01.12.2014 | Research | Ausgabe 1/2014 Open Access

# Prediction of microstructure evolution during multi-stand shape rolling of nickel-base superalloys

- Zeitschrift:
- Integrating Materials and Manufacturing Innovation > Ausgabe 1/2014

## Electronic supplementary material

## Competing interests

## Authors’ contributions

## Background

^{″}(Ni3(Nb,Ti)), γ

^{′}, δ, and carbides (MC and M6C). The high temperature strength of Alloy 718 is derived essentially from the coherent γ

^{″}and to a smaller extent from γ

^{′}. The presence of the other precipitates improves hot-working to produce very fine-grained billet structures. However, modeling the complex precipitation processes requires a more detailed understanding of precipitation kinetics than is presently available. A mechanism-based model considering the δ phase precipitate effects has been discussed in literature by Thomas, et al. [4]. The proposed phenomenological model in this work results in good agreement with the observed microstructure as long as it is applied to materials where the initial microstructure does not contain excessive δ phase since these precipitates retard metadynamic recrystallization (MDRX). The proposed model is recommended when the rolling occurs at temperatures above δ solvus as observed in this work. Due to this, the current work aims to find the average grain sizes from recrystallization processes alone and does not take into account the precipitation of phases. The results are discussed and conclusions and future direction are provided in the last two sections of this paper.

## Methods

### FE formulation of multi-stand rolling

_{D}and Q refer to the stiffness contribution for velocity and pressure, respectively, the FE formulation of Equation 2 leads to

### Microstructure evolution and algorithm

#### Microstructure theory

#### Microstructure evolution

#### Empirical modeling

#### Computations

#### Formulations

#### Microstructure algorithm

_{st}, develops two grain families primarily characterized by the average volume recrystallized (F) and the average grain size (D).

_{strex}, X

_{st}) that represent the instantaneous grain size due to recrystallization laws and the recrystallization fraction, respectively. The non-recrystallized portion is the strained portion that is characterized by (d

_{stst}, (1−X

_{st})) which are functions of the recrystallized subgroup characteristics. A similar analogy is applicable to the recrystallized grain family subgroups and are characterized by (d

_{rexrex}, X

_{rex}) for the recrystallized portion and (d

_{rexst}, (1−X

_{rex})) for the strained portions. These families are expected to evolve during the deformation in a pass and during interpass. Then, a weighted averaging algorithm is applied to calculate D

_{rex}and F

_{rex}prior to the achievement of critical deformation parameters during the next pass. D

_{st}and F

_{st}are calculated as follows:

### Implementation

#### Dynamic recrystallization

^{0}=0, and therefore,

#### Metadynamic recrystallization

_{0.5}) at which the recrystallization is 50% complete. The general expression is similar to Equation 9, and the curve is a similar sigmoidal curve, except that the independent strain variable $\overline{\epsilon}$ is replaced by time variable t indicating the fact that MDRX evolves with time. The time at which 50% recrystallization occurs can be expressed in general by the following equation:

#### Static grain growth

_{ggr}, t

_{ggr}, and Q

_{ggr}can be found from literature. The grain growth typically occurs during the long interpasses and during hold times at the end of the rolling process.

## Results and discussion

### Cooling

#### Four-stand analysis with air cooling at the end of the 4th stand

#### Sixteen-stand analysis with air and water cooling at the end of the 16th stand

### Streamline results

_{st}is set to zero if all the grains at this location are fully recrystallized.

#### Final observations for the four-stand analysis

_{rex}) at the beginning of the deformation during the 4th stand and the recrystallized fraction (X

_{rex}) at the end of the air cooling analysis after the 4th stand are depicted in Figure 11a,b, respectively. F

_{rex}is calculated based on the averaging algorithm proposed in the ‘Formulations’ subsection. It can be observed that there are very few portions in the cross section near the surface that are not fully recrystallized.

#### Final observations for the 16-stand analysis

_{rex}and d

_{rexrex}are shown in Figure 14. The grain size distribution is almost uniform around 20 µm (see Figure 14a) for the D

_{rex}which characterizes the overall grain size distribution while the recrystallized grains (represented by d

_{rexrex}in Figure 14b) show slightly larger grains close to the center, since the center does not cool quickly. The actual microstructure observed at the end of cooling after 16-stand rolling of the considered material is shown in Figure 15. The calculated microstructure results are in good agreement with the observed microstructure.

### Discussion

## Conclusions

## Nomenclature

^{∗}Potential Energy Functional

_{t}Traction-specified boundary

_{i}Velocity component

_{ii}Summation implied over the index i implied on any quantity a

^{T}Transpose of the quantity a

_{D}Global stiffness matrix for velocity

_{r}Heat flux due to radiation

_{h}Heat flux due to convection

_{i}i th component of velocity

_{rex}Fraction of recrystallized grains due to averaging

_{rex}Grain size of recrystallized grains due to averaging

_{st}Grain size of the strained grains due to averaging

_{st}Fraction of strained grains due to averaging

_{strex}Instantaneous recrystallized grain size evaluated using the equations for the strained grain family

_{stst}Instantaneous strained grain size evaluated using the equations for the strained grain family

_{rexrex}Instantaneous recrystallized grain size evaluated using the equations for the recrystallized grain family

_{rexst}Instantaneous strained grain size evaluated using the equations for the recrystallized grain family

_{α}Exponent for calculating the strained grain size

_{0.5}for MDRX

_{0.5}

_{0.5}for MDRX

_{1}Temperature-specific DRX or MDRX parameter to evaluate X and ${\stackrel{\u0304}{\epsilon}}_{0.5}$ for DRX and t

_{0.5}for MDRX

_{2}Temperature-specific DRX por MDRX parameter to evaluate X and ${\stackrel{\u0304}{\epsilon}}_{0.5}$ for DRX and t

_{0.5}for MDRX

_{h}Activation energy for DRX and MDRX in evaluating respectively the ${\stackrel{\u0304}{\epsilon}}_{0.5}$ and t

_{0.5}

_{d}Activation energy for DRX and MDRX in evaluating the steady state grain size

_{0.5}during MDRX

_{1}Temperature-specific DRX parameter to evaluate ${\stackrel{\u0304}{\epsilon}}_{p}$

_{2}Temperature-specific DRX parameter to evaluate ${\stackrel{\u0304}{\epsilon}}_{p}$

_{dynctop}Fraction to calculate the critical strain ${\stackrel{\u0304}{\epsilon}}_{c}$

_{0}Initial grain size used in evaluating the microstructure variables

_{xdrx}Exponent to calculate the instantaneous grain size due to DRX

_{xdrxst}Exponent to calculate the instantaneous strained grain size due to DRX

_{0.5}Time required to achieve 50% recrystallization during MDRX

_{v}Virtual time

_{1}MDRX parameter to evaluate the steady-state grain size

_{2}MDRX parameter to evaluate the steady-state grain size

_{xmdrx}Exponent to calculate the instantaneous grain size due to MDRX

_{xmdrxst}Exponent to calculate the instantaneous strained grain size due to MDRX

_{ggr}Grain size during static grain growth

_{ggr}Exponent denoting the type of grain growth law (quadratic or cubic)

_{ggr}Grain growth parameter which is material specific

_{ggr}Activation energy for grain growth