Acta Mechanica Sinica

, Volume 35, Issue 5, pp 1021–1032 | Cite as

Improved method for zero-energy mode suppression in peridynamic correspondence model

  • Ji Wan
  • Zhuang Chen
  • Xihua ChuEmail author
  • Hui Liu
Research Paper


The peridynamic correspondence model provides a general formulation to incorporate the classical local model and, therefore, helps to solve mechanical problems with discontinuities easily. But it suffers from zero-energy mode instability in numerical implementation due to the approximation of deformation gradient tensor. To suppress zero-energy modes, previous stabilized methods were generally more based on adding a supplemental force state derived from bond-based peridynamic theory, which requires a bond-based peridynamic micro-modulus. In this work, we present an improved stabilized method where the stabilization force state is derived directly from the peridynamic correspondence model. Hence, the bond-based peridynamic micro-modulus is abandoned. This improved method needs no extra constant to control the magnitude of stabilization force state and it is suitable for either isotropic or anisotropic materials. Several examples are presented to demonstrate its performance in simulating crack propagation, and numerical results show its efficiency and effectiveness.

Graphical abstract


Peridynamic correspondence model Zero-energy modes Anisotropic material Damage 



This work was supported by the National Natural Science Foundation of China (Grants 11472196, 11172216 and 11772237).


  1. 1.
    Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Madenci, E., Oterkus, E.: Peridynamic Theory and Its Applications. Springer, New York (2014)CrossRefGoogle Scholar
  3. 3.
    Gerstle, W., Sau, N., Silling, S.: Peridynamic modeling of concrete structures. Nucl. Eng. Des. 237, 1250–1258 (2007)CrossRefGoogle Scholar
  4. 4.
    Oterkus, E., Madenci, E.: Peridynamic analysis of fiber-reinforced composite materials. J. Mech. Mater. Struct. 7, 45–84 (2015)CrossRefGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Zhou, X.P., Gu, X.B., Wang, Y.T.: Numerical simulations of propagation, bifurcation and coalescence of cracks in rocks. Int. J. Rock Mech. Min. Sci. 80, 241–254 (2015)CrossRefGoogle Scholar
  7. 7.
    Oterkus, E., Guven, I., Madenci, E.: Fatigue failure model with peridynamic theory. In: 2010 12th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems, Las Vegas, June 2–5 (2010)Google Scholar
  8. 8.
    Hu, Y.L., Madenci, E.: Peridynamics for fatigue life and residual strength prediction of composite laminates. Compos. Struct. 160, 169–184 (2017)CrossRefGoogle Scholar
  9. 9.
    Zhu, F., Zhao, J.: A peridynamic investigation on crushing of sand particles. Géotechnique. 69, 529–540 (2019)Google Scholar
  10. 10.
    Fan, H., Bergel, G.L., Li, S.: A hybrid peridynamics–SPH simulation of soil fragmentation by blast loads of buried explosive. Int. J. Impact Eng. 87, 14–27 (2016)CrossRefGoogle Scholar
  11. 11.
    Madenci, E., Dorduncu, M., Barut, A., et al.: A state-based peridynamic analysis in a finite element framework. Eng. Fract. Mech. 195, 104–128 (2018)CrossRefGoogle Scholar
  12. 12.
    Parks, M.L., Lehoucq, R.B., Plimpton, S.J., et al.: Implementing peridynamics within a molecular dynamics code. Comput. Phys. Commun. 179, 777–783 (2007)CrossRefGoogle Scholar
  13. 13.
    Kaushik, D., Kaushik, B.: Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. J. Mech. Phys. Solids 54, 1811–1842 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Roy, P., Pathrikar, A., Deepu, S.P., et al.: Peridynamics damage model through phase field theory. Int. J. Mech. Sci. 128–129, 181–193 (2017)CrossRefGoogle Scholar
  15. 15.
    Bessa, M.A., Foster, J.T., Belytschko, T., et al.: A meshfree unification: reproducing kernel peridynamics. Comput. Mech. 53, 1251–1264 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ganzenmüller, G.C., Hiermaier, S., May, M.: On the similarity of meshless discretizations of peridynamics and smooth-particle hydrodynamics. Comput. Struct. 150, 71–78 (2014)CrossRefGoogle Scholar
  17. 17.
    Silling, S.A., Epton, M., Weckner, O., et al.: Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Silling, S.A., Lehoucq, R.B.: Peridynamic theory of solid mechanics. Adv. Appl. Mech. 44, 73–168 (2010)CrossRefGoogle Scholar
  19. 19.
    Jiang, T., Ren, J.L., Lu, W.G., et al.: A corrected particle method with high-order Taylor expansion for solving the viscoelastic fluid flow. Acta Mech. Sin. 33, 20–39 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Littlewood, D.J.: Simulation of dynamic fracture using peridynamics, finite element modeling, and contact. In: ASME 2010 International Mechanical Engineering Congress and Exposition, Vancouver, November 12–18 (2010)Google Scholar
  21. 21.
    Littlewood, D.J.: A nonlocal approach to modeling crack nucleation in AA 7075-T651. In: ASME 2011 International Mechanical Engineering Congress and Exposition, Denver, November 11–17 (2011)Google Scholar
  22. 22.
    Tupek, M.R., Radovitzky, R.: An extended constitutive correspondence formulation of peridynamics based on nonlinear bond-strain measures. J. Mech. Phys. Solids 65, 82–92 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Breitenfeld, M.S., Geubelle, P.H., Weckner, O., et al.: Non-ordinary state-based peridynamic analysis of stationary crack problems. Comput. Methods Appl. Mech. Eng. 272, 233–250 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Wu, C.T.: Kinematic constraints in the state-based peridynamics with mixed local/nonlocal gradient approximations. Comput. Mech. 54, 1255–1267 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wu, C.T., Ren, B.: A stabilized non-ordinary state-based peridynamics for the nonlocal ductile material failure analysis in metal machining process. Comput. Methods Appl. Mech. Eng. 291, 197–215 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yaghoobi, A., Mi, G.C.: Higher-order approximation to suppress the zero-energy mode in non-ordinary state-based peridynamics. Comput. Struct. 188, 63–79 (2017)CrossRefGoogle Scholar
  27. 27.
    Du, Q., Tian, X.C.: Stability of nonlocal dirichlet integrals and implications for peridynamic correspondence material modeling. SIAM J. Appl. Math. 78, 1536–1552 (2018)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Gu, X., Madenci, E., Zhang, Q.: Revisit of non-ordinary state-based peridynamics. Eng. Fract. Mech. 190, 31–52 (2018)CrossRefGoogle Scholar
  29. 29.
    Chen, H.: Bond-associated deformation gradients for peridynamic correspondence model. Mech. Res. Commun. 90, 34–41 (2018)CrossRefGoogle Scholar
  30. 30.
    Chen, H., Spencer, B.W.: Peridynamic bond-associated correspondence model: Stability and convergence properties. Int. J. Numer. Methods Eng. 117, 713–727 (2019)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Luo, J., Sundararaghavan, V.: Stress-point method for stabilizing zero-energy modes in non-ordinary state-based peridynamics. Int. J. Solids Struct. 150, 197–207 (2018)CrossRefGoogle Scholar
  32. 32.
    Silling, S.A.: Stability of peridynamic correspondence material models and their particle discretizations. Comput. Methods Appl. Mech. Eng. 322, 42–57 (2017)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Li, P., Hao, Z.M., Zhen, W.Q.: A stabilized non-ordinary state-based peridynamic model. Comput. Methods Appl. Mech. Eng. 339, 262–280 (2018)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Nicely, C., Tang, S., Qian, D.: Nonlocal matching boundary conditions for non-ordinary peridynamics with correspondence material model. Comput. Methods Appl. Mech. Eng. 338, 463–490 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Engineering MechanicsWuhan UniversityWuhanChina

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