Homogenization of the Neumann problem for the Lamé equations of linear elasticity in domains with a periodic system of channels of small length

Abstract

The problem of homogenization is considered for the solutions of the Neumann problem for the Lamé system of plane elasticity in two-dimensional domains with channels that have the form of rectilinear cylinders of length ε ^q (ε is a small positive parameter, q = const > 0) and radius a _ɛ. The bases of the channels form an ε-periodic structure on the hyperplane {x ∈ ℝ²: x ₁ = 0} and their number is equal to N _ɛ= O(ɛ⁻¹) as ε → 0. Under the limit condition lim $\mathop {\lim }\limits_{\varepsilon \to 0} a_\varepsilon \varepsilon ^{ - 1 - q} = \beta = const \geqslant 0$ on the parameters characterizing the geometry of the domain, the weak H ¹-limit of the generalized solution of this problem is found.