In this paper, a novel phase field (PF) model for fracture is developed in the framework of strain gradient elasticity. The strain energy decomposition methods initially proposed for linear elastic fracture problems are extended to the gradient elasticity situation. The PF model is numerically implemented through Abaqus subroutine UEL (user defined element) with nine-node quatrilateral C¹-continous element. The finite element implementation is validated by studying the stress field near the static crack tip. When the length-scale parameter of the gradient elasticity is zero, the stress field is consistent with the analytical solution predicted by linear elastic fracture mechanics (LEFM). For non-zero length-scale parameters, the Cauchy stress at the crack tip is found to be non-singular and the numerical results exhibit less mesh sensitivity in the crack tip region compared with the linear elastic case. After that, four different strain energy decomposition methods (Method I-IV) are compared and discussed by studying Mode I fracture behavior under tensile load. It is found that Method I can properly characterize the toughening effect of gradient elasticity. Based on Method I, the influences of the fracture length-scale parameter l_c and the strain gradient length-scale parameter l on the characteristics of the smeared crack model and the load–displacement curves are systematically discussed. The specimen size effect of Mode I crack propagation is also investigated. It is found that the length-scale parameter l has a larger influence on the load–displacement curve as the specimen size decreases. Finally, crack propagation behavior in pure shear test, three point bending test and tensile test of a notched plate with hole is investigated. It is demonstrated that the proposed PF model can well predict the curvilinear crack propagation path and properly characterize the effect of gradient elasticity. Thus, the PF model developed in this paper can be applied to study complex fracture behaviors in the framework of gradient elasticity, where the effect of internal structure on the mechanical responses become significant.

在本文中，在应变梯度弹性的框架内开发了一种新型的断裂相场（PF）模型。最初为线弹性断裂问题提出的应变能分解方法扩展到梯度弹性情况。PF 模型通过 Abaqus 子程序 UEL（用户定义单元）与九节点四边形 C ^{1 进行}数值实现- 连续元素。通过研究静态裂纹尖端附近的应力场来验证有限元实现。当梯度弹性的长度尺度参数为零时，应力场与线弹性断裂力学（LEFM）预测的解析解一致。对于非零长度尺度参数，裂纹尖端的柯西应力被发现是非奇异的，与线弹性情况相比，数值结果显示裂纹尖端区域的网格敏感性较低。之后，通过研究拉伸载荷下的模式 I 断裂行为，比较和讨论了四种不同的应变能分解方法（方法 I-IV）。发现方法Ⅰ可以很好地表征梯度弹性的增韧作用。基于方法一，裂缝长度尺度参数的影响l _c和应变梯度长度尺度参数l对涂抹裂纹模型和载荷-位移曲线的特性进行了系统讨论。还研究了模式 I 裂纹扩展对试样尺寸的影响。发现长度尺度参数l随着试样尺寸的减小，对载荷-位移曲线的影响更大。最后，研究了带孔缺口板在纯剪切试验、三点弯曲试验和拉伸试验中的裂纹扩展行为。结果表明，所提出的 PF 模型可以很好地预测曲线裂纹扩展路径并正确表征梯度弹性的影响。因此，本文开发的 PF 模型可用于研究梯度弹性框架下的复杂断裂行为，其中内部结构对力学响应的影响变得显着。