Homogenization in open sets with holes

Let Q^r be a cylindrical bar with r cylindrical cavities having generators parallel to those of Q^r. Let Ω be the cross-section of the bar, Ω1 the cross-section of the domain occupied by the material and $\text{Ω}{}^{\mathrm{i}}\text{(i = 1,…, r)}$ the cross- sections of the cavities:

$\text{Ω}\text{?}{}^{\mathrm{i}}\text{? Ω}\text{Ω}\text{?}{}^{\mathrm{i}}\text{∩}\text{Ω}\text{?}{}^{\mathrm{k}}\text{= φ, i ≠ k}$

. The study of the elastic torsion of this bar leads to the following problem [see 2., 3., 267–320)]:

$\text{Δ?}{}_{\mathrm{r}}\text{+ 2μα = 0 in Ω1}$

$\text{?}{}_{\mathrm{r¦?\Omega }}\text{= 0}$

(1)

$\text{?}{}_{\mathrm{r}}\text{=}\text{constant on}\text{?Ω}{}^{\mathrm{i}}\text{; i = 1,…, r}$

where μ is the shear modulus of the material, α is the angle of twist and $\text{?}{}_{\mathrm{r}}$ 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 $\text{?}{}_{\mathrm{r}}$ has a limit $\text{?}{}_{\mathrm{\infty }}\text{as}\text{r → + ∞}$, and if so, the equation satisfied by this limit. This is an “homogenization” problem — the heterogeneous bar Q^r is replaced by a homogeneous one, the response of which under torsion approximates as closely as possible that of Q^r. 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)].