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Applied Mathematics and Mechanics

, Volume 20, Issue 1, pp 51–58 | Cite as

An exact solution of crack problems in piezoelectric materials

  • Gao Cunfa
  • Fan Weixun
Article

Abstract

An assumption that the normal component of the electric displacement on crack faces is thought of as being zero is widely used in analyzing the fracture mechanics of piezoelectric materials. However, it is shown from the available experiments that the above assumption will lead to erroneous results. In this paper, the two-dimensional problem of a piezoelectric material with a crack is studied based on the exact electric boundary condition on the crack faces. Stroh formalism is used to obtain the closed-form solutions when the material is subjected to uniform loads at infinity. It is shown from these solutions that: (i) the stress intensity factor is the same as that of isotropic material, while the intensity factor of the electric displacement depends on both material properties and the mechanical loads, but not on the electric load. (ii) the energy release rate in a piezoelectric material is larger than that in a pure elastic-anisotropic material, i. e., it is always positive, and independent of the electric loads. (iii) the field solutions in a piezoelectric material are not related to the dielectric constant of air or vacuum inside the crack.

Key words

piezoelectric material plane problem crack energy release rate exact solution 

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References

  1. [1]
    Parton V Z. Fracture mechanics of piezoelectric materials [J].Acta Astronautic, 1976,3 (9): 671–683MATHCrossRefGoogle Scholar
  2. [2]
    Pak Y E. Crack extension force in a piezoelectric material [J].ASME J Appl Mech, 1990,57 (3): 647–653MATHCrossRefGoogle Scholar
  3. [3]
    Suo Z, Kuo C M, Barnett D M, Willis J R. Fracture mechanics for piezoelectric ceramics [J].J Mech Phys Solids, 1992,40 (4): 739–765MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    Sosa H A. On the fracture mechanics of piezoelectric solids [J].Int J Solids Structures, 1992,29 (21): 2613–2622MATHCrossRefGoogle Scholar
  5. [5]
    Pak Y E. Linear electro-elastic fracture mechanics of piezoelectric materials [J].Int J Fracture, 1992,54 (1): 79–100Google Scholar
  6. [6]
    Pak Y E, Tobin A. On electric field effects in fracture of piezoelectric materials [J]. AMD-Vol, 161/MD-Vol. 42,Mechanics of Electromagnetic Materials and Structure, ASME, 1993, 51–62Google Scholar
  7. [7]
    Sosa H A. Crack problems in piezoelectric ceramics [J]. AMD-Vol. 161/MD-Vol. 42,Mechanics of Electromagnetic Materials and Structure, ASME, 1993, 63–75Google Scholar
  8. [8]
    Dunn M L. The effect of crack face boundary conditions on the fracture mechanics of piezoelectric solids [J].Eng Fracture Mech, 1994,48 (1): 25–39CrossRefGoogle Scholar
  9. [9]
    Park S B, Sun C T. Effect of electric field on fracture of piezoelectric ceramic [J].Int J Fracture, 1995,70 (3): 203–216CrossRefGoogle Scholar
  10. [10]
    Beom H G, Atluri S N. Near-tip fields and intensity factors for interfacial cracks in disimilar anisotropic piezoelectric media [J].Int J Fracture, 199675 (2): 163–183CrossRefGoogle Scholar
  11. [11]
    Zhang T Y, Tong P. Fracture mechanics for a mode III crack in a piezoelectric material [J].Int J Solids Structures, 1996,33 (3): 343–359MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    Kogan L, Hui C Y, Molkov V. Stress and induction field of a spheroidal inclusion or a penny-shaped crack in a transversely isotropic piezoelectric material [J].Int J solids Structures, 1996,33 (19): 2719–2737MATHCrossRefGoogle Scholar
  13. [13]
    Yu S W, Qin Q H. Damage analysis of thermopiezoelectric properties: Part I—Crack tip singularities [J].Theor Appl Fract Mech, 1996,25 (3): 263–277CrossRefGoogle Scholar
  14. [14]
    Qin Q H, Yu S W. An arbitrarily-oriented plane crack terminating at the interface between dissimilar piezoelectric materials [J].Int J Solids Structures, 1997,34 (5): 581–590MATHCrossRefGoogle Scholar
  15. [15]
    Du Shanyi, Liang Jun, Han Jiecal. The coupled solution of a rigid line inclusion and a crack in anisotropic piezoelectric solids [J].Acta Mechanica Sinica, 1995,27 (5): 544–550 (in Chinese)Google Scholar
  16. [16]
    Yang Xiaoxiang, Kuang Zhenbang. The calculation of piezoelectric materials with a mixed mode crack [J],Acta Mechanicia Sinica, 1997,29 (3): 314–322 (in Chinese)Google Scholar
  17. [17]
    Sosa H A, Khutoryansky N. New developments concerning piezoelectric materials with defects [J].Int J Solids Structures, 1996,33 (23): 3399–3414MATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    Chung M Y, Ting T C T. Piezoelectric solids with an elliptic inclusion or hole [J].Int J Solids Structures, 1996,33 (23): 3343–3361MATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    Suo Z. Singularities. Interfaces and cracks in dissimilar anisotropic media [J].Proc R Soc Lond, 1990, A427: 331–358MATHMathSciNetCrossRefGoogle Scholar
  20. [20]
    Lothe J. Integral formalism for surface waves in piezoelectric crystals: Existence considerations [J].J Appl Phys, 1976,47 (5): 1799–1807CrossRefGoogle Scholar
  21. [21]
    Muskhelishvili N I.Some Basic Problems of Mathematical Theory of Elasticity [M]. Leyden: Noordhoof, 1975Google Scholar
  22. [22]
    Wangsness R K.Electromagnetic Fields [M]. New York: John Wiley & Sons, 1979Google Scholar

Copyright information

© Editorial Committee of Applied Mathematics and Mechanics 1999

Authors and Affiliations

  • Gao Cunfa
    • 1
  • Fan Weixun
    • 1
  1. 1.Department of AircraftNanjing University of Aeronautics & AstronauticsNanjingP. R. China

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