Skip to main content
SpringerLink
Log in

The Emerging of Stress Triaxiality and Lode Angle in Both Solid and Damage Mechanics: A Review

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract—

The review paper covers an overview of the early and current research related to models for both solid and damage mechanics. It addresses the most well-known phenomenological metal plasticity and ductile fracture models in these fields for monotonic loading conditions. The paper commences with a comprehensive literature review outlining the history and current state of the art of the plasticity and ductile fracture models. Then, the paper explains the principal stresses, showing how they represent a metal yield surface. Because most yield functions involve the stress invariant, an extended explanation of stress invariants’ space is extensively described. Moreover, a list of coupled and non-coupled plasticity models and ductile fracture criteria are thoroughly explained chronologically to show the emerging of stress triaxiality and Lode angle in models for solid and damage mechanics. The models presented in this paper assume the material’s isotropy, homogeneity, and elastic-plastic behavior. Finally, two comparison tables demonstrating the most well-known phenomenal plasticity and fracture models for continuum mechanics are chronologically listed.

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.

Similar content being viewed by others

REFERENCES

  1. J. S. Rao, Turbomachine Blade Vibration (New Age International, New Delhi, 1991)

    Google Scholar 

  2. N. F. Rieger, “Blade fatigue,” in Proc. Tech. Comm. Rotor Dynamics, Sixth IFToMM Congress (New Delhi, India, 1983), p. 66.

  3. N. S. Vyas, J. S. Rao, “Fatigue life estimation procedure for a turbine blade under transient loads,” in Proceedings of the ASME 1992 International Gas Turbine and Aeroengine Congress and Exposition, Vol. 5: Manufacturing Materials and Metallurgy; Ceramics; Structures and Dynamics; Controls, Diagnostics and Instrumentation; Education. Cologne, Germany. June 1–4, 1992 (ASME, 1992), 92-GT-078, V005T14A007.https://doi.org/10.1115/92-GT-078

  4. P. Mešt’áneka, “Low cycle fatigue analysis of a last stage steam turbine blade,” Appl. Computat. Mech. 2 (1), 71–82, (2008).

    Google Scholar 

  5. M. Algarni, “Notch factor correction using stress triaxiality of plane-stress state in high-cycle fatigue,” Int. J. Fatigue. 128, 105204 (2019). https://doi.org/10.1016/j.ijfatigue.2019.105204

    Article  Google Scholar 

  6. Y. Bai and T. Wierzbicki, “A new model of metal plasticity and fracture with pressure and Lode dependence,” Int. J. Plast. 24 (6), 1071–1096 (2008).https://doi.org/10.1016/j.ijplas.2007.09.004

    Article  MATH  Google Scholar 

  7. G. Gruben, O. S. Hopperstad, and T. Børvik, “Evaluation of uncoupled ductile fracture criteria for the dual-phase steel Docol 600DL,” Int. J. Mech. Sci. 62 (1), 133–146 (2012). https://doi.org/10.1016/j.ijmecsci.2012.06.009

    Article  Google Scholar 

  8. M. Achouri, G. Germain, P. Dal Santo, and D. Saidane, “Experimental characterization and numerical modeling of micromechanical damage under different stress states,” Mater. Des. 50, 207–222 (2013). https://doi.org/10.1016/j.matdes.2013.02.075

    Article  Google Scholar 

  9. K. Nahshon, J.W. Hutchinson, “Modification of the Gurson model for shear failure,” Eur. J. Mech. A-Solids 27 (1), 1–17 (2008). https://doi.org/10.1016/j.euromechsol.2007.08.002

    Article  ADS  MATH  Google Scholar 

  10. L. Xue, “Constitutive modeling of void shearing effect in ductile fracture of porous materials,” Eng. Fract. Mech. 75 (11), 3343–3366 (2008). https://doi.org/10.1016/j.engfracmech.2007.07.022

    Article  Google Scholar 

  11. B. Erice and F. Gálvez, “A coupled elastoplastic-damage constitutive model with Lode angle dependent failure criterion,” Int. J. Solids Struct. 51 (1), 93–110 (2014). https://doi.org/10.1016/j.ijsolstr.2013.09.015

    Article  Google Scholar 

  12. G. R. Johnson, and W.H. Cook, “A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures,” In Proceedings of the 7th International Symposium on Ballistics, The Hague, Netherlands, 9–21 April 1983 (American Defense Preparedness Association, 1983), Vol. 21, No. 1, pp. 541–547.

  13. S. Bari and T. Hassan, “Anatomy of coupled constitutive models for ratcheting simulation,” Int. J. Plast. 16 (3), 381–409 (2000). https://doi.org/10.1016/S0749-6419(99)00059-5

    Article  MATH  Google Scholar 

  14. S. Bari and T. Hassan, “An advancement in cyclic plasticity modeling for multiaxial ratcheting simulation,” Int. J. Plast. 18 (7), 873–894 (2002). https://doi.org/10.1016/S0749-6419(01)00012-2

    Article  MATH  Google Scholar 

  15. S. Bari and T. Hassan, “Kinematic hardening rules in uncoupled modeling for multiaxial ratcheting simulation,” Int. J. Plast. 17 (7), 885–905 (2001). https://doi.org/10.1016/S0749-6419(00)00031-0

    Article  MATH  Google Scholar 

  16. J. L. Chaboche, “Constitutive equations for cyclic plasticity and cyclic viscoplasticity,” Int. J. Plast. 5 (3), 247–302 (1989). https://doi.org/10.1016/0749-6419(89)90015-6

    Article  MATH  Google Scholar 

  17. J. L. Chaboche, K. Dang Van, and G. Cordier, Modelization of the Strain Memory Effect on the Cyclic Hardening of 316 Stainless Steel (IASMiRT, Berlin, 1979).

  18. J. L. Chaboche, “On some modifications of kinematic hardening to improve the description of ratchetting effects,” Int. J. Plast. 7 (7), 661–678 (1991). https://doi.org/10.1016/0749-6419(91)90050-9

    Article  Google Scholar 

  19. J. L. Chaboche “Time-independent constitutive theories for cyclic plasticity,” Int. J. Plast. 2 (2), 149–188 (1986). https://doi.org/10.1016/0749-6419(86)90010-0

    Article  MATH  Google Scholar 

  20. D. L. McDowell, “Stress state dependence of cyclic ratchetting behavior of two rail steels,” Int. J. Plast. 11 (4), 397–421 (1995). https://doi.org/10.1016/S0749-6419(95)00005-4

    Article  Google Scholar 

  21. Y. Jiang and H. Sehitoglu, “Modeling of cyclic ratchetting plasticity, part I: development of constitutive relations,” J. Appl. Mech. 63 (3), 720–725 (1996). https://doi.org/10.1115/1.2823355

    Article  ADS  MATH  Google Scholar 

  22. Y. Jiang and H. Sehitoglu, “Modeling of cyclic ratchetting plasticity, part II: comparison of model simulations with experiments,” J. Appl. Mech. 63 (3), 726–733 (1996). https://doi.org/10.1115/1.2823356

    Article  ADS  MATH  Google Scholar 

  23. X. Chen, R. Jiao, and K. S. Kim, “On the Ohno–Wang kinematic hardening rules for multiaxial ratcheting modeling of medium carbon steel,” Int. J. Plast. 21 (1), 161–184 (2005). https://doi.org/10.1016/j.ijplas.2004.05.005

    Article  MATH  Google Scholar 

  24. X. Chen and R. Jiao. “Modified kinematic hardening rule for multiaxial ratcheting prediction,” Int. J. Plast. 20 (4), 871–898 (2004). https://doi.org/10.1016/j.ijplas.2003.05.005

    Article  Google Scholar 

  25. L. Xue, “Damage accumulation and fracture initiation in uncracked ductile solids subject to triaxial loading,” Int. J. Solids Struct. 44 (16), 5163–5181 (2007). https://doi.org/10.1016/j.ijsolstr.2006.12.026

    Article  MATH  Google Scholar 

  26. L. Xue and T. Wierzbicki, “Numerical simulation of fracture mode transition in ductile plates,” Int. J. Solids Struct. 46 (6), 1423–1435 (2009). https://doi.org/10.1016/j.ijsolstr.2008.11.009

    Article  MATH  Google Scholar 

  27. Y. Bai and T. Wierzbicki, “Application of extended Mohr–Coulomb criterion to ductile fracture,” Int. J. Fract. 161 (1), 1–20 (2010). https://doi.org/10.1007/s10704-009-9422-8

    Article  MATH  Google Scholar 

  28. G. R. Johnson and W.H. Cook, “Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures,” Eng. Fract. Mech. 21 (1), 31–48 (1985). https://doi.org/10.1016/0013-7944(85)90052-9

    Article  Google Scholar 

  29. L. Xue and T. Wierzbicki, “Ductile fracture initiation and propagation modeling using damage plasticity theory,” Eng. Fract. Mech. 75 (11), 3276–3293 (2008).

    Article  Google Scholar 

  30. Z. Xue, M. G. Pontin, F. W. Zok, and J. W. Hutchinson, “Calibration procedures for a computational model of ductile fracture,” Eng. Fract. Mech. 77 (3), 492–509 (2010). https://doi.org/10.1016/j.engfracmech.2009.10.007

    Article  Google Scholar 

  31. J. L. Chaboche, “Continuum damage mechanics: Part II—Damage growth, crack initiation, and crack growth,” J. Appl. Mech. 55 (1), 65–72 (1988). https://doi.org/10.1115/1.3173662

    Article  ADS  Google Scholar 

  32. J. L. Chaboche and G. Rousselier, “On the plastic and viscoplastic constitutive equations—Part I: Rules developed with internal variable concept,” J. Pressure Vessel Technol. 105 (2), 153–158 (1983).

    Article  Google Scholar 

  33. Y. Bao, “Dependence of ductile crack formation in tensile tests on stress triaxiality, stress and strain ratios,” Eng. Fract. Mech. 72 (4), 505–522 (2005). https://doi.org/10.1016/j.engfracmech.2004.04.012

    Article  Google Scholar 

  34. F. A. McClintock, “A criterion for ductile fracture by the growth of holes,” J. Appl. Mech. 35 (2), 363–371 (1968). https://doi.org/10.1115/1.3601204

    Article  ADS  Google Scholar 

  35. J. R. Rice and D. M. Tracey, “On the ductile enlargement of voids in triaxial stress fields,” J. Mech. Phys. Solids 17 (3), 201–217 (1969). https://doi.org/10.1016/0022-5096(69)90033-7

    Article  ADS  Google Scholar 

  36. J. W. Hancock, and A. C. Mackenzie, “On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states,” J. Mech. Phys. Solids 24 (2), 147–160 (1976).

    Article  ADS  Google Scholar 

  37. J. W. Hancock and D. K. Brown, “On the role of strain and stress state in ductile failure,” J. Mech. Phys. Solids 31 (1), 1–24 (1983).

    Article  ADS  Google Scholar 

  38. T. S. Cao, J. M. Gachet, P. Montmitonnet, and P.O. Bouchard, “A Lode-dependent enhanced Lemaitre model for ductile fracture prediction at low stress triaxiality,” Eng. Fract. Mech. 124–125, 80–96 (2014). https://doi.org/10.1016/j.engfracmech.2014.03.021

    Article  Google Scholar 

  39. J. M. Gachet, G. Delattre, and P. O. Bouchard, “Fracture mechanisms under monotonic and non-monotonic low Lode angle loading,” Eng. Fract. Mech. 124–125, 121–141 (2014). https://doi.org/10.1016/j.engfracmech.2014.04.009

    Article  Google Scholar 

  40. Y. J. Liu, Q. Sun, X. L. Fan, and T. Suo, “A stress-invariant based multi-parameters ductile progressive fracture model,” Mat. Sci. Eng. A-Struct. 576, 337–345 (2013). https://doi.org/10.1016/j.msea.2013.04.013

    Article  Google Scholar 

  41. R. Ghajar, G. Mirone, and A. Keshavarz, “Sensitivity analysis on triaxiality factor and lode angle in ductile fracture,” J. Mech. 29, 177–184 (2013). https://doi.org/10.1017/jmech.2012.125

    Article  Google Scholar 

  42. X. Gao, T. Wang, and J. Kim, “On ductile fracture initiation toughness: Effects of void volume fraction, void shape and void distribution,” Int. J. Solids Struct. 42 (18–19), 5097–5117 (2005). https://doi.org/10.1016/j.ijsolstr.2005.02.028

    Article  MATH  Google Scholar 

  43. X. Gao and J. Kim, “Modeling of ductile fracture: Significance of void coalescence,” Int. J. Solids Struct. 43 (20), 6277–6293 (2006). https://doi.org/10.1016/j.ijsolstr.2005.08.008

    Article  MATH  Google Scholar 

  44. X. Gao, T. Zhang, M. Hayden, and C. Roe, “Effects of the stress state on plasticity and ductile failure of an aluminum 5083 alloy,” Int. J. Plast. 25 (12), 2366–2382 (2009). https://doi.org/10.1016/j.ijplas.2009.03.006

    Article  Google Scholar 

  45. M. Algarni, PhD Thesis (Univ. of Central Florida, Orlando, 2017).

  46. S. Ghazali, M. Algarni, Y. Bai, and Y. Choi, “A study on the plasticity and fracture of the AISI 4340 steel alloy under different loading conditions and considering heat-treatment effects,” Int. J. Fract. 225 (1), 69–87 (2020). https://doi.org/10.1007/s10704-020-00466-y

    Article  Google Scholar 

  47. Y. Li, and T. Wierzbicki, “Prediction of plane strain fracture of AHSS sheets with post-initiation softening,” Int. J. Solids Struct. 47(17), 2316–2327 (2010). https://doi.org/10.1016/j.ijsolstr.2010.04.028

    Article  MATH  Google Scholar 

  48. M. Dunand, D. Mohr, “On the predictive capabilities of the shear modified Gurson and the modified Mohr–Coulomb fracture models over a wide range of stress triaxialities and Lode angles,” J. Mech. Phys. Solids. 59(7), 1374–1394 (2011). https://doi.org/10.1016/j.jmps.2011.04.006

    Article  ADS  MATH  Google Scholar 

  49. M. Luo, and T. Wierzbicki, “Numerical failure analysis of a stretch-bending test on dual-phase steel sheets using a phenomenological fracture model,” Int. J. Solids Struct. 47 (22–23), 3084–3102 (2010). https://doi.org/10.1016/j.ijsolstr.2010.07.010

    Article  MATH  Google Scholar 

  50. Y. Li and T. Wierzbicki, “Mesh-size effect study of ductile fracture by non-local approach,” in Proceedings of the SEM Annual Conference June 1-4, 2009, Albuquerque, New Mexico USA (Curran Associates, Inc., New York, 2009), Vol. 1, pp. 292–301.

  51. M. Luo, M. Dunand, and D. Mohr, “Experiments and modeling of anisotropic aluminum extrusions under multi-axial loading – Part II: Ductile fracture,” Int. J. Plast. 32–33, 36–58 (2012). https://doi.org/10.1016/j.ijplas.2011.11.001

    Article  Google Scholar 

  52. Y. Bai, and T. Atkins, “Tension and shear cracking during indentation of ductile materials by opposed wedges,” Eng. Fract. Mech. 96, 49–60 (2012). https://doi.org/10.1016/j.engfracmech.2012.06.014

    Article  Google Scholar 

  53. Y. Bai, PhD Thesis (Massachusetts Institute of Technology, Cambridge, 2008).

  54. L. Malcher, F. M. Andrade, J. M. César de Sá, and F.X. Andrade, “Numerical integration algorithm of a new model for metal plasticity and fracture including pressure and lode angle dependence,” Int. J. Mater. Form 2 (1), 443–446 (2009). https://doi.org/10.1007/s12289-009-0525-6

    Article  Google Scholar 

  55. S. K. Iyer, and C. J. Lissenden, “Multiaxial constitutive model accounting for the strength-differential in Inconel 718,” Int. J. Plast. 19 (12), 2055–2081 (2003). https://doi.org/10.1016/S0749-6419(03)00059-7

    Article  MATH  Google Scholar 

  56. D. C. Drucker, “Relation of experiments to mathematical theories of plasticity,” J. Appl. Mech. 16 (4), 349–357 (1949). https://doi.org/10.1115/1.4010009

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. W. Prager and D. C. Drucker, “Soil mechanics and plastic analysis or limit design,” Q. Appl. Math. 10 (2), 157–165 (1952). https://doi.org/10.1090/qam/48291

    Article  MathSciNet  MATH  Google Scholar 

  58. T. Kobayashi, J. W. Simons, C. S. Brown, and D. A. Shockey, “Plastic flow behavior of Inconel 718 under dynamic shear loads,” Int. J. Impact Eng. 35 (5), 389–396 (2008). https://doi.org/10.1016/j.ijimpeng.2007.03.005

    Article  Google Scholar 

  59. M. Becker and H. P. Hackenberg, “A constitutive model for rate dependent and rate independent inelasticity: Application to IN718,” Int. J. Plast. 27 (4), 596–619 (2011). https://doi.org/10.1016/j.ijplas.2010.08.005

    Article  MATH  Google Scholar 

  60. J. M. Pereira and B. A. Lerch, “Effects of heat treatment on the ballistic impact properties of Inconel 718 for jet engine fan containment applications,” Int. J. Impact Eng. 25 (8), 715–733 (2001). https://doi.org/10.1016/S0734-743X(01)00018-5

    Article  Google Scholar 

  61. B. Erice, M. J. Pérez-Martín, and F. Gálvez, “An experimental and numerical study of ductile failure under quasi-static and impact loadings of Inconel 718 nickel-base superalloy,” Int. J. Impact Eng. 69, 11–24 (2014). https://doi.org/10.1016/j.ijimpeng.2014.02.007

    Article  Google Scholar 

  62. J. P. Meyer and J. F. Labuz, “Linear failure criteria with three principal stresses,” Int. J. Rock Mech. Min. Sci. 60, 180–187 (2013). https://doi.org/10.1016/j.ijrmms.2012.12.040

    Article  Google Scholar 

  63. B. E. Echávarri, PhD Thesis (Universidad Politécnica de Madrid, Madrid, 2012).

  64. H. É. Tresca, “Compte rendu de la soirée scientifique du 29 octobre 1864 au Conservatoire des arts et métiers,” Annales du Conservatoire des arts et métiers 5, 205–288 (1864).

  65. R. Von Mises, “Göttinger Nachrichten,” Math. Phys. Klasse 4, 582–592 (1913).

    Google Scholar 

  66. P. W. Bridgman, Studies in Large Plastic Flow and Fracture with Special Emphasis on the Effects of Hydrostatic Pressure (McGraw-Hill, USA, 1954).

    MATH  Google Scholar 

  67. P. W. Bridgman, Studies in Large Plastic Flow and Fracture, 2nd ed. (Harvard University Press, USA, 1964).

    Book  Google Scholar 

  68. J. Lubliner, Plasticity Theory (Macmillan Publ. Comp., New York, 2008).

    MATH  Google Scholar 

  69. J. P. Bardet, “Lode dependences for isotropic pressure-sensitive elastoplastic materials,” J. Appl. Mech. 57 (3), 498–506 (1990). https://doi.org/10.1115/1.2897051

    Article  ADS  Google Scholar 

  70. P. Menetrey, and K. J. Willam, “Triaxial failure criterion for concrete and its generalization,” Struct. J. 92 (3), 311–318 (1995). https://doi.org/10.14359/1132

    Article  Google Scholar 

  71. L. M. Kachanov, “Time of the rupture process under creep conditions,” Izv. Akad. Nauk. SSR Otd. Tekh. Nauk 8, 26–31 (1958).

    Google Scholar 

  72. F. A. Leckie, and D. R. Hayhurst, “Creep rupture of structures,” Proc. Roy. Soc. London. A. Math. Phys. Sci. 340 (1622), 323–347 (1974).

    ADS  MATH  Google Scholar 

  73. D. S. Neto, D. Peric, and D.R. Owen, Computational Methods for Plasticity: Theory and Applications (John Wiley & Sons, United Kingdom, 2011).

    Google Scholar 

  74. J. Lemaitre, “A three-dimensional ductile damage model applied to deep-drawing forming limits,” in Proceedings of the 4th International Congress on the Mechanical (ICM-4), Stockholm, Sweden (Stockholm, 1983), Vol. 2, pp. 1059–1065.

  75. J. Lemaitre and H. Lippmann, A Course on Damage Mechanics, Vol. 2. (Springer, Berlin, 1996)

    Book  Google Scholar 

  76. M. L. Wilkins, R. D. Streit, and J. E. Reaugh, Cumulative-Strain-Damage Model of Ductile Fracture: Simulation and Prediction of Engineering Fracture Tests (Lawrence Livermore National Lab., San Leandro, CA (USA), 1980).

    Book  Google Scholar 

  77. M. L. Wilkins, Computer Simulation of Dynamic Phenomena (Springer-Verlag Berlin Heidelberg, New York, 1999).

    Book  Google Scholar 

  78. A. Brownrigg, W. A. Spitzig, O. Richmond, D. Teirlinck, et al., “The influence of hydrostatic pressure on the flow stress and ductility of a spheroidized 1045 steel,” Acta Mater. 31 (8), 1141–1150 (1983).

    Article  Google Scholar 

  79. M. Brünig, “Numerical simulation of the large elastic–plastic deformation behavior of hydrostatic stress-sensitive solids,” Int. J. Plast. 15 (11), 1237–1264 (1999). https://doi.org/10.1016/S0749-6419(99)00042-X

    Article  MATH  Google Scholar 

  80. O. Richmond and W. A. Spitzig, “Pressure dependence and dilatancy of plastic flow,” in IUTAM Conference, Theoretical and Applied Mechanics, Proc. 15th International Congress of Theoretical and Applied Mechanics, Amsterdam, 1980 (North-Holland Publishers, Amsterdam, 1980), pp. 377–386.

  81. W. A. Spitzig and O. Richmond, “The effect of pressure on the flow stress of metals,” Acta Metall. 32 (3), 457–463 (1984). https://doi.org/10.1016/0001-6160(84)90119-6

    Article  Google Scholar 

  82. M. Algarni, Y. Bai, and Y. Choi, “A study of Inconel 718 dependency on stress triaxiality and Lode angle in plastic deformation and ductile fracture,” Eng. Fract. Mech. 147, 140–157 (2015). https://doi.org/10.1016/j.engfracmech.2015.08.007

    Article  Google Scholar 

  83. M. Algarni, Y. Choi, and Y. Bai, “A unified material model for multiaxial ductile fracture and extremely low cycle fatigue of Inconel 718,” Int. J. Fatigue 96, 162–177 (2017). https://doi.org/10.1016/j.ijfatigue.2016.11.033

    Article  Google Scholar 

  84. J. Rottler and M. O. Robbins, “Yield conditions for deformation of amorphous polymer glasses,” Phys. Rev. E 64 (5), 051801 (2001). https://doi.org/10.1103/PhysRevE.64.051801

    Article  ADS  Google Scholar 

  85. N. Bonora, and A. Ruggiero, “Micromechanical modeling of ductile cast iron incorporating damage. Part I: Ferritic ductile cast iron,” Int. J. Solids Struct. 42(5–6), 1401–1424 (2005). https://doi.org/10.1016/j.ijsolstr.2004.07.025

    Article  MATH  Google Scholar 

  86. N. Bonora, D. Gentile, A. Pirondi, and G. Newaz, “Ductile damage evolution under triaxial state of stress: theory and experiments,” Int. J. Plast. 21 (5), 981–1007 (2005). https://doi.org/10.1016/j.ijplas.2004.06.003

    Article  MATH  Google Scholar 

  87. A. Pirondi and N. Bonora, “Modeling ductile damage under fully reversed cycling,” Computat. Mater. Sci. 26, 129–141 (2003). https://doi.org/10.1016/S0927-0256(02)00411-1

    Article  Google Scholar 

  88. J. Lemaitre and J. L. Chaboche, Mechanics of Solid Materials (Cambridge Uni. Press, Cambridge, 1990).

    Book  Google Scholar 

  89. J. Lemaitre, “Coupled elasto-plasticity and damage constitutive equations,” Comput. Meth. Appl. Mech. Eng. 51 (1), 31–49 (1985). https://doi.org/10.1016/0045-7825(85)90026-X

    Article  ADS  MATH  Google Scholar 

  90. J. Lemaitre, “A continuous damage mechanics model for ductile fracture,” J. Eng. Mater. Technol. 107 (1), 83–89 (1985). https://doi.org/10.1115/1.3225775

    Article  Google Scholar 

  91. A. L. Gurson, “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). https://doi.org/10.1115/1.3443401

    Article  Google Scholar 

  92. V. Tvergaard, “Influence of voids on shear band instabilities under plane strain conditions,” Int. J. Fract. 17 (4), 389–407 (1981). https://doi.org/10.1007/BF00036191

    Article  Google Scholar 

  93. V. Tvergaard, “On localization in ductile materials containing spherical voids,” Int. J. Fract. 18 (4), 237–252 (1982). https://doi.org/10.1007/BF00015686

    Article  Google Scholar 

  94. V. Tvergaard and A. Needleman, “Analysis of the cup-cone fracture in a round tensile bar,” Acta Metar. 32 (1), 157-169 (1984). https://doi.org/10.1016/0001-6160(84)90213-X

    Article  Google Scholar 

  95. C. C. Chu and A. Needleman, “Void nucleation effects in biaxially stretched sheets,” J. Eng. Mater. Technol. 102 (3), 249–256 (1980). https://doi.org/10.1115/1.3224807

    Article  Google Scholar 

  96. M. G. Cockcroft and D. J. Latham, “Ductility and the workability of metals,” J. Inst. Metals. 96 (1), 33–39 (1968).

    Google Scholar 

  97. T. Børvik, O. S. Hopperstad, T. Berstad, and M. Langseth, “A computational model of viscoplasticity and ductile damage for impact and penetration,” Eur. J. Mech. A-Solids 20 (5), 685–712 (2001). https://doi.org/10.1016/S0997-7538(01)01157-3

    Article  MATH  Google Scholar 

  98. F. Neukamm, M. Feucht, and A. Haufe, “Considering damage history in crashworthiness simulations,” in Proc. 7 European LS-DYNA Conference, Salzburg, 2009 (DYNAMore, 2009).

  99. F. Neukamm, M. Feucht, and M. Bischoff, “On the Application of Continuum Damage Models to Sheet Metal Forming Simulations,” in X International Conference on Computational Plasticity COMPLAS X, Barcelona, Spain, 2009 (CIMNE, Barcelona, 2009), pp. 1–4.

  100. F. Neukamm, M. Feucht, and A. Haufe, “Consistent damage modelling in the process chain of forming to crashworthiness simulations,” in 7th German LS-DYNA Anwenderforum, Bamberg 2008 (DYNA, 2008), pp. 11–20.

  101. F. X. Andrade, M. Feucht, A. Haufe, and F. Neukamm, “An incremental stress state dependent damage model for ductile failure prediction,” Int. J. Fract. 200 (1–2), 127–150 (2016). https://doi.org/10.1007/s10704-016-0081-2

Download references

Author information

Authors and Affiliations

  1. King Abdulaziz University, Mechanical Engineering Department, Faculty of Engineering, P.O. Box 344, 21911, Rabigh, Saudi Arabia

    Mohammed Algarni & Mohammed Zwawi

  2. University of Jeddah, Mechanical and Material Engineering Department, Faculty of Engineering, P.O. Box 80327, 21589, Jeddah, Saudi Arabia

    Sami Ghazali

Authors
  1. Mohammed Algarni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sami Ghazali
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohammed Zwawi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammed Algarni.

Additional information

Corresponding author at the Mechanical Engineering Dept. - Rabigh, King Abdulaziz University, Mohammed Algarni, Ph.D. <malgarni1@kau.edu.sa>

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Algarni, M., Ghazali, S. & Zwawi, M. The Emerging of Stress Triaxiality and Lode Angle in Both Solid and Damage Mechanics: A Review. Mech. Solids 56, 787–806 (2021). https://doi.org/10.3103/S0025654421050058

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654421050058

Keywords:

Search

Navigation