Limit models for thin heterogeneous structures with high contrast

[摘要] We investigate two linear conductivity problems, with strongly contrasting conductivity, in a thin heterogeneous cylinder with a small cross-section of radius h(n), n in N. In this cylinder we distinguish an inner cylindrical core (C) over cap (n) with cross-section of radius r(n) << h(n) and its complementary annulus (I) over cap (n) and we treat two complementary cases. In the first case we consider a low conductivity of order delta(2)(n) in the core (C) over cap (n) and a conductivity of order 1 in the annulus (I) over cap (n); the opposite situation in the second case. We study the asymptotic behavior of these problems with three small parameters: h(n), r(n), and delta(n), as h(n) -> 0, r(n) -> 0, r(n)/h(n) -> 0, and delta(n) -> 0. In the first case we prove that the inner core has not any influence on the limit behavior. In the second case, we pinpoint three different limit regimes depending on the ratio mu = lim(n) delta(n)/h(n), according to mu = 0, 0 < mu < +infinity, or mu = +infinity. We obtain L-2-strong convergence for the solution and its gradient. We examine the limit problems and compare them with other models. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-11-25

[关键词] Thin structure;Contrasting conductivity;Degenerating equation;Correctors