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 (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. Bouchitte, I. Fonseca and L. Mascarenhas, A global method for relaxation, Arch. Ration. Mech. Anal. 145 (1998), no. 1, 51-98] and a recent Korn-type inequality in GSBD(p), cf. [F. Cagnetti, A. Chambolle and L Scardia, Korn and Poincare-Korn inequalities for functions with a small jump set, preprint (2020)]. Our general strategy also allows to generalize integral representation results in SBDp, obtained in dimension two [S. Conti, M. Focardi and E lurlano, Integral representation for functionals defined on SBDP in dimension two, Arch. Ration. Mech. Anal. 223 (2017), no. 3, 1337-1374], to higher dimensions, and to revisit results in the framework of generalized special functions of bounded variation (GSBV(p)).

Automated summary

This paper proves an integral representation formula for a class of energies defined on the space of generalized special functions of bounded deformation (GSBDP) in arbitrary space dimensions. These functionals are relevant in modeling linear elastic solids with surface discontinuities, such as fracture, damage, surface tension between different elastic phases, or material voids. The approach is based on the global method for relaxation and a recent Korn-type inequality in GSBD(p). Furthermore, the general strategy allows for the generalization of integral representation results in SBDp to higher dimensions and revisiting results in the framework of generalized special functions of bounded variation (GSBV(p)).