Macro–micro dual scale crack model linked by a restraining stress zone with a linear distribution

Material in a macro-crack tip zone would suffer to some extent damage under loading condition. This can be described by a restraining stress zone in which the distribution of restraining stresses would depend on the material damage. In order to consider the microscopic effects, a micro-notch is attached to the macro-crack tip. Thus, a macro–micro dual scale damage model is established. A linear distribution of restraining stresses is assumed in this work including three cases. Cases A, B and C are, respectively, a right triangle shape, uniform distribution and left triangle shape. Muskhelishivili approach is employed to solve the problem. Obtained are the stress intensity factors (SIF) both at the macro- and micro-scale, crack opening displacement (COD) and strain energy densities (SED) in the analytical form. Numerical calculations are performed. The results show that there is a complicated interaction between the macro-crack and micro-defect. In addition, the distribution of restraining stresses has a large influence on the SIF, COD and SED.