Calculus of Variations and Geometric Measure Theory

V. Crismale - M. Friedrich - F. Solombrino

Integral representation for energies in linear elasticity with surface discontinuities

created by solombrin on 14 May 2020
modified on 11 Nov 2020


Published Paper

Inserted: 14 may 2020
Last Updated: 11 nov 2020

Journal: Advances in Calculus of Variations
Year: 2020
Doi: 10.1515/acv-2020-0047

ArXiv: 2005.06866 PDF


In this paper we prove an integral representation formula for a general class of energies defined on the space of generalized special functions of bounded deformation ($GSBD^p$) in arbitrary space dimensions. Functionals of this type naturally arise in the modeling of linear elastic solids with surface discontinuities including phenomena as fracture, damage, surface tension between different elastic phases, or material voids. Our approach is based on the global method for relaxation devised in Bouchittè et al. '98 and a recent Korn-type inequality in $GSBD^p$ (Cagnetti-Chambolle-Scardia '20). Our general strategy also allows to generalize integral representation results in $SBD^p$, obtained in dimension two (Conti-Focardi-Iurlano '16), to higher dimensions, and to revisit results in the framework of generalized special functions of bounded variation ($GSBV^p$).