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

Authors

DOI:

https://doi.org/10.15407/dopovidi2022.03.029

Keywords:

inelastic deformation, boundary-value problem, irreversible strains, iteration process, convergence and accuracy of the successive approximations

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.

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References

Ortega, J. M. & Rheinboldt, W. C. (1970). Iterative Solution if Nonlinear Equations in Several Variables. New York; London: Academic Press.

Il'yushin, A. A. (1963). Fundamentals of the General Mathematical Theory of Plasticity. Moscow: Izd. Akad. Nauk SSSR (in Russian).

Birger, I. A. (1951). Some general methods for solving problems of the theory of plasticity. Prikl. Matem. Mekh., 15, No. 6, pp. 765-770 (in Russian).

Temis, Yu. M. (1982). Convergence of the method of variable elastic parameters for numerical solution of problems of plasticity by the finite element method, in: Applied Problems of Strength and Plasticity: Statics and Dynamics of Deformable Systems, Moscow, pp. 21-34 (in Russian).

Umanskii, S. E. (1983). Optimization of approximate methods for solving boundary value problems in

mechanics. Kyiv: Naukova Dumka (in Russian).

Chirkov, A. Yu. (2005). Iteration algorithms for solving boundary-value problems of the theory of small elasticplastic strains on the basis of the mixed finite element method. Strength Mater. 37, pp. 310-322. https://doi.org/10.1007/s11223-005-0044-8

https://doi.org/10.1007/s11223-005-0044-8

Chirkov, O. Yu. (2021). The Correctness of the Radiation Creep Equations that Take into Account Stress and Accumulated Irreversible Deformation in the Radiation Model Swelling of the Irradiated Material. Dopov. Nac. akad. nauk Ukr., 2021, No. 4, pp. 36-45 (in Ukrainian) https://doi.org/10.15407/dopovidi2021.04.036

https://doi.org/10.15407/dopovidi2021.04.036

Published

02.07.2022

How to Cite

Chirkov О. . (2022). Convergence of the one-step iteration process in the tasks of inelastic deformation mechanics considering the loading history. Reports of the National Academy of Sciences of Ukraine, (3), 29–38. https://doi.org/10.15407/dopovidi2022.03.029