Unsteady thermoelastic-diffusive vibrations of a Bernoulli-Euler beam on an elastic foundation

This paper addresses the problem of unsteady vibrations of a Bernoulli–Euler beam, taking into account the relaxation effects in thermal and diffusion processes. The original mathematical model comprises a system of equations describing unsteady bending vibrations, incorporating both heat and mass transfer. These equations are derived from the general model of thermoelastic diffusion in a continuum, using the variational D'Alembert principle. As an example, a simply supported three-component beam composed of an alloy of zinc, copper, and aluminum is considered. The interaction among the mechanical, thermal, and diffusion fields in the beam is investigated under a mechanical load distributed along its length. The influence of relaxation effects on the kinetics of heat and mass transfer is analyzed. The solution is presented both in analytical form and as spatiotemporal 3D graphs of displacement, temperature increment, and concentration increment fields.