A closed form solution to the antiplane problem of doubly periodic cracks of unequal size in piezoelectric materials

The problem of doubly periodic cracks in piezoelectric materials under far-field antiplane mechanical load and inplane electric load is investigated, where the fundamental cell contains four parallel and unequal cracks. By combining the elliptical function theory and conformal mapping technique, an exact solution in closed form to this problem is obtained. Based on this solution, the exact formulae for the crack tip field intensity factors and the effective electroelastic moduli of such cracked piezoelectric materials are derived. Numerical examples are graphically presented to show the interesting mechanical and electrical coupling phenomena induced by multicrack interactions, especially to reveal the shielding and intensifying effects of small cracks on the main cracks and to quantify the loss of electroelastic properties due to the presence of such arrays of microcracks. New and existing solutions, such as those for doubly periodic cracks in purely elastic materials and a single row or a single stack of periodical unequal cracks in piezoelectric materials are obtained as special cases of the present solution. The present solution can provide benchmark results for other numerical and approximate methods.