Crack extension force and rate of mechanical work of fracture in linear dielectrics and piezoelectrics

The physical significance of the crack extension force produced by mechanical loads and electric fields in linear dielectric and piezoelectric materials is examined using simple thermodynamic arguments. General expressions are derived for the crack extension force in dielectrics and piezoelectrics, under mechanical and electrical loads, in terms of the measurable parameters elastic compliance and electric capacitance. It is shown that the crack extension force produced by an electric field on an impermeable crack is always negative and it is argued that under combined electromechanical loads the total crack extension force in a piezoelectric cannot be separated into a mechanical component and an electrical component. Expressions for the crack extension force in terms of mechanical and electrical intensity factors are also given. Their derivation from available solutions for the electromechanical field at the crack tip (in a transversely isotropic material of crystal class 6mm) is presented in detail to emphasize the physical significance of the coefficients that appear in the final expressions. In the light of these results, existing experimental observations that appear to be inconsistent with theoretical expectations are re-examined. The suggestion that the crack extension force is not a valid parameter to characterize the fracture behaviour of ferroelectrics. is justified on physical grounds. Its importance is discussed and the rate of mechanical work of fracture is proposed as a more suitable parameter for those cases where the electric field does not produce dielectric rupture, nor degradation of the material at the crack tip.