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Scattering of elastic waves by an arbitrary small imperfection in the surface of a half-space
0022-5096192 $5.00+0.00 f‘ 1992Pergamon PressLtd
J. /Mech.Phys. Solids Vol. 40. No. 8, pp. 1683-1706,1992 Printedin Great Britain.
SCATTERING OF ELASTIC WAVES BY AN ARBITRARY SMALL IMPERFECTION IN THE SURFACE OF A HALF-SPACE I.
Department
of Mathematics,
D.
ABRAHAMS
Keele University,
Keele, Staffs. ST5 .5BG, U.K
and G. R. Department
of Mathematics,
WICKHAM
The University
(Received 25 September
of Manchester,
199 1 ; in revised form
Manchester
18 February
M 13 9PL, U.K.
1992)
ABSTRACT THIS PAPER, the first of two articles,
is concerned with the scattering of elastic waves by arbitrary surfacebreaking or near surface defects in an isotropic half-plane. We present an analytical solution, by the method of matched asymptotic expansions, when the parameter E, which is the ratio of a typical length scale of the imperfection to the incident radiation’s wavelength, is small. The problem is formulated for a general class of small defects, including cracks, surface bumps and inclusions. and for arbitrary incident waves. As a straightforward example of the asymptotic scheme we specialize the defect to a two-dimensional circular void or protrusion, which breaks the free surface, and assume Rayleigh wave excitation; this inner problem is exactly solvable by conformal mapping methods. The displacement field is found uniformly to leading order in E, and the Rayleigh waves which are scattered by the crack are explicitly determined. In the second article we use the method given here to tackle the important problem of an inclined edge-crack. In that work we show that the scattered field can be found to any asymptotic order in a straightforward manner, and in particular the Rayleigh wave coefficients are given to O(E’).
1.
INTRODUCTION
LINEAR HARMONIC WAVE SCATTERING in a semi-infinite elastic body is a subject which has, through its numerous applications, received a considerable amount of analytical and numerical study in recent years. Even in the simplest case, of a homogeneous isotropic material, there are few closed form analytical solutions. In particular, if the body has inclusions or surface defects, see Fig. 1, none are known. The principal reason for this is that wave motion in such bodies is coupled at the free surface and the associated differential equations and boundary conditions do not admit a simple continuation formula such as the Schwarz reflection principle available in the corresponding classical problems of acoustics in fluids. Accordingly most of the quantitative work has centred on the numerical treatment of boundary integral equations, see for example ZHANG and ACHENBACH (1988a, b), MENDELSOHN et al. (1980), BRIND and ACHENBACH (1981) or infinite algebraic systems, GREGORY (1967). As far as
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I. D. ABRAHAMS and G. R.
WICKHAM
obtaining explicit analytical estimates is concerned, the usual methods for an infinite medium (separation of variables and integral equations) do not in general, carry over in a simple way. Consequently, when the scatterer is in or near the surface, only integral equation techniques have so far been successfully applied to the simplest cases (WICKHAM, 1977, 1980). When the influence of the free surface is small (DATTA and SABINA, 1986) or a small surface irregularity belongs to a special class (SABINA and WILLIS, 1977), the method of matched asymptotic expansions has proved fruitful. Here, we introduce a general and systematic approach based on the method of matched asymptotic expansions to obtain analytical estimates for the field scattered by small surface irregularities occupying a compact region. There are several notable difficulties for such problems which means that the work presented here is not just ;I direct extension of the analysis reviewed by DATTA and SARINA (19X6). Firstly. wc require a way of postulating the form of the outer expansion. In acoustic wave propagation in fluids, or SH waves in solids, it is relatively simple to gauge the form of the sources of the sound on physical grounds (SABINA and WILLIS, 1975). in elasticity, on the other hand, the coupling between the shear and compressional motions at the scattering boundaries makes such heuristic judgements less certain. For example, the expansion of the local tractions in the in-plane model posed by SABINA and WILLIS (1977) does not include variations perpendicular to the free surface. That this is true for all surface irregularities is not immediately obvious to the present authors, but is confirmed for compact scatterers by our rigorous expansion derived in Section 4. Another technical difficulty associated with any outer expansion is the determination of its analytical behaviour as we approach the inner region. Thirdly, the inner problems for any particular geometry are themselves intricate, although given that in general they may be posed using the apparatus of classical elastostatics, e.g. the Airy stress function. there is a reasonable selection of tractable cases of physical significance. SABINA and WILIJS (1977) confined their attention to a class of non-compact irregularities which are solvable in the inner region by conformal transformation, whereas we concern ourselves with compact bodies with sharp edges. The contrast between the usual formulation of elastostatic and dynamic problems raises our fourth dilemma, namely, to what physical quantities should we apply the asymptotic matching principle‘? WC show by specific example that it is sufficient in half-space problems to match on the dilatation only. This work is presented in two parts: the first is largely expository of our general approach while part two, henceforth denoted by II, concentrates on the detailed calculation for an inclined edge crack (ABRAHAMS and WITKHAM, 1992). Section 2 of this first paper gives a general formulation of the scattering problem for an ensemble of voids and inclusions in or near the free surface of the half-space. The following section introduces integral representation for the Lami: potentials 4. $ for the gcncral case. These relate the scattered field (4‘, $‘) to the unknown components of stress and displacement on the scattering obstacles and allow us. in Section 4, to derive rigorously the form of the outer expansions. For example, when the scatterer is a single void, it is readily seen that the outer field for 4 is, to leading order, a compressional source at the origin. Similarly. if the scatterer is a rigid obstacle, this appears to a distant observer to exert a point body force at the origin on the elastic material. The method in Section 4 provides a systematic way of developing the outer field to any order without
Scatteringfrom a surface imperfection
1685
cross-referencing the inner field and, in Section 5, the two are related through a statement of the asymptotic matching principle. We demonstrate the approach of this paper by reference to a simple specific example, namely the scattering by a small surface irregularity which takes the shape of an arc of a circle and which breaks the surface; i.e. the circular “bite” (see Fig. 3) or the mathematically equivalent circular protrusion. In Section 6 we specialize the results of Section 4 to this case and in Section 7 we solve the inner problem. Here, we are able to determine explicitly the appropriate elastostatic field using the well known bipolar transformation. Finally, in Section 8, we use the asymptotic solution obtained for this example to calculate the reflected and transmitted Rayleigh waves as a function of the depth of the centre of the arc. This result shows how the scattering varies with the geometry of the irregularity at low frequencies. This paper provides an introduction to the application of matched asymptotic expansions to an intrinsically difficult class of problems and we have tried, by way of what is probably the simplest example, to demonstrate its workings without overburdening the reader with the algebraic details. The circular “bite” example calculation discussed here has a number of applications in geophysics and in the ultrasonic detection of surface imperfections. Further, the asymptotic result will certainly be of use in testing the output from purely numerical codes for solving this class of problem. In the following, part II, we provide a full description of the calculation for an inclined edge crack. This is a particularly important physical example and has some interesting mathematical features worthy of separate and detailed treatment. In particular, the inner problem gives rise to an unusually tricky Wiener-Hopf problem first discussed by KHRAPKOV (1971). Further, we perform the analysis up to third order in that example since only then does the inner solution depend on the inertia terms in the differential equation.
2.
FORMULATION OF THE BOUNDARY VALUE PROBLEM
We consider the two-dimensional problem of a homogeneous isotropic elastic halfspace, occupying the region y > 0, - co < x < co, whose surface is free from applied tractions and which contains an edge crack or submerged void or inclusion close to or in the surface. We develop a general theory for this problem but will later specify the surface imperfection to a circular defect (Section 6) and to an edge crack (pai-t II) as examples. The defect is irradiated by a time harmonic disturbance, with radian frequency w, travelling from infinity (see Fig. 1). The latter may take the form of a Rayleigh wave travelling in the positive x direction, say, or a compressional or shear body wave. For small amplitude incident waves we know that the steady state elastic displacement vector u = (u, P) satisifes the partial differential equation ,~V*u+(/Z+~)graddivu+p,,o~u
= 0,
(2.1)
where V2 is the two-dimensional Laplacian operator, A and @are the Lamk constants, pa is the material density, and the sinusoidal time dependence in u has been suppressed for brevity. On the free surface, JI = 0, of the material, the boundary conditions are
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I. D. ABRAHAMS and
G. R. WICKHAM
FIG. 1. The general physical configuration showing small defects such as cracks, rigid inclusions protrusions confined to a semi-circle of radius CIcentred at the origin.
cri2(X, 0) = 0,
and surface
(2.2)
where o,,(-u,y) is the stress tensor and, throughout, we shall identify the x and J directions with the subscripts 1 and 2 respectively. The boundary conditions on the defect depend on its physical nature. In particular, if it is a cavity or crack then again the tractions vanish, i.e. G’,~L’~ = 0,
rEdC,
(2.3)
where v = (V,, v2) is the outward normal vector on the void boundary dC and r is the position vector. For a fixed rigid inclusion, say, the displacements vanish U, = 0,
(2.4)
on the boundary ret3D. The analysis we shall present here is trivially extended to more general boundary conditions though in the interest of clarity of exposition we shall omit such details. Now, if we express the displacement vector in the form ” = “(‘I +“(3)
(2.5)
where u(j) is the incident field chosen so that it satisfies (2.1) and (2.2), then we require the scattered displacement, u(“)’to satisfy a radiation condition at infinity. Physically, this amounts to demanding that the Lame potentials (&‘“‘, e’“‘), where uc3) = grad (4”)(x, y)) fcurl
(@V”‘(x,y))
(2.6)
and i is a unit vector such that the I,2 and i directions form a right handed set, consists of outgoing Rayleigh and body waves as r = (x*+y’) I,‘* + co. Mathematically, we write
1687
Scatteringfrom a surfaceimperfection
4(S) =
!$
Ai
!$ A(@_&+.
,i%-Y(~o).~+
I)(‘)
=
B+ eiik~+-6(k~)y+2
2
(2.7)
0
0
B(d)&
+o
(2.8)
0
0
as Y+ co, where here the upper (+) and lower (-) alternatives refer to x > 0 and x < 0 respectively, k2 = poo2/(A+2p), K2 = pow2/,u, 8 is measured clockwise from they axis (Fig. I), and U. is a typical displacement magnitude in the body. Also y. = y(ko) = ,/k;-k2, and k. is the positive
do = 6(k,) = Jk;-K2,
real root of the Rayleigh R(a) = (2r~-
k. > K > k,
(2.9)
secular determinant
K2)’ -4a2y(c()6(a).
(2.10)
Note that the definitions of y and 6 as functions in the whole x plane are given in Appendix A (A.6). It is convenient at this point to introduce the complementary expression to R(a), S(a) = (2~ - K2)2 +4cx2y(c()6(a),
(2.11)
which is required in Appendix A. The dimensionless far-field amplitudes A+, B,, A(8) and B(B) of the scattered Rayleigh, compression and shear waves respectively, are the quantities of physical interest in this study, and in the case of the Rayleigh waves, we have
Bi -_= A, Finally, we shall that the solution
3.
T
Pi-K*) 2ik,6,
.
(2.12)
we remark that if our defect has a sharp corner or edge, like the tip of a crack, require that the displacements are finite there. GREGORY(1974) has shown radiation and edge conditions as posed here are sufficient to ensure that if a of the boundary value problem exists then it is unique.
EXACT INTEGRAL REPRESENTATION FOR THE DISPLACEMENT POTENTIALS
To determine the form of the field far from the defect it is first expedient to obtain an exact representation of the displacement potentials everywhere in terms of boundary integrals over its surface. This is readily achieved by introducing two fundamental solutions of (2.1) for the perfect, or non-defective. half-space which also satisfy (2.2) over the whole of y = 0. These are the well known compressional and shear sources (LAPWOOD, 1949) defined by the potential pairs (W, ‘I”) and (Cp”, Y”) respectively. Their explicit forms, expressed as Fourier integrals, are given in Appendix A. If we denote a field point as r, and a source point as r’, then the potential pairs satisfy (V’ + k2)W(r, r’) = di,6(r - r’), (V2 + K’)Yj(r,
r’) = S,,b(r-r’)
(3.1)
I. D. ABRAHAMS and G. R. WICKHAM
1688
in .y 3 0, where j = P or S, 6,, is the Kronecker delta, and 6(r -r’) generalized function. On the free surface, JI = 0, these potentials satisfy
is the usual
(3.2)
(3.3) forj = P or S, and are outgoing for large r in the sense of (2.7) and (2.8). In (3.2) and (3.3), Xz(r, r’) is the stress tensor defined by (Q,‘, Y’), and similarly for Cc. and the corresponding displacement vector is
(3.4) We can now apply the Betti-Rayleigh reciprocal formula to the region D enclosed by the contour C (aC c C) shown in Fig. 2. With r’ inside C, we obtain
,uK2~‘“‘(r’) =
sc
[ul”(r)Xc(r,
r’) - Up(r, r’)oj;“(r)]v, dl,
r’ E D,
(3.5)
where al;’ is the scattered stress tensor, dl is an infinitesimal line segment of C, and v is the outward unit normal to the boundary. The free surface boundary conditions and the radiation conditions ensure that in the limit as the radius of the large arc R + cc (see Fig. 2) the contour of integration in (3.5) may simply be replaced by G’C, the boundary of the defect. A second application of the reciprocal formula, this time to u(‘)(r), and Up(r, r’) over the region interior to aC with r’ fixed outside dC gives
X
FIG. 2. Contour
of integration
for the application
of the Betti -Rayleigh
reciprocal
theorem
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Scatteringfrom a surfaceimperfection
0 =
Bc [uj”(r)Cg(r,
r’ ED.
r’) - Up(r, r’)ojj’(r)]v.idl,
(3.6)
s Adding
this last result to (3.5) gives finally pK2+(3)(r’)
=
s
u,(r)Cc(r, r’)v, dl,
r’ ED,
(3.7)
BC
which expresses the unknown scattered potential 4” in terms of the total displacement on dC. Similarly, if the defect is modelled by a rigid inclusion welded to the elastic material, then the corresponding result is pK*qj(“)(r’)
= -
s
oj,(r) Up@, r’)v, dl,
Pn
More generally, for an ensemble n = 0,1,2,. . , , we obtain pK*4(“)(r’)
= 1
s
of inclusions
u,(r)CG(r, r’)vj df
do,,,
r’ ED.
(3.8)
m = 0, 1,2,. . . and cavities
oJr)Up(r,
r’)v, dl,
r’ ED.
K’,,,
(3.9)
In ?(’n,
A somewhat different class of problem is produced when one considers a surface irregularity which protrudes into the region y < 0. The simplest example would be a hill or mound of material identical in elastic properties to that in y > 0. In this case the argument leading to (3.7) must be modified to the extent that r’ must be placed in y > 0 outside the circle of smallest radius which completely encloses the defect. The reason for this is that the fundamental potentials may only be analytically continued across y = 0 into the whole (x,y) plane cut from the point (x’,y’) to the point at infinity, see BRIND (198 1) and BRINDand WICKHAM (1991). The restriction on r’ is not, however, material to the arguments which follow in Section 4. Finally in this section, we remark that the corresponding representations for @‘)(r’) are obtained in exactly the same way using the fundamental shear source. The results are identical to (3.7), (3.8) and (3.9) with “P” replaced by “S” and @‘) replaced by Cd * .
4.
GENERAL OUTER EXPANSION OF THE SCATTERED FIELD
As remarked earlier, we know that the fundamental potential pairs (a’, Yp) and (@, Y”) may be analytically continued across the free surface into the whole (x, y)plane cut from the point (x’, -y’) to a point at infinity in y < 0. It follows that Z$(r, r’) and Up(r, r’) have Taylor series expansions about Irj = 0 which converge inside the circle Irl < ]r’l. Applying this result to (3.7) and interchanging the order of integration, we find that
1. D. ABRAHAMS and
I690
pK24”‘(r’)
= C(l(0, r’)
G. R. WICKHAM
ui(r)v,xc dl
ul(r)vi dl+ 2:; X:(0, r’) s iC
s iC
h
s
u,(r)v,x,x,dl+ Cl(
.'1, r'= Ir'l > u,
(4.1)
where a is the radius of the smallest circumscribing circle of the defective region (see Fig. 1). NOW we wish to solve the general boundary value problem, or equivalently the integral equation (4.1), when the imperfection is small compared to the incident wavelength 2n/k, i.e. we define the dimensionless parameter E = ka<< 1.
(4.2)
Our approach here is the method of matched asymptotic expansions and so it is first necessary to identify inner and outer regions in which we expect to obtain different asymptotic solutions. In the vicinity of the defects the dominant length scale is u, whilst away from that region, in the outer field where kx = O(l), ky = O(l), the appropriate lengthscale is the wavelength. Thus, we define inner and outer variables according to x = x/a,
R = (X2+
Y = y/a.
Y’)“‘,
R = (X, Y)
(4.3)
and x = kx, respectively
J, = ky,
v = (2’ +J?‘) ‘:2,
i; = (_f,p)
(4.4)
which gives ?= ER.
(4.5)
Also, if Uio is a typical amplitude of the incident displacement then it is appropriate to scale all physical scattered displacements, without loss of generality, as u() = FU,V ’ 1
(4.6)
because we expect these displacements to be an order of magnitude smaller than the forcing. Note that V is dimensionless. We now rewrite (4.1) using inner coordinates for the integration variable and outer coordinates for the observation variable; we will also employ the convention that overbarred (outer) variables are functions of r and capitalized (inner) variables are functions of R. Then the dimensionless scattered potential may be written as $(“)(i’) = E2C1,2)~1:‘(j;‘)+E3CI::~j:k)(r’)+C4CIPk:~IPk)l(F’)+ +EDj:‘~j:)(i;‘)+&ZD~~~~~((r’)+E3D1Pk),~jPk),(F’)+ where
the constant
tensors
Cc”’ r,k...i, D$...,
of rank n are given as
‘.‘,
(4.7)
from a surface imperfection
Scattering
1691 (4.9)
1
D!?’
UI”(R)v,X,...X,dL, Ilk...’ = (n - 2)! s >c u.
and the dimensionless
(4.10)
function (4.11)
is the (n-2)th spatial derivative of the stress tensor. Note that the infinitesimal displacement on LJC has been written as dl = adL in (4.9) and (4.10). The expansion (4.7) is the required outer solution of the boundary value problem for an unloaded surface defect such as a crack or stress free protrusion and it shows that, to leading source at the origin. The higher order order, 4 (‘) behaves as a point compressional terms represent successive corrections to this solution. An analogous expression to (4.7) can easily be found when the defect is a general inclusion, i.e. the displacements on the boundary may be specified rather than the stresses. The functions $(;j,..,(r’) are written in Appendix A, in terms of two fundamental Fourier integrals which are given explicitly ; we will use these expressions to later deduce the outward propagating Rayleigh waves on the free surface y = 0. The constants, D$..,,, are known for any given forcing, but C$,, ,, expressed in (4.9) as weighted integrals of the unknown scattered displacement over the surface(s) of the defect, are yet to be determined. We must examine the solution in the inner region to obtain asymptotic expansions of these latter coefficients. Before we concentrate on a particular example, it is interesting to determine the number of unknown constants which, in general, must be evaluated for a specified number of terms in the expansion of (4.7). To leading order, for instance, although there are four constants C1;2), the boundary conditions (3.2) and (3.3). state that @.“1 = $(2:) = 6’2’1= 0. Hence we need only find C(,:‘. Similarly, to order c3 we have &?I = &‘,‘, = $il, = 0 and by adding the coefficients of $(,?/2 and ~$(i:)~together there are in total five constants to find. To order .z4 there are altogether 12 coefficients to determine. For simple shaped defects this number is usually reduced, and in II we show that the edge-crack geometry gives five unknown outer coefficients.
5.
THE MATCHING PRINCIPLE
The unknown coefficients in the outer expansion (4.7) must be determined from an examination of the region around the defect. We will generate an asymptotic expansion of this inner field in later sections (Section 7 here and Section 3 of 11) for specific models. The relationship between the inner and outer asymptotic expansions, called the matching principle, will enable the constants to be found. This principle is stated here. We require the expansion of the outer (compressional) potential as the field point
I692 F’
I. D. ABRAHAMS and Cr. R. WICKHAM
tends to zero. It is convenient
to introduce
a notation
for the asymptotic
expansions
as
$5) (f’) = @“‘(f’) + 0(&u-+’ log c)
(5.1)
where c$“’ is the expansion up to an including terms of O(E’) and O(c” log c). Note that, for the matching principle employed herein (see CRICHTON and LEPPINGTON. 1973) the C”log F terms must always be grouped with the an terms. We then change from outer to inner coordinates F’ + ER’, expand in 6 and truncate after terms of order Eh, say, to give $“‘.6)(R’). Similarly, in the following sections we shall obtain an asymptotic expansion of a quantity, which is effectively a scattered potential, in the inner region, which to order &hmay be written QCh’(R’). We then rewrite this in outer coordinates, re-expand and truncate after terms of O(E’) to give @“.“‘(F’). The matching principle [a modified form of that given by VAN DYKE (1964)] is -(Q&IE @.a) 4
for any
We need to be specific here as to what the inner to. As will be seen, the inner region displacements therefore cannot be written in terms of potentials. governing equation that the outer scattered - (k/U,)k- 2V’$(s)(r) which may be written in inner
a, j,.
(5.2)
variable @ actually corresponds satisfy elastostatic equations and However, we know from the potential $‘)(F) is equal to variables simply as the dilatation
- V)
6(J) = -(V,
(5.3)
where
v, = {a/ax, ajar).
(5.4)
We will define the stress in the inner region in terms of the inner displacement inner variables X, Y. Therefore the sum of principal stresses is
V and
(5.5) in which 0 is the non-dimensional
stress invariant
scaled on
go = A’kn, for reasons
(S.6)
that will be made clear in Section 7. Hence 0 = 2(k,/k)(z2
- l)V, . V,
where z = K/k, and we may match inner and outer problems by defining the inner matching potential
WV
= -
WJ 2FI
1>
O(R).
(5.7) [using the principle
(5.2)]
(5.8)
Scattering
6.
from a surface
imperfection
I693
THE OUTER EXPANSION FOR A CIRCULAR DEFECT
To illustrate the proposed method we will now concentrate attention on the imperfection illustrated in Fig. 3. This is a circular void or protrusion (of the host material) whose boundary intersects the free surface at x = fa, denoted by the letters B and D respectively, and which has arbitrary radius. The perpendicular distance from the centre of the circle. 0’, to the free surface will be taken as ad and so the radius is u &+T’. The angle DO’B is 2c( as shown, so d = cot CI,
(6.1)
and the radius is u/sin c(. It is easy to see that as CIincreases from zero the circular void decreases in radius until, at CI= 71/2, it is a semicircular “bite” of radius a. This case has already been discussed briefly by the authors (ABRAHAMS and WICKHAM, 1991), and can also be tackled by multipole methods (GREGORY and AUSTIN, 1990). The bite continues to shrink (although the radius increases) as CIincreases, and when M = rr the defect vanishes. We therefore expect nonscattering at this particular value! We interpret a protrusion as the case when cx> Z. When CI= 3rc/2 the protrusion is semicircular with radius a. This is most easily understood by replacing a with 271-LX, which reverses the sign of d and measures the new CIacute when 0’ lies in y = 0. This is the most natural coordinate system in this case. To obtain the leading order outer expansion we need to specify the incident wave field. We take this as a right travelling Rayleigh wave of the form
“(I)
=
FIG. 3. The model problem of a circular void (a < a) or protrusion
(a > z) which intersects at the points B (x = a) and D (x = -a).
(6.2)
the free surface
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ABRAHAMS and G. R. WICKHAM
I. D.
where U0 is the amplitude of the displacement and B+/A+, do, yo, k, are given in (2.9)-(2.12). The incident displacement must be expanded about the origin and then substituted into the integrand of (4.10) to determine D$,! ,. To do this, it is most suitable in view of the shape of the defect, to introduce an alternative inner polar coordinate system based at 0’. We write the relationship between the original inner coordinates (4.3) and the new ones as X = p sin [I,
(6.3)
Y= d+pcos~, and so the outward
pointing
(6.4)
vector on the surface is v = -(sin/,‘,cos/I).
Substituting “(‘1 =
(6.3), (6.4) into (6.2) and expanding 2k,,K26,
_.!!!L 4k&
- iK2(2k; - K2) >
(6.5) gives
(1 + &i(k,/k)p sin p) K’(2k;-K2)
CUO 4kk&,
i -2ikodo(2k’
- K2)
(d+p
cos B) + 0(&Q.
(6.6)
This may be used in (4.10) to give D’,2/ =
iK’ = -&Sk
,I
2(7c-a)+sin --si.F
where the O(1) term is identically
zero. Similarly
iK2(2ki 002
-
2r ~ +O(c2),
(6.7)
we find
- K2) 2(7c- a) + sin 2a sin' a
8k&
’
(6.9)
along with Difi = D,‘:\ E 0. As has already been mentioned, @,22),$(22, 65’2, &‘/,, are all zero and so only the terms (6.7))(6.9) need be considered. Our observations made earlier regarding the case LY= n are borne out here, as all the D$ , are zero at this angle. The remaining (unknown) term at O(c’) is J7- a Cc,‘! = -cosec
(a)
s U-F
V”’ sin p dfl,
where V”’ is the leading order inner scattered displacement. We can therefore the leading order expansion, using the notation of the last section, as
(6.10) write
Scattering
from a surface
I695
imperfection
in which K* 2(7c- cc)+ sin 2~ D = 4k2 -TnTa0
(6.12)
and C = Cc,:’ - i(k,/k)D.
(6.13)
To perform matching we rewrite r’ in terms of R’ and expand, regrouping like terms in E, The details of the expansion of the $tl,,,i functions are given in Appendices A and B, and here we merely quote [from (A.1 1) and (A.8)] that (6.14) and also (6.15) whilst @j2 = 0(1/e).
Therefore
to leading
order (6.11) gives (6.16)
and all that remains
is to evaluate
7.
the constant
C by examination
of the inner region.
THE INNER EXPANSION
The model defect was introduced in Section 6, and illustrated in Fig. 3. In inner coordinates (X, Y), defined in (4.3), the distance between the end points of the imperfection boundary, BD, is 2, and from (6.3), (6.4) the radius of the void (protrusion) is p = cosec c(, and d = cot CI. The governing equation (2.1) for the dimensionless displacement V(R) is, in inner coordinates, ,Llv;v+@+/L)V,(V,~V)+~*(~~+2~)V
= 0,
(7.1)
where Vy = (d/8X, d/dY} and E = ka. We expect that the dimensionless have an expansion of the form v = V’O’+ O(E) and the related as
stresses
(based on inner displacements
displacements
(7.2) and coordinates)
@;, = 0@~’ + O(E), where we recall that a. = pk/ko so that gi,(‘I is also dimensionless.
are written
(7.3) On the surface of
1. D. ABKAHAMS and G. R. WICKHAM
1696
the circular coordinates
imperfection we know that the total normal (6.3), (6.4) based at 0’ are
and shear stresses, written
&” = - u):I,> 0,’ (0 e$ = - “{j/J 3
in
(7.4) (7.5)
on the boundary p = cosec LX,a-n < fl < 71-E. In Section 6 we stated that the forcing is an incident Rayleigh wave (6.2), from which we can calculate the right hand sides of (7.4) and (7.5). The leading order boundary value problem can now be summarized. V”” satisfies (7.1) with 6;set to zero, the stresses on the defect are D$) = I0 sin* /I, 0$
(7.6)
= I, sin [j cos p,
(7.7)
where I,, = -2i(r’-
1),
(7.8)
for p = cosec M. x--71 < p < IC--CI and the stresses o(F$.,~$1, vanish on the free surface Y = 0, IX/ > 1. We insist that the displacements remain finite everywhere in the elastic material, and we also suppose that there are no tractions applied at infinity. Thus V’“’ must be o( l/R) as R 4 cc. We solve this elastostatic problem by the method of conformal transformation, after first introducing an Airy stress function and complex variables, following GREEN and ZERNA (1954). If we write the Airy stress function as s, defined as E g$2 = -YY,
(7.9)
G(>?$= z-XXI
(7.10)
C(J)G’XY -
_sXr
(7.11)
and introduce (7.12)
I? = x+iy, then the divergence
of (7.1) gives (7.13)
where ? is the complex
conjugate
Z(s,Y) = :
of z. A solution [i(z-f),f(z,Y)+(zZ-
of this equation l)g(5,2)],
is (7.14)
in which ,f and g are as yet arbitrary harmonic functions. It is the fact that B can be reduced to two independent harmonic functions which allows us to use conformal transformation methods. We choose the mapping 2: which takes any point infinite strip
on the domain
coth (+<) cf the elastic material
(7.15) to a single point in the
Scattering
I697
from a surface imperfection 03q3
(7.16)
-r,
(7.17)
i = t+iy.
The points A, B, C, D, shown in Fig. 3, are mapped to [ = 0, n3 + iv, --iz, - co + iv respectively, i.e. the circular arc of the defect boundary is mapped to the line q = - CI and the plane surface is taken to y = 0. Substituting the new variable E,, = (cash 5-cos and noting
that ,f, ,9, being harmonic,
(7.18)
r/)E
satisfy (7.19)
allows us to write the general a,(<,?!)
=;
form of the solution
(7.14) as
; {[F,(s) sinh (ST) + F*(s) cash (sv)] sin y s ‘X + [G,(s) sinh (sq) + G2(s) cash (sq)] cos q> emrsi ds
by Fourier transform techniques. We now impose the boundary plane Y = 0, or equivalently 17= 0. These are, after integrating changing coordinates, that z = E,, = 0, Y,
Pj = 0.
(7.20)
conditions on the (7.10), (7.11), and
(7.21)
Hence G*(S) = 0, F,(s) = -sG,(s).
(7.22)
The boundary conditions on the “bite” are here much more difficult to derive than for the case considered by Green and Zerna. Firstly we write the relationship between the stresses, in radial coordinates centred on 0’ (6.3), (6.4) and E. We integrate these using (7.6) and (7.7) to yield (7.23)
az
af
= / p=coseca
ps B+cos4
cos @
0
sin2 CI
.
(7.24)
In deriving these terms, the integration constant was chosen to give finite energy at the edges Band D. Secondly, from the mapping (7.15) we can obtain the relationships
as ap and
(7.25) p=cosecl
I. D. ABRAHAMS and
1698
G.
R. WICKHAM 2
(-0s
/j+cos
c(
=
_-Y-a-.-
.
(7.26)
cash i”- cos a Therefore
we can determine
the boundary
conditions
on the defect, namely (7.27)
(7.28) These can be Fourier transformed Omitting all details we obtain
explicitly
in order to determine
F,(s) and G,(s).
sinh (SM)sinh [.s(n - a)] + s’ sin’ Mcash (sz) - .s sin CIcos u sinh (sz) J’,@) = _ ~~~-~---_---____-,--.----~---~-(smh* (SK) -s2 sin2 CC) smh (UC) (7.29) G,(s) =
_ +‘!?? _ ~~ . smh- (RX)- s2 sm’ cx
(7.30)
The solution has now been obtained and all that remains is to generate its far-field behaviour. For matching purposes we require the sum of the principal stresses, which in terms of the Airy Stress Function is (7.31)
0 = Exx+Zyr. This may be found by transforming operator of (7.20). We find
to < coordinates
and then taking
the Laplacian
where
= sinh (sy){G, (s)(l +s2 fis +cosh(,sy){sF,(s)(cos
cos q sinh <-s’cos
r/ cash l-
1)-G,(.
q cash 5) +isF,(s)
F)(:\ cash <+ is2 sinh t) sin n}.
The leading order matching potential Qcol, (5.8), is proportional required as R + co. This is deduced from (7.32) after changing and after establishing that as ]z[ --f MI, the mapping (7.15) is [ --. So, in polar coordinates
we write
2 ?
sin y sinh 0 (7.33)
to 0 and its form is to outer coordinates
(7.34)
1699
Scatteringfrom a surfaceimperfection
ir’ z
=
e-rfl’
__
E
which corresponds
)
r’ = O(l),
(7.35)
to (5, ‘1) - (sin Q’, -cos
and these forms are substituted to be
(7.36)
0’) +, I”
into (7.33) and expanded.
The leading
term is found
(7.37) where the integral on the right hand side is a real function of CI. We will use this in the next section to deduce the unknown outer coefficient and the scattered Rayleigh waves.
8.
DETERMINATION OF THE SCATTERED RAYLEICH WAVE AND CLOSING REMARKS
We showed, in the particular example chosen in Section 6, that there is just one unknown coefficient in the outer solution, (6.12), to leading order. This can now be determined by applying the matching condition, (5.2), at first order : 4 Therefore
(2.0)= @(02)~ -
(8.1)
(6.16) and (7.37) give
where F,(s) is written in (7.29), r = K/k, and r0 = K/k,. Hence, we can now deduce the reflected and transmitted Rayleigh waves scattered by the circular defect. We wish to obtain A + and A ~, the surface wave coefficients of the compressional potential expressed in (2.7), and note that the incident forcing wave (6.2) has been chosen so that A? is unity. To determine A, we firstly close the integration contours in (A.8) and (A.9) in the upper (lower) half planes according to x < 0 (x > 0). The residue contribution from the simple pole, which corresponds to a real zero of the Rayleigh determinant (2. lo), gives rise to the surface waves. These expressions are then differentiated to yield the Rayleigh wave components of $\:‘, . . , @,4:,,(F), (A. 1 l)-(A.22), and from these the outer expansion (6. I 1) is known. We obtain {(2--2;)2(1 to leading
order, where D is written
in (6.12), 6, = Q’?&
f l)Tr;t}D
K2, and
(8.3)
1700
I. D. AHKAHAMS and
R’(k
0
)
=
G. R. WICKHAM
d”(?! da
63.4) cr-k,,
The first observation we can make about the above result is that Ai are purely imaginary [.%(C) = 01. This would suggest at first sight that the total transmitted wave, which is the sum of the incident plus scattered wave, has energy flux greater than that of the incident wave in .Y < O! However, it is easy to show that terms of O(E’) contribute to the energy at the same order as those calculated here. Therefore, these leading order Rayleigh wave coefficients (8.3) are useful for determining displacements and stresses, but for energy calculations the matching must be performed to second order. It is interesting to examine how the scattering coefficients alter as the size of the defect increases or decreases. Therefore in Fig. 4 we plot that part of A+ and A_ written in square brackets in (8.3), as a function of the angle M(which is displayed in Fig. 3). For 0 < IX< rc the imperfection is a void, and for 7-r< CI< 2n it is a protrusion with angle DO’B = 2x-x. As already mentioned, the scattered field must vanish when x = n, and this is confirmed in the figure. The angle x is not taken too close to 0 or 271otherwise the void or protrusion would become very large, thus violating our basic assumption in the paper. The elastic constants are chosen so that r = 1.5 and hence r0 = 0.971494. The most interesting feature of the graph is that a void has A,>A, whilst a protrusion results in the transmission coefficient A, being less than the reflection coefficient A ~. However, more importantly, the magnitude of the transmitted wave A + is less (greater) than the magnitude of the reflected wave A for c( angles less (greater) than approximately 2.22 radians. Note that Fig. 4 remains of the same form for any values of the elastic constants. This concludes our discussion of the circular defect example. The method introduced in this paper has been applied successfully, and, after matching to leading order, the Rayleigh surface wave terms have been examined. There are a number of general comments to be made regarding our method, but we have chosen to reserve these for the concluding section of Paper II. There, we shall present a full demonstration to third order, where the inner geometry is both particularly difficult and of physical
FE. 4. Graph
of the square
bracketed
terms of A i written
in (8.3) plotted
against
the void angle (x.
Scattering
interest. without
from a surface
1701
imperfection
The aim of this paper has been to give a clear exposition choosing a difficult inner problem to obfuscate the solution.
of the method,
ACKNOWLEDGEMENTS Part of this work was carried out whilst Abrahams was in receipt of a Nuffield Science Research Fellowship, He is grateful to the Nuffield Foundation for this generous award.
REFERENCES ABRAHAMS, 1. D. and WICKHAM, G. R. ARRAHAMS, I. D. and WICKHAM, G. R. BRIND, R. J. BRIND, R. J. and ACHENBACH, J. D. BRIND, R. J. and WICKHAM, G. R. CRIGHTON, D. G. and LEPPINGTON, F. G. DATTA, S. K. and SABINA, F. J.
1991
Proc. Inst. Acoust. 13(2), 103.
1992
.I. Me&
1981 1981
Ph.D. thesis. Manchester. J. Sound Vib. 78. 555.
1991
Proc. Roy. Sot. Lond. A433, 101.
1973
Proc. Roy. Sot. Lond. A335, 313.
1986
In High and Low Frequency
Phys. Solids 40, 1107.
Asymptotics
(edited
by V. K. VARADAN and V. V. VARADAN). North-Holland, Amsterdam. GREEN, A. E. and ZERNA. W. GREGORY, R. D. GREGORY, R. D. GREGORY, R. D. and AUSTIN, D. M. KHRAPKOV, A. A. LAPWOOD, E. R. MENDELSOHN, D. A., ACHENBACH, J. D. and KEER, L. M. SABINA, F. J. and WILLIS, J. R. SABINA, F. J. and WILLIS, J. R. VAN DYKE, M.
1954
1967 1974 1990
Theoretical Elasticity. Proc. Camb. Phil. Sot. Proc. Camb. Phil. Sot. Quart. J. Mech. Appl.
Clarendon Press, Oxford. 63, 1341. 77,385. Math. 43, 293.
1971 1949 1980
Prikl. Mat. Mekh. 35, 677. Phil. Trans. Roy. Sot. Land. A242, 63. Wave Motion 2, 277.
1975 1977 1964
Geophys. J. Roy. Astr. Sot. 42, 685. J. Geophys. 43,401. Perturbation Methods in Fluid Mechanics.
Aca-
demic Press, New York, WATSON, G. N.
1966
A Treatise on the Theory of‘ Bessel Functions, 2nd Edn. Cambridge University Press, Cam-
WICKHAM, G. R. WICKHAM, G. R. ZHANG, C. and ACHENBACH, J. D. ZHANG, C. and ACHENBACH, J. D.
1977 1980 1988a
Proc. Camb. Phil. Sot. 81, 291. Quart. J. Mech. Appl. Math. 33,409. Wave Motion 10,365.
1988b
(iltrasonics
bridge.
26, 132.
1702
1. D. ABRAHAMSand G. R.
WKKHAM
APPENDIX A : FUNDAMENTAL POTENTIALS Here we present notation employed
the potential pairs (@‘, Y’) and (a*, Y”), satisfying by BRIND and WICKHAM (1991). These are
(3.1)-(3.3),
using the
(A.2)
(A.3)
where R(a) and S(z) are written in (2.10) and (2.11) and T(X) = 4ix(2r’The square root functions
of the complex y(a) = ($-k2)“2,
K’).
(A.5)
variable CI,Y(E) and C?(X),arc defined by 6(c() = (El-K’)’
1
(A.6)
with branches chosen to satisfy v(O) = - ik, S(O) = - iK. The contours of the Fourier inverse integrals lie along the real line but are indented above the pole at u = -k, and the branch points at x = -K, -k and below the pole at CI= k, and the branch points at CI= K, k. The Hankel function of the first kind N’,“(klr-r’l) may also be expressed in the same form, i.e. (A.7) We require the above potentials in order to construct the outer field functions &;i ,(F’) from the derivatives of the stresses X:(0, F/k), (4.11). Rather than work with (a’, ul’) it is, however, more convenient to define the dimensionless potentials
(A.8) (A.9) formed from a combination of expressions (A.l)-(A.4) set at r = 0. The integration path 01 both integrals is the same as that defined previously. Then it is straightforward to show from (A. IO) and equation
(4.1 I) that (changing
? to r for convenience) (A.1 I)
(A.12) whereas
from a surface
Scattering
i’,‘,‘,o
Further
imperfection
= ,‘; (2~;+++~&
1703
i-?(+$,&~.
(A.13)
we find that fjq:,,(q
-c4)
dlllz(f)=
-f
= 16$T-l,
;;
f$,((O),
(A. 14)
~~[(2~~+i:)(4(r’-I)~~+T2(12-2))1C*rl(i),
@,4’,,(f) = _*
(A.15)
(2” - 1) a2 _IzT[i’+2+4gM~.
(A.16)
z-
Equation
(3.2) allows us to express the non-zero
&M)
= 242
c$~, potentials
g
&4?),2@ = -2r5;
(2 $
q%&,(f) = 8q
$(2%
as
++,(i).
(A.17)
++,(i),
ox
(A.18)
+~~)45 A(kj 3
(A. 19)
and similarly from (3.3) @J2(f) = p_ a rii4(r’-l)$ Tz c’x
(
49i?j2(f)
=
-f
2’;
1‘++4(r'-I)~$
(
1 ,
$n(i),
(A.20)
(A.21)
fjA(i),
>
$(,42,,(f) = 4 (?I’>
(A.22)
z2 The following appendix will be concerned field point tends to the origin.
with the expansion
of these c#$? , terms as the
APPENDIX B: NEAR FIELD EXPANSIONSOF 4n AND 4B
To obtain a near field expansion as (f --t 0) of the potentials (A.1 ])-(A. 12) it is necessary to expand the fundamental solutions #A(T) and q5#(i) which are defined in (AX) and (A.9) respectively. We follow the method of BRIND and WICKHAM (1991) by first mapping the complex a-plane onto the complex t-plane according to the one-many transformation CI= -cosht, so that one copy of the contour
of integration
in the E-plane is mapped
(B.1) into the contour
%‘”
I704
I. D. ABKAHAMSand G. R. WICKHAM
co
.__1 CO
-To-T
T To
shown in Fig. 5. WC cut the complex t-plane along an infinite array of parallel line segments joining each pair of branch points at t = i T+ inn, where T = cash ’ T. T = K/k, and y1is any integer. These arc the branch points of A(t) = O( -k cash t)/k = (cosh’t-
T’)“~ = -sinht(l-imh:,)‘.
(B.2)
and clearly we must choose the right factor of the last expression to have a positive real part on the contour % ,,. Further. we choose A(.(t)to be negative imaginary as WCapproach the branch cut corresponding to n = 0 from above, and, as we approach the cut corresponding to IZ= 1 from below. We further specify that A(t) is periodic with period 2ni in the whole cut t-plant. With this definition for A(t). we may write (B.3) where R,,(f) = (2 cash’ t--r’)‘-4
cash’ f sinh tA(t).
(B.4)
An exactly analogous formula holds for 4,?(~, 0). Now the contour of integration in (B.3) may be closed by the two loops f’ shown in Fig. 5 and it is easily verified that R”(t) has simple zeros at i T,,+ inn. where T,, = cash ’ (s/t,,). to = K/k,,. It follows that (B.5) where C, is the sum of the residues at the four poles i_ T,, + in and f T,, + 2in and .#,, is a closed loop around the nth branch cut. WC now evaluate the contributions from each contour scgmcnt. Firstly note that. for .8(f) sufficiently large and positive. we may write
(B.6) where h2,( are the coefficients
in the Taylor series of
about u = 0. Hence the contribution
from -r-
is
Scattering
1705
from a surface imperfection
(B.8) However, each of the integrals on the right hand side of this last result may be evaluated using the contour integral representation of the Bessel function given on page 178of WaTsoN (1966) ; we find that c#I:(?, 0) = i i hz,, :~ [en”~‘n) J, (?)J =Z,r. 01 ,I_ 0 Similarly. the contribution of e’ to yield
from -r^
may be evaluated
(PI 05 0) = ; i
,I 0
Next
calculate
WC
the contributions
4’,5ll’ +_/$I = L +
s
by expanding
(B.9) the integrand
in powers
~[e~~(~-~~ J,.(F)],._,,,.
h,,z 0t
to $‘,,‘I, @y’ from g ,, &z respectively
’ _T,ql(t)[(t+in)em
(B.lO)
; WCfind that
l’WItl(!+1~~) + (t+ 2j7() ciSi,+I,+,(,,] dt,
(B.11)
where &&)_!L
sinh t(2 cash’ t--ZZ)Zj’~B-coshz ~~~_~~__~__~_~,~~~~~~~~ 4 (2cosh’t-~~)~+16cosh“tsmh~t(z’-cosh’t)
t
~~~~,
(B.12)
Equation (B. I 1) is easily expanded as a Fourier series in 0 on the interval (- 7c/2,7r/2) ; explicitly displaying only those terms that are even in 0, we have +y’ + q$i2’ =
-II i
E,,in’
n - 1110
s ’
-/
gn(t)qM(t) dt J,,,(F) cos r&fan
odd function
of 8, (B.13)
where E,, = 2, c,,, = I, WI> 0 and rwc
mn
rl,3V(t)= 2t cos mm-cash mt+n sin ~ smh mt 2 2 Similarly,
(B.14)
denoting (B.15)
we find that c,,inr~nT(TO)J,,(~)cos &+an Finally, collecting
all terms together
odd function
of 0.
(B.16)
yieldsl(B.17)
where (B.18) Performing
an identical
procedure
for dB, we obtain
f Since q5Ais an even function of 8, the odd terms must sum to zero
I706
I.
where C&are the coefficients
of the Taylor series of
D.
ABKAHAMS and G.
R. WNXHAM
and
with t,,,(t) = 2t sin ?2: sinh mt--n
cos “1” cash nztt,
(B.22)
and sinh’ t cash’ t&‘~&Sh’t
,~-/B(~j_
(2cosh’t-z’)“+16cosh4tsinhZt(r2-cosh’t)’
(B.23)
We have thus provided formulae for all the coefficients in the cylindrical wave expansions of tia, $H; in the determination of the inner expansion of the outer field we shall require the explicit values of the first few h,,,, d,,, ; these are h = e:-‘,-’ 0 4n h ZniA, = 0,
d2,=0,
d,=
m = 0, 1,2,.
m=0,1,2
I (3r4-52’f4) -2n-7-1F-.
(B.24)
’
T
,...,
,
(B.25)
(B.28) (B.29)
It is now easy to see how an asymptotic expansion of each &. ,(F) potential can be obtained. The power series expansion for J,,(f) is substituted into (B.17), (B.19), and after performing the differentiations with respect to order and rearranging we can obtain expansions of @,,, b,,. Further differentiations with respect to X according to the formulae in (A. 1 l)-(A.22) product the required expansions, which are then used in (4.7) to obtain 6”)(i) to any required order. This procedure was carried out with the aid of the algebraic manipulation package MAPLE on an Amdahl 5890 computer.
J. /Mech.Phys. Solids Vol. 40. No. 8, pp. 1683-1706,1992 Printedin Great Britain.
SCATTERING OF ELASTIC WAVES BY AN ARBITRARY SMALL IMPERFECTION IN THE SURFACE OF A HALF-SPACE I.
Department
of Mathematics,
D.
ABRAHAMS
Keele University,
Keele, Staffs. ST5 .5BG, U.K
and G. R. Department
of Mathematics,
WICKHAM
The University
(Received 25 September
of Manchester,
199 1 ; in revised form
Manchester
18 February
M 13 9PL, U.K.
1992)
ABSTRACT THIS PAPER, the first of two articles,
is concerned with the scattering of elastic waves by arbitrary surfacebreaking or near surface defects in an isotropic half-plane. We present an analytical solution, by the method of matched asymptotic expansions, when the parameter E, which is the ratio of a typical length scale of the imperfection to the incident radiation’s wavelength, is small. The problem is formulated for a general class of small defects, including cracks, surface bumps and inclusions. and for arbitrary incident waves. As a straightforward example of the asymptotic scheme we specialize the defect to a two-dimensional circular void or protrusion, which breaks the free surface, and assume Rayleigh wave excitation; this inner problem is exactly solvable by conformal mapping methods. The displacement field is found uniformly to leading order in E, and the Rayleigh waves which are scattered by the crack are explicitly determined. In the second article we use the method given here to tackle the important problem of an inclined edge-crack. In that work we show that the scattered field can be found to any asymptotic order in a straightforward manner, and in particular the Rayleigh wave coefficients are given to O(E’).
1.
INTRODUCTION
LINEAR HARMONIC WAVE SCATTERING in a semi-infinite elastic body is a subject which has, through its numerous applications, received a considerable amount of analytical and numerical study in recent years. Even in the simplest case, of a homogeneous isotropic material, there are few closed form analytical solutions. In particular, if the body has inclusions or surface defects, see Fig. 1, none are known. The principal reason for this is that wave motion in such bodies is coupled at the free surface and the associated differential equations and boundary conditions do not admit a simple continuation formula such as the Schwarz reflection principle available in the corresponding classical problems of acoustics in fluids. Accordingly most of the quantitative work has centred on the numerical treatment of boundary integral equations, see for example ZHANG and ACHENBACH (1988a, b), MENDELSOHN et al. (1980), BRIND and ACHENBACH (1981) or infinite algebraic systems, GREGORY (1967). As far as
1683
1684
I. D. ABRAHAMS and G. R.
WICKHAM
obtaining explicit analytical estimates is concerned, the usual methods for an infinite medium (separation of variables and integral equations) do not in general, carry over in a simple way. Consequently, when the scatterer is in or near the surface, only integral equation techniques have so far been successfully applied to the simplest cases (WICKHAM, 1977, 1980). When the influence of the free surface is small (DATTA and SABINA, 1986) or a small surface irregularity belongs to a special class (SABINA and WILLIS, 1977), the method of matched asymptotic expansions has proved fruitful. Here, we introduce a general and systematic approach based on the method of matched asymptotic expansions to obtain analytical estimates for the field scattered by small surface irregularities occupying a compact region. There are several notable difficulties for such problems which means that the work presented here is not just ;I direct extension of the analysis reviewed by DATTA and SARINA (19X6). Firstly. wc require a way of postulating the form of the outer expansion. In acoustic wave propagation in fluids, or SH waves in solids, it is relatively simple to gauge the form of the sources of the sound on physical grounds (SABINA and WILLIS, 1975). in elasticity, on the other hand, the coupling between the shear and compressional motions at the scattering boundaries makes such heuristic judgements less certain. For example, the expansion of the local tractions in the in-plane model posed by SABINA and WILLIS (1977) does not include variations perpendicular to the free surface. That this is true for all surface irregularities is not immediately obvious to the present authors, but is confirmed for compact scatterers by our rigorous expansion derived in Section 4. Another technical difficulty associated with any outer expansion is the determination of its analytical behaviour as we approach the inner region. Thirdly, the inner problems for any particular geometry are themselves intricate, although given that in general they may be posed using the apparatus of classical elastostatics, e.g. the Airy stress function. there is a reasonable selection of tractable cases of physical significance. SABINA and WILIJS (1977) confined their attention to a class of non-compact irregularities which are solvable in the inner region by conformal transformation, whereas we concern ourselves with compact bodies with sharp edges. The contrast between the usual formulation of elastostatic and dynamic problems raises our fourth dilemma, namely, to what physical quantities should we apply the asymptotic matching principle‘? WC show by specific example that it is sufficient in half-space problems to match on the dilatation only. This work is presented in two parts: the first is largely expository of our general approach while part two, henceforth denoted by II, concentrates on the detailed calculation for an inclined edge crack (ABRAHAMS and WITKHAM, 1992). Section 2 of this first paper gives a general formulation of the scattering problem for an ensemble of voids and inclusions in or near the free surface of the half-space. The following section introduces integral representation for the Lami: potentials 4. $ for the gcncral case. These relate the scattered field (4‘, $‘) to the unknown components of stress and displacement on the scattering obstacles and allow us. in Section 4, to derive rigorously the form of the outer expansions. For example, when the scatterer is a single void, it is readily seen that the outer field for 4 is, to leading order, a compressional source at the origin. Similarly. if the scatterer is a rigid obstacle, this appears to a distant observer to exert a point body force at the origin on the elastic material. The method in Section 4 provides a systematic way of developing the outer field to any order without
Scatteringfrom a surface imperfection
1685
cross-referencing the inner field and, in Section 5, the two are related through a statement of the asymptotic matching principle. We demonstrate the approach of this paper by reference to a simple specific example, namely the scattering by a small surface irregularity which takes the shape of an arc of a circle and which breaks the surface; i.e. the circular “bite” (see Fig. 3) or the mathematically equivalent circular protrusion. In Section 6 we specialize the results of Section 4 to this case and in Section 7 we solve the inner problem. Here, we are able to determine explicitly the appropriate elastostatic field using the well known bipolar transformation. Finally, in Section 8, we use the asymptotic solution obtained for this example to calculate the reflected and transmitted Rayleigh waves as a function of the depth of the centre of the arc. This result shows how the scattering varies with the geometry of the irregularity at low frequencies. This paper provides an introduction to the application of matched asymptotic expansions to an intrinsically difficult class of problems and we have tried, by way of what is probably the simplest example, to demonstrate its workings without overburdening the reader with the algebraic details. The circular “bite” example calculation discussed here has a number of applications in geophysics and in the ultrasonic detection of surface imperfections. Further, the asymptotic result will certainly be of use in testing the output from purely numerical codes for solving this class of problem. In the following, part II, we provide a full description of the calculation for an inclined edge crack. This is a particularly important physical example and has some interesting mathematical features worthy of separate and detailed treatment. In particular, the inner problem gives rise to an unusually tricky Wiener-Hopf problem first discussed by KHRAPKOV (1971). Further, we perform the analysis up to third order in that example since only then does the inner solution depend on the inertia terms in the differential equation.
2.
FORMULATION OF THE BOUNDARY VALUE PROBLEM
We consider the two-dimensional problem of a homogeneous isotropic elastic halfspace, occupying the region y > 0, - co < x < co, whose surface is free from applied tractions and which contains an edge crack or submerged void or inclusion close to or in the surface. We develop a general theory for this problem but will later specify the surface imperfection to a circular defect (Section 6) and to an edge crack (pai-t II) as examples. The defect is irradiated by a time harmonic disturbance, with radian frequency w, travelling from infinity (see Fig. 1). The latter may take the form of a Rayleigh wave travelling in the positive x direction, say, or a compressional or shear body wave. For small amplitude incident waves we know that the steady state elastic displacement vector u = (u, P) satisifes the partial differential equation ,~V*u+(/Z+~)graddivu+p,,o~u
= 0,
(2.1)
where V2 is the two-dimensional Laplacian operator, A and @are the Lamk constants, pa is the material density, and the sinusoidal time dependence in u has been suppressed for brevity. On the free surface, JI = 0, of the material, the boundary conditions are
I686
I. D. ABRAHAMS and
G. R. WICKHAM
FIG. 1. The general physical configuration showing small defects such as cracks, rigid inclusions protrusions confined to a semi-circle of radius CIcentred at the origin.
cri2(X, 0) = 0,
and surface
(2.2)
where o,,(-u,y) is the stress tensor and, throughout, we shall identify the x and J directions with the subscripts 1 and 2 respectively. The boundary conditions on the defect depend on its physical nature. In particular, if it is a cavity or crack then again the tractions vanish, i.e. G’,~L’~ = 0,
rEdC,
(2.3)
where v = (V,, v2) is the outward normal vector on the void boundary dC and r is the position vector. For a fixed rigid inclusion, say, the displacements vanish U, = 0,
(2.4)
on the boundary ret3D. The analysis we shall present here is trivially extended to more general boundary conditions though in the interest of clarity of exposition we shall omit such details. Now, if we express the displacement vector in the form ” = “(‘I +“(3)
(2.5)
where u(j) is the incident field chosen so that it satisfies (2.1) and (2.2), then we require the scattered displacement, u(“)’to satisfy a radiation condition at infinity. Physically, this amounts to demanding that the Lame potentials (&‘“‘, e’“‘), where uc3) = grad (4”)(x, y)) fcurl
(@V”‘(x,y))
(2.6)
and i is a unit vector such that the I,2 and i directions form a right handed set, consists of outgoing Rayleigh and body waves as r = (x*+y’) I,‘* + co. Mathematically, we write
1687
Scatteringfrom a surfaceimperfection
4(S) =
!$
Ai
!$ A(@_&+.
,i%-Y(~o).~+
I)(‘)
=
B+ eiik~+-6(k~)y+2
2
(2.7)
0
0
B(d)&
+o
(2.8)
0
0
as Y+ co, where here the upper (+) and lower (-) alternatives refer to x > 0 and x < 0 respectively, k2 = poo2/(A+2p), K2 = pow2/,u, 8 is measured clockwise from they axis (Fig. I), and U. is a typical displacement magnitude in the body. Also y. = y(ko) = ,/k;-k2, and k. is the positive
do = 6(k,) = Jk;-K2,
real root of the Rayleigh R(a) = (2r~-
k. > K > k,
(2.9)
secular determinant
K2)’ -4a2y(c()6(a).
(2.10)
Note that the definitions of y and 6 as functions in the whole x plane are given in Appendix A (A.6). It is convenient at this point to introduce the complementary expression to R(a), S(a) = (2~ - K2)2 +4cx2y(c()6(a),
(2.11)
which is required in Appendix A. The dimensionless far-field amplitudes A+, B,, A(8) and B(B) of the scattered Rayleigh, compression and shear waves respectively, are the quantities of physical interest in this study, and in the case of the Rayleigh waves, we have
Bi -_= A, Finally, we shall that the solution
3.
T
Pi-K*) 2ik,6,
.
(2.12)
we remark that if our defect has a sharp corner or edge, like the tip of a crack, require that the displacements are finite there. GREGORY(1974) has shown radiation and edge conditions as posed here are sufficient to ensure that if a of the boundary value problem exists then it is unique.
EXACT INTEGRAL REPRESENTATION FOR THE DISPLACEMENT POTENTIALS
To determine the form of the field far from the defect it is first expedient to obtain an exact representation of the displacement potentials everywhere in terms of boundary integrals over its surface. This is readily achieved by introducing two fundamental solutions of (2.1) for the perfect, or non-defective. half-space which also satisfy (2.2) over the whole of y = 0. These are the well known compressional and shear sources (LAPWOOD, 1949) defined by the potential pairs (W, ‘I”) and (Cp”, Y”) respectively. Their explicit forms, expressed as Fourier integrals, are given in Appendix A. If we denote a field point as r, and a source point as r’, then the potential pairs satisfy (V’ + k2)W(r, r’) = di,6(r - r’), (V2 + K’)Yj(r,
r’) = S,,b(r-r’)
(3.1)
I. D. ABRAHAMS and G. R. WICKHAM
1688
in .y 3 0, where j = P or S, 6,, is the Kronecker delta, and 6(r -r’) generalized function. On the free surface, JI = 0, these potentials satisfy
is the usual
(3.2)
(3.3) forj = P or S, and are outgoing for large r in the sense of (2.7) and (2.8). In (3.2) and (3.3), Xz(r, r’) is the stress tensor defined by (Q,‘, Y’), and similarly for Cc. and the corresponding displacement vector is
(3.4) We can now apply the Betti-Rayleigh reciprocal formula to the region D enclosed by the contour C (aC c C) shown in Fig. 2. With r’ inside C, we obtain
,uK2~‘“‘(r’) =
sc
[ul”(r)Xc(r,
r’) - Up(r, r’)oj;“(r)]v, dl,
r’ E D,
(3.5)
where al;’ is the scattered stress tensor, dl is an infinitesimal line segment of C, and v is the outward unit normal to the boundary. The free surface boundary conditions and the radiation conditions ensure that in the limit as the radius of the large arc R + cc (see Fig. 2) the contour of integration in (3.5) may simply be replaced by G’C, the boundary of the defect. A second application of the reciprocal formula, this time to u(‘)(r), and Up(r, r’) over the region interior to aC with r’ fixed outside dC gives
X
FIG. 2. Contour
of integration
for the application
of the Betti -Rayleigh
reciprocal
theorem
1689
Scatteringfrom a surfaceimperfection
0 =
Bc [uj”(r)Cg(r,
r’ ED.
r’) - Up(r, r’)ojj’(r)]v.idl,
(3.6)
s Adding
this last result to (3.5) gives finally pK2+(3)(r’)
=
s
u,(r)Cc(r, r’)v, dl,
r’ ED,
(3.7)
BC
which expresses the unknown scattered potential 4” in terms of the total displacement on dC. Similarly, if the defect is modelled by a rigid inclusion welded to the elastic material, then the corresponding result is pK*qj(“)(r’)
= -
s
oj,(r) Up@, r’)v, dl,
Pn
More generally, for an ensemble n = 0,1,2,. . , , we obtain pK*4(“)(r’)
= 1
s
of inclusions
u,(r)CG(r, r’)vj df
do,,,
r’ ED.
(3.8)
m = 0, 1,2,. . . and cavities
oJr)Up(r,
r’)v, dl,
r’ ED.
K’,,,
(3.9)
In ?(’n,
A somewhat different class of problem is produced when one considers a surface irregularity which protrudes into the region y < 0. The simplest example would be a hill or mound of material identical in elastic properties to that in y > 0. In this case the argument leading to (3.7) must be modified to the extent that r’ must be placed in y > 0 outside the circle of smallest radius which completely encloses the defect. The reason for this is that the fundamental potentials may only be analytically continued across y = 0 into the whole (x,y) plane cut from the point (x’,y’) to the point at infinity, see BRIND (198 1) and BRINDand WICKHAM (1991). The restriction on r’ is not, however, material to the arguments which follow in Section 4. Finally in this section, we remark that the corresponding representations for @‘)(r’) are obtained in exactly the same way using the fundamental shear source. The results are identical to (3.7), (3.8) and (3.9) with “P” replaced by “S” and @‘) replaced by Cd * .
4.
GENERAL OUTER EXPANSION OF THE SCATTERED FIELD
As remarked earlier, we know that the fundamental potential pairs (a’, Yp) and (@, Y”) may be analytically continued across the free surface into the whole (x, y)plane cut from the point (x’, -y’) to a point at infinity in y < 0. It follows that Z$(r, r’) and Up(r, r’) have Taylor series expansions about Irj = 0 which converge inside the circle Irl < ]r’l. Applying this result to (3.7) and interchanging the order of integration, we find that
1. D. ABRAHAMS and
I690
pK24”‘(r’)
= C(l(0, r’)
G. R. WICKHAM
ui(r)v,xc dl
ul(r)vi dl+ 2:; X:(0, r’) s iC
s iC
h
s
u,(r)v,x,x,dl+ Cl(
.'1, r'= Ir'l > u,
(4.1)
where a is the radius of the smallest circumscribing circle of the defective region (see Fig. 1). NOW we wish to solve the general boundary value problem, or equivalently the integral equation (4.1), when the imperfection is small compared to the incident wavelength 2n/k, i.e. we define the dimensionless parameter E = ka<< 1.
(4.2)
Our approach here is the method of matched asymptotic expansions and so it is first necessary to identify inner and outer regions in which we expect to obtain different asymptotic solutions. In the vicinity of the defects the dominant length scale is u, whilst away from that region, in the outer field where kx = O(l), ky = O(l), the appropriate lengthscale is the wavelength. Thus, we define inner and outer variables according to x = x/a,
R = (X2+
Y = y/a.
Y’)“‘,
R = (X, Y)
(4.3)
and x = kx, respectively
J, = ky,
v = (2’ +J?‘) ‘:2,
i; = (_f,p)
(4.4)
which gives ?= ER.
(4.5)
Also, if Uio is a typical amplitude of the incident displacement then it is appropriate to scale all physical scattered displacements, without loss of generality, as u() = FU,V ’ 1
(4.6)
because we expect these displacements to be an order of magnitude smaller than the forcing. Note that V is dimensionless. We now rewrite (4.1) using inner coordinates for the integration variable and outer coordinates for the observation variable; we will also employ the convention that overbarred (outer) variables are functions of r and capitalized (inner) variables are functions of R. Then the dimensionless scattered potential may be written as $(“)(i’) = E2C1,2)~1:‘(j;‘)+E3CI::~j:k)(r’)+C4CIPk:~IPk)l(F’)+ +EDj:‘~j:)(i;‘)+&ZD~~~~~((r’)+E3D1Pk),~jPk),(F’)+ where
the constant
tensors
Cc”’ r,k...i, D$...,
of rank n are given as
‘.‘,
(4.7)
from a surface imperfection
Scattering
1691 (4.9)
1
D!?’
UI”(R)v,X,...X,dL, Ilk...’ = (n - 2)! s >c u.
and the dimensionless
(4.10)
function (4.11)
is the (n-2)th spatial derivative of the stress tensor. Note that the infinitesimal displacement on LJC has been written as dl = adL in (4.9) and (4.10). The expansion (4.7) is the required outer solution of the boundary value problem for an unloaded surface defect such as a crack or stress free protrusion and it shows that, to leading source at the origin. The higher order order, 4 (‘) behaves as a point compressional terms represent successive corrections to this solution. An analogous expression to (4.7) can easily be found when the defect is a general inclusion, i.e. the displacements on the boundary may be specified rather than the stresses. The functions $(;j,..,(r’) are written in Appendix A, in terms of two fundamental Fourier integrals which are given explicitly ; we will use these expressions to later deduce the outward propagating Rayleigh waves on the free surface y = 0. The constants, D$..,,, are known for any given forcing, but C$,, ,, expressed in (4.9) as weighted integrals of the unknown scattered displacement over the surface(s) of the defect, are yet to be determined. We must examine the solution in the inner region to obtain asymptotic expansions of these latter coefficients. Before we concentrate on a particular example, it is interesting to determine the number of unknown constants which, in general, must be evaluated for a specified number of terms in the expansion of (4.7). To leading order, for instance, although there are four constants C1;2), the boundary conditions (3.2) and (3.3). state that @.“1 = $(2:) = 6’2’1= 0. Hence we need only find C(,:‘. Similarly, to order c3 we have &?I = &‘,‘, = $il, = 0 and by adding the coefficients of $(,?/2 and ~$(i:)~together there are in total five constants to find. To order .z4 there are altogether 12 coefficients to determine. For simple shaped defects this number is usually reduced, and in II we show that the edge-crack geometry gives five unknown outer coefficients.
5.
THE MATCHING PRINCIPLE
The unknown coefficients in the outer expansion (4.7) must be determined from an examination of the region around the defect. We will generate an asymptotic expansion of this inner field in later sections (Section 7 here and Section 3 of 11) for specific models. The relationship between the inner and outer asymptotic expansions, called the matching principle, will enable the constants to be found. This principle is stated here. We require the expansion of the outer (compressional) potential as the field point
I692 F’
I. D. ABRAHAMS and Cr. R. WICKHAM
tends to zero. It is convenient
to introduce
a notation
for the asymptotic
expansions
as
$5) (f’) = @“‘(f’) + 0(&u-+’ log c)
(5.1)
where c$“’ is the expansion up to an including terms of O(E’) and O(c” log c). Note that, for the matching principle employed herein (see CRICHTON and LEPPINGTON. 1973) the C”log F terms must always be grouped with the an terms. We then change from outer to inner coordinates F’ + ER’, expand in 6 and truncate after terms of order Eh, say, to give $“‘.6)(R’). Similarly, in the following sections we shall obtain an asymptotic expansion of a quantity, which is effectively a scattered potential, in the inner region, which to order &hmay be written QCh’(R’). We then rewrite this in outer coordinates, re-expand and truncate after terms of O(E’) to give @“.“‘(F’). The matching principle [a modified form of that given by VAN DYKE (1964)] is -(Q&IE @.a) 4
for any
We need to be specific here as to what the inner to. As will be seen, the inner region displacements therefore cannot be written in terms of potentials. governing equation that the outer scattered - (k/U,)k- 2V’$(s)(r) which may be written in inner
a, j,.
(5.2)
variable @ actually corresponds satisfy elastostatic equations and However, we know from the potential $‘)(F) is equal to variables simply as the dilatation
- V)
6(J) = -(V,
(5.3)
where
v, = {a/ax, ajar).
(5.4)
We will define the stress in the inner region in terms of the inner displacement inner variables X, Y. Therefore the sum of principal stresses is
V and
(5.5) in which 0 is the non-dimensional
stress invariant
scaled on
go = A’kn, for reasons
(S.6)
that will be made clear in Section 7. Hence 0 = 2(k,/k)(z2
- l)V, . V,
where z = K/k, and we may match inner and outer problems by defining the inner matching potential
WV
= -
WJ 2FI
1>
O(R).
(5.7) [using the principle
(5.2)]
(5.8)
Scattering
6.
from a surface
imperfection
I693
THE OUTER EXPANSION FOR A CIRCULAR DEFECT
To illustrate the proposed method we will now concentrate attention on the imperfection illustrated in Fig. 3. This is a circular void or protrusion (of the host material) whose boundary intersects the free surface at x = fa, denoted by the letters B and D respectively, and which has arbitrary radius. The perpendicular distance from the centre of the circle. 0’, to the free surface will be taken as ad and so the radius is u &+T’. The angle DO’B is 2c( as shown, so d = cot CI,
(6.1)
and the radius is u/sin c(. It is easy to see that as CIincreases from zero the circular void decreases in radius until, at CI= 71/2, it is a semicircular “bite” of radius a. This case has already been discussed briefly by the authors (ABRAHAMS and WICKHAM, 1991), and can also be tackled by multipole methods (GREGORY and AUSTIN, 1990). The bite continues to shrink (although the radius increases) as CIincreases, and when M = rr the defect vanishes. We therefore expect nonscattering at this particular value! We interpret a protrusion as the case when cx> Z. When CI= 3rc/2 the protrusion is semicircular with radius a. This is most easily understood by replacing a with 271-LX, which reverses the sign of d and measures the new CIacute when 0’ lies in y = 0. This is the most natural coordinate system in this case. To obtain the leading order outer expansion we need to specify the incident wave field. We take this as a right travelling Rayleigh wave of the form
“(I)
=
FIG. 3. The model problem of a circular void (a < a) or protrusion
(a > z) which intersects at the points B (x = a) and D (x = -a).
(6.2)
the free surface
I694
ABRAHAMS and G. R. WICKHAM
I. D.
where U0 is the amplitude of the displacement and B+/A+, do, yo, k, are given in (2.9)-(2.12). The incident displacement must be expanded about the origin and then substituted into the integrand of (4.10) to determine D$,! ,. To do this, it is most suitable in view of the shape of the defect, to introduce an alternative inner polar coordinate system based at 0’. We write the relationship between the original inner coordinates (4.3) and the new ones as X = p sin [I,
(6.3)
Y= d+pcos~, and so the outward
pointing
(6.4)
vector on the surface is v = -(sin/,‘,cos/I).
Substituting “(‘1 =
(6.3), (6.4) into (6.2) and expanding 2k,,K26,
_.!!!L 4k&
- iK2(2k; - K2) >
(6.5) gives
(1 + &i(k,/k)p sin p) K’(2k;-K2)
CUO 4kk&,
i -2ikodo(2k’
- K2)
(d+p
cos B) + 0(&Q.
(6.6)
This may be used in (4.10) to give D’,2/ =
iK’ = -&Sk
,I
2(7c-a)+sin --si.F
where the O(1) term is identically
zero. Similarly
iK2(2ki 002
-
2r ~ +O(c2),
(6.7)
we find
- K2) 2(7c- a) + sin 2a sin' a
8k&
’
(6.9)
along with Difi = D,‘:\ E 0. As has already been mentioned, @,22),$(22, 65’2, &‘/,, are all zero and so only the terms (6.7))(6.9) need be considered. Our observations made earlier regarding the case LY= n are borne out here, as all the D$ , are zero at this angle. The remaining (unknown) term at O(c’) is J7- a Cc,‘! = -cosec
(a)
s U-F
V”’ sin p dfl,
where V”’ is the leading order inner scattered displacement. We can therefore the leading order expansion, using the notation of the last section, as
(6.10) write
Scattering
from a surface
I695
imperfection
in which K* 2(7c- cc)+ sin 2~ D = 4k2 -TnTa0
(6.12)
and C = Cc,:’ - i(k,/k)D.
(6.13)
To perform matching we rewrite r’ in terms of R’ and expand, regrouping like terms in E, The details of the expansion of the $tl,,,i functions are given in Appendices A and B, and here we merely quote [from (A.1 1) and (A.8)] that (6.14) and also (6.15) whilst @j2 = 0(1/e).
Therefore
to leading
order (6.11) gives (6.16)
and all that remains
is to evaluate
7.
the constant
C by examination
of the inner region.
THE INNER EXPANSION
The model defect was introduced in Section 6, and illustrated in Fig. 3. In inner coordinates (X, Y), defined in (4.3), the distance between the end points of the imperfection boundary, BD, is 2, and from (6.3), (6.4) the radius of the void (protrusion) is p = cosec c(, and d = cot CI. The governing equation (2.1) for the dimensionless displacement V(R) is, in inner coordinates, ,Llv;v+@+/L)V,(V,~V)+~*(~~+2~)V
= 0,
(7.1)
where Vy = (d/8X, d/dY} and E = ka. We expect that the dimensionless have an expansion of the form v = V’O’+ O(E) and the related as
stresses
(based on inner displacements
displacements
(7.2) and coordinates)
@;, = 0@~’ + O(E), where we recall that a. = pk/ko so that gi,(‘I is also dimensionless.
are written
(7.3) On the surface of
1. D. ABKAHAMS and G. R. WICKHAM
1696
the circular coordinates
imperfection we know that the total normal (6.3), (6.4) based at 0’ are
and shear stresses, written
&” = - u):I,> 0,’ (0 e$ = - “{j/J 3
in
(7.4) (7.5)
on the boundary p = cosec LX,a-n < fl < 71-E. In Section 6 we stated that the forcing is an incident Rayleigh wave (6.2), from which we can calculate the right hand sides of (7.4) and (7.5). The leading order boundary value problem can now be summarized. V”” satisfies (7.1) with 6;set to zero, the stresses on the defect are D$) = I0 sin* /I, 0$
(7.6)
= I, sin [j cos p,
(7.7)
where I,, = -2i(r’-
1),
(7.8)
for p = cosec M. x--71 < p < IC--CI and the stresses o(F$.,~$1, vanish on the free surface Y = 0, IX/ > 1. We insist that the displacements remain finite everywhere in the elastic material, and we also suppose that there are no tractions applied at infinity. Thus V’“’ must be o( l/R) as R 4 cc. We solve this elastostatic problem by the method of conformal transformation, after first introducing an Airy stress function and complex variables, following GREEN and ZERNA (1954). If we write the Airy stress function as s, defined as E g$2 = -YY,
(7.9)
G(>?$= z-XXI
(7.10)
C(J)G’XY -
_sXr
(7.11)
and introduce (7.12)
I? = x+iy, then the divergence
of (7.1) gives (7.13)
where ? is the complex
conjugate
Z(s,Y) = :
of z. A solution [i(z-f),f(z,Y)+(zZ-
of this equation l)g(5,2)],
is (7.14)
in which ,f and g are as yet arbitrary harmonic functions. It is the fact that B can be reduced to two independent harmonic functions which allows us to use conformal transformation methods. We choose the mapping 2: which takes any point infinite strip
on the domain
coth (+<) cf the elastic material
(7.15) to a single point in the
Scattering
I697
from a surface imperfection 03q3
(7.16)
-r,
(7.17)
i = t+iy.
The points A, B, C, D, shown in Fig. 3, are mapped to [ = 0, n3 + iv, --iz, - co + iv respectively, i.e. the circular arc of the defect boundary is mapped to the line q = - CI and the plane surface is taken to y = 0. Substituting the new variable E,, = (cash 5-cos and noting
that ,f, ,9, being harmonic,
(7.18)
r/)E
satisfy (7.19)
allows us to write the general a,(<,?!)
=;
form of the solution
(7.14) as
; {[F,(s) sinh (ST) + F*(s) cash (sv)] sin y s ‘X + [G,(s) sinh (sq) + G2(s) cash (sq)] cos q> emrsi ds
by Fourier transform techniques. We now impose the boundary plane Y = 0, or equivalently 17= 0. These are, after integrating changing coordinates, that z = E,, = 0, Y,
Pj = 0.
(7.20)
conditions on the (7.10), (7.11), and
(7.21)
Hence G*(S) = 0, F,(s) = -sG,(s).
(7.22)
The boundary conditions on the “bite” are here much more difficult to derive than for the case considered by Green and Zerna. Firstly we write the relationship between the stresses, in radial coordinates centred on 0’ (6.3), (6.4) and E. We integrate these using (7.6) and (7.7) to yield (7.23)
az
af
= / p=coseca
ps B+cos4
cos @
0
sin2 CI
.
(7.24)
In deriving these terms, the integration constant was chosen to give finite energy at the edges Band D. Secondly, from the mapping (7.15) we can obtain the relationships
as ap and
(7.25) p=cosecl
I. D. ABRAHAMS and
1698
G.
R. WICKHAM 2
(-0s
/j+cos
c(
=
_-Y-a-.-
.
(7.26)
cash i”- cos a Therefore
we can determine
the boundary
conditions
on the defect, namely (7.27)
(7.28) These can be Fourier transformed Omitting all details we obtain
explicitly
in order to determine
F,(s) and G,(s).
sinh (SM)sinh [.s(n - a)] + s’ sin’ Mcash (sz) - .s sin CIcos u sinh (sz) J’,@) = _ ~~~-~---_---____-,--.----~---~-(smh* (SK) -s2 sin2 CC) smh (UC) (7.29) G,(s) =
_ +‘!?? _ ~~ . smh- (RX)- s2 sm’ cx
(7.30)
The solution has now been obtained and all that remains is to generate its far-field behaviour. For matching purposes we require the sum of the principal stresses, which in terms of the Airy Stress Function is (7.31)
0 = Exx+Zyr. This may be found by transforming operator of (7.20). We find
to < coordinates
and then taking
the Laplacian
where
= sinh (sy){G, (s)(l +s2 fis +cosh(,sy){sF,(s)(cos
cos q sinh <-s’cos
r/ cash l-
1)-G,(.
q cash 5) +isF,(s)
F)(:\ cash <+ is2 sinh t) sin n}.
The leading order matching potential Qcol, (5.8), is proportional required as R + co. This is deduced from (7.32) after changing and after establishing that as ]z[ --f MI, the mapping (7.15) is [ --. So, in polar coordinates
we write
2 ?
sin y sinh 0 (7.33)
to 0 and its form is to outer coordinates
(7.34)
1699
Scatteringfrom a surfaceimperfection
ir’ z
=
e-rfl’
__
E
which corresponds
)
r’ = O(l),
(7.35)
to (5, ‘1) - (sin Q’, -cos
and these forms are substituted to be
(7.36)
0’) +, I”
into (7.33) and expanded.
The leading
term is found
(7.37) where the integral on the right hand side is a real function of CI. We will use this in the next section to deduce the unknown outer coefficient and the scattered Rayleigh waves.
8.
DETERMINATION OF THE SCATTERED RAYLEICH WAVE AND CLOSING REMARKS
We showed, in the particular example chosen in Section 6, that there is just one unknown coefficient in the outer solution, (6.12), to leading order. This can now be determined by applying the matching condition, (5.2), at first order : 4 Therefore
(2.0)= @(02)~ -
(8.1)
(6.16) and (7.37) give
where F,(s) is written in (7.29), r = K/k, and r0 = K/k,. Hence, we can now deduce the reflected and transmitted Rayleigh waves scattered by the circular defect. We wish to obtain A + and A ~, the surface wave coefficients of the compressional potential expressed in (2.7), and note that the incident forcing wave (6.2) has been chosen so that A? is unity. To determine A, we firstly close the integration contours in (A.8) and (A.9) in the upper (lower) half planes according to x < 0 (x > 0). The residue contribution from the simple pole, which corresponds to a real zero of the Rayleigh determinant (2. lo), gives rise to the surface waves. These expressions are then differentiated to yield the Rayleigh wave components of $\:‘, . . , @,4:,,(F), (A. 1 l)-(A.22), and from these the outer expansion (6. I 1) is known. We obtain {(2--2;)2(1 to leading
order, where D is written
in (6.12), 6, = Q’?&
f l)Tr;t}D
K2, and
(8.3)
1700
I. D. AHKAHAMS and
R’(k
0
)
=
G. R. WICKHAM
d”(?! da
63.4) cr-k,,
The first observation we can make about the above result is that Ai are purely imaginary [.%(C) = 01. This would suggest at first sight that the total transmitted wave, which is the sum of the incident plus scattered wave, has energy flux greater than that of the incident wave in .Y < O! However, it is easy to show that terms of O(E’) contribute to the energy at the same order as those calculated here. Therefore, these leading order Rayleigh wave coefficients (8.3) are useful for determining displacements and stresses, but for energy calculations the matching must be performed to second order. It is interesting to examine how the scattering coefficients alter as the size of the defect increases or decreases. Therefore in Fig. 4 we plot that part of A+ and A_ written in square brackets in (8.3), as a function of the angle M(which is displayed in Fig. 3). For 0 < IX< rc the imperfection is a void, and for 7-r< CI< 2n it is a protrusion with angle DO’B = 2x-x. As already mentioned, the scattered field must vanish when x = n, and this is confirmed in the figure. The angle x is not taken too close to 0 or 271otherwise the void or protrusion would become very large, thus violating our basic assumption in the paper. The elastic constants are chosen so that r = 1.5 and hence r0 = 0.971494. The most interesting feature of the graph is that a void has A,>A, whilst a protrusion results in the transmission coefficient A, being less than the reflection coefficient A ~. However, more importantly, the magnitude of the transmitted wave A + is less (greater) than the magnitude of the reflected wave A for c( angles less (greater) than approximately 2.22 radians. Note that Fig. 4 remains of the same form for any values of the elastic constants. This concludes our discussion of the circular defect example. The method introduced in this paper has been applied successfully, and, after matching to leading order, the Rayleigh surface wave terms have been examined. There are a number of general comments to be made regarding our method, but we have chosen to reserve these for the concluding section of Paper II. There, we shall present a full demonstration to third order, where the inner geometry is both particularly difficult and of physical
FE. 4. Graph
of the square
bracketed
terms of A i written
in (8.3) plotted
against
the void angle (x.
Scattering
interest. without
from a surface
1701
imperfection
The aim of this paper has been to give a clear exposition choosing a difficult inner problem to obfuscate the solution.
of the method,
ACKNOWLEDGEMENTS Part of this work was carried out whilst Abrahams was in receipt of a Nuffield Science Research Fellowship, He is grateful to the Nuffield Foundation for this generous award.
REFERENCES ABRAHAMS, 1. D. and WICKHAM, G. R. ARRAHAMS, I. D. and WICKHAM, G. R. BRIND, R. J. BRIND, R. J. and ACHENBACH, J. D. BRIND, R. J. and WICKHAM, G. R. CRIGHTON, D. G. and LEPPINGTON, F. G. DATTA, S. K. and SABINA, F. J.
1991
Proc. Inst. Acoust. 13(2), 103.
1992
.I. Me&
1981 1981
Ph.D. thesis. Manchester. J. Sound Vib. 78. 555.
1991
Proc. Roy. Sot. Lond. A433, 101.
1973
Proc. Roy. Sot. Lond. A335, 313.
1986
In High and Low Frequency
Phys. Solids 40, 1107.
Asymptotics
(edited
by V. K. VARADAN and V. V. VARADAN). North-Holland, Amsterdam. GREEN, A. E. and ZERNA. W. GREGORY, R. D. GREGORY, R. D. GREGORY, R. D. and AUSTIN, D. M. KHRAPKOV, A. A. LAPWOOD, E. R. MENDELSOHN, D. A., ACHENBACH, J. D. and KEER, L. M. SABINA, F. J. and WILLIS, J. R. SABINA, F. J. and WILLIS, J. R. VAN DYKE, M.
1954
1967 1974 1990
Theoretical Elasticity. Proc. Camb. Phil. Sot. Proc. Camb. Phil. Sot. Quart. J. Mech. Appl.
Clarendon Press, Oxford. 63, 1341. 77,385. Math. 43, 293.
1971 1949 1980
Prikl. Mat. Mekh. 35, 677. Phil. Trans. Roy. Sot. Land. A242, 63. Wave Motion 2, 277.
1975 1977 1964
Geophys. J. Roy. Astr. Sot. 42, 685. J. Geophys. 43,401. Perturbation Methods in Fluid Mechanics.
Aca-
demic Press, New York, WATSON, G. N.
1966
A Treatise on the Theory of‘ Bessel Functions, 2nd Edn. Cambridge University Press, Cam-
WICKHAM, G. R. WICKHAM, G. R. ZHANG, C. and ACHENBACH, J. D. ZHANG, C. and ACHENBACH, J. D.
1977 1980 1988a
Proc. Camb. Phil. Sot. 81, 291. Quart. J. Mech. Appl. Math. 33,409. Wave Motion 10,365.
1988b
(iltrasonics
bridge.
26, 132.
1702
1. D. ABRAHAMSand G. R.
WKKHAM
APPENDIX A : FUNDAMENTAL POTENTIALS Here we present notation employed
the potential pairs (@‘, Y’) and (a*, Y”), satisfying by BRIND and WICKHAM (1991). These are
(3.1)-(3.3),
using the
(A.2)
(A.3)
where R(a) and S(z) are written in (2.10) and (2.11) and T(X) = 4ix(2r’The square root functions
of the complex y(a) = ($-k2)“2,
K’).
(A.5)
variable CI,Y(E) and C?(X),arc defined by 6(c() = (El-K’)’
1
(A.6)
with branches chosen to satisfy v(O) = - ik, S(O) = - iK. The contours of the Fourier inverse integrals lie along the real line but are indented above the pole at u = -k, and the branch points at x = -K, -k and below the pole at CI= k, and the branch points at CI= K, k. The Hankel function of the first kind N’,“(klr-r’l) may also be expressed in the same form, i.e. (A.7) We require the above potentials in order to construct the outer field functions &;i ,(F’) from the derivatives of the stresses X:(0, F/k), (4.11). Rather than work with (a’, ul’) it is, however, more convenient to define the dimensionless potentials
(A.8) (A.9) formed from a combination of expressions (A.l)-(A.4) set at r = 0. The integration path 01 both integrals is the same as that defined previously. Then it is straightforward to show from (A. IO) and equation
(4.1 I) that (changing
? to r for convenience) (A.1 I)
(A.12) whereas
from a surface
Scattering
i’,‘,‘,o
Further
imperfection
= ,‘; (2~;+++~&
1703
i-?(+$,&~.
(A.13)
we find that fjq:,,(q
-c4)
dlllz(f)=
-f
= 16$T-l,
;;
f$,((O),
(A. 14)
~~[(2~~+i:)(4(r’-I)~~+T2(12-2))1C*rl(i),
@,4’,,(f) = _*
(A.15)
(2” - 1) a2 _IzT[i’+2+4gM~.
(A.16)
z-
Equation
(3.2) allows us to express the non-zero
&M)
= 242
c$~, potentials
g
&4?),2@ = -2r5;
(2 $
q%&,(f) = 8q
$(2%
as
++,(i).
(A.17)
++,(i),
ox
(A.18)
+~~)45 A(kj 3
(A. 19)
and similarly from (3.3) @J2(f) = p_ a rii4(r’-l)$ Tz c’x
(
49i?j2(f)
=
-f
2’;
1‘++4(r'-I)~$
(
1 ,
$n(i),
(A.20)
(A.21)
fjA(i),
>
$(,42,,(f) = 4 (?I’>
(A.22)
z2 The following appendix will be concerned field point tends to the origin.
with the expansion
of these c#$? , terms as the
APPENDIX B: NEAR FIELD EXPANSIONSOF 4n AND 4B
To obtain a near field expansion as (f --t 0) of the potentials (A.1 ])-(A. 12) it is necessary to expand the fundamental solutions #A(T) and q5#(i) which are defined in (AX) and (A.9) respectively. We follow the method of BRIND and WICKHAM (1991) by first mapping the complex a-plane onto the complex t-plane according to the one-many transformation CI= -cosht, so that one copy of the contour
of integration
in the E-plane is mapped
(B.1) into the contour
%‘”
I704
I. D. ABKAHAMSand G. R. WICKHAM
co
.__1 CO
-To-T
T To
shown in Fig. 5. WC cut the complex t-plane along an infinite array of parallel line segments joining each pair of branch points at t = i T+ inn, where T = cash ’ T. T = K/k, and y1is any integer. These arc the branch points of A(t) = O( -k cash t)/k = (cosh’t-
T’)“~ = -sinht(l-imh:,)‘.
(B.2)
and clearly we must choose the right factor of the last expression to have a positive real part on the contour % ,,. Further. we choose A(.(t)to be negative imaginary as WCapproach the branch cut corresponding to n = 0 from above, and, as we approach the cut corresponding to IZ= 1 from below. We further specify that A(t) is periodic with period 2ni in the whole cut t-plant. With this definition for A(t). we may write (B.3) where R,,(f) = (2 cash’ t--r’)‘-4
cash’ f sinh tA(t).
(B.4)
An exactly analogous formula holds for 4,?(~, 0). Now the contour of integration in (B.3) may be closed by the two loops f’ shown in Fig. 5 and it is easily verified that R”(t) has simple zeros at i T,,+ inn. where T,, = cash ’ (s/t,,). to = K/k,,. It follows that (B.5) where C, is the sum of the residues at the four poles i_ T,, + in and f T,, + 2in and .#,, is a closed loop around the nth branch cut. WC now evaluate the contributions from each contour scgmcnt. Firstly note that. for .8(f) sufficiently large and positive. we may write
(B.6) where h2,( are the coefficients
in the Taylor series of
about u = 0. Hence the contribution
from -r-
is
Scattering
1705
from a surface imperfection
(B.8) However, each of the integrals on the right hand side of this last result may be evaluated using the contour integral representation of the Bessel function given on page 178of WaTsoN (1966) ; we find that c#I:(?, 0) = i i hz,, :~ [en”~‘n) J, (?)J =Z,r. 01 ,I_ 0 Similarly. the contribution of e’ to yield
from -r^
may be evaluated
(PI 05 0) = ; i
,I 0
Next
calculate
WC
the contributions
4’,5ll’ +_/$I = L +
s
by expanding
(B.9) the integrand
in powers
~[e~~(~-~~ J,.(F)],._,,,.
h,,z 0t
to $‘,,‘I, @y’ from g ,, &z respectively
’ _T,ql(t)[(t+in)em
(B.lO)
; WCfind that
l’WItl(!+1~~) + (t+ 2j7() ciSi,+I,+,(,,] dt,
(B.11)
where &&)_!L
sinh t(2 cash’ t--ZZ)Zj’~B-coshz ~~~_~~__~__~_~,~~~~~~~~ 4 (2cosh’t-~~)~+16cosh“tsmh~t(z’-cosh’t)
t
~~~~,
(B.12)
Equation (B. I 1) is easily expanded as a Fourier series in 0 on the interval (- 7c/2,7r/2) ; explicitly displaying only those terms that are even in 0, we have +y’ + q$i2’ =
-II i
E,,in’
n - 1110
s ’
-/
gn(t)qM(t) dt J,,,(F) cos r&fan
odd function
of 8, (B.13)
where E,, = 2, c,,, = I, WI> 0 and rwc
mn
rl,3V(t)= 2t cos mm-cash mt+n sin ~ smh mt 2 2 Similarly,
(B.14)
denoting (B.15)
we find that c,,inr~nT(TO)J,,(~)cos &+an Finally, collecting
all terms together
odd function
of 0.
(B.16)
yieldsl(B.17)
where (B.18) Performing
an identical
procedure
for dB, we obtain
f Since q5Ais an even function of 8, the odd terms must sum to zero
I706
I.
where C&are the coefficients
of the Taylor series of
D.
ABKAHAMS and G.
R. WNXHAM
and
with t,,,(t) = 2t sin ?2: sinh mt--n
cos “1” cash nztt,
(B.22)
and sinh’ t cash’ t&‘~&Sh’t
,~-/B(~j_
(2cosh’t-z’)“+16cosh4tsinhZt(r2-cosh’t)’
(B.23)
We have thus provided formulae for all the coefficients in the cylindrical wave expansions of tia, $H; in the determination of the inner expansion of the outer field we shall require the explicit values of the first few h,,,, d,,, ; these are h = e:-‘,-’ 0 4n h ZniA, = 0,
d2,=0,
d,=
m = 0, 1,2,.
m=0,1,2
I (3r4-52’f4) -2n-7-1F-.
(B.24)
’
T
,...,
,
(B.25)
(B.28) (B.29)
It is now easy to see how an asymptotic expansion of each &. ,(F) potential can be obtained. The power series expansion for J,,(f) is substituted into (B.17), (B.19), and after performing the differentiations with respect to order and rearranging we can obtain expansions of @,,, b,,. Further differentiations with respect to X according to the formulae in (A. 1 l)-(A.22) product the required expansions, which are then used in (4.7) to obtain 6”)(i) to any required order. This procedure was carried out with the aid of the algebraic manipulation package MAPLE on an Amdahl 5890 computer.