Skip to main content
SpringerLink
Log in

A new numerical approach for determination of the Lemaitre’s ductile damage parameter in bulk metal forming processes

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

The damage of materials is the progressive or unexpected deterioration of mechanical strength because of loadings, thermal or chemical effects. The micromechanical damage process of ductile materials is generally studied by the continuum damage mechanics (CDM). One of the most well-known damage models is the Lemaitre’s ductile damage criterion. This model only requires one material-dependent parameter to represent damage evolution. In this investigation first, a novel numerical approach is proposed to determine the Lemaitre’s ductile damage parameter. Then, a user-defined material subroutine founded on the Lemaitre’s ductile damage model is developed. Following, numerical results are achieved for a standard round tensile test specimen. Finally, to validate the suggested method, experimental tests are carried out and compared with the numerical results. The comparison reveals a good agreement and excellent correlation between the numerical and practical results. Hence, it is concluded that the offered numerical approach can accurately determine the Lemaitre’s ductile damage parameter as well as the damage behavior of ductile metals.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Abbreviations

\(\mathrm{A}\) :

Total cross section area of RVE

\({\mathrm{A}}_{D}\) :

Damaged area of RVE

\(D\) :

Damage variable

\({D}_{1C}\) :

Critical damage parameter in tension

\({e}_{u}\) :

Elongation

\(E\) :

Young’s modulus

\(\tilde{E }\) :

Effective elasticity modulus

\(f\) :

Yield surface function

\(\mathrm{G}\) :

Shear modulus

\(\mathbf{I}\) :

Second-order identity tensor

\(K\) :

Hardening coefficient

\(\mathrm{K}\) :

Bulk modulus

\(n\) :

Hardening power

\(p\) :

Hydrostatic stress

\(r\) :

Lemaitre’s ductile damage parameter

\(\mathrm{R}\) :

Isotropic hardening function

\(s\) :

Lemaitre’s damage power parameter

\({\varvec{S}}\) :

Deviatoric stress tensor

\(\nu\) :

Poisson’s ratio

\(Y\) :

Damage strain energy release rate

\({\varvec{\varepsilon}}\) :

Strain tensor

\({{\varvec{\varepsilon}}}^{d}\) :

Deviatoric strain tensor

\({{\varvec{\varepsilon}}}^{e}\) :

Elastic strain tensor

\({\varepsilon }^{p}\) :

Plastic strain

\({{\varvec{\varepsilon}}}^{p}\) :

Plastic strain tensor

\({\varepsilon }^{v}\) :

Volumetric strain

\({\varepsilon }_{eq}^{p}\) :

Equivalent plastic strain

\({\varepsilon }_{pd}\) :

Threshold plastic strain

\({\dot{{\varvec{\varepsilon}}}}^{p}\) :

Plastic strain rate tensor

\(\dot{\gamma }\) :

Plastic consistency parameter

\(\eta\) :

Stress triaxiality

\(\rho\) :

Density

\({\varvec{\sigma}}\) :

Stress tensor of virgin material

\(\stackrel{\sim }{{\varvec{\sigma}}}\) :

Effective stress tensor

\({\sigma }_{eq}\) :

Von Mises equivalent stress

\({\sigma }_{f}\) :

Fracture stress

\({\sigma }_{u}\) :

Ultimate stress

\({\sigma }_{y}^{0}\) :

Initial yield stress

\(\psi\) :

Potential dissipation function

\({\psi }_{D}\) :

Damage component of potential dissipation function

\({\psi }_{P}\) :

Plastic component of potential dissipation function

References

  1. Lemaitre, J.: A course on damage mechanics. Springer, Cham (2012)

    MATH  Google Scholar 

  2. Kachanov, L.M.: On the time to failure under creep conditions. Izv. AN SSSR Otd. Tekhn. Nauk. 8, 26–31 (1958)

    Google Scholar 

  3. Rabotnov, Y.N.: A model of an elastic-plastic medium with delayed yield. J. Appl. Mech. Tech. Phys. 9(3), 265–269 (1968)

    Article  Google Scholar 

  4. McClintock, F.A.: A criterion for ductile fracture by the growth of holes. Int. J. Fract. Mech. (1968). https://doi.org/10.1007/BF00188939

    Article  Google Scholar 

  5. Cockcroft, M.G., Latham, D.J.: Ductility and the workability of metals. J. Inst. Met. 96(1), 33–39 (1968)

    Google Scholar 

  6. Rice, J.R., Tracey, D.M.: On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids 17(3), 201–217 (1969)

    Article  Google Scholar 

  7. Gurson, A.L.: Continuum theory of ductile rupture by void nucleation and growth: part I—Yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol. 99(1), 2–15 (1977)

    Article  Google Scholar 

  8. Wilkins, M. L., Streit, R. D., & Reaugh, J. E. (1980). Cumulative-strain-damage model of ductile fracture: simulation and prediction of engineering fracture tests (No. UCRL-53058). Lawrence Livermore National Lab., CA (USA); Science Applications, Inc., San Leandro, CA (USA)

  9. Tvergaard, V.: On localization in ductile materials containing spherical voids. Int. J. Fract. 18(4), 237–252 (1982)

    Google Scholar 

  10. Tvergaard, V.: Ductile fracture by cavity nucleation between larger voids. J. Mech. Phys. Solids 30(4), 265–286 (1982)

    Article  MathSciNet  Google Scholar 

  11. Needleman, A., Tvergaard, V.: An analysis of ductile rupture in notched bars. J. Mech. Phys. Solids 32(6), 461–490 (1984)

    Article  Google Scholar 

  12. Chaboche, J.L.: Anisotropic creep damage in the framework of continuum damage mechanics. Nucl. Eng. Des. 79(3), 309–319 (1984)

    Article  Google Scholar 

  13. Lemaitre, J.: A continuous damage mechanics model for ductile fracture. J. Eng. Mater. Technol. 107(1), 83–89 (1985)

    Article  Google Scholar 

  14. Johnson, G.R., Cook, W.H.: Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng. Fract. Mech. 21(1), 31–48 (1985)

    Article  Google Scholar 

  15. Benallal, A., Billardon, R., Doghri, I.: An integration algorithm and the corresponding consistent tangent operator for fully coupled elastoplastic and damage equations. Commun. Appl. Numer. Methods 4(6), 731–740 (1988)

    Article  Google Scholar 

  16. Steinmann, P., Miehe, C., Stein, E.: Comparison of different finite deformation inelastic damage models within multiplicative elastoplasticity for ductile materials. Comput. Mech. 13(6), 458–474 (1994)

    Article  Google Scholar 

  17. Doghri, I.: Numerical implementation and analysis of a class of metal plasticity models coupled with ductile damage. Int. J. Numer. Meth. Eng. 38(20), 3403–3431 (1995)

    Article  Google Scholar 

  18. Dhar, S., Sethuraman, R., Dixit, P.M.: A continuum damage mechanics model for void growth and micro crack initiation. Eng. Fract. Mech. 53(6), 917–928 (1996)

    Article  Google Scholar 

  19. De. Souza Neto, E.A., Peric, D.: A computational framework for a class of fully coupled models for elastoplastic damage at finite strains with reference to the linearization aspects. Comput. Methods Appl. Mech. Eng. 130(1–2), 179–193 (1996)

    Article  Google Scholar 

  20. La. Rosa, G., Mirone, G., Risitano, A.: Effect of stress triaxiality corrected plastic flow on ductile damage evolution in the framework of continuum damage mechanics. Eng. Fract. Mech. 68(4), 417–434 (2001)

    Article  Google Scholar 

  21. De. Souza Neto, E.A.: A fast, one-equation integration algorithm for the Lemaitre ductile damage model. Commun. Numer. Methods Eng. 18(8), 541–554 (2002)

    Article  Google Scholar 

  22. Brünig, M.: An anisotropic ductile damage model based on irreversible thermodynamics. Int. J. Plast 19(10), 1679–1713 (2003)

    Article  Google Scholar 

  23. Hooputra, H., Gese, H., Dell, H., Werner, H.: A comprehensive failure model for crashworthiness simulation of aluminium extrusions. Int. J. Crashworthiness 9(5), 449–464 (2004)

    Article  Google Scholar 

  24. Bonora, N., Gentile, D., Pirondi, A., Newaz, G.: Ductile damage evolution under triaxial state of stress: theory and experiments. Int. J. Plast 21(5), 981–1007 (2005)

    Article  Google Scholar 

  25. Bai, Y., Wierzbicki, T.: A new model of metal plasticity and fracture with pressure and lode dependence. Int. J. Plast 24(6), 1071–1096 (2008)

    Article  Google Scholar 

  26. Rezaiee-Pajand, M., Kazemiyan, M.S., Aftabi, S.A.: Static damage identification of 3D and 2D frames. Mech. Based Des. Struct. Mach. 42(1), 70–96 (2014)

    Article  Google Scholar 

  27. Zhai, X., Wang, Y., Sun, Z.: Damage model and damage assessment for single-layer reticulated domes under exterior blast load. Mech. Based Des. Struct. Mach. 47(3), 319–338 (2019)

    Article  Google Scholar 

  28. Mehditabar, A., Rahimi, G.H.: Cyclic elastoplastic responses of thick-walled FG pipe behaves as a power law function considering damage evolution. Mech. Based Des. Struct. Mach. (2020). https://doi.org/10.1080/15397734.2020.1768112

    Article  Google Scholar 

  29. Simo, J.C., Hughes, T.J.R.: Interdisciplinary applied mathematics. In: Bloch, A., Epstein, C.L., Goriely, A., Greengard, L. (eds.) Mechanics and materials, computational inelasticity. Springer-Verlag, New York (1998)

    MATH  Google Scholar 

  30. Mashayekhi, M., Ziaei-Rad, S., Parvizian, J., Niklewicz, J., Hadavinia, H.: Ductile crack growth based on damage criterion: experimental and numerical studies. Mech. Mater. 39(7), 623–636 (2007)

    Article  Google Scholar 

  31. Haji Aboutalebi, F., Farzin, M., Poursina, M.: Numerical simulation and experimental validation of a ductile damage model for DIN 1623 St14 steel. Int. J. Adv. Manuf. Technol. 53(1–4), 157–165 (2011)

    Article  Google Scholar 

  32. Haji Aboutalebi, F., Farzin, M., Mashayekhi, M.: Numerical predictions and experimental validations of ductile damage evolution in sheet metal forming processes. Acta Mech. Solida Sin. 25(6), 638–650 (2012)

    Article  Google Scholar 

  33. Haji Aboutalebi, F., Roostaei, M., Safaie, P.: Empirical determination of Lemaitre's ductile damage parameters for thin sheet metals by variations of the elasticity modulus. Steel Res. Int. 83(Special Issue), 1195–1198 (2012)

    Google Scholar 

  34. Cao, T.S., Gachet, J.M., Montmitonnet, P., Bouchard, P.O.: A Lode-dependent enhanced Lemaitre model for ductile fracture prediction at low stress triaxiality. Eng. Fract. Mech. 124, 80–96 (2014)

    Article  Google Scholar 

  35. Papasidero, J., Doquet, V., Mohr, D.: Ductile fracture of aluminum 2024–T351 under proportional and non-proportional multi-axial loading: Bao-Wierzbicki results revisited. Int. J. Solids Struct. 69, 459–474 (2015)

    Article  Google Scholar 

  36. Gerke, S., Zistl, M., Brünig, M.: Experiments and numerical simulation of damage and fracture of the X0-specimen under non-proportional loading paths. Eng. Fract. Mech. 224, 106795 (2020)

    Article  Google Scholar 

  37. Autay, R., Koubaa, S., Wali, M., Dammak, F.: Numerical implementation of coupled anisotropic plasticity-ductile damage in sheet metal forming process. J. Mech. 34(4), 417–430 (2018)

    Article  Google Scholar 

  38. Ghorbel, O., Koubaa, S., Mars, J., Wali, M., Dammak, F.: Non associated-anisotropic plasticity model fully coupled with isotropic ductile damage for sheet metal forming applications. Int. J. Solids Struct. 166, 96–111 (2019)

    Article  Google Scholar 

  39. Ghorbel, O., Mars, J., Koubaa, S., Wali, M., Dammak, F.: Coupled anisotropic plasticity-ductile damage: modeling, experimental verification, and application to sheet metal forming simulation. Int. J. Mech. Sci. 150, 548–560 (2019)

    Article  Google Scholar 

  40. Bringas, J.E.: Handbook of comparative world steel standards. ASTM, United States (2002)

    Google Scholar 

  41. Standard E8/E8M-16a (2016). Standard test methods for tension testing of metallic materials. An American National Standard. AASHTO No.: T68. pp. 8

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Isfahan, 81746-73441, Isfahan, Iran

    A. R. Shamshiri, F. Haji Aboutalebi & M. Poursina

Authors
  1. A. R. Shamshiri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Haji Aboutalebi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Poursina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Haji Aboutalebi.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shamshiri, A.R., Haji Aboutalebi, F. & Poursina, M. A new numerical approach for determination of the Lemaitre’s ductile damage parameter in bulk metal forming processes. Arch Appl Mech 91, 4163–4177 (2021). https://doi.org/10.1007/s00419-021-01998-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-021-01998-y

Keywords

Search

Navigation