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
3.2. Finding
3.3. Calculating
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 , , , 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 .
- 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 .
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
- This limit can be expressed by the Taylor series expansion of hyperbolic sine function:
- For this case, we can use the exponential definition of hyperbolic sine function:
Appendix A.2. Power Law Model: Z = A′σn′
Appendix A.3. Exponential Model: Z = A″ exp(βσ)
Appendix B. Artificial Neural Networks (ANNs)
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No. | Parameter | Description | X | Y |
---|---|---|---|---|
1 | slope | |||
2 | slope | |||
3 | — | — | ||
4 | n | slope | ||
5 | Q | slope | ||
6 | A | intercept |
ST (,n) | Power-Law | Exponential | ||||
---|---|---|---|---|---|---|
0.066 | 0.068 | 0.068 | 0.070 | 0.059 | 0.150 | |
0.986 | 0.986 | 0.983 | 0.980 | 0.986 | 0.945 |
Method 1 | Method 2 | |
---|---|---|
0.033 | 0.059 | |
0.995 | 0.983 |
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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
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 StyleSolhjoo, 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