# A lower bound for the void coalescence load in nonlinearly elastic solids

## TL;DR

This paper establishes a lower bound for the load at which voids in nonlinearly elastic solids coalesce, showing under certain conditions that cavities remain circular as their size approaches zero, with implications for material failure analysis.

## Contribution

It provides a mathematical lower bound for void coalescence loads in nonlinear elastic solids and characterizes cavity shapes during growth under specific constraints.

## Key findings

- Cavities remain circular in the small-void limit under certain load conditions.
- A lower bound for the void coalescence load is derived.
- The shape of cavities is controlled by their proximity to boundaries or other cavities.

## Abstract

The problem of the sudden growth and coalescence of voids in elastic media is considered. The Dirichlet energy is minimized among incompressible and invertible Sobolev deformations of a two-dimensional domain having $n$ microvoids of radius $\varepsilon$. The constraint is added that the cavities should reach at least certain minimum areas $v_{1},...,v_{n}$ after the deformation takes place. They can be thought of as the current areas of the cavities during a quasistatic loading, the variational problem being the way to determine the state to be attained by the elastic body in a subsequent time step. It is proved that if each $v_{i}$ is smaller than the area of a disk having a certain well defined radius, which is comparable to the distance, in the reference configuration, to either the boundary of the domain or the nearest cavity (whichever is closer), then there exists a range of external loads for which the cavities opened in the body are circular in the $\varepsilon \rightarrow 0$ limit.