Extensive experiments over the decades unequivocally point to a pronounced scale-dependency of plastic deformation in metals. This observation is fairly general, and broadly speaking, strengthening against deformation is observed with the decrease in the size of a relevant geometrical feature of the material, e.g., the thickness of a thin film. The classical theory of plasticity is size-independent, and this has spurred extensive research into an appropriate continuum theory to elucidate the observed size effects. This pursuit has led to the emergence of strain gradient plasticity, along with its numerous variants, as the paradigm of choice. Recognizing the constrained shear of a thin metallic film as the model problem to understand the observed size-effect, all conventional (and reasonable candidate) theories of strain gradient plasticity predict a scaling of yield strength that inversely varies with the film thickness . Experimental findings indicate a considerably diminished scaling, the magnitude of which can exhibit significant variation based on processing conditions or even the mode of deformation. As an example, the scaling exponent as low as has been observed for as-deposited copper thin films. Two perspectives have been posited to explain this perplexing anomaly. Kuroda and Needleman (2019) argue that the conventional boundary conditions used in strain gradient plasticity theory are not meaningful for the canonical constrained thin film problem and propose a physically motivated alternative. Dahlberg and Ortiz (2019) contend that the intrinsic differential calculus structure of all strain gradient plasticity theories will invariably lead to the incorrect (or rather inadequate) explanation of the size-scaling. They propose a fractional strain gradient plasticity framework where the fractional derivative order is a material property that correlates with the scaling exponent. In this work, we present an alternative approach that complements the existing explanations. We create a statistical mechanics model for interacting microscopic units that deform and yield with the rules of , and plastic yielding is treated as a phase transition. We coarse-grain the model to precisely elucidate the microscopic interactions that can lead to the emergent size-effects observed experimentally. Specifically, we find that depending on the nature of the long-range microscopic interactions, the emergent coarse-grained theory can be of fractional differential type or alternatively a form of integral nonlocal model. Our theory, therefore, provides a partial (and microscopic) justification for the fractional derivative model and makes clear the precise microscopic interactions that must be operative for a continuum plasticity theory to be a valid phenomenological descriptor for capturing the correct size-scale dependency.

中文翻译：

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可塑性统计力学：阐明异常尺寸效应和涌现的分数非局部连续体行为


几十年来的大量实验明确指出金属塑性变形具有明显的尺度依赖性。这一观察结果相当普遍，并且从广义上讲，随着材料相关几何特征尺寸（例如薄膜厚度）的减小，观察到抗变形强化。经典的可塑性理论与尺寸无关，这促使人们对适当的连续介质理论进行广泛的研究，以阐明观察到的尺寸效应。这种追求导致了应变梯度可塑性及其众多变体的出现，作为选择的范式。认识到薄金属薄膜的约束剪切是理解观察到的尺寸效应的模型问题，所有传统的（和合理的候选）应变梯度塑性理论都预测屈服强度的缩放与薄膜厚度成反比。实验结果表明，缩放比例显着减少，其大小可能会根据加工条件甚至变形模式而表现出显着变化。例如，缩放指数与沉积铜薄膜观察到的一样低。人们提出了两种观点来解释这种令人困惑的异常现象。 Kuroda 和 Needleman（2019）认为，应变梯度塑性理论中使用的传统边界条件对于规范约束薄膜问题没有意义，并提出了一种物理驱动的替代方案。 Dahlberg 和 Ortiz (2019) 认为，所有应变梯度塑性理论的内在微分计算结构都将不可避免地导致对尺寸缩放的错误（或者说不充分）解释。 他们提出了一种分数应变梯度塑性框架，其中分数导数阶数是与比例指数相关的材料属性。在这项工作中，我们提出了一种替代方法来补充现有的解释。我们为相互作用的微观单元创建了一个统计力学模型，这些单元按照 的规则变形和屈服，并且塑性屈服被视为相变。我们对模型进行粗粒度化，以精确阐明可能导致实验观察到的出现的尺寸效应的微观相互作用。具体来说，我们发现，根据长程微观相互作用的性质，新兴的粗粒度理论可以是分数阶微分类型，也可以是积分非局部模型的形式。因此，我们的理论为分数阶导数模型提供了部分（和微观）合理性，并明确了连续塑性理论必须有效的精确微观相互作用，才能成为捕获正确尺寸尺度依赖性的有效现象学描述符。