For an isotropic elastic material under a two-dimensional plane-stress or plane-strain deformation that involves a closed boundary of a given geometry Γ, it is a common practice to use a conformal mapping to map Γ to a circle of unit radius. For an anisotropic elastic material, the mapping produces the correct solution to the physical problem when Γ is an ellipse. There are published (and unpublished) papers that report solutions to anisotropic elasticity problems with a nonelliptic curve Γ. They all fail to solve the physical problem correctly. The purpose of this paper is to show the common errors made by many researchers, and to alert newcomers on the subject so that they do not fall into the same trap. [S0021-8936(00)02403-X]

Issue Section:
Technical Papers
Keywords:
elasticity, errors, elastic deformation
Topics:
Anisotropy, Elasticity, Errors, Deformation
1.
Muskhelishvili, N. I., 1953, Some Basic Problems of the Mathematical Theory of Elasticity (translation by J. R. M. Radok, Noordhoff, Groningen).
2.
Lekhnitskii, S. G., 1950, Theory of Elasticity of an Anisotropic Body, Gostekhizdat, Moscow (in Russian), Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day, San Francisco (in English, 1963), and Mir Pub. Moscow (in English, 1981).
3.
Stroh
,
A. N.
,
1958
, “
Dislocations and Cracks in Anisotropic Elasticity
,”
Philos. Mag.
,
3
, pp.
625
–
646
.
4.
Savin, G. N., 1961, Stress Concentration Around Holes, Pergamon Press, London.
5.
Ting, T. C. T., 1996, Anisotropic Elasticity: Theory and Applications, Oxford University Press, New York.
6.
Lekhnitskii, S. G., 1957, Anisotropic Plates, 2nd Ed., Gostekhizdat, Moscow (in Russian), (translation by S. W. Tsai and T. Cheron, Gordon and Breach, New York (1968, 1984, 1987).
7.
Churchill, R. V., 1948, Introduction to Complex Variables and Applications, McGraw-Hill, New York.
8.
Ting
,
T. C. T.
,
1994
, “
On Anisotropic Elastic Materials That Possess Three Identical Stroh Eigenvalues as do Isotropic Materials
,”
Q. Appl. Math.
,
52
, pp.
363
–
375
.
9.
Ting
,
T. C. T.
,
1996
, “
Existence of an Extraordinary Degenerate Matrix N for Anisotropic Elastic Materials
,”
Q. J. Mech. Appl. Math.
,
49
, pp.
405
–
417
.
10.
Yin
,
W.-L.
,
2000
, “
Deconstructing Plane Anisotropic Elasticity, Part I: The Latent Structure of Lekhnitskii’s Formalism
,”
Int. J. Solids Struct.
,
37
, pp.
5257
–
5276
.
11.
Yin
,
W.-L.
,
2000
, “
Deconstructing Plane Anisotropic Elasticity, Part II: Stroh’s Formalism Sans Frills
,”
Int. J. Solids Struct.
,
37
, pp.
5277
–
5296
.
12.
Ting
,
T. C. T.
,
2000
, “
Anisotropic Elastic Materials That Uncouple Antiplane and Inplane Displacements but Not Antiplane and Inplane Stresses, and Vice Versa
,”
Math. Mech. Solids
,
5
, pp.
139
–
156
.
Copyright © 2000
 by ASME
You do not currently have access to this content.