分形裂纹的断裂参数

Fracture Parameters of Fractal Cracks

基于分形几何学,推导了含Ⅰ型分形单裂纹的无限大平板在单向受拉时的临界开裂应力、断裂韧性、裂纹扩展力和断裂能的理论公式.以标准Koch分形曲线构造裂纹的边界,并由分形区域周长与面积的关系推导出平板的弹性应变能.利用Griffith断裂准则得到材料开裂的临界应力,并在此基础上给出了断裂韧性、裂纹扩展力以及断裂能的表达式,从而将G判据推广到分形裂纹情形,并对其进行数值计算以分析各参数的影响.结果表明:材料的断裂韧性与裂纹长度呈反向关系;裂纹的量测尺度将影响断裂韧性与分形维数的关系;裂纹越粗糙、长度越长,材料的断裂能越大.文中还证明了裂纹扩展的非光滑性.

In this paper, first, four parameters including the critical fracture stress, the fracture toughness, the crack extension force and the fracture energy of an infinite plate with a solitary fractal crack of mode I under one- way tension are investigated based on the fraetal geometry. Next, the boundary of the crack domain is constructed with standard Koch fractal curve, and the elastic strain energy of the plate is deduced according to the perimeter- area relationship of the fractal crack domain. Then, the critical fracture stress of the plate is obtained according to the Griffith fracture criterion, and theoretical equations of the fracture toughness, the crack extension force and the fracture energy are consequently derived, which are utilized to generalize the G criterion to the fractal crack case. Finally, numerical simulations are conducted to analyze the influences of various factors on the four above-mentioned parameters. The results indicate that the fracture toughness of the material negatively correlates with the crack length, that the relationship between the fracture toughness and the fractal dimension varies with the yardstick length, and that the fracture energy increases with the length and roughness of the crack. In addition, the growth of a crack is proved unsmooth in the paper.