By means of the Hankel transform and dual-integral equations, the nonlinear response of a penny-shaped dielectric crack with a permittivity in a transversely isotropic piezoelectric ceramic is solved under the applied tensile stress and electric displacement . The solution is given through the universal relation, , regardless of the electric boundary conditions of the crack, where is the effective electric displacement of the crack medium, and and are the electric displacement and the stress intensity factors, respectively. The proportional constant has been derived and found to have the characteristics: (i) for an impermeable crack it is equal to ; (ii) for a permeable one it is only a function of the ceramic property; and (iii) for a dielectric crack with a finite it depends on the ceramic property, the itself, and the applied and . The latter dependence makes the response of the dielectric crack nonlinear. This nonlinear response is found to be further controlled by a critical state , through which all the versus curves must pass, regardless of the value of . When , the response of an impermeable crack serves as an upper bound, whereas that of the permeable one serves as the lower bound, and when the situation is exactly reversed. The response of a dielectric crack with any always lies within these bounds. Under a negative , our solutions further reveal the existence of a critical , given by , and a critical , given by ( depends only on the ceramic property), such that when or when , the effective will still remain positive in spite of the negative .
Nonlinear Behavior and Critical State of a Penny-Shaped Dielectric Crack in a Piezoelectric Solid
Chiang, C., and Weng, G. J. (July 13, 2006). "Nonlinear Behavior and Critical State of a Penny-Shaped Dielectric Crack in a Piezoelectric Solid." ASME. J. Appl. Mech. September 2007; 74(5): 852–860. https://doi.org/10.1115/1.2712227
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