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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. a b s t r a c tA new semi-phenomenological model is developed based on a mean-field description of the TRIP behavior for the simulation of multiaxial loads. This model intends to reduce the number of internal variables of crystalline models that cannot be used for the moment in metal forming simulations. Starting from local and crystallographic approaches, the mean-field approach is obtained at the phase level with the concept of Mean Instantaneous Transformation Strain (MITS) accompanying martensitic transformation. Within the framework of the thermodynamics of irreversible processes, driving forces, martensitic volume fraction evolution and an expression of the TRIP effect are determined for this new model. A classical self-consistent scheme is used to model the behavior of multiphased TRIP steels. The model is tested for cooling at constant loads and for multiaxial loadings at constant temperatures. The predictions reproduce the increase in ductility, the dynamic softening effect and the multiaxial behavior of a multiphased TRIP steel.

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Mechanical behaviour modelling of the heterogeneous materials is one of important fields of physical metallurgy. The available solving methods can be classified in three main families. The first one is based on classical mixture rule widely used because very simple but it is not realistic or needs arbitrary fitting parameter to improve the prediction. The second one is based on homogenisation techniques like self-consistent method. The third one is based on finite element description of either a unit cell or a representative microstructure. It is able to be more accurate but much more difficult and heavy to perform, and therefore less used. To keep easy running mixture law modelling but to avoid any fitting parameter and improve prediction accuracy, an equi-incremental mechanical work has been assumed in each constituent in stressstrain mixture law.Despite the unrealistic iso-strain assumption (the strains in each constituent are taken equal, so-called Voigt assumption in elasticity and Taylor assumption in plasticity) and a weak predictability the simple stress mixture law is so far the most widely used way to quantify the heterogeneous material behaviour.In the case of two constituents, this law is expressed as:re 1 À F: r 1 e F: r 2 eWhere F is the volume fraction of the second phase, e is the strain and r, r 1 , r 2 are respectively the flow stress of the aggregate and of the two constituents.The prediction accuracy becomes very degraded when the constituents exhibit very different behaviour laws (strength and ductility). [1±3] To improve this previous approach, authors proposed to introduce a phenomenological non linear function of volume fraction in equation one. The first one was Gladman using a power law volume fraction function: [3] re 1 À F n : r 1 e F n : r 2 e(2)A second approach keeps the linear volume fraction mixture law for stress but applies the same mixture law to the strains. Such an intermediate mixture law was first proposed by Tomota et al.: [4] re 1 À F: r 1 e 1 À Á F: r 2 e 2 À Á (3a)e 1 À F: e 1 F: e 2 (3b)where e 1 is the average strain in the first constituent and e 2 the average strain in the second one. This modelling seems more realistic than the previous one but needs always a fitting parameter. Indeed, it should be noted that Eq. 3 does not indicate anything about the absolute amount of the stress and strain transfer. That is the reason why several authors have proposed to fix the following quantity: [2,5] r 2 À r 1 e 2 À e 1 6 7 bwhere b is an arbitrary fitting constant with 0 £ b £ ¥ (Fig. 1). Fig. 1. Graphic illustration of stress-strain mixture law and b parameter.Fig. 2. Calculated Young modulus versus volume fraction with IsoW assumption, selfconsistent, Hashin-Shtrikman bounds, upper and lower bound models (E 1 =100 GPa and E 2 =1000 GPa).

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