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

International Journal of Engineering Science 48 (2010) 275–289

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International Journal of Engineering Science
journal homepage: www.elsevier.com/locate/ijengsci

On formulas for the Rayleigh wave velocity in pre-strained elastic
materials subject to an isotropic internal constraint
Pham Chi Vinh *, Pham Thi Ha Giang
Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam

a r t i c l e

i n f o

Article history:
Received 16 June 2009
Received in revised form 13 August 2009
Accepted 27 September 2009
Available online 31 October 2009
Communicated by K.R. Rajagopal
Keywords:
Rayleigh waves
Rayleigh wave velocity
Prestrain
Prestress
Isotropic internal constraint

a b s t r a c t
In the present paper, formulas for the velocity of Rayleigh waves propagating along principal directions of prestrain of an elastic half-space subject to a pure homogeneous prestrain, and an isotropic internal constraint have been derived using the theory of cubic

equation. They have simple algebraic form, and hold for any strain-energy function and
any isotropic constraint. In undeformed state, these formulas recover the exact value of
the Rayleigh wave speed in incompressible isotropic elastic materials. Some specific cases
of strain-energy function and isotropic constraint are considered, and the corresponding
formulas become totally explicit in terms of the parameters characterizing the material
and the prestrains. The necessary and sufficient conditions for existence of Rayleigh wave
are examined in detail. The use of obtained formulas for nondestructive evaluation of prestrains and prestresses is discussed.
Ó 2009 Elsevier Ltd. All rights reserved.

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 Rahman and Barber [3], but for a limited range of values of the parameter  ¼ ð1 À 2mÞ=ð2 À 2mÞ, where m is Poisson’s ratio, by
using the theory of cubic equations. Employing Riemann problem theory Nkemzi [4] derived a formula for the velocity of
Rayleigh waves expressed as a continuous function of  for any range of values. It is rather cumbersome [5] and the final
result as printed in his paper is incorrect [6]. Malischewsky [6] obtained a formula for the speed of Rayleigh waves for
any range of values of  by using Cardan’s formula together with trigonometric formulas for the roots of a cubic equation
and MATHEMATICA. It is expressed as a continuous function of . In Malischewsky’s paper [6] it is not shown, however,
how Cardan’s formula together with the trigonometric formulas for the roots of the cubic equation are used with MATHEMATICA to obtain the formula. A detailed derivation of this formula was given by Vinh and Ogden [7] together with an
* Corresponding author. Tel.: +84 4 5532164; fax: +84 4 8588817.
E-mail address: (P.C. Vinh).
0020-7225/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijengsci.2009.09.010

276

P.C. Vinh, P.T. Ha Giang / International Journal of Engineering Science 48 (2010) 275–289

alternative formula. For non-isotropic materials, for some special cases of compressible monoclinic materials with symmetry
plane x3 = 0, formulas for the Rayleigh wave speed have been found by Ting [8] and Destrade [5] as the roots of quadratic
equations, while for incompressible orthotropic materials an explicit formula has been given by Ogden and Vinh [9] based
on the theory of cubic equations. Further, in a recent papers [10,11] Vinh and Ogden have obtained explicit formulas for the
Rayleigh wave speed in compressible orthotropic elastic solids.
Nowadays pre-stressed materials have been widely used. Nondestructive evaluation of prestresses of structures before
and during loading (in the course of use) becomes necessary and important, and the Rayleigh wave is a convenient tool
for this task, see for example [12–15]. In these studies (also in [16,17]), for evaluating prestresses by the Rayleigh wave,
the authors have established the (approximate) formulas for the relative variation of the Rayleigh wave velocity [12–15]
or its variation ([16,17]). They are linear in terms of the prestrains (or prestresses), thus they are very convenient in use.
However, since these formulas are derived by using the perturbation method they are only valid for enough small prestrains.
They are no longer to be applicable when prestrains are not small.
The main purpose of this paper is to find exact formulas for the velocity of Rayleigh waves propagating in a uniformly prestrained elastic half-space subject to an isotropic internal constraint. The wave propagation direction is one of the principal
axes of prestrain. Since these formulas are exact and valid for any range of prestrain, they will be very significant in practical
applications, especially for the nondestructive evaluation of prestresses of structures. It is noted that there have been many
papers dedicated to the theory of elasticity with internal constraints, see for example [18–23] and references therein. In
[18,19], the authors discussed universal relations and solutions for isotropic homogeneous elastic materials subject to a general isotropic internal constraint. The investigation [20] explored the relationship between isotropic constraints and the
associated constraint manifolds. The studies [21,22] developed equations for a small deformation superimposed on a finite
deformation of isotropic elastic materials with isotropic constraints, and some applications of these equations for small
amplitude waves. The paper [23] was about the Stroh formalism for a generally constrained and prestressed elastic material.
The authors derived the corresponding integral representation for the surface-impedance tensor and explained how it can be
used, together with a matrix Riccati equation, to (numerically) calculate the surface-wave speed. It is noted that, however,
this investigation did not lead to any formula for the (Rayleigh) surface-wave velocity.
The paper is organized as follows: the derivation of the secular equation of Rayleigh waves is presented briefly in Section

2. Formulas for the velocity of Rayleigh wave propagating in the principal directions of prestrain are derived in Section 3.
They hold for any strain-energy function and any isotropic internal constraint. In this section are also established the necessary and sufficient conditions for unique existence of Rayleigh wave. In Section 4, some specific cases of strain-energy
function and isotropic constraint are considered. Some remarks on the use of the obtained formulas for nondestructive evaluation of prestress are made in Section 5.
2. Secular equation
In this section we first summarize the basic equations which govern small amplitude time-dependent motions superimposed upon a large static primary deformation, under the assumption of plane strain elasticity subjected to an isotropic
internal constraint, and then derive the secular equation of the wave. This secular equation coincides with the one obtained
recently by Destrade and Scott [24] by a different way.
We consider an unstressed isotropic hyperelastic body corresponding to the half-space X 2 P 0 and we suppose that the
deformed configuration is obtained by application of a pure homogeneous strain of the form:

x1 ¼ k 1 X 1 ;

x2 ¼ k2 X 2 ;

x3 ¼ k3 X 3 ;

ki ¼ const;

ki > 0;

i ¼ 1; 2; 3:

ð1Þ

In its deformed configuration the body, therefore, occupies the region x2 > 0 with the boundary x2 ¼ 0.
Suppose that the material is subject to an isotropic internal constraint, written as [24]:

Cðk1 ; k2 ; k3 Þ ¼ 0;

ð2Þ

where C is a symmetric function of the principal stretches ki . We restrict attention to the case:

Ci > 0;

where Ci ¼ @ C=@ki :

ð3Þ

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

Nii ¼ J À1 ki Ci ;

ðno sumÞ;

ð4Þ

 is of the diagonal form with non-zero
where J ¼ k1 k2 k3 . The Cauchy stress tensor associated to the static deformed state r
components [24]:

r ii ¼ JÀ1 ki W i þ PNii ðno sumÞ;

ð5Þ

where the strain-energy Wðk1 ; k2 ; k3 Þ is a symmetric function of ki , i.e. its value is left unchanged by any permutation of
k1 ; k2 ; k3 , W i ¼ @W=@ki , and P is determined as follows:

P¼À

W2

C2

 22 ¼ 0;
if r

otherwise P ¼

 22 À k2 W 2
Jr
:
k2 C2

ð6Þ


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P.C. Vinh, P.T. Ha Giang / International Journal of Engineering Science 48 (2010) 275–289

 22 ¼ 0. We consider a plane motion in the ðx1 ; x2 Þ-plane with displaceAs in [24], in this paper we are interested in the case r
ment components u1 ; u2 ; u3 such that:

ui ¼ ui ðx1 ; x2 ; tÞ;

i ¼ 1; 2;

u3  0;

ð7Þ

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

€1 ;
u
s11;1 þ s21;2 ¼ q

€2 ;
u
s12;1 þ s22;2 ¼ q

ð8Þ

 is mass density of the material at the static deformed state, a superposed dot signifies differentiation with respect to
where q
t, commas indicate differentiation with respect to spatial variables xi , and [21,24]:

sij ¼ BÃijkl ul;k þ pNij

ð9Þ

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

BÃijkl

BÃijkl

are given by:

e ijkl :
¼ Bijkl þ P B

ð10Þ

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

JBiijj ¼ ki kj W ij ;

JBijij ¼

e iijj ¼ ki kj Cij ;
JB

e ijij ¼
JB

ki W i À kj W j
k2i À k2j
ki Ci À kj Cj
k2i À k2j

k2i ði – jÞ;

k2i ði – jÞ;

JBijji ¼ JBijij À ki W i ;

e ijji ¼ J B

e ijij À ki Ci ;
JB

ð11Þ

ð12Þ

e ijij defined
where W ij ¼ @ 2 W=@ki @kj . Note that there is no summation over i or j in the formulas (11) and (12), and JBijij , J B
e ijij are given by:
when i–j, ki –kj . In the case where i – j, ki ¼ kj , JBijij , J B

JBijij ¼

1
ðJB À JBiijj þ ki W i Þ;
2 iiii

e ijij ¼
JB

1 e
e iijj þ ki Ci Þ:
ðJ B iiii À J B
2

ð13Þ

Note that:

BÃiijj ¼ BÃjjii

BÃijji ¼ BÃjiij

ðno sumÞ:

ð14Þ

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

N11 u1;1 þ N22 u2;2 ¼ 0:

ð15Þ

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

s21 ¼ s22 ¼ 0 at x2 ¼ 0:

ð16Þ

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

p;

um ! 0 as x2 ! þ1;

Since tensor

A > 0;

BÃijkl

m ¼ 1; 2:

ð17Þ

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

C > 0;

Bþ

pffiffiffiffiffiffi
AC > 0;

ð18Þ

where

A ¼ ðk1 k2 C1 C2 ÞÀ1 BÃ1212 ; C ¼ ðk1 k2 C1 C2 ÞÀ1 BÃ2121 ;
1
B ¼ ½ðk1 C1 ÞÀ2 BÃ1111 þ ðk2 C2 ÞÀ2 BÃ2222 Š À ðk1 k2 C1 C2 ÞÀ1 ðBÃ1122 þ BÃ1221 Þ:
2

ð19Þ

Now we consider a surface Rayleigh wave propagating in the x1 -direction with the velocity v and the wave number k. Then,
u1 ; u2 , p and smn ðm; n ¼ 1; 2Þ are sought in the form:

uj ¼ U j ðkx2 Þeikðx1 Àv tÞ ðj ¼ 1; 2Þ;

smn ¼ kSmn ðkx2 Þeikðx1 Àv tÞ ;

p ¼ kQ ðkx2 Þeikðx1 Àv tÞ ;

ðm; n ¼ 1; 2Þ;

ð20Þ

2

where i ¼ À1. Introducing (20)1, (20)3 into (8) and (15) and taking into account (7) yield:

 v 2 U1 ;
iS11 þ S021 ¼ Àq
 v 2 U2 ;
iS12 þ S022 ¼ Àq
ik1 C1 U 1 þ k2 C2 U 02 ¼ 0
in which the prime indicates the derivative with respect to y ¼ kx2 . Substituting (20) into (9) gives:

ð21Þ


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P.C. Vinh, P.T. Ha Giang / International Journal of Engineering Science 48 (2010) 275–289
Ã

Ã

S11 ¼ iB1111 U 1 þ BÃ1122 U 02 þ Q N11 ;

S21 ¼

BÃ2121 U 01

þ

Ã
iB1221 U 2 ;

S22 ¼

S12 ¼ BÃ1221 U 01 þ iB1212 U 2 ;
Ã
iB1122 U 1

þ

BÃ2222 U 02

ð22Þ

þ Q N22 :

Now we introduce a new variable z given by:

z¼

k 1 C1
y:
k 2 C2

ð23Þ

Substituting (22) into (21) leads a system of three differential equations for three unknown functions U 1 ; U 1 ; Q of z, namely:

8
 2
h
i
>
 v 2 U 1 þ iN11 Q ¼ 0;
BÃ2121 kk12 CC12 U 001 þ BÃ1221 kk12 CC12 À BÃ1111 þ kk12 CC12 BÃ1122 þ q
>
>
>
< 

 2
0
Ã
Ã
Ã
0
Ã
k1 C1
k1 C1
k1 C1
k1 C1
 2
À

B
À
B
i
B
>
2222 k2 C2
1221 k2 C2
1122 k2 C2 U 1 þ ðB1212 À qv ÞU 2 À k2 C2 N 22 Q ¼ 0;
>
>
>
:
iU 1 þ U 02 ¼ 0:

ð24Þ

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

U 1 ¼ A1 eÀsz ;

U 2 ¼ A2 eÀsz ;

Q ¼ A3 eÀsz ;

ð25Þ

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

ReðsÞ > 0

ð26Þ

in order to ensure the decay condition (17). Introducing (25) into (24) yields a homogeneous system of three linear equations
for A1 , A2 , A3 , and vanishing its determinant leads the characteristic equation that defines s, namely:

cà s4 À ð2bà À q v 2 Þs2 þ ðaà À q v 2 Þ ¼ 0;

ð27Þ


2
k 1 C1
C2
BÃ2121 ¼ 12 aà ;
k 2 C2
C2

2
k 1 C1 Ã
k1 C1
2bà ¼ BÃ1111 À 2
ðB1221 þ BÃ1122 Þ þ
BÃ2222 :
k 2 C2
k2 C2

ð28Þ

where

aà ¼ BÃ1212 ; cà ¼

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

aà > 0; cà > 0; bà þ

pffiffiffiffiffiffiffiffiffiffi

aà cà > 0:

ð29Þ

From (27) we have:

s21 þ s22 ¼

v2
2bà À q

cÃ

;

s21 s22 ¼

aà À q v 2
:
cÃ

ð30Þ

The roots s21 , s22 of the quadratic Eq. (27) for s2 are either both real (and, if so, both positive because of positive real parts of s1 ,
s2 ) or they are a complex conjugate pair. In both case: s21 s22 > 0. Therefore, by (30)2 and cà > 0 we have:

 v 2 < aà :
0
ð31Þ

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

S21 ¼ S22 ¼ 0 at z ¼ 0:

ð32Þ

Let s1 , s2 be two roots of (27) satisfying (26). Then, the general solution of (24) that decays at þ1 is:

8
Às1 z
þ B2 eÀs2 z ;
>
< U 1 ¼ B1 e
iB1 Às1 z
þ iBs22 eÀs2 z ;
U 2 ¼ s1 e
>
:
Às1 z
þ B2 C 2 eÀs2 ;

Q ¼ B1 C 1 e

ð33Þ

where B1 , B2 are constant and:

Cj ¼

"
#

2
i
k 1 C1
k1 C1
k1 C1
v2 ;
ÀBÃ1111 þ BÃ1122
þ BÃ2121
s2j þ BÃ1221
þq
k 2 C2
k2 C2
k2 C2
N11

j ¼ 1; 2:

ð34Þ

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





k 1 C1 Ã
k 1 C1 Ã
s2 cà s21 þ
B1221 B1 þ s1 cà s22 þ
B1221 B2 ¼ 0;
k 2 C2
k 2 C2




k 1 C1 Ã
k C
Ã
2
Ã
2
 v À c s1 B1 þ 2bà þ 1 1 BÃ1221 À q
 v 2 À cà s22 B2 ¼ 0:
2b þ
B1221 À q
k 2 C2
k2 C2

ð35Þ


P.C. Vinh, P.T. Ha Giang / International Journal of Engineering Science 48 (2010) 275–289

279

Making to zero the determinant of the system (35) provides the secular equation that determines the Rayleigh wave velocity.
Taking into account (30), after some algebra and removal of a factor s1 À s2 , the secular equation is of the form:
1

cà ðaà À q v 2 Þ þ ð2bà þ 2dà À q v 2 Þ½cà ðaà À q v 2 ފ2 ¼ dÃ2 ;

ð36Þ

where

dà ¼

k1 C1 Ã
k2 C2 Ã
B
¼
c:
k2 C2 1221 k1 C1

ð37Þ

Eq. (36) is desired secular equation, which coincides with the result obtained recently by Destrade and Scott [24] using the
displacement potential. Note that Eq. (36) is also the secular equation in the case s1 ¼ s2 , as pointed out in [24].

3. Formulas for the Rayleigh wave velocity
We define the variable x by:

;
x ¼ v 2 =c22 ðx > 0Þ where c22 ¼ cà =q

ð38Þ

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





rffiffiffiffiffiffiffiffiffiffiffiffiffi
aÃ
dÃ2
À x ¼ Ã2 :
Ã



aÃ
2bà þ 2dÃ
Àx þ
Àx
Ã
c
cÃ

c

ð39Þ

c

From (31) and (38) it follows:

0
aÃ
:
cÃ

ð40Þ

Now we introduce a new variable gà defined by:

gà ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffi
aÃ
À x:
Ã

ð41Þ

c

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

f ðgÃ Þ ¼ gÃ3 þ gÃ2 þ agà À b ¼ 0;

ð42Þ

where

a¼

2bà þ 2dà À aÃ

c

Ã

;

b¼

dÃ2

cÃ2

ð43Þ

:

From (40) and (41) it implies:

0 < gà <

pffiffiffiffiffiffiffiffiffiffiffiffi
aà =cà :

pffiffiffiffiffiffiffiffiffiffiffiffi
Note that the transformation (41) is a 1 À 1 mapping that maps ð0; a =c Þ into 0; aà =cà .
Ã

Ã



ð44Þ

Remark 1. It is clear that a Rayleigh wave exists () Eq. (39) has a positive real solution x and Re½si ðxފ > 0 ði ¼ 1; 2Þ. It is
shown in [24] that: (i) If 2bà > aà , then Re½si ðxފ > 0 ði ¼ 1; 2Þ if and only if x 2 ð0; x1 Þ; (ii) If 2bà 6 aà , then
Re½si ðxފ > 0 ði ¼ 1; 2Þ if and only if x 2 ð0; x2 Þ; where:

aÃ
bÃ
x1 ¼ Ã ; x 2 ¼ 2 Ã À 1 þ
c
c

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
aà À 2bÃ
ð< x1 Þ:
þ1
Ã

c

ð45Þ

Note that
Ã

g ð0Þ ¼

rffiffiffiffiffiÃ

a
^Ã ¼
; gà ðx1 Þ ¼ 0; gà ðx2 Þ ¼ g
cÃ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aà 2bÃ
À Ã þ 1 À 1:
Ã

c

c

ð46Þ

From (1)5, (3), (29)2, (37) it deduces dà > 0 ) b ¼ dÃ2 =cÃ2 > 0, therefore:

f ð0Þ ¼ Àb < 0:

ð47Þ

Lemma 1. Eq. (42) has a unique solution in the interval ð0; þ1Þ:
Proof. If a P 0 then f 0 ðgÃ Þ ¼ 3gÃ2 þ 2gà þ a > 0 8 gà > 0, i.e. f ðgÃ Þ is strictly increasingly monotonous in the interval ð0; þ1Þ.
Since f ð0Þ < 0 and f ðþ1Þ ¼ þ1, it is clear that Eq. (42) has a unique solution in the interval ð0; þ1Þ.


280

P.C. Vinh, P.T. Ha Giang / International Journal of Engineering Science 48 (2010) 275–289

If a < 0 then equation f 0 ðgÃ Þ ¼ 0 has distinct solutions denoted by gÃmax , gÃmin , and gÃmax < 0, gÃmin > 0. Since f ðgÃ Þ strictly
decreasingly monotonous in ðgÃmax ; gÃmin Þ, gÃmax < 0 < gÃmin and f ð0Þ < 0, we have:

f ðgÃ Þ < 0 8 gà 2 ð0; gÃmin Š:

ð48Þ

This implies that equation (42) has no root in ð0; g
Since f ðg Þ strictly increasingly monotonous in ðg
and f ðþ1Þ ¼ þ1, it follows that Eq. (42) has a unique solution in the interval ð0; þ1Þ. h
Ã
min Š.

Ã

Ã
min ; þ1Þ,

f ðg

Ã
min Þ

<0

Proposition 1
(i) If:

rffiffiffiffiffià 

a
> 0;
cÃ

f

ð49Þ

 qffiffiffiffi
Ã
then Eq. (42) has a unique root in the interval 0; acÃ
 qffiffiffiffi
Ã
(ii) Otherwise, Eq. (42) has no root in the interval 0; acà .

Proof
(i) Suppose (49) be satisfied. With the help of Lemma 1, from (49) and f ð0Þ < 0 it implies that (42) has a unique root in the
 qffiffiffiffi

Ã
interval 0; acà .
qffiffiffiffi
aà 6 0. Since f ðþ1Þ ¼ þ1, it follows that Eq. (42) has a root in the interval
(ii) If (49) does not hold, i.e. f
cÃ
 qffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffi
Ã
½ aà =cà ; þ1Þ. This implies, by Lemma 1, that Eq. (42) has no root in the interval 0; acà . h

Proposition 2
 pffiffiffiffiffiffiffiffiffiffiffiffi
(i) Let 2bà > aà . Then Eq. (42) has a unique solution in the interval 0; aà =cà if:

f

rffiffiffiffiffià 

a
> 0;
cÃ

ð50Þ

otherwise, it has no solution in this interval.
pffiffiffiffiffiffiffiffiffiffiffiffi
^ à ; aà =cÃ Þ if:
(ii) Let 2bà 6 aà . Then Eq. (42) has a unique solution in the domain ðg

rffiffiffiffiffià 
f

a
^ Ã Þ < 0;
> 0 and f ðg
cÃ

ð51Þ

otherwise, it has no solution in this domain.
Proof
(i) It is deduced from the Proposition 1. Note that the Proposition
1 holds for both cases: 2bà > aà and 2bà 6 aà .
pffiffiffiffiffiffiffiffiffiffiffiffi
Ã
Ã
Ã
6ffiffiffiffiffiffiffiffiffiffiffiffi
a . It is not difficult to verify that 0 6 g^ < aà =cà . If (51) is satisfied,
(2i) Suppose that 2b p
then Eq. (42) has a solution in
pffiffiffiffiffiffiffiffiffiffiffiffi
Ã
à =cÃ Þ 6 0 (f ðg
^ Ã Þ P 0Þ, because
^
a
this
domain.

If
f
ð
the domain ðg ; aà =cà Þ. By Lemma 1, it has a unique solution inp
ffiffiffiffiffiffiffiffiffiffiffiffi
à =cà ; þ1Þ ðð0; g
^ à ŠÞ. By Lemma 1, it has no root in
a
f ðþ1Þ ¼ þ1 ðf ð0Þ
<
0Þ,
it
implies
that
Eq.
(42)
has
a
solution
in
½
pffiffiffiffiffiffiffiffiffiffiffiffi
^ à ; aà =cà Þ. h
the interval ðg
From Proposition 1, Remark 1 and the fact:

 rffiffiffiffiffià 

gà 2 0;



rffiffiffiffiffi

a
aÃ
^Ã;
() x 2 ð0; x1 Þ; gà 2 g
() x 2 ð0; x2 Þ;
Ã
c
cÃ

ð52Þ

we have immediately the following theorem.
Theorem 1
(i) A Rayleigh wave exists if and only if either:

2bà > aÃ

and f

rffiffiffiffiffià 

a
> 0;
cÃ

ð53Þ

P.C. Vinh, P.T. Ha Giang / International Journal of Engineering Science 48 (2010) 275–289

281

or

2bà 6 aà ;

f

rffiffiffiffiffià 

a
^ Ã Þ < 0;
> 0 and f ðg
cÃ

ð54Þ

is satisfied.
(ii) When a Rayleigh wave exists, it is unique.
It should be noted that the conditions (53) and (54) were stated in [24], but without explanation.
Proposition 3. If Eq. (42) has two or three distinct real roots, then the root corresponding to the Rayleigh wave, denoted by gÃr , is
the largest root.
Proof. Suppose Eq. (42) has two or three distinct real roots. According to Lemma 1, only one of them is positive. This positive
root of Eq. (42) is the largest root and it corresponds to the Rayleigh wave. h
Lemma 2. If equation:

f 0 ðgÃ Þ ¼ 3gÃ2 þ 2gà þ a ¼ 0;
has two distinct real roots g

Ã
max ;

Ã
min Þ

f ðg

g

Ã
min

ð55Þ
(g

Ã
max


Ã
min ),

then:

< 0:

ð56Þ

Proof. Suppose f 0 ðgÃ Þ ¼ 0 has two distinct real roots gÃmax , gÃmin (gÃmax < gÃmin ). Since:

2
3

gÃmax þ gÃmin ¼ À ;

ð57Þ

it follows:

gÃmax < 0:

ð58Þ

If gÃmax < 0 < gÃmin , then f ðgÃmin Þ < f ð0Þ < 0 because f ðgÃ Þ strictly decreasingly monotonous in the interval ðgÃmax ; gÃmin Þ. If
gÃmin 6 0, then also f ðgÃmin Þ 6 f ð0Þ < 0 because f ðgÃ Þ strictly increasingly monotonous in ðgÃmin ; þ1Þ. In both cases we have
f ðgÃmin Þ < 0. h
Now in order to solve the cubic Eq. (42) we introduce the new variable z defined as:

1
z ¼ gà þ :
3

ð59Þ

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

z3 À 3q2 z þ r ¼ 0:

ð60Þ

where

q2 ¼

1
ð1 À 3aÞ;
9

r¼

2 1
À a À b;
27 3

ð61Þ

here q2 may be negative. It is noted from the geometrical point that r ¼ f ðgÃN Þ where N is the point of inflexion of the cubic
curve f ¼ f ðgà Þ.
Our task is now to find the real solution zr of Eq. (60) which is corresponding to gÃr by the relation (59). As gÃr is 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]):

z1 ¼ S þ T;
1
1 pffiffiffi

z2 ¼ À ðS þ TÞ þ i 3ðS À TÞ;
2
2
1
1 pffiffiffi
z3 ¼ À ðS þ TÞ À i 3ðS À TÞ;
2
2

ð62Þ

2

where i ¼ À1 and:

S¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
3
R þ D;

D ¼ R2 þ Q 3 ;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
3
R À D;
1
R ¼ À r; Q ¼ Àq2 :

2
T¼

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

ð63Þ


282

P.C. Vinh, P.T. Ha Giang / International Journal of Engineering Science 48 (2010) 275–289

(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 SÃ ,
where SÃ is 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) zr is given by:

zr ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
3
q2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Rþ Dþp
pffiffiffiffi ;
3

Rþ D

ð64Þ

in which each radical is understood as complex roots taking its principle value.
Case 1: D > 0
If D > 0, then by Remark 3, Eq. (60) has a unique real solution, so it is zr , given by the first of (62), in particular:

zr ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi 3
pffiffiffiffi
3
Rþ Dþ RÀ D

ð65Þ

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

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi 3
pffiffiffiffi
3
3
3
3
R þ D R À D ¼ R2 À D ¼ ÀQ 3 ¼ ÀðÀq2 Þ3 ¼ q2 ;

ð66Þ

therefore

zr ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
pffiffiffiffi
3
3
R þ D þ q2
R þ D;

ð67Þ

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:

Rþ

pffiffiffiffi
D > 0:

ð68Þ
0

0

Indeed, consider equation f ðg Þ ¼ 3g þ 2g þ a ¼ 0. Its discriminant is D ¼ 1 À 3a. If D > 0, then f ¼ 0 has two distinct
real roots gÃmax ; gÃmin (gÃmax < gÃmin ). Because Eq. (60) has a unique real solution, it follows that f ðgÃmax Þ:f ðgÃmin Þ > 0. By Lemma
2, f ðgÃmin Þ < 0, therefore: f ðgÃmax Þ < 0; f ðgÃmin Þ < 0: This implies that f ðgÃN Þ < 0 ) r < 0. If D0 6 0, then f 0 ðgÃ Þ P 0 8 gà ) f ðgà Þ
Ã
strictly increasingly monotonous
pffiffiffiffi in ðÀ1; þ1Þ ) f ðgN Þ ¼ f ðÀ1=3Þ < f ð0Þ < 0 ) r < 0. Thus, in both cases we have: r < 0.
Since R ¼ À 12 r ) R > 0 ) R þ D > 0, and (68) is proved.
Case 2: D ¼ 0
When D ¼ 0, then according to Remark 3 Eq. (60) has two distinct real roots. In this case equation f 0 ðgÃ Þ ¼ 0 has also two
distinct real roots ) f ðgÃmin Þ < 0, according Lemma 2, and D0 > 0 ) q2 ¼ 19 D0 > 0. On view of f ðgÃmin Þ < 0 and the fact that
equation (60) has two distinct real roots, it deduces that f ðgÃmax Þ ¼ 0. From f ðgÃmin Þ < 0 and f ðgÃmax Þ ¼ 0 it follows that
r ¼ f ðgÃN Þ < 0. From (63)3,4,5, D ¼ 0, r < 0 we have:
0

R ¼ jqj3 > 0;

Ã

Ã2

Ã

0

r ¼ À2jqj3 :

ð69Þ
2

3

Taking into account (69)2 Eq. (60) becomes: z3 À 3jqj z À 2jqj ¼ 0, and its roots are: z1 ¼ 2jqj, z2 ¼ Àjqj (double root)
) zr ¼ z1 ¼ 2jqj because zr is the largest according to Proposition 3. In the other hand, using (69)1 and D ¼ 0 it is easy to verify that:

2jqj ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
pffiffiffiffi
3
3
Rþ D
R þ D þ q2

ð70Þ

in which each radical is understood as complex roots taking its principle value. The formula (64) again valid for the case
D ¼ 0.
Case 1: D < 0
If D < 0, then according to Remark 3, Eq. (60) has three distinct real roots, and zr is the largest root by Proposition 3. By
arguments presented in [10] (p. 255) one can show that, in this case the largest root zr of Eq. (60) is given by:

zr ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi 3
pffiffiffiffi

3
R þ D þ R À D;

ð71Þ

within which each radical is understood
pffiffiffiffiffiffiffiffi as the complex root taking its principal value. By h ð2 ð0; pÞÞ we denote the phase
angle of the complex number R þ i ÀD. It is not difficult to verify that:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
3
R þ D ¼ jqjeih ;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
3
R À D ¼ jqjeÀih ;

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

ð72Þ


P.C. Vinh, P.T. Ha Giang / International Journal of Engineering Science 48 (2010) 275–289

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
3
q2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
RÀ D¼ p
pffiffiffiffi :
3
Rþ D

283

ð73Þ

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 v 2 =cà is given by:
sionless squared velocity xr ¼ q

xr ¼

C22
À
C21

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi 12
pffiffiffiffi
3
3
Rþ DÀ

R þ D þ q2
3

ð74Þ

in which radicals are understood as complex roots taking the principal values, R, D, q2 defined by:

1
ð1 À 3aÞ;
9
1
1
1
R¼ aþ bÀ
;
6
2
27
1 3
1 2 1 2
1
1
D¼
a À
a þ b À
b þ ab;
27
108
4
27

6

q2 ¼

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 from Theorem 1, (28)2, (41), (59) and (64). h
For deriving (75), the formulas (61) and (63)3,4,5 are employed. Note that the formula (74) holds for a general strainenergy function and an arbitrary isotropic internal constraint.
In the undeformed state (k1 ¼ k2 ¼ k3 ¼ 1; P ¼ 0) we have (see also [24]): aà ¼ bà ¼ cà ¼ dà ¼ l (the shear modulus)
) a ¼ 3, b ¼ 1, C22 =C21 ¼ aà =cà ¼ 1 ) q2 ¼ À8=9, R ¼ 26=27, D ¼ 44=27. Introducing these results into (74) yields:

0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12
pffiffiffiffiffiffi
8
1C
B 3 26 2 11
xr ¼ 1 À @
þ pffiffiffi À qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À A ;
27 3 3 9 3 26 þ 2ppffiffiffiffi
11
ffiffi 3
27

ð76Þ

3 3

which coincides with the exact value of the Rayleigh wave speed in incompressible linear isotropic elastic solids [9]. Approximate value of xr given by (76) is 0.9126, the classical value of the Rayleigh wave velocity in incompressible isotropic elastic
materials [27].
 kk ¼ 0. By xðikÞ
Now we consider the half-space xk P 0 ðk 2 f1; 2; 3gÞ, and suppose that r
r ði 2 f1; 2; 3g; i – kÞ we denote the
ðikÞ
velocity of Rayleigh wave propagating in the xi -direction and attenuating in the xk -direction. It is not difficult to see that xr
are defined by:

xðikÞ
r

C2
¼ k2 À
Ci

!
,qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffi 1 2
3
3
ðikÞ
ðikÞ
ðikÞ
ðikÞ
2
R þ D þ qðikÞ
R þ D À

3

ð77Þ

in which radicals are understood as complex roots taking the principal values, RðikÞ , DðikÞ , q2ðikÞ defined by:

1
ð1 À 3aðikÞ Þ;
9
1
1
1
;
RðikÞ ¼ aðikÞ þ bðikÞ À
6
2
27
1 3
1 2
1 2
1
1
DðikÞ ¼
a À
a þ b À
bðikÞ þ aðikÞ bðikÞ ;
27 ðikÞ 108 ðikÞ 4 ðikÞ 27
6
q2ðikÞ ¼

ð78Þ

here:

aðikÞ ¼

2bÃðikÞ þ 2dÃðikÞ À aÃðikÞ

c

Ã
ðikÞ

aÃðikÞ ¼ BÃikik ; cÃðikÞ ¼
2bÃðikÞ ¼ BÃiiii À 2

2
i
2
k

;

C Ã
aðikÞ ;
C

bðikÞ ¼

dÃ2

ðikÞ

cÃ2
ðikÞ

;
ð79Þ



ki Ci Ã
ki Ci
ðB þ BÃiikk Þ þ
kk Ck ikki
kk Ck

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

2

BÃkkkk :


284

P.C. Vinh, P.T. Ha Giang / International Journal of Engineering Science 48 (2010) 275–289

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 constraints [24]: those of incompressibility, Bell, constant area, and Ericksen. For seeking simplicity, we confine ourself to the
ð12Þ

case of plane strain: k3 ¼ 1. Note that xr  xr , Rð12Þ  R, Dð12Þ  D. . .
4.1. Incompressible materials
For incompressible materials we have:

C ¼ k1 k2 k3 À 1 ¼ 0:

ð80Þ

The incompressibility constraint is often used for the modelling of finite deformations of rubber-like materials and shows
good correlation with experiment, see for example [25, Chapter 7]. Suppose that the underlying deformation of the halfspace corresponds to strain plane with k3 ¼ 1, then (80) simplifies to:

C ¼ k1 k2 À 1 ¼ 0:

ð81Þ

It follows form (81):

C1 ¼ k2 ;

C11 ¼ C22 ¼ 0;

C2 ¼ k1 ;

C12 ¼ 1:

ð82Þ

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

W¼

1
lðk21 þ k22 þ k23 À 3Þ:
2

ð83Þ

In the case of strain plane with k3 ¼ 1, it is reduced to:

W¼

l
2

ðk21 þ k22 À 2Þ:

ð84Þ

It is readily to see that:

W 1 ¼ lk1 ;

W 2 ¼ lk2 ;

W 11 ¼ W 22 ¼ l;

W 12 ¼ 0:

ð85Þ

From (6)1, (10), (11), (12), (28), (37), (82), (85) and taking into account k1 k2 ¼ 1 we have:

aà ¼ lk2 ; cà ¼ dà ¼

l
k

;
2



1
2bà ¼ l k2 þ 2 ;
k

ð86Þ

here we write k1 ¼ k ðk > 0Þ. Introducing (86) into (43) leads to a ¼ 3, b ¼ 1, and then using (75) provides:

R¼

26
;
27

D¼

44
;

27

8
q2 ¼ À :
9

ð87Þ

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

0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12
pffiffiffiffiffiffi
8
1
B 3 26 2 p11
ffiffiffi À qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À C
xr ¼ k 4 À @
þ
A :
27 3 3 9 3 26 þ 2ppffiffiffiffi
11
ffiffi 3
27

ð88Þ

3 3

Note that one can obtain the result (7.11) in [29] by multiplying two sides of (88) by kÀ2 . The formula (88) can be obtained

 2 ¼ 0. From (86) it implies: 2bà > aà . It is not difficult to verify that the inequation (53)2 is
from (78) in [30] by putting r
equivalent to:

k6 þ k4 þ 3k2 À 1 > 0;

ð89Þ

and its solution is:

k > kð1Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
usffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u 3 26 2pffiffiffiffiffiffi
11
8
1
u
¼t
þ pffiffiffi À qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À % 0:5437:
27 3 3 9 3 26 þ 2ppffiffiffiffi
3
11
ffiffi
27
3 3

ð90Þ

Thus, from the Theorem 2, we have the following theorem.
Theorem 3. Suppose that the incompressible elastic half-space is subject to the homogeneous initial deformations (1) with k3 ¼ 1,
and the strain-energy function W is given by (83). If k1 ¼ k > kð1Þ (defined by (90)), then there exists a unique surface Rayleigh
wave propagating in the half-space whose dimensionless squared velocity xr is given by (88). For the values of k so that 0 < k 6 kð1Þ
the surface Rayleigh wave does not exist.
Fig. 1 shows the dependence of the dimensionless squared velocity xr of the Rayleigh wave on the parameter k in the
interval ð0:6; 2Þ.


285

P.C. Vinh, P.T. Ha Giang / International Journal of Engineering Science 48 (2010) 275–289

4.2. Bell’s constraint
The Bell constraint was found to hold experimentally over countless trials on polycrystalline annealed solids, including
aluminum, brass, copper, and mild steel, see [31, Chapter 2]. For a Bell constrained material we have [22,24]:

C ¼ k1 þ k2 þ k3 À 3 ¼ 0:

ð91Þ

For a specific example of the Bell material we take [24]:

W ¼ d2 ðk1 k2 þ k2 k3 þ k3 k1 À 3Þ;

ð92Þ

where d2 < 0 is a material constant. Noted that (92) comes from (4.5) in [24] with d3 ¼ 0. On view of k3 ¼ 1, (91) and (92)
become:

C ¼ k1 þ k2 À 2 ¼ 0;

W ¼ d2 ðk1 k2 þ k2 þ k1 À 3Þ;

ð93Þ

where 0 < kk < 2; k ¼ 1; 2. It follows from (93):

C1 ¼ C2 ¼ 1;

Cij ¼ 0;

W 1 ¼ d2 ð1 þ k2 Þ;

W 2 ¼ d2 ð1 þ k1 Þ;

W 11 ¼ W 22 ¼ 0;

ð94Þ

W 12 ¼ d2 :

From (6)1, (10), (11), (12), (28), (37), (94), and taking into account k1 þ k2 À 2 ¼ 0 we have:

aà ¼ bà ¼ cà ¼

d2 k2
;
2ðk À 2Þ

dà ¼ À

d2 k
;
2

0 < k < 2;

ð95Þ

here we write k ¼ k1 . On use of (95) into (43) yields:

a¼



4
À1 ;
k

b¼


2
2
À1 ;
k

ð96Þ

and then introducing (96) into (75) gives:

q2 ¼



4
3
1À
;
9
k

R¼

2
k

2

À

4
8
þ
;
3k 27

D¼

4
2

k



1

k

2

À


20
4
þ
:
27k 27

ð97Þ

From (74) and (97) and C1 ¼ C2 ¼ 1 it follows:

32
2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u

,u
u
u
3 2
3 2
4
8 2 1
20
4 4
3
4
8 2 1
20
4 17
6t
t
þ
À 5 :
xr ¼ 1 À 4
À
À
À
À
þ
þ
þ

1À
þ
þ
þ
k
k2 3k 27 k k2 27k 27 9
k2 3k 27 k k2 27k 27 3

ð98Þ

16
14
12
10

x

r

8
6
4
2
0
0.6

0.8

1

1.2

1.4

1.6

1.8

2

λ
Fig. 1. Dependence of the dimensionless squared velocity xr of the Rayleigh wave on k in the interval (0.6, 2). The elastic material is incompressible, its
strain-energy function is given by (83), k3 ¼ 1.


286

P.C. Vinh, P.T. Ha Giang / International Journal of Engineering Science 48 (2010) 275–289

It follows from (95) that 2bà > aà . Taking into account aà =cà ¼ 1 and (93)1, it is easy to show that the solution of (53)2 is
0:5 < k < 2.
On view of the Theorem 2, we have the following the theorem.
Theorem 4. Suppose that the strain-energy function W of the Bell material (91), is given by (92), and the elastic half-space is
subject to the homogeneous initial deformations (1) with k3 ¼ 1. If 0:5 < k1 ¼ k < 2, then there exists a unique surface Rayleigh
wave propagating in the half-space, and the dimensionless squared velocity xr is given by (98). For the values of k 2 ð0; 0:5Š the
surface Rayleigh wave does not exist.
Fig. 2 shows the dependence of the dimensionless squared velocity xr on k in the interval ð0:6; 2Þ for this case.
4.3. Areal constraint
The areal (or constant area) constraint has the form [24]:

C ¼ k1 k2 þ k1 k2 þ k1 k2 À 3 ¼ 0:

ð99Þ

The areal (or constant area) constraint has the interpretation that a material cubic in the reference configuration with edges
parallel to the principal axes of strain retains the same total surface area after deformation [32]. For the areal materials, we
take:

W ¼ d1 ðk1 þ k2 þ k3 À 3Þ;

ð100Þ

where d1 > 0 is a material constant. Noted that (100) originates from (4.8) in [24] with d3 ¼ 0. On view of k3 ¼ 1, (99) and
(100) reduce to:

C ¼ k1 k2 þ k1 þ k2 À 3 ¼ 0;

W ¼ d1 ðk1 þ k2 À 3Þ:

ð101Þ

It implies from (101):

C1 ¼ 1 þ k2 ;

C2 ¼ 1 þ k 1 ;

W 1 ¼ W 2 ¼ d1 ;

C11 ¼ C22 ¼ 0;

C12 ¼ 1;

ð102Þ

W ij ¼ 0:

From (6)1, (10), (11), (12), (28), (37), (102), and taking into account (101)1, it deduces:

aà ¼
dà ¼

d1 k2 ð1 þ kÞ
2

ð3 À kÞð3 þ k Þ

cà ¼

;

4d1 k
ð1 þ kÞ2 ð3 þ k2 Þ

16d1 k2
ð1 þ kÞ ð3 À kÞð3 þ k2 Þ

2bà ¼

;

3

;

8d1 k2
ð1 þ kÞð3 À kÞð3 þ k2 Þ

ð103Þ
;

where k ¼ k1 , and 0 < k < 3. Substituting (103) into (43) leads to:

1

0.9

0.8

x 0.7
r
0.6

0.5

0.4
0.6

0.8

1

1.2

1.4

1.6

1.8

2

λ
Fig. 2. Dependence of the dimensionless squared Rayleigh wave velocity xr on k in the interval (0.6, 2) for the Bell materials. The strain-energy function is
given by (92), k3 ¼ 1.


287

P.C. Vinh, P.T. Ha Giang / International Journal of Engineering Science 48 (2010) 275–289

a¼

ð1 þ kÞ
ð24 À k þ 5k2 À 3k3 À k4 Þ;
16k

b¼

ð1 þ kÞ2 ð3 À kÞ2

16k2

ð104Þ

:

On view of (103) and 0 < k < 3 we have:

2bà > aÃ

pffiffiffi
if 0 < k < 2 2 À 1;

2bà 6 aÃ

pffiffiffi
if 2 2 À 1 6 k < 3;

ð105Þ

pffiffiffiffiffiffiffiffiffiffiffiffi
^ à ¼ ðk þ 3Þðk À 1Þ=4 À 1. With 0 < k < 3, Eq. (53)2 is equivalent to 3k2 þ 4k À 3 > 0, therefore, takand aà =cà ¼ ð1 þ kÞ2 =4, g
pffiffiffiffiffiffi
pffiffiffi
ing
into
account
(105),
the
solution of (53) is: ð 13 À 2Þ=3 < k < 2 2 À

pffiffiffi
p1.
ffiffiffiffiffiffiIt is not difficult to see that the solution of (54) is
2 2 À 1 6 k < 3. Thus, the values of k satisfying either (53) or (54) is ð 13 À 2Þ=3 < k < 3. From this and the Theorem 2 we
have:
Theorem 5. Suppose that the strain-energy function W of the areal material
pffiffiffiffiffiffi (99), is given by (100), and the elastic half-space is
subject to the homogeneous initial deformations (1) with k3 ¼ 1. If ð 13 À 2Þ=3 < k < 3, then there exists a unique surface
Rayleigh wave propagating in the half-space, and the dimensionless squared velocity xr is given by:

xr ¼

ð1 þ kÞ4
À
16

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi 12
pffiffiffiffi
3
3
;
R þ D þ q2 = R þ D À
3

ð106Þ

pffiffiffiffiffiffi
where R, D, q2 defined by (75), in which a, b are given by (104). For the values of k 2 ð0; ð 13 À 2Þ=3Š the surface Rayleigh wave

does not exist.
The dependence of the dimensionless squared velocity xr on k in the interval ð0:55; 2Þ for this case is shown in Fig. 3.
4.4. Ericksen’s constraint
The Ericksen constraint is of the form:

C ¼ k21 þ k21 þ k21 À 3 ¼ 0:

ð107Þ

It was proposed by Ericksen [33] to model the behaviour of certain twinned elastic crystals. For the Ericksen materials, the
strain energy function is chosen as follows:

W ¼ D2 ðk21 k22 þ k22 k23 þ k23 k21 À 3Þ;

ð108Þ

where D2 < 0 is a material constant. Noted that (108) originates from (4.11) in [24] with D3 ¼ 0. On view of k3 ¼ 1, (107) and
(108) simplifies to:

C ¼ k21 þ k22 À 2 ¼ 0;

W ¼ D2 ðk21 k22 þ k21 þ k22 À 3Þ:

ð109Þ

It implies from (109):

C1 ¼ 2k1 ;

C2 ¼ 2k2 ; C11 ¼ C22 ¼ 2; C12 ¼ 0;

W 1 ¼ 2D2 k1 ð1 þ k22 Þ; W 2 ¼ 2D2 k2 ð1 þ k21 Þ;
W 11 ¼ 2D2 ð1 þ k22 Þ; W 22 ¼ 2D2 ð1 þ k21 Þ; W 12 ¼ 4D2 k1 k2 :

ð110Þ

5
4.5
4
3.5
3

x r 2.5
2
1.5
1
0.5
0

0.6

0.8

1

1.2

λ

1.4

1.6

1.8

2

Fig. 3. Dependence of the dimensionless squared Rayleigh wave velocity xr on k in the interval (0.55, 2) for the areal materials, the strain-energy function is
given by (100), k3 ¼ 1.


288

P.C. Vinh, P.T. Ha Giang / International Journal of Engineering Science 48 (2010) 275–289

From (6)1, (10), (11), (12), (28), (37), (110), and taking into account (109)1, we have:

2D k3

2
ffi ; cà ¼ À
aà ¼ dà ¼ À pffiffiffiffiffiffiffiffiffiffiffiffiffi
2

2Àk

where we write k ¼ k1 and 0 < k <





2
2
a¼ 5À 2
À
1
;
k
k2

2D2 k5
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
ð2 À k2 Þ 2 À k2

4D2 kð1 À 2k2 Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
2 À k2

ð111Þ

pffiffiffi
2. On use of (111) into (43) yields:


b¼

2bà ¼

2

2

À1
k2

ð112Þ

;

and then introducing (112) into (75) gives:




1 1
2
2
4
10
À1 ; R¼ 4À
þ
5À 2
;
2
9 3
27
k
k
3k
1024 1 2672 1 1088 1 1936 1 64 1
64 1

148
À
þ
À
þ
À
À
D¼
:
27 k2
27 k4
9 k6
27 k8
3 k10 27 k12
27

q2 ¼

ð113Þ

^ Ã ¼ 2=k2 À 3. On view of these and noting that f ð2=k2 À 3Þ ¼ À4, it is not difficult
It follows from (111) that aà =cà ¼ 2=k2 À 1, g
pffiffiffi
to verify that the values of k satisfying either (53) or (54) is kð2Þ < k < 2, where kð2Þ % 0:6578.
Thus we have the following theorem.
Theorem 6. Suppose that the strain-energy function W of the Ericksen material (107),
pffiffiffi is given by (108), and the elastic half-space
is subject to the homogeneous initial deformations (1) with k3 ¼ 1. If kð2Þ < k < 2, where kð2Þ % 0:6578, then there exists a
unique surface Rayleigh wave propagating in the half-space, and the dimensionless squared velocity xr is given by:

xr ¼

2
k

À1À
2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
pffiffiffiffi 12
3
3
R þ D þ q2 = R þ D À
;
3

ð114Þ

where R, D, q2 defined by (113). For the values of k 2 ð0; kð2Þ Š the surface Rayleigh wave does not exist.
Fig. 4 shows the dependence of the dimensionless squared velocity xr on k in the interval ð0:66; 1:3Þ for this case.
Note that, when k ¼ 1, the dimensionless squared Rayleigh wave velocities of four cases considered above take a the same
value that is given by (76).
5. Conclusions and remarks
In this paper, we consider the propagation of Rayleigh waves along principal directions of prestrain of a deformed isotropic elastic half-space subject to an isotropic internal constraint, and have been derived the formulas for the wave velocity
using the theory of cubic equation. They are explicit and hold for any strain-energy function and any isotropic constraint.
In undeformed state, these formulas return to the exact value of the Rayleigh wave speed in incompressible isotropic elastic
materials.

1.5

1

x

r

0.5

0

0.7

0.8

0.9
1
Plot of VR4 in (0.66 1.35)

1.1

1.2

1.3

λ
Fig. 4. Dependence of the dimensionless squared Rayleigh wave velocity xr on k in the interval (0.66, 1.3) for the Ericksen materials, the strain-energy
function is given by (108), k3 ¼ 1.

P.C. Vinh, P.T. Ha Giang / International Journal of Engineering Science 48 (2010) 275–289

289

Since obtained formulas are valid for any range of prestrain, they will be significant in practical applications, especially for
the nondestructive evaluation of prestresses of structures. In relation to the use of these formulas for the determination of
prestresses and prestrains we emphasize the following points:
ðikÞ

(i) For a given material, xr are functions of two of three principal stretches due to (2). Suppose that they are functions of
k1 and k2 .
(ii) Let the half-space X 2 P 0 be subjected to the pure homogeneous prestrain (1) and the isotropic constraint (2), and
r 22 ¼ 0. Let the material of the half-space be given. In order to evaluate the prestrains and the prestresses we do as
follows:
ð12Þ
ð32Þ
1. First, we define xr and xr by laser techniques [34], for example.
2. Then the principal stretches k1 , k2 are determined using two following equations:

uð12Þ ðk1 ; k2 Þ ¼ xð12Þ
; uð32Þ ðk1 ; k2 Þ ¼ xð32Þ
:
r
r

ð115Þ

 11 , r
 33 are determined

The third principal stretches k3 is calculated by the relation (2). Finally, the principal Cauchy stresses r
by:

r 11 ¼ ðW 1 C2 À W 2 C1 Þ

k1
;
J C2

r 33 ¼ ðW 3 C2 À W 2 C3 Þ

k3
;
J C2

ð116Þ

which are originated from (5).

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[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
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