Anisotropic energies for the modeling of cavitation in nonlinear elasticity

Abstract

We study free-discontinuity functionals in nonlinear elasticity, where discontinuities correspond to the phenomenon of cavitation. The energy comprises two terms: a volume term accounting for the elastic energy; and a surface term concentrated on the boundaries of the cavities in the deformed configuration that depends on their unit normal. First, we prove the existence of energy-minimizing deformations. While the treatment of the volume term is standard, that of the surface term relies on the regularity of inverse deformations, their weak continuity properties, and Ambrosio's lower semicontinuity theorem for special functions of bounded variation. Additionally, we identify sufficient conditions for minimality by employing outer variations and applying the formula for the first derivative of the anisotropic perimeter.