DSpace at VNU: On formulas for the Rayleigh wave velocity in pre-strained elastic materials subject to an isotropic internal constraint

DSpace at VNU: On formulas for the Rayleigh wave velocity in pre-strained elastic materials subject to an isotropic inte...

On formulas for the Rayleigh wave velocity in pre-strained elastic

materials subject to an isotropic internal constraint

Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam

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 prin-cipal directions of prestrain of an elastic half-space subject to a pure homogeneous pre-strain, 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 pre-strains and prestresses is discussed

Ó 2009 Elsevier Ltd All rights reserved

1 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 Rah-man and Barber[3], but for a limited range of values of the parameter¼ ð1  2mÞ=ð2  2mÞ, wheremis 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 offor 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 ofby 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 MATHEM-ATICA to obtain the formula A detailed derivation of this formula was given by Vinh and Ogden [7]together with an

0020-7225/$ - see front matter Ó 2009 Elsevier Ltd All rights reserved.

* Corresponding author Tel.: +84 4 5532164; fax: +84 4 8588817.

E-mail address: pcvinh@vnu.edu.vn (P.C Vinh).

Contents lists available atScienceDirect

International Journal of Engineering Science

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / i j e n g s c i

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 pre-strained 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 gen-eral 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

2 Formulas for the velocity of Rayleigh wave propagating in the principal directions of prestrain are derived in Section3 They hold for any strain-energy function and any isotropic internal constraint In this section are also established the nec-essary and sufficient conditions for unique existence of Rayleigh wave In Section4, some specific cases of strain-energy function and isotropic constraint are considered Some remarks on the use of the obtained formulas for nondestructive eval-uation of prestress are made in Section5

2 Secular equation

In this section we first summarize the basic equations which govern small amplitude time-dependent motions superim-posed 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 X2P0 and we suppose that the deformed configuration is obtained by application of a pure homogeneous strain of the form:

x1¼ k1X1; x2¼ k2X2; x3¼ k3X3; ki¼ const; ki>0; i ¼ 1; 2; 3: ð1Þ

In its deformed configuration the body, therefore, occupies the region x2>0 with the boundary x2¼ 0

Suppose that the material is subject to an isotropic internal constraint, written as[24]:

whereCis a symmetric function of the principal stretches ki We restrict attention to the case:

The constraint(2)create the workless reaction tensor N[21]whose non-zero components are:

where J ¼ k1k2k3 The Cauchy stress tensor associated to the static deformed state ris of the diagonal form with non-zero components[24]:



where the strain-energy Wðk1;k2;k3Þ is a symmetric function of ki, i.e its value is left unchanged by any permutation of

k1;k2;k3, Wi¼ @W=@ki, and P is determined as follows:

P ¼ W2

C if r22¼ 0; otherwise P ¼J r22 k2W2

As in[24], in this paper we are interested in the case r22¼ 0 We consider a plane motion in the ðx1;x2Þ-plane with displace-ment components u1;u2;u3such that:

where t is the time Then, in the absence of body forces the equations governing infinitesimal motion are[21,24]:

s11;1þ s21;2¼ q€u1; s12;1þ s22;2¼ q€u2; ð8Þ

where qis mass density of the material at the static deformed state, a superposed dot signifies differentiation with respect to

t, commas indicate differentiation with respect to spatial variables xi, and[21,24]:

in which p represents the increment in P, and the components of the fourth order elasticity tensor B

ijklare given by:

B

Non-zero components of tensors B and eB are[21,24,25]:

JBiijj¼ kikjWij; JBijij¼kiWi kjWj

k2i  k2j

k2iði – jÞ; JBijji¼ JBijij kiWi; ð11Þ

JeBiijj¼ kikjCij; JeBijij¼kiCi kjCj

k2i  k2j k

2

i ði – jÞ; JeBijji¼ JeBijij kiCi; ð12Þ

where Wij¼ @2W=@ki@ j Note that there is no summation over i or j in the formulas(11)and(12), and JBijij, JeBijijdefined when i–j, ki–kj In the case where i – j, ki¼ kj, JBijij, JeBijijare given by:

JBijij¼1

2ðJBiiii JBiijjþ kiWiÞ; JeBijij¼1

2ðJeBiiii JeBiijjþ kiCiÞ: ð13Þ

Note that:

The incremental constraint of(2)is[21,24]:

Since the surface of the half-space is free of traction, we have:

In addition to Eqs.(8), (9) and (15)and the boundary condition(16), the decay condition is required, namely:

Since tensor B

ijklis strongly elliptic[25], it follows that[24]:

A > 0; C > 0; B þ ffiffiffiffiffiffi

AC

p

where

A ¼ ðk1k2C1C2Þ1B

1212; C ¼ ðk1k2C1C2Þ1B

2121;

B ¼1

2½ðk1C1Þ

2

B

1111þ ðk2C2Þ2B

2222  ðk1k2C1C2Þ1ðB

1122þ B

Now we consider a surface Rayleigh wave propagating in the x1-direction with the velocityvand the wave number k Then,

u1;u2, p and smnðm; n ¼ 1; 2Þ are sought in the form:

uj¼ Ujðkx2Þeikðx 1 vtÞðj ¼ 1; 2Þ; p ¼ kQðkx2Þeikðx 1 vtÞ;

where i2¼ 1 Introducing(20)1, (20)3into(8) and (15)and taking into account(7)yield:

iS11þ S021¼ q v2U1;

iS12þ S022¼ q v2U2;

ik1C1U1þ k2C2U0

2¼ 0

ð21Þ

in which the prime indicates the derivative with respect to y ¼ kx Substituting(20)into(9)gives:

S11¼ iB1111U1þ B1122U02þ Q N11; S12¼ B1221U01þ iB1212U2;

S21¼ B

2121U0

1þ iB1221U2; S22¼ iB1122U1þ B

2222U0

Now we introduce a new variable z given by:

z ¼k1C1

k2C2

Substituting(22)into(21)leads a system of three differential equations for three unknown functions U1;U1;Q of z, namely:

B

2121 k1

C 1

k 2 C 2

 2

U00

1þ B1221k1

C 1

k 2 C 2 B1111þk 1 C 1

k 2 C 2B

1122þ q v2

U1þ iN11Q ¼ 0;

i B2222 k 1 C1

k 2 C2

 2

 B1221k 1 C1

k 2 C2 B1122k 1 C1

k 2 C2

U01þ ðB1212 q v2ÞU2k 1 C1

k 2 C2N22Q0¼ 0;

iU1þ U02¼ 0:

8

>

>

>

>

ð24Þ

The solution of(24)is sought in the form:

where Ak(k ¼ 1; 2; 3) are constant and:

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:

where

a¼ B1212; c¼ k1C1

k2C2

 2

B

2121¼C

2 1

C2a; 2b¼ B1111 2k1C1

k2C2

ðB1221þ B1122Þ þ k1C1

k2C2

 2

B2222:

ð28Þ

From(18), (19) and (28)it follows:

a>0; c>0; bþ ffiffiffiffiffiffiffiffiffiffi

ac

p

From(27)we have:

s2þ s2¼2b



 q v2

c ; s2s2¼a q v2

The roots s2, s2of the quadratic Eq.(27)for s2are 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: s2s2>0 Therefore, by(30)2andc>0 we have:

From(16) and (20)3it deduces:

Let s1, s2be two roots of(27)satisfying(26) Then, the general solution of(24)that decays at þ1 is:

U1¼ B1es 1 zþ B2es 2 z;

U2¼iB1

s1es 1 zþiB2

s2es 2 z;

Q ¼ B1C1es 1 zþ B2C2es 2;

8

>

where B1, B2are constant and:

Cj¼ i

N11

B1111þ B1122k1C1

k2C2

þ B2121 k1C1

k2C2

 2

s2

j þ B1221k1C1

k2C2

þ q v2

Introducing(22)3,4(23), (33), (34)into(32)yields a homogeneous system of two linear equations for B1;B2, namely:

s2 cs2þk1C1

k2C2

B 1221

B1þ s1 cs2þk1C1

k2C2

B 1221

B2¼ 0;

2b

þk1C1

k C B



1221 q v2cs2

B1þ 2bþk1C1

kC B



1221 q v2cs2

B2¼ 0:

ð35Þ

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 s2, the secular equation is of the form:

cđa q v2

ỡ ợ đ2bợ 2d q v2

ỡơcđa q v2

where

dỬk1C1

k2C2

B

1221Ửk2C2

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 s1Ử s2, as pointed out in[24]

3 Formulas for the Rayleigh wave velocity

We define the variable x by:

x Ửv2=c2

and in terms of x Eq.(36)is written as:

a

c x

ợ 2b

ợ 2d

  ffiffiffiffiffiffiffiffiffiffiffiffiffia

c x

r

Ửd

2

From(31) and (38)it follows:

0 < x <a

Now we introduce a new variablegdefined by:

gỬ

ffiffiffiffiffiffiffiffiffiffiffiffiffi

a

c x

r

Then Eq.(39)now becomes[24]:

where

a Ử2b



ợ 2da

2

From(40) and (41)it implies:

0 <g< ffiffiffiffiffiffiffiffiffiffiffiffi

a=c

p

Note that the transformation(41)is a 1  1 mapping that maps đ0;a=cỡ into 0; ffiffiffiffiffiffiffiffiffiffiffiffi

a=c

p

Remark 1 It is clear that a Rayleigh wave exists () Eq.(39)has a positive real solution x and Reơsiđxỡ > 0 đi Ử 1; 2ỡ It is shown in [24] that: (i) If 2b>a, then Reơsiđxỡ > 0 đi Ử 1; 2ỡ if and only if x 2 đ0; x1ỡ; (ii) If 2b6a, then Reơsiđxỡ > 0 đi Ử 1; 2ỡ if and only if x 2 đ0; x2ỡ; where:

x1Ửa

c; x2Ử 2 b



c 1 ợ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a 2b

Note that

gđ0ỡ Ử

ffiffiffiffiffi

a

c

r

; gđx1ỡ Ử 0; gđx2ỡ Ử ^gỬ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a

c2b



c ợ 1

s

From(1)5, (3), (29)2, (37)it deduces d>0 ) b Ử d2=c2>0, therefore:

Lemma 1 Eq.(42)has a unique solution in the interval đ0; ợ1ỡ:

Proof If a P 0 then f0đgỡ Ử 3g2ợ 2gợ a > 08g>0, i.e f đgỡ is strictly increasingly monotonous in the interval đ0; ợ1ỡ Since f đ0ỡ < 0 and f đợ1ỡ Ử ợ1, it is clear that Eq.(42)has a unique solution in the interval đ0; ợ1ỡ

If a < 0 then equation f0đgỡ Ử 0 has distinct solutions denoted byg

max,g min, andg

max<0,g

min>0 Since f đgỡ strictly decreasingly monotonous in đg

max;g minỡ,g max<0 <g

minand f đ0ỡ < 0, we have:

f đgỡ < 0 8 g2 đ0;g

This implies that equation(42)has no root in đ0;g

min Since f đgỡ strictly increasingly monotonous in đg

min;ợ1ỡ, f đg

minỡ < 0 and f đợ1ỡ Ử ợ1, it follows that Eq.(42)has a unique solution in the interval đ0; ợ1ỡ h

Proposition 1

(i) If:

f

ffiffiffiffiffi

a

c

r

then Eq.(42)has a unique root in the interval 0; ffiffiffiffi

a 

c 

q

(ii) Otherwise, Eq.(42)has no root in the interval 0; ffiffiffiffi

a 

c 

q

Proof

(i) Suppose(49)be satisfied With the help ofLemma 1, from(49)and f đ0ỡ < 0 it implies that(42)has a unique root in the interval 0; ffiffiffiffia

c 

q

(ii) If (49) does not hold, i.e f ffiffiffiffia

c 

q

60 Since f đợ1ỡ Ử ợ1, it follows that Eq (42) has a root in the interval

ơ ffiffiffiffiffiffiffiffiffiffiffiffi

a=c

p

;ợ1ỡ This implies, byLemma 1, that Eq.(42)has no root in the interval 0; ffiffiffiffi

a 

c 

q

h

Proposition 2

(i) Let 2b>a Then Eq.(42)has a unique solution in the interval 0; ffiffiffiffiffiffiffiffiffiffiffiffi

a=c

p

if:

f

ffiffiffiffiffi

a

c

r

otherwise, it has no solution in this interval

(ii) Let 2b6a Then Eq.(42)has a unique solution in the domain đ^g; ffiffiffiffiffiffiffiffiffiffiffiffi

a=c

p

ỡ if:

f

ffiffiffiffiffi

a

c

r

otherwise, it has no solution in this domain

Proof

(i) It is deduced from theProposition 1 Note that theProposition1 holds for both cases: 2b>aand 2b6a (2i) Suppose that 2b6a It is not difficult to verify that 0 6 ^g< ffiffiffiffiffiffiffiffiffiffiffiffi

a=c

p If(51)is satisfied, then Eq.(42)has a solution in the domain đ^g; ffiffiffiffiffiffiffiffiffiffiffiffi

a=c

p

ỡ ByLemma 1, it has a unique solution in this domain If f đ ffiffiffiffiffiffiffiffiffiffiffiffi

a=c

p

ỡ 6 0 (f đ^gỡ P 0ỡ, because

f đợ1ỡ Ử ợ1 đf đ0ỡ < 0ỡ, it implies that Eq.(42)has a solution in ơ ffiffiffiffiffiffiffiffiffiffiffiffi

a=c

p

;ợ1ỡ đđ0; ^gỡ ByLemma 1, it has no root in the interval đ^g; ffiffiffiffiffiffiffiffiffiffiffiffi

a=c

p

ỡ h FromProposition 1, Remark 1and the fact:

g2 0;

ffiffiffiffiffi

a

c

r

() x 2 đ0; x1ỡ; g2 g^;

ffiffiffiffiffi

a

c

r

we have immediately the following theorem

Theorem 1

(i) A Rayleigh wave exists if and only if either:

2b>a and f

ffiffiffiffiffi

a

c

r

2b6a; f

ffiffiffiffiffi

a

c

r

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 3 If Eq.(42)has two or three distinct real roots, then the root corresponding to the Rayleigh wave, denoted byg

r, is the largest root

Proof Suppose Eq.(42)has two or three distinct real roots According toLemma 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 2 If equation:

has two distinct real rootsg

max;g min(g max<g min), then:

f ðg

Proof Suppose f0ðgÞ ¼ 0 has two distinct real rootsg

max,g min(g max<g min) Since:

g

maxþg

min¼ 2

it follows:

g

If g

max<0 <g

min, then f ðg

minÞ < f ð0Þ < 0 because f ðgÞ strictly decreasingly monotonous in the interval ðg

max;g minÞ If

g

min60, then also f ðg

minÞ 6 f ð0Þ < 0 because f ðgÞ strictly increasingly monotonous in ðg

min;þ1Þ In both cases we have

f ðg

minÞ < 0 h

Now in order to solve the cubic Eq.(42)we introduce the new variable z defined as:

z ¼gþ1

In terms of z, Eq.(42)is of the form:

where

q2

¼1

9ð1  3aÞ; r ¼

2

27

1

here q2may be negative It is noted from the geometrical point that r ¼ f ðg

NÞ where N is the point of inflexion of the cubic curve f ¼ f ðgÞ

Our task is now to find the real solution zrof Eq.(60)which is corresponding tog

rby the relation(59) Asg

ris the largest root of Eq.(42), zr is the largest one of Eq.(60)in the case that it has two or three distinct real roots By theory of cubic equation, three roots of Eq.(60)are given by the Cardan’s formula as follows (see[26]):

z1¼ S þ T;

z2¼ 1

2ðS þ TÞ þ

1

2i

ffiffiffi 3

p

ðS  TÞ;

z3¼ 1

2ðS þ TÞ 

1

2i

ffiffiffi 3

p

ðS  TÞ;

ð62Þ

where i2¼ 1 and:

S ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi

D p

3

q

; T ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R  ffiffiffiffi D p

3

q

;

D ¼ R2

þ Q3; R ¼ 1

2r; Q ¼ q

2:

ð63Þ

Remark 2 In relation to these formulas we emphasize two points:

(i) The cubic root of a real, negative number is taken as the negative real root.

(ii) If the argument in S is complex we take the phase angle in T as the negative of the phase angle in S, such as T ¼ S, where Sis the complex conjugate value of S

Remark 3 The nature of three roots of Eq.(60)depends on the sign of its discriminant D, in particular: If D > 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)zris given by:

zr¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi

D p

3

q

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi D p

3

in which each radical is understood as complex roots taking its principle value

Case 1: D > 0

If D > 0, then byRemark 3, Eq.(60)has a unique real solution, so it is zr, given by the first of(62), in particular:

zr¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi

D p

3

q

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R  ffiffiffiffi D p

3

q

ð65Þ

in which the radicals are understood as real ones From(63)we have:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi

D

p

3

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R  ffiffiffiffi D p

3

q

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R2 D

3

p

¼

ffiffiffiffiffiffiffiffiffiffi

Q3

3

q

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðq2Þ3

3

q

therefore

zr¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi

D p

3

q

þ q2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi D p

3

q

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:

R þ ffiffiffiffi

D

p

Indeed, consider equation f0ðgÞ ¼ 3g2þ 2gþ a ¼ 0 Its discriminant isD0¼ 1  3a IfD0>0, then f0¼ 0 has two distinct real rootsg

max;g

min(g

max<g min) Because Eq.(60)has a unique real solution, it follows that f ðg

maxÞ:f ðg minÞ > 0 ByLemma

2, f ðg

minÞ < 0, therefore: f ðg

maxÞ < 0; f ðg

minÞ < 0: This implies that f ðg

NÞ < 0 ) r < 0 IfD060, then f0ðgÞ P 08g) f ðgÞ strictly increasingly monotonous in ð1; þ1Þ ) f ðg

NÞ ¼ f ð1=3Þ < f ð0Þ < 0 ) r < 0 Thus, in both cases we have: r < 0 Since R ¼ 1r ) R > 0 ) R þ ffiffiffiffi

D p

>0, and(68)is proved

Case 2: D ¼ 0

When D ¼ 0, then according toRemark 3 Eq (60)has two distinct real roots In this case equation f0ðgÞ ¼ 0 has also two distinct real roots ) f ðg

minÞ < 0, accordingLemma 2, andD0>0 ) q2¼1D0>0 On view of f ðg

minÞ < 0 and the fact that equation (60) has two distinct real roots, it deduces that f ðg

maxÞ ¼ 0 From f ðg

minÞ < 0 and f ðg

maxÞ ¼ 0 it follows that

r ¼ f ðg

NÞ < 0 From(63)3,4,5, D ¼ 0, r < 0 we have:

Taking into account (69)2 Eq (60) becomes: z3 3jqj2z  2jqj3

¼ 0, and its roots are: z1¼ 2jqj, z2¼ jqj (double root) ) zr¼ z1¼ 2jqj because zris the largest according toProposition 3 In the other hand, using(69)1and D ¼ 0 it is easy to ver-ify that:

2jqj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi

D p

3

q

þ q2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi D p

3

q

ð70Þ

in which each radical is understood as complex roots taking its principle value The formula(64)again valid for the case

D ¼ 0

Case 1: D < 0

If D < 0, then according toRemark 3, Eq.(60)has three distinct real roots, and zris the largest root byProposition 3 By arguments presented in[10](p 255) one can show that, in this case the largest root zrof Eq.(60)is given by:

zr¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi

D p

3

q

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R  ffiffiffiffi D p

3

q

within which each radical is understood as the complex root taking its principal value By h ð2 ð0;pÞÞ we denote the phase angle of the complex number R þ i ffiffiffiffiffiffiffiffi

D p It is not difficult to verify that:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi

D

p

3

q

¼ jqjeih;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R  ffiffiffiffi D p

3

q

where each radical is understood as the complex root taking its principal value It follows from(72)that:

R  ffiffiffiffi

D

p

3

q

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi D p

3

By substituting(73)into(71)we obtain(64), and the validity of(64)is proved for the case D < 0

We are now in the position to state the following theorem

Theorem 2

(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-sionless squared velocity xr¼ q v2=cis given by:

xr¼C

2

C2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi D p

3

q

þ q2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi D p

3

q

1 3

ð74Þ

in which radicals are understood as complex roots taking the principal values, R, D, q2defined by:

q2¼1

9ð1  3aÞ;

R ¼1

6a þ

1

2b 

1

27;

D ¼ 1

27a

3 1

108a

2þ1 4 2

 1

27b þ

1

6ab;

ð75Þ

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 fromTheorem 1,(28)2, (41), (59) and (64) h

For deriving(75), the formulas(61) and (63)3,4,5are employed Note that the formula(74) holds for a general strain-energy function and an arbitrary isotropic internal constraint

In the undeformed state (k1¼ k2¼ k3¼ 1; P ¼ 0) we have (see also[24]):a¼ b¼c¼ d¼l(the shear modulus) ) a ¼ 3, b ¼ 1,C2=C2¼a=c¼ 1 ) q2¼ 8=9, R ¼ 26=27, D ¼ 44=27 Introducing these results into(74)yields:

xr¼ 1 

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 26

27þ

2 ffiffiffiffiffiffi 11 p

3 ffiffiffi 3 p

3

s

9 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

26þ2 pffiffiffiffi11

3 pffiffi3

3

3

0

B

1 C 2

which coincides with the exact value of the Rayleigh wave speed in incompressible linear isotropic elastic solids[9] Approx-imate value of xrgiven by(76)is 0.9126, the classical value of the Rayleigh wave velocity in incompressible isotropic elastic materials[27]

Now we consider the half-space xkP0 ðk 2 f1; 2; 3gÞ, and suppose that rkk¼ 0 By xðikÞr ði 2 f1; 2; 3g; i – kÞ we denote the velocity of Rayleigh wave propagating in the xi-direction and attenuating in the xk-direction It is not difficult to see that xðikÞr are defined by:

xðikÞ

r ¼C

2

k

C2i



ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

RðikÞþ ffiffiffiffiffiffiffiffiffi

DðikÞ p

3

q

þ q2 ðikÞ

, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

RðikÞþ ffiffiffiffiffiffiffiffiffi

DðikÞ p

3

q

1 3

!2

ð77Þ

in which radicals are understood as complex roots taking the principal values, RðikÞ, DðikÞ, q2

ðikÞdefined by:

q2

ðikÞ¼1

9ð1  3aðikÞÞ;

RðikÞ¼1

6 ðikÞþ1

2bðikÞ 1

27;

DðikÞ

¼ 1

27a

3 ðikÞ 1

108a

2 ðikÞþ1

4b

2 ðikÞ 1

27bðikÞþ

1

6aðikÞbðikÞ;

ð78Þ

here:

aðikÞ¼2b



ðikÞþ 2d

ðikÞa ðikÞ

c ðikÞ

; bðikÞ¼d

2 ðikÞ

c2 ðikÞ

;

a

ðikÞ¼ Bikik; c

ðikÞ¼C

2 i

C2 k

a ðikÞ; 2b

ðikÞ¼ B

iiii 2kiCi

kkCk

ðB ikkiþ B iikkÞ þ kiCi

kkCk

 2

B kkkk:

ð79Þ

There is no summation over i or k in(79)

4 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 con-straints[24]: those of incompressibility, Bell, constant area, and Ericksen For seeking simplicity, we confine ourself to the case of plane strain: k3¼ 1 Note that xð12Þr  xr, Rð12Þ

 R, Dð12Þ D

4.1 Incompressible materials

For incompressible materials we have:

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 half-space corresponds to strain plane with k3¼ 1, then(80)simplifies to:

It follows form(81):

For a specific example, we take the neo-Hookean strain-energy function, namely[28,29]:

W ¼1

In the case of strain plane with k3¼ 1, it is reduced to:

W ¼l

2ðk

2

It is readily to see that:

From(6)1, (10), (11), (12), (28), (37), (82), (85)and taking into account k1k2¼ 1 we have:

a¼lk2; c¼ d¼l

k2; 2b¼l k2þ1

k2

here we write k1¼ k ðk > 0Þ Introducing(86)into(43)leads to a ¼ 3, b ¼ 1, and then using(75)provides:

R ¼26

27; D ¼

44

27; q

2¼ 8

From(74), (82) and (87)it follows:

xr¼ k4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 26

27þ

2 ffiffiffiffiffiffi 11 p

3 ffiffiffi 3 p

3

s

9 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

26þ2 pffiffiffiffi11

3 pffiffi3

3

3

0

B

1 C 2

Note that one can obtain the result (7.11) in[29]by multiplying two sides of(88)by k2 The formula(88)can be obtained from (78) in[30]by putting r2¼ 0 From(86)it implies: 2b>a It is not difficult to verify that the inequation(53)2is equivalent to:

and its solution is:

k > kð1Þ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 26

27þ

2 ffiffiffiffiffiffi 11 p

3 ffiffiffi 3 p

3

s

9 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

26þ2 pffiffiffiffi11

3 pffiffi3

3

3

v

u

Thus, from theTheorem 2, we have the following theorem

Theorem 3 Suppose that the incompressible elastic half-space is subject to the homogeneous initial deformations(1)with k3¼ 1, and the strain-energy function W is given by(83) If k1¼ k > kð1Þ(defined by(90)), then there exists a unique surface Rayleigh wave propagating in the half-space whose dimensionless squared velocity xris given by(88) For the values of k so that 0 < k 6 kð1Þ

the surface Rayleigh wave does not exist

Fig 1shows the dependence of the dimensionless squared velocity xr of the Rayleigh wave on the parameter k in the interval ð0:6; 2Þ