#### Abstract

The influence of deformation conditions on the critical damage factor of AZ31 magnesium alloy was analyzed in this paper. Physical experiments and numerical simulation were used to study the critical damage factor. Compression test was carried out using a Gleeble 1500 device at temperatures between 250°C and 400°C, as well as strain rates from 0.01 s^{−1} to 1 s^{−1}. True stress-strain curves of samples were obtained. Based on experimental data, an Arrhenius constitutive model was constructed. Material performance parameters and constitutive model were inputted into the finite element program DEFORM. Simulation results show that the maximum damage appears on the outer edge of the upsetting drum, and damage softening behavior is more sensitive to strain rate. According to the concept of damage sensitive rate, its values were computed. The intersection of line fitted and horizontal axis was obtained in the fracture step, and its relative maximum damage value was as the critical damage factor. The distribution of the critical damage value shows that it is not a constant but fluctuates within the range of 0.1445–0.3759, and it is more sensitive to strain rate compared with temperature.

#### 1. Introduction

Magnesium alloys are widely used in many industrial sectors because of their good mechanical properties, high strength to weight ratio, and good formability compared with other metallic alloys. However, these alloys have limited workability at room temperature owing to insufficient slip systems. Thus, bulk forming, such as extrusion, forging, and rolling, is usually conducted at high temperature, where additional slip systems are activated [1–7]. The emergence of ductile fracture in many metal forming processes is a restriction factor [8, 9]. Metal material toughness damage occurs when materials in plastic deformation have a fracture tendency of physical quantity, especially when damage is up to the critical value, or when the material is broken [10]. At present, few local and foreign studies have focused on the critical damage factor by material effects of shape conditions [11–13]. In the process of studying the critical damage factor, the Cockroft–Lathem is widely used to predict the crackle in volume forming [14, 15]. Some scholars believe that critical damage factors are related to material metallurgical properties and cannot change with the deformation process conditions. Besides, the constitutive model plays an important role in hot forming process which is helpful to choose hot deformation parameters and forming forces. At the same time, it is the premise for the plastic forming process which is used for numerical simulation software.

The present work aimed at material high-temperature flow stress performance by a Gleeble 1500 thermomechanical simulator. Based on experimental data, a constitutive model of flow stress was built and used to improve operation accuracy. Some scholars have previously imported the true stress-strain into software, but some errors exist in the operation process because experimental data fluctuate heavily. In this research, material performance parameters and high-temperature constitutive model were inputted into DEFORM to study the critical damage factors. And the maximum damage values were regarded as the critical damage factors. Thus, the law of the critical damage factor was obtained at temperatures between 250 and 400°C as well as strain rates between 0.01 and 1 s^{−1}. Hot forming processes were also controlled and optimized to avoid some defects.

#### 2. Experiments

AZ31 magnesium alloys were used in the experiment. AZ31 magnesium alloy ingots (chemical composition (wt.%): 2.94% Al; 0.87% Zn; 0.57% Mn; 0.0027% Fe; 0.00112% Si; 0.0008% Cu; 0.0005% Ni; and balance Mg) were cut from cylindrical ingots with a diameter of 10 mm and length of 12 mm, and 12 samples were prepared for testing in all. The high-temperature flow stress test was carried out using a Gleeble 1500 machine to obtain stress-strain data at different temperatures. Magnesium alloy is as cast. All the samples were subjected to rapid water quenching at a temperature of 400°C for 12 hours. The height to diameter ratios of specimens are generally less than 1.5, so the impact on the one-way compressive strength is the smallest.

Conditions for isothermal uniaxial compression were as follows: strain rates from 0.01 s^{−1} to 1 s^{−1} and temperatures from 250°C to 400°C. Gleeble 1500 was used to achieve constant true strain rates during compression. The maximum deformation degree was 60%. Graphite powder mixed with grease was used as lubricant in all experiments. Specimens deformed up to a true strain and then quenched in water. Load-stroke data were converted into true stress-true strain curves using standard equations. A specimen was heated quickly at the heating rate of 1°C·s^{−1} to deformation temperature, and holding time was set to 5 min to reduce and eliminate the temperature gradient within the specimen.

#### 3. Results and Discussion

The true strain-stress curves at 250, 300, 350, and 400°C for different strain rates are shown in Figures 1(a)–1(d), respectively. Flow stress increases when decreasing temperature or increasing strain rate. With true strain increasing, true stress gets to the maximum value because of work hardening mechanism, and at this stage, dynamic recovery and recrystallization just take place in some parts of magnesium alloy.

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Dynamic recrystallization is believed to be responsible for flow-softening behavior because it involves the nucleation of grains in the initial stages of deformation followed by grain-boundary migration. When a steady state is reached, the rates of nucleation and migration balance each other, and the effect of any difference in the initial microstructure is wiped out [16]. Besides, the true strain-stress curves are not smooth curves but thin zigzag curves. Mutual competition between dynamic recrystallization and hot work hardening results in strengthening and softening interaction.

##### 3.1. Construction of Constitutive Model

Flow strain and flow stress depend on the heating temperature of materials which can be inducted from the true stress-strain curve. Meanwhile, these above parameters are related to material performance virtually. Especially, metal high deformation is a process of thermal activation, and deformation temperature and strain rate affect flow stress which is expressed by the Arrhenius equation. Based on experimental data, a high-temperature constitutive model was built with Arrhenius model theory [17–19]:where , , and are stress rate, strain rate, and deformation activation energy, respectively. , , and represent absolute temperature, stress exponent, and molar gas constant; and are constants related to material performance.

The hyperbolic sine model of Thaler was expressed as follows:

At , the above trinomial item is negligible, so , and the relative error <4.2%. At , item is negligible, so , and the relative error is less than 1.9%. Thus, the Arrhenius equation of can be simplified as a linear or exponential function as follows:where , , and are expressed as , , and , respectively.

Because the four parameters , , , and are constant, the logarithm of both sides of formulas (3) and (4) can be taken:where is equal to , andwhere is equal to . Finally, and are confirmed, that is, and .

The peak stress values under different conditions were regarded as strain-stress, and the relationship curves of and were fitted, as showed in Figure 2.

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As shown in Figure 2(a), is obtained by the linear slope reciprocal of the mean value, that is, . As shown in Figure 2(b), is obtained by the linear slope reciprocal of the mean value, that is, . So, .

Taking the logarithm of both sides of formula (1) leads towhere and are expressed as the linear slopes of and , respectively.

The peak stress values under different conditions were regarded as strain-stress, and the relationship curves of and were fitted, as showed Figure 3.

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The linear slope reciprocal of the mean value was obtained, and these values were substituted into formula (9). Accordingly the hot deformation activation energy () was obtained, that is, .

Zener et al. [20, 21] discovered that material deformation conditions (deformation temperature, strain rate, etc.) directly affect the true stress-strain, so they introduced the parameter . Taking the logarithm of both sides of formula (10) leads to

Based on formula (11), the data of were fitted as shown in Figure 4, and and were obtained, that is, and .

Finally, experimental data were fitted, and the high-temperature constitutive model was obtained as follows:

Then, this model was inputted into user-defined materials’ database in FEM software for further numerical computation.

##### 3.2. Critical Damage Factor

Ductile damage fracture is a very complicated process closely related to process and material parameters. Cockcroft and Latham [18] have put forward a criterion through a dimensionless stress concentration factor, which generalizes unit volume plastic work. The equation is as follows:where , , , and are the equivalent stress, equivalent plastic-strain increment, crack equivalent strain, and maximum tensile strain, respectively. is a material constant. Cockcroft and Latham thought that is an important indicator to measure material failure, and its limit value is the critical damage factor.

Other researchers have put forward the notion of the critical damage factor [22]. The ratio between the unit time increment of the damage incremental and existing cumulative damage quantity is expressed aswhere , , and are the damage sensitive rate, unit time damage incremental, and existing cumulative damage quantity, respectively.

###### 3.2.1. Numerical Simulation of Thermal Compression Test

Material true stress-strain curves under different temperatures and Arrhenius constitutive model were inputted into material library of the Finite Element Program DEFORM to simulate the hot compression test. The diagram of deformation under plastic pressure is shown in Figure 5. In order to simulate the thermal simulation experiment, the damage distributions of the magnesium alloy at different temperatures and different strain rates were analyzed, and the relationships between strain rate and temperature and damage factor were further studied.

###### 3.2.2. Cockcroft–Latham of the Critical Damage Factor and Damage Sensitive Rate

The result shows that the minimum damage value always appears in the billet central zone, while the maximum damage value always appears on the outer edge of the upsetting drum. The distribution regularities of the outer drum are shown in Figure 6. Distribution regularities of the maximum damage value are not obvious, and the curves are very close to each other under different temperatures at a strain rate of 0.01 s^{−1}. However, distributions of the damage value become fluctuated at strain rates of 0.1 s^{−1} and 1 s^{−1}, and the maximum damage values decrease when temperature gets higher at these two strain rates. This phenomenon is called damage softening, which is more sensitive to strain rate.

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The outer edge of the upsetting drum is also regarded as the research object. The ratio between unit time increment of the damage incremental and existing cumulative damage quantity is shown in Figure 7. It is found that damage sensitive rate always decreases quickly in the first 5 s, and the damage sensitive rate fluctuates between 0.5 s and 1.5 s until it approaches zero.

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Damage sensitive rate curves are also amplified locally in Figure 7. And in this amplified figures, local curves do not approach zero under different strain rates of 0.01 s^{−1}, 0.1 s^{−1}, and 1 s^{−1}. But local curves should be fitted to reach zero in order to reveal the compression step of the damage sensitive rate. Meanwhile, existing cumulative damage values are determined by numerical simulation results, and these values are the critical damage factor. According to this method, distribution regularities of the damage value are determined at different temperatures and strain rates (Figure 8). The critical damage factor is found to be inconstant and varies from 0.1445 to 0.3759.

The critical damage factor will decrease obviously with the increasing of strain rate at the same temperature. Critical damage factor will change little with the increasing of temperature at the same strain rate.

#### 4. Conclusions

In this paper, AZ31 was regarded as the research object. True stress-strain curves were obtained by the high-temperature tensile test. Based on these experimental data, some parameters were solved, and a high-temperature constitutive model was constructed. The Arrhenius constitutive model was inputted into material library of the Finite Element Program DEFORM. Simulation accuracy is substantially improved by the constitutive equation. Then, a hot compression test simulation is proceeded. Finally, critical damage values were obtained at temperatures between 250°C and 400°C and strain rates between 0.01 s^{−1} and 1 s^{−1}. The maximum damage value always appears at the outer edge of the upsetting drum. Distribution regularities of the maximum damage value regularly increases with increased temperature at a strain rate of 0.01 s^{−1}. However, distribution regularities of the damage value are poor at a strain rate of 0.1 s^{−1} and 1 s^{−1}. Distribution regularities of the damage values are determined at different temperatures and strain rates. The critical damage factor is not a constant and fluctuates from 0.1445 to 0.3759.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The authors thank the National Natural Science Foundation of Guangdong Province (no. 2017A030310621) for financial support of this research.