A theoretical evaluation of plasticity hardening algorithms for nonproportional loadings


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.

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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


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


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


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


Stress tensor


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


Mróz translation vector


Garud translation vector


Symbol denoting inner product


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


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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

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