Energetically consistent boundary conditions for electromechanical fracture

Chad M Landis

doi:10.1016/j.ijsolstr.2004.05.062

Abstract

Energetically consistent crack face boundary conditions are formulated for cracks in electromechanical materials. The model assumes that the energy of the solid can be computed from standard infinitesimal deformation theory and that the opening of the crack faces creates a capacitive gap that can store electrical energy. The general derivation of the crack face boundary conditions is carried out for non-linear but reversible constitutive behavior of both the solid material and the space filling the gap. It is shown that a simple augmentation of the J-integral can be used to determine the energy release rate for crack advance with these boundary conditions. The energetically consistent boundary conditions are then applied to the Griffith crack problem in a polar linear piezoelectric solid and used to demonstrate that the energy release rate computed near the crack tip is equivalent to the total energy release rate for the solid-gap system as computed from global energy changes. A non-linear constitutive law is postulated for the crack gap as a model for electrical discharge and the effects of the breakdown field on the energy release rate are ascertained.

Highlights

This work reports on results that have been developed in greater detail in Landis (2004)
(3) When the crack opens electric fields can permeate the crack medium and electrical energy can be stored within the crack
(5) Electric field components within the crack medium parallel to the crack faces are negligible compared to the electric field normal to the crack

Summary

Background

This work reports on results that have been developed in greater detail in Landis (2004). A second system is proposed with the crack gap removed, and in its place tractions and surface charge densities are applied to the crack surfaces In order for these two systems to be equivalent, the variations of the total electrical enthalpy of these systems must be identical for arbitrary variations of the crack face displacement and electric potential. Such stresses that occur due to displacements and electrical effects are common in more general studies on large deformation behavior of electrically active materials and are referred to as Maxwell stresses It is shown in Landis (2004) that the general energetically consistent boundary conditions on the crack faces can be derived as, ω+. More specific forms for these boundary conditions will be given for the special cases of perfect linear dielectric crack gap behavior and an idealized model for electrical discharge within the crack gap

The linear dielectric and discharging crack gaps

The Griffith crack in a poled linear piezoelectric solid

G H11σ 2πa

Discussion