量子化塑性变形
Quantized plastic deformation
AI总结：
本文扩展介观张量模型MTM，引入矩阵值自旋变量与离散准自动机描述量子化塑性变形，并以混合最小化算法加速，再现二维晶体位错雪崩的临界幂律统计。
摘要：
在工程晶体塑性中，非弹性机制对应于宏观应变空间中的张量型零能谷。这些能谷的平坦性质与如下事实相矛盾：模拟晶格不变剪切的塑性滑移本质上是离散的。最近，晶体塑性的介观张量模型（MTM）实现了二者的调和；该模型引入周期性调制的能谷，同时以几何上精确的方式捕捉塑性滑移在晶体学上的特定方面。本文扩展了MTM框架：其原始形式看似离散化的非线性弹性理论，而我们显式引入塑性变形概念。由此产生的模型包含一个新颖的矩阵值自旋变量，用于表示量子化塑性畸变，其率无关演化可由离散（准）自动机描述。所提出的MTM重构带来显著计算加速，这得益于使用稳健且高效的混合Gauss-Newton–Cauchy能量最小化算法。为说明新方法的有效性，我们给出详细案例研究，聚焦经典连续介质理论无法触及的晶体塑性方面。因此，我们提供有力证据，表明重构后的MTM完全足以处理准静态加载下塑性响应的间歇性。特别地，数值实验表明，与二维方形晶体塑性屈服相关的位错雪崩统计呈现幂律尾，其临界指数与一般理论考虑所预测的值相符，也与离散位错动力学（DDD）模拟中独立观测到的结果一致。
英文摘要：
In engineering crystal plasticity inelastic mechanisms correspond to tensorial zero-energy valleys in the space of macroscopic strains. The flat nature of such valleys is in contradiction with the fact that plastic slips, mimicking lattice-invariant shears, are inherently discrete. A reconciliation has recently been achieved in the mesoscopic tensorial model (MTM) of crystal plasticity, which introduces periodically modulated energy valleys while also capturing in a geometrically exact way the crystallographically-specific aspects of plastic slips. In this paper, we extend the MTM framework, which in its original form had the appearance of a discretized nonlinear elasticity theory, by explicitly introducing the concept of plastic deformation. The ensuing model contains a novel matrix-valued spin variable, representing the quantized plastic distortion, whose rate-independent evolution can be described by a discrete (quasi-)automaton. The proposed reformulation of the MTM leads to a considerable computational speedup associated with the use of a robust and efficient hybrid Gauss-Newton--Cauchy energy minimization algorithm. To illustrate the effectiveness of the new approach, we present a detailed case-study focusing on the aspects of crystal plasticity that are beyond reach for the classical continuum theory. Thus, we provide compelling evidence that the re-formulated MTM is fully adequate to deal with the intermittency of plastic response under quasi-static loading. In particular, our numerical experiments show that the statistics of dislocational avalanches, associated with plastic yield in 2D square crystals, exhibits a power-law tail with a critical exponent matching the value predicted by general theoretical considerations and also independently observed in discrete-dislocation-dynamics (DDD) simulations.
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