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Three-dimensional wave propagation analysis of a smoothly heterogeneous solid
J. Mech. Phyv. Solids, Vol. 43, No. 4, pp. 533-551, I995
Copyright 3’1 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-5096’95 $9.50+0.00
Pergamon 0022~5096(94)0008&8
THREE-Dr~ENSIONAL WAVE PROPAGATION ANALYSIS OF A SMOOTHLY HETEROGENEOUS SOLID R. Y. S. PAK and B. B. GUZINA Department
of Civil, Environmental
and Architectural CO 80309-0428.
Engineering. U.S.A.
University
of Colorado.
Boulder.
ABSTRACT A method of analysis is presented for three-dimensional wave propagation problems of a verticallyheterogeneous half-space with a linear shear wave velocity profile. With the aid of a displacement-potential representation. Hankel transforms and Fourier decompositions. the dynamic response of the semi-infinite solid to an arbitrarily distributed buried source is shown to admit integral representation in terms of modified Bessel functions. Specific aspects of the problem such as the multiple poles along the inversion path on the complex plane and the characteristics of the wave propagation in the vertical direction are elucidated. Apart from its intrinsic interest, the solution can be degenerated to ring-load and point-load Green’s functions which are fundamental to boundary integral equation formulations.
1. INTRODUCTION Wave propagation in a smoothly-heterogeneous solid is a topic of engineering interest owing to its relevance to a number of applications. In material and solid mechanics, a spatial gradient of the material modulus can be caused by di~erential temperature distribution in the solid, or realized in the chemical process of substrate formation in modern micro-electronic manufacturing (Hirano et al., 1988 ; Ozturk and Erdogan. 1993). In earthquake engineering and soil dynamics, an analytical framework for the problem is important to a rational assessment of the dynamic performance of foundations and structures which rest on sites with a gradual stiffness variation resulting from the natural deposition process as well as gravity-induced stress conditions (Hardin and Drnevich, 1972). Because of the complexity of the underlying physical phenomenon and mathematical analysis, most of the existing treatments on the dynamics of a vertically-heterogeneous medium have been focused on the study of specific wave forms and their propagation. On Love waves. for instance, there are the works of Meissner (1921, 1926), Aichi (1922), Jeffreys (1928), Bateman (1928) Matuzawa (1929), Wilson (1942) and Sato (1952). The characteristics of Rayleigh waves in such kinds of solids have been considered by Honda (193 I), Stoneley (1934), Pekeris (1935), Newlands (I 950) and Vardoulakis and Vrettos (1988). A description of some of the mathematical aspects of such investigations can be found in Ewing et Q!. (1957) and Brekhovskikh (1960). For general plane problems in a verticallyheterogeneous solid, the formulation of Gupta (1966) is of interest wherein the 533
534
R. Y. S. PAK and B. B. GUZINA
reflection of plane elastic waves in a medium with a linearly-varying wave velocity is examined. A few approximate techniques for studying surface wave propagation in an inhomogeneous medium whose stiffness has an arbitrary depth-dependence can be found in Aki and Richards (1980). As to the treatment of general elastodynamic boundary-value problems involving vertically-heterogeneous media with smooth variations, however, only asymptotic or approximate analyses for some simple cases have been attempted as in Awojobi (1972, 1973) and Gazetas (1980). In this paper, a method is presented which can be used to tackle three-dimensional wave propagation problems for a vertically-heterogeneous half-space with a linear shear wave velocity profile. With the aid of a displacement-potential representation, Hankel transforms and Fourier decompositions, it is illustrated that the dynamic response of the semi-infinite solid to an arbitrarily distributed buried source can be expressed as integrals of modified Bessel functions. Apart from its intrinsic interest, the solution can be degenerated to ring-load and point-load Green’s functions for use in various boundary integral equation formulations for engineering problems (Brebbia ef ul., 1984, Pak and Ji, 1994).
2. MATHEMATICAL
FORMULATION
2.1. A method qfpotentials In the linear theory of elastic wave propagation in an isotropic solid, the governing field equations are the Cauchy’s first law of motion, the constitutive law, and the strain~isplacement relation : V*2+f
= pii,
(1)
z = A tr (y)I + 2py,
(2)
y = ;(vu+w).
(3)
Here u is the displacement vector, z is the stress tensor, y is the strain tensor, p is the mass density, and 1 and ~1are the Lame’s constants. For the case where ;i and p are varying functions of space, the foregoing equations can be combined into @+2p)V(V~u)-pVxVxu+(v*U)V~+(VU+VU=)Vp+f
= pii,
(4)
which is valid in a general coordinate system. In what follows, attention will be focused on the dynamic response of a half-space with a constant mass density and Lame moduli whose variation can be expressed as ~(z)=~(z)=~~(l+~~)~,
~20,
b>O,
(5)
in a cylindrical coordinate system (r, 8, z). The foregoing stipulation yields a Poisson’s ratio of l/4 and a linear shear wave velocity profile which is the case of interest in this communication. For such a medium, the displacement equation of motion (4) can, in the absence of body forces, be reduced to 3~v(v~u)-~vxvxu+(v~u)v,u+(vu+vuT)v~
= pii.
(6)
Wave propagation in a heterogeneous solid
535
For the solution of this class of problems, it is useful to write, as an extension of Pak (1987) and Gupta (1966), the displacement field in the form of u=pV
-jcp( ,@,z) +~Vx(pX(r,B,;)e,+llVxrl(~,U.z)q). ilr
(7)
)
In (7), e, is the unit vector in the z-direction, rp is the pseudo-dilatationa potential, and x and y are the pseudo-distortional potentials. If p is independent of z, (7) will degenerate to the displacement-potential representation in Pak (1987) for a homogeneous elastic medium. For time-harmonic motion with circular frequency w, one may write the displacement field of the heterogeneous half-space as
u(r, H,z, t) =
um(u,8, A,t) - z&r, U,,) e”“’ = u(r, Q,z) err”’ I:::::_~:::~ - ~~~~:~~:~~
(8)
and other quantities of interest in similar complex notations, By virtue of representation (7), the equation of motion (6) will be satisfied provided that v’tp+?%;(?Jy, = 0, V”I+k;(Z)‘l
v*x+&
= 0,
(9) (10)
2 J&(Z)X = 0,
where k;tzj =
Ozp
(12)
w2p
(13)
3&(1 +hz)’ ’ kj(z) =
&(l +bz)*
One may identify (9) with the primary (P) waves, and (10) and (11) with secondary (S) waves. It should be noted, however, that such P-waves are not purely irrotational, nor such S-waves purely equivoluminal. Similar observation was made by Gazetas (1980) regarding the plane strain problem. One should also keep in mind that the phase velocities of such waves are generally frequency-dependent as heterogeneous media are intrinsically dispersive in nature. 2.2. General solution of the governing equations In cylindrical coordinates, the governing equations (9), (IO) and (I 1) can be written explicitly as (14)
536
R. Y. S. PAK and B. B. GUZINA
y;m81virirz
of the completeness of the angular eigenfunctions -n < 6 < TLwith respect to the class of solutions under consideration, one may express cp(r,e,z) = E:,“=_mp)m(~,~)eimff,
r(r, 0,z>= CiTY, __qm(r, z) eime, x(r, 0, z) = X2=__exnAr, z) elm’,
(17)
and u,(r, 8,~) = X2= -caurm(r,2) elm’, ue(r,6,z) = EZ= -a31kg_(r,~)eime, zfZ(r,e,z) = Cg= _~~~,(r,z)e’~~,etc. With an appeal to (17) and the orthogonality (15) and (16) yield
(18)
of (eImH),the governing equations (14),
(I91
G-9)
for each m. For further reduction, transform
one may make use of the mth order Hankel
whose inversion formula is
In the above, J, is the Bessel function of the first kind of order m. Application of the foregoing transform to (19)-(21) leads to a set of ordinary differential equations
Wave propagation in a heterogeneous solid
d’@,m 7 +(k,2zH2)@mm
d’fm” -+-_ dz2
2b
dx’,” dz
l+bz
537
= 0,
+(k;(Z)--2)f;
= 0.
(26)
By means of a change of variable 1 +bz
x=r-
b
’
(27)
and the definition of an auxiliary function F(x) = x-“‘@z({, z), the general solution of (24) can be found to be
where
a=
1 --___ W2P 4 3pOb2’
J
(29)
In the above, A, and B, are integration constants, while Z,(X) and K,(x) are the modified Bessel functions of the first and second kind, respectively. Their order c1can be either a real (in the case of low frequency) or an imaginary number. Analogously, the general solution of (25) can be written as qm”(&z) =(l +bz)“’
(c~,,,,(,~)+D,(C)K,(i~))
(30)
where (31) Finally, by defining the auxiliary function G(x) = x”‘fz(<, z) where x is given by (27), one finds that the general solution of (26) can be expressed as x:(&z)
=(l+bz))“2
(Z.(5)Z~(,lihhi)+~~*(n~~(r~)).
(3.2)
2.3. Boundary conditions for a buried source problem Although the present formulation rules out body forces, the action of an arbitrarily distributed source on the plane z = s in a half-space can be represented as a set of
538
R. Y. S. PAK and B. B. GUZINA
BodyForce Field
Fig. 1 A buried source in a heterogeneous
elastic half-space.
prescribed stress discontinuities across the corresponding planar region (see Fig. 1). In the treatment of this class of problems, it is convenient to view the half-space as being composed of Region Z {z < S} and Region ZZ{z > s} as indicated in Fig. 1, and to represent the distributed body-force field as a general discontinuity of stresses across z = s (see Pak, 1987) in the form of
(33) while requiring the displacements to be continuous everywhere. In the above, P(r, e), Q(Y,0) and R(r, 0) are the prescribed body-force distributions in the radial, angular and axial directions which can be expanded as
To render the foregoing problem specific for a half-space with a traction-free surface, one would require cZZcr,e,o)
= ez,(r, e,o) = z,e(y, 8, 0) = 0.
(35)
Finally, to ensure the formulation is physically meaningful, one should impose the regularity condition that the solution contains no incoming waves from infinity and that it vanishes as z -+ cc. It is important to observe, however, that the transformed solutions (28), (30) and (32) represent standing waves in the z-direction. As a result, the first part of the regularity condition is automatically satisfied by the functional space derived.
Wave propagation in a heterogeneous solid
539
2.4. Transfbrmed displacement- and stress-potential relations
From (7), one finds that the components of the displacement field can be expressed Xi
(36) In terms of the Hankel transforms, they are equi~dlent to
(37) With the aid of the constitutive relations can be stated as
relations (2), the transformed
stress-potential
Likewise, the boundary conditions translate to z1’,,(5,0) = 0, ~_~~-‘(5,O)-i~~~_‘(~,O) = 0, ~~“~‘(5,0)+i~~+1(g,0) for the traction-free
= 0,
surface at -7= 0, %;,(5,5-) = K;,(<,s+),
(39)
540
R. Y. S. PAK and B. B. GUZINA
~~~‘(5,s~)-i~~~‘(5,s~)
= ~~~-‘(5,s+)-i~~-‘(5,s+),
~~,+‘(5,s-)+i~~s”,,+‘(5,s-)
= ~~“(5,s+)+izlo”+‘(5,~+),
for the continuity
of displacements
at z = s, and
~~-m(r,~-)-~~:J(5J+) (TE;‘(4,sP)-iTsP
= Z,(5),
~‘(LSP))-(G;‘(5,s+)-i%zmg_‘(L~+))
(~~~‘(5,s~)+i~~,t’(r,s~))-(~~~~‘(5,sf)+i~~,ms_f’(5,S+)) for the discontinuity
conditions
(40)
=X,(5), = Y,(t),
(41)
on the stresses. In the above,
-%I(0 = &X5), L(5)
= Z%‘(5)
K(5)
=
p”,“+‘(~)+@,“+‘(S).
Consistent with the regularity condition (28), (30) and (32) can be expressed as GXLZ)
-i@-‘(5),
at infinity,
the relevant
potential
solutions
= (1 +~z)“2(~!n(~)Z&)+~t,(S)ZUx)),
%X&z) = (1 +~V2(C~(5)Z&)
+D!‘z)K&&)),
Xm”((,z) = (l+Z=-‘~2(Ef,(5)Z&)+Ff,(S)K&)), in Region
(42)
(43)
Z, and
K(r,z)
= (1+w”2mtvx4,
fmr,4
= (1+w”2RxvqX),
fm”(&z) = (1 +bz)-“2F:(S)Z$(x),
(44)
in Region ZZ. In the above, x is given by (27) and a and /3 are defined in (29) and (31), respectively. 2.5. Solution of the buried-source
problem
According to (43) and (44), the transformed Fourier components of displacements and stresses can be expressed in terms of the nine unknown coefficients conAi, B;, . . . , D,“, F,“. With the aid of the relations (37) and (38), the boundary ditions (39)-(41) provide the nine equations required for the solution of the unknown coefficients. In terms of the transforms of the prescribed loading, the solution for the Fourier components of the displacement field can be expressed as l4-:“,
= Q,(<,z;s;b)p
xlil-- Y, %I
+Q&,z;s;b)% PO
Wave propagation in a heterogeneous solid
+fdc+
I (Y, wq:
1(x, i)
541
2P+5 _iu~5+Iw--
X1,:2Y1'21 +H(z-.s) [
L(YK&)
6($3c12)
+ Z/S(Y) K, + I (-4
=
H(s-zf
,y
112
Y
-l/2
i
xp1:2
(46)
-112 b y
-W-:)W~+,(I,&i)
+fqz-s)
I[
28+5 ~~~+kX)-I
+Y;+‘hJ4
PI,2 y2(&z;s;h)
’
’
ivd”)KB(Y)l
W+1
(x9 i)zB(i)K,~(Y)ra-~+r.4p5+‘(i,
i>l>
-l/2
y b
i VdY)f$(X)),
(47)
R. Y. S. PAK and B. B. GUZINA
542
-py’/25
y,(<,z;s;b) = m--z)
b(7-3a2)i
L(eL,l(Y)
II [ ,.pyl12~ +b(+ x 28+5
+47(-wp(Y)
yjy
[
20(---3 ~%+,(Yd-l
1
+Y$+‘(x,x)
~zt(vmxww) 3a2)A([) i
[
ye+,(Y,Y)-l
1
[-(fi~~+~(I,i>+fi)+(f3~~+I(i,i)+f4)~~+I(r,r)l
(48)
Wave propagation in a heterogeneous solid
543
(49)
and
+fs%+
I (Y, im:
2c(--3
21_w:+
f (x, 0
[ Ii2
I (x, xl - 1
.
.I/2 i
+H(z-s)
x b(:
J -
34
Il,(Y)K,&)
y+C+‘(y,y) In the foregoing equations, (29) and (31), respectively,
(50)
H denotes
the Heaviside
function,
CIand /? are given by
and
i=$
(51)
x = i(1 +bz), y = i(1 +bs), 2’ihY)
K, W = ~ MY) ’
(52)
544
R. Y. S. PAK and B. B. GUZINA
(53) fi = 4~4+(6a2-4+t-2~2-10a+6~-1)~2 -t-(3a2/J2 +6a*p-6cQ2
-+x2 - 12cQ+;p2 +++;a-$>,
(54)
f2 =(4tl-6)[3+(-6t12+12~-;)[,
(55)
.A =(4~+10)~3~(6~2+12~-~)~,
(56)
f4 = -4c4 - IZC”,
(57)
fs = -413-21f12+;i, NC) =.f,a:+ I (5, W$+ I CL0 -t-.f+C+ 1CLi)+_fS$+
(58) 1@I,0 +.L
(59)
On substituting the inverted Fourier components of displacements (45) into the corresponding angular eigenfun~tion expansions (l&I, the formal three-dimensional solution of the general buried-source problem under consideration can be obtained. Due to the roots of A, however, some of the functions defined in (46)-(50) have multiple poles along the formal path of integration for the inverse transform. As will be shown in the next section, the sense of the contour integration is uniquely defined by the branch points and branch cuts corresponding to the case of b + 0. For o = 0, it can be shown that the analytical solutions degenerate to the static case for the heterogeneous half-space without the need of asymptotic analyses. 2.6. Analytical charac~e~i~aiio~of a weakly i~homoge~eous medium To examine the limiting behavior of the solution as b tends to zero which implies a homogeneous half-space, it is necessary to determine the character of modified Bessel functions with large real arguments (<( 1 -I-bz)/b) and large imaginary orders CIand j3. To this end, one could note that
(60) where
For the ensuing analysis, the following identities are also useful Iv(vx) = e- ‘/2v*iJy(ivx),
Wavepropagation in a heterogeneous solid
545
(‘m(Y)
Bra& Cutsfor the Square-root Function
BranchCutforthe ~g~~c axon -
o-
I-
. Re(y)
Complexy-Plane
Fig. 2 Branchcuts in the complex y-plane.
K,.(vx) = Te’~2’.‘Hi’)(iv.u),
Z,._, (vx) = e-‘/2”ni.J,,(ivx) [i +iE],
~~‘)‘(il?~) rC,,_,(v-x) = _ ~er~2VZij$~,~)(ivx)i +i-.L..__ H(l’(ivx) I’ 1’ [
--n < arg(vx) d ; )
(62)
where H$“(vx) stands for the Hankel function, and ““’ denotes differentiation with respect to the argument. With reference to the asymptotic expansions of Bessel and Hankel functions for large arguments and orders (Abramovitz, 1972), one finds
J”(vY) _pnv)Jo
I!'(1 _y2)--1/4
_ +v)-“‘(1
_y2)W
H:s”(vy) ~ ___2i(2~y-l/2(1 H!,“‘(vy) _ ;(2nv)-
e-2wJ~,
e-2/3i'3'2,.,
_y*)--1!4
l/2(] _y2)‘!4
e2135*3'z~s,
e2/35’3”y
(63) In the above expansions, [* is the auxiliary variable defined by the following mapping (64) where dq = eeixi2,/‘~. As indicated in Olver (1954), the three branch cuts in this mapping have to be chosen in such a way that 5* is real when y is positive. Specifically, the branch cut associated with the logarithmic function in (64) in this study is set along the negative y-axis (see Fig. 2) while the other two branch cuts
R. Y. S. PAK and B. B. GUZINA
546
t WQ Branch Cuts Path of Integration A -kso -k@
0
kfi
_
Complex f -Plane
ko
)
IW4,
5 Branch Cuts
Fig. 3 Branch cuts in the complex t-plane for small values of b
associated with the square-root functions are chosen such that the one originating from y = 1 extends vertically through the lower half-plane and the other originating from F = - 1 extends vertically through the upper half-plane. In order to find the limiting forms of
it is useful to define v=
1+&i
b ’
s=f!klE kii-b’
where k stands for kPOor kSo.By virtue of these definitions, the branches in the complex y-plane in Fig. 2 for small values of b correspond to the branch cuts in the complex t-plane as indicated in Fig. 3. To be consistent with the choice of branches and to stay within the domain of validity of (63), it is proper to deform the formal path of integration for the inversion integral with respect to the wave numbers g around the singularities in the upper half-plane as shown in the figure. Furthermore, to ensure the solution is analytically continuous with respect to b, one should maintain the contour integration in the first quadrant for all values of the inhomogeneity parameter and avoid any sudden switching of the branches. Upon expanding the auxiliary variable 5* with respect to b about 0 in the form of 3*3/2
_
jlog(“‘J7)-
it can be shown that
;~j&+&,/j?++;)b,
(66)
Wave propagation in a heterogeneous solid
547
Finally, upon direct substitution of the above expansions into (46)-(50), the solution for the homogeneous half-space (Pak, 1987) with h = /coand the Poisson’s ratio equal to 0.25 can be recovered as
I
+
y2([,z;s)
=
-l-(e-hdt +e-h4), Wh
(69)
(70)
548
R. Y. S. PAK and B. B. GUZINA
3325*-e)
-k-(t)
+ahPhe-U++~h~)), (5’e-UQ+~hS)
(72)
+ 2~*%(2? -ks)’ (e -(j3,;+a&+e-(&Y+““Z’), (73) k.:R - (5) where R * = (2c2 - k,2)’ rt:452a&,,
(74)
fib =((*-k;)“*,
(75)
k, = w&%%&,
k.9= w&
(76)
d, = Iz-$1,
d2 = z+s,
(77)
01~ =(+k;)i'z,
sgn(z-s)
=
+1,z>s i -l,z
I ’
(78)
In view of the correspondence of the two solutions, one may refer to the function A as the “heterogeneous counterpart” of the Rayleigh wave function R-. It should be observed, however, that A leads to multiple poles, rather than a single one along the formal path of integration. In addition, such poles correspond not only to Rayleigh, but also to Love as well as Stoneley waves, for a continuously-heterogeneous medium can be considered as a limiting case of a layered medium. With appropriate specifications of the internal source distributions P(u,@), Q(r,0) and R(r,B), Green’s functions for a variety of load configurations can be derived and evaluated as in Pak (1987) and Pak and Ji (1994). As illustrations, the radial displacements of a solid half-space with a linear-varying shear wave velocity in the vertical direction due to an embedded uniformly distributed lateral body-force field acting in the 0 = O-direction over the circular region II, = {(r, 8, z)l Y< a, z = s> are presented in Figs 4-7. The results are given along the z-axis (i.e. r = 0) for the case s = 0 {a surface source, Figs 4 and 5) and the case s = 8a (a buried source, Figs 6 and 7) at a dimensionless frequency of 0.5. The solutions for a heterogeneous medium with b = 0.05/a and b = 0.15/a are compared with the corresponding solution for a homogeneous solid labeled as “‘b = 0.00”. Consistent with the vertically increasing stiffness of a heterogeneous medium, the differences between the two formulations are more pronounced for the case of a buried source. From the displays, it is also apparent that the waves in a solid with higher b (i.e. the stiffer medium) are characterized by increased wave lengths and reduced magnitudes.
Wave propagation in a heterogeneous solid
S4Y
3.5
8
2.5
81 3
1.5
r$ a
0.5
---. ----
-0.5 0.0
4.0
bQEO.00 b*a = 0.05 b*a=O.H
8.0
12.0
16.0
z/a Fig. 4 Displacement U, under unit lateral force on II, at r = 0, s = 0 : real part. 0.5 /
I r/a = 0.0 sla = 0.0
$$
0.0 [p-y-
wo,&&=
, ,-=/.
_---__ __--L;,-“_
4.0
---.
----
b*a=O.QQ
---.
b*a=O.OS b*a=0.15
-1.0 1 0.0
0.5
8.0
12.0
16.0
Ja Fig. 5 Displacement U, under unit lateral force on 17, at r = 0, s = 0: imaginary part. 1.8 1.4
% .8_ 1.0 1 ’ $
$
i
t
r/a = 0.0 s/a = 8.0
!8’:
4.0
8.0
;:
------. -
b*a=O.OO b*a=0.05 b*a=0.15
06
. 0.2 -0.2 ’ 0.0
12.0
16.0
Fig. 6 Displacement U, under unit lateral force on II, at r = 0, s = 8a : real part.
3. CONCLUSIONS By a method of displacement potentials, a general solution for the dynamic response of a vertically-heterogeneous elastic half-space to the action of an arbitrary, time-
550
R. Y. S. PAK and B. B. GUZINA 0.50 ------. -
r/a = 0.0 3 &
.Z
0.25
-
-0.25 -
-0.50 0.0
_
b*a=0.15
I’--,,
0.00 . “‘--:2.
5
dz
s/a = 8.0
b*a=O.m b*a=0.05
,.er
“;-_ __ ‘\ ---_ ____---‘\ ‘\
‘\
. ..____.=
4.0
__/Lt------ , <’
/’ /’ -‘llxZ/&&
8.0 z/a
12.0
= 0.5 16.0
Fig. 7 Displacement U, under unit lateral force on B, at r = 0, s = 8a: imaginary part
harmonic, finite buried source is derived analytically. The result for a homogeneous half-space is shown to be recoverable in the limit as the inhomogeneity parameter tends to zero. Some specific aspects of the analysis such as the multiple poles along the formal path of inversion integrals and the characteristics of the wave propagation in the vertical direction are elucidated. With the aid of Fourier or Laplace transforms, the method can be extended to the treatment of transient problems in a straightforward manner.
ACKNOWLEDGEMENT The support of the National gratefully acknowledged.
Science Foundation
during the course of this investigation
is
REFERENCES Abramovitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. Dover Publications, Inc., New York. Aichi, K. (1922) On the transversal seismic waves travelling upon the surface of heterogeneous material. Proc. Math. Sot. Japan 4, 137-142. Aki, K. and Richards, P. G. (1980) Quantitative Seismology : Theory and Methods, Vol. 1. W. H. Freeman & Co., New York. Awojobi, A. 0. (1972) Vertical vibration of a rigid circular foundation on Gibson soil. Geotechnique 22(2), 333-343. Awojobi, A. 0. (1973) Vibration of rigid bodies on non-homogeneous semi-infinite elastic media. Q. J. Mech. Appl. Math. 26,483-498. Bateman, H. (1928) Transverse seismic waves on the surface of a semi-infinite solid composed of heterogeneous material. Bull. Amer. Math. Sot. 34, 343-348. Brebbia, C. A., Telles, J. C. F. and Wrobel, L. C. (1984) Boundary Element Techniques : Theory and Applications in Engineering. Springer-Verlag, New York. Brekhovskikh, L. M. (1960) Waves in Layered Media. Academic Press, New York. Ewing, W. M., Jardetsky, W. S. and Press, F. (1957) Elastic Waves in La_veredMedia. McGrawHill Book Co., New York.
Wave propagation in a heterogeneous solid
551
Gazetas, G. (1980) Static and dynamic displacements of foundations on heterogeneous multilayered soils. Geotechnique 30(2) 150-176. Gupta, R. N. (1966) Reflection of elastic waves from a linear transition layer. Bull. Se&m. Sock. Amer. 56(2) 51 l-526. Hardin, B. 0. and Drnevich, V. P. (1972) Shear modulus and damping in soils : design equations and curves. J. Soil Mech. Found. Engng Div., AXE 98, 6688692. Hirano, T., Yamada, T., Teraki, J., Niino, M. and Kumakawa, A. (1988) A study on a functionally gradient material design system for a thrust chamber. Proc. 16th Int. Symp. Spuce Tech. Sci., Sapparo, Japan. Honda, H. (1931) Kisyo-syusi [9]. Vol. 6, p. 237 (in Japanese). Jeffreys, H. (1928) The effect on love waves of heterogeneity in the lower layer. Month1.y Notices Roy. Astron. Sot. : Geophys. Suppl. 2, 101-l 11. Matuzawa, T. (1929) Observation of some recent earthquakes and their time-distance curves. Bull. Eurth. Res. Inst. (Tokyo) 6, 2255228. Meissner, E. (1921) Elastische oberflachenwellen mit dispersion in einem inhomogenen medium. Vierteljuhresschr. Naturforsch. Ges. Zurich. 66, 181-195. Meissner, E. (1926) Elastische oberflachen-querwellen. Proc. Secondintern. Congr. Appl. Me&. (Zurich) 3-l 1. Newlands, M. (1950) Rayleigh waves in a two-layer heterogeneous medium. Monthly Notices Rev. Astron. Sot. : Geophys. Suppl. 6, 1099125. Olver, F. W. J. (1954) The asymptotic expansion of bessel functions of large order. Phil. Trans. Roy. Sot. London A247, 328-368. Ozturk, M. and Erdogan, F. (1993) The axisymmetric crack problem in a nonhomogeneous medium. J. Appl. Mech., ASME 60,406413. Pak, R. Y. S. (1987) Asymmetric wave propagation in an elastic half-space by a method of potentials. J. Appl. Mech., ASME 54, 121-126. Pak, R. Y. S. and Ji, F. (1994) Mathematical boundary integral equation analysis of an embedded shell under axisymmetric excitations. Int. J. Numer. Meth. Engng 37(14), 25012520. Pekeris, C. L. (1935) The propagation of Rayleigh waves in heterogeneous media. Physics 6, 133-138. Sato, Y. (1952) Study on surface waves. V: Love waves propagated upon heterogeneous medium. Bull. Eurth. Res. Inst. (Tokyo) 30, 1-12. Stoneley, R. (1934) The transmission of Rayleigh waves in a heterogeneous medium. Monthly~ Notices Roy. Astron. Sot. : Geophys. Suppl. 3,2222232. Vardoulakis, I. and Vrettos, Ch. (1988) Dispersion law of Rayleigh-type waves in a compressible gibson half-space. Int. J. Numer. Analyt. Meth. Geomech. 12, 639-655. Wilson, J. T. (1942) Surface waves in a heterogeneous medium. Bull. Seism. Sot. Amer. 32, 2977304.
Copyright 3’1 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-5096’95 $9.50+0.00
Pergamon 0022~5096(94)0008&8
THREE-Dr~ENSIONAL WAVE PROPAGATION ANALYSIS OF A SMOOTHLY HETEROGENEOUS SOLID R. Y. S. PAK and B. B. GUZINA Department
of Civil, Environmental
and Architectural CO 80309-0428.
Engineering. U.S.A.
University
of Colorado.
Boulder.
ABSTRACT A method of analysis is presented for three-dimensional wave propagation problems of a verticallyheterogeneous half-space with a linear shear wave velocity profile. With the aid of a displacement-potential representation. Hankel transforms and Fourier decompositions. the dynamic response of the semi-infinite solid to an arbitrarily distributed buried source is shown to admit integral representation in terms of modified Bessel functions. Specific aspects of the problem such as the multiple poles along the inversion path on the complex plane and the characteristics of the wave propagation in the vertical direction are elucidated. Apart from its intrinsic interest, the solution can be degenerated to ring-load and point-load Green’s functions which are fundamental to boundary integral equation formulations.
1. INTRODUCTION Wave propagation in a smoothly-heterogeneous solid is a topic of engineering interest owing to its relevance to a number of applications. In material and solid mechanics, a spatial gradient of the material modulus can be caused by di~erential temperature distribution in the solid, or realized in the chemical process of substrate formation in modern micro-electronic manufacturing (Hirano et al., 1988 ; Ozturk and Erdogan. 1993). In earthquake engineering and soil dynamics, an analytical framework for the problem is important to a rational assessment of the dynamic performance of foundations and structures which rest on sites with a gradual stiffness variation resulting from the natural deposition process as well as gravity-induced stress conditions (Hardin and Drnevich, 1972). Because of the complexity of the underlying physical phenomenon and mathematical analysis, most of the existing treatments on the dynamics of a vertically-heterogeneous medium have been focused on the study of specific wave forms and their propagation. On Love waves. for instance, there are the works of Meissner (1921, 1926), Aichi (1922), Jeffreys (1928), Bateman (1928) Matuzawa (1929), Wilson (1942) and Sato (1952). The characteristics of Rayleigh waves in such kinds of solids have been considered by Honda (193 I), Stoneley (1934), Pekeris (1935), Newlands (I 950) and Vardoulakis and Vrettos (1988). A description of some of the mathematical aspects of such investigations can be found in Ewing et Q!. (1957) and Brekhovskikh (1960). For general plane problems in a verticallyheterogeneous solid, the formulation of Gupta (1966) is of interest wherein the 533
534
R. Y. S. PAK and B. B. GUZINA
reflection of plane elastic waves in a medium with a linearly-varying wave velocity is examined. A few approximate techniques for studying surface wave propagation in an inhomogeneous medium whose stiffness has an arbitrary depth-dependence can be found in Aki and Richards (1980). As to the treatment of general elastodynamic boundary-value problems involving vertically-heterogeneous media with smooth variations, however, only asymptotic or approximate analyses for some simple cases have been attempted as in Awojobi (1972, 1973) and Gazetas (1980). In this paper, a method is presented which can be used to tackle three-dimensional wave propagation problems for a vertically-heterogeneous half-space with a linear shear wave velocity profile. With the aid of a displacement-potential representation, Hankel transforms and Fourier decompositions, it is illustrated that the dynamic response of the semi-infinite solid to an arbitrarily distributed buried source can be expressed as integrals of modified Bessel functions. Apart from its intrinsic interest, the solution can be degenerated to ring-load and point-load Green’s functions for use in various boundary integral equation formulations for engineering problems (Brebbia ef ul., 1984, Pak and Ji, 1994).
2. MATHEMATICAL
FORMULATION
2.1. A method qfpotentials In the linear theory of elastic wave propagation in an isotropic solid, the governing field equations are the Cauchy’s first law of motion, the constitutive law, and the strain~isplacement relation : V*2+f
= pii,
(1)
z = A tr (y)I + 2py,
(2)
y = ;(vu+w).
(3)
Here u is the displacement vector, z is the stress tensor, y is the strain tensor, p is the mass density, and 1 and ~1are the Lame’s constants. For the case where ;i and p are varying functions of space, the foregoing equations can be combined into @+2p)V(V~u)-pVxVxu+(v*U)V~+(VU+VU=)Vp+f
= pii,
(4)
which is valid in a general coordinate system. In what follows, attention will be focused on the dynamic response of a half-space with a constant mass density and Lame moduli whose variation can be expressed as ~(z)=~(z)=~~(l+~~)~,
~20,
b>O,
(5)
in a cylindrical coordinate system (r, 8, z). The foregoing stipulation yields a Poisson’s ratio of l/4 and a linear shear wave velocity profile which is the case of interest in this communication. For such a medium, the displacement equation of motion (4) can, in the absence of body forces, be reduced to 3~v(v~u)-~vxvxu+(v~u)v,u+(vu+vuT)v~
= pii.
(6)
Wave propagation in a heterogeneous solid
535
For the solution of this class of problems, it is useful to write, as an extension of Pak (1987) and Gupta (1966), the displacement field in the form of u=pV
-jcp( ,@,z) +~Vx(pX(r,B,;)e,+llVxrl(~,U.z)q). ilr
(7)
)
In (7), e, is the unit vector in the z-direction, rp is the pseudo-dilatationa potential, and x and y are the pseudo-distortional potentials. If p is independent of z, (7) will degenerate to the displacement-potential representation in Pak (1987) for a homogeneous elastic medium. For time-harmonic motion with circular frequency w, one may write the displacement field of the heterogeneous half-space as
u(r, H,z, t) =
um(u,8, A,t) - z&r, U,,) e”“’ = u(r, Q,z) err”’ I:::::_~:::~ - ~~~~:~~:~~
(8)
and other quantities of interest in similar complex notations, By virtue of representation (7), the equation of motion (6) will be satisfied provided that v’tp+?%;(?Jy, = 0, V”I+k;(Z)‘l
v*x+&
= 0,
(9) (10)
2 J&(Z)X = 0,
where k;tzj =
Ozp
(12)
w2p
(13)
3&(1 +hz)’ ’ kj(z) =
&(l +bz)*
One may identify (9) with the primary (P) waves, and (10) and (11) with secondary (S) waves. It should be noted, however, that such P-waves are not purely irrotational, nor such S-waves purely equivoluminal. Similar observation was made by Gazetas (1980) regarding the plane strain problem. One should also keep in mind that the phase velocities of such waves are generally frequency-dependent as heterogeneous media are intrinsically dispersive in nature. 2.2. General solution of the governing equations In cylindrical coordinates, the governing equations (9), (IO) and (I 1) can be written explicitly as (14)
536
R. Y. S. PAK and B. B. GUZINA
y;m81virirz
of the completeness of the angular eigenfunctions -n < 6 < TLwith respect to the class of solutions under consideration, one may express cp(r,e,z) = E:,“=_mp)m(~,~)eimff,
r(r, 0,z>= CiTY, __qm(r, z) eime, x(r, 0, z) = X2=__exnAr, z) elm’,
(17)
and u,(r, 8,~) = X2= -caurm(r,2) elm’, ue(r,6,z) = EZ= -a31kg_(r,~)eime, zfZ(r,e,z) = Cg= _~~~,(r,z)e’~~,etc. With an appeal to (17) and the orthogonality (15) and (16) yield
(18)
of (eImH),the governing equations (14),
(I91
G-9)
for each m. For further reduction, transform
one may make use of the mth order Hankel
whose inversion formula is
In the above, J, is the Bessel function of the first kind of order m. Application of the foregoing transform to (19)-(21) leads to a set of ordinary differential equations
Wave propagation in a heterogeneous solid
d’@,m 7 +(k,2zH2)@mm
d’fm” -+-_ dz2
2b
dx’,” dz
l+bz
537
= 0,
+(k;(Z)--2)f;
= 0.
(26)
By means of a change of variable 1 +bz
x=r-
b
’
(27)
and the definition of an auxiliary function F(x) = x-“‘@z({, z), the general solution of (24) can be found to be
where
a=
1 --___ W2P 4 3pOb2’
J
(29)
In the above, A, and B, are integration constants, while Z,(X) and K,(x) are the modified Bessel functions of the first and second kind, respectively. Their order c1can be either a real (in the case of low frequency) or an imaginary number. Analogously, the general solution of (25) can be written as qm”(&z) =(l +bz)“’
(c~,,,,(,~)+D,(C)K,(i~))
(30)
where (31) Finally, by defining the auxiliary function G(x) = x”‘fz(<, z) where x is given by (27), one finds that the general solution of (26) can be expressed as x:(&z)
=(l+bz))“2
(Z.(5)Z~(,lihhi)+~~*(n~~(r~)).
(3.2)
2.3. Boundary conditions for a buried source problem Although the present formulation rules out body forces, the action of an arbitrarily distributed source on the plane z = s in a half-space can be represented as a set of
538
R. Y. S. PAK and B. B. GUZINA
BodyForce Field
Fig. 1 A buried source in a heterogeneous
elastic half-space.
prescribed stress discontinuities across the corresponding planar region (see Fig. 1). In the treatment of this class of problems, it is convenient to view the half-space as being composed of Region Z {z < S} and Region ZZ{z > s} as indicated in Fig. 1, and to represent the distributed body-force field as a general discontinuity of stresses across z = s (see Pak, 1987) in the form of
(33) while requiring the displacements to be continuous everywhere. In the above, P(r, e), Q(Y,0) and R(r, 0) are the prescribed body-force distributions in the radial, angular and axial directions which can be expanded as
To render the foregoing problem specific for a half-space with a traction-free surface, one would require cZZcr,e,o)
= ez,(r, e,o) = z,e(y, 8, 0) = 0.
(35)
Finally, to ensure the formulation is physically meaningful, one should impose the regularity condition that the solution contains no incoming waves from infinity and that it vanishes as z -+ cc. It is important to observe, however, that the transformed solutions (28), (30) and (32) represent standing waves in the z-direction. As a result, the first part of the regularity condition is automatically satisfied by the functional space derived.
Wave propagation in a heterogeneous solid
539
2.4. Transfbrmed displacement- and stress-potential relations
From (7), one finds that the components of the displacement field can be expressed Xi
(36) In terms of the Hankel transforms, they are equi~dlent to
(37) With the aid of the constitutive relations can be stated as
relations (2), the transformed
stress-potential
Likewise, the boundary conditions translate to z1’,,(5,0) = 0, ~_~~-‘(5,O)-i~~~_‘(~,O) = 0, ~~“~‘(5,0)+i~~+1(g,0) for the traction-free
= 0,
surface at -7= 0, %;,(5,5-) = K;,(<,s+),
(39)
540
R. Y. S. PAK and B. B. GUZINA
~~~‘(5,s~)-i~~~‘(5,s~)
= ~~~-‘(5,s+)-i~~-‘(5,s+),
~~,+‘(5,s-)+i~~s”,,+‘(5,s-)
= ~~“(5,s+)+izlo”+‘(5,~+),
for the continuity
of displacements
at z = s, and
~~-m(r,~-)-~~:J(5J+) (TE;‘(4,sP)-iTsP
= Z,(5),
~‘(LSP))-(G;‘(5,s+)-i%zmg_‘(L~+))
(~~~‘(5,s~)+i~~,t’(r,s~))-(~~~~‘(5,sf)+i~~,ms_f’(5,S+)) for the discontinuity
conditions
(40)
=X,(5), = Y,(t),
(41)
on the stresses. In the above,
-%I(0 = &X5), L(5)
= Z%‘(5)
K(5)
=
p”,“+‘(~)+@,“+‘(S).
Consistent with the regularity condition (28), (30) and (32) can be expressed as GXLZ)
-i@-‘(5),
at infinity,
the relevant
potential
solutions
= (1 +~z)“2(~!n(~)Z&)+~t,(S)ZUx)),
%X&z) = (1 +~V2(C~(5)Z&)
+D!‘z)K&&)),
Xm”((,z) = (l+Z=-‘~2(Ef,(5)Z&)+Ff,(S)K&)), in Region
(42)
(43)
Z, and
K(r,z)
= (1+w”2mtvx4,
fmr,4
= (1+w”2RxvqX),
fm”(&z) = (1 +bz)-“2F:(S)Z$(x),
(44)
in Region ZZ. In the above, x is given by (27) and a and /3 are defined in (29) and (31), respectively. 2.5. Solution of the buried-source
problem
According to (43) and (44), the transformed Fourier components of displacements and stresses can be expressed in terms of the nine unknown coefficients conAi, B;, . . . , D,“, F,“. With the aid of the relations (37) and (38), the boundary ditions (39)-(41) provide the nine equations required for the solution of the unknown coefficients. In terms of the transforms of the prescribed loading, the solution for the Fourier components of the displacement field can be expressed as l4-:“,
= Q,(<,z;s;b)p
xlil-- Y, %I
+Q&,z;s;b)% PO
Wave propagation in a heterogeneous solid
+fdc+
I (Y, wq:
1(x, i)
541
2P+5 _iu~5+Iw--
X1,:2Y1'21 +H(z-.s) [
L(YK&)
6($3c12)
+ Z/S(Y) K, + I (-4
=
H(s-zf
,y
112
Y
-l/2
i
xp1:2
(46)
-112 b y
-W-:)W~+,(I,&i)
+fqz-s)
I[
28+5 ~~~+kX)-I
+Y;+‘hJ4
PI,2 y2(&z;s;h)
’
’
ivd”)KB(Y)l
W+1
(x9 i)zB(i)K,~(Y)ra-~+r.4p5+‘(i,
i>l>
-l/2
y b
i VdY)f$(X)),
(47)
R. Y. S. PAK and B. B. GUZINA
542
-py’/25
y,(<,z;s;b) = m--z)
b(7-3a2)i
L(eL,l(Y)
II [ ,.pyl12~ +b(+ x 28+5
+47(-wp(Y)
yjy
[
20(---3 ~%+,(Yd-l
1
+Y$+‘(x,x)
~zt(vmxww) 3a2)A([) i
[
ye+,(Y,Y)-l
1
[-(fi~~+~(I,i>+fi)+(f3~~+I(i,i)+f4)~~+I(r,r)l
(48)
Wave propagation in a heterogeneous solid
543
(49)
and
+fs%+
I (Y, im:
2c(--3
21_w:+
f (x, 0
[ Ii2
I (x, xl - 1
.
.I/2 i
+H(z-s)
x b(:
J -
34
Il,(Y)K,&)
y+C+‘(y,y) In the foregoing equations, (29) and (31), respectively,
(50)
H denotes
the Heaviside
function,
CIand /? are given by
and
i=$
(51)
x = i(1 +bz), y = i(1 +bs), 2’ihY)
K, W = ~ MY) ’
(52)
544
R. Y. S. PAK and B. B. GUZINA
(53) fi = 4~4+(6a2-4+t-2~2-10a+6~-1)~2 -t-(3a2/J2 +6a*p-6cQ2
-+x2 - 12cQ+;p2 +++;a-$>,
(54)
f2 =(4tl-6)[3+(-6t12+12~-;)[,
(55)
.A =(4~+10)~3~(6~2+12~-~)~,
(56)
f4 = -4c4 - IZC”,
(57)
fs = -413-21f12+;i, NC) =.f,a:+ I (5, W$+ I CL0 -t-.f+C+ 1CLi)+_fS$+
(58) 1@I,0 +.L
(59)
On substituting the inverted Fourier components of displacements (45) into the corresponding angular eigenfun~tion expansions (l&I, the formal three-dimensional solution of the general buried-source problem under consideration can be obtained. Due to the roots of A, however, some of the functions defined in (46)-(50) have multiple poles along the formal path of integration for the inverse transform. As will be shown in the next section, the sense of the contour integration is uniquely defined by the branch points and branch cuts corresponding to the case of b + 0. For o = 0, it can be shown that the analytical solutions degenerate to the static case for the heterogeneous half-space without the need of asymptotic analyses. 2.6. Analytical charac~e~i~aiio~of a weakly i~homoge~eous medium To examine the limiting behavior of the solution as b tends to zero which implies a homogeneous half-space, it is necessary to determine the character of modified Bessel functions with large real arguments (<( 1 -I-bz)/b) and large imaginary orders CIand j3. To this end, one could note that
(60) where
For the ensuing analysis, the following identities are also useful Iv(vx) = e- ‘/2v*iJy(ivx),
Wavepropagation in a heterogeneous solid
545
(‘m(Y)
Bra& Cutsfor the Square-root Function
BranchCutforthe ~g~~c axon -
o-
I-
. Re(y)
Complexy-Plane
Fig. 2 Branchcuts in the complex y-plane.
K,.(vx) = Te’~2’.‘Hi’)(iv.u),
Z,._, (vx) = e-‘/2”ni.J,,(ivx) [i +iE],
~~‘)‘(il?~) rC,,_,(v-x) = _ ~er~2VZij$~,~)(ivx)i +i-.L..__ H(l’(ivx) I’ 1’ [
--n < arg(vx) d ; )
(62)
where H$“(vx) stands for the Hankel function, and ““’ denotes differentiation with respect to the argument. With reference to the asymptotic expansions of Bessel and Hankel functions for large arguments and orders (Abramovitz, 1972), one finds
J”(vY) _pnv)Jo
I!'(1 _y2)--1/4
_ +v)-“‘(1
_y2)W
H:s”(vy) ~ ___2i(2~y-l/2(1 H!,“‘(vy) _ ;(2nv)-
e-2wJ~,
e-2/3i'3'2,.,
_y*)--1!4
l/2(] _y2)‘!4
e2135*3'z~s,
e2/35’3”y
(63) In the above expansions, [* is the auxiliary variable defined by the following mapping (64) where dq = eeixi2,/‘~. As indicated in Olver (1954), the three branch cuts in this mapping have to be chosen in such a way that 5* is real when y is positive. Specifically, the branch cut associated with the logarithmic function in (64) in this study is set along the negative y-axis (see Fig. 2) while the other two branch cuts
R. Y. S. PAK and B. B. GUZINA
546
t WQ Branch Cuts Path of Integration A -kso -k@
0
kfi
_
Complex f -Plane
ko
)
IW4,
5 Branch Cuts
Fig. 3 Branch cuts in the complex t-plane for small values of b
associated with the square-root functions are chosen such that the one originating from y = 1 extends vertically through the lower half-plane and the other originating from F = - 1 extends vertically through the upper half-plane. In order to find the limiting forms of
it is useful to define v=
1+&i
b ’
s=f!klE kii-b’
where k stands for kPOor kSo.By virtue of these definitions, the branches in the complex y-plane in Fig. 2 for small values of b correspond to the branch cuts in the complex t-plane as indicated in Fig. 3. To be consistent with the choice of branches and to stay within the domain of validity of (63), it is proper to deform the formal path of integration for the inversion integral with respect to the wave numbers g around the singularities in the upper half-plane as shown in the figure. Furthermore, to ensure the solution is analytically continuous with respect to b, one should maintain the contour integration in the first quadrant for all values of the inhomogeneity parameter and avoid any sudden switching of the branches. Upon expanding the auxiliary variable 5* with respect to b about 0 in the form of 3*3/2
_
jlog(“‘J7)-
it can be shown that
;~j&+&,/j?++;)b,
(66)
Wave propagation in a heterogeneous solid
547
Finally, upon direct substitution of the above expansions into (46)-(50), the solution for the homogeneous half-space (Pak, 1987) with h = /coand the Poisson’s ratio equal to 0.25 can be recovered as
I
+
y2([,z;s)
=
-l-(e-hdt +e-h4), Wh
(69)
(70)
548
R. Y. S. PAK and B. B. GUZINA
3325*-e)
-k-(t)
+ahPhe-U++~h~)), (5’e-UQ+~hS)
(72)
+ 2~*%(2? -ks)’ (e -(j3,;+a&+e-(&Y+““Z’), (73) k.:R - (5) where R * = (2c2 - k,2)’ rt:452a&,,
(74)
fib =((*-k;)“*,
(75)
k, = w&%%&,
k.9= w&
(76)
d, = Iz-$1,
d2 = z+s,
(77)
01~ =(+k;)i'z,
sgn(z-s)
=
+1,z>s i -l,z
I ’
(78)
In view of the correspondence of the two solutions, one may refer to the function A as the “heterogeneous counterpart” of the Rayleigh wave function R-. It should be observed, however, that A leads to multiple poles, rather than a single one along the formal path of integration. In addition, such poles correspond not only to Rayleigh, but also to Love as well as Stoneley waves, for a continuously-heterogeneous medium can be considered as a limiting case of a layered medium. With appropriate specifications of the internal source distributions P(u,@), Q(r,0) and R(r,B), Green’s functions for a variety of load configurations can be derived and evaluated as in Pak (1987) and Pak and Ji (1994). As illustrations, the radial displacements of a solid half-space with a linear-varying shear wave velocity in the vertical direction due to an embedded uniformly distributed lateral body-force field acting in the 0 = O-direction over the circular region II, = {(r, 8, z)l Y< a, z = s> are presented in Figs 4-7. The results are given along the z-axis (i.e. r = 0) for the case s = 0 {a surface source, Figs 4 and 5) and the case s = 8a (a buried source, Figs 6 and 7) at a dimensionless frequency of 0.5. The solutions for a heterogeneous medium with b = 0.05/a and b = 0.15/a are compared with the corresponding solution for a homogeneous solid labeled as “‘b = 0.00”. Consistent with the vertically increasing stiffness of a heterogeneous medium, the differences between the two formulations are more pronounced for the case of a buried source. From the displays, it is also apparent that the waves in a solid with higher b (i.e. the stiffer medium) are characterized by increased wave lengths and reduced magnitudes.
Wave propagation in a heterogeneous solid
S4Y
3.5
8
2.5
81 3
1.5
r$ a
0.5
---. ----
-0.5 0.0
4.0
bQEO.00 b*a = 0.05 b*a=O.H
8.0
12.0
16.0
z/a Fig. 4 Displacement U, under unit lateral force on II, at r = 0, s = 0 : real part. 0.5 /
I r/a = 0.0 sla = 0.0
$$
0.0 [p-y-
wo,&&=
, ,-=/.
_---__ __--L;,-“_
4.0
---.
----
b*a=O.QQ
---.
b*a=O.OS b*a=0.15
-1.0 1 0.0
0.5
8.0
12.0
16.0
Ja Fig. 5 Displacement U, under unit lateral force on 17, at r = 0, s = 0: imaginary part. 1.8 1.4
% .8_ 1.0 1 ’ $
$
i
t
r/a = 0.0 s/a = 8.0
!8’:
4.0
8.0
;:
------. -
b*a=O.OO b*a=0.05 b*a=0.15
06
. 0.2 -0.2 ’ 0.0
12.0
16.0
Fig. 6 Displacement U, under unit lateral force on II, at r = 0, s = 8a : real part.
3. CONCLUSIONS By a method of displacement potentials, a general solution for the dynamic response of a vertically-heterogeneous elastic half-space to the action of an arbitrary, time-
550
R. Y. S. PAK and B. B. GUZINA 0.50 ------. -
r/a = 0.0 3 &
.Z
0.25
-
-0.25 -
-0.50 0.0
_
b*a=0.15
I’--,,
0.00 . “‘--:2.
5
dz
s/a = 8.0
b*a=O.m b*a=0.05
,.er
“;-_ __ ‘\ ---_ ____---‘\ ‘\
‘\
. ..____.=
4.0
__/Lt------ , <’
/’ /’ -‘llxZ/&&
8.0 z/a
12.0
= 0.5 16.0
Fig. 7 Displacement U, under unit lateral force on B, at r = 0, s = 8a: imaginary part
harmonic, finite buried source is derived analytically. The result for a homogeneous half-space is shown to be recoverable in the limit as the inhomogeneity parameter tends to zero. Some specific aspects of the analysis such as the multiple poles along the formal path of inversion integrals and the characteristics of the wave propagation in the vertical direction are elucidated. With the aid of Fourier or Laplace transforms, the method can be extended to the treatment of transient problems in a straightforward manner.
ACKNOWLEDGEMENT The support of the National gratefully acknowledged.
Science Foundation
during the course of this investigation
is
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