Constitutive Modeling for Flow Stress Behavior of Nimonic 80A Superalloy During Hot Deformation Process

Introduction

During hot forming process, the deformed metal is liable to undergo work hardening (WH), dynamic recovery (DRV) and dynamic recrystallization (DRX). WH results mainly from the increase of dislocation density. DRV is a softening process due to the dislocation rearrangement. DRX accumulates and converts the dislocation energy into the activation energy of the formation of new strain-free grains. Flow stress of metal is a direct consequence of microstructural changes involving the three metallurgical phenomena during hot deformation process [1]. Recently, numerical simulation technology has been widely applied to the study of metal forming process. Accurate constitutive models for DRX behavior and flow stress are the keys to employ computer code to model the response of materials [1–3]. Considerable previous works [1–7] on the constitutive models for the DRX behavior and the flow stress of metals have been reported.

Nimonic 80A is a nickel-based superalloy extensively employed for high-temperature and high-strength components of aerospace, marine engineering and power generation, etc., due to its excellent creep, fatigue and corrosion resistance [8–12]. Some works have been performed on the microstructure of Nimonic 80A. Jeong et al. [13] obtained a set of mathematical models to describe the recrystallization and grain growth behaviors of Nimonic 80A and predicted the temperature, effective strain and microstructural evolution during hot closed die forging. Bombac et al. [14] defined optimal hot forming characteristics of Nimonic 80A using laboratory compression tests and established a mathematical expression connecting mean grain size and true stress. Tian et al. [15] determined microstructural changes during hot deformation as a function of strain and strain rate by investigating the DRX and DRV of Nimonic 80A. However, little attention has been paid to the study on the hot deformation characteristics and the flow stress of Nimonic 80A, and the related data are still rare. Further studies on the constitutive model for flow stress of Nimonic 80A are still essential.

In the present work, isothermal compression tests of Nimonic 80A were performed at different temperatures and strain rates to investigate the hot deformation characteristics of the alloy. The constitutive models for DRX behavior and flow stress of Nimonic 80A by considering the effects of deformation temperature, strain rate and strain were established. The accuracy of the developed constitutive model was further evaluated.

Experimental procedures

The alloy (Nimonic 80A) used in this investigation was provided in the form of the rolled bar by Dongbei Special Steel Group. Its chemical composition is shown in Table 1. To evaluate the hot deformation characteristics, cylindrical compression specimens with a diameter of 8 mm and a height of 12 mm were machined to perform isothermal compression tests. In order to reduce the effect of friction between the anvils and the end faces of the specimens during the compression process, the end faces of the specimens were polished.

The isothermal compression tests were performed on a Gleeble-1500 thermomechanical simulator. Figure 1 shows a schematic diagram of the compression process. Before compression, the specimens were heated to 1,200°C at the rate of 10°C/s and held for 300 s to make sure that the initial grain sizes of all the specimens were in conformance with each other. Then, the specimens were cooled at the rate of 10°C/s to testing temperature (1,000°C, 1,050°C, 1,100°C, 1,150°C and 1,200°C) and held for 30 s to eliminate temperature gradient. Subsequently, the specimens were deformed to the strain of 0.8 at the testing temperature and the strain rates of 0.01, 0.1, 1 and 5 s^–1. The specimens were quenched immediately after each test to preserve the grain boundary of the elevated temperature austenite microstructures. Finally, the specimens were sectioned along the longitudinal axis. The longitudinal sections of the specimens were polished and then etched (the etchant is 100 ml HCl + 10 ml H₂SO₄ + 5 g CuSO₄). The austenite grain size of the specimens was measured by optical microscopy.

Figure 1:

Schematic diagram of compression process.

Constitutive equations characterizing flow stress

At present, many researchers have utilized two-stage equations to model flow stress of metals. The following equations were proposed to model WH in DRV region and strain softening in DRX region of metals [1–5]:

in which σ_wh is the flow stress when DRV is the only softening mechanism, σ₀, σ_s and σ_ss represent the initial stress, the saturation stress and the steady-state stress, respectively, Ω is the DRV parameter, ε is the strain, ε_c is the critical strain for the onset of DRX, X_dyn is the DRX volume fraction. X_dyn can be expressed as follows [3–5]:

in which ε_p is the peak strain, k_d and n_d are the material constants. In order to construct the constitutive model for flow stress, the above parameters need to be determined. σ₀, σ_s, σ_ss, Ω, ε_c and ε_p are usually regressed from deformation temperature and strain rate [1–5].

The Arrhenius equation is widely employed to describe the relationships between deformation temperature, strain rate and flow stress at elevated temperature. The Arrhenius equation can be written as follows [16–18]:

where

in which ε˙ is the strain rate, Q is the DRX activation energy, R is the universal gas constant, T is the absolute temperature, σis the flow stress for a given strain, A, β, α and n are the material constants.

In order to reflect the effects of deformation temperature and strain rate on the deformation characteristics, Zener–Hollomon parameter (Z) is introduced in an exponential equation as follows [19]:

The stress–strain data obtained from isothermal compression tests under different deformation conditions can be used to determine the material constants and the DRX activation energy in eqs. (4) and (5).

Results and discussion

Analysis of flow stress curves

The true stress–strain curves of Nimonic 80A at various temperatures with strain rates of 0.01, 0.1, 1 and 5 s^–1 are shown in Figure 2. Most of the curves exhibit a single peak stress and peak strain followed by a gradual fall to a steady-state stress, which indicates that a typical flow softening feature presents during the entire deformation process due to the occurrence of WH, DRV and DRX [3–7]. At the initial stage of the deformation process, the stress increases to a peak σ_p due to the dominance of WH. When the deformation exceeds the peak strain, the stress decreases gradually to a steady-state stress with increasing strain as softening caused by DRV and DRX overtaking WH. As the temperature increases or the strain rate decreases, the peak stress and the steady-state stress decrease, and the stress rises to the peak stress at smaller strain. Furthermore, it is more and more difficult to observe the steady-state stress with the increasing of strain rate from the stress–strain curves. The steady-state stress can be reached at larger strain. Compared with the stress–strain curves of C–Mn steel [3, 20–22], the features of the stress–strain curves for Nimonic 80A are more noticeable. It indicates that DRX is more difficult to complete. In addition, it is found from Figure 2(d) that the shape of the stress–strain curve at the temperature of 1,000°C and the strain rate of 5 s^–1 is different from the others at the strain rate of 5 s^–1, although the experiment under the deformation condition had been repeated twice. The deformation mechanism at the temperature of 1,000°C and the strain rate of 5 s^–1 is different from the others. Deformation twinning is a significant deformation mechanism under the deformation condition [23, 24]. So the flow stress curve data at the temperature of 1,000°C and the strain rate of 5 s^–1 were not used to develop the constitutive model for flow stress of Nimonic 80A. The values of peak stress, peak strain, initial stress, saturation stress and steady-state stress for the flow stress curves at different temperatures and strain rates were given in Table 2.

Figure 2:

Stress–strain curves at various temperatures with strain rates of (a) 0.01 s^–1, (b) 0.1 s^–1, (c) 1 s^–1, (d) 5 s^–1.

Table 2:

Values of parameters of the flow stress curves under different deformation conditions.

Temperature (°C)	Strain rate (s–1)	σb (MPa)	εp	σ0 (MPa)	σs (MPa)	σss (MPa)
1,000	0.01	142	0.135	101	146	110
	0.1	218	0.169	-	220	174
	1	315	0.200	198	293	-
1,050	0.01	106	0.125	76	107	87
	0.1	167	0.147	117	164	134
	1	246	0.180	-	250	214
	5	325	0.206	-	327	-
1,100	0.01	91	0.117	69	92	69
	0.1	135	0.140	93	136	105
	1	200	0.167	136	204	167
	5	265	0.188	172	270	-
1,150	0.01	66	0.100	52	73	51
	0.1	110	0.12	75	112	83
	1	-	0.152	109	-	135
	5	212	0.172	-	220	185
1,200	0.01	60	0.095	49	63	43.6
	0.1	95	0.117	68	94	67
	1	140	0.136	98	142	111
	5	184	0.154	127	188	152

Parameter fitting processes for the constitutive equations

Activation energy and Zener–Hollomon parameter

In order to calculate DRX activation energy, α in eq. (4) needs to be determined. α is the optimized factor. In order to determine the value of α accurately, the value of α was estimated from the minimum residual sum of squares in this paper.

For the peak stress σ_p, eq. (4) can be written as the hyperbolic sine law and taking natural logarithms on both sides of the equation:

(6)lnsinhασp=1nlnε˙+QnRT−lnAn

Equation (6) can be regarded as a multiple regression equation, when the observed values of ln[sinh(ασ_p)] constitute the values of the dependent variable, the lnε˙ and 1/T are the settings of the independent variables, 1/n and Q/nR are the regression coefficients, and ln A/n is constant. In order to determine the value of α, the error sum of squares (SSE) between the measured values of ln[sinh(ασ_p)] and the calculated ones was calculated. The relationship between SSE and α is illustrated in Figure 3. It is not difficult to obtain the value of α as 2.23 × 10^–3MPa^–1, when SSE is the minimum.

Figure 3:

Relationship between SSE and α.

In order to determine the value of n, the relationships between lnε˙ and ln[sinh(ασ_p)] can be obtained for all the deformation temperatures from eq. (6), as shown in Figure 4. The value of n is determined by the linear slope, and the average value of n is 5.366.

Figure 4:

Relationships between lnε˙ and ln[sinh(ασ_p)].

In order to determine the value of Q, for a particular strain rate condition, differentiating eq. (6) and the following equation can be derived:

(7)Q=Rndlnsinhασpd1/T

According to the linear relationships between ln[sinh(ασ_p)] and 1/T at different strain rates shown in Figure 5, the average value of the slope is 8,232.510. Then, the average value of Q is determined as 367.3 kJ/mol.

Figure 5:

Relationships between ln[sinh(ασ_p)] and 1/T.

Substituting Q into eq. (5), Z parameter can be obtained as

(8)Z=ε˙exp3.673×103RT

Determination of σ₀, σ_s, σ_ss and Ω

For the initial stress σ₀ (low stress level), the following equation can be obtained by taking the natural logarithm of both sides of eq. (4):

(9)lnA+nlnσ0=lnZ

For the saturation stress σ_s and the steady-state stress σ_ss (all stress levels), the following equations can be obtained by taking the natural logarithm of both sides of eq. (4):

(10)lnA+nlnsinhασs=lnZ

(11)lnA+nlnsinhασss=lnZ

The initial stress, the saturation stress and the steady-state stress have been expressed through a linear regression as shown in Figure 6. Therefore, the three stresses of Nimonic 80A can be obtained as follows:

(12)σ0=0.896Z0.156

(13)σs=448.431 sinh−11.322×10−3Z0.183

(14)σss=448.431 sinh−15.153×10−4Z0.206

The DRV parameter Ω is related to temperature and strain rate. Ω was determined by substituting the stress and strain values (low stress level) obtained from the flow curves of Nimonic 80A into eq. (1). The dependence of Ω on temperature and strain rate has been described through a multiple linear regression. Therefore, the DRV parameter can be expressed as follows:

(15)Ω=136.677ε˙−0.067exp−2.744×103/T

Figure 6:

Regressed (a) initial stress, (b) saturation stress and (c) steady-state stress under various conditions.

Constitutive equations of DRX

DRX is characterized by some parameters, such as the critical strain ε_c, the peak strain ε_p, the DRX volume fraction X_dyn and the DRX grain size D_dyn.

The critical strain for the onset of DRX is calculated using the following equation [25]:

(16)εc=0.83εp

It is widely accepted that the peak strain during deformation process is strongly related to temperature and strain rate [26–28]. The peak strain can be described as follows:

(17)εp=A1Zk

where A₁ and k are the material constants. Through regression analysis, the dependence of the peak strain on Z parameter is fitted from the relationship between ln ε_p and ln Z in Figure 7. So the expression of the peak strain can be written as

(18)εp=1.227×10−2Z0.081

It is difficult to determine the DRX volume fraction by microstructural observations accurately, because the method is particularly useful for unstable microstructure where metallographic measurements are difficult to carry out. Generally, the flow stress curves can provide some information about that [28]. Based on eq. (2), DRX volume fraction can be derived as follows:

(19)Xdyn=σwh−σσs−σss, ε > εc

where σ_s and σ_ss are obtained from the flow stress curves, as shown in Table 2. σ_wh can be calculated from eq. (1).

Figure 7:

Relationship between ln ε_p and ln Z.

According to eq. (3), the value of k_d and n_d is determined by the linear intercept and slope from linear relationship of ln[–ln(1 – X_dyn)] and ln[(ε– ε_c)/ε_p] as shown in Figure 8. Thus, the DRX volume fraction can be expressed by the following equation:

(20)Xdyn=1−exp−0.108ε−εcεp2.191

Some authors [2–5, 10, 17, 18] also describe the DRX volume fraction of metal during hot deformation process with a form similar to eq. (3). In general, the value of n_d is in the vicinity of 2. However, compared with the reported [3, 5, 10, 17, 18], it can be found that DRX is more difficult to complete with the increase of the value of k_d.

Figure 8:

Relationship between ln[–ln(1 – X_dyn)] and ln[(ε– ε_c)/ε_p].

Recrystallized microstructures are also one of important DRX behaviors, which have an important influence on the final mechanical properties of products. Figure 9 shows the optical microstructures on the section of specimens deformed to a strain of 0.8 under different deformation conditions. The deformation microstructures depend closely on the deformation temperature and the strain rate. The fine and strain-free grains have completely replaced the original grains at high temperature and low strain rate as shown in Figure 9(a), which implies that fully DRX has taken place in this specimen [28, 29]. With the increase of the strain rate and the decrease of the temperature, the DRX volume fraction decreases, the grains were obviously fined as shown in Figure 9(b) and (c). The original grains have been flattened and elongated perpendicular to the deformation direction. Additionally, some fine and strain-free grains were observed between the grain boundaries of the original grains and form the necklace structure, which indicates that DRX occurs partially in the specimens [29–31]. When at 1,000°C and 1 s^–1, DRX also occurred partially, although the flattened and elongated original grains were the dominant feature and the fine recrystallized grains are too small to observe in Figure 9(d). Furthermore, the average DRX grain size of each deformed specimen was measured by quantitative metallographic methods.

Figure 9:

Microstructures of Nimonic 80A under different conditions: (a) 0.01 s^–1, 1,200°C, (b) 5 s^–1, 1,150°C, (c) 0.01 s^–1, 1,000°C, (d) 1 s^–1, 1,000°C.

The relationship between the deformation conditions and the DRX grain size D_dyn was obtained through multiple linear regressions. The expression of D_dyn can be expressed as

(21)Ddyn=6.333×107ε˙−0.228exp−2.117×104/T

Model verification

According to the determined parameters (Z and σ₀, σ_s, σ_ss, Ω, X_dyn), the constitutive model for the flow stress of Nimonic 80A can be obtained. Comparisons between the experimental flow stress values of Nimonic 80A and the predicted ones were carried out to evaluate the accuracy of the developed constitutive model, as shown in Figure 10. It can be seen that the flow stress values predicted by the constitutive model are in good agreement with the experimental ones, except for the flow stress at the temperature of 1,000°C and the strain rate of 5 s^–1. It can be believed that the developed constitutive model is not fit for the description of the flow stress behavior at the temperature of 1,000°C and the strain rate of 5 s^–1. The accuracy of the predicted values is further verified via employing standard statistical parameter, correlation coefficient (R). R is expressed as follows:

(22)R=∑i=1N(Ei−E¯)(Pi−P¯)∑i=1N(Ei−E¯)2∑i=1n(Pi−P¯)2

where E_i and P_i are the experimental value and the predicted value, respectively, Eˉ and Pˉ are the average value of E_i and P_i, respectively, N is the number of data employed in the verification.

Figure 10:

Comparisons between experimental flow stress of Nimonic 80A and predicted ones at various temperatures with strain rates of (a) 0.01 s^–1, (b) 0.1 s^–1, (c) 1 s^–1, (d) 5 s^–1.

The experimental flow stress values and the predicted ones at the strains of 0.1, 0.3, 0.5 and 0.7 were collected. Using eq. (21), R is determined as 0.998. The plot of experimental flow stress values and predicted values is shown in Figure 11. Similarly, the plot of experimental grain size and predicted one is shown in Figure 12, R is determined as 0.987. The results indicate that the proposed constitutive model could give an accurate prediction for the flow stress of Nimonic 80A.

Figure 11:

Correlation between experimental flow stress values of Nimonic 80A and predicted ones from the developed constitutive model.

Figure 12:

Correlation between experimental grain size of Nimonic 80A and the predicted one from the developed constitutive model.

Conclusions

1. The hot deformation characteristics of Nimonic 80A alloy were investigated by isothermal compression tests in the temperature range of 1,000–1,200°C and the strain rate range of 0.01–5 s^–1. The stress–strain curves show that as the temperature increases or the strain rate decreases, the peak stress and the steady-state stress decrease and the stress rises to the peak stress at smaller strain.

2. By regression analysis of the experimental data, the DRX activation energy for Nimonic 80A was determined to be 367.3 kJ/mol. Then, the constitutive models for DRX behavior and flow stress of Nimonic 80A incorporating the effects of temperature, strain rate and strain were developed to model the DRX and the flow stress of Nimonic 80A.

3. Comparisons between experimental flow stress values of Nimonic 80A and predicted ones were carried out. The accuracy of predicted values is further verified via employing standard statistical parameter. It can be concluded that the developed constitutive model could give an accurate estimate for the flow stress of Nimonic 80A.