DSpace at VNU: On formulas for the Rayleigh wave velocity in pre-stressed compressible solids

Wave Motion 48 (2011) 614–625

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Wave Motion
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / wave m o t i

On formulas for the Rayleigh wave velocity in pre-stressed
compressible solids
Pham Chi Vinh ⁎
Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam

a r t i c l e

i n f o

Article history:
Received 22 November 2010
Received in revised form 9 April 2011
Accepted 22 April 2011
Available online 4 May 2011
Keywords:
Rayleigh waves
Rayleigh wave velocity
Prestresses
Pre-strains
Compressible

a b s t r a c t
In this paper, formulas for the velocity of Rayleigh waves in compressible isotropic solids
subject to uniform initial deformations are derived using the theory of cubic equation. They are

explicit, have simple algebraic forms, and hold for a general strain energy function. Unlike the
previous investigations where the derived formulas for Rayleigh wave velocity are
approximate and valid for only small enough values of pre-strains, this paper establishes
exact formulas for Rayleigh wave velocity being valid for any range of pre-strains. When the
prestresses are absent, the obtained formulas recover the Rayleigh wave velocity formula for
compressible elastic solids. Since obtained formulas are explicit, exact and hold for any range of
pre-strains, they are good tools for evaluating nondestructively prestresses of structures.
© 2011 Elsevier B.V. All rights reserved.

1. Introduction
Elastic surface waves, discovered by Rayleigh [1] more than 120 years ago for compressible isotropic elastic solids, have been
studied extensively and exploited in a wide range of applications such as those in seismology, acoustics, geophysics,
telecommunications and materials science. 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 addressed by Adams et al. [2].
For the Rayleigh wave, its speed is a fundamental quantity which is of great interest to researchers in various fields of science. It
is discussed in almost every survey and monograph on the subject of surface acoustic waves in solids. It also involves Green's
function for many elastodynamic problems for a half-space. Therefore, explicit formulas for the Rayleigh wave speed are clearly of
practical as well as theoretical interest.
In 1995, the first formula for the Rayleigh wave speed in compressible isotropic elastic solids was obtained by Rahman and Barber [3]
by using the theory of cubic equations. As this formula is defined by two different expressions depending on the sign of the discriminant of
the cubic Rayleigh equation, it is not convenient to apply it to inverse problems. Employing the Riemann problem theory, Nkemzi [4]
derived a formula for the velocity of Rayleigh waves expressed as a continuous function of γ=μ/(λ+2μ), where λ and μ are the usual
Lame constants. It is rather cumbersome [5] and the final result as printed in his paper is incorrect [6]. Malischewsky [6] obtained a
formula, given by one expression, for the speed of Rayleigh waves by using Cardan's formula together with trigonometric formulas for the
roots of a cubic equation and MATHEMATICA. In [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 alternative formula. For incompressible orthotropic materials, an explicit formula has been
given by Ogden and Vinh [8] based on the theory of cubic equations. The explicit formulas for the Rayleigh wave speed in compressible
orthotropic elastic solids were obtained later by Vinh and Ogden [9], Vinh and Ogden [10].

⁎ Tel.: + 84 4 35532164; fax: + 84 4 38588817.
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0165-2125/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.wavemoti.2011.04.015


P.C. Vinh / Wave Motion 48 (2011) 614–625

615

Nowadays pre-stressed materials have been widely used. Nondestructive evaluation of prestresses of structures before and
during loading (in the course of use) is necessary and important, and the Rayleigh wave is a convenient tool for this task, see for
example: Makhort [11,12], Hirao et al.[13], Husson [14], Delsanto and Clark [15], Dyquennoy et al. [16,17], and Hu et al. [18]. In
these studies, for evaluating prestresses by the Rayleigh wave, the authors have established, or used, the approximate formulas for
the Rayleigh wave velocity (see also: Tanuma and Man [19]; Song and Fu [20]). They are linear in terms of the pre-strains (or
prestresses), thus they are very convenient to use. However, since these formulas were derived by using the perturbation method
they are only valid for small enough pre-strains. They are no longer applicable when pre-strains in materials are not small enough.
Recently, formulas for the velocity of Rayleigh wave propagating in pre-strained isotropic elastic solids which are incompressible
or are subject to a general internal constraint have been obtained by Vinh [21], and Vinh and Giang [22].
Since pre-stressed compressible material is used widely in practical, exact formulas for the Rayleigh wave velocity for that
material are necessary and significant. The main purpose of this paper is to provide exact formulas for the Rayleigh wave velocity
for compressible isotropic solids subject to a homogeneous initial deformation. These formulas are explicit, have a simple algebraic
form, hold for a general strain-energy function, and are valid for any range of pre-strains. They are therefore powerful tools for
evaluating nondestructively prestresses in structures.
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 compressible plane strain elasticity, and then derive the secular
equation of Rayleigh waves in pre-stressed compressible elastic solids. For details, the reader is referred to the papers by Dowaikh
and Ogden [23].

We consider an unstressed body of compressible isotropic elastic material corresponding to the half-space X2 ≤ 0 and we
suppose that the deformed configuration is obtained by application of a pure homogeneous strain of the form:
x1 = λ1 X1 ; x2 = λ2 X2 ; x3 = λ3 X3 ; λi = const; i = 1; 2; 3

ð1Þ

where λi N 0, i = 1, 2, 3, are the principal stretches of the deformation. In its deformed configuration the body, therefore, occupies
the region x2 b 0 with the boundary x2 = 0. We consider a plane motion in the (x1, x2)-plane with displacement components u1, u2,
u3 such that ui = ui(x1, x2, t), i = 1, 2, u3 ≡ 0, where t is the time. Then in the absence of body forces the equations governing
infinitesimal motion, expressed in terms of displacement components ui, are [23]:
::
A1111 u1;11 + A2121 u1;22 + ðA1122 + A2112 Þu2;12 = ρ u1

ð2Þ

::
ðA1221 + A2211 Þu1;12 + A1212 u2;11 + A2222 u2;22 = ρ u2
where ρ is the mass density of the material in the deformed state, a superposed dot signifies differentiation with respect to t,
commas indicate differentiation with respect to spatial variables xi, Aijkl are components of the fourth order elasticity tensor
defined as follows [23,24]:
2

JAiijj = λi λj

JAijij

∂ W
∂λi ∂λj

ð3Þ

!
8


>
∂W
∂W
λ2i
>
>
−λ
;
i≠j; λi ≠λj
λ
>
i
j
<
2
2
∂λi
∂λj λi −λj
=

>1


>
∂W

>
>
:
JAiiii −JAiijj + λi
;
i≠j; λi = λj
2
∂λi

JAijji = JAjiij = JAijij −λi

∂W
ði≠jÞ
∂λi

ð4Þ

ð5Þ

for i, j ∈ 1, 2, 3, W = W(λ1, λ2, λ3) is the strain-energy function per unit volume in unstressed state, J = λ1λ2λ3, all other
components being zero. Note that no sum on repeated indices in formulas (3)–(5). The principal Cauchy stresses given by: Jσi = λi
∂W/∂λi (see [23–25]). In the stress-free configuration Eqs. (3)–(5) reduce to:
Aiiii = λ + 2μ; Aiijj = λði≠jÞ; Aijij = Aijji = μ ði≠jÞ

ð6Þ

Equations of motion (Eq. (2)) are taken together with the boundary conditions of zero incremental traction, which are
expressed as:
A2121 u1;2 + A2112 u2;1 = 0; A1122 u1;1 + A2222 u2;2 = 0 on x2 = 0

ð7Þ


616

P.C. Vinh / Wave Motion 48 (2011) 614–625

For seeking the simplicity we use the notations (see also Dowaikh and Ogden [23]):
αij = αji = JAiijj ði; j = 1; 2Þ; γ1 = JA1212 ; γ2 = JA2121 ; γT = JA2112

ð8Þ

In terms of these notations Eq. (2) becomes:
::
α11 u1;11 + γ2 u1;22 + ðα12 + γT Þu2;12 = ρ0 u1 ;

::
ðα12 + γT Þu1;12 + γ1 u2;11 + α22 u2;22 = ρ0 u2

ð9Þ

and boundary conditions (7) are of the form:
γ2 u1;2 + γT u2;1 = 0; α12 u1;1 + α22 u2;2 = 0 on x2 = 0

ð10Þ

here ρ0 = Jρ is the mass density of the material in the (natural) undeformed configuration. From the strong-ellipticity condition of
system (2), αij, γi are required to satisfy the inequalities [23,25]:
α11 N 0; α22 N 0; γ1 N 0; γ2 N 0

ð11Þ

We now consider a time-harmonic wave propagating along the x1-principal direction and set:
uj = Aj exp ½iksx2 + ikðx1 −ct ފ; j = 1; 2

ð12Þ

where k is the wave number, c is the wave speed, A1, A2 are constants. For the decay of ui at x2 = − ∞ it requires Ims b 0. Substituting
Eq. (12) into Eq. (9) yields a homogeneous system of two linear equations for A1, A2, and vanishing its determinant leads to
quadratic equation for s 2, namely:
4

2

b4 s + 2b2 s + b0 = 0

ð13Þ








2
2
2
2
2

γ1 −ρ0 c :
b4 = α22 γ2 ; 2b2 = α22 α11 −ρ0 c + γ2 γ1 −ρ0 c −ðα12 + γT Þ ; b0 = α11 −ρ0 c

ð14Þ

where

From Eqs. (13) and (14) we have:
2
s1

+

2 2
s1 s2

2
s2

=

=−





2
2
2

α22 α11 −ρ0 c + γ2 γ1 −ρ0 c −ðα12 + γT Þ
α22 γ2




α11 −ρ0 c2 γ1 −ρ0 c2
α22 γ2

:

ð15Þ

ð16Þ

Proposition 1. If a Rayleigh wave in pre-stressed compressible elastic solids exists, then its velocity c has to be subjected to the
inequalities:
2

0 b ρ0 c b minðα11 ; γ1 Þ

ð17Þ

Proof. Setting Y = s 2, then Eq. (13) becomes:
2

b4 Y + 2b2 Y + b0 = 0:

ð18Þ

By Δ we denote the discriminant of Eq. (18), and Y1, Y2 are its roots. (i) If Δ ≥ 0 then Y1, Y2 are real. This ensures Y1, Y2 are
pffiffiffiffiffi
negative, otherwise, for example s1 = Y1 is a real number, so that its imaginary part is zero. This contradicts the requirement
Ims b 0. From Eq. (16) and the fact Y1Y2 N 0 it follows either α11 − ρ0c 2 N 0, γ1 − ρ0c 2 N 0 or α11 − ρ0c 2 b 0, γ1 − ρ0c 2 b 0. Suppose that
α11 − ρ0c 2 b 0, γ1 − ρ0c 2 b 0. Then, from Eq. (15) and taking into account Eq. (11), it deduces Y1 + Y2 N 0, but this contradicts the
observed above fact that Y1 b 0, Y2 b 0, so we have α11 − ρ0c 2 N 0, γ1 − ρ0c 2 N 0. (ii) If Δ b 0, then Y1 = Y 2 , hence Y1Y2 = |Y1| 2 N 0. In the
other hand, it is not difficult to verify that:
h



i2
Δ = α22 α11 −ρ0 c2 −γ2 γ1 −ρ0 c2
h



i
−2ðα12 + γT Þ2 α22 α11 −ρ0 c2 + γ2 γ1 −ρ0 c2 + ðα12 + γT Þ4 :

ð19Þ


P.C. Vinh / Wave Motion 48 (2011) 614–625

617

From Eq. (16) and the fact Y1Y2 N 0 it follows that α11 − ρ0c 2 and γ1 − ρ0c 2 have the same sign. Suppose they are all negative.
Then from Eq. (19) it follows that Δ ≥ 0. But this contradicts the assumption that Δ b 0. Hence, both α11 − ρ0c 2 and γ1 − ρ0c 2 must
be positive, and the proof is completed. It is noted that the inequalities (Eq. (17)) were mentioned by Dowaikh and Ogden [23], but

without a detail explanation.
Proposition 2. Let s1, s2 be two roots of the characteristic Eq. (13), and satisfy the condition Ims b 0, then s1s2 b 0, and:
v
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffi
u
u α −ρ c2 γ −ρ c2
0
1
0
t 11
:
s1 s2 = −
α22 γ2

ð20Þ

Proof. Indeed, if the discriminant Δ of the quadratic Eq. (13) for s 2 is non-negative, then its two roots must be negative in order
that Ims b 0 is to be satisfied. In this case the pair s1, s2 are of the form: s1 = − ir1, s2 = − ir2 where r1, r2 are positive. If Δ b 0, the
quadratic Eq. (13) for s 2 has two conjugate complex roots, and in order to ensure the condition Ims b 0: s1 = t − ir, s2 = − t − ir
where r is positive and t is a real number. In both cases, s1s2 is a negative real number, and therefore it is given by Eq. (20) due to
Eqs. (11), (16) and (17). Note that, Hayes and Rivlin [26] using a different notation, assumed that is1 and is2 are real. In general is1
and is2 are complex numbers, therefore this is not a valid assumption (see also Dowaikh and Ogden [23]).
Let α12 + γ* ≠ 0 (the case α12 + γ* = 0 will be considered latter). Suppose that s1, s2 are the roots of Eq. (13) satisfying Ims b 0
and s1 ≠ s2. Then, displacement field of the Rayleigh wave is:
h
i
h
i

iks x
iks x
iðkx −ct Þ
iks x
iks x
iðkx −ct Þ
; u2 = q1 C1 e 1 2 + q2 C2 e 2 2 e 1
u1 = C1 e 1 2 + C2 e 2 2 e 1

ð21Þ

where the constants C1, C2 are to be defined by the boundary conditions (10), q1, q2 determined by:
qm =

ρ0 c2 + γ2 s2m −α11
À
Á
; m = 1; 2:
α12 + γT sm

ð22Þ

Substitution of Eq. (21) into the boundary conditions (10) yields a pair of equations for C1 and C2. For non-trivial solutions the
determinant of coefficients must vanish. After some algebra we obtain:
γ2 α12 ðs1 −s2 Þ−γ2 α22 s1 s2 ðq1 −q2 Þ + α12 γT ðq1 −q2 Þ−α22 γT q1 q2 ðs1 −s2 Þ = 0:

ð23Þ

Using Eq. (22) it is not difficult to verify that:

q1 −q2 =



2
α11 −ρ0 c −γ2 s1 s2 ðs1 −s2 Þ
ðα12 + γT Þs1 s2

; q1 q2 =

α11 −ρ0 c2
:
α22 s1 s2

ð24Þ

By substituting Eq. (24) into Eq. (23), and taking into account Eq. (20), and then removing the factor (s1 − s2), we obtain (see
also Dowaikh and Ogden [23]):
#
#qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffiffi"
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" 



γ2
2
2
2
2

2
α11 −ρ0 c γ2 γ1 −ρ0 c −γT +
γ1 −ρ0 c2 = 0:
α α −ρ0 c −α12
α22 22 11

ð25Þ

When s1 = s2 = s and Ims b 0, the form of solution (21) must be replaced by:
sy iðkx1 −ct Þ

u1 = ½C1 + C2 yŠe e

sy iðkx1 −ct Þ

; u2 = ½C3 + C4 yŠe e

; y = ikx2

where C3 =−C1[γ2s2 +(α11 −ρ0c2)]/[(α12 +γ*)s] +C2[(α11 −ρ0c2)−γ2s2]/[(α12 +γ*)s2], C4 =−C2[γ2s2 +(α11 −ρ0c2)]/[(α12 +γ*)s],
but it can still be shown that the secular equation is given by Eq. (25). Thus Eq. (25) is the secular equation Rayleigh waves for the
case α12 + γ* ≠ 0.
Now we consider the case α12 + γ* = 0. For this case, the Eq. (2) decouple form each other, and we have:
u1 = Aexp ðiks1 x2 Þexp½iðkx1 −ct ފ; u2 = Bexpðiks2 x2 Þexp½iðkx1 −ct ފ

ð26Þ

where:
ffi
ffi

qÀffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qÀffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Á
Á
s1 = −i α11 −ρ0 c2 = γ2 ; s2 = −i γ1 −ρ0 c2 = α22 :

ð27Þ

The secular equation is derived by substituting Eqs. (26) and (27) into the boundary conditions (10), and it is:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
γ2 α22

qÀffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ÁÀ
Áffi
2
α11 −ρ0 c2 γ1 −ρ0 c2 −α12 = 0

ð28Þ


618

P.C. Vinh / Wave Motion 48 (2011) 614–625

3. Explicit formulas for the Rayleigh wave velocity in pre-stressed compressible solids
As in the undeformed state: γ1 b α11 (γ1 = μ, α11 = λ + 2μ), we first suppose that γ1 b α11. The cases γ1 = α11 and γ1 N α11 are
noted in Remark 6. Introducing (dimensionless) parameters:

a = 1−

γ2T
α α
α212
γ
; b = 11 22 ; d = 1−
; θ= 1
γ1 γ2
γ1 γ2
α11 α22
α11

ð29Þ

then, secular Eq. (25) is equivalent to:
pffiffiffipffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi
ða−xÞ 1−θx + b 1−xðd−θxÞ = 0

ð30Þ

where x = ρ0c 2/γ1 being the dimensionless (squared) velocity of Rayleigh waves. From 0 b γ1 b α11 we have 0 b θ b 1, and from
Eq. (17) it follows 0 b x b 1. By xr we denote a root of Eq. (30) satisfying 0 b x b 1. On introducing the variable t defined by:
t=

rffiffiffiffiffiffiffiffiffiffiffiffiffi
1−θx
1−x

ð31Þ

Eq. (30) becomes:
3

ð1−aÞt +

pffiffiffi
pffiffiffi
2
bðθ−dÞt + ðaθ−1Þt + bθðd−1Þ = 0:

ð32Þ

It follows from 0 b x b 1 and Eq. (31) that 1 b t b + ∞. It is noted that Eq. (31) is a 1–1 mapping from (0, 1) to (1, + ∞). By tr we
denote a solution of Eq. (32) satisfying 1 b t b + ∞. It is obvious that:

tr =

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
1−θxr
1−tr
:
; xr =
2
1−xr
θ−tr

ð33Þ

If γ* ≠ 0, on view of Eq. (29)1, a ≠ 1, and Eq. (32) is rewritten as:
3

2

F ðt Þ≡t + a2 t + a1 t + a0 = 0

ð34Þ

where:
pffiffiffi
pffiffiffi
aθ−1
bθðd−1Þ
bðθ−dÞ
; a2 =
; a1 =
:
a0 =
1−a
1−a
1−a

ð35Þ

When γ* = 0 (→ a = 1), Eq. (32) degenerates into a quadratic equation, namely:
pffiffiffi
pffiffiffi
2
φ1 ðt Þ≡ bðθ−dÞt + ðθ−1Þt + bθðd−1Þ = 0:

ð36Þ

The main result of the paper is the following theorem:
Theorem 1. (formulas for the velocity):
Let γ1 b α11. If there exists a Rayleigh wave propagating along the x1-direction, and attenuating in the x2-direction, in a compressible
elastic half-space subject to a homogeneous initial deformation (Eq. (1)), then it is unique, and its velocity is determined as follows:
(i) If α12 + γ* ≠ 0 and γ* ≠ 0, then xr is given by Eq. (33)2 in which:
1
tr = − a2 +
3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
3
R+ D+

a2 −3a1
p2ffiffiffiffi
pffiffiffiffi
9 3R+ D

ð37Þ

where each radical is understood as the complex root taking its principal value, a1, a2 are given by Eq. (35), and:
R=−
D=

h pffiffiffi
i

pffiffiffi
pffiffiffi
1
2
3
9 bð1−aθÞðθ−dÞð1−aÞ + 27 bθðd−1Þð1−aÞ + 2b bðθ−dÞ
54ð1−aÞ3

h
i
1
2
3
2
2
3
2
2
2
4b θðd−1Þðθ−dÞ −bðaθ−1Þ ðθ−dÞ + 4ðaθ−1Þ ð1−aÞ−18bθðd−1Þðaθ−1Þðθ−dÞð1−aÞ + 27bθ ðd−1Þ ð1−aÞ :
4
108ð1−aÞ

ð38Þ


P.C. Vinh / Wave Motion 48 (2011) 614–625

619

(ii) If α12 + γ* ≠ 0 and γ* = 0, then xr is given by Eq. (33)2 where:
tr =

pffiffiffiffi
ð1−θÞ + Δ
2
pffiffiffi
; Δ = ð1−θÞ + 4bθð1−dÞðθ−dÞ:
2 bðθ−dÞ

ð39Þ

(iii) If α12 + γ* = 0, the velocity is calculated by:
2

ρ0 c =

α11 + γ1 1
−
2
2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðα11 −γ1 Þ2 + 4α412 = α22 γ2 :

ð40Þ

Theorem 1 is deduced from Propositions 3–5 (case: α12 + γ* ≠ 0 and γ* ≠ 0), Proposition 6 (case: α12 + γ* ≠ 0 and γ* = 0) and
Proposition 7 (case: α12 + γ* = 0) that will be proved below. Note that the results for the cases γ1 = α11 and γ1 N α11 are mentioned
in Remark 6, and we have similar formulas for the velocity of Rayleigh waves propagating along the xk-direction, and attenuating

in the xm-direction (k, m = 1, 2, 3, k ≠ m).
3.1. Case α12 + γ* ≠ 0 and γ* ≠ 0 (Propositions 3–5)
Since 1 − a N 0, d − 1 ≤ 0, aθ − 1 b 0, it follows from Eq. (35) that a0 ≤ 0, a1 b 0. From Eq. (34) and a0 ≤ 0 we have F(0) ≤ 0. Also
from Eq. (34) it follows F′(t) = 3t 2 + 2a2t + a1. As the discriminant of the equation F′(t) = 0 is 4(a22 − 3a1) N 0 (noting that a1 b 0),
this equation has always two distinct real roots denoted by tmin (at which F(t) has a local minima) and tmax (at which F(t) has a
local maxima). Since tmin. tmax = a1/3 b 0, hence we have:
tmax b 0 b tmin

ð41Þ

Proposition 3. Suppose that α12 + γ* ≠ 0 and γ* ≠ 0. Then the Eq. (34) has a unique root in the interval (1, + ∞) if:
a+

pffiffiffi
bd N 0

ð42Þ

otherwise, it has no solution belonging to the interval (1, + ∞).
Proof. i) From F(0) ≤ 0 and the fact that the function F(t) is strictly discreasingly monotonous in the interval (tmax, tmin), so in (0, tmin)
by Eq. (41), it deduces that F(t) b 0 ∀ t ∈ (0, tmin]. As F(t) is strictly increasingly monotonous in the intervals (tmin, + ∞), F(tmin) b 0,
F(+ ∞) = + ∞, Eq. (34) has exactly one root in the interval (0,
by tr. It is clear that tr falls into the interval (1, + ∞) if
 + ∞),

pffiffiffidenoted
and only if F(1) b 0. From Eq. (34) we have F ð1Þ = ðθ−1Þ a + bd = ð1−aÞ. Since θ − 1 b 0, 1 − a N 0, it is clear that F(1) b 0 is
equivalent to the condition (42). The proof is finished.
From the above arguments, we have immediately the following proposition.
Proposition 4. Suppose α12 + γ* ≠ 0, γ* ≠ 0 and Eq. (42) holds. If Eq. (34) has two or three distinct real roots, then tr is the largest root.

Proposition 5. Suppose α12 + γ* ≠ 0, γ* ≠ 0, and Eq. (42) holds. Then, the (dimensionless squared) velocity xr of Rayleigh waves in
pre-stressed compressible is defined by Eq. (33)2 in which tr is given by:
1
tr = − a2 +
3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
pffiffiffiffi
3
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R+ D+ p
pffiffiffiffi
3
R+ D

ð43Þ

where each radical is understood as the complex root taking its principal value,




2
2
3
q = a2 −3a1 = 9; R = 9a1 a2 −27a0 −2a2 = 54



3
2 2
2
3
D = 4a0 a2 −a1 a2 −18a0 a1 a2 + 27a0 + 4a1 = 108

ð44Þ

and ak, k = 0, 1, 2 are expressed in terms of four dimensionless parameters θ, a, b, d by Eq. (35).
Note that a22 − 3a1 N 0 due to a1 b 0, q is therefore a positive real number.
Proof. We recall that, with assumptions α12 + γ* ≠ 0, γ* ≠ 0, the secular equation of Rayleigh waves is Eq. (34), and if Eq. (42)
holds, Eq. (34) has a unique root, namely tr, in the interval (1, + ∞), according to Proposition 3, and by Proposition 4, in the case
that Eq. (34) has two or three distinct real roots, tr is the largest root. We now find an explicit formula for tr. To do that we
introduce new variable z given by:
z=t+

1
a :
3 2

ð45Þ


620

P.C. Vinh / Wave Motion 48 (2011) 614–625

In terms of z Eq. (34) becomes:
3

2

z −3q z + r = 0

ð46Þ

where q 2 is defined by Eq. (44)1 and:


3
r = 2a2 −9a1 a2 + 27a0 = 27:

ð47Þ

Our task is now to find the real solution zr of Eq. (46) which is related to tr by the relation Eq. (45). As tr is the largest root of
Eq. (34), zr is the largest one of Eq. (46) in the case that it has two or three distinct real roots.
By theory of cubic equation, three roots of Eq. (46) are given by the Cardan's formula as follows (see Cowles and Thompson
[27]):
1
1 pffiffiffi
1
1 pffiffiffi
z1 = S + T; z2 = − ðS + T Þ + i 3ðS−T Þ; z3 = − ðS + T Þ− i 3ðS−T Þ
2
2
2
2

ð48Þ

where i 2 = − 1 and:

S=

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
pffiffiffiffi
3
3
1
2
3
2
R + D; T = R− D; D = R + Q ; R = − r; Q = −q :
2

ð49Þ

Remark 1. In relation to these formulas we emphasize
two points: (i) The cube root of a negative real number is taken as the real
pffiffiffiffi
negative root. (ii) If, in the expression for S, R + D is complex, the phase angle in T is taken as the negative of the phase angle in S,
so that T = S*, where S* is the complex conjugate of S.
Remark 2. The nature of three roots of Eq. (46) depends on the sign of its discriminant D, in particular: If D N 0, then Eq. (46) has
one real root and two complex conjugate roots; if D = 0, the equation has three real roots, at least two of which are equal; if D b 0,
then it has three real distinct roots.
We now show that in each case the largest real root of Eq. (46) zr is given by:
zr =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
3
q2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R+ D+ p
pffiffiffiffi
3
R+ D

ð50Þ

in which each radical is understood as the complex root taking its principle value, q 2, R, D are given by Eq. (44). It is noted that one
can obtain Eqs. (44)2 and (44)3 by substituting the expressions for q 2 defined by Eq. (44)1 and r given by Eq. (47) into Eqs. (49)3–
(49)5. Now we examine the distinct cases dependent on the values of D in order to prove Eq. (50).
(i) If D N 0, then Eq. (46), according to Remark 2, has a unique real root, so it is zr, given by Eq. (48)1 in which the radicals are
understood as real ones. As Eq. (46) has a unique real root, F(tmin). F(tmin) N 0, otherwise, Eq. (46) has two or three real roots.
As proved above, in the Proposition 3, F(tmin) b 0, so we have F(tmax) b 0. This leads to F(tN) b 0, where tN is the abscissa of the
pointpof
ffiffiffiffi inflexion N of the cubic curve y = F(t). Since r = F(tN), it follows that r b 0, or equivalently, R N 0. This yields:
R + D N 0. In view of this inequality and:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
3
q2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R− D = p
pffiffiffiffi
3
R+ D

ð51Þ

formula (48)1 coincides with (50). That means formula (50) is true for this case.
(ii) If D = 0, analogously as above, it is not difficult to observe that r b 0, or equivalently, R N 0. When D = 0 we have R 2 =
− Q 3 = q 6 ⇒ R = q 3 ⇒ r = − 2R = − 2q 3, so Eq. (46) becomes z 3 − 3q 2z − 2q 3 = 0 whose roots are: z1 = 2q, z2 = − q (double
root). This yields zr = 2q, since it is the largest root. From Eq. (50) and taking into account q N 0, D = 0, it follows zr = 2q. This
shows the validity of Eq. (50).
(3i) If D b 0, then Eq. (46) has three distinct real roots, and according to Proposition 4, zr is the largest root. By arguments
presented in Ref.[9] (p.255) one can show that, in this case the largest root zr of Eq. (46) is given by:
zr =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
pffiffiffiffi
3
3
R + D + R− D

ð52Þ

within which each radical is understood
as the complex root taking its principal value. By 3θ (∈ (0, π)) we denote the phase
pffiffiffiffiffiffiffiffi
angle of the complex number R + i −D. It is not difficult to verify that:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
pffiffiffiffi

3
3
iθ
−iθ
R + D = qe ;
R− D = qe

ð53Þ


P.C. Vinh / Wave Motion 48 (2011) 614–625

621

where each radical is understood as the complex root taking its principal value. It follows from Eq. (53) that:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
pffiffiffiffi
3
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R− D = p
pffiffiffiffi
3
R + D:

ð54Þ

By substituting Eq. (54) into Eq. (52) we obtain Eq. (50), and the validity of Eq. (50) is proved. From Eqs. (45) and (50) we
obtain Eq. (43). The proof of Proposition 5 is completed. Note that one can obtain Eq. (38) by substituting Eq. (35) into

Eq. (44).
Remark 3
i) From the above arguments we have:
1
tr = − a2 +
3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
pffiffiffiffi
3
3
R + D + R− D

ð55Þ

where R, D defined by Eq. (38).
ii) It follows from Eqs. (30) and (31) that the dimensional Rayleigh wave velocity xr can be expressed by:
pffiffiffi
bd
pffiffiffi
tr + bθ

atr +

xr =

ð56Þ

where tr is given by Eq. (43) or Eq. (55) in which R, D defined by Eq. (38). The formula (56) contains only the first power of
tr, it is thus somewhat simpler than Eq. (33)2.
iii) When the prestresses are absent, by Eqs. (6), (8), and (29) we have:
2

θ = γ; a = 0; b = 1 = γ ; d = 4γð1−γÞ:

ð57Þ

Using Eqs. (35) and (57) provides:
2

a0 = −ð1−2γÞ ; a1 = −1; a2 = 4γ−3:

ð58Þ

Substituting Eq. (58) into Eqs. (44)2 and (44)3 (or Eq. (57) into Eq. (38)), and after some manipulation we obtain:




2
3
2
2
3
R = 2 27−90γ + 99γ −32γ = 27; D = 4ð1−γÞ 11−62γ + 107γ −64γ = 27:

ð59Þ

From Eqs. (56) and (57) and taking into account Eqs. (55) and (58)3, it deduces:

ð

4
xr = 4ð1−γÞ 2− γ +
3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
pffiffiffiffi
3
3
R + D + R− D

Þ

−1

ð60Þ

in which R, D are given by Eq. (59). This is the formula for the Rayleigh wave speed in compressible isotropic elastic solids
that was already derived by Vinh and Ogden [7]. Interestingly that, Eq. (60) (along with Eq. (51)) provides a simple
expression for dimensionless squared Rayleigh wave slowness sr = 1/xr for compressible isotropic elastic solids,
namely:
sr =

"
#

pffiffiffiffiffiffiffiffiffiffiffi
1
4
3 + ð4γ−3Þ2
ffiffiffiffiffiffiffiffiffiffi
p
2− γ + 3 V ðγÞ +
4ð1−γÞ
3
9 3 VðγÞ

ð61Þ

where:
V ðγÞ =




2 
2
2
3
2
3
27−90γ + 99γ −32γ + pffiffiffi ð1−γÞ 11−62γ + 107γ −64γ
27
3 3

ð62Þ

iv) From Eq. (55) and the proof 3i) of Proposition 5, it is obvious that, in the case D b 0, xr can also be calculated by a real
expression, namely:
xr =

1−tr2
1
R
; tr = − a2 + 2qcosθ; cos3θ = 3 ; 3θ ∈ ð0; πÞ:
3
q
θ−tr2

ð63Þ


622

P.C. Vinh / Wave Motion 48 (2011) 614–625

For unstressed solids Eq. (63) becomes (see also Vinh and Ogden [7]):
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u 
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uμ 1−tr2
ð3−4γÞ
2
t
À

Á ; tr =
+ cosθ ð4γ−3Þ2 + 3
cr =
2
3
3
ρ γ−tr
27R
cos3θ = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Â
Ã3ffi ; 3θ ∈ ð0; π = 2Þ
ð4γ−3Þ2 + 3

ð64Þ

ð65Þ

where cr is the Rayleigh wave speed, R is given by Eq. (59)1.
v) Since the nearly incompressible materials (see, for example, Rogerson and Murphy [28]; Kobayashi and Vanderby [29]) are
a special class of the compressible material, the obtained formulas therefore hold for them.
3.2. Case α12 + γ* ≠ 0, γ* = 0 (Proposition 6)
When γ* = 0, then a = 1, and Eq. (32) is equivalent to the following equation in the interval (1, + ∞):
pffiffiffi
pffiffiffi
2
φ1 ðt Þ≡ bðθ−dÞt + ðθ−1Þt + bθðd−1Þ = 0

ð66Þ

Proposition 6. i) If:

θ−d N 0; 1 +

pffiffiffi
bd N 0

ð67Þ

then Eq. (66) has a unique root in the interval (1, + ∞), and it is given by:

tr =

ð1−θÞ +

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1−θÞ2 + 4bθð1−dÞðθ−dÞ
pffiffiffi
2 bðθ−dÞ

ð68Þ

In this case, the Rayleigh wave velocity xr is given by Eqs. (33)2 and (68).
ii) If Eq. (67) is not valid, then Eq. (66) has no root belonging to the interval (1, + ∞). The Rayleigh wave does not exist
in this case.
pffiffiffi
Remark 4. Using the facts: θ−1b0; bθðd−1Þb0, it is not difficult to verify the following: a) If θ − d N 0, then Eq. (66) has two
different roots t1, t2 and t1 b 0 b t2. b)p
Ifffiffiffiθ − d b 0, then the roots t1, t2 of Eq. (66), if exist, must satisfy t1 ≤ t2 b 0. c) If θ − d = 0, Eq. (66)
has only one root, namely: t1 = − bθ b 0.
Proof

pffiffiffi
i) Suppose Eq. (67) holds. From Eqs. (66), (67) and 0 b θ b 1 we have bðθ−dÞφ1 ð1Þb0. This inequality ensures that Eq. (66) has
two different roots t1, t2: t1 b 1 b t2, i.e. Eq. (66) has a unique root, namely t2, in the interval (1, + ∞). Since t2 is the bigger root of
Eq. (66), it is thus given by Eq. (68).
ii) It is clearpthat,
ii), we have to examine only the four following cases: (1) θ − d b 0; (2) θ − d = 0;
ffiffiffi in order to verify the conclusion
pffiffiffi
(3) 1 + bd b 0 and θ−d N 0; (4) 1 + bd = 0 and θ−d N 0.
+ From b), c) of Remark 4, p
itffiffiffiis clear thatpii)
ffiffiffi is true for the cases (1) and (2).
+ It is easy to see that 1 + bd b 0 (1 + bd = 0) is equivalent to φ1(1) N 0 (φ1(1) = 0). By these facts, the validity of ii) for
the cases (3) and (4) are deduced from a) of Remark 4.

Remark 5. The condition (42) is equivalent to (5.19) in [23], but without the equality, and Eq. (67) is equivalent to the conditions
(5.33) and (5.34) in [23], but also without the equality.
3.3. Case α12 + γ* = 0 (Proposition 7)
It is not difficult to verify that:
Proposition 7. Let α12 + γ* = 0. Then Eq. (28) has a unique solution satisfying: 0 b ρ0c 2 b γ1 if:
4

α12 ≠ 0 and γ1 γ2 α11 α22 − α12 N 0

ð69Þ


P.C. Vinh / Wave Motion 48 (2011) 614–625

623

otherwise, Eq. (28) has no root satisfying: 0 b ρ0c 2 b γ1. In the case that Eq. (69) is satisfied, the (unique) solution of the Eq. (28)
satisfying: 0 b ρ0c 2 b γ1 is given by:
2

ρ0 c =

α11 + γ1 1
−
2
2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðα11 −γ1 Þ2 + 4α412 = α22 γ2

ð70Þ

It is readily to see that Theorem 1 is deduced from Propositions 3–7.


pffiffiffi
pffiffiffi 
Remark 6. It is not difficult to verify that when γ1 =α11, i.e. θ=1, the Rayleigh wave velocity is given by xr = a + d b = 1 + b
(using Eq. (30)), and for the case γ1 N α11, the velocity of Rayleigh waves xr =ρ0c2/α11 is also determined by Theorem 1 in which α11, α22,
α12 are replaced respectively by γ1, γ2, γ*, and inversely.
4. An example: solid and foam rubbers
As an example, we consider a half-space X2 b 0 with the traction-free surface (i.e. σ2 = 0), and in the plane-strain deformation
(λ3 = 1), its strain-energy function is given by (see Murphy and Destrade [30]):
W=



μ
  2ð −1Þ = 
I−2 +
J
−1
2
1−

ð71Þ

where:
2
2
 −1
I = λ1 + λ2 ; J = λ1 λ2 ; λ2 = λ1 ; 0 b  b 1

ð72Þ

According to Murphy and Destrade [30], the solid and foam rubbers are well characterized by this strain-energy function. From
Eqs.(3)–(5), (8) and (71), and (72), it is not difficult to verify that:
h
i
2ð −2Þ
2ð −1Þ
; α22 = ð2μ =  Þλ1
α11 = μλ21 1 + ð2 =  −1Þλ1
2ð −1Þ

α12 = ð1− Þα22 ; γ1 = μλ21 ; γ2 = μλ1

; γT = γ2 :

ð73Þ

Using Eqs. (29), (35) and (73) yields:
pffiffiffi
2 2ð1− Þ2
2
; a1 = −
i
2ð −2Þ
2ð −2Þ 3 = 2
 + ð2− Þλ1
 + ð2− Þλ1
pffiffiffi
2ð2 −3Þ

a2 = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; θ =
:
2ð −2Þ
2ð −2Þ

+
ð
2−
Þλ1
 + ð2− Þλ1

a0 = − h

ð74Þ

From Eqs. (11)2, (11)4, (73)3, (73)6 and the assumption 0 b  b 1, it is clear that α12 N 0, γ* N 0. By Theorem 1, the dimensionless
squared velocity xr of Rayleigh waves is given by Eqs. (33)2 and (37), in which q 2, R, D are determined by Eq. (44), and a0, a1, a2 and
θ are calculated by Eq. (74).
When  tends to zero, it follows from Eq. (74):
6

4

2

a0 = −λ1 ; a1 = −λ1 ; a2 = −3λ1 ; θ = 0:

ð75Þ

By using Eqs. (33)2, (37), (44) and (75) we have:
1
xr = 1− 2 4 ; m = 1 +
m λ1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffiffi
rffiffiffiffiffiffiffi
3
3
2 11

2 11
+ 2−
≈ 3:3830:
2+
3 3
3 3

ð76Þ

This is the formula of the dimensionless Rayleigh wave speed for the case  → 0, i.e. the case of incompressible material, because
pffiffiffiffiffi
 → 0 leads to λ2 = λ1− 1, or equivalently J = 1. It follows immediately from Eq. (76) that λ1 must be bigger than 1 = m ≈ 0:5437 in
À
pffiffiffiffiffiÃ
order to ensure that 0 b xr b 1. This means that for the values of λ1 belong to the interval 0; 1 = m , the Rayleigh wave does not
exist. On view of Eqs. (73)4, (76) and xr = ρc 2/γ1 we have a different form of Eq. (76), namely:
c2
1
μ
2
2
= λ1 − 2 2 ; c2 =
ρ
m λ1
c22

ð77Þ


624

P.C. Vinh / Wave Motion 48 (2011) 614–625

1
0.9
0.8
0.7
0.6

(a)

x1/2 0.5
r

(d)
(b)

0.4

(c)

0.3
0.2
0.1
0
0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

λ1
pffiffiffiffiffi
Fig. 1. Dependence of the dimensionless velocity xr of the Rayleigh wave on λ1 for different values of :  = 0 (line (a)),  = 0.5 (line (b)),  = 0.8 (line (c)),  = 1
(line (d)). The elastic material is characterized by the strain-energy function (71) and (72).

Taking λ1 = 1 we obtain from Eq. (77) the exact value, denoted by xr0, of the dimensionless squared velocity of Rayleigh waves
propagating in incompressible elastic solids (without pre-strains), namely:

xr0

1
= 1− 2 ; m = 1 +
m

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffiffi

rffiffiffiffiffiffiffi
3
3
2 11
2 11
2+
+ 2−
3 3
3 3

ð78Þ

whose approximate value is 0.9126, agreeing with the classical result (see Ewing et al. [31]). Some other exact expressions of xr0
have been derived recently by Malischewsky [32], Ogden and Vinh [8], Vinh [21].
We now consider the second limiting case when  → 1. It follows from Eq. (74) that when  → 1, ak, k = 0, 1, 2 and θ reach the
following expressions:
pffiffiffi
2λ1
2λ21
λ21
ffi; θ =
; a2 = − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
a0 = 0; a1 = −
2
1 + λ1
1 + λ21
1 + λ2

ð79Þ

1

By using Eqs. (33)2, (37), (44) and (79) we arrive at:

xr = 1−

1
; n ≈ 4:2359; λ1 N 0:4859:
nλ21

ð80Þ

pffiffiffiffiffi
Fig. 1 shows the plots of the dimensionless Rayleigh wave velocity xr for different values of the compressibility parameter . It
pffiffiffiffiffi
says that the effect of  on xr is not considerable. Note that Fig. 1 is almost identical to Fig. 2 in Ref. [30].
5. Conclusions
In this paper, exact formulas for the Rayleigh wave velocity in compressible isotropic solids with homogeneous initial
deformations are derived employing the theory of cubic equation. They are explicit and have simple algebraic forms. They hold for
any strain-energy function and are valid for any range of pre-strains. From the obtained results, we can go back to the formula of
Rayleigh wave velocity for unstressed compressible elastic solids, and reach the exact value of dimensionless squared Rayleigh
wave speed c 2/c22 for unstressed incompressible elastic solids as well. Since obtained formulas are explicit, exact and hold for any
range of pre-strains, they will be significant in practical applications, especially in the nondestructive evaluation of prestresses of
structures.


P.C. Vinh / Wave Motion 48 (2011) 614–625

625

Acknowledgments
The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under
grant no. 107.02-2010.07.
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