Effect of rare-earth metals on the hot strength of HSLA steels

APPENDIX

Theoretical modelling of the flow curves

It is well known that during the high-temperature deformation of austenite, two opposite mechanisms are controlling the resulting flow stress. Firstly, the work hardening mechanism due to the dislocation generation during the deformation process, which can also be aided by other hardening factors such as fine grain sizes or presence of second phase particles. Secondly, the softening mechanism caused by dynamic recovery followed by dynamic recrystallization. The combination of these opposite mechanisms leads to a steady state or saturation stress (see Fig. Al). In case that dynamic recristallization takes place, once a maximum stress is reached in the flow curve an additional softening is apparent. Both softening mechanisms are illustrated in Fig. Al, as well as the two different types of dynamic recrystallization traditionally observed: single- and multiple-peak types. It is worth noting that dynamic recrystallization of the cyclic mode is associated to grain coarsening, while the single peak one is associated to grain refinement.

Fig. Al

Sketch of the flow curves under high temperature deformation conditions, at constant strain rate. Description of the two types of dynamic recrystallization.

The modelling of the latter flow curve can be done by steps. One can assume that only work hardening and dynamic recovery are operating up to the peak stress. Physically-based models are readily available to describe such step. In the case of dynamic recrystallization, no clear models are still accepted in the literature. This inconvenience can be overpassed by assuming that the softening in the flow curve is directly proportional to the recrystallized volume fraction, which in turn follows an Avrami kinetics.

First step: The flow curve up to the peak stress

The stress component due to the dislocation glide can be related to the dislocation density according to the classical following expression:

where b is the Burgers vector, α' is a proportionality constant close to l, μ is the shear modulus and p is the density of mobile dislocations. This latter density evolves during a hot-deformation process as follows [13, 19, 21, 22, 23, 24, 25]:

The first term describes the change in dislocation density due to dislocation generation by work hardening processes, while the second term refers to dislocations recovered due to the softening mechanism of dynamic recovery. The combination of Eqs. (Al) and (A2) with the work hardening rate Θ=∂σ/ε|ε˙,T gives a complete set of expressions to predict the dislocation density variation during deformation at high temperature, provided that dynamic recovery is the only softening mechanism taking place.

Diverse models proposing laws for 0 can be found in the literature [22, 23]. However, the theory proposed by Estrin and Mecking [13] and Bergström [24, 25] gives the most satisfactory results. They assume that the mean free path of dislocations is geometrically imposed and, consequently, the rate of dislocations generated, which is proportional to the mean free path, is also constant. Like Kocks [23] they also assume that dynamic recovery follows a first order kinetic. The latter hypothesis transforms Eq. (A2) into:

The integration of the latter expression is relatively easy by assuming that the work hardening term (U) and the softening term (Ω ) are independent of the strain s. Taking ρ = ρ_o for ε = 0 results:

Introducing now Eq. (A1) into the latter formula, the modeling equation for the stress – strain curve can be derived:

or alternatively:

where

and

being σ_m the saturation stress in the flow curve in the absence of dynamic recrystallization (see Fig. A1), equal to σ_ss in case of softening solely by dynamic recovery, and very close to σ_p when dynamic recrystallization is occurring.

Second step: The flow curve after the peak stress

The modeling of dynamic recrystallization can be accomplished by treating it as a solid state transformation [9, 22, 26]. In this case, the kinetics of dynamic recrystallization can be represented by an Avrami equation. For this purpose, it is assumed that the mechanical softening is directly proportional to the recrystallized volume fraction X. In this way, the constitutive equation that applies after the initiation of dynamic recrystallization is:

where σ_ss is the steady-state stress at large strains. The latter in turn follows the Avrami equation:

or using the time for 50 % of recrystallization, t_{50 %} :

where B and k are constants associated with the nucleation mechanism (k) and the nucleation and growth rates (B).

Equations (A6) to (A11) can be used to represent the flow behavior of a metallic material, although some rate equations must also be provided. These are the ones that specify the saturation, peak and steady-state stresses, the work hardening and dynamic recovery terms and the recrystallization kinetics parameters. When these are available, the flow stress can be expressed as an explicit function of the temperature, strain rate and strain.