Numerical Modeling of Plane Elastoplastic Problems in Strains
DOI:
https://doi.org/10.71310/pcam.1_71.2026.06Keywords:
compatibility condition, plasticity, strains, displacement, finite-difference equations, stress, elimination and iteration methodsAbstract
In this article, within the framework of the Saint-Venant compatibility conditions, plane problems of plasticity theory formulated in terms of strains are presented, aimed at investigating the stress-strain state of a rectangular plate under elastoplastic deformation. The numerical implementation of the formulated problem is carried out using the finite difference method, on the basis of which grid equations for the displacement and strain components at the nodes of the rectangular computational domain are constructed. Spatial discretization enables the transition from the differential form of the boundary-value problem to a system of algebraic finite-difference equations suitable for stable and efficient numerical solution. In the development of the finite-difference scheme, boundary conditions of various types are taken into account, adequately reflecting real loading regimes and fixation conditions of the plate. The system of finite-difference equations with respect to strains is solved using the alternating direction method, which ensures an increase in computational efficiency and improved convergence when solving problems with pronounced nonlinearity. The application of this method contributes to reducing approximation errors while maintaining high accuracy of the obtained results. A detailed comparison of numerical solutions obtained in formulations based on displacements and strains under identical boundary and loading conditions is performed. The analysis demonstrates their qualitative and quantitative agreement, manifested in the close correspondence of strain and stress distributions throughout the entire domain of the rectangular plate.
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