Homogenization of problems of elasticity theory on composite structures with thin armoring

The paper considers the problems of elasticity theory on a flat slab armored by a periodic thin mesh or in a three-dimensional body armored by a periodic thin box structure. The composite medium depends on two small mutually related geometric parameters; one of them controls the periodicity cell and the other controls the thickness of the armoring structure. It is proved that the homogenization of the indicated problems is classical. In doing so, one applies V. V. Zhikov’s approach (“Zhikov measure approach”) together with the two-scale convergence method. Preliminarily, the paper studies the peculiarities of the two-scale convergence with the variable composite measure and also the Sobolev spaces of elasticity theory with variable composite measure. The obtained compactness principle (an analog of the Rellich theorem) in these spaces made it possible to prove the Hausdor. convergence of the spectrum of the problem studied. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.