Homogenization in open sets with holes

AbstractLet Qr be a cylindrical bar with r cylindrical cavities having generators parallel to those of Qr. Let Ω be the cross-section of the bar, Ω∗ the cross-section of the domain occupied by the material and Ωi(i = 1,…, r) the cross- sections of the cavities: Ω̄i ⊂ Ω Ω̄i ∩ Ω̄k = φ, i ≠ k. The study of the elastic torsion of this bar leads to the following problem [see Lanchon (Thèse, Paris, 1972; J. Mécanique, 13 (1974), 267–320)]: Δƒr + 2μα = 0 in Ω∗ƒr¦∂Ω = 0 (1) ƒr = constant on ∂Ωi; i = 1,…, r where μ is the shear modulus of the material, α is the angle of twist and ƒr represents the stress function. In this paper the problem (1) with an increasing number of holes which are distributed periodically is considered. One would like to know if ƒr has a limit ƒ∞as r → + ∞, and if so, the equation satisfied by this limit. This is an “homogenization” problem — the heterogeneous bar Qr is replaced by a homogeneous one, the response of which under torsion approximates as closely as possible that of Qr. A more general problem will be studied and the case of elastic torsion will be obtained as an application. The proof uses the energy method [see Lions (Collège de France, 1975–1977), Tartar (Collège de France, 1977)] and extension theorems. A related problem is the homogenization of a perforated plate [cf. Duvaut (to appear)]