Mikheil Basheleishvili

Solution of the Basic Contact Problem of Statics of Elastic Mixtures

abstract:
A finite domain $D_1$ and an infinite domain $D_0$ are considered with the common boundary $S$ having H\"{o}lder continuous curvature. $D_1$ and $D_0$ are filled with isotropic elastic mixtures. In $D_1$ and $D_0$ $u^{(1)}$ and $u^{(0)}$ are displacement vectors while $T^{(1)}u^{(1)}$ and $T^{(2)}u^{(2)}$ are stress vectors. The main contact problem considered in the paper may be formulated as follows: in the domains $D_1$ and $D_0$, find regular vectors $u^{(1)}$ and $u^{(0)}$ satisfying on the boundary $S$ the conditions
\begin{align*}
(u^{(1)})^{+}-(u^{(0)})^{-} & =f, \\
(T^{(1)}u^{(1)})^{+}-(T^{(2)}u^{(2)})^{-} & = F,
\end{align*}
where $f$ and $F$ are given vectors. A uniqueness theorem is proved for this problem. A Fredholm system of integral equations is derived for the problem. An existence theorem is proved for the main contact problem via investigation of the latter system.

Mathematics Subject Classification: 74B05

Key words and phrases: Simple and double layer potentials, Fredholm system of integral equations of second kind