The stress state of the rock mass with spherical cavity

Moscow State University of Civil Engineering, Yaroslavskoye shosse, 26, 129337, Moscow, Russia

^* Corresponding author: asv@mgsu.ru

Abstract

One of the main hypotheses accepted in the mechanics of deformable solids is the assumption of the homogeneity of materials. This means that all mechanical characteristics of the material (modulus of elasticity, Poisson’s ratio, yield strength, relaxation parameters, etc.) are constant over the volume of the body, in other words, these characteristics are constants. This hypothesis makes it possible not to take into account the natural inhomogeneity of materials at the microlevel - the presence of various fractions in composite materials (concrete, fiberglass, etc.), crystal lattice defects, etc. Examples can be given when various physical phenomena (temperature field, radiation exposure, explosive impact, etc.) lead to a change in the mechanical characteristics along the body. These changes can be quite significant. So, for example, in the presence of high-gradient temperature fields, the deformation characteristics of materials at different points of the body can change dozens of times. Thus, when calculating and designing structures, it is necessary to take into account such macro heterogeneity, since it leads to a significant change in the stress-strain state of bodies. This article considers the problem associated with the continuous inhomogeneity of materials. It means such a heterogeneity that arose in the process of creating an underground cavity with the help of an explosion. In contrast to the classical mechanics of a deformable solid body, the problems of which are reduced to differential equations with constant coefficients, in the mechanics of continuously inhomogeneous bodies we deal with equations with variable coefficients, which greatly complicates their solution. In this case, depending on the type of inhomogeneity functions—functions that describe the change in mechanical characteristics along the coordinates—differential equations turn out to be significantly different.