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

Abstract

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.