Energy estimates and cavity interaction for a critical-exponent cavitation model

TL;DR

This paper analyzes the asymptotic behavior of energy and cavity interactions in a critical-exponent cavitation model, revealing how cavity shape and placement influence the energy landscape and cavity coalescence.

Contribution

It provides new estimates for the renormalized energy in a cavitation model at the critical exponent, linking cavity geometry and positioning to energy minimization.

Findings

Cavities tend to be spherical and well separated or close and coalesced.

Energy estimates depend on cavity size, shape, and distance to boundary.

Results align with numerical simulations and fracture mechanics theories.

Abstract

We consider the minimization of $\int_{\Omega_{\ep}} |D\vec u|^p \dd\vec x$ in a perforated domain $\Omega_{\ep}:= \Omega \setminus \bigcup_{i=1}^M B_{\ep}(\vec a_i)$ of $\R^n$, among maps $\vec u \in W^{1,p}(\Omega_{\ep}, \R^n)$ that are incompressible ($\det D\vec u\equiv 1$), invertible, and satisfy a Dirichlet boundary condition $\vec u= \vec g$ on $\partial \Omega$. If the volume enclosed by $\vec g (\partial \Omega)$ is greater than $|\Omega|$, any such deformation $\vec u$ is forced to map the small holes $B_{\ep}(\vec a_i)$ onto macroscopically visible cavities (which do not disappear as $\ep\to 0$). We restrict our attention to the critical exponent $p=n$, where the energy required for cavitation is of the order of $\sum_{i=1}^M v_i |\log \ep|$ and the model is suited, therefore, for an asymptotic analysis ($v_1,..., v_M$ denote the volumes of the cavities). In the spirit of the analysis of vortices in Ginzburg-Landau theory, we obtain estimates for the "renormalized" energy $\frac{1}{n}\int_{\Omega_{\ep}} |\frac{D\vec u}{\sqrt{n-1}}|^p \dd\vec x - \sum_i v_i |\log \ep|$, showing its dependence on the size and the shape of the cavities, on the initial distance between the cavitation points $\vec a_1,..., \vec a_M$, and on the distance from these points to the outer boundary $\partial \Omega$. Based on those estimates we conclude, for the case of two cavities, that either the cavities prefer to be spherical in shape and well separated, or to be very close to each other and appear as a single equivalent round cavity. This is in agreement with existing numerical simulations, and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence.