Semi-analytical finite element method for solving three-dimensional problems of thermo-visco-elasto-plasticity of prismatic bodies with material damage consideration

Abstract

The paper presents a methodology for solving three-dimensional problems of thermo-visco-elastoplasticity of prismatic bodies using the semi-analytical finite element method. An approach to modeling geometrically complex heterogeneous structures subjected to spatially and temporally varying mechanical and thermal loads is described. It is noted that accounting for nonlinear deformation processes, such as plasticity and creep, requires step-by-step iterative algorithms. The use of nonhomogeneous skew prismatic finite elements is proposed, in which the formulas account for the variability of the metric tensor. This makes it possible to significantly reduce the number of unknowns and improve the accuracy of approximating the stress–strain state. An algorithm for solving thermo-visco-elastoplastic problems has been developed, which includes two iterative cycles: an inner cycle for solving the system of linear equations and an outer cycle for the nonlinear problem. The approach is based on the Newton–Kantorovich iterative procedure. A distinctive feature of the method is the use of displacement extrapolation at the current step based on the results of the previous step, which ensures faster convergence. An example of modeling the deformation of a non-uniformly heated cube is presented. For this example, the semi-analytical finite element method results showed a discrepancy of less than 5% compared with known reference data, confirming the reliability of the method. The analysis shows that the maximum stresses are concentrated in the center of the cube, where a state close to hydrostatic compression is realized, while plastic deformations reduce the stress level by up to 35%. The application of the algorithm with displacement extrapolation made it possible to reduce the number of iterations by more than half and decrease computational costs by 1.5–3 times. The obtained results confirm the efficiency and accuracy of the proposed approach for three-dimensional thermo-visco-elastoplastic problems of complex prismatic bodies.