Plane problem of discrete environment mechanics

Abstract

Many engineering problems related to the design of structures and machines, the mathematical description of technological processes, etc., are reduced to the need to solve a plane problem for materials with a significant effect of internal friction on their deformation. Such materials include a large class of materials in which the compressive strength is greater than tensile. These are composite materials, concretes, rocks, soils, granular, loose, highly fractured materials, as well as structurally heterogeneous materials in which rigid and strong particles are interconnected by weaker layers. The laws of deformation and destruction of such materials differ significantly from elastic ones. A feature of these laws is an increase in resistance to shear deformations and an increase in the strength of materials with an increase in the magnitude of compressive stresses. This is associated with the influence of internal Coulomb friction on the process of their deformation in the limiting and boundary stages.

The need to formulate and solve a special boundary value problem for materials with significant internal friction is because the results of solving problems using models of elasticity and plasticity differ significantly from experimental data. The difference increases when approaching the limiting state of discrete materials and depends significantly on the structure of the material and operating conditions.

The boundary value problem of the mechanics of a deformable solid is formulated as a system of equations of three types: static, geometric, and physical. For all linear and physically nonlinear problems, provided the deformations are small, the first two groups of equations remain the same. Thus, these differences can be attributed to the inconsistency of the accepted in the calculations of physical relations "stress - strain" and the real laws of deformation of these materials, which are more complex rheological objects than structurally homogeneous solids, liquids or gases.

The article uses an approach where the material is immediately considered as quasi-continuous, and the physical equations are based on the experimentally obtained relationships between the invariants of the stress and strain tensors, which consider the influence of both molecular connectivity and internal Coulomb friction.