Numerical analysis of processes of thermoplastic deformation of axisymmetric bodies with regard for unloading

Abstract

We investigate the thermoelastoplastic state of an isotropic homogeneous medium under the action of temperature loads. A mathematical model of plastic flow is considered. We propose a method for constructing an unconditionally stable numerical scheme of the finite-element method for the solution of such problems. The process of propagation of the primary and secondary unloading zones in a body is shown. The time distribution of the intensity of plastic strain at a corner point of the body is presented. Results obtained without regard for the dependence of the yield strength on temperature and with regard for it are investigated.A number of works, a review of which is presented in [1, 6-8, 9, 12], are devoted to the investigation of thermoplastic deformation in bodies of revolution. A series of works [3,5,11,12] is devoted to the use of the finite-element method and Newton-Raphson-Kantorovich methods for the solution of spatial problems on studying the thermoelastoplastic stress-strain state [3,5,11,12]. Methods for computing these problems were developed using an explicit step-by-step integration scheme, which is numerically unstable and requires very small steps. In [10], a systematic approach to the numerical computation of plastic flow processes and construction of a numerical scheme of the Newton iteration process was developed. In [2,13], the implicit numerically stable scheme of intermediate points is used for the solution of problems of force elastoplastic deformation. In the proposed work, this scheme is used to improve the accuracy and efficiency of determination of thermoplastic strains.The aim of this work is also to investigate the character of unloading in an axisymmetric thick-walled body of revolution for different dependences of the limit of elasticity on temperature. For this purpose, the nonstationary problem of heat conduction [4] is solved by the Newton-Kantorovich method. On the basis of obtained temperature fields, the problem of thermoplastic deformation is reduced to a sequence of nonlinear boundaryvalue problems each of which is linearized by the Newton method. The solution of the linearized problem is performed on the basis of schemes of the finite-element method. Finite elements are constructed relative to a some basic surface. This makes it possible to solve easily the problem of identification of spatial finite elements. On the basis of the described method, the problem of investigation of the propagation of plastic and unloading zones in an axisymmetric thick-walled body of revolution is solved. The obtained results indicate the reliability and efficiency of the proposed method. Statement of the Problem and Main RelationsConsider the deformation process of a solid isotropic body located in a volume V bounded by a surface S and subjected to force and thermal loads that do not lead to loss of its stability. Let us use a mathematical