On the plastic zone size of solids containing doubly periodic rectangular-shaped arrays of cracks under longitudinal shear

• The elastic-plastic problem of doubly periodic cracks is solved.
• The influence of plasticity around the crack tip is addressed.
• A highly accurate approach is put forward by avoiding double infinite summation.
• A new identity suitable for periodic cracks research is put forward first and proved.
• An analytical formula is obtained for calculating SIF with a reliable precision.

Multiple cracks interaction plays an important role in fracture behavior of materials. A number of studies have been devoted to analytical and numerical analyses of the doubly periodic arrays of cracks. A very natural and highly accurate solution procedure is proposed to describe the interaction effect among the doubly periodic rectangular-shaped arrays of cracks. The proposed solution is implemented in the framework of continuously distributed dislocation model and singular integral equation approach. The accuracy of this solution is proved through a comparison of results from the present simulation and known closed form solutions. Further, the interaction effects among the periodic cracks on the plastic zone size and crack tip opening displacement are studied. It is found that the interaction distance among the vertical and horizontal periodic cracks is quite different.