Mathematical modeling and analysis of the thermomechanical behavior of a bimetallic spherical element during technological processing with an electromagnetic impulse

Abstract

The initial-boundary problem of thermomechanics for a bimetallic spherical element under
homogenous non-stationary electromagnetic action is formulated. The azimuthal component of the
magnetic field strength vector, temperature, and the radial component of the displacement vector were
selected as key functions. To find the key functions, the approximation of their distributions in the
constituent layers of the spherical element by quadratic polynomials in the radial variable is proposed.
This approximation makes it possible to accurately satisfy the specified boundary conditions both on
the surfaces of this element and on the surface of the connection of the component layers, where ideal
electromagnetic, thermal and mechanical contacts take place. The coefficients of the polynomials
approximating the key functions in the component layers of the spherical element are given by a linear
combination of functions describing all given boundary conditions and integral characteristics of the
key functions on both component layers. As a result, the original initial-boundary value problems for
the defining functions are reduced to Cauchy problems for their integral (summed over the package of
layers) characteristics. With the use of Laplace transformation in time, the general solutions of these
problems under homogeneous non-stationary electromagnetic action are recorded. The change in time
of radial and azimuthal stresses and stress intensity was numerically analyzed by technological
processing with an electromagnetic impulse, as well as the performance and properties of the contact
connection of the bimetallic spherical element were investigated.