# Revisiting the Common Practice of Sellars and Tegart’s Hyperbolic Sine Constitutive Model

## Abstract

**:**

## 1. Introduction

## 2. A Brief Overview of the ST Model

## 3. Determining the Constants of the Model

#### 3.1. Finding ${n}^{\prime}$

#### 3.2. Finding $\beta $

#### 3.3. Calculating $\alpha $

#### 3.4. Finding n

#### 3.5. Finding Q

#### 3.6. Finding A

## 4. Results and Discussion

#### 4.1. Initial Results of the ST Model

#### 4.2. Revising the ST Model

#### 4.2.1. Method 1

#### 4.2.2. Method 2

#### 4.3. The Performance of the Revised Model

## 5. Final Remarks

- The first common practice is to define all constants of the model, including ${n}^{\prime}$, $\beta $, $\alpha $, and n, as functions of strain. This can be achieved by taking their temperature-averaged values at any given strain. The current work reasoned that this practice can result in unjustifiable activation energies. Moreover, the results show that defining these fitting parameters as functions of strain does not effectively increase the accuracy of the model. Alternatively, it is suggested to consider these four parameters simply as constants, independent of other controlling parameters (including strain), by taking their total average values.
- The other common practice is to take the activation energy simply as a function of strain. Considering that the activation energy of the hyperbolic sine model is its only physically meaningful parameter, it would be reasonable to extract and express its values with more care. Aligned with this idea, a revised method for the ST model is suggested, in which the activation energy can be defined as $Q(\epsilon ,\dot{\epsilon})$.
- The other assumption, made by Sellars and Tegart [27], is on considering the proportionality constant A to be a temperature-independent constant; however, such assumption cannot be justified as A must be free to be defined arbitrarily for its form and dependence. The current work suggests a method to express this constant of proportionality as $A(\epsilon ,\dot{\epsilon},T)$.

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Hyperbolic Sine Model and Its Limiting Cases

#### Appendix A.1. Hyperbolic Sine Model: Z = A(sinh(ασ))^{n}

- $\alpha \sigma \to {0}^{+}$This limit can be expressed by the Taylor series expansion of hyperbolic sine function:$$\begin{array}{l}\phantom{\rule{-5.em}{0ex}}sinh\left(x\right)=\sum _{k=0}^{+\infty}\frac{{x}^{2k+1}}{(2k+1)!}\end{array}$$$$\begin{array}{l}Z\propto {\left(\right)}^{\alpha}n\end{array}$$$$\begin{array}{l}\phantom{\rule{1.em}{0ex}}\alpha \sigma \to {0}^{+}\Rightarrow \alpha \sigma \gg \frac{{\left(\alpha \sigma \right)}^{3}}{3!}+\frac{{\left(\alpha \sigma \right)}^{5}}{5!}+...\end{array}$$$$\begin{array}{l}\phantom{\rule{-11.em}{0ex}}\Rightarrow Z\propto {\sigma}^{{n}^{\prime}}.\end{array}$$
- $\alpha \sigma \to +\infty $For this case, we can use the exponential definition of hyperbolic sine function:$$\begin{array}{l}sinh\left(x\right)=\frac{{e}^{x}-{e}^{-x}}{2}\end{array}$$$$\begin{array}{l}\phantom{\rule{-1.em}{0ex}}Z\propto {\left(\right)}^{{e}^{\alpha \sigma}}n\end{array}$$$$\begin{array}{l}\phantom{\rule{2.em}{0ex}}\alpha \sigma \to +\infty \Rightarrow {e}^{\alpha \sigma}\gg {e}^{-\alpha \sigma}\end{array}$$$$\begin{array}{l}\phantom{\rule{-3.em}{0ex}}\Rightarrow Z\propto {e}^{{n}^{\u2033}\alpha \sigma}.\end{array}$$

#### Appendix A.2. Power Law Model: Z = A′σ^{n}′

**Figure A1.**(

**a**) The linear plots used for estimating the activation energy and the proportionality constant for the power law model. (

**b**) The obtained $\dot{\epsilon}$-averaged ${Q}^{{}^{\prime}}$ and $ln\left({A}^{{}^{\prime}}\right)$.

**Figure A2.**The correlation between ${\sigma}_{\mathrm{m}}$ and ${\sigma}_{\mathrm{p}}$ using the power law model.

#### Appendix A.3. Exponential Model: Z = A″ exp(βσ)

**Figure A3.**(

**a**) The linear plots used for estimating the activation energy and the proportionality constant for the exponential model. (

**b**) The obtained $\dot{\epsilon}$-averaged ${Q}^{{}^{\u2033}}$ and $ln({A}^{{}^{\prime \prime}})$.

**Figure A4.**The correlation between ${\sigma}_{\mathrm{m}}$ and ${\sigma}_{\mathrm{p}}$ using the exponential model.

## Appendix B. Artificial Neural Networks (ANNs)

**Figure A5.**The distributions of (

**a**) $MARE$ and (

**b**) ${r}^{2}$ for 100 trained ANNs using the Bayesian regularization backpropagation with a simple architecture: two hidden layers, each with five nodes.

**Figure A6.**The comparison between the measured stresses and their estimations using a trained ANN. The stress–strain curves are shown for different temperatures and strain rates of (

**a**) $0.01\phantom{\rule{3.33333pt}{0ex}}{\mathrm{s}}^{-1}$, (

**b**) $0.1\phantom{\rule{3.33333pt}{0ex}}{\mathrm{s}}^{-1}$, and (

**c**) $1\phantom{\rule{3.33333pt}{0ex}}{\mathrm{s}}^{-1}$. (

**d**) The correlation between ${\sigma}_{\mathrm{m}}$ and ${\sigma}_{\mathrm{p}}$.

## References

- Solhjoo, S.; Vakis, A.I.; Pei, Y.T. Two phenomenological models to predict the single peak flow stress curves up to the peak during hot deformation. Mech. Mater.
**2017**, 105, 61–66. [Google Scholar] [CrossRef] - Groen, M.; Solhjoo, S.; Voncken, R.; Post, J.; Vakis, A.I. FlexMM: A standard method for material descriptions in FEM. Adv. Eng. Softw.
**2020**, 148, 102876. [Google Scholar] [CrossRef] - Saravanan, L.; Velmurugan, K.; Venkatachalapathy, V. Hot deformation behavior and ANN modeling of an aluminium hybrid nanocomposite. Mater. Today Proc.
**2021**, 47, 6594–6599. [Google Scholar] [CrossRef] - Ashtiani, H.R.; Shayanpoor, A. Hot Deformation Characterization of Pure Aluminum Using artificial neural network (ANN) and Processing Map Considering Initial Grain Size. Met. Mater. Int.
**2021**, 27, 5017–5033. [Google Scholar] [CrossRef] - Wang, T.; Chen, Y.; Ouyang, B.; Zhou, X.; Hu, J.; Le, Q. Artificial neural network modified constitutive descriptions for hot deformation and kinetic models for dynamic recrystallization of novel AZE311 and AZX311 alloys. Mater. Sci. Eng. A
**2021**, 816, 141259. [Google Scholar] [CrossRef] - Li, X.; Zhou, X.G.; Cao, G.M.; Xu, S.H.; Wang, Y.; Liu, Z.Y. Machine Learning Hot Deformation Behavior of Nb Micro-alloyed Steels and Its Extrapolation to Dynamic Recrystallization Kinetics. Metall. Mater. Trans. A
**2021**, 52, 3171–3181. [Google Scholar] [CrossRef] - Mecking, H.; Kocks, U. Kinetics of flow and strain-hardening. Acta Metall.
**1981**, 29, 1865–1875. [Google Scholar] [CrossRef] - Kocks, U.; Mecking, H. Physics and phenomenology of strain hardening: The FCC case. Prog. Mater. Sci.
**2003**, 48, 171–273. [Google Scholar] [CrossRef] - Estrin, Y.; Mecking, H. A unified phenomenological description of work hardening and creep based on one-parameter models. Acta Metall.
**1984**, 32, 57–70. [Google Scholar] [CrossRef] - Zerilli, F.J.; Armstrong, R.W. Dislocation-mechanics-based constitutive relations for material dynamics calculations. J. Appl. Phys.
**1987**, 61, 1816–1825. [Google Scholar] [CrossRef] - Svyetlichnyy, D.S. The coupled model of a microstructure evolution and a flow stress based on the dislocation theory. ISIJ Int.
**2005**, 45, 1187–1193. [Google Scholar] [CrossRef] - Svyetlichnyy, D.S.; Majta, J.; Nowak, J. A flow stress for the deformation under varying condition—Internal and state variable models. Mater. Sci. Eng. A
**2013**, 576, 140–148. [Google Scholar] [CrossRef] - Voyiadjis, G.Z.; Almasri, A.H. A physically based constitutive model for fcc metals with applications to dynamic hardness. Mech. Mater.
**2008**, 40, 549–563. [Google Scholar] [CrossRef] - Najafizadeh, A.; Jonas, J.J. Predicting the critical stress for initiation of dynamic recrystallization. ISIJ Int.
**2006**, 46, 1679–1684. [Google Scholar] [CrossRef] - Solhjoo, S. Analysis of flow stress up to the peak at hot deformation. Mater. Des.
**2009**, 30, 3036–3040. [Google Scholar] [CrossRef] - Solhjoo, S. Determination of critical strain for initiation of dynamic recrystallization. Mater. Des.
**2010**, 31, 1360–1364. [Google Scholar] [CrossRef] - Ebrahimi, R.; Zahiri, S.; Najafizadeh, A. Mathematical modelling of the stress–strain curves of Ti-IF steel at high temperature. J. Mater. Process. Technol.
**2006**, 171, 301–305. [Google Scholar] [CrossRef] - Solhjoo, S. Determination of flow stress under hot deformation conditions. Mater. Sci. Eng. A
**2012**, 552, 566–568. [Google Scholar] [CrossRef] - Shafiei, E.; Ebrahimi, R. A new constitutive equation to predict single peak flow stress curves. J. Eng. Mater. Technol.
**2013**, 135, 011006. [Google Scholar] [CrossRef] - Ebrahimi, R.; Solhjoo, S. Characteristic points of stress-strain curve at high temperature. Int. J. Iron Steel Soc. Iran
**2007**, 4, 24–27. [Google Scholar] - Solhjoo, S. Determination of flow stress and the critical strain for the onset of dynamic recrystallization using a hyperbolic tangent function. Mater. Des. (1980–2015)
**2014**, 54, 390–393. [Google Scholar] [CrossRef] - Solhjoo, S. Determination of flow stress and the critical strain for the onset of dynamic recrystallization using a sine function. arXiv
**2014**, arXiv:1405.0196. [Google Scholar] - Chen, F.; Feng, G.; Cui, Z. Mathematical modeling of critical condition for dynamic recrystallization. Procedia Eng.
**2014**, 81, 486–491. [Google Scholar] [CrossRef] - Varela-Castro, G.; Cabrera, J.M.; Prado, J.M. Critical Strain for Dynamic Recrystallisation. The Particular Case of Steels. Metals
**2020**, 10, 135. [Google Scholar] [CrossRef] - Johnson, G.R.; Cook, W.H. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng. Fract. Mech.
**1985**, 21, 31–48. [Google Scholar] [CrossRef] - Khan, A.S.; Huang, S. Experimental and theoretical study of mechanical behavior of 1100 aluminum in the strain rate range 10- 5- 104s- 1. Int. J. Plast.
**1992**, 8, 397–424. [Google Scholar] [CrossRef] - Sellars, C.M.; McTegart, W. On the mechanism of hot deformation. Acta Metall.
**1966**, 14, 1136–1138. [Google Scholar] [CrossRef] - Jonas, J.; Sellars, C.; Tegart, W.M. Strength and structure under hot-working conditions. Metall. Rev.
**1969**, 14, 1–24. [Google Scholar] [CrossRef] - Zener, C.; Hollomon, J.H. Effect of strain rate upon plastic flow of steel. J. Appl. Phys.
**1944**, 15, 22–32. [Google Scholar] [CrossRef] - Garofalo, F. An empirical relation defining the stress dependence of minimum creep rate in metals. Trans Met. Soc AIME
**1963**, 227, 351–355. [Google Scholar] - Tello, K.E.; Gerlich, A.P.; Mendez, P.F. Constants for hot deformation constitutive models for recent experimental data. Sci. Technol. Weld. Join.
**2010**, 15, 260–266. [Google Scholar] [CrossRef] - Trimble, D.; O’Donnell, G. Constitutive modelling for elevated temperature flow behaviour of AA7075. Mater. Des.
**2015**, 76, 150–168. [Google Scholar] [CrossRef] - Chen, F.; Feng, G.; Cui, Z. New constitutive model for hot working. Metall. Mater. Trans. A
**2016**, 47, 1229–1239. [Google Scholar] [CrossRef] - Uvira, J.L.; Jonas, J. Hot Compression of Armco Iron and Silicon Steel. Ph.D. Thesis, McGill University Libraries, Montréal, QC, Canada, 1967. [Google Scholar]
- Rieiro, I.; Ruano, O.A.; Eddahbi, M.; Carsı, M. Integral method from initial values to obtain the best fit of the Garofalo’s creep equation. J. Mater. Process. Technol.
**1998**, 78, 177–183. [Google Scholar] [CrossRef] - Shokry, A.; Gowid, S.; Kharmanda, G.; Mahdi, E. Constitutive models for the prediction of the hot deformation behavior of the 10% Cr steel alloy. Materials
**2019**, 12, 2873. [Google Scholar] [CrossRef] - Ahamed, H.; Senthilkumar, V. Hot deformation behavior of mechanically alloyed Al6063/0.75 Al2O3/0.75 Y2O3 nano-composite—A study using constitutive modeling and processing map. Mater. Sci. Eng. A
**2012**, 539, 349–359. [Google Scholar] [CrossRef] - Solhjoo, S. Hot Deformation Fitting Tool. 2021. Available online: https://zenodo.org/record/5512531 (accessed on 18 August 2021).
- Momeni, A. The physical interpretation of the activation energy for hot deformation of Ni and Ni–30Cu alloys. J. Mater. Res.
**2016**, 31, 1077–1084. [Google Scholar] [CrossRef] - Gujrati, R.; Gupta, C.; Jha, J.S.; Mishra, S.; Alankar, A. Understanding activation energy of dynamic recrystallization in Inconel 718. Mater. Sci. Eng. A
**2019**, 744, 638–651. [Google Scholar] [CrossRef] - Bayesian Regularization Backpropagation. Available online: https://www.mathworks.com/help/deeplearning/ref/trainbr.html (accessed on 30 June 2022).

**Figure 1.**(

**a**) The linear plots for determining ${n}^{\prime}$, and (

**b**) the values of ${n}^{\prime}$ and their temperature-averaged determined as functions of strain. The discrete symbols of the “averaging” set shows the obtained values from the measured values, and the continuous line represents the fitted polynomial.

**Figure 2.**(

**a**) The linear plots for determining $\beta $, and (

**b**) determined values of $\beta $ and their T-averaged as functions of strain.

**Figure 3.**The values of $\alpha $ calculated from the T-averaged values of $\beta $ and ${n}^{\prime}$ as a function of strain.

**Figure 4.**(

**a**) The linear relationships for determining n. For this plot, $\alpha $ is defined as a function of $\epsilon $, and (

**b**) the temperature-averaged values of n by imposing different values of $\alpha $.

**Figure 5.**The linear plots for determining Q. For this plot, $\alpha $ and n are considered as functions of $\epsilon $.

**Figure 6.**The estimated activation energies as functions of strain and strain rate, with $\alpha $ and n being (

**a**) functions of strain, and (

**b**) constants.

**Figure 7.**The estimated values of $ln\left(A\right)$ as functions of strain and strain rate, for $\alpha $ and n treated as (

**a**) functions of strain, and (

**b**) constants.

**Figure 8.**The correlation between the measured (${\sigma}_{\mathrm{m}}$) and predicted (${\sigma}_{\mathrm{p}}$) stresses using the two sets of constants, where $\alpha $, n are considered both as functions of strain (green circles) and constants (red diamonds). The continuous black line represents a perfect correlation.

**Figure 9.**The obtained activation energies (unfilled-symbols) and their estimations (continuous lines) as functions of strain and strain rate using method 1.

**Figure 10.**The obtained activation energies as $Q(\epsilon ,\dot{\epsilon},T)$ using method 2 for three randomly selected sets of strain rate and temperature.

**Figure 11.**The results of the revised ST model with method 1. The stress–strain curves studied in this work at different temperatures and three different strain rates of (

**a**) $0.01{\mathrm{s}}^{-1}$, (

**b**) $0.1{\mathrm{s}}^{-1}$, and (

**c**) $1{\mathrm{s}}^{-1}$. The unfilled symbols and the continuous lines represent the ${\sigma}_{\mathrm{m}}$ and ${\sigma}_{\mathrm{p}}$, respectively. (

**d**) the correlation between ${\sigma}_{\mathrm{m}}$ and ${\sigma}_{\mathrm{p}}$.

**Table 1.**The conventional 6-step procedure for estimating the parameters of Equation (2) at any given strain.

No. | Parameter | Description | X | Y |
---|---|---|---|---|

1 | ${n}^{\prime}$ | slope ${}^{\left(\mathrm{a}\right)}$ | $ln\left(\sigma \right)$ | $ln\left(\dot{\epsilon}\right)$ |

2 | $\beta $ | slope | $\sigma $ | $ln\left(\dot{\epsilon}\right)$ |

3 | $\alpha $ | $=\beta /{n}^{\prime}$ | — | — |

4 | n | slope | $ln(sinh(\alpha \sigma \left)\right)$ | $ln\left(\dot{\epsilon}\right)$ |

5 | Q | slope | $1/RT$ | $nln(sinh(\alpha \sigma \left)\right)$ |

6 | A${}^{\left(\mathrm{b}\right)}$ | intercept ${}^{\left(\mathrm{c}\right)}$ | $1/RT$ | $nln(sinh(\alpha \sigma \left)\right)$ |

^{(a)}In calculation of slopes (ΔY/ΔX), it is common to assume a linear relation between X and Y over all data points.

^{(b)}Commonly, coefficient A is expressed as ln(A).

^{(c)}ln(A) can be expressed in terms of intercept (y

_{0}) and strain rate as $\mathrm{ln}\left(A\right)\text{}=\text{}{y}_{0}\text{}-\text{}\mathrm{ln}\left(\dot{\epsilon}\right)$.

**Table 2.**The goodness of fit assessed by $MARE$ (Equation (4a)) and ${r}^{2}$ (Equation (4b)) between ${\sigma}_{\mathrm{m}}$ and ${\sigma}_{\mathrm{p}}$ for the four sets of the ST model and its limiting power-law and exponential models. See Appendix A for the details of the limiting cases.

ST ($\mathit{\alpha}$,n) | Power-Law | Exponential | ||||
---|---|---|---|---|---|---|

$(v,v)$ | $(v,c)$ | $(c,v)$ | $(c,c)$ | |||

$MARE$ | 0.066 | 0.068 | 0.068 | 0.070 | 0.059 | 0.150 |

${r}^{2}$ | 0.986 | 0.986 | 0.983 | 0.980 | 0.986 | 0.945 |

**Table 3.**The performance of the revisited model by methods 1 and 2, assessed by $MARE$ (Equation (4a)) and ${r}^{2}$ (Equation (4b)).

Method 1 | Method 2 | |
---|---|---|

$MARE$ | 0.033 | 0.059 |

${r}^{2}$ | 0.995 | 0.983 |

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**MDPI and ACS Style**

Solhjoo, S.
Revisiting the Common Practice of Sellars and Tegart’s Hyperbolic Sine Constitutive Model. *Modelling* **2022**, *3*, 359-373.
https://doi.org/10.3390/modelling3030023

**AMA Style**

Solhjoo S.
Revisiting the Common Practice of Sellars and Tegart’s Hyperbolic Sine Constitutive Model. *Modelling*. 2022; 3(3):359-373.
https://doi.org/10.3390/modelling3030023

**Chicago/Turabian Style**

Solhjoo, Soheil.
2022. "Revisiting the Common Practice of Sellars and Tegart’s Hyperbolic Sine Constitutive Model" *Modelling* 3, no. 3: 359-373.
https://doi.org/10.3390/modelling3030023