Mathematical Model of Rheological Processes of Composite Materials Deformation

Inelastic materials, in particular, composites are widely used in industry. This necessitates the development and research of mathematical models to describe the rheological properties of such materials at different temperatures and types of loads. The principle of heredity for the study of composite materials leads to the construction of the most general equations that take into account hereditary effects, the influence of speed, types of loading, operating conditions, etc. such prominent relations are integral equations of the Voltaire type. According to the law of inheritance and the principle of superposition, the total deformation of a body consists of the instantaneous deformation, which is determined by the stress acting at a given moment in time and related to it by Hooke's law, and the inherited deformation. The heredity principle is the most general principle that can be the basis of studies of the rheological properties of inelastic materials. The article formulates and describes a mathematical model based on this principle, which describes the behavior of composite materials in different loads and temperatures modes. The purpose of the article is to develop an adequate model of the environment, quite simple and convenient for describing the rheological properties of composites under conditions of various loads and elevated temperatures. Objectives of the article: to present the developed model for describing the rheological properties of composite anisotropic materials; show the method of determining viscosity and temperature parameters for inelastic materials; to demonstrate the possibility of predicting the behavior of composites in different modes of loads and temperatures. The mathematical model is an integral equation with a creep nucleus. The nucleus of the model is chosen in the form of the Abel nucleus. The parameters of the Abel nucleus are determined by experiments on samples of anisotropic composites. The loading mode can be any, for example, stretching the sample at different load speeds, or stress relaxation at constant deformations. As determined, temperature heredity is not significant. The behavior of the material is determined only by the value of temperature at a given time. The temperature effect function can be selected as a normal degree dependence. The model is determined by three parameters. The parameter determination problem can be solved analytically, for simple load cases. More complex load cases, require the use of computer technology and methods of approximate calculations. The advantage of the proposed model is the possibility to transfer certain parameters from one load mode to another. The basic relations are given; the method of determining the parameters of the hereditary model is described; theoretical and experimental research is presented; the possibility of predicting the behavior of composite materials at different temperatures is shown.