Introduction

The microstructure of aluminum alloys in the process of hot deformation is controlled primarily by dynamic recrystallization. Therefore, it is very important for technologists to analyze the occurrence of dynamic recrystallization in hot deformation, because this is an effective tool for controlling the mechanical properties during treatment of the metal by lowering the flow stress and the grain size [1–4].

It is interesting to develop a mathematical model of dynamic recrystallization for quantitative analysis of deformation processes [5,6]. To design such a model we should obtain three characteristics from flow curves of the alloy, i.e., the critical strain εc, the strain due to the peak flow εp, and the static strain εs. The value of εp can be measured directly on flow curves at a maximum stress. To derive a model of the occurrence of dynamic recrystallization in a production process we should predict εc and εs [2–7].

Some works suggest mathematical relationships for computing critical and static parameters of dynamic recrystallization. The key moment in the description of the kinetics of dynamic recrystallization is measurement of the volume fraction of the dynamically recrystallized material [8,9]. It is hard to determine the proportion of recrystallized volume under various deformation conditions from the change in the microstructure, because it requires much metallographic study and many measurements. It is required to find a fast and efficient method of observation of the kinetics of recrystallization in the process of hot deformation.

The aim of the present work was to determine the values of εc and εs by measuring the degree of strain hardening under various conditions of deformation of aluminum alloy 3003 and to develop a method for computing the fraction of recrystallized structure X d in the studied material.

Methods of Study

We studied aluminum alloy 3003 containing (in wt.%) 0.58 Si, 0.62 Fe, 0. 068 Cu, 1.09 Mn, 0.03 Mg, 0.006 Ti, 0.008 Zn, 0.007 Ni, the remainder Al. The alloy was melted in an F97-116 resistance furnace with a graphite crucible [10]; the mass of the melted material was 60 kg. Ingots 110 × 20 × 70 mm in size were cast into metallic molds and subjected to solution annealing at 510°C for 20 h. Then we performed axisymmetric hot compressive deformation at a rate of 0.01 − 10.0 sec − 1 and reduction ratio 50% (ε ≈ 0.7) at a temperature of 573–773 K in a Gleeble-1500 testing machine. The deformation process was controlled with the help of a computer with automatic recording of the data.

Results and Discussion

The flow curves of alloy 3003 in coordinates “true stress − true strain” are presented in Fig. 1. It can be seen from the behavior of the curves, most of which have a single maximum, that the alloy undergoes dynamic recrystallization. The effect of the deformation rate on the flow curves taken at 400°C is presented in Fig. 1a . We see that the flow stress grows upon increase in the deformation rate, which implies a positive coefficient of strain hardening of alloy 3003 under these conditions. The dynamic recrystallization manifests itself as lowering of the flow right after the peak at deformation rates of 0.01 − 10.0 sec − 1. The effect of the temperature on the flow curves at a deformation rate of 1 sec − 1 is presented in Fig. 1b . It can be seen that the flow stress increases upon decrease in the deformation temperature. Analysis of the flow curves shows that the critical and static strains depend on the test conditions, which will be considered below.

Fig. 1
figure 1

Typical flow curves for aluminum alloy 3003 in coordinates “true stress (σ) − true strain (e)” obtained at different rates and temperatures of deformation (indicated at the curves): a) t test = 400°C; b ) \( \dot{e} \) = 0.1 sec − 1.

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The degree of strain hardening (θ), which corresponds to a specific strain value, can be computed using the flow curves (Fig. 1) from an equation

$$ \uptheta =\frac{{d\upsigma}}{{d\upvarepsilon}}\approx \frac{{\Delta \upsigma}}{{\Delta \upvarepsilon}}. $$
(1)

The critical condition for the start of dynamic recrystallization is described by the relation

$$ \frac{\partial }{{\partial \mathrm{z}}}\left( {-\frac{{\partial \uptheta}}{{\partial \upsigma}}} \right)=0. $$
(2)

Having computed the rate of strain hardening we plotted its dependence on the flow stress (Fig. 2) for deformation rates from 0.1 to 10 sec − 1 and deformation temperatures from 300 to 500°C. In accordance with the approach described in [7,11,12] the abscissa of the inflection point on the θ − σ curves corresponds to the critical stress (σc ) of the start of dynamic recrystallization. We used the data of Fig. 2. to compute the derivative of the rate of strain hardening with respect to the stress as a function of the flow stress for different deformation rates and temperatures (Fig. 3). The minimums on these curves correspond to the critical stress σc . Then the critical strain εc can be determined by marking the critical stress on the primary stress-strain curves [13].

Fig. 2
figure 2

Rate of growth of strain hardening θ as a function of stress σ in tests of aluminum alloy 3003 at different rates and temperatures of deformation (indicated at the curves): a) t test = 400°C; b ) \( \dot{e} \) = 0.1 sec − 1.

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Fig. 3
figure 3

Dependence of ∂θ/∂σ of flow stress for different rates and temperatures of deformation: a) t test = 400°C; b ) \( \dot{e} \) = 0.1 sec − 1.

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Figure 4 presents the rate of strain hardening as a function of true strain at different rates and temperatures of deformation. We used these curves to determine the strain at which the rate of dynamic softening is the highest. It can be seen that θ attains a maximum and then decreases to a minimum (negative) value. The points of minimum on these curves correspond to the strains at which the rate of dynamic softening has the highest (maximum) value [13]. The strain corresponding to the first zero value (θ = 0) is the peak strain. If θ tends to zero again upon further growth of the deformation, the dynamic recrystallization develops by a periodic law. The strain corresponding to the first zero value of the rate of strain hardening θ can be described as static strain. It is obvious that the static strain increases upon growth in the deformation rate and decrease in the deformation temperature. Then we may conclude that lowering of the deformation rate and growth of the deformation temperature will bring the softening due to the dynamic recrystallization to dynamic balance at a lower strain. Dynamic recrystallization depends primarily on the rate and temperature of the deformation, and this dependence is describable by the Zener − Hollomon parameter Z, i.e.,

$$ Z=\dot{e}\exp \left( {{Q \left/ {RT } \right.}} \right), $$
(3)

where \( \dot{e} \) is the deformation rate, T is the deformation temperature, Q = 156.71 J/mole is the activation energy of the hot deformation, and R = 8.314 J/(mole ∙ K) is the gas constant.

Fig. 4
figure 4

Rate of strain hardening (θ) as a function of true strain (e) of aluminum alloy 3003 at different rates and temperatures of deformation (indicated at the curves): a) t test = 400°C; b ) \( \dot{e} \) = 0.1 sec − 1.

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All the computed parameters of dynamic recrystallization are presented in Table 1. These parameters can be used for obtaining linear regression dependences of the maximum stress σmax , the strain εp at σmax , the static strain εs , and the critical strain εc as a function of ln Z. It can be seen from Fig. 5 that all these parameters grow upon increase in ln Z. Since Z decreases upon growth in the temperature and decrease in the deformation rate, the static (εs ) and critical (εc ) strains decrease gradually, which means that the dynamic recrystallization may develop easier and finish faster.

Table 1 Experimental Parameters Obtained by Analyzing Flow Stress Curves of Aluminum Alloy 3003
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Fig. 5
figure 5

Dependences of parameters of flow curves on ln Z (see Table 1).

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We used the experimental data presented in the Tableto plot the diagram of dynamic recrystallization (Fig. 6) that may be divided into three regions, i.e., A, B, and C. It can be seen that complete dynamic recrystallization develops upon growth in the strain. When the parameter Z decreases, the critical strain declines slower than the static strain (i.e., upon growth in the temperature and lowering of the rate of the deformation). This shows that the critical strain is less sensitive to parameter Z than the static strain. The lower the parameter Z, the easier the dynamic recrystallization occurs. This is favorable for refining of grains during deformation and improving the mechanical behavior of the material.

Fig. 6
figure 6

Diagram of dynamic recrystallization: A ) dynamic recrystallization does not occur; B ) partial recrystallization; C ) complete recrystallization.

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According to the data of [14] the proportion of the dynamically recrystallized material can be computed using the parameters of stress − strain curves from the equation

$$ {X_d}=\frac{{{\upsigma_{\mathrm{rec}}}-{\upsigma_{\mathrm{DRX}}}}}{{{\upsigma_{\mathrm{rec}}}-{\upsigma_{\mathrm{s}}}}}, $$
(4)

where σrec is the instantaneous flow stress under the assumption that the only mechanism of softening is the dynamic recrystallization, σDRX and σs are the instantaneous and steady flows for the occurrence of dynamic recrystallization, respectively. The values of σrec , σDRX and σs are presented schematically by the σ − ε curve in Fig. 7. The values of σDRX and σs can be determined from the σ − ε curves presented in Fig. 1. The value of σrec can be determined by the mathematical model given in [15] by extrapolation of the initial part of the σ − ε curve, where the strain is lower than the critical value. This part of the curve is described as follows in [15]:

$$ \frac{{\upsigma -{\upsigma_{0.2 }}}}{{{\upsigma_{\mathrm{ss}}}- {\upsigma_{0.2 }}}}={{\left( {1-{e^{{-C\upvarepsilon }}}} \right)}^M}, $$
(5)

where σss is the steady flow stress under the conditions of dynamic retrogression and C and M are constants.

Fig. 7
figure 7

Scheme of a stress − strain curve.

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Fig. 8
figure 8

Recrystallization curves (X d ) proportion of recrystallization) at different deformation conditions.

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We obtained dependences for the fraction of recrystallization X d on the true strain with the help of Eqs. (4) and (5). It can be seen that X d increases upon decrease in the rate of the deformation and growth in the temperature.

As a rule, it is assumed that the proportion of crystallization can be described by the Avrami equation

$$ {X_d}=1-\exp \left[ {-{k_1}{{{\left( {\frac{{\upvarepsilon -{\upvarepsilon_{\mathrm{c}}}}}{{{\upvarepsilon_{\mathrm{p}}}}}} \right)}}^{{{k_2}}}}} \right], $$
(6)

where k 1 and k 2 are constants that can be found by nonlinear regression analysis of Eq. (6) on the basis of the data of the Tableand of Fig. 6. The results are represented by an equation with correlation coefficient r = 0.9623, i.e.,

$$ {X_d}=1-\exp \left[ {-1.5726{{{\left( {\frac{{\upvarepsilon -{\upvarepsilon_{\mathrm{c}}}}}{{{\upvarepsilon_{\mathrm{p}}}}}} \right)}}^{2.5234 }}} \right]. $$
(7)

Thus, the model makes it possible to compute the proportion of dynamically recrystallized material for aluminum alloy 3003 as a function of the temperature, of the strain, and of the deformation rate.

Conclusions

1. The parameters of flow curves of aluminum alloy 3003 (σmax , εp , εs , εc ) grow linearly upon increase in ln Z (Z is the Zener − Hollomon parameter). Accordingly, the dynamic recrystallization starts and ends at a lower strain, which is favorable for refining of deformed grains and improving of mechanical characteristics of the material.

2. The experimental parameters have been used to plot a diagram of dynamic recrystallization of aluminum alloy 3003.

3. The proportion of the dynamically recrystallized volume of the alloy (X d ) at different deformation conditions can be computed from flow curves. We have determined the constants in the Avrami equation for the kinetics of dynamic recrystallization of aluminum alloy 3003.

We are sincerely grateful to the Department of Education of the Fujian Province (China) for the financial support within the Natural Sciences Foundation (Project E061004) and the Science and Technologies Foundation (Project JA08249).