**3. DRX critical strain and DRX kinetic model**

#### **3.1. The initiation of DRX**

The similar flow behavior of as-cast AZ80 magnesium alloy with as-extruded 7075 aluminum alloy is illustrated in Fig. 4a~d. Both deformation temperature and strain rate have considerable influence on the flow stress of AZ80 magnesium alloy. From the true stress-strain curves in Fig. 4a~d, it also can be seen that the stress evolution with strain exhibits three distinct stages. At the first stage where work hardening (WH) predominates and cause dislocations to polygonize into stable subgrains, flow stress exhibits a rapid increase to a critical value with increasing strain, meanwhile the stored energy in the grain boundaries originates from a large difference in dislocation density within subgrains or grains and grows rapidly to DRX activation energy. When the critical driving force is attained, new grains are nucleated along the grain boundaries, deformation bands and dislocations, resulting in equiaxed DRX grains. At the second stage, flow stress exhibits a smaller and smaller increase until a peak value or an inflection of work-hardening rate, which shows that the thermal softening due to DRX and dynamic recovery (DRV) becomes more and more predominant, then it exceeds WH. At the third stage, two types of curve variation tendency can be generalized as following: decreasing gradually to a steady state with DRX softening (573~673 K & 0.01 s-1, 623~673 K & 0.1 s-1, 573 K & 1 s-1,, 673 K & 1 s-1), and decreasing continuously with significant DRX softening (523 K &

66 Recent Developments in the Study of Recrystallization

0.01 s-1, 523~573 K & 0.1 s-1, 523 K & 1 s-1,, 623 K & 1 s-1, 523~673 K & 10 s-1) [12-14].

(a) (b)

(c) (d)

**Figure 4.** True stress-strain curves of as-cast AZ80 magnesium alloy obtained by Gleeble 1500 under different defor‐

mation temperatures with strain rates (a) 0.01 s-1, (b) 0.1 s-1, (c) 1 s-1, (d) 10 s-1.

From the true compressive stress-strain data of as-extruded 42CrMo high-strength steel shown in Fig. 5a~d, the values of the strain hardening rate ( *θ* =d*σ* / d*ε* ) were calculated. The critical conditions for the onset of DRX can be attained when the value of | −d*θ* / d*σ* | , where strain hardening rate *θ* =d*σ* / d*ε* , reaches the minimum which corresponds to an inflection of d*σ* / d*ε* versus *σ* curve. In this study, analysis of inflections in the plot of d*σ* / d*ε* versus *σ* up to the peak point of the true stress-strain curve has been performed to reveal whether DRX occurs. Results confirm that the d*σ* / d*ε* versus *σ* curves have characteristic inflections as shown in Fig. 6a~d, which indicates that DRX is initiated at corresponding deformation conditions. The critical stress to initiation can be identified, and hence the corresponding critical strain to initiation can be obtained from true stress-strain curve. As a result, the values of critical strain and peak stress at different deformation conditions were shown in Table.1, from which it can be seen that the critical strain and critical stress depend on temperature and strain rate nonlinearly, and it is summarized that *ε*<sup>c</sup> / *ε*<sup>p</sup> =0.165~0.572 , *σ*<sup>c</sup> / *σ*<sup>p</sup> =0.645~0.956 [15].

**Strain rate (s-1)**

True strain

True stress (MPa)

σp

ε<sup>c</sup> / ε<sup>p</sup>

σ

εc

εp

σc

**Table 1.** Values of σ<sup>c</sup> / σp , εc , σc and εp at different deformation conditions.

**3.2. Arrhenius equation for flow behavior with DRX**

**Temperature (K) 1123 1198 1273 1348**

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69

0.01 -0.177 -0.160 -0.113 -0.083 0.1 -0.336 -0.172 -0.140 -0.142 1 -0.146 -0.135 -0.077 -0.059 10 -0.259 -0.359 -0.247 -0.218

Characterization for Dynamic Recrystallization Kinetics Based on Stress-Strain Curves

0.01 -0.400 -0.309 -0.236 -0.355 0.1 -0.818 -0.436 -0.318 -0.264 1 -0.545 -0.591 -0.455 -0.355 10 -0.564 -0.627 -0.600 -0.573

0.01 -100.875 -70.523 -45.244 -35.122 0.1 -137.038 -89.748 -63.469 -42.362 1 -153.988 -122.629 -76.918 -55.127 10 -215.180 -203.734 -131.429 -105.296

0.01 -119.111 -83.558 -57.221 -40.714 0.1 -155.026 -110.287 -77.083 -53.280 1 -187.522 -151.374 -114.790 -85.495 10 -235.405 -213.198 -149.191 -122.623

0.01 0.441 0.519 0.476 0.234 0.1 0.411 0.394 0.439 0.540 1 0.268 0.229 0.170 0.165 10 0.459 0.572 0.412 0.381

0.01 0.847 0.844 0.791 0.863 0.1 0.884 0.814 0.823 0.795 1 0.821 0.810 0.670 0.645 10 0.914 0.956 0.881 0.859

It is known that the thermally activated stored energy developed during deformation controls softening mechanisms which induce different DRX softening and work-hardening. The activation energy of DRX, an important material parameter, determines the critical conditions for DRX initiation. So far, several empirical equations have been proposed to determine the deformation activation energy and hot deformation behavior of metals. The most frequently used one is Arrhenius equation which designs a famous Zener-Hollomon parameter, *σ*p , to

Fig. 6 d d versus plots up to the peak points of the true stress-strain curves under different deformation temperatures with strain rates (a) 0.01 s-1, (b) 0.1 s-1, (c) 1 s-1, (d) 10 s-1. **Figure 6.** Formula: Eqn012.wmf>versus dσ / dε plots up to the peak points of the true stress-strain curves under differ‐ ent deformation temperatures with strain rates (a) 0.01 s-1, (b) 0.1 s-1, (c) 1 s-1, (d) 10 s-1.

Strain rate (s-1) Temperature (K)

<sup>p</sup> at different deformation conditions.

1123 1198 1273 1348

0.01 -0.177 -0.160 -0.113 -0.083 0.1 -0.336 -0.172 -0.140 -0.142 1 -0.146 -0.135 -0.077 -0.059

Table 1 Values of c

True strain c

 , <sup>c</sup> , p and 


**Table 1.** Values of σ<sup>c</sup> / σp , εc , σc and εp at different deformation conditions.

#### **3.2. Arrhenius equation for flow behavior with DRX**

conditions for the onset of DRX can be attained when the value of | −d*θ* / d*σ* | , where strain hardening rate *θ* =d*σ* / d*ε* , reaches the minimum which corresponds to an inflection of d*σ* / d*ε* versus *σ* curve. In this study, analysis of inflections in the plot of d*σ* / d*ε* versus *σ* up to the peak point of the true stress-strain curve has been performed to reveal whether DRX occurs. Results confirm that the d*σ* / d*ε* versus *σ* curves have characteristic inflections as shown in Fig. 6a~d, which indicates that DRX is initiated at corresponding deformation conditions. The critical stress to initiation can be identified, and hence the corresponding critical strain to initiation can be obtained from true stress-strain curve. As a result, the values of critical strain and peak stress at different deformation conditions were shown in Table.1, from which it can be seen that the critical strain and critical stress depend on temperature and strain rate

nonlinearly, and it is summarized that *ε*<sup>c</sup> / *ε*<sup>p</sup> =0.165~0.572 , *σ*<sup>c</sup> / *σ*<sup>p</sup> =0.645~0.956 [15].


plots up to the peak points of the true stress-strain curves under

<sup>p</sup> at different deformation conditions.

1123 1198 1273 1348

0.01 -0.177 -0.160 -0.113 -0.083 0.1 -0.336 -0.172 -0.140 -0.142 1 -0.146 -0.135 -0.077 -0.059

*θ*=d*σ*/d*ε* (MPa)

(a) (b)

*θ*=d*σ*/d*ε* (MPa)

(c) (d)

different deformation temperatures with strain rates (a) 0.01 s-1, (b) 0.1 s-1, (c) 1 s-1, (d) 10 s-1.

**Figure 6.** Formula: Eqn012.wmf>versus dσ / dε plots up to the peak points of the true stress-strain curves under differ‐

Strain rate (s-1) Temperature (K)

1s 0 -30 -60 -90 -120 -150 -180

1348K 1273K 1198K 1123K

*σ* (MPa)

<sup>1200</sup> -1

1348K 1273K 1198K 1123K

0 -40 -80 -120 -160 -200 -240 -280

*σ* (MPa)


0.1s

> 10s

0 -20 -40 -60 -80 -100 -120

1123K

0.01s

*σ* (MPa)

<sup>1200</sup> -1

1348K 1273K 1198K 1123K

0 -35 -70 -105 -140 -175 -210

*σ* (MPa)

 , <sup>c</sup> , p and 

ent deformation temperatures with strain rates (a) 0.01 s-1, (b) 0.1 s-1, (c) 1 s-1, (d) 10 s-1.

Table 1 Values of c

True strain c

1348K 1273K 1198K

68 Recent Developments in the Study of Recrystallization

Fig. 6 d dversus

*θ*=d*σ*/d*ε* (MPa)

*θ*=d*σ*/d*ε* (MPa)

> It is known that the thermally activated stored energy developed during deformation controls softening mechanisms which induce different DRX softening and work-hardening. The activation energy of DRX, an important material parameter, determines the critical conditions for DRX initiation. So far, several empirical equations have been proposed to determine the deformation activation energy and hot deformation behavior of metals. The most frequently used one is Arrhenius equation which designs a famous Zener-Hollomon parameter, *σ*p , to

represent the effects of the temperatures and strain rate on the deformation behaviors, and then uncovers the approximative hyperbolic law between *Z* parameter and flow stress [15].

$$Z = \dot{\varepsilon} \exp(\mathbf{Q}\_l^\prime \mathbf{R} \mathbf{T}) \tag{1}$$

$$
\dot{\varepsilon} = AF(\sigma) \exp(-Q/RT) \tag{2}
$$

Where,

where *<sup>F</sup>* (*σ*)={ <sup>|</sup>*<sup>σ</sup>* <sup>|</sup> *<sup>n</sup> <sup>α</sup>* <sup>|</sup>*<sup>σ</sup>* <sup>|</sup> <0.8 exp(*β* |*σ* |) *α* |*σ* | >1.2 sinh(*α* |*σ* |) *<sup>n</sup>* for all *σ* is the strain rate (s-1), *ε*˙ is the universal gas constant (8.31

J mol-1 K-1), *R* is the absolute temperature (K), *T* is the activation energy of DRX (kJ mol-1), *Q* is the flow stress (MPa) for a given stain, *σ* , *A* and *α* are the material constants ( *n* ).

#### *3.2.1. Calculation of material constant α = β / n*

For the low stress level ( *n* ), substituting the power law of *ασ* <0.8 into Eq. (2) and taking natural logarithms on both sides of Eq. (2) give

$$
\ln \dot{\varepsilon} = \ln A + n \ln \left| \sigma \right| - Q/RT \tag{3}
$$

ln ln e

0

50

100

True stress (MPa)


*σ*|

**Figure 8.** The relationships between α =β / *n* =0.00913 and |σ | .

150

200

250

20.0

54.4

ln

**Figure 7.** The relationships between *n* and lnσ .

*σ*| |(MPa) 147.8

as 0.07558 MPa-1. Thus, another material constant *β* MPa-1.

*3.2.3. Calculation of DRX activation energy lnε˙*

following

 s

Then, ln*ε*˙ =ln*A* + *β* |*σ* | −*Q* / *RT* . The peak stresses at different temperatures and strain rates can be identified for the target stresses with high level. The linear relationships between *β* =dln*ε*˙ / |*σ* | and |*σ* | at different temperatures were fitted out as Fig. 8. The mean value of all the slope rates is accepted as the inverse of material constant ln*ε*˙ , thus *β* value is obtained

0.02 0.14 1.00 7.39

Strain rate in ln scale lne&(s -1)

0.02 0.14 1.00 7.39

1123K 1198K 1273K 1348K

lne&(s -1)

1123K 1198K 1273K 1348K

e- 1.0

Characterization for Dynamic Recrystallization Kinetics Based on Stress-Strain Curves

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71

For all the stress level (including low and high stress levels), Eq.(2) can be represented as the

& - *A Q RT* (4)

Then, ln*ε*˙ =ln*A* + *n*ln|*σ* | −*Q* / *RT* . In 2010, Quan et al. [17] plotted the relationships between the true stress and true strain of 42CrMo high-strength steel in ln-ln scale under different temperatures and strain rates, and hence found a true strain range of -0.08~-0.18 including part of the first stage and the second stage described in the previous, in which all the stresses increase gradually with almost the same ratios. Therefore, this true strain range was accepted as a steady WH stage corresponding to low stress level. In further, Quan et al. [17] fitted the relationships between the stress and the strain rate as the true strain was -0.14, and then found almost equally linear relationships which revealed that the influence of temperature was very small. Thus, it can be deduced that to evaluate the material constant *n* =dln*ε*˙ / dln|*σ* | of Arrhenius equation, the stress-strain data in the true strain range of -0.08~-0.18 contribute to the minimum calcu‐ lation tolerance. Here true strain *n* was chose. Fig. 7 shows the relationships between *ε* = −0.1 and ln|*σ* | for ln*ε*˙ under different temperatures. The linear relationship is observed for each temperature and the slope rates are almost similar with each other. The mean value of all the slope rates is accepted as the inverse of material constant *ε* = −0.1 , thus *n* value is obtained as 8.27780.

#### *3.2.2. Calculation of material constant lnε˙*

For the high stress level ( *β* ), substituting the exponential law of *α* |*σ* | >1.2 into Eq. (2) and taking natural logarithms on both sides of Eq. (2) give

**Figure 7.** The relationships between *n* and lnσ .

represent the effects of the temperatures and strain rate on the deformation behaviors, and then uncovers the approximative hyperbolic law between *Z* parameter and flow stress [15].

J mol-1 K-1), *R* is the absolute temperature (K), *T* is the activation energy of DRX (kJ mol-1), *Q*

For the low stress level ( *n* ), substituting the power law of *ασ* <0.8 into Eq. (2) and taking

 s

Then, ln*ε*˙ =ln*A* + *n*ln|*σ* | −*Q* / *RT* . In 2010, Quan et al. [17] plotted the relationships between the true stress and true strain of 42CrMo high-strength steel in ln-ln scale under different temperatures and strain rates, and hence found a true strain range of -0.08~-0.18 including part of the first stage and the second stage described in the previous, in which all the stresses increase gradually with almost the same ratios. Therefore, this true strain range was accepted as a steady WH stage corresponding to low stress level. In further, Quan et al. [17] fitted the relationships between the stress and the strain rate as the true strain was -0.14, and then found almost equally linear relationships which revealed that the influence of temperature was very small. Thus, it can be deduced that to evaluate the material constant *n* =dln*ε*˙ / dln|*σ* | of Arrhenius equation, the stress-strain data in the true strain range of -0.08~-0.18 contribute to the minimum calcu‐ lation tolerance. Here true strain *n* was chose. Fig. 7 shows the relationships between *ε* = −0.1 and ln|*σ* | for ln*ε*˙ under different temperatures. The linear relationship is observed for each temperature and the slope rates are almost similar with each other. The mean value of all the slope rates is accepted as the inverse of material constant *ε* = −0.1 , thus *n* value is obtained as

For the high stress level ( *β* ), substituting the exponential law of *α* |*σ* | >1.2 into Eq. (2) and

is the flow stress (MPa) for a given stain, *σ* , *A* and *α* are the material constants ( *n* ).

ln ln ln

e

& exp( ) (1)

is the strain rate (s-1), *ε*˙ is the universal gas constant (8.31

& - *A n Q RT* (3)

& *AF Q RT* ( )exp( ) - (2)

*Z Q RT* e

> s

e

Where,

8.27780.

*3.2.2. Calculation of material constant lnε˙*

taking natural logarithms on both sides of Eq. (2) give

where *<sup>F</sup>* (*σ*)={ <sup>|</sup>*<sup>σ</sup>* <sup>|</sup> *<sup>n</sup> <sup>α</sup>* <sup>|</sup>*<sup>σ</sup>* <sup>|</sup> <0.8

70 Recent Developments in the Study of Recrystallization

*3.2.1. Calculation of material constant α = β / n*

natural logarithms on both sides of Eq. (2) give

exp(*β* |*σ* |) *α* |*σ* | >1.2 sinh(*α* |*σ* |) *<sup>n</sup>* for all *σ*

**Figure 8.** The relationships between α =β / *n* =0.00913 and |σ | .

$$
\ln \dot{\varepsilon} = \ln A + \beta \left| \sigma \right| - Q/RT \tag{4}
$$

Then, ln*ε*˙ =ln*A* + *β* |*σ* | −*Q* / *RT* . The peak stresses at different temperatures and strain rates can be identified for the target stresses with high level. The linear relationships between *β* =dln*ε*˙ / |*σ* | and |*σ* | at different temperatures were fitted out as Fig. 8. The mean value of all the slope rates is accepted as the inverse of material constant ln*ε*˙ , thus *β* value is obtained as 0.07558 MPa-1. Thus, another material constant *β* MPa-1.

#### *3.2.3. Calculation of DRX activation energy lnε˙*

For all the stress level (including low and high stress levels), Eq.(2) can be represented as the following

$$
\ln \dot{\varepsilon} = \ln A + n \left[ \ln \sinh(\alpha \left| \sigma \right|) \right] - Q / RT \tag{5}
$$

*3.2.4. Construction of constitutive equation*

Substituting *ε*˙ =2.44154×10<sup>25</sup> sinh(0.00913|*σ* |)

(7), thus, the the flow stress can be expressed as

evolution can be predicted by the following equation [20].

equation, means that *m* depends on strain, strain rate and temperature.

expressed as

s

e

**3.3. DRX kinetic model**

Substituting 1 / *T* , *α* , *n* and four sets of *Q* , *ε*˙ and *T* into Eq. (5), the mean value of material constant *σ* is obtained as 2.44154×1025 s-1. Thus, the relationship between *A* , *ε*˙ and *T* can be

25 8.27780 3

2.44154 10 sinh(0.00913 ) exp (599.73210 10 ) 8.31 <sup>é</sup> ù é- *<sup>T</sup>*<sup>ù</sup> <sup>ë</sup> û ë <sup>û</sup> & (7)

{ } 25 1/8.27780 25 2/8.27780 1/2

109.52903ln [ / (2.44154 10 )] {[ / (2.44154 10 )] 1} *Z Z* (8)

During thermoplastic deformation process, dislocations continually increase and accumulate to such an extent that at a critical strain, DRX nucleus would form and grow up near grain boundaries, twin boundaries and deformation bands. It is well known that the conflicting effects coexist between the multiplication of dislocation due to continual hot deformation and the annihilation of dislocation due to DRX. When work-hardening corresponding to the former and DRX softening corresponding to the later are in dynamic balance, flow stress will keep constant with increasing strain, meanwhile deformation comes to a steady stage in which complete DRX grains have equiaxed shape and keep constant size. In common, the kinetics of DRX can be described in terms of normal S-curves of the recrystallized volume expressed as a function of time. In a constant strain rate, time can be replaced by strain and recrystallized volume fraction can be expressed by modified Avrami equation. Thus, the kinetics of DRX

> DRX 1 exp{ ( <sup>c</sup> ) \* } *<sup>m</sup> X* - - - é ù

ee

where *X*DRX =1−exp{ − (*ε* −*ε*c) / *ε* \* *<sup>m</sup>*} is the volume fraction of dynamic recrystallized grain and *X*DRX is Avrami's constant. This expression, which is modified from the Avrami's

The true stress-strain curve data after the peak stress point were adopted to calculate DRX softening rate ( *X*DRX versus *θ* =d*σ* / d*ε* ) plots, and the results were shown as Fig. 10a~d. The maximum softening rate corresponds to the negative peak of such plot. The strain for maxi‐ mum softening rate, *σ* , identified from Fig. 10a~d, and the critical strain, *ε* \* , identified from Fig. 10a~d can be considered with a power function of dimensionless parameter, *ε*<sup>c</sup> (Fig. 11a~b).

 e

ë û (9)

8.27780 exp −(599.73210×10<sup>3</sup>

Characterization for Dynamic Recrystallization Kinetics Based on Stress-Strain Curves

) / 8.31*T* into Eq.

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73

 s

If ln*ε*˙ =ln*A* + *n* lnsinh(*α* |*σ* |) −*Q* / *RT* is constant, there is a linear relationship between *ε*˙ and lnsinh(*α* |*σ* |) , and Eq. (5) can be rewritten as

$$Q = Rn \left\{ \mathbf{d} \left[ \ln \sinh(\alpha \| \sigma \|) \right] \right\} \left\{ \mathbf{d} (1 / T) \right\} \tag{6}$$

The peak stresses at different temperatures and strain rates can be identified for the present target stresses. The linear relationships between *Q* =*Rn*{d lnsinh(*α* |*σ* |) / d(1 / *T* )} and lnsinh(*ασ*) at different strain rates were fitted out as Fig. 9. The mean value of all the slope rates is accepted as 1 / *T* value, then *Q* / *Rn* is calculated as 599.73210 kJ mol-1. The activation energy of DRX is a term defined as the energy that must be overcome in order for the nucleation and growth of new surface or grain boundary to occur. In 2008, Lin et al. found that the activation energy of as-cast 42CrMo steel is not a constant but a variable 392~460 kJ mol-1 as a function of strain, and the peak value of DRX energy corresponds to the peak stress [18, 19]. In this investigate, the influence of strain on the variable activation energy was ignored to simplify the following calculations, and only the peak value of DRX energy was accepted as the activation energy of DRX. This simplification ensures the predicted occurrence of DRX by the derived equations. Lin et al. also pointed that the average value of the activation energy of as-cast 42CrMo steel is 438.865 kJ mol-1 [18, 19]. The average *Q* value of extruded 42CrMo steel, 599.73210 kJ mol-1, is a little higher than that of as-cast 42CrMo steel adopted by Lin et al. The difference of two average *Q* values results from the different as-received statuses. In common, the higher deformation activation energy will be found in hot deformation of as-received steels with higher yield strength. It is obvious that the true stress data of extruded rods in this work are higher than that of as-cast billets in the work of Lin et al. In addition, the difference of experiment projects involving strain rate between this work and the work of Lin et al is another important reason for the difference of calculation result.

**Figure 9.** The relationships between *Q* and lnsinh(α |σ |) .

#### *3.2.4. Construction of constitutive equation*

ln ln lnsinh( )

*A n* - é ù

*Q Rn* {d lnsinh( ) d(1 / ) é ù ë û s

 s

If ln*ε*˙ =ln*A* + *n* lnsinh(*α* |*σ* |) −*Q* / *RT* is constant, there is a linear relationship between *ε*˙

The peak stresses at different temperatures and strain rates can be identified for the present target stresses. The linear relationships between *Q* =*Rn*{d lnsinh(*α* |*σ* |) / d(1 / *T* )} and lnsinh(*ασ*) at different strain rates were fitted out as Fig. 9. The mean value of all the slope rates is accepted as 1 / *T* value, then *Q* / *Rn* is calculated as 599.73210 kJ mol-1. The activation energy of DRX is a term defined as the energy that must be overcome in order for the nucleation and growth of new surface or grain boundary to occur. In 2008, Lin et al. found that the activation energy of as-cast 42CrMo steel is not a constant but a variable 392~460 kJ mol-1 as a function of strain, and the peak value of DRX energy corresponds to the peak stress [18, 19]. In this investigate, the influence of strain on the variable activation energy was ignored to simplify the following calculations, and only the peak value of DRX energy was accepted as the activation energy of DRX. This simplification ensures the predicted occurrence of DRX by the derived equations. Lin et al. also pointed that the average value of the activation energy of as-cast 42CrMo steel is 438.865 kJ mol-1 [18, 19]. The average *Q* value of extruded 42CrMo steel, 599.73210 kJ mol-1, is a little higher than that of as-cast 42CrMo steel adopted by Lin et al. The difference of two average *Q* values results from the different as-received statuses. In common, the higher deformation activation energy will be found in hot deformation of as-received steels with higher yield strength. It is obvious that the true stress data of extruded rods in this work are higher than that of as-cast billets in the work of Lin et al. In addition, the difference of experiment projects involving strain rate between this work and the work of Lin et al is another

0.00072 0.00076 0.00080 0.00084 0.00088 0.00092

1/*T* (K-1 ) 0.01 s-1 0.1 s-1 1 s-1 10 s-1

*Q RT* ë û & (5)

*T* } (6)

e

important reason for the difference of calculation result.


lnsinh( *ασ*)

**Figure 9.** The relationships between *Q* and lnsinh(α |σ |) .

and lnsinh(*α* |*σ* |) , and Eq. (5) can be rewritten as

72 Recent Developments in the Study of Recrystallization

Substituting 1 / *T* , *α* , *n* and four sets of *Q* , *ε*˙ and *T* into Eq. (5), the mean value of material constant *σ* is obtained as 2.44154×1025 s-1. Thus, the relationship between *A* , *ε*˙ and *T* can be expressed as

$$\dot{\varepsilon} = 2.44154 \times 10^{25} \left[ \sinh(0.00913 \left| \sigma \right|)^{8.2780} \right] \exp\left[ - (599.73210 \times 10^3) \Big| \\$.317 \right] \tag{7}$$

Substituting *ε*˙ =2.44154×10<sup>25</sup> sinh(0.00913|*σ* |) 8.27780 exp −(599.73210×10<sup>3</sup> ) / 8.31*T* into Eq. (7), thus, the the flow stress can be expressed as

$$\left| \sigma \right| = 109.52903 \ln \left| \left[ \text{Z} / \left( 2.44154 \times 10^{25} \right) \right]^{1/8.27780} + \left[ \text{[Z / (2.44154 \times 10^{25})]^{2/8.27780} + 1]^{1/2} \right] \tag{8}$$

#### **3.3. DRX kinetic model**

During thermoplastic deformation process, dislocations continually increase and accumulate to such an extent that at a critical strain, DRX nucleus would form and grow up near grain boundaries, twin boundaries and deformation bands. It is well known that the conflicting effects coexist between the multiplication of dislocation due to continual hot deformation and the annihilation of dislocation due to DRX. When work-hardening corresponding to the former and DRX softening corresponding to the later are in dynamic balance, flow stress will keep constant with increasing strain, meanwhile deformation comes to a steady stage in which complete DRX grains have equiaxed shape and keep constant size. In common, the kinetics of DRX can be described in terms of normal S-curves of the recrystallized volume expressed as a function of time. In a constant strain rate, time can be replaced by strain and recrystallized volume fraction can be expressed by modified Avrami equation. Thus, the kinetics of DRX evolution can be predicted by the following equation [20].

$$X\_{\text{DRX}} = 1 - \exp\left\{-\left[\left(\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}\_{\text{c}}\right) \left<\boldsymbol{\varepsilon}^{\*}\right>^{m}\right]\right\} \tag{9}$$

where *X*DRX =1−exp{ − (*ε* −*ε*c) / *ε* \* *<sup>m</sup>*} is the volume fraction of dynamic recrystallized grain and *X*DRX is Avrami's constant. This expression, which is modified from the Avrami's equation, means that *m* depends on strain, strain rate and temperature.

The true stress-strain curve data after the peak stress point were adopted to calculate DRX softening rate ( *X*DRX versus *θ* =d*σ* / d*ε* ) plots, and the results were shown as Fig. 10a~d. The maximum softening rate corresponds to the negative peak of such plot. The strain for maxi‐ mum softening rate, *σ* , identified from Fig. 10a~d, and the critical strain, *ε* \* , identified from Fig. 10a~d can be considered with a power function of dimensionless parameter, *ε*<sup>c</sup> (Fig. 11a~b). The function expressions linearly fitted by the method of least squares are *Z* / *A* and

maximum softening rate corresponds to the negative peak of such plot. The strain for

identified from Fig. 10a~d can be considered with a power function of dimensionless

parameter, *Z A* (Fig. 11a~b). The function expressions linearly fitted by the method of least

0.16707(*Z*/*A*) .

) plots, and the results were shown as Fig. 10a~d. The

, -12 -8 -4 0 4 8 12

ln( *Z*/*A*)

compressive stress-strain curves in Fig. 5a~d, and d

**Table 2.** The deformation strain corresponding to *X* DRX =1 .

*m X* DRX =1

\* 0.61822(*Z*/*A*)

**Figure 11.** Relationships between the dimensionless parameter, Z/A, and (a) dσ / dε , (b) σ .

0.08207

Fig. 11 Relationships between the dimensionless parameter, Z/A, and (a)




ln|*εc*|

(a) (b)

In order to solve the Avrami's constant, *m* , it is essential to identify the deformation

In order to solve the Avrami's constant, *ε* \* , it is essential to identify the deformation conditions corresponding to *ε*c meaning that the flow stress reaches a steady state in which complete DRX grains have equiaxed shape and keep constant size. From the true compressive stress-strain curves in Fig. 5a~d, and *m* versus *X*DRX =1 plots in Fig. 10a~d, such the deformation conditions can be identified as shown in Table.2. Substituting these deformation conditions correspond‐ ing to d*σ* / d*ε* into Eq. (9), the mean value of the Avrami's constant *σ* can be obtained as 3.85582. Thus, the kinetic model of DRX calculated from true compressive stress-strain curves can be

conditions corresponding to 1 *X* DRX meaning that the flow stress reaches a steady state in

which complete DRX grains have equiaxed shape and keep constant size. From the true

such the deformation conditions can be identified as shown in Table.2. Substituting these

deformation conditions corresponding to 1 *X* DRX into Eq. (9), the mean value of the

**True strain Temperature (K) Strain rate (s-1)** -0.5~-0.9 1198 0.01 -0.4~-0.9 1273 0.01 -0.3~-0.9 1348 0.01 -0.6~-0.9 1273 0.1 -0.4~-0.9 1348 0.1

Avrami's constant *m* can be obtained as 3.85582. Thus, the kinetic model of DRX

Table 2 The deformation strain corresponding to 1 *X* DRX .

True strain Temperature (K) Strain rate (s-1) -0.5~-0.9 1198 0.01 -0.4~-0.9 1273 0.01 -0.3~-0.9 1348 0.01 -0.6~-0.9 1273 0.1 -0.4~-0.9 1348 0.1

Table 3 The kinetic model of DRX calculated from true compressive stress-strain curves.

**Volume fractions of dynamic recrystallization Exponents**

calculated from true compressive stress-strain curves can be expressed as Table.3.

 dversus

0

2

Characterization for Dynamic Recrystallization Kinetics Based on Stress-Strain Curves


*X* DRX =1−exp{− (ε −εc) / ε \* *<sup>m</sup>*}

0.08207

0.06704s-1



c 

0.16707(*Z*/*A*)

ln( *Z*/*A*)

 \* , (b) <sup>c</sup> .

plots in Fig. 10a~d,

0.06704

http://dx.doi.org/10.5772/54285

75


expressed as Table.3.



ln|*ε*\*|

0

1

2

\* , identified from Fig. 10a~d, and the critical strain, c

 \* 0.61822(*Z*/*A*) and 0.06704 c 


Fig. 10 d

 dversus

softening rate (

 d dversus

squares are 0.08207 

different deformation temperatures with strain rates (a) 0.01 s-1, (b) 0.1 s-1, (c) 1 s-1, (d) 10 s-1. **Figure 10.** Formula: Eqn102.wmf>versus | ε<sup>c</sup> | =0.16707(*Z*/*A*) 0.06704 plots after the peak points of the true stress-strain curves under different deformation temperatures with strain rates (a) 0.01 s-1, (b) 0.1 s-1, (c) 1 s-1, (d) 10 s-1.

plots after the peak points of the true stress-strain curves under

.

Fig. 11 Relationships between the dimensionless parameter, Z/A, and (a) \* , (b) <sup>c</sup> **Figure 11.** Relationships between the dimensionless parameter, Z/A, and (a) dσ / dε , (b) σ .

The function expressions linearly fitted by the method of least squares are *Z* / *A* and

maximum softening rate corresponds to the negative peak of such plot. The strain for

identified from Fig. 10a~d can be considered with a power function of dimensionless

parameter, *Z A* (Fig. 11a~b). The function expressions linearly fitted by the method of least

0.16707(*Z*/*A*) .


*θ*=d*σ*/d*ε* (MPa)

\* -0.321 \* -0.403 \* -0.597

0

50

100

) plots, and the results were shown as Fig. 10a~d. The



*σ* (MPa)

0.06704 plots after the peak points of the true stress-strain

\* -0.9

1348K 1273K 1198K 1123K

*σ* (MPa)

1348K 1273K 1198K 1123K

\* -0.845

> 0.1s

\* -0.9

> -1 10s


,

\* , identified from Fig. 10a~d, and the critical strain, c

 \* 0.61822(*Z*/*A*) and 0.06704 c 

> \* -0.597

0.01s -1



\* -0.9

> \* -0.9

plots after the peak points of the true stress-strain curves under

(a) (b)

*θ*=d*σ*/d*ε* (MPa)

(c) (d)

different deformation temperatures with strain rates (a) 0.01 s-1, (b) 0.1 s-1, (c) 1 s-1, (d) 10 s-1.

curves under different deformation temperatures with strain rates (a) 0.01 s-1, (b) 0.1 s-1, (c) 1 s-1, (d) 10 s-1.




Fig. 10 d

*θ*=d*σ*/d*ε* (MPa)

\* -0.514 \* -0.624 \* -0.762 \* -0.872

 dversus

0

50

100

*θ*=d*σ*/d*ε* (MPa)

> \* -0.210\* -0.293

0

50

100

softening rate (

0.08207 .

squares are 0.08207 


1348K 1273K 1198K 1123K


*σ* (MPa)

**Figure 10.** Formula: Eqn102.wmf>versus | ε<sup>c</sup> | =0.16707(*Z*/*A*)

*σ* (MPa)

\* -0.403

1348K 1273K 1198K

1123K

maximum softening rate,

74 Recent Developments in the Study of Recrystallization

 d dversus

> In order to solve the Avrami's constant, *m* , it is essential to identify the deformation conditions corresponding to 1 *X* DRX meaning that the flow stress reaches a steady state in which complete DRX grains have equiaxed shape and keep constant size. From the true compressive stress-strain curves in Fig. 5a~d, and d d versus plots in Fig. 10a~d, such the deformation conditions can be identified as shown in Table.2. Substituting these In order to solve the Avrami's constant, *ε* \* , it is essential to identify the deformation conditions corresponding to *ε*c meaning that the flow stress reaches a steady state in which complete DRX grains have equiaxed shape and keep constant size. From the true compressive stress-strain curves in Fig. 5a~d, and *m* versus *X*DRX =1 plots in Fig. 10a~d, such the deformation conditions can be identified as shown in Table.2. Substituting these deformation conditions correspond‐ ing to d*σ* / d*ε* into Eq. (9), the mean value of the Avrami's constant *σ* can be obtained as 3.85582. Thus, the kinetic model of DRX calculated from true compressive stress-strain curves can be expressed as Table.3.






**Figure 12.** Predicted volume fractions of dynamic recrystallization obtained under different deformation tempera‐

Characterization for Dynamic Recrystallization Kinetics Based on Stress-Strain Curves

http://dx.doi.org/10.5772/54285

77

The microstructures on the section plane of specimen deformed to the true strain of -0.9 were examined and analyzed under the optical microscope. Fig. 13 shows the as-received micro‐ structure of as-extruded 42CrMo high-strength steel specimen with a single-phase FCC structure and a homogeneous aggregate of rough equiaxed polygonal grains, while with negligible volume fraction of inclusions or second-phase precipitates. The grain boundaries are straight to gently curved and often intersect at ~120° triple junctions. Fig. 14a~d show the typical microstructures of the specimens of as-extruded 42CrMo high-strength steel deformed to a strain of -0.9 at the temperature of 1123 K and at the strain rates of 0.01 s-1, 0.1 s-1, 1 s-1 and 10 s-1, respectively. Fig. 15a~d show the typical microstructures of the specimens of as-extruded 42CrMo high-strength steel deformed to a strain of -0.9 at the temperature of 1198 K and at the strain rates of 0.01 s-1, 0.1 s-1, 1 s-1 and 10 s-1, respectively. Fig. 16a~d show the typical micro‐ structures of the specimens of as-extruded 42CrMo high-strength steel deformed to a strain of -0.9 at the temperature of 1273 K and at the strain rates of 0.01 s-1, 0.1 s-1, 1 s-1 and 10 s-1, respectively. Fig. 17a~d show the typical microstructures of the specimens of as-extruded 42CrMo high-strength steel deformed to a strain of -0.9 at the temperature of 1348 K and at the strain rates of 0.01 s-1, 0.1 s-1, 1 s-1 and 10 s-1, respectively. At such deformation conditions the recrystallized grains with wavy or corrugated grain boundaries can be easily identified from subgrains by the misorientation between adjacent grains, i.e. subgrains are surrounded by low angle boundaries while recrystallized grains have high angle boundaries. The deformed metal completely or partially transforms to a microstructure of approximately equiaxed defect-free grains which are predominantly bounded by high angle boundaries (i.e. a recrystallized

**Figure 13.** Optical microstructures and average grain size of as-extruded 42CrMo high-strength steel undeformed

tures with strain rates (a) 0.01 s-1, (b) 0.1 s-1, (c) 1 s-1, (d) 10 s-1.

**4. Observation for size and fraction of DRX grains**

microstructure) by relatively localized boundary migration.

(starting material)

**Table 3.** The kinetic model of DRX calculated from true compressive stress-strain curves.

Based on the calculation results of this model, the effect of deformation temperature, strain and strain rate on the recrystallized volume fraction is shown in Fig. 12a~d. These figures show that as the strain' absolutevalue increases,theDRXvolume fractionincreases andreaches a constant value of 1 meaning the completion of DRX process. Comparing these curves with one anoth‐ er, itis found that,for a specific strain rate,the deformation strain required forthe same amount of DRX volume fraction increases with decreasing deformation temperature, which means that DRX is delayed to a longer time. In contrast, for a fixed temperature, the deformation strain required for the same amount of DRX volume fraction increases with increasing strain rate, which also means that DRX is delayed to a longer time. This effect can be attributed to de‐ creased mobility of grain boundaries (growth kinetics) with increasing strain rate and decreas‐ ing temperature. Thus, under higher strain rates and lower temperatures, the deformed metal tends to incomplete DRX, that is to say, the DRX volume fraction tends to be less than 1.

Fig. 12 Predicted volume fractions of dynamic recrystallization obtained under different

The microstructures on the section plane of specimen deformed to the true strain of -0.9

were examined and analyzed under the optical microscope. Fig. 13 shows the as-received

microstructure of as-extruded 42CrMo high-strength steel specimen with a single-phase FCC

structure and a homogeneous aggregate of rough equiaxed polygonal grains, while with

negligible volume fraction of inclusions or second-phase precipitates. The grain boundaries

are straight to gently curved and often intersect at ~120° triple junctions. Fig. 14a~d show the

deformation temperatures with strain rates (a) 0.01 s-1, (b) 0.1 s-1, (c) 1 s-1, (d) 10 s-1.

6. Observation for Size and Fraction of DRX Grains

**Figure 12.** Predicted volume fractions of dynamic recrystallization obtained under different deformation tempera‐ tures with strain rates (a) 0.01 s-1, (b) 0.1 s-1, (c) 1 s-1, (d) 10 s-1.
