DSpace at VNU: On Rayleigh waves in incompressible orthotropic elastic solids

Ogdena Department of Mathematics, University of Glasgow, Glasgow G12 8QW, United Kingdom Pham Chi Vinh Faculty of Mathematics, Mechanics and Informatics, Hanoi National University, 334 N

On Rayleigh waves in incompressible orthotropic elastic solids

Ray W Ogdena)

Department of Mathematics, University of Glasgow, Glasgow G12 8QW, United Kingdom

Pham Chi Vinh

Faculty of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnam

共Received 7 June 2003; accepted for publication 4 November 2003兲

In this paper the secular equation for the Rayleigh wave speed in an incompressible orthotropic elastic solid is obtained in a form that does not admit spurious solutions It is then shown that inequalities on the material constants that ensure positive definiteness of the strain–energy function guarantee existence and uniqueness of the Rayleigh wave speed Finally, an explicit formula for the

Rayleigh wave speed is obtained © 2004 Acoustical Society of America.

关DOI: 10.1121/1.1636464兴

I INTRODUCTION

Rayleigh共surface兲 waves were first studied by Rayleigh

共1885兲 for a compressible isotropic elastic solid The

exten-sion of surface wave analysis to anisotropic elastic materials

has been the subject of many studies; see, for example,

Stoneley 共1963兲; Chadwick and Smith 共1977兲; Royer and

Dieulesaint共1984兲; Mozhaev 共1995兲; Destrade 共2001a兲; Ting

共2002a, c兲, and references contained therein Some recent

work has focused on incompressible anisotropic elastic

sol-ids 共Nair and Sotiropoulos, 1997, 1999; Destrade, 2001b;

Destrade et al., 2002兲, while application to materials,

com-pressible or incomcom-pressible, subject to prestress has also

at-tracted considerable attention 共Dowaikh and Ogden, 1990,

1991; Chadwick, 1997, for example兲

Since Green’s functions for many elastodynamic

prob-lems for a half-space require the solution of the secular

equa-tion for Rayleigh waves, formulas for the Rayleigh wave

speed in various elastic media are clearly of practical as well

as theoretical interest A formula for the Rayleigh wave

speed in compressible isotropic solids was first obtained by

Rahman and Barber 共1995兲 for a limited range of values of

the parameter ⑀⬅␮/共␭⫹2␮兲, where ␭ and ␮ are the Lame´

constants For any range of values of ⑀a formula was

ob-tained by Nkemzi 共1997兲; see also Malischewsky 共2000兲

Recently, for some special cases of compressible monoclinic

materials with symmetry plane x3⫽0, formulas for the

Ray-leigh wave speed were found by Ting 共2002b兲 and Destrade

共2003兲 as the roots of quadratic equations

Rayleigh waves in incompressible orthotropic elastic

materials were examined recently by Destrade 共2001b兲

De-strade used the method of first integrals proposed by

Mozhaev 共1995兲 and found a form of the secular equation

He used this to prove that Rayleigh waves exist and are

unique in these materials for all values of the relevant

mate-rial constants However, the form of the secular equation

obtained by use of Mozhaev’s method necessarily admits

spurious solutions Thus, the analysis of Destrade requires

some modification The secular equation for Rayleigh waves

in incompressible orthotropic materials presented recently by

Destrade et al.共2002兲 also admits spurious solutions

The aim of the present paper is to obtain a formula for the Rayleigh wave speed in an incompressible orthotropic elastic material For this purpose a form of the secular equa-tion that does not admit spurious soluequa-tions is required

In Sec II the basic equations and notation are presented for describing motion in an incompressible orthotropic elas-tic material We consider a half-space whose boundary is a symmetry plane of the material Since the equations for time-harmonic waves propagating parallel to the boundary of this half-space decouple into a plane motion, in the plane defined

by the half-space normal and the direction of propagation, and a motion normal to that plane 共see, for example, De-strade, 2001b兲, it suffices to consider the plane strain case In Sec III the secular equation for Rayleigh waves is derived in the desired form and it is shown how it relates to the form given by Destrade共2001b兲 Existence and uniqueness results are presented in Sec IV In particular, it is shown that Ray-leigh waves exist and are unique in an incompressible ortho-tropic elastic solid, provided the inequalities

␥⬅c66⬎0, ␦⬅c11⫹c22⫺2c12⬎0 共1兲

are satisfied, where c11, c12, c22, and c66 are material con-stants associated with the considered plane, which, in

Carte-sian coordinates, is taken to be the (x1,x2) plane These inequalities are necessary and sufficient for the strain energy 共specialized to the considered plane兲 to be positive definite

It is also easy to show that they are necessary and sufficient for strong ellipticity to hold for the considered motion

An explicit formula for the Rayleigh speed is derived in Sec V by using the theory of cubic equations Corresponding results for the compressible theory will be discussed in a separate paper

II BASIC EQUATIONS

Let (x1,x2,x3) be Cartesian coordinates and consider an

orthotropic elastic material occupying the half-space x2⬍0,

with traction-free boundary x2⫽0, which is a plane of

sym-a 兲Electronic mail: rwo@maths.gla.ac.uk

metry for the orthotropy We consider a plane motion in the

(x1,x2) plane with displacement components (u1,u2,u3)

such that

u i ⫽u i 共x1,x2,t 兲, i⫽1,2, u3⬅0, 共2兲

where t is time The nonzero components ⑀i j of the

infini-tesimal strain tensor are given by

⑀i j⫽1共u i, j ⫹u j,i 兲, i, j⫽1,2, 共3兲

where a comma signifies partial differentiation with respect

to spatial variables

For an incompressible material, we have

from which we deduce the existence of a scalar function,

denoted ␺(x1,x2,t), such that

For the considered motion the relevant components of

the stress are given by, for example, Nair and Sotiropoulos

共1997兲 Thus,

␴11⫽⫺p⫹c11⑀11⫹c12⑀22,

␴22⫽⫺p⫹c12⑀11⫹c22⑀22, ␴12⫽2c66⑀12, 共6兲

where␴i j , i ⫽1, 2, are components of the stress tensor, c i j

are elastic constants of the material in standard notation, and

p ⫽p(x1,x2,t) is the hydrostatic pressure associated with the

incompressibility constraint Note that, in general, ␴33⫽0,

but we shall not need to use this component of stress

For the strain energy to be positive definite for the

con-sidered plane motion, the elastic constants c i jin Eq.共6兲 must

satisfy the inequalities given in Eq.共1兲

In the absence of body forces the relevant components

of the equation of motion are

␴11,1⫹␴12,2⫽␳u¨1, ␴12,1⫹␴22,2⫽␳u¨2, 共7兲

where dots denote partial differentiation with respect to t,

and␳is the mass density of the material

Use of Eqs.共3兲, 共5兲, and 共6兲 in Eq 共7兲 and elimination of

p by cross differentiation leads to an equation for␺, namely

␥␺,1111⫹2␤␺,1122⫹␥␺,2222⫽␳共␺¨

,11⫹␺¨

,22兲, 共8兲 where

and␥and␦are defined by Eq.共1兲

In terms of the stress components the traction-free

boundary conditions are written

Using Eqs.共3兲, 共5兲, 共6兲, and the first of 共7兲, Eq 共10兲 can be

expressed as conditions on␺ This requires differentiation of

␴22 with respect to x1 so as to facilitate elimination of the

term in p, as in Dowaikh and Ogden 共1990兲 for Rayleigh

waves on a prestrained half-space of incompressible

isotro-pic elastic material The resulting boundary conditions are

␥共␺,22⫺␺,11兲⫽0,

␥共␺,222⫺␺,112兲⫹␦␺,112⫺␳␺¨

,2⫽0 on x2⫽0 共11兲

We shall also require that

III RAYLEIGH WAVES: SECULAR EQUATION

We now consider harmonic waves propagating in the x1

direction, and we write␺in the form

␺共x1,x2,t兲⫽␾共y兲exp关ik共x1⫺ct兲兴, 共13兲

where k is the wave number, c is the wave speed, y ⫽kx2, and the function␾is to be determined

Substitution of Eq 共13兲 into Eq 共8兲 yields

␥␾⬙⬙⫺共2␤⫺␳c2兲␾⬙⫹共␥⫺␳c2兲␾⫽0, 共14兲 where, in Eq.共14兲 and the following, a prime on␾indicates

differentiation with respect to y.

In terms of ␾the boundary conditions共11兲 become

␾⬙共0兲⫹␾共0兲⫽0, ␥␾⵮共0兲⫹共␥⫺␦⫹␳c2兲␾⬘共0兲⫽0,

共15兲

in the first of which we have omitted the factor ␥ on the assumption that␥⫽0 From Eq 共12兲 we also require that

Thus, the problem is reduced to solving Eq 共14兲 with the boundary conditions 共15兲 and 共16兲 The general solution for

␾(y ) that satisfies the condition共16兲 is

where A and B are constants, while s1 and s2 are the solu-tions of the equation

␥s4⫺共2␤⫺␳c2兲s2⫹共␥⫺␳c2兲⫽0, 共18兲 with positive real parts

From Eq.共18兲 it follows that

s12⫹s2 2

⫽共2␤⫺␳c2兲/␥, s12s22⫽共␥⫺␳c2兲/␥ 共19兲

If the roots s12 and s22 of the quadratic Eq.共18兲 are real,

then they must be positive to ensure that s1 and s2 can have

a positive real part If they are complex then they are

conju-gate In either case the product s12s22 must be positive and hence a real共surface兲 wave speed c satisfies the inequalities

Note that the limiting wave speed such that ␳c2⫽␥ is the speed of a shear body wave, not a surface wave

Substituting Eq.共17兲 into the boundary conditions 共15兲,

we obtain the equations

共1⫹s1

2兲A⫹共1⫹s2

2兲B⫽0,

共21兲

关␥共s1

2⫹1兲⫹␳c2⫺␦兴s1A⫹关␥共s2

2⫹1兲⫹␳c2⫺␦兴s2B⫽0,

for A and B For nontrivial solution the determinant of

coef-ficients of the system共21兲 must vanish After removal of the

factor (s1⫺s2), this yields

␥共s1

2⫹s2

2⫹s1

2s22兲⫹共␦⫺␳c2兲s1s2⫹␥⫺␦⫹␳c2⫽0 共22兲 Use of Eq 共19兲 in Eq 共22兲 then leads to

共␦⫺␳c2兲冑1⫺␳c2/␥⫺␳c2⫽0, 共23兲

which is the required secular equation for the wave speed

through␳c2 Note that for this equation to have a real

solu-tion for c it is necessary, in addisolu-tion to Eq. 共20兲, that the

inequality

must hold Note that a solution of Eq 共22兲 with s1⫽s2 is

admissible, but it can be shown that this corresponds to a real

surface wave if and only if ␦⫽32␥/9 and␳c2⫽␦/4, a

solu-tion that is given by共23兲 for this very restricted combination

of material constants Of course, in this case, the solution

共17兲 requires modification

By rearranging Eq 共23兲 and squaring to eliminate the

square root, we obtain the secular equation derived by

De-strade共2001b兲, which, in the present notation, can be written

共␦⫺␳c2兲2共1⫺␳c2/␥兲⫽共␳c2兲2 共25兲

Destrade共2001b兲 used the method of first integrals proposed

by Mozhaev 共1995兲 and needed to impose the restriction

␳c2⫽␦, which in our derivation is automatically satisfied

through Eq 共24兲 Using this equation, Destrade concluded

that a unique Rayleigh wave exists in an incompressible

orthotropic elastic material for any values of␥ and␦

How-ever, Eq.共25兲 may have spurious solutions for ␳c2 that are

not solutions of Eq 共23兲, and it is therefore advisable to

avoid drawing conclusions on the basis of Eq.共25兲

IV EXISTENCE AND UNIQUENESS OF RAYLEIGH

WAVES

We now show that the inequalities␥⬎0 and␦⬎0 jointly

ensure the existence and uniqueness of a Rayleigh wave For

this purpose it is convenient to introduce the new variable

␩⫽冑1⫺␳c2/␥ so that the secular equation共23兲 may be

re-written as

f共␩兲⬅␩3⫹␩2⫹共␦/␥⫺1兲␩⫺1⫽0, 0⬍␩⬍1 共26兲

Then

which guarantees that Eq.共26兲 has at least one solution in the

interval共0,1兲

We also have

f⬘共␩兲⫽3␩2⫹2␩⫹␦/␥⫺1, f⬙共␩兲⬎0 共␩⬎0兲 共28兲

If␦⭓␥then it follows that f⬘(␩)⬎0 for␩⬎0 and hence f is

monotonic increasing for␩⬎0 In this case the solution for␩

is unique If, on the other hand, 0⬍␦⬍␥ then f⬘(0)⬍0

Thus, f has a maximum for ␩⬍0 and a minimum for␩⬎0

By the inequality in Eq.共28兲 f therefore decreases to a

mini-mum as␩ increases from 0, and thereafter increases

mono-tonically Hence, the solution is also unique in this case

We therefore conclude that in an incompressible

ortho-tropic elastic half-space there exists a unique Rayleigh wave

provided the material constants satisfy the conditions 共1兲,

which ensure that the strain–energy function is positive

defi-nite for the considered plane strain restriction We note in

passing that if␦⭐0 then it can be seen immediately that Eq

共23兲 has no real nonzero solution for c whatever the sign of

␥, although it is not physically meaningful to admit nonposi-tive values of these constants

V A FORMULA FOR THE WAVE SPEED

In this section we derive an explicit formula for the wave speed, given that ␥⬎0, ␦⬎0, by seeking the unique root,␩0say, of Eq.共26兲 in the interval 共0,1兲 The wave speed

c is then given by

␳c2⫽␥共1⫺␩0

We now show that the cubic Eq.共26兲 has only one real root, namely ␩0, the other two being complex

According to the theory of cubic equation 共see, for ex-ample, Cowles and Thompson, 1947 or Abramowitz and Ste-gun 1974兲, the nature of the three roots of the cubic

␩3⫹a2␩2⫹a1␩⫹a0⫽0, 共30兲

is determined by the sign of the discriminant D defined by

where R and Q are given in terms of the coefficients a0, a1,

a2 by

R⫽ 1

54共9a1a2⫺27a0⫺2a2

3兲, Q⫽1共3a1⫺a2

2兲 共32兲

If D⬎0, Eq 共30兲 has one real root and two complex

conju-gate roots If D⫽0, the equation has three real roots, at least

two of which are equal If D⬍0, Eq 共30兲 has three distinct

real roots In the first case (D⬎0) the single real root␩0 is given by Cardano’s formula 共Cowles and Thompson, 1947;

Abramowitz and Stegun, 1974兲 in the form

␩0⫽⫺1

3a2⫹共R⫹冑D兲1/3⫹共R⫺冑D兲1/3 共33兲 For the secular equation in the form Eq 共26兲, we have

and hence

where⌬⫽␦/␥ Using Eq.共35兲 in Eq 共31兲, it is easy to verify that

D⫽ 1

It is clear from Eq.共36兲 that D⬎0 provided ⌬⬎0 Thus, Eq.

共30兲 has only one real root, necessarily within the required range of values

Use of Eqs.共34兲, 共35兲, and 共36兲 in Eq 共33兲 leads to

␩0⫽1

3关⫺1⫹冑3

关9⌬⫹16⫹3冑3冑⌬共4⌬2⫺13⌬⫹32兲兴/2

⫹冑3 关9⌬⫹16⫺3冑3冑⌬共4⌬2⫺13⌬⫹32兲兴/2兴

共37兲 From Eqs 共29兲 and 共37兲 the speed c of the Rayleigh

wave is given by

9关⫺1

⫹冑3 关9⌬⫹16⫹3冑3冑⌬共4⌬2⫺13⌬⫹32兲兴/2

⫹冑3 关9⌬⫹16⫺3冑3冑⌬共4⌬2⫺13⌬⫹32兲兴/2兴2

共38兲 For an 共incompressible兲 isotropic material c11⫽c22,

c11⫺c12⫽2␮, and c66⫽␮, where ␮ is the classical shear

modulus, and hence, by Eq 共1兲, ⌬⫽4 In this case the

for-mula共38兲 specializes to

␳c2/␥⫽1⫺1

9关冑3

6冑33⫹26⫺冑3

6冑33⫺26⫺1兴2 共39兲 This is approximately 0.9126, which is the classical value for

an incompressible isotropic elastic solid 共see, for example,

Ewing et al., 1957兲

In Fig 1 a plot of ␳c2/␥ against ⌬共⬎0兲 based on Eq

共38兲 is shown in order to illustrate the dependence of the

wave speed on the ratio of material constants The wave

speed is very small for small ⌬ and increases rapidly as ⌬

increases, reaching its isotropic value for⌬⫽4 and then

ap-proaching an asymptotic value with␳c2/␥→1 as ⌬ becomes

very large The asymptotic limit corresponds to a wave speed

equal to the shear wave speed Note that␦may be interpreted

as a shear modulus of the material; indeed, in the isotropic

case␦⫽2␮, where␮is the Lame´ shear modulus Thus, the

limit ⌬→0 共which is not applicable for isotropic materials兲

corresponds to a material with one vanishingly small shear

modulus Similarly,␥is a shear modulus and, if␦⫽0, in the

limit␥→0 we have ⌬→⬁ Thus, we have interpretations for

the two extreme values of⌬

ACKNOWLEDGMENTS

The work is partly supported by the Ministry of

Educa-tion and Training of Vietnam and completed during a visit of

the second author to the Department of Mathematics,

Uni-versity of Glasgow, UK

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FIG 1 Plot of ␳c2 / ␥ against ⌬共⬎0兲 from Eq 共38兲.