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In the 1940s, Fitzgerald developed a modelling technique for yielding surfaces and the thin-walled shells formed by such yielding that is used as the basis of the finite element method for modelling the deformation of shells. This finite element method is an approximate method for solving the implicit differential equations of the non-linear stress distribution. It was originally developed by Mulock and it was further refined by Bowers and Rodrigues for rotational problems while Gilbob served as a consultant involved in the development. The method consists of several steps, mainly related to the appropriate finite element solution of an initial step [11].
More recently, the Dixon-Blytt group used the power spectral density method (PSDM) and applied it to rate-of-strain models of fracture of plastically deforming rocks [12]. The method can be used to predict the fracture energy KIc for a single rectangular crack or for a crevice of prescribed shape. The method yielded values of KIc for different crevices which were comparable to those predicted by a scaling based on the elastic energies of the two well-separated parts of the crack plane. When a triaxial test condition was applied, the Maxwell-Hookean model produced KIc values which were independent of crack length, but not of the aspect ratio R of the crevice. This is due to the fact that the stress field of the material is not isotropic. The numerical resolution of the problem is, however, only in the plane of the crack.
The method of displacive analysis was employed by Foti to evaluate the strain energy release rate in a given volume at the crack tip and the energy dissipated on the crack plane. The question of the nonlinearity of the deformation of the crack tip in the Brinell sense was dealt with in terms of the change of the effective gradient for large changes in misorientation so that the plane of the crack may not be flat.
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