Junction in a thin multi-domain for nonsimple grade two materials in BH
Rita Ferreira, José Matias, Elvira Zappale
TL;DR
The paper analyzes a thin multi-domain comprised of a vertical rod on a horizontal disk, governed by a bulk second-order energy density $W(D^2U)$ with linear growth. By scaling the two shrinking components at the same rate, the authors derive a Gamma-limit on a fixed domain that reduces to a union of a $1$-dimensional and an $(N-1)$-dimensional region, introducing four limit fields $(u^a,u^b,\xi^a,\xi^b)$. The limit energy is given by $E[(u^a,u^b),(\xi^a,\xi^b)]=K^a[(u^a,\xi^a)]+ qK^b[(u^b,\xi^b)]$, with $q=\lim_{n}\frac{h_n}{r_n^{N-1}}$, and the analysis hinges on $BH$/$BV$-type functionals, area-strict convergence, and $\mathcal{A}$-quasiconvex relaxation. The results show the limit problem is uncoupled for $N\ge 3$ and partially coupled for $N=2$, highlighting how boundary and transmission constraints between thin components shape the reduced model. This work extends prior models by explicitly incorporating second-order gradient effects and boundary-induced penalties, providing a rigorous pathway from fully 3D theories to effective lower-dimensional descriptions in martensitic and related thin-structure materials.
Abstract
We consider a thin multi-domain of $\mathbb R^N$, with $N\geq 2$, consisting of a vertical rod upon a horizontal disk. In this thin multi-domain, we introduce a bulk energy density of the kind $W(D^2U)$, where $W$ is a continuous function with linear growth at $\infty$ and $D^2U$ 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 $N\geq 3$, and ``partially" coupled if $N=2$.