A partial contact zone model is developed for the stress and electric displacement fields due to the obstruction of a uniform heat flux by an electrically impermeable crack in piezoelectric materials. Green's function method is used to reduce the problem to a set of singular integral equations that are solved in closed form. When the crack is assumed to be traction free, the crack opening displacement is found to be negative over one-half of the crack unless a sufficiently large far field tensile stress is superposed. The problem is reformulated assuming a contact zone at one crack tip. The extent of this zone, the stress and electric displacement intensity factors at each crack tip are obtained as functions of the applied mechanical stress and heat flux.
Zhang, A. B., & Wang, B. (2012). Contact zone approach for an electrically impermeable crack in piezoelectric materials under thermal-mechanical loading. Journal of Thermal Stresses, 35(9), 749-765. https://doi.org/10.1080/01495739.2012.689231