Junction in a thin multi-domain for nonsimple grade two materials in BH

Abstract

We consider a thin multi-domain in $R^N$, with $N\ge 2$, consisting of a vertical rod on top of a horizontal disk made of non-simple grade-two materials or multiphase ones. In this thin multi-domain, we consider a classical hyperelastic energy and complement it by adding an interfacial energy with a bulk density of the kind $W\left(D^{2}U\right)$, where $W$ is a continuous function with linear growth at $\infty$ and $D^{2}U$ denotes the Hessian tensor of a vector-valued function $U$ that represents a deformation of the multi-domain. Considering suitable boundary conditions on the admissible deformations and assuming that the two volumes tend to zero at the same rate, we prove that the limit model is well posed in the union of the limit domains, with dimensions 1 and $N-1$, respectively, and its limiting energy keeps memory of the original full dimensional trace constraints in a more accurate way than previous related models in the literature. Moreover, we show that the limit problem is uncoupled if $N\ge 3$, and “partially” coupled if $N=2$.