Nonlinear Behavior and Critical State of a Penny-Shaped Dielectric Crack in a Piezoelectric Solid

By means of the Hankel transform and dual-integral equations, the nonlinear response of a penny-shaped dielectric crack with a permittivity $κ0$ in a transversely isotropic piezoelectric ceramic is solved under the applied tensile stress $σzA$ and electric displacement $DzA$⁠. The solution is given through the universal relation, $Dc∕σzA=KD∕KI=MD∕Mσ$⁠, regardless of the electric boundary conditions of the crack, where $Dc$ is the effective electric displacement of the crack medium, and $KD$ and $KI$ are the electric displacement and the stress intensity factors, respectively. The proportional constant $MD∕Mσ$ has been derived and found to have the characteristics: (i) for an impermeable crack it is equal to $DzA∕σzA$⁠; (ii) for a permeable one it is only a function of the ceramic property; and (iii) for a dielectric crack with a finite $κ0$ it depends on the ceramic property, the $κ0$ itself, and the applied $σzA$ and $DzA$⁠. The latter dependence makes the response of the dielectric crack nonlinear. This nonlinear response is found to be further controlled by a critical state $(σc,DzA)$⁠, through which all the $Dc$ versus $σzA$ curves must pass, regardless of the value of $κ0$⁠. When $σzA<σc$⁠, the response of an impermeable crack serves as an upper bound, whereas that of the permeable one serves as the lower bound, and when $σzA>σc$ the situation is exactly reversed. The response of a dielectric crack with any $κ0$ always lies within these bounds. Under a negative $DzA$⁠, our solutions further reveal the existence of a critical $κ*$⁠, given by $κ*=−RDzA$⁠, and a critical $D*$⁠, given by $D*=−κ0∕R$ (⁠$R$ depends only on the ceramic property), such that when $κ0>κ*$ or when $∣DzA∣<∣D*∣$⁠, the effective $Dc$ will still remain positive in spite of the negative $DzA$⁠.