A two-dimensional nonstationary problem of elastic diffusion for an orthotropic one-component layer

Abstract

A 2 D nonstationary problem of an orthotropic one-component elastic layer, accounting for diffusion, is considered. A locally balanced model of elastic diffusion is used, which includes a coupled set of equations of elastic body motion and a mass-transfer equation. On the layer boundaries normal displacement, tangential stress and diffusive flow are assumed. At an initial time, the layer is in an undisturbed state.

The solution is sought in an integral form, as a double convolution in time and along the space coordinate of Green functions and the right-hand sides of the boundary equations. To construct a solution of the initial problem, Laplace time transform, exponential Fourier transform along the space coordinate in the direction of the layer surface, reduction to zero boundary conditions, and expansion into incomplete Fourier series along a coordinate directed through the depth of the layer are successively used. Thus, the initial problem is reduced to a set of linear algebraic equations, from which forms of the sought functions are found. It is shown that the necessary condition of the applicability of the present algorithm is orthotropy of the medium in question. Inversion of Laplace transform of the sought functions is reduced to computing the originals of the rational functions, which is done analytically, using deductions. It is found that all Green functions are even (odd) relative to Fourier transform parameters, which makes it possible, when the right-hand sides of the boundary conditions are even (odd), to reduce the inversion of exponential Fourier transform to the inversion of sinus-, cosine-transform. In that case, quadrature formulas of medium rectangles are used to find the originals of Fourier transformants.

An example is considered, where surface perturbation is modeled with Heaviside time function and a Gaussian along the space coordinate. A homogenized medium of interlaced layers of aluminum and copper is used as a model of an orthotropic medium. The computational results are presented both analytically and in the form of 3D diagrams of the diffusant concentration increment and displacement vector components.