Phase-Field Modeling of Fracture in Ferroelectric Materials


This paper presents a family of phase-field models for the coupled simulation of the microstructure formation and evolution, and the nucleation and propagation of cracks in single and polycrystalline ferroelectric materials. The first objective is to introduce a phase-field model for ferroelectric single crystals. The model naturally couples two existing energetic phase-field approaches for brittle fracture and ferroelectric domain formation and evolution. Simulations show the interactions between the microstructure and the crack under mechanical and electromechanical loadings. Another objective of this paper is to encode different crack face boundary conditions into the phase-field framework since these conditions strongly affect the fracture behavior of ferroelectrics. The smeared imposition of these conditions are discussed and the results are compared with that of sharp crack models to validate the proposed approaches. Simulations show the effects of different conditions and electromechanical loadings on the crack propagation. In a third step, the model is modified by introducing a crack non-interpenetration condition in the variational approach to fracture accounting for the asymmetric behavior in tension and compression. The modified model makes it possible to explain anisotropic crack growth in ferroelectrics under the Vickers indentation loading. This model is also employed for the fracture analysis of multilayer ferroelectric actuators, which shows the potential of the model for future applications. The coupled phase-field model is also extended to polycrystals by introducing realistic polycrystalline microstructures in the model. Inter- and trans-granular crack propagation modes are observed in the simulations. Finally, and for completeness, the phase-field theory is extended to the simulation of the propagation of conducting cracks under purely electrical loading and to the three-dimensional simulation of crack propagation in ferroelectric single crystals. Salient features of the crack propagation phenomenon predicted by the simulations of this paper are directly compared with experimental observations.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29


  1. 1.

    Abdollahi A, Arias I (2011a) Phase-field modeling of the coupled microstructure and fracture evolution in ferroelectric single crystals. Acta Mater 59(12):4733–4746

    Google Scholar 

  2. 2.

    Abdollahi A, Arias I (2011b) Phase-field simulation of anisotropic crack propagation in ferroelectric single crystals: effect of microstructure on the fracture process. Model Simul Mater Sci Eng 19(7):074010

    Google Scholar 

  3. 3.

    Abdollahi A, Arias I (2012a) Crack initiation patterns at electrode edges in multilayer ferroelectric actuators. Smart Mater Struct 21(9):094011

    Google Scholar 

  4. 4.

    Abdollahi A, Arias I (2012b) Numerical simulation of intergranular and transgranular crack propagation in ferroelectric polycrystals. Int J Fract 174(1):3–15

    Google Scholar 

  5. 5.

    Abdollahi A, Arias I (2012c) Phase-field modeling of crack propagation in piezoelectric and ferroelectric materials with different electromechanical crack conditions. J Mech Phys Solids 60:2100–2126

    MathSciNet  Google Scholar 

  6. 6.

    Abdollahi A, Arias I (2013) Conducting crack propagation driven by electric fields in ferroelectric ceramics. Acta Mater 61(19):7087–7097

  7. 7.

    Abdollahi A, Arias I (2014) Three-dimensional simulation of crack propagation in ferroelectric polycrystals: effect of combined toughening mechanisms (Acta Mater 65:106–117).

  8. 8.

    Aburatani H, Harada S, Uchino K, Furuta A, Fuda Y (1994) Destruction mechanisms in ceramic multilayer actuators. JPN J Appl Phys 1(33):3091–3094

    Google Scholar 

  9. 9.

    Amor H, Marigo JJ, Maurini C (2009) Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J Mech Phys Solids 57(8):1209–1229

    MATH  Google Scholar 

  10. 10.

    Arias I, Serebrinsky S, Ortiz M (2006) A phenomenological cohesive model of ferroelectric fatigue. Acta Mater 54(4):975–984

    Google Scholar 

  11. 11.

    Benes M, Chalupecky V, Mikula K (2004) Geometrical image segmentation by the Allen–Cahn equation. Appl Numer Math 51(2–3):187–205

    MATH  MathSciNet  Google Scholar 

  12. 12.

    Beom HG, Atlurib SN (2003) Effect of electric fields on fracture behavior of ferroelectric ceramics. J Mech Phys Solids 51:1107–1125

    MATH  Google Scholar 

  13. 13.

    Beom HK, Youn SK (2004) Electrical fracture toughness for a conducting crack in ferroelectric ceramics. Int J Solids Struct 41(1):145–157

    MATH  Google Scholar 

  14. 14.

    Beom HG, Jeong KM, Park JY, Lin S, Kim GH (2009) Electrical failure of piezoelectric ceramics with a conductive crack under electric fields. Eng Fract Mech 76:2399–2407

    Google Scholar 

  15. 15.

    Bourdin B, Francfort GA, Marigo JJ (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826

    MATH  MathSciNet  Google Scholar 

  16. 16.

    Bourdin B (2007) Numerical implementation of the variational formulation for quasi-static brittle fracture. Interfaces Free Bound 9:411–430

    MATH  MathSciNet  Google Scholar 

  17. 17.

    Bourdin B, Francfort GA, Marigo JJ (2008) The variational approach to fracture. J Elast 91(1–3):5–148

    MATH  MathSciNet  Google Scholar 

  18. 18.

    Camacho GT, Ortiz M (1996) Computational modelling of impact damage in brittle materials. Int J Numer Methods Eng 33(20–22):2899–2938

    MATH  Google Scholar 

  19. 19.

    Dadvand P, Rossi R, Onate E (2010) An object-oriented environment for developing finite element codes for multi-disciplinary applications. Arch Comput Methods Eng 17(3):253–297

    MATH  Google Scholar 

  20. 20.

    Dayal K, Bhattacharya K (2007) A real-space non-local phase-field model of ferroelectric domain patterns in complex geometries. Acta Mater 55(6):1907–1917

    Google Scholar 

  21. 21.

    Deeg WFJ (1980) The analysis of dislocation, crack, and inclusion problems in piezoelectric solids. PhD thesis, Stanford University

  22. 22.

    Devonshire AF (1949) Theory of barium titanate 1. Philos Mag 40(309):1040–1063

    Google Scholar 

  23. 23.

    Devonshire AF (1951) Theory of barium titanate 2. Philos Mag 42(333):1065–1079

    MATH  Google Scholar 

  24. 24.

    Elhadrouz M, Zineb TB, Patoor E (2006) Finite element analysis of a multilayer piezoelectric actuator taking into account the ferroelectric and ferroelastic behaviors. Int J Eng Sci 44:996–1006

    Google Scholar 

  25. 25.

    Faber KT, Evans AG (1983) Intergranular crack-deflection toughening in silicon-carbide. J Am Ceram Soc 66(6): C94–C96

  26. 26.

    Fan D, Chen LQ (1997) Computer simulation of grain growth using a continuum field model. Acta Mater 45(2):611–622

    MathSciNet  Google Scholar 

  27. 27.

    Fang D, Jiang Y, Li S, Sun CT (2007) Interactions between domain switching and crack propagation in poled BaTiO\(_3\) single crystal under mechanical loading. Acta Mater 55(17):5758–5767

    Google Scholar 

  28. 28.

    Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342

    MATH  MathSciNet  Google Scholar 

  29. 29.

    Fu R, Zhang TY (2000) Effects of an electric field on the fracture toughness of poled lead zirconate titanate ceramics. J Am Ceram Soc 83(5):1215–1218

    Google Scholar 

  30. 30.

    Fu R, Qian CF, Zhang TY (2000) Electrical fracture toughness for conductive cracks driven by electric fields in piezoelectric materials. Appl Phys Lett 76(1):126–128

    Google Scholar 

  31. 31.

    Furuta A, Uchino K (1993) Dynamic observation of crack-propagation in piezoelectric multilayer actuators. J Am Ceram Soc 76:1615–1617

    Google Scholar 

  32. 32.

    Gao HJ, Zhang TY, Tong P (1997) Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J Mech Phys Solids 45(4):491–510

    Google Scholar 

  33. 33.

    Gehrig F, Jelitto H, Schneider GA (2008) Fracture criterion for a conducting crack in poled PZT-PIC 151 investigated by stable crack growth. Acta Mater 56(2):222–229

    Google Scholar 

  34. 34.

    Gong X, Suo Z (1996) Reliability of ceramic multilayer actuators: a nonlinear finite element simulation. J Mech Phys Solids 44:751–769

    Google Scholar 

  35. 35.

    Grah M, Alzebdeh K, Sheng PY, Vaudin MD, Bowman KJ, Ostoja-Starzewski M (1996) Brittle-intergranular failure in 2D microstructures: experiments and computer simulations. Acta Mater 44(10):4003–4018

    Google Scholar 

  36. 36.

    Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond Ser A–221:163–198

    Google Scholar 

  37. 37.

    Gu H, Wang J (2013) The influence of crack face electrical boundary conditions on the nonlinear behavior of ferroelectric single crystal. Smart Mater Struct 22(6):065001

    Google Scholar 

  38. 38.

    Guo XH, Fang DN, Soh AK, Kim HC, Lee JJ (2006) Analysis of piezoelectric ceramic multilayer actuators based on an electro-mechanical coupled meshless method. Acta Mech Sin 22:34–39

    MATH  Google Scholar 

  39. 39.

    Hackemann S, Pfeiffer W (2003) Domain switching in process zones of pzt: characterization by microdiffraction and fracture mechanical methods. J Eur Ceram Soc 23(1):141–151

    Google Scholar 

  40. 40.

    Hakim V, Karma A (2009) Laws of crack motion and phase-field models of fracture. J Mech Phys Solids 57(2):342–368

    MATH  Google Scholar 

  41. 41.

    Han X, Li X, Mao SX (2002) Toughening and weakening in ferroelectric ceramics by domain-switching process under mixed electric and mechanical loading. Metall Mater Trans A 33(9):2835–2845

    Google Scholar 

  42. 42.

    Hao TH, Shen ZY (1994) A new electric boundary-condition of electric fracture-mechanics and its applications. Eng Fract Mech 47(6):793–802

    Google Scholar 

  43. 43.

    Hao TH, Gong X, Suo Z (1996) Fracture mechanics for the design of ceramic multilayer actuators. J Mech Phys Solids 44:23–48

    Google Scholar 

  44. 44.

    Haug A, McMeeking RM (2006) Cracks with surface charge in poled ferroelectrics. Eur J Mech A Solids 25(1):24–41

    MATH  Google Scholar 

  45. 45.

    Heyer V, Schneider GA, Balke H, Drescher J, Bahr HA (1998) A fracture criterion for conducting cracks in homogeneously poled piezoelectric PZT-PIC 151 ceramics. Acta Mater 46(18):6615–6622

    Google Scholar 

  46. 46.

    Huber JE (2005) Micromechanical modelling of ferroelectrics. Curr Opin Solid State Mater Sci 9(3):100–106

    Google Scholar 

  47. 47.

    Hwang SC, Lynch CS, McMeeking RM (1995) Ferroelectric/ferroelastic interactions and a polarization switching model. Acta Mater 43(5):2073–2084

    Google Scholar 

  48. 48.

    Jacqmin D (1999) Calculation of two-phase Navier–Stokes flows using phase-field modeling. J Comput Phys 155(1):96–127

    MATH  MathSciNet  Google Scholar 

  49. 49.

    Jeong KM, Beom HG (2004) Conducting crack growth in ferroelectric ceramics subjected to electric loading. Smart Mater Struct 13(2):275

    Google Scholar 

  50. 50.

    Jiang Y, Zhang Y, Liu B, Fang D (2009) Study on crack propagation in ferroelectric single crystal under electric loading. Acta Mater 57:1630–1638

    Google Scholar 

  51. 51.

    Jones LJ, Motahari SM, VarlioglU M, Lienert U, Bernier JV, Hoffman M, Uestuendag E (2007) Crack tip process zone domain switching in a soft lead zirconate titanate ceramic. Acta Mater 55(16):5538–5548

    Google Scholar 

  52. 52.

    Kamlah M (2001) Ferroelectric and ferroelastic piezoceramics: modeling of electromechanical hysteresis phenomena. Contin Mech Thermodyn 13(4):219–268

    MATH  Google Scholar 

  53. 53.

    Kamlah M, Bohle U (2001) Finite element analysis of piezoceramic components taking into account ferroelectric hysteresis behavior. Int J Solids Struct 38:605–633

    MATH  Google Scholar 

  54. 54.

    Kanzig W (1957) Ferroelectrics and antiferroelectrics. In: Seitz F, Das TP, Turnbull D, Hahn EL (eds) Solid state physics. Academic Press, New York, p 5

    Google Scholar 

  55. 55.

    Kim SB, Kim DY, Kim JJ, Cho SH (1990) Effect of grain-size and poling on the fracture mode of lead zirconate titanate ceramics. J Am Ceram Soc 73:161–163

    Google Scholar 

  56. 56.

    Koh JH, Jeong SJ, Ha MS, Song JS (2004) Electric field induced fracture mechanism and aging of piezoelectric behavior in Pb(MgNb)O-3–Pb(ZrTi)O-3 multilayer ceramic actuators. Ceram Int 30:1863–1867

    Google Scholar 

  57. 57.

    Krill CE, Chen LQ (2002) Computer simulation of 3-D grain growth using a phase-field model. Acta Mater 50(12):3057–3073

    Google Scholar 

  58. 58.

    Kueck AM, Kim DK, Ramasse QM, De Jonghe LC, Ritchie RO (2008) Atomic-resolution imaging of the nanoscale origin of toughness in rare-earth doped SiC. Nano Lett 8:2935–2939

    Google Scholar 

  59. 59.

    Kuna M (2010) Fracture mechanics of piezoelectric materials: where are we right now? Eng Fract Mech 77(2):309–326

    MathSciNet  Google Scholar 

  60. 60.

    Landau L (1937) On the theory of phase transitions. Gordon and Breach, New York

    Google Scholar 

  61. 61.

    Landis CM (2003) On the fracture toughness of ferroelastic materials. J Mech Phys Solids 51:1347–1369

    MATH  Google Scholar 

  62. 62.

    Landis CM (2004a) Energetically consistent boundary conditions for electromechanical fracture. Int J Solids Struct 41(22–23):6291–6315

    MATH  Google Scholar 

  63. 63.

    Landis CM (2004b) Non-linear constitutive modeling of ferroelectrics. Curr Opin Solid State Mater Sci 8:59–69

    Google Scholar 

  64. 64.

    Li XF, Tang GJ (2003) Electroelastic analysis for a piezoelectric layer with surface electrodes. Mech Research Commun 30:345–351

    MATH  Google Scholar 

  65. 65.

    Li YL, Cross LE, Chen LQ (2005) A phenomenological thermodynamic potential for BaTiO\(_3\) single crystals. J Appl Phys 98(064):101

    Google Scholar 

  66. 66.

    Li WY, McMeeking RM, Landis CM (2008) On the crack face boundary conditions in electromechanical fracture and an experimental protocol for determining energy release rates. Eur J Mech A Solids 27(3):285–301

    MATH  Google Scholar 

  67. 67.

    Li W, Landis CM (2011) Nucleation and growth of domains near crack tips in single crystal ferroelectrics. Eng Fract Mech 78(7):1505–1513

    Google Scholar 

  68. 68.

    Li Q, Kuna M (2012) Evaluation of electromechanical fracture behavior by configurational forces in cracked ferroelectric polycrystals. Comput Mater Sci 57:94–101

    Google Scholar 

  69. 69.

    Linder C, 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 60:882–903

    MathSciNet  Google Scholar 

  70. 70.

    Lines M, Glass A (1979) Principles and applications of ferroelectrics and related materials. Clarendon Press, Oxford

    Google Scholar 

  71. 71.

    Liu TQ, Oates WS, Wan S, Lynch CS (2005) Crack initiation at electrode edges in PZN-4.5PT single crystals. J Intell Mater Syst Struct 16:373–379

    Google Scholar 

  72. 72.

    Lucato SLDE, Lupascu DC, Kamlah M, Rodel J, Lynch CS (2001) Constraint-induced crack initiation at electrode edges in piezoelectric ceramics. Acta Mater 49:2751–2759

    Google Scholar 

  73. 73.

    Lucato SLDE, Bahr HA, Pham VB, Lupascu DC, Balke H, Rodel J, Bahr U (2002) Electrically driven cracks in piezoelectric ceramics: experiments and fracture mechanics analysis. J Mech Phys Solids 50:2333–2353

    MATH  Google Scholar 

  74. 74.

    Lucato SLDE, Bahr HA, Pham VB, Lupascu DC, Balke H, Rodel J, Bahr U (2003) Crack deflection in piezoelectric ceramics. J Eur Ceram Soc 23:1147–1156

    Google Scholar 

  75. 75.

    Lynch CS, Chen L, Suo Z, McMeeking RM, Yang W (1995) Crack-growth in ferroelectric ceramics driven by cyclic polarization switching. J Intell Mater Syst Struct 6:191–198

    Google Scholar 

  76. 76.

    Lynch CS (1998) Fracture of ferroelectric and relaxor electro-ceramics: influence of electric field. Acta Mater 46(2):599–608

    Google Scholar 

  77. 77.

    McMeeking RM (1999) Crack tip energy release rate for a piezoelectric compact tension specimen. Eng Fract Mech 64(2):217–244

    Google Scholar 

  78. 78.

    McMeeking RM, Landis CM (2002) A phenomenological multi-axial constitutive law for switching in polycrystalline ferroelectric ceramics. Int J Eng Sci 40(14):1553–1577

    MATH  MathSciNet  Google Scholar 

  79. 79.

    McMeeking RM (2004) The energy release rate for a griffith crack in a piezoelectric material. Eng Fract Mech 71(7–8): 1149–1163

  80. 80.

    Meschke F, Kolleck A, Schneider GA (1997) R-curve behaviour of BaTiO\(_3\) due to stress-induced ferroelastic domain switching. J Am Ceram Soc 17(9):1143–1149

    Google Scholar 

  81. 81.

    Meschke F, Raddatz O, Kolleck A, Schneider GA (2000) R-curve behavior and crack-closure stresses in barium titanate and (Mg, Y)-PSZ ceramics. J Am Ceram Soc 83(2):353–361

    Google Scholar 

  82. 82.

    Miehe C, Welschinger F, Hofacker M (2010) A phase field model of electromechanical fracture. J Mech Phys Solids 58(10):1716–1740

    MATH  MathSciNet  Google Scholar 

  83. 83.

    Miehe C, Welschinger F, Hofacker M (2010) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int J Numer Methods Eng 83(10):1273–1311

    MATH  MathSciNet  Google Scholar 

  84. 84.

    Moes N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150

    MATH  Google Scholar 

  85. 85.

    Oliver J, Huespe AE, Pulido MDG, Chaves E (2002) From continuum mechanics to fracture mechanics: the strong discontinuity approach. Eng Fract Mech 69(2):113–136

    Google Scholar 

  86. 86.

    Park SB, Sun CT (1995) Fracture criteria for piezoelectric ceramics. J Am Ceram Soc 78(6):1475–1480

    Google Scholar 

  87. 87.

    Parton VZ (1976) Fracture mechanics of piezoelectric materials. Acta Astronaut 3(9–10):671–683

    MATH  Google Scholar 

  88. 88.

    Peco C, Rosolen A, Arroyo M (2013) An adaptive meshfree method for phase-field models of biomembranes. Part ii: a lagrangian approach for membranes in viscous fluids. J Comput Phys 249:320–336

    MATH  MathSciNet  Google Scholar 

  89. 89.

    Pisarenko GG, Chushko VM, Kovalev SP (1985) Anisotropy of fracture toughness of piezoelectric ceramics. J Am Ceram Soc 68(5):259–265

    Google Scholar 

  90. 90.

    Pojprapai S, Jones LJ, Vodenitcharova T, Bernier JV, Hoffman M (2011) Investigation of the domain switching zone near a crack tip in pre-poled lead zirconate titanate ceramic via in situ X-ray diffraction. Scr Mater 64(1):1–4

    Google Scholar 

  91. 91.

    Pritchard J, Bowen CR, Lowrie F (2001) Multilayer actuators: review. Brit Ceram Trans 100:265–273

    Google Scholar 

  92. 92.

    Qian TY, Tong CF, Zhang P (1998) Linear electro-elastic analysis of a cavity or a crack in a piezoelectric material. Int J Solids Struct 35(17):2121–2149

    MATH  Google Scholar 

  93. 93.

    Rajapakse RKND, Zeng X (2001) Toughening of conducting cracks due to domain switching. Acta Mater 49:877–885

    Google Scholar 

  94. 94.

    Ratz A, Ribalta A, Voigt A (2006) Surface evolution of elastically stressed films under deposition by a diffuse interface model. J Comput Phys 214(1):187–208

    MathSciNet  Google Scholar 

  95. 95.

    Ricoeur A, Kuna M (2003) Influence of electric fields on the fracture of ferroelectric ceramics. J Eur Ceram Soc 23(8):1313–1328

    Google Scholar 

  96. 96.

    Rosato C, Miehe D, Linder 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):141–161

    MathSciNet  Google Scholar 

  97. 97.

    Rosolen A, Peco C, Arroyo M (2013) An adaptive meshfree method for phase-field models of biomembranes. Part i: approximation with maximum-entropy basis functions. J Comput Phys. doi:10.1016/

  98. 98.

    Ru CQ, Mao X, Epstein M (1998) Electric-field induced interfacial cracking in multilayer electrostrictive actuators. J Mech Phys Solids 46:1301–1318

    MATH  MathSciNet  Google Scholar 

  99. 99.

    Ru CQ (2000) Exact solution for finite electrode layers embedded at the interface of two piezoelectric half-planes. J Mech Phys Solids 48:693–708

    MATH  MathSciNet  Google Scholar 

  100. 100.

    Schneider GA, Heyer V (1999) Influence of the electric field on vickers indentation crack growth in BaTiO\(_3\). J Eur Ceram Soc 19(6–7):1299–1306

    Google Scholar 

  101. 101.

    Schneider GA (2007) Influence of electric field and mechanical stresses on the fracture of ferroelectrics. Annu Rev Mater Res 37:491–538

    Google Scholar 

  102. 102.

    Schrade D, Mueller R, Xu BX, Gross D (2007) Domain evolution in ferroelectric materials: a continuum phase field model and finite element implementation. Comput Methods Appl Mech Eng 196(41–44):4365–4374

    MATH  Google Scholar 

  103. 103.

    Sekerka RF (2004) Morphology: from sharp interface to phase field models. J Cryst Growth 264(4):530–540

    Google Scholar 

  104. 104.

    Sheng JS, Landis CM (2007) Toughening due to domain switching in single crystal ferroelectric materials. Int J Fract 143(2):161–175

    MATH  Google Scholar 

  105. 105.

    Shieh J, Huber JE, Fleck NA (2006) Fatigue crack growth in ferroelectrics under electrical loading. J Eur Ceram Soc 26:95–109

    Google Scholar 

  106. 106.

    Shindo Y, Murakami H, Horiguchi K, Narita F (2002) Evaluation of electric fracture properties of piezoelectric ceramics using the finite element and single-edge precracked-beam methods. J Am Ceram Soc 85(5):1243–1248

    Google Scholar 

  107. 107.

    Shu YC, Bhattacharya K (2001) Domain patterns and macroscopic behaviour of ferroelectric materials. Philos Mag B 81(12):2021–2054

    Google Scholar 

  108. 108.

    Song YC, Soh AK, Ni Y (2007) Phase field simulation of crack tip domain switching in ferroelectrics. J Phys D Appl Phys 40(4):1175–1182

    Google Scholar 

  109. 109.

    Steinbach I (2009) Phase-field models in materials science. Model Simul Mater Sci Eng 17(7):073001

    MathSciNet  Google Scholar 

  110. 110.

    Su Y, Landis CM (2007) Continuum thermodynamics of ferroelectric domain evolution: theory, finite element implementation, and application to domain wall pinning. J Mech Phys Solids 55(2):280–305

    MATH  MathSciNet  Google Scholar 

  111. 111.

    Sukumar N, Srolovitz DJ, Baker TJ, Prevost JH (2003) Brittle fracture in polycrystalline microstructures with the extended finite element method. Int J Numer Methods Eng 56:2015–2037

    MATH  Google Scholar 

  112. 112.

    Sun CT, Park SB (2000) Measuring fracture toughness of piezoceramics by vickers indentation under the influence of electric fields. Ferroelectrics 248(1–4):79–95

    Google Scholar 

  113. 113.

    Suo Z (1991) Mechanics concepts for failure in ferroelectric ceramics. Am Soc Mech Eng 24:1–6

    Google Scholar 

  114. 114.

    Suo Z (1993) Models for breakdown-resistant dielectric and ferroelectric ceramics. J Mech Phys Solids 41:1155–1176

    Google Scholar 

  115. 115.

    Tieresten HF (1969) Linear piezoelectric plate vibrations. Plenum Press, New York

    Google Scholar 

  116. 116.

    Tobin AG, Pak YE (1993) Effects of electric fields on fracture behavior of pzt ceramics. Proc SPIE Smart Struct Mater 1916:78–86

    Google Scholar 

  117. 117.

    Uchino K, Takahashi S (1996) Multilayer ceramic actuators. Curr Opin Solid State Mater Sci 1:698–705

    Google Scholar 

  118. 118.

    Uchino K, Furuta A (1992) Destruction mechanism of multilayer ceramic actuators. In: Liu M, Safari A, Kingon A, Haertling G (eds): Proceedings of 8th IEEE International Symposium Applications of Ferroelectrics (ISAF 92), pp 195–198

  119. 119.

    Vendik OG, Zubko SP (2000) Ferroelectric phase transition and maximum dielectric permittivity of displacement type ferroelectrics \((\text{ Ba }_x\text{ Sr }_{1-x}\text{ TiO }_3)\). J Appl Phys 88:5343–5350

    Google Scholar 

  120. 120.

    Verhoosel CV, Gutierrez MA (2009) Modelling inter- and transgranular fracture in piezoelectric polycrystals. Eng Fract Mech 76(6):742–760

    Google Scholar 

  121. 121.

    Wang X (2005) Phase field models and simulations of vesicle bio-membranes. PhD thesis, Pennsylvania State University

  122. 122.

    Wang HY, Singh RN (1997) Crack propagation in piezoelectric ceramics: effects of applied electric fields. J Appl Phys 81(11):7471–7479

    Google Scholar 

  123. 123.

    Wang J, Landis CM (2006) Effects of in-plane electric fields on the toughening behavior of ferroelectric ceramics. J Mech Mater Struct 1(6):1075–1095

    Google Scholar 

  124. 124.

    Wang J, Zhang TY (2007) Phase field simulations of polarization switching-induced toughening in ferroelectric ceramics. Acta Mater 55(7):2465–2477

    Google Scholar 

  125. 125.

    Wang YL, Tagantsev AK, Damjanovic D, Setter N, Yarmarkin VK, Sokolov AI, Lukyanchuk IA (2007) Landau thermodynamic potential for BaTiO\(_3\). J Appl Phys 101(104):115

    Google Scholar 

  126. 126.

    Wang J, Kamlah M (2009) Three-dimensional finite element modeling of polarization switching in a ferroelectric single domain with an impermeable notch. Smart Mater Struct 18(104):008

  127. 127.

    Wang J, Kamlah M (2010) Effect of electrical boundary conditions on the polarization distribution around a crack embedded in a ferroelectric single domain. Eng Fract Mech 77(18): 3658–3669

  128. 128.

    Westrain I, Oates WS, Lupascu DC, Roedel J, Lynch CS (2007) Mechanism of electric fatigue crack growth in lead zirconate titanate. Acta Mater 55:301–312

    Google Scholar 

  129. 129.

    Westram I, Lupascu D, Roedel J, Laskewitz B, Kamlah M (2007) Electric-field-induced crack initiation from a notch in a ferroelectric ceramic. J Am Ceram Soc 90:2849–2854

    Google Scholar 

  130. 130.

    Wilson ZA, Borden MJ, Landis CM (2013) A phase-field model for fracture in piezoelectric ceramics. Int J Fract .doi:10.1007/s10704-013-9881-9

  131. 131.

    Winzer SR, Shankar N, Ritter AP (1989) Designing cofired multilayer electrostrictive actuators for reliability. J Am Ceram Soc 72:2246–2257

    Google Scholar 

  132. 132.

    Xu XP, Needleman A (1994) Numerical simulations of fast crack-growth in brittle solids. J Mech Phys Solids 42(9):1397–1434

    MATH  Google Scholar 

  133. 133.

    Xu BX, Schrade D, Mueller R, Gross D (2009) Micromechanical analysis of ferroelectric structures by a phase field method. Comput Mater Sci 45(3):832–836

    Google Scholar 

  134. 134.

    Xu BX, Schrade D, Gross D, Mueller R (2010) Phase field simulation of domain structures in cracked ferroelectrics. Int J Fract 165:163–173

    MATH  Google Scholar 

  135. 135.

    Yang W, Suo Z (1994) Cracking In ceramic actuators caused by electrostriction. J Mech Phys Solids 42:649–663

    Google Scholar 

  136. 136.

    Yang W, Zhu T (1998) Switch-toughening of ferroelectrics subjected to electric fields. J Mech Phys Solids 46(2):291–311

    MATH  Google Scholar 

  137. 137.

    Yang L, Dayal K (2011) Effect of lattice orientation, surface modulation, and applied fields on free-surface domain structure in ferroelectrics. Acta Mater 59(17):6594–6603

    Google Scholar 

  138. 138.

    Yang L, Dayal K (2012) Microstructure and stray electric fields at surface cracks in ferroelectrics. Int J Fract 174:17–27

    Google Scholar 

  139. 139.

    Ye RQ, He LH (2001) Electric field and stresses concentrations at the edge of parallel electrodes in piezoelectric ceramics. Int J Solids Struct 38:6941–6951

    MATH  Google Scholar 

  140. 140.

    Zhang TY, Wang TH, Zhao MH (2003) Failure behavior and failure criterion of conductive cracks (deep notches) in thermally depoled PZT-4 ceramics. Acta Mater 51(16):4881–4895

    Google Scholar 

  141. 141.

    Zhang TY, Gao CF (2004) Fracture behaviors of piezoelectric materials. Theor Appl Fract Mech 41(1–3):339–379

    Google Scholar 

  142. 142.

    Zhang TY, Liu GN, Wang Y (2004) Failure behavior and failure criterion of conductive cracks (deep notches) in piezoelectric ceramics II: experimental verification. Acta Mater 52:2025–2035

  143. 143.

    Zhang W, Bhattacharya K (2005a) A computational model of ferroelectric domains. Part i: model formulation and domain switching. Acta Mater 53(1):185–198

    Google Scholar 

  144. 144.

    Zhang W, Bhattacharya K (2005b) A computational model of ferroelectric domains. Part ii: grain boundaries and defect pinning. Acta Mater 53(1):199–209

  145. 145.

    Zhang TY, Liu G, Wang T, Tong P (2007) Application of the concepts of fracture mechanics to the failure of conductive cracks in piezoelectric ceramics. Eng Fract Mech 74(7):1160–1173

    Google Scholar 

  146. 146.

    Zhao XJ, Liu B, Fang DN (2010) Study on electroelastic field concentration around the electrode tip in multilayer ferroelectric actuators of two designs and their optimizations. Int J Plast 26:533–548

    MATH  Google Scholar 

  147. 147.

    Zhu T, Yang W (1997) Toughness variation of ferroelectrics by polarization switch under non-uniform electric field. Acta Mater 45(11):4695–4702

    Google Scholar 

  148. 148.

    Zhu T, Yang W (1999) Fatigue crack growth in ferroelectrics driven by cyclic electric loading. J Mech Phys Solids 47(1):81–97

    MATH  Google Scholar 

Download references


The authors gratefully acknowledge the support of the Ministerio de Ciencia e Innovación (DPI2011-26589).

Author information



Corresponding author

Correspondence to Irene Arias.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Abdollahi, A., Arias, I. Phase-Field Modeling of Fracture in Ferroelectric Materials. Arch Computat Methods Eng 22, 153–181 (2015).

Download citation


  • Ferroelectricity
  • Piezoelectricity
  • Fracture
  • Phase-field models
  • Polycrystals
  • Finite element analysis
  • Domain switching