A crack tip stress field coupled with damage of stationary mode I crack is studied by mathematical and numerical methods. The damage effect is incorporated into the power-law creep constitutive equation by using the strain equivalence principle and the evolution of the cumulative damage is described by the multi-axial Kachanov-Rabotnov kinetic equation.
Based on the regular perturbation approach the interaction between crack and damage has been investigated in case of small damage region around the crack tip. Taking a dimensionless coefficient as a perturbation parameter, which characterizes the concerned damage scale, a series of asymptotic solution is obtained with the zeros order approximation being the HRR type of solution. According to the solution, in the small region around a crack tip the HRR type singularity still dominates the solution with damage effect as only a perturbation.
In case of large damage in the region very near the crack tip, numerical procedure do not work so easily because of the localization of damage. Thus, analytical procedures is employed to obtain the locally asymptotic structure of the solution that applies to the regime very near the crack tip. Within this regime the damage effect dominates the local zone. A coordinate perturbation approach is used to deal with the asymptotic behavior of crack tip stress field. And eigenvalues which represent the crack tip singularity are obtained with variation of material constants. According to the solution, stress vanish with coordinate r approaches zero due to damage relaxation to them. Consequently no stress singularity occurs and stress approach zero according to r(5-2 where s is a positive eigenvalue larger than 2. Eigenvalue shows a large dependency on the creep exponent and damage constants.
Since global analytical solution is actually impossible to attain for this problem, finite element method is used. For this purpose, finite element program which can perform the structural analysis with viscoplastic constitutive equation having arbitrary scalar and tensor state variables is developed and FEM program is used to investigate the crack tip stress field with damage coupling. In analysis, CCP type specimen is considered. According to result, at the crack tip stress vanish and then increase with the increasing of r and finally they reach their corresponding maximum at some position from the crack tip. After that stresses decrease with the increase of r approximately according to the law predicted by HRR solution. The damage distribution on the other hand, shows its maximum value at the crack tip and apart from the crack tip, mesovolume element of the medium persist in a small damage state. As the time continues to increase the damage constantly accumulates in the crack tip region.
