Strain dependent rate equation to predict elevated temperature flow behavior of modified 9Cr-1Mo (P91) steel

Materials Science and Engineering A 528 (2011) 1071–1077

Contents lists available at ScienceDirect

Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea

Strain dependent rate equation to predict elevated temperature flow behavior of modified 9Cr-1Mo (P91) steel Dipti Samantaray, C. Phaniraj ∗ , Sumantra Mandal, A.K. Bhaduri Materials Technology Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, Tamil Nadu, India

a r t i c l e

i n f o

Article history: Received 6 August 2010 Received in revised form 21 September 2010 Accepted 13 October 2010

Keywords: Ferritic steel Constitutive analysis Flow stress Shear modulus Diffusivity

a b s t r a c t True stress–strain data from isothermal hot compression tests on modified 9Cr-1Mo steel over a wide range of temperature (1173–1373 K) and strain rate (0.001–100 s−1 ) were employed for constitutive analysis following the sine-hyperbolic Arrhenius equation. The correction for shear modulus and diffusivity in the constitutive equation showed a clear deviation from power-law at higher stresses and this was accounted for by considering the contribution from pipe diffusion. The stress dependence was found ˙ L = constant [sinh(˛L /G)]nh , where DL is lattice diffusivity, G is to obey rate equation of the form ε/D shear modulus and, ˛L and nh are constants. After incorporating the influence of strain on material constants, the developed constitutive equation could predict flow stress in the strain rate range 0.1–100 s−1 at all temperatures with very good correlation and generalization. Though deviation in prediction was observed at lower strain rates 0.001 and 0.01 s−1 , a higher correlation coefficient (R = 0.99) and a lower average absolute relative error (7.3%) for the entire investigated hot working domain revealed that the prediction of flow stress was satisfactory. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The advent of new generation power plants with an increased thermal efficiency has led to the development of modified versions of 9Cr-1Mo ferritic steel such as P91 and P92 steels [1–3]. Modified 9Cr-1Mo (P91) steel is an important material of choice for steam generator applications because of excellent creep properties, fracture toughness and corrosion resistance. The material flow behavior during hot deformation is complex and, hardening and softening mechanisms are affected by temperature as well as strain rate [4,5]. The knowledge of the processing window in a strain rate–temperature space for stable and unstable deformation is useful for safe workability of materials [6]. The fabrication of final component involves processing of the steel through various thermo-mechanical treatments and is therefore necessary to understand the hot deformation behavior through constitutive modeling [7–18]. Further, the simulation of metal forming processes using finite element analysis relies on the accurate knowledge of hot deformation behavior as influenced by the process parameters [12]. In the literature [9–18], most widely accepted constitutive equation in the hot working domain is the sine-hyperbolic Arrhenius type expression that relates flow stress ˙ and temperature (T). This was first pointed out (), strain rate (ε)

∗ Corresponding author. Tel.: +91 44 27480118; fax: +91 44 27480075. E-mail address: [email protected] (C. Phaniraj). 0921-5093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2010.10.036

by Sellars and Tegart [9] and Jonas et al. [10] for hot deformation at high strain rates. Since then, sine-hyperbolic description of Zener–Hollomon parameter (Z), i.e. temperature compensated strain rate, has been employed for the prediction of flow stress in the hot working range. These are given below. Z = ε˙ exp

Q  RT

(1)

 Q

ε˙ = A[sinh(˛)]n exp −

RT

(2)

where Q is the apparent activation energy for hot deformation, A, n and ˛ are constants. The sine-hyperbolic function is suitable in the entire stress range, as it reduces to power-law at low stresses (valid for ˛ < 0.8) and exponential at higher stress limits (applicable for ˛ > 1.2). The constitutive analysis in hot working for austenitic stainless steels, carbon and alloy steels, ferritic steel and Al alloys has been comprehensively reviewed by McQueen and Ryan [11]. The influence of strain in the above constitutive equation has been incorporated by many researchers [8,13–18] for prediction of flow behavior and this is done by determining the constants, and expressing them as a function of strain (i.e. strain compensation). Further, a revised constitutive equation by suitably modifying the Zener–Hollomon parameter (i.e. strain rate compensation) for prediction of flow stress has been adopted by Lin et al. [16] and Mandal et al. [17]. In the literature, very limited efforts have been directed to understand and predict the elevated temperature flow behavior of modified 9Cr-1Mo steel. In our recent work [15], it was shown

1072

D. Samantaray et al. / Materials Science and Engineering A 528 (2011) 1071–1077

that Eq. (2) with compensation for strain could precisely predict the flow curves of P91 steel. In the present study, the analysis is extended to consider the correction for shear modulus and diffusivity. The distinct deviation from power-law behavior observed at higher stresses is examined by taking in to account the contribution of dislocation pipe diffusion to overall diffusivity. The objective of the present study is to develop a suitable rate equation that can be used to evaluate and predict the influence of process parameters on high temperature flow behavior of modified 9Cr-1Mo ferritic steel. Towards this, isothermal hot compression tests were conducted over a wide range of strain rate and temperature. True stress–true plastic strain experimental data obtained were analyzed and the validity of the developed constitutive equation was examined over the investigated strain rate and temperature range. 2. Experimental details Modified 9Cr-1Mo steel used in the present study was in the normalized (1323 K for 1 h) and tempered (1033 K for 3 h) condition. The chemical composition of the steel (in wt.%) is as follows: 0.102C, 0.374Mn, 8.946Cr, 0.914Mo, 0.0227Si, 0.064Cu, 0.107Ni, 0.182V, 0.03Al, 0.075Nb, 0.003Ti, 0.0594N, and balance Fe. The compression test specimens of cylindrical geometry with 10 mm diameter and 15 mm height were machined from the as-received plate. The specimens were prepared with a very good surface finish of the order of  (N6 grade) so as to exclude any surface effects on the flow behavior. A small hole of 0.8 mm diameter and 5 mm depth was drilled at mid height of the specimen to insert a thermocouple, which was used to measure the actual temperature of the specimen. Isothermal hot compression tests were conducted using a computer controlled servo-hydraulic testing machine (DARTEC, Stourbridge, UK) with a maximum load capacity of 100 kN. The machine is equipped with a control system to impose exponential decay of the actuator speed to obtain constant true strain rates. A resistance heating split furnace with SiC heating elements was used to surround the platens and specimen. The specimens were coated with a borosilicate glass paste that acted as a lubricant as well as protective coating. The tests were conducted at temperatures ranging from 1173–1373 K at intervals of 50 K and at constant true strain rates of 0.001, 0.01, 0.1, 1, 10, and 100 s−1 . A Nicolet transient recorder was used to record the adiabatic temperature rise during hot deformation. Standard equations were used to convert the loadstroke data to true stress–true strain data. The elastic region was subtracted from the true stress–strain curve to get true stress–true plastic strain in the range 0.1–0.5. Flow stress data obtained at different processing conditions were corrected for adiabatic temperature rise, if any, by linear interpolation (point to point) between ln  and 1/T, where T is the absolute test temperature. It may be noted that the appearance of the specimens before and after testing showed no significant barreling on testing [15]. 3. Results and discussion In our previous study on P91 steel [15], for a given strain, the average value of material constants ˛ and n were evaluated following Eq. (2) from the plots of ln [sinh(˛)] vs. ln ε˙ by iterating ˛ values that brings (˛) in to the correct range yielding parallel linear lines for different temperatures. Knowing ˛ and n, from the plots of ln ˙ accordingly Q and ln A were deter[sinh(˛)] against 1/T and ln ε, mined. The prediction of flow stress was performed by considering the influence of strain on material constants. The apparent activation energy Q was found to vary in the range 369–391 kJ/mol [15] which is higher than that of lattice diffusion (QL = 270 kJ/mol) for ␥Fe [19]. These results may also be interpreted as Q = 380 ± 11 kJ/mol. The higher apparent activation energy is generally rationalized in

terms of variation of modulus with temperature and resisting (or internal) stress associated with dislocation structure [2,20]. Furthermore, a single Q value determined (at a particular strain) for the entire stress range [15] that follows power-law at lower stresses, while exponential dependence at higher stresses could possibly imply that the deformation may be governed by a single mechanism for both the stress regimes. This seems reasonable, since modified 9Cr-1Mo steel being higher stacking fault energy material, dynamic recovery may be sufficiently pronounced and it could be presumed that dynamic recrystallization may not be a predominant restoration mechanism. This was also supported by the observations that the true stress–strain behavior of P91 steel [15] did not reveal distinct flow softening after reaching a peak or undulations. These details along with the various factors that affect hot deformation behavior have been discussed by Kassner and co-workers [21–24]. The analysis of results in terms of diffusivity and shear modulus is given in the following section. 3.1. Constitutive equations for flow stress prediction: diffusivity and modulus It may be noted that the flow stress values are after correction for deformation heating and no significant barreling was observed as the specimens were compressed till true strain of 0.5. Flow stresses obtained at a given strain for different strain rates (0.001–100 s−1 ) and temperatures (1173–1373 K) were analyzed in terms of normalized shear modulus and lattice diffusion coefficient, to determine the material constants. For the data analyzed in the testing temperature range 1173–1373 K, modified 9Cr-1Mo steel would be austenitic and the data for lattice diffusivity, Burger’s vector and shear modulus were taken for ␥-Fe. Lattice diffusivity DL (in m2 /s) for iron in austenite at a given temperature was obtained [25] from DL = 6.8 × 10−6 exp(−17TM /T), where TM = 1803 K is the melting point for modified 9Cr-1Mo steel. The shear modulus G at temperature T was calculated according to G = G0 [1 + {(T − 300)/TM } × (TM /G0 )(dG/dT)] using the data for ␥-Fe [19] with G0 = 8.1 × 104 MPa, and the term (TM /G0 )(dG/dT) = −0.91. ˙ /DL Gb) vs. /G is shown in Fig. 1(a) The bi-logarithmic plot of (εkT for a typical strain = 0.2, where k is Boltzman constant and the magnitude of Burger’s vector b = 2.58 × 10−10 m for ␥-Fe [19]. It can be seen from the figure that the power-law holds good at lower stresses and a gradual deviation from linearity is observed as stress increases. It also clearly depicts that the data obtained at various strain rates and temperatures fall on to a single curve. It is noticed ˙ /DL Gb) that the deviation from power-law occurs nearly at (εkT = 3 × 10−6 . Such plots were drawn for different strains (0.1–0.5) at intervals of 0.05 and revealed similar observations. Accordingly in the low stress range, the data can be fitted by the well known Dorn power-law equation [26–30] ε˙ =

AL DL Gb kT

  nL G

(3)

where nL is obtained as the slope of the straight line in Fig. 1(a) for those data points lower than the cut-off value of 3 × 10−6 for the ordinate and nL = 5.05 at strain = 0.2. The lattice diffusion is well accepted mechanism for plastic flow in the power-law regime. On the other hand, in the high stress exponential (power-law breakdown) region, it is still a subject of debate. The exact mechanism in the power-law breakdown regime is not yet well understood. Sherby and co-workers [25,30–34] have suggested that the powerlaw breakdown could be associated with two contributing factors: (a) increased contribution from dislocation pipe diffusion and (b) excess vacancy generation. On the contrary, according to other researchers, the exponential regime has been suggested [35–37] to be related to a transition from the diffusion-controlled dislocation motion to glide-controlled thermally activated dislocation

D. Samantaray et al. / Materials Science and Engineering A 528 (2011) 1071–1077

a

10 10 10

ε kT/DLGb

10

.

10 10 10 10 10 10

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

10

b

10 10

ε kT/DeffGb

10 10 10

.

10 10 10

ε = 0.2

1173 1223 1273 1323 1373

-3

σ/G

10

-2

-3

-4

-5

1173 K 1223 K 1273 K 1323 K 1373 K

ε = 0.2

-6

neff = 5.438

-7

-8

-9

-10

10

-3

10

σ/G

-2

˙ /DL Gb) vs. log (/G) and (b) log (εkT ˙ /Deff Gb) vs. log (/G) at Fig. 1. Plots of (a) log (εkT strain = 0.2 for various strain rates (0.001–100 s−1 ) and temperatures (1173–1373 K).

mechanism. Nix and Ischner [38] have put forward a model based on thermally activated glide in the subgrain interior and diffusioncontrolled recovery at the subgrain walls. The detailed discussion on the five power law creep and its break down at higher stresses is dealt with comprehensively by Kassner and Perez-Prado [23,24]. At higher stresses, the deviation from power-law could be accounted for by considering the contribution of dislocation pipe diffusion to the overall lattice diffusivity and the effective diffusion coefficient (Deff ) can be given as [25,34]



  2

Deff = DL fL + DP fP = DL + 7.03125

G



DP ,

(4)

where DP is the dislocation pipe diffusion coefficient and, f L and f P are the fraction of atoms participating in lattice and pipe diffusion, with the term f L essentially equal to unity and f P = 50 (/E)2 [25,34]. The value of DP for ␥-Fe is taken as DP = 10−4 exp(−11TM /T) m2 /s [25,34]. The data shown in Fig. 1(a) were re-plotted replacing DL with Deff and the results are shown in Fig. 1(b) typically for strain = 0.2. The plot shows that the data could be described by a straight line after incorporating the correction for pipe diffusivity at higher stresses and the slope gives the value of neff = 5.438 at strain = 0.2. The reason for the scatter in the plot is not quite clear

1073

to us and it may be due to marginal scatter in the experimental data, and also because that the precise value of Deff is not known (to our best of knowledge) for the modified 9Cr-1Mo (P91) steel. Such plots (i.e. Fig. 1(b)) were drawn for various strain levels (0.1–0.5) at intervals of 0.05 and similar observation was found to be true at all strains. These trends suggest that the data could be represented by taking in to account the effective diffusion coefficient. It may be mentioned that the well known Garofalo sine-hyperbolic [39] empirical expression for creep has been derived by Weertman [40], Barrett and Nix [41] by associating the high stress regime with the presence of a high vacancy concentration [23,24,30] and Barrett–Nix theory is based on non-conservative (climb) motion of jogs (edge) on screw dislocations. It is still unclear whether the mechanism of plasticity changes in the exponential higher stress regime [23,24]. Further, it has been mentioned that well defined subgrain boundaries that form from dislocation reaction (perhaps as a consequence of dynamic recovery) suggesting that substantial dislocation climb is at least occurring in power-law break down regime [23]. A comment on the contribution of pipe diffusion for the results presented in Fig. 1(b) is in order. The lattice diffusion dominates at higher temperatures and lower stresses, whereas pipe diffusion contribution becomes important at lower temperatures and higher stresses [19,23,24,32]. The contribution of lattice and pipe diffusion to Deff (Eq. (4)) depends on both temperature and dislocation density [23,24,32]. In their study on deformation behavior of Fe–C alloys at high temperatures (0.7–0.9 TM ) and strain rates (1–100 s−1 ), Leuser et al. [25] have reported that at the high value of /G = 2.66 × 10−3 , the contribution from dislocation pipe diffusion dominates at all temperatures below 1448 K, whereas, at the low value of /G = 1.6 × 10−3 , pipe diffusion is dominant at relatively lower temperatures (<1233 K). In the present work, the test temperatures range from 1173 to 1373 K (0.65–0.76 TM ) and /G values are in the range from 9.43 × 10−4 to 7.74 × 10−3 . Further, the deviation from power-law occurs nearly at /G = 3.8 × 10−3 (Fig. 1(a)) for the cut-off value of 3 × 10−6 for the ordinate. On a close examination of the data with those observations reported by Leuser et al. [25], it seems reasonable to assume that the deviation from power-law at higher stresses could be explained by considering the contribution of dislocation pipe diffusion to the effective diffusivity. In further plots and in arriving at the rate equation given below (Eq. (5)), we have preferred to use DL (not Deff ) since the scatter was comparatively higher for the plots with Deff . Based on the ˙ /DL Gb) vs. /G is discussions so far, our attempts to plot log (εkT shown typically for a strain of 0.2 in Fig. 2 and it is observed that the linearity is valid at higher stresses and is gradually lost as the ˙ /DL Gb) stress decreases. For the values above the cut-off, i.e. (εkT ≥ 3 × 10−6 , the data could be described by a straight line and from the slope, ˇL can be obtained as ˇL = (2.303 × slope) giving ˛L = ˇL /nL . For a strain of 0.2, with nL = 5.05 and ˇL = 1911.4, value of ˛L = 378.52. Such plots were drawn for different strains (0.1–0.5) at intervals of 0.05 and similar observations were found to be true at all strains. A linear behavior for the entire stress regime as described by sine-hyperbolic function is typically shown in Fig. 3 for strain = 0.2 and accordingly, it can be expressed as ε˙ =



Ah DL Gb sinh kT

 ˛  nh L G

(5)

˙ /DL Gb) vs. ln sinh[(˛L /G)] as shown in In the plot of ln(εkT Fig. 3, nh is the slope of the line and ln(Ah ) is the intercept; and for strain = 0.2, nh = 4.861 and ln(Ah ) = −16.005. Such plots were drawn for different strains (0.1–0.5) at an interval 0.05. It has also been verified that the data in Fig. 1(a) and Fig. 2 could be described by the sine-hyperbolic fit according to Eq. (5). The values of nL , neff and nh along with ˇL obtained at various strains are listed in Table 1 and the average values of stress exponents can be calculated. From the

1074

D. Samantaray et al. / Materials Science and Engineering A 528 (2011) 1071–1077

10 10

10

nh calculated

ε = 0.2

-2

rd

-3

6.0 2

nh = 5.83665 - 7.42893 ε + 14.87628 ε - 9.2667 ε

-4 -5

3

5.5 5.0

-6

ε

.

10

3 order polynomial fit

6.5

L

kT/D Gb

10

7.0

-1

nh

10

10 10 10

-7

4.5

1173 1223 1273 1323 1373

-8 -9

-10

4.0 0.1

0.004

0.006

0.008

0.3

0.4

0.5

True strain

10

0.002

0.2

Fig. 4. Variation of nh with true strain for the strain range (0.1–0.5) at intervals of 0.05 and the coefficients of the polynomial are given in the figure.

0.010

σ/G ˙ /DL Gb) vs. (/G) at strain = 0.2 for different strain rates Fig. 2. Plot of log (εkT (0.001–100 s−1 ) and temperatures (1173–1373 K). Table 1 Summary of nL , neff and nh along with ˇL values determined at various strains for strain rate range (0.001–100 s−1 ) and temperature range (1173–1373 K). True strain

nL

neff

nh

ˇL

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

5.457 5.236 5.050 5.058 4.975 5.034 5.055 5.090 5.137

5.769 5.598 5.438 5.399 5.289 5.287 5.259 5.239 5.270

5.238 5.021 4.861 4.781 4.689 4.670 4.652 4.650 4.689

1989.4 1942.9 1911.4 1844.8 1794.8 1778.6 1768.3 1804.7 1875.9

average values of nL = 5.12, neff = 5.39 and nh = 4.81, it is discerning to note that these stress exponents are nearly the same and are in close agreement with n = 5.22 observed for P91 steel [15]. It may be mentioned that the hot compression test data obtained for the

ε = 0.2

-2

entire strain rate–temperature domain fall on to a single line (Fig. 3) or on to a single master curve as seen in Fig. 1(a) and Fig. 2 independent of which of the laws describes the correlation between stress and strain rate; these results are in similar vein with those observed by Cadek [42] for creep behavior of Al for a wide range of stresses. Fig. 4 shows the variation of nh with strain and, the variation of ˛L and ln(Ah ) with strain is given in Fig. 5. With increasing strain, ˛L and ln(Ah ) showed opposite trends. From the average value of G = 41.36 × 103 MPa for the data at 1173–1373 K, it is seen that ˛L /G values range from 0.00845 to 0.00915 corresponding to ˛ values observed for P91 steel [15]. 3.2. Prediction of flow stress: compensation for strain It is assumed that the influence of strain on the elevated temperature flow behavior is insignificant and thereby is not considered in Eq. (5). However it is seen that ˛L , Ah show significant variation with strain (Fig. 5) and hence compensation for strain needs to be incorporated in to the constitutive equation while predicting flow stress. This is done by assuming that the material constants (˛L , Ah and nh ) vary with strain and for P91 steel, it is observed that they are described by the best fit 3rd order polynomial with very good correlation and generalization. The coefficients of the polynomial are given in the respective figures (Figs. 4 and 5) and the

-4

400

αL

ln (Ah) 3 order polynomial fit rd

.

380 -10 -12

-15.7

360

-14

1173 K 1223 K 1273 K 1323 K 1373 K

-16 -18 -20

-15.8 340

1 ln (sinh(αLσ/G))

2

-16.0 -16.1

300 0

-15.9

320

-22 -1

-15.5 -15.6

nh = 4.861

αL

ln (ε k T/DLG b)

-8

-15.4

ln (Ah)

-6

3

˙ /DL Gb) vs. ln [sinh(˛L /G)] at strain = 0.2 for various strain rates Fig. 3. Plot of ln (εkT (0.001–100 s−1 ) and temperatures (1173–1373 K).

0.1

0.2

0.3

0.4

0.5

-16.2

True strain Fig. 5. Variation of ˛L and ln Ah with true strain. The polynomials for ˛L and ln Ah are given as ˛L = 309.1614 + 864.0863ε − 3519.9957ε2 + 4037.7778ε3 and ln Ah = 14.1901 − 16.5769ε + 45.1268ε2 − 39.0168ε3 .

D. Samantaray et al. / Materials Science and Engineering A 528 (2011) 1071–1077

a

150

-1

Experimental predicted

Strain rate = 0.001 s

c

350

1173 K

100

1223 K

75

1273 K 50

1323 K 1373 K

25

Strain rate = 1 s

1173 K

250

1223 K

200

1273 K 1323 K

150

1373 K

100 50

0 0.1

0.2

0.3

0.4

0.5

0.1

0.6

0.2

True strain

b

-1

Experimental predicted

300

True stress (MPa)

True stress (MPa)

125

1075

300

-1

Experimental predicted

0.3

0.4

0.5

0.6

True strain Strain rate = 0.1 s

d

500

250

-1

Experimental predicted

450

Strain rate = 100 s

1223 K 150 1273 K 1323 K

100

1373 K 50

True stress (MPa)

True stress (MPa)

400 1173 K

200

1173 K

350

1223 K

300

1273 K 1323 K

250

1373 K 200 150 100

0 0.1

0.2

0.3

0.4

0.5

0.6

0.1

0.2

0.3

0.4

0.5

0.6

True strain

True strain

Fig. 6. Predicted flow curves and its comparison with those experimentally obtained at strain rates of (a) 0.001 s−1 , (b) 0.1 s−1 , (c) 1 s−1 and (d) 100 s−1 in the temperature range 1173–1373 K.

constants can be evaluated for a given strain. Knowing the evaluated material constants and on rearranging Eq. (5), the modulus compensated flow stress (/G) at each strain for a given strain rate and temperature can be predicted following



 1 sinh−1 = ˛L G

 εkT  n1  ˙ h Ah DL Gb

,

(6)

since DL at different temperatures as well as Boltzman constant (k) and Burgers vector (b) are known. Further, knowing G for the test temperature, flow stress can be determined. It may be noted that as mentioned before, in arriving at Eq. (6) (or Eq. (5)) for prediction of flow stress, we have preferred DL (instead of Deff ) due to higher scatter with Deff and the precise Deff value unknown for the present steel. Further, Eq. (6) with DL is simple to evaluate the flow stress, but on the other hand if Deff is used, then it can be seen that both sides of the equation would contain the /G term and solving it for predicting the flow stress becomes more complicated. 3.3. Verification of constitutive equation Typical predicted flow curves (following Eq. (6)) and their comparison with those experimentally obtained are shown in Fig. 6(a)–(d) for the data at lower (0.001 s−1 ), intermediate (0.1 and 1 s−1 ) and higher (100 s−1 ) strain rates in the temperature range 1173–1373 K. It was observed that the predicted flow stresses are

in good agreement with those experientially measured for strain rates 0.1–100 s−1 (Fig. 6(b)–(d)). Whereas it was not that satisfactory at lower strain rates 0.01 and 0.001 s−1 (0.001 s−1 in Fig. 6(a)) and the predicted flow stresses were lower than the measured values at 1173 and 1223 K. The reason for this is not quite clear to us. It has been reported [21] that the decrease in average Taylor factor causing textural softening could occur in torsion testing, but in contrast to this, during compression the average Taylor factor may increase with plastic strain causing some amount of hardening in the absence of recrystallization. If this is regarded as the possible explanation for the underestimation in flow stress prediction at lower strain rates (1173 and 1223 K at 0.001 and 0.01 s−1 ), then it may be argued that the increase in average Taylor factor should take place at other temperatures (1273–1373 K) and higher strain rates (0.1–100 s−1 ) as well. Possibly at higher temperatures, the thermal softening may underplay the effects due to increase in average Taylor factor with strain. It may be recalled (Section 3) that based on the nature of true stress–strain curves and higher stacking fault energy of P91 steel, it was presumed that dynamic recrystallization may not be a predominant restoration mechanism. However, detailed analysis of work hardening is necessary to establish this, since the hardening rate would show a sudden decrease at an increasing rate at the onset of dynamic recrystallization [21,22]. Further work on these aspects relating to changes in average Taylor factor with strain and dynamic recrystallization is needed for reaching a firm conclusion.

1076

D. Samantaray et al. / Materials Science and Engineering A 528 (2011) 1071–1077

Predicted flow stress (MPa)

400

[sinh(˛L /G)]nh and the material constants were accordingly obtained at various strains. The constants (˛L , Ah and nh ) were found to be strain dependent and they were described by a 3rd order polynomial with very good correlation and generalization. The constitutive equation (incorporating strain compensation) was found to predict the flow stress very well in the strain rate range (0.1–100 s−1 ) at all temperatures, whereas a notable deviation in prediction was observed at lower strain rates viz. 0.001 and 0.01 s−1 . However, statistical analysis of the predicted flow stress data with a higher correlation coefficient (R = 0.99) and a lower average absolute relative error (7.3%) for the entire investigated hot working domain revealed that the prediction of flow stress was satisfactory.

R = 0.99

350 300 250 200 150 100 Flow stress data Best linear fit

50 0 0

50

100

150

200

250

300

350

Acknowledgement

400

Experimental flow stress (MPa) Fig. 7. Correlation between the predicted and experimentally obtained flow stresses for the strain range 0.1–0.5 (at intervals of 0.05) over the entire strain rate (0.001–100 s−1 ) and temperature range (1173–1373 K).

The authors sincerely thank Mr. S. Sasidhara, Department of Materials Engineering, Indian Institute of Science, Bangalore for his valuable help during compression testing. References

The standard statistical parameters such as correlation coefficient (R) and average absolute relative error () were also determined for the data at various strains (0.1–0.5) at intervals of 0.05 for the entire strain rate–temperature regime according to expressions given below R=

i=N i (exp − ¯ exp )(pi − ¯ p ) i=1 i=N i 2 i=N i (exp − ¯ exp ) i=1



(7) 2



i i 1  exp − p  × 100,  i N exp  i=N

=

(p − ¯ p ) i=1

(8)

i=1

where  exp is the experimental flow stress,  p is the predicted flow stress, ¯ exp and ¯ p are the mean values of  exp and  p , respectively, and N is the total number of data points. The correlation coefficient provides information on the strength of linear relationship between the observed and the computed values, whereas  is computed through a term by term comparison of the relative error. It can be seen from Fig. 7 that a good correlation (R = 0.99) is obtained between the experimental and the predicted data. The average absolute relative error was found to be 7.3% for the entire range of strain, strain rate and temperature. These statistical analyses emphasize the good predictability of the developed constitutive equation for strain rates (0.001–100 s−1 ) in the temperature range (1173–1373 K) for modified 9Cr-1Mo steel. Further, it signifies that the formulated constitutive equation could be employed to analyze and optimize the thermo-mechanical processes. 4. Conclusions The flow stress data obtained from isothermal hot compression tests on modified 9Cr-1Mo (P91) steel over a wide range of temperature (1173–1373 K), stain rate (0.001–100 s−1 ) and strain (0.1–0.5) were examined in terms of diffusivity and modulus correction, and constitutive analysis was performed. The conclusions from this study are given below. The plots of strain rate normalized by diffusivity against flow stress normalized with shear modulus showed distinct deviation from power-law behavior at higher stresses and the deviation could be accounted for by considering the contribution of dislocation pipe diffusion to the effective diffusivity. The stress and temperature dependence for P91 steel could ˙ L = constant be represented by the rate equation of the form ε/D

[1] P.J. Ennis, A. Zielinska-Lipiec, O. Wachter, A. Czyrska-Filemonowicz, Acta Mater. 45 (1997) 4901–4907. [2] B.K. Choudhary, C. Phaniraj, K. Bhanu Sankara Rao, S.L. Mannan, ISIJ Int. Suppl. 41 (2001) S73–S80. [3] K. Kimura, Y. Toda, H. Kushima, K. Sawada, Int. J. Pres. Ves. Piping 87 (2010) 282–288. [4] W.J. Dan, W.G. Zhang, S.H. Li, Z.Q. Lin, Comput. Mater. Sci. 40 (2007) 101–107. [5] C. Sommitsch, R. Sievert, T. Wlanis, B. Gu¨nther, V. Wieser, Comput. Mater. Sci. 39 (2007) 55–64. [6] S.V.S. Narayana Murthy, B. Nageswara Rao, B.P. Kashyap, Int. Mater. Rev. 45 (2000) 15–26. [7] S. Mandal, P.V. Sivaprasad, S. Venugopal, K.P.N. Murthy, B. Raj, Mater. Sci. Eng. A 485 (2008) 571–580. [8] Y.C. Lin, G. Liu, Comput. Mater. Sci. 48 (2010) 54–58. [9] C.M. Sellars, W.J.McG. Tegart, Acta Metall. 14 (1966) 1136–1138. [10] J. Jonas, C.M. Sellars, W.J.McG. Tegart, Metall. Rev. 14 (1969) 1–24. [11] H.J. McQueen, N.D. Ryan, Mater. Sci. Eng. A 322 (2002) 43–63. [12] K.P. Rao, Y.K.D.V. Prasad, E.B. Hawbolt, J. Mater. Process. Technol. 56 (1996) 908–917. [13] F.A. Slooff, J. Zhou, J. Duszczyk, L. Katgerman, Scr. Mater. 57 (2007) 759–762. [14] Y.C. Lin, M.-S. Chen, J. Zhang, Mater. Sci. Eng. A 499 (2009) 88–92. [15] D. Samataray, S. Mandal, A.K. Bhaduri, Mater. Des. 31 (2010) 981–984. [16] Y.C. Lin, M.-S. Chen, J. Zhong, Comput. Mater. Sci. 42 (2008) 470–477. [17] S. Mandal, V. Rakesh, P.V. Sivaprasad, S. Venugopal, K.V. Kasiviswanathan, Mater. Sci. Eng. A 500 (2009) 114–121. [18] J. Castellanos, I. Rieiro, M. Carsi, O.A. Ruano, J. Mater. Sci. (2010), doi:10.1007/s10853-1010-4610-5. [19] H.J. Frost, M.F. Ashby, Deformation Mechanism Maps: The Plasticity and Creep of Metals and Ceramics, Pergamon Press, Oxford, UK, 1992. [20] D.V.V. Satyanarayana, G. Malakondaiah, D.S. Sarma, Mater. Sci. Eng. A 323 (2002) 119–128. [21] M.E. Kassner, J. Pollard, E. Evangelista, E. Cerri, Acta Metall. Mater. 42 (1994) 3223–3230. [22] M.E. Kassner, S.R. Barrabes, Mater. Sci. Eng. A 410–411 (2005) 152–155. [23] M.E. Kassner, M.-T. Perez-Prado, Prog. Mater. Sci. 45 (2000) 1–102. [24] M.E. Kassner, M.-T. Perez-Prado, Fundamentals of Creep in Metals and Alloys, 2nd ed., Elsevier, Amsterdam, 2008. [25] D.R. Leuser, C.K. Syn, J.D. Whittenberger, M. Carsi, O.A. Ruano, O.D. Sherby, Mater. Sci. Eng. A 317 (2001) 101–107. [26] A.K. Mukherjee, J.E. Bird, J.E. Dorn, Trans. Am. Soc. Metals 62 (1969) 155–179. [27] J.E. Bird, A.K. Mukherjee, J.E. Dorn, in: D.G. Brandon, A. Rosen (Eds.), Quantitative Relation Between Properties and Microstructures, Israel University Press, Jerusalem, 1969, pp. 255–342. [28] A.K. Mukherjee, Mater. Sci. Eng. A 322 (2002) 1–22. [29] C. Phaniraj, Creep Behaviour of AISI 304 Stainless Steel—Analysis and Implications of First Order Kinetics, Ph.D. Thesis, Indian Institute of Technology, Bombay, 1997. [30] O.D. Sherby, P.M. Burke, Prog. Mater. Sci. 13 (1967) 325–390. [31] O.D. Sherby, Young, in: J.C.M. Li, A.K. Mukherjee (Eds.), Rate Processes in Plastic Deformation of Materials, American Society for Materials, Materials Park, OH, 1975, p. 497. [32] S.L. Robinson, O.D. Sherby, Acta Metall. 17 (1969) 109–125. [33] H. Luthy, A.K. Miller, O.D. Sherby, Acta Metall. 28 (1980) 169–178. [34] O.A. Ruano, O.D. Sherby, Mater. Sci. Eng. 64 (1984) 61–66. [35] A. Arieli, A.K. Mukerjee, in: B. Wilshire (Ed.), Proc. Int. Conf. Creep, Swansea, UK, 1981, pp. 97–111.

D. Samantaray et al. / Materials Science and Engineering A 528 (2011) 1071–1077 [36] S.V. Raj, T.G. Langdon, Acta Metall. Mater. 39 (1991) 1823– 1832. [37] S.V. Raj, A.D. Freed, Mater. Sci. Eng. A 283 (2000) 196–202. [38] W.D. Nix, B. Ilschner, in: P. Haasen, et al. (Eds.), Strength of Metals and Alloys, vol. 3, Pergamon Press, London, 1980, p. 1503.

1077

[39] F. Garofalo, Fundamentals of Creep and Creep-Rupture in Metals, McMillan, New York, 1965, pp. 50–54. [40] J. Weertman, J. Appl. Phys. 28 (1957) 362–364. [41] C.R. Barrett, W.D. Nix, Acta Metall. 13 (1965) 1247–1258. [42] J. Cadek, Creep in Metallic Materials, Elsevier, Amsterdam, 1988, pp. 52–64.