Integral representation for energies in linear elasticity with surface discontinuities

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 GSBDp) 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 [G. Bouchitté, I. Fonseca and L. Mascarenhas, A global method for relaxation, Arch. Ration. Mech. Anal. 145 1998, 1, 51-98] and a recent Korn-type inequality in GSBD p GSBDp, cf. [F. Cagnetti, A. Chambolle and L. Scardia, Korn and Poincaré-Korn inequalities for functions with a small jump set, preprint 2020]. Our general strategy also allows to generalize integral representation results in SBD pSBDp, obtained in dimension two [S. Conti, M. Focardi and F. Iurlano, Integral representation for functionals defined on SBD p SBDp in dimension two, Arch. Ration. Mech. Anal. 223 2017, 3, 1337-1374], to higher dimensions, and to revisit results in the framework of generalized special functions of bounded variation (GSBV pGSBVp).