#
Constitutive Models for Dynamic Strain Aging in Metals: Strain Rate and Temperature Dependences on the Flow Stress^{ †}

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Constitutive Models

#### 2.1. Athermal and Thermal Stresses

#### 2.2. DSA-Induced Stress

#### 2.2.1. Proposed Model I (PM I)

#### 2.2.2. Proposed Model II (PM II)

## 3. Model Validation and Calibration

#### 3.1. Athermal and Thermal Stressesl

#### 3.2. DSA-Induced Stress

#### 3.2.1. Proposed Model I (PM I)

#### 3.2.2. Proposed Model II (PM II)

#### 3.2.3. Strain Rate Effect on the DSA Stress

## 4. Comparison between the Model Predictions (VA Model, Proposed Model I, and Proposed Model II) and the Experimental Measurements

## 5. Strain Rate Sensitivity

## 6. Conclusions

- Dynamic strain aging, which is characterized by the bell-shaped hardening in stress-temperature curves, appears under both quasi-static and dynamic loadings. As the strain rate increases, this bell-shaped hardening moves to elevated temperature region and the magnitude of hardening reduces.
- The VA model is not able to predict the bell-shaped hardening.
- The proposed model II shows an excellent agreement with the experimental results at both low and high strain rates, whereas the proposed model I fails to capture them at high strain rates.
- The negative strain rate sensitivity due to DSA is well captured by the proposed model II unlike the VA model.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Experimental stress-temperature graphs for Q235B steel for different strain rates ($\dot{\epsilon}$) and strain levels: (

**a**) $\dot{\epsilon}=0.001{s}^{-1}$, (

**b**) $\dot{\epsilon}=0.02{s}^{-1}$, (

**c**) $\dot{\epsilon}=800{s}^{-1}$, and (

**d**) $\dot{\epsilon}=7000{s}^{-1}$ [1]. Dynamic strain aging (DSA) is observed in all cases.

**Figure 3.**Profiles of model predictions (lines) and experimental data (dots) according to the temperature variation: (

**a**) lower yield stress, (

**b**) $U-A$ and (

**c**) $\mathsf{\Omega}$ [13].

**Figure 4.**The athermal flow stress-strain curve from the experiments [1] and the proposed model.

**Figure 5.**The thermal flow stress versus temperature curves from the experiments [1] and the VA model (Equation (15)) with (

**a**) $\dot{\epsilon}=0.001{s}^{-1}$, (

**b**) $\dot{\epsilon}=0.02{s}^{-1}$, (

**c**) $\dot{\epsilon}=800{s}^{-1}$, and (

**d**) $\dot{\epsilon}=7000{s}^{-1}$.

**Figure 6.**The plots of ${a}_{D}$ and ${b}_{D}$ versus ${\epsilon}_{p}$. Dots for both of the parameters are obtained from the experimental data [1]. The corresponding trend lines are displayed using a power law form.

**Figure 7.**The plot of $\mathcal{W}$ versus ${\dot{\epsilon}}_{p}$. Dots for both of the parameters are obtained from the experimental data [1]. The corresponding trend lines are displayed using a power law form.

**Figure 8.**The DSA-induced flow stress versus temperature curves from the experiments [1] and the proposed model I with (

**a**) $\dot{\epsilon}=0.001{s}^{-1}$, (

**b**) $\dot{\epsilon}=0.02{s}^{-1}$, (

**c**) $\dot{\epsilon}=800{s}^{-1}$, and (

**d**) $\dot{\epsilon}=7000{s}^{-1}$.

**Figure 9.**Comparisons between model predictions from the VA and proposed model I and experimental data from [1] on the total true stress versus temperature responses at (

**a**) $\epsilon =0.1$, (

**b**) $\epsilon =0.2$, (

**c**) $\epsilon =0.3$, and (

**d**) $\epsilon =0.4$. Quasi-static loading with $\dot{\epsilon}=0.001{s}^{-1}$ is applied.

**Figure 10.**Comparisons between model predictions from the VA and proposed model I and experimental data from [1] on the total true stress versus temperature responses at (

**a**) $\epsilon =0.1$, (

**b**) $\epsilon =0.2$, (

**c**) $\epsilon =0.3$. and (

**d**) $\epsilon =0.4$. Quasi-static loading with $\dot{\epsilon}=0.02{s}^{-1}$ is applied.

**Figure 11.**Comparisons between model predictions from the VA and proposed model I and experimental data from [1] on the total true stress versus temperature responses at (

**a**) $\epsilon =0.05$ and (

**b**) $\epsilon =0.1$. Dynamic loading with $\dot{\epsilon}=800{s}^{-1}$ is applied.

**Figure 12.**Comparisons between model predictions from the VA and proposed model I and experimental data from [1] on the total true stress versus temperature responses at (

**a**) $\epsilon =0.1$, (

**b**) $\epsilon =0.2$, and (

**c**) $\epsilon =0.3$. Dynamic loading with $\dot{\epsilon}=7000{s}^{-1}$ is applied.

**Figure 13.**The DSA-induced flow stress versus temperature curves from the experiments [1] and the proposed model II with (

**a**) $\dot{\epsilon}=0.001{s}^{-1}$, (

**b**) $\dot{\epsilon}=0.02{s}^{-1}$, (

**c**) $\dot{\epsilon}=800{s}^{-1}$, and (

**d**) $\dot{\epsilon}=7000{s}^{-1}$.

**Figure 14.**Comparisons between model predictions from the VA and proposed model II and experimental data from [1] on the total true stress versus temperature responses at (

**a**) $\epsilon =0.1$, (

**b**) $\epsilon =0.2$, (

**c**) $\epsilon =0.3$, and (

**d**) $\epsilon =0.4$. Quasi-static loading with $\dot{\epsilon}=0.001{s}^{-1}$ is applied.

**Figure 15.**Comparisons between model predictions from the VA and proposed model II and experimental data from [1] on the total true stress versus temperature responses at (

**a**) $\epsilon =0.1$, (

**b**) $\epsilon =0.2$, (

**c**) $\epsilon =0.3$, and (

**d**) $\epsilon =0.4$. Quasi-static loading with $\dot{\epsilon}=0.02{s}^{-1}$ is applied.

**Figure 16.**Comparisons between model predictions from the VA and proposed model II and experimental data from [1] on the total true stress versus temperature responses at (

**a**) $\epsilon =0.05$ and (

**b**) $\epsilon =0.1$. Dynamic loading with $\dot{\epsilon}=800{s}^{-1}$ is applied.

**Figure 17.**Comparisons between model predictions from the VA and proposed model II and experimental data from [1] on the total true stress versus temperature responses at (

**a**) $\epsilon =0.1$, (

**b**) $\epsilon =0.2$, and (

**c**) $\epsilon =0.3$. Dynamic loading with $\dot{\epsilon}=7000{s}^{-1}$ is applied.

**Figure 19.**True stress-true strain curves from experimental measurement [1], predictions by the VA model, proposed model (PM) I, and PM II with $\dot{\epsilon}=0.001{s}^{-1}$: (

**a**)$\text{}T=93,153,223$ and $373K$ (

**b**)$T=473,523,573,623$ and $893K$.

**Figure 20.**True stress-true strain curves from experimental measurement [1], predictions by the VA model, PM I, and PM II with $\dot{\epsilon}=0.02{s}^{-1}$: (

**a**)$\text{}T=93,153,289$ and $373K$ (

**b**)$T=473,573$ and $673K$.

**Figure 21.**True stress-true strain curves from experimental measurement [1], predictions by the VA model, PM I, and PM II with $\dot{\epsilon}=800{s}^{-1}$: (

**a**)$\text{}T=300,773$ and $873K$ (

**b**)$T=973,1073$ and $1173K$.

**Figure 22.**True stress-true strain curves from experimental measurement [1], predictions by the VA model, PM I, and PM II with $\dot{\epsilon}=7000{s}^{-1}$: (

**a**)$\text{}T=300,573$ and $773K$ (

**b**)$T=873,973$ and $1073K$.

**Figure 23.**PM I flow stress surfaces according to variation of temperature and strain level with (

**a**) $\dot{\epsilon}=0.001{s}^{-1}$, (

**b**) $\dot{\epsilon}=0.02{s}^{-1}$, (

**c**) $\dot{\epsilon}=800{s}^{-1}$, and (

**d**) $\dot{\epsilon}=7000{s}^{-1}$. The experimental data are from [1].

**Figure 24.**The PM II flow stress surfaces according to variation of temperature and strain level with (

**a**) $\dot{\epsilon}=0.001{s}^{-1}$, (

**b**) $\dot{\epsilon}=0.02{s}^{-1}$, (

**c**) $\dot{\epsilon}=800{s}^{-1}$, and (

**d**) $\dot{\epsilon}=7000{s}^{-1}$. The experimental data are from [1].

**Figure 25.**True stress versus strain rate graphs with three temperatures ($93\text{}K$, $473\text{}K$, and $873\text{}K$ ) at $\epsilon =0.1$. The experimental data are from [1].

${Y}_{a}\text{}\left(MPa\right)$ | ${B}_{1}\text{}\left(MPa\right)$ | ${n}_{1}\text{}(-)$ | ${Y}_{d}\text{}\left(MPa\right)$ | ${B}_{2}\text{}\left(MPa\right)$ | ${n}_{2}\text{}(-)$ | ${\dot{\epsilon}}_{p}^{0}\text{}\left({s}^{-1}\right)$ |

$0$ | $166$ | $0.18$ | $100$ | $1800$ | $0.15$ | $1.0$ |

${\beta}_{1}^{Y}$($1/K$) | ${\beta}_{2}^{Y}$($1/K$) | ${\beta}_{1}^{H}$($1/K$) | ${\beta}_{2}^{H}$($1/K$) | $p\text{}(-)$ | $q\text{}(-)$ | |

$5.0\times {10}^{-4}$ | $4.7\times {10}^{-5}$ | $9.0\times {10}^{-4}$ | $5.5\times {10}^{-5}$ | $0.51$ | $1.65$ |

${\overline{a}}_{D}\text{}\left(MPa\right)$ | ${\stackrel{=}{a}}_{D}\text{}\left(MPa\right)$ | ${n}_{3}\text{}(-)$ | $\dot{\zeta}\text{}\left({s}^{-1}\right)$ | ${T}_{1}\text{}\left(K\right)$ | ${T}_{2}\text{}\left(K\right)$ | $\eta \text{}(-)$ | ${\epsilon}_{p}^{0}\text{}(-)$ |

$-27$ | $10$ | $0.20$ | $6.5\times {10}^{10}$ | $-17,000$ | $-4100$ | $-0.35$ | $1.0$ |

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**MDPI and ACS Style**

Song, Y.; Garcia-Gonzalez, D.; Rusinek, A.
Constitutive Models for Dynamic Strain Aging in Metals: Strain Rate and Temperature Dependences on the Flow Stress. *Materials* **2020**, *13*, 1794.
https://doi.org/10.3390/ma13071794

**AMA Style**

Song Y, Garcia-Gonzalez D, Rusinek A.
Constitutive Models for Dynamic Strain Aging in Metals: Strain Rate and Temperature Dependences on the Flow Stress. *Materials*. 2020; 13(7):1794.
https://doi.org/10.3390/ma13071794

**Chicago/Turabian Style**

Song, Yooseob, Daniel Garcia-Gonzalez, and Alexis Rusinek.
2020. "Constitutive Models for Dynamic Strain Aging in Metals: Strain Rate and Temperature Dependences on the Flow Stress" *Materials* 13, no. 7: 1794.
https://doi.org/10.3390/ma13071794