Mech 62, 250–252 Rayleigh, Lord ͑1885͒ ‘‘On waves propagated along the plane surface of an elastic solid,’’ Proc R Soc London, Ser A 17, –11 Royer, D., and Dieulesaint, E ͑1984͒ ‘‘Rayleigh wave velocity and displacement in orthorhombic, tetragonal, hexagonal and cubic crystals,’’ J Acoust Soc Am 75, 1438 –1444 Stoneley, R ͑1963͒ ‘‘The propagation of surface waves in an elastic medium with orthorhombic symmetry,’’ Geophys J R Astron Soc 8, 176 – 186 Ting, T C T ͑2002a͒ ‘‘An explicit secular equation for surface waves in an elastic material of general anisotropy,’’ Q J Mech Appl Math 55, 297– 311 Ting, T C T ͑2002b͒ ‘‘A unified formalism for elastostatics or steady state motion of compressible or incompressible anisotropic elastic materials,’’ Int J Solids Struct 39, 5427–5445 Ting, T C T ͑2002c͒ ‘‘Explicit secular equations for surface waves in monoclinic materials with symmetry plane at x ϭ0, x ϭ0 or x ϭ0,’’ Proc R Soc London, Ser A 458, 1017–1031 R W Ogden and P Chi Vinh: Rayleigh waves in orthotropic elastic solids 533 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms Download to IP: 142.104.240.194 On: Tue, 16 Dec 2014 09:51:40 ... surface waves in monoclinic materials with symmetry plane at x ϭ0, x ϭ0 or x ϭ0,’’ Proc R Soc London, Ser A 458, 1017–1031 R W Ogden and P Chi Vinh: Rayleigh waves in orthotropic elastic solids. .. decreases to a minimum as ␩ increases from 0, and thereafter increases monotonically Hence, the solution is also unique in this case We therefore conclude that in an incompressible orthotropic elastic. .. conditions on ␺ This requires differentiation of ␴ 22 with respect to x so as to facilitate elimination of the term in p, as in Dowaikh and Ogden ͑1990͒ for Rayleigh waves on a prestrained half-space