Junction in a Thin Multidomain for a Fourth Order Problem

Abstract

We consider a thin multidomain of ℝN, 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(D2U), where W is a convex function with growth p ∈ ]1,+∞[, and D2U 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.

A remark on the junction in a thin multi-domain: the non convex case

“…We point out that, as proven in [17], the limit problem is partially coupled by the junction condition:…”

“…Some remarks, essentially dealing with multi-structures in R N , for N > 3, both for gradient and hessian cases, are given in the last section. There is also a final remark showing that arguing as in Proposition 4.1 the limit energy can be better described also in the convex case, through a fewer number of limit functions than it has been done in [13] and [17]. …”

“…It is a sequel of previous works investigating junction conditions in thin multi-domains modelled either by convex second order bulk energies or by convex energies depending on the first order derivatives of the displacement, see [17] and [13], respectively. The bulk of presented proofs is mainly concerned about the second order case, and we present the result in the gradient case just in the last section, the motivation being that, though not a corollary the gradient case can be given in a more classical framework.…”

Asymptotic analysis of the eigenvalues of a Laplacian problem in a thin multidomain

“…It appears in the othonormal conditions: 10) where δ h,k is the Kronecker's delta. Moreover, it explicitly intervenes in the corrector result (1.6), and, by the previous othonormal conditions, also in the corrector result (1.5).…”

“…Then, by making use of the method of oscillating test functions, introduced by L. Tartar in [17], by applying some results obtained by A. Gaudiello, B. Gustafsson, C. Lefter and J. Mossino in [6] and [7] and by adapting the techniques used by M. Vanninathan in [16], we derive the limit eigenvalue problem and the limit of the rescaled basis, as n → +∞, in the case h n ≃ r For the study of thin multi-structures we refer to [2], [3], [4], [11], [12], [13], [14] and the references quoted therein. For a thin multi-structure as considered in this paper, we refer to [5], [6], [7], [8], [9] and [10]. For the study of the spectrum of the Laplace operator in a thin tube with a Dirichlet condition on its boundary we refer to [1].…”