Unsteady bending of the orthotropic cantilever Bernoulli–Euler beam with the relaxation of diffusion fluxes

Abstract

AbstractThe unsteady bending problem of a cantilever elastic‐diffusion homogeneous orthotropic Bernoulli–Euler beam is considered, taking into account the relaxation of diffusion fluxes. The solution to the problem is sought using the method of equivalent boundary conditions. For this, an auxiliary problem is additionally considered. The solution to the auxiliary problem is obtained using the integral Laplace transform in time and the trigonometric Fourier series expansions in spatial coordinates. Next, relations are constructed connecting the right‐hand sides of the boundary conditions of the original and the auxiliary problems. These relations represent a system of Voltaire integral equations of the 1st kind. This system is solved numerically using quadrature formulas. The limit transitions to the static problem of elastic diffusion, as well as to the unsteady elastic problem for a cantilever beam, are investigated. Using a three‐component material as an example, a numerical study of unsteady mechanical and diffusion fields interaction in an orthotropic beam is done. The calculation results are presented in the form of graphs. They show the time and coordinate dependencies of the sought displacement fields and components concentration increments of the continuum. At the end of the publication, the main conclusions about the coupling effect of the stress–strain state and mass transfer in the beam are represented.