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

We consider a thin multi-domain of ${\mathbb{R}}^N$, with $N\ge 2$, consisting of a vertical rod upon a horizontal disk. In this thin multi-domain, we introduce a bulk energy density of the kind $W(D^{2}U)$, where $W$ is a continuous function with linear growth at $\mathrm{\infty }$ 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 with 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. Moreover, we show that the limit problem is uncoupled if $N\ge 3$, and ``partially" coupled if $N=2$.