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

TL;DR

This paper analyzes the asymptotic behavior of a thin, multidomain hyperelastic structure allowing for fractures, deriving a limit model as the domain's dimensions tend to zero, with implications for fracture mechanics in reduced dimensions.

Contribution

It provides a rigorous asymptotic analysis of a hyperelastic multidomain with fractures, deriving a well-posed limit model in lower-dimensional domains.

Findings

Limit model is well-posed in 1D and 2D domains.

Convergence results for the energy and configurations.

Framework for fracture analysis in thin structures.

Abstract

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 $W(\nabla U)$, where $W$ is a Borel function with linear growth and $\nabla U$ denotes the gradient of the displacement, i.e. a vector valued function $U:\Omega \to \mathbb R^3$. 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 $2$.