论文标题
$γ$ - 线性弹性中的自由诊断问题：同质化和放松
$Γ$-convergence for free-discontinuity problems in linear elasticity: Homogenization and relaxation
论文作者
论文摘要
我们分析了与表面不连续性的线性弹性固体建模时产生的自由透视功能序列的$γ$ - 连接，包括现象为断裂，损伤或物质空隙。我们证明了相对于$γ$ - convergence的紧凑性，并代表$γ$ - 限制的限制形式，该形式在有界变形的广义特殊功能（$ GSBD^p $）上定义的积分形式。我们从渐近细胞公式来识别积分，并证明体积和表面贡献之间的非相互作用。最终，我们研究了相应的边界价值问题的序列，并显示最小值和最小化器的收敛性。特别是，我们的技术可以表征$ GSBD^p $上功能的放松，并涵盖经典的定期均质化案例。
We analyze the $Γ$-convergence of sequences of free-discontinuity functionals arising in the modeling of linear elastic solids with surface discontinuities, including phenomena as fracture, damage, or material voids. We prove compactness with respect to $Γ$-convergence and represent the $Γ$-limit in an integral form defined on the space of generalized special functions of bounded deformation ($GSBD^p$). We identify the integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions. Eventually, we investigate sequences of corresponding boundary value problems and show convergence of minimum values and minimizers. In particular, our techniques allow to characterize relaxations of functionals on $GSBD^p$, and cover the classical case of periodic homogenization.
