In phase-field models the damage evolution problem is considered as a minimisation problem of a Griffith-like energy functional, governed by the principles of irreversibility, stability and energy balance. Herein, we consider phase-field models characterised by different degradation and energy dissipation functions. With a proper choice for the characteristic functions in phase-field models, it is possible to reproduce cohesive fracture in a one-dimensional setting. We consider a one-dimensional bar with stress softening, which exhibits homogeneous deformations provided that the length of the bar length is below a state-dependent critical value. Otherwise, the bar will lose stability and show a localised response. It appears that the phase-field method can partially reproduce the response of a cohesive zone model, for instance the traction–separation law, but not all aspects of the model, like the dissipated energy. For a one-dimensional problem, the crack nucleation load varies smoothly from that predicted by a strength criterion to that of a toughness criterion for different lengths of the bar. We have compared the one-dimensional results with the numerical solutions in a two-dimensional setting, which yielded a very good agreement.

在相场模型中，损伤演化问题被认为是类格里菲斯能量泛函的最小化问题，受不可逆性、稳定性和能量平衡的原则支配。在这里，我们考虑以不同退化和能量耗散函数为特征的相场模型。With a proper choice for the characteristic functions in phase-field models, it is possible to reproduce cohesive fracture in a one-dimensional setting. 我们考虑具有应力软化的一维杆，它表现出均匀变形假设钢筋长度低于状态相关的临界值。否则，条形图将失去稳定性并显示局部响应。相场方法似乎可以部分再现内聚区模型的响应，例如牵引-分离定律，但不是模型的所有方面，如耗散能量。对于一维问题，对于不同长度的钢筋，裂纹形核载荷从强度准则预测的值到韧性准则的预测值平滑变化。我们将一维结果与二维设置中的数值解进行了比较，结果非常吻合。