The paper provides a brief review of multilevel models (mostly, two-level models) that allow the description of deformation of single- and polycrystals and attendant microstructure evolution. Particular attention is given to models of texture formation. The paper covers models that are based on a statistical determination of characteristics of higher levels by appropriate characteristics of lower levels.

The paper continues the review of works on the development and application of multiscale physical models to analyzing the behavior of single- and polycrystals under severe plastic deformation. Unlike mathematical models of plasticity, this class of models operates with meso- and microstructures of deformed material and allows analysis of their evolution. The recently much used so-called direct models that are based on crystal plasticity theories and numerical methods are discussed.

The formulation of geometrically nonlinear boundary value problems is a necessary component in modeling real technological processes of thermomechanical material processing, wherein the main issues ar

Widespread approaches to generalizing geometrically linear constitutive relations to the case of large displacement gradients have been considered. These approaches are based on the replacement of the

The development of new structural materials (primarily metals and alloys) and their processing technologies for the manufacture of high-performance products is and will be the focus of many research areas and industries. In recent decades, the complex problems of material design and processing have been solved using various mathematical models, many of which are based on macrophenomenological continuum theories of elastoplasticity. However, it is known that the physical and mechanical properties of metals as well as the performance characteristics of metal products are almost fully determined by their meso- and microstructures, whose evolution cannot be described by the above theories. This gap has been successfully filled in the last 15–20 years by multilevel models that explicitly describe the physical mechanisms of inelastic deformation and their carriers causing structural changes in the material at various structural scale levels. Multilevel models can be considered an effective tool for the implementation of the main principles, approaches and methods of physical mesomechanics, developed in the works of Victor Panin et al. This paper discusses the structure and constitutive equations, hypotheses, classification, applications and limitations of multilevel models used to describe the thermomechanical processing of metals and alloys. Since a large number of technological processes for the manufacture of high-performance parts and components involve severe plastic deformation, special attention is paid to taking into account geometric nonlinearity in the relationships included in the model. Application examples of our original models are given, in particular, for describing superplastic deformation, recrystallization processes, the effect of external and internal crystallite boundaries on the deformation of polycrystalline samples, and for complex loading analysis. The further development of multilevel models with the involvement of deeper structural scale levels into the models is discussed.

This paper reports the research results on the continuum theory of damage which goes back to the works of Kachanov, Gurson, and Rabotnov. In these models, internal variables that generally have different mathematical structure are explicitly introduced to constitutive relations. The internal variables describe a non-oriented (using scalar damage parameters) or oriented (using different-order tensors) damage distribution in the material. Then, a fracture criterion is introduced based on mechanical or thermodynamic considerations. Models of this type are still most frequently used in the structural analysis of strength of some materials (e.g., composites). Since damage nucleation and growth are closely related to strain localization, consideration is given to formulations and methods for analyzing the stability of inelastic deformation processes. Much attention is given to the effect of the finite element mesh on simulation results, to solution algorithms for such problems, and to the possibilities of using non-local constitutive models. The studies that use gradient models are also included, because damage formation is associated with sharp spatial variations of kinematic and/or dynamic characteristics which must be described by non-classical constitutive relations (gradient, non-local, micromorphic continuum).