On the convergence of the energy, stress tensors, and eigenvalues in homogenization problems of elasticity. (English) Zbl 0572.73059

Oleinik, O. A.; Shamaev, A. S.; Yosifian, G. A.

Z. Angew. Math. Mech. 65, 13-17 (1985).

In the framework of homogenization problems the authors study the convergence of the energy integrals, stress tensors and eigenvalue for elastic problems. The domain with smooth boundary belongs to a class of perforated domains. The medium is supposed nonhomogeneous and porous elastic with a periodic structure of period \(\epsilon\) which tends to zero. Starting from the estimates of the solutions in the norm \(L^ 2\) and, using correctors, in \(H^ 1\), the convergence of energy integrals is valuated. Moreover, using correctors, an estimate is furnished for the difference between the stress tensors of the problems with \(\epsilon >0\) and \(\epsilon =0\). Some inequality is also obtained for the frequencies of free vibrations.

Reviewer: M.Codegone

MSC:

74H45	Vibrations in dynamical problems in solid mechanics
74E05	Inhomogeneity in solid mechanics
35J25	Boundary value problems for second-order elliptic equations

Keywords:

L(sup 2)-norm; H(sup 1)-norm; homogenization problems; convergence of the energy integrals; stress tensors; eigenvalue for elastic problems; domain with smooth boundary; class of perforated domains; porous elastic; periodic structure; estimates of the solutions; correctors; inequality; frequencies of free vibrations

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