Simulation of Dynamic Recrystallization Behavior under Hot Isothermal Compressions for as-extruded 3Cr20Ni10W2 Heat-Resistant Alloy by Cellular Automaton Model

Introduction

3Cr20Ni10W2, as a kind of heat-resistant alloys, has excellent heat-resistance and high corrosion resistance, high strength, and superior creep strength. For this reason, 3Cr20Ni10W2 is primarily used in manufacture of turbines components and exhaust valves on diesel engine [1, 2]. It is well known that mechanical properties of the metal are mainly decided by microstructure such as grain size to a larger extent. Usually, the three typical metallurgical phenomena during the hot deformation, namely work hardening (WH), dynamic recovery (DRV), and dynamic recrystallization (DRX), has a decisive influence on grain size, of which DRX is generally recognized a phenomenon that new high-angle, mobile grain boundaries generated and migrated, in other words, DRX as a kind of mechanism of grain refinement has dominant influence on the final grain size. However, it is tedious to observe the microstructural evolution process during grain growth by physical experiments. Therefore, more and more attention is paid to develop an efficient method to simulate the grain size evolution when the hot deformation occurred to the metal.

In recent years, two approaches are commonly supposed to simulate the DRX with the development of computer simulation technology, i. e. the Monte Carlo (MC) method and the cellular automaton (CA) method [3]. For instance, MC method was used to study microstructural evolution such as static recrystallization, DRX and grain growth [4, 5, 6]. However, grain growth kinetics during DRX was not simulated by this method. Subsequently, CA model gets noticed gradually, which provides an approach describing the growth kinetics and microstructure evolution during the hot deformation process [7]. In other words, the CA model can describe the DRX and the grain growth graphically. Specifically, the CA is a mathematical method which is generally adopted to reveal the complex system evolution rules in the discrete space-time by an application of deterministic or probabilistic transformation rules to the location of a lattice. Every cell is not isolated, and its present state and next state are decided by the neighbor cells based on the above mathematical arithmetic [8, 9] Meanwhile dislocation density in the CA algorithm acted as a very significant role in nucleation and microstructural evolution of DRX during hot deformation, and then the modified LJ model, which has been proved that it is suitable to simulate the dislocation density evolution in detail [10], recommended by GOURDET and MONTHEULLET [11] was adopted to describe the dislocation densities evolution. For example, Chen et al. [12] presented a CA model to predict the DRX behaviors of 30Cr2Ni4MoV rotor steel during hot deformation. In order to overcome the difficulty inaccurately determining the DRX nucleation parameters for low carbon steel, Jin and Cui [13] developed a flow stress-based inverse analysis method, which was implemented by coupling a CA model with an adaptive response surface method. In addition, E. Afshari et al. studied the kinetics of static recrystallization and distribution of recrystallized grain size by means of a cellular automata model together with the stored energy calculated by the deformation analysis [14]. Yan-Xing Liu et al. [15] Study of DRX in a Ni-based superalloy by experiments and CA model.

In present paper, DRX model and the model for flow stress of as-extruded 3Cr20Ni10W2 were established based on the true stress–strain data, and then the various parameters in the dislocation density model, recovery model and nucleation and growth of DRX were determined respectively. Subsequently, the grain size evolution under different deformation conditions was modeled through the present CA model.

Experimental procedure

The specific experimental has already been reported [16]. Briefly, the chemical compositions of 3Cr20Ni10W2 heat-resisting alloy are (wt.%) C 0.25, Si 1, Cr 20, Ni 10, Mn 1, W 2, Fe (balance). The experimental samples with same diameter of 10 mm and height of 12 mm were subjected to the designed experimental steps, detailed, the samples were heated to a specified temperature at a heating rate of 1 K/s by the Gleeble 1500 machine, and held at the specified temperature for 180 s to ensure a uniform starting temperature and decrease the material anisotropy. All the twenty specimens were compressed to a fixed true strain of 0.9163 at four different temperatures of 1203 K, 1253 K, 1303 K 1353 K and 1403 K, and four different strain rates of 0.01 s⁻¹, 0.1 s⁻¹, 1 s⁻¹ and 10 s⁻¹ and then a series of strain–stress data was collected. Subsequently, the microstructures on the section plane of specimen deformed to the true strain of −0.91 were examined and analyzed under the optical microscope. By means of the metallography, the grain size under different deformation conditions was counted [17].

Analysis of strain–stress curve flow behavior

As shown in Figure 1 (a~d), the true compressive stress-strain curves of 3Cr20Ni10W2 heat-resistant alloy described the stress change with the variation of strain under the different strain rate and temperature. After analysis of these curves in whole, it can be found that, for all of specimens, the flow stress decreases monotonically with different softening rates. Specifically, for a fixed strain rate, the flow stress decreases obviously with the increased temperature, and the flow stress increases with the increased strain rate at a fixed temperature.

Figure 1:

True stress–strain curves of 3Cr20Ni10W2 alloy at different strain rates and temperatures (a) 0.01 s⁻¹, 1203~1403 K; (b) 0.1 s⁻¹, 1203~1403 K; (c) 1 s⁻¹, 1203~1403 K; (d) 10 s⁻¹, 1203~1403 K.

In addition, the previous study [16] divided the stress evolution into three distinct stages including WH, DRX, and DRV, and the detail type classification is concluded as following: decreasing gradually to a steady state with DRV softening (1403 K & 0.01 s⁻¹, 1353~1403 K & 0.1 s⁻¹, 1403 K & 1 s⁻¹), and decreasing continuously with significant DRX softening (1203~1303 K & 0.01 s⁻¹, 1203~1303 K & 0.1 s⁻¹, 1203~1353 K & 1 s⁻¹, 1203~1403 K & 10 s⁻¹).

The establishment of a cellular automata model (CA)

CA model is gradually utilized in studying microstructure evolution [15, 18, 19]. In this method, the discrete lattice geometry is square. Subsequently, the simulated area was defined, the number of rows and columns of discrete lattice points in the microstructure was set as 200×200 and every cell in the discrete lattice is 2 um, thus this would result in a microstructure simulation of a region of material 400×400 um. The boundary conditions is periodic wrap-around by default, then the neighbor type was set as the Moore`s, in addition, the radius of the neighbor is 1, thus, the adjacent cell is 8. For an annealed metal, the dislocation density was typically taken as 0.01. The dislocation density for each region of microstructure is calculated as a function of hardening and recovery with respect to strain. The type of nucleation conditions for new grains was threshold dislocation density and probability. In the end, the initial grain size was input and the value is 63.9 um.

DRX volume fraction model and grain size model

Initiation of DRX

As mentioned above, twelve deformation conditions with distinct DRX characteristic corresponding to three temperatures of 1203 K, 1253 K and 1303 K, and four strain rates of 0.01 s⁻¹, 0.1 s⁻¹, s⁻¹ and 10 s⁻¹ were selected, Based on the strain–stress data of the twelve curves, the strain hardening rate dσ/dε versus σ curves were plotted as shown in Figure 2, it is recognized that the critical conditions for DRX can be determined when the minimum value of −dθ/dσ appeared, where the θ was defined as θ=dσ/dε [28]. According to this method, the critical strains (εc) and critical stresses under these twelve deformation conditions were calculated, and the linear relationship between εc and the strain for the peak stress (εp) is summarized as the eq. (15) [29]. As for the εp, it is usually recognized that the value of εp is controlled by strain rate (ε˙), deformation activation energy (Q₁), the initial grain size (d₀), and deformation temperature (T), and this relationship is expressed as eq. (15) [30, 31].

(15)εc=αεp

Figure 2:

dσ/dε versus σ plots up to the peak points of the true stress–strain curves at different deformation temperatures and strain rates of: (a) 0.01 s⁻¹, (b) 0.1 s⁻¹, (c) 1 s⁻¹, (d) 10 s⁻¹.

where α is a material constant.

(16)εp=α1d0n1ε˙m1exp(Q1/RT)

where a₁, n₁ and m₁ are material constants, R is the universal gas constant (8.31 J mol⁻¹ K⁻¹). In this work, the effect of initial grain size on the peak strain was without consideration, so eq. (16) can be described to eq. (17).

(17)εp=α1ε˙m1exp(Q1/RT)

(1) Calculation of material constant a

As shown in Figure 3, in the basis of the values of εc and εp under different temperatures and strain rates, the relationships between εc and εp were fitted linearly. The slope value of this line is the value of a, and its value was 0.1069.

Figure 3:

The relationships between εc and εp.

(2) Determination of m₁, Q₁ and a₁

Take natural logarithms on both sides of eq. (17), the eq. (18) is obtained.

(18)lnεp=lnα1+m1lnε˙+Q/RT

Obviously, m1=∂lnεp/∂lnε˙. As shown in Figure 4, the relationships between lnεp and lnε˙ under different temperatures were fitted linearly. Every slope value of fitted line under a certain temperature and strain rates is the value of m₁. Observed from Figure 4, the slope values of the three fitted lines under different strain rates were nearly equal. Therefore, the mean value of all the three slope values is recognized as the value of m₁, and value of m₁ was 0.113.

Figure 4:

The relationships between ∂lnεp and ∂lnε˙.

From eq. (17), it is noticed that Q1=R∂lnεp/∂(1/T). As shown in Figure 5, the linear relationships under different strain rates between lnεp and 1/T were described. The slope value under different strain rates was value of ∂lnεp/∂(1/T). It is observed that the four slopes were almost equal. Therefore the value of material constant Q₁ is equal to the mean value of the four slopes, and its value is 32251.664 J mol⁻¹. Afterwards, m₁ and Q₁ were substituted into eq. (18) to calculate twelve values of material constant a₁ under the deformation conditions above. Finally a₁ was calculated as 0.0082.

Figure 5:

The relationships between ∂lnεp and 1/T.

DRX kinetic model

It is well known that nucleation and growth of DRX started when the dislocations increased and accumulated to a critical degree. Generally, the well-known Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation as shown eq. (19) is widely used to describe the relationships between DRX fraction and true strain [28, 29, 30, 31].

(19)XDRX=1−exp−βdε−εcε0.5kd

where XDRX is the volume fraction of DRX grains, ε is the true strain. ε0.5 is the strain when DRX volume fraction reached 50%, βd and kd are the Avrami material constants. In addition, an empirical model as shown in the below [29, 31] was introduced to calculate the value of ε0.5.

(20)ε0.5=α2d0n2ε˙m2exp(Q2/RT)

where Q2 is the activation energy for recrystallization, and α2, n2 and m₂ are all material constants.

Although DRX can be observed from the metallography, it is tedious to carry out extensive and quantitative metallography measurements, which results in a hard work to calculate of DRX volume fraction under different deformation conditions according to microstructures. However, flow stress of metal during hot deformation is an actual and indirect reflection of microstructure evolution, therefore, based on the further analysis of the true stress–strain data XDRX can be determined. In this work, eq. (21) [32, 33] shown in the below is used to calculate the XDRX.

(21)DDRX=(σdrvx)2−(σdrxx)2(σdrvss)2−(σdrxss)2

where σdrvss is final steady state stress, and σdrvx is the and the transient state stress on the true stress–strain curve of ideal DRV (as shown in Figure 6); σdrxss is the final steady state stress, and σdrxx is the transient state stress on actual DRX true stress–strain curve (as shown in Figure 6) plotted from a series of isothermal compression test data.

Figure 6:

Plot of θ−σ employed to determine σdrvss by the intercept with the horizontal axis.

The method to determine the value of σdrvss is shown below. Firstly, the critical stress (σc) corresponding to the εc can be obtained, and the εc has been already determined above. Thus the θ corresponding to σc was determined, based on the physical meaning, the σdrvss is defined as a value, which is the value corresponding to point D, in addition, it is obvious that the working-hardening behavior prior to σc characterizes σdrvss [34]. For the values of σdrvx, σdrxx, σdrvss and σdrxss, which can be determined by dσ/dε versus σ curves and the true stress–strain curves under different deformation conditions. Thus, the XDRX and the ε0.5 can be calculated. Figure 7 describes the predicted DRX volume fractions under the different deformation conditions. Observed from Figure 7, it can be concluded that the deformation strain required for the same amount of DRX volume fraction decreases with the increased deformation temperature under a condition of fixed strain rate, which means that a low temperature is a poor condition for DRX, meanwhile, the deformation strain corresponding to the same amount of DRX volume fraction decreases with the decreased strain rate under a fixed temperature. This effect can be attributed to increased mobility of grain boundaries (growth kinetics) with the decreased strain rate and the increased temperature.

Figure 7:

Predicted volume fractions of dynamic recrystallization obtained at different deformation temperatures and strain rates of: (a) 0.01 s⁻¹, (b) 0.1 s⁻¹, (c) 1 s⁻¹, (d) 10 s⁻¹.

(1) Calculation of material constants: βd, kd

Taking natural logarithm on both sides of eq. (19) gives eq. (22).

(22)ln[−ln(1−XDRX)]ε0.5=lnβd+kdln[(ε−εc)/ε0.5]

Therefore, kd=∂ln[−ln(1−XDRX)]/∂ln[(ε−εc)/ε0.5]. Take the values of the parameters into the ln[−ln(1−XDRX)] and ln[(ε−εc)/ε0.5], then a series first derivatives values of ln[−ln(1−XDRX)] versus ln[(ε−εc)/ε0.5] were obtained, which can be fitted linearly as shown in Figure 8. Obviously, the value of the kdis the slope of this fitted line, and here its value was obtained as 1.6286. Then kd and a series of XDRX, ε0.5, εc and ε are substituted into eq. (8), and then the mean value of material constant βd was calculated as 0.9885.

Figure 8:

The relationships between ln[−ln(1−XDRX)] and ln[(ε−εc)/ε0.5].

(2) Calculation of material constants: m_2,α2, Q2

Taking natural logarithm on both sides of eqs (20), equation (20) can be represented as eq. (23).

(23)lnε0.5=lnα2+n2lnd0+m2lnε˙+Q2/RT

Obviously, m2=∂lnε0.5/∂lnε˙. As shown in Figure 9, through linear match, the relationships between lnε0.5 and lnε˙ under different temperatures were clear. It is found that the slope values of the three fitted lines were basic equal. Therefore, the value of the material constant, m2, is recognized as the mean value of the three slope values, so its value was calculated as 0.0294.

Figure 9:

The relationships between lnε0.5 and lnε.

Observed from eq. (23), it is deduced that Q2=R∂lnε0.5/∂(1/T). As shown in Figure 10, through linear match, the relationships between lnε0.5 and 1/T under different strain rates appeared. It is found that the slope values of the four fitted lines were basic equal. Similarly, the value of the material constant, Q2, is recognized as the mean value of the four slope values, so its value was calculated as 36651 J mol⁻¹.

Figure 10:

The relationships between lnε0.5 and 1/T.

In present paper, the n2 was set as 0 because that the effect of initial grain size is without consideration. Then take m2 and Q2 into eq. (23), twelve values of material constant, α2, under different deformation conditions can be calculated, and the value of α2 is accepted as the mean value of these twelve values, namely 0.0141.

Grain growth model

The average austenite grain size of 3Cr20Ni10W2 under different temperature and strain rate has been published elsewhere [17], Based on these grain size, an empirical equation in Avrami type of the materials was utilized in this work, which is shown as following:

(24)ddrex=α3d0n3ε˙m3exp(Q3/RT)

where R is the universal gas constant (8.31 J mol⁻¹ K⁻¹), a₃, n₃ and m₃ are material constants. In present work, the influence of different strain rate and temperature on the grain size was mainly investigated, so the eq. (24) can be simplified to eq. (25).

(25)ddrex=α3ε˙m3exp(Q3/RT)

1. Calculation of material constant m₃, Q3, α3

Take natural logarithms on both sides of eq. (25), the eq. (26) is obtained.

(26)lnddrex=lnα3+m3lnε˙+(Q3/RT)

The calculation method similar to the calculation for m₁, Q1, α1 was conducted to determine the values of m₃, Q3, α3. The values are −0.0365, −36252.9816 J mol⁻¹ and 750.892 respectively.

Results and discussion

Microstructural evolution of DRX by CA simulation

Figure 11 (a~g) illustrates the grain size evolution of the 3Cr20Ni10W2 sample at the temperature of 1203 K and at the strain rate of 0.01 s⁻¹ by means of the CA simulation. It is observed from Figure 11 (a) that the recrystallization fraction increases with the increased simulation time. Moreover, Figure 11 shows that the mean grain size decreases gradually with the increased simulation time. Therefore, it is concluded that the grain size becomes smaller due to the DRX during the hot compression process. In addition, the recrystallized grains firstly occurred at the grain boundaries in the initial microstructure when the DRX starts, subsequently, these new grains grow up with the increase of deformation. Meanwhile, the new crystal nuclei occur at the grain boundaries of the recrystallized grains formed previously as shown in Figure 11 (e), in other words, it can be summarized that the new recrystallized grains arise repeatedly until the plastic deformation ends as shown in Figure 11 (g), and DRX is precisely characterized by repeated nucleation and finite growth of the recrystallized grains.

Figure 11:

Microstructure evolution of the 3Cr20Ni10W2 at the deformation temperature of 1203 K and strain rate of 0.01 s⁻¹.

Prediction of grain size by CA simulation

It is well know that the deformation temperatures and the strain rates have an important influence on the mean grain size during DRX. Based on CA simulation results, the effect of the deformation temperatures and the strain rates on the mean grain size is obtained. As shown in Figures 12 and 13, it is concluded that the mean grain size increases with the increased deformation temperature at a fixed strain rate, but decreases with the increased strain rate at a fixed temperature. This is because that both the decrease of deformation temperature and the increase of strain rate can cause the increase in the dislocation density, which contributes to accelerating DRX. Therefore, the recrystallized grains take place more repeatedly in the same deformation time.

Figure 12:

Prediction of microstructure and the simulated grain size at the deformation temperature of 1203 K and strain rates of: (a) 0.01 s⁻¹ (b) 0.1 s⁻¹ (c) 1 s⁻¹ (d) 10 s⁻¹.

Figure 13:

Prediction of microstructure and the simulated grain size at the fixed strain rate of 0.01 s⁻¹ and deformation temperatures of: (a) 1203 K (b) 1253 K (c) 1303 K.

Conclusions

Based on the analysis about the hot isothermal compressions strain–stress curves of the for as-extruded 3Cr20Ni10W2, the experimental data obtained from hot compression tests in the temperature range of 1203–1303 K with an interval of 50 K and strain rate range of 0.01–10 s⁻¹ is adopted. After the establishment of the CA model, the DRX behavior of the 3Cr20Ni10W2 was studied, meanwhile the absolute average relative error indicates that the present CA model is able to investigate the microstructure evolution and calculate the grain size accurately during hot compression of 3Cr20Ni10W21 alloy. The main conclusions are summarized as following:

1. Analysis on the hot isothermal compressions strain–stress curves of the 3Cr20Ni10W2 indicates that DRX occurred under the deformation conditions including 1203~1303 K & 0.01 s⁻¹, 1203~1303 K & 0.1 s⁻¹, 1203~1353 K & 1 s⁻¹ and 1203~1403 K & 10 s⁻¹. Therefore, the temperature range of 1203–1303 K with an interval of 50 K and strain rate range of 0.01–10 s⁻¹ is adopted to establish the CA model.

2. Dislocation density model, recovery model, the DRX nucleation and growth model and DRX volume fraction model and grain size model are established successively. During this process, the value of different coefficient was determined, which lay a foundation for the subsequent CA model.

3. The simulated results shows that the recrystallized grains firstly occur at the grain boundaries of the grains in the initial microstructure when the DRX starts, subsequently, these new grains grow up with the increase of deformation. In addition, the mean grain size increases with the increased deformation temperature at a fixed strain rate, but decreases with the increased strain rate at a fixed temperature. The average absolute relative error also indicates that the present CA model is able to simulate the DRX behavior and calculate the final grain size accurately.