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On formulas for the Rayleigh wave velocity in pre-stressedcompressible solids Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh

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On formulas for the Rayleigh wave velocity in pre-stressed

compressible solids

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

Article history:

Received 22 November 2010

Received in revised form 9 April 2011

Accepted 22 April 2011

Available online 4 May 2011

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

Keywords:

Rayleigh waves

Rayleigh wave velocity

Prestresses

Pre-strains

Compressible

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 variousﬁelds 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, theﬁrst 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 ofγ=μ/(λ+2μ), where λ and μ are the usual Lame constants It is rather cumbersome[5]and theﬁnal 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.

E-mail address: pcvinh@vnu.edu.vn

Contents lists available atScienceDirect

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

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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 weﬁrst 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=λ1X1; x2=λ2X2; x3=λ3X3; λi= const; i = 1; 2; 3 ð1Þ whereλiN0, i=1, 2, 3, are the principal stretches of the deformation In its deformed conﬁguration the body, therefore, occupies the region x2b0 with the boundary x2= 0 We consider a plane motion in the (x1, x2)-plane with displacement components u1, u2,

u3such 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]:

A1111u1 ;11+ A2121u1 ;22+ Að 1122+ A2112Þu2 ;12=ρ::u1

A1221+ A2211

ð Þu1;12+ A1212u2;11+ A2222u2;22=ρ::u2

ð2Þ

whereρ 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, Aijklare components of the fourth order elasticity tensor

deﬁned as follows[23,24]:

JAiijj=λiλj ∂2

W

∂λi∂λj

ð3Þ

JAijij=

λi∂W

∂λi−λj∂W

∂λj

!

λ2 i

λ2

i−λ2 j

; i≠j; λi≠λj

1

2 JAiiii−JAiijj+λi∂W

∂λi

; i≠j; λi=λj

8

>

ð4Þ

JAijji= JAjiij= JAijij−λi∂W

∂λi

i≠j

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=λ i≠jð Þ; Aijij= Aijji=μ i≠jð Þ ð6Þ Equations of motion (Eq.(2)) are taken together with the boundary conditions of zero incremental traction, which are expressed as:

A u1;2+ A u2;1= 0; A u1;1+ A u2;2= 0 on x = 0 ð7Þ

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For seeking the simplicity we use the notations (see also Dowaikh and Ogden[23]):

αij=αji= JAiijjði; j = 1; 2Þ; γ1= JA1212; γ2= JA2121; γT= JA2112 ð8Þ

In terms of these notations Eq.(2)becomes:

α11u1;11+γ2u1;22+ðα12+γTÞu2;12=ρ0u::

1; ðα12+γTÞu1;12+γ1u2;11+α22u2;22=ρ0u::

and boundary conditions(7)are of the form:

γ2u1;2+γTu2;1= 0; α12u1;1+α22u2;2= 0 on x2= 0 ð10Þ hereρ0= Jρ is the mass density of the material in the (natural) undeformed conﬁguration From the strong-ellipticity condition of system(2),αij,γiare required to satisfy the inequalities[23,25]:

We now consider a time-harmonic wave propagating along the x1-principal direction and set:

uj= Ajexp iksx½ 2+ ik xð1−ctÞ; j = 1; 2 ð12Þ where k is the wave number, c is the wave speed, A1, A2are constants For the decay of uiat x2=−∞ it requires Imsb0 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 s2, namely:

where

b4=α22γ2; 2b2=α22α11−ρ0c2

+γ2γ1−ρ0c2

− αð 12+γTÞ2

; b0=α11−ρ0c2

γ1−ρ0c2

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

s21+ s22=−α22 α11−ρ0c

2

+γ2γ1−ρ0c2

− αð 12+γTÞ2

s21s22= α11−ρ0c2

γ1−ρ0c2

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

Proof Setting Y = s2, then Eq.(13)becomes:

ByΔ we denote the discriminant of Eq.(18), and Y1, Y2are its roots (i) IfΔ≥0 then Y1, Y2are real This ensures Y1, Y2are negative, otherwise, for example s1= ffiffiffiffiffi

Y1

p

is a real number, so that its imaginary part is zero This contradicts the requirement Imsb0 From Eq.(16)and the fact Y1Y2N0 it follows either α11−ρ0c2N0, γ1−ρ0c2N0 or α11−ρ0c2b0, γ1−ρ0c2b0 Suppose that

α11−ρ0c2b0, γ1−ρ0c2b0 Then, from Eq.(15)and taking into account Eq.(11), it deduces Y1+ Y2N0, but this contradicts the observed above fact that Y1b0, Y2b0, so we have α11−ρ0c2N0, γ1−ρ0c2N0 (ii) If Δb0, then Y1= Y2, hence Y1Y2= |Y1|2N0 In the other hand, it is not difﬁcult to verify that:

Δ = α22α11−ρ0c2

−γ2γ1−ρ0c2

−2 αð +γTÞ2

α α −ρc2

+γγ−ρ c2

+ðα +γTÞ4

:

ð19Þ

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From Eq.(16)and the fact Y1Y2N0 it follows that α11−ρ0c2andγ1−ρ0c2have the same sign Suppose they are all negative Then from Eq.(19)it follows thatΔ≥0 But this contradicts the assumption that Δb0 Hence, both α11−ρ0c2andγ1−ρ0c2must

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, s2be two roots of the characteristic Eq.(13), and satisfy the condition Imsb0, then s1s2b0, and:

s1s2=−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

α11−ρ0c2

γ1−ρ0c2

α22γ2

v

u

Proof Indeed, if the discriminantΔ of the quadratic Eq.(13)for s2is non-negative, then its two roots must be negative in order that Imsb0 is to be satisﬁed In this case the pair s1, s2are of the form: s1=−ir1, s2=−ir2where r1, r2are positive IfΔb0, the quadratic Eq.(13)for s2has two conjugate complex roots, and in order to ensure the condition Imsb0: s1= t−ir, s2=−t−ir where r is positive and t is a real number In both cases, s1s2is 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 is1and is2are real In general is1

and is2are complex numbers, therefore this is not a valid assumption (see also Dowaikh and Ogden[23])

Letα12+γ*≠0 (the case α12+γ*= 0 will be considered latter) Suppose that s1, s2are the roots of Eq.(13)satisfying Imsb0 and s1≠s2 Then, displacementﬁeld of the Rayleigh wave is:

u1= C1eiks1 x2

+ C2eiks2 x2

ei kxð 1 −ct Þ; u2= q1C1eiks1 x2

+ q2C2eiks2 x2

ei kxð 1 −ct Þ ð21Þ where the constants C1, C2are to be deﬁned by the boundary conditions(10), q1, q2determined by:

qm=ρ0c2+γ2s2m−α11

α12+γT

Substitution of Eq.(21)into the boundary conditions(10)yields a pair of equations for C1and C2 For non-trivial solutions the determinant of coefﬁcients must vanish After some algebra we obtain:

γ2α12ðs1−s2Þ−γ2α22s1s2ðq1−q2Þ + α12γTðq1−q2Þ−α22γTq1q2ðs1−s2Þ = 0: ð23Þ Using Eq.(22)it is not difﬁcult to verify that:

q1−q2= α11−ρ0c2−γ2s1s2

s1−s2

ð Þ

α12+γT

ð Þs1s2 ; q1q2= α11−ρ0c2

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]):

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

α11−ρ0c2

q "

γ2γ1−ρ0c2

−γ2 T

# +

ffiffiffiffiffiffiffiffi

γ2

α22

r "

α22α11−ρ0c2

−α2 12

# ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

γ1−ρ0c2

q

When s1= s2= s and Imsb0, the form of solution(21)must be replaced by:

u1= C½ 1+ C2yesy

ei kxð 1 −ct Þ;u2= C½ 3+ C4yesy

ei kxð 1 −ct Þ; 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 caseα12+γ*≠0

Now we consider the caseα12+γ*= 0 For this case, the Eq.(2)decouple form each other, and we have:

u1= Aexp iksð 1x2Þexp i kx½ð 1−ctÞ; u2= Bexp iksð 2x2Þexp i kx½ð 1−ctÞ ð26Þ where:

s1=−i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiα11−ρ0c2

= γ2

q

; s2=−i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγ1−ρ0c2

= α22

q

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

γα

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

α −ρc2

γ−ρ c2

q

−α2

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3 Explicit formulas for the Rayleigh wave velocity in pre-stressed compressible solids

As in the undeformed state:γ1bα11(γ1=μ, α11=λ+2μ), we ﬁrst suppose that γ1bα11 The casesγ1=α11andγ1Nα11are noted inRemark 6 Introducing (dimensionless) parameters:

a = 1− γ

2

T

γ1γ2; b =α11α22

γ1γ2 ; d = 1− α

2 12

α11α22; θ = γ1

then, secular Eq.(25)is equivalent to:

a−x

ð Þpffiffiffiffiffiffiffiffiffiffiffiffiffi1−θx+ ffiffiffi

b

p ffiffiffiffiffiffiffiffiffiffi

1−x

p

d−θx

where x =ρ0c2/γ1being the dimensionless (squared) velocity of Rayleigh waves From 0bγ1bα11we have 0bθb1, and from

Eq.(17)it follows 0bxb1 By xrwe denote a root of Eq.(30)satisfying 0bxb1 On introducing the variable t deﬁned by:

t =

ffiffiffiffiffiffiffiffiffiffiffiffiffi

1−θx

1−x

r

ð31Þ

Eq.(30)becomes:

1−a

ð Þt3

+ ffiffiffi

b

p

θ−d

ð Þt2

+ að θ−1Þt +pffiffiffib

It follows from 0bxb1 and Eq.(31)that 1btb+∞ It is noted that Eq.(31)is a 1–1 mapping from (0, 1) to (1,+∞) By trwe denote a solution of Eq.(32)satisfying 1btb+∞ It is obvious that:

tr=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1−θxr

1−xr

s

; xr= 1−t2

r

θ−t2 r

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

F tð Þ≡t3

where:

a0=

ffiffiffi

b

p

θ d−1ð Þ

1−a ; a1= aθ−1

1−a; a2=

ffiffiffi b p θ−d

ð Þ

Whenγ*= 0 (→a=1), Eq.(32)degenerates into a quadratic equation, namely:

φ1ð Þ≡t pffiffiffib

θ−d

ð Þt2

+ðθ−1Þt +pffiffiffib

The main result of the paper is the following theorem:

Theorem 1 (formulas for the velocity):

Letγ1bα11 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) Ifα12+γ*≠0 and γ*≠0, then xris given by Eq.(33)2in which:

tr=−13a2+

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R + ffiffiffiffi D p

3

q

+ a

2

−3a1

9 ffiffiffiffi R

3

p + ffiffiffiffi D

p

where each radical is understood as the complex root taking its principal value, a1, a2are given by Eq.(35), and:

R =− 1

54 1ð −aÞ3 9 ffiffiffi

b

p

1−aθ

ð Þ θ−dð Þ 1−að Þ + 27pffiffiffib

θ d−1ð Þ 1−að Þ2

+ 2b ffiffiffi b p θ−d

ð Þ3

108 1−að Þ4 4b2θ d−1ð Þ θ−dð Þ3

−b aθ−1ð Þ2

θ−d

ð Þ2 + 4 aθ−1ð Þ3

1−a

ð Þ−18bθ d−1ð Þ aθ−1ð Þ θ−dð Þ 1−að Þ + 27bθ2

d−1

ð Þ2 1−a

ð Þ2

:

ð38Þ

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(ii) Ifα12+γ*≠0 and γ*= 0, then xris given by Eq.(33)2where:

tr=ð1−θÞ +pffiffiffiffiΔ

2 ffiffiffi b p θ−d

ð Þ ; Δ = 1−θð Þ

2

(iii) Ifα12+γ*= 0, the velocity is calculated by:

ρ0c2=α11+γ1

2 −12 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðα11−γ1Þ2+ 4α4

12= α22γ2

q

Theorem1is deduced fromPropositions 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 casesγ1=α11andγ1Nα11are mentioned

inRemark 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 Caseα12+γ*≠0 and γ*≠0 (Propositions 3–5)

Since 1−aN0, d−1≤0, aθ−1b0, it follows from Eq.(35)that a0≤0, a1b0 From Eq.(34)and a0≤0 we have F(0)≤0 Also from Eq.(34)it follows F′(t)=3t2+ 2a2t + a1 As the discriminant of the equation F′(t)=0 is 4(a2−3a1)N0 (noting that a1b0), 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= a1/3b0, hence we have:

Proposition 3 Suppose thatα12+γ*≠0 and γ*≠0 Then the Eq.(34)has a unique root in the interval (1, +∞) if:

a + ffiffiffi

b

p

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)b0∀ t ∈(0, tmin] As F(t) is strictly increasingly monotonous in the intervals (tmin, +∞), F(tmin)b0, F(+∞) =+ ∞, Eq.(34)has exactly one root in the interval (0, +∞), denoted by tr It is clear that trfalls into the interval (1, +∞) if and only if F(1)b0 From Eq.(34)we have F 1ð Þ = θ−1ð Þ a +pffiffiffib

d

= 1−að Þ Since θ−1b0, 1−aN0, it is clear that F(1) b0 is equivalent to the condition(42) The proof isﬁnished

From the above arguments, we have immediately the following proposition

Proposition 4 Supposeα12+γ*≠0, γ*≠0 and Eq.(42)holds If Eq.(34)has two or three distinct real roots, then tris the largest root Proposition 5 Supposeα12+γ*≠0, γ*≠0, and Eq.(42)holds Then, the (dimensionless squared) velocity xrof Rayleigh waves in pre-stressed compressible is deﬁned by Eq.(33)2in which tris given by:

tr=−13a2+

3

q

+ q

2

3

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

q2= a2−3a1

= 9; R = 9a 1a2−27a0−2a3

= 54

D = 4a0a3−a2

a2−18a0a1a2+ 27a2+ 4a3

and ak, k = 0, 1, 2 are expressed in terms of four dimensionless parametersθ, a, b, d by Eq.(35)

Note that a2−3a1N0 due to a1b0, q is therefore a positive real number

Proof We recall that, with assumptionsα12+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 toProposition 3, and byProposition 4, in the case that Eq.(34)has two or three distinct real roots, tris the largest root We nowﬁnd an explicit formula for tr To do that we introduce new variable z given by:

z = t + 1

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In terms of z Eq.(34)becomes:

z3−3q2

where q2is deﬁned by Eq.(44)1and:

r = 2a3−9a1a2+ 27a0

Our task is now toﬁnd the real solution zrof Eq.(46)which is related to trby the relation Eq.(45) As tris the largest root of

Eq.(34), zris 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]):

z1= S + T; z2=−12ðS + TÞ + 12i ffiffiffi

3

p

S−T

ð Þ; z3=−12ðS + TÞ−12i ffiffiffi

3

p

S−T

where i2=−1 and:

S =

R + ffiffiffiffi

D p

3

q

; T =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R−pffiffiffiffiD

3

q

; D = R2

+ Q3; R = −1

2r; Q = −q2

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 negative root (ii) If, in the expression for S, R + ffiffiffiffi

D p

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 DN0, 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 Db0, then it has three real distinct roots

We now show that in each case the largest real root of Eq.(46)zris given by:

zr=

R + ffiffiffiffi

D p

3

q

+ q

2

3

in which each radical is understood as the complex root taking its principle value, q2, R, D are given by Eq.(44) It is noted that one can obtain Eqs.(44)2and(44)3by substituting the expressions for q2deﬁned by Eq.(44)1and 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 DN0, then Eq.(46), according toRemark 2, has a unique real root, so it is zr, given by Eq.(48)1in which the radicals are understood as real ones As Eq.(46)has a unique real root, F(tmin) F(tmin)N0, otherwise, Eq.(46)has two or three real roots

As proved above, in theProposition 3, F(tmin)b0, so we have F(tmax)b0 This leads to F(tN)b0, where tNis the abscissa of the point of inﬂexion N of the cubic curve y=F(t) Since r=F(tN), it follows that rb0, or equivalently, RN0 This yields:

R + ffiffiffiffi

D

p

N 0 In view of this inequality and:

R−pffiffiffiffiD

3

q

= q

2

3

formula(48)1coincides 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 rb0, or equivalently, RN0 When D=0 we have R2=

−Q3= q6⇒R=q3⇒r=−2R=−2q3, so Eq.(46)becomes z3−3q2z−2q3= 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 qN0, D=0, it follows zr= 2q This shows the validity of Eq.(50)

(3i) If Db0, then Eq.(46)has three distinct real roots, and according toProposition 4, zris the largest root By arguments presented in Ref.[9](p.255) one can show that, in this case the largest root zrof Eq.(46)is given by:

zr=

3

q

+

R−pffiffiffiffiD

3

q

ð52Þ within which each radical is understood as the complex root taking its principal value By 3θ (∈(0, π)) we denote the phase angle of the complex number R + i ffiffiffiffiffiffiffiffi

−D

p It is not difﬁcult to verify that:

R + ffiffiffiffi

D

p

3

q

= qeiθ;

R−pffiffiffiffiD

3

q

Trang 8

where each radical is understood as the complex root taking its principal value It follows from Eq.(53)that:

R−pffiffiffiffiD

3

q

= q

2

3

p

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) Theproof of Proposition 5is completed Note that one can obtain Eq.(38)by substituting Eq.(35)into

Eq.(44)

Remark 3

i) From the above arguments we have:

tr=−13a2+

3

q

+

R−pffiffiffiffiD

3

q

ð55Þ

where R, D deﬁned by Eq.(38)

ii) It follows from Eqs.(30) and (31)that the dimensional Rayleigh wave velocity xrcan be expressed by:

xr=atr+

ffiffiffi b p d

tr+ ffiffiffi b p

where tris given by Eq.(43)or Eq.(55)in which R, D deﬁned by Eq.(38) The formula(56)contains only theﬁrst 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:

θ = γ; a = 0; b = 1 = γ2

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

a0=− 1−2γð Þ2

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

R = 2 27−90γ + 99γ2

−32γ3

= 27; D = 4 1−γð Þ2

11−62γ + 107γ2

−64γ3

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

xr= 4 1ð −γÞð2−43γ +

3

q

+

R−pffiffiffiffiD

3

q

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= 1

4 1ð −γÞ 2−

4

3γ +p3 ffiffiffiffiffiffiffiffiffiffiffiVð Þγ

+ 3 + 4ð γ−3Þ2

9 ffiffiffiffiffiffiffiffiffiffi

Vð Þγ

3

p

#

"

ð61Þ where:

Vð Þ =γ 272 27−90γ + 99γ2

−32γ3

+ 2

3 ffiffiffi 3

p 1−γð Þ 11−62γ + 107γ2

−64γ3

ð62Þ

iv) From Eq.(55)and theproof 3i) of Proposition 5, it is obvious that, in the case Db0, xrcan also be calculated by a real expression, namely:

xr= 1−t2

r

θ−t2; tr=−13a2+ 2qcosθ; cos3θ = R

Trang 9

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

cr=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

μ 1−t2 r

ρ γ−t2 r

v u

; tr= ð3−4γÞ

3 +

2

3cosθ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið4γ−3Þ2+ 3

q

ð64Þ

cos3θ = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi27R

4γ−3

ð Þ2+ 3

where cris 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 Caseα12+γ*≠0, γ*= 0 (Proposition 6)

Whenγ*= 0, then a = 1, and Eq.(32)is equivalent to the following equation in the interval (1, +∞):

φ1ð Þ≡t pffiffiffib

θ−d

ð Þt2

+ðθ−1Þt +pffiffiffib

Proposition 6 i) If:

θ−d N 0; 1 +pffiffiffib

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

tr= ð1−θÞ +qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1−θÞ2+ 4bθ 1−dð Þ θ−dð Þ

2 ffiffiffi b p θ−d

In this case, the Rayleigh wave velocity xris given by Eqs.(33)2and(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

Remark 4 Using the facts:θ−1b0; pffiffiffib

θ d−1ð Þb0, it is not difﬁcult to verify the following: a) If θ−dN0, then Eq.(66)has two different roots t1, t2and t1b0bt2 b) Ifθ−db0, then the roots t1, t2of Eq.(66), if exist, must satisfy t1≤t2b0 c) If θ−d=0, Eq.(66)

has only one root, namely: t1=−pffiffiffib

θ b 0

Proof

i) Suppose Eq.(67)holds From Eqs.(66), (67)and 0bθb1 we havepffiffiffib

θ−d

ð Þφ1ð Þb0 This inequality ensures that Eq.1 (66)has two different roots t1, t2: t1b1bt2, i.e Eq.(66)has a unique root, namely t2, in the interval (1, +∞) Since t2is the bigger root of

Eq.(66), it is thus given by Eq.(68)

ii) It is clear that, in order to verify the conclusion ii), we have to examine only the four following cases: (1)θ−db0; (2) θ−d=0; (3) 1 + ffiffiffi

b

p

db 0 and θ−d N 0; (4) 1 +pffiffiffib

d = 0 andθ−d N 0

+ From b), c) ofRemark 4, it is clear that ii) is true for the cases (1) and (2)

+ It is easy to see that 1 + ffiffiffi

b

p

db 0 (1 +pffiffiffib

d = 0) is equivalent toφ1(1)N0 (φ1(1) = 0) By these facts, the validity of ii) for the cases (3) and (4) are deduced from a) ofRemark 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 Caseα12+γ*= 0 (Proposition7)

It is not difﬁcult to verify that:

Proposition 7 Letα12+γ*= 0 Then Eq.(28)has a unique solution satisfying: 0bρ0c2bγ1if:

α ≠ 0 and γγ α α − α4

Trang 10

otherwise, Eq.(28)has no root satisfying: 0bρ0c2bγ1 In the case that Eq.(69)is satisﬁed, the (unique) solution of the Eq.(28)

satisfying: 0bρ0c2bγ1is given by:

ρ0c2=α11+γ1

2 −12 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðα11−γ1Þ2+ 4α4

12= α22γ2

q

ð70Þ

It is readily to see that Theorem1is deduced fromPropositions 3–7

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 ffiffiffi

b p

= 1 + pffiffiffib (using Eq.(30)), and for the caseγ1Nα11, the velocity of Rayleigh waves xr=ρ0c2/α11is also determined by Theorem1in whichα11,α22,

α12are replaced respectively byγ1,γ2,γ*, and inversely

4 An example: solid and foam rubbers

As an example, we consider a half-space X2b0 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

2I−2 + 1− J2ð−1 Þ = −1 ð71Þ where:

I =λ2

+λ2

; J = λ1λ2; λ2=λ1−1; 0 bb 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:

α11=μλ2

1 + 2ð = −1Þλ2 ð−2 Þ

1

; α22= 2ð μ =Þλ2 ð−1 Þ

1

α12= 1ð −Þα22; γ1=μλ2

; γ2=μλ2 ð−1 Þ

Using Eqs.(29), (35) and (73)yields:

a0=− 2

ffiffiffi 2 p

1−

ð Þ2

+ 2−ð Þλ2 ð−2 Þ

1

h i3 = 2; a1=− 2

+ 2−ð Þλ2 ð−2 Þ

1

a2=

ffiffiffi 2 p

2 −3

ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

+ 2−ð Þλ2 ð−2 Þ

1

+ 2−ð Þλ2 ð−2 Þ

1

:

ð74Þ

From Eqs.(11)2,(11)4,(73)3,(73)6and the assumption 0bb1, it is clear that α12N0, γ*N0 By Theorem1, the dimensionless squared velocity xrof Rayleigh waves is given by Eqs.(33)2and(37), in which q2, R, D are determined by Eq.(44), and a0, a1, a2and

θ are calculated by Eq.(74)

When tends to zero, it follows from Eq.(74):

a0=−λ6

; a1=−λ4

; a2=−3λ2

By using Eqs.(33)2,(37), (44) and (75)we have:

xr= 1− 1

m2λ4; m = 1 +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 +2 3

ffiffiffiffiffiffi 11 3

r

3

s

+

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2−23

ffiffiffiffiffiffi 11 3

r

3

s

This is the formula of the dimensionless Rayleigh wave speed for the case→0, i.e the case of incompressible material, because

→0 leads to λ2=λ1 − 1, or equivalently J = 1 It follows immediately from Eq.(76)thatλ1must be bigger than 1=pffiffiffiffiffim

≈ 0:5437 in order to ensure that 0bxrb1 This means that for the values of λ1belong to the interval 0 ; 1 =pffiffiffiffiffim

, the Rayleigh wave does not exist On view of Eqs.(73)4,(76)and xr=ρc2

/γ1we have a different form of Eq.(76), namely:

c2

c2 =λ2

1− 1

m2λ2; c2

2= μ

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