Dislocation-density-based constitutive modelling of ...

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Dislocation-density-based constitutive modelling of tensile flow and workhardening behaviour of P92 steel a

J. Christopher & B.K. Choudhary

a

a

Deformation and Damage Modeling Section, Mechanical Metallurgy Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, Tamil Nadu, India Published online: 18 Aug 2014.

Click for updates To cite this article: J. Christopher & B.K. Choudhary (2014) Dislocation-density-based constitutive modelling of tensile flow and work-hardening behaviour of P92 steel, Philosophical Magazine, 94:26, 2992-3016, DOI: 10.1080/14786435.2014.944608 To link to this article: http://dx.doi.org/10.1080/14786435.2014.944608

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Philosophical Magazine, 2014 Vol. 94, No. 26, 2992–3016, http://dx.doi.org/10.1080/14786435.2014.944608

Dislocation-density-based constitutive modelling of tensile flow and work-hardening behaviour of P92 steel J. Christopher and B.K. Choudhary* Deformation and Damage Modeling Section, Mechanical Metallurgy Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, Tamil Nadu, India (Received 7 March 2014; accepted 1 July 2014) The flow and work-hardening behaviour of tempered martensitic P92 steel have been investigated using phenomenological constitutive model in the temperature range 300–873 K for the strain rates ranging from 3.16 × 10−5 to 1.26 × 10−3 s−1. The analysis indicated that the hybrid model reduced to Estrin–Mecking (E–M) one-internal-variable model at intermediate and high temperatures. Further, the analysis also indicated that dislocation dense martensite lath/cell boundaries and precipitates together act as effective barriers to dislocation glide in P92 steel. The flow behaviour of the steel was adequately described by the E–M approach for the range of temperatures and strain rates examined. Three distinct temperature regimes have been obtained for the variations in work-hardening parameters with respect to temperature and strain rate. Signatures of dynamic strain ageing in terms of the anomalous variations in work-hardening parameters at intermediate temperatures and the dominance of dynamic recovery at high temperatures have been observed. The evaluation of activation energy suggested that deformation is controlled by the dominance of cross-slip of dislocations at room and intermediate temperatures, and climb of dislocations at high temperatures. Keywords: P92 steel; tensile work hardening; Estrin–Mecking approach; dynamic recovery

1. Introduction 9Cr–0.5Mo–1.8W–V–Nb steel (designated as P92 steel according to ASTM standards [1]) is an improved version of 9% Cr-containing martensitic steels developed by the addition of 1.8 wt.% tungsten with reduced molybdenum in the P91 steel modified by the addition of strong carbide/nitride-forming elements niobium and vanadium along with controlled nitrogen [2–4]. The development of P92 steel for steam generator (SG) applications in ultra super critical power plants with increased operating steam temperatures and pressures was aimed to achieve higher efficiency and better environmental protection [2]. The selection of P92 steel for SG applications originates from its low thermal expansion coefficient and high resistance to stress-corrosion cracking in water– steam systems compared with austenitic stainless steels and superior elevated temperature mechanical properties than the alternate P9 and P91 steels [2–4]. The microstructure of 9% Cr steels in the normalized and tempered condition consists of *Corresponding author. Email: [email protected] © 2014 Taylor & Francis


Philosophical Magazine

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tempered lath martensite with hierarchical microstructure comprising of prior austenite grains (PAGs), packets, blocks, sub-blocks and laths [5,6] along with precipitates such as M23C6 carbides at the boundaries, and Nb- and V-rich MX-type carbonitrides/nitrides in the intralath matrix regions/subgrain interiors [7–12]. The superior elevated temperature mechanical properties of P92 steel is derived mainly from higher solid solution strengthening due to the addition of tungsten, initial high dislocation density accrued from austenite to martensitic transformation, hardening due to hierarchical boundaries and precipitation hardening from M23C6 carbides and MX-type carbonitrides/nitrides [13]. It has been reported that the presence of tungsten improves the stability of primary and secondary precipitates along with dislocation substructure with lower rate of recovery [14]. Detailed investigation on the tensile properties of P92 steel indicated three distinct temperature regimes in the variations of flow stress/strength values, work hardening rate and tensile ductility with respect to temperature and strain rate [15]. At intermediate temperatures, P92 steel exhibits dynamic strain ageing (DSA) manifested by the occurrence of serrated flow along with the anomalous variations in tensile properties. The anomalous variations in tensile properties are observed in terms of plateaus/peaks in flow stress/strength values and average work-hardening rate, negative strain rate sensitivity on flow stress/strength values and ductility minima in the DSA temperature regime. At high temperatures, the rapid decrease in flow stress/strength values and work-hardening rate, and increase in ductility with increase in temperature and decrease in strain rate indicated the dominance of dynamic recovery. P92 steel exhibited transgranular ductile fracture characterized by the presence of dimples originating from microvoid coalescence for the wide range of temperatures and strain rates [15]. In addition to the tensile properties with respect to temperature and strain rate, the flow and work-hardening characteristics attract continued scientific interest for understanding the physical phenomena controlling the deformation in metals and alloys. It also attracts technological importance for improved material processing as well as the safe performance of the components during service. An appropriate description of flow and work-hardening becomes necessary for determining the stress–strain field distribution for damage tolerant design of structural components using finite element analysis. Several empirical relations have been proposed to describe the tensile flow and work-hardening behaviour of metals and alloys [16–21]. These flow relationships lack physical significance, since the flow stress (σ) is described as a function of an external variable, i.e. true plastic strain (εp) with additional relevant terms defining the workhardening characteristics. In order to provide better description of flow and work-hardening behaviour, Kocks and Mecking (K–M) [22,23] proposed a phenomenological model based on the micromechanical state of the material defined by dislocation density as an internal state variable. The K–M approach has also been the basis for further developments towards microstructural-based constitutive descriptions of deformation behaviour of metals and alloys [24,25]. Subsequent to K–M approach, new modulus accounting for the influence of different geometrical obstacles such as grain boundaries and precipitates to dislocation mean free path was successfully introduced by Estrin and Mecking [26]. The applicability of K–M approach has been extensively used for describing tensile flow and work-hardening behaviour of simple face centered cubic (FCC) metals and alloys [22,23,26,27] and complex martensitic 9% Cr steels [28–30]. The monotonic deformation of Alloy 800H, Inconel 617 and tempered martensitic steels


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J. Christopher and B.K. Choudhary

has been successfully examined using Estrin–Mecking (E–M) models [31–35]. In the present study, the tensile flow and work-hardening behaviour of tempered martensitic P92 steel have been examined in the framework of dislocation dynamics-based Kocks– Mecking–Estrin (K–M–E) one-internal-variable approach [22,23,26] in the wide range of temperatures and strain rates. An attempt has been made to investigate the role of geometrical obstacles such as hierarchical boundaries and precipitates present in P92 steel on the evolution of total dislocation density with plastic strain. The influence of temperature and strain rate on the work-hardening parameters associated with the oneinternal-variable approach for steel is presented. In addition to above, the dynamic recovery model proposed by Bergström and Hallén [36,37] has been employed for the evaluation of activation energy for cross-slip and climb of dislocations towards the analysis of dynamic recovery parameter obtained from K–M–E approach. 2. Experimental stress–strain data The experimental tensile stress–strain data obtained on P92 steel pipe of dimension 260 mm outer diameter and 25 mm wall thickness have been used in this investigation [15]. The chemical composition of the steel (in wt.%) confirming to ASTM standards [1] was as follows: Fe–0.12C–0.54Mn–0.22Si–0.007S–0.013P–9.2Cr–0.54Mo–1.6W– 0.2V–0.07Nb–0.051N–0.002B. Specimen blanks of 12 mm diameter and 60 mm length machined in the longitudinal direction of the pipe were subjected to normalizing at 1338 K for 2 h followed by air cooling and tempering at 1053 K for 2 h followed by air cooling. The PAG size measured by linear intercept method was found to be 22 μm. Tensile tests were performed on button-head cylindrical specimens of 26 mm gauge length and 4 mm gauge diameter machined from the heat-treated specimen blanks in air using Instron 1195 universal testing machine equipped with a three-zone temperature control furnace and a stepped-load suppression unit. Tests were performed over a temperature range 300–873 K employing nominal strain rates of 3.16 × 10−5, 3.16 × 10−4 and 1.26 × 10−3 s−1. The test temperatures were controlled within ±2 K in all the tests. Engineering stress–strain data were recorded using a data acquisition system attached to the tensile test system. During tensile deformation, P92 steel exhibited serrated flow characterized by the occurrence of types D, A, A + B and D + C serrations at intermediate temperatures in the range 523–673 K [15]. For measuring engineering stress values from the serrated stress–strain curves, a uniform approach based on the classification of serrations was adopted [38,39]. The average stress values considered for different types of serrations have been described in detail in Ref. [15] and are typically shown as broken lines in Figure 1. True stress–true plastic strain data were evaluated using a computer program from the engineering stress–strain data up to the maximum load values. Since no strain gauge was used for the investigation, the cross-head displacement was taken as the sum of the deformation in the specimen and extension due to machine frame and load-train assembly. The initial linear portion of engineering stress–strain data was contributed by the displacement due to elastic deformation in the specimen, machine frame and load-train assembly. The combination of these elastic strains was subtracted from the total strain by appropriately using the slope of the initial linear portion for the calculation of plastic strain. Stress and plastic strain data were used to determine the true stress (σ)–true plastic strain (εp).



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Figure 1. Averaging scheme for engineering stress–strain curves showing different types of serrations at intermediate temperatures in P92 steel. Average stress–strain values considered for evaluating true stress–true plastic strain are shown by broken lines.

3. Analytical framework In the one-internal-variable K–M approach [22,23], the rate of evolution of total dislocation density with strain originates from the two competitive processes, i.e. the storage of dislocations (hardening) and annihilation/rearrangement of dislocations (recovery), which is superimposed in an additive manner. The evolution of total dislocation density (ρ) with true plastic strain (εp) is expressed as þ dq dq dq ¼M ; dep dep dep

(1)

where M is the Taylor factor, dρ+/dεp is the rate of dislocation accumulation and dρ−/ dεp is the rate of dynamic recovery. The dislocation storage rate is inversely related to the dislocation mean free path (L) and is described as dqþ 1 ; ¼ bL dep

(2)

where b is the Burgers vector. The rate of dynamic recovery is determined either by the cross-slip of screw dislocations at low temperatures or by the diffusion-controlled climb of edge dislocations at high temperatures. The boundary between the two temperature

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regimes generally falls between one-half and two-third of temperatures of melting point. The rate of dynamic recovery is expressed as


dq ¼ k2 q; dep

(3)

where k2 is the recovery coefficient. The generalized equation for the rate of dislocation density evolution with plastic strain can be obtained by substituting Equations (2) and (3) in Equation (1) and is expressed as dq 1 k2 q : ¼M dep bL

(4)

According to the K–M one-internal-variable formulation [22,23], dislocation structure is treated as the only obstacle to the moving dislocation, and regardless of its arrangement, the mean free path (L) is assumed to be directly proportional to 1/ρ0.5. Therefore, Equation (4) can be written as 0:5 dq k1 q k2 q ; ¼M dep b

(5)

where k1 is the dislocation accumulation or hardening parameter. Estrin and Mecking [26] disregarded the assumption that the mean free path of dislocation is proportional to 1/ρ0.5, when the spacing between impenetrable obstacles such as particles/precipitates or the grain boundaries is much smaller than the obstacles generated due to dislocation structure itself. An appropriate assumption for this condition is the mean free path that is considered to be a constant in the dislocation accumulation term in Equation (2), and the dislocation density evolution with strain, i.e. Equation (4) is written as dq k k2 q ; ¼M dep b

(6)

where k is a constant and is equal to 1/L. This model is popularly known as modified or E–M one-internal-variable model [26]. In order to incorporate the immobilizing effects of both type of obstacles, i.e. the dislocations and the impenetrable geometrical obstacles such as particles and the boundaries, the term 1/L in Equation (2) is assumed as a linear combination of the inverse spacing of these obstacles and expressed as 1 ¼ k þ k1 q0:5 : L

(7)

The resulting evolution equation for dislocation density with strain is given as a hybrid of Equations (5) and (6) [26] as dq k þ k1 q0:5 k2 q : ¼M dep b

(8)


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The macroscopic flow strength of material related to the microscopic internal variable, i.e. dislocation density is expressed as


r ¼ r0 þ M albq0:5 ;

(9)

where α is the constant and μ is the shear modulus. The term σ0 is arising from the strength contributions other than dislocation density such as solutes, grain boundaries and precipitates. The work-hardening rate (θ = dσ/dεp) can be derived from the differential form of Equation (9) as dr M alb dq ¼ : dep 2q0:5 dep

(10)

The flow behaviour of materials can be examined from the numerical integration of work-hardening rate given in Equation (10) coupled with dislocation density evolution equation along with appropriate initial boundary conditions, i.e. the initial dislocation density (ρi) and the initial stress (σI). The initial stress for ρi as initial dislocation density can be viewed from the generalized flow stress relation given in Equation (9) as rI ¼ r0 þ M albq0:5 i . In the present investigation, it was considered that the initial stress (σI) is equivalent to the true yield strength at εp = 0.001. The choice of dislocation density evolution equation depends on the initial microstructure of the material under consideration. The microstructure of P92 steel consists of initial high dislocation density inside lath, hierarchical boundaries in terms of PAGs and martensite packets, blocks, sub-blocks and laths [5,6] and large amount of precipitates. In view of this, the hybrid model for the evolution of total dislocation density with strain has been assumed to be more appropriate and therefore, the hybrid model has been chosen initially for the work-hardening analysis in this study. The coupled first-order differential forms of Equations (8) and (10) were integrated by the fourth-order Runge–Kutta method and the evolution of dislocation density and flow stress with strain was estimated. The unknown constants in the differential equations such as initial stress (σI), dislocation accumulation constants (k1 and k) and dynamic recovery parameter (k2) were optimized using interior-point algorithm by fitting predicted true stress–true plastic strain with experimental σ–εp data. The dislocation accumulation constant (k) was assumed to be the average dislocation mean free path arising from the combination of both the precipitates and the hierarchical boundaries. Dislocation density of the order of 1014 m−2 has been reported for tempered martensitic steels [13,40]. Based on the transmission electron microscopic examination, the total dislocation density inside lath close to 1014 m−2 has been observed for Grade 91 steel [41]. In P92 steel, dislocation density of 2 × 1014 m−2 has been obtained in both the normalized and tempered condition and the specimen deformed to low plastic strain close to 0.001 [4]. In the present study, the initial total dislocation density value of 2 × 1014 m−2 inside lath has been chosen for the analysis in P92 steel. The goodness of fit was judged by low reduced χ2 value obtained during non-linear optimization. The reduced χ2 is expressed as

Reduced v2 ¼

n X i¼1

2 ri;experimental ri;predicted =n p;

(11)


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where n is the total number of experimental points and p is the number of fitting parameters. In Equation (11), the term n–p is defined as the degree of freedom. Low reduced χ2 values signify the minimum difference between the predicted and experimental stress values. At all the temperatures and strain rates, the values of constants were obtained from the best-fit true stress–true plastic strain data. At 300 K, the dislocation accumulation parameter (k1) has been successfully optimized to a specific value. Contrary to this, the dislocation accumulation parameter, k1, falls less than 10−4 and convergence to a specific value is not observed at intermediate and high temperatures. The sensitivity of this parameter has been examined by adjusting the values of dislocation accumulation parameter, k1, in the range from 10−8 to 0.1 and simultaneously evaluating the reduced χ2 values without changing the other optimized parameters. The influence of the variations in k1 values on reduced χ2 is shown for the strain rate 3.16 × 10−4 s−1 in Figure 2 as an example. It can be seen that any deviation in k1 values from the optimized value of k1 = 0.0162 results in the sharp increase in reduced χ2 values at 300 K. This convergence in k1 value with low reduced χ2 value obtained for 300 K is not observed for the temperature range 473–873 K (Figure 2). Further, the other parameters in the hybrid model such as σI, k and k2 converge to the specific value at all the temperature and strain rate conditions. Figure 3 shows an example for the sensitivity of k parameter with low reduced χ2 for the strain rate 3.16 × 10−4 s−1. The low values of dislocation accumulation constant, k1, without convergence in the temperature range 473–873 K clearly provide the evidence of the dominance of k/b over k1ρ0.5/b at intermediate and high temperatures. In view of this, the dislocation storage term k1ρ0.5/b in Equation (8) has been disregarded in the present analysis for P92 steel. Accordingly, the modified or E–M one-internal-variable model for total dislocation density evolution with strain, i.e. Equation (6) coupled with Equation (10) has been used for describing the macroscopic true stress–true plastic strain behaviour in the entire temperature range

Figure 2. Variations of reduced χ2 values with dislocation accumulation parameter (k1) at 300, 648 and 823 K for the strain rate 3.16 × 10−4 s−1.



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Figure 3. Variations of reduced χ2 values with dislocation accumulation parameter (k) at 300, 648 and 823 K for the strain rate 3.16 × 10−4 s−1.

300–873 K. The analysis in this framework resulted in the optimized values of all the parameters, i.e. σI, k and k2 obtained from the convergence to the respective specific values with low reduced χ2. It is important to mention that the analytical framework proposed for the plastic flow stress (σd)–plastic strain (εp) relationship by Spätig et al. [34] based on E–M model can also be conveniently used for the work-hardening analysis. 4. Results 4.1. Influence of temperature and strain rate on true stress–true plastic strain behaviour Experimental true stress (σ)–true plastic strain (εp) data obtained for the temperature range 300–873 K in P92 steel is typically shown as double logarithmic plots of σ vs. εp for the strain rate 3.16 × 10−4 s−1 in Figure 4. True stress–true plastic strain at room and intermediate temperatures (300–723 K) was characterized by curvilinear behaviour with large positive stress deviations at low strains from the extrapolated linear σ–εp data at high strains. At high temperatures in the range 773–873 K, σ–εp displayed nearly linear behaviour with saturation in flow stress at high strains. The variations in flow stress with respect to temperature exhibited three distinct temperature regimes characterized by a decrease in flow stress with increasing temperature from 300 to 473 K (Regime-I) followed by insignificant variations in the stress values at intermediate temperatures in the range 523–723 K (Regime-II) and a rapid decrease in flow stress at high temperatures above 723 K (Regime-III). The influence of strain rate on σ–εp behaviour at 300, 648 and 823 K representing room, intermediate and high temperatures, respectively, are shown in Figure 5. Insignificant variation in the flow stress with respect to strain rate can be observed at 300 K. At intermediate temperatures, a marginal increase in the flow


3000


stress values with decrease in strain rate indicating negative strain rate sensitivity on flow stress can be seen at 648 K. At high temperatures, a rapid decrease in flow stress with decrease in strain rate has been observed at 823 K. Like flow stress, true uniform plastic strain also exhibited three temperature regimes characterized by a decrease with increasing temperature in Regime-I followed by a marginal decrease in the values at intermediate temperatures (Regime-II) and a sharp reduction in uniform plastic strain values at high temperatures (Regime-III) as shown in Figure 4. 4.2. Prediction of flow and work-hardening behaviour True stress vs. true plastic strain data at different temperatures and strain rates have been predicted using the optimized parameters, σI, k and k2 with initial dislocation density of 2 × 1014 m−2 in the E–M one-internal-variable model described by the numerical integration of coupled differential Equations (6) and (10). Typical true stress vs. true plastic strain data at 300, 648 and 823 K representing flow behaviour in the low, intermediate and high temperatures, respectively, for the strain rate of 3.16 × 10−4 s−1 are presented in Figure 6. The respective best fit σ–εp curves obtained using E–M one-internal-variable approach are superimposed as full lines. Good agreement between the predicted and experimental σ–εp can be seen in Figure 6. This is also supported by the low reduced χ2 values obtained for different temperatures and strain rates shown in Figure 7. These observations clearly suggest the applicability of E–M one-internal-variable approach for P92 steel. The different stages of work hardening in P92 steel has been examined using the variations in instantaneous work-hardening rate (θ = dσ/dεp) as a function of true stress for various temperatures and strain rates. Instantaneous work-hardening rate was evaluated using centred difference method as dr ¼h¼ dep

rðiþ1Þ rðiÞ rðiÞ rði1Þ 1 þ ; 2 epðiþ1Þ epðiÞ epðiÞ epði1Þ

(12)

where subscript i refers to the individual σ–εp data points. The variations in θ as a function of σ for 300, 648 and 823 K representing low, intermediate and high temperatures, respectively, for the strain rate of 3.16 × 10−4 s−1 are shown in Figure 8 along with the derivative of the predicted σ–εp obtained using E–M model as full lines. At all the temperature and strain rate conditions, experimental θ–σ data exhibited two-stage work hardening characterized by an initial rapid decrease in θ at low stresses (referred to as transient stage or, TS) followed by a gradual decrease in θ at high stresses (referred to as stage-III). It can be clearly seen that the predicted θ vs. σ follows the experimental θ vs. σ data more closely at high stresses represented by a gradual decrease in θ in stageIII. The observed two-stage work hardening in P92 steel is in agreement with those reported for P9 and P91 steels [29,30,42]. 4.3. Variations of work-hardening parameters with temperatures and strain rates The variations of work-hardening parameters obtained from the best fit σ–εp data associated with E–M one-internal variable approach with temperatures and strain rates are



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presented through Figures 9–11. In order to examine the influence of temperature alone on the initial stress, the shear modulus (μ) compensated initial stress has been used. The shear modulus is obtained as μ = E/2(1 + υ), where υ is the Poisson’s ratio, and the temperature dependence of Young’s modulus (E) has been obtained from the French nuclear design code RCC-MR for P91 steel [43]. The variations of normalized initial stress (σI/μ) with temperature displayed a marginal decrease in σI/μ with increase in temperature from 300 to 473 K followed by plateaus/peaks at intermediate temperatures and a rapid decrease in σI/μ values at high temperatures (Figure 9). The influence of strain rate on σI/μ is reflected in the negative strain rate sensitivity at intermediate temperatures and a decrease in σI/μ with decrease in strain rate at high temperatures. The variations of normalized initial stress with temperature and strain rate indirectly reflects the temperature and strain rate dependence of σ0/μ values (i.e. r0 =l ¼ rI =l M aq0:5 i , where ρi is a fixed value). The variations in dislocation accumulation constant, k, with temperature for the three strain rates displayed a marginal increase from 300 to 723 K followed by rapid increase in the k values at high temperatures (Figure 10). A marginal increase in k at the lowest strain rate at intermediate temperatures and a systematic increase in k with decrease in strain rates at high temperatures can be seen in Figure 10. Like k, dynamic recovery parameter, k2, also exhibited a marginal increase in values in the temperature range 300–723 K followed by rapid increase at high temperatures (Figure 11). The influence of strain rate on k2 at room and intermediate temperatures has been insignificant, but at high temperatures (T > 723 K), systematic increase in k2 with decrease in strain rate is observed. Based on the observed variations in the dislocation accumulation constant, k, and dynamic recovery parameter, k2, with respect to temperature and strain rate, the three temperature regimes obtained for σ–εp (Figures 4 and 5) and σI (Figure 9) can be broadly treated as two temperature regimes comprising of (a) room and intermediate temperatures and (b) high temperatures.

Figure 4. Influence of temperature on true stress (σ)–true plastic strain (εp) behaviour of P92 steel at a strain rate 3.16 × 10−4 s−1.


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Figure 5. Influence of strain rate on true stress (σ)–true plastic strain (εp) behaviour of P92 steel at 300, 648 and 823 K.

Bergström and Hallén [36,37] derived the constitutive model for dynamic recovery parameter with an assumption that the dynamic recovery is assisted by cross-slip of screw dislocations at low temperatures, whereas at high temperatures, dislocation climb dominates over cross-slip mechanism. The dominance of cross-slip at low temperatures and dislocation climb at high temperatures is shown schematically in Figure 12. Accordingly, the two components of dynamic recovery parameter as cross-slip and climb dominated recovery regimes with temperature are expressed in generalized form as k2 ¼ k2;Cross-slip þ k2;Climb

(13)

The term k2,Cross-slip is given as

k2;Cross-slip

Qc ¼ C exp ; RT

(14)

where C is a constant, R is the universal gas constant and Qc is the activation energy for cross-slip. The term k2,Climb in Equation (13) is given as

k2;Climb

Qm ¼ A exp ; 3RT

(15)

where A is a constant and Qm is the activation energy for vacancy diffusion. Following the assumption that cross-slip dominates at room and intermediate temperatures (300–723 K), the k2 parameter was fitted with Equation (14), and the values for C and Qc were evaluated. The values of k2,Climb at high temperatures above 723 K were then computed as k2,Climb = k2 − k2,Cross-slip by appropriately using the extrapolated k2,Cross-slip values for different temperatures. Following this, the values of A and Qm are



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evaluated from the best-fit k2,Climb vs. T using Equation (15). The values for C, Qc, A and Qm are summarized in Table 1 for different strain rates. The average values of activation energy for cross-slip mechanism as Qc = 4.47 kJ mol−1 (46.3 × 10−3 eV/atom) and for climb mechanism as Qm = 377 kJ mol−1 have been obtained. The typical variations of recovery parameter components, i.e. k2,Cross-slip and k2,Climb with temperature for the strain rate 3.16 × 10−4 s−1 are shown in Figure 13. A transition in the controlling deformation mechanism from cross-slip at room and intermediate temperatures to climb mechanism at high temperatures can be clearly seen in Figure 13. The predicted dynamic recovery parameter values using Bergström and Hallén [36,37] approach for different temperatures and strain rates are also superimposed as broken lines in Figure 11. A good agreement between predicted and experimental k2 values is discernible in Figure 11. 4.4. Prediction of dislocation density and flow stress at saturation The evolution of dislocation density towards steady state or, saturation value of ρsat can be obtained for condition dρ/dεp = 0 in Equation (6) and the ρsat can be expressed as qsat ¼

k : bk2

(16)

The variations in the computed saturation dislocation density ρsat with temperature and strain rates are shown in Figure 14. A marginal decrease in the dislocation density values at saturation was observed in the temperature range 300–723 K followed by a rapid decrease at high temperatures. At high temperatures, a systematic decrease in the dislocation density with decrease in strain rate can be seen. The saturation stress has been interpreted as a state of constancy arising due to the equilibrium between dislocation generation, and its annihilation and rearrangement to low energy configuration. Accordingly, the normalized saturation stress (σsat/μ) can be evaluated for condition when dislocation density approaches to its steady-state value as rsat r0 M albq0:5 sat ¼ þ : l l l

(17)

The term σ0 in Equation (17) can be obtained as r0 ¼ rI M albq0:5 i :

(18)

Table 1. The calculated values of constants and activation energy for dynamic recovery model proposed by Bergström and Hallén [36,37]. Strain rate, s−1 1.26 × 10−3 3.16 × 10−4 3.16 × 10−5

C

Qc, kJ mol−1

A

Qm, kJ mol−1

43.61 43.07 52.97

4.38 4.34 4.67

8.15 × 109 9.74 × 109 1.23 × 109

397 394 341


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The variations of normalized saturation stress computed using Equation (17) with temperature for the three strain rates are presented in Figure 15. σsat/μ displayed three distinct temperature regimes characterized by a marginal decrease with increase in the temperature from 300 to 473 K followed by plateaus/peaks at intermediate temperature and a rapid decrease in high temperatures. A marginal increase in σsat/μ with decrease in strain rate indicating negative stain rate sensitivity at intermediate temperatures and a decrease in the values with decreasing strain rate at high temperatures are observed. 5. Discussion 5.1. Applicability of E–M one-internal-variable approach Accurate prediction and explanation of true stress–true plastic strain behaviour is the key to the applicability of the theory of work hardening to metal and alloys. In the tempered lath martensitic steel consisting of hierarchical boundaries, it has been reported that the plastic flow originates from the preferentially oriented martensitic laths and propagates to the sub-block boundaries [44]. Since, sub-block boundaries display a misorientation of about 10°, it has been suggested that the slip would be restricted by the sub-block boundaries at low strains followed by the block boundaries with progress in further deformation. This implies that the size of effective geometrical obstacles unit changes with increase in the plastic deformation [44]. Apart from the influence of hierarchical boundaries present in P92-tempered martensitic steel, the presence of large amount of fine precipitates also provides other effective geometrical obstacles for the glide of dislocations along with forest dislocations. It is difficult to incorporate the influence of the individual geometrical obstacles on dislocation accumulation and annihilation separately. In view of this, the dislocation accumulation parameter k in Equation (8) is assumed to represent the inverse average geometrical obstacle spacing for dislocation glide. The work-hardening behaviour of P92 steel was initially examined in the framework of the hybrid model comprising of both the dislocation accumulation parameter k due to geometrical obstacles and k1 for total dislocation population given in Equations (8) and (10) for the evolution of total dislocation density and flow stress with strain. The observed low values of dislocation accumulation constant, k1, without convergence in the temperature range 473–873 K (Figure 2) clearly indicated the dominance of dislocation storage term k/b over k1ρ0.5/b at intermediate and high temperatures. This also suggested that the hierarchical boundaries and large amount of precipitates act as effective barriers to the glide of dislocations in tempered martensitic steel. Discounting k1ρ0.5/b term in Equation (8), the hybrid model reduces to the E–M one-internal-variable model, where k represents the global influence of hierarchical boundaries and precipitates on dislocation density evolution with strain. Accordingly, coupled differential equations expressed in Equations (6) and (10) have been successfully used to describe the macroscopic true stress–true plastic strain behaviour in the entire temperature range 300–873 K. This is clearly demonstrated by the excellent agreement obtained between predicted and experimental macroscopic true stress–true plastic strain data for the range of temperatures and strain rates examined in P92 steel (Figures 6(a)–(c)). This is also reflected in the low reduced χ2 values obtained for various temperatures and strain rates shown in Figure 7. The marginal higher values of reduced χ2 (i.e. χ2 > 10) observed in

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Figure 6. True stress (σ)–true plastic strain (εp) obeying E–M model at (a) 300, (b) 648 and (c) 823 K for P92 steel at a strain rate 3.16 × 10−4 s−1. Symbols represent experimental data and the best-fit stress–strain data predicted by E–M model are shown by full lines.


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Figure 7. Reduced χ2 values at different temperatures and strain rates for E–M model.

two experimental conditions (Figure 7) result from the increased tendency towards linearity in log σ–log εp at the transition temperatures between intermediate and high temperatures. The applicability of E–M one-internal-variable model is further demonstrated by comparing the predicted initial stress (σI) with true yield strength (σY) at εp = 0.001 and the predicted true ultimate tensile stress (σPre,UTS) with experimental true ultimate tensile strength (σExp,UTS) in Figure 16(a) and (b), respectively. The true ultimate tensile stress (σPre,UTS) at different temperatures has been predicted using Equation (19) as rPre;UTS ¼ r0 þ

M albq0:5 sat 0:5

ð2ðk2 M Þ1 þ 1Þ

:

(19)

Equation (19) was obtained by combining Equations (9) and (10) with Considère’s criterion as θ = σ. An excellent correlation obtained between the predicted and experimental measured values for the range of temperatures and strain rates in P92 steel can be seen in Figure 16. It has been demonstrated that the one-internal-variable-models adequately describe the deformation behaviour in monotonic conditions without abrupt changes in the deformation rate or the deformation path [24,25]. It has been also shown that the oneinternal-variable models fail to account for the short transients associated with the beginning of plastic flow during monotonic deformation [24,25,45]. In the present investigation, the derivative form of the predicted σ–εp data, i.e. θ vs. σ clearly demonstrates that the E–M one-internal-variable model is not able to predict the TS of work hardening at low stresses (Figures 8(a)–(c)). It has been suggested that the incorporation of a second internal variable accounting for the transients during monotonic deformation in the one-interval-variable model would provide an improved description of tensile flow behaviour [45]. In view of this, a two-internal-variable formulation [24,25,45] was

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Figure 8. Variations of instantaneous work-hardening rate (θ) as a function of true stress (σ) at (a) 300, (b) 648 and (c) 823 K for P92 steel at a strain rate 3.16 × 10−4 s−1. The derivative of best-fit true stress–true plastic strain data obtained for E–M model are shown as full lines.

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devised by distinguishing the total dislocation density in terms of mobile and forest dislocation density. The one-internal-variable model is obtained as a special case of the two-internal-variable approach. 5.2. Influence of temperature and strain rate on work-hardening behaviour P92 steel exhibited three distinct temperature regimes of low (300–473 K), intermediate (523–723 K) and high (773–873 K) temperatures in the variations in tensile flow and work-hardening parameters with temperature and strain rate (Figures 4, 5, 9 and 15). The three temperature regimes are also reflected in the variations of calculated dislocation density at saturation ρsat with respect to temperature and strain rate (Figure 14). In the low temperature regime, the observed marginal decrease in flow stress/strength values results from the ease of deformation due to easy glide and cross-slip activity with increase in temperature from 300 to 473 K. This is also reflected in the marginal increase in values of dynamic recovery parameter, k2. The observed plateaus/peaks in normalized initial and saturation stresses (Figures 9 and 15) along with the grouping of true stress–true plastic strain data in a narrow band at intermediate temperatures (Figure 4) clearly indicate the anomalous flow and work hardening due to the occurrence of DSA manifested by the presence of serrated flow (Figure 1) in the steel. The negative strain rate sensitivity on initial and saturation stress values indicates other manifestations of DSA at intermediate temperatures (Figures 9 and 15). Following measurement of activation energy for serrated flow, diffusion of interstitial carbon atoms has been suggested to be responsible for DSA in P92 steel [15]. It has been proposed that suzuki segregation of solute atoms towards the stacking fault ribbon of an extended dislocation could be enhanced significantly during deformation at serrated flow temperatures [46,47]. Following solute segregation, it becomes more difficult for extended screw dislocations to cross-slip owing to the increase in their dissociation width.

Figure 9. Variations of normalized initial stress (σI)/μ with temperature and strain rate.



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Figure 10. Variations of dislocation accumulation parameter (k) with temperature and strain rate.

Therefore, an enhanced work-hardening rate is expected as observed in type 304 stainless steel deformed at 673 K [48]. The increased dislocation width for dislocations and larger stacking faults with increased number density has been shown following deformation in the DSA regime in superalloy, which results in the reduction of stacking fault energy with increased propensity to DSA [47]. It has been suggested that when solutes interact with mobile dislocation, it not only decreases the mobility of mobile dislocations, but also results in the reduction in the ease of rearrangement of stored

Figure 11. Variations of dynamic recovery parameter (k2) with temperature and strain rate. Symbols represent experimental values and the predicted values of k2 using Bergström and Hallén model [36,37] are shown as broken lines.


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Figure 12. Schematic representation of dynamic recovery parameter (k2) as components of crossslip and climb dominated temperature regions [37].

dislocations [49]. These investigations clearly indicate reduced dynamic recovery in the DSA temperature regime. The variations in the dislocation accumulation parameter, k, and the recovery parameter, k2, with respect to temperature have shown only marginal difference between room and intermediate temperatures (Figures 10 and 11). The observed marginal increase in k and k2 in the temperature range 300–723 K for all the strain rates suggests that the dislocation sub-structural behaviour in its totality is not very different in the two temperature regimes. This is also reflected in the only marginal decrease in the normalized initial

Figure 13. Variations of dynamic recovery parameter (k2) components with temperature for P92 steel at a strain rate 3.16 × 10−4 s−1.



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Figure 14. Variations of saturation dislocation density (ρsat) with temperature and strain rate.

and saturation stress values in the temperature range 300–723 K (Figures 9 and 15). Contrary to this, a rapid decrease in the flow stresses (Figures 4 and 5) with respect to temperature and strain rate indicated the dominance of dynamic recovery at high temperatures. This is also reflected in rapid decrease in the initial stress, saturation stress (which indirectly denotes the decrease of k/k2 ratio in Equation (16)) and increase in the dynamic recovery parameter k2 with increase in temperature and decrease in strain rate at high temperatures (Figures 9, 11 and 15). The value of k2 can be viewed to be associated with the rate-controlling deformation mechanisms as d2ρ/dρ · dεp=−Mk2

Figure 15. Variations of normalized saturation stress (σsat)/μ with temperature and strain rate.


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Figure 16. Comparison between (a) predicted initial stress (σI) and true yield strength (σY) at εp = 0.001 and (b) predicted true ultimate tensile stress (σPre,UTS) with experimental true ultimate tensile strength (σExp,UTS) at different temperatures and strain rates.

using Equation (6). As expected, the observed low k2 values indicate the dominance of cross-slip of dislocations at room and intermediate temperatures (Figure 11). A change in recovery mechanism from cross-slip to dislocation climb results in high k2 values at high temperatures. It was also reported that the dynamic recovery parameter k2 is directly proportional to critical annihilation distance (statistically defined as the maximum distance between the parallel slip planes of two attractive edge or screw dislocations that can mutually annihilate each other) [50–52]. This suggests that with increasing k2 value, probability of annihilation of dislocation increases. The rapid increase in k2 at high temperatures is also reflected in the evolution of total dislocation



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density with plastic strain obtained using Equation (6). The evolution of dislocation density with plastic strain at room, intermediate and high temperatures is shown in Figure 17. The rapid evolution of dislocation density towards steady state value at high temperatures can be seen at 823 and 873 K in Figure 17. This is in contrast to the observed variations in the evolution of dislocation density with plastic strain showing only marginal variations at room (300 K) and intermediate temperatures (573 and 648 K). The observed decrease in saturation dislocation density (Figure 14) with increasing temperature and decreasing strain rate implies the dominance of climb-controlled deformation process at high temperatures. The low value of activation energy of 4.47 kJ mol−1 (46.3 × 10−3 eV/atom) obtained at room and intermediate temperatures in P92 steel is comparable to the reported values of activation energy for cross-slip phenomenon in FCC metals having high stacking fault energy and 18–8 stainless steel [37]. Following Bergström and Hallén [36], the activation energy Q = 754 kJ mol−1 (i.e. Q = 2 × Qm) can be assigned for high temperatures. The activation energy value of 754 kJ mol−1 obtained for deformation process at high temperatures is higher than the activation energy of 250 kJ mol−1 for self-diffusion in BCC Fe, but is comparable to the apparent activation energy values in the range 468–719 kJ mol−1 reported for climb-controlled creep deformation in 9% Cr-tempered martensitic steels in the high stress regime [53–56]. The high values of activation energy have been rationalized by incorporating resisting/back stress concept into power-law creep relationship resulting in the true activation energy close to self diffusion in P9 and P91 steels [54,56]. The distinct values of activation energy obtained for the temperature range 300–723 K and at high temperatures is consistent with the observed variations in k and k2 with respect to temperature and strain rate (Figures 10 and 11). The low k values observed at room and intermediate temperatures can be correlated with the effective obstacle spacing arising from the dislocation cell/martensite

Figure 17. Evolution of predicted dislocation density with true plastic strain for different temperatures at a strain rate 3.16 × 10−4 s−1.


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lath size and the distribution of fine precipitates in the hierarchal microstructure in P92 steel. An estimate of mean free path in the range 0.3–0.6 μm from the k values obtained at room and intermediate temperatures is comparable to the experimentally observed values of cell size/lath width in P92 and F82H steels [4,57]. These observations suggest that the effective dislocation storage distance can be equivalent up to the cell/lath size in P92 steel. Contrary to this, low mean free path in the range 0.2–0.1 μm has been obtained from high k values observed at high temperatures. At high temperatures, applied stress is low enough for dislocations to sweep the obstacles such as fine precipitates in P92 steel and as a consequence, this can lead to low mean free path observed in the temperature range 773–873 K. Further, enhanced dynamic recovery in terms of annihilation during glide of dislocations can add to reduced mean free path observed at high temperatures. 6. Conclusions Detailed analysis towards flow and work-hardening behaviour of P92 steel in the wide range of temperatures and strain rates indicated that the hybrid model reduced to E–M one-internal-variable model. Parameter sensitivity analysis over optimization results clearly demonstrated the dominance of dislocation accumulation term k/b over k1ρ0.5/b in the hybrid model at intermediate and high temperatures. This implies that the average mean free path arising from the dislocation dense lath/cell boundaries and precipitates act as effective barriers to glide of dislocations over total dislocation population inside the lath in P92 steel. The variations in work-hardening parameters associated with the E–M model with temperature and strain rate exhibited three distinct temperature regimes. Signatures of DSA at intermediate temperatures and dominance of dynamic recovery at high temperatures have been observed. Measurement of activation energy based on recovery model suggested that the dynamic recovery is controlled by the dominance of cross-slip of dislocations at room and intermediate temperatures (300–723 K) and climb of dislocations at high temperatures (773–873 K). References [1] Standard Specification for Seamless Ferritic and Austenitic Alloy-Steel Boiler, Superheater, and Heat-Exchanger Tubes, ASTM Standards A213/A213M–11a, ASTM International, West Conshohocken, PA, 2011. [2] R.L. Klueh, Int. Mater. Rev. 50 (2005) p.287. [3] F. Masuyama, Int. J. Pres. Ves. Piping 84 (2007) p.53. [4] P.F. Giroux, F. Dalle, M. Sauzay, J. Malaplate, B. Fournier and A.F. Gourgues-Lorenzon, Mater. Sci. Eng. A 527 (2010) p.3984. [5] S. Morito, H. Tanaka, R. Konishi, T. Furuhara and T. Maki, Acta Mater. 51 (2003) p.1789. [6] S. Morito, H. Saito, T. Ogawa, T. Furuhara and T. Maki, ISIJ Inter. 45 (2005) p.91. [7] P.J. Ennis, A. Zielinska-Lipiec, O. Wachter and A. Czyrska-Filemonowicz, Acta Mater. 45 (1997) p.4901. [8] F. Masuyama, ISIJ Int. 41 (2001) p.612. [9] K. Sawada, K. Kubo and F. Abe, Mater. Sci. Eng. A 319–321 (2001) p.784. [10] V. Sklenicka, K. Kucharova, M. Svoboda, L. Kloc, J. Bursık and A. Kroupa, Mater. Charact. 51 (2003) p.35.



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