DSpace at VNU: On formulas for the velocity of Rayleigh waves in prestrained incompressible elastic solids

Pham Chi Vinh1Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnam e-mail: pcvinh@vnu.edu.vn On Formulas fo

Pham Chi Vinh1

Faculty of Mathematics, Mechanics and

Informatics, Hanoi University of Science,

334, Nguyen Trai Street, Thanh Xuan,

Hanoi, Vietnam e-mail: pcvinh@vnu.edu.vn

On Formulas for the Velocity of Rayleigh Waves in Prestrained Incompressible Elastic Solids

In the present paper, formulas for the velocity of Rayleigh waves in incompressible isotropic solids subject to a general pure homogeneous prestrain are derived using the theory of cubic equation They have simple algebraic form and hold for a general strain-energy function The formulas are concretized for some specific forms of strain-strain-energy function They then become totally explicit in terms of parameters characterizing the material and the prestrains These formulas recover the (exact) value of the dimension-less speed of Rayleigh wave in incompressible isotropic elastic materials (without pre-strain) Interestingly that, for the case of hydrostatic stress, the formula for the Rayleigh wave velocity does not depend on the type of strain-energy function.

关DOI: 10.1115/1.3197139兴

Keywords: Rayleigh waves, Rayleigh wave velocity, prestrains, prestresses, incompressible

1 Introduction

Elastic surface waves in isotropic elastic solids, discovered by

Lord Rayleigh 关1兴 more than 120 years ago, have been studied

extensively and exploited in a wide range of applications in

seis-mology, acoustics, geophysics, telecommunication 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 on modern life and

many things that we take for granted today, stretching from

mo-bile phones through to the study of earthquakes, as stressed by

Adams et al.关2兴

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 Furthermore, it also involves Green’s

function for many elastodynamic problems for a half-space,

ex-plicit formulas for the Rayleigh wave speed are clearly of

practi-cal, as well as theoretical interest

In 1995, a first formula for the Rayleigh wave speed in

com-pressible isotropic elastic solids have been obtained by Rahman

and Barber关3兴, but for a limited range of values of the parameter

⑀=␮/共␭+2␮兲, where ␭ and ␮ are the usual Lame constants, by

using the theory of cubic equations Employing Riemann problem

theory Nkemzi关4兴 derived a formula for the velocity of Rayleigh

waves expressed as a continuous function of⑀ for any range of

values It is rather cumbersome关5兴, and the final result, as printed

in his paper, is incorrect关6兴 Malischewsky 关6兴 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 andMATHEMATICA It is expressed as

a continuous function of⑀ In Malischewsky’s paper 关6兴 it is not

shown, however, how Cardan’s formula together with the

trigono-metric formulas for the roots of the cubic equation are used with

MATHEMATICAto obtain the formula A detailed derivation of this

formula was given by Pham and Ogden关7兴 together with an

al-ternative formula For nonisotropic materials, for some special cases of compressible monoclinic materials with symmetry plane, formulas for the Rayleigh wave speed have been found by Ting 关8兴 and Destrade 关5兴 as the roots of quadratic equations, while for incompressible orthotropic materials an explicit formula has been given by Ogden and Pham关9兴 based on the theory of cubic equa-tions Furthermore, in recent papers 关10,11兴 Pham and Ogden have obtained explicit formulas for the Rayleigh wave speed in compressible orthotropic elastic solids

Nowaday prestressed materials have been widely used Nonde-structive evaluation of prestresses of structures before and during loading共in the course of use兲 becomes necessary and important, and the Rayleigh wave is a convenient tool for this task, see, for example, Refs 关12–15兴 In these studies 共also in Refs 关16,17兴兲, for evaluating prestresses by the Rayleigh wave, the authors have established the共approximate兲 formulas for the relative variation in the Rayleigh wave velocity关12,15兴 or its variation 关16,17兴兲 They are linear in terms of the prestrains共or prestresses兲, thus, they are very convenient to 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 pre-strains are not small

The main purpose of this paper is to find共exact兲 formulas for the velocity of Rayleigh waves in incompressible isotropic elastic materials subject to a general pure homogeneous prestrain by us-ing the theory of cubic equation Since they are valid for any range of prestrain, they will provide a powerful tool for the non-destructive evaluation of prestresses of structures

The paper is organized as follows The derivation of the secular equation of Rayleigh waves in a half-space of incompressible iso-tropic material subject to a generally pure homogeneous prestrain

is presented briefly in Sec 2 The formulas for the Rayleigh wave velocity are derived in Sec 3 In this section the necessary and sufficient conditions for the unique existence of the dimensionless

Rayleigh wave speed x rare also established In Sec 4, concreti-zation of formulas is carried out for a number of particular strain-energy functions, and the obtained formulas are then totally ex-plicit with respect to the parameters characterizing the material and the prestrains It is noted that, for the case of hydrostatic stress, the formula for Rayleigh wave velocity does not depend on the type of strain-energy function

1

Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the

J OURNAL OF A PPLIED M ECHANICS Manuscript received February 9, 2009; final

manu-script received June 15, 2009; published online December 10, 2009 Review

con-ducted by Professor Sridhar Krishnaswamy.

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

incompressibility, and then present briefly the derivation of the

secular equation of Rayleigh waves in prestrained elastic solids

For more details, the reader is referred to the paper by Dowaikh

and Ogden关18兴

We consider an unstressed body corresponding to the half-space

X2ⱕ0 and we suppose that the deformed configuration is obtained

by application of a pure homogeneous strain of the form

x1=␭1X1, x2=␭2X2, x3=␭3X3, ␭i= const, i = 1,2,3

共1兲 where␭i ⬎0, i=1,2,3 are the principal stretches of the

deforma-tion In its deformed configuration the body, therefore, occupies

the region x2⬍0 with the boundary x2= 0

We consider a plane motion in the共x1, x2兲-plane with

displace-ment components u1, u2, and u3such that

u i = u i 共x1,x2,t 兲, i = 1,2, u3⬅ 0 共2兲

where t is the time Then in the absence of body forces the

equa-tions governing infinitesimal motion, expressed in terms of

dis-placement components u i, are

B1111u1,11+共B1122+ B2112兲u2,21+ B2121u1,22− p,1ⴱ=␳u¨1

共B1221+ B2211兲u1,12+ B1212u2,11+ B2222u2,22− p,2ⴱ=␳u¨2 共3兲

where pⴱis a time-dependent pressure increment,␳ is mass

den-sity of the material, a superposed dot signifies differentiation with

respect to t, commas indicate differentiation with respect to spatial

variables x i , B ijkl is a component of the fourth order elasticity

tensor defined as follows:

B iijj=␭i␭j

⳵2W

⳵␭i⳵ ␭j

共4兲

B ijij=冦 冉␭i

⳵W

⳵␭i

−␭j

⳵W

⳵␭j冊 ␭i2

␭i

2, 共i ⫽ j,␭ i⫽ ␭j兲 1

2冉B iiii − B iijj+␭i

⳵W

⳵␭i冊 共i ⫽ j,␭ i=␭j兲 冧 共5兲

B ijji = B jiij = B ijij−␭i

⳵W

⳵␭i

for i , j 苸1,2,3, W=W共␭1,␭2,␭3兲 共noting that ␭1␭2␭3= 1兲 is the

strain-energy function per unit volume, all other components

be-ing zero In the stress-free configuration, Eqs.共4兲–共6兲 reduce to

B iiii = B ijij=␮共i ⫽ j兲, B iijj = B ijji= 0共i ⫽ j兲 共7兲 where␮ is the shear modulus of the material in that configuration

Equation of motion 共3兲 are taken together with the boundary

conditions of zero incremental traction, which are expressed as

B2121u1,2+共B2121−␴2兲u2,1= 0 on x2= 0

共B1122− B2222− p 兲u1,1− pⴱ= 0 on x2= 0 共8兲

where p denotes a static pressure in the prestressed equilibrium

state,␴i 共i=1,2,3兲 are the principal Cauchy stresses given by

␴i=␭i

⳵W

⳵␭i

For an incompressible material, we have

From Eq.共10兲 we deduce the existence of a function␺ of x1, x2,

and t such that

Elimination of pⴱfrom Eq.共3兲 and use of Eq 共11兲 then yield an equation for␺ having the form

where

␣ = B1212, ␥ = B2121, 2␤ = B1111+ B2222− 2B1122− 2B1221

共13兲

It is noted from the strong-ellipticity condition of system共3兲 that

␣, ␤, and ␥ are required to satisfy the inequalities

Differentiation of the second of Eq.共8兲 with respect to x1followed

by use of Eqs.共3兲, 共9兲, and 共11兲 then allows Eq 共8兲 to be put in the form

␥共␺,22−␺,11兲 +␴2␺,11= 0 on x2= 0 共2␤ + ␥ − ␴2兲␺,112+␥␺,222−␳␺¨,2= 0 on x2= 0 共15兲

We now consider a wave propagating in the x1-direction For this

to be a surface wave, the displacement u1and u2and, hence,␺

must decay to zero as x2→−⬁ We therefore take␺ to have the form

␺ = 共Ae ks1x2+ Be ks2x2兲exp共kx1−␻t兲 共16兲

where A and B are constants, ␻ is the wave frequency, k is the wavenumber, c is the wave speed, and s1and s2are roots of the equation

␥s4−共2␤ − ␳c2兲s2+␣ − ␳c2= 0 共17兲 From Eq.共17兲 we have

s1+ s2=共2␤ − ␳c2兲/␥, s1s2=共␣ − ␳c2兲/␥ 共18兲 For decay of␺ as x2→−⬁,s1, s2are required to have positive real

parts The roots s1and s2of the quadratic Eq.共17兲 are either both real共and, if so, both positive because of positive real parts of s1

and s2兲 or they are a complex conjugate pair In either case: s1

2

s22

⬎0 and so, by Eq 共18兲 and␥⬎0

Substitution of Eq.共16兲 into the boundary condition 共15兲 yields

共␥s1+␥ − ␴2兲A + 共␥s2+␥ − ␴2兲B = 0

共2␤ + ␥ − ␴2−␳c2−␥s1兲s1A +共2␤ + ␥ − ␴2−␳c2−␥s2兲s2B = 0

共20兲 For nontrivial solution of Eq.共20兲 for A and B, the determinant of

coefficients must vanish After some algebra and after using Eq

共18兲, removal of a factor s1− s2leads to

␥共␣ − ␳c2兲 + 共2␤ + 2␥ − 2␴2−␳c2兲关␥共␣ − ␳c2兲兴1/2=␥ⴱ2 共21兲 where ␥ⴱ=␥−␴2 It is noted that vanishing of the factor s1− s2

yields a trivial solution Equation 共21兲 is the secular equation,

which determines the speed c of propagation of surface共Rayleigh兲 waves of the type considered It follows from Eq.共19兲 that the

Rayleigh wave speed c has to satisfy the inequalities

0⬍ c2⬍ c2

3 Formulas for the Rayleigh Wave Velocity in Pre-strained Incompressible Solids

In order to proceed it is convenient to introduce three dimen-sionless parameters defined as follows:

␦1=␥/␣, ␦2=␤/␣, ␦3=␥ⴱ/␣ 共23兲

It is noted from Eqs.共14兲 and 共23兲 that

␦1⬎ 0 共24兲

We also define the variable x by

and in terms of the new variable x, Eq.共21兲 is of the form

␦1共1 − x兲 +冑␦1共2␦2+ 2␦3− x兲冑1 − x =␦3 共26兲

It follows from Eqs.共22兲 and 共25兲

On introducing the variable t given by

Eq.共26兲 becomes

F 共t兲 ⬅ t3+冑␦1t2+共2␦2+ 2␦3− 1兲t − ␦3

2

It follows from Eqs.共27兲 and 共28兲

It is noted that Eq.共28兲 is a 1-1 mapping from 共0,1兲 to itself From

Eq.共29兲 it follows

F⬘共t兲 = 3t2+ 2冑␦1t + 2␦2+ 2␦3− 1 共31兲

It is clear from Eqs.共24兲 and 共31兲 that if the equation F⬘共t兲=0 has

two distinct real roots, denoted by tmax, tmin共tmax⬍tmin兲, then

tmax+ tmin= − 2冑␦1/3 ⬍ 0 共32兲 Since the cubic equation共29兲 will be equivalently reduced to a

quadratic one, in the interval共0,1兲 when␦3= 0, we examine

sepa-rately the cases:␦3= 0 and␦3⫽0 By x rwe denote the solution of

Eq.共26兲 satisfying Eq 共27兲, and call it the dimensionless Rayleigh

wave velocity

For Case 1␦3= 0, we have the following proposition

PROPOSITION1

共i兲 Suppose ␦3= 0, then Eq (29) has a unique root in the

interval (0,1) if and only if

−冑␦1⬍ 2␦2⬍ 1 共33兲

共ii兲 Let␦3= 0 and Eq.共33兲 holds, then the dimensionless

Ray-leigh wave velocity is defined by following formula:

x r=关4 − 共冑␦1− 8␦2+ 4 −冑␦1兲2兴/4 共34兲

Proof.

共i兲 Let␦3= 0, then Eq.共29兲, in the interval 共0,1兲, is equivalent

to

␾共t兲 ⬅ t2+冑␦1t + 2␦2− 1 = 0 共35兲

Note that the coefficient of t2in the expression for␾共t兲

is a positive number

• “⇐:” suppose Eq 共33兲 holds, then we have

It follows from the first part of Eq.共36兲 that Eq 共35兲 has

two distinct real roots t1and t2and

By the second part of Eq.共36兲 we deduce

From Eqs.共37兲 and 共38兲 we conclude that Eq 共35兲, thus,

Eq.共29兲 has a unique solution in the interval 共0,1兲

• “⇒:” suppose that Eq 共29兲, thus, Eq 共35兲 has a unique

solution, namely, t2in the interval共0,1兲, then Eq 共35兲 has

two distinct real roots t1, t2, otherwise

2t2= −冑␦1⬍ 0 共39兲

But this is impossible because t2⬎0 Since two 共real兲 roots of Eq.共35兲 are related by

we have共noting that t2⬎0兲

From Eq.共41兲 and noting that the coefficient of t2in the expression for␾共t兲 is a positive number, it follows

From Eq.共42兲 we obtain Eq 共33兲 and the proof of 共i兲 is finished

共ii兲 Suppose␦3= 0 and Eq.共33兲 holds According to 共i兲, in this case, Eq 共29兲 has only one root denoted by t r, in the interval共0,1兲 By the proof of 共i兲, t r = t2the bigger root of

Eq.共35兲, thus, it is defined by

t r=冑␦1− 8␦2+ 4 −冑␦1

From Eq.共28兲

and Eq.共34兲 is deduced from Eqs 共43兲 and 共44兲

Case 2.␦3⫽0

PROPOSITION2 Suppose␦3⫽0, then Eq (29) has a unique root

in the interval (0,1) if and only if

F共1兲 =冑␦1+ 2␦2+ 2␦3− ␦3

2

Proof Let␦3⫽0, it follows from Eq 共29兲 that F共0兲⬍0.

If d = 2␦2+ 2␦3− 1ⱖ0, then from Eq 共31兲 F⬘共t兲⬎0 for ∀t⬎0.

Thus Eq.共29兲 has a unique root in the interval 共0,1兲 if and only

F共1兲⬎0

If d ⬍0, then the equation F⬘共t兲=0 has two distinct 共real兲 roots

tmax, tmin, and tmax⬍0⬍tmin Since F 共0兲⬍0 and F共t兲 is strictly

decreasingly monotonous in 共tmax, tmin兲, it follows that F共t兲

⬍0∀t苸共0,tmin兴, i.e., the equation F共t兲=0 has no root in the

in-terval共0,tmin兴 Since F共t兲 is strictly increasingly monotonous in

the interval共tmin, +⬁兲, it is strictly increasingly monotonous in the interval 共tmin, 1兲 This and F共tmin兲⬍0 yield that Eq 共29兲 has a unique solution in the interval 共0,1兲 if and only F共1兲⬎0 The

proof is completed

Remark 1.

共i兲 Inequality 共45兲 is equivalent to 共6.9兲 in Ref 关18兴, namely

共␣ − ␥兲␥ + 2冑␣␥ + 2␴2共␥ −冑␣␥兲 − ␴2⬎ 0 共46兲 and it gives

␥ −冑␣␥ −冑2冑␣␥共␤ +冑␣␥兲 ⬍ ␴2⬍␥ −冑␣␥

+冑2冑␣␥共␤ +冑␣␥兲

共47兲 共ii兲 Inequality 共33兲 is equivalent to 共6.17兲 in Ref 关18兴 without the left-hand equality

From the proof of the Proposition 2, we have immediately the following proposition

PROPOSITION 3 Suppose␦3⫽0 and F共1兲⬎0 If Eq (29) has two or three distinct real roots, then the root corresponding to the Rayleigh wave, say t r , is the largest root.

By introducing the notations

a0= − ␦3

2

冑␦1, a1= 2␦2+ 2␦3− 1, a2=冑␦1 共48兲

Eq.共29兲 becomes

F 共t兲 ⬅ t3+ a2t2+ a1t + a0= 0 共49兲

In terms of the variable z given by

Eq.共49兲 has the form

where

q2=共a2

2− 3a1兲/9, r = 共2a2

3− 9a1a2+ 27a0兲/27 共52兲

It should be noted that here q2can be negative

Our task is now to find the real solution z rof Eq.共51兲, that is

related to t r by the relation共50兲 As t ris the largest root of Eq

共49兲, z ris the largest one in Eq.共51兲 in the case that it has two or

three distinct real roots By theory of cubic equation, three roots

of Eq.共51兲 are given by the Cardan’s formula as follows 共see Ref

关19兴兲:

z1= S + T

z2= −12共S + T兲 +1

2i冑3共S − T兲

z3= −12共S + T兲 −1

2i冑3共S − T兲 共53兲

where i2= −1 and

S =冑3

R +冑D, T =冑3

R −冑D

D = R2+ Q3, R = −12r, Q = − q2 共54兲

Remark 2 In relation to these formulas we emphasize two

points:

共i兲 the cubic root of a real negative number is taken as the

negative real root

共ii兲 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 = Sⴱ,

where Sⴱis the complex conjugate value of S.

Remark 3 The nature of three roots of Eq.共51兲 depends on the

sign of its discriminant D, in particular: If D⬎0, then Eq 共51兲 has

one real root and two complex conjugate roots; if D = 0, the

equa-tion 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.共51兲

z ris given by

z r=冑3

R +冑D + q

2

冑3

in which each radical is understood as complex roots taking its

principle value, and

R = 共9a1a2− 27a0− 2a2兲/54

D = 共4a0a2− a1a2− 18a0a1a2+ 27a0+ 4a1兲/108 共56兲

where a i , i = 0 , 1 , 2 expressed in terms of the dimensionless

param-eters␦i , i = 1 , 2 , 3 by Eq.共48兲

First, we note that one can obtain Eq.共56兲 by substituting Eq

共52兲 into Eq 共54兲 Now we examine the distinct cases dependent

on the values of q2in order to prove Eq.共55兲

Case 1 q2⬍0

If q2⬍0⇒Q⬎0⇒D=R2+ Q3⬎0 This ensures that

共i兲 冑D ⬎兩R兩⇒冑D + R⬎0

共ii兲 Eq 共51兲, by the Remark 3, has a unique real root, so it is

z ris given by the first of Eq.共53兲, in particular

z r=冑3

R +冑D +冑3

here the radicals are understood as real ones Since

冑3

R −冑D 冑3

R +冑D =冑3

R2− D =冑3

共− Q兲3= − Q = q2

共58兲

we have

冑3

R −冑D = q

2

冑3

In view of Eqs.共57兲 and 共59兲, R+冑D⬎0 and the fact that for a positive real number, its real cubic root and complex cubic root taking its principal value are the same, the validity of Eq.共55兲 is clear

Case 2 q2= 0

When q2= 0, F⬘共t兲ⱖ0∀t苸共−⬁,+⬁兲, so function F共t兲 is strictly

increasingly monotonous 共−⬁,+⬁兲 Since a2=冑␦1⬎0⇒−a2/3

⬍0⇒r=F共−a2/3兲⬍F共0兲ⱕ0⇒r⬍0⇒R⬎0 In view of q2= 0,

Eq 共51兲 has a unique real 冑3

2R, so z r=冑3

2R In other hand, q2

= 0⇒Q=0⇒D=R2⇒R=+冑D Using this and q2= 0, from Eq

共55兲 we have: z r=冑32R Thus, formula共55兲 is valid for this case

Case 3 q2⬎0

We recall that in this case function F 共t兲 attains maximum and minimum values at tmax and tmin共tmax⬍tmin兲, and they are sub-jected to Eq.共32兲

共i兲 If D⬎0, then Eq 共51兲, according to the Remark 3, has a unique real root, so it is z rgiven by Eq.共57兲 in which the radicals are understood as real ones The use of Eq.共32兲

and the fact r = F 共−a2/3兲, it is not difficult to prove that

r ⬍0 or equivalently R⬎0 This leads to: R+冑D⬎0 In view of this inequality and Eq 共59兲, formula 共57兲 coin-cides with Eq.共55兲 This means formula 共55兲 is true

共ii兲 If D=0, analogously as above, it is not difficult to observe that r ⬍0, or equivalently, R⬎0 When D=0 we have

R2= −Q3=兩q兩6⇒R=兩q兩3⇒r=−2R=−2兩q兩3, so Eq.共51兲 be-comes

z3− 3兩q兩z2− 2兩q兩3= 0 共60兲

whose roots are z1= 2兩q兩 and z2= −兩q兩 共double root兲 This yields z r= 2兩q兩 according to the Proposition 3 From Eq 共55兲 and taking into account q2⬎0, D=0, it follows z r

= 2兩q兩 This shows the validity of Eq 共55兲.

共iii兲 If D⬍0, then Eq 共51兲 has three distinct real roots, and according to Proposition 3, z ris the largest root By argu-ments presented in Ref.关10兴 共p 255兲 one can show that, in

this case, the largest root z rof Eq.共51兲 is given by

z r=冑3

R +冑D +冑3

within which each radical is understood as the complex root taking its principal value By␪共苸共0兲,␲兲 we denote

the phase angle of the complex number R + i冑−D It is not

difficult to verify that

冑3

R +冑D = 兩q兩e i␪, 冑R −冑D = 兩q兩e −i␪ 共62兲 where each radical is understood as the complex root taking its principal value It follows from Eq.共62兲 that

冑3

R −冑D = q

2

冑3

By substituting Eq 共63兲 into Eq 共61兲 we obtain Eq 共55兲, and the validity of Eq 共55兲 is proved

We are now in the position to state the following proposition.

PROPOSITION 4 Suppose␦3⫽0 and Eq (45) holds Then, the

dimensionless velocity x r of Rayleigh waves in prestrained

incom-pressible solids is given by

x r= 1 −冋 冑3

R +冑D + q

2

冑3

R +冑D

−1

3a2册2

共64兲

in which each radical is understood as complex roots taking its

principle value, q2, and D and R are given by Eqs (52) and (56).

Proof Formula共64兲 is deduced from Eq 共28兲, 共50兲, and 共55兲

Remark 4.

共i兲 According to the Propositions 1, 2, and 4, the

dimension-less velocity x rof the Rayleigh waves is defined by either

Eq 共34兲 or Eq 共64兲, depending on the values of ␦3 In particular, if␦3= 0, x ris calculated by Eq.共34兲, provided that Eq.共33兲 holds; if␦3⫽0, it is given by Eq 共64兲 pro-vided that Eq.共45兲 is valid It is stressed that the formulas 共34兲 and 共64兲 are valid for a general strain-energy function

W.

共ii兲 The dimensionless Rayleigh wave velocity x ris a

continu-ous function of two dimensionless parameters ␦1and ␦2 for the case␦3= 0共see Eq 共34兲兲, and of three dimension-less parameters␦1,␦2, and␦3for the case␦3⫽0 共see Eq

共64兲兲

共iii兲 When the half-space is unstressed, according to Eq 共7兲:

␣=␤=␥=␮, thus in view of Eq 共23兲 and ␴2= 0 it follows

Since␦3= 1⫽0, x r is given by Eq 共64兲, as remarked above From Eqs.共48兲, 共52兲, 共56兲, 共64兲, and 共65兲 we obtain the exact value of the dimensionless Rayleigh wave veloc-ity for the incompressible linear elastic solids 共without prestresses兲, namely

where

␩0=冉26

27+

2

3冑11

3冊1/3

−8

9冉26

27+

2

3冑11

3冊−1/3

−1 3 共67兲 The result, Eqs 共66兲 and 共67兲, was first obtained by Ogden and Pham关9兴 in 2004 By Eq 共67兲, the approxi-mate value of ␩0 is 0.2956 共see also Ref 关18兴兲, so the

approximate value of x r0is 0.9126 This agrees with the classical result for the incompressible linear elasticity共see, e.g., Ref 关20兴兲 An alternative expression for x r0 was found by Malischewsky关21兴 in 2000, namely

x r0=23共4 +冑3

− 17 + 3冑33 −冑3

17 + 3冑33兲 共68兲 which yields a simpler representation of␩0

2

␩0= 1 −23共4 +冑3

− 17 + 3冑33 −冑3

17 + 3冑33兲 共69兲

It is clear from Eqs 共34兲 and 共64兲 that the Rayleigh wave velocity depends on the type of the strain-energy

function W, in general Interestingly there is a special case

for which the formula for the Rayleigh wave velocity does

not depend on the type of W This is the case of

hydro-static stress 共see Ref 关18兴兲, when ␭1=␭2=␭3= 1 and ␴1

=␴2=␴3=␴ In this case, as indicated by Dowaikh and Ogden关18兴,␣=␤=␥=␮ thus␦1=␦2= 1 It is not difficult

to verify that in this case the dimensionless Rayleigh wave

velocity x ris defined by

x r= 1 −冋 冑3

R +冑D −2

9

共1 + 3␦3兲

冑3

R +冑D

−1

3册2 , − 1⬍␦3

where␦3= 1 −␴¯, ␴¯=␴/␮, and

R =共3␦3+ 1兲共9␦3− 3␦3+ 7兲/54, D = 共␦3+ 1兲2共27␦3

It is noted that x r does not define at ␦3= 0共␴¯=1兲 be-cause Eq 共33兲 is not valid for this case 共noting ␦2= 1兲

Figure 1 shows the dependence of x ron␦3in the interval

共⫺1,3兲 Note that x ris␨ in Ref 关18兴

Taking␦3= 1共␴¯=0兲 in Eqs 共70兲 and 共71兲, we again obtain the exact value of the dimensionless Rayleigh wave velocity for the incompressible linear elastic solids, which is defined by Eqs.共66兲 and共67兲

4 Formulas for Particular Strain-Energy Functions

In this section we concretize the formulas 共34兲 and 共64兲 for some specific strain-energy functions, which were considered in Ref.关18兴 For seeking simplicity, we confine ourself to the case of plane strain

neo-Hookean strain-energy function, we have共see Ref 关18兴兲

W =12␮共␭1

It is noted that since␭1␭2␭3= 1,␭1−2␭3−2=␭2 When the underlying deformation of the half-space corresponds to strain plane with

␭3= 1, we write␭1=␭, ␭3=␭−1, and

With the use of Eqs.共4兲–共6兲, 共13兲, and 共73兲 we have

␣ = ␮␭2, ␤ =1

2␮冉␭2+ 1

␭2冊, ␥ = ␮

Using Eqs.共23兲 and 共74兲 provides

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

δ3

xr

Fig 1 Dependence of dimensionless Rayleigh velocity x ron

␦3 « „−1,3… for the case of hydrostatic stress

␦1= 1

␭4, ␦2=1

2冉1 + 1

␭4冊, ␦3= 1

␭4−␴¯2

␭2, ␴¯2=␴2/␮

共75兲 Since 2␦2= 1 + 1/␭4⬎1 共noting that ␭⬎0兲, condition 共33兲 is not

satisfied, thus, x rdoes not define at␦3= 0共␴¯2= 1/␭2兲 For the

val-ues of␴¯2such that␴¯2⫽1/␭2共␦3⫽0兲, x ris expressed by Eq.共64兲

provided that Eq.共45兲 is valid

From Eqs.共48兲 and 共75兲 it follows

a0= −冉1

␭3−␴¯2

␭冊2 , a1= 3

␭4−2␴¯2

␭2, a2= 1

␭2 共76兲 Substituting Eq.共76兲 into Eqs 共52兲 and 共56兲, and after some

ma-nipulation, we have

q2= 2

9␭2冉3␴¯2− 4

␭2冊

R =冉27␴¯2

␭2 −72␴¯2

␭4 +52

␭6冊/54

D =冉176

␭12 −448␴¯2

␭10 +424␴¯2

␭8 −176␴¯2

␭6 +27␴¯2

␭4 冊/108 共77兲 Finally, in view of Eqs 共64兲, 共76兲, and 共77兲, the dimensionless

Rayleigh wave velocity x ris defined by the following formula:

x r= 1 −冋 冑3

R +冑D + 2

9␭2

共3␴¯2− 4/␭2兲

冑3

R +冑D

− 1

3␭2册2

共78兲

where R and D are given by Eq.共77兲 By Eqs 共45兲 and 共75兲 it

follows

␭−2−␭−1− 1 −␭ ⬍␴¯2⬍ ␭−2+␭−1− 1 +␭ 共79兲 Figure 2 shows dependence of the dimensionless Rayleigh

ve-locity x ron␭苸关0.72 1.5兴 for different given values of␴¯2for the

case␦3⫽0

Now we turn our attention to a special case in which␴¯2= 0 For

this case, it follows from Eq.共77兲 that

R = 26

27␭6, D = 44

and by Eq.共78兲 we obtain

x r= 1 − 1

␭4␩0, ␭ ⬎␩0 /2 共81兲

It is noted that this case was numerically examined by Dowaikh and Ogden关18兴 Here the explicit expressions for the

dimension-less Rayleigh wave velocity x r is obtained Figure 3 shows the

plot of x ras a function of␭, defined by Eq 共81兲

strain-energy function is of the form共see Ref 关18兴兲

W = 2␮共␭1+␭1−1␭3−1+␭3− 3兲 共82兲

In the plane strain␭3= 1, so that

Here we write␭1=␭

From Eqs.共4兲–共6兲, 共13兲, and 共83兲 we have

␣ =2␮␭3

␭2+ 1, ␤ = 2␮␭

␭2+ 1, ␥ = 2␮

␭共␭2+ 1兲 共84兲 Using Eqs.共23兲 and 共84兲 yields

␦1= 1

␭4, ␦2= 1

␭2, ␦3= 1

␭4−共␭2+ 1兲␴¯2

共a兲 If␦3⫽0 and F共1兲⬎0, then x ris given by Eq.共64兲 Using Eqs.共48兲 and 共85兲 gives

a0= −冉1

␭3−共␭2+ 1兲␴¯2

2␭2 冊2

, a1= 2

␭4+ 2

␭2−共␭2+ 1兲␴¯2

␭3

− 1, a2= 1

Substituting Eq.共86兲 into Eqs 共52兲 and 共56兲, and after some manipulation we have

q2=1

3冉共␭2+ 1兲␴¯2

␭3 + 1 − 2

␭2− 5

3␭4冊 共87兲

R =冉43

␭6+18

␭4− 9

␭2−36共␭2+ 1兲␴¯2

␭5 +27共␭2+ 1兲2␴¯2

2

4␭4 冊/54

共88兲

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.72

(a) (b)

x r

λ

Fig 2 Dependence of the dimensionless Rayleigh velocity x r

on ␭«†0.72 1.5‡ for different given values of␴¯2 :␴¯2 = 0 „line a…,

␴¯2 = 0.2 „line b…,␴¯2 = −0.5 „line c…, and␴¯2 = −1 „line d… for the case

␦3Å 0; W =␮„␭2 +␭−2 − 2 …/2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

x r

λ

η01/2

Fig 3 Plot of x ron ␭«†␩0 1/2 3 ‡ for neo-Hookean strain-energy function and␴2 = 0

D =冋87

␭12+124

␭10+30

␭8−60

␭6−25

␭4+24

␭2− 4

−共␭2+ 1兲␴¯2

2␭3 冉296

␭8 +256

␭6 −32

␭4−96

␭2+ 24冊

+共␭2+ 1兲2␴¯2

4␭6 冉358

␭4 +132

␭2 − 66冊−22共␭2+ 1兲3␴¯2

␭9 +27共␭2+ 1兲4␴¯2

Finally, in view of Eqs.共64兲, 共86兲, and 共87兲, x ris ex-pressed by the following formula:

x r= 1 −冋 冑3

R +冑D +1

9

3共␭−1+␭−3兲␴¯2+ 3 − 6␭−2− 5␭−4

冑3

R +冑D

− 1

3␭2册2

共90兲

where R and D are given by Eqs.共88兲 and 共89兲 From Eqs.共45兲 and 共85兲 it is deduced that

2␭−1共1 − 3␭2兲/共␭2+ 1兲 ⬍␴¯2⬍ 2␭−1 共91兲 When␴¯2= 0, Eqs.共88兲 and 共89兲 simplify to

R =冉43

␭6+18

␭4− 9

␭2冊/54

D =冉87

␭12+124

␭10 +30

␭8−60

␭6−25

␭4+24

␭2− 4冊/108 共92兲

and x ris given by

x r= 1 −冋 冑3

R +冑D +1

9

共3 − 6␭−2− 5␭−4兲

冑3

R +冑D

− 1

3␭2册2

, ␭ ⬎冑13 共93兲

in which R and D are calculated by Eq.共92兲

Figure 4 shows dependence of the dimensionless Rayleigh

ve-locity x ron␭苸关0.72 1.5兴 for different given values of␴¯2for the

case␦3⫽0

共b兲 If ␦3= 0共␴¯2= 2/␭共␭2+ 1兲兲 and Eq 共33兲 holds, then x ris expressed by Eq.共34兲 In particular

x r= 1 − 1

4␭4共冑1 + 4␭2共␭2− 2兲 − 1兲2, ␭ ⬎冑2 共94兲

The dependence of x ron␭共冑2⬍␭⬍4兲 for this case is shown in Fig 5

strain-energy function is of the form共see Ref 关18兴兲

W = 8␮共␭1 1/2+␭1 −1/2␭3 −1/2+␭3 1/2− 3兲 共95兲

In the plane strain␭3= 1, W becomes

Here we write␭1=␭

From Eq.共4兲–共6兲, 共13兲, and 共96兲 we have

␣ =冑␭共␭ + 1兲共␭4␮␭42+ 1兲, ␤ =

␮共− ␭4+ 2␭3+ 2␭2+ 2␭ − 1兲

冑␭共␭ + 1兲共␭2+ 1兲 , ␥

= ␣

Using Eqs.共23兲 and 共97兲 yields

␦1= 1

␭4, ␦2=1

4冉− 1 +2

␭+

2

␭2+ 2

␭3− 1

␭4冊, ␦3= 1

␭4− e, e =␴2

␣ 共98兲

共a兲 If␦3⫽0 and F共1兲⬎0, then x ris given by Eq.共64兲 Using Eqs.共48兲 and 共98兲 gives

a0= −冉1

␭3−␭e冊2

, a1=1

␭+

1

␭2+ 1

␭3+ 3

2␭4− 2e

−3

2, a2=

1

Substituting Eq.共99兲 into Eqs 共52兲 and 共56兲, and after some manipulation, we have

q2=1

9冉6e +9

2−

3

␭−

3

␭2− 3

␭3− 7

2␭4冊

R =冉77

2␭6+ 9

␭5+ 9

␭4+ 9

␭3−9共8e + 3/2兲

␭2 + 27␭2e2冊/54

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(a) (b)

0.72

λ

x r

Fig 4 Dependence of dimensionless Rayleigh velocity x r on

␭«†0.72 1.5‡ for different given values of ␴¯2 : ␴¯2 = 0 „line a…,

␴¯2 = 0.2 „line b…,␴¯2 = −0.5 „line c…, and␴¯2 = −1 „line d… for the case

␦3Å 0; W = 2␮„␭+␭ −1 − 2…

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

λ

2 /µ

Fig 5 Plot of␨=␳c2 /␮as a function of␭„冑2 < ␭<4… for the case

␦3 = 0 and the Varga strain-energy function

D =␭4冋245

␭16 +168

␭15 +236

␭14 +320

␭13 −952e

␭12 +共28 − 416e兲

␭11

−共96 + 512e兲

␭10 −共252 + 608e兲

␭9 +共− 15 + 336e + 1300e2兲

␭8 +共− 20 + 96e + 264e2兲

␭7 +共36 + 192e + 264e2兲

␭6 +共108 + 288e + 264e2兲

␭5

−共54 + 216e + 396e2+ 704e3兲

␭4 + 108e4册/432 共100兲 Finally, in view of Eqs.共64兲 and 共100兲, x ris expressed

by the following formula:

x r= 1 −冋 冑3

R +冑D

+1 9

共6e + 9/2 − 3/␭ − 3/␭2− 3/␭3− 3.5/␭4兲

冑3

R +冑D

− 1

3␭2册2 共101兲

where R and D are given by Eq.共100兲 By Eqs 共47兲,

共97兲, and the fourth of Eq 共98兲 e is subjected to

1 −␭

␭4 −共1 + ␭兲

冑2␭3 冑−␭2+ 4␭ − 1 ⬍ e ⬍1 −␭

␭4 +共1 + ␭兲

冑2␭3 冑−␭2+ 4␭ − 1 共102兲

or the bounds on␴¯2=␴2/␮ are

4共1 − ␭兲

冑␭共␭2+ 1兲⫾

2冑2␭ 共␭2+ 1兲冑−␭2+ 4␭ − 1 共103兲 For␴2= 0,␭ is subject to

−␭3+ 3␭2+ 2␭ − 2 ⬎ 0 共104兲

Figure 6 shows dependence of x r on ␴¯2苸关−1 1兴 for different

given values of␭ for the case␦3⫽0

共b兲 When␦3= 0共e=1/␭4兲 and Eq 共33兲 holds, x ris expressed

by Eq.共34兲, in particular

x r= 1 − 1

4␭4共冑6␭4− 4␭3− 4␭2− 4␭ + 3 − 1兲2

共105兲

in which␭ must satisfy the following two inequalities:

− 1 + 2/␭ + 4/␭2+ 2/␭3− 1/␭4⬎ 0 ⬎ − 3 + 2/␭ + 2/␭2 + 2/␭3− 1/␭4

Plots of␨=␳c2/␮ and ␨s=␣/␮ as functions of ␭ for ␴2= 0 are shown in Fig 7

Finally, we note that formulas共78兲, 共90兲, and 共101兲 all lead to

x r= 1 −␩0 at ␭ = 1, ␴2= 0 共106兲 where␩0defined by Eq.共67兲 This means these formulas all re-cover the共exact兲 value of the dimensionless speed of the Rayleigh wave in incompressible isotropic elastic materials 共without pre-strain兲

5 Conclusions and Remarks

In this paper, formulas for the Rayleigh wave velocity in in-compressible isotropic solids subject to uniform initial deforma-tion are derived using the theory of cubic equadeforma-tion They have a simple algebraic form, valid for any range of prestrain and hold for a general strain-energy function The Rayleigh wave velocity

is expressed by two different formulas depending on that␦3= 0 or

␦3⫽0 These formulas are concretized for a number of forms of strain-energy function, and the obtained formulas express the Ray-leigh wave velocity as totally explicit continuous functions of the principle stretches of the deformation␭iand the stress␴2 For the case of hydrostatic stress, the Rayleigh wave velocity is expressed

by a simple formula that does not include the strain-energy func-tion

The obtained formulas will provide a good tool for the nonde-structive evaluation of prestresses of structures In relation to the use of these formulas, we emphasize the following points

共i兲 By x r 共ik兲 共i,k=1,2,3,i⫽k兲 we denote the velocity of the Rayleigh wave propagating in the x i-direction and

decay-ing in the x k -direction, then x r 共ik兲 is defined by formulas

similar to Eqs.共34兲 and 共64兲 For example, if␦3共32兲⫽0 共i.e.,

−10 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(d)

(a)

(b) (c)

x r

σ2

Fig 6 Dependence of dimensionless Rayleigh velocity x r on

␴¯2 « †−1 1‡ for different given values of ␭: ␭=1 „line a…, ␭=0.8

„line b…, ␭=1.5 „line c…, and ␭=1.7 „line d… for the case␦3 Å 0;

0 1 2 3 4 5

0.59

λ

ζ

ζs

ζ

3.4

Fig 7 Plots of␨=␳c2 /␮„solid line… and␨s=␣/␮„dashed line…

as functions of ␭ for␴2= 0 and m = 1 / 2 strain-energy function

␴2⫽␥共32兲= B

2323兲 then x r共32兲defined by formula共64兲, along with Eqs.共23兲, 共48兲, 共52兲, and 共56兲, in which

␣共32兲= B

and␥ⴱ共32兲=␥共32兲−␴2 Formula共107兲 is derived from Eq

共13兲 in which the index 1 of B ijklis replaced by 3 Note

that, for a given material x r 共ik兲 is a function of two of the

three principal stretches共because ␭1␭2␭3= 1兲 and␴k, i.e.,

x r 共ik兲 = f 共ik兲共␭1,␭2,␴k 兲 共i,k = 1,2,3,i ⫽ k兲 共108兲 共ii兲 If ␴2= 0 and x r共12兲, x

r

共32兲are known共by laser techniques, for example兲, then ␭1,␭2are determined from two following equations:

f共12兲共␭1,␭2,0兲 = x r共12兲, f共32兲共␭1,␭2,0兲 = x r共32兲 共109兲 and then␴1and␴3are calculated from

␴1=␭1

⳵W

⳵␭1

−␭2

⳵W

⳵␭2 , ␴3=␭3

⳵W

⳵␭3

−␭2

⳵W

⳵␭2 共110兲 which are originated from Eq.共9兲 Note that from Eq

共9兲 it follows

␴1−␴2=␭1

⳵W

⳵␭1

−␭2

⳵W

⳵␭2

␴2−␴3=␭2

⳵W

⳵␭2

−␭3

⳵W

⳵␭3

␴3−␴1=␭3

⳵W

⳵␭3

−␭1

⳵W

Also note that when ␴2= 0, ␦3共12兲⫽0, and ␦3共32兲⫽0,

therefore, x r共12兲and x

r

共32兲are defined by Eq.共64兲 A similar situation will be met when␴1= 0 or␴3= 0

When ␴k ⫽0, k=1,2,3, in order to determine ␭1,␭2, ␴1, ␴2,

and␴3, we have to use five equations, two of which come from

Eq.共111兲, for example

␭1

⳵W

⳵␭1

−␭2

⳵W

⳵␭2

=␴1−␴2, ␭3

⳵W

⳵␭3

−␭2

⳵W

⳵␭2

=␴3−␴2

共112兲

and the others are originated from the formulas of x r 共ik兲, for

in-stance

f共12兲共␭1,␭2,␴2兲 = x r共12兲, f共32兲共␭1,␭2,␴2兲 = x r共32兲, f共21兲共␭1,␭2,␴1兲

Here x r共12兲, x

r

共32兲, and x

r

共21兲 are known 共by measurement tech-niques兲

共iii兲 If from two equations, Eq 共112兲 and ␭1␭2␭3= 1, we can obtain analytical expressions ␭1=␭1共␴1,␴2,␴3兲 and ␭2

=␭2共␴1,␴2,␴3兲, then by introducing them into Eq 共108兲,

x r 共ik兲 is expressed as a function of the prestresses ␴k , k

= 1 , 2 , 3 However, such an analytical inversion of Eq 共112兲 and ␭1␭2␭3= 1 is not always possible共see also Ref 关22兴, p 150兲

References

关1兴 Rayleigh, L., 1885, “On Waves Propagating Along the Plane Surface of an

Elastic Solid,” Proc R Soc London, 17, pp 4–11.

关2兴 Adams, S D M., Craster, R V., Williams, D P., 2007, “Rayleigh Waves

Guided by Topography,” Proc R Soc London, Ser A, 463, pp 531–550.

关3兴 Rahman, M., and Barber, J R., 1995, “Exact Expression for the Roots of the

Secular Equation for Rayleigh Waves,” ASME J Appl Mech., 62, pp 250–

252.

关4兴 Nkemzi, D., 1997, “A New Formula for the Velocity of Rayleigh Waves,”

Wave Motion, 26, pp 199–205.

关5兴 Destrade, M., 2003, “Rayleigh Waves in Symmetry Planes of Crystals:

Ex-plicit Secular Equations and Some ExEx-plicit Wave Speeds,” Mech Mater., 35,

pp 931–939.

关6兴 Malischewsky, P G., 2000, “Comment to ‘A New Formula for Velocity of Rayleigh Waves’ by D Nkemzi 关Wave Motion 26 共1997兲 199–205兴,” Wave

Motion, 31, pp 93–96.

关7兴 Pham, C V., and Ogden, R W., 2004, “On Formulas for the Rayleigh Wave

Speed,” Wave Motion, 39, pp 191–197.

关8兴 Ting, T C T., 2002, “A Unified Formalism for Elastostatics or Steady State Motion of Compressible or Incompressible Anisotropic Elastic Materials,” Int.

J Solids Struct., 39, pp 5427–5445.

关9兴 Ogden, R W., and Pham, C V., 2004, “On Rayleigh Waves in Incompressible

Orthotropic Elastic Solids,” J Acoust Soc Am., 115共2兲, pp 530–533.

关10兴 Pham, C V., and Ogden, R W., 2004, “Formulas for the Rayleigh Wave Speed

in Orthotropic Elastic Solids,” Arch Mech., 56共3兲, pp 247–265.

关11兴 Pham, C V., and Ogden, R W., 2005, “On a General Formula for the Rayleigh

Wave Speed in Orthotropic Elastic Solids,” Meccanica, 40, pp 147–161.

关12兴 Hirao, M., Fukuoka, H., and Hori, K., 1981, “Acoustoelastic Effect of

Ray-leigh Surface Wave in Isotropic Material,” ASME J Appl Mech., 48, pp.

119–124.

关13兴 Delsanto, P P., and Clark, A V., 1987, “Rayleigh Wave Propagation in

De-formed Orthotropic Materials,” J Acoust Soc Am., 81共4兲, pp 952–960.

关14兴 Duquennoy, M., Ouaftouh, M., and Ourak, M., 1999, “Ultrasonic Evaluation

of Stresses in Orthotropic Materials Using Rayleigh Waves,” NDT & E Int.,

32, pp 189–199.

关15兴 Duquennoy, M., Devos, D., Ouaftouh, M., Lochegnies, D., and Roméro, E.,

2006, “Ultrasonic Evaluation of Residual Stresses in Flat Glass Tempering: Comparing Experimental Investigation and Numerical Modeling,” J Acoust.

Soc Am., 119共6兲, pp 3773–3781.

关16兴 Tanuma, K., and Man, C.-S., 2006, “Pertubation Formula for Phase Velocity of

Rayleigh Waves in Prestressed Anisotropic Media,” J Elast., 85, pp 21–37.

关17兴 Song, Y Q., and Fu, Y B., 2007, “A Note on Perturbation Formulae for the Surface-Wave Speed Due to Perturbations in Material Properties,” J Elast.,

88, pp 187–192.

关18兴 Dowaikh, M A., and Ogden, R W., 1990, “On Surface Waves and Deforma-tions in a Pre-Stressed Incompressible Elastic Solids,” IMA J Appl Math.,

44, pp 261–284.

关19兴 Cowles, W H., and Thompson, J E., 1947, Algebra, Van Nostrand, New York 关20兴 Ewing, W M., Jardetzky, W S., and Press, F., 1957, Elastic Waves in Layered Media, McGraw-Hill, New York.

关21兴 Malischewsky, P G., 2000, “Some Special Solutions of Rayleigh’s Equation

and the Reflections of Body Waves at a Free Surface,” Geofis Int., 39, pp.

155–160.

关22兴 Novozhilov, V V., 1961, Theory of Elasticity, Pergamon, Oxford.