Numerical and exact solutions to the deformations of unbounded domains induced by time‐accelerated thermal conduction and electromagnetic induction

Abstract It is widely recognized that the thermal conduction phenomena are significantly influenced by both the temperature and the internal structure of materials. For example, in a variety of material species that have impurities and/or pores which reflect the complexity of fractal structure, thermal conductivity can deviate from the classical Fourier behavior. In the presence of a uniform applied magnetic field, the current study develops a fractional approach to the dynamical and/or quasistatic coupled theory of magnetothermoelasticity based on a crossover law of heat conduction. This crossover law illustrates how the thermal conductivity transitions from lower values in the short‐time range to higher values in the long‐time range, that is, a variable thermal conductivity with time‐accelerated increment. Additionally, it examines how such a crossover heat conduction model affects the displacement, and the stresses generated in an unbounded Hookean domain. An initial‐value problem, based on three initial conditions imposed on the temperature, the displacement, and the displacement time rate, is considered. Analytical solutions for the coupled, uncoupled, and quasistatic theories of magnetothermoelasticity are obtained in the Laplace domain. Then we use the Riemann sum series to derive numerical results for the temperature, the hydrostatic stress, and the displacement. Exact solutions, in terms of the Fox H‐function, are derived for the quasistatic case when the electromagnetic induction is neglected. It is found that the conservation of thermal energy is satisfied only in the uncoupled dynamical/quasistatic theory. The electromagnetic force presented acts as a damping source for deformation in the absence of mechanical wave speed.