Abstract
Abstract 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 {\mathrm{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 [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 {\mathrm{GSBD}^{p}} , 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 p {\mathrm{SBD}^{p}} , obtained in dimension two [S. Conti, M. Focardi and F. Iurlano, Integral representation for functionals defined on SBD p \mathrm{SBD}^{p} 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 p {\mathrm{GSBV}^{p}} ).
Highlights
Integral representation results are a fundamental tool in the abstract theory of variational limits by Γ-convergence or in relaxation problems
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
In this paper we contribute to this topic by proving an integral representation result for a general class of energies arising in the modeling of linear elastic solids with surface discontinuities
Summary
Integral representation results are a fundamental tool in the abstract theory of variational limits by Γ-convergence or in relaxation problems (see [32]). Already in dimension two, the extension of [28] to the case where only a control of type (2) is available is no straightforward task This is a fundamental difference with respect to the BV-theory where problems for generalized functions of bounded variation can be reconducted to SBV by a perturbation trick (see for instance [21]): one considers a small perturbation of the functional, depending on the jump opening, to represent functionals on SBVp. by letting the perturbation parameter vanish and by truncating functions suitably, the representation can be extended to GSBVp. the trick of reducing problem (2) to (1) is not expedient in the linearly elastic context and does not allow to deduce an integral representation result in GSBDp from the one.
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