Asymptotic analysis of a clamped thin multidomain allowing for fractures and discontinuities

We consider a thin multidomain of ${\mathbb{R}}^3,$ consisting of a vertical rod upon a horizontal disk. The equilibrium configurations of the thin hyperelastic multidomain, allowing for fracture and damage, are described by means of a bulk energy density of the kind $W(\mathrm{\nabla }U)$, where $W$ is a Borel function with linear growth and $\mathrm{\nabla }U$ denotes the gradient of the displacement, i.e. a vector valued function $U:\mathrm{\Omega }\to {\mathbb{R}}^3$. By assuming that the two volumes tend to zero, under suitable boundary conditions and loads, and suitable assumptions of the rate of convergence of the two volumes, we prove that the limit model is well posed in the union of the limit domains, with dimensions, respectively, $1$ and $2$.