Page 425 - Biaxial Multiaxial Fatigue and Fracture

Cyclic Behaviour of a Duplex Stainless Steel Under Multiaxial Loading: Experiments and Modelling 409

CONSTITUTIVE MODELING OF CYCLIC BEHAVIOR

Different constitutive laws were used with the aim at checking their abilities to describe the
experimental data presented above. The base model for the simulations is a cyclic plasticity
model with one non-linear isotropic hardening rule and two non-linear kinematic hardening
rules, initially proposed by Armstrong and Frederick [28] (model NLK). It has been shown that
the non-linear Armstrong-Frederick rule does not consider the extra-hardening induced by non-
proportional loadings in tension-torsion tests conducted on austenitic stainless steels [ 17-28].
Therefore, the three other models tested, called non-proportional models, are derived from this
base model and propose modifications either of the isotropic or of the kinematic rules to
improve the description of non-proportional hardening.
For all these models, the elastic behaviour takes the following form for an orthotropic
material under tension-torsion loading:
- - Ee = (4)
where

5 is the total strain tensor, 5' is the elastic strain tensor, gp is the plastic strain tensor and 4
-
represents the elasticity matrix. E is the Young modulus, v is the Poisson ratio and G is the
shear modulus.

The three non-proportional models include the equations of base model NLK. The only
alteration consists in an addition of strengthening variables taking into account the extra-
hardening. The corresponding constitutive equations are presented below and in the following
four tables.

Models NP1 and NP2 use a non-proportionality parameter A defined by Benallal and
Marquis [20]. The non-proportionality of the loading is characterised by the angle between
-
tensors X and $. -
A = 1 - cos'a (6)

(7)
In its first version, model Np1 considered the circle path as the most hardening, this
hypothesis has then been invalidated by numerous experimental results [S, 10-121. We used
therefore the version modified by Calloch which removes this hypothesis [ 10, 121.
The non-proportionality parameter affects either the isotropic hardening (model NPl) [ 121,
or the kinematic hardening (model NP2) [25].

The last model was proposed by Tanaka and uses another parameter A to take into account
the non-proportionality of the loading (model TANA) [26]. Tanaka expressed the plastic strain
tensor - gP as a vector EP in a five-dimensional deviatoric space.