The internal state variable formalism

Because the process conditions inevitably vary during industrial processing of metals, the internal state variable method was originally proposed to model the non-isothermal transformation behaviour. If the phenomena of the microstructure evolution can be represented in the form of some differential equations about the state variables variation with respect to time, then using some appropriate numerical codes that have been developed commercially and available easily at present, the process history, in principle, can be represented by these state variables whose values are calculated by integrating step-wise throughout the process.

Modelling of microstructure evolution explicitly in differential form has been the basis for most classical theories of work hardening and annealing. The internal state variables are now physically meaningful quantities that can, at least in principle, be measured by electron microscopy (dislocation density etc.).

developments of this approach can now benefit from the recent advances in microscopy, such as semiautomatic Electron Back Scatter Diffraction (EBSD), which enables substructures to be quantified with far greater speed and precision (Hurley and Humphreys 2003a, 2003b). Differential physically based state variable models have the potential to follow complex process histories and provide a means of conveying microstructure explicitly from one processing stage to the next.

Physically-based internal state variable models have been reported extensively by these research groups (Nes et al at the Norwegian University of Science and Technology in Trondheim, Sellars et al at the University of Sheffield, UK and Sheppard’s group at Bournemouth University, UK, after establishment at Imperial College, UK). There are three separate modelling tasks: (a) describing the evolution of the deformation substructure, in particular the subgrain size, dislocation density and subgrain boundary misorientation; (b) relating substructure parameters to flow stress; and (c) predicting recrystallisation behaviour.

The Trondheim group have proposed and further developed a three-parameter approach to model the metal plasticity (Marthinsen and Nes 1997, 2001; Nes 1997;

Nes and Marthinsen 2002; Nes et al. 2000). Their result was concluded as a work hardening model during plastic deformation of FCC metals and alloys. Based on a statistical approach to the problem of athermal storage of dislocations, the model combines the solution for the dislocation storage problem with models for dynamic recovery of network dislocations and sub-boundary structures. Finally a general state variable description is obtained. The model includes the effects resulting from variations (a) stacking fault energy (b) grain size (c) solid solution

content (d) particle size and volume fraction. Although the Trondheim group called their model a ‘unified theory of deformation’, controversy exists. Some workers (Shercliff and Lovatt 1999) doubt their ‘unified model’ cannot be straightforwardly applied in a practical context because it introduces many adjustable parameters. Marthinsen and Nes (2001; 2002) later argued the large number of tuning parameters is not a problem and their model can deal with processing conditions under any combination of constant strain rate and temperature or more complex transient conditions. However from the fact that nearly no researchers adopted their model in the FE simulation except only one paper can be found by themselves in a conference (Marthinsen et al. 2003) it is safe to say too many adjustable parameters, at least, do hamper its wide application in its integration with FE simulation. Furthermore, it appears that any attempt to unify this theory with materials which dynamically recrystallise has not been successful.

The Sheffield group have approached hot working of aluminium alloys from a background of FE analysis of the transient nature of the deformation history in flat rolling, in terms of temperature, strain rate and strain path. Models for predicting the evolution of internal state variables such as internal dislocation density, subgrain size and misorientation between subgrains, as well as subsequent recrystallisation behaviour are developed for both constant and transient deformation conditions (Baxter et al. 1999; Zhu and Sellars 2000). It should be noted that although great efforts have been invested by the Sheffield group to carry out experiments to get the mathematical expressions and to finally validate these models they used very simple FEM that is not capable of structure prediction.

Until recently have they begun to resolve proper problems, mainly rolling (Talamantes-Silva et al. 2009; Zhu et al. 2003).

Despite the criticism that the physically based models from the Sheffield have mainly been concentrated on a specific alloy (Al-1%Mg) and developed from

experiments utilising plain strain compression (PSC), in which plastic strains greater than 2 are difficult to achieve and the interaction of recrystallisation and precipitation has not been considered in detail in the models (Jones and Humphreys 2003), the Sheffield models assisted researchers to study microstructure evolution because the nature of these models focus on the transient nature of the metal processing and they have the advantage of less number of tuning parameters and convenience to be integrated into commercial FEM codes.

Sheppard’s group has done a considerable amount of pioneering work in this respect. After the success in applying these models into rolling simulation (Duan and Sheppard 2002b, 2003a; Sheppard and Duan 2002) researching attention was turned to its integration with the FEM simulation of more complex extrusion process (Duan and Sheppard 2003b; Duan et al. 2004; Flitta and Sheppard 2004;

Flitta et al. 2007; Peng and Sheppard 2004 ; Sheppard and Velay 2007). In this work suitable models will be chosen and adapted for extrusion and its post-treatment simulation by user-subroutine interface of the commercial FEM software Forge2009^®.

2.6.4.3 Modelling dislocation substructure changes