pp 1–14 | Cite as

An equivalent von Mises stress and corresponding equivalent plastic strain for elastic–plastic ordinary peridynamics

  • Mojtaba Asgari
  • Mohammad Ali KouchakzadehEmail author


Simple formulas for calculating equivalent von Mises stress and von Mises effective plastic strain in an elastic–plastic ordinary peridynamic analysis are proposed. The equivalent von Mises stress is calculated by equating the deviatoric part of strain energy obtained from classical continuum mechanics and peridynamics. The effective plastic strain is proposed so that it reduced to uniaxial plastic strain in uniaxial tension test. Two example problems of the plate with a hole and a central crack under tension are considered to verify the validity of the proposed formulas. The plots of von Mises stress, equivalent plastic strain, plastic zone area and horizontal and vertical displacements are extracted and compared with the results obtain from the finite element analysis. The results show the good accuracy of the peridynamics in predicting the above mentioned parameters as well as the validity of the suggested formulas in predicting von Mises stress and equivalent plastic strain.


Peridynamics Plasticity von Mises stress Plastic strain 



This work was supported by the Sharif University of Technology [Grant Number QA961027].

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Jiang X-W, Wang H (2018) Ordinary state-based peridynamics for open-hole tensile strength prediction of fiber-reinforced composite laminates. J Mech Mater Struct 13:53–82MathSciNetCrossRefGoogle Scholar
  2. 2.
    Zhou W, Liu D, Liu N (2017) Analyzing dynamic fracture process in fiber-reinforced composite materials with a peridynamic model. Eng Fract Mech 178:60–76CrossRefGoogle Scholar
  3. 3.
    Yaghoobi A, Chorzepa MG (2017) Fracture analysis of fiber reinforced concrete structures in the micropolar peridynamic analysis framework. Eng Fract Mech 169:238–250CrossRefGoogle Scholar
  4. 4.
    Hu Y, Madenci E, Phan N (2017) Peridynamics for predicting damage and its growth in composites. Fatigue Fract Eng Mater Struct 40:1214–1226CrossRefGoogle Scholar
  5. 5.
    Sun C, Huang Z (2016) Peridynamic simulation to impacting damage in composite laminate. Compos Struct 138:335–341CrossRefGoogle Scholar
  6. 6.
    Oterkus E, Madenci E (2012) Peridynamic analysis of fiber-reinforced composite materials. J Mech Mater Struct 7:45–84CrossRefGoogle Scholar
  7. 7.
    Hu W, Ha YD, Bobaru F (2012) Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites. Comput Methods Appl Mech Eng 217–220:247–261MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen Z, Bobaru F (2015) Peridynamic modeling of pitting corrosion damage. J Mech Phys Solids 78:352–381ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    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
  10. 10.
    Zhou X, Wang Y, Qian Q (2016) Numerical simulation of crack curving and branching in brittle materials under dynamic loads using the extended non-ordinary state-based peridynamics. Eur J Mech A Solids 60:277–299MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cheng Z, Zhang G, Wang Y, Bobaru F (2015) A peridynamic model for dynamic fracture in functionally graded materials. Compos Struct 133:529–546CrossRefGoogle Scholar
  12. 12.
    Hafezi MH, Alebrahim R, Kundu T (2017) Peri-ultrasound for modeling linear and nonlinear ultrasonic response. Ultrasonics 80:47–57CrossRefGoogle Scholar
  13. 13.
    Tao Y, Tian X, Du Q (2017) Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations. Appl Math Comput 305:282–298MathSciNetzbMATHGoogle Scholar
  14. 14.
    Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48:175–209ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88:151–184MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gu X, Madenci E, Zhang Q (2018) Revisit of non-ordinary state-based peridynamics. Eng Fract Mech 190:31–52CrossRefGoogle Scholar
  17. 17.
    Ballarini R, Diana V, Biolzi L, Casolo S (2018) Bond-based peridynamic modelling of singular and nonsingular crack-tip fields. Meccanica 53:3495–3515MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tupek MR, Radovitzky R (2014) An extended constitutive correspondence formulation of peridynamics based on nonlinear bond-strain measures. J Mech Phys Solids 65:82–92ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Macek RW, Silling SA (2007) Peridynamics via finite element analysis. Finite Elem Anal Des 43:1169–1178MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ladányi G, Jenei I (2008) Analysis of plastic peridynamic material with RBF meshless method. Pollack Periodica 3:65–77CrossRefGoogle Scholar
  21. 21.
    Celik E, Guven I, Madenci E (2011) Simulations of nanowire bend tests for extracting mechanical properties. Theoret Appl Fract Mech 55:185–191CrossRefGoogle Scholar
  22. 22.
    Guven I, Zelinski BJ (2015) Peridynamic modeling of damage and fracture in EM windows and domes. In: SPIE defense + security, SPIE, 2015, p 10Google Scholar
  23. 23.
    Oterkus E, Diyaroglu C, Zhu N, Oterkus S, Madenci E (2015) Utilization of peridynamic theory for modeling at the nano-scale. In: Baillin X, Joachim C, Poupon G (eds) Nanopackaging: from nanomaterials to the atomic scale. Advances in atom and single molecule machines. Springer, Cham, pp 1–16Google Scholar
  24. 24.
    Ahadi A, Hansson P, Melin S (2016) Indentation of thin copper film using molecular dynamics and peridynamics. Proc Struct Integr 2:1343–1350CrossRefGoogle Scholar
  25. 25.
    Lee J, Liu W, Hong J-W (2016) Impact fracture analysis enhanced by contact of peridynamic and finite element formulations. Int J Impact Eng 87:108–119CrossRefGoogle Scholar
  26. 26.
    Yolum U, Taştan A, Güler MA (2016) A peridynamic model for ductile fracture of moderately thick plates. Proc Struct Integr 2:3713–3720CrossRefGoogle Scholar
  27. 27.
    Warren TL, Silling SA, Askari A, Weckner O, Epton MA, Xu J (2009) A non-ordinary state-based peridynamic method to model solid material deformation and fracture. Int J Solids Struct 46:1186–1195CrossRefzbMATHGoogle Scholar
  28. 28.
    Foster JT, Silling SA, Chen WW (2010) Viscoplasticity using peridynamics. Int J Numer Meth Eng 81:1242–1258zbMATHGoogle Scholar
  29. 29.
    Littlewood DJ (2010) Simulation of dynamic fracture using peridynamics, finite element modeling, and contact. In: ASME 2010 international mechanical engineering congress and exposition, American Society of Mechanical Engineers, pp 209–217Google Scholar
  30. 30.
    Littlewood DJ (2011) A nonlocal approach to modeling crack nucleation in AA 7075-T651. In: ASME 2011 international mechanical engineering congress and exposition, American Society of Mechanical Engineers, pp 567–576Google Scholar
  31. 31.
    Tupek MR, Rimoli JJ, Radovitzky R (2013) An approach for incorporating classical continuum damage models in state-based peridynamics. Comput Methods Appl Mech Eng 263:20–26ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    O’Grady J, Foster J (2014) Peridynamic beams: a non-ordinary, state-based model. Int J Solids Struct 51:3177–3183CrossRefGoogle Scholar
  33. 33.
    Sun S, Sundararaghavana V (2014) A peridynamic implementation of crystal plasticity. Int J Solids Struct 51:3350–3360CrossRefGoogle Scholar
  34. 34.
    Amani J, Oterkus E, Areias P, Zi G, Nguyen-Thoi T, Rabczuk T (2016) A non-ordinary state-based peridynamics formulation for thermoplastic fracture. Int J Impact Eng 87:83–94CrossRefGoogle Scholar
  35. 35.
    Lai X, Liu LS, Liu QW, Cao DF, Wang Z, Zhai PC (2015) Slope stability analysis by peridynamic theory. Appl Mech Mater 744–746:584–588CrossRefGoogle Scholar
  36. 36.
    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 291:197–215ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Yu Y, Liu S-S, Zhao S-L, Yu Z (2016) The nonlinear inplane behavior and progressive damage modeling for laminate by peridynamics. In: ASME 2016 international mechanical engineering congress and exposition, American Society of Mechanical Engineers, p V001T003A054Google Scholar
  38. 38.
    Rahaman MM, Roy P, Roy D, Reddy JN (2017) A peridynamic model for plasticity: micro-inertia based flow rule, entropy equivalence and localization residuals. Comput Methods Appl Mech Eng 327:369–391ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Sami A (2017) Peridynamic analysis of residual stress around a cold expanded fastener hole. Int J Appl Innov Eng Manage 6:1–7Google Scholar
  40. 40.
    Lai X, Liu L, Li S, Zeleke M, Liu Q, Wang Z (2018) A non-ordinary state-based peridynamics modeling of fractures in quasi-brittle materials. Int J Impact Eng 111:130–146CrossRefGoogle Scholar
  41. 41.
    Mitchell JA (2011) A nonlocal, ordinary, state-based plasticity model for peridynamics. Sandia Report SAND2011-3166, Sandia National Laboratories, AlbuquerqueGoogle Scholar
  42. 42.
    Lammi CJ, Vogler TJ (2014) A Nonlocal peridynamic plasticity model for the dynamic flow and fracture of concrete. Sandia National Laboratories (SNL-CA), Livermore, CA (United States)Google Scholar
  43. 43.
    Madenci E, Oterkus S (2016) Ordinary state-based peridynamics for plastic deformation according to von Mises yield criteria with isotropic hardening. J Mech Phys Solids 86:192–219ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Silling SA (2016) Introduction to peridynamics. In: Bobaru F, Foster JT, Geubelle PH, Silling SA (eds) Handbook of peridynamic modeling. Chapman and Hall/CRC, Boca Raton, pp 63–98Google Scholar
  45. 45.
    Chen W, Zhu F, Zhao J, Li S, Wang G (2018) Peridynamics-based fracture animation for elastoplastic solids. Comput Graph Forum 37:112–124CrossRefGoogle Scholar
  46. 46.
    Madenci E, Oterkus E (2013) Peridynamic theory and its applications. Springer, New YorkzbMATHGoogle Scholar
  47. 47.
    Zhou XP, Shou YD, Berto F (2018) Analysis of the plastic zone near the crack tips under the uniaxial tension using ordinary state-based peridynamics. Fatigue Fract Eng Mater Struct 41:1159–1170CrossRefGoogle Scholar
  48. 48.
    de Souza Neto EA, Peric D, Owen DRJ (2011) Computational methods for plasticity: theory and applications. Wiley, New YorkGoogle Scholar
  49. 49.
    Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83:1526–1535CrossRefGoogle Scholar
  50. 50.
    Le QV, Chan WK, Schwartz J (2014) A two-dimensional ordinary, state-based peridynamic model for linearly elastic solids. Int J Numer Meth Eng 98:547–561MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringSharif University of TechnologyTehranIran

Personalised recommendations