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

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Abstract

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.

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References

  1. Aranson, I.S., Kalatsky, V.A., Vinokur, V.M.: Continuum field description of crack propagation. Phys. Rev. Lett. 85, 118–121 (2000)

    Article  ADS  Google Scholar 

  2. Karma, A., Kessler, D.A., Levine, H.: Phase-field model of mode III dynamic fracture. Phys. Rev. Lett. 87, 045501 (2001)

    Article  ADS  Google Scholar 

  3. Henry, H., Levine, H.: Dynamic instabilities of fracture under biaxial strain using a phase field model. Phys. Rev. Lett. 93, 105504 (2004)

    Article  ADS  Google Scholar 

  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). https://doi.org/10.1007/s00161-018-0685-z

    Article  Google Scholar 

  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). https://doi.org/10.1016/j.ijplas.2018.07.005

    Article  Google Scholar 

  6. Levitas, V.I., Javanbakht, M.: Phase transformations in nanograin materials under high pressure and plastic shear: nanoscale mechanisms. Nanoscale 6, 162–166 (2014)

    Article  ADS  Google Scholar 

  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). https://doi.org/10.1016/j.commatsci.2015.10.037

    Article  Google Scholar 

  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). https://doi.org/10.1016/j.commatsci.2018.07.034

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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). https://doi.org/10.1016/j.jmps.2015.05.005

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  13. Levitas, V.I., Javanbakht, M.: Surface-induced phase transformations: multiple scale and mechanics effects and morphological transitions. Phys. Rev. Lett. 107, 175701 (2011)

    Article  ADS  Google Scholar 

  14. Levitas, V.I., Javanbakht, M.: Phase field approach to interaction of phase transformation and dislocation evolution. Appl. Phys. Lett. 102, 251904 (2013)

    Article  ADS  Google Scholar 

  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). https://doi.org/10.1016/j.jmps.2015.05.009

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  17. Javanbakht, M., Levitas, V.I.: Nanoscale mechanisms for high-pressure mechanochemistry: a phase field study. J. Mater. Sci. 53(19), 13343–13363 (2018)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  21. Placidi, L.: A variational approach for a nonlinear 1-dimensional second gradient continuum damage model. Contin. Mech. Thermodyn. 27, 623–638 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  22. Levitas, V.I.: Thermodynamically consistent phase field approach to phase transformations with interface stresses. Acta Mater. 61, 4305–4319 (2013). https://doi.org/10.1016/j.actamat.2013.03.034

    Article  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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). https://doi.org/10.1039/C4SM01129E

    Article  ADS  Google Scholar 

  25. Chuang, T.J.: Effect of surface tension on the toughness of glass. J. Am. Ceram. Soc. 70, 160–164 (1987)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  27. Placidi, L., Barchiesi, E.: Energy approach to brittle fracture in strain-gradient modelling. Proc. R. Soc. A 474, 20170878 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  29. Placidi, L., Misra, A., Barchiesi, E.: Simulation results for damage with evolving microstructure and growing strain gradient moduli. Contin. Mech. Thermodyn. (2018). https://doi.org/10.1007/s00161-018-0693-z

    Article  ADS  MathSciNet  Google Scholar 

  30. Porter, D.A., Easterling, K.E., Sherif, M.: Phase Transformations in Metals and Alloys. (Revised Reprint). CRC Press, Boca Raton (2009)

    Book  Google Scholar 

  31. Cuomo, M.: Continuum damage model for strain gradient materials with applications to 1D examples. Contin. Mech. Thermodyn. (2018). https://doi.org/10.1007/s00161-018-0698-7

    Article  ADS  MathSciNet  Google Scholar 

  32. Cuomo, M.: Continuum model of microstructure induced softening for strain gradient materials. Math. Mech. Solids (2018). https://doi.org/10.1177/1081286518755845

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mechanical Engineering, Sharif University of Technology, Tehran, 11365-11155, Iran

    Hossein Jafarzadeh & Gholam Hossein Farrahi

  2. Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, 84156-83111, Iran

    Mahdi Javanbakht

Authors
  1. Hossein Jafarzadeh
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  2. Gholam Hossein Farrahi
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  3. Mahdi Javanbakht
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Correspondence to Gholam Hossein Farrahi.

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Communicated by Francesco dell’Isola.

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Jafarzadeh, H., Farrahi, G.H. & Javanbakht, M. Phase field modeling of crack growth with double-well potential including surface effects. Continuum Mech. Thermodyn. 32, 913–925 (2020). https://doi.org/10.1007/s00161-019-00775-1

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  • DOI: https://doi.org/10.1007/s00161-019-00775-1

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