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Equilibrium Configurations for Epitaxially Strained Films and Material Voids in Three-Dimensional Linear Elasticity
Archive for Rational Mechanics and Analysis ( IF 2.5 ) Pub Date : 2020-04-25 , DOI: 10.1007/s00205-020-01525-3
Vito Crismale , Manuel Friedrich

We extend the results about the existence of minimizers, relaxation, and approximation proven by Bonnetier and Chambolle (SIAM J Appl Math 62:1093–1121, 2002), Chambolle and Solci (SIAM J Math Anal 39:77–102, 2007) for an energy related to epitaxially strained crystalline films, and by Braides et al. (ESAIM Control Optim Calc Var 13:717–734, 2007) for a class of energies defined on pairs of function-set. We study these models in the framework of three-dimensional linear elasticity, where a major obstacle to overcome is the lack of any a priori assumption on the integrability properties of displacements. As a key tool for the proofs, we introduce a new notion of convergence for $$(d{-}1)$$ ( d - 1 ) -rectifiable sets that are jumps of $${ GSBD}^p$$ GSBD p functions, called $$\sigma ^p_{\mathrm{sym}}$$ σ sym p -convergence.

中文翻译:

三维线弹性外延应变膜和材料空隙的平衡构型

我们扩展了 Bonnetier 和 Chambolle (SIAM J Appl Math 62:1093–1121, 2002)、Chambolle 和 Solci (SIAM J Math Anal 39:77–102, 2007) 证明的关于极小化、松弛和近似的存在的结果,用于与外延应变晶体薄膜相关的能量,由 Braides 等人提出。(ESAIM Control Optim Calc Var 13:717–734, 2007)用于定义在函数集对上的一类能量。我们在三维线性弹性的框架内研究这些模型,其中需要克服的一个主要障碍是缺乏关于位移的可积性属性的任何先验假设。作为证明的关键工具,我们为 $$(d{-}1)$$ ( d - 1 ) - 可纠正集合引入了一个新的收敛概念,这些集合是 $${ GSBD}^p$$ GSBD p 的跳跃函数,称为 $$\sigma ^p_{\mathrm{sym}}$$ σ sym p -convergence。
更新日期:2020-04-25
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