Study on Critical Damage Factor and the Constitutive Model Including Dynamic Recrystallization Softening of AZ80 Magnesium Alloy

Quantities AZ80 magnesium alloy billets were compressed with 60 % height reduction on hot process simulator at 473, 523, 573, 623, 673, 723 K under strain rates of 0.001, 0.01, 0.1, 1 and 10 s. In order to predict the occurrence of surface fracture, the critical damage factor based on the Cockcroft-Latham equation were obtained by analysing the results of the corresponding finite element calculation. The results show that the critical damage factor at 523, 573, 623, 673 K under strain rates of 0.001, 0.01, 0.1 and 1 s is not a constant but varies in a range from 0.1397 to 0.4653. Meanwhile, a constitutive model with a few parameters is used to characterize the dynamic recrystallization strain softening of AZ80 alloy, which comprehensively reflect the effects of the deformation temperature, strain and strain rate on the flow stress.


Introduction
An important concern on metal-forming processes is whether the desired deformation can be accomplished without fracture in the workpiece.In industrial practice, however, the empirical know-how of the designer is decisive for the fracture-free quality of the products, but often requires very costly trial-and-error [1][2].Thus, there is a critical need for predicting and preventing fracture, which is a major feature of the forming processes and the quality of the products.Most bulk metal-forming processes may be limited by ductile fracture e.g. the occurrence of internal or surface fracture in the workpiece.If it was possible to predict the conditions of the deforming workpiece which lead to fracture; it may be feasible to choose appropriate process conditions and to modify the forming processes in order to produce sound products [3][4].
For the metal charactering dynamic recrystallization, because of its complexity of flow curves, flow stress model should not only reflect the rising of curve caused by the work hardening, but also reflect the dropping and then tending to smooth stage of curve caused by the dynamic recrystallization.Many researchers find it difficult to adopt an independent equation using a few parameters to establish the flow stress model [5][6].
Magnesium alloy is the lightest mental material available for industrial application now, and has many good mechanical properties, such as high specific strength, excellent performance on vibration damping and electromagnetic shielding, and high specific stiffness, etc.Thus, high-performance magnesium has so far been widely used in the fields like top aerospace industry, automobile industry and military industry.Serious lack of some corresponding research about the plastic deformation of magnesium alloy at elevated temperatures, such as predicting and preventing fracture, determining the processing parameters, etc. made magnesium alloy component processed by plastic deformation to be rare [7].
In this study, for AZ80 magnesium alloy, the relations between critical damage factor and temperature and strain rate were analyzed based on a series of compressing tests data and damage data by numerical computation.Meanwhile, an equation with a few parameters to characterize the flow stress model of the AZ80 alloy, which comprehensively reflect the deformation temperature, strain and strain rate on flow stress, and including dynamic recrystallization strain softening.

Experimental materials and methods
Mg-8.9wt%Al-0.53wt%Znalloy ingots are adopted as the materials for experiment.Firstly, intercept along the axial direction from the ingot a piece of slab with a thickness of 15 mm.Secondly, heat the slab to 355±5 K and insulate it for 16 hours and air-cool it to room temperature.A computer-controlled, servo-hydraulic Gleeble3800 machine was used for compression testing.The homogenized ingot was scalped to diameter of 10 mm and height of 15 mm with grooves on both sides filled with machine oil mingled using graphite powder as lubrication during isothermal hot compression tests at 473, 523, 573, 623, 673 and 723 K with a strain rate of 0.001, 0.01, 0.1, 1 and 10 s -1 , respectively.

True stress-true strain curves
The true compressive stress-strain curves of AZ80 magnesium alloy are shown in Fig. 1.The flow stress as well as the shape of the flow curves is sensitively dependent on temperature and strain rate.For all of specimens, after initial yielding, the flow stress decreases monotonically with different softening rates.For a specific strain rate, the flow stress decreases markedly with the increase of temperature, while for a specific temperature, the flow stress increases markedly with the increase of strain rate.Only for strain rates of 10 s - 1 and 1 s -1 , fracture is seen to occur after a dynamic softening.

Critical damage factor of AZ80 alloy based on Cockcroft and Latham damage 4.1. Cockcroft and Latham damage
Cockcroft and Latham, based on cumulative damage theory, developed a damage computation module which has been applied successfully to a variety of loading situations [8][9].The damage in plastic deformation is defined by Cockcroft and Latham as an amount of work that the ratio of maximum tensile stress * σ to effective stress σ carries out through the applied equivalent strain ε in a metal-working process, i.e.

Determination of critical damage factor of AZ80 alloy
The Cockcroft-Latham constant, C max (critical damage factor), is dependent on the same material parameters that forming limits are dependent on.Metallurgical properties such as the microstructure, alloy constants, grain size, grain form, and non-metallic inclusion content, whilst having a small effect on the strength and the hardness, have a significant effect on the critical damage factor.In order to predict the occurrence of surface fracture, the value of the Cockcroft-Latham equation expressed by means of Eq.( 1) is calculated at the integration point inside the elements.Furthermore, a concept about incremental ratio of Cockcroft-Latham damage in plastic deformation (m dam ) is brought out and defined as the ratio of the damage increment at one step (ΔD) to the accumulated value (D accum ) [10]: True stress-strain data collected at different temperature under different strain rate from the thermal compression experiment of AZ80 alloy were inputted into the custom materials library of the finite element software Deform-3D.Using Deform-3D software make the thermal physical simulation compression experiments reappear.
According to cumulative damage theory, the damage value a moment ago is less than that a moment later during a compassing process.Therefore, the maximum value will appear at the last simulation step, but it does not mean fracture step.Fig. 2 shows the damage distribution at the last step (height reduction of 60%) at 573 K and strain rate of 0.0l s -1 .From the simulation results, it can be seen that the maximum damage value appears in the region of upsetting drum, while the minimal value appears in the middle region.Fig. 3. shows this incremental ratio varying during the upsetting processes at different temperatures and different strain rates.It can be seen that m dam decreases to the trough point rapidly, then it has a slight increase, soon after which it decreases to zero gradually.To find the fracture time (step), the point arrays after step 100 are picked out from each incremental ratio varying curve and fitted linearly (as shown in Fig. 3).The intersection of line fitted and horizontal axis is obtained and it is made certain as the fracture step.Then, the simulation with only 115 steps is continued to run until the fracture.The maximum cumulative damage value at last step or the critical damage factor is computed.By this way, the critical damage factor of AZ80 magnesium alloy can be achieved.The results show (as shown in Tab..I) that the critical damage factor at 523,573,623,673 K under strain rates of 0.001, 0.01, 0.1 and 1s -1 is not a constant but varies in a range from 0.1397-0.4653.

The Phenomenological Constitutive Model including Dynamic Recrystallization Softening
One of the most significant characteristics of the high temperature deformation is that the deformation is controlled by a thermal activation process.Sellars [11] consider that the steady-state flow stress of the thermal deformation depends on the deformation temperature and the strain rate, which can be express by the creep equation: Where Z is temperature compensating strain rate factor, i.e. the Zener-Hollomon parameter; ε& is the strain rate; Q is the deformation activation energy; σ is the steady-state flow stress; R is the gas constant, 8.31 J•mol -1 •K -1 ; T is the temperature; A, α , n are undetermined constants.

The relationship among peak stress, peak strain and Z
The researchers [12] regard Eq.( 3) as following if the stress value is low: ( ) The researchers [12] regard Eq.( 3) as following if the stress value is high: / , n α β ′ = (6) An expression of Q is as follows [12]: From the study of McQueen [11], it is feasible to regard the peak stress p σ as σ for calculations of Eq.( 4) and Eq.( 5).So, n′ can be obtained by plotting ln ln p

Q
When calculating the softening caused by the dynamic recrystallization during hot deformation, we would use the peak strain P ε , in order to establish the following expression at any time, according to the relationship between ln ln p Z ε − (see Fig. 7), it is suitable to regard the peak strain and Z as follows: ( ) Where C 1 is 46.979;C 2 is 8.772.

The relationship between flow stress and stress and strain peak
In order to analyze the softening caused by the dynamic recrystallization, considering the relationship between the difference of stress and stress peak and the strain at the moment, this study found that if the strain is multiplied by a coefficient and all values are logarithmic, shown in Fig. 8a    According to the law of Fig. 8a, in order to facilitate the following derivation, a 'softening-factor' was introduced, which was defined as [13]: Theoretically, the equation with two unknowns can be solved just needing two equations.There are so many data obtained from hot compression tests that ζ and * η to be fitted by the least squares regression method to fit.The fitting results: ζ is 0.6987; when T ≤ 573 K and ε& ≥ 0.1 s -1 , λ = 1.5943; under other conditions, λ = 1.4265.According to the formula (11), when ε& = 0.01 s -1 , the comparison between the predicted flow stress and the experimental data is shown in Fig. 9.The Fig. 9 shows that predicted values form the model is in good agreement with the experimental values .When the strain is relatively small, due to dynamic recrystallization is still not enough, the prediction accuracy will be reduced.

Conclusion
(1) The critical damage factor of AZ80 alloy based on Cockcroft and Latham damage at 523,573,623,673 K under strain rates of 0.001, 0.01, 0.1 and 1 s -1 is not a constant but varies in a range from 0.1397-0.4653.
(2) In order to take the influence of strain, deformation temperature and strain rate on flow stress of AZ80 alloy into account, a constitutive model of the AZ80 magnesium alloy with dynamic recrystallization on the basis of the creep equation was established, as follows: ( )

εFig. 1 .
is the total equivalent strain at the end of forming process.The magnitude of C cannot exceed a maximum value C max (critical damage factor) to failure.By comparing C with C max , the risk of material failure during processing is assessed.Compressive true stress-strain curves for as-cast AZ80 alloy at different test.

Fig. 3 .
Fig. 3. Incremental ratio varying of Cockcroft-Latham damage during compressing process at different temperatures and different strain rates.
and are undetermined constants.The approximate n′ α is:

Fig.Fig. 4 .
Fig. 4a); β can be obtained by plotting ln . It can be seen from the figure at the different temperature and strain rate conditions, distribution curve was a quadratic function approximately.According to Lagrange mean value theorem, use the least square to fit a ratio between the value of ln ln p σ σ − and ln ζε , then Fig. 8b can be obtained.It can be found that a curve linear relationship occurs at the different temperature and strain rate.
analysis, it can be found that, when ζ is less than or equal to 0.7, * η and ε meet the following equation well:

Fig. 9 .
Fig. 9. Comparison of modeling calculated results with experimental result for flow stress of AZ80 alloy dashed curves denotes modeling calculated results).
Critical damage factor at different temperatures and different strain rates.