裂纹扩展分析的无网格流形方法

运用无网格流形方法求解裂纹扩展问题。该方法利用单位分解法和有限覆盖技术建立形函数。形函数的建立不受域内不连续面的影响,可较好地求解裂纹问题。尤其当这种不连续面变得复杂时,更能显示该方法的优点。对于应变局部化问题,该方法的形函数构造较其他方法更为有效,避免了其他方法在建立试函数时不能考虑不连续尖端的缺点。与传统的数值流形方法相比,无网格流形方法的有限覆盖形状更加灵活。它可以用一系列节点的影响域来建立有限覆盖和单位分解函数,具有无网格方法的特性,从而摆脱了传统数值流形方法中网格所带来的困难。与目前的无网格方法相比,由于采用了有限覆盖技术,试函数的构造不受域内不连续面的影响,克服了原有的无网格方法在处理不连续问题时所遇到的困难。在求解裂纹扩展问题时,弹性力学基本方程的弱形式采用加权残数法获得。最后利用无网格流形方法追踪岩体试件在复杂应力状态下裂纹扩展过程。在此给出了数值算例,并将计算结果与实验结果进行对比,说明该方法的正确性和可行性。

A MESHLESS MANIFOLD METHOD FOR CRACK PROPAGATION ANALYSIS

A meshless manifold method (MMM) is presented to analyze the problems of crack propagation, especially the advantages of MMM for irregular cracks. The shape functions in this method are formed by the partition of unity and the finite cover technology,so the shape functions are not affected by discontinuous domains and crack problems can be more properly treated. For strain localization problems,the shape functions can be more effectively established,compared with other methods in which the tips of the discontinuous cracks are not considered. Compared with the conventional numerical manifold method,the shapes of the finite covers can be selected more easily. The finite covers and the partition of unity functions are formed by using the influence domains of a series of nodes with an advantages over the mesh-based numerical manifold method. Compared with the conventional meshless methods,the test functions are not influenced by the discontinuities in the solution domain since finite cover technology is used to overcome some difficulties inherented in the conventional meshless methods. In this paper,the meshless manifold method (MMM) is applied to analyze crack growth in rock samples. The weak solution of the partial differential equation for elasticity are derived using the method of weighted residuals (MWR). Finally,a problem with crack growth under complex stress state is solved with the MMM,the numerical results agree well with the test data,and the validity and accuracy of the MMM are demonstrated.