DSpace at VNU: Formulas for the Rayleigh wave speed in orthotropic elastic solids

Recently, a formula for the Rayleigh wave speed in an isotropic elastic half-space has been given by Malischewsky and a detailed derivation given by the present authors.. For non-isotrop

DOI 10.1007/s11012-005-1603-6

On the Rayleigh Wave Speed in Orthotropic Elastic Solids

PHAM CHI VINH and R W OGDEN1,∗

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

1Department of Mathematics, University of Glasgow Glasgow G12 8QW, UK

(Received: 22 June 2004; accepted in revised form: 7 January 2005)

Abstract Recently, a formula for the Rayleigh wave speed in an isotropic elastic half-space has been

given by Malischewsky and a detailed derivation given by the present authors This study deals with the generalization of this formula to orthotropic elastic materials and Malischewsky’s formula is recov-ered as a special case The formula is obtained using the theory of cubic equations and is expressed

as a continuous function of three dimensionless material parameters.

Key words: Rayleigh waves, Wave speed, Surface waves, Orthotropy.

1 Introduction

Recently, there has been considerable interest in obtaining explicit formulas for the Rayleigh wave speed in an elastic half-space For isotropic materials such formu-las have been given by Rahman and Barber [1], Nkemzi [2] and Malischewsky [3]

In obtaining his formula Malischewsky used Cardan’s formula for the solution of

a cubic equation together with Mathematica A detailed derivation of this formula was given by Pham and Ogden [4] together with an alternative formula See also the recent analysis of Malischewsky [5]

For non-isotropic materials, for some special cases of compressible monoclinic

materials with symmetry plane x3= 0, formulas for the Rayleigh wave speed have been found by Ting [6] and Destrade [7] as the roots of quadratic equations, while for incompressible orthotropic materials an explicit formula has been given by Ogden and Pham [8] based on the theory of cubic equations Further, in a recent paper [9]

we have obtained explicit formulas for the Rayleigh wave speed in compressible or-thotropic elastic solids One of the formulas is analogous to that of Malischewsky in the isotropic case, but we were not able to establish its validity for all relevant ranges

of values of the material parameters

The main purpose of the present paper therefore is to provide a generalization

of Malischewsky’s formula for compressible orthotropic materials that is valid for all appropriate ranges of values of the material parameters We consider a compress-ible elastic body possessing a stress-free configuration of semi-infinite extent in which

∗Author for correspondence Tel.: +44-141-330-4550; Fax: +44-141-330-4111; e-mail: rwo@maths. gla.ac.uk.

the material exhibits orthotropic symmetry The boundary of the body is taken to be parallel to the (001) mirror plane of the material and we choose rectangular

Carte-sian axes (x1, x2, x3) such that the x3 direction is normal to the boundary, the body

occupies the region x3
0 and the Rayleigh wave propagates in the (x1, x3) plane and decouples from any transverse motions (which are not considered here); (see, for example, [10, 11])

We recall that for an orthotropic material with the symmetry planes coinciding with the Cartesian coordinate planes the stress-strain relation may be written in the

standard compact form σ i =c ij e j , i, j ∈{1, , 6}, where σ i , e i are the stress and strain

components and c ij = c j i the elastic constants (c ij = 0 for i = j when i = 4, 5 or 6) In terms of the tensor components σ ij , e ij we have

σ i = σ ii , i = 1, 2, 3, σ4= σ23, σ5 = σ13, σ6 = σ12, (1)

e i = e ii , i = 1, 2, 3, e4= 2e23, e5= 2e13, e6= 2e12. (2) For the considered specialization, however, the relevant material constants (those

appearing in the equation of motion) are just c11, c33, c55, c13 Necessary and sufficient conditions for the strain energy of the material (under the considered restriction) to

be positive definite are

c ii > 0, i = 1, 3, 5, c11c33− c2

It is convenient to pursue the analysis in terms of three dimensionless material parameters, defined by

α = c33/c11, γ = c55/c11, δ = 1 − c2

so that, in accordance with (3),

These parameters may also be expressed in terms of other elastic constants For

example, if ν13, ν31 are the Poisson’s ratios in the (x1, x3) plane then

while γ =δG13/E1, where G13 is the shear modulus associated with the (x1, x3) plane

and E1 the Young’s modulus for the x1 direction

A generalization of Malischewsky’s formula for the Rayleigh wave speed is

obtained for values of these parameters such that either α  1, 0 < δ < 1, 0 < γ < α

or 0 < α < 1, 0 < δ < 1, 0 < γ < α, γ (α − 1) + 2αδ > 0 In this formula, which is based

on the theory of cubic equations, each cubic root takes its principal value We also obtain an alternative formula in which the cubic roots take their secondary values Formulas for the Rayleigh wave speed for the remaining ranges of the parameter val-ues are also investigated For the case of isotropy (for which Malischewsky’s formula

applies) the values of the parameters are such that α = 1, 0 < γ < 3/4, δ = 4γ (1 − γ ),

as we will note in Section 3

2 The Secular Equation

The equations of motion have been examined in detail previously (see, for example, [9] and references therein) and are not therefore repeated here We begin with the form of the secular equation given by Chadwick [10], namely

c55 − ρc2 

c213− c33

c11 − ρc2

where c is the Rayleigh wave speed and ρ the mass density of the material As

dis-cussed previously (see, for example, [9]), the wave speed must satisfy the inequalities inequality

Note that for c11, c33, c55, c13 satisfying (3), Chadwick [10] showed that equation (7) has a unique (real) solution satisfying (8) and corresponding to a surface wave

We now introduce the notation

Then, from (8), we deduce that

0< x < 1
σ if 0 < c55
c11,

where, for convenience, we have also introduced the notation

It follows that

Equation (7) can now be written in the form

√

with x ∈ (0, ˆσ ) and the parameters α, γ, δ satisfying the inequalities (5).

3 A Formula for the Rayleigh Wave Speed

From equation (13), after squaring and rearranging, we obtain the cubic equation

for x.

PROPOSITION 1 In the interval (0, σ∗), where σ∗=min{1, σ δ}, equation (14) has a

Proof According to Chadwick [10], in the interval (0, ˆσ ), equation (13) has a unique real solution x0 corresponding to the Rayleigh wave From (13) it follows that

x0∈ (0, σ∗ By (5), 0 < δ < 1, and hence (0, σ∗ ⊂ (0, ˆσ ) Thus, in the interval (0, σ∗ ,

equation (13) has a unique real solution x0, and since in this interval equations (13) and (14) are equivalent, the proposition is established

For the values of α and γ such that α = γ , equation (14) may be written

where

a0=ασ2δ2

α − 1 + 2ασ δ

From (15) and (16), we then have

F1( 0)=ασ2δ2

M = M(α, γ, δ) denote a point of E We define the following subsets of E:

 = {M ∈ E : α > 0, γ > 0, 0 < δ < 1},

1 = {M ∈  : α > γ }, 2= {M ∈  : α < γ },

3 = {M ∈  : α = γ }, 4= {M ∈ 1: γ  1},

5 = {M ∈ 1: 0 < γ < 1 },

11 = {M ∈ 1: d < 0 }, 12= {M ∈ 1: α  1, d > 0},

13= {M ∈ 1: α < 1, d > 0, (α − 1)γ + 2αδ > 0},

14= {M ∈ 1: α < 1, d > 0, (α − 1)γ + 2αδ < 0},

 = {M ∈ 1: d = 0}, 1= {M ∈  : (α − 1)γ + 2αδ > 0},

In (18) we have used the notation

where a1 and a2 are given by (16) Note that 4d is the discriminant of the quadratic

F1(x) Thus, when γ = α and d
0, F1(x) is a monotone function and equation (15)

has a unique real root Note also that a1> 0 in 1 and that a2= 0 when (α − 1)γ + 2αδ = 0; therefore, for a point M ∈ 1 such that (α − 1)γ + 2αδ = 0, d < 0, i.e M ∈ 11

and hence M ∈ .

We now state a theorem concerning a formula for the Rayleigh wave speed

THEOREM 1 In the region ∗= 11∪ 12∪ 13∪ 1, x0, and hence the Rayleigh

wave speed c, with x0= ρc2/c55, is given by

ρc2/c55=α − 1 + 2ασδ

3(α − γ ) + sign(−d)

3



sign( −d)[R +√D]− 3



where the (complex) roots take their principal values, the principal argument of a

complex w, Arg w, is taken in the interval ( −π, π], and R and D are given by

R=
9a1a2− 27a0− 2a3

2



/ 54,

D=
4a0a23− a2

1a22− 18a0a1a2+ 27a2

0+ 4a3 1



in terms of a0, a1, a2, as defined in (16)

For the case of an isotropic material we have c11= c33= λ + 2µ, c55= µ, c13= λ and hence α = 1, δ = 4γ (1 − γ ), γ = µ/(λ + 2µ), 0 < γ < 3/4, where λ and µ are the

Lam´e moduli From these and equations (16), (19) and (21) we obtain

d = 48(γ − 1/6), R = 8(45γ − 17)/27,

Using (22) it is easy to show that formula (20) reduces to Malischewsky’s formula given in [3], namely

3



4−3

h3(η) + sign[h4(η)]3

sign[h4(η) ]h2(η)



where the functions h i (η), i = 1, 2, 3, 4, are given by

with η = γ in our notation.

We also note that (23) can be rewritten in the form (20) in the region ω∗= {M :

α = 1, δ = 4γ (1 − γ ), 0 < γ < 3/4} ⊂ ∗, in which c55= µ, γ = η and d, R and D are

specialized according to (22) Thus, the form of (23) does not change when passing

from ω∗ to ∗

In order to prove Theorem 1 we need a number of lemmas

LEMMA 1 (a)  ∩ 3= ∅; (b)  ∩ 2= ∅; (c)  ∩ 4= ∅.

a1 < 0 in 2 it follows that d > 0 ∀M ∈ 2 Hence the result (c) From (17), since

From Lemma 1, therefore, the surface  is located in 5 We also note that the

set 2 is in 5, with the values of α necessarily restricted according to 0 < α < 1.

We now introduce the sets defined by

 (121) = {M : α  1, 0 < γ < 1, 0 < δ < 1, d > 0},

 (122) = {M : α > 1, α > γ  1, 0 < δ < 1, d > 0},

∗∗=  ( 1)

It is clear that 12=  ( 1)

12∪  ( 2)

12, and from (17) we have

F1( 0) < 0, F1( 1) > 0, F1(σ∗) >0 in ∗∗. (26)

LEMMA 2 The set ∗∗ is a connected set.

α = α0> 0, where α0 is a constant Then, ∗∗= α0>0G(α0) We shall show below

that G(α0) defines a region in the (δ, γ ) plane, as illustrated as in Figure 1.

From Figure 1 we see that G(α0) is a connected set, and G(α0) contains the set defined by

T (α0) ={M ∈ G(α0): 2/3 < δ < 1, 0 < γ < 1 if α0 1, 0 < γ < α0 if 0 < α0<1}.

(27)

It is clear that the strip

α0>0T (α0) is a connected set Thus, two arbitrary

points M1(α1, γ1, δ1) and M2(α2, γ2, δ2) in ∗∗ can be connected by a simple curve

in the strip Hence, the set ∗∗ is connected

We now establish the property of G(α0) stated above Let 1(α0) denote the

inter-section 1∩ P (α0) From (16) and (19) it can be seen that, in 1(α0), the equation

d = 0 may be written as a quadratic equation for γ , namely

g(γ ) ≡ [(α0− 1)2+ 6α0δ]γ2− α0δ(4− 3δ + 2α0)γ+ α2

where δ ∈ (0, 1) is considered as a parameter and d > 0 if and only if g(γ ) > 0 We

0)

We note the following facts, that may easily be verified

(i) For any given positive value of α0, equation (28) has negative discriminant for

δ ∈ (2/3, 1) and therefore has no real solution for such values of δ On the other hand, it has two distinct positive real roots, denoted γ1, γ2(> γ1) , for δ ∈(0, 2/3), and has a unique (double) positive real root, denoted γ0, when δ = 2/3.

0) is located in 5(α0)= 5∩ P (α0) (according to Lemma 1), and

it is a continuous curve

Figure 1 Plot of the curve d 0) , in (δ, γ ) space with γ (vertical axis) against δ (horizontal axis) for (a) α0 > 1, (b) α0 = 1, (c) 0 < α0 < 1 In (a) and (c) the curve encloses the region d < 0; in (b) the curve and the γ axis enclose the region d < 0 In (c) the line defined by (α0 − 1)γ + 2α0 δ= 0

is also shown; it cuts the curve d = 0 at its maximum point δ = δ0 ≡ (1 − α0 )/ 2, γ = α0 Within the

square (0, 1) × (0, 1) in (a) and (b) the connected region outside the curve d = 0 is the set G(α0 ).

In (c), G(α0) is the region in the rectangle (0, 1) × (0, α0 ) outside the curve d= 0 and below the line

(α −1)γ +2α0 δ =0 Note that d is not defined at the point (δ0 , α ), which is shown as a filled circle.

(iii) For α0 1, 0 < γ1< γ2< 1 for all δ ∈ (0, 2/3] Note that g(1) = (α0− 1 − α0δ)2+

3α0δ2> 0 for all α0> 0, δ > 0 For 0 < α0< 1 we have 0 < γ1< γ2 < α0 for all

δ ∈ (0, 2/3], δ = (1 − α0)/ 2 When δ = (1 − α0)/ 2, γ2= α0 The point M(α0, α0, (1−

α0)/ 2) belongs to the line (α0− 1)γ + 2α0δ 0)

(iv) For any α0= 1, γ1 and γ2 tend to zero as δ → 0 For α0= 1, γ1 tends to zero

while γ2 tends to 1 as δ→ 0

(v) Clearly, g(γ ) < 0 for all γ ∈ (γ1, γ2) This means that, in respect of Figure 1(a)

0) , while for Figure 1(b) d < 0 in the domain bounded

0) and the γ axis.

The facts (i)–(v) show that the set G(α0) has the structure shown in Figure 1, and the proof of Lemma 2 is completed

LEMMA 3 (a) Let r =−2R; then r =F1(xN), where xN is the inflection point on the

calculation that (a) is true From (15), we have

When d > 0, F1(x) has two distinct real zeros, namely xmin, xmax From (16) and the

definition of ∗∗, it is clear that the following inequalities hold in ∗∗:

xminxmax= a1/ 3 > 0, xmin+ xmax= −2a2/ 3 > 0. (31) Hence, (c)

LEMMA 4 In ∗∗, R < 0 if D  0.

M1 ∈ ∗∗ such that D(M1) 0 but R(M1) 0 If R(M1)= 0 then r(M1)= 0 Since

at M1 Thus, by Remark 1(iii) below as will be shown shortly, it follows that D < 0 This contradicts the assumption D(M1)  0 Next, consider R(M1) >0 (and hence

we deduce that equation (15) has two distinct real roots in the interval (0, σ∗ This

contradicts Proposition 1 Thus D(M1) >0

It is not difficult to verify that the point M2(1, 3/4, 3/4) ∈ ∗∗ and D(M2) <0.

Since M1, M2∈ ∗∗, then by Lemma 2 we can connect M1 and M2 by a simple

con-tinuous curve, which we denote by L12∈ ∗∗ Since D is a continuous function on

L12 and D(M1) > 0, D(M2) < 0, there must exist a point M0∈ L12, M0= M1, M2 such

that D(M0)= 0 and D(M) > 0 for all M ∈ L10 (except M0), where L10 is the part

of L12 connecting M1 and M0 Analogously to above arguments, one can see that

R does not vanish at any point M ∈ L10 Since R is a continuous function on L10 and R(M1) >0, then R(M) > 0 for all M ∈ L10 Hence R(M0) >0, i.e r(M0) <0 This

together d(M0) > 0, D(M0)= 0, (26), (29) and Lemma 3(a), (b) shows that equation

(15) has two distinct real roots in the interval (0, σ∗ But this contradicts Proposi-tion 1, and the proof of Lemma 4 is therefore completed

We are now in a position to prove Theorem 1

Proof of Theorem 1 In terms of the variable z defined by

equation (15) becomes

where

q2= (a2

By the theory of cubic equation the three roots of equation (33) are given by the Cardan’s formula (see, for example, [12])

z1 = S + T , z2= −1

2(S + T ) +1

2i

√

3(S − T ), z3= −1

2(S + T ) −1

2i

√

3(S − T ), (35)

where i2= −1,





and D is given by (16) and (21) It is noted that R and D in (36) are given by the

values defined in (21)

neg-ative result (ii) When D < 0 and hence R+√D is complex, then T = S∗, where S∗ is

the complex conjugate of S (iii) The nature of three roots of equation (33) depends

on the sign of the discriminant D In particular, if D > 0, equation (33) has one real root and two complex conjugate roots; if D= 0, it has three real roots, at least two

of which are equal; if D < 0, it has three distinct real roots.

Let z0 denote the real root of equation (33) corresponding to x0 (defined in Prop-osition 1) and the Rayleigh wave speed given by (20) In order to prove Theorem 1

we examine the distinct cases associated with different subsets of ∗

First, we consider 11 On 11, we have d < 0 From (36) 3,5 , and the fact that D >

0 we have

Since D > 0 equation (33) has a unique real solution, namely z0= z1, given by (35)1 and (36), in which the radicals are understood as real roots Since the value of the

real root of a positive real number coincides with the principal value of its corre-sponding complex root, it is clear that inequalities (37) together with (16), (32) ensure that (20) is valid

Second, on 1 we make use of the following lemma

LEMMA 5 On 1, R
0.

This result is established below

If R < 0, then, by (36) 3,5 , D > 0, so that equation (33) has a unique real solution and (20) is valid If R = 0 then D = 0, and it is clear from (35), (36) that equation (33) has a unique (triple) real root z0= 0 and in this case (20) is also valid

We now show that R
0 on 1 Suppose that M0(α0, γ0, δ0) is an arbitrary point

of 1, so that d(M0)= 0 If R(M0) >0, then D(M0) >0 by (36)3 Since D is a con-tinuous function in the open set 5⊃ 1 (according to Lemma 1), and M0∈ 5,

there exists a sufficiently small neighborhood U0(M0)= {M : (α − α0)2+ (γ − γ0)2+

Since R is continuous on 5⊃ U, and M0 is a boundary point of U , we conclude that R(M0)
0 But this contradicts the assumption that R(M0) >0

Next, noting that 12∪13=∗∗∪ ( 2)

12, we examine formula (20) on ∗∗ and  (122)

separately

On ∗∗, by definition, d > 0.

If D > 0, equation (33) has a unique real solution, namely z0= z1, given by (35)1

and (36) Since, by Lemma 4, R < 0 and d > 0, it follows from (36) 3,5 that

− R+√D

Taking into account (16), (32), (38) and the fact that the value of the real root of a positive real number coincides with the principal value of its corresponding complex root, it is easy to see that (20) is valid

If D = 0, then, by Lemma 4, R < 0 Taking into account (36)3−5, we have r=

−2R = 2q3(q > 0), and equation (33) becomes

whose solutions are q (double root) and −2q Bearing in mind that equation (15) has

a unique real solution in the interval (0, σ∗ , it follows from (26)1 and (29) that the Rayleigh wave speed is determined from the smallest real root of (15) in this case,

and thus z0 is the smallest real root of equation (33), i.e z0= −2q and (20) is

appli-cable

In the case D < 0 equation (33) has three distinct real roots, and hence so does

equation (15) From Proposition 1, (26)1 and (29), it is clear that x0 is the

small-est real root of equation (15), and thus z0 is the smallest real root of (33) When

(square) roots can take one of three (two) possible values such that T = S∗ In our

case we take the principal value and we shall indicate that z2, as expressed by (35)2

is the smallest real root of (33) Throughout the remainder of this section, for sim-plicity, we take the complex roots as their principal values

From (36) we have

S=3

Let 3θ denote the principal argument of R+ i√−D Since √−D > 0, 3θ ∈ (0, π), and the phase angle corresponding to the principal value of S is θ ∈ (0, π/3) From (40)

this implies that |S| = q, and hence S and T can be expressed as

where θ ∈ (0, π/3) satisfies the equation

which is obtained by substituting z = S + T = 2q cos θ into equation (33) Note that

in the interval (0, π/3).

From (35) and (41) it may be verified that

z1= 2q cos θ, z2= 2q cos(θ + 2π/3), z3= 2q cos(θ + 4π/3). (43)

With reference to (43), taking into account that θ ∈ (0, π/3), it is clear that

z1> z3> z2, i.e z2 is the smallest real root of (33) Therefore,

It is clear that to prove (20) for the case d > 0, D < 0, we need the equality

−3

−R +√D− 3



where the roots are complex roots taking their principal values Indeed, we have

Arg(R+√D) = 3θ, Arg(R −√D) = −3θ, 3θ ∈ (0, π) being the solution of (42) Thus,

Arg[−(R +√D)]=3θ −π, Arg[−(R −√D)]=−3θ +π Note that by Arg w we denote the principal argument of the complex number w Since |R ±√D |=q3 it follows that 3



−(R +√D) = qe i(θ −π/3) , 3



From (46) it follows that:

−3



−R +√D− 3



−(R +√D) = −2q cos (θ − π/3) = 2q cos (θ + 2π/3) (47) and (45) is established

On  (122) we have γ  1, and hence 0 < σ∗<1, and from (17) we have

F1( 0) < 0, F1(σ∗) > 0, F1( 1)
0 in  ( 2)

It follows from (48) that on  (122) (15) has at least two distinct real roots, and therefore

D
0 and, on account of Proposition 1, x0 is the smallest real root It is noted that

when equation (15) has two distinct real roots, i.e D = 0, then R < 0 By arguments