The tree-level viscoelastic model: analysis of the influence of the packing defect energy to the response of materials under complex loading

The development of new and improvement of existing modes of thermomechanical treatment of metals and alloys in present conditions is impossible without development of appropriate mathematical models, that allow determining material characteristics during technological processes. Constitutive equations are the core, the main components that determine the quality of such models. Macrophenomenological theories of plasticity relying on processing the results of experiments on macrosamples, have become widespread as such in solving applied problems of solid mechanics. Taking into account the need to describe the memory of processes, the equations of this class have a complicated mathematical structure, require expensive tests (generally speaking, for complex loading) for each material, due to which they are not universal. In the past 15-20 years, constitutive models based on the introduction of internal state variables, of a multilevel approach, and physical theories of inelasticity (plasticity, viscoplasticity) became very popular. Models of this class are focused on describing the evolving structure (including microstructure), which ultimately determines the physical and mechanical properties of materials and constructions. As the physical mechanisms and their carriers are identical for wide classes of materials, the models of this class have significant versatility, including the prediction of behavior of new, not yet existing materials, to study the physical mechanisms of the occurrence of various effects, observed in macro experiments. Hardening is one of interesting effects observed in experiments on complex (including cyclic) loading (as compared to directional loading) of samples, made of various metals and alloys, arising from a significant evolution of the microstructure. Empirical data analysis made it possible to establish that the tendency to manifest this effect is usually experienced by metals and alloys with a low stacking fault energy (SFE). The paper provides a brief analysis of the experimental work and mathematical models describing the response of a material to complex deformation. It is noted that macrophenomenological theories do not allow one to describe in an explicit form the evolution of the microstructure and the carriers of plastic deformation and hardening mechanisms, thus they do not provide an opportunity to explain the physical reasons for the above effects. The purpose of this work is to develop, study and implement a multilevel elasto-visco-plastic model that allows describing the evolution of crystal lattice defects in materials with different SFE under different thermomechanical processing, different strengthening mechanisms at different structural-scale levels. In the framework of constructing a constitutive model, special attention is paid to the development of a submodel, focused on description of the evolution dislocations and barrier densities on slip systems. Kinetic equations for dislocation densities on slip systems make it possible to analyze the nucleation of dislocations due to the activation of Frank - Read sources, annihilation of dislocations of different signs on one slip system, interaction of split dislocations of intersecting slip systems with the formation of barriers. Relations for the description of hardening are given, taking into account the current density of dislocations and barriers. The general structure of the model and the relationship between the parameters of submodels of different levels are considered. An algorithm and a program of implementing the model were developed, the evolution of dislocation densities on slip systems was analyzed, and the intensity of hardening and the formation of barriers on split dislocations were obtained depending on the type of loading.