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.

中文翻译：

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量子化塑性变形


在工程中，晶体塑性非弹性机制对应于宏观应变空间中的张量零能谷。这种山谷的平坦性质与模仿晶格不变剪切的塑性滑移本质上是离散的这一事实相矛盾。最近在晶体塑性的介观张量模型（MTM）中实现了协调，该模型引入了周期性调制的能谷，同时还以几何精确的方式捕获了塑性滑移的晶体学特定方面。在本文中，我们通过明确引入塑性变形的概念，扩展了 MTM 框架，其原始形式具有离散非线性弹性理论的外观。随后的模型包含一个新颖的矩阵值自旋变量，表示量化的塑性畸变，其与速率无关的演化可以通过离散（准）自动机来描述。所提出的 MTM 重新表述可显着提高与使用稳健且高效的混合高斯-牛顿-柯西能量最小化算法相关的计算速度。为了说明新方法的有效性，我们提出了一个详细的案例研究，重点关注经典连续介质理论无法达到的晶体可塑性方面。因此，我们提供了令人信服的证据，表明重新制定的 MTM 完全足以处理准静态载荷下塑性响应的间歇性。 特别是，我们的数值实验表明，与二维方形晶体中的塑性屈服相关的位错雪崩的统计数据表现出幂律尾部，其临界指数与一般理论考虑所预测的值相匹配，并且也在离散位错中独立观察到。动力学（DDD）模拟。