JUNCTION IN A THIN MULTIDOMAIN FOR A FOURTH ORDER PROBLEM

Abstract

We consider a thin multidomain of ℝ N , N ≥ 2, consisting (e.g. in a 3D setting) of a vertical rod upon a horizontal disk. In this thin multidomain we introduce a bulk energy density of the kind W(D 2 U), where W is a convex function with growth p ∈ ]1,+∞[, and D 2 U denotes the Hessian tensor of a scalar (or vector-valued) function U. By assuming that the two volumes tend to zero with the same rate, under suitable boundary conditions, we prove that the limit model is well-posed in the union of the limit domains, with dimensions, respectively, 1 and N - 1. Moreover, we show that the limit problem is uncoupled if [Formula: see text], "partially" coupled if [Formula: see text], and coupled if N - 1 < p. The main result is applied in order to derive the equilibrium configuration of two joint beams, T-shaped, clamped at the three endpoints and subject to transverse loads. The main result is also applied in order to describe the equilibrium configuration of a wire upon a thin film with contact at the origin, when the thin structure is filled with a martensitic material.