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

Abstract

A theoretical study on the problem of piezoelectric materials with a nano elliptic cavity considering surface effect subjected to far‐field antiplane mechanical load and inplane electric load is reported. Surface effect is introduced based on the theory of Gurtin‐Murdoch surface/interface model. A rigorous whole‐field solution is presented by using the complex variable elasticity theory and the conformal mapping method, in terms of which closed‐form solution of the stress/electric displacement fields and the stress/electric displacement intensity factors is obtained. The results reveal that the stress/electric displacement fields and the stress/electric displacement intensity factors are size dependent when the size of the cavity/crack is on the order of nanometer. The present solution approaches to the classical electroelastic results with the increase of the size of the nano elliptic cavity and crack. The influences of the elliptic cavity shape ratio on the stress and electric displacement concentration factors are discussed.

Surface effect on a nano‐elliptical hole or nano‐crack in magnetoelectroelastic materials under antiplane shear

“…When the size of nano‐crack is large enough (i.e., a > 100 nm), the size‐dependent field intensity factors are close to the classical elasticity results. It is noted that the results of PE materials are consistent with the previous ones …”

“…The conformal mapping function is introduced as follows which maps the exterior of the ellipse in the z ‐plane on the exterior of the circle with radius L 0 in the ζ ‐plane as shown in Figure (b), where , and .…”

“…We choose PZT‐5 and CoFe 2 O 4 as the matrix, whose material properties of the matrix and surface layer are listed in Tables and to analyze the surface effects on the field intensity factors near the nano‐crack tip in PE and PM materials under remotely anti‐plane mechanical and in‐plane electrical and magnetic loadings, respectively, as shown in Figures –.…”

“…To describe the surface/interface effects of nanostructures, the Gurtin‐Murdoch surface/interface model theory was proposed by Gurtin and Murdoch . Based on the Gurtin‐Murdoch surface/surface model, Xiao et al . studied the influence of surface/interface effects on the field intensity factors of a crack in piezoelectric materials by means of conformal mapping function.…”

“…[2,3] To describe the surface/interface effects of nanostructures, the Gurtin-Murdoch surface/interface model theory was proposed by Gurtin and Murdoch. [4][5][6] Based on the Gurtin-Murdoch surface/surface model, Xiao et al [7,8] studied the influence of surface/interface effects on the field intensity factors of a crack in piezoelectric materials by means of conformal mapping function. Nan and Wang [9] studied the fracture behavior of an anti-plane crack in the piezoelectric materials with surface effects by applying the singular integral method.…”