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International Journal of Engineering Science 48 (2010) 275–289 Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci On formulas for the Rayleigh wave velocity in pre-strained elastic materials subject to an isotropic internal constraint Pham Chi Vinh *, Pham Thi Ha Giang Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam a r t i c l e i n f o Article history: Received 16 June 2009 Received in revised form 13 August 2009 Accepted 27 September 2009 Available online 31 October 2009 Communicated by K.R Rajagopal Keywords: Rayleigh waves Rayleigh wave velocity Prestrain Prestress Isotropic internal constraint a b s t r a c t In the present paper, formulas for the velocity of Rayleigh waves propagating along principal directions of prestrain of an elastic half-space subject to a pure homogeneous prestrain, and an isotropic internal constraint have been derived using the theory of cubic equation They have simple algebraic form, and hold for any strain-energy function and any isotropic constraint In undeformed state, these formulas recover the exact value of the Rayleigh wave speed in incompressible isotropic elastic materials Some specific cases of strain-energy function and isotropic constraint are considered, and the corresponding formulas become totally explicit in terms of the parameters characterizing the material and the prestrains The necessary and sufficient conditions for existence of Rayleigh wave are examined in detail The use of obtained formulas for nondestructive evaluation of prestrains and prestresses is discussed Ó 2009 Elsevier Ltd All rights reserved Introduction Elastic surface waves in isotropic elastic solids, discovered by Rayleigh [1] more than 120 years ago, have been studied extensively and exploited in a wide range of applications in seismology, acoustics, geophysics, telecommunications industry and materials science, for example It would not be far-fetched to say that Rayleigh’s study of surface waves upon an elastic half-space has had fundamental and far-reaching effects upon modern life and many things that we take for granted today, stretching from mobile phones through to the study of earthquakes, as stressed by Adams et al [2] For the Rayleigh wave, its speed is a fundamental quantity which interests researchers in seismology and geophysics, and in other fields of physics and the material sciences It is discussed in almost every survey and monograph on the subject of surface acoustic waves in solids Further, it also involves Green’s function for many elastodynamic problems for a half-space, explicit formulas for the Rayleigh wave speed are clearly of practical as well as theoretical interest In 1995, a first formula for the Rayleigh wave speed in compressible isotropic elastic solids have been obtained by Rahman and Barber [3], but for a limited range of values of the parameter ẳ 2mị=2 2mị, where m is Poisson’s ratio, by using the theory of cubic equations Employing Riemann problem theory Nkemzi [4] derived a formula for the velocity of Rayleigh waves expressed as a continuous function of for any range of values It is rather cumbersome [5] and the final result as printed in his paper is incorrect [6] Malischewsky [6] obtained a formula for the speed of Rayleigh waves for any range of values of by using Cardan’s formula together with trigonometric formulas for the roots of a cubic equation and MATHEMATICA It is expressed as a continuous function of In Malischewsky’s paper [6] it is not shown, however, how Cardan’s formula together with the trigonometric formulas for the roots of the cubic equation are used with MATHEMATICA to obtain the formula A detailed derivation of this formula was given by Vinh and Ogden [7] together with an * Corresponding author Tel.: +84 5532164; fax: +84 8588817 E-mail address: pcvinh@vnu.edu.vn (P.C Vinh) 0020-7225/$ - see front matter Ó 2009 Elsevier Ltd All rights reserved doi:10.1016/j.ijengsci.2009.09.010 276 P.C Vinh, P.T Ha Giang / International Journal of Engineering Science 48 (2010) 275–289 alternative formula For non-isotropic materials, for some special cases of compressible monoclinic materials with symmetry plane x3 = 0, formulas for the Rayleigh wave speed have been found by Ting [8] and Destrade [5] as the roots of quadratic equations, while for incompressible orthotropic materials an explicit formula has been given by Ogden and Vinh [9] based on the theory of cubic equations Further, in a recent papers [10,11] Vinh and Ogden have obtained explicit formulas for the Rayleigh wave speed in compressible orthotropic elastic solids Nowadays pre-stressed materials have been widely used Nondestructive evaluation of prestresses of structures before and during loading (in the course of use) becomes necessary and important, and the Rayleigh wave is a convenient tool for this task, see for example [12–15] In these studies (also in [16,17]), for evaluating prestresses by the Rayleigh wave, the authors have established the (approximate) formulas for the relative variation of the Rayleigh wave velocity [12–15] or its variation ([16,17]) They are linear in terms of the prestrains (or prestresses), thus they are very convenient in use However, since these formulas are derived by using the perturbation method they are only valid for enough small prestrains They are no longer to be applicable when prestrains are not small The main purpose of this paper is to find exact formulas for the velocity of Rayleigh waves propagating in a uniformly prestrained elastic half-space subject to an isotropic internal constraint The wave propagation direction is one of the principal axes of prestrain Since these formulas are exact and valid for any range of prestrain, they will be very significant in practical applications, especially for the nondestructive evaluation of prestresses of structures It is noted that there have been many papers dedicated to the theory of elasticity with internal constraints, see for example [18–23] and references therein In [18,19], the authors discussed universal relations and solutions for isotropic homogeneous elastic materials subject to a general isotropic internal constraint The investigation [20] explored the relationship between isotropic constraints and the associated constraint manifolds The studies [21,22] developed equations for a small deformation superimposed on a finite deformation of isotropic elastic materials with isotropic constraints, and some applications of these equations for small amplitude waves The paper [23] was about the Stroh formalism for a generally constrained and prestressed elastic material The authors derived the corresponding integral representation for the surface-impedance tensor and explained how it can be used, together with a matrix Riccati equation, to (numerically) calculate the surface-wave speed It is noted that, however, this investigation did not lead to any formula for the (Rayleigh) surface-wave velocity The paper is organized as follows: the derivation of the secular equation of Rayleigh waves is presented briefly in Section Formulas for the velocity of Rayleigh wave propagating in the principal directions of prestrain are derived in Section They hold for any strain-energy function and any isotropic internal constraint In this section are also established the necessary and sufficient conditions for unique existence of Rayleigh wave In Section 4, some specific cases of strain-energy function and isotropic constraint are considered Some remarks on the use of the obtained formulas for nondestructive evaluation of prestress are made in Section Secular equation In this section we first summarize the basic equations which govern small amplitude time-dependent motions superimposed upon a large static primary deformation, under the assumption of plane strain elasticity subjected to an isotropic internal constraint, and then derive the secular equation of the wave This secular equation coincides with the one obtained recently by Destrade and Scott [24] by a different way We consider an unstressed isotropic hyperelastic body corresponding to the half-space X P and we suppose that the deformed configuration is obtained by application of a pure homogeneous strain of the form: x1 ¼ k X ; x2 ¼ k2 X ; x3 ¼ k3 X ; ki ¼ const; ki > 0; i ¼ 1; 2; 3: ð1Þ In its deformed configuration the body, therefore, occupies the region x2 > with the boundary x2 ¼ Suppose that the material is subject to an isotropic internal constraint, written as [24]: Cðk1 ; k2 ; k3 ị ẳ 0; 2ị where C is a symmetric function of the principal stretches ki We restrict attention to the case: Ci > 0; where Ci ¼ @ C=@ki : ð3Þ The constraint (2) create the workless reaction tensor N [21] whose non-zero components are: Nii ¼ J À1 ki Ci ; ðno sumÞ; ð4Þ is of the diagonal form with non-zero where J ¼ k1 k2 k3 The Cauchy stress tensor associated to the static deformed state r components [24]: r ii ¼ J1 ki W i ỵ PNii no sumị; 5ị where the strain-energy Wðk1 ; k2 ; k3 Þ is a symmetric function of ki , i.e its value is left unchanged by any permutation of k1 ; k2 ; k3 , W i ¼ @W=@ki , and P is determined as follows: P¼À W2 C2 22 ¼ 0; if r otherwise P ¼ 22 À k2 W Jr : k2 C2 ð6Þ 277 P.C Vinh, P.T Ha Giang / International Journal of Engineering Science 48 (2010) 275–289 22 ¼ We consider a plane motion in the ðx1 ; x2 Þ-plane with displaceAs in [24], in this paper we are interested in the case r ment components u1 ; u2 ; u3 such that: ui ¼ ui x1 ; x2 ; tị; i ẳ 1; 2; u3 0; ð7Þ where t is the time Then, in the absence of body forces the equations governing infinitesimal motion are [21,24]: ; u s11;1 ỵ s21;2 ẳ q ; u s12;1 ỵ s22;2 ẳ q 8ị is mass density of the material at the static deformed state, a superposed dot signifies differentiation with respect to where q t, commas indicate differentiation with respect to spatial variables xi , and [21,24]: sij ẳ Bijkl ul;k ỵ pNij ð9Þ in which p represents the increment in P, and the components of the fourth order elasticity tensor BÃijkl Bijkl are given by: e ijkl : ẳ Bijkl ỵ P B ð10Þ e are [21,24,25]: Non-zero components of tensors B and B JBiijj ¼ ki kj W ij ; JBijij ¼ e iijj ¼ ki kj Cij ; JB e ijij ¼ JB ki W i À kj W j k2i À k2j ki Ci À kj Cj k2i À k2j k2i ði – jÞ; k2i ði – jÞ; JBijji ¼ JBijij À ki W i ; e ijji ¼ J B e ijij À ki Ci ; JB 11ị 12ị e ijij dened where W ij ẳ @ W=@ki @kj Note that there is no summation over i or j in the formulas (11) and (12), and JBijij , J B e ijij are given by: when i–j, ki –kj In the case where i – j, ki ¼ kj , JBijij , J B JBijij ẳ JB JBiijj ỵ ki W i Þ; iiii e ijij ¼ JB e e iijj ỵ ki Ci ị: J B iiii J B 13ị Note that: Biijj ẳ Bjjii Bijji ẳ BÃjiij ðno sumÞ: ð14Þ The incremental constraint of (2) is [21,24]: N11 u1;1 ỵ N22 u2;2 ẳ 0: 15ị Since the surface of the half-space is free of traction, we have: s21 ¼ s22 ¼ at x2 ¼ 0: ð16Þ In addition to Eqs (8), (9) and (15) and the boundary condition (16), the decay condition is required, namely: p; um ! as x2 ! ỵ1; Since tensor A > 0; Bijkl m ẳ 1; 2: 17ị is strongly elliptic [25], it follows that [24]: C > 0; Bỵ p AC > 0; 18ị where A ẳ k1 k2 C1 C2 ị1 B1212 ; C ẳ k1 k2 C1 C2 ị1 B2121 ; B ẳ ẵk1 C1 ị2 B1111 ỵ k2 C2 ị2 B2222 k1 k2 C1 C2 ị1 B1122 ỵ B1221 ị: 19ị Now we consider a surface Rayleigh wave propagating in the x1 -direction with the velocity v and the wave number k Then, u1 ; u2 , p and smn ðm; n ẳ 1; 2ị are sought in the form: uj ¼ U j ðkx2 Þeikðx1 Àv tÞ ðj ¼ 1; 2ị; smn ẳ kSmn kx2 ịeikx1 v tị ; p ¼ kQ ðkx2 Þeikðx1 Àv tÞ ; ðm; n ¼ 1; 2ị; 20ị where i ẳ Introducing (20)1, (20)3 into (8) and (15) and taking into account (7) yield: v U1 ; iS11 ỵ S021 ẳ q v U2 ; iS12 ỵ S022 ẳ q ik1 C1 U ỵ k2 C2 U 02 ¼ in which the prime indicates the derivative with respect to y ¼ kx2 Substituting (20) into (9) gives: ð21Þ 278 P.C Vinh, P.T Ha Giang / International Journal of Engineering Science 48 (2010) 275–289 à à S11 ẳ iB1111 U ỵ B1122 U 02 ỵ Q N11 ; S21 ẳ B2121 U 01 ỵ iB1221 U ; S22 ¼ S12 ¼ BÃ1221 U 01 þ iB1212 U ; à iB1122 U þ B2222 U 02 22ị ỵ Q N22 : Now we introduce a new variable z given by: z¼ k C1 y: k C2 ð23Þ Substituting (22) into (21) leads a system of three differential equations for three unknown functions U ; U ; Q of z, namely: 2 h i > v U ỵ iN11 Q ẳ 0; B2121 kk12 CC12 U 001 ỵ B1221 kk12 CC12 B1111 ỵ kk12 CC12 B1122 ỵ q > > > < 2 à à à à k1 C1 k1 C1 k1 C1 k1 C1 À B À B i B > 2222 k2 C2 1221 k2 C2 1122 k2 C2 U ỵ B1212 qv ÞU À k2 C2 N 22 Q ¼ 0; > > > : iU ỵ U 02 ẳ 0: ð24Þ The solution of (24) is sought in the form: U ¼ A1 eÀsz ; U ¼ A2 esz ; Q ẳ A3 esz ; 25ị where Ak (k ẳ 1; 2; 3) are constant and: Resị > ð26Þ in order to ensure the decay condition (17) Introducing (25) into (24) yields a homogeneous system of three linear equations for A1 , A2 , A3 , and vanishing its determinant leads the characteristic equation that defines s, namely: cà s4 À ð2bà À q v ịs2 ỵ a q v ị ẳ 0; 27ị 2 k C1 C2 B2121 ẳ 12 aà ; k C2 C2 2 k C1 à k1 C1 2bà ¼ BÃ1111 À ðB1221 þ BÃ1122 Þ þ BÃ2222 : k C2 k2 C2 28ị where a ẳ B1212 ; c ẳ From (18), (19) and (28) it follows: aà > 0; cà > 0; b ỵ p a c > 0: 29ị From (27) we have: s21 ỵ s22 ẳ v2 2b À q cà ; s21 s22 ¼ aà À q v : cà ð30Þ The roots s21 , s22 of the quadratic Eq (27) for s2 are either both real (and, if so, both positive because of positive real parts of s1 , s2 ) or they are a complex conjugate pair In both case: s21 s22 > Therefore, by (30)2 and cà > we have: v < aà : 0 < U ¼ B1 e iB1 s1 z ỵ iBs22 es2 z ; U ẳ s1 e > : s1 z ỵ B2 C es2 ; Q ẳ B1 C e 33ị where B1 , B2 are constant and: Cj ¼ " # 2 i k C1 k1 C1 k1 C1 v2 ; B1111 ỵ B1122 ỵ B2121 s2j ỵ B1221 ỵq k C2 k2 C2 k2 C2 N11 j ẳ 1; 2: 34ị Introducing (22)3,4 (23), (33), (34) into (32) yields a homogeneous system of two linear equations for B1 ; B2 , namely: k C1 à k C1 à s2 cà s21 ỵ B1221 B1 ỵ s1 c s22 ỵ B1221 B2 ¼ 0; k C2 k C2 k C1 à k C à v c s1 B1 ỵ 2b ỵ 1 B1221 q v c s22 B2 ẳ 0: 2b ỵ B1221 q k C2 k2 C2 ð35Þ P.C Vinh, P.T Ha Giang / International Journal of Engineering Science 48 (2010) 275–289 279 Making to zero the determinant of the system (35) provides the secular equation that determines the Rayleigh wave velocity Taking into account (30), after some algebra and removal of a factor s1 À s2 , the secular equation is of the form: cà ðaà À q v ị ỵ 2b ỵ 2d q v ịẵc a q v ị2 ẳ d2 ; 36ị where d ẳ k1 C1 k2 C2 B ẳ c: k2 C2 1221 k1 C1 37ị Eq (36) is desired secular equation, which coincides with the result obtained recently by Destrade and Scott [24] using the displacement potential Note that Eq (36) is also the secular equation in the case s1 ¼ s2 , as pointed out in [24] Formulas for the Rayleigh wave velocity We define the variable x by: ; x ¼ v =c22 x > 0ị where c22 ẳ c =q 38ị and in terms of x Eq (36) is written as: rffiffiffiffiffiffiffiffiffiffiffiffiffi aà dÃ2 À x ¼ Ã2 : a 2b ỵ 2d x ỵ x c cà c ð39Þ c From (31) and (38) it follows: 0 i ẳ 1; 2ị It is shown in [24] that: (i) If 2bà > aà , then Reẵsi xị > i ẳ 1; 2ị if and only if x ð0; x1 Þ; (ii) If 2b a , then Reẵsi xị > i ¼ 1; 2Þ if and only if x ð0; x2 ị; where: a b x1 ẳ ; x ẳ ỵ c c s! a 2b < x1 ị: ỵ1 c 45ị Note that g 0ị ẳ r a ^ ẳ ; g x1 ị ẳ 0; g x2 ị ẳ g c s a 2b ỵ 1: à c c ð46Þ From (1)5, (3), (29)2, (37) it deduces dà > ) b ¼ dÃ2 =cÃ2 > 0, therefore: f 0ị ẳ b < 0: 47ị Lemma Eq (42) has a unique solution in the interval 0; ỵ1ị: Proof If a P then f g ị ẳ 3g2 ỵ 2g ỵ a > gà > 0, i.e f ðgÃ Þ is strictly increasingly monotonous in the interval 0; ỵ1ị Since f 0ị < and f ỵ1ị ẳ ỵ1, it is clear that Eq (42) has a unique solution in the interval 0; ỵ1ị 280 P.C Vinh, P.T Ha Giang / International Journal of Engineering Science 48 (2010) 275–289 If a < then equation f g ị ẳ has distinct solutions denoted by gÃmax , gÃmin , and gÃmax < 0, gÃmin > Since f ðgÃ Þ strictly decreasingly monotonous in ðgÃmax ; gÃmin Þ, gÃmax < < gÃmin and f ð0Þ < 0, we have: f ðgÃ Þ < gà ð0; gÃmin : ð48Þ This implies that equation (42) has no root in ð0; g Since f ðg Þ strictly increasingly monotonous in g and f ỵ1ị ẳ ỵ1, it follows that Eq (42) has a unique solution in the interval ð0; þ1Þ h à à à ; þ1Þ, f ðg Ã Þ 0; cà f ð49Þ qffiffiffiffi à then Eq (42) has a unique root in the interval 0; acà qffiffiffiffi à (ii) Otherwise, Eq (42) has no root in the interval 0; acà Proof (i) Suppose (49) be satisfied With the help of Lemma 1, from (49) and f ð0Þ < it implies that (42) has a unique root in the qffiffiffiffi à interval 0; acà qffiffiffiffi aà Since f ỵ1ị ẳ ỵ1, it follows that Eq (42) has a root in the interval (ii) If (49) does not hold, i.e f cà qffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi à ½ aà =c ; ỵ1ị This implies, by Lemma 1, that Eq (42) has no root in the interval 0; acà h Proposition pffiffiffiffiffiffiffiffiffiffiffiffi (i) Let 2bà > aà Then Eq (42) has a unique solution in the interval 0; aà =cà if: f rffiffiffiffiffià a > 0; cà ð50Þ otherwise, it has no solution in this interval pffiffiffiffiffiffiffiffiffiffiffiffi ^ à ; aà =cÃ Þ if: (ii) Let 2bà aà Then Eq (42) has a unique solution in the domain ðg rffiffiffiffiffià f a ^ Ã Þ < 0; > and f ðg cà ð51Þ otherwise, it has no solution in this domain Proof (i) It is deduced from the Proposition Note that the Proposition holds for both cases: 2bà > aà and 2bà aà pffiffiffiffiffiffiffiffiffiffiffiffi à à à 6ffiffiffiffiffiffiffiffiffiffiffiffi a It is not difficult to verify that g^ < aà =cà If (51) is satisfied, (2i) Suppose that 2b p then Eq (42) has a solution in pffiffiffiffiffiffiffiffiffiffiffiffi à à =cÃ Þ (f ðg ^ Ã Þ P 0Þ, because ^ a this domain If f ð the domain ðg ; aà =cÃ Þ By Lemma 1, it has a unique solution inp =c ; ỵ1ị 0; g ^ Þ By Lemma 1, it has no root in a f ỵ1ị ẳ ỵ1 f 0ị < 0ị, it implies that Eq (42) has a solution in ½ pffiffiffiffiffiffiffiffiffiffiffiffi ^ à ; aà =cÃ Þ h the interval ðg From Proposition 1, Remark and the fact: rffiffiffiffiffià gà 0; rffiffiffiffiffi a aà ^Ã; () x ð0; x1 Þ; gà g () x ð0; x2 Þ; à c cà ð52Þ we have immediately the following theorem Theorem (i) A Rayleigh wave exists if and only if either: 2bà > aà and f rffiffiffiffiffià a > 0; cà ð53Þ P.C Vinh, P.T Ha Giang / International Journal of Engineering Science 48 (2010) 275–289 281 or 2bà aà ; f rffiffiffiffiffià a ^ Ã Þ < 0; > and f ðg cà ð54Þ is satisfied (ii) When a Rayleigh wave exists, it is unique It should be noted that the conditions (53) and (54) were stated in [24], but without explanation Proposition If Eq (42) has two or three distinct real roots, then the root corresponding to the Rayleigh wave, denoted by gÃr , is the largest root Proof Suppose Eq (42) has two or three distinct real roots According to Lemma 1, only one of them is positive This positive root of Eq (42) is the largest root and it corresponds to the Rayleigh wave h Lemma If equation: f g ị ẳ 3g2 ỵ 2g þ a ¼ 0; has two distinct real roots g à max ; Ã Þ f ðg g à ð55Þ (g à max 0, then (60) has one real root and two complex conjugate roots; if D ¼ 0, the equation has three real roots, at least two of which are equal; if D < 0, then it has three real distinct roots We now show that in each case the largest real root of Eq (60) zr is given by: zr ẳ q p q2 Rỵ Dỵp p ; Rỵ D 64ị in which each radical is understood as complex roots taking its principle value Case 1: D > If D > 0, then by Remark 3, Eq (60) has a unique real solution, so it is zr , given by the first of (62), in particular: zr ẳ q q p p Rỵ Dỵ R D 65ị in which the radicals are understood as real ones From (63) we have: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi 3 3 R þ D R À D ¼ R2 À D ¼ Q ẳ q2 ị3 ẳ q2 ; 66ị therefore zr ẳ q q p p 3 R ỵ D ỵ q2 R ỵ D; 67ị where the radicals are understood as real ones Since the real cubic root of a positive number is the same as its complex cubic root taking the principal value, in order to prove (64) we will demonstrate that: Rỵ p D > 0: 68ị 0 Indeed, consider equation f g ị ẳ 3g ỵ 2g ỵ a ẳ Its discriminant is D ¼ À 3a If D > 0, then f ¼ has two distinct real roots gÃmax ; gÃmin (gÃmax < gÃmin ) Because Eq (60) has a unique real solution, it follows that f ðgÃmax Þ:f ðgÃmin Þ > By Lemma 2, f ðgÃmin Þ < 0, therefore: f ðgÃmax Þ < 0; f ðgÃmin Þ < 0: This implies that f ðgÃN Þ < ) r < If D0 0, then f ðgÃ Þ P gà ) f ðgà ị strictly increasingly monotonous p in 1; ỵ1ị ) f gN ị ẳ f 1=3ị < f 0ị < ) r < Thus, in both cases we have: r < Since R ¼ À 12 r ) R > ) R ỵ D > 0, and (68) is proved Case 2: D ¼ When D ¼ 0, then according to Remark Eq (60) has two distinct real roots In this case equation f g ị ẳ has also two distinct real roots ) f ðgÃmin Þ < 0, according Lemma 2, and D0 > ) q2 ¼ 19 D0 > On view of f ðgÃmin Þ < and the fact that equation (60) has two distinct real roots, it deduces that f gmax ị ẳ From f gmin ị < and f gmax ị ẳ it follows that r ẳ f gN ị < From (63)3,4,5, D ¼ 0, r < we have: R ¼ jqj3 > 0; à Ã2 à r ẳ 2jqj3 : 69ị Taking into account (69)2 Eq (60) becomes: z3 À 3jqj z À 2jqj ¼ 0, and its roots are: z1 ¼ 2jqj, z2 ¼ Àjqj (double root) ) zr ¼ z1 ¼ 2jqj because zr is the largest according to Proposition In the other hand, using (69)1 and D ¼ it is easy to verify that: 2jqj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi p 3 Rỵ D R ỵ D ỵ q2 ð70Þ in which each radical is understood as complex roots taking its principle value The formula (64) again valid for the case D ¼ Case 1: D < If D < 0, then according to Remark 3, Eq (60) has three distinct real roots, and zr is the largest root by Proposition By arguments presented in [10] (p 255) one can show that, in this case the largest root zr of Eq (60) is given by: zr ẳ q q p p R ỵ D þ R À D; ð71Þ within which each radical is understood pffiffiffiffiffiffiffiffi as the complex root taking its principal value By h ð2 ð0; pÞÞ we denote the phase angle of the complex number R ỵ i D It is not difcult to verify that: q p R ỵ D ¼ jqjeih ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi R À D ¼ jqjeÀih ; where each radical is understood as the complex root taking its principal value It follows from (72) that: ð72Þ P.C Vinh, P.T Ha Giang / International Journal of Engineering Science 48 (2010) 275–289 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi q2 R Dẳ p p : Rỵ D 283 ð73Þ By substituting (73) into (71) we obtain (64), and the validity of (64) is proved for the case D < We are now in the position to state the following theorem Theorem (i) Suppose that either (53) or (54) is satisfied Then there is a unique surface Rayleigh wave propagating along the x1 -direction, in an elastic medium subject to homogeneous initial deformations (1), and an isotropic internal constraint (2) Its dimen v =cà is given by: sionless squared velocity xr ¼ q xr ¼ C22 C21 q q p 12 p 3 Rỵ D R ỵ D ỵ q2 74ị in which radicals are understood as complex roots taking the principal values, R, D, q2 defined by: ð1 À 3aÞ; 1 Rẳ aỵ b ; 27 2 1 D¼ a a ỵ b b ỵ ab; 27 108 27 q2 ẳ 75ị where a, b are determined by (43) (ii) If both (53) and (54) are not satisfied, then the surface Rayleigh wave does not exist Proof It follows from Theorem 1, (28)2, (41), (59) and (64) h For deriving (75), the formulas (61) and (63)3,4,5 are employed Note that the formula (74) holds for a general strainenergy function and an arbitrary isotropic internal constraint In the undeformed state (k1 ¼ k2 ¼ k3 ¼ 1; P ¼ 0) we have (see also [24]): aà ¼ bà ¼ cà ¼ dà ¼ l (the shear modulus) ) a ¼ 3, b ¼ 1, C22 =C21 ¼ aà =cà ¼ ) q2 ¼ À8=9, R ¼ 26=27, D ¼ 44=27 Introducing these results into (74) yields: 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 pffiffiffiffiffiffi 1C B 26 11 xr ẳ @ ỵ p q A ; 27 3 26 ỵ 2pp 11 ffiffi 27 ð76Þ 3 which coincides with the exact value of the Rayleigh wave speed in incompressible linear isotropic elastic solids [9] Approximate value of xr given by (76) is 0.9126, the classical value of the Rayleigh wave velocity in incompressible isotropic elastic materials [27] kk ¼ By xðikÞ Now we consider the half-space xk P ðk f1; 2; 3gÞ, and suppose that r r ði f1; 2; 3g; i – kÞ we denote the ðikÞ velocity of Rayleigh wave propagating in the xi -direction and attenuating in the xk -direction It is not difficult to see that xr are defined by: xikị r C2 ẳ k2 Ci ! ,q q pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 3 ðikÞ ðikÞ ðikÞ ðikÞ R ỵ D ỵ qikị R ỵ D ð77Þ in which radicals are understood as complex roots taking the principal values, RðikÞ , DðikÞ , q2ðikÞ defined by: ð1 À 3aðikÞ Þ; 1 ; Rikị ẳ aikị ỵ bikị 27 2 1 Dikị ẳ a a ỵ b bikị ỵ aikị bikị ; 27 ðikÞ 108 ðikÞ ðikÞ 27 q2ðikÞ ẳ 78ị here: aikị ẳ 2bikị ỵ 2dikị aikị c ikị aikị ẳ Bikik ; cikị ẳ 2bikị ¼ BÃiiii À 2 i k ; C aikị ; C bikị ẳ d2 ikị c2 ikị ; 79ị ki Ci ki Ci B ỵ Biikk ị ỵ kk Ck ikki kk Ck There is no summation over i or k in (79) 2 BÃkkkk : 284 P.C Vinh, P.T Ha Giang / International Journal of Engineering Science 48 (2010) 275–289 Formulas for particular strain-energy functions and internal constraints In this section we concretize the formula (74) for some specific strain-energy functions in the case of four isotropic constraints [24]: those of incompressibility, Bell, constant area, and Ericksen For seeking simplicity, we confine ourself to the 12ị case of plane strain: k3 ẳ Note that xr xr , Rð12Þ R, Dð12Þ D 4.1 Incompressible materials For incompressible materials we have: C ¼ k1 k2 k3 À ¼ 0: ð80Þ The incompressibility constraint is often used for the modelling of finite deformations of rubber-like materials and shows good correlation with experiment, see for example [25, Chapter 7] Suppose that the underlying deformation of the halfspace corresponds to strain plane with k3 ¼ 1, then (80) simplifies to: C ¼ k1 k2 À ¼ 0: ð81Þ It follows form (81): C1 ¼ k2 ; C11 ¼ C22 ¼ 0; C2 ẳ k1 ; C12 ẳ 1: 82ị For a specific example, we take the neo-Hookean strain-energy function, namely [28,29]: Wẳ lk21 ỵ k22 ỵ k23 3ị: ð83Þ In the case of strain plane with k3 ¼ 1, it is reduced to: W¼ l ðk21 þ k22 À 2Þ: ð84Þ It is readily to see that: W ¼ lk1 ; W ¼ lk2 ; W 11 ¼ W 22 ¼ l; W 12 ¼ 0: ð85Þ From (6)1, (10), (11), (12), (28), (37), (82), (85) and taking into account k1 k2 ¼ we have: aà ¼ lk2 ; cà ¼ dà ¼ l k ; 2bà ¼ l k2 ỵ ; k 86ị here we write k1 ¼ k ðk > 0Þ Introducing (86) into (43) leads to a ¼ 3, b ¼ 1, and then using (75) provides: R¼ 26 ; 27 D¼ 44 ; 27 q2 ẳ : 87ị From (74), (82) and (87) it follows: 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 pffiffiffiffiffiffi B 26 p11 ffiffiffi À qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À C xr ẳ k @ ỵ A : 27 3 26 ỵ 2pp 11 27 ð88Þ 3 Note that one can obtain the result (7.11) in [29] by multiplying two sides of (88) by kÀ2 The formula (88) can be obtained ¼ From (86) it implies: 2bà > aà It is not difficult to verify that the inequation (53)2 is from (78) in [30] by putting r equivalent to: k6 ỵ k4 ỵ 3k2 > 0; 89ị and its solution is: k > kð1Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi usffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 26 2p 11 u ẳt ỵ pffiffiffi À qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À % 0:5437: 27 3 26 ỵ 2pp 11 27 3 ð90Þ Thus, from the Theorem 2, we have the following theorem Theorem Suppose that the incompressible elastic half-space is subject to the homogeneous initial deformations (1) with k3 ¼ 1, and the strain-energy function W is given by (83) If k1 ẳ k > k1ị (dened by (90)), then there exists a unique surface Rayleigh wave propagating in the half-space whose dimensionless squared velocity xr is given by (88) For the values of k so that < k kð1Þ the surface Rayleigh wave does not exist Fig shows the dependence of the dimensionless squared velocity xr of the Rayleigh wave on the parameter k in the interval ð0:6; 2Þ 285 P.C Vinh, P.T Ha Giang / International Journal of Engineering Science 48 (2010) 275–289 4.2 Bell’s constraint The Bell constraint was found to hold experimentally over countless trials on polycrystalline annealed solids, including aluminum, brass, copper, and mild steel, see [31, Chapter 2] For a Bell constrained material we have [22,24]: C ẳ k1 ỵ k2 ỵ k3 ẳ 0: 91ị For a specic example of the Bell material we take [24]: W ¼ d2 k1 k2 ỵ k2 k3 ỵ k3 k1 3Þ; ð92Þ where d2 < is a material constant Noted that (92) comes from (4.5) in [24] with d3 ¼ On view of k3 ¼ 1, (91) and (92) become: C ẳ k1 ỵ k2 ẳ 0; W ẳ d2 k1 k2 ỵ k2 ỵ k1 À 3Þ; ð93Þ where < kk < 2; k ¼ 1; It follows from (93): C1 ¼ C2 ¼ 1; Cij ¼ 0; W ¼ d2 ð1 ỵ k2 ị; W ẳ d2 ỵ k1 Þ; W 11 ¼ W 22 ¼ 0; ð94Þ W 12 ¼ d2 : From (6)1, (10), (11), (12), (28), (37), (94), and taking into account k1 ỵ k2 ¼ we have: aà ¼ bà ¼ cà ¼ d2 k2 ; 2ðk À 2Þ dà ¼ À d2 k ; < k < 2; ð95Þ here we write k ¼ k1 On use of (95) into (43) yields: a¼ À1 ; k bẳ 2 ; k 96ị and then introducing (96) into (75) gives: q2 ¼ 1À ; k R¼ k ỵ ; 3k 27 Dẳ k k 20 ỵ : 27k 27 ð97Þ From (74) and (97) and C1 ¼ C2 ¼ it follows: 32 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ,u u u 3 20 4 20 17 6t t ỵ : xr ¼ À À À À À þ þ þ 1À þ þ þ k k2 3k 27 k k2 27k 27 k2 3k 27 k k2 27k 27 ð98Þ 16 14 12 10 x r 0.6 0.8 1.2 1.4 1.6 1.8 λ Fig Dependence of the dimensionless squared velocity xr of the Rayleigh wave on k in the interval (0.6, 2) The elastic material is incompressible, its strain-energy function is given by (83), k3 ¼ 286 P.C Vinh, P.T Ha Giang / International Journal of Engineering Science 48 (2010) 275–289 It follows from (95) that 2bà > aà Taking into account aà =cà ¼ and (93)1, it is easy to show that the solution of (53)2 is 0:5 < k < On view of the Theorem 2, we have the following the theorem Theorem Suppose that the strain-energy function W of the Bell material (91), is given by (92), and the elastic half-space is subject to the homogeneous initial deformations (1) with k3 ¼ If 0:5 < k1 ¼ k < 2, then there exists a unique surface Rayleigh wave propagating in the half-space, and the dimensionless squared velocity xr is given by (98) For the values of k ð0; 0:5 the surface Rayleigh wave does not exist Fig shows the dependence of the dimensionless squared velocity xr on k in the interval ð0:6; 2Þ for this case 4.3 Areal constraint The areal (or constant area) constraint has the form [24]: C ẳ k1 k2 ỵ k1 k2 ỵ k1 k2 ẳ 0: 99ị The areal (or constant area) constraint has the interpretation that a material cubic in the reference configuration with edges parallel to the principal axes of strain retains the same total surface area after deformation [32] For the areal materials, we take: W ẳ d1 k1 ỵ k2 ỵ k3 3ị; 100ị where d1 > is a material constant Noted that (100) originates from (4.8) in [24] with d3 ¼ On view of k3 ¼ 1, (99) and (100) reduce to: C ẳ k1 k2 ỵ k1 ỵ k2 ẳ 0; W ẳ d1 k1 ỵ k2 3ị: 101ị It implies from (101): C1 ẳ ỵ k2 ; C2 ẳ ỵ k ; W ¼ W ¼ d1 ; C11 ¼ C22 ¼ 0; C12 ¼ 1; ð102Þ W ij ¼ 0: From (6)1, (10), (11), (12), (28), (37), (102), and taking into account (101)1, it deduces: aà ¼ dà ¼ d1 k2 ỵ kị kị3 ỵ k ị c ẳ ; 4d1 k ỵ kị2 þ k2 Þ 16d1 k2 ð1 þ kÞ ð3 À kị3 ỵ k2 ị 2b ẳ ; ; 8d1 k2 ỵ kị3 kị3 ỵ k2 ị 103ị ; where k ¼ k1 , and < k < Substituting (103) into (43) leads to: 0.9 0.8 x 0.7 r 0.6 0.5 0.4 0.6 0.8 1.2 1.4 1.6 1.8 λ Fig Dependence of the dimensionless squared Rayleigh wave velocity xr on k in the interval (0.6, 2) for the Bell materials The strain-energy function is given by (92), k3 ¼ 287 P.C Vinh, P.T Ha Giang / International Journal of Engineering Science 48 (2010) 275289 aẳ ỵ kị 24 k ỵ 5k2 3k3 k4 ị; 16k bẳ þ kÞ2 ð3 À kÞ2 16k2 ð104Þ : On view of (103) and < k < we have: 2bà > aà pffiffiffi if < k < 2 À 1; 2bà aà pffiffiffi if 2 À k < 3; ð105Þ pffiffiffiffiffiffiffiffiffiffiffiffi ^ à ẳ k ỵ 3ịk 1ị=4 With < k < 3, Eq (53)2 is equivalent to 3k2 þ 4k À > 0, therefore, takand aà =cà ẳ ỵ kị2 =4, g p p ing into account (105), the solution of (53) is: ð 13 À 2Þ=3 < k < 2 À pffiffiffi p1 ffiffiffiffiffiffiIt is not difficult to see that the solution of (54) is 2 À k < Thus, the values of k satisfying either (53) or (54) is ð 13 À 2Þ=3 < k < From this and the Theorem we have: Theorem Suppose that the strain-energy function W of the areal material pffiffiffiffiffiffi (99), is given by (100), and the elastic half-space is subject to the homogeneous initial deformations (1) with k3 ¼ If ð 13 À 2Þ=3 < k < 3, then there exists a unique surface Rayleigh wave propagating in the half-space, and the dimensionless squared velocity xr is given by: xr ẳ ỵ kị4 16 q q p 12 p 3 ; R ỵ D ỵ q2 = R ỵ D 106ị p where R, D, q2 defined by (75), in which a, b are given by (104) For the values of k ð0; ð 13 À 2Þ=3 the surface Rayleigh wave does not exist The dependence of the dimensionless squared velocity xr on k in the interval ð0:55; 2Þ for this case is shown in Fig 4.4 Ericksen’s constraint The Ericksen constraint is of the form: C ẳ k21 ỵ k21 ỵ k21 ẳ 0: 107ị It was proposed by Ericksen [33] to model the behaviour of certain twinned elastic crystals For the Ericksen materials, the strain energy function is chosen as follows: W ẳ D2 k21 k22 ỵ k22 k23 ỵ k23 k21 3ị; 108ị where D2 < is a material constant Noted that (108) originates from (4.11) in [24] with D3 ¼ On view of k3 ¼ 1, (107) and (108) simplifies to: C ¼ k21 ỵ k22 ẳ 0; W ẳ D2 k21 k22 ỵ k21 ỵ k22 3ị: 109ị It implies from (109): C1 ¼ 2k1 ; C2 ¼ 2k2 ; C11 ¼ C22 ¼ 2; C12 ¼ 0; W ẳ 2D2 k1 ỵ k22 ị; W ẳ 2D2 k2 ỵ k21 ị; W 11 ẳ 2D2 ỵ k22 ị; W 22 ẳ 2D2 ỵ k21 ị; W 12 ẳ 4D2 k1 k2 : ð110Þ 4.5 3.5 x r 2.5 1.5 0.5 0.6 0.8 1.2 λ 1.4 1.6 1.8 Fig Dependence of the dimensionless squared Rayleigh wave velocity xr on k in the interval (0.55, 2) for the areal materials, the strain-energy function is given by (100), k3 ¼ 288 P.C Vinh, P.T Ha Giang / International Journal of Engineering Science 48 (2010) 275–289 From (6)1, (10), (11), (12), (28), (37), (110), and taking into account (109)1, we have: 2D k3 ffi ; cà ¼ À aà ¼ dà ¼ À pffiffiffiffiffiffiffiffiffiffiffiffiffi 2Àk where we write k ¼ k1 and < k < 2 a¼ 5À À ; k k2 2D2 k5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð2 À k2 Þ À k2 4D2 kð1 À 2k2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; À k2 ð111Þ pffiffiffi On use of (111) into (43) yields: b¼ 2bà ¼ 2 À1 k2 ð112Þ ; and then introducing (112) into (75) gives: 1 2 10 ; Rẳ ỵ ; 27 k k 3k 1024 2672 1088 1936 64 64 148 ỵ ỵ Dẳ : 27 k2 27 k4 k6 27 k8 k10 27 k12 27 q2 ẳ 113ị ^ ẳ 2=k2 On view of these and noting that f ð2=k2 À 3ị ẳ 4, it is not difcult It follows from (111) that aà =cà ¼ 2=k2 À 1, g pffiffiffi to verify that the values of k satisfying either (53) or (54) is kð2Þ < k < 2, where kð2Þ % 0:6578 Thus we have the following theorem Theorem Suppose that the strain-energy function W of the Ericksen material (107), pffiffiffi is given by (108), and the elastic half-space is subject to the homogeneous initial deformations (1) with k3 ¼ If kð2Þ < k < 2, where kð2Þ % 0:6578, then there exists a unique surface Rayleigh wave propagating in the half-space, and the dimensionless squared velocity xr is given by: xr ¼ k À1À qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q p p 12 3 R ỵ D ỵ q2 = R ỵ D ; 114ị where R, D, q2 defined by (113) For the values of k ð0; kð2Þ the surface Rayleigh wave does not exist Fig shows the dependence of the dimensionless squared velocity xr on k in the interval ð0:66; 1:3Þ for this case Note that, when k ¼ 1, the dimensionless squared Rayleigh wave velocities of four cases considered above take a the same value that is given by (76) Conclusions and remarks In this paper, we consider the propagation of Rayleigh waves along principal directions of prestrain of a deformed isotropic elastic half-space subject to an isotropic internal constraint, and have been derived the formulas for the wave velocity using the theory of cubic equation They are explicit and hold for any strain-energy function and any isotropic constraint In undeformed state, these formulas return to the exact value of the Rayleigh wave speed in incompressible isotropic elastic materials 1.5 x r 0.5 0.7 0.8 0.9 Plot of VR4 in (0.66 1.35) 1.1 1.2 1.3 λ Fig Dependence of the dimensionless squared Rayleigh wave velocity xr on k in the interval (0.66, 1.3) for the Ericksen materials, the strain-energy function is given by (108), k3 ¼ P.C Vinh, P.T Ha Giang / International Journal of Engineering Science 48 (2010) 275–289 289 Since obtained formulas are valid for any range of prestrain, they will be significant in practical applications, especially for the nondestructive evaluation of prestresses of structures In relation to the use of these formulas for the determination of prestresses and prestrains we emphasize the following points: ðikÞ (i) For a given material, xr are functions of two of three principal stretches due to (2) Suppose that they are functions of k1 and k2 (ii) Let the half-space X P be subjected to the pure homogeneous prestrain (1) and the isotropic constraint (2), and r 22 ¼ Let the material of the half-space be given In order to evaluate the prestrains and the prestresses we as follows: ð12Þ ð32Þ First, we define xr and xr by laser techniques [34], for example Then the principal stretches k1 , k2 are determined using two following equations: uð12Þ ðk1 ; k2 ị ẳ x12ị ; u32ị k1 ; k2 ị ¼ xð32Þ : r r ð115Þ 11 , r 33 are determined The third principal stretches k3 is calculated by the relation (2) Finally, the principal Cauchy stresses r by: r 11 ¼ ðW C2 À W C1 ị k1 ; J C2 r 33 ẳ ðW C2 À W C3 Þ k3 ; J C2 ð116Þ which are originated from (5) References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] L Rayleigh, On waves propagating along the plane surface of an elastic solid, Proc Roy Soc Lond A 17 (1885) 4–11 S.D.M Adams et al., Rayleigh waves guided by topography, Proc Roy Soc Lond A 463 (2007) 531–550 M Rahman, J.R Barber, Exact expression for the roots of the secular equation for Rayleigh waves, ASME J Appl Mech 62 (1995) 250–252 D Nkemzi, A new formula for the velocity of Rayleigh waves, Wave Motion 26 (1997) 199–205 M Destrade, Rayleigh waves in symmetry planes of crystals: explicit secular equations and some explicit wave speeds, Mech Mater 35 (2003) 931– 939 P.G Malischewsky, Comment to A new formula for velocity of Rayleigh waves by D Nkemzi [Wave Motion 26 (1997) 199–205], Wave Motion 31 (2000) 93–96 P.C Vinh, R.W Ogden, On formulas for the Rayleigh wave speed, Wave Motion 39 (2004) 191–197 T.C.T Ting, A unified formalism for elastostatics or steady state motion of compressible or incompressible anisotropic elastic materials, Int J Solids Struct 39 (2002) 5427–5445 R.W Ogden, P.C Vinh, On Rayleigh waves in incompressible orthotropic elastic solids, J Acoust Soc Am 115 (2) (2004) 530–533 P.C Vinh, R.W Ogden, Formulas for the Rayleigh wave speed in orthotropic elastic solids, Ach Mech 56 (3) (2004) 247–265 P.C Vinh, R.W Ogden, On a general formula for the Rayleigh wave speed in orthotropic elastic solids, Meccanica 40 (2005) 147–161 M Hirao, H Fukuoka, K Hori, Acoustoelastic effect of Rayleigh surface wave in isotropic material, J Appl Mech 48 (1981) 119–124 P.P Delsanto, A.V Clark, Rayleigh wave propagation in deformed orthotropic materials, J Acoust Soc Am 81 (4) (1987) 952–960 M Dyquennoy, M Ouaftouh, M Ourak, Ultrasonic evaluation of stresses in orthotropic materials using Rayleigh waves, NDT & E Int 32 (1999) 189– 199 M Dyquennoy, D Devos, M Ouaftouh, Ultrasonic evaluation of residual stresses in flat glass tempering: comparing experimental investigation and numerical modeling, J Acoust Soc Am 119 (6) (2006) 3773–3781 K Tanuma, C-S Man, Perturbation formula for phase velocity of Rayleigh waves in prestressed anisotropic media, J Elasticity 85 (2006) 21–37 Y.Q Song, Y.B Fu, A note on perturbation formulae for the surface-wave speed due to perturbations in material properties, J Elasticity 88 (2007) 187– 192 E Pucci, G Saccomandi, Universal relations in constrained elasticity, Math Mech Solids (1996) 207–217 E Pucci, G Saccomandi, Universal generalized plane deformations in constrained elasticity, Math Mech Solids (1998) 201–216 M.M Carroll, On isotropic constraints, Int J Eng Sci 47 (2009) 1142–1148 P Chadwick, A.M Whitworth, P Borejko, Basic theory of small-amplitude waves in a constrained elastic body, Arch Ration Mech Anal 87 (1985) 339–354 M Beatty, M.A Hayes, Deformations of an elastic, internally constrained material Part 3: Small superimposed deformations and waves, Z.A.M.P 46 (1995) S72–S106 (Special issue) R.T Edmondson, Y.B Fu, Stroh formulation for a generally constrained and pre-stressed elastic material, Int J Non-Linear Mech 44 (2009) 530–537 M Destrade, N.H Scott, Surface waves in a deformed isotropic hyperelastic material subject to an isotropic internal constraint, Wave Motion 40 (2004) 347–357 R.W Ogden, Non-Linear Elastic Deformations, Ellis Horwood, Chichester, 1984 W.H Cowles, J.E Thompson, Algebra, D Van Nostrand Company, New York, 1947 W.M Ewing, W.S Jardetzky, F Press, Elastic Waves in Layered Media, McGraw-Hill, New York, 1957 M.A Dowaikh, R.W Ogden, Interface waves and deformations in pre-stressed elastic media, Proc Roy Soc Lond A 433 (1991) 313–328 M.A Dowaikh, R.W Ogden, On surface waves and deformations in a pre-stressed incompressible elastic solids, IMA J Appl Math 44 (1990) 261–284 P.C Vinh, On formulas for the velocity of Rayleigh waves in pre-strained incompressible elastic solids, ASME J Appl Mech 77 (2010), in press, doi:10.1115/1.3197139 M.F Beatty, Hyperelastic Bell materials, retrospection, experiment, theory, in: Y.B Fu, R.W Ogden (Eds.), NonLinear Elasticity: Theory and Applications, Cambridge University Press, London, 2001, pp 58–96 M Bosi, M.C Salvatori, Some non homogeneous deformations for a special class of isotropic constrained materials, Rend Mat VII 16 (1996) 689–713 J.L Ericksen, Constitutive theory for some constrained elastic crystals, Int J Solids Struct 22 (1986) 951–964 M Dyquennoy, M Ouaftouh, M Ourak, Stress state evaluation of laminated aluminum alloy sheet by surface acoustic waves, Rev Prog Quantitative Nondestruct Evaluat 17 (1998) 1621–1625 ... for the wave velocity using the theory of cubic equation They are explicit and hold for any strain-energy function and any isotropic constraint In undeformed state, these formulas return to the. .. internal constraint The investigation [20] explored the relationship between isotropic constraints and the associated constraint manifolds The studies [21,22] developed equations for a small deformation... is satisfied Then there is a unique surface Rayleigh wave propagating along the x1 -direction, in an elastic medium subject to homogeneous initial deformations (1), and an isotropic internal constraint