Unsteady thermoelastic diffusion vibrations of the Bernoulli - Euler beam under the action of a distributed transverse load

The paper deals with the problem of unsteady vibrations of the Bernoulli - Euler beam, taking into account relaxation of temperature and diffusion processes. The original mathematical model includes a system of equations of non-stationary bending oscillations of the beam taking into account heat and mass transfer, which is obtained from the general model of thermoelastic diffusion for continuum using variational D'Alembert principle. Based on the obtained equations, the statement of the initial-boundary problem concerning bending of the hinged orthotropic beam, which is under the action of thermo-elastic diffusion perturbations distributed on the surface, is formulated. Solutions of the problem of unsteady thermoelastic diffusion vibrations of the beam are sought in integral form. The kernels of integral representations are Green’s functions, for finding of which decompositions into trigonometric Fourier series and Laplace transformation over time are used. Laplace transformants of Green's functions are represented through the rational functions of Laplace transformation parameter. Transition into the space of the originals is carried out analytically using deductions and tables of operational calculus. Analytical expressions for Green functions of the problem under consideration are obtained. On the example of a simple supported three-component beam made of an alloy of zinc, copper, and aluminum, which is under the influence of mechanical load distributed along the length, the interaction of mechanical, temperature and diffusion fields is investigated. The influence of relaxation effects on the kinetics of heat and mass transfer is analyzed. The solution is presented in analytical form and in the form of graphs of the dependence of the desired fields of movement, temperature increments, and increments of concentration of medium components on time and coordinates. In conclusion, the main conclusions concerning influence of field connectivity and relaxation effects on the stress-strain state and heat and mass transfer in the bendable beam are given.