Anti-plane Green's functions and cracks for piezoelectric material with couple stress and electric field gradient effects

Abstract

This research is concerned with the anti-plane strain problems of polarized ceramics with both the couple stress and electric field gradient effects. This theory possesses two characteristic lengths l 1 and l 2 which are determined explicitly. In addition the two characteristic lengths can be either positive real or complex conjugate with positive real part. We first investigate the electroelastic field induced by a static line force and a line charge. It is found that the displacement and the electric potential are regular at the point where the line force and line charge are located. We then consider the near-tip asymptotic electroelastic field for a mode III crack. The analysis demonstrates that the near-tip asymptotic electroelastic field is governed by two parameters B and D. The total stresses and in-plane electric displacements exhibit the stronger r −3/2 singularity near the crack tip; while the couple stresses, the out-of-plane electric displacement, and those associated with electric quadrupole densities exhibit the weaker r −1/2 singularity near the crack tip.

A rigorous solution for the piezoelectric materials containing elliptic cavity or crack with surface effect

“…Substituting Eqs. (17) and (26) into Eq. (7), the overall stress field and electric displacement field in the matrix can be expressed as…”

“…By using an integral transform technique, Singh, Rokne, Dhaliwal, and Vrbik considered the problem of an antiplane crack situated in the interface of two bonded dissimilar graded piezoelectric half‐spaces under the permeable crack assumption, and the mechanical and electrical properties of the half‐spaces were considered for a class of functional forms for which the equilibrium equation has analytic solutions. Wang, Pan, and Feng concerned with the anti‐plane strain problems of polarized ceramics with both the couple stress and electric field gradient effects. Zhang, Zhou, and Li studied the behavior of four parallel non‐symmetric permeable cracks with different lengths in a functionally graded piezoelectric material plane subjected to anti‐plane shear stress loading by the Schmidt method.…”

Microscale size effects on the electromechanical coupling in piezoelectric material for anti-plane problem

“…Substitution of equation ( 27) into equation ( 28) will lead to the following set of coupled homogeneous partial differential equations for w and ϕ ϕ χ ϕ χ ρ ϕ In order to solve the above set of coupled differential equations, we consider the following eigenvalue problem [37]: Then equation (30) possesses two positive real eigenvalues given by The two eigenvectors associated with the two eigenvalues are Equation (30) also has two complex conjugate eigenvalues. Since displacement w and potential ϕ cannot be a complex number, we ignore this case.…”