Averaging of a system of elasticity theory with almost periodic coefficients

V. V. Zhikov ; O. A. Oleinik

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (1982), pp. 62-70

Résumé

We prove, for the linear elasticity system $L_\varepsilon(u^\varepsilon)=f$ with coefficients of the form $a_{ij,kh}\bigl(\frac{x}{\varepsilon}\bigr)$ where $\varepsilon$ is a small parameter, $\varepsilon$ is a positive constant and $a_{ij,kh}(y)$ is a Bohr's almost periodic function, that $u_\varepsilon\to u$ as $\varepsilon\to 0$ in the norm of $L^2(\Omega)$, $\hat L(u)=f$ in $\Omega$, $u_\varepsilon=0$, on the boundary of $\Omega$ and $\hat L(u)=f$ is an elasticity system with constant coefficients. The strain tensor also converges as $\varepsilon\to 0$ to the strain tensor of the homogenized system $\hat L(u)=f$.