# Calculation of Deformation-Related Quantities in a Hot-Rolling Process

^{*}

## Abstract

**:**

## 1. Introduction

_{0}(1 + B$\overline{\u03f5}$)

^{n}. Alexander [3,8] numerically solved Orowan’s equation for an arbitrary variable-yield stress along the contact arc and also considered the flow-stress approximation based on Swift’s stress-strain hardening law. Bland and Ford [9] generally considered the strain-dependent flow stress and evaluated the error introduced by replacing it with the average flow stress ${\sigma}_{FM}$. There are also other strain-dependent flow models describing, specifically, strain hardening which is the research by Holomon, Ludwik, and Voce [10]. Other flow models describe only the strain softening or mixed strain hardening/softening above a certain strain [10]. Note that the above-mentioned models are approximative. On the other hand, an exact strain-dependent flow rule $\sigma \left(\epsilon \right)$ enables an exact description of the strain-dependent flow rule, regardless of the strain hardening, softening, or any mixed modes. Thus, using a single function (TF) that accurately describes the flow stress as a function of strain can, in some situations, be an advantage. Additionally, using the same TF for an exact calculation of other statistical values, such as the average flow stress ${\sigma}_{FM}$, brings additional benefits.

## 2. Materials and Methods

#### 2.1. Experimental

^{−1}, respectively. From the resulting bars, cylindrical samples of 5 × 10 mm were machined, with the cylinder axis in the rolling direction. A deformational dilatometer apparatus, TA Instruments 805A/D (TA Instruments, New Castle, PA, USA), was used for the determination of the true stress vs. true strain curves, shown in Figure 1. The compression tests were performed with a constant strain rate to obtain the material behavior at a specified strain rate. Outside of compressive tests, tensile and torsion tests were used [2]. The experiments were performed at different temperatures (1000, 1050, 1100, 1150, and 1200 °C) and strain rates (0.01, 0.1, 1, and 10 s

^{−1}), with the original results being presented in earlier published work [15]. Standard, non-lubricated 0.1 mm thick Mo plates were placed on the sample contact surfaces. The experimentally obtained true stress vs. true strain curves were not friction corrected. Numerical optimizations were performed in Octave [22], while some other computations used MATLAB [23].

#### 2.2. Transfer Function and Differential Equation

_{3}… a

_{0}are coefficients of the numerator polynomial, b

_{3}… b

_{0}are coefficients of the denominator polynomial, and TF (s) is defined in the Laplace domain. The independent variable in the Laplace transform is usually time. For a visualization, Equation (2) is transformed in the time domain by recalling the relation

_{3}= 0 and b

_{3}= 0) similarly leads to a second-order ordinary differential equation. TFs in the Laplace domain of form (2) are, therefore, in the time domain, equivalent to the linear ordinary differential Equation (3).

#### 2.3. Replacing Time with Strain in the Laplace Transform

#### 2.4. Identification of G(s) Parameters by Optimization

_{3}, a

_{2}, a

_{1}, a

_{0}, b

_{3}, b

_{2}, b

_{1}, b

_{0}} are determined so that the difference between the measured stress ${\sigma}_{m}\left(t\right)$ and the calculated stress ${\sigma}_{c}\left(t\right)$ along the whole $\sigma \left(t\right)$ curve is minimized. The calculated ${\sigma}_{c}\left(t\right)$ stress can be defined as ${\sigma}_{c}\left(t\right)={\mathcal{L}}^{-1}\left\{G\left(s\right)\mathcal{L}\right({\epsilon}_{m}\left(t\right))$.

_{3}, a

_{2}, a

_{1}, a

_{0}, b

_{3}, b

_{2}, b

_{1}, b

_{0}}. The procedure should be performed for each call during the optimization with a varying set of parameters {a

_{3}, a

_{2}, a

_{1}, a

_{0}, b

_{3}, b

_{2}, b

_{1}, b

_{0}}. The optimization procedure is more problematic. In this case, the ‘lsqnonlin’ optimization function in Octave is used. Several initialization options offered by ‘lsqnonlin’ are applied, while neither the upper nor the lower bounds are used. In most cases, no stable convergence towards the measured stress curve can be observed, thus requiring many trials with various settings and initial optimization points. When the TF parameters are determined for one temperature vs. strain rate condition ($T,\dot{\epsilon}$), the TF parameters of ($T,\dot{\epsilon}$) might be a very good initial value of the TF parameters for neighboring pairs (${T}_{1},{\dot{\epsilon}}_{1}$), although unfortunately, it may not necessarily be so.

## 3. Results

#### 3.1. Calculation of Average Stress ${\sigma}_{FM}$ during Hot Rolling and Work Done per Unit Deformation

#### 3.2. Calculation of Flow Stress Derivative

^{−}

^{1}and represent G′(s) ($\sigma \left(\u03f5\right)$), the derivative of G′(s) ($d\sigma /d\u03f5$), and the curve of mean flow stress as a function of strain ${\sigma}_{FM}$. Based on distinct G′(s), we can obtain an exact and causal expression of (1) the derivative of the flow stress, (2) the average flow stress ${\sigma}_{FM}$, and (3) the work done per unit deformation based on a single G′(s) expression and as continuous functions along the whole deformation range. The same procedure can be used for the determination of these quantities at the remaining strain rates and temperatures.

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Tightly matching measured and calculated true stress vs. true strain curves for temperatures (1000, 1050, 1100, 1150, and 1200 °C) and strain rates (0.01, 0.1, 1, and 10 s

^{−1}).

**Figure 2.**True stress vs. true strain curve ${G}_{\epsilon}\left(\epsilon \right)$ (

**A**) for 1000 °C and a strain rate 0.01 s

^{−1}, work done per unit deformation (

**B**) and calculated average flow stress ${\sigma}_{FM}$ (

**C**).

**Figure 3.**Flow curve ${G}^{\prime}\left(s\right)\stackrel{\mathcal{L},{\mathcal{L}}^{-1}}{\leftrightarrow}\sigma (\u03f5)$ at (1000 °C, 0.01 s

^{−1}), flow stress derivative with respect to strain $d\sigma /d\u03f5\left(\u03f5\right)$ and average flow stress ${\sigma}_{FM}\left(\u03f5\right)$. All three quantities are determined along whole deformation range as continuous functions of strain $\u03f5$ based on G′(s).

C | Si | Mn | Cr | V |
---|---|---|---|---|

0.5 | 0.30 | 0.95 | 1.00 | 0.15 |

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**MDPI and ACS Style**

Vode, F.; Malej, S.; Tehovnik, F.; Arh, B.; Podgornik, B.
Calculation of Deformation-Related Quantities in a Hot-Rolling Process. *Materials* **2023**, *16*, 2787.
https://doi.org/10.3390/ma16072787

**AMA Style**

Vode F, Malej S, Tehovnik F, Arh B, Podgornik B.
Calculation of Deformation-Related Quantities in a Hot-Rolling Process. *Materials*. 2023; 16(7):2787.
https://doi.org/10.3390/ma16072787

**Chicago/Turabian Style**

Vode, Franci, Simon Malej, Franc Tehovnik, Boštjan Arh, and Bojan Podgornik.
2023. "Calculation of Deformation-Related Quantities in a Hot-Rolling Process" *Materials* 16, no. 7: 2787.
https://doi.org/10.3390/ma16072787