International Journal of Fracture

, Volume 214, Issue 1, pp 49–68 | Cite as

A peridynamic failure analysis of fiber-reinforced composite laminates using finite element discontinuous Galerkin approximations

  • Bo Ren
  • C. T. WuEmail author
  • Pablo Seleson
  • Danielle Zeng
  • Dandan Lyu
Original Paper


This paper presents a discontinuous Galerkin weak form for bond-based peridynamic models to predict the damage of fiber-reinforced composite laminates. To represent the anisotropy of a laminate in a peridynamic model, a lamina is simplified as a transversely isotropic medium under a plane stress condition. The laminated structure is modeled by stacking the surface mesh layers along the thickness direction according to the laminate sequence. To avoid a mesh dependence on either the fiber orientation or the discretization, the spherical harmonic expansion theory is employed to construct a function for the micro-elastic modulus in terms of the bond-fiber angle. The laminate material is decomposed into an isotropic matrix material part and a transversely isotropic material part. The bond stiffness can be evaluated using the engineering material constants, based on the equivalence between the elastic energy density in the peridynamic theory and the elastic energy density in the classic continuum mechanics theory. Benchmark tests are conducted to verify the proposed model. Numerical results illustrate that the convergence of simulations with different horizon sizes and meshes can be achieved. In terms of damage analysis, the proposed model can capture the dynamic process of the complex coupling of the inner-layer and delamination damage modes.


Peridynamic theory Finite element method Failure analysis Discontinuous Galerkin method Fiber-reinforced composites 



This work was sponsored by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory (ORNL), managed by UT-Battelle, LLC, for the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. The authors would also like to thank Dr. John O. Hallquist from Livermore Software Technology Corporation for his support to this research. Helpful discussions with Dr. Stewart Silling from Sandia National Laboratories, New Mexico, as well as with Dr. Mazdak Ghajari from Imperial College are gratefully acknowledged.


  1. Azdoud Y, Han F, Lubineau G (2014) The morphing method as a flexible tool for adaptive local/non-local simulation of static fracture. Comput Mech 54:711–722CrossRefGoogle Scholar
  2. Belytschko T, Lin JI (1987) A three-dimensional impact-penetration algorithm with erosion. Int J Impact Eng 5:111–127CrossRefGoogle Scholar
  3. Bobaru F, Hu W (2012) The meaning, selection, and use of the peridynamic horizon and its relation to crack branching in brittle materials. Int J Fract 176:215–222CrossRefGoogle Scholar
  4. Chen X, Gunzburger M (2011) Continuous and discontinuous finite element methods for a peridynamics model of mechanics. Comput Methods Appl Mech Eng 200:1237–1250CrossRefGoogle Scholar
  5. Chen JS, Wu CT, Belytschko T (2000) Regularization of material instabilities by meshfree approximations with intrinsic length scales. Int J Numer Methods Eng 47:1303–1322CrossRefGoogle Scholar
  6. Foster JT, Silling SA, Chen WW (2010) Viscoplasticity using peridynamics. Int J Numer Methods Eng 81:1242–1258Google Scholar
  7. Gerstle W, Sau N, Silling SA (2007) Peridynamic modeling of concrete structures. Nucl Eng Des 237:1250–1258CrossRefGoogle Scholar
  8. Ghajari M, Iannucci L, Curtis P (2014) A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media. Comput Methods Appl Mech Eng 276:431–452CrossRefGoogle Scholar
  9. Han F, Lubineau G, Azdoud Y, Askari A (2016) A morphing approach to couple state-based peridynamics with classical continuum mechanics. Comput Methods Appl Mech Eng 301:336–358CrossRefGoogle Scholar
  10. Hu YL, Madenci E (2016) Bond-based peridynamic modeling of composite laminates with arbitrary fiber orientation and stacking sequence. Compos Struct 153:139–175CrossRefGoogle Scholar
  11. Hu W, Ha YD, Bobaru F (2011) Modeling dynamic fracture and damage in a fiber-reinforced composite lamina with peridynamics. Int J Multiscale Comput Eng 9(6):707–726CrossRefGoogle Scholar
  12. Hu W, Ha YD, Bobaru F (2012) Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites. Comput Methods Appl Mech Eng 217–220:247–261CrossRefGoogle Scholar
  13. Hu YL, Yu Y, Wang H (2014) Peridynamic analytical method for progressive damage in notched composite laminates. Compos Struct 108:801–810CrossRefGoogle Scholar
  14. Hu YL, Carvalho NV, Madenci E (2015) Peridynamic modeling of delamination growth in composite laminates. Compos Struct 132:610–620CrossRefGoogle Scholar
  15. Johnson GR, Stryk RA (1987) Eroding interface and improved tetrahedral element algorithms for high-velocity impact computations in three dimensions. Int J Impact Eng 5:411–421CrossRefGoogle Scholar
  16. Kaminski MM (2005) Computational mechanics of composite materials. Springer-Verlag, LondonGoogle Scholar
  17. Kilic B, Madenci E, Ambur DR (2006) Analysis of brazed single-lap joints using the peridynamics theory. In: 47th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference. Newport, Rhode IslandGoogle Scholar
  18. Kröner E (1967) Elasticity theory of materials with long range cohesive forces. Int J Solids Struct 3:731–742CrossRefGoogle Scholar
  19. Lai X, Ren B, Fan H, Li S, Wu CT, Regueiro RA, Liu L (2015) Peridynamics simulations of geomaterial fragmentation by impulse loads. Int J Numer Anal Methods Geomech 39:1304–1330CrossRefGoogle Scholar
  20. Lehoucq RB, Silling SA (2008) Force flux and the peridynamic stress tensor. J Mech Phys Solids 56:1566–1577CrossRefGoogle Scholar
  21. Liu Z, Bessa MA, Liu WK (2016) Self-consistent clustering analysis: an efficient multi-scale scheme for inelastic heterogeneous materials. Comput Methods Appl Mech Eng 306:319–341CrossRefGoogle Scholar
  22. Madenci E, Oterkus E (2014) Peridynamic theory and its applications. Springer, New YorkCrossRefGoogle Scholar
  23. Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150CrossRefGoogle Scholar
  24. Oterkus E (2010) Peridynamic theory for modeling three-dimensional damage growth in metallic and composite structures. Ph.D. Dissertation, Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZGoogle Scholar
  25. Ren B, Li S (2012) Modeling and simulation of large-scale ductile fracture in plates and shells. Int J Solids Struct 49:2373–2393CrossRefGoogle Scholar
  26. Ren B, Wu CT, Askari E (2017) A 3D discontinuous Galerkin finite element method with the bond-based peridynamics model for dynamic brittle failure analysis. Int J Impact Eng 99:14–25CrossRefGoogle Scholar
  27. Seleson P, Littlewood DJ (2016) Convergence studies in meshfree peridynamic simulations. Comput Math Appl 71:2432–2448CrossRefGoogle Scholar
  28. Seleson P, Parks ML (2011) On the role of the influence function in the peridynamic theory. Int J Multiscale Comput Eng 9(6):689–706CrossRefGoogle Scholar
  29. Seleson P, Gunzburger M, Parks ML (2013) Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains. Comput Methods Appl Mech Eng 266:185–204CrossRefGoogle Scholar
  30. Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48:175–209CrossRefGoogle Scholar
  31. Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83:1526–1535CrossRefGoogle Scholar
  32. Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88:151–184CrossRefGoogle Scholar
  33. Wu CT, Ren B (2015) A stabilized non-ordinary state-based peridynamics for the nonlocal ductile material failure analysis in metal machining process. Comput Methods Appl Mech Eng 291:197–215CrossRefGoogle Scholar
  34. Wu CT, Ma N, Takada K, Okada H (2016) A meshfree continuous-discontinuous approach for the ductile fracture modeling in explicit dynamics analysis. Comput Mech 58:391–409CrossRefGoogle Scholar
  35. Xu XP, Needleman A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solid 42:1397–1434CrossRefGoogle Scholar
  36. Xu J, Askari A, Weckner O, Silling SA (2008) Peridynamic analysis of impact damage in composite laminates. J Aerosp Eng 21:187–194CrossRefGoogle Scholar
  37. Zhang B (2016) Grain-scale computational modeling of quasi-static and dynamic loading on natural soils. Ph.D. Thesis, Department of Civil, Environmental, and Architectural Engineering, University of Colorado at BoulderGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Bo Ren
    • 1
  • C. T. Wu
    • 1
    Email author
  • Pablo Seleson
    • 2
  • Danielle Zeng
    • 3
  • Dandan Lyu
    • 4
  1. 1.Livermore Software Technology CorporationLivermoreUSA
  2. 2.Computer Science and Mathematics DivisionOak Ridge National LaboratoryOak RidgeUSA
  3. 3.Ford Motor Company, Ford Research and Innovation CenterDearbornUSA
  4. 4.Department of Civil and Environmental EngineeringThe University of CaliforniaBerkeleyUSA

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