This article proposes a micropolar phase-field model for size-dependent brittle fracture in solids under electro-mechanical loading conditions. Considering displacement, micro-rotation, electric potential, and phase-field variable as the kinematic descriptors and employing the virtual power principle, we derive a set of coupled governing partial differential equations (PDEs) for size-dependent solids. Invoking the first and second laws of thermodynamics, we determine the constitutive relations for the thermodynamic fluxes. Carrying out the finite element implementation of the derived governing PDEs using the open-source Gridap package in Julia, we demonstrate the efficacy of the proposed phase-field model through a few representative numerical examples. Especially the importance of the proposed model in incorporating the effect of relative rotation, i.e., the difference between macro- and micro-rotation, on the response of solids under electro-mechanical loading is shown that may not be possible with the existing non-local models such as strain-gradient or couple-stress approaches. To capture the experimentally observed size effects in solids under electro-mechanical loading, the proposed model does not demand higher-order continuity of the field variables, unlike a typically used strain gradient model. To demonstrate the efficacy of the proposed model, we have compared our results against demanding experimental and numerical benchmark results available in the literature. We provide a parametric study to unravel the effect of different micropolar material parameters on the electro-mechanical response of a brittle solid. Interestingly, the proposed micropolar model is less sensitive to the phase-field length scale than the conventional non-polar phase-field models.

中文翻译：

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尺寸依赖性机电断裂的微极相场模型


本文提出了一种微极相场模型，用于在机电载荷条件下固体中尺寸依赖性的脆性断裂。以位移、微旋转、电势和相场变量为运动学描述符，并采用虚功率原理，我们推导出了一组与尺寸相关的固体的耦合控制偏微分方程 （PDE）。调用热力学的第一定律和第二定律，我们确定了热力学通量的本构关系。使用 Julia 中的开源 Gridap 包对派生的治理 PDE 进行有限元实现，我们通过一些具有代表性的数值示例证明了所提出的相场模型的有效性。特别是所提出的模型在考虑相对旋转（即宏观和微观旋转之间的差异）对机电载荷下固体响应的影响方面的重要性，表明了现有的非局部模型（如应变梯度或耦合应力方法）可能无法实现。为了捕捉在机电载荷下实验观察到的固体尺寸效应，与通常使用的应变梯度模型不同，所提出的模型不需要场变量的高阶连续性。为了证明所提出的模型的有效性，我们将结果与文献中要求苛刻的实验和数值基准结果进行了比较。我们提供了一项参数研究，以揭示不同微极材料参数对脆性固体机电响应的影响。有趣的是，与传统的非极性相场模型相比，所提出的微极模型对相场长度尺度的敏感性较低。