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Research Papers

A Complex Variable Solution Based Analysis of Electric Displacement Saturation for a Cracked Piezoelectric Material

[+] Author and Article Information
Christian Linder

Department of Civil and
Environmental Engineering,
Stanford University,
Stanford, CA 94305
e-mail: linder@stanford.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 13, 2014; final manuscript received June 4, 2014; accepted manuscript posted June 11, 2014; published online June 26, 2014. Editor: Yonggang Huang.

J. Appl. Mech 81(9), 091006 (Jun 26, 2014) (10 pages) Paper No: JAM-14-1213; doi: 10.1115/1.4027834 History: Received May 13, 2014; Revised June 04, 2014; Accepted June 11, 2014

This paper presents an analysis of the effect of electric displacement saturation for a failing piezoelectric ceramic material based on a complex variable solution of a Mode III and a Mode I crack. This particular electric nonlinearity is caused by a reduction of the ionic movement in the material in the presence of high electric fields. Total and strain energy release rates are computed for varying far field stresses, electric displacements, and electric fields and compared for cases without and with full electric displacement saturation to further advance the understanding of failure initiation in piezoelectric ceramics.

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Figures

Grahic Jump Location
Fig. 1

The electromechanical boundary value problem (BVP) of piezoelectric ceramics. The mechanical loading ρbi,t¯i and corresponding decomposition of the mechanical boundary ∂B = ∂uB∪∂tB¯ is shown on the left whereas the electrical loading ρe,q¯ and corresponding decomposition of the boundary ∂B = ∂ϕB∪∂qB¯ is illustrated on the right.

Grahic Jump Location
Fig. 2

Illustration of the influence of electric displacement saturation on the electric displacement versus electric field relation in piezoelectric ceramics (left), where ds is the saturated electric displacement. Illustration of the strip saturation model proposed in Ref. [21] confined to a strip ahead of the crack tip (right).

Grahic Jump Location
Fig. 3

Complex variable solution of a center crack of 2 units loaded in Mode III within the framework of linear piezoelectricity. Illustration of the total energy release rate (left column) and the strain energy release rate (right column) versus σ23∞ under constant electric displacements d2∞ = 0 and d2∞ = ± 2 × 10-4 C/m2 (top row) and versus d2∞ under constant mechanical loadings σ23∞ = 0 and σ23∞ = ±0.4 MPa (bottom row) without (solid lines) and with full (dashed lines) electric displacement saturation.

Grahic Jump Location
Fig. 4

Complex variable solution of a center crack of 2 units loaded in Mode III within the framework of linear piezoelectricity. Illustration of the total energy release rate (left column) and the strain energy release rate (right column) versus σ23∞ under constant electric fields e2∞ = 0 and e2∞ = ±20 kV/cm (top row) and versus e2∞ under constant mechanical loadings σ23∞ = 0 and σ23∞ = ±0.4 MPa (bottom row) without (solid lines) and with full (dashed lines) electric displacement saturation.

Grahic Jump Location
Fig. 5

Complex variable solution of a center crack of 2 units loaded in Mode I within the framework of linear piezoelectricity. Illustration of the total energy release rate (left column) and the strain energy release rate (right column) versus σ33∞ under constant electric displacements d3∞ = 0 and d3∞ = ±2 × 10-4 C/m2 (top row) and versus d3∞ under constant mechanical loadings σ33∞ = 0 and σ33∞ = ±0.4 MPa (bottom row) without (solid lines) and with full (dashed lines) electric displacement saturation.

Grahic Jump Location
Fig. 6

Complex variable solution of a center crack of 2 units loaded in Mode I within the framework of linear piezoelectricity. Illustration of the total energy release rate (left column) and the strain energy release rate (right column) versus σ33∞ under constant electric fields e3∞ = 0 and e3∞ = ±20 kV/cm (top row) and versus e3∞ under constant mechanical loadings σ33∞ = 0 and σ33∞ = ±0.4 MPa without (solid lines) and with full (dashed lines) electric displacement saturation.

Grahic Jump Location
Fig. 7

Complex variable solution of a center crack of 2 units loaded in Mode III and Mode I within the framework of linear piezoelectricity. Illustration of the total energy release rate and the strain energy release rate for Mode III versus e2∞ under a constant mechanical loading σ23∞ = 0.4 MPa on the left and for Mode I versus e3∞ under a constant mechanical loading σ33∞ = 0.4 MPa on the right without (solid lines) and with full (dashed lines) electric displacement saturation.

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