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Continuum Damage Mechanics Approach to Predict Creep Behaviour of Modified 9Cr-1Mo Ferritic Steel at 873 K
Available online at www.sciencedirect.com
Procedia Engineering 55 (2013) 798 – 804
6th International Conference on n Creep, Fatigue and Creep-Fatigue Interaction [CF-6]
Continuum Damage Mechannics Approach to Predict Creep Beehaviour of Modified 9C Cr-1Mo Ferritic Steel at 873 K J. Christophera,b, G. Sainathb, V V.S. Srinivasanb, E. Isaac Samuelb, B.K. Choudharyb∗, M.D D. Mathewb, T. Jayakumarb b
a Safety Research Institute, Atom mic Energy Regulatory Board, Kalpakkm-603102, Tamil Nadu, India Metallurgy and Materials Group, Indira Gandhi Centre for Atomic Research, Kalpakkam-603102, Tamil Nadu, India
Abstract The paper deals with the prediction of creep behavioour of modified 9Cr-1Mo ferritic steel at 873 K using continuum damage mechanics (CDM) approach. Creep data as well as the quantitattive microstructural information available on heat treated and crreep tested conditions in literature have been used for the analysis. The maaterial parameters were then optimised by matching the theoretiically generated creep strain-time curves with experimental curves at 110 and 140 MPa corresponding to low and high stress regimess, respectively. Using optimised material parameters, creep behaviour in teerms of creep strain-time data, minimum creep rate and time tto rupture at different stress levels have been predicted. In general, good agrreement between predicted and experimental data has been obtaained. ©©2013 Authors. Published by Elsevier Ltd. Ltd. Selection and/or peer-review under responsibility of the Indira Gandhi 2013The The Authors. Published by Elsevier Selection andAtomic peer-review under responsibility of the Indira Gandhi Centre for Atomic Research. Centre for Research Keywords: Creep behaviour; modified 9Cr-1Mo steel; continnuum damage mechanics
1. Introduction Modified 9Cr-1Mo steel is an important highh temperature material for steam generator (SG) applications in thermal and nuclear power industries. The modified version is obtained by the addition of strong carbide/nitride form ming elements such Nb and V and nitrogen in plain 9Cr-1Mo steeel and is also known as 9Cr-1Mo-V-Nb steel designatedd as P91 or T91 by ASTM standards. The steel derives its creep streength mainly from fine distribution of stable MX precipittates. The choice of 9Cr-1Mo steel for all the SG components in sodium cooled fast reactors (SFRs) is based on low thermal expansion coefficient and high resistance to stress corrosiion cracking in water-steam systems compared to austennitic stainless steels and superior elevated temperature mechanical properties than the alternate 2.25Cr-1Mo steel. The 10 1 5 h creep rupture strength of modified 9Cr-1Mo steel has been foound to be higher than 2.25Cr-1Mo and 9Cr-1Mo steels, and remains higher or equal to that of type 304 SS up to 898 K [1]. Further, the long term creep properties of modified 9Cr--1Mo steel has been observed to be superior than the conventional ssteels with Cr content up to 9%Cr, i.e. plain 9Cr-1Mo stteel and 12Cr-1Mo1W-0.3V steel [2,3]. Modified 9Cr-1Mo steel exxhibits distinct high and low stress regimes in terms of sttress dependence of minimum creep rate and rupture life. In low strress regime, the steel displays degradation in long-term creep c properties due to decrease in dislocation density, dislocation substructure coarsening and coarsening of M23C6 carbid de with increase in Mo due to nucleation creep exposure. Apart from these, decrease in solid solution strengthening caused by the depletion of M w increase in creep exposure add in decreasing long-term creep strength of and growth of secondary Laves phase Fe2Mo with the steel. In addition to above, degradation in lonng-term creep properties has been ascribed to heterogeneo ous and preferential
∗
Corresponding author: E-mail address: [email protected]
1877-7058 © 2013 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of the Indira Gandhi Centre for Atomic Research. doi:10.1016/j.proeng.2013.03.334
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recovery in the vicinity of prior austenite grain boundaries (PAGBs) and formation and growth of a complex nitride Cr(V,Nb)N, the Z-phase at expense of fine V(C,N) and Nb(C,N) precipitate has been reported for the steel [4,5]. Z-phase coarsens fast compared to V(C,N) and Nb(C,N) precipitates and directly contributes to the reduction in creep rupture strength by removal of key precipitation strengthening elements such as Cr, V, Nb and N from the matrix. In the present investigation, physically based Continuum Damage Mechanics (CDM) approach has been used to predict creep behaviour of modified 9Cr-1Mo steel. In this microstructural based CDM approach, damage caused by coarsening of dislocation networks and subgrains together with coarsening of precipitates has been considered. The above damage evolution rates are coupled with kinetic creep law containing microstructural parameters in order to account for total damage and for prediction of creep curves at 873 K for different stress levels. The results obtained from CDM analysis in terms of creep strain-time data, minimum creep rate and time to rupture at different stress levels have been compared with the experimentally observed values. 2. CDM Approach Dyson and Osgerby [6] proposed a new kinetic creep law containing microstructural parameters for particle-strengthened alloys as
ε = ε0 (1 + Dd ) sinh( and
σ (1 − H ) ) σ 0,i (1 − DP )
(1)
H H = (h / σ )(1 − * )ε . H
ε0 is the characteristic strain rate and is a composites of fundamental material properties such as particle and subgrain volume fraction, initial dislocation density, dislocation jog density and appropriate matrix diffusivity. Dd is a nondimensional damage parameter for the damage due to network dislocation coarsening. This damage parameter Dd is given as D d = (ρ N / ρ N ,i ) − 1 for 0 ≥ Dd ≥ − 1,
(2)
where ȡN and ȡN,i are the dislocation density at any strain and initial dislocation density, respectively. The damage parameter DP is used to define damage due to particle coarsening and is expressed as
D P = 1 − (D 0 / D) for 0 ≤ DP ≤ 1,
(3)
where D0 and D are the initial particle size and size of the particle at any time t, respectively. H is the normalised kinematic back stress defined as
H = σk /σ ,
(4)
where σk is a measure of strain-induced stress distribution between hard subgrain boundary and soft matrix. H* is the limiting value of H. The upper bound to stress redistribution is achieved upon H reaching H* value. σ0,i is a normalising stress related to the dislocation-particle interaction. The constant h is the effective modulus. The kinetic rate equations of the maximum attainable normalised kinematic stress, dislocation coarsening damage, particle coarsening damage and applied stress are given below.
H * (1 − H * ) H * = (1 + Dd ) −1 D d 2
ρ ρ D d = k 2 ( SS ) 0.5 (1 + Dd ) 0.5 (1 − (1 + Dd ) −1 (( N ,i ) 0.5 )ε ρ N ,i
' = K P (1 − D ) 4 D P P 3
σ = σε .
ρ SS
(5)
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The constant k2 is a dislocation annihilation parameter. The constant k2 is derived from Kocks–Mecking phenomenological approach [7] and is related to the rate at which steady state is reached. K'p is related to rate constant for particle coarsening. The evolution rate for the normalised maximum value of the kinematic back stress depends on the rate of subgrain coarsening. Semba et. al. [8] correlated the subgrain coarsening rate in terms of evolution of network dislocation density kinetics. 3. Creep data for modified 9Cr-1Mo Steel Creep data on normalised and tempered modified 9Cr-1Mo steel (MGC heat) conforming to ASTM standards used in this investigation was obtained from Ref. [9,10]. Creep strain-time data at stress levels 70-200 MPa for creep test durations in the range 41-80550 h have been used. 4. Initial parameters: estimation and optimisation The parameters σ0,i, K'p , ȡSS, ȡN,i, k2, ε 0 and H ( 0 ) are required for CDM analysis. The initial material parameters for CDM approach have been obtained from the quantitative microstructural data for heat treated as well as creep tested conditions [11-17]. The term σ0,i is expressed [11] as *
σ 0,i = kTσ orowan /(αGb3 ) ,
(6)
where k is a Boltzmann constant, T is temperature in Kelvin, b is burger vector, α is a constant of the order of unity. The σOrowan is expressed as
σ orowan = αMGb / L ,
(7)
where M is a Taylor factor and L is mean inter-particle spacing. By substituting σOrowan in Eq. (6) σ0,i is obtained as
σ 0,i = MkT /( Lb 2 ) .
(8)
It has been demonstrated that in modified 9Cr-1Mo steel, the presence of MX particles contributes significantly towards low inter-particle spacing together with large M23C6 precipitates. Therefore, effective inter-particle spacing resulting from both MX and M23C6 precipitates are considered. The effective inter-particle spacing, Leff is obtained as [18]
1 1 1 . = + Leff LM 23C6 L MX
(9)
Typical mean inter-particle spacing of 260 and 320 nm for M23C6 and MX particles, respectively has been for ferriticmartensitic steel [12]. Effective inter-particle spacing using Eq. (9) is estimated as 143 nm. Oruganti et. al. [13] reported MX spacing of about 79 nm with 7 nm of diameter and 1 nm in thickness in a ferritic-martensitic steel. By taking the values 260 nm for M23C6 and 79 nm for MX particles, Leff is obtained as 60 nm. By taking M = 3, K = 1.38 × 10−23 J mol−1 K−1 and Leff in the range 60-143 nm, the values of σ0,i in the range 3.5-8 MPa are obtained at 873 K. The term σ0,i.(1 – DP) results in decrease in normalised stress value with the coarsening of precipitates and consequently creep rate increases as envisaged in Eq. (1). The constant K'p in Eq. (5) is related to damage due to particle coarsening damage obtained by the Ostwald ripening for M23C6 and MX particles. Particle coarsening in the framework of Ostwald ripening is described as
D 3 = D03 + 8K P t ,
(10)
where KP is rate constant. The differential form of Eq. (10) can be substituted in time derivative of Eq. (3) to yield
8K P D P = (1 − DP ) 4 . 3D03
(11)
From Eq. (5) and Eq. (11), K'p can be obtained as K'p = 8Kp/(D0)3. In the present work, the Ostwald ripening data for M23C6 and MX derived by Hald and Korcakova have been used [14]. The values of Kp are taken approximately as 3.6 × 10−26 and 3.6 × 10−29 m3/hr for M23C6 and MX, respectively. Using these Kp values and initial diameters of M23C6 and MX as 100 and ' 40 nm, respectively, K P values of 2.88 × 10−4 (1/hr) for M23C6 and 4.5 × 10−6 (1/hr) for MX are obtained. The network
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dislocation coarsening damage parameters are evaluated using Kocks-Mecking approach [7] for dislocation evolution with respect to strain. This is expressed as
dρ 0 .5 = k1 ρ N − k 2 ρ N , dε
(12)
where k1and k2 are dislocation storage and annihilation parameters. As network dislocation density approaches steady state, i.e. dρ / dε = 0 , one can easily derive the saturation dislocation density value as ȡSS = (k1/k2)2. Therefore, Eq. (11) can be written as
dρ 0.5 = k1 ρ N (1 − ( ρ N / ρ SS )) 0.5 . dε
(13)
Following integration of Eq. (13), ρN can be obtained as
ρ N = (ρSS
0.5
− (ρSS
0.5
− ρN , i ) exp(−(k 2 / 2)ε))2 . 0.5
(14)
An attempt has been made to fit the published experimental data on dislocation density variation with strain [15] using Eq. (14). Figure 1 shows the variations in dislocation density with strain for 98.1 MPa at 923 K [15] along with the best-fit line obtained using Kocks-Mecking relation (i.e. Eq. 14). The network dislocation coarsening parameters are obtained as ȡSS = 1.0719 × 1013 m−2, ȡN,i = 3.21 × 1014 m−2 and k2 = 99.57. Various microscopic parameters are required to estimate ε0 value [8]. However, one can evaluate ε0 value by the relation
ε 0 = ε at t =0.1h / sinh(σ / σ 0,i ) .
(15)
In the above equation (i.e., Eq. 15), the microstructural damage is assumed as negligible as time approaches zero. For the numerical purpose, strain rate of 0.1 hr has been used to calculate ε0 in this investigation. ε0 value as 1.7 × 10−13 at applied stress of 140 MPa for an average value of σ0,i = 5.75 is obtained. In order to calculate effective modulus, the value of h can be evaluated as h = φ sg E where E is young’s modulus and φsg is volume fraction of subgrain boundaries in 9Cr-1Mo steel. By taking E = 158 MPa and by assuming φsg = 0.33 the value of h = 5.21 × 104 is obtained [8]. For analysis, the initial value * of H ( 0 ) is taken as 0.4. In general, the value of H* for precipitation hardened materials has been reported in the range 0.3 0.45 [11,13,14,16,17].
Fig. 1. The variation of experimentally measured dislocation density with strain [15] along with best-fit line using Kocks-Mecking approach.
The creep strain-time data at applied stress of 110 and 140 MPa are used for optimisation of material parameters. Using * the above creep strain-time data as the basis, the parameters σ0,i, K'p , k2, ε0 and H ( 0 ) are optimised. The value of ȡSS and ȡN,i are kept same as initial value. The optimisation algorithm was developed to provide minimum least square error function and this can be given as m
n
LS = ¦ (¦ ( i =1
j =1
− εexp (ε pred j j ) ε
exp j
)2 ) ,
(16)
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where m and n denotes the number of creep curves and number of creep stain vs. time data points, respectively. The superscripts ‘pred’ and ‘exp’ denote the numerically predicted and experimental strain values, respectively. The best set of parameters can be obtained by adjusting initial parameters to reach a minimum of this above merit function. The experimental cut-off strain of 6% for both the stresses has been taken for optimisation of material parameters. The initial value or range of different material parameters obtained from literature and optimised final parameters used in CDM analysis are presented in Table 1. It can be seen that the optimised parameters are close to the initial parameters which are derived from the quantitative microstructural information available in literature [11-17]. The value of optimised k2 is two times lower than the initial value. This difference in k2 can arise from the fact that the initial estimation of k2 was performed for 923 K, whereas the optimised value of k2 is obtained at 873 K. The optimised value of K'p is in between the initial parameters used for M23C6 and MX. This suggests that the interaction of dislocations with both the M23C6 and MX play a major role during creep deformation of modified 9Cr-1Mo steel. Table 1. CDM parameters initial value or range obtained from literature and optimised final parameters. CDM Parameters
Unit
Parameter value or range [11-17]
Optimised parameter
σ0,i
MPa
3.5 - 8
7.2
K p'
h−1
ȡSS ȡN,i k2
ε0 H*( 0 )
h
m m
2.88 × 10−4 – 4.5 × 10−6
4.3785 × 10−5
−2
1.0719 × 10
1.0719 × 1013
−2
3.21 × 10
14
13
3.21 × 1014
-
99.57
44.5
h−1
1.7 × 10−13
1.7635 × 10−11
-
0.4
0.35
MPa
5.21 × 10
4
1.0342 × 104
5. Creep strain-time prediction using CDM Approach A fourth order Runge-Kutta method has been used to solve the equation set (1) and (5). The program reads optimised material parameter set from a user-defined file and simulates the creep test for any given set of stress. A graphic user interface displays graphically the results of computed creep strain-time trajectories along with experimental data. The experimental creep strain vs. time for 110 and 140 MPa along with predicted creep strain-time data has been shown in Fig. 2. It can be seen that the predicted creep curves follow closely experimental data at 110 and 140 MPa. Creep strain-time data has also been predicted using optimised material parameter set at 100 and 120 MPa and compared with the respective experimental creep curves in Fig. 3. Good match between predicted and experimentally observed creep curves is obtained at 100 and 120 MPa. These observations suggest that the optimised material parameters and their incorporation into the evolution relationship in CDM approach are appropriate for the stress range 100-140 MPa.
Fig. 2. Comparison of predicted and experimental creep strain–time data for 110 and 140 MPa at 873 K.
Fig. 3. Comparison of predicted and experimental creep strain– time data for 100 and 120 MPa at 873 K.
J. Christopher et al. / Procedia Engineering 55 (2013) 798 – 804
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Creep strain-time data has been also predicted at relatively high and low stresses. Figure 4 shows a comparison of predicted creep curves with corresponding experimental data at 160 and 200 MPa. It can be clearly seen that the evolution of creep strain is significantly higher than experimental creep strain-time data at both stress conditions. This results in lower rupture lives than the respective experimental values at high stress levels of 160 and 200 MPa. It is important to point out that in the present analysis; particle coarsening is incorporated in the damage evolution equation. In reality, it is not expected that the particle coarsening will be dominant at lower rupture lives less than 1000 h. In addition to this, steady state dislocation density has been considered based on the values and rate equation at 98.1 MPa and this leads to more dislocation network coarsening damage at high stresses. In reality, steady state dislocation density is expected to higher than the value considered for analysis, since, dislocation density is proportional to applied stress. A combination of both results in overestimation of creep strain with time and lower rupture lives. On the other hand, at low stress level of 70MPa, the evolution of creep strain is significantly lower than experimental creep strain-time data and this result in higher predicted rupture life (Fig. 5). Significant decrease in number density of MX particles after 30000 h has been reported for the steel due to the formation and growth of complex nitride Cr(V,Nb)N, the Z-phase [10]. The decrease in long-term creep strength has been attributed to growth of Z-phase at the expense of beneficial MX precipitates. The present CDM analysis does not consider the rate of dissolution of MX and microstructural degradation due to formation and growth of deleterious Z phase. The observed under-estimation of evolution of creep strain with time and higher rupture life at 70 MPa can result from the above mentioned reasons (Fig. 5). The lower and higher predicted rupture lives at high and low stresses, respectively, can also be seen in Fig. 6. At intermediate stress regimes, good match between predicted and experimental values can be seen. Figure 7 demonstrates a good agreement for minimum creep rates between predicted and experimental values.
Fig. 4. Comparison of predicted and experimental creep strain–time data for 160 and 200 MPa at 873 K.
Fig. 6. Comparison of CDM predictions with experimental data for applied stress vs time to rupture at 873 K.
Fig. 5. Comparison of predicted and experimental creep strain–time data for 70 MPa at 873 K.
Fig. 7. Comparison of CDM predictions with experimental data of minimum creep rate vs. applied stress at 873K.
6. Conclusions Microstructural based CDM model has been successfully used to predict the creep behaviour of modified 9Cr-1Mo steel. Using the optimised material parameters, a good agreement between predicted and experimental values have been observed
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for stresses in the range 100-140 MPa. The discrepancies between the predicted and experimental values have been observed at high and low stresses. An attempt is being made to incorporate the influence of Z-phase kinetics into the CDM approach to achieve reliability in long-term creep life prediction.
References [1] S.L.Mannan, S.C.Chetal, B.Raj, S.B.Bhoje, Selection of materials for prototype fast breeder reactor, Trans. Indian Inst. Met., (2003)56:155-78. [2] R.L.Klueh, Elevated-temperature ferritic and martensitic steels and their applications to future nuclear reactors, Int. Mater. Rev., 50(2005)287-310. [3] K.Kimura, H.Kushima, F.Abe, K.Yagi, Inherent creep strength and long term creep strength properties of ferritic steels, Mater. Sci. Engg. A, (1997)234–236:1079-82. [4] K.Sawada, K.Suzuki, H.Kushima, M.Tabuchi, K.Kimura, Effect of tempering on Z-phase formation and creep strength in 9Cr-1Mo-V-Nb-N steel, Mater. Sci. Engg. A, (2008);480:558-63. [5] H.K.Danielsen, J.Hald, Behaviour of Z phase in 9-12%Cr steels, Energy Mater. 1(2006)49-57. [6] B.F.Dyson, S.Osgerby, Modelling and analysis of creep deformation and fracture in a 1Cr1/2Mo Ferritic Steel. NPL Report DMA A116 1993. [7] Y.Estrin, H.Mecking, A unified phenomenological description of work hardening and creep based on one parameter models, Acta Metall., 32(1984)57-70. [8] H.Semba, B.Dyson, M.McLean, Microstructural-based creep modelling of a 9%Cr martensitic steel, Mater. High Temp., (2008)25:131-37. [9] K.Kimura, H.Kushima, K.Sawada, Long-term creep deformation property of modified 9Cr-1Mo steel, Mater. Sci. Engg. A, (2009)510-511:58-63. [10] K.Sawada, H.Kushima, M.Tabuchi, K.Kimura, Microstructural degradation of Gr. 91 steel during creep under low stress, Mater. Sci. Engg. A, (2011)528:5511-8. [11] B.F.Dyson, M.McLean, Microstructural evolution and its effects on the creep performance of high temperature alloys, In: A.Strang, J.Cawley, G.W.Greenwood, editors, Microstructural Stability of Creep Resistant Alloys for High Temperature Plant Applications, London: Institute of Materials; (1998) p. 371-93. [12] K.Maruyama, K.Sawada, J.Koike, Strengthening mechanisms of creep resistant tempered martensitic steel, ISIJ Int., (2001)41:641-53. [13] R.Oruganti, M.Karadge, S.Swaminathan, Damage mechanics-based creep model for 9-10%Cr ferritic steels, Acta Mater., (2011)59:2145-55. [14] J.Hald, L.Korcakova, Precipitate stability in creep resistant ferritic steels-experimental investigations and modelling, ISIJ Int., (2003)43:420-7. [15] K.Sawada, M.Takeda, K.Maruyama, R.Ishii, M.Yamada, Y.Nagae, R.Komine, Effect of W on recovery of lath structure during creep of high chromium martensitic steels, Mater. Sci. Engg. A, (1999)267 (1999)19-25. [16] B.Dyson, The use of CDM in materials modelling and component creep life prediction, J. Pres. Ves. Technol., (2000)122 :281-96. [17] S.Hore, R.N.Ghosh, Computer simulation of the high temperature creep behaviour of Cr–Mo steels, Mater. Sci. Engg. A, (2011)528:6095-102. [18] J.Purmensky, V.Foldyna, Development of low alloy modified chromium steels strengthened by nanoparticles, Acta Metallurgica Slovaca, (2007)13:94-100.
Procedia Engineering 55 (2013) 798 – 804
6th International Conference on n Creep, Fatigue and Creep-Fatigue Interaction [CF-6]
Continuum Damage Mechannics Approach to Predict Creep Beehaviour of Modified 9C Cr-1Mo Ferritic Steel at 873 K J. Christophera,b, G. Sainathb, V V.S. Srinivasanb, E. Isaac Samuelb, B.K. Choudharyb∗, M.D D. Mathewb, T. Jayakumarb b
a Safety Research Institute, Atom mic Energy Regulatory Board, Kalpakkm-603102, Tamil Nadu, India Metallurgy and Materials Group, Indira Gandhi Centre for Atomic Research, Kalpakkam-603102, Tamil Nadu, India
Abstract The paper deals with the prediction of creep behavioour of modified 9Cr-1Mo ferritic steel at 873 K using continuum damage mechanics (CDM) approach. Creep data as well as the quantitattive microstructural information available on heat treated and crreep tested conditions in literature have been used for the analysis. The maaterial parameters were then optimised by matching the theoretiically generated creep strain-time curves with experimental curves at 110 and 140 MPa corresponding to low and high stress regimess, respectively. Using optimised material parameters, creep behaviour in teerms of creep strain-time data, minimum creep rate and time tto rupture at different stress levels have been predicted. In general, good agrreement between predicted and experimental data has been obtaained. ©©2013 Authors. Published by Elsevier Ltd. Ltd. Selection and/or peer-review under responsibility of the Indira Gandhi 2013The The Authors. Published by Elsevier Selection andAtomic peer-review under responsibility of the Indira Gandhi Centre for Atomic Research. Centre for Research Keywords: Creep behaviour; modified 9Cr-1Mo steel; continnuum damage mechanics
1. Introduction Modified 9Cr-1Mo steel is an important highh temperature material for steam generator (SG) applications in thermal and nuclear power industries. The modified version is obtained by the addition of strong carbide/nitride form ming elements such Nb and V and nitrogen in plain 9Cr-1Mo steeel and is also known as 9Cr-1Mo-V-Nb steel designatedd as P91 or T91 by ASTM standards. The steel derives its creep streength mainly from fine distribution of stable MX precipittates. The choice of 9Cr-1Mo steel for all the SG components in sodium cooled fast reactors (SFRs) is based on low thermal expansion coefficient and high resistance to stress corrosiion cracking in water-steam systems compared to austennitic stainless steels and superior elevated temperature mechanical properties than the alternate 2.25Cr-1Mo steel. The 10 1 5 h creep rupture strength of modified 9Cr-1Mo steel has been foound to be higher than 2.25Cr-1Mo and 9Cr-1Mo steels, and remains higher or equal to that of type 304 SS up to 898 K [1]. Further, the long term creep properties of modified 9Cr--1Mo steel has been observed to be superior than the conventional ssteels with Cr content up to 9%Cr, i.e. plain 9Cr-1Mo stteel and 12Cr-1Mo1W-0.3V steel [2,3]. Modified 9Cr-1Mo steel exxhibits distinct high and low stress regimes in terms of sttress dependence of minimum creep rate and rupture life. In low strress regime, the steel displays degradation in long-term creep c properties due to decrease in dislocation density, dislocation substructure coarsening and coarsening of M23C6 carbid de with increase in Mo due to nucleation creep exposure. Apart from these, decrease in solid solution strengthening caused by the depletion of M w increase in creep exposure add in decreasing long-term creep strength of and growth of secondary Laves phase Fe2Mo with the steel. In addition to above, degradation in lonng-term creep properties has been ascribed to heterogeneo ous and preferential
∗
Corresponding author: E-mail address: [email protected]
1877-7058 © 2013 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of the Indira Gandhi Centre for Atomic Research. doi:10.1016/j.proeng.2013.03.334
799
J. Christopher et al. / Procedia Engineering 55 (2013) 798 – 804
recovery in the vicinity of prior austenite grain boundaries (PAGBs) and formation and growth of a complex nitride Cr(V,Nb)N, the Z-phase at expense of fine V(C,N) and Nb(C,N) precipitate has been reported for the steel [4,5]. Z-phase coarsens fast compared to V(C,N) and Nb(C,N) precipitates and directly contributes to the reduction in creep rupture strength by removal of key precipitation strengthening elements such as Cr, V, Nb and N from the matrix. In the present investigation, physically based Continuum Damage Mechanics (CDM) approach has been used to predict creep behaviour of modified 9Cr-1Mo steel. In this microstructural based CDM approach, damage caused by coarsening of dislocation networks and subgrains together with coarsening of precipitates has been considered. The above damage evolution rates are coupled with kinetic creep law containing microstructural parameters in order to account for total damage and for prediction of creep curves at 873 K for different stress levels. The results obtained from CDM analysis in terms of creep strain-time data, minimum creep rate and time to rupture at different stress levels have been compared with the experimentally observed values. 2. CDM Approach Dyson and Osgerby [6] proposed a new kinetic creep law containing microstructural parameters for particle-strengthened alloys as
ε = ε0 (1 + Dd ) sinh( and
σ (1 − H ) ) σ 0,i (1 − DP )
(1)
H H = (h / σ )(1 − * )ε . H
ε0 is the characteristic strain rate and is a composites of fundamental material properties such as particle and subgrain volume fraction, initial dislocation density, dislocation jog density and appropriate matrix diffusivity. Dd is a nondimensional damage parameter for the damage due to network dislocation coarsening. This damage parameter Dd is given as D d = (ρ N / ρ N ,i ) − 1 for 0 ≥ Dd ≥ − 1,
(2)
where ȡN and ȡN,i are the dislocation density at any strain and initial dislocation density, respectively. The damage parameter DP is used to define damage due to particle coarsening and is expressed as
D P = 1 − (D 0 / D) for 0 ≤ DP ≤ 1,
(3)
where D0 and D are the initial particle size and size of the particle at any time t, respectively. H is the normalised kinematic back stress defined as
H = σk /σ ,
(4)
where σk is a measure of strain-induced stress distribution between hard subgrain boundary and soft matrix. H* is the limiting value of H. The upper bound to stress redistribution is achieved upon H reaching H* value. σ0,i is a normalising stress related to the dislocation-particle interaction. The constant h is the effective modulus. The kinetic rate equations of the maximum attainable normalised kinematic stress, dislocation coarsening damage, particle coarsening damage and applied stress are given below.
H * (1 − H * ) H * = (1 + Dd ) −1 D d 2
ρ ρ D d = k 2 ( SS ) 0.5 (1 + Dd ) 0.5 (1 − (1 + Dd ) −1 (( N ,i ) 0.5 )ε ρ N ,i
' = K P (1 − D ) 4 D P P 3
σ = σε .
ρ SS
(5)
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The constant k2 is a dislocation annihilation parameter. The constant k2 is derived from Kocks–Mecking phenomenological approach [7] and is related to the rate at which steady state is reached. K'p is related to rate constant for particle coarsening. The evolution rate for the normalised maximum value of the kinematic back stress depends on the rate of subgrain coarsening. Semba et. al. [8] correlated the subgrain coarsening rate in terms of evolution of network dislocation density kinetics. 3. Creep data for modified 9Cr-1Mo Steel Creep data on normalised and tempered modified 9Cr-1Mo steel (MGC heat) conforming to ASTM standards used in this investigation was obtained from Ref. [9,10]. Creep strain-time data at stress levels 70-200 MPa for creep test durations in the range 41-80550 h have been used. 4. Initial parameters: estimation and optimisation The parameters σ0,i, K'p , ȡSS, ȡN,i, k2, ε 0 and H ( 0 ) are required for CDM analysis. The initial material parameters for CDM approach have been obtained from the quantitative microstructural data for heat treated as well as creep tested conditions [11-17]. The term σ0,i is expressed [11] as *
σ 0,i = kTσ orowan /(αGb3 ) ,
(6)
where k is a Boltzmann constant, T is temperature in Kelvin, b is burger vector, α is a constant of the order of unity. The σOrowan is expressed as
σ orowan = αMGb / L ,
(7)
where M is a Taylor factor and L is mean inter-particle spacing. By substituting σOrowan in Eq. (6) σ0,i is obtained as
σ 0,i = MkT /( Lb 2 ) .
(8)
It has been demonstrated that in modified 9Cr-1Mo steel, the presence of MX particles contributes significantly towards low inter-particle spacing together with large M23C6 precipitates. Therefore, effective inter-particle spacing resulting from both MX and M23C6 precipitates are considered. The effective inter-particle spacing, Leff is obtained as [18]
1 1 1 . = + Leff LM 23C6 L MX
(9)
Typical mean inter-particle spacing of 260 and 320 nm for M23C6 and MX particles, respectively has been for ferriticmartensitic steel [12]. Effective inter-particle spacing using Eq. (9) is estimated as 143 nm. Oruganti et. al. [13] reported MX spacing of about 79 nm with 7 nm of diameter and 1 nm in thickness in a ferritic-martensitic steel. By taking the values 260 nm for M23C6 and 79 nm for MX particles, Leff is obtained as 60 nm. By taking M = 3, K = 1.38 × 10−23 J mol−1 K−1 and Leff in the range 60-143 nm, the values of σ0,i in the range 3.5-8 MPa are obtained at 873 K. The term σ0,i.(1 – DP) results in decrease in normalised stress value with the coarsening of precipitates and consequently creep rate increases as envisaged in Eq. (1). The constant K'p in Eq. (5) is related to damage due to particle coarsening damage obtained by the Ostwald ripening for M23C6 and MX particles. Particle coarsening in the framework of Ostwald ripening is described as
D 3 = D03 + 8K P t ,
(10)
where KP is rate constant. The differential form of Eq. (10) can be substituted in time derivative of Eq. (3) to yield
8K P D P = (1 − DP ) 4 . 3D03
(11)
From Eq. (5) and Eq. (11), K'p can be obtained as K'p = 8Kp/(D0)3. In the present work, the Ostwald ripening data for M23C6 and MX derived by Hald and Korcakova have been used [14]. The values of Kp are taken approximately as 3.6 × 10−26 and 3.6 × 10−29 m3/hr for M23C6 and MX, respectively. Using these Kp values and initial diameters of M23C6 and MX as 100 and ' 40 nm, respectively, K P values of 2.88 × 10−4 (1/hr) for M23C6 and 4.5 × 10−6 (1/hr) for MX are obtained. The network
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dislocation coarsening damage parameters are evaluated using Kocks-Mecking approach [7] for dislocation evolution with respect to strain. This is expressed as
dρ 0 .5 = k1 ρ N − k 2 ρ N , dε
(12)
where k1and k2 are dislocation storage and annihilation parameters. As network dislocation density approaches steady state, i.e. dρ / dε = 0 , one can easily derive the saturation dislocation density value as ȡSS = (k1/k2)2. Therefore, Eq. (11) can be written as
dρ 0.5 = k1 ρ N (1 − ( ρ N / ρ SS )) 0.5 . dε
(13)
Following integration of Eq. (13), ρN can be obtained as
ρ N = (ρSS
0.5
− (ρSS
0.5
− ρN , i ) exp(−(k 2 / 2)ε))2 . 0.5
(14)
An attempt has been made to fit the published experimental data on dislocation density variation with strain [15] using Eq. (14). Figure 1 shows the variations in dislocation density with strain for 98.1 MPa at 923 K [15] along with the best-fit line obtained using Kocks-Mecking relation (i.e. Eq. 14). The network dislocation coarsening parameters are obtained as ȡSS = 1.0719 × 1013 m−2, ȡN,i = 3.21 × 1014 m−2 and k2 = 99.57. Various microscopic parameters are required to estimate ε0 value [8]. However, one can evaluate ε0 value by the relation
ε 0 = ε at t =0.1h / sinh(σ / σ 0,i ) .
(15)
In the above equation (i.e., Eq. 15), the microstructural damage is assumed as negligible as time approaches zero. For the numerical purpose, strain rate of 0.1 hr has been used to calculate ε0 in this investigation. ε0 value as 1.7 × 10−13 at applied stress of 140 MPa for an average value of σ0,i = 5.75 is obtained. In order to calculate effective modulus, the value of h can be evaluated as h = φ sg E where E is young’s modulus and φsg is volume fraction of subgrain boundaries in 9Cr-1Mo steel. By taking E = 158 MPa and by assuming φsg = 0.33 the value of h = 5.21 × 104 is obtained [8]. For analysis, the initial value * of H ( 0 ) is taken as 0.4. In general, the value of H* for precipitation hardened materials has been reported in the range 0.3 0.45 [11,13,14,16,17].
Fig. 1. The variation of experimentally measured dislocation density with strain [15] along with best-fit line using Kocks-Mecking approach.
The creep strain-time data at applied stress of 110 and 140 MPa are used for optimisation of material parameters. Using * the above creep strain-time data as the basis, the parameters σ0,i, K'p , k2, ε0 and H ( 0 ) are optimised. The value of ȡSS and ȡN,i are kept same as initial value. The optimisation algorithm was developed to provide minimum least square error function and this can be given as m
n
LS = ¦ (¦ ( i =1
j =1
− εexp (ε pred j j ) ε
exp j
)2 ) ,
(16)
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where m and n denotes the number of creep curves and number of creep stain vs. time data points, respectively. The superscripts ‘pred’ and ‘exp’ denote the numerically predicted and experimental strain values, respectively. The best set of parameters can be obtained by adjusting initial parameters to reach a minimum of this above merit function. The experimental cut-off strain of 6% for both the stresses has been taken for optimisation of material parameters. The initial value or range of different material parameters obtained from literature and optimised final parameters used in CDM analysis are presented in Table 1. It can be seen that the optimised parameters are close to the initial parameters which are derived from the quantitative microstructural information available in literature [11-17]. The value of optimised k2 is two times lower than the initial value. This difference in k2 can arise from the fact that the initial estimation of k2 was performed for 923 K, whereas the optimised value of k2 is obtained at 873 K. The optimised value of K'p is in between the initial parameters used for M23C6 and MX. This suggests that the interaction of dislocations with both the M23C6 and MX play a major role during creep deformation of modified 9Cr-1Mo steel. Table 1. CDM parameters initial value or range obtained from literature and optimised final parameters. CDM Parameters
Unit
Parameter value or range [11-17]
Optimised parameter
σ0,i
MPa
3.5 - 8
7.2
K p'
h−1
ȡSS ȡN,i k2
ε0 H*( 0 )
h
m m
2.88 × 10−4 – 4.5 × 10−6
4.3785 × 10−5
−2
1.0719 × 10
1.0719 × 1013
−2
3.21 × 10
14
13
3.21 × 1014
-
99.57
44.5
h−1
1.7 × 10−13
1.7635 × 10−11
-
0.4
0.35
MPa
5.21 × 10
4
1.0342 × 104
5. Creep strain-time prediction using CDM Approach A fourth order Runge-Kutta method has been used to solve the equation set (1) and (5). The program reads optimised material parameter set from a user-defined file and simulates the creep test for any given set of stress. A graphic user interface displays graphically the results of computed creep strain-time trajectories along with experimental data. The experimental creep strain vs. time for 110 and 140 MPa along with predicted creep strain-time data has been shown in Fig. 2. It can be seen that the predicted creep curves follow closely experimental data at 110 and 140 MPa. Creep strain-time data has also been predicted using optimised material parameter set at 100 and 120 MPa and compared with the respective experimental creep curves in Fig. 3. Good match between predicted and experimentally observed creep curves is obtained at 100 and 120 MPa. These observations suggest that the optimised material parameters and their incorporation into the evolution relationship in CDM approach are appropriate for the stress range 100-140 MPa.
Fig. 2. Comparison of predicted and experimental creep strain–time data for 110 and 140 MPa at 873 K.
Fig. 3. Comparison of predicted and experimental creep strain– time data for 100 and 120 MPa at 873 K.
J. Christopher et al. / Procedia Engineering 55 (2013) 798 – 804
803
Creep strain-time data has been also predicted at relatively high and low stresses. Figure 4 shows a comparison of predicted creep curves with corresponding experimental data at 160 and 200 MPa. It can be clearly seen that the evolution of creep strain is significantly higher than experimental creep strain-time data at both stress conditions. This results in lower rupture lives than the respective experimental values at high stress levels of 160 and 200 MPa. It is important to point out that in the present analysis; particle coarsening is incorporated in the damage evolution equation. In reality, it is not expected that the particle coarsening will be dominant at lower rupture lives less than 1000 h. In addition to this, steady state dislocation density has been considered based on the values and rate equation at 98.1 MPa and this leads to more dislocation network coarsening damage at high stresses. In reality, steady state dislocation density is expected to higher than the value considered for analysis, since, dislocation density is proportional to applied stress. A combination of both results in overestimation of creep strain with time and lower rupture lives. On the other hand, at low stress level of 70MPa, the evolution of creep strain is significantly lower than experimental creep strain-time data and this result in higher predicted rupture life (Fig. 5). Significant decrease in number density of MX particles after 30000 h has been reported for the steel due to the formation and growth of complex nitride Cr(V,Nb)N, the Z-phase [10]. The decrease in long-term creep strength has been attributed to growth of Z-phase at the expense of beneficial MX precipitates. The present CDM analysis does not consider the rate of dissolution of MX and microstructural degradation due to formation and growth of deleterious Z phase. The observed under-estimation of evolution of creep strain with time and higher rupture life at 70 MPa can result from the above mentioned reasons (Fig. 5). The lower and higher predicted rupture lives at high and low stresses, respectively, can also be seen in Fig. 6. At intermediate stress regimes, good match between predicted and experimental values can be seen. Figure 7 demonstrates a good agreement for minimum creep rates between predicted and experimental values.
Fig. 4. Comparison of predicted and experimental creep strain–time data for 160 and 200 MPa at 873 K.
Fig. 6. Comparison of CDM predictions with experimental data for applied stress vs time to rupture at 873 K.
Fig. 5. Comparison of predicted and experimental creep strain–time data for 70 MPa at 873 K.
Fig. 7. Comparison of CDM predictions with experimental data of minimum creep rate vs. applied stress at 873K.
6. Conclusions Microstructural based CDM model has been successfully used to predict the creep behaviour of modified 9Cr-1Mo steel. Using the optimised material parameters, a good agreement between predicted and experimental values have been observed
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for stresses in the range 100-140 MPa. The discrepancies between the predicted and experimental values have been observed at high and low stresses. An attempt is being made to incorporate the influence of Z-phase kinetics into the CDM approach to achieve reliability in long-term creep life prediction.
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