An approach to numerical estimating the stability of multilevel constitutive models

Abstract

Multilevel constitutive models of materials give the possibility to explicitly describe the mechanisms of inelastic deformation, evolution of a material structure and changes in the physical and mechanical properties of materials determined by their chemical composition and their internal structure. Therefore, these models seem to be very effective for improving metal processing and forming techniques. A study of the solutions (response change histories) obtained using constitutive models under perturbation of input data (influence history and initial conditions) and model operator is actual due to the fact that the material mechanical characteristics (including the lower scale level properties), physical processes occurring during deformation (for example, acts of interaction of defect structures at the microscale level) and the resulting influences (produced by stochastic boundary conditions) are stochastic in nature. Finding the solution to this problem is particularly important when researchers need to justify the use of new constitutive models for describing modern technological processes of thermomechanical treatment, in particular, those focused on creation of functional materials. The disadvantages of traditional analytical approaches (Lyapunov methods) taken to analyze the stability of multilevel material models have been briefly discussed. The definition of the solution stability is introduced; in contrast to the traditional definition, it takes into account the parametric perturbation of operators and the perturbation of the history of influences, which determine the right-hand side of the system of equations. A procedure for the model stability numerical assessment includes consideration of solutions stability for various values of the parameters that determine the operator and input data. The description of the program of computational experiments for the implementation of the proposed approach is presented. This program can be used to study various perturbations of initial conditions, the influence history, the operator, as well as to analyze the norms of their deviations and the integral norm of deviation of perturbed solutions from the base ones obtained in calculations with unperturbed parameters.