Iso‐work Increment Assumption for Heterogeneous Material Behaviour Modelling

Abstract

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).

A model for the microstructure behaviour and strength evolution in lath martensite

“…However, the partitioning behaviour in these steels is still not fully understood [56]. Bouaziz and Buessler [57] have proposed an alternative approach to describe plasticity in multiphase materials based on the assumption that the mechanical work on each constituent is equal: r 1 e 1 ¼ r 2 e 2 , where r 1;2 and e 1;2 are the stress and strain induced on each phase, respectively. This model is consistent with the strain partitioning behaviour in dual-phase steels in which ferrite displays lower strength and higher ductility, whereas in martensite most of the external work is due to its higher strength.…”

Microstructure evolution during dynamic discontinuous recrystallization in particle-containing Cu

“…For simplicity, a Taylor assumption is employed in the present homogenization scheme, wherebẏ ε p eff =ε p eff,i is assumed as the elastic strains are negligible. It can be noted that alternative formulations exist, such as the iso-work assumption suggested in [64]. A more rigorous homogenization procedure would be to consider a multilevel formulation, where the macroscopic deformation is enforced as a periodic boundary condition on the microlevel RVE.…”

A new flow behaviour model for the austenite at high temperature

“…-a linear mixture law in terms of the volume fractions of recrystallized and unrecrystallized austenite; -a same mixture law applied to the strains; -the condition of equal plastic work increment in each constituent called IsoW proposed by reference [4].…”