Analysis Based on a Two-Dimensional Mathematical Model of the Thermo-Stressed State of a Copper Plate During Its Induction Heat Treatment

A two-dimensional physical and mathematical model is proposed for determining the components of the stress tensor and stress intensity in an electroconductive plate of rectangular cross-section under the action of an unsteady electromagnetic field. The two-dimensional thermomechanics problem for such a plate is formulated. The determining functions are the component of the magnetic field intensity vector tangent to the plate bases, temperature, and components of the stress tensor. The methodology for solving the formulated thermomechanics problem by approximating all the determining functions by the thickness variable with cubic polynomials is developed. As a result, the original two-dimensional problems related to the determining functions are reduced to one-dimensional problems on the integral characteristics of these functions. To solve the obtained systems of equations for the integral characteristics of the determining functions, a finite integral transformation in the transverse variable is used. During induction heating of a copper plate by a homogeneous quasi-steady-state electromagnetic field, the change in the stress intensity depending on the Fourier time, as well as its distribution across the plate cross-section depending on the parameters of induction heating and the Biot criterion, is numerically analyzed.