The Green's function for the general anisotropic solid has been the subject of several studies. Here a variation of a standard integral transform approach allows the transient Green's function to be expressed in a somewhat different form. This alternative form is less compact, but features explicit integrals of functions in terms of polar and azimuthal angles defined with respect to the principal basis coordinates. Dimensionless expressions for the three anisotropic wave speeds are also given in terms of these angles, and sample calculations presented that show wave speed dependence on propagation direction. Some standard formalisms of anisotropic elasticity are not invoked, but similar terms are identified in the course of the analysis, and help define the solution expressions.

References

References
1.
Kelvin
,
L.
,
1848
, “
Note on the Integration of the Equations of Equilibrium of an Elastic Solid
,”
Cambridge and Dublin Mathematical Journal
,
3
, pp.
87
89
.
2.
Ting
,
T. C. T.
,
1996
,
Anisotropic Elasticity
,
Oxford University
,
New York
.
3.
Sneddon
,
I. N.
,
1972
,
The Use of Integral Transforms
,
McGraw-Hill
,
New York
.
4.
Synge
,
J. L.
,
1957
,
The Hypercircle in Mathematical Physics
,
Cambridge University
,
Cambridge, UK
.
5.
Stroh
,
A. N.
,
1958
, “
Dislocations and Cracks in Anisotropic Elasticity
,”
Philos. Mag.
,
3
, pp.
625
646
.10.1080/14786435808565804
6.
Stroh
,
A. N.
,
1962
, “
Steady State Problems in Anisotropic Elasticity
,”
J. Math. Phys.
,
41
, pp.
77
103
.
7.
Barnett
,
D. M.
, and
Lothe
,
J.
,
1973
, “
Synthesis of the Sextic and the Integral Formalism for Dislocations, Green's Function and Surface Waves in Anisotropic Elastic Solids
,”
Phys. Norv.
,
7
, pp.
13
19
.
8.
Ting
,
T. C. T.
, and
Lee
,
V.
,
1997
, “
The Three-Dimensional Elastostatic Green's Function for General Anisotropic Linear Elastic Solids
,”
Q.J. Mech. Appl. Mathematics
,
50
, pp.
407
426
.10.1093/qjmam/50.3.407
9.
Wang
,
C.-Y.
, and
Achenbach
,
J.D.
,
1994
, “
Elastodynamic Fundamental Solutions for Anisotropic Solids
,”
Geophys. J. Int.
,
118
, pp.
384
392
.10.1111/j.1365-246X.1994.tb03970.x
10.
Willis
,
J. R.
,
1973
, “
Self-Similar Problems in Elastodynamics
,”
Philos. Trans. R. Soc. London, Ser. A
,
274
, pp.
435
491
.10.1098/rsta.1973.0073
11.
Payton
,
R. G.
,
1983
,
Elastic Wave Propagation in Transversely Isotropic Media
Martinus Nijhoff
,
The Hague
.
12.
Wang
,
C.-Y.
, and
Achenbach
,
J. D.
,
1995
, “
Three-Dimensional Time-Harmonic Elastodynamic Green's Functions for Anisotropic Solids
,”
Proc. R. Soc. London, Ser. A
,
449
, pp.
441
458
.10.1098/rspa.1995.0052
13.
Willis
,
J. R.
,
1980
, “
Polarization Approach to the Scattering of Elastic Waves –I. Scattering by a Single Inclusion
,”
J. Mech. Physics Solids
,
28
, pp.
287
305
.10.1016/0022-5096(80)90021-6
14.
Van der Pol
,
B.
, and
Bremmer
,
H.
, 1950,
Operations Based on the Two-Sided Laplace Integral
,
Cambridge University
,
Cambridge
, UK.
15.
Barber
,
J. R.
, and
Ting
,
T. C. T.
,
2007
, “
Three-Dimensional Solutions for General Anisotropy
,”
J. Mech. Phys. Solids
,
55
, pp.
1993
2006
.10.1016/j.jmps.2007.02.002
16.
Hohn
,
F. E.
,
1965
,
Elementary Matrix Algebra
,
MacMillan
,
New York
.
17.
Abramowitz
,
A.
, and
Stegun
,
I. A.
, eds.,
1972
,
Handbook of Mathematical Functions
,
Dover
,
New York
.
18.
Jones
,
R. M.
,
1999
,
Mechanics of Composite Materials
,
2nd ed.
,
Brunner-Routledge
,
New York
.
19.
Crandall
,
S. H.
, and
Dahl
,
N. C.
, eds.,
1959
,
An Introduction to the Mechanics of Solids
,
McGraw-Hill
,
New York
.
You do not currently have access to this content.