A theoretical evaluation of plasticity hardening algorithms for nonproportional loadings

Summary

Incremental plasticity theories are being incorporated into many engineering numerical analyses. There are two basic categories of incremental plasticity algorithms, (i) multiple surface such as proposed by Mróz and Garud, and (ii) the Armstrong-Frederick type as modified by Chaboche et al. Engineering bounds on the general applicability of these models for cyclic loading have not been the subject of a detailed investigation. Similar first order stress-strain results are obtained for proportional loadings when either category of hardening rules is chosen.

A π-circle loading path in deviatoric stress space was identified as a severe test of the algorithms. A closed form solution can be obtained for this loading path, so that the phenomena noted cannot be attributed to numerical errors. While this loading is highly idealized, analogous results are also noted for 90 degree out-of-phase tension-torsion. Multiple surface and the associated two surface models with a stationary bounding surface are shown to have both theoretical and numerical problems for the severe cyclic loading under consideration. The Armstrong-Frederick class of models demonstrates a diminished sensitivity to modeling parameters, as well as a single-valued stress-strain representation for both severe nonproportional loading paths investigated.

This is a preview of subscription content, access via your institution.

Abbreviations

b :

Magnitude of backstress

c, r :

Material constants in the Armstrong-Frederick hardening rule

c (i),r (i) :

Material constants in the Chaboche hardening rule (i=1,2,...,M)

d :

Prefix denoting infinitesimal increment or differentiation

f :

Yield surface function

f (i) :

A surface in the multiple surface models

h :

Plastic modulus function

I :

Unit tensor or Kronecker Delta

k :

Yield stress in simple shear

K′:

Cyclic strength coefficient

M :

An integer denoting the number of surfaces for a multiple surface model or the number of terms in the backstress expansion for an Armstrong-Frederick model

n′:

Cyclic strain hardening exponent

n :

Unit exterior normal to the yield surface at the stress state

p :

Equivalent plastic strain

R :

Yield surface radius in deviatoric stress space

R b :

Bounding surface radius in deviatoric stress space

R (i) :

Radius of theith surface in the multiple surface type hardening rules

S :

Deviatoric stress tensor

\(\bar S\) :

Scalar representation for stress

α:

Total backstress tensor

α (i) :

ith backstress tensor (i=1,2,...,M) in the Armstrong-Frederick type models or the center of theith surface in the Mróz multiple surface type models

ε p :

Plastic strain tensor

Δ ε p/2:

Scalar representation for plastic strain

ε x ε p x :

Axial plastic strain for tension-torsion loading

γ p :

Shear plastic strain for tension-torsion loading

σ:

Stress tensor

θ:

Angle between the translation direction of the yield surface and the exterior normal at the stress state on the yield surface

ν:

Mróz translation vector

ν′:

Garud translation vector

∶:

Symbol denoting inner product

A‖:

Representing an invariant of a tensorA defined by\(||A|| = \sqrt {A:A} \)

References

  1. [1]

    Mróz, Z.: On the description of anisotropic workhardening. J. Mech. Phys. Solids15, 163–175 (1967).

    Google Scholar 

  2. [2]

    Mróz, Z.: An attempt to describe the behavior of metals under cyclic loads using a more general workhardening model. Acta Mech.7, 199–212 (1969).

    Google Scholar 

  3. [3]

    Iwan, W. D.: On a class of models for the yielding behaviour of continuous and composite systems. ASME J. Appl. Mech.34, 612–617 (1967).

    Google Scholar 

  4. [4]

    Hunsaker, B. Jr., Vaughan, D. K., Stricklin, J. A.: A comparison of the capability of four hardening rules to predict a material's plastic behavior. ASME J. Press. Vessel Tech.98, 66–74 (1976).

    Google Scholar 

  5. [5]

    Lamba, H. S., Sidebottom, O. M.: Cyclic plasticity for nonproportional paths: part I: cyclic hardening, erasure of memory, and subsequent strain hardening experiments. ASME J. Eng. Mater. Tech.100, 96–103 (1978).

    Google Scholar 

  6. [6]

    Lamba, H. S., Sidebottom, O. M.: Cyclic plasticity for nonproportional paths: part II: comparison with predictions of three incremental plasticity models. ASME J. Eng. Mater. Tech.100, 104–111 (1978).

    Google Scholar 

  7. [7]

    Garud, Y. S.: Multiaxial fatigue of metals. Ph.D Dissertation, Department of Mechanical Engineering, Stanford University, 1981.

  8. [8]

    Garud, Y. S.: Prediction of stress-strain response under general multiaxial loading. Mechanical Testing for Deformation Model Development, ASTM STP 765 (Rohde, R. W., Swearengen, J. C., eds.), pp. 223–238. Philadelphia: American Society for Testing and Materials 1982.

    Google Scholar 

  9. [9]

    Chaboche, J. L.: A new constitutive framework to describe limited ratchetting effects. In: Advances in plasticity: Proceedings of Plasticity '89 (Khan, A. S., Tokuda, M., eds.), pp. 211–214. 2nd Int. symposium on Plasticity and its Current Applications, 1989, Mie, Japan. New York: Pergamon Press 1989.

    Google Scholar 

  10. [10]

    Chang, K. C., Lee, G. C.: Constitutive relations of structure steel under nonproportional loading. J. Eng. Mech.112, 806–820 (1986).

    Google Scholar 

  11. [11]

    Iwata, K.: A new constitutive model for cyclic plasticity. In: Structural mechanics in reactor technology (Shibata, H., ed.), Vol. L, L22/4. 1th Int. Conf. Struct. Mech. Reactor Technol., Tokyo, Japan, 1991. Atomic Energy Society of Japan and The International Association for Structural Mechanics in Reactor Technology 1991.

  12. [12]

    Lu, W. Y., Mohamed, Z. M.: A two-surface plasticity theory and its application to multiaxial loading. Acta Mech.69, 43–57 (1987).

    Google Scholar 

  13. [13]

    McDowell, D. L.: A two surface model for transient nonproportional cyclic plasticity, part I: development of appropriate equations. ASME J. Appl. Mech.52, 298–302 (1985).

    Google Scholar 

  14. [14]

    McDowell, D. L.: A two surface model for transient nonproportional cyclic plasticity, part II: comparison of theory with experiments. ASME J. Appl. Mech.52, 303–308 (1985).

    Google Scholar 

  15. [15]

    Takahashi, Y., Ogata, T.: Description of nonproportional cyclic plasticity of stainless steel by a two-surface model. ASME J. Appl. Mech.58, 623–630 (1991).

    Google Scholar 

  16. [16]

    Tanaka, E., Murakami, S., Ooka, M.: Constitutive modeling of cyclic plasticity in non-proportional loading conditions. In: Constitutive laws for engineering materials: theory and applications. vol. I (Sedai, C. S., Krempl, E., Kiousis, P. D., Kundu, T., eds.), pp. 639–646. 2nd Int. Conf. Constitutive Laws for Engng. Mater.: Theory and Applications, Arizona, U.S.A. New York: Elsevier 1987.

    Google Scholar 

  17. [17]

    Bruhns, O. T., Pape, A.: A three surface model in nonproportional cyclic plasticity. In: Advances in constitutive laws for engineering materials (Fan, J., Murakami, S., eds.), vol. 2, pp. 703–708. Int. Conf. Constitutive Laws for Eng. Mater., Chongqing, China, 1989. Beijing: Int. Academic Publ. 1989.

    Google Scholar 

  18. [18]

    Bruhns, O. T., Lehmann, T., Pape, A.: On the description of transient cyclic hardening behavior of mild steel CK 15. Int. J. Plasticity8, 331–359 (1992).

    Google Scholar 

  19. [19]

    Chaboche, J. L.: A new kinematic hardening rule with discrete memory surfaces. Rech. Aérosp.4, 49–69 (1989).

    Google Scholar 

  20. [20]

    Mróz, Z.: On generalized kinematic hardening rule with memory of maximum prestress. J. Méch. Appl.5, 241–260 (1981).

    Google Scholar 

  21. [21]

    Mróz, Z.: Hardening and degradation rules for metals under monotonic and cyclic loading. ASME J. Eng. Mater. Tech.105, 113–118 (1983).

    Google Scholar 

  22. [22]

    Chu, C.-C.: A three dimensional model of anisotropic hardening in metals and its applications to the analysis of sheet metal formability. J. Mech. Phys. Solids32, 197–212 (1984).

    Google Scholar 

  23. [23]

    Köttgen, V. B., Seeger, T.: A Masing type integration of the Mróz model for some non-proportional stress-controlled paths. ASME J. Eng. Mater. Tech. (submitted).

  24. [24]

    Hashiguchi, K.: Mechanical requirements and structures of cyclic plasticity models. Int. J. Plasticity9, 721–748 (1993).

    Google Scholar 

  25. [25]

    McDowell, D. L.: Evaluation of intersection conditions for two-surface plasticity theory. Int. J. Plasticity5, 29–50 (1989).

    Google Scholar 

  26. [26]

    Tseng, N. T., Lee, G. C.: Simple plasticity model of the two-surface type. J. Eng. Mech.109, 795–810 (1983).

    Google Scholar 

  27. [27]

    Jiang, Y., Sehitoglu, H.: Comments on the Mróz multiple surface plasticity models. Int. J. Solids Struct. (in press).

  28. [28]

    Armstrong, P. J., Frederick, C. O.: A mathematical representation of the multiaxial Bauschinger effect. Report RD/B/N 731, Central Electricity Generating Board, 1966.

  29. [29]

    Bower, A. F.: Some aspects of plastic flow, residual stress and fatigue due to rolling and sliding contact. Ph.D. Dissertation, Emmanuel College, Department of Engineering, University of cambridge, 1987.

  30. [30]

    Bower, A. F.: Cyclic hardening properties of hard-drawn copper and rail steel. J. Mech. Phys. Solids37, 455–470 (1989).

    Google Scholar 

  31. [31]

    Chaboche, J. L., Dang Van, K., Cordier, G.: Modelization of the strain memory effect on the cyclic hardening of 316 stainless steel. In: Structural mechanics in reactor technology. Trans. 5th Int. Conf. Struct. Mech. in Reactor Technol., Div. L, Berlin, L11/3 (1979).

  32. [32]

    Chaboche, J. L.: On some modifications of kinematic hardening to improve the description of ratchetting effects. Int. J. Plasticity7, 661–687 (1991).

    Google Scholar 

  33. [33]

    Ohno, N., Wang, J.-D.: Kinematic hardening rules with critical state of dynamic recovery: part I — formulation and basic features for ratchetting behavior. Int. J. Plasticity9, 375–390 (1993).

    Google Scholar 

  34. [34]

    Jiang, Y., Sehitoglu, H.: Modeling of cyclic ratchetting plasticity: part 1 — development of constitutive equations. ASME J. Appl. Mech. (in press).

  35. [35]

    Prager, W.: The theory of plasticity: a survey of recent achievements. Proc. Inst. Mech. Eng.169, 41–57 (1955).

    Google Scholar 

  36. [36]

    Ziegler, H.: A modification of Prager's hardening rule. Q. Appl. Mech.17, 55–65 (1959).

    Google Scholar 

  37. [37]

    Drucker, D. C.: A more fundamental approach to plastic stess-strain relations. Proc. 1st U.S. National Congress of Appl. Mech., ASME, pp. 487–491 (1951).

  38. [38]

    Dafalias, Y. F.: A novel bounding surface constitutive law for the monotonic and cyclic hardening response of metals. Trans. 6th Int. Conf. Struct. Mech. in Reactor Technol., L3/4, Paris, France, L3/4, 1981.

  39. [39]

    Dafalias, Y. F., Popov, E. P.: A model of nonlinearly hardening materials for complex loading. Acta Mech.21, 173–192 (1975).

    Google Scholar 

  40. [40]

    Dafalias, Y. F., Popov, E. P.: Plastic internal variables formalism of cyclic plasticity. ASME J. Appl. Mech.43, 645–651 (1976).

    Google Scholar 

  41. [41]

    Krieg, R. D.: A practical two surface plasticity theory. ASME J. Appl. Mech.42, 641–646 (1975).

    Google Scholar 

  42. [42]

    McDowell, D. L.: An experimental study of the structure of constitutive equations for nonproportional cyclic plasticity. ASME J. Eng. Mater. Tech.107, 307–315 (1985).

    Google Scholar 

  43. [43]

    McDowell, D. L., Moyar, G. J.: Effects of non-linear kinematic hardening on plastic deformation and residual stresses in rolling line contact. Wear114, 19–37 (1991).

    Google Scholar 

  44. [44]

    Fatemi, A., Stephens, R. I.: Cyclic deformation of 1045 steel under in-phase and 90 deg out-of-phase axial-torsional loading conditions. In: Multiaxial fatigue; analysis and experiments, AE-14 (Leese, G. E., Socie, D., eds.), pp. 139–147. Warrendale: Society of Automotive Engineers 1989.

    Google Scholar 

  45. [45]

    Ohno, N., Wang, J. D.: Transformation of a nonlinear kinematic hardening rule to a multisurface form under isothermal and nonisothermal conditions. Int. J. Plasticity7, 879–891 (1991).

    Google Scholar 

  46. [46]

    Chaboche, J. L., Rousselier, G.: On the plastic and viscoplastic constitutive equations — part I: rules developed with internal variable concept. ASME J. Pressure Vessel Tech.105, 153–158 (1983).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Jiang, Y., Kurath, P. A theoretical evaluation of plasticity hardening algorithms for nonproportional loadings. Acta Mechanica 118, 213–234 (1996). https://doi.org/10.1007/BF01410518

Download citation

Keywords

  • Cyclic Loading
  • Closed Form Solution
  • Deviatoric Stress
  • Numerical Error
  • Basic Category