Explanation of fracture phenomena by crack extension analysis using extended finite element method

扩展有限元法（X-FEM）是Belytcheko等人于1996年首次提出的用于线弹性断裂力学的方法，以避免在裂纹扩展过程中裂纹尖端附近的网格重新划分。扩展有限元法的核心思想是在节点连续位移中加入不连续位移的未知变量，使裂纹能够在单元中扩展。本文针对岩土材料断裂问题，提出了一种适用于弹塑性材料的扩展有限元法（X-FEM）。本文分析了Drucker-Prager弹塑性材料中含中心裂纹的矩形板在双轴向压缩载荷作用下裂纹尖端附近的应力分布。弹塑性材料分析采用隐式返回映射算法，该算法采用Newton-Raphson法的增量迭代法，计算精度高，大大减少了计算时间。在求解该接触问题时，我们引入了一种新的方法，该方法与扩展有限元法相一致。摩擦力假设为弹塑性材料，滑移准则为库仑定律。摩擦问题也在隐式返回映射算法中进行了分析，从而使接触问题和弹塑性材料问题成为Newton-Raphson方法的同一隐式格式的同时结合增量和迭代的方法。计算结果与经典弹塑性有限元解的可靠解吻合得很好。因此，对于未来，我们预计这种新方法的实际应用，如滑坡。

Extended finite element method (X-FEM) was first developed by Belytcheko et al. in 1996 for the linear elastic fracture mechanics to avoid awkward remeshing near a crack tip as the crack grows. The key idea of the extended finite element method is that the unknown variables of discontinuous displacement are added to the nodal continuous displacements so that the crack can extend across the element. The method was, however, restricted only to the cracks in the linear elastic material.With the aim of analyzing the fracture of geomaterials, we develop a new extended finite element method(X-FEM) available to an elastic-plastic material. We analyzes the stress distributions near a crack tip for the Drucker-Prager elastic-plastic material in a rectangular plate with a centered crack under two axial compressive loads. The elastic-plastic material is analyzed by the implicit return mapping algorithm, which gives a high accuracy and much reduces CPU time by using an incremental and iterative method of the Newton-Raphson method.Under the compressive loads there exist friction forces along the crack faces. In solving this contact problem, we introduce a new method which conform to the extended finite element method. The friction forces are assumed to follow the elastic-perfect plastic material with Coulomb law for the slip criterion. The friction problem is also analyzed in the context of an implicit return mapping algorithm.Thus the contact problem and the elastic-plastic material become a simultaneous combining incremental and iterative method of the same implicit scheme of the Newton-Raphson method. This makes the algorithm very simple.The stress distributions near the crack tip obtained by the simple X-FEM agree very well with the reliable solutions in the classical FEM solution for the elastic-plastic material. For the future, therefore, we anticipate a practical application of this new method such as a land slide.