A mathematical model of the deformation and failure of rock materials

Abstract

A mathematical model is proposed for the deformation and failure of rock materials in which the bulk plastic deformation rate includes hydrostatic and dilatancy components. The latter is taken to be proportional to the rates of change of the stress tensor invariants. Shear is described by plastic flow theory relationships analogous to /1–3/. The model relationships do not take account of effects associated with the influence of the strain rate on the fracture process. Therefore, the model proposed can be considered as the limit, dynamic or static. It is conceivable that the quantitative measure of the effects taken into account and the specific form of the main dependences for the two limit models will be distinct for an identical material. Within the framework of the model proposed, quantitative data are presented on the mechanical characteristics of different mountain rocks obtained on the basis of the authors' statistical treatment of static test results published in the literature. A number of mathematical mountain rock models have been proposed /1, 4–9/ that reflect available experimental data on their deformation and fracture under static and dynamic loads to some degree /10–22/. Within the framework of these models the mountain rocks are considered as a continuous medium in the elastic and plastic stages of operation as well as in the fractured state. It is assumed here that considerable changes occur in the state of stress of such a medium at distances considerably exceeding the block dimensions into which the medium is separated in the developed fracture stage. As is seen from the experimental data /10–22/, the important effects observed during fracture are the change in strength and the occurrence of additional porosity in connection with the appearance of block disintegration, the appearance of so-called dilatancy of the medium. It is assumed in the formulation of the mathematical models for such media in /6–9/ that the rate of the dilatancy component of the bulk strain is proportional to the shear rate. The rate of the dilatancy component of the bulk strain in /7, 9/ depends additionally on the first invariant of the stress tensor.