Published Paper

Inserted: 12 jan 2004
Last Updated: 18 feb 2009

Journal: MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES.
Volume: 16
Number: 12
Pages: 1-32
Year: 2006

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 $W(D^2U)$, where $W$ is a convex function with finite growth $p>1$, and $D^2U$ denotes the Hessian tensor of a scalar (or vector valued) function $U$. By assuming that the two volumes tend to zero with 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 $\displaystyle{{N-1}<p}$. 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.

Keywords: Dimensional Reduction, junction, thin multidomain, foruth order operator