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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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References

Jaffe, H., 1958, “Piezoelectric Ceramics,” J. Am. Ceram. Soc., 41(11), pp. 494–498. [CrossRef]
Nowacki, W., 1979, “Foundations of Linear Piezoelectricity,” Electromagnetic Interactions in Elastic Solids, H.Parkus, ed. Springer, Vienna, pp. 105–157.
Griffith, A., 1921, “The Phenomena of Rupture and Flow in Solids,” Philos. Trans. R. Soc. London, Ser. A, 221(582–593), pp. 163–198. [CrossRef]
Irwin, G. R., 1956, “Onset of Fast Crack Propagation in High Strength Steel and Aluminum Alloys,” U.S. Naval Research Laboratory, Washington, DC, NRL Report No. 4763.
Irwin, G. R., 1957, “Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate,” ASME J. Appl. Mech., 24(3), pp. 361–364.
Rice, J., 1968, “A Path-Independent Integral and the Approximate Analysis of Strain Concentrations by Notches and Cracks,” ASME J. Appl. Mech., 35(2), pp. 379–386. [CrossRef]
Li, S., Linder, C., and Foulk III, J., 2007, “On Configurational Compatibility and Multiscale Energy Momentum Tensors,” J. Mech. Phys. Solids, 55(5), pp. 980–1000. [CrossRef]
Eshelby, J. D., Read, W. T., and Shockley, W., 1953, “Anisotropic Elasticity With Applications to Dislocation Theory,” Acta Metall., 1(3), pp. 251–259. [CrossRef]
Lekhnitskii, S. G., 1950, Theory of Elasticity of an Anisotropic Elastic Body, Gostekhizdat, Moscow.
Stroh, A. N., 1958, “Dislocations and Cracks in Anisotropic Elasticity,” Philos. Mag., 3(30), pp. 625–646. [CrossRef]
Parton, V. Z., 1976, “Fracture Mechanics of Piezoelectric Materials,” Acta Astronaut., 3(9–10), pp. 671–683. [CrossRef]
Pak, Y. E., 1990, “Crack Extension Force in a Piezoelectric Material,” ASME J. Appl. Mech., 57(3), pp. 647–653. [CrossRef]
Pak, Y. E., 1992, “Linear Electro-Elastic Fracture Mechanics of Piezoelectric Materials,” Int. J. Fract. Mech., 54(1), pp. 79–100. [CrossRef]
Suo, Z., Kuo, C., Barnett, D., and Willis, J., 1992, “Fracture Mechanics for Piezoelectric Ceramics,” J. Mech. Phys. Solids, 40(4), pp. 739–765. [CrossRef]
Park, S., 1994, “Fracture Behavior of Piezoelectric Materials,” Ph.D. thesis, Purdue University, West Lafayette, IN.
Tobin, A. G., and Pak, Y. E., 1993, “Effect of Electric Fields on Fracture Behavior of PZT Ceramics,” Proc. SPIE, 1916, pp. 78–86. [CrossRef]
Park, S., and Sun, C. T., 1995, “Fracture Criteria for Piezoelectric Ceramics,” J. Am. Ceram. Soc., 78(6), pp. 1475–1480. [CrossRef]
Wang, H., and Singh, R. N., 1997, “Crack Propagation in Piezoelectric Ceramics: Effects of Applied Electric Fields,” J. Appl. Phys., 81(11), pp. 7471–7479. [CrossRef]
Fu, R., and Zhang, T. Y., 2000, “Effects of an Electric Field on the Fracture Toughness of Poled Lead Zirconate Titanate Ceramics,” J. Am. Ceram. Soc., 83(5), pp. 1215–1218. [CrossRef]
Park, S., and Sun, C. T., 1995, “Effect of Electric Field on Fracture of Piezoelectric Ceramics,” Int. J. Fract. Mech., 70(3), pp. 203–216. [CrossRef]
Gao, H., Zhang, T., and Tong, P., 1997, “Local and Global Energy Release Rates for an Electrically Yielded Crack in a Piezoelectric Ceramic,” J. Mech. Phys. Solids, 45(4), pp. 491–510. [CrossRef]
Linder, C., and Miehe, C., 2012, “Effect of Electric Displacement Saturation on the Hysteretic Behavior of Ferroelectric Ceramics and the Initiation and Propagation of Cracks in Piezoelectric Ceramics,” J. Mech. Phys. Solids, 60(5), pp. 882–903. [CrossRef]
Linder, C., 2012, “An Analysis of the Exponential Electric Displacement Saturation Model in Fracturing Piezoelectric Ceramics,” Tech. Mech., 32(1), pp. 53–69, available at: [CrossRef]
Simo, J. C., Oliver, J., and Armero, F., 1993, “An Analysis of Strong Discontinuities Induced by Strain-Softening in Rate-Independent Inelastic Solids,” Comput. Mech., 12(5), pp. 277–296. [CrossRef]
Linder, C., and Armero, F., 2007, “Finite Elements With Embedded Strong Discontinuities for the Modeling of Failure in Solids,” Int. J. Numer. Methods Eng., 72(12), pp. 1391–1433. [CrossRef]
Linder, C., and Armero, F., 2009, “Finite Elements With Embedded Branching,” Finite Elem. Anal. Des., 45(4), pp. 280–293. [CrossRef]
Armero, F., and Linder, C., 2008, “New Finite Elements With Embedded Strong Discontinuities in the Finite Deformation Range,” Comput. Methods Appl. Mech. Eng., 197(33–40), pp. 3138–3170. [CrossRef]
Armero, F., and Linder, C., 2009, “Numerical Simulation of Dynamic Fracture Using Finite Elements With Embedded Discontinuities,” Int. J. Fract. Mech., 160(2), pp. 119–141. [CrossRef]
Linder, C., and Raina, A., 2013, “A Strong Discontinuity Approach on Multiple Levels to Model Solids at Failure,” Comput. Methods Appl. Mech. Eng., 253, pp. 558–583. [CrossRef]
Linder, C., and Zhang, X., 2013, “A Marching Cubes Based Failure Surface Propagation Concept for Three-Dimensional Finite Elements With Non-Planar Embedded Strong Discontinuities of Higher-Order Kinematics,” Int. J. Numer. Methods Eng., 96(6), pp. 339–372. [CrossRef]
Linder, C., Rosato, D., and Miehe, C., 2011, “New Finite Elements With Embedded Strong Discontinuities for the Modeling of Failure in Electromechanical Coupled Solids,” Comput. Methods Appl. Mech. Eng., 200(1–4), pp. 141–161. [CrossRef]
Linder, C., and Zhang, X., 2014, “Three-Dimensional Finite Elements With Embedded Strong Discontinuities to Model Failure in Electromechanical Coupled Materials,” Comput. Methods Appl. Mech. Eng., 273, pp. 143–160. [CrossRef]
Schröder, J., and Gross, D., 2004, “Invariant Formulation of the Electromechanical Enthalpy Function of Transversely Isotropic Piezoelectric Materials,” Arch. Appl. Mech., 73(8), pp. 533–552. [CrossRef]
Barnett, D. M., and Lothe, J., 1975, “Dislocations and Line Charges in Anisotropic Piezoelectric Insulators,” Phys. Status Solidi B, 67(1), pp. 105–111. [CrossRef]
Yang, W., and Suo, Z., 1994, “Cracking in Ceramic Actuators Caused by Electrostriction,” J. Mech. Phys. Solids, 42(4), pp. 649–663. [CrossRef]
Lynch, C. S., Yang, W., Collier, L., Suo, Z., and McMeeking, R. M., 1995, “Electric Field Induced Cracking in Ferroelectric Ceramics,” Ferroelectrics, 166(1), pp. 11–30. [CrossRef]
Hao, T., Gong, X., and Suo, Z., 1996, “Fracture Mechanics for the Design of Ceramic Multilayer Actuators,” J. Mech. Phys. Solids, 44(1), pp. 23–48. [CrossRef]
Jona, F., and Shirane, G., 1993, Ferroelectric Crystals, Vol. 1, Dover Publications, New York.
Dugdale, D. S., 1960, “Yielding of Steel Sheets Containing Slits,” J. Mech. Phys. Solids, 8(2), pp. 100–104. [CrossRef]
Barenblatt, G., 1962, “The Mathematical Theory of Equilibrium Cracks in Brittle Fracture,” Adv. Appl. Mech., 7, pp. 55–129. [CrossRef]

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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