International Journal of Non-Linear Mechanics 47 (2012) 128–134 Contents lists available at ScienceDirect International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm Uniqueness of Stoneley waves in pre-stressed incompressible elastic media Pham Chi Vinh Ã, Pham Thi Ha Giang Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnam a r t i c l e i n f o a b s t r a c t Available online April 2011 The main aim of this paper is to prove, for the general case, the uniqueness of Stoneley waves propagating along the bonded interface of two pre-stressed incompressible elastic half-spaces In order to that the authors have used the complex function method By this approach, it is shown that the secular equation of Stoneley waves in pre-stressed incompressible elastic half-spaces has at most one solution in the complex plane This says that if a Stoneley wave exists, then it is unique & 2011 Elsevier Ltd All rights reserved Keywords: Stoneley waves Pre-stressed incompressible elastic halfspaces The uniqueness Secular equation Holomorphic function Introduction Interfacial waves traveling along the welded plane boundary of two different isotropic elastic half-spaces were first investigated by Stoneley [1] in 1924 He derived the secular equation of the wave, and showed by means of examples that such interfacial waves not always exist Subsequent studies by Sezawa and Kanai [2] and Scholte [3,4] focused on the range of existence of Stoneley waves Scholte [4] found the equations expressing the boundaries of the existence domain that were in complete agreement with the corresponding curves numerically obtained by Sezawa and Kanai [2] for the case of Poisson solids Their studies showed that the restriction on material constants that permit the existence of Stoneley waves are rather severe However, Sezawa and Kanai and Scholte did not prove the uniqueness of Stoneley waves This question was settled by Barnett et al [5] for two general anisotropic half-spaces with a welded interface The propagation of Stoneley waves in anisotropic media was also studied by Stroh [6] and Lim et al [7] Much of the early attentions to Stoneley waves was directed toward geophysical applications Latter studies have indicated that interfacial waves may prove to be useful probes for the nondestructive evaluations (see [8,9]) Nowadays pre-stressed materials have been widely used The non-destructive evaluation of prestresses of structures before and during loading (in the course of use) is necessary and important, and the Stoneley wave is a convenient tool for this task However, few investigations have been done for the subject of propagation of Stoneley waves at the interface between two welded pre-stressed half-spaces In papers [10,11] Chadwick and Jarvis considered Stoneley waves propagating in an arbitrary direction parallel to the interface and restricted attention to two half-spaces of the same incompressible neo-Hookean material subject to different homogeneous pure à Corresponding author Tel.: ỵ 84 5532164; fax: ỵ 84 8588817 E-mail address: pcvinh@vnu.edu.vn (P.C Vinh) 0020-7462/$ - see front matter & 2011 Elsevier Ltd All rights reserved doi:10.1016/j.ijnonlinmec.2011.03.014 strains, and the principal axes of strain in the two half-spaces aligned Dasgupta [12] investigated the effect of initial stress on the range of existence of Stoneley waves in neo-Hookean incompressible materials Dunwoody [13] has investigated certain aspects of the interfacial wave problem for pre-stressed compressible elastic half-spaces In paper [14] Dowaikh and Ogden examined the propagation of Stoneley waves along the bonded interface of two incompressible isotropic elastic half-spaces subject to pure homogeneous strains with one principal axis of strain normal to the interface and others having a common orientation The wave was assumed to propagate along a principal axis By detailed analysis of the dispersion equation, the authors have derived general sufficient conditions for the existence of a Stoneley wave and, in the case of biaxial deformations, necessary and sufficient conditions for the existence of a unique interfacial wave However, the question of uniqueness for the general case has been left, and it has not yet had an answer so far, to the best of the authors’ knowledge The main purpose of this paper is to settle this question The tool that is employed to that is complex function theory Using this tool it is shown that the secular equation of Stoneley waves propagating along the bonded interface of two pre-stressed incompressible elastic half-spaces has at most one solution in the complex plane This means that if a Stoneley exists, then it is unique The complex function method In this section we present the complex function method by using it to establish the uniqueness of Stoneley waves for a special case when the corresponding strain-energy functions are of neo-Hookean type [15]: W¼ m 2 2 ðl1 ỵ l2 ỵ l3 3ị, W ẳ m 2 2 l1 ỵ l2 ỵ l3 3ị 1ị where m, lk k ẳ 1,2,3ị and m , lk k ẳ 1,2,3ị are the Lame constants, the principal stretches of deformation of the half-spaces B P.C Vinh, P Thi Ha Giang / International Journal of Non-Linear Mechanics 47 (2012) 128–134 and Bà , respectively Chadwick and Jarvis [10,11] and Dowaikh and Ogden [14] have examined the existence of Stoneley waves for this case, but the question of uniqueness is still open so far Note that in this paper we use the notations presented in the paper [14], and same quantities related to B and Bà have the same symbol but are systematically distinguished by an asterisk if pertaining to Bà For this case the secular equation of Stoneley waves is of the form (see [14, Eq (4.59)]) 2 g Z ỵ Z þ 3ZÀ1Þ þ g ðZ þ Z þ3Z À1Þ þ 2gg 1Zị1Z ị ỵ gg Z ỵ Z ị1 ỵ Zị1 ỵ Z ị ẳ where s arc2 Zẳ , g Zà ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aà Àrà c2 , à 2ị a ẳ ml21 g g ẳ ml22 , a ¼ mà lÃ2 gà ¼ mà lÃ2 , ð3Þ c is the velocity of Stoneley waves that is subject to o c ominfct ,ctà g ð4Þ ct (ctà ) is the speed of transverse wave in the half-space B (Bà ) defined as ct2 ¼ a=r, ctÃ2 ¼ aà =rà ð5Þ r (rà ) is the mass density of the half-space B (Bà ) Without loss of generality we can suppose that ct rctà We now introduce the notations x ¼ ct2 =c2 (the dimensionless squared slowness of Stoneley waves), b ¼ ct2 =ctÃ2 ð0 ob r 1Þ From (4) it follows that x41 ð6Þ and in terms of x the quantities Z, Z defined by ð3Þ1,2 become sffiffiffiffiffi rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi rffiffiffi xÀ1 xÀb a aà , dẳ Zẳd , d ẳ 7ị , Z ẳd x x g g Introducing 7ị1,2 into (2), the secular equation of Stoneley waves now is of the form in terms of x ð ZbÞ: pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi xÀb xÀ1 ð8Þ f ðxÞ q1 xị p ỵq2 xị p ỵ q3 xị x1 xb ỵ q4 xị ẳ x x where Ã2 Ã2 2 Ã2 Ã2 129 Denote L ¼ L1 [ L2 with L1 ¼ ½0,b, L2 ¼ ẵb,1, S ẳ fz A C, z= 2Lg, Nz0 ị ¼ fz A S : o jzÀz0 j o eg, e is a sufficient small positive number, z0 is some point of the complex plane C If a function fðzÞ is holomorphic in O & C we write fðzÞ A HðOÞ From (10) it is not difficult to show that the function f(z) has the properties: (f1) (f2) (f3) (f4) (f5) f ðzÞ A HðSÞ f(z) is bounded in N(1) f zị ẳ Oz1=2 ị as z-0 f zị ẳ OA1 z ỵA0 ị as jzj-1 (A0 ,A1 are constant) f(z) is continuous on L from the left and from the right (see [16]) with the boundary values f þ ðtÞ (the right boundary value of f(z)), f À ðtÞ (the left boundary value of f(z)) defined as follows: ( f1 ðtÞ, t A L1 f ðtÞ ¼ ð11Þ f27 ðtÞ, t A L2 where fkÀ ðtÞ ẳ fkỵ tị, k ẳ 1,2 12ị the bar indicates the complex conjugate, and p p p pp f1ỵ tị ẳ q4 tịq3 tị 1t bt ỵ iẵq1 tị 1t þq2 ðtÞ bÀt = t , t A L1 # p " p pp tb 1t f2ỵ tị ẳ q4 tị ỵq2 tị p ỵ i q1 tị p ỵ q3 ðtÞ 1Àt tÀb , t t t A L2 13ị tị fkỵ p fk tịị is the right (left) boundary value of f(z) on Note that Lk and i ¼ À1 In order to prove Proposition we will establish the following propositions: Proposition Eq (10) is equivalent to equation P(z)¼0 in the region S [ f0g [ f1g, where P(z) is a first-order polynomial of z whose coefcients not vanish simultaneously, i.e.: Pzị ẳ A^ z ỵ A^ , 2 A^ ỵ A^ a0 14ị q1 xị ẳ edẵe3 ỵ d ị ỵd 1ịxedẵed ỵ d b Proposition Equation P(z)¼0 has at most one solution in the complex plane C q2 xị ẳ d ẵ3 ỵ d ị ỵ ed 1ịxd ẵed ỵ d b q3 xị ẳ 4edd 9ị 2 2 q4 xị ẳ ẵe2 d 1ị ỵd 1ị ỵ ed ỵ d ỵ 2ịx 2 2 ẵe2 d ỵ d b ỵed ỵ bd ị, eẳ g g Now, in the complex plane C we consider the equation: pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi zb z1 10ị f zị q1 zị p ỵq2 zị p ỵ q3 zị z1 zb ỵ q4 zị ¼ z z pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi where the functions qk ðzÞ are defined by (9), z, zÀ1, zÀb are chosen as the principal branches of the corresponding square roots Note that Eq (10) coincides with Eq (8) for the real values of z bigger than 1, we can thus call it the complex form of the real equation (8) We will prove the following proposition: Proposition In the complex plane C, Eq (10) has at most one solution From Proposition 1, we have immediately: Theorem For pre-stressed incompressible elastic half-spaces of neo-Hookean material, if a Stoneley wave exists, then it is unique Proposition is implied directly from Propositions 2, 3, and the fact that Eq (10) has no solution in interval (0, 1) due to the discontinuity of f(z) on this interval Proof of Propositions 2, Now we introduce function g(t) (t A L) as follows: ỵ f1 tị > > > > f À ðtÞ , t A L1 < ð15Þ gðtÞ ẳ f2ỵ tị > > > , t A L2 > : f ðtÞ From (13) and (15) it is obvious that f ỵ tị ẳ gtịf ðtÞ, tAL Consider the function GðzÞ defined as Z loggtị dt Gzị ẳ 2pi L tz It is not difficult to verify that (g1 ) GðzÞ A HðSÞ, (g2 ) G1ị ẳ 0, 16ị 17ị 130 P.C Vinh, P Thi Ha Giang / International Journal of Non-Linear Mechanics 47 (2012) 128134 (g3 ) Gzị ẳ 1=2ịlogz ỵ O0 zị, z A N0ị, Gzị ẳ O1 zị, z A N1ị, where O0 ðzÞ ðO1 ðzÞÞ bounded in Nð0Þ ðNð1ÞÞ and takes a defined value at z¼ (z ¼1) It is noted that (g3 ) comes from the fact (see [16]): logg0ị ẳ ip, logg1ị ẳ 18ị Introduce a new function Fzị dened by Fzị ẳ expGzị 19ị It is implied from ðg1 Þ2ðg3 Þ that: (f1 ) (f2 ) (f3 ) (f4 ) FðzÞ A HðSÞ, FðzÞ a 8z A S, Fzị ẳ O1ị as jzj-1, Fzị ¼ zÀ1=2 expO0 ðzÞ for z A Nð0Þ, FðzÞ ¼ expO1 ðzÞ, z A Nð1Þ From the Plemelj formula [16], the function FðzÞ is seen directly to satisfy the boundary condition: ỵ F tị ẳ gtịF tị, t A L ð20Þ We now consider the function Y(z) defined by Yzị ẳ f zị=Fzị 21ị From (f1)(f4), (16), f1 ịf4 Þ and (20), (21), it follows that: (y1) YðzÞ A HSị, (y2) Yzị ẳ OA1 z ỵ A0 ị as jzj-1, (y3) Y(z) is bounded in N(0) [due to (f3) and the first of (f4 )] and in N(1), (y4) Y ỵ tị ẳ Y tị, t A L Properties (y1) and (y4) of the function Y(z) show that Y(z) is holomorphic in entire complex plane C, with the possible exception of points: z¼0 and By (y3) these points are removable singularity points and it may be assumed that the function Y(z) is holomorphic in the entire complex plane C (see [17]) Thus, by the generalized Liouville theorem [17] and taking account into (y2) we have Yzị ẳ PðzÞ In this section we will prove the uniqueness of Stoneley waves for the general case by employing the complex function method presented in the previous section The general case means the case of two general incompressible isotropic elastic half-spaces subjected to any initial homogeneous deformations The principal axes of strain are aligned in the considered plane of motion with one axis normal to the interface For the general case, the secular equation of Stoneley waves is of the form (see [14, Eq (3.14)]) p 2b ỵ2gXị gaXị ỵ gaXịg2 p ỵ 2b ỵ 2g X ị g a X ị ỵ g a X ịg2 p p ỵ 2ẵg gaXịẵg g a X ị p p q p ẵ g aXị ỵ ga X ị 2bX ỵ gaXị q p 25ị 2b X ỵ g a X ị ẳ where a, b, g, a , b , gà are (constant) material parameters defined by the formulas (2.6) and (2.7) in [14], X ¼ rc2 , X à ¼ rà c2 , c is the velocity of Stoneley waves subjected to oc o minfct ,ct g, ct ẳ a=r, ct ẳ a =r 26ị As above, we can assume, without loss of generality, that ct rctà , then ob r 1, where b ¼ ct2 =ctÃ2 Note that a, g, aà , gà are strictly positive (see [14, (2.10a, b)]) We introduce the following notations: g b gà bà , n ¼ , m à ¼ à , nà ¼ à a a a a a ct2 d ¼ à , x ¼ ðx 1Þ a c m¼ ð27Þ Then, in terms of these notations Eq (25) can be written as follows: " pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi xÀb xÀ1 xÀ1 q1 xị p ỵq2 xị p ỵq3 xị x1 xb þ q4 ðxÞ þ q5 ðxÞ pffiffiffi x x x vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 v !2 u pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi#u pffiffiffiffiffiffiffiffiffi u xb u xb p x1 p p ỵ m et p ỵ m e ẳ ỵ q6 xị p t x x x ð22Þ where P(z) is a polynomial of order in terms of z: Pzị ẳ A^ z ỵ A^ From (21) and (22) we have f zị ẳ FzịPzị 23ị Since Fzị a0 8z A S (by ðf2 Þ), and FðzÞ-1 as z-0, Fð1Þ a (by ðf4 Þ), from (23) it is implied that Eq (10) is equivalent to equation P(z)¼ in the region S [ f0g [ f1g Coefficients A^ and A^ cannot vanish simultaneously, because if A^ ẳ A^ ẳ 0-Pzị 0, then f ðzÞ according to (23) But this contradicts the statement (f3) 2 Proposition is proved From the fact A^ ỵ A^ a 0, we have immediately Proposition & à à Remark For the biaxial deformations (l1 ¼ l2 , l1 ¼ l2 ), the secular equation of Stoneley waves is Eq (2) (see [14]) in which sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rc2 rà c à Z ẳ , Z ẳ 24ị a The uniqueness of Stoneley waves for the general case a i.e in terms of x the secular equation of Stoneley waves is of the à form (8), where qk xị are dened by (9) in which d ẳ d ¼ Proposition therefore hold for this case Thus, for biaxial à à deformations (l1 ¼ l2 , l1 ¼ l2 ), if a Stoneley wave exists, it must be unique This fact has also been proved by Dowaikh and Ogden [14] by another way ð28Þ where pffiffiffiffiffi pffiffiffiffiffi q1 xị ẳ 2d mẵdm ỵ nịm xd2 m p p q2 xị ẳ m m ỵn dmịxb m , p q3 xị ẳ 2d mm q4 xị ẳ ẵd2 m1mị ỵm 1m ị ỵ 2dmm xd2 m ỵbm Þ pffiffiffiffiffiffiffi q5 ðxÞ ¼ d mà x, pffiffiffiffiffi q6 xị ẳ d mx, e ẳ ỵ m2n, e ẳ ỵ m 2n 29ị Note that when the two half-spaces are made of neo-Hookean material (2n ¼ þ m, 2nà ¼ 1þ mà -e ¼ eà ¼ 0, see [14]), the lefthand side of Eq (28) differs from the left-hand side of Eq (8) by a Ã4 à positive factor d , where d defined by ð7Þ4 In the complex plane C Eq (28) takes the form f zị f1 zị ỵ f2 zịf3 zịf3 zị ẳ 30ị where p p pp zb z1 f1 zị ẳ q1 zị p ỵ q2 zị p þ q3 ðzÞ zÀ1 zÀb þ q4 ðzÞ z z p p zb z1 f2 zị ẳ q5 zị p þ q6 ðzÞ pffiffiffi , z z vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u p u z1 p p ỵ m e f3 zị ¼ t z P.C Vinh, P Thi Ha Giang / International Journal of Non-Linear Mechanics 47 (2012) 128–134 ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u p u p zb p ỵ m e f3 zị ẳ t 31ị z p p pffiffiffiffiffiffiffiffiffi where z, zÀ1, zÀb are chosen as the principal branches of the corresponding square roots, the functions qk ðzÞ are defined by (29) Note that Eq (30) coincides with Eq (28) for the real values of z bigger than We will prove the following proposition: 131 y0 B1 Proposition In the complex plane C, Eq (30) has at most one solution A x0 a From Proposition 4, we have immediately: Theorem For pre-stressed incompressible elastic half-spaces, if a Stoneley wave exists, then it is unique In order to prove Proposition 4, first we need to know the set L of discontinuity points of the function f(z) and their right and left boundary values f ỵ ðtÞ and f À ðtÞ on L, and the values of the ratio f ỵ tị=f tị at the ends of L Thus, we have to know those of the functions f1 ðzÞ, f2 ðzÞ, f3 ðzÞ, f3à ðzÞ due to (30) B2 Fig The image S0 of mapping Z ¼ pffiffiffiffiffiffiffiffiffi pffiffiffi zÀ1= z, z A S, S0 ẳ fZ ẳ x0 ỵ iy0 : x0 Z 0g v1 B1 3.1 The set of discontinuity points of f1 ðzÞ, f2 ðzÞ It is clear that: Lemma (i) The functions f1(z) are discontinuous on the set 7 L ẳ L1 [ L2 , L1 ẳ ẵ0,b, L2 ẳ ẵb,1, and f11 , f12 are given by (12) and (13) in which qk ðtÞ defined by (29) (ii) The set of discontinuity points of f2 ðzÞ is also L ¼ L1 [ L2 , and: pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi tÀb 7i 1Àt 7 f21 ¼ pffiffi ẵq5 tị 1t ỵ q6 tị bt , f22 ẳ q6 ðtÞ pffiffi iq5 ðtÞ pffiffi t t t ð32Þ A a1/2 u1 B2 (iii) The following equalities hold: ỵ f11 0ị f11 0ị ẳ ỵ f21 0ị ẳ 1, f21 0ị ỵ f12 1ị f ỵ 1ị ẳ 22 1ị ẳ ỵ f12 1ị f22 33ị ỵ By fkm (fkm ) we denote the right (left) boundary values of fk ðzÞ on Lm 3.2 The set of discontinuity points of f3(z) p p 3.2.1 The image of Zzị ẳ z1= z By S we denote the set obtained by removing from the complex C the interval L ẳ ẵ0, p (see Fig 1), by S0 we denote the pffiffiffi image of the function Zzị ẳ z1= z, z A S, and ẵa,b ỵ (ẵa,b ), a,b being real numbers, is the set of points z A ẵa,b : z ẳ t ỵi0 ỵ , t A ẵa,b (z A ẵa,b : z ẳ t ỵi0 , t A ẵa,b) Then, it is clear that S0 ẳ fZ ẳ x0 ỵiy0 : x0 Z0g, see Fig 2, in which: the point Q ð1,0Þ is converted to the point A, the point ỵ (0 ) is converted to B1 (B2), L þ (LÀ ) is mapped onto AB1 (AB2) y Fig The image S1 (non-shaded region) of mapping x1 ðZÞ ẳ p Z ỵ a, a Z 0, Z A S0 p 3.2.2 The image of x1 Zị ẳ Z ỵa,a Z0 As S0 ẳ fZ ẳ x0 ỵiy0 : x0 Z 0g, according to Section 3.2.1, it is not pffiffiffiffiffiffiffiffiffiffiffi difficult to verify that the image of the function x1 Zị ẳ Z ỵ a, a Z0, denoted by S1 , is the ‘‘non-shaded region’’ in Fig Here, AB1 A S0 (AB2 A S0 ) is mapped onto AB1 A S1 (AB2 A S1 ), and AB1 A S1 , AB2 A S1 are expressed by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p > < u1 tị ẳ a2 ỵ t2 ỵ aị=2 q 34ị p > : v1 tị ẳ a2 ỵ t2 aị=2, t Z x1 ẳ u1 ỵiv1 , the sign ỵ ( À ’’) is corresponding to AB1 (AB2) It is readily seen that the set of discontinuity points of the function x1 Zzịị is the interval Lẳ[0,1], and v ! v rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u u1 u1 1 þ t t a þ À1 þ a þ i a2 ỵ 1a , x1 tị ẳ t t ỵ x tị ẳ x1 tị 35ị From (35) it is readily seen that Q Fig The complex plane C with the cut L¼ [0 1] x x1ỵ 0ị ẳ eip=2 , x 0ị x1ỵ 1ị ẳ ỵ1 x 1ị 36ị p 3.2.3 The image of x2 Zị ẳ Za, oa o1 Similarly, it is not difficult to show that the image of the function p x2 Zị ẳ Za, o a o 1, denoted by S2, is the ‘‘non-shaded region’’ 132 P.C Vinh, P Thi Ha Giang / International Journal of Non-Linear Mechanics 47 (2012) 128–134 in Fig Here, ẵ0,a ỵ A S0 (ẵ0,a A S0 ) is mapped onto 0A1 A S2 (0A2 A S2 ), and AB1 A S0 (AB2 A S0 ) is mapped onto A1 B1 A S2 (A2 B2 A S2 ), and A1 B1 A S2 , A2 B2 A S2 are expressed by q p > < u2 tị ẳ a2 ỵ t2 aị=2 q 37ị p > : v2 tị ẳ a2 ỵ t2 ỵ aị=2, t Z0 v3 x2 ẳ u2 ỵiv2 , the sign ỵ () is corresponding to A1 B1 (A2 B2 ) It is readily seen that the function x2 ðZðzÞÞ is discontinuous on the interval B1 A u3 a1/2 L ¼ L1 [ L2 , L1 ¼ ½0,1, L2 ¼ ½1,1=ð1Àa2 Þ (see Fig 5), and vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u r ! u r u1 u1 1 ỵ t t 2 a ỵ 1a ỵ i a ỵ ỵ a x21 tị ẳ t t p ỵ x22 tị ẳ it, rt r a, B2 ỵ x 2k tị ẳ x2k tị, k ẳ 1,2 ð38Þ Fig The image of mapping x3 ðZÞ (non-shaded region): x3 Zị ẳ aZ 0, p 40, Z A S0 From (38) it is obvious that ỵ x21 0ị x21 0ị ỵ x21 1=1a2 ịị ẳ ỵ1 x21 1=1a2 ịị ẳ eip=2 , 39ị p 3.2.4 The image of x3 Zị ẳ Z ỵa ỵip, a Z 0, p ispaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi real number It is clear that the image of the function x3 Zị ẳ Z þa þip, a Z0, p being any real number is S1 Here, AB1 ðAB2 Þ A S0 is mapped onto v2 B1 a1/2 s q a2 ỵ p2 þ a =2, u3A ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þp2 Àa =2 v3A ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v3A ¼ a2 ỵ p2 a =2 if p o if p Z0 ð40Þ The function x3 ðZðzÞÞ is discontinuous on the interval L¼[0,1], and if p Z0: rt r ð41Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1ffi v u 0v u 0v > rffiffiffiffiffiffiffiffiffiffi!2 rffiffiffiffiffiffiffiffiffiffi!2 u u > u >u u > 1@u 1 u u > ta2 ỵ p > ỵ aA þ it @ta2 þ pÀ À1 ÀaA > > t2 t t > > > > > > > > > > rt r1 > < 1ỵ p2 x3 tị ẳ v v v 1ffi v u 0u u 0u > rffiffiffiffiffiffiffiffiffiffi!2 rffiffiffiffiffiffiffiffiffiffi!2 > u u > > u1@u 1 u1@u > t t 2 A A > t a ỵ p a ỵ p ỵ a it a > > t t > > > > > > > > > > > : 0r t o ỵ p2 42ị u2 A2 B2 Fig The image S2 (non-shaded region) of mapping x2 Zị ẳ o 1, Z A S0 p Za, o a for p o 0: y AB1 AB2 ị A S1 (see Fig 6), where A ẳ ðu3A ,v3A Þ, and: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v u 0v u 0v rffiffiffiffiffiffiffiffiffiffiffi!ffi2 rffiffiffiffiffiffiffiffiffiffiffi!ffi2 u u u u u u u 1 @u ỵ t t 2 @ A t t ỵ a ỵ i aA x3 tị ẳ a ỵ pỵ a ỵ pỵ t t A1 a1/2 p Z ỵ a ỵ ip, Q 1/(1a2) R x vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1ffi v u 0v u 0v > rffiffiffiffiffiffiffiffiffiffi!2 rffiffiffiffiffiffiffiffiffiffi!2 u u > u > >u 1@u 1@u u u > t t 2 A A > a ỵ pỵ a ỵ pỵ ỵ a ỵ it a > > t2 t t > > > > > > > > > > > < r t o ỵ p2 ỵ x3 tị ẳ v v u 0v u 0v >u rffiffiffiffiffiffiffiffiffiffi!2 rffiffiffiffiffiffiffiffiffiffi!2 u u > u > > 1@u 1 u1@u >u t t þ pþ þ pþ A A > t t a a ỵ a i a 1 > > t t > > > > > > > > > > > : ỵ p2 r t r 1, ð43Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1ffi u rffiffiffiffiffiffiffiffiffiffi!2 u 0v u rffiffiffiffiffiffiffiffiffiffi!2 u u u u1@u u1@u À t t aA x3 tị ẳ t a ỵ p þaAÀit a þ pÀ t t Fig The complex plane C with the cut L ẳ ẵ01=1a2 Þ, 0o ao rt r ð44Þ P.C Vinh, P Thi Ha Giang / International Journal of Non-Linear Mechanics 47 (2012) 128134 ỵ x3ỵ 1ị ẳ ỵ1 x 1ị 45ị f3 zị ẳ f31 zịf32 zị f32 zị ẳ q p p Zzị ỵ m e Lemma 3à If eà o 0, then the set of discontinuity points of the function f3à ðzÞ is [0, b], and the boundary values f3 ỵ tị, f3 tị are þ À calculated by using (41)–(44) in which x3 ðtÞ, x3 tị, a, 1/t are ỵ respectively replaced by x3 ðtÞ, x3 ðtÞ, aà , b=t and ð48Þ for the second possibility, f3ỵ tị, f3 tị are calculated by using (38) and f3ỵ 0ị ẳ 1, f3 0ị f3ỵ 1=1a2 ịị ẳ1 f3 1=1a2 ịị 49ị Case 2: eo 0: If eo 0, then f3(z) takes the form (46) where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffi ð50Þ f31 ðzÞ ẳ Zzị ỵ m ỵi jej, f32 zị ẳ Zzị þ mÀi jej The function f31 ðzÞ (f32 ðzÞ) is therefore the function x3 ðZðzÞÞ with pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi a ¼ m 0, p ¼ jej (a ¼ m, p ¼ À jej) From the results of Section 3.2.4, one can see that: Lemma If eo 0, then the set of discontinuity points of the function f3(z) is [0, 1], and the boundary values f3ỵ tị, f3 tị are calculated by using (41)(44) and f3ỵ 0ị f3 0ị ẳ 1, f3ỵ 1ị f3 1ị ẳ1 f3 ỵ b=1a2 ịị ẳ1 f3 b=1a2 ịị 47ị Lemma If eZ 0, then the set of discontinuity points of the pffiffiffi pffiffiffiffiffi function f3(z) is either [0, 1] or ½0,1=ð1Àa2 ị (0 o a ẳ e m o 1), ỵ and for the first possibility, the boundary values f3 ðtÞ, f3 tị are calculated by using (35) and f3ỵ 1ị ẳ1 f3 1ị 52ị f3 ỵ 0ị ẳ 1, f3 ð0Þ It is readily seen that f31(z) is the function x1 Zzịị with p p a ẳ m ỵ e 40 (noting that m 0, see, for example (2.10a, b) pffiffiffiffiffi pffiffiffi in [14]) If mÀ e Z0 then f32(z) is the function x1 ðZðzÞÞ with pffiffiffiffiffi pffiffiffi a ¼ mÀ e Z 0, otherwise it is the function x2 Zzịị with p p p a ẳ e m, o a o1 Note that, since ag ỵ b (see [14, pffiffiffi pffiffiffiffiffi (2.10c)]), it follows that a ¼ eÀ m o As the set of discontinuity points of the functions x1 ðZðzÞÞ and x2 ðZðzÞÞ are respectively the intervals [0, 1] and ẵ0,1=1a2 ị as shown above in Sections 3.2.2 and 3.2.3, we have the following conclusion: f3ỵ 0ị ẳ 1, f3 0ị f3 ỵ bị ẳ1 f3 bị 46ị where q p p Zzị ỵ m ỵ e, f3 ỵ 0ị ẳ 1, f3 0ị for the second possibility, f3 ỵ tị, f3 tị are calculated by using (38) in ỵ ỵ which x2k ðtÞ, x2k ðtÞ, a, 1=t are respectively replaced by x2k ðtÞ, x2k ðtÞ, aà , b/t and: 3.2.5 The set of discontinuity points of f3(z) Case 1: eZ 0: If eZ 0, from (31) it follows: f31 ðzÞ ¼ À are calculated by using (35) in which x1 tị, x1 tị, a, 1/t are ỵ respectively replaced by x1 ðtÞ, x1 ðtÞ, aà , b/t and From (41)(44) we have x3ỵ 0ị ẳ eip=2 , x 0ị 133 f3 ỵ 0ị ẳ 1, f3 0ị f3 ỵ bị ẳ1 f3 bị 53ị 54ị p p Remark 2à (i) If either eà o0 or eà Z and mà À eà Z 0, then à f3 ðzÞ is discontinuous on [0, b], otherwise (i.e eà Z and p p it is discontinuous on ẵ0,1=1a2 ị mà À eà o 0) pffiffiffiffiffi pffiffiffiffiffiffiffi (0 oaà ¼ eà À mà o1) (ii) When f3à ðzÞ is discontinuous on [0, b], we have the evaluations (52) at the end of this interval, when it is discontinuous on ½0,1=ð1Àa2à Þ, the evaluations (53) hold 3.2.7 The set of discontinuity points of f(z) From Lemmas 1–3, 2à ,3à , (30) and taking into account the fact o br we have Proposition (i) The set L of discontinuity points of f(z) is one of the three following intervals: [0, 1], ẵ0,1=1a2 ị, ẵ0,b=1a2 ị, p p p p where o a ¼ eÀ m o 1, oaà ¼ eà À mà o1 For the first possibility: f3ỵ 1ị f ỵ 0ị ẳ 1, ẳ1 f ð0Þ f3À ð1Þ ð55Þ For the second possibility: f ỵ 0ị ẳ 1, f 0ị f3ỵ 1=1a2 ÞÞ ¼1 f3À ð1=ð1Àa2 ÞÞ ð56Þ For the last possibility: f ỵ 0ị ẳ 1, f 0ị f3ỵ b=1a2 ịị ẳ1 f3 b=1a2 ịị 57ị 51ị Remark From Lemmas and it follows that: pffiffiffiffiffi pffiffiffi (i) If either e o0 or e Z and mÀ e Z 0, then f3 ðzÞ is discontinpffiffiffiffiffi pffiffiffi uous on [0, 1], otherwise (i.e e Z and mÀ e o0) it is pffiffiffi pffiffiffiffiffi discontinuous on ẵ0,1=1a ị (0 o a ẳ e m o 1) (ii) When f3 ðzÞ is discontinuous on [0, 1], we have the evaluations (48) at the ends of this interval, when it is discontinuous on ẵ0, 1=1a2 ị, the evaluations (49) hold f3à ðzÞ 3.2.6 The set of discontinuity points of For f3à ðzÞ we have the results similar to Lemmas and for f3(z), in which m, n, e, a, f3(z) are replaced by mà , nà , eà , aà , f3à ðzÞ In particular, we have: Lemma 2à If eà Z0, then the set of discontinuity points of the pffiffiffiffiffi pffiffiffiffiffiffiffi function f3à ðzÞ is either [0, b] or ẵ0,b=1a2 ị (0 o a ẳ e m ỵ o1), and for the rst possibility, the boundary values f3 ðtÞ, f3ÃÀ ðtÞ (ii) The boundary values f ỵ tị and f tị are calculated by the formulas mentioned in Lemmas1–3, 2à , 3à 3.3 The proof of Proposition From (29)–(31) and Proposition 5, it is not difficult to see that: Proposition The function f(z) has the properties: (f1) f ðzÞ A HSị, S ẳ fz A C, z= 2Lg, L is one of the three intervals: [0, 1], pffiffiffi pffiffiffiffiffi pffiffiffiffiffi ẵ0,1=1a2 ị, ẵ0, b=1a2 ị, oa ẳ e m o1, o aà ¼ eà À pffiffiffiffiffiffiffi mà o (f2) f(z) is bounded in N(1), Nð1=ð1Àa2 ÞÞ and Nb=1a2 ịị (f3) f zị ẳ Oz1=2 ị as z-0 (f4) f zị ẳ OA1 z ỵA0 ị as jzj-1 (A0 ,A1 are constant) (f5) f(z) is continuous on L from the right and from the left with the boundary values f ỵ tị, f tị that are calculated by the formulas mentioned in Lemmas1–3, 2à , 3à 134 P.C Vinh, P Thi Ha Giang / International Journal of Non-Linear Mechanics 47 (2012) 128–134 Following the same procedure carried out in Section we arrive at the following propositions: References Proposition Eq (30) is equivalent to equation P(z)¼0 in the region S [ ftwo ends of Lg, where P(z) is a first-order polynomial of z whose coefficients not vanish simultaneously, i.e: [1] R Stoneley, Elastic waves at the surface of separation of two solids, Proc R Soc London A (1924) 416–428 [2] K Sezawa, K Kanai, The range of possible existence of Stoneley waves, and some related problems, Bull Earthq Res Inst Tokyo Univ 17 (1939) 1–8 [3] J.G Scholte, On the Stoneley wave equation, Proc Kon Acad Sci Amsterdam 45 (1942) 159–164 [4] J.G Scholte, The range of existence of Rayleigh and Stoneley waves, Mon Not R Astron Soc Geophys Suppl (1947) 120–126 [5] D.M Barnett, J Lothe, S.D Gavazza, M.J.P Musgrave, Consideration of the existence of interfacial (Stoneley) waves in bonded anisotropic elastic halfspaces Proc R Soc London A 412 (1985) 153–166 [6] A.N Stroh, Steady state problems in anisotropic elasticity, J Math Phys 41 (1962) 77–103 [7] T.C Lim, M.J.P Musgrave, Nature 225 (1970) 372 [8] D.A Lee, D.M Corbly, IEEE Trans Sonics Ultrason 24 (1977) 206–212 [9] S Rokhlin, M Hefet, M Rosen, J Appl Phys 51 (1980) 3579–3582 [10] P Chadwick, D.A Jarvis, Interfacial waves in a pre-strain neo-Hookean body I Biaxial state of strain, Q J Mech Appl Math 32 (1979) 387–399 [11] P Chadwick, D.A Jarvis, Interfacial waves in a pre-strain neo-Hookean body II Triaxial state of strain, Q J Mech Appl Math 32 (1979) 401–418 [12] A Dasgupta, Effect of high initial stress on the propagation of Stoneley waves at the interface of two isotropic elastic incompressible media, Indian J Pure Appl Math 12 (1981) 919–926 [13] J Dunwoody, Elastic interfacial standing waves, in: M.F McCarthy, M.A Hayes (Eds.), Elastic Waves PropagationNorth-Holland, Amsterdam, 1989, pp 107–112 [14] M.A Dowaikh, R.W Ogden, Interfacial waves and deformations in prestressed elastic media, Proc R Soc London A 433 (1991) 313–328 [15] R.W Ogden, Non-linear Elastic Deformations, Ellis Horwood, Chichester, 1984 [16] N.I Muskhelishvili, Singular Integral Equations, Noordhoff-Groningen, 1953 [17] N.I Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity, Noordhoff, Netherlands, 1963 Pzị ẳ A^ z ỵ A^ , 2 A^ ỵ A^ a0 58ị Proposition Equation P(z)¼0 has at most one solution in the complex plane C Proposition is deduced immediately from Propositions and 8, and the fact that Eq (30) has no solution in the set of internal points of L due to the discontinuity of f(z) in this set The proof of Proposition is completed Conclusions In this paper, the uniqueness question of Stoneley waves propagating along the bonded interface of two pre-stressed incompressible elastic half-spaces has been settled for the general case To this end the authors have employed the complex function method, and showed that in the complex plane the secular equation of Stoneley waves in pre-stressed incompressible elastic half-spaces has at most one solution This yields immediately that if a Stoneley wave exists, then it is unique Acknowledgment The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no 107.02-2010.07 ... Triaxial state of strain, Q J Mech Appl Math 32 (1979) 401–418 [12] A Dasgupta, Effect of high initial stress on the propagation of Stoneley waves at the interface of two isotropic elastic incompressible. .. points of L due to the discontinuity of f(z) in this set The proof of Proposition is completed Conclusions In this paper, the uniqueness question of Stoneley waves propagating along the bonded interface... complex plane the secular equation of Stoneley waves in pre-stressed incompressible elastic half-spaces has at most one solution This yields immediately that if a Stoneley wave exists, then it