Analytical Solutions Of Multiscale Plane Crack Model Based On The Restraining Stress Zone

Material in the vicinity of the crack tip may be damaged under the high stress intensification. Under the influence of external environment, microscopic defects, etc., the damage form of the material in the crack tip may be diversified. Micro-defects to some extent determine the direction of crack propagation, even structural failure. The interaction between the macro-and micro-scale increases the uncertainty of the crack tip. Studying different damage distribution of the crack tip is essential to understand the mechanism of crack propagation. Connecting the material damage with the crystal defects in the crack tip zone, multi-scale restraining stress zone model has been successfully used for solving the static, dynamic crack and fatigue crack problems, etc. By assuming a linear distribution of the restraining stresses, a multi-scale restraining stress zone model is established. The main work is as follows:(1) The basic theories and methods of plane crack problems are described in detail.(2) The solution of the singularity of the crack tip is described with different boundary condition of the V-notch. The singularities of fixed-fixed, free-free and free-fixed boundary conditions are displayed.(3) Considering the free-free boundary condition of the V-notch and using the restraining stress zone to connect the main macro-crack and micro-defect, a multi-scale plane crack model based on restraining stress zone is established. The model includes two different scales, say macro-and micro-scale. By application of Muskhelishivili’s method, the analytical solutions of the multi-scale plane crack model are derived.(4) Assume that the restraining stresses have a linear distribution. Three different distributions of restraining stresses are considered in the mode I crack tip zone. Case A has a right triangle shape. Case B is the uniform distribution. Case C has a left triangle shape. By application of the analytical approach, the expressions of stress intensity factor (SIF) and crack opening displacement (COD) for three different cases are derived. By numerical calculations, the interactions between the main macro-crack and micro-defect are investigated. In addition, the distribution of strain energy density in the front of the crack tip is also calculated, The results show that the singularity of V-notch, the variation and distribution of restraining stresses determine the length of the restraining stress zone. And, they also have a significant effect on SIF, COD and the distribution of strain energy density.