Development of the method with enhanced accuracy for solving problems from the theory of thermo-pseudoelastic-plasticity - Petrov, Alexander; Chernyakov, Yuriy; Steblyanko, Pavel; Demichev, Konstantin; Haydurov, Vladislav (Eastern-European Journal of Enterprise Technologies, 2018)
Complex behavior of bodies made from pseudoelastic and pseudoelastic-plastic materials requires the development of specialized algorithms for calculating the stressed-deformed state. This work reports the developed numerical method with enhanced accuracy for solving the multidimensional non-stationary problems from the theory of thermo-elastic-plasticity for bodies made from pseudoelastic and pseudoelastic-plastic materials. This method of component-wise splitting, which is based on the application of the new expression for two-dimensional spline-functions, made it possible to improve the accuracy of calculations by two orders of magnitude. Subject to the same accuracy when calculating by the classic finite-difference method, a given method allows us to obtain results faster, due to the choice of larger steps of integration based on coordinates. This leads to the two orders of magnitude reduction in the number of applied nodes in the spatial grid, which appears important and useful from a practical point of view. We recorded basic equations. These include the equation of heat conductivity, the equation of motion, geometric correlations. When constructing the physical correlations, it was assumed that the deformation at a point is represented as the sum of the elastic component, a jump in deformation during phase transition, plastic deformation, and the deformation caused by temperature changes. The boundary and initial conditions are stated in a general form. We have experimentally substantiated a variant of the phenomenological model for the behavior of a material possessing shape memory. This model implies a possibility to quantify complex interactions between stresses, temperature, deformation, and the speed of loading a material, which are suitable for modeling at the continuum level as well. Based on it, we have resolved a qualitatively new class of two-dimensional non-stationary problems for materials possessing shape memory when then unknown magnitudes are sought in the form of two-dimensional strained splines.