Variational formulation of the thermal deformation problems of electrically conductive bodies in an electromagnetic field

Abstract

The paper discusses issues concerning the thermal deformation of electrically conductive bodies under the action of the electromagnetic field. Similar problems arise, for example, in the analysis of induction heating processes. Transient electromagnetic field leads to heat release in electrically conductive bodies, and the change in temperature fields leads to a change of stress-strain state of a body. The creation of methods for the quantitative analysis of the stress-strain state of bodies under the action of an electromagnetic field is an urgent scientific problem because such an analysis allows us to evaluate the performance and durability of various structural elements. The modern approach dictates the need to consider three related problems: the problem of spatio-temporal distribution analysis of the electromagnetic field, transient heat-transfer problem and the problem of stress-strain analysis. The analysis of real technical and technological systems can only be done using appropriate numerical methods. In this case, the most universal is the finite element method, which has proven itself both in the analysis of the deformable bodies mechanics and in the analysis of various multiphysical problems. The usage of the finite element method requires an appropriate mathematical formulation of the problem. The mathematical problem formulation in variational form is considered in this article. Examples of corresponding functionals that allow finding solutions to a problem by finite element method are presented in the article. The functionals describing the transient distribution of the electromagnetic field are constructed based on the using of the concept of scalar electric and vector magnetic potentials. The influence of the electromagnetic field on the temperature distribution and the deformation process is taken into account by introducing distributed heat sources and distributed electromagnetic forces. The operation of varying the solution functions – potentials, temperature and displacements – makes it possible to obtain a system of resolving algebraic equations of the finite element method.