Fracture failure is a major concern in mechanical engineering, particularly for piezoelectric materials. In contrast to other numerical methods, phase field method has significant advantages in addressing fracture progress. It can automatically track crack surfaces through ordered parameter evolution, which is versatile for modeling complex fracture behaviors. However, previous studies on phase field fracture in piezoelectric solids have primarily focused on brittle ceramics with small deformation. In recent years, flexible piezoelectrics with high stretchability have already been achieved in industrial production. These materials exhibit obvious nonlinear characteristics during deformation, which renders the traditional assumption of small deformations inadequate for predicting their fracture behaviors. In this work, we propose a finite deformation phase field fracture model for flexible piezoelectric materials, building upon the established nonlinear electromechanical material model. The numerical framework is carried out in the commercial software ABAQUS via a user element subroutine. Both single–pass staggered algorithm (SPSA) and residual-controlled staggered algorithm with even-odd iteration split (RCSA-EO) are employed to solve coupled electro-mechanical-phase field governing equations. The proposed model is validated through comparisons with analytical solutions and existing literature. Moreover, the developed numerical framework effectively explains the nonlinear fracture behavior observed in experiments conducted on Polyvinylidene fluoride (PVDF), a flexible piezoelectric material with a large failure strain. Numerical simulations are also performed to demonstrate the influence of the applied electric field on electromechanical fracture behavior. The results highlight that the specific impact of electric fields depends on material parameters, geometric parameters, and boundary conditions. The developed model is capable of making accurate and realistic predictions of fracture in flexible piezoelectric materials. This is particularly important for evaluating the reliability and safety of flexible piezoelectric devices.

中文翻译：

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柔性压电材料机电断裂的有限变形相场模型

断裂失效是机械工程中的一个主要问题，特别是对于压电材料。与其他数值方法相比，相场方法在解决断裂进展方面具有显着的优势。它可以通过有序参数演化自动跟踪裂纹表面，这对于复杂断裂行为的建模来说是通用的。然而，以往对压电固体相场断裂的研究主要集中在小变形的脆性陶瓷上。近年来，具有高拉伸性的柔性压电材料已经实现工业化生产。这些材料在变形过程中表现出明显的非线性特征，这使得传统的小变形假设不足以预测其断裂行为。在这项工作中，我们基于已建立的非线性机电材料模型，提出了柔性压电材料的有限变形相场断裂模型。数值框架是在商业软件 ABAQUS 中通过用户元素子程序执行的。采用单通道交错算法（SPSA）和偶奇迭代分裂的残差控制交错算法（RCSA-EO）来求解耦合机电相场控制方程。通过与解析解和现有文献的比较来验证所提出的模型。此外，所开发的数值框架有效地解释了在聚偏二氟乙烯（PVDF）（一种具有大失效应变的柔性压电材料）上进行的实验中观察到的非线性断裂行为。还进行了数值模拟，以证明所施加的电场对机电断裂行为的影响。结果强调，电场的具体影响取决于材料参数、几何参数和边界条件。开发的模型能够对柔性压电材料的断裂进行准确而现实的预测。这对于评估柔性压电器件的可靠性和安全性尤为重要。