Phase field modeling of crack growth with double-well potential including surface effects

  • Hossein Jafarzadeh
  • Gholam Hossein FarrahiEmail author
  • Mahdi Javanbakht
Original Article


A thermodynamically consistent phase field approach for fracture including surface stresses is presented. The surface stresses which are distributed inside a finite width layer at the crack surface are introduced as a result of employing some geometrical nonlinearities, i.e., by defining energy terms per unit volume at the current time and evaluating gradient of the order parameter in the current configuration. A double-well barrier term is included in the structure of the Helmholtz free energy which allows one to use the energy terms per unit volume at the current time when introducing the surface stresses in the current approach. Thus, the surface stresses are introduced in a similar way to the interfacial stresses in phase transformations. The differences in the modeling of crack growth with considering the surface stresses and without it are discussed. It is shown how the surface stresses affect the stress fields and consequently the crack nucleation and propagation. The finite element method is utilized to solve the coupled equations of mechanics and crack phase field. It is emphasized that the surface stresses affect the driving force for both the crack nucleation and propagation by disturbing the momentum balance. Thus, a different external loading is required in the presence of the surface stresses.


Phase field Crack propagation Surface stress and energy Continuum thermodynamics 


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  1. 1.
    Aranson, I.S., Kalatsky, V.A., Vinokur, V.M.: Continuum field description of crack propagation. Phys. Rev. Lett. 85, 118–121 (2000)ADSCrossRefGoogle Scholar
  2. 2.
    Karma, A., Kessler, D.A., Levine, H.: Phase-field model of mode III dynamic fracture. Phys. Rev. Lett. 87, 045501 (2001)ADSCrossRefGoogle Scholar
  3. 3.
    Henry, H., Levine, H.: Dynamic instabilities of fracture under biaxial strain using a phase field model. Phys. Rev. Lett. 93, 105504 (2004)ADSCrossRefGoogle Scholar
  4. 4.
    Farrahi, G.H., Javanbakht, M., Jafarzadeh, H.: On the phase field modeling of crack growth and analytical treatment on the parameters. Contin. Mech. Thermodyn. (2018). Google Scholar
  5. 5.
    Levitas, V.I., Jafarzadeh, H., Farrahi, G.H., Javanbakht, M.: Thermodynamically consistent and scale-dependent phase field approach for crack propagation allowing for surface stresses. Int. J. Plast. 111, 1–35 (2018). CrossRefGoogle Scholar
  6. 6.
    Levitas, V.I., Javanbakht, M.: Phase transformations in nanograin materials under high pressure and plastic shear: nanoscale mechanisms. Nanoscale 6, 162–166 (2014)ADSCrossRefGoogle Scholar
  7. 7.
    Javanbakht, M., Barati, E.: Martensitic phase transformations in shape memory alloy: phase field modeling with surface tension effect. Comput. Mater. Sci. 115, 137–144 (2016). CrossRefGoogle Scholar
  8. 8.
    Mirzakhani, S., Javanbakht, M.: Phase field-elasticity analysis of austenite-martensite phase transformation at the nanoscale: finite element modeling. Comput. Mater. Sci. 154, 41–52 (2018). CrossRefGoogle Scholar
  9. 9.
    Javanbakht, M., Levitas, V.I.: Phase field approach to dislocation evolution at large strains: computational aspects. Int. J. Solids Struct. 82, 95–110 (2016)CrossRefGoogle Scholar
  10. 10.
    Javanbakht, M., Levitas, V.I.: Interaction between phase transformations and dislocations at the nanoscale. Part 2: phase field simulation examples. J. Mech. Phys. Solids 82, 164–185 (2015)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Levitas, V.I., Javanbakht, M.: Interaction between phase transformations and dislocations at the nanoscale. Part 1: general phase field approach. J. Mech. Phys. Solids 82, 287–319 (2015). ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Levitas, V.I., Javanbakht, M.: Surface tension and energy in multivariant martensitic transformations: phase-field theory, simulations, and model of coherent interface. Phys. Rev. Lett. 105, 165701 (2010)ADSCrossRefGoogle Scholar
  13. 13.
    Levitas, V.I., Javanbakht, M.: Surface-induced phase transformations: multiple scale and mechanics effects and morphological transitions. Phys. Rev. Lett. 107, 175701 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    Levitas, V.I., Javanbakht, M.: Phase field approach to interaction of phase transformation and dislocation evolution. Appl. Phys. Lett. 102, 251904 (2013)ADSCrossRefGoogle Scholar
  15. 15.
    Levitas, V.I., Javanbakht, M.: Thermodynamically consistent phase field approach to dislocation evolution at small and large strains. J. Mech. Phys. Solids 82, 345–366 (2015). ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Javanbakht, M., Levitas, V.I.: Phase field simulations of plastic strain-induced phase transformations under high pressure and large shear. Phys. Rev. B 94, 214104 (2016)ADSCrossRefGoogle Scholar
  17. 17.
    Javanbakht, M., Levitas, V.I.: Nanoscale mechanisms for high-pressure mechanochemistry: a phase field study. J. Mater. Sci. 53(19), 13343–13363 (2018)ADSCrossRefGoogle Scholar
  18. 18.
    Rinaldi, A., Placidi, L.: A microscale second gradient approximation of the damage parameter of quasi-brittle heterogeneous lattices. ZAMM J. Appl. Math. Mech. 94, 862–877 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Math. Mech. Solids 20, 887–928 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dell’Isola, F., Seppecher, P., Della Corte, A.: The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. Proc. R. Soc. A 471, 20150415 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Placidi, L.: A variational approach for a nonlinear 1-dimensional second gradient continuum damage model. Contin. Mech. Thermodyn. 27, 623–638 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Levitas, V.I.: Thermodynamically consistent phase field approach to phase transformations with interface stresses. Acta Mater. 61, 4305–4319 (2013). CrossRefGoogle Scholar
  23. 23.
    Levitas, V.I.: Phase field approach to martensitic phase transformations with large strains and interface stresses. J. Mech. Phys. Solids 70, 154–189 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Liu, T., Long, R., Hui, C.-Y.: The energy release rate of a pressurized crack in soft elastic materials: effects of surface tension and large deformation. Soft Matter 10, 7723–7729 (2014). ADSCrossRefGoogle Scholar
  25. 25.
    Chuang, T.J.: Effect of surface tension on the toughness of glass. J. Am. Ceram. Soc. 70, 160–164 (1987)CrossRefGoogle Scholar
  26. 26.
    Dell’Isola, F., Corte, A.D., Giorgio, I.: Higher-gradient continua: the legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Math. Mech. Solids 22, 852–872 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Placidi, L., Barchiesi, E.: Energy approach to brittle fracture in strain-gradient modelling. Proc. R. Soc. A 474, 20170878 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Placidi, L., Misra, A., Barchiesi, E.: Two-dimensional strain gradient damage modeling: a variational approach. Zeitschrift für angewandte Mathematik und Physik 69, 56 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Placidi, L., Misra, A., Barchiesi, E.: Simulation results for damage with evolving microstructure and growing strain gradient moduli. Contin. Mech. Thermodyn. (2018).
  30. 30.
    Porter, D.A., Easterling, K.E., Sherif, M.: Phase Transformations in Metals and Alloys. (Revised Reprint). CRC Press, Boca Raton (2009)Google Scholar
  31. 31.
    Cuomo, M.: Continuum damage model for strain gradient materials with applications to 1D examples. Contin. Mech. Thermodyn. (2018).
  32. 32.
    Cuomo, M.: Continuum model of microstructure induced softening for strain gradient materials. Math. Mech. Solids (2018).

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mechanical EngineeringSharif University of TechnologyTehranIran
  2. 2.Department of Mechanical EngineeringIsfahan University of TechnologyIsfahanIran

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