Fundamental solution of the incompatibility problem in three-dimensional infinite anisotropic elasticity theory

Abstract

The decomposition of the strain field into its deformation and its incompatible parts which is unique in an infinite elastic medium is used to derive a fourth-rank Green's function tensorPabij (fundamental solution) of the incompatibility problem of linear anisotropic elasticity theory. This fundamental solution is homogeneous of degree — 1 and non-unique. For the calculation of internal stresses only one of the possible fundamental solutions is needed. In order to solve the integration problem, it is convenient to decompose the fundamental solution into two parts. The one part consists of a particular solution of the incompatibility condition (Eq. (3)) and is obtained in a straightforward manner. The other part results from a deformation part of the strain. The integration of this part is worked out by using the theory of residues. In the case of hexagonal crystal symmetry an explicit solution is presented.