We investigate the conditions for crack nucleation in variational gradient damage models used as phase-field models of brittle and cohesive fracture. Viewing crack nucleation as a structural stability problem, we analyze how solutions with diffuse damage become unstable and bifurcate towards localized states, representing the smeared version of cracks. We consider gradient damage models with a linear softening response, incorporating distinct softening parameters for the spherical and deviatoric modes. These parameters are employed to adjust the peak pressure and shear stress, resulting in an equivalent cohesive behavior. Through analytical and numerical second-order stability and bifurcation analyses, we characterize the crack nucleation conditions in quasi-static, rate-independent evolutions governed by a local energy minimization principle. We assess the stability of crack development, determining whether it is preceded by a stable phase with diffuse damage or not. Our results quantitatively characterize the classical transition between brittle and cohesive-like behaviors. A fully analytical solution for a one-dimensional problem provides a clear illustration of the complex bifurcation and instability phenomena, underpinning their connection with classical energetic arguments. The stability analysis under multi-axial loading reveals a fundamental non-trivial influence of the loading mode on the critical load for crack nucleation. We show that volumetric-dominated deformation mode can remain stable in the softening regime, thus delaying crack nucleation after the peak stress. This feature depends only on the properties of the local response of the material and is insensitive to structural scale effects. Our findings disclose the subtle interplay among the regularization length, the material’s cohesive length-scale, structural size, and the loading mode to determine the crack nucleation conditions and the effective strength of phase-field models of fracture.

中文翻译：

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断裂变分相场模型中的稳定性和裂纹成核：长度尺度和应力多轴性的影响


我们研究了用作脆性和内聚断裂相场模型的变分梯度损伤模型中裂纹成核的条件。将裂纹成核视为结构稳定性问题，我们分析了具有扩散损伤的解决方案如何变得不稳定并向局部状态分叉，代表裂纹的涂抹版本。我们考虑具有线性软化响应的梯度损伤模型，并结合球形和偏模模式的不同软化参数。这些参数用于调整峰值压力和剪切应力，从而产生等效的内聚行为。通过解析和数值二阶稳定性和分岔分析，我们描述了受局部能量最小化原理控制的准静态、速率无关演化中的裂纹成核条件。我们评估裂纹发展的稳定性，确定其之前是否存在具有扩散损伤的稳定阶段。我们的结果定量地描述了脆性行为和内聚性行为之间的经典转变。一维问题的完全解析解提供了复杂分岔和不稳定现象的清晰说明，支撑了它们与经典能量论证的联系。多轴加载下的稳定性分析揭示了加载模式对裂纹形核临界载荷的重要影响。我们表明，体积主导的变形模式可以在软化状态下保持稳定，从而延迟峰值应力后的裂纹形核。该特征仅取决于材料局部响应的特性，并且对结构尺度效应不敏感。 我们的研究结果揭示了正则化长度、材料的内聚长度尺度、结构尺寸和加载模式之间的微妙相互作用，以确定裂纹成核条件和断裂相场模型的有效强度。