基于最短路径的三维网格曲面裂纹生成模拟

裂纹模拟已经成为计算机图形学的研究热点之一，本文引入断裂力学中裂纹尖端应力场的概念，从力和能量的角度研究脆性材料裂纹的扩展过程。首先，分析了复合型裂纹尖端应力场在二维极坐标系下的分布特点，运用最大周向应力准则和Griffith能量释放率准则推导出裂纹扩展的临界条件。其次，提出广义极坐标系，采用源点到三维网格模型各顶点的测地距离作为极径，以便保证获取到网格模型各顶点的准确位置信息，然后将三维网格模型的几何信息转化到广义极坐标系。最后，根据裂纹扩展的临界条件，在恒载荷作用下确定裂纹扩展的方向，分析裂纹扩展过程中能量和速率的变化，运用Dijkstra算法追踪裂纹在三维网格模型上的扩展路径，实现了三维网格结构上曲面裂纹的模拟。

关键词：断裂力学；应力场；广义极坐标；精确测地路径
      
Abstract:
The crack simulation has become one of the hot topics in computer graphics. In this paper, the concept of crack tip stress field in fracture mechanics is introduced, and the crack propagation process of brittle materials is studied from the angle of force and energy. First, the distribution characteristics of the stress field of the composite crack tip in the two-dimensional polar coordinate system are analyzed. The critical conditions of the crack propagation are derived by using the maximum circumferential stress criterion and the Griffith energy release rate criterion. Secondly, the generalized polar coordinate system is proposed, which uses the geodesic distance of each vertex from the source point to the 3D mesh model as the pole path, so as to ensure the accurate location information of each vertex of the grid model, and then transform the geometric information of the 3D mesh model into the generalized polar coordinate system. Finally, according to the critical condition of crack propagation, the direction of crack propagation is determined under the constant load, the change of energy and speed in the process of crack propagation is analyzed. The Dijkstra algorithm is used to trace the propagation path of the crack on the 3D mesh model, and the simulation of the surface crack on the three-dimensional grid structure is realized.

Key words: fracture mechanics; stress field; generalized polar coordinates; exact geodesic path

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