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The Indentation of an Anisotropic Half Space by a Rigid Punch

Published online by Cambridge University Press:  09 April 2009

D. L. Clements
D. L. Clements
Affiliation:
Department of Mathematics University of Melbourne
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Eshelby et al. [1] and Stroh [2] have developed the theory of anisotropic elasticity for a three dimensional state of stress in which the stress is independent of one of the Cartesian coordinates. Various problems involving dislocations in an infinite anisotropic medium are solved in the first paper, while Stroh considers dislocation problems as well as determining the stresses round a crack subjected to an arbitrary non-uniform applied stress. In this note, which follows their treatment, we consider the problem of determining the stresses produced by the indentation of the plane surface of an anisotropic half space by a rigid punch. Problems of this type have been solved by Green and Zerna [3], Lekhnitskii [4], Brilla [5], Gallin [6] and Milne-Thomson [7] but if we take Cartesian coordinates x1, x2, x3 and let the stress be independent of x3, then these authors all assume the x1x2 plane to be one of elastic symmetry. The solution presented in this paper does not require this assumption so that it has a more general application than has been the case with previous solutions to problems of this type. The first part of the analysis given here is for general anisotropy, but in order to obtain a solution to the punch problem by the method of this paper, it is necessary to consider only a particular class of anisotropic materials. The indentation of such materials by a circular block is discussed in section 4 and the results are used in section 5 to examine the case when the circular block is on a transversely isotropic half space.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 1 , February 1971 , pp. 75 - 82
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Eshelby, J. D., Read, W. T. and Shockley, W., ‘Anisotropic elasticity with applications to dislocation theory’, Acta Met. 1 (1953), 251–259.CrossRefGoogle Scholar
[2]Stroh, A. N., ‘Dislocations and cracks in anisotropic elasticity’, Phil. Mag. 3 (1958), 625–646.CrossRefGoogle Scholar
[3]Green, A. E. and Zerna, W., Theoretical elasticity (O.U.P. 1st edition, 1954), Chapter 9.Google Scholar
[4]Lekhnitskii, S. G., Theory of elasticity of an anisotropic elastic body (Holden-Day, 1963), 131–141.Google Scholar
[5]Brilla, J., ‘Contact problems of an elastic anisotropic half plane’ (in Roumanian), Studii si Cercetari Mecan. Apl. Inst. Mecan. Appl., Acad. Rep. Pop. Romine 12, (1961), 959–987.Google Scholar
[6]Galin, L. A., Contact problems in the theory of elasticity (Translation by North Carolina State College Department of Mathematics, 1961), 90–111.Google Scholar
[7]Milne-Thomson, L. M., Plane elastic systems (Springer-Verlag, 1960), Chapter 7.CrossRefGoogle Scholar