Junction in a Thin Multidomain for a Fourth Order Problem

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 1 < p ≤ N−1 2 , “partially” coupled if N−1 2 < p ≤ N − 1, 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.