International Journal of Fracture

, Volume 201, Issue 2, pp 157–170 | Cite as

Bond-based peridynamics: a quantitative study of Mode I crack opening

  • Patrick DiehlEmail author
  • Fabian Franzelin
  • Dirk Pflüger
  • Georg C. Ganzenmüller
Original Paper


This paper shows a new approach to estimate the critical traction for Mode I crack opening before crack growth by numerical simulation. For quasi-static loading, Linear Elastic Fracture Mechanics predicts the critical traction before crack growth. To simulate the crack growth, we used bond-based peridynamics, a non-local generalization of continuum mechanics. We discretize the peridynamics equation of motion with a collocation by space approach, the so-called EMU nodal discretization. As the constitutive law, we employ the improved prototype micro brittle material model. This bond-based material model is verified by the Young’s modulus from classical theory for a homogeneous deformation for different quadrature rules. For the EMU-ND we studied the behavior for different ratios of the horizon and nodal spacing to gain a robust value for a large variety of materials. To access this wide range of materials, we applied sparse grids, a technique to build high-dimensional surrogate models. Sparse grids significantly reduce the number of simulation runs compared to a full grid approach and keep up a similar approximation accuracy. For the validation of the quasi-static loading process, we show that the critical traction is independent of the material density for most material parameters. The bond-based IPMB model with EMU nodal discretization seems very robust for the ratio \({\delta }/{\varDelta X}=3\) for a wide range of materials, if an error of 5 % is acceptable.


Bond-based peridynamics EMU-ND Critical traction Sparse grids 


  1. Agwai A, Guven I, Madenci E (2011) Predicting crack propagation with peridynamics: a comparative study. Int J Fract 171(1):65–78CrossRefGoogle Scholar
  2. Berger M, Bokhari S (1987) A partitioning strategy for nonuniform problems on multiprocessors. IEEE Trans Comput C-36(5):570–580Google Scholar
  3. Bobaru F, Duangpanya M (2012) A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities. J Comput Phys 231(7):2764–2785CrossRefGoogle Scholar
  4. Bobaru F, Yang M, Alves LF, Silling SA, Askari E, Xu J (2009) Convergence, adaptive refinement, and scaling in 1d peridynamics. Int J Numer Methods Eng 77(6):852–877CrossRefGoogle Scholar
  5. Breitenfeld M (2014) Quasi-static non-ordinary state-based peridynamics for the modeling of 3d fracture. PhD thesis, University of Illinois at Urbana-ChampaignGoogle Scholar
  6. Bungartz HJ, Griebel M (2004) Sparse grids. Acta Numer 13:1–123CrossRefGoogle Scholar
  7. Chen X, Gunzburger M (2011) Continuous and discontinuous finite element methods for a peridynamics model of mechanics. Comput Methods Appl Mech Eng 200(912):1237–1250CrossRefGoogle Scholar
  8. Cook R, Young W (1998) Advanced mechanics of materials, 2nd edn. Prentice Hall, Englewood CliffsGoogle Scholar
  9. Diehl P (2012) Implementing peridynamics theory on GPU. Diplomarbeit, Institute of Parallel and Distributed Systems, University of StuttgartGoogle Scholar
  10. Du Q, Tian X (2015) Robust discretization of nonlocal models related to peridynamics. In: Griebel M, Schweitzer MA (eds) Meshfree methods for partial differential equations VII, Lecture notes in computational science and engineering, vol 100, Springer, pp 97–113Google Scholar
  11. Emmrich E, Weckner O (2007) The peridynamic equation and its spatial discretisation. Math Model Anal 12(1):17–27CrossRefGoogle Scholar
  12. Franzelin F, Diehl P, Pflüger D (2015) Non-intrusive uncertainty quantification with sparse grids for multivariate peridynamic simulations. In: Griebel M, Schweitzer MA (eds) Meshfree methods for partial differential equations VII, Lecture notes in computational science and engineering, vol 100, Springer, pp 115–143Google Scholar
  13. Ganzenmüller GC, Hiermaier S, May M (2014) Improvements to the prototype micro-brittle linaear elasticity model of peridynamics. In: Griebel M, Schweitzer MA (eds) Meshfree methods for partial differential equations VII, vol 100. Lecture notes in computational science and engineering. SpringerGoogle Scholar
  14. Ha YD, Bobaru F (2010) Studies of dynamic crack propagation and crack branching with peridynamics. Int J Fract 162(1–2):229–244CrossRefGoogle Scholar
  15. Ha YD, Bobaru F (2011) Characteristics of dynamic brittle fracture captured with peridynamics. Eng Fract Mech 78(6):1156–1168CrossRefGoogle Scholar
  16. Hu W, Ha YD, Bobaru F (2010) Numerical integration in peridynamics. Technical report, University of Nebraska-LincolnGoogle Scholar
  17. Hu W, Ha YD, Bobaru F (2012a) Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites. Comput Methods Appl Mech Eng 217220:247–261CrossRefGoogle Scholar
  18. Hu W, Ha YD, Bobaru F, Silling SA (2012b) The formulation and computation of the nonlocal j-integral in bond-based peridynamics. Int J Fract 176(2):195–206CrossRefGoogle Scholar
  19. Kilic B, Madenci E (2010) Coupling of peridynamic theory and the finite element method. J Mech Mater Struct 5(5):707–733CrossRefGoogle Scholar
  20. Liu W, Hong J (2012) Discretized peridynamics for linear elastic solids. Comput Mech 50(5):579–590CrossRefGoogle Scholar
  21. Macek RW, Silling SA (2007) Peridynamics via finite element analysis. Finite Elem Anal Des 43(15):1169–1178CrossRefGoogle Scholar
  22. Pflüger D (2010) Spatially adaptive sparse grids for high-dimensional problems. Verlag Dr. Hut, MünchenGoogle Scholar
  23. Pflüger D (2012) Spatially adaptive refinement. In: Garcke J, Griebel M (eds) Sparse grids and applications, Lecture notes in computational science and engineering. Springer, Berlin, pp 243–262Google Scholar
  24. Plimpton S (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput Phys 117:1–19Google Scholar
  25. Sarego G, Le QV, Bobaru F, Zaccariotto M, Galvanetto U (2016) Linearized state-based peridynamics for 2-D problems. Int J Numer Methods Eng. doi: 10.1002/nme.5250
  26. Seleson P (2014) Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations. Comput Methods Appl Mech Eng 282:184–217CrossRefGoogle Scholar
  27. Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48(1):175–209CrossRefGoogle Scholar
  28. Silling SA (2011) A coarsening method for linear peridynamics. Int J Multiscale Com Eng 9(6):609–621CrossRefGoogle Scholar
  29. Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83:1526–1535CrossRefGoogle Scholar
  30. Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic states and constititive modeling. J Elast 88:151–184CrossRefGoogle Scholar
  31. Silling SA, Weckner O, Askari E, Bobaru F (2010) Crack nucleation in a peridynamic solid. Int J Fract 162(1–2):219–227CrossRefGoogle Scholar
  32. Thompson AP, Plimpton SJ, Mattson W (2009) General formulation of pressure and stress tensor for arbitrary many-body interaction potentials under periodic boundary conditions. J Chem Phys 131(15):154107CrossRefGoogle Scholar
  33. Tian X, Du Q (2013) Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations. SIAM J Numer Anal 51(6):3458–3482CrossRefGoogle Scholar
  34. Weckner O, Emmrich E (2005) Numerical simulation of the dynamics of a nonlocal, inhomogeneous, infinite bar. J Comput Appl Mech 6(2):311–319Google Scholar
  35. Weiner JH (2002) Statistical mechanics of elasticity, 2nd edn. Dover, New YorkGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Patrick Diehl
    • 1
    Email author
  • Fabian Franzelin
    • 2
  • Dirk Pflüger
    • 2
  • Georg C. Ganzenmüller
    • 3
  1. 1.Institut für Numerische SimulationBonnGermany
  2. 2.IPVS/SGSStuttgartGermany
  3. 3.Fraunhofer EMIFreiburgGermany

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