We investigate a quasi-static tensile fracture in nonlinear strain-limiting solids by coupling with the phase-field approach. A classical model for the growth of fractures in an elastic material is formulated in the framework of linear elasticity for deformation systems. This linear elastic fracture mechanics (LEFM) model is derived based on the assumption of small strain. However, the boundary value problem formulated within the LEFM and under traction-free boundary conditions predicts large singular crack-tip strains. Fundamentally, this result is directly in contradiction with the underlying assumption of small strain. In this work, we study a theoretical framework of nonlinear strain-limiting models, which are algebraic nonlinear relations between stress and strain. These models are consistent with the basic assumption of small strain. The advantage of such framework over the LEFM is that the strain remains bounded even if the crack-tip stress tends to the infinity. Then, employing the phase-field approach, the distinct predictions for tensile crack growth can be governed by the model. Several numerical examples to evaluate the efficacy and the performance of the model and numerical algorithms structured on finite element method are presented. Detailed comparisons of the strain, fracture energy with corresponding discrete propagation speed between the nonlinear strain-limiting model and the LEFM for the quasi-static tensile fracture are discussed.

我们通过与相场方法耦合研究非线性应变限制固体中的准静态拉伸断裂。在变形系统的线性弹性框架中制定了弹性材料中裂缝增长的经典模型。这种线弹性断裂力学 (LEFM) 模型是基于小应变假设推导出来的。然而，在 LEFM 和无牵引边界条件下制定的边界值问题预测大的奇异裂纹尖端应变。从根本上说，这个结果与小应变的基本假设直接矛盾。在这项工作中，我们研究了非线性应变限制模型的理论框架，这是应力和应变之间的代数非线性关系。这些模型符合小应变的基本假设。这种框架相对于 LEFM 的优势在于，即使裂纹尖端应力趋于无穷大，应变仍保持有界。然后，采用相场方法，可以通过模型控制拉伸裂纹扩展的不同预测。给出了几个数值例子来评估模型的有效性和性能，以及基于有限元方法构建的数值算法。讨论了非线性应变限制模型和准静态拉伸断裂的 LEFM 之间应变、断裂能量与相应离散传播速度的详细比较。该模型可以控制对拉伸裂纹扩展的不同预测。给出了几个数值例子来评估模型的有效性和性能，以及基于有限元方法构建的数值算法。讨论了非线性应变限制模型和准静态拉伸断裂的 LEFM 之间应变、断裂能量与相应离散传播速度的详细比较。该模型可以控制对拉伸裂纹扩展的不同预测。给出了几个数值例子来评估模型的有效性和性能，以及基于有限元方法构建的数值算法。讨论了非线性应变限制模型和准静态拉伸断裂的 LEFM 之间应变、断裂能量与相应离散传播速度的详细比较。