МОДЕЛИРОВАНИЕ НЕСТАЦИОНАРНЫХ СВЯЗАННЫХ МЕХАНОДИФФУЗИОННЫХ ПРОЦЕССОВ В ИЗОТРОПНОМ СПЛОШНОМ ЦИЛИНДРЕ

Abstract

We considered the one-dimensional polar-symmetric problem of stress-strain state determining of a continuum isotropic multicomponent cylinder. The cylinder is affected by unsteady surface elastic diffusive perturbations. The coupled system of elastic diffusion equations in the polar coordinate system is used as a mathematical model. The problem solution is sought in the integral form and is represented as convolutions of Green's functions with functions defining surface elastodiffusive perturbations. Mechanical loads and diffusion fields are considered as external influences. We used the Laplace transform by time, and Fourier series expansion in first kind Bessel functions to find the Green's functions. To calculate the coefficients of these series, we obtained formulas for transforming differential operators of the first, second, and third orders using the Hankel integral transform on a segment, which allowed us to reduce the initial boundary-value problem of mechanodiffusion to a system of linear algebraic equations. Laplace transforms of Green's functions are represented through rational functions of the Laplace transform parameter. The Laplace transform inversion is done analytically due to residues and operational calculus tables. As a result, Analytical expressions of surface Green's functions are obtained for the considering problem. Numerical study of the mechanical and diffusion fields interaction in a continuous isotropic cylinder is performed. We used two-component material as an example. The cylinder is under pressure uniformly distributed over the surface. The solution is presented in analytical form and in the form of three-dimensional graphs of the desired displacement fields and concentration increments as functions of time and radial coordinate. This calculation example allows us to demonstrate the coupling effect of mechanical and diffusion fields. It manifests itself as a change in the concentrations of the continuum components under the influence of external unsteady surface pressure.