线性无关高阶数值流形法在断裂力学中的应用

APPLICATION OF THE LINEARLY INDEPENDENT HIGH-ORDER NUMERICAL MANIFOLD METHOD IN FRACTURE MECHANICS

数值流形方法(NMM)实现了对连续和非连续问题的统一求解,非常适合应用于求解断裂力学问题。但传统高阶NMM,例如采用一次多项式作为其局部位移函数时,存在线性相关问题,这又在一定程度上限制了NMM的进一步发展和应用。因此,通过在物理片上引入一种新的局部位移函数以及用于模拟裂纹尖端应力场奇异性的位移函数,建立新的NMM求解体系,尝试消除线性相关问题和求解线弹性断裂力学问题。数值算例结果表明:(1)该方法有效地解决了线性相关问题。(2)对于典型的线弹性断裂力学问题,即便在网格密度较小时也能够精确地计算出裂纹尖端的应力强度因子。(3)研究区域内插值点处的应力是连续的。(4)定义在非奇异物理片上自由度具有明确的物理含义,且第3到5个自由度恰好是所对应插值点处的应变分量,可直接获得此处的应力,减少了计算量。最后,所建议方法可以很容易地推广到其他基于单位分解理论的方法中。

The numerical manifold method(NMM) has succeeded in providing a unified solution to continuum and discontinuum problems and therefore it is highly suitable for solving fracture mechanics problems. However, the conventional high-order NMM using the first order polynomial as the local displacement function has the problem of linear dependence, which restricts to a certain degree its further development and application. A new NMM framework was established in this research by introducing a new localized displacement function, as well as a special displacement function for modeling the stress singularity around crack tips. A new paradigm that eliminates the problem of linear dependence is then derived to solve linear elastic fracture mechanics problems. The numerical examples show that: (1) The proposed method successfully eliminates the problem of linear dependence;(2) For classic linear elastic fracture problems, the stress intensity factors at the crack tip can be calculated accurately even if the mesh is relatively sparse;(3) The stress function at interpolation points inside the physical domain is continuous;(4) All the degrees of freedom defined on non-singular physical patches are physically meaningful, with the third to the fifth being the strain components at the interpolation point of the patch. As a result, the stress components at the interpolation point can be directly obtained. Finally, the proposed approach can be easily extended to other methods based on the theory of the partition of unity. © 2015, Science Press. All right reserved.