Exact solution to the antiplane problem of doubly periodic conducting rigid line inclusions of unequal size in piezoelectric materials

Abstract

Key words Doubly periodic array, rigid line inclusion, antiplane problem, stress and electric displacement intensity factors, effective electroelastic moduli.The problem of a doubly periodic array of conducting rigid line inclusions in piezoelectric materials under far-field antiplane mechanical load and inplane electric load is investigated, where the fundamental cell contains four rigid line inclusions of unequal size. An exact solution to the problem is presented by employing the conformal mapping technique and the elliptical function theory. The closed form formulae for the stress and electrical displacement intensity factors at the rigid line inclusion tip and the effective electroelastic moduli of such composites are derived. Many new and existing solutions can be regarded as the special or degenerated cases. Numerical examples are provided to show the interesting electroelastic interaction phenomenon induced by multiple rigid line inclusions. The present work is helpful in understanding the optimization mechanism of some naturally occurring composites and in designing novel materials by direct engineering of their microstructure.

Doubly Periodic Cracks in the Anisotropic Medium with the Account of Contact of Their Faces

“…Chen et al (2003) have studied various multiple crack problems in elasticity. Xiao et al (2011) obtained the closed-form solution for stress and electric displacement intensity factors and effective properties of piezoelectric materials with a doubly periodic set of conducting rigid line inclusions. Malits (2010) studied the doubly periodic arrays of rigid line inclusions in an elastic solid.…”

An Analysis of Mode III Doubly Periodic Crack-Tip Field of Orthotropic Composite Materials

“…Chen and Lee [4], Chen and Lin [5], Dong and Lee [6], Yan and Jiang [7], Yan et al [8], and Tong et al [9] used the eigen function expansion variational method to solve the doubly periodic cracks and analyze the interaction of stress intensity factor in the different cracks arrangement of the corresponding rule. In 2011, Xiao et al [10] obtained the exact solution to the antiplane problem of doubly periodic conducting rigid line inclusions of unequal size in piezoelectric materials. Chang et al [11] discussed the anti-plane problem of doubly periodic cracks in the infinite homogeneous piezoelectric materials using conformal mapping and elliptic function and calculated the stress intensity factor and stress field in a closed form solution.…”