Energy Estimates and Cavity Interaction for a Critical‐Exponent Cavitation Model

We consider the minimization of $\int_{\Omega _\varepsilon } {|D{\bf u}|^p } d{\bf x}$ in a perforated domain $\Omega _\varepsilon : = \Omega \backslash \cup _{i = 1}^M B_\varepsilon ({\bf a}_i )$ of $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^n$ among maps $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf u} \in W^{1,p} (\Omega _\varepsilon ,\R^n )$ that are incompressible (det $D{\bf u} \equiv 1$ ) and invertible, and satisfy a Dirichlet boundary condition u = g on ∂Ω. If the volume enclosed by g (∂Ω) is greater than |Ω|, any such deformation u is forced to map the small holes B ε ( a i ) onto macroscopically visible cavities (which do not disappear as ε → 0). We restrict our attention to the critical exponent p = n , where the energy required for cavitation is of the order of $\sum\nolimits_{i = 1}^M {v_i |\log \varepsilon |}$ 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$${1 \over n}\int\limits_{\Omega _\varepsilon } {\left| {{{D{\bf u}} \over {\sqrt {n - 1} }}} \right|^p } {\rm d}{\bf x} - \sum\limits_i {v_i |\log \varepsilon |} ,$$ showing its dependence on the size and the shape of the cavities, on the initial distance between the cavitation points a 1 ,…, a M , and on the distance from these points to the outer boundary ∂Ω. 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. © 2012 Wiley Periodicals, Inc.