应变梯度塑性作为混合生长非线性位错能的$Γ$极限
Strain-Gradient Plasticity as the $Γ$-Limit of a Nonlinear Dislocation Energy with Mixed Growth
摘要：
本文利用$Γ$ -收敛方法，从无限长圆柱晶体位错的半离散模型导出了宏观应变梯度塑性。在现有的工作相比，我们认为能源次二次增长接近的位错。这允许处理存储的弹性能量，而不需要引入特定的截止半径。作为证明补紧性的主要工具，我们给出了非零$\operatorname{curl}$域的几何刚性结果的一个推广。其中一个主要的组成部分是对发散在临界索伯列夫空间中的$L^1$ -函数的精细分解结果。
英文摘要：
In this paper a we derive by means of $Γ$-convergence a macroscopic strain-gradient plasticity from a semi-discrete model for dislocations in an infinite cylindrical crystal. In contrast to existing work, we consider an energy with subquadratic growth close to the dislocations. This allows to treat the stored elastic energy without the need to introduce an ad-hoc cut-off radius. As the main tool to prove a complementing compactness statement, we present a generalized version of the geometric rigidity result for fields with non-vanishing $\operatorname{curl}$. A main ingredient is a fine decomposition result for $L^1$-functions whose divergence is in certain critical Sobolev spaces.
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