Two-component elastic diffusion half-space under the influence of time-dependent perturbation

Abstract

The problem of finding stress-strain state of the two-component half-space (for example alloy), with surface parallel to plane $x_1Ox_2$ of Cartesian coordinate system. The physics-mechanical processes in the Cartesian coordinate system are modeled by equations $$C_{3333}\frac{\partial^2u_3}{\partial x_3^2}=\rho\frac{\partial^2u_3}{\partial t^2}+\alpha_{33}^{\left(1\right)}\frac{\partial \eta^{\left(1\right)}}{\partial x_3}+\alpha_{33}^{\left(2\right)}\frac{\partial\eta^{\left(2\right)}}{\partial x_3},$$ $$D_{33}^{\left(q\right)}\frac{\partial^2\eta^{\left(q\right)}}{\partial x_3^2}=\frac{\partial \eta^{\left(q\right)}}{\partial t}+\Lambda_{3333}^{\left(q\right)}\frac{\partial^3u_3 }{\partial x_3^3}\quad\left(q=1,2\right) $$ where $t$ – time; $x_3$ – Cartesian coordinates; $u_3$ – displacements vector component $Ox_3$; $\eta^{\left(q\right)}=n^{\left(q\right)}-n_0^{\left(q\right)}$ – are the concentrations variation; $n_0^{\left(q\right)}$ – initial components concentrations; $n^{\left(q\right)}$ – are the concentrations; $C_{3333}$ – components of elastic constants tensor; $\rho$ – medium density; $\alpha_{33}^{\left(q\right)}$ – components of tensor, which is defined by crystal structure type and characterizing the relative volume change due to diffusion; $D_{33}^{\left(q\right)}$ – components of diffusion tensor; $R$ – is the universal gas constant; $T_0$ – temperature. Coefficients $\Lambda_{3333}^{\left(q\right)}$ are defined by the following formulas $$\Lambda_{3333}^{\left(q\right)}=\frac{n_0^{\left(q\right)}\alpha_{33}^{\left(q\right)}D_{33}^{\left(q\right)}}{RT_0}$$ It is assumed that at the surface of the half-space $x_3=0$ the displacement and diffusion flows are set $$\left. u_3\right|_{x_3=0}=f_3\left(t\right),\quad\left. J^{\left(q\right)}\right|_{x_3=0}=f_q\left(t\right),$$ $$u_3=O\left(1\right),\quad J^{\left(q\right)}=O\left(1\right)\quad\left(x_3\rightarrow\infty\right)$$ where $J^{\left(q\right)}=\Lambda_{3333}^{\left(q\right)}\cfrac{\partial^2u_3}{\partial x_3^2}-D_{33}^{\left(q\right)}\cfrac{\partial\eta^{\left(q\right)}}{\partial x_3}$ – diffusion flows. The initial terms are assumed zero: $$\left.u_3\right|_{t=0}=0,\quad\left.\frac{\partial u_3 }{\partial t}\right|_{t=0}=0,\quad\left.\eta^{\left(q\right)}\right|_{t=0}=0.$$ The solving algorithm is based on using integral Laplace transform and Fourier transform. The received Laplace transforms are rational functions of conversion parameter. Laplace transforms originals are sought by the second decomposition theorem of operational calculus. For Fourier inversion using a numerical algorithm based on the use of quadrature formulas Philo. The final solution is received as integral convolution. The cases of constant diffusion flows at the surfaces are shown.