Uniqueness of Stoneley waves in pre-stressed incompressible elastic media

July 19, 2017 | Autor: Pham Vinh | Categoría: Mechanical Engineering, Civil Engineering, Applied Mathematics, Wave propagation

Share Embed

Laporkan tautan ini

Descripción

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 , 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 1 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 do 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

1. 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 do 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 Corresponding author. Tel.: þ 84 4 5532164; fax: þ 84 4 8588817.

E-mail address: [email protected] (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 do 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.

2. 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

2

2

ðl1 þ l2 þ l3 3Þ,

W ¼

m 2

2

2

2

ðl1 þ l2 þ l3 3Þ

ð1Þ

where m, lk ðk ¼ 1,2,3Þ and m , lk ðk ¼ 1,2,3Þ 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 B , 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 B have the same symbol but are systematically distinguished by an asterisk if pertaining to B . For this case the secular equation of Stoneley waves is of the form (see [14, Eq. (4.59)]) 2

3

2

2

3

2

g ðZ þ Z þ 3Z1Þ þ g ðZ þ Z þ3Z 1Þ þ 2gg ð1ZÞð1Z Þ þ gg ðZ þ Z Þð1 þ ZÞð1 þ Z Þ ¼ 0 where sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ arc2 Z¼ ,

g

Z ¼

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a r c2 ,

ð2Þ

a ¼ ml21

g

g ¼ ml22 , a ¼ m l2 g ¼ m l2 1 , 2

ð3Þ

c is the velocity of Stoneley waves that is subject to 0 o c ominfct ,ct g

ð4Þ

ct (ct )

is the speed of transverse wave in the half-space B (B ) deﬁned as ct2 ¼ a=r,

ct2 ¼ a =r

ð5Þ

r (r ) is the mass density of the half-space B (B ). Without loss of generality we can suppose that ct rct . We now introduce the notations x ¼ ct2 =c2 (the dimensionless squared slowness of Stoneley waves), b ¼ ct2 =ct2 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 x1 xb a a , d¼ Z¼d , d ¼ ð7Þ , Z ¼d x x g g

Introducing ð7Þ1,2 into (2), the secular equation of Stoneley waves now is of the form in terms of x ð 4 1 ZbÞ: pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃ xb x1 ð8Þ f ðxÞ q1 ðxÞ pﬃﬃﬃ þq2 ðxÞ pﬃﬃﬃ þ q3 ðxÞ x1 xb þ q4 ðxÞ ¼ 0 x x where 2

2

2

2

2

2

2

2

129

Denote L ¼ L1 [ L2 with L1 ¼ ½0,b, L2 ¼ ½b,1, S ¼ fz A C, z= 2Lg, Nðz0 Þ ¼ fz A S : 0 o jzz0 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 ðzÞ ¼ Oðz1=2 Þ as z-0. f ðzÞ ¼ OðA1 z þA0 Þ 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 ðtÞ (the left boundary value of f(z)) deﬁned as follows: ( 7 f1 ðtÞ, t A L1 f 7 ðtÞ ¼ ð11Þ f27 ðtÞ, t A L2 where fk ðtÞ ¼ fkþ ðtÞ,

k ¼ 1,2

ð12Þ

the bar indicates the complex conjugate, and pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃ f1þ ðtÞ ¼ q4 ðtÞq3 ðtÞ 1t bt þ i½q1 ðtÞ 1t þq2 ðtÞ bt = t , t A L1 # pﬃﬃﬃﬃﬃﬃﬃﬃﬃ " pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃ tb 1t f2þ ðtÞ ¼ q4 ðtÞ þq2 ðtÞ pﬃﬃ þ i q1 ðtÞ pﬃﬃ þ q3 ðtÞ 1t tb , t t t A L2

ð13Þ ðtÞ fkþﬃﬃﬃﬃﬃﬃﬃ p

ðfk ðtÞÞ is the right (left) boundary value of f(z) on Note that Lk and i ¼ 1. In order to prove Proposition 1 we will establish the following propositions: Proposition 2. 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 coefﬁcients do not vanish simultaneously, i.e.: PðzÞ ¼ A^ 1 z þ A^ 0 ,

2 2 A^ 0 þ A^ 1 a0

ð14Þ

q1 ðxÞ ¼ ed½eð3 þ d Þ þðd 1Þxed½ed þ d b

Proposition 3. Equation P(z)¼0 has at most one solution in the complex plane C.

q2 ðxÞ ¼ d ½ð3 þ d Þ þ eðd 1Þxd ½ed þ d b q3 ðxÞ ¼ 4edd

ð9Þ

2

2

2

2

q4 ðxÞ ¼ ½e2 ðd 1Þ þðd 1Þ þ eðd þ d þ 2Þx 2

2

2

2

½e2 d þ d b þeðd þ bd Þ,

e¼

g g

Now, in the complex plane C we consider the equation: pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃ zb z1 ð10Þ f ðzÞ q1 ðzÞ pﬃﬃﬃ þq2 ðzÞ pﬃﬃﬃ þ q3 ðzÞ z1 zb þ q4 ðzÞ ¼ 0 z z pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ where the functions qk ðzÞ are deﬁned by (9), z, z1, zb 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 1. In the complex plane C, Eq. (10) has at most one solution. From Proposition 1, we have immediately: Theorem 1. For pre-stressed incompressible elastic half-spaces of neo-Hookean material, if a Stoneley wave exists, then it is unique.

Proposition 1 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, 3. Now we introduce function g(t) (t A L) as follows: 8 þ f1 ðtÞ > > > > f ðtÞ , t A L1 < 1 ð15Þ gðtÞ ¼ f2þ ðtÞ > > > , t A L2 > : f ðtÞ 2 From (13) and (15) it is obvious that f þ ðtÞ ¼ gðtÞf ðtÞ,

tAL

Consider the function GðzÞ deﬁned as Z 1 loggðtÞ dt GðzÞ ¼ 2pi L tz It is not difﬁcult to verify that (g1 ) GðzÞ A HðSÞ, (g2 ) Gð1Þ ¼ 0,

ð16Þ

ð17Þ

130

P.C. Vinh, P. Thi Ha Giang / International Journal of Non-Linear Mechanics 47 (2012) 128–134

(g3 ) GðzÞ ¼ ð1=2Þlogz þ O0 ðzÞ, z A Nð0Þ, GðzÞ ¼ O1 ðzÞ, z A Nð1Þ, where O0 ðzÞ ðO1 ðzÞÞ bounded in Nð0Þ ðNð1ÞÞ and takes a deﬁned value at z¼ 0 (z ¼1). It is noted that (g3 ) comes from the fact (see [16]): loggð0Þ ¼ ip,

loggð1Þ ¼ 0

ð18Þ

Introduce a new function FðzÞ deﬁned by

FðzÞ ¼ expGðzÞ

ð19Þ

It is implied from ðg1 Þ2ðg3 Þ that: (f1 ) (f2 ) (f3 ) (f4 )

FðzÞ A HðSÞ, FðzÞ a 0 8z A S, FðzÞ ¼ Oð1Þ as jzj-1, FðzÞ ¼ z1=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: þ

F ðtÞ ¼ gðtÞF ðtÞ, t A L

ð20Þ

We now consider the function Y(z) deﬁned by YðzÞ ¼ f ðzÞ=FðzÞ

ð21Þ

From (f1)–(f4), (16), ðf1 Þ–ðf4 Þ and (20), (21), it follows that: (y1) YðzÞ A HðSÞ, (y2) YðzÞ ¼ OðA1 z þ A0 Þ 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 þ ðtÞ ¼ Y ðtÞ, 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 1. 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 YðzÞ ¼ 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)]) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð2b þ2gXÞ gðaXÞ þ gðaXÞg2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ ð2b þ 2g X Þ g ða X Þ þ g ða X Þg2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ 2½g gðaXÞ½g g ða X Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ½ g ðaXÞ þ gða X Þ 2bX þ 2 gðaXÞ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð25Þ 2b X þ 2 g ða X Þ ¼ 0

where a, b, g, a , b , g are (constant) material parameters deﬁned by the formulas (2.6) and (2.7) in [14], X ¼ rc2 , X ¼ r c2 , c is the velocity of Stoneley waves subjected to 0 oc o minfct ,ct g,

ct ¼ a=r,

ct ¼ a =r

ð26Þ

As above, we can assume, without loss of generality, that ct rct , then 0 ob r 1, where b ¼ ct2 =ct2 . Note that a, g, a , g are strictly positive (see [14, (2.10a, b)]). We introduce the following notations:

g b g b , n ¼ , m ¼ , n ¼ a a a a a ct2 d ¼ , x ¼ 2 ðx 4 1Þ a c

m¼

ð27Þ

Then, in terms of these notations Eq. (25) can be written as follows: " pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃ xb x1 x1 q1 ðxÞ pﬃﬃﬃ þq2 ðxÞ pﬃﬃﬃ þq3 ðxÞ 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 pﬃﬃﬃﬃﬃﬃﬃ x1 pﬃﬃﬃﬃﬃ pﬃﬃﬃ þ m et pﬃﬃﬃ þ m e ¼ 0 þ q6 ðxÞ pﬃﬃﬃ t x x x

ð22Þ

where P(z) is a polynomial of order 1 in terms of z: PðzÞ ¼ A^ 1 z þ A^ 0 . From (21) and (22) we have f ðzÞ ¼ FðzÞPðzÞ

ð23Þ

Since FðzÞ a0 8z A S (by ðf2 Þ), and FðzÞ-1 as z-0, Fð1Þ a 0 (by ðf4 Þ), from (23) it is implied that Eq. (10) is equivalent to equation P(z)¼ 0 in the region S [ f0g [ f1g. Coefﬁcients A^ 0 and A^ 1 cannot vanish simultaneously, because if A^ 0 ¼ A^ 1 ¼ 0-PðzÞ 0, then f ðzÞ 0 according to (23). But this contradicts the statement (f3). 2 2 Proposition 2 is proved. From the fact A^ 0 þ A^ 1 a 0, we have immediately Proposition 3. &

Remark 1. 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 r c 2 Z ¼ 1 , Z ¼ 1 ð24Þ

a

3. 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 form (8), where qk ðxÞ are deﬁned by (9) in which d ¼ d ¼ 1. Proposition 1 therefore hold for this case. Thus, for biaxial 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 ðxÞ ¼ 2d m½dðm þ nÞm xd2 m pﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃ q2 ðxÞ ¼ 2 m ðm þn dmÞxb m ,

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ q3 ðxÞ ¼ 2d mm

q4 ðxÞ ¼ ½d2 mð1mÞ þm ð1m Þ þ 2dmm xðd2 m þbm Þ pﬃﬃﬃﬃﬃﬃﬃ q5 ðxÞ ¼ d m x,

pﬃﬃﬃﬃﬃ q6 ðxÞ ¼ d mx,

e ¼ 1 þ m2n,

e ¼ 1 þ m 2n ð29Þ

Note that when the two half-spaces are made of neo-Hookean material (2n ¼ 1 þ m, 2n ¼ 1þ m -e ¼ e ¼ 0, see [14]), the lefthand side of Eq. (28) differs from the left-hand side of Eq. (8) by a 4 positive factor d , where d deﬁned by ð7Þ4 . In the complex plane C Eq. (28) takes the form f ðzÞ f1 ðzÞ þ f2 ðzÞf3 ðzÞf3 ðzÞ ¼ 0

ð30Þ

where pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃ zb z1 f1 ðzÞ ¼ q1 ðzÞ pﬃﬃﬃ þ q2 ðzÞ pﬃﬃﬃ þ q3 ðzÞ z1 zb þ q4 ðzÞ z z pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ zb z1 f2 ðzÞ ¼ q5 ðzÞ pﬃﬃﬃ þ q6 ðzÞ pﬃﬃﬃ , z z

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ !2 u pﬃﬃﬃﬃﬃﬃﬃﬃﬃ u z1 pﬃﬃﬃﬃﬃ pﬃﬃﬃ þ m e f3 ðzÞ ¼ t z

P.C. Vinh, P. Thi Ha Giang / International Journal of Non-Linear Mechanics 47 (2012) 128–134

ﬃ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ !2 u pﬃﬃﬃﬃﬃﬃﬃﬃﬃ u ﬃﬃﬃﬃﬃﬃ ﬃ p zb pﬃﬃﬃ þ m e f3 ðzÞ ¼ t ð31Þ z pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ where z, z1, zb 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 1. We will prove the following proposition:

131

y0 B1

0

Proposition 4. In the complex plane C, Eq. (30) has at most one solution.

A

x0

a

From Proposition 4, we have immediately: Theorem 2. 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 þ ðtÞ and f ðtÞ on L, and the values of the ratio f þ ðtÞ=f ðtÞ at the ends of L. Thus, we have to know those of the functions f1 ðzÞ, f2 ðzÞ, f3 ðzÞ, f3 ðzÞ due to (30).

B2 Fig. 2. The image S0 of mapping Z ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ z1= z, z A S, S0 ¼ fZ ¼ x0 þ iy0 : x0 Z 0g.

v1 B1

3.1. The set of discontinuity points of f1 ðzÞ, f2 ðzÞ It is clear that: Lemma 1. (i) The functions f1(z) are discontinuous on the set 7 7 L ¼ L1 [ L2 , L1 ¼ ½0,b, L2 ¼ ½b,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 tb 7i 1t 7 7 f21 ¼ pﬃﬃ ½q5 ðtÞ 1t þ q6 ðtÞ bt , f22 ¼ q6 ðtÞ pﬃﬃ 7 iq5 ðtÞ pﬃﬃ t t t ð32Þ

0

A

a1/2

u1

B2

(iii) The following equalities hold: þ f11 ð0Þ f11 ð0Þ

¼

þ f21 ð0Þ ¼ 1, f21 ð0Þ

þ f12 ð1Þ f þ ð1Þ ¼ 22 ð1Þ ¼ þ 1 f12 ð1Þ f22

ð33Þ

þ 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) pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ 3.2.1. The image of ZðzÞ ¼ z1= z By S we denote the set obtained by removing from the complex C the interval L ¼ ½0, p 1ﬃﬃﬃﬃﬃﬃﬃﬃﬃ (see Fig. 1), by S0 we denote the pﬃﬃﬃ image of the function ZðzÞ ¼ z1= z, z A S, and ½a,b þ (½a,b ), a,b being real numbers, is the set of points z A ½a,b : z ¼ t þi0 þ , t A ½a,b (z A ½a,b : z ¼ t þi0 , t A ½a,b). Then, it is clear that S0 ¼ fZ ¼ x0 þiy0 : x0 Z0g, see Fig. 2, in which: the point Q ð1,0Þ is converted to the point A, the point 0 þ (0 ) is converted to B1 (B2), L þ (L ) is mapped onto AB1 (AB2).

y

Fig. 3. The image S1 (non-shaded region) of mapping x1 ðZÞ ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z þ a, a Z 0, Z A S0 .

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3.2.2. The image of x1 ðZÞ ¼ Z þa,a Z0 As S0 ¼ fZ ¼ x0 þiy0 : x0 Z 0g, according to Section 3.2.1, it is not pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ difﬁcult to verify that the image of the function x1 ðZÞ ¼ Z þ a, a Z0, denoted by S1 , is the ‘‘non-shaded region’’ in Fig. 3. 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 8 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > < u1 ðtÞ ¼ ð a2 þ t2 þ aÞ=2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð34Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > : v1 ðtÞ ¼ 7 ð a2 þ t2 aÞ=2, t Z 0

x1 ¼ u1 þiv1 , the sign ‘‘ þ’’ (‘‘ ’’) is corresponding to AB1 (AB2). It is readily seen that the set of discontinuity points of the function x1 ðZðzÞÞ is the interval L¼[0,1], and ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! v rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! u u u1 u1 1 1 þ t t 2 a þ 1 þ a þ i a2 þ 1a , x1 ðtÞ ¼ 2 t 2 t þ x 1 ðtÞ ¼ x1 ðtÞ

ð35Þ

From (35) it is readily seen that

0

1 Q

Fig. 1. The complex plane C with the cut L¼ [0 1].

x

x1þ ð0Þ ¼ eip=2 , x 1 ð0Þ

x1þ ð1Þ ¼ þ1 x 1 ð1Þ

ð36Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3.2.3. The image of x2 ðZÞ ¼ Za, 0 oa o1 Similarly, it is not difﬁcult to show that the image of the function pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x2 ðZÞ ¼ Za, 0 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. 4. Here, ½0,a þ A S0 (½0,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 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > < u2 ðtÞ ¼ ð a2 þ t2 aÞ=2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð37Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > : v2 ðtÞ ¼ 7 ð a2 þ t2 þ aÞ=2, t Z0

v3

x2 ¼ u2 þiv2 , the sign ‘‘þ’’ (‘‘’’) is corresponding to A1 B1 (A2 B2 ). It is readily seen that the function x2 ðZðzÞÞ is discontinuous on the interval

0

B1 A u3

a1/2

L ¼ L1 [ L2 , L1 ¼ ½0,1, L2 ¼ ½1,1=ð1a2 Þ (see Fig. 5), and vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ !ﬃ u rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! u rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u1 u1 1 1 þ t t 2 2 a þ 1a þ i a þ 1 þ a x21 ðtÞ ¼ 2 t 2 t pﬃﬃﬃ þ x22 ðtÞ ¼ it, 0 rt r a,

B2

þ x 2k ðtÞ ¼ x2k ðtÞ, k ¼ 1,2

ð38Þ Fig. 6. The image of mapping x3 ðZÞ (non-shaded region): x3 ðZÞ ¼ aZ 0, p 40, Z A S0 .

From (38) it is obvious that þ x21 ð0Þ x21 ð0Þ

þ x21 ð1=ð1a2 ÞÞ ¼ þ1 x21 ð1=ð1a2 ÞÞ

¼ eip=2 ,

ð39Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3.2.4. The image of x3 ðZÞ ¼ Z þa þip, a Z 0, p ispaﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ real number It is clear that the image of the function x3 ðZÞ ¼ 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

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a2 þ p2 þ a =2, u3A ¼

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a2 þp2 a =2 v3A ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ v3A ¼ a2 þ p2 a =2

if p o 0

if p Z0

ð40Þ

The function x3 ðZðzÞÞ is discontinuous on the interval L¼[0,1], and if p Z0:

0 rt r 1

ð41Þ

8 vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1ﬃ v u 0v u 0v > rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2 1 u u > u >u u > 1@u 1 1 1 u u > ta2 þ p > 1 þ aA þ it @ta2 þ p 1 aA > > t2 t 2 t > > > > > > > > 1 > > rt r1 > < 1þ p2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ x3 ðtÞ ¼ v vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1ﬃ v u 0u u 0u > rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2 1 > u u > > u1@u 1 1 u1@u > t t 2 2 A A > t a þ p a þ p 1 þ a it 1 a > > t 2 t > 2 > > > > > > > 1 > > > : 0r t o 1 þ p2

ð42Þ

u2

0 A2

B2

Fig. 4. The image S2 (non-shaded region) of mapping x2 ðZÞ ¼ o 1, Z A S0 .

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Za, 0 o a

for p o 0:

y

0

AB1 ðAB2 Þ A S1 (see Fig. 6), where A ¼ ðu3A ,v3A Þ, and:

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 v u 0v u 0v rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!ﬃ2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!ﬃ2 1 u u u u u u u 1 1 1 @u 1 þ t t 2 2 @ A t t 1 þ a þ i 1 aA x3 ðtÞ ¼ a þ pþ a þ pþ 2 t 2 t

A1

−a1/2

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z þ a þ ip,

1 Q

1/(1−a2) R

x

8 vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1ﬃ v u 0v u 0v > rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2 1 u u > u > >u 1@u 1 1@u 1 u u > t t 2 2 A A > a þ pþ a þ pþ 1 þ a þ it 1 a > > t2 t 2 t > > > > > > > > 1 > > > < 0 r t o 1 þ p2 þ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ x3 ðtÞ ¼ v ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1ﬃ v u 0v u 0v >u rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2 1 u u > u > > 1@u 1 1 u1@u >u t t 2 þ pþ 2 þ pþ A A > t t a a þ a i a 1 1 > > 2 t 2 t > > > > > > > > 1 > > > : 1 þ p2 r t r 1,

ð43Þ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0v ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1ﬃ u rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2 1 u 0v u rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2 u u u u1@u 1 u1@u 1 t 2 t 2 1 aA x3 ðtÞ ¼ t a þ p 1 þaAit a þ p 2 t 2 t Fig. 5. The complex plane C with the cut L ¼ ½01=ð1a2 Þ, 0o ao 1.

0 rt r 1

ð44Þ

P.C. Vinh, P. Thi Ha Giang / International Journal of Non-Linear Mechanics 47 (2012) 128–134 þ

x3þ ð1Þ ¼ þ1 x 3 ð1Þ

ð45Þ

f3 ðzÞ ¼ f31 ðzÞf32 ðzÞ

f32 ðzÞ ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃ ZðzÞ þ m e

Lemma 3 . If e o 0, then the set of discontinuity points of the function f3 ðzÞ is [0, b], and the boundary values f3 þ ðtÞ, f3 ðtÞ are þ calculated by using (41)–(44) in which x3 ðtÞ, x3 ðtÞ, a, 1/t are þ respectively replaced by x3 ðtÞ, x3 ðtÞ, a , b=t and

ð48Þ

for the second possibility, f3þ ðtÞ, f3 ðtÞ are calculated by using (38) and f3þ ð0Þ ¼ 1, f3 ð0Þ

f3þ ð1=ð1a2 ÞÞ ¼1 f3 ð1=ð1a2 ÞÞ

ð49Þ

Case 2: eo 0: If eo 0, then f3(z) takes the form (46) where qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ ð50Þ f31 ðzÞ ¼ ZðzÞ þ m þi jej, f32 ðzÞ ¼ ZðzÞ þ mi jej The function f31 ðzÞ (f32 ðzÞ) is therefore the function x3 ðZðzÞÞ with pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ a ¼ m 4 0, p ¼ jej (a ¼ m, p ¼ jej). From the results of Section 3.2.4, one can see that: Lemma 3. If eo 0, then the set of discontinuity points of the function f3(z) is [0, 1], and the boundary values f3þ ðtÞ, f3 ðtÞ are calculated by using (41)–(44) and f3þ ð0Þ f3 ð0Þ

¼ 1,

f3þ ð1Þ f3 ð1Þ

¼1

f3 þ ðb=ð1a2 ÞÞ ¼1 f3 ðb=ð1a2 ÞÞ

ð47Þ

Lemma 2. If eZ 0, then the set of discontinuity points of the pﬃﬃﬃ pﬃﬃﬃﬃﬃ function f3(z) is either [0, 1] or ½0,1=ð1a2 Þ (0 o a ¼ e m o 1), þ and for the ﬁrst possibility, the boundary values f3 ðtÞ, f3 ðtÞ are calculated by using (35) and f3þ ð1Þ ¼1 f3 ð1Þ

ð52Þ

f3 þ ð0Þ ¼ 1, f3 ð0Þ

It is readily seen that f31(z) is the function x1 ðZðzÞÞ with pﬃﬃﬃﬃﬃ pﬃﬃﬃ a ¼ m þ e 40 (noting that m 4 0, see, for example (2.10a, b) pﬃﬃﬃﬃﬃ pﬃﬃﬃ in [14]). If m e Z0 then f32(z) is the function x1 ðZðzÞÞ with pﬃﬃﬃﬃﬃ pﬃﬃﬃ a ¼ m e Z 0, otherwise it is the function x2 ðZðzÞÞ with pﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ a ¼ e m, 0 o a o1. Note that, since ag þ b 4 0 (see [14, pﬃﬃﬃ pﬃﬃﬃﬃﬃ (2.10c)]), it follows that a ¼ e m o 1. As the set of discontinuity points of the functions x1 ðZðzÞÞ and x2 ðZðzÞÞ are respectively the intervals [0, 1] and ½0,1=ð1a2 Þ as shown above in Sections 3.2.2 and 3.2.3, we have the following conclusion:

f3þ ð0Þ ¼ 1, f3 ð0Þ

f3 þ ðbÞ ¼1 f3 ðbÞ

ð46Þ

where qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃ ZðzÞ þ m þ e,

f3 þ ð0Þ ¼ 1, f3 ð0Þ

for the second possibility, f3 þ ðtÞ, f3 ðtÞ are calculated by using (38) in þ þ which x2k ðtÞ, x2k ðtÞ, a, 1=t are respectively replaced by x2k ðtÞ, x2k ðtÞ, a , 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

are calculated by using (35) in which x1 ðtÞ, x1 ðtÞ, a, 1/t are þ respectively replaced by x1 ðtÞ, x1 ðtÞ, a , b/t and

From (41)–(44) we have

x3þ ð0Þ ¼ eip=2 , x 3 ð0Þ

133

f3 þ ð0Þ ¼ 1, f3 ð0Þ

f3 þ ðbÞ ¼1 f3 ðbÞ

ð53Þ

ð54Þ

pﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ Remark 2 . (i) If either e o0 or e Z 0 and m e Z 0, then f3 ðzÞ is discontinuous on [0, b], otherwise (i.e. e Z 0 and pﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ it is discontinuous on ½0,1=ð1a2 Þ m e o 0) pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃ (0 oa ¼ e m o1). (ii) When f3 ðzÞ is discontinuous on [0, b], we have the evaluations (52) at the end of this interval, when it is discontinuous on ½0,1=ð1a2 Þ, the evaluations (53) hold. 3.2.7. The set of discontinuity points of f(z) From Lemmas 1–3, 2 ,3 , (30) and taking into account the fact 0 o br 1 we have Proposition 5. (i) The set L of discontinuity points of f(z) is one of the three following intervals: [0, 1], ½0,1=ð1a2 Þ, ½0,b=ð1a2 Þ, pﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃ where 0 o a ¼ e m o 1, 0 oa ¼ e m o1. For the ﬁrst possibility: f3þ ð1Þ f þ ð0Þ ¼ 1, ¼1 f ð0Þ f3 ð1Þ

ð55Þ

For the second possibility: f þ ð0Þ ¼ 1, f ð0Þ

f3þ ð1=ð1a2 ÞÞ ¼1 f3 ð1=ð1a2 ÞÞ

ð56Þ

For the last possibility: f þ ð0Þ ¼ 1, f ð0Þ

f3þ ðb=ð1a2 ÞÞ ¼1 f3 ðb=ð1a2 ÞÞ

ð57Þ

ð51Þ

Remark 2. From Lemmas 2 and 3 it follows that: pﬃﬃﬃﬃﬃ pﬃﬃﬃ (i) If either e o0 or e Z 0 and m e Z 0, then f3 ðzÞ is discontinpﬃﬃﬃﬃﬃ pﬃﬃﬃ uous on [0, 1], otherwise (i.e. e Z 0 and m e o0) it is pﬃﬃﬃ pﬃﬃﬃﬃﬃ 2 discontinuous on ½0,1=ð1a Þ (0 o a ¼ 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 ½0, 1=ð1a2 Þ, the evaluations (49) hold. f3 ðzÞ

3.2.6. The set of discontinuity points of For f3 ðzÞ we have the results similar to Lemmas 2 and 3 for f3(z), in which m, n, e, a, f3(z) are replaced by m , n , e , a , f3 ðzÞ. In particular, we have: Lemma 2 . If e Z0, then the set of discontinuity points of the pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃ function f3 ðzÞ is either [0, b] or ½0,b=ð1a2 Þ (0 o a ¼ e m þ o1), and for the ﬁrst possibility, the boundary values f3 ðtÞ, f3 ðtÞ

(ii) The boundary values f þ ðtÞ and f ðtÞ are calculated by the formulas mentioned in Lemmas1–3, 2 , 3 . 3.3. The proof of Proposition 4 From (29)–(31) and Proposition 5, it is not difﬁcult to see that: Proposition 6. The function f(z) has the properties: (f1) f ðzÞ A HðSÞ, S ¼ fz A C, z= 2Lg, L is one of the three intervals: [0, 1], pﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ ½0,1=ð1a2 Þ, ½0, b=ð1a2 Þ, 0 oa ¼ e m o1, 0 o a ¼ e pﬃﬃﬃﬃﬃﬃﬃ m o 1. (f2) f(z) is bounded in N(1), Nð1=ð1a2 ÞÞ and Nðb=ð1a2 ÞÞ. (f3) f ðzÞ ¼ Oðz1=2 Þ as z-0. (f4) f ðzÞ ¼ OðA1 z þA0 Þ 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 þ ðtÞ, f ðtÞ that are calculated by the formulas mentioned in Lemmas1–3, 2 , 3 .

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 2 we arrive at the following propositions:

References

Proposition 7. 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 do 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. 5 (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.

PðzÞ ¼ A^ 1 z þ A^ 0 ,

2 2 A^ 0 þ A^ 1 a0

ð58Þ

Proposition 8. Equation P(z)¼0 has at most one solution in the complex plane C. Proposition 4 is deduced immediately from Propositions 7 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 4 is completed.

4. 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.

Lihat lebih banyak...

Uniqueness of Stoneley waves in pre-stressed incompressible elastic media

Descripción

Comentarios