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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

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, DcσzA=KDKI=MDMσ, 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 MDMσ has been derived and found to have the characteristics: (i) for an impermeable crack it is equal to DzAσzA; (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*=κ0R (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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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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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