Unsteady bending of a cantilevered Euler-Bernoulli beam with diffusion

Abstract

We considered the problem of unsteady direct bending of an isotropic homogeneous elastodiffusive cantilevered Euler-Bernoulli beam. For the mathematical formulation of the problem, we use the closed system of equations of transverse unsteady beam vibrations with inner diffusion. The formulation is based on the model of elastic diffusion using the d'Alember variational principle. It is assumed that the deflections of the beam are small and the hypothesis of flat sections is fulfilled. The Euler-Bernoulli hypothesis is valid for section rotations. A solution to the problem is sought using the method of equivalent boundary conditions, which allows a transition from the initial formulation with arbitrary boundary conditions to the problem of the same type and with the same domain geometry. First, an auxiliary problem is solved using the integral Laplace transform in time and trigonometric Fourier series. Then, some relations connecting the boundary conditions right-hand sides of the original and auxiliary problems are constructed. These relations are the Volterra integral equations of the first kind. For solving this system, quadrature formulas of an average rectangle are used. Finally, the solution of the original problem is represented in the form of the convolution of the Green's functions for the auxiliary problem with the functions determined by solving the system of the Volterra integral equations. The interaction between unsteady mechanical and diffusion fields is analyzed using an isotropic beam as an example. Graphs showing the dependence of the displacement fields and concentration increments on time and coordinates are given. Analysis of the results obtained led to conclusion that the coupling action of mechanical and diffusion fields affects the stress-strain state and mass transfer in a beam