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

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 $\upsilon_1, \ldots, \upsilon_n$ 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 $\upsilon_i$ 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 $\varepsilon\to 0$ limit. In light of the results by Sivalonagathan & Spector and Henao & Serfaty that cavities always prefer to have a circular shape (unless prevented to do so by the constraint of incompressibility), our theorem suggests that the elongation and coalescence of the cavities experimentally and numerically observed for large loads can only take place after all the cavities have attained a volume comparable to the space they have available in the reference configuration. Based on the previous work of Henao & Serfaty, who apply the Ginzburg–Landau theory for superconductivity to the cavitation problem, this paper shows how the study of the interaction of the cavities is connected to the following more basic question: for what cavitation sites $a_1,\ldots, a_n$ and areas $v_1, \ldots, v_n$ does there exist an incompressible and invertible deformation producing cavities of those areas originating from those points. In order to use the incompressible flow of Dacorogna & Moser to answer that question, it is necessary to study first how do the elliptic regularity estimates for the Neumann problem in domains with circular holes depend on the domain geometry.