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 Bouchitte 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 ($GSBV^p$).

中文翻译：

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具有表面不连续性的线性弹性能量的积分表示

在本文中，我们证明了定义在任意空间维度的有界变形的广义特殊函数（$GSBD^p$）空间上的一类一般能量的积分表示公式。这种类型的泛函在具有表面不连续性的线性弹性固体建模中自然出现，包括断裂、损坏、不同弹性相之间的表面张力或材料空隙等现象。我们的方法基于 Bouchitte 等人设计的全局松弛方法。'98 和最近在 $GSBD^p$ 中的 Korn 型不等式（Cagnetti-Chambolle-Scardia '20）。我们的一般策略还允许将在第二维 (Conti-Focardi-Iurlano '16) 中获得的 $SBD^p$ 中的积分表示结果推广到更高的维度，