Convergence of the one-step iteration process in the tasks of inelastic deformation mechanics considering the loading history

Abstract

The paper presents the one-step iteration process of the solution to nonlinear boundary tasks of inelastic deformation mechanics considering the loading history. Under such circumstances, the stress-strain state depends on the loading history. The process of deformation should be observed within the entire investigated time interval. The deformation process consists of several calculation stages. At each stage, the boundary task is presented in the form of a nonlinear operator equation in the Hilbert plane. The initial strains in the equation are the temperature, structural, and accumulated irreversible ones at the beginning of the loading stage. The irreversible strains depend on the deformation process and are determined considering the loading history. The analysis of convergence between the iteration methods of the solution to the nonlinear boundary tasks, which consider the deformation history of loading, involves the repeatability of the successive approximations for the current loading stage. The known assessments of the convergence of the elastic solution methods and variable elasticity parameters do not consider the error in the calculation of initial strains, which do not depend on the inelastic deformation history. They are determined using the approximated solution to the boundary tasks at the preliminary stages of the loading by iteration methods. In practice, at each loading stage, the approximate equation is solved instead of the output boundary task. The solution to the approximate equation involves the error in the calculation of irreversible strains from the calculation results at the preliminary loading stages. Therefore, the a priori estimates of the convergence between the elastic solution methods and variable elasticity parameters define the convergence of the successive approximations for the solution to this approximate equation. This paper describes some aspects of the convergence analysis of the one-step iteration process, as well as the assessment of the repeatability of the successive approximations considering the loading history.