Modelling dislocation substructure changes

Figure 2.2 A schematic representation of the microstructure; cell diameter, δ, cell wall thickness, h, cell wall dislocation density ρb and dislocation density within the

cells ρi (after Nes 1997)

Dislocation theory is an important tool to study the mechanism of metal plasticity.

Dislocation substructure can be described by three internal state variables:

dislocation density (ρ), subgrain size (δ) and misorientation across subgrain boundaries (θ). To have a clear understanding of these variables, a schematic representation of substructure was taken from the literature (Nes 1997).

During the early stage of deformation, dislocation multiplication occurs, and the total dislocation density = + increases from 10 ~10 to 10 ~10 at the commencement of macroscopic flow. Dislocations move and interact with one another to form tangles. This terminates in a cellular structure with the dislocations clustering tightly into the cell walls separating dislocation free regions. As deformation proceeds, continues increasing and attains a constant value of approximately 10 when the steady state regime is reached. The cellular structures are replaced completely by the formation of subgrains due to the additional dislocation reactions. Subgrains can be regarded as an extension of a cellular structure in that the dislocations are arranged in the form of planar networks in subgrain boundaries, while the cellular boundaries consist of three-dimensional network and tangles of dislocation. The ability to form a cellular or subgrain structure depends on several factors: the stacking fault energy, the applied stress, the strain, the temperature and the presence or absence of obstacles.

A notable feature of subgrains is they are equiaxed and maintain their equilibrium size and shape in the steady state even at very large strains whereas the grains are always elongated in the direction of the extension. There are two interpretations for this phenomenon. The first considers that sub-boundaries are constantly migrating in such a way as to keep the substructure equiaxed. The second possible interpretation is by the repeated unravelling of the sub-boundaries and the subsequent reformation of new sub-boundaries at locations which keep their average spacing and dislocation density constant, termed ‘repolygonisation’ (Jonas

et al. 1969).

2.6.4.3.2 Dislocation density evolution modelling

For steady state deformation a generally accepted equation that represents the relation between the internal dislocation density and subgrain size is written as

= (2.17)

It was first proposed by Holt (1970) from his experiment observations.

In contrast with the consensus reached for the calculation of the dislocation density at steady state deformation, different dislocation evolution models can be found in different work-hardening models such as the Mechanical-Threshold-Strength (MTS) model, the Microstructure Metal Plasticity (MMP) model and the Three Internal Variable (3IV) model reviewed by Holmedal et al (Hirsch 2006, p.129). But their purpose is to try to find a flow stress formulation that can include effects due to variations in solid solution level, particle contents, grain size, etc.

from a microstructure view point, which inevitably involves many tuning parameters. The difficulty is increased in this problem because some of the parameters are not as yet defined that an intelligent guess must be utilised.

By contrast, Sellars and Zhu’s model (Baxter et al. 1999; Sellars and Zhu 2000;

Zhu et al. 1997) developed from an FE background that doesn’t have a large number of tuning parameter seems more suitable to be integrated with FEM. In their model, the internal dislocation density has two components, that is, the so-called ‘random’ dislocation and the ‘geometrically necessary’ dislocation density ,

= + (2.18)

This model will be discussed in detail in Chapter 5.

2.6.4.3.3 Subgrain size modelling

It is generally accepted that the subgrain size δ can be directly related to the temperature-compensated strain-rate or Zener-Hollomon parameter, Z, by the following equation

= + (2.19)

where , , are constants.

A good subgrain size fit could be obtained by varying exponent within a range between -1.25 and -0.35 (Zaidi and Sheppard 1983). In fact, many researchers (Castle 1992; Chanda et al. 2000; Subramaniyan 1989) have chosen = −1 to produce accepted results. This is because in hot working range the subgrain size ranges obtainable are very small when compared with the range of value.

It should be noted that equation (2.19) can be modified into different forms (Jonas et al. 1969; Nes et al. 1994; Sheppard and Raghunathan 1989) to predict steady state subgrain size. But they all use more extra statistically-defined parameters from experimental data and would not be discussed.

It should also be emphasised that equation (2.19) is not valid for prediction of subgrain size in a transient deformation. In contrast with the well recognised relationship for subgrain size during steady-state deformation, there is still a lack of quantitative relationship to relate the subgrain size with the deformation parameters in a transient deformation. Nevertheless, the Trondheim group

(Marthinsen and Nes 1997) and Sheffield group (Sellars and Zhu 2000; Zhu et al.

1997) have proposed their equations respectively to model the subgrain size evolution during transient state deformation. The Trondheim group approached the subgrain size problem including the effects of different microstructures which naturally need corresponding parameters to describe. As a result, to use this model those parameters must be derived from experimental data or from reasonable estimation, which greatly increased the difficulty in its integration with FEM. On the other hand, the model from the Sheffield group is calculated from a background of FE analysis of the transient nature of hot deformation history in terms of temperature, strain rate and strain path. In their model, the subgrain size evolution have been explicitly expressed in a differential form, so even though its physical basis is limited to some extent, using this evolution law reasonable results have been achieved in hot working simulation, at least, in rolling simulation (Ahmed et al. 2005; Duan 2001). However there is a great lack of research in prediction of subgrain size by FEM in aluminium extrusion. The few attempts (Dashwood et al. 1996; Duan and Sheppard 2003b) carried out on this topic either used only empirical steady state equation or were limited to simple rod extrusion.

This exactly highlights that using physically-based model to simulate complex shape extrusion is of significance.

2.6.4.3.4 Misorientation change modelling

Misorientation could be the least well-characterised microstructure variable probably because of the obvious experimental technique constraints. Some work concerned about the high purity aluminium is shown in Figure 2.3 (Nes and Marthinsen 2002) that clearly shows that the average boundary misorientation increases rapidly with strain, reaching about 3º at a strain of about 1, after that it remains constant up to strain as high as 4.

Figure 2.3 Sub-boundary misorientation vs. strain

Sheffield group (Zhu and Sellars 2000) have proposed another relationship for Al-1%Mg alloy during transient deformation conditions in differential form:

= 1

( − ) (2.20)

where is a characteristic strain. In their study, the predicted results agree well with the experiments for the increasing and constant strain rate conditions while the discrepancy became larger during decreasing strain rate condition. But this is not the case for extrusion; therefore, equation (2.20) has the potential to predict the misorientation that will be investigated further in Chapter 5.

Boundary misorientation (º)

Strain ( ) Rolling reduction (%)

3. Thermal-mechanical simulation of the extrusion process