JUSTIFICATION OF INVARIANT RELATIONSHIPS AND FORMULATION OF A MATHEMATICAL MODEL OF THE MECHANISM ELEMENTS INTERACTION WITH A GRANULAR ENVIRONMENT

Abstract

One of the most commonly used deformation characteristics of granular materials is the shear modulus G, which, unlike elastic and plastic bodies, depends on both deformations  and the magnitude of compressive stress . Therefore, it should be considered not as a material constant, but as a parameter of the deformation surface . Its value depends on the achieved level of stress-strain state and can be determined only based on the results of laboratory tests.

The analysis of experimental studies, as well as data published in the literature, allowed us to substantiate the relations that form the basis of the mathematical model, describing the regularities of the change in the shape and volume of a discrete medium. It is suggested to describe the regularity of the shape change in the system of axes  by a power dependence that approximates the experimental data and is convenient for calculations.

The mathematical model should be constructed in such a way that, by continuously or discretely changing the parameters influencing the interaction process, the behavior of the system can be predicted. In such a formulation, it allows determining the power parameters necessary for calculating a machine or mechanism, the power of the engine, describing the features of their functioning in different conditions, and designing in detail the structures of the working bodies and the power part of the machine.

The article considers only a fragment of the general model - a model of interaction with a discrete environment of mechanism individual elements. The basis for modeling is the proposed representation of the process in the form of an interaction problem of machine elements (mechanism) with a non-uniform environment with significant internal friction. The interaction process of machine elements with the environment can be most fully evaluated by solving the contact interaction problem of an elastic element with the environment. The boundary conditions are formulated in such a way that the modeling problem describes the features of the technological process as accurately as possible.

The peculiarity of our nonlinear problem is that the deformation laws of the material of the technological discrete medium are described by complex nonlinear relations between the invariants of the stress and strain tensors, which take into account the impact of internal friction and dilatancy. In such a formulation, the deformation parameters of the medium depend on the achieved stressed state level at each of its points, therefore the problem can be classified as a physically nonlinear boundary value problem of an inhomogeneous area.