One of the fundamental but unsolved problems in the phase-field theory is how to obtain the energy density expression based on an arbitrary cohesive law. In this paper, we propose a phase-field model with the energy density determined by the cohesive law with an arbitrary form, which builds a rigorous link between the phase-field theory and the cohesive zone theory. Using the proposed method, the energy density in the phase-field model for any cohesive law can be determined from a unified formula. As examples, we derive the explicit energy density expressions for the linear, exponential, polynomial, and Cornelissen cohesive laws. Five numerical examples are presented to show the effectiveness of the proposed method, including a uniaxial tension test for mode-I failure, a uniaxial compression test for mode-II failure, a mixed-mode fracture test, the mode-I failure in a wedge splitting test, and the mixed-mode failure of an L-shaped panel. In the first three numerical examples, the predicted mode-I, mode-II and mixed-mode cohesive laws are extracted and compared with the target analytical cohesive laws. Excellent agreements are observed. In the last two numerical examples, the predicted global responses and the crack paths are in good agreement with the experimental results.

相场理论中一个基本但尚未解决的问题是如何基于任意内聚定律获得能量密度表达式。在本文中，我们提出了一种具有由内聚定律以任意形式确定的能量密度的相场模型，该模型在相场理论和内聚区理论之间建立了严格的联系。使用所提出的方法，可以从统一公式确定任何内聚律的相场模型中的能量密度。作为示例，我们导出了线性，指数，多项式和Cornelissen内聚定律的显式能量密度表达式。给出了五个数值示例，以证明所提方法的有效性，包括针对I型破坏的单轴拉伸试验，针对II型破坏的单轴压缩试验，混合模式断裂试验，楔形劈裂测试中的I型故障以及L形面板的混合模式故障。在前三个数值示例中，提取预测的模式I，模式II和混合模式内聚律，并将其与目标分析内聚律进行比较。观察到了很好的协议。在最后两个数值示例中，预测的整体响应和裂纹路径与实验结果非常吻合。