The Indentation of an Anisotropic Half Space by a Rigid Punch

Abstract

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.