Variational formulation of some problems concerning fracture

Abstract

i. Problems concerning fracture are of great practical importance in mining and other branches of technology.The problem of fracture is closely associated with the general problem of plastic deformation of a continuous medium.We will consider an approach to fracture problems which follows from a successive development of one class of models of plasticity. We will limit ourselves to the case of plane deformation.In soil mechanics and rock mechanics we are faced with the problem of generalizing plasticity models for materials with internal friction.It is known that in plastic conditions, depending on the first invarlant of the stress tensor, the characteristics of the stress field are nonorthogonal.The construction of kinematic equations for such media leads to the necessity of introducing a new kinematic variable which represents the possible difference between the shears on the sllp areas from different families [i]. Further generalization to the class of hardening or weakening plastic materials reveals that the measure of shear on the slip area must have the dimensions of a length [2]. Thus, we must introduce parameters with the dimensions of a length, characterizing the discreteness of the network of sllp lines.Some of the equations of state take the form of relations between the stresses and the slips, with the dimensions of length; these are expressed in terms of the gradients of the displacements, a new kinematic variable, and the discreteness parameters of the network of sllp lines.Thus the continuous model [2] allows for the discreteness of the slip lines, and consequently occupies an intermediate position between the classical models of plasticity assuming infinitely close sllp lines and the fracture models in which the behavior of one or several essentially discrete cracks is considered (including shear cracks).If the stress--slip diagram has a descending branch, then the solution in terms of the equations in [2] can lead to the result that the sllp lines, along which load relief does not occur and the sllp increases, become essentially discrete.It is evident that such a transition to discrete lines is observed experimentally and is the cause of qualitatively new deformation effects [3,4].Thus, the equations of plasticity [2] lead to a model of the material with essentially discrete fracture surfaces on which tangential, and possibly normal, displacement discontinuities are associated with the stresses.This model borders on the class of models which are developed directly for cracks and in which account is taken of the interaction between their flanks [5][6][7][8].Let us consider some variational formulations of the boundary-value problems. (One of the possible variational approaches to problems of fracture was discussed in [9].) Let S denote the deformed region, bounded by the perimeter P. Let the region be divided by a line of possible discontinuity of the displacements L. Let us introduce Cartesian coordinates Ox~xa.The indices + and -will denote symbols pertaining to the regions to the right and left...