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

Abstract

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.