This dissertation focuses on the main methods of research of behavior of structural elements made of functionally inhomogeneous materials under large deformations. Functionally heterogeneous materials (FHMs) or intelligent materials are widely used in science and technology. One of the representatives of this class of materials is shape memory materials or materials with the property of pseudo-elasticity. Their physical or mechanical properties differ sharply from the behavior of ordinary structural, heat-resistant or tooling materials or polymeric materials. The mechanical behavior depends to a large extent on the external conditions (temperature, pressure) and the prehistory of their change. Materials that exhibit shape memory, pseudo-elasticity, and pseudo-elastic-plasticity properties are usually referred to as nasties: NiTi AgCd, AuCd, CuAlNi, CuSn, CuZn, FePt, MnCu, FeMnSi, CoNiAl, CoNiGa, NiFeGa, TiPd, NiTi, NiTiNb, NiMnGa . Materials with shape memory (MPM) at low temperatures under load accumulate deformation, and after heating are fully or partially capable of shape recovery. These alloys can be the basis of composite materials that are more or less capable of shape recovery. The main mechanism in such processes is the martensitic transformation between solid phases, which can occur with a relatively small change in temperature.
In our opinion, the study of such a problem is the development of a new section of the mechanics of a rigid deformable body. To model the behavior of such structural elements, it uses geometric nonlinearity, a new nonlinear phenomenological model of the behavior of shape memory materials, the apparatus of integro-differential partial differential equations, and an improved method of component-wise splitting.
The dissertation consists of an introduction, five chapters, a conclusion, a list of references, and appendices.
The first chapter analyzes the literature on the topic of the thesis research. The analysis of existing models of phase transformations, which can take place in functionally inhomogeneous materials, has been carried out. The criteria by which phase transformations in materials can be classified are analyzed. The methods for numerical solution of problems for bodies with pseudo-elastic-plastic materials are considered. The problems of the thesis research are formulated.
In the second section, a complete system of equations is written down and a method for determining the elastic-plastic nonstationary stress-strain state of three-dimensional bodies with functionally inhomogeneous materials at large deformations is developed. A physical formulation of the problem of conception is made.
To solve the above problem, it is necessary to determine the temperature, three components of the displacement velocity vector, six components of the stress tensor, and six components of the strain tensor. This means that it is necessary to determine sixteen unknown functions of time and three coordinates. For this purpose a complete system of integrodifferential equations with geometric nonlinearity in mind has been formed. It consists of the equation of motion, geometric and physical equations, and the equation of heat conduction, which are satisfied under certain initial and boundary conditions.
Using an approach based on the idea of a component-wise splitting, a new variant of the method of reducing a three-dimensional geometrically nonlinear nonstationary problem of thermoelastic-plasticity to a sequentially solvable system of three one-dimensional problems is proposed.
Key words: functionally inhomogeneous material, shape memory, large deformations, nonlinear phenomenological model, pseudo-plasticity theory, two-dimensional splines.
