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TECHNICAL PAPERS

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

[+] Author and Article Information
Chun-Ron Chiang

Department of Power Mechanical Engineering, National Tsing Hua University, Hsin Chu 30013, Taiwancrchiang@pme.nthu.edu.tw

George J. Weng1

Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, NJ 08903weng@jove.rutgers.edu

1

Corresponding author.

J. Appl. Mech 74(5), 852-860 (Jul 13, 2006) (9 pages) doi:10.1115/1.2712227 History: Received May 30, 2006; Revised July 13, 2006

## 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 $DzA$.

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## Figures

Figure 1

The effective electric displacement, Dc, of the crack medium versus the applied stress, σzA, for PZT-4. The permittivity κ0 inside the crack was 8.85×10−12C∕Vm.

Figure 2

A schematic plot on the influence of permittivity, κ0, of the crack medium to the effective electric displacement, Dc, versus the applied stress, σzA relation. Regardless of the value of κ0 all the curves must pass through the critical state (σc,DzA). When σzA<σc, the responses of the impermeable and permeable cracks will serve as the upper and lower bounds, respectively, and when σzA>σc the situation is reversed. The response of a dielectric crack with any κ0 always lies within these bounds.

Figure 3

A quantitative assessment for PZT-4 on the influence of permittivity κ0 of the crack medium to the effective electric displacement, Dc, versus the applied stress, σzA relation. The result with vacuum (κ0=8.85×10−12C∕Vm), lying between those of κ0=10−12C∕Vm and 10−11C∕Vm, is seen to be far away from those of the permeable and impermeable cracks.

Figure 4

A schematic plot on the influence of permittivity, κ0, of the crack medium to the effective electric displacement, Dc, versus the applied stress, σzA, relation under a negative DzA. The critical (σc,DzA) state in this case only exists at the origin. There exists a critical κ* for κ0, beyond which Dc will remain positive in spite of the negative DzA.

Figure 5

A quantitative assessment for PZT-4 on the influence of permittivity κ0 of the crack medium to the effective electric displacement, Dc, versus the applied stress, σzA, relation under a negative DzA.

Figure 6

A quantitative assessment for PZT-4 on the influence of the negative electric load, DzA, to the nonlinear relation of Dc versus σzA. There exists a critical D*, below which the effective Dc will remain positive in spite of the negative DzA.

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