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

扩展有限元法（X-FEM）是由Belytcheko等人于1996年首次提出的用于线弹性断裂力学的扩展有限元法，以避免裂纹扩展过程中裂纹尖端附近的网格重新划分。扩展有限元法的关键思想是在节点连续位移中加入不连续位移的未知变量，使裂纹可以跨单元扩展。然而，该方法仅限于线弹性材料中的裂纹。为了分析岩土材料的断裂，提出了一种适用于弹塑性材料的扩展有限元法。本文分析了带中心裂纹矩形板中Drucker-Prager弹塑性材料在两轴向压缩载荷作用下裂纹尖端附近的应力分布。采用隐式返回映射算法对弹塑性材料进行分析，该算法采用Newton-Raphson方法的增量迭代方法，不仅精度高，而且大大减少了CPU时间。在压缩载荷作用下，沿裂纹面存在摩擦力。在求解这一接触问题时，我们引入了一种符合扩展有限元法的新方法。在滑移准则上假定摩擦力遵循库仑定律的弹塑性材料。在隐式返回映射算法的背景下分析了摩擦问题。因此，接触问题和弹塑性材料成为牛顿-拉夫逊法同一隐式格式的增量与迭代相结合的同时求解方法。这使得算法非常简单。简单X-FEM计算得到的裂纹尖端附近应力分布与弹塑性材料经典有限元解的可靠解吻合较好。因此，对于未来，我们期待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.