宏微观双尺度运动裂纹模型面内拉伸下的解析解

Analytical solution of macro/micro dual scales moving crack model under in-plane tension

研究裂纹动态扩展中宏微观因素相互作用机制与微观裂尖区的钝化效应。平面拉伸状态下,宏观主裂纹以恒定速度运动。通过一个介观约束应力过渡区,将宏观主裂纹与微观裂尖区相连接,由此建立了一个宏微观双尺度运动裂纹模型。应用弹性动力学与复变函数理论,分别在宏观与微观尺度下对该模型进行解析求解,获得了解析解。通过裂纹张开位移从宏观到微观的连续性条件与宏微观应力场协调条件,将两个不同尺度下的解相耦合,获得了计算宏微观损伤区特征长度的显式表达式。研究表明,运动裂纹的宏观应力场仍具有通常的r-1/2的奇异性。由于微观裂尖的钝化,微观应力场奇异性的阶次有所降低,与宏观应力场相比具有弱奇异性。双尺度运动裂纹模型中,可允许裂纹运动速度达到剪切波速,解除了经典运动裂纹理论中裂纹速度不能超过Rayleigh波速的限制。数值结果表明,介观损伤过渡区与裂尖微观损伤区尺寸,及裂纹张开位移等,与裂纹运动速度、材料性质、约束应力比、裂尖钝化角度等因素有关。

The interaction between macroscopic and microscopic factors and the blunting effect of crack tip during the crack dynamic propagation were investigated.A macroscopic crack moves with a constant speed under the in-plane tension.A microscopic V-notch tip is attached to the main crack through a mesoscopic restraining stress transition zone,and so a macro/micro dual scales moving crack model was thus developed.The problem was solved in accordance with the elastic dynamics and complex variable function theory and the analytical solution was then obtained.Two solutions under the macroscopic and microscopic scales were coupled by application of the continuity condition of crack opening displacement from macro to micro scale and the consistence condition of stress fields under two different scales.Two explicit equations to determine the mesoscopic and microscopic damage zone sizes were presented.It is shown that the macroscopic stress field of a moving crack has a normal r（1/2 singularity while comparatively,the microscopic stress field exhibits a weaker singularity due to the microscopic blunting effect of the crack tip.The crack moving speed can reach the shear wave speed in the dual scales moving crack model.Therefore,the limitation that the crack moving speed can not exceed the Rayleigh wave speed in the classical moving crack theory can be cancelled.The numerical results show that the mesoscopic damage zone size,microscopic crack tip zone size and crack opening displacement depend on the crack moving speed,material property,restraining stress ratio and blunting angle of the crack tip,etc.