Mixed projection-mesh scheme of the finite-element method to solve boundary-value problems describing the non-isothermal processes of elastoplastic deformation

A mixed projection-mesh scheme for solving a boundary-value problem of thermal plasticity is formulated in a quasi-static statement when the process of non-isothermal elastoplastic deformation of a body is a sequence of equilibrium states. In this case, the stress-strain state depends on the loading history, and the process of inelastic deformation is to be observed over the whole time interval under study. The correctness and convergence of the mixed approximations for stresses, strains and displacements are investigated as applied to the solution of nonlinear boundary-value problems that describe the non-isothermal processes of active loading taking into account the initial strains dependent on the history of deformation and heating. The properties of the projecting operators are studied in detail, and on this basis, the condition that ensures the existence, uniqueness and stability of solution is formulated. The results of the analysis of special formulas of the interpolation-type numerical integration are presented, the use of which considerably simplifies the computation procedure for solving equations of the mixed method. The convergence and accuracy estimations are based on the results of the theory of the generalized boundary-value problems and methods of the functional analysis. According to the estimations obtained, the accuracy of solution of a finite-dimensional problem at the initial stages of loading should be sufficient to avoid the effect of increase of the first coefficients in the expansion of the total error on the accuracy of solution of the elastoplastic problem at the subsequent stages of loading. __________ Translated from Problemy Prochnosti, No. 3, pp. 87–117, May–June, 2007.     

Analysis of Boundary-Value Problems of the Theory of Small Elastoplastic Strains,which Allows for Hydrostatic Stress and Stress Deviator Type

The author considers some versions of the deformation theory of plasticity, which takes into account the influence of hydrostatic stress and stress deviator type on the mechanical properties of a medium for the case of proportional loading. The emphasis is on the generalized statement of a nonlinear boundary-value problem and investigation of its solvability. Conditions have been determined that ensure the existence, uniqueness, and continuous dependence of the generalized solution on the loads applied.__________Translated from Problemy Prochnosti, No. 2, pp. 107 – 135, March – April, 2005     

Construction of Constitutive Relationships of the Deformation Theory for Simple in Noll's Sense Hardening Materials with Elastoplastic Behavior

Based on the theory of simple hardening materials with elastoplastic behavior, the general constitutive relationships of the deformation theory of plasticity are mathematically strictly constructed for arbitrary continuous, piece-wise continuously differentiable deformation trajectories, any strains and symmetry types of the material properties. Two conditions under which this is possible are considered. The approaches to a strict specialization of general constitutive relationships of the deformation theory of plasticity have been developed by imposing restrictions on the material strains, deformation processes and properties. In this case, the restrictions on the properties of materials formalize the data obtained in the experimental investigations. A series of both new and known constitutive relationships have been constructed that are arranged into a hierarchy according to the level of complexity of the response to deformation. The area of applicability of the derived physical equations has been defined. Special attention has been given to the modeling of finite and infinitesimal strains of isotropic materials. __________ Translated from Problemy Prochnosti, No. 6, pp. 35 – 49, November – December, 2005.     

Simulation of the fading memory of the form of trajectory in the theory of simple materials with elastoplastic behavior. Part 2. Infinitely small strains

For the processes of deformation close to proportional and slightly different from the stress- and strain-free configurations, in which plastic strains are formed immediately after the application of loading and monotonically increase in the process of deformation, we develop a mathematical theory of rigorous construction and specialization of determining relations for hardening elastoplastic materials with fading memory of the first order form of trajectory. The strains are regarded as infinitely small. The type of symmetry of the material is arbitrary. We use the determining relations of the linear theory of elastoplasticity for finite strains established earlier by the author. The condition of smallness of the measures of strains for the entire history is accepted. Special attention is given to the case of isotropic materials. The conditions of reduction of the constructed relations to one of the existing versions of the endochronic theory of plasticity are established.Translated from Problemy Prochnosti, No. 6, pp. 87–98, November–December, 2004