Bernoulli-Euler Beam Unsteady Bending Model with Consideration of Heat and Mass Transfer

The article describes the problem of unsteady vibrations of a Bernoulli-Euler beam taking into account the relaxation of temperature and diffusion processes. The initial mathematical model includes a system of equations for unsteady bending vibrations of the beam with consideration of heat and mass transfer. This model is obtained from the general model of thermomechanodiffusion for continuum using the D'Alembert's variational principle. The solution of the problem is obtained in the integral form. The kernels of the integral representations are Green's functions. For finding of Green's functions the expansion into trigonometric Fourier series and Laplace transform in time are used. The calculation example is investigated for a freely supported three-component beam made of zinc, copper and aluminum alloy under the action of unsteady bending moments, including the interaction of mechanical, temperature and diffusion fields.