The multiple scattering of shear waves and dynamic stress in a semi-infinite functionally graded material with a circular cavity is investigated, and the analytical solution of this problem is derived. The analytical solutions of wave fields are expressed by employing the wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary condition of the cavity. The image method is used to satisfy the traction-free boundary condition of the material structure. As an example, the numerical solution of the dynamic stress concentration factors around the cavity is also presented. The effects of the buried depth of the cavity, the incident wave number, and the nonhomogeneity parameter of materials on the dynamic stress concentration factors are analyzed. Analyses show that when the nonhomogeneity parameter of materials is $<0$, it has less influence on the maximum dynamic stress around the cavity; however, it has greater influence on the distribution of dynamic stress around the cavity. When the nonhomogeneity parameter of materials is $>0$, it has greater influence on both the maximum dynamic stress and the distribution of dynamic stress around the cavity, especially in the case that the buried depth is comparatively small.

1.
Gray
,
L. J.
,
Kaplan
,
T.
, and
Richardson
,
J. D.
, 2003, “
Green’s Functions and Boundary Integral Analysis for Exponentially Graded Materials: Heat Conduction
,”
ASME J. Appl. Mech.
0021-8936,
70
, pp.
543
549
.
2.
Kuo
,
H.-Y.
, and
Chen
,
T.
, 2005, “
Steady and Transient Green’s Functions for Anisotropic Conduction in an Exponentially Graded Solid
,”
Int. J. Solids Struct.
0020-7683,
42
, pp.
1111
1128
.
3.
Rice
,
J. M.
, and
,
M. H.
, 1984, “
Propagation and Scattering of SH-Waves in Semi-Infinite Domains Using a Time-Dependent Boundary Element Method
,”
ASME J. Appl. Mech.
0021-8936,
51
, pp.
641
645
.
4.
Li
,
C.
, and
Weng
,
G. J.
, 2000, “
Dynamic Stress Intensity Factor of a Cylindrical Interface Crack With a Functionally Graded Interflayer
,”
Mech. Mater.
0167-6636,
33
, pp.
325
333
.
5.
Ueda
,
S.
, 2001, “
The Surface Crack Problem for a Layered Plate With a Functionally Graded Nonhomogeneous Interface
,”
Int. J. Fract.
0376-9429,
110
, pp.
189
204
.
6.
Roussseau
,
C.-E.
, and
Tippur
,
H. V.
, 2001, “
,”
Int. J. Solids Struct.
0020-7683,
38
, pp.
7839
7856
.
7.
Chan
,
Y.-S.
,
Paulino
,
G. H.
, and
Fannjiang
,
A. C.
, 2001, “
The Crack Problem for Nonhomogeneous Materials Under Antiplane Shear Loading—A Displacement Based Formulation
,”
Int. J. Solids Struct.
0020-7683,
38
, pp.
2989
3005
.
8.
,
J.
,
,
V.
, and
Zhang
,
C.
, 2005, “
A Meshless Local Boundary Integral Equation Method for Dynamic Anti-Plane Shear Crack Problem in Functionally Graded Materials
,”
Eng. Anal. Boundary Elem.
0955-7997,
29
, pp.
334
342
.
9.
Pao
,
Y. H.
, and
Chao
,
C. C.
, 1964, “
Diffractions of Flexural Waves by a Cavity in an Elastic Plate
,”
AIAA J.
0001-1452,
2
(
11
), pp.
2004
2010
.
10.
Kung
,
G. C. S.
, 1964, “
Dynamical Stress Concentration in an Elastic Plate
,” M.S. thesis, Cornell University, Ithaca.
11.
Klyukin
,
I. I.
,
Klyukin
,
I. I.
,
Satedkim
,
B. H.
, and
Tirbln
,
D. C.
, 1964, “
Scattering of Flexural Waves by Antivibrators on a Plate
,”
Sov. Phys. Acoust.
0038-562X,
10
, pp.
49
53
.
12.
Hayir
,
A.
, and
Bakirtas
,
I.
, 2004, “
A Note on Plate Having a Circular Cavity Excited by Plane Harmonic SH Waves
,”
J. Sound Vib.
0022-460X,
271
, pp.
241
255
.
13.
Stratton
,
J. A.
, 1941,
Electromagnetic Theory
,
McGraw-Hill
,
New York
, pp.
372
374
.
14.
Pao
,
Y.-H.
, and
Mow
,
C. C.
, 1973,
Diffraction of Elastic Waves and Dynamic Stress Concentrations
,
Crane
,
Russak, New York
, Chap. 4.