# Hot Deformation Behavior of a 2024 Aluminum Alloy Sheet and its Modeling by Fields-Backofen Model Considering Strain Rate Evolution

^{1}

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## Abstract

**:**

^{−1}and temperature range of 375–450 °C. In order to analyze the deformation behavior with higher accuracy, a digital image correlation (DIC) system was applied to determine the strain distribution during hot tensile tests. Local stress-strain curves for different local points on the specimens were calculated. The strain rate evolution of each point during the tensile tests was investigated under different deformation conditions. Then, an improved Fields–Backofen (FB) model, taking into account the local strain rate evolution instead of the fixed strain rate, was proposed to describe the constitutive behaviors. It has been found that obvious non-uniform strain distribution occurred when the true strain was larger than 0.3 during hot tensile tests. The strain rate distribution during deformation was also non-uniform. It showed increasing, steady, and decreasing variation tendencies for different points with the increasing of strain, which led to the local flow stress being different at different local points. The flow stresses predicted by the improved FB model showed good agreement with experimental results when the strain rate evolutions of local points during tensile tests were considered. The prediction accuracy was higher than that of traditional FB models.

## 1. Introduction

^{−1}. Although they take the non-uniform deformation into account, the strain rate was still regarded as a constant during hot tensile tests. In fact, the non-uniform strain distribution results in the changing of strain rate in the measurement area, which cannot be ignored, due to the strain rate sensitivity for hot deformation. Thus, if the flow stress is calculated by the local strain measured using the DIC system, the local strain rate evolution must be given at the same time. Only in this way can the flow behavior of metals at high temperatures be described accurately.

## 2. Materials and Methods

^{−1}, respectively. After the furnace reached the set temperature, the specimens were fixed on the clamp and held for 20 min to gain a uniform temperature distribution before tensile tests. To obtain the strain distribution, a white under coating was painted on the sample to enhance the image contrast when preparing the speckles. However, it fell off easily under large deformation at high temperatures, resulting in the failure of strain measurement. Thus, a dull finish was adopted instead of the white under coating. After that, the DIC specimen was prepared by painting black speckles on the specimen with heat-resistant paint.

## 3. Results

#### 3.1. Non-uniform Flow Behavior

#### 3.1.1. Strain Distribution

_{max}= 0.15, 0.30, 0.45, and 0.60, respectively. It can be seen that, as the displacement of the clamp increased, the strain in the reduced section of the specimen increased. The specimen deformed almost uniformly within the reduced section of the specimen before the maximum strain reached to 0.3. After that, the strain distribution became more and more inhomogeneous with the deformation increase. Localized strain distribution was clearly observed in the middle area when the maximum strain increased to 0.6. The strain distribution could not be analyzed accurately when the strain was greater than 0.6, because the speckle on the specimen failed to be recognized at larger deformation conditions. The preparation of speckles on the specimen needs to be further improved to obtain the strain distribution for a larger deformation.

_{max}to ε

_{max}. The percentage of uniform deformation area was calculated by dividing the length of the uniform deformation area by the instant length of the reduced section, as given in Figure 4b. The percentage of uniform deformation area was 73.0% when ε

_{max}was 0.15. It decreased to 36.4% when ε

_{max}increased to 0.6. The percentage of uniform deformation area had a linear regression with ε

_{max}, which indicated that the inhomogeneity of deformation increased with the proceeding of strain.

#### 3.1.2. Local Stress-Strain Curves

_{0}is the initial cross section area and ε is the instant local true strain at local point measured by the DIC system.

#### 3.1.3. Strain Rate Evolution

^{−1}for Point 1 while it was 0.015 s

^{−1}for Point 3. The great difference of the strain rate at different points was the result of the strong inhomogeneous deformation on the specimen. The decrease of strain rate of Point 3 indicated that deformation did not occur at this point, as localized deformation concentrated at Point 1.

#### 3.2. Improved Fields-Backofen Constitutive Model

#### 3.2.1. Improved FB Model with Varied Strain Rates and its Determination

^{−1}. Thus, the original FB model is a monotonic increasing function, by which the softening behavior cannot be described.

#### 3.2.2. Verification of the Improved FB Models

^{−1}.

## 4. Conclusions

^{−1}was studied by hot tensile tests. The local flow stress curves and the corresponding strain rate evolutions for different local points were analyzed. An improved FB constitutive model in consideration of strain rate evolutions of local points was proposed to describe the flow behavior. The conclusions are as follows:

- (1)
- The strain distribution during hot tensile tests was non-uniform on the reduced section of the sample. Concentrated strain distribution was observed when the maximum strain on the specimen was greater than 0.3. The percentage of uniform deformation area decreased linearly as the strain increased.
- (2)
- The local flow stress curves determined by different local points exhibited hardening, steady, and softening behaviors, respectively, resulting from the different strain rate evolutions during the tensile tests. The flow stress exhibited a hardening behavior when the strain rate increased with the proceeding of straining. It kept steady when the strain rate changed slightly. Obvious softening behavior was observed when the strain rate decreased with the increase of strain.
- (3)
- The improved FB constitutive model considering strain rate evolutions of local points showed a good agreement with experimental results. The hardening and softening behavior of the flow stress can be well predicted when varied strain rates of local points were considered in the equation. The improved FB model can describe the deformation behavior for continuously varied strain rate, resulting in an extended application of the model.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 4.**Non-uniform deformation behavior in the reduced section at 90 mm/min crosshead speed: (

**a**) strain distribution at different stages; (

**b**) percentage of uniform deformation area.

**Figure 7.**Stress-strain curves of specimens deformed at different temperatures and crosshead speeds: (

**a**) 375°C; (

**b**) 400°C; (

**c**) 425°C; (

**d**) 450°C.

**Figure 9.**Comparisons of the flow stresses predicted by the improved FB models and the experimental flow stresses for each point at different temperatures: (

**a**) 375 °C; (

**b**) 400 °C; (

**c**) 425 °C; (

**d**) 450 °C.

**Figure 10.**Accuracy evaluation of the established FB model: (

**a**) verification for Point 2; (

**b**) verification for Point 1 and Point 3.

Cu | Mg | Mn | Fe | Si | Zn | Ti | Ni | Al |
---|---|---|---|---|---|---|---|---|

4.78 | 1.56 | 0.57 | 0.24 | 0.11 | 0.2 | 0.1 | 0.1 | Bal |

Temperature (°C) | K | n | m |
---|---|---|---|

375 °C | 93.789 | 0.0346 | 0.139 |

400 °C | 89.502 | 0.0305 | 0.153 |

425 °C | 81.618 | 0.0228 | 0.164 |

450 °C | 74.961 | 0.0144 | 0.180 |

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**MDPI and ACS Style**

He, Z.; Wang, Z.; Lin, Y.; Fan, X.
Hot Deformation Behavior of a 2024 Aluminum Alloy Sheet and its Modeling by Fields-Backofen Model Considering Strain Rate Evolution. *Metals* **2019**, *9*, 243.
https://doi.org/10.3390/met9020243

**AMA Style**

He Z, Wang Z, Lin Y, Fan X.
Hot Deformation Behavior of a 2024 Aluminum Alloy Sheet and its Modeling by Fields-Backofen Model Considering Strain Rate Evolution. *Metals*. 2019; 9(2):243.
https://doi.org/10.3390/met9020243

**Chicago/Turabian Style**

He, Zhubin, Zhibiao Wang, Yanli Lin, and Xiaobo Fan.
2019. "Hot Deformation Behavior of a 2024 Aluminum Alloy Sheet and its Modeling by Fields-Backofen Model Considering Strain Rate Evolution" *Metals* 9, no. 2: 243.
https://doi.org/10.3390/met9020243