Influence of Deformation Conditions on the Critical Damage Factor of AZ 31 Magnesium Alloy

2e 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 to 1 s. True stressstrain 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. 2e 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.2e 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.


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.us, bulk forming, such as extrusion, forging, and rolling, is usually conducted at high temperature, where additional slip systems are activated [1][2][3][4][5][6][7].e 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][12][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.
e 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.us, 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.

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.e 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.e 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.e 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 stresstrue 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.

Results and Discussion
e 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.
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 2 Advances in Materials Science and Engineering 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.

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 hightemperature constitutive model was built with Arrhenius model theory [17][18][19]: where σ, _ ε, and Q are stress rate, strain rate, and deformation activation energy, respectively.K, n, and R represent absolute temperature, stress exponent, and molar gas constant; α and A are constants related to material performance.
e hyperbolic sine model (sin h(x) � (e x − e −x )/2) of aler was expressed as follows: At x < 0.8, the above trinomial item is negligible, so sin h(x) ≈ x, and the relative error <4.2%.At x > 1.2, e −x item is negligible, so sin h(x) ≈ e −x /2, and the relative error is less than 1.9%.us, the Arrhenius equation of sin h(x) can be simplified as a linear or exponential function as follows: where A 1 , sA 2 , and β are expressed as A 1 � Aσn 1 , A 2 � (A/2)n 1 , and β � αn 1 , respectively.
Because the four parameters Q, R, T, and A are constant, the logarithm of both sides of formulas ( 3) and (4) can be taken: where B 1 is equal to ln A 1 − (Q/RT), and where B 2 is equal to ln A 1 − (Q/RT).Finally, n 1 and β are confirmed, that is, n 1 � z ln _ ε/z ln σ and β � z ln _ ε/zσ.e peak stress values under different conditions were regarded as strain-stress, and the relationship curves of σ − ln _ ε and ln σ − ln _ ε were fitted, as showed in Figure 2. As shown in Figure 2(a), n 1 is obtained by the linear slope reciprocal of the mean value, that is, n 1 � 7.101.As shown in Figure 2(b), β is obtained by the linear slope reciprocal of the mean value, that is, Taking the logarithm of both sides of formula (1) leads to e linear slope reciprocal of the mean value was obtained, and these values were substituted into formula (9).Accordingly the hot deformation activation energy (Q) was obtained, that is, Q � 203107.713J/mol.
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 Z. Taking the logarithm of both sides of formula (10) leads to Based on formula (11), the data of ln Z − ln[sin h(ασ)] were fitted as shown in Figure 4, and n and ln A were obtained, that is, n � 10.  en, this model was inputted into user-defined materials' database in FEM software for further numerical computation.4 Advances in Materials Science and Engineering

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.e equation is as follows: where σ, dε, ε, and σ are the equivalent stress, equivalent plastic-strain increment, crack equivalent strain, and maximum tensile strain, respectively.C 1 is a material constant.
Cockcroft and Latham thought that C 1 is an important indicator to measure material failure, and its limit value C 1 is the critical damage factor.Other researchers have put forward the notion of the critical damage factor [22]. e ratio between the unit time increment of the damage incremental and existing cumulative damage quantity is expressed as where m dam , ΔA, and A accum are the damage sensitive rate, unit time damage incremental, and existing cumulative damage quantity, respectively.

Numerical Simulation of ermal Compression Test.
Material true stress-strain curves under di erent temperatures and Arrhenius constitutive model were inputted into material library of the Finite Element Program DEFORM to simulate the hot compression test.e diagram of deformation under plastic pressure is shown in Figure 5.In order to simulate the thermal simulation experiment, the Advances in Materials Science and Engineering 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.

Cockcroft-Latham of the Critical Damage Factor and Damage Sensitive Rate.
e 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.e 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.is phenomenon is called damage softening, which is more sensitive to strain rate.
e outer edge of the upsetting drum is also regarded as the research object.e 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.
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.
6 Advances in Materials Science and Engineering values are the critical damage factor.According to this method, distribution regularities of the damage value are determined at di erent temperatures and strain rates (Figure 8).e critical damage factor is found to be inconstant and varies from 0.1445 to 0.3759.e 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.

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.e Arrhenius constitutive model was inputted into material library of the Finite Element Program DEFORM.Simulation accuracy is substantially improved by the constitutive equation.en, 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 .e 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 di erent temperatures and strain rates.
e critical damage factor is not a constant and uctuates from 0.1445 to 0.3759.

1 )Figure 8 :
Figure 8: Temperature and strain rate e ect on the critical damage value of AZ31 magnesium alloy.
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