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A Peridynamic Model of Fracture Mechanics with Bond-Breaking


We present a new formulation of a peridynamic model for brittle fracture that incorporates a properly defined bond-breaking rule which leads to a dynamic system of time-dependent differential integral equations having both spatial nonlocal/nonlinear interactions and temporal memory/history dependence. The dynamic system is shown to be well-posed through rigorous mathematical analysis. Its effectiveness in simulating crack propagation in a two dimensional brittle material is also demonstrated.

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  1. 1.

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Askari, E., Bobaru, F., Lehoucq, R.B., Parks, M.L., Silling, S.A., Weckner, O.: Peridynamics for multiscale materials modeling. J. Phys. Conf. Ser. 125, 012078 (2008)

    Article  Google Scholar 

  3. 3.

    Aguiar, A.R., Fosdick, R.: A constitutive model for a linearly elastic peridynamic body. Math. Mech. Solids 19(5), 502–523 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Silling, S., Weckner, O., Askari, E., Bobaru, F.: Crack nucleation in a peridynamic solid. Int. J. Fract. 162(1), 219–227 (2010)

    Article  MATH  Google Scholar 

  5. 5.

    Oterkus, E., Madenci, E.: Peridynamic analysis of fiber-reinforced composite materials. J. Mech. Mater. Struct. 7(1), 45–84 (2012)

    Article  Google Scholar 

  6. 6.

    Bobaru, F., Duangpanya, M.: The peridynamic formulation for transient heat conduction. Int. J. Heat Mass Transf. 53(19), 4047–4059 (2010)

    Article  MATH  Google Scholar 

  7. 7.

    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 

  8. 8.

    Hu, W., Ha, Y.D., Bobaru, F.: Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites. Comput. Methods Appl. Mech. Eng. 217, 247–261 (2012)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E.: Peridynamic states and constitutive modeling. J. Elast. 88(2), 151–184 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Aksoylu, B., Parks, M.L.: Variational theory and domain decomposition for nonlocal problems. Appl. Math. Comput. 217(14), 6498–6515 (2011)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J.: Nonlocal Diffusion Problems. Mathematical Surveys and Monographs, vol. 165. Am. Math. Soc., Providence (2010)

    Book  MATH  Google Scholar 

  12. 12.

    Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Models Methods Appl. Sci. 23, 493–540 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 56, 676–696 (2012)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Mengesha, T., Du, Q.: The bond-based peridynamic system with Dirichlet type volume constraint. Proc. R. Soc. Edinb. A 144, 161–186 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Mengesha, T., Du, Q.: Nonlocal constrained value problems for a linear peridynamic Navier equation. J. Elast. 116(1), 27–51 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Emmrich, E., Weckner, O.: On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity. Commun. Math. Sci. 5(4), 851–864 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Emmrich, E., Puhst, D.: Well-posedness of the peridynamic model with Lipschitz continuous pairwise force function. Commun. Math. Sci. 11(4), 1039–1049 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Lipton, R.: Dynamic brittle fracture as a small horizon limit of peridynamics. J. Elast. 117(1), 21–50 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Mengesha, T., Du, Q.: Characterization of function spaces of vector fields and an application in nonlinear peridynamics. Nonlinear Anal. 140, 82–111 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Tian, X., Du, Q.: A class of high order nonlocal operators. Arch. Ration. Mech. Anal. 222(3), 1521–1553 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Emmrich, E., Puhst, D.: A short note on modeling damage in peridynamics. J. Elast. 123(2), 245–252 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Bobaru, F., Zhang, G.: Why do cracks branch? A peridynamic investigation of dynamic brittle fracture. Int. J. Fract. 196(1–2), 59–98 (2015)

    Article  Google Scholar 

  23. 23.

    Parks, M., Plimpton, S., Lehoucq, R., Silling, S.A.: Peridynamics with LAMMPS: a user guide. Sandia National Laboratory Report, SAND2008-0135, Albuquerque, New Mexico (2008)

  24. 24.

    Mengesha, T., Du, Q.: Analysis of a scalar nonlocal peridynamic model with a sign changing kernel. Discrete Contin. Dyn. Syst., Ser. B 18(5), 1415–1437 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

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

    Article  Google Scholar 

  26. 26.

    Fosdick, R., Royer-Carfagni, G.: The constraint of local injectivity in linear elasticity theory. Proc. R. Soc. Lond. A, Math. Phys. Eng. Sci. 457, 2167–2187 (2001)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Hale, J.K.: Functional differential equations. In: Analytic Theory of Differential Equations, pp. 9–22. Springer, Berlin (1971)

    Chapter  Google Scholar 

  28. 28.

    Wu, J.: Theory and Applications of Partial Functional Differential Equations, vol. 119. Springer, Berlin (2012)

    Google Scholar 

  29. 29.

    Gajewski, H., Gröger, K., Zacharias, K.: Nonlinear Operator Equations and Operator Differential Equations, p. 336. Mir, Moscow (1978)

    Google Scholar 

  30. 30.

    Du, Q.: Local limits and asymptotically compatible discretizations. In: Handbook of Peridynamic Modeling, pp. 87–108. CRC Press, Boca Raton (2016).

    Google Scholar 

  31. 31.

    Tian, X., Du, Q.: Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations. SIAM J. Numer. Anal. 51, 3458–3482 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Tian, X., Du, Q.: Asymptotically compatible schemes and applications to robust discretization of nonlocal models. SIAM J. Numer. Anal. 52, 1641–1665 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Parks, M., Littlewood, D., Mitchell, J., Silling, S.A.: Peridigm users’ guide v1.0.0. Sandia Rep. 2012-7800, Sandia National Laboratories, Albuquerque, New Mexico (2012)

  34. 34.

    Tao, Y., Tian, X., Du, Q.: Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations. Appl. Math. Comput. 305, 282–298 (2017)

    MathSciNet  Google Scholar 

  35. 35.

    Aubin, J.P., Cellina, A.: Differential Inclusions: Set-Valued Maps and Viability Theory. Springer, New York (1984)

    Book  MATH  Google Scholar 

  36. 36.

    Deimling, K.: Multivalued Differential Equations, vol. 1. de Gruyter, Berlin (1992)

    Book  MATH  Google Scholar 

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We thank Dr. Stewart Silling, Prof. Florin Bobaru and Dr. Pablo Seleson for their helpful discussions on the subject. We also thank Prof. Robert Lipton for bringing [21] to our attention.

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Correspondence to Qiang Du.

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Du, Q., Tao, Y. & Tian, X. A Peridynamic Model of Fracture Mechanics with Bond-Breaking. J Elast 132, 197–218 (2018).

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  • Peridynamics
  • Fracture
  • Bond-breaking
  • Well-posedness
  • Energy decay
  • Nonlocal model
  • Memory effect
  • Crack propagation
  • Convergence of numerical simulations

Mathematics Subject Classification

  • 74H20
  • 74H25
  • 45K05
  • 47G10
  • 74R10