Phase-field methods for fracture have been integrated with plasticity for better describing constitutive behaviours. In most of the previous phase-field models, however, the length-scale parameter must be interpreted as a material property in order to match the material strength in experiments. This study presents a phase-field model for fracture coupled with plasticity for quasi-brittle materials with emphasis on insensitivity of the length-scale parameter. The proposed model is formulated using variational principles and implemented numerically in the finite element framework. The effective yield stress is calibrated to vary with the length-scale parameter such that the tensile strength remains the same. Moreover, semi-analytical solutions are derived to demonstrate that the length-scale parameter has a negligible effect on the stress–displacement curve. Five representative examples are considered here to validate the phase-field model for fracture in quasi-brittle materials. The simulated force–displacement curves and crack paths agree well with the corresponding experimental results. Importantly, it is found that the global structural response is insensitive to the length scale though it may influence the size of the failure zone. In most cases, a large length-scale parameter can be used for saving the computational cost by allowing the use of a coarse mesh. On the other hand, a sufficiently small length-scale parameter can be selected to prevent overly diffusive damage, making it possible for the proposed phase-field model to simulate the fracture behaviour with $$ \varGamma $$ -convergence.

中文翻译：

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弹塑性固体中的相场断裂：准脆性材料的长度尺度不敏感模型

断裂的相场方法已与塑性相结合，以更好地描述本构行为。然而，在大多数以前的相场模型中，长度尺度参数必须被解释为材料属性，以便在实验中匹配材料强度。本研究提出了一种用于准脆性材料的断裂与塑性相结合的相场模型，重点是长度尺度参数的不敏感性。所提出的模型是使用变分原理制定的，并在有限元框架中以数值方式实现。有效屈服应力被校准为随长度尺度参数而变化，从而使抗拉强度保持不变。而且，推导出半解析解来证明长度尺度参数对应力-位移曲线的影响可以忽略不计。这里考虑了五个有代表性的例子来验证准脆性材料断裂的相场模型。模拟的力-位移曲线和裂纹路径与相应的实验结果非常吻合。重要的是，发现整体结构响应对长度尺度不敏感，尽管它可能会影响破坏区的大小。在大多数情况下，通过允许使用粗网格，可以使用大的长度尺度参数来节省计算成本。另一方面，可以选择足够小的长度尺度参数来防止过度扩散损伤，