Kinetic relations and local energy balance for LEFM from a nonlocal peridynamic model

Abstract

A simple nonlocal field theory of peridynamic type is applied to model brittle fracture. The kinetic relation for the crack tip velocity given by Linear Elastic Fracture Mechanics (LEFM) is recovered directly from the nonlocal dynamics, this is seen both theoretically and in simulations. An explicit formula for the change of internal energy inside a neighborhood enclosing the crack tip is found for the nonlocal model and applied to LEFM.

This is a preview of subscription content, log in to check access.

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

References

  1. Atkinson C, Eshelby JD (1968) The flow of energy into the tip of a moving crack. Int J Fract 4:3–8

    Article  Google Scholar 

  2. Anderson TL (2005) Fracture mechanics: fundamentals and applications, 3rd edn. Taylor & Francis, Boca Raton

    Google Scholar 

  3. Bouchbinder E, Goldman T, Fineberg J (2014) The dynamics of rapid fracture: instabilities, nonlinearities and length scales. Rep Prog Phys 77(4):046501

    Article  Google Scholar 

  4. Freund LB (1972) Energy flux into the tip of an extending crack in an elastic solid. J Elast 2:341–349

    Article  Google Scholar 

  5. Freund B (1990) Dynamic fracture mechanics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge

    Google Scholar 

  6. Freund B, Clifton RJ (1974) On the uniqueness of plane elastodynamic solutions for running cracks. J Elast 4:293–299

    Article  Google Scholar 

  7. Goldman T, Livne A, Fineberg J (2010) Acquisition of inertia by a moving crack. Phys Rev Lett 104(11):114301

    Article  Google Scholar 

  8. Ha YD, Bobaru F (2010) Studies of dynamic crack propagation and crack branching with peridynamics. Int J Fract 162:229–244

    Article  Google Scholar 

  9. Hu W, Ha YD, Bobaru F, Silling S (2012) The formulation and computation of the nonlocal J-integral in bond-based peridynamics. Int J Fract 176:195–206

    Article  Google Scholar 

  10. Irwin G R (1967) Constant speed, semi-infinite tensile crack opened by a line force. Lehigh University Memorandum

  11. Jha PK, Lipton, (2020) Finite element convergence for state-based peridynamic fracture models. Commun Appl Math Comput 2:93–128

  12. Jha PK, Lipton R (2019b) Numerical convergence of finite difference approximations for state based peridynamic fracture models. Comput Meth Appl Mech Eng 351:184–225. https://doi.org/10.1016/j.cma.2019.03.024

    Article  Google Scholar 

  13. Kostrov BV, Nikitin LV (1970) Some general problems of mechanics of brittle fracture. Arch Mech Stosowanej. 22:749–775

    Google Scholar 

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

    Article  Google Scholar 

  15. Lipton R (2016) Cohesive dynamics and brittle fracture. J Elast 124(2):143–191

    Article  Google Scholar 

  16. Lipton R, Jha P K (2020). Plane elastodynamic solutions for running cracks as the limit of double well nonlocal dynamics. arXiv:2001.00313

  17. Mott NF (1948) Fracture in mild steel plates. Engineering 165:16–18

    Google Scholar 

  18. Nillison F (1974) A note on the stress singularity at a non-uniformly moving crack tip. J Elast 4:293–299

    Article  Google Scholar 

  19. Ravi-Chandar K (2004) Dynamic fracture. Elsevier, Oxford

    Google Scholar 

  20. Rice JR (1968) A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 9:379–386

    Article  Google Scholar 

  21. Rice JR (1968) Mathematical analysis in the mechanics of fracture. Fracture: An advanced treatise, vol II. Academic Press, New York, p 191

    Google Scholar 

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

    Article  Google Scholar 

  23. Sih GC (1968) Some elastodynamic problems of cracks. Int J Fract Mech 4:51–68

    Article  Google Scholar 

  24. Sih GC (1970) Dynamic aspects of crack propagation. Inelastic Behavior of Solids, McGraw-Hill, pp 607–633

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  27. Silling SA, Lehoucq RB (2010) Peridynamic theory of solid mechanics. Adv Appl Mech 44:73–168

    Article  Google Scholar 

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

  29. Slepian Y (2002) Models and phenomena in fracture mechanics foundations of engineering mechanics. Springer, Berlin

    Google Scholar 

  30. Willis JR (1975) Equations of motion for propagating cracks. The mechanics and physics of fracture. The Metals Society, New York, pp 57–67

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Robert P. Lipton.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This material is based upon work supported by the U. S. Army Research Laboratory and the U. S. Army Research Office under contract/grant number W911NF1610456.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jha, P.K., Lipton, R.P. Kinetic relations and local energy balance for LEFM from a nonlocal peridynamic model. Int J Fract 226, 81–95 (2020). https://doi.org/10.1007/s10704-020-00480-0

Download citation

Keywords

  • Fracture
  • Peridynamics
  • LEFM
  • Fracture toughness
  • Stress intensity
  • Local power balance