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Peridynamic formulation of the mean stress and incubation time fracture criteria and its correspondence to the classical Griffith’s approach

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Abstract

Peridynamic formulations of the mean stress and incubation time fracture models are discussed in the paper. Contemporary fracture simulations using the peridynamic theory often rely on critical bond stretch fracture criterion which is known to operate similar to energy-based fracture models. The energy-based fracture criteria—both in classical Griffith’s and Irwin’s form—are known to be powerful tools for fracture simulations and analysis. However, a number of experimentally observed dynamic fracture effects cannot be captured by these models, e.g. rate sensitivity of the material toughness. Thus, coupling of peridynamic approach with alternative stress-based fracture models would possibly broaden the peridynamics applicability. Here implementation technique of the aforementioned fracture model is discussed and its results for the case of a dynamically propagating crack with relatively low velocity due to quasistatic load appear to be in good agreement with the classical energy release rate approach.

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References

  1. Griffith, A.A.: VI. The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. Ser. A, Contain. Pap. Math. Phys. Character 221(582–593), 163–198 (1921)

  2. Kashtanov, A., Petrov, Yu.V.: Fractal models in fracture mechanics. Int. J. Fract. 128, 271–276 (2004)

    Article  MATH  Google Scholar 

  3. Ma, C.C., Freund, L.B.: The extent of the stress intensity factor field during crack growth under dynamic loading conditions. ASME J. Appl. Mech. 53, 303–310 (1986). https://doi.org/10.1115/1.3171756

    Article  ADS  Google Scholar 

  4. Ravi-Chandar, K., Knauss, W.G.: An experimental investigation into dynamic fracture: III. On steady-state crack propagation and crack branching. Int. J. Fract. 26(2), 141–154 (1984)

  5. Kalthoff, J.F.: On some current problems in Experimental Fracture Dynamics, pp. 11–25. California Institute of Technology, Workshop on Dynamic Fracture (1983)

  6. Petrov, Y.V., Utkin, A.A.: Dependence of the dynamic strength on loading rate. Sov. Mater. Sci. 25(2), 153–156 (1989). https://doi.org/10.1007/BF00780499

    Article  Google Scholar 

  7. Petrov, Y.V., Morozov, N.F.: On the modeling of fracture of brittle solids. J. Appl. Mech. 61, 710–712 (1994). https://doi.org/10.1115/1.2901518

    Article  ADS  Google Scholar 

  8. Silling, S.A., Askari, E.: A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83(17–18), 1526–1535 (2005)

    Article  Google Scholar 

  9. Ignatiev, M.O., Petrov, Y.V., Kazarinov, N.A.: Simulation of dynamic crack initiation based on the peridynamic numerical model and the incubation time criterion. Tech. Phys. 66(3), 422–425 (2021)

    Article  Google Scholar 

  10. Ignatev, M., Kazarinov, N., Petrov, Yu.: Peridynamic modelling of the dynamic crack initiation. Procedia Struct. Integr. 28C, 1657–1661 (2020)

    Google Scholar 

  11. Parks, M.L., Littlewood, D.J., Mitchell, J.A., Silling, S.A.: Peridigm Users’ Guide, Tech. Report SAND2012-7800, Sandia National Laboratories (2012)

  12. Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48(1), 175–209 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Silling, S.A., et al.: Peridynamic states and constitutive modeling. J. Elast. 88(2), 151–184 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Madenci, E., Oterkus, E.: Peridynamic Theory and its Applications. Springer, New York (2014)

    Book  MATH  Google Scholar 

  15. Bobaru, F., Foster, J.T., Geubelle, P.H., Silling, S.A. (eds.): Handbook of Peridynamic Modeling. CRC Press (2016)

  16. Oterkus, E., Oterkus, S., Madenci, E. (eds.): Peridynamic Modeling, Numerical Techniques, and Applications. Elsevier (2021)

  17. 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(8), 887–928 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dell’Isola, F., Della, Corte A., Giorgio, I.: Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Math. Mech. Solids 22(4), 852–872 (2017)

  19. Marconi, V.I., Jagla, E.A.: Diffuse interface approach to brittle fracture. Phys. Rev. E 71, 036110 (2005)

    Article  ADS  Google Scholar 

  20. Randles, P., Libersky, L.: Recent improvements in SPH modeling of hypervelocity impact. Int. J. Impact Eng. 20, 525–532 (1997)

    Article  Google Scholar 

  21. http://www.peridynamics.org/what-is-peridynamics

  22. Placidi, E.L., Barchiesi, E., Misra, A., Timofeev, D.: Micromechanics-based elasto-plastic-damage energy formulation for strain gradient solids with granular microstructure. Contin. Mech. Thermodyn. 33, 2213–2241 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  23. Timofeev, D., Barchiesi, E., Misra, A., Placidi, L.: Hemivariational continuum approach for granular solids with damage-induced anisotropy evolution. Math. Mech. Solids 26(5), 738–770 (2021). https://doi.org/10.1177/1081286520968149

    Article  MathSciNet  MATH  Google Scholar 

  24. Neuber, H.: Kerbspannungslehre (Notch Stresses). Verlag von Julius Springer, Berlin (1937)

    Book  Google Scholar 

  25. Novozhilov, V.V.: On a necessary and sufficient criterion for brittle strength: PMM vol. 33, no. 2, 1969, 212–222. J. Appl. Math. Mech. 33(2), 201–210 (1969)

  26. Petrov, Y.V.: On “quantum” nature of dynamic fracture of brittle solids. Dokl. Akad. Nauk. USSR 321, 66–68 (1991)

  27. Maksimov, V., Barchiesi, E., Misra, A., Placidi, L., and Timofeev, D.: Two-dimensional analysis of size effects in strain gradient granular solids with damage-induced anisotropy evolution. J. Eng. Mech. 04021098-1 (2021)

  28. De Angelo, M., Spagnuolo, M., D’Annibale, F., et al.: The macroscopic behavior of pantographic sheets depends mainly on their microstructure: experimental evidence and qualitative analysis of damage in metallic specimens. Contin. Mech. Thermodyn. 31, 1181–1203 (2019). https://doi.org/10.1007/s00161-019-00757-3

    Article  ADS  Google Scholar 

  29. dell’Isola, F., Seppecher, P., Alibert, J.J., et al.: Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Contin. Mech. Thermodyn. 31(4), 851–884 (2019). https://doi.org/10.1007/s00161-018-0689-8

    Article  ADS  MathSciNet  Google Scholar 

  30. Gruzdkov, A., Sitnikova, E., Morozov, N., Petrov, Y.: Thermal effect in dynamic yielding and fracture of metals and alloys. Math. Mech. Solids 14, 72–87 (2009). https://doi.org/10.1177/1081286508092603

    Article  MathSciNet  MATH  Google Scholar 

  31. Petrov, Y.V., Karihaloo, B.L., Bratov, V.V., Bragov, A.M.: Multi-scale dynamic fracture model for quasi-brittle materials. Int. J. Eng. Sci. 61, 3–9 (2009). https://doi.org/10.1016/j.ijengsci.2012.06.004

    Article  MATH  Google Scholar 

  32. Bratov, V., Petrov, Y.: Application of incubation time approach to simulate dynamic crack propagation. Int. J. Fract. 146, 53–60 (2007). https://doi.org/10.1007/s10704-007-9135-9

    Article  Google Scholar 

  33. Evstifeev, A., Kazarinov, N., Petrov, Yu.V., Witek, L., Bednarz, A.: Experimental and theoretical analysis of solid particle erosion of a steel compressor blade based on incubation time concept. Eng. Fail. Anal. 87, 15–21 (2018). https://doi.org/10.1016/j.engfailanal.2018.01.006

    Article  Google Scholar 

  34. Hattori, G., Trevelyan, J., Coombs, W.M.: A non-ordinary state-based peridynamics framework for anisotropic materials. Comput. Methods Appl. Mech. Eng. 339, 416–442 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Stenström, C., Eriksson, K.: The J-contour integral in peridynamics via displacements. Int. J. Fract. 216(2), 173–183 (2019)

    Article  Google Scholar 

  36. Ha, Y.D., Bobaru, F.: Studies of dynamic crack propagation and crack branching with peridynamics. Int. J. Fract. 162(1–2), 229–244 (2010)

    Article  MATH  Google Scholar 

  37. Kazarinov, N., Bratov, V., Petrov, Y.V.: Simulation of dynamic crack propagation under quasistatic loading. In: Applied Mechanics and Materials, vol. 532, pp. 337–341. Trans Tech Publications Ltd (2014)

  38. Fineberg, J., Gross, S.P., Marder, M., Swinney, H.L.: Instability in the propagation of fast cracks. Phys. Rev. B 45(10), 5146 (1992)

    Article  ADS  Google Scholar 

  39. Le, Q.V., Bobaru, F.: Surface corrections for peridynamic models in elasticity and fracture. Comput. Mech. 61(4), 499–518 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the Russian Foundation for Basic Research (No. 19-31-60037). Sections 1 and 5 were made by Yuri Petrov within the framework of the Russian Science Foundation project (No. 22-11-00091). The authors are grateful to Mikhnovich I.V. for assistance in writing a program code.

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Authors and Affiliations

  1. Saint Petersburg State University, Saint Petersburg, Russia

    M. O. Ignatiev, Y. V. Petrov & N. A. Kazarinov

  2. Emperor Alexander I Saint Petersburg State Transport University, Saint Petersburg, Russia

    Y. V. Petrov & N. A. Kazarinov

  3. Institute of Problems of Mechanical Engineering, Saint Petersburg, Russia

    E. Oterkus

  4. University of Strathclyde, Glasgow, UK

    N. A. Kazarinov

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  1. M. O. Ignatiev
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  2. Y. V. Petrov
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  3. N. A. Kazarinov
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  4. E. Oterkus
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Correspondence to N. A. Kazarinov.

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Communicated by Andreas Öchsner.

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Ignatiev, M.O., Petrov, Y.V., Kazarinov, N.A. et al. Peridynamic formulation of the mean stress and incubation time fracture criteria and its correspondence to the classical Griffith’s approach. Continuum Mech. Thermodyn. 35, 1523–1534 (2023). https://doi.org/10.1007/s00161-022-01159-8

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