Symmetry properties of elastodynamic wave fields and their application to space-time scattering theory

SYMMETRY PROPERTIES OF ELASTODYNAMIC

WAVE FIELDS AND THEIR APPLICATION TO

SPACE-TIME SCATTERING THEORY

R. DU CLOUX

LEVENSBERICHT

De s a m e n s t e l l e r van d i t p r o e f s c h r i f t werd op 8 januari 1958 in Rijswijk

geboren. Na de middelbare school doorlopen te hebben, studeerde h i j van

1976 t o t 1982 e l e c t r o t e c h n i e k aan de Technische U n i v e r s i t e i t Delft.

Vervolgens was hij van 1982 t o t 1986 aan de Afdeling der E l e c t r o t e c h n i e k

van de T e c h n i s c h e U n i v e r s i t e i t Delft verbonden a l s wetenschappelijk

a s s i s t e n t . De onderzoekingen die h i j aan de Technische Universiteit Delft,

onder l e i d i n g van p r o f . d r . A.W. Grootendorst en p r o f . d r . i r . A.T. de Hoop,

heeft v e r r i c h t , hebben geleid t o t de samenstelling van d i t p r o e f s c h r i f t .

Sinds 1 augustus 1986 i s de s a m e n s t e l l e r werkzaam als wetenschappelijk

medewerker b i j het Philips Natuurkundig Laboratorium te Eindhoven.

Dit proefschrift is goedgekeurd door de promotoren prof.dr. A.W. Grootendorst en prof.dr. ir. A.T. de Hoop.

1\

SYMMETRY PROPERTIES OF ELASTODYNAMIC

WAVE FIELDS AND THEIR APPLICATION TO

SPACE-TIME SCATTERING THEORY

Proefschrift ter verkrijging van de graad van doctor aan de Technische

Universiteit Delft, op gezag van de Rector Magnificus, prof.dr. J.M. Dirken,

in het openbaar te verdedigen ten overstaan van een commissie aangewezen

door het College van Dekanen op 18 december 1986 te 16.00 uur door

Renó du Cloux,

geboren te Rijswijk,

elektrotechnisch ingenieur.

TR diss ^

1521

J

TABLE OF CONTENTS

INTRODUCTION

CHAPTER 1. BASIC RELATIONS IN ELASTODYNAMIC RADIATION AND SCATTERING THEORY

1.1. Elastodynamic Field Equations 1.2. Constitutive Relations

1.3. Boundary Conditions

1.1. Exchange Of Elastodynamic Energy

1.5. Elastodynamic Field Theory In The Time-Laplace Transform Domain

1.6. The Time-Laplace Transform-Domain Elastodynamic Reciprocity Theorem

CHAPTER 2. SYMMETRY CONSIDERATIONS IN ELASTODYNAMIC SCATTERING THEORY

2.1. Formulation Of The Scattering Problem 2.2. Symmetry Groups

2.3. Invariance Theorem

2.3-1. Invariance of the elastodynamic field equations 2.3.2. Invariance of the boundary conditions

2.3-3. Invariance of the initial condition 2.3.1. Conclusion

2.1. Decomposition Of The Elastodynamic Field 2.5. Examples

2.5.1. Example 1 2.5.2. Example 2

-Contents-CHAPTER 3. ELASTODYNAMIC WAVE PROPAGATION IN A STRATIFIED MEDIUM 63

3.1. Description Of The Configuration 61 3.2. Description Of The Elastodynaraic Field 67 3.3. Representation Of The Elastodynamic Field 75

3.1. Recursive Algorithm 81 3.1.1. Scattering matrix of a homogeneous layer 81

3.1.2. Scattering matrix of an interface 83

3.1.3. Forward recursion 86 3.1.1. Backward recursion 91 3.1.5. Source condition 95 3.1.6. Conclusion 96

CHAPTER 1. THE TRANSFORM-DOMAIN SCATTERING OPERATOR OF A

SEMI-INFINITE PLANE SCREEN 105 1.1. Formulation Of The Scattering Problem 106

1.2. Symmetry Considerations 115 1.3. The Scattering By A Semi-infinite Rigid Baffle 121

1.3-1. Odd scattering problem 121 1.3.2. Even scattering problem 135 1.3-3. Factorization of the kernel function 117

1.1. The Scattering By A Semi-infinite Void Crack 153

1.1.1. Even scattering problem 155 1.1.2. Odd scattering problem 162 1.1.3. Factorization of the kernel function 170

1.5. The Scattering By A Semi-infinite Fluid-filled Crack 175

1.5.1. Even scattering problem 175 1.5.2. Odd scattering problem 177

1.6. Conclusion 178

CHAPTER 5. ELASTODYNAMIC RADIATION FROM AN IMPULSIVE POINT SOURCE OF

EXPANSION IN THE PRESENCE OF A SEMI-INFINITE PLANE SCREEN 183

5.1. Formulation Of The Scattering Problem 181 5.2. Integral Representation Of The Incident Field 189

C o n t e n t s

-5 . 1 . Space-Time R e p r e s e n t a t i o n Of The I n c i d e n t F i e l d 202 5 . 5 . Space-Time Representation Of The Scattered F i e l d 213

5 . 5 . 1 . The s c a t t e r e d c o n g r e s s i o n a l wave 215

5 . 5 . 2 . The s c a t t e r e d shear wave 232

5 . 6 . T o t a l Wave Motion 251 5 . 7 . N u m e r i c a l R e s u l t s 260

APPENDIX A . EIGENWAVE REPRESENTATION OF THE ELASTODYNAMIC FIELD 2 7 9

A . 1 . R e p r e s e n t a t i o n Of The E l a s t o d y n a m i c F i e l d 2 7 9

A . 2 . C h o i c e Of T h e E i g e n w a v e s 287 A. 3 . The E l a s t o d y n a m i c F i e l d G e n e r a t e d By A H o r i z o n t a l

D i s t r i b u t i o n Of S o u r c e s 2 9 8

APPENDIX B . DIFFRACTION OF A PULSED, SPHERICAL, SCALAR WAVE BY A

SEMI-INFINITE PLANE SCREEN 3 0 3 B . 1 . F o r m u l a t i o n Of The S c a t t e r i n g P r o b l e m 3 0 1 B . 2 . I n t e g r a l R e p r e s e n t a t i o n Of The I n c i d e n t F i e l d 307 B . 3 . I n t e g r a l R e p r e s e n t a t i o n Of The S c a t t e r e d Wave 309 B . I . S p a c e - T i m e R e p r e s e n t a t i o n Of The I n c i d e n t Wave 319 B . 5 . S p a c e - T i m e R e p r e s e n t a t i o n Of The S c a t t e r e d Wave 3 2 8 B . 6 . T o t a l Wave M o t i o n 3145 REFERENCES 351

-Introduction-and e x h i b i t s p a r t i c u l a r symmetry p r o p e r t i e s . Both the number of

constituents and t h e i r symmetry p r o p e r t i e s a r e r e l a t e d t o the symmetry

group a t hand. In Chapter 3, t h e wave motion in a s t r a t i f i e d medium,

consisting of a s t a c k of plane h o r i z o n t a l , homogeneous, i s o t r o p i c and

p e r f e c t l y e l a s t i c l a y e r s , i s considered. The wave motion i s generated by

an impulsive point source which i s taken t o be l o c a t e d e i t h e r in t h e

i n t e r f a c e between two d i f f e r e n t l a y e r s or in t h e i n t e r i o r of a layer.

Taking advantage of the invariance of the c o n f i g u r a t i o n r e l a t i v e t o time

and with respect to a translation in a horizontal plane, the elastodynamic

f i e l d q u a n t i t i e s a r e c o n s e c u t i v e l y s u b j e c t e d t o a t i m e - L a p l a c e

transformation and to a horizontal space-Fourier transformation. The

time-Laplace and horizontal space-Fourier transform-domain elastodynamic f i e l d

i s r e p r e s e n t e d i n t e r m s of t h e eigenwaves r e l a t i v e t o the v e r t i c a l

coordinate which i s o r i e n t e d along the normal t o the i n t e r f a c e s . The

eigenwaves turn out to be upward and downward propagating waves. I n

Appendix A, i t i s shown that the eigenwaves can be chosen such t h a t , in t h e

sense of t h e elastodynamic r e c i p r o c i t y theorem, they form an orthogonal

s e t . Moreover, the set of eigenwaves is shown to be physically complete i n

view of t h e fact t h a t the elastodynamic f i e l d generated by an a r b i t r a r y

h o r i z o n t a l d i s t r i b u t i o n of s o u r c e s can be expanded in terms of t h e s e

eigenwaves. The eigenwave representation of the elastodynamic f i e l d proves

to be identical with i t s c l a s s i c a l decomposition i n t o compressional (P-)

waves, v e r t i c a l l y p o l a r i z e d shear (SV-) waves and horizontally polarized

shear (SH-) waves (see e.g. Aki and Richards, 1980, pp. 215-216). Based on

the s c a t t e r i n g matrices pertaining to a layer and to an interface between

two adjacent layers, a r e c u r s i v e a l g o r i t h m i s developed from which t h e

elastodynamic f i e l d q u a n t i t i e s everywhere i n t h e c o n f i g u r a t i o n can be

calculated (see also Kennett, 1974 and 1979). As a consequence of t h e

particular representation of the transform-domain elastodynamic f i e l d , t h i s

algorithm has a f a i r l y simple shape. In Chapter H, the transform-domain

scattering-matrix operators of a semi-infinite r i g i d (immovable) baffle and

of a semi-infinite crack (either void or f l u i d - f i l l e d ) are c a l c u l a t e d . The

use of t h e eigenwave expansion of Chapter 3 and Appendix A, and t h e

a p p l i c a t i o n of the symmetry c o n s i d e r a t i o n s of C h a p t e r 2 l e a d s to a

-Introduction-r e l a t i v e l y simple -Introduction-rep-Introduction-resentation of the scatte-Introduction-ring-mat-Introduction-rix ope-Introduction-rato-Introduction-r of the

screens under consideration. The scattering problem i s separated i n t o two

i n d e p e n d e n t subproblems. Each of these i s formulated in terras of a

singular integral equation of Cauchy type, in which the diffraction matrix

of the screen is the unknown quantity. As a consequence of the particular

representation of the transform-domain elastodynamic f i e l d , the kernel of

t h i s i n t e g r a l equation i s a diagonal matrix. Accordingly, i t s solution i s

obtainable by the a p p l i c a t i o n of an e s s e n t i a l l y s c a l a r Wiener-Hopf (or

f a c t o r i z a t i o n ) method (see e.g. Noble, 1958, pp. 36-38 and Weinstein, 1969,

pp. 1 7 - 2 2 ) . A t t e n t i o n i s given t o the n u m e r i c a l e v a l u a t i o n of t h e

r e s u l t i n g f a c t o r s . Also, an approximation of the kernel function i s

i n t r o d u c e d t h a t allows i t s f a c t o r s to be obtained by i n s p e c t i o n ( c f .

Kol t e r , 1954 and C a r r i e r , 1959). In Chapter 5, the case where the

semi-i n f semi-i n semi-i t e screen semi-i s embedded semi-in an otherwsemi-ise homogeneous, semi-i s o t r o p semi-i c and

p e r f e c t l y e l a s t i c solid of i n f i n i t e extent is investigated in d e t a i l . The

elastodynamic field in the configuration i s generated by an impulsive point

s o u r c e of expansion ( t h i s monopole transducer serves as a model for an

e x p l o s i v e s o u r c e ) . The t i m e - L a p l a c e and h o r i z o n t a l s p a c e - F o u r i e r

transform-domain elastodynamic f i e l d i s c a l c u l a t e d with the aid of the

theory developed in Chapters 3 and 4. Subsequently, t h e s p a c e - t i m e

r e p r e s e n t a t i o n of the elastodynamic field everywhere in the configuration

is obtained with the aid of an amended version of the Cagniard-De Hoop

method for s o l v i n g s e i s m i c pulse problems (see De Hoop 1960, Miklowltz,

1978, p. 302 and Aki and Richards, 1980, p. 224). Characteristic for t h i s

amended v e r s i o n of the Cagniard-De Hoop method i s t h a t the orthogonal

transformation of the h o r i z o n t a l space-Fourier transform v a r i a b l e s i s

a v o i d e d . This allows a n a t u r a l d i s t i n c t i o n between the wave motions

perpendicular to the edge of the screen and the wave motions p a r a l l e l to

the edge of the s c r e e n . In the f i n a l expressions only integrals over a

f i n i t e range occur that can be computed with any degree of accuracy. In

Appendix B, the method of solution i s , by way of i l l u s t r a t i o n , applied to

the diffraction of a p u l s e d , s p h e r i c a l , s c a l a r wave by a s e m i - i n f i n i t e

plane screen (see a l s o Cagniard, 1936).

CHAPTER 1

BASIC RELATIONS IN ELASTODYNAMIC RADIATION AND SCATTERING THEORÏ

. . . t h e three fundamental concepts

of p h y s i c s . . . : i n i t i a l conditions,

laws of nature, invariance

p r i n c i p l e s . . . (E.P. Wigner)

In t h i s chapter, the basic equations of linearized e l a s t i c wave propagation

are summarized. In the process of l i n e a r i z a t i o n i t i s assumed t h a t the

amplitudes of the wave motion are so small that, to a sufficient degree of

accuracy, the f i r s t - o r d e r terms account for the elastodynamic e f f e c t s we

want t o i n v e s t i g a t e . The r e l e v a n t equations are s t a t e d without any

p h y s i c a l background. For the l a t t e r , the reader i s r e f e r r e d to t h e

l i t e r a t u r e (see e . g . Thurston, 1961 and Achenbach, 1973. pp. 17-21 and

p. 46). A formal introduction to the theory of linear elastodynamics can

be found in Gurtin (1972).

1.1. Elastodynamic Field Equations

A point in t h r e e - d i m e n s i o n a l Euclidean space R

3

i s r e f e r r e d to by i t s

c o o r d i n a t e s {x,, x

2

, x

3

} r e l a t i v e to a f i x e d , o r t h o g o n a l , C a r t e s i a n

r e f e r e n c e frame with o r i g i n 0 and the three, mutually perpendicular base

vectors of unit length {e,, e

2

, e , } . In the given o r d e r , e , , e

a

and e ,

form a right-handed system. The time coordinate is denoted by t . Vectors

and tensors are denoted in bold face. In particular, x = x ^ , + x

2

e

2

+ x

3

e

s

denotes the p o s i t i o n v e c t o r . Whenever the s u b s c r i p t n o t a t i o n for

Chapter 1 | Basic r e l a t i o n s

( C a r t e s i a n ) v e c t o r s and t e n s o r s i s u s e d , t h e summation c o n v e n t i o n i s u n d e r s t o o d f o r r e p e a t e d s u b s c r i p t s ; t h e l a t i n s u b s c r i p t s (except f o r " t " ) run through the v a l u e s 1, 2, and 3.

The r e l e v a n t p h y s i c a l q u a n t i t i e s a r e l i s t e d in Table 1 . 1 . In any domain where t h e e l a s t o d y n a m i c f i e l d q u a n t i t i e s a r e c o n t i n u o u s l y d i f f e r e n t i a b l e , t h e y s a t i s f y t h e ( l i n e a r i z e d ) e l a s t o d y n a m i c f i e l d e q u a t i o n s , ( 1 . 1 . D 9 j i (T l J ♦ T ^ ) - at P l - - rV ; 1 , ( 1 . 1 . 2 ) i O . V j + 3 ^ ) - 8t e . . = nV ; i j Table 1.1. Elastodynamic f i e l d q u a n t i t i e s .

Quantity Type Symbol S i - u n i t

particle velocity volume density of linear momentum stress strain vector vector tensor of rank 2 tensor of rank 2 V

P

T

e

ms ' Ns = kgm~2s_ 1 Pa d i m e n s i o n l e s s volume d e n s i t y of body force vector. Nm" time rate of body deformation tensor of rank 2

A vector is to be identified with a tensor of rank 1

In (1.1.1), the summation convention is understood with respect to the dummy subscript j (i.e. 3. i (T^. + T ^ ) = l' 3 * (T, . + T ^ ) ) , while

Chapter 1 | Basic r e l a t i o n s

i is a free subscript ( i . e . i e {1, 2, 3 ) ) . In (1.1.2), both i and j are t o

be i n t e r p r e t e d as f r e e s u b s c r i p t s . F u r t h e r , 3 d e n o t e s p a r t i a l

d i f f e r e n t i a t i o n with respect to the s p a t i a l coordinate x. and 9. refers to

p a r t i a l differentiation with respect to the time coordinate t . Equation

(1.1.1) i s the (linearized) equation of motion; in i t , only the symmetrical

part of the s t r e s s i s present. This i s the case when no body torques a r e

p r e s e n t in the e l a s t i c s o l i d under consideration. A discussion of wave

propagation in media in which body torques do a c t , the s o - c a l l e d Cosserat

media, can be found in Teodorescu (1975, pp.162-169). Equation (1.1.2) i s

the (linearized) equation of deformation r a t e , where the time r a t e of body

deformation i s assumed to be symmetric, i . e .

( 1 . 1 . 3 ) ftV;ij = nV ; J i .

Then, equation ( 1 . 1 . 2 ) is the symmetrical part of a more general equation

in which 9,v occurs. For common media, and those are the ones that we

c o n s i d e r , the a n t i - s y m m e t r i c a l p a r t of 9. v. does not interact with the

symmetrical part of the s t r e s s .

1.2. Constitutive Relations

The material properties of the media under consideration are s p e c i f i e d by

the constitutive r e l a t i o n s . These i n t e r r e l a t e the volume density of linear

momentum, the s t r a i n , the p a r t i c l e velocity and the s t r e s s . The mechanical

behaviour of the media at hand i s accounted for by their inertia properties

and their deformation p r o p e r t i e s . The i n e r t i a p r o p e r t i e s are expressed

through the constitutive r e l a t i o n

( 1 . 2 . 1 ) P i = P l J Vj ,

in which p is the tensorial volume density of mass (Si-unit kgm- 3). The

deformation properties relate the strain at any instant and point of observation to the stress at the same instant and the same position. For

Chapter 1 | Basic r e l a t i o n s

time-invariant and p e r f e c t l y e l a s t i c media, the r e l e v a n t c o n s t i t u t i v e

r e l a t i o n i s

( 1 . 2 . 2 ) e . . = s . . i d + T ) , i j iJPq PQ QP

in which 3 denotes the t e n s o r i a l compliance (Si-unit P a

- 1

) . Inversely,

(1.2.2) can be rewritten as

( 1 . 2 . 3 ) * < ! „ ♦ x j , ) - c1 J p q ep q ,

in which c denotes the tensorial stiffness (Si-unit Pa). If the medium i s

v l s c o - e l a s t i c ( i . e . not instantaneously reacting but showing r e l a x a t i o n ) ,

(1.2.2) - (1.2.3) have to be appropriately modified (see e . g . Leitman and

Fisher, 1973). The media for which (1.2.1) - (1.2.3) hold will be referred

to as perfect s o l i d s . In general, the constitutive coefficients depend on

position; the solid is then denoted as inhomogeneous. In a domain in space

where the c o n s t i t u t i v e c o e f f i c i e n t s a r e independent of t h e p o i n t of

observation, the solid is called homogeneous.

From (1.2.2) and ( 1 . 2 . 3 ) , we observe t h a t the s t i f f n e s s and t h e

compliance satisfy the relation

( 1 . 2 . 4 ) c , , s = i ( 6 . 64 + 5 , 6. ) .

i j p q p q r s 2 ir js is jr

Since the compliance (or the stiffness) i n t e r r e l a t e s two symmetric tensors,

the number of independent components i s at most 36. For an i s o t r o p i c

m a t e r i a l the p r o p e r t i e s at any position are direction independent. Then,

the volume density of mass has only one e s s e n t i a l component, while t h e

number of independent compliance (or stiffness) coefficients reduces to two

(see e.g. Love, 1959, pp.101-105). In that case, we have

( 1 . 2 . 5 ) Pt J - p J j j ,

( 1 . 2 . 6 ) s. . = A 6. . 6 + M ( 6 . 6 . + 5 . 6 . ) i j p q i j pq i p j q i q JP

Chapter 1 | Basic relations

and

( K 2-7 ) Cijpq = X 6i j 6PQ + " ( 6i p 6 J q + 6iq V '

in which p i s the scalar volume density of mass, while \ and \i are the Lamê coefficients (Si-unit Pa). Substitution of (1.2.6) - (1.2.7) into (1.2.11) yields ( 1 . 2 . 8 ) A = -o/Y and ( 1 . 2 . 9 ) M = i ( 1 + a)/Y . I n ( 1 . 2 . 8 ) a n d ( 1 . 2 . 9 ) , we h a v e i n t r o d u c e d P o i s s o n ' s r a t i o <j ( d i m e n s i o n l e s s ) , d e f i n e d through (1 . 2 . 1 0 ) o = X / 2(A + y)

and Young's modulus Y (or modulus of longitudinal elasticity, Si-unit P a ) , defined through

(1 .2.11) Y = u (3X + 2p)/(A + u) .

In contrast to the commonly used Lamê coefficients, we consider Poisson's r a t i o and Young's modulus as the fundamental e l a s t i c parameters of an i s o t r o p i c s o l i d . The l a t t e r are closely related to experiments by which the e l a s t i c p r o p e r t i e s of an i s o t r o p i c solid can be determined. The r e l a t i o n s between the different parameters are listed in Gurtin (1972, p . 79).

Chapter 1 | Basic relations

1.3. Boundary Conditions

The elastodynamic field equations (1.1.1) - (1.1.2) have to be supplemented by boundary conditions across an interface of two media with different material properties. On physical grounds, the field quantities are at most allowed t o jump across an interface by a f i n i t e amount. To derive the relevant boundary conditions, equations (1.1.1) - (1.1.2) are r e w r i t t e n in an i n t e g r a l form in which no spatial derivatives of the field quantities occur. This form is obtained as

(1.3.D }3 v v. i (T l J ♦ x..) dA . |vOt P l - fV ; 1) dV

and

(1.3.2) ja v * (V j + V i ) dA = f ^ . * nV ;. . ) d V ,

in which V is a bounded subdomain of R3, enclosed by a piecewise smooth

boundary 3V with outward normal of unit length v (see Figure 1.1).

Fig. 1.1. Geometry to which the integral form of the elastodynamic field equations applies.

Chapter 1 | Basic relations

Consider now t h e two-sided ( t i m e - i n d e p e n d e n t ) surface S between t h e media 1 and 2, with t h e u n i t v e c t o r n along i t s normal p o i n t i n g from medium 1 t o medium 2. The i n t e g r a l r e l a t i o n s ( 1 . 3 . 1 ) and ( 1 . 3 . 2 ) a r e applied t o a s m a l l c y l i n d e r , a r o u n d an a r b i t r a r y p o i n t i n t h e s u r f a c e S ( s e e Figure 1 . 2 ) .

maximum

diameter

Fig. 1.2. Surface of discontinuity in material properties S.

We account for the possible presence of a surface density of surface force f„ (Si-unit Nm~2) and of a time rate of surface deformation n„

(SI-unit B I S "1) . When we let, consecutively, the height and the maximum

diameter of the cross-section of the cylinder tend to zero, and apply in the latter limit the mean-value theorem of integral calculus, we end up with

(1.3.3) (ti)2 - (t.), = -fS ;. on S

Chapter 1 | Basic r e l a t i o n s and ( 1 . 3 . 4 ) i ( n . v . + n . vi ) 2 - i ( nl V. + nJv1)1 - ^ . . on S, w h e r e we h a v e i n t r o d u c e d t h e v e c t o r i a l t r a c t i o n t ( S i - u n i t P a ) , defined through ( 1 . 3 . 5 ) t4 - i (T l j ♦ T . . ) n . on S.

Note t h a t i t follows from ( 1 . 3 . 3 ) - ( 1 . 3 - 4 ) t h a t in t h e absence of s u r f a c e s o u r c e s , b o t h the t r a c t i o n and the p a r t i c l e v e l o c i t y a r e c o n t i n u o u s a c r o s s S.

In T a b l e 1 . 2 , a summary i s g i v e n of boundary c o n d i t i o n s a c r o s s a two-sided s u r f a c e of d i s c o n t i n u i t y in m a t e r i a l p r o p e r t i e s . Also i n c l u d e d a r e t h e e x p l i c i t b o u n d a r y c o n d i t i o n s t h a t have t o be p r e s c r i b e d on boundaries of a s p e c i f i c t y p e .

Table 1.2. Elastodynamic boundary c o n d i t i o n s a t boundary S.

Type of boundary Boundary c o n d i t i o n

Firm c o n t a c t at i n t e r f a c e v. and t . continuous a c r o s s S

l l

S l i d i n g c o n t a c t a t i n t e r f a c e n . v . and n . t . continuous a c r o s s S t . - ( n . t . ) n + 0 on S

i J J i Immovable rigid boundary _{v •* 0 on S}

Chapter 1 | Basic r e l a t i o n s

1.4. Exchange Of Elastodynamic Energy

We i n v e s t i g a t e the exchange of elastodynamic energy between a c e r t a i n p a r t of a medium and i t s s u r r o u n d i n g s . Let t h e p a r t of t h e medium u n d e r c o n s i d e r a t i o n be l o c a t e d in t h e bounded domain V, i n t e r i o r t o the plecewise s m o o t h , c l o s e d s u r f a c e 8V and l e t the u n i t v e c t o r along t h e normal t o 3V, p o i n t i n g away from V, be denoted by v. A subdomain V of V i s o c c u p i e d

sr c by the elastodynamic sources (see Figure 1.3).

Fig. 1.3. Geometry to which the energy relation applies.

It is assumed here that the sources present in V start to act at src

t = t0 ; p r i o r t o t h i s i n s t a n t no e l a s t o d y n a m i c f i e l d i s p r e s e n t in t h e

c o n f i g u r a t i o n ( i n i t i a l c o n d i t i o n ) . M u l t i p l i c a t i o n of ( 1 . 1 . 1 ) by v. and of ( 1 . 1 . 2 ) by £ ( T , , + T i - ) a n d a d d i t i o n of t h e r e s u l t i n g e q u a t i o n s y i e l d ,

a f t e r i n t e g r a t i o n over t h e domain V and i n t e g r a t i o n with r e s p e c t t o time t , t h e r e s u l t

( 1 . 1 . 1 ) W = W u

s r c men "rad

Chapter 1 | Basic relations in which ( 1 ,

" -

2 ) W s r c =

l t - o l

[ f

v ; i

v

i "

ft

v;ij

i ( T

i J

+

^ j i

) ] d V d t

'

0 s r c d e n o t e s t h e amount of e n e r g y d e l i v e r e d by t h e s o u r c e d u r i n g t h e t i m e i n t e r v a l ( t0, t ) , ( 1

- " -

3 ) W

mch * \l

n

L t

v

i

3

fPi

+

*

( T

i j

+ T

j i

) 3

t .

e

i j J

d V d t

'

t o""U " d e n o t e s t h e m e c h a n i c a l e n e r g y gone i n t o t h e e l a s t o d y n a m i c f i e l d in t h e medium in V during the time i n t e r v a l ( t0, t ) and

I' I

( 1 . 4 . 4 ) W . = S . v . d A d t '

r a d jt „ - 0 J3V J J

denotes the amount of energy radiated away through the boundary 3V during the time interval (t„, t ) . In (1.4.4), we have introduced the vectorial surface density of elastodynamic power flow (acoustic Poynting vector) S

(Si-unit W m- 2) , defined as (1.4.5) S = - i(x + T ) vt on S. W i t h t h e c o n s t i t u t i v e r e l a t i o n s ( 1 . 2 . 1 ) and ( 1 . 2 . 2 ) , and t h e a d d i t i o n a l assumption t h a t t h e medium i n V i s r e c i p r o c a l , i . e . ( 1 . 4 . 6 ) p . . = p . . and s . , = s . . , Hi j *Ji i j p q p q i j ' e q u a t i o n ( 1 . 4 . 3 ) can be r e w r i t t e n as ( 1 . 4 . 7 ) W . = W, . + W ' , , mch kin def in which

Chapter 1 | Basic relations

(1 .4.8) W, . = w, , dV kin Jv kin

denotes the kinetic energy stored in the medium in V during the time. interval (t0, t) and

(1

-"'

9) W

def * J

v W

def

dV

d e n o t e s t h e deformation energy stored in the medium in V during the time

interval ( t

0

, t ) . In ( 1 . 4 . 8 ) , we have introduced the volume d e n s i t y of

k i n e t i c energy

(1.4.10) w, . = i v. p. . v. .

_kin

2 i *ij j

In ( 1 . 4 . 9 )

f

we have introduced the volume density of deformation energy

1.4.1D wd e f - J [ K T . J ♦ t j l) si J p q * ( tp q ♦ ,q p) ]

In the d e r i v a t i o n of ( 1 . 4 . 7 ) - ( 1 . 4 . 1 1 ) , we have used the fact that the

medium i s at r e s t at t = t

o

- 0 .

1.5. Elastodynamic Field Theory In The Time-Laplace Transform Domain

In t h i s s e c t i o n , we d i s c u s s the elastodynamic f i e l d e q u a t i o n s , t h e

c o n s t i t u t i v e r e l a t i o n s and the boundary c o n d i t i o n s in the time-Laplace

t r a n s f o r m domain or s - d o m a i n . The r e a s o n for t h i s i s t h a t t h e

c o n f i g u r a t i o n s t h a t we consider are linear and time invariant. In the s

-domain, the time c o o r d i n a t e has been e l i m i n a t e d and a f i e l d problem in

space remains in which the transform v a r i a b l e s occurs as a parameter.

Causality of the f i e l d is accounted for by taking Re(s) > 0, and r e q u i r i n g

t h a t the transforms of a l l causal field quantities are analytic functions

of s in the r i g h t - h a l f of the s-plane ( i . e . in { s e C | 0 < Re(s) <■>,-«>

< Im(s) < »}). In a number of wave-propagation problems, the transform

Chapter 1 | Basic r e l a t i o n s

v a r i a b l e s i s p r o f i t a b l y chosen t o be r e a l and p o s i t i v e . In o t h e r

problems, s i s chosen t o be imaginary, i . e . s = iw, where u (real and

p o s i t i v e ) denotes the angular f r e q u e n c y . The l a t t e r c h o i c e i s of

importance for the complex steady-state representation of sinusoidally in

time oscillating fields (the complex time factor being e x p ( i u t ) ) .

It i s assumed that the elastodynamic sources s t a r t to act at t = t

0

;

prior to t h i s instant the elastodynamic field vanishes i d e n t i c a l l y . This

i n i t i a l condition represents the principle of causality. The elastodynamic

field quantities and the elastodynamic source strengths are subjected to a

one-sided Laplace transformation with respect to the time interval ( t

0

, » ) .

The relevant transforms are denoted by a circumflex:

(1.5.1) {v, p, T, e}(x,s) = exp(-st) {v, p, x, e}(x,t) dt

J

t

0

- 0

and

( 1 . 5 . 2 ) { fv , ftv)(x,s)

r

Jt0- o

exp(-st) {f

y

, ft

v

Hx,t) dt

A possible growth in time of the source s t r e n g t h s i s r e s t r i c t e d to be of

e x p o n e n t i a l order at i n f i n i t y . Consequently, the l e f t - h a n d s i d e s of

(1.5.1) (1.5.2) are analytic functions of s in the r i g h t h a l f of the s

-plane ( i . e . in { s e C | 0 < Re(s) <»,-<■>< Im(s) < =>}). Moreover, the

transformed quantities in the left-hand sides of (1.5.1) - (1.5.2) are r e a l

v a l u e d f o r r e a l and p o s i t i v e values of s. On account of Schwarz' s

reflection principle of complex function t h e o r y , the r e l e v a n t transforms

then take on complexconjugate values in conjugate complex points of the s

-plane. Relative to (1.5.1) - ( 1 . 5 . 2 ) , we inversely have

Chapter 1 | Basic relations

(1.5.3) (v, p, T , e}(x,t)

[(2ivi)-i«

exp(st) {v, p, T, e}(x,s) ds]

x

t i

. „,-,(t)

and

when t

t t

e

(1.5.4) {f

y

, h

v

)(x,t)

ri»

[ (2-trl)-

1

f exp(st)

[t

y

, RyKx.s) ds]

X ( t

^(t)

when t ^ t

0

,

where we have introduced the characteristic function x

T

> pertaining to the

real interval I, which is defined as

1 when t e I,

(1.5.5) x

x

(t) =

{

0 when t i l .

I n ( 1 . 5 . 3 ) - ( 1 . 5 . 1 ) i t h a s been taken i n t o account t h a t the transformed q u a n t i t i e s a r e a n a l y t i c f u n c t i o n s of s in t h e r i g h t h a l f of the s - p l a n e . The a p p l i c a t i o n of ( 1 . 5 . 1 ) - ( 1 . 5 . 2 ) t o t h e e l a s t o d y n a m i c f i e l d e q u a t i o n s ( 1 . 1 . 1 ) - ( 1 . 1 . 2 ) y i e l d s ( 1 . 5 . 6 ) 3 . * (T l J +x . . ) - s P l= - fV ; 1 , ( 1 . 5 . 7 ) i O j V j + 3j V l) - a e j j - ftV;1J , 19

Chapter 1 | Basic r e l a t i o n s

where i t has-been taken i n t o account t h a t the elastodynamic f i e l d v a n i s h e s a t t = t0 - 0. Equations ( 1 . 5 . 6 ) - ( 1 . 5 . 7 ) a r e t h e e l a s t o d y n a m i c f i e l d

e q u a t i o n s in the time-Laplace transform domain.

The time-Laplace transform-domain c o n s t i t u t i v e r e l a t i o n s follow from ( 1 . 2 . 1 ) - ( 1 . 2 . 3 ) :

( 1 . 5 . 8 ) P i - P y V j

and

( 1 . 5 . 9 )

*U

ijpq (T„„pq + T ) , qp

where i t has been t a k e n i n t o a c c o u n t t h a t t h e media a r e l i n e a r and t i m e i n v a r i a n t in t h e i r elastodynamic b e h a v i o u r .

The boundary c o n d i t i o n s t h a t have been d i s c u s s e d i n S e c t i o n 1.3 a p p l y t o t i m e - i n v a r i a n t b o u n d a r i e s . C o n s e q u e n t l y , t h e s e b o u n d a r y c o n d i t i o n s a r e d i r e c t l y t r a n s f e r r e d t o t h e time-Laplace transform domain. The r e s u l t s are given in Table 1.3.

Table 1.3. Time-Laplace transform-domain boundary c o n d i t i o n s a t boundary S.

Type of boundary Boundary c o n d i t i o n

Firm c o n t a c t a t i n t e r f a c e v. and t . continuous a c r o s s S

S l i d i n g c o n t a c t a t i n t e r f a c e n . v . and n . t . continuous a c r o s s S »i i ~ i l

t - ( n . t ) n.-> 0 on S

Immovable rigid boundary v + 0 o n S

Chapter 1 | Basic r e l a t i o n s

1.6. The Time-Laplace Transform-Domain Elastodynamic R e c i p r o c i t y Theorem

In t h i s s e c t i o n , we d i s c u s s the time-Laplace t r a n s f o r m - d o m a i n r e c i p r o c i t y t h e o r e m . A r e c i p r o c i t y ' t h e o r e m i s a b i l i n e a r r e l a t i o n between the f i e l d q u a n t i t i e s in two n o n - i d e n t i c a l p h y s i c a l s t a t e s t h a t can o c c u r i n one and t h e same domain i n s p a c e . The B e t t i - R a y l e i g h r e c i p r o c i t y theorem a p p l i e s to a t i m e - i n v a r i a n t c o n f i g u r a t i o n . T h e r e f o r e , the r e c i p r o c i t y theorem i s f o r m u l a t e d i n t h e t i m e - L a p l a c e t r a n s f o r m d o m a i n . I t s t i m e - d o m a i n c o u n t e r p a r t f o l l o w s u p o n a p p l i c a t i o n of t h e i n v e r s e t i m e - L a p l a c e t r a n s f o r m a t i o n .

Let V be a bounded subdomain of R3, enclosed by a p i e c e w i s e smooth

boundary 8V with normal n, of u n i t l e n g t h , p o i n t i n g away from V ( s e e Figure 1 . 4 ) . The two e l a s t o d y n a m i c s t a t e s t h a t can o c c u r i n t h e domain V a r e r e f e r r e d t o as "A" and "B", r e s p e c t i v e l y . S t a t e A i s c h a r a c t e r i z e d by t h e

*A *A "A "A A A f i e l d q u a n t i t i e s {v , p , T , e }, the c o n s t i t u t i v e parameters {p , s ) and

"A "A

t h e s o u r c e d i s t r i b u t i o n s {fv , ftvJ. Likewise, s t a t e B i s c h a r a c t e r i z e d by

^ R **R **R R R R

the field quantities {v , p , t , e }, the constitutive parameters {p , a ) AR * R

and t h e s o u r c e d i s t r i b u t i o n s {f„ , ft..}. Neither the media nor the source d i s t r i b u t i o n s need be t h e same in the two s t a t e s . In s t a t e A t h e s-domain elastodynamic f i e l d e q u a t i o n s a r e given by (see ( 1 . 5 . 6 ) - ( 1 . 5 . 7 ) )

(1

-

6

-

1> 9

J *

( ?

i j

+

V '

3

P\ " "'},! '

(1.6.2) i O ^ V V V ^ ^ i j ' ^ l j '

and t h e c o n s t i t u t i v e r e l a t i o n s by (see ( 1 . 5 . 8 ) - ( 1 . 5 . 9 ) ) ( 1 . 6 . 3 ) P i = P t J v . , ( 1 . 6 . 4 ) eA - sA * ( TA + TA ) . i j i j p q pq qp

In s t a t e B t h e s-domain elastodynamic f i e l d e q u a t i o n s a r e given by

( 1 . 6 . 5 ) 3 j ^ \ . ♦ x ^ - s p \ - -fvB;1 ,

Chapter 1 | Basic r e l a t i o n s

( 1 . 6 . 6 ) i O ^ + S j V ^ ) - s ê6^ = ftj;ij ,

and t h e c o n s t i t u t i v e r e l a t i o n s by

d . 6 . 7 ) PB 1- PB 1 Jv Bj.

(1

-

6

-

8) ; B

u "

s B

i

J P q

^

BM + ;

V " •

If surfaces of discontinuity in the material properties are present in V, (1.6.1) - (1.6.2) in state A and (1.6.5) - (1.6.6) in state B are supplemented by boundary conditions according to Table 1.3

*R * R * R

- i • < ' - 6 . 2 > * y i < TB 1 J + ^Bj i

Multiplying ( 1 . 6 . 1 ) by v , , ( 1 . 6 . 2 ) by i ( x , , + T . . ) , ( 1 . 6 . 5 ) by

■ki and ( 1 . 6 . 6 ) by * (T A ♦ • ! *

o b t a i n the expression

v , and ( 1 . 6 . 6 ) by £(x ,t + x ,,) and combining the r e s u l t i n g e q u a t i o n s , we

, , , - . . r~A , , " B "B , ~B ,,*A "A >i ( 1 . 6 . 9 ) 3j [v i i ( T t. + x J t) - v . i (T . j + x . . ) ] :A ; B : B , "A A "A , rV ; i V i -f tV ; i j *( T i j + T J i ' - fB vA + nA MxB + ? ) rV ; i V i " v ; i j *U i j X j i; ' provided t h a t ( 1 . 6 . 1 0 ) p\ vB. - ;B 1 vki

"A *B *B . x "B ,,"A "A ,. n

" 6 i j "( T i j + T Ji> + 6 i j '( T i j + T j i) = ° *

The q u a n t i t y i n t h e l e f t - h a n d s i d e of ( 1 . 6 . 9 ) d e s c r i b e s t h e i n t e r a c t i o n between t h e two s t a t e s A and B. This i n t e r a c t i o n i s only d e p e n d e n t on t h e s o u r c e d i s t r i b u t i o n s of s t a t e s A and B, p r o v i d e d t h a t t h e media in t h e s t a t e s A and B a r e r e l a t e d t o each o t h e r such t h a t ( 1 . 6 . 1 0 ) i s s a t i s f i e d .

Chapter 1 | Basic relations

Fig. 1.1. The states A and B to which the Betti-Rayleigh reciprocity theorem applies.

Chapter 1 | Basic r e l a t i o n s

The use of ( 1 . 6 . 3 ) - ( 1 . 6 . 1 ) and of ( 1 . 6 . 7 ) - ( 1 . 6 . 8 ) in ( 1 . 6 . 1 0 ) show t h a t the l a t t e r i s e q u i v a l e n t to'

, , c 1 , \ B A B A

( 1 . 6 . 1 1 ) p , . = p . , and s . . = s . . .

When the media in the state A and the state B are one and the same, equation (1.6.11) implies that the constitutive parameters obey the relations p. . = p., and s. . = s . .. In that case the medium is called

ij ji ljpq pqij

reciprocal. Note that in particular an isotropic material (see (1.2.5) -(1.2.7)) is reciprocal. Equation (1.6.9) is the local form of the Betti-Rayleigh reciprocity theorem.

The integration of (1.6.9) over the domain V and the use of Gauss' theorem lead to (1

-

6

-

12)

l

9 V

[ * \ *

(

*

B

u

+ ;

V -

;

\ ^

A

i j

+ ;

V

]

»j

dA r: A ; B : B " A . " A . Jv L V ; l 1 V ; i j 2 i j j i ' - f,, . v . + n„ . . i( T . . + T . . ) I dV . V;i i V;ij 2 V i j j i/ J

Equation (1.6.12) is the global form of the Betti-Rayleigh reciprocity theorem.

CHAPTER 2

SYMMETRY CONSIDERATIONS IN ELASTODYNAMIC SCATTERING THEORY

l i s ( l e s enfants dans l e u r s jeux)

s o n t . . . amoureux des regies e t de l a

symêtrie (J. de La Bruyère)

In the realm of elastodynamic field theory, the term "symmetry" i s used t o

i n d i c a t e both g e o m e t r i c a l symmetry ( i . e . symmetry in the s p a t i a l

c o n s t e l l a t i o n of sources and boundaries) and m a t e r i a l symmetry ( i . e .

symmetry in the m a t e r i a l p r o p e r t i e s of the media). In general, both of

t h e s e w i l l a f f e c t t h e elastodynamic f i e l d in the c o n f i g u r a t i o n under

consideration.

Empty space has a very high degree of symmetry: every point i s l i k e

any o t h e r , and at no point t h e r e i s an i n t r i n s i c difference between the

s e v e r a l d i r e c t i o n s . This type of symmetry i s described by t h e r e a l

o r t h o g o n a l group 0 ( 3 ) . I n s e r t i o n of a geometrical object in space

r e s t r i c t s the symmetry to the group of mappings t h a t leave t h i s object

i n v a r i a n t . P h y s i c a l l y , t h i s object is f i l l e d with a medium, which again

imposes r e s t r i c t i o n s on the symmetry at hand, except, by d e f i n i t i o n , when

the medium i s homogeneous and isotropic.

In elastodynamic scattering theory, we consider a geometrical object

f i l l e d with a m a t e r i a l medium which shows c o n t r a s t with respect to i t s

surroundings. The symmetry of t h i s configuration is described by the group

of t r a n s f o r m a t i o n s t h a t map the object onto i t s e l f and that leave the

c o n s t i t u t i v e r e l a t i o n s of the media both in the s c a t t e r i n g object and i n

the embedding i n v a r i a n t . This group will be referred to as the symmetry

Chapter 2 | Symmetry considerations

group of the scattering configuration. As a consequence, the elastodynamic

f i e l d can be w r i t t e n as a superposition of a number of c o n s t i t u e n t s , each

of which exhibits particular symmetry properties and i s the s o l u t i o n of a

s e p a r a t e s c a t t e r i n g problem. Both the number of independent f i e l d

constituents and t h e i r symmetry p r o p e r t i e s are r e l a t e d to the symmetry

group of the scattering configuration.

In t h i s chapter, we r e s t r i c t ourselves to the special case where the

geometrical symmetry group constitutes a subgroup of the material symmetry

group. In that case, the symmetry group of the s c a t t e r i n g c o n f i g u r a t i o n

coincides with the geometrial symmetry group. . For an extensive treatment

of m a t e r i a l symmetries, the reader i s r e f e r r e d to T r u e s d e l l and Noil

(1965).

2 . 1 . Formulation Of The Scattering Problem

In an embedding of i n f i n i t e extent (three-dimensional Euclidean space R

3

),

a bounded o b j e c t i s present whose p r o p e r t i e s d i f f e r from t h o s e of i t s

surroundings. The object occupies the domain D inside a bounded, piecewise

smooth, closed surface 3D, with outward normal of u n i t length n. The

unbounded domain exterior to 3D is denoted as D, i . e . D = R

3

\(D U 3D) (see

Figure 2 . 1 ) .

We s h a l l d i s c u s s four types of scattering objects: (i) penetrable

objects, ( i i ) impenetrable, p e r f e c t l y r i g i d , immovable o b j e c t s , ( i i i )

i m p e n e t r a b l e p e r f e c t l y compliant o b j e c t s (voids) and (iv) o b j e c t s of

vanishing t h i c k n e s s ( s c r e e n s ) , such as r i g i d b a f f l e s , void c r a c k s and

f l u i d - f i l l e d c r a c k s . Let {v, p, T, e} denote the elastodynamic field

quantities in D, while in D they are denoted as {v, p, x, i } . The relevant

boundary conditions are l i s t e d below (see also Table 1.2).

Chapter 2 | Symmetry c o n s i d e r a t i o n s Incident field 9 a Y Scattered field

E

9 a Y Incident field

Fig. 2.1. The scattering configuration.

Chapter 2 | Symmetry considerations

(i) Penetrable objects. The particle velocity and the traction are continuous across 3D. Hence,

(2.1.1) v = v and t = t on 3D,

where the tractions on 3D are given by (see (1.3-5)),

(2.1.2) ti = i(ti + T ) n and -^ = i(x. + T.J) n on 3D.

(ii) Perfectly rigid, Immovable objects. The particle velocity vanishes upon approaching 3D via D. Hence,

(2.1.3) vi = 0 on 3D.

(ill) Perfectly compliant objects. The traction vanishes upon approaching 3D via 5. Hence,

(2.1.1) t, = 0 on 3D,

where the traction on 3D follows from (2.1.2).

(iv) Objects of vanishing thickness (screens). A screen is a two-sided, piecewise smooth, surface E with a, piecewise smooth, closed boundary curve C. The two sides of the surface are denoted by E and E , respectively. The unit normal to E is denoted by n , while n~ refers to the unit normal to E . The direction of n is connected to the orientation of C according to the right-hand rule (see Figure 2.2).

For a rigid baffle, the particle velocity vanishes on both faces of the screen. Hence,

(2.1.5) v - 0 on E*.

For a void crack, the traction vanishes on both faces of the screen. Hence,

Chapter 2 | Symmetry considerations

(2.1.6) on Z±.

For a f l u i d - f i l l e d crack, the normal component of the particle velocity and

the normal component of the t r a c t i o n are continuous across the s c r e e n ,

while the t a n g e n t i a l part of the t r a c t i o n vanishes on both faces of the

screen. Hence,

ni vi = ni *l on Z ,

( 2 . 1 . 7 ) 'n* t1 = n* t^ on Z ,

t . - (n* t . ) n* = 0 on Z±.

thicknass-»0

Fig. 2.2. Scattering object of vanishing thickness ("screen").

Chapter 2 | Symmetry considerations

Returning to the scattering problem, in accordance with the standard procedure in scattering theory, we introduce the incident field {v , p ,

T1, e } as the field that would be present in the configuration if the

scattering object showed no contrast with respect to its surroundings. The difference between the actual field and the incident field present in the configuration is everywhere denoted as the scattered field (see also Table 2.1),

(2.1.8)

, s s s s, , i i i i, „

{v , p , T , e } - lv-v , p-p , T-T , e-e } in D,

{Vs, ps, TS, e3} - {v-v1, p-p1, T - T1, i-e1} in D.

The incident field {v

ui

i

p " , T ~ , e } i s defined for a l l t e R and h i t s the o b s t a c l e a t t = t ~ . T h u s , in t h e domain D, t h e i n c i d e n t f i e l d vanishes i d e n t i c a l l y for a l l t e T = R\(T U 8T), where

( 2 . 1 . 9 ) { t t < t <

Table 2.1. Nomenclature of the different field constituents in the configuration.

Field quantities Type Spatial support Temporal support v, p, T, e total field v, p, T, e total field i i i i v ,p ,T ,e incident field s s s s v ,P ,T ,e scattered field R3\3D

Chapter 2 | Symmetry considerations

3 3 3 3

Since the scattered field {v , p , T , e ) is in a causal relationship to the incident field, it vanishes identically for all (x,t) e R' « T (initial condition).

Next, we specify the material properties of the media that are present in the configuration. The medium surrounding the scattering object is taken as simple as possible, viz. a homogeneous and isotropic perfect solid. Then, its elastodynamic behaviour is characterized by the scalar volume density of mass p, Poisson's ratio a and Young's modulus Y (of.

Section 1.2). Hence, p = p v. and (2.1.10)

5

u - -

( 5 / ï ) 6

ij V

+ [(1 + 5 ) / ?

l *

( ï

u

+

V •

where p, a and Y are (real and positive) constants. As regards the scattering object, we only assume that it is a perfect solid (see Section 1.2) and leave open the possibility that it is inhomogeneous and/or anisotropic. Hence,

(2.1.11) p, = p.. v. and e.. = s.. £ ( T + T ) .

i Hij J ij ijPQ PQ qP

The scattering problem is now formulated as follows. The incident field (generated by sources in D ) , satisfies the elastodynamic field equations (1.1.1) - (1.1.2) together with the constitutive relations (2.1.10) in the entire three-dimensional space R3. Given the incident

field, the scattered field is to be determined from the elastodynamic field equations (1.1.1) - (1.1.2) together with the constitutive relations (2.1.10) in D and (2.1.11) in D such that both the initial condition and relevant boundary condition are satisfied.

Although we do not make any use of operator theory in our analysis, we do make use of its notational convenience. As a first step, we introduce the set of matrices

(2.1.12) T,(R3) x T2(R3) = ) T | T = [t1 , t2 J1

Chapter 2 | Symmetry c o n s i d e r a t i o n s

where t1 e T ^ R3) i s any t e n s o r of rank one ( v e c t o r ) and where t2 e T2(R3)

i s any t e n s o r of rank two, b o t h d e f i n e d in t h r e e - d i m e n s i o n a l E u c l i d e a n s p a c e R3. As b e f o r e , t h e s u b s c r i p t s i and j i n ( 2 . 1 . 1 2 ) run through t h e

v a l u e s 1,2 and 3; j being f r e e when i i s f i x e d . F i n a l l y , "T" r e f e r s t o t h e t r a n s p o s e d m a t r i x . Now, t h e elastodynamic f i e l d q u a n t i t i e s a r e c o l l e c t e d in a matrix F according t o

( 2 . 1 . 1 3 ) F = [v. , - H-t + T ) ] ' , F e T,(R3) * T2(R3)

Likewise, the matrix of volume sources is introduced as

(2.1.14) Qv - [fV ; i , nV ; l j] ' , Qv e T,(R3) x T2(R3) .

The elastodynamic field equations (1.1.1) - (1.1.2), together with the constitutive relations (1.2.1) - (1.2.2) are now written as the matrix equation

(2.1.15) E F - Qv ,

where the square matrix E of tensor operators is given by

E : T^R3) x T2(R3) •» T,(R3) x T2(R3) , (2.1.16) E1' E1 2 ip ipq E2 1., E2 2 . iJP iJPQ in which (2.1.17) T,(R3) ■» T,(R3) , E" i p t y * ' ^ " fip(x) 3t t'p^.tJ '

Chapter 2 | Symmetry c o n s i d e r a t i o n s E1 2 : T2(R3) ->• T , ( R3) , ( 2 . 1 . 1 8 ) E,

\pa *W

Xtt) =

*

(

Vi

P + 3

p V

t 2

pQ

U > t )

'

E2 1 : T , ( R3) - T2( R3) , ( 2 . 1 . 1 9 ) and E2 2 : T2(R3) + T2(R3) , ( 2 . 1 . 2 0 ) E2 2. . t2 ( x , t ) - s . . (x) 3. t2 ( x , t ) . u p q m ijpq t pq Equations ( 2 . 1 . 1 5 ) - ( 2 . 1 . 2 0 ) c o n s t i t u t e t h e s t a r t i n g p o i n t of our a n a l y s i s of symmetry p r o p e r t i e s of the elastodynamic f i e l d .

2 . 2 . Symmetry Groups

In t h e p r e s e n t s e c t i o n , we i n t r o d u c e t h e n o t i o n of symmetry group ( o r t r a n s f o r m a t i o n group) in e l a s t o d y n a m i c s .

Let G be the ( g e o m e t r i c a l ) symmetry g r o u p of D ( s e e M i l l e r , 1972, p p . 8 - 1 3 ) . S i n c e D i s of bounded e x t e n t , G c o n s i s t s of t h e orthogonal t r a n s f o r m a t i o n s i n t h r e e - d i m e n s i o n a l E u c l i d e a n s p a c e R3 t h a t l e a v e D

i n v a r i a n t . T h u s , G i s a s u b g r o u p of t h e r e a l o r t h o g o n a l group 0 ( 3 ) . S p e c i f i c a l l y , t h e s e t r a n s f o r m a t i o n s can e i t h e r be r o t a t i o n s around a f i x e d a x i s , r e f l e c t i o n s in a plane o r r e f l e c t i o n s in a point (see Weyl, 1952, p p . 1 2 7 - 1 3 5 ) . The e l e m e n t s of G a r e d e n o t e d a s g o r a s g , w i t h a e' (1 , 2 , . . . , | G|} , where | G | d e n o t e s the o r d e r (or c a r d i n a l i t y ) of the group. L e t , f u r t h e r , [d(g) ] denote t h e matrix r e p r e s e n t a t i o n of g r e l a t i v e t o t h e c h o s e n C a r t e s i a n r e f e r e n c e frame. In g e n e r a l , the image under g (or a q u a n t i t y a s s o c i a t e d w i t h g) i s denoted by a prime. Hence,

Chapter 2 | Symmetry considerations

gQ : D - D

(2.2.1) x' = g X

a with a e {1,2,...,|G|}.

X'i " ^ i j Xj

Each of the matrices [d(g ) ] is orthogonal, i.e.

(2.2.2) d ( ga)p i d ( ga)p. - 6y with a e {1,2,...,I G| }.

Of t h e s e t of m a t r i c e s {[d(g ) ] | o = 1 , 2 , . . . , | G | } a l l m a t r i c e s a r e n o n

-s i n g u l a r ( c f . ( 2 . 2 . 2 ) ) and moreover they -s a t i -s f y t h e r e l a t i o n

( 2 . 2 . 3 ) d ( ga)i p d ( g6)p. = d ( gag6 ) i J with a , 6 e { 1 , 2 , . . . , | G | } ,

where the product of the transformations g g. is defined in the usual way:

a 3

f i r s t gD i s a p p l i e d t o t h e operand, followed by g . I n accordance with t h e

p a s t a n d a r d terminology i n group t h e o r y , t h e s e t of m a t r i c e s { [ d ( g ) ] | o = 1 , 2 , . . . , |G| } , w i t h t h e u s u a l m a t r i x m u l t i p l i c a t i o n , t h e r e f o r e forms a r e p r e s e n t a t i o n of t h e symmetry group. In t h e a n a l y s i s , we need a c e r t a i n r e p r e s e n t a t i o n of t h e symmetry g r o u p G i n terms of o p e r a t o r s t h a t o p e r a t e on t h e e n t i t i e s a s s o c i a t e d with t h e elastodynamic f i e l d under c o n s i d e r a t i o n : F and Qv , r e s p e c t i v e l y . To

t h i s e n d , f o r each g e G, a s q u a r e m a t r i x U(g) of t e n s o r o p e r a t o r s i s defined through U(g ) : T,(R3) x T2(R3) - T,(R3) * T2(R3) ( 2 . 2 . H ) _{w i t h a e { 1 , 2 | G | } ,}

u(g

a

)

u»<g

a

>

l p

o

0 U2 2( g ) . . a i j p q

Chapter 2 | Symmetry considerations

in which

ü

, 1

( g

a

) : T , ( R

3

) - T , ( R

3

)

(2.2.5) t

l

' - U

l l

( g

Q

) t

l

with

a e {1,2,...,|G|}

t''.(x',t) - d ( g

a

)

i p

t'

p

(x,t)

and

U " ( g

a

) : T

2

(R

3

) - Ti(R

3

)

(2.2.6) t

2

' = U

2 2

(g ) t

2

with a e {1 ,2,...,|G|}.

t«' (x'.t) - (Kg ) d(g ) t

2

(x,t)

ij a ip a jq pq

As before, in (2.2.5) (2.2.6) we have put x' = g x. Equations (2.2.4)

-(2 2.6) express the relation between T and T' = U(g)T in the geometrically

equivalent points of observation x and x', respectively. In Figure 2.3,

the action of the operator U ' M g ) on the vector t' is illustrated. As an

example we have taken the symmetry group of a triangular prism of finite

height.

Algebraically, the set of operators {U(g ) | a = 1 , 2 , . . . , | G | } is a

representation of the symmetry group G, since they satisfy the relation

(2.2.7) U(g

o

) U(g

g

) = U(g

a

g

B

) with a, B e {l,2 | G | }

-From (2.2.1) - ( 2 . 2 . 6 ) , it follows that (2.2.7) is equivalent to

(2.2.8) U

l ,

( g

a

) U'Mgg) = U " ( g

a

g

g

) with a, 8 e {1,2 |c|}

and

Chapter 2 | Symmetry c o n s i d e r a t i o n s

F i g . 2 . 3 . R e l a t i o n between t1 and t1 in t h e p o i n t s of

o b s e r v a t i o n x and x ' , r e s p e c t i v e l y ; x ' i s t h e r o t a t i o n of x through n around t h e symmetry a x i s CF

( c r o s s s e c t i o n of a t r i a n g u l a r prism of f i n i t e h e i g h t ) .

(2.2.9) U22(g ) U2 2(g„) = 0 " ( g g J with a, 8 E { l , 2 I Gl }.

01 p Ct p

The a l g e b r a i c p r o p e r t i e s of t h e s e t of o p e r a t o r s {u(g ) | a = 1 , 2 , . . . , | G | } a r e an immediate c o n s e q u e n c e of t h e f a c t t h a t t h e s e t of m a t r i c e s { [ d ( g ) ] | o = 1 , 2 , . . . , | G | } forms a r e p r e s e n t a t i o n of t h e symmetry group G. To p r o v e ( 2 . 2 . 8 ) , l e t x ' = g . x and a c c o r d i n g l y t1 = ü " ( g J t ' .

L i k e w i s e , l e t x ' ' = g x ' and a c c o r d i n g l y t1 = Ul I( g ) tl . A p p l i c a t i o n of

Chapter 2 | Symmetry considerations

(2.2.10) t ' ^ x ' - . t ) = d ( g

a

)

l p

t '

p

( x ' , t )

d

V i p *(g

B

)p

q

t

l q

( x , t )

-

d ( 8

«

8

B

)

i q

t

V

X

'

t )

'

where we have used the r e l a t i o n ( 2 . 2 . 3 ) . To prove ( 2 . 2 . 9 ) , l e t x ' = g x

t P

and accordingly t

2

= U

2 2

(g„)t

2

. Likewise, l e t x'* = g x' and accordingly

11 i P a

t* = U

22

(g ) t

2

. Application of (2.2.6) twice r e s u l t s in

(2.2.11) t

2 l j (

x " . t ) = d ( g

a

)

i p

d ( g

a

)

J q

t ^

p q

( x ' , t )

"

d ( 8

a

}

l p « « a ' j q

d

V p r « V q s

l

\ s

U

'

l )

=

d

W i r ^ a V j s ^ r s ^ ^ '

where we have used the r e l a t i o n ( 2 . 2 . 3 ) . Since by assumption x ' ' = g g.x

ii ii a P

and t1 = Ul l( g ) Ul l( gQ) t » and t2 = U2 2(g ) U2 2( g J t2, where t1 and t2

ct p ct p

are a r b i t r a r y , the r e l a t i o n s (2.2.8) and (2.2.9) have been established.

A n a l y t i c a l l y , t h e s e t of o p e r a t o r s {U(g ) | a = 1 , 2 , . . . , | G | }

i n t e r a c t s with the d i f f e r e n t i a t i o n o p e r a t o r s 3. and 3 . , r e s p e c t i v e l y .

Obviously, the operators U(g ) and 3 commute; hence,

( 2 . 2 . 1 2 ) 3. Ul l( g ) = Ul l( g ) . 3 . with a e {1,2 |G|}

I* Ct i p Ct i p v

and

(2.2.13) 3

t

"

2 2

( 8

a

)

i j p q

= « " ( « ^ i j p q

3

t

w i t h a e

I

1

'

2

I

G

U

-The operator products in (2.2.12) - (2.2.13) are defined in the usual way:

for example in the left-hand side of (2.2.12) f i r s t 3 i s applied t o the

o p e r a n d , followed by U ' ' ( g ) . In our a n a l y s i s , we f u r t h e r need the

properties

Chapter 2 | Symmetry c o n s i d e r a t i o n s ( 2 . 2 . 1 1 a ) 3j U2 2( Sa)i j p q - u"(8a)i p 3q w i t ' h a e { l , 2 | G | } , ( 2 . 2 . 1 1 b ) 3. U2 2( ga)j i q p = u , 1( ga)l p 3q with a e {1.2 N } and with a e {l ,2 | G | } , with a e {l ,2 | G | }. Again, in ( 2 . 2 . 1 1 ) - ( 2 . 2 . 1 5 ) the p r o d u c t s of t h e o p e r a t o r s a r e d e f i n e d i n i

t h e u s u a l way. To p r o v e ( 2 . 2 . 1 1 a ) , we put x ' = gx and a c c o r d i n g l y t2 =

U2 2( g ) t2 for any g e G. F u r t h e r , d i f f e r e n t i a t i o n w i t h r e s p e c t t o x ' i s

denoted as 3 ' . A p p l i c a t i o n of ( 2 . 2 . 5 ) - ( 2 . 2 . 6 ) r e s u l t s i n

( 2 . 2 . 1 6 ) 3 ' j t / ^ x ' . t ) = d ( g )j p 3p d ( g )i r d ( g )j s t2 r s( x , t )

= d ( g )i r 3s ^ r s ^ - t ) '

where we have used t h e r e l a t i o n ( 2 . 2 . 2 ) . To prove ( 2 . 2 . 1 5 a ) , we p u t x ' =

i gx a n d a c c o r d i n g l y tl = U " ( g ) t1 f o r a n y g e G. As b e f o r e , d i f f e r e n t i a t i o n with r e s p e c t t o x ' . i s d e n o t e d a s 3 ' . A p p l i c a t i o n of ( 2 . 2 . 5 ) - ( 2 . 2 . 6 ) r e s u l t s in ( 2 . 2 . 1 7 ) 3 ' j t1' ( x \ t ) = d(g) 3 <Kg) tl ( x , t ) " ^ i p <"8>J q 3p t *q( x , t ) .

In ( 2 . 2 . 1 6 ) and ( 2 . 2 . 1 7 ) we have used t h e p r o p e r t y

( 2 . 2 . 1 8 ) 3 ^ = d ( ga)l p 3p with a e (1,2 | G | } ,

which implies that the operator 3 behaves like a tensor of rank one: 3' = Ull(g )3, with a e {1,2,...,|G|}. Since in (2.2.16) and (2.2.17) tl and t2

(2.2.15a) 3 U"(g ).„

°a ljpq p

(2.2.15b) 3,

Chapter 2 | Symmetry considerations

are a r b i t r a r y , the r e l a t i o n s (2.2.14a) and (2.2.15a) have been established.

The r e l a t i o n s (2.2.11b) and (2.2.15b) follow from (2.2.14a) and (2.2.15a),

respectively, with the aid of the property

(2.2.19) U

2 2

( g

a

)

i J p q

= "

2 2

( 8

a

)

j i q p

with a e {1,2 | G | } ,

which is easily inferred from ( 2 . 2 . 6 ) .

2 . 3 . Invariance Theorem

In the present s e c t i o n , we show that the scattering problem as formulated

in S e c t i o n 2.1 i s (form) i n v a r i a n t under the symmetry group of the

s '

scattering configuration. By t h i s we mean that the s c a t t e r e d f i e l d F

U(g)F i s a s s o c i a t e d with a s c a t t e r i n g problem similar to the one with

which F i s associated. The difference being that the former corresponds

i ' i

to the i n c i d e n t f i e l d F = U(g)F , which i s generated by the source

d i s t r i b u t i o n Q

v

' = U(g) Q

v

, while the l a t t e r corresponds t o the incident

1

f i e l d F , g e n e r a t e d by the source d i s t r i b u t i o n Q„. In our analysis, we

r e s t r i c t ourselves to the special case where the geometrical symmetry group

c o n s t i t u t e s a subgroup of the material symmetry group. In that case, the

symmetry group of the s c a t t e r i n g c o n f i g u r a t i o n c o i n c i d e s with t h e

geometrical symmetry group G. The proof consists of three p a r t s . F i r s t l y ,

the elastodynamic operator E i s shown to be invariant under the geometrical

symmetry group, provided t h a t the c o n s t i t u t i v e r e l a t i o n s (2.1.10) and

(2.1.11) show the required (material) symmetry p r o p e r t i e s . Secondly, the

i n v a r i a n c e of t h e boundary c o n d i t i o n s of the four types of scattering

problems (cf. Section 2.1) is established. Thirdly, the i n i t i a l c o n d i t i o n

is invariant under the symmetry group.

2.3.1 ■ Invariance of the elastodynamic field equations

The elastodynamic operator E i s invariant under the symmetry group G, i . e .

Chapter 2 | Symmetry considerations

(2.3.1) E U(ga) = U ( ga) E wi th a e {1 , 2 , .. . , | G | },

provided that the constitutive parameters (cf. (2.1.10) - (2.1.11)) satisfy the (material) symmetry relations

(2.3.2) P l J( x ' ) = d ( ga)l p d ( ga)J q Pp q( x ) with a e {1,2 |Gj}

and

(2.3.3) sl j k l( x ' ) - d ( ga)i p d ( ga) .q d ( ga)k r d ( ga)l s sp q r s( x )

with a e {l ,2 |G| },

in which x' = g x. Equations (2.3.2) - (2.3-3) express that the material properties of the medium should be identical in the two geometrically equivalent points of observation x and x', respectively. In a homogeneous and isotropic perfect solid, (2.3.2) and (2.3-3) are always satisfied. Thus, in particular the property holds in the embedding D.

It is seen from (2.1.16) and (2.2.4) that (2.3.1) is equivalent to

(2.3.-4) E>» U'Mg ) = U"(g ) E>1

a a with a e (1,2,...,|G|},

(2.3.5) E1 2 U"(gQ) = U1,(gQ) E1 2 with a e {1,2,...,|G|},

(2.3.6) E2 1 Ul,(ga) = U22(ga) E2 1 with a e {l ,2, . . ., |G|

and

(2.3.7) E2 2 U22(g ) = U22(g ) E2 2 with a e {l ,2 |G| }.

To prove (2.3.4), we use (2.1.17), (2.2.5), (2.2.12) and (2.3.2) to obtain for any g e G (as before, x' = gx and t1 = Ull(g)tl)

Chapter 2 | Symmetry considerations

(2.3.8) pip(x*) 3t t''p(x-.t) = d(g)ir d(g)p 3 prs(x) 3t d(g)p q t'q(x,t)

where we have used the property (2.2.2). To prove (2.3.5), we use (2.1.18) and (2.2.11) to obtain for any g e G

(2.3.9) E»»lpq U » ( g )p q r 3 = * (3q «lp ♦ 3p 6lq) U » ( g )N r s

= i Ui p U " ( g )p p 3a + 6lq U - ( g )q s 3r]

-U'»(g)l pi O s 6pr + 3p 6ps)

u lg;ip prs

To prove (2.3.6), we use (2.1.19) and (2.2.15) to obtain for any g e G

( 2 . 3 . 1 0 ) E »1 J p U > M g )p q - i O j 6i p + 3 i 6J p) U - ( g )p q - * " i pU , , ( 8 )J p r Qar + SJ pU , , ( 8 )l p r « l8r3 " U 2 2 ( g )i j p r * ( 3r «pa. + 3p W ' U"( g )U p r E" p rq • To p r o v e ( 2 . 3 - 7 ) , we use ( 2 . 1 . 2 0 ) , ( 2 . 2 . 6 ) , ( 2 . 2 . 1 3 ) and ( 2 . 3 - 3 ) t o o b t a i n f

for any g e G ( a s b e f o r e , x ' = gx and t * = U " ( g ) t * )

( 2 . 3 - 1 1 ) sl j p qC x ' ) 3t t » 'p q( x - . t ) - d ( g )i k d ( g )j l d ( g )p m d ( g )q n sk l m n( x )

• 3t d ( 8 ) Pr d ( 8 )q s t 2r s( x'f c )

■ d ( g )i k d ( g )i X sklmn(x) 3t t ^ ( x , t ) ,

Chapter 2 | Symmetry c o n s i d e r a t i o n s

where-we have used the p r o p e r t y ( 2 . 2 . 2 ) . Since in ( 2 . 3 . 8 ) and ( 2 . 3 . 1 1 ) t1

and t2 are a r b i t r a r y , r e l a t i o n s ( 2 . 3 . 4 ) - ( 2 . 3 . 7 ) have been e s t a b l i s h e d .

2 . 3 . 2 . I n v a r i a n c e of t h e boundary c o n d i t i o n s

The boundary c o n d i t i o n s imposed a t 3D a r e i n v a r i a n t under t h e symmetry group G. As the boundary 3D remains i n v a r i a n t under t h e symmetry g r o u p G, i t i s e a s i l y seen t h a t t h e normal n of u n i t length p o i n t i n g i n the outward d i r e c t i o n of D i s i n v a r i a n t as w e l l , i . e .

( 2 . 3 . 1 2 ) n ^ x ' ) = d(gQ) n (x) on 3D a e { l , 2 | G | } ,

where x ' = g x. As a c o n s e q u e n c e , t h e r e l a t i o n ( 2 . 1 . 2 ) b e t w e e n t h e t r a c t i o n and t h e s t r e s s on 3D i s i n v a r i a n t under t h e symmetry group ( i n Figure 2.4 t h i s i s i l l u s t r a t e d for t h e symmetry group of a t r i a n g u l a r prism of f i n i t e h e i g h t ) . Indeed, for any g e G,

( 2 . 3 - 1 3 ) t ' ^ x ' . t ) = d ( g ) .p tp( x , t )

"

d ( g )

i p *

( T

pq

+

V

( x

'

t }

V

X)

" d ( g )iP d ( g )j q *( Tp q + V U , t ) d ( 6 )j r " r( x )

= è ( x ' + T ' .i) ( x ' , t ) n . ( x ' ) on 3D,

where we have adopted the notation of (2.2.5) - (2.2.6).

The four types of boundary conditions considered in Section 2.1 are dealt with below.

(i) Penetrable objects. From (2.1.1) and (2.2.5), it follows that for any g E G

Chapter 2 | Symmetry considerations

2.4. The tractions t and t' in the points of observation x and x', respectively; x' is the rotation of x through IT around the symmetry axis AD (cross section of a triangular prism of finite height).

v'^x'.t) - d(g)l p vp(x.t)

■

d ( g )

i

P

V

x , t )

= v'^x' ,t) on 3D

Chapter 2 | Symmetry c o n s i d e r a t i o n s

( 2 . 3 . 1 5 ) t ' ^ x ' . t ) = d(g) t ( x , t )

"

d ( g )

ip V * ' "

= t ' . ( x ' . t ) on 3D.

(ii) Perfectly rigid, immovable objects. From (2.1.3) and (2.2.5), it

follows that for any g e G

(2.3.16) v'^x'.t) - d(g).

p

v (x,t)

= 0 on 3D.

(Ill) Perfectly compliant objects. From (2.1.1) and (2.2.5), it follows

that for any g e G

(2.3.17) t y x ' . t ) = d(g) t (x,t)

= 0 on 3D.

(iv) Objects of vanishing thickness (screens). We distinguish between

planar screens and non-planar screens. For the first class, the plane in

which the screen is located is itself a geometrical symmetry plane. In the

present analysis, we confine ourselves to the planar screens. For the

non-planar screens, the analysis runs along the same lines as under ( i ) , (ii)

and (ill).

We have transformations that map

l~ onto itself and the Reflection

in the plane of the screen, denoted by r, that maps E~ onto E . For the

first, the normal vectors are invariant,

(2.3.18) n

±i

(gx) = d(g) n

±p

(x) on

z

±

,

Chapter 2 | Symmetry considerations

(2.3.19) n^rx) =

d

(

r

)

i p n

V

x ) on l±

>

where i t should be remembered t h a t r = r ~ ' . In both c a s e s , the o r i e n t a t i o n o f t h e b o u n d a r y c u r v e C r e m a i n s u n c h a n g e d . The two t y p e s of t r a n s f o r m a t i o n s a r e v i s u a l i z e d in Figure 2 . 5 , where the screen c o n s i s t s of a t r i a n g u l a r prism of vanishing h e i g h t .

In t h e p r e s e n t s u b s e c t i o n , we only d e a l w i t h t h e r e f l e c t i o n . For t h e o t h e r t r a n s f o r m a t i o n s , the a n a l y s i s runs along t h e same l i n e s as under

( i ) , ( i i ) and ( i i i ) . As a c o n s e q u e n c e of ( 2 . 3 - 1 9 ) , we o b t a i n f o r t h e t r a c t i o n on both f a c e s of t h e screen ( 2 . 3 . 2 0 ) t ' ^ x ' . t ) = d ( r ) t ( x , t )

■

d ( r )

i p *

( T P

q

+

V

U > t )

"V

X ) " d<r )i p d ( r )j q *( Tp q + V ( X' t } d ( r )j r " VX ) = i(.T' i. + T ' . . ) ( x ' , t ) n+. ( x ' ) when x ' e E+,

where we have put x ' = r x . For a r i g i d (immovable) b a f f l e , we have from ( 2 . 1 . 5 ) and ( 2 . 2 . 5 )

( 2 . 3 . 2 1 ) v ' . ( x ' . t ) = d ( r ) vp( x , t )

For a void crack, we have from (2.1.6) and (2.2.5)

(2.3-22) t*.(x',t) = d(r) t (x,t)

when x' e

I'

= 0 when x' e Z .