1. Introduction

Aluminum matrix composites (AMCs) are increasingly used in aerospace, aircraft, automotive industries due to the advantages such as light weight, good wear resistance, high specific strength and low thermal expansion coefficient^{1}^{-}^{3}. So far, the main widely used reinforcing particles for AMCs are Ni_{2}O_{3}^{4}, Al_{2}O_{3}^{3}, ZrB_{2}^{5}, SiC^{1}^{,}^{6} and TiB_{2}^{7} etc. Among these strengthened Al-matrix composites, the main research goals of previous research reports were at obtaining good adhesion property at the interface of particles and matrix and mature fabrication processes. However, the hot deformation behavior of the composites was few reported.

Hot deformation behaviors of alloys were normally studied using constitutive models and processing maps^{8}^{-}^{10}. Until now, many constitutive models to describe the hot flow behaviors of alloys were developed. As early as in 1966, C.M. Sellars and W.J. McTegart proposed the nowadays most widely used Arrhenius model. Subsequently, G.R. Johnson and W.H. Cook firstly proposed the Johnson-Cook model for describing severe plastic deformation behavior of metal in 1983^{11} and F.J. Zerilli and R.W. Armstrong proposed the Zerilli-Armstrong model based on dislocation mechanics in 1986^{12}. As one of the most classical phenomenological models, Johnson-Cook model has been widely used due to its comprehensive advantages of simplicity and high precision^{13}^{-}^{15}. Nevertheless, the original Johnson-Cook model cannot describe the flow behavior of most alloys with high precision and researchers normally like to modify the model to better suit the flow behavior of a certain alloy^{15}^{-}^{17}. Hence, it is possible to develop a better modified Johnson-Cook model for 2219/TiB_{2} Al-matrix composite since this field has been rarely reported. Meanwhile, processing map has become a mature method to identify the intrinsic workability of alloys^{18}^{-}^{21}, but it was either seldom used in the investigations of 2219/TiB_{2} Al-matrix composite. To better understand the hot deformation behavior and intrinsic workability of 2219/TiB_{2} Al-matrix composite, it is important to construct the suitable constitutive model and processing map for it. Besides, investigating the effect of ceramic particles in the microstructure evolution during hot deformation of 2219/TiB_{2} Al-matrix composite is also necessary.

In this study, the original Johnson-Cook constitutive model for 2219/TiB_{2} Al-matrix composite was firstly constructed and the prediction precision was evaluated. After analyzing the main factors limiting the precision of the model, a modified Johnson-Cook model was proposed to better describing the flow behavior of this composite. Meanwhile, the processing map at the strain of 0.9 was developed to identify the intrinsic workability of this composite. Subsequently, the microstructure evolution was analyzed by optical microscopy and used to verify the processing map, at the same time, the microstructure evolution during hot deformation was studied.

2. Experimental

The as-received material is a homogenized ingot 2219/TiB_{2} Al-matrix composite with chemical compositions (wt.%) of 5.8-6.8Cu, 0.2-0.4Mn, 0.2Si, 0.1-0.25Zr, 0.3Fe, 0.02Mg, 0.1Zn, 0.02-0.1V, balance Al, and the addition of 6vol.% TiB_{2} particles. The optical microscopy (OM) and scanning electronic microscopy (SEM) microstructures of the alloy under as-received condition are shown in Figure 1a and 1b. Also, the EDS mapping of B and Ti elements are shown in Figure 1c and 1d. To prepare for hot compression tests, the specimens were cut into cuboid with dimensions of 10mm×15mm×20mm. Hot compression tests were conducted on a Gleeble-3500 isothermal simulator at the temperature range of 300~500°C with an interval of 50°C and the strain rate range of 0.01~10s^{-1} and the height reduction of 60%. To reduce the effect of friction, graphite flakes were added between the anvils and the largest end faces of specimens. Also, to ensure uniformed temperature distribution before compressing, specimens were heated by 10°C/s to a certain temperature and held 180s. The nominal stress strain data were automatically obtained by a computer-assisted monitor and interpreted into true stress strain data.

3. Results and Discussion

3.1 True stress strain curves

Figure 2 shows the measured true stress strain curves of 2219/ TiB_{2} Al-matrix composite at different conditions. It can be seen that true stress varies with the deformation temperature and strain rate distinctly. Higher deformation temperature and lower strain rate will result in lower true stress, a conventional explain for this phenomenon is that the nucleation of dynamic recrystallization is easier to process under higher temperature and lower strain rate^{22}^{-}^{24}. Besides, the true stress increases dramatically at the initial stage of deformation and decreases with the increasing strain, which is caused by the predomination of work hardening at the beginning of deformation and the subsequent effect of dynamic recovery and dynamic recrystallization becoming stronger with the increasing strain.

3.2 Construction of Johnson-Cook model

As an empirical model, Johnson-Cook constitutive model was firstly proposed by Gordon R. Johnson and William H. Cook in 1983 to describe the relationship between flow stress and deformation parameters including strain, strain rate and temperature for the large plastic deformation of alloys^{11}. It was expressed as following:

where σ,
*T* is the current absolute temperature, *T _{m}* is the melting temperature, here,

*T*=650°C,

_{m}*T*is the reference temperature, respectively.

_{ref}*A*is the yielding stress at reference condition,

*B*is strain hardening coefficient,

*C*is strain rate hardening coefficient,

*n*is strain hardening exponent and

*m*is temperature softening exponent.

According to Equation 1, it is clear that a reference condition needs to be set before the calculation of this model. In our study, the reference condition was set as the deformation temperature of 300°C and strain rate of 0.01s^{-1}. As shown in Figure 3, at the reference condition, the yielding stress (namely *A*) of this composite is about 80MPa.

Under the reference temperature and strain rate, the effect of strain rate and temperature on flow curves could be ignored and Equation 1 can be denoted as:

Taking natural logarithm of both sides:

According to Equation 3, it is obvious that ln(σ-*A*) is linearly relate to lnε. Hence, the values of *n* and *B* were determined to be -0.8835 and 2.0495 by linear fitting respectively.

Under reference temperature condition, Equation 1 can be denoted as:

Equation 4 can be converted into:

Similarly, *σ/(A+Bε ^{n})* and

*C*-value was determined to be 0.1512 by linear fitting as shown in Figure 4.

To obtain the value of *m*, Equation 1 can be expressed as:

Where

Taking natural logarithm of the both sides of Equation 6:

According to the linear fitting of ln*Y* and ln*T** under the strain of 0.3 as shown in Figure 5, the *m*-value was finally determined to be 0.6859.

Here, after obtaining the material constants, the Johnson-Cook model for 2219/ TiB_{2} Al-matrix composite was constructed as:

Figure 6 shows the comparison between predictions of constructed Johnson-Cook model and experimental results. It can be seen that the predictions are much higher than the experimental results at lower strain condition and close to the experimental results at higher strain condition. Moreover, at lower strain rate condition, e.g. 0.01s^{-1} and 0.1s^{-1}, the prediction accuracy is relative better than that at higher strain rate condition, the error of prediction precision increases with the increasing strain rate.

3.3 Modification of Johnson-Cook model

As discussed above, the original Johnson-Cook model can describe the flow behavior of 2219/ TiB_{2} Al-matrix composite to some extent, but the prediction precision is not satisfying. The main reasons of lower precision are: 1) according to the original calculation process, linearly fitting ln(σ-*A*) and lnε will obtain negative *n*-value under low strain conditions, which will result in opposite variation tendency of flow stress and greater error; 2) the accuracy of linear fitting *σ/(A+Bε ^{n})* and

*C*-value better; 3) The precision of predicted flow stress becomes lower with the increasing strain rate, which can be attributed to the

*m*-value which is a certain value under different strain rate. Obviously,

*m*should be a function about strain rate.

Due to the larger error of the result of linear fitting ln(σ-*A*) and lnε as shown in Figure 7a, it is better to choose polynomial fitting. Here, a fifth order polynomial fitting were taken to better fit the relationship between ln(σ-*A*) and lnε. The determination coefficient R^{2} rises from 0.44949 of linear fitting to 0.99044. As a result, the following expression was obtained:

Where *f*(ε) = ln(σ-*A*).

Similarly, to better fit the relationship between *σ/(A+Bε ^{n})* and

^{2}-value rises from 0.91347 to 0.9379. At the same time, the following expression was obtained:

Moreover, the *m*-values at the strain of 0.3 and different strain rate were calculated. As shown in Figure 7c, the relationship between *m* and

Here, the modified Johnson-Cook model was developed as following:

where

3.4 Prediction precision analysis of the modified model

Figure 8 shows the comparison between experimental result and predicted result of modified Johnson-Cook model. Compared with Figure 6, it is distinct that the new predicted results are more accurate, especially at low strain conditions. To better analyze the accuracies of the two models, relative coefficient (*R*) expressed as Equation 13 and absolute average relative error (*AARE*) expressed as Equation 14 were introduced. As shown in Figure 9, the *R*-value and *AARE*-value of original Johnson-Cook model are 0.94731 and 10.3747% respectively and the two indicators of modified model are 0.9852 and 6.4415% respectively.

where *E _{i}* and

*P*are experimental result and predicted result respectively,

_{i}3.4 Processing map

The intrinsic workability of a metal means its maximum ability of deformation without being fractured during hot processing^{25}^{,}^{26}. Usually, this ability is characterized by method of processing map which can identify the stable and instable hot working region and help researchers to understand the deformation mechanism and microstructure evolution mechanism of a metal during hot deformation. It is widely accepted that constructing processing maps based on dynamic materials model (DMM) is effective and precise, which was firstly done by Prasad et al.^{27}. Prasad et al.^{27} believe that the hot working process is a power dissipation process, the total absorbed power is mainly dissipated by plastic deformation and the structural variation of microstructure during plastic deformation, and this process was denoted as:

Where *P* is the total absorbed power, *G* content represents the absorbed power by plastic deformation and *J* co-content represents the absorbed power by microstructure evolution. At a giving temperature and strain rate, *J* co-content can be denoted as^{21}^{,}^{26}:

Where *m* is strain rate sensitivity index denoted as Equation 17. The value of *m* can reflect the deformation mechanism of a metal especially the ability of superplastic deformation during hot forming^{28}^{,}^{29}.

To calculate the value of *m*, a third order polynomial function can be used to fit the relationship of
^{30}^{-}^{32}:

The m-value can be obtained by taking the derivation of Equation 18:

Where *k*_{1~4} are material constants. At a certain strain and deformation temperature, the values of *k*_{1~4} can be calculated by third order polynomial fitting the relationship of
*k*_{1~4}, the *m*-values can be calculated and its 3D response surface map is shown in Figure 10b.

Under ideal linear dissipation condition, *J* co-content reached its maximum value:

The power dissipation efficiency (*η*) is expressed as:

The value of power dissipation efficiency reveals different deformation mechanism, e.g. dynamic recrystallization (DRX), dynamic recovery (DRV), adiabatic shear band and crack. Generally, higher *η*-value reveals higher possibility that DRV and DRX operate. After calculating the *η*-values under different conditions, the power dissipation maps at the strain of 0.5 and 0.9 were constructed as shown in Figure 11. The power dissipation maps reflect the workability of this alloy to some extent by means of *η*-values. However, to better understand the working stability of the alloy during hot processing, the instability situations need to be considered and instability criterion needs to be introduced. The common instability criterions for plastic deformation include Gegel’s instability criterion^{33}, Prasad’s instability criterion^{34}, Ziegler’s instability criterion^{35}, Alexzander’s instability criterion^{36} and Murty’s instability criterion^{37}. Here, we choose Prasad’s instability criterion which is expressed as^{27}^{,}^{34}:

Where *ξ* is the instability coefficient and it indicates the occurrence of instability when it is negative. The instability maps at the strain of 0.5 and 0.9 were constructed as shown in Figure 12 after the determination of *ξ*-values.

By overlapping the instability maps on the power efficiency maps, the processing maps at the strain of 0.5 and 0.9 were obtained as shown in Figure 13. It can be seen that, the *η*-values are higher at the conditions of 350°C/0.1s^{-1}, 450-500°C/0.1s^{-1}, 400~430°C/0.01s^{-1}, which reveals that DRX and DRV are more tend to happen. Due to the addition of ceramic particles which offer more particles for nucleation, the continuous dynamic recrystallization (CDRX) are more likely to occur. Hence, CDRX operates at this temperature area. Besides, the grey areas in Figure 13 are instability areas which distribute in the regions of high strain rate. As a result, one of the intrinsic optimum workable areas is at the temperature range of 300~400°C and strain rate range of 0.01~0.1s^{-1}, another area is at the temperature range of 420~500°C and the strain rate range of 0.01~1s^{-1}.

3.5 Microstructure analysis

Figure 14a and 14b show the microstructures of the deformed center of specimens at the temperature of 350°C and strain rates of 0.01s^{-1} and 10s^{-1}, respectively. It is obvious that the flow net in Figure 14a distributed uniformly. As a comparison, a deformation band with an angle of approximately 45° to the compression axis can be seen in Figure 14b, this structure is usually believed to be shear band which is a typical defect of deformed microstructure. Here, the microstructures coincide with the results of processing map.

Figure 15 shows the microstructures at the temperature ranges of 350~500°C with the strain rate of 1 s^{-1}, it can be seen that θ-phase reduces with the increasing temperature. As shown in Figure 15d, θ-phase was vanished at the temperature of 500°C because it was dissolved in the α-phase matrix, meanwhile, the grain boundary becomes vaguer which coincides with the phase diagram of Al-Cu alloy. Besides, TiB_{2} particles can also be seen in Figure 15 and they usually concentrated in grain boundary.

4. Conclusions

1.
The hot deformation behavior of 2219/TiB_{2} Al-matrix composite was greatly influenced by deformation temperature and strain rate. Also, deformation temperature and strain rate have an interaction effect on flow stress;

2. A new modified Johnson-Cook model was proposed because the original Johnson-Cook model has greater error after error analysis. As a result, the AARE-value of new model was reduced 37.91% comparing with original one and the model was denoted as:

where

3.
Processing map at the strain of 0.9 for 2219/ TiB_{2} Al-matrix composite was constructed. There are two stable regions located at the temperature range of 300~400°C with strain rate range of 0.01~0.1s^{-1} and temperature range of 420~500°C with strain rate range of 0.01~1s^{-1}.

4. The microstructures are coinciding with the prediction of instability area in processing map. Also, the content of θ-phase reduces with the increasing temperature.