SBH-ellipticity of the relaxed interfacial energy density in the context of second-order structured deformations
A. C. Barroso, J. Matias, E. Zappale
TL;DR
This work addresses the relaxation of a bulk-plus-surface energy defined on second-order structured deformations in the $SBH$ setting. Employing the global method for relaxation, it proves that the relaxed functional admits an integral representation with densities $f$ and $g$, and that the relaxed surface density $g$ satisfies an $SBH$-ellipticity condition. The results establish linear growth bounds for the relaxed densities and connect the interfacial energy to a convexity-type property, enhancing the mathematical foundation for multi-scale material models involving bending and disarrangements. These findings align with and extend previous first-order ellipticity results, providing a rigorous tool for analyzing phase transitions and interfacial energies in complex materials.
Abstract
Starting from an energy comprised of both a bulk term and a surface term, set in the space of special functions of bounded hessian, $SBH$, a relaxation problem in the context of second-order structured deformations was studied in Fonseca-Hagerty-Paroni. It was shown, via the global method for relaxation, that the relaxed functional admits an integral representation and the relaxed energy densities were identified. In this paper we show that, under certain hypotheses on the original densities, the corresponding relaxed energy densities verify the same type of growth conditions and the surface energy density satisfies a specific ``convexity-type'' property, i.e. it is $SBH$-elliptic.