Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Analysis of Cracks in Piezoelectric Solids with Consideration of Electric Field and Strain Gradients

  • Jan Sladek
  • Vladimir Sladek
  • Michael WünscheEmail author
Living reference work entry
First Online:
DOI: https://doi.org/10.1007/978-3-662-53605-6_237-1
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Synonyms

Crack analysis; Electric field and strain gradient; FEM; J-integral; Size effect phenomenon

Definitions

Due to the progress in nanotechnology, the size of the electronic components and devices has been significantly reduced in recent years. If the dimension of the structure is of the same order of magnitude as the material length scale, the classical electromechanical coupling theory of piezoelectricity fails to describe the observed size-dependent phenomenon. Discrete atomistic methods such as molecular dynamics simulations can be utilized to analyze nano-sized structures. However, the required computational cost of such methods can be too high to investigate realistic engineering problems. An alternative and less expensive possibility is the formulation of an advanced continuum mechanics model which takes the size effect phenomenon into account, where the strain and electric field gradients are included in the constitutive equations.

Introduction

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Notes

Acknowledgements

The authors acknowledge the support by the Slovak Science and Technology Assistance Agency registered under number APVV-14-0216, VEGA 1/0145/17 and the Slovak Academy of Sciences Project (SASPRO) 0106/01/01.

References

  1. Aravas N, Giannakopoulos AE (2009) Plane asymptotic crack-tip solutions in gradient elasticity. Int J Solids Struct 46:4478–4503CrossRefGoogle Scholar
  2. Argyris JH, Fried I, Scharpf DW (1968) The tuba family of plate elements for the matrix displacement method. Aeronaut J 72:701–709CrossRefGoogle Scholar
  3. Baskaran S, He X, Chen Q, Fu JF (2011) Experimental studies on the direct flexoelectric effect in alpha-phase polyvinylidene fluoride films. Appl Phys Lett 98:242901CrossRefGoogle Scholar
  4. Buhlmann S, Dwir B, Baborowski J, Muralt P (2002) Size effects in mesiscopic epitaxial ferroelectric structures: increase of piezoelectric response with decreasing feature-size. Appl Phys Lett 80:3195–3197CrossRefGoogle Scholar
  5. Catalan G, Lubk A, Vlooswijk AHG, Snoeck E, Magen C, Janssens A, Rispens G, Rijnders G, Blank DHA, Noheda B (2011) Flexoelectric rotation of polarization in ferroelectric thin films. Nat Mater 10:963–967CrossRefGoogle Scholar
  6. Cross LE (2006) Flexoelectric effects: charge separation in insulating solids subjected to elastic strain gradients. J Mater Sci 41:53–63CrossRefGoogle Scholar
  7. Exadaktylos G (1998) Gradient elasticity with surface energy: mode-I crack problem. Int J Solids Struct 35:421–456MathSciNetCrossRefGoogle Scholar
  8. Exadaktylos G, Vardoulakis I, Aifantis E (1996) Cracks in gradient elastic bodies with surface energy. Int J Fract 79: 107–119CrossRefGoogle Scholar
  9. Fannjiang AC, Chan YS, Paulino GH (2002) Strain gradient elasticity for antiplane shear cracks: a hypersingular integrodifferential equation approach. SIAM J Appl Math 62:1066–1091MathSciNetCrossRefGoogle Scholar
  10. Gellmann R, Ricoeur A (2012) Some new aspects of boundary conditions at cracks in piezoelectrics. Arch Appl Mech 82:841–852CrossRefGoogle Scholar
  11. Georgiadis HG, Grentzelou CG (2006) Energy theorems and the J integral in dipolar gradient elasticity. Int J Solids Struct 43:5690–5712MathSciNetCrossRefGoogle Scholar
  12. Gitman I, Askes H, Kuhl E, Aifantis E (2010) Stress concentrations in fractured compact bone simulated with a special class of anisotropic gradient elasticity. Int J Solids Struct 47:1099–1107CrossRefGoogle Scholar
  13. Hadjesfandiari AR (2013) Size-dependent piezoelectricity. Int J Solids Struct 50:2781–2791CrossRefGoogle Scholar
  14. Hao TH, Shen ZY (1994) A new electric boundary condition of electric fracture mechanics and its applications. Eng Fract Mech 47:793–802CrossRefGoogle Scholar
  15. Harden J, Mbanga B, Eber N, Fodor-Csorba K, Sprunt S, Gleeson JT, Jakli A (2006) Giant flexoelectricity of bent-core nematic liquid crystals. Phys Rev Lett 97:157802CrossRefGoogle Scholar
  16. Hu SL, Shen SP (2009) Electric field gradient theory with surface effect for nano-dielectrics. CMC Comput Mater Continua 13:63–87Google Scholar
  17. Huang Y, Zhang L, Guo TF, Hwang KC (1997) Mixed mode near-tip fields for cracks in materials with strain-gradient effects. J Mech Phys Solids 45:439–465CrossRefGoogle Scholar
  18. Karlis GF, Tsinopoulos SV, Polyzos D, Beskos DE (2007) Boundary element analysis of mode I and mixed mode (I and II) crack problems of 2-D gradient elasticity. Comput Methods Appl Mech Eng 196:5092–5103CrossRefGoogle Scholar
  19. Kogan ShM (1964) Piezoelectric effect during inhomogeneous deformation and acoustic scattering of carriers in crystals. Sov Phys Solid State 5:2069–2070Google Scholar
  20. Landau LD, Lifshitz EM (1984) Electrodynamics of continuous media, 2nd edn. Butterworth-Heinemann, Oxford, pp 358–371CrossRefGoogle Scholar
  21. Landis CM (2004) Energetically consistent boundary conditions for electromechanical fracture. Int J Solids Struct 41:6291–6315CrossRefGoogle Scholar
  22. Majdoub MS, Sharma P, Cagin T (2008) Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect. Phys Rev B 77:1–9CrossRefGoogle Scholar
  23. Maugin GA (1980) The method of virtual power in continuum mechanics: applications to coupled fields. Acta Mech 35:1–80MathSciNetCrossRefGoogle Scholar
  24. Meyer RB (1969) Piezoelectric effects in liquid crystals. Phys Rev Lett 22:918–921CrossRefGoogle Scholar
  25. Radi E (2003) Strain gradient effects on steady-state crack growth in linear hardening materials. J Mech Phys Solids 51:543–573MathSciNetCrossRefGoogle Scholar
  26. Sharma P, Maranganti R, Sharma ND (2006) Electromechanical coupling in nanopiezoelectric materials due to nanoscale nonlocal size effects: green function solution and embedded inclustions. Phys Rev B 74:014110CrossRefGoogle Scholar
  27. Shen SP, Hu SL (2010) A theory of flexoelectricity with surface effect for elastic dielectrics. J Mech Phys Solids 58:665–677MathSciNetCrossRefGoogle Scholar
  28. Shi MX, Huang Y, Hwang KC (2000) Fracture in a higher-order elastic continuum. J Mech Phys Solids 48: 2513–2538CrossRefGoogle Scholar
  29. Shvartsman VV, Emelyanov AY, Kholkin AL, Safari A (2002) Local hysteresis and grain size effects in Pb(Mg1/3Nb2/3)O-SbTiO3. Appl Phys Lett 81: 117–119CrossRefGoogle Scholar
  30. Sladek J, Sladek V, Stanak P, Zhang Ch, Wünsche M (2011) An interaction integral method for computing fracture parameters in functionally graded magnetoelectroelastic composites. CMC Comput Mater Continua 586:1–34Google Scholar
  31. Sladek J, Sladek V, Stanak P, Zhang Ch, Tan CL (2017) Fracture mechanics analysis of size-dependent piezoelectric solids. Int J Solids Struct 113:1–9CrossRefGoogle Scholar
  32. Vardoulakis I, Exadaktylos G, Aifantis E (1996) Gradient elasticity with surface energy: mode-III crack problem. Int J Solids Struct 33:4531–4559CrossRefGoogle Scholar
  33. Wang GF, Yu SW, Feng XQ (2004) A piezoelectric constitutive theory with rotation gradient effects. Eur J Mech A Solids 23:455–466CrossRefGoogle Scholar
  34. Wei Y (2006) A new finite element method for strain gradient theories and applications to fracture analyses. Eur J Mech A Solids 25:897–913MathSciNetCrossRefGoogle Scholar
  35. Yaghoubi ST, Mousavi SM, Paavola J (2017) Buckling of centrosymmetric anisotropic beam structures within strain gradient elasticity. Int J Solids Struct 109:84–92CrossRefGoogle Scholar
  36. Yang XM, Hu YT, Yang JS (2004) Electric field gradient effects in anti-plane problems of polarized ceramics. Int J Solids Struct 41:6801–6811CrossRefGoogle Scholar
  37. Yang J (2004) Effects of electric field gradient on an antiplane crack in piezoelectric ceramics. Int J Fract 127:L111–L116CrossRefGoogle Scholar
  38. Zhu W, Fu JY, Li N, Cross LE (2006) Piezoelectric composite based on the enhanced flexoelectric effects. Appl Phys Lett 89:192904CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Construction and Architecture, Slovak Academy of SciencesBratislavaSlovakia

Section editors and affiliations

  • Eduard-Marius Craciun
    • 1
  1. 1.Faculty of Mechanical, Industrial and Maritime Engineering"Ovidius" University of ConstantaConstantaRomania