International Journal of Plasticity 158 (2022) 103411
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International Journal of Plasticity
journal homepage: www.elsevier.com/locate/ijplas
Microstructure-sensitive modeling of high temperature creep in
grade-91 alloy
Mariyappan Arul Kumar *, Laurent Capolungo
Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, United States
A R T I C L E I N F O
A B S T R A C T
Keywords:
Mechanistic model
Creep
Microstructure
Deformation mechanism
Steels
Predicting the effects of microstructure on high-temperature creep responses of steel components
is critical to minimize the risks of failure and maximize economic viability in the energy sector. In
this work, a recently developed advanced mechanistic constitutive model is employed to study
the effect of microstructure on the creep responses and to rationalize experimentally reported
variability in the performances of grade-91 alloy. Hundreds of experimental creep tests from the
literature are used to assess the predictability of the model. The model proposed is shown to
accurately predict the steady-state creep rates and also the less commonly considered, yet
important, primary-to-secondary transients for a wide range of creep conditions. Further, the
model is capable of extrapolating the creep response of grade 91 alloy even outside of the cali­
bration regime. Using this model, the roles of initial microstructure described in terms of grain
size, dislocation density, and precipitate content, on the creep responses are investigated. The
effect of initial microstructure on the steady-state creep rates and creep-strain is found to vary
with stress and temperature. Grain size plays a significant role in the low-stress creep, where the
diffusional creep is dominant. On the other hand, the effects of dislocation densities and pre­
cipitate content are significant in the dislocation-plasticity-dominated high-stress and tempera­
ture regimes. Furthermore, by comparing model predictions against a large experimental creep
database, it is found that variability in the creep responses can be rationalized on the basis of
differences in the initial microstructure. Overall, this work provides a robust pathway for
microstructure-aware engineering design, verification, and validation of metallic components for
energy applications.
1. Introduction
Ferritic-martensitic grade-91 alloys, modified 9Cr-1Mo steel alloyed with V, Nb and N, are widely used in energy sectors, including
nuclear and fossil industries (Abe, 2016, 2022; Du et al., 2020). The engineering components of grade-91 alloys are mostly operated at
high temperature and low-to-moderate stress regimes. Under these conditions, the dimensional changes of the structural components
are primarily associated with thermal creep (Frost and Ashby, 1982). Thus, the development of a modeling framework to understand
and predict the thermal creep mechanisms and their relationship to the mechanical properties of steels is essential to assess the
performance and life of the components in service and as a way to enhance its applicability for different applications.
The thermomechanical processes that are commonly employed to shape the engineering components of grade-91 alloy develop
* Corresponding author.
E-mail address: marulkr@lanl.gov (M.A. Kumar).
https://doi.org/10.1016/j.ijplas.2022.103411
Received 18 June 2022; Received in revised form 24 August 2022;
Available online 1 September 2022
0749-6419/Published by Elsevier Ltd.
International Journal of Plasticity 158 (2022) 103411
M.A. Kumar and L. Capolungo
complex hierarchal microstructures (Abe, 2008, 2022; Du et al., 2020; Golański et al., 2017b; Li and Mitchell, 2013; Li and Chen, 2020;
Maruyama et al., 2001; Panait et al., 2010b; Peansukmanee et al., 2022; Ragab et al., 2022; Riedlsperger et al., 2020; Smith et al.,
2020; Yamamoto and Ortolani, 2021). Often, the polycrystalline material will be crystallographically textured with equiaxed grains
comprising precipitates, sub-grain boundaries, and dislocation structures. The addition of Mo and W in grade-91 alloys favors the
precipitation of Fe2(Mo, W) Laves phase and M23C6 carbides, where M denotes the metal elements (e.g., Cr) and C the carbon atom
(Abe, 2008; Golański et al., 2017b; Li and Mitchell, 2013; Maruyama et al., 2001). The M23C6 carbides are mostly distributed along the
prior austenite grain and sub-grain boundaries (lath structures). In turn, the M23C6 carbides govern the evolution of sub-grain
boundaries via controlling their size and dislocation content, which will strongly affect the dislocation mediated plasticity. Thus,
the addition of Mo and W strengthens the material in three ways: solid solution strengthening, precipitate hardening, and
sub-boundary hardening (Abe, 2008; Maruyama et al., 2001). Further, the presence of V, Nb, and N in grade-91 alloy leads to the
precipitation of MX-type carbides or nitrides (Abe, 2008; Du et al., 2020; Golański et al., 2017b; Maruyama et al., 2001; Panait et al.,
2010b). These MX precipitates are mostly formed inside the sub-grain structures and act as obstacles for moving dislocations. Here, M
is the metal element (e.g., V or Nb), and X can be of C or N.
In metals, the plastic straining and the creep relaxation processes are accommodated by a combination of various plastic dissipative
mechanisms such as dislocation glide, dislocation climb, and vacancy-mediated diffusion (Frost and Ashby, 1982). Activation of
deformation mechanisms depends on the operating conditions (stress and temperature), environment, internal stresses due to prior
loading history, and microstructure (Jiang et al., 2022; Tallman et al., 2020, 2021; Wen et al., 2017, 2020; Zhao et al., 2018a, 2018b).
The vacancy-driven diffusion process is dominant at higher temperatures and lower stresses (Coble, 1963; Frost and Ashby, 1982). On
the other hand, the dislocation motion is dominant at moderate-to-high stresses for all temperatures. The migration of dislocations is
strongly governed by the internal microstructures via their interaction with the existing obstacles (Bacon et al., 2009; Fournier et al.,
2011; Kim et al., 2020; Lehtinen et al., 2016; Lüthi et al., 2018; Morra et al., 2009; Yamamoto and Ortolani, 2021). In grade-91 alloy,
the following three different microstructural obstacles are commonly encountered: forest dislocations, solutes, and precipitates. The
effects of stress and temperature on the rates at which different types of obstacles are bypassed are specific to each obstacle. Thus, a
unified constitutive model with explicit stress, temperature, and microstructure dependence is necessary to simulate the plastic
deformation of metals under extreme environments. Most commonly used empirical or phenomenological creep models do not
explicitly track the relative contribution of distinct deformation mechanisms as a function microstructure state during loading (Basirat
et al., 2012; Drexler et al., 2018; Horstemeyer and Bammann, 2010; Ragab et al., 2021; Svyetlichnyy et al., 2013). Thus, these models
are not capable of capturing these microstructural effects on the creep responses explicitly, and also cannot be used in the stress and
temperature conditions outside of the calibration regime (Ayubali et al., 2021; Becker and Hackenberg, 2011; Cai et al., 2019; Jiang
and Dasgupta, 2021; Petersmann et al., 2019; Ragab et al., 2021, 2022; Salifu et al., 2020; Zhao et al., 2017).
Naturally, validation of constitutive models primarily relies on the availability and quality of the experimental measurements. For
grade-91 alloy, the available experimental creep data vary significantly from source to source. For example, for the same chemical
composition, the experimentally reported steady-state creep rate (in s− 1) for the creep of 600 ◦ C/100 MPa is: 6.65 × 10− 11 (Sklenička
et al., 2003), 1.53 × 10− 10 (Kloc and Sklenička, 1997), 4.37 × 10− 10 (Haney et al., 2009) and 1.07 × 10− 9 (Guguloth and Roy, 2017).
For the same 600 ◦ C/100 MPa creep condition, the steady-state creep rates in the NIMS database vary from 6.11 × 10− 11 s− 1 to 3.61
×10− 10 s− 1 (Sawada et al., 2014). The reported steady-state creep rates vary more than an order of magnitude. One can postulate that
this significant discrepancy in the creep responses is likely to be associated with the following three key factors: (i) the variation in the
initial microstructure due to prior thermomechanical process; (ii) the differences in the internal stress state; (iii) differences in the
experimental procedure such as mechanical testing setup, data collection method and frequency, and post-processing of the data. The
latter aspect cannot be the primary reason for the variation in the NIMS database since it is from a single source. The internal stress
state and its distribution in the creep sample can potentially alter the overall creep responses. These local stresses are not available for
the simulated creep tests and are also not easy to measure experimentally. So, for the sake of simplicity, here we assume that the stress
distribution in the sample is homogeneous. Thus, we hypothesize that the observed variability in the creep responses of the grade-91
alloy is primarily due to the differences in the initial microstructure due to thermomechanical processing. For several decades
numerous studies have investigated the connection between the microstructure and the creep responses (Christopher and Choudhary,
2019; Messner et al., 2018; Munday et al., 2020; Murchú et al., 2017; Nassif et al., 2019; Ragab et al., 2021). However, to the best of
our knowledge, none of the existing literature connects the observed variation in the creep responses with the initial microstructure.
In this work, a recently developed microstructure-sensitive mechanistic constitutive model which explicitly captures the stress and
temperature dependence is extended to test the aforementioned hypothesis (Wen et al., 2017, 2020). This constitutive model is
implemented within an effective medium visco-plastic self-consistent (VPSC) modeling approach (Lebensohn and Tome, 1993). The
model parameters are calibrated using the experimental data (Basirat et al., 2012; Kimura et al., 2009; Shrestha et al., 2012). Using the
said model and a large experimental database consisting of more than 281 creep tests, the following key questions are investigated: (i)
can the model be used to extrapolate the response of grade-91 beyond the domain used for model calibration? (ii) what is the effect of
initial microstructure on the creep responses of the grade-91 alloy? (iii) is it possible to rationalize the observed variability in the
experimental data on the basis of the initial microstructure fingerprint? This study finds that the employed mechanistic model is
capable of extrapolating beyond the calibration regime. Through a systematic investigation, this study reveals that slight changes in
the initial microstructure can vary the creep rates by orders of magnitude, and it helps to explain the variations in the experimental
data. Further, the effect of each initial microstructure on the creep responses varies with imposed stress and temperature. For instance,
the effect of initial dislocation density and precipitate content is significant in the high stress and temperature regime where dislo­
cation plasticity is dominant. On the other hand, grain size plays a critical role in the low stress and temperature regime where the
2
International Journal of Plasticity 158 (2022) 103411
M.A. Kumar and L. Capolungo
diffusional creep mechanism is dominant.
The paper is structured as follows. In Section 2, the employed constitutive model is presented. Model setup, parameters calibration,
and validations are presented in Section 3.1. The effect of initial microstructure on creep rates and creep life of the material is studied
in Sections 3.2 and 3.3, respectively. Finally, the key findings are summarized in Section 4.
2. Mechanistic constitutive model
The proposed advanced mechanistic constitutive model is primarily based on the following studies (Bieberdorf et al., 2021;
Munday et al., 2020; Wen et al., 2017, 2020). This constitutive model is sensitive to the stress, temperature, and microstructural
features on the activation of appropriate plastic dissipative mechanisms. The constitutive model is embedded in a mean-field vis­
co-plastic self-consistent (VPSC) framework (Lebensohn and Tome, 1993), in which a polycrystal is represented by a collection of
crystallographic orientations (grains) with associated volume fractions and the appropriate initial texture. Each grain is regarded as a
visco-plastic inclusion embedded in, and interacting with, a “homogeneous effective medium” (HEM), which has the average prop­
erties of the polycrystalline aggregate. The macroscopic response of the polycrystal results from the contribution of each grain and the
visco-plastic compliance of the HEM is given by a self-consistent condition applied to the grain averages. Note that the initial elastic
contribution resulting from creep loading of the material to the overall creep strain and creep rates is minimal, and so the plastic strain
rate from the VPSC model is directly correlated with experimentally reported creep curves. At the same time, the elasticity should be
fully accounted for to comprehensively model the primary creep response, particularly in highly anisotropic media. At the single
crystal level, the total plastic strain rate is assumed to be accommodated by a combination of dislocation motion and vacancy diffusion.
Accordingly, the plastic strain rate at every grain is expressed as:
(1)
ε̇p = ε̇d + ε̇Diff
ε̇d and ε̇Diff refer to the plastic strain rate due to dislocation motion and diffusion process (e.g., Coble creep), respectively. As mentioned
in Section 1, in the studied grade 91 alloy M23C6 precipitates are primarily located at the GBs, hindering the GB sliding and grain
growth processes (Sun and Jonas, 1994; Wu and Koul, 1995). Thus, in this work, GB sliding is not accounted for. The strain rate due to
dislocation motion is given by the individual shear contributions of all the slip systems:
∑
s
ε̇d =
msij γ̇
(2)
s
s
Here, m and γ̇ is the symmetric part of the Schmid tensor and average shear rate of the slip system, s within a grain. Since the strain
rate and stress within each VPSC grain are homogenous, following the work of Wang et al. (2017), the heterogeneity within each grain
is represented statistically via the internal stress distribution. Accordingly, the average shear rate of the slip system s in the grain
domain is written as:
s
∫∞
s
(3)
γ̇s (τs ) P(τs − τs )dτs
γ̇ =
− ∞
Where γ̇s is the shear rate of one sub-material point. τs is the local resolved shear stress. τs = σ : ms denotes the mean resolved
shear stress in one grain, where σ is the deviatoric stress tensor. Note that the sub-material point here refers to a small volume inside the
VPSC grain. The probability distribution function P(τs − τs ), representing the volume fraction of sub-material points with τs , is rep­
resented by the Gaussian distribution as,
)
(
1
(τs − τs )2
(4)
P(τs − τs ) = √̅̅̅̅̅̅̅̅̅ exp −
2V 2
2π V
Here V is the variance of the resolved shear stress, and it can be related to the dislocation density. However, for the sake of
simplicity, V is assumed to be constant throughout the creep simulations, even though the dislocation density evolves with plastic
deformation.
2.1. Dislocation shear rate
Following the Orowan’s equation, the local shear rate on slip system, s, due to dislocation motion (γ̇s ) is calculated as,
(5)
γ̇s = ρsci bs vs . sign(τs )
here bs is the magnitude of the Burgers vector. ρsci is the density of mobile dislocations present in the cell interior. vs is the dislocation
velocity, which depends on the mean spacing between obstacles (dislocation mean free path, λs ) and the time a dislocation spends
traveling between obstacles, given by the sum of the waiting time at obstacles (tws ) and the travel time within the interspacing (tts ):
vs =
λs
tws + tts
(6)
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International Journal of Plasticity 158 (2022) 103411
M.A. Kumar and L. Capolungo
In grade-91 alloy, quasi-spherical MX precipitates reside inside the sub-grains, and also rod-like M23C6 precipitates are located
primarily at the sub-grain and grain boundaries (Abe, 2008; Golański et al., 2017a; Maruyama et al., 2001; Murchú et al., 2017; Panait
et al., 2010a). The MX type precipitates directly control the dislocation motion within a sub-grain compared to M23C6 precipitates. In
addition to the precipitates, dislocation junctions are also present inside the sub-grains (Golański et al., 2017a; Murchú et al., 2017).
The moving dislocations can potentially interact with both the dislocation junctions and precipitates simultaneously. Thus, here the
bypassing of these obstacles is treated as a parallel circuit. Accordingly, the effective mean free path for dislocation motion is written as
the geometric mean of the interspacing for individual obstacles:
1
1
1
= s+ s
λs
λρ λMX
(7)
Here, λsρ and λsMX denote the dislocation mean free path for dislocation obstacles and MX precipitates, respectively. Following
(Franciosi and Zaoui, 1982), the mean dislocation interspacing associated with dislocation-dislocation interactions is written as,
√∑
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
1
′
′
(8)
=
αss ρsci
λsρ
s
where αss refers to the latent hardening matrix. Following the geometric configuration of obstacles on a plane, the mean interspacing
for MX precipitates is written as,
′
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
1
= hMX NMX dMX
λsMX
(9)
Here hMX , NMX and dMX is the trapping coefficient, number density, and size of MX precipitates, respectively.
The traveling time depends on the dislocation traveling distance (λs ) and its traveling speed (vt ). Thus, it is expressed as: tts = λs /vt .
Since the traveling time is significantly less than the waiting time, in this work, the dislocation traveling velocity is assumed to be the
√̅̅̅̅
shear wave velocity, Cs = ρμ . Accordingly, the traveling time is approximated as tts = λs /Cs . The terms μ and ρm refer to the tem­
m
perature dependent shear modulus and mass density, respectively. The waiting time depends on the available bypassing mechanisms
s
to overcome an obstacle. In general, dislocations can overcome the obstacles either by thermally activated glide, tw,i,g
(e.g., junction
s
unzipping and Orowan bypass) or climb assisted glide process, tw,i,cg
(Arzt and Rösler, 1988). These two mechanisms can co-occur, so
the total waiting time can be calculated by visualizing the bypass process as a parallel circuit. Accordingly, the effective waiting time
for each obstacle type i (= ρ for dislocations and MX for precipitates) is expressed as,
1
1
1
= s + s
s
tw,i
tw,i,g ftw,i,cg
(10)
s
s
Here, tw,i,g
and tw,i,cg
represent the waiting time for the thermally activated glide and climb-assisted glide processes, respectively,
which are discussed later in detail. The factor f (≥ 1) is introduced as a mode-specific fitting parameter to account for the additional
waiting time required to detach the dislocations from obstacles after the climb-over process. The mean waiting time of slip system s
s
s
while accounting for both the obstacles (dislocation type and MX precipitates) is given by the average of tw,
ρ and tw,MX as,
(11)
s
s
tws = Pρ tw,
ρ + (1 − Pρ )tw,MX
Here Pρ is the probability that a dislocation encounters other dislocation type obstacle, and 1 − Pρ is the probability of encountering
MX precipitates.
2.2. Waiting time for thermally activated glide
Following Kocks-type activation enthalpy law (Kocks et al., 1975), the waiting time to overcome an obstacle either by junction
unzipping or Orowan mechanisms is,
(
(
( s )p )q )
1
ΔG0,i
|τ |
exp
sgn(τs ) if |τs | < τsc
1
−
s
s
kT
τsc
tw,i,g
= { νi
(12)
0
if |τs | ≥ τsc
here ΔG0,i is the thermal activation energy without any external stress. k is Boltzmann constant. T is the absolute temperature.
p(0 < p ≤ 1) and q(1 < q ≤ 2) are the exponent parameters related to the shape of the obstacle’s resistance profile. νsi is the attempt
frequency. For the dislocation junction unzipping process, the attempt frequency depends on the dislocation travel velocity (vt = Cs ),
an entropy factor (χ e ), and the average length of the vibrating dislocation segment that is represented here as the dislocation free mean
path (λs ) (Granato et al., 1964). Accordingly, the attempt frequency for dislocation type obstacles is written as: νsi = χ e Cs /λs . The
attempt frequency for the precipitate-type obstacles is assumed to be constant. The terms τs and τsc in the above equation refer to the
local resolved shear stress and the Critical Resolved Shear Stress (CRSS), respectively.
The intrinsic frictional resistance to the dislocation motion and resistance due to precipitates, solute pinning, dislocation clusters
4
International Journal of Plasticity 158 (2022) 103411
M.A. Kumar and L. Capolungo
within the cell, and immobile dislocations present in the cell wall ρCW (through long-range stress field) constitute the CRSS value for
each slip system. Apart from the intrinsic frictional resistance τs0 , other resistance terms are expressed via a non-linear superposition
law. Accordingly, the CRSS for slip system s is written as,
(
)1/m
(13)
̅
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
∑ ss′ s′
With τρ = μbs
α (ρci + ρscw′ ) denotes the hardening due to dislocation structures and its interactions. τh denotes the strength­
τsc = τs0 + τmρ + τmh
s
ening due to both MX and M23C6 precipitates and solutes. τh varies with temperatures.
2.3. Waiting time for climb assisted glide
The waiting time for the climb process to overcome an obstacle can be determined from the mean climb velocity of edge dislo
Table 1
The calibrated constitutive model parameter values for P91 steel. * Parameters are estimated within the range given in the references.
Parameters
Equation #
b
5
2.48 × 10
ρm
μ
6
13
8000.0
103, 572 − 48T
Kg/m3
MPa/K
D0v
Evm
g0
g1
g2
g3
rd
r∞
16
16
18
18
18
18
15
15
14
7.87 ⋅ 10− 7
0.6
1.724
− 1.2 × 10− 4
− 2.79 × 10− 8
− 5.93 × 10− 11
2b
200b
2b
m 2 s− 1
eV
eV
eV/K
eV/K2
eV/K3
m
m
m
dcρ
dcP
f
14
10
12
12
13
14
p
q
m
Re
αss0
′
8
τ0
12
12
13
τP
13
ΔG0,ρ
ΔG0,MX
υG,MX
χe
k1
k2
k3
k4
n0
ε̇0
ADiff
QDiff
V
Dg
ρsci
ρscw
dMX
NMX
hMX
λsg
12
12
21
22
23
24
22
22
19
19
4
19
5
13
9
9
9
23
Value
Unit
m
− 10
dMX /2
8.2
0.7
1.4
2.0
10%
m
–
–
–
–
–
0.7 (s = s ’);
0.05 (s 1 s ’)
3.62
6.87
0.0
◦
393 MPa at 550 C
◦
359 MPa at 600 C
322 MPa at 650 ◦ C
306 MPa at 700 ◦ C
1.2 × 1010
1.0
0.1
87.0
5 × 10− 8
0.15
3.5
1
1.1 × 10−
0.35
1000
9.2 × 10−
4 × 1012
1 × 1013
37 × 10−
3 × 1020
1
0.5 × 10−
6
5
Gaffard (2004)
Mendelev and Mishin (2009)
Geers et al. (2014), Gu et al. (2015)
–
Sugiyama et al. (2021)
–
Queyreau et al. (2009)
eV
eV
MPa
Fitted
MPa
Fitted
MPa2
m
9
–
Sobie et al. (2015)
s− 1
MPa− 1s− 1K
eV
6
Dingley and Hale (1966)
–
Fitted
Kocks et al. (1975)
s− 1
–
–
–
m− 1
–
–
8
Refs.
m−
m−
m
2
m−
–
m
3
2
Gilbert et al. (2013)
Sobie et al. (2017)*
Fitted
Estrin (1998)*
–
Fitted
Kimura et al. (2011)
Basirat et al. (2012), Panait et al. (2010b)*
Shrestha et al. (2012)
Maruyama et al. (2001), Shrestha et al. (2012) *
Maruyama et al. (2001), Panait et al. (2010b)*
International Journal of Plasticity 158 (2022) 103411
M.A. Kumar and L. Capolungo
cations and the average distance to be climbed to bypass. Accordingly, the average waiting time for edge dislocations is expressed as,
⃒ s⃒
⃒v ⃒
1
c
=
R
(14)
e
s
tw,i,cg
dic
Here, Re denotes the proportion of edge dislocations that contribute to the climb process. dci is the distance to climb to bypass the
obstacles. Following (Wen et al., 2017), the climb velocity vsc is expressed as,
[
( s )]
fc Ω
2πDv Cv0 − Cv0 exp − kTb
s
vc =
(15)
bln(r∞/rd )
Here Ω ≈ b3 is the atomic volume. fcs is the climb component of the Peach-Koehler force. Dv is the vacancy diffusivity, and it is
expressed as (Mendelev and Mishin, 2009):
/ )
(
Dv = D0v exp − Emv kT
(16)
Where Evm is the vacancy migration energy. D0v is the exponential prefactor for vacancy diffusivity. C0v is the equilibrium vacancy
concentration at temperature T in the bulk of the crystal, and it is given by (Mendelev and Mishin, 2009; Was, 2016),
(
/ )
( / )
Cv0 = exp Sfv k exp − Efv kT
(17)
Where Svf and Evf are the vacancy formation entropy and energy, respectively, and it is written as a function of temperature as
(Mendelev and Mishin, 2009),
(18)
Efv = go − g2 T 2 − 2g3 T 3 Svf = − g1 − 2g2 T − . 3g3 T 2
The gx coefficients are listed in Table 1. The terms rd and r∞ in climb velocity equation denote the radii of inner and outer
boundaries for the cylindrical control volume defined around the dislocation line.
2.4. Diffusional creep
Diffusional creep contributes to the plastic deformation in polycrystals via the migration of point defects inside the grain/crystal
and along the grain or sub-grain boundaries, and also via grain boundary sliding (Frost and Ashby, 1982). In the VPSC framework,
grain and grain boundaries are not simulated explicitly, and also, the grain boundary-related material properties are not known for the
steel alloy system considered. Thus, for the sake of simplicity, following the works of (Coble, 1963; Herring, 1950; Mott and Nabarro,
1948), in this work diffusional creep model is simplified as,
( Diff )
ADiff σij
Q
ε̇Diff
=
exp
−
(19)
ij
TD3g
kT
Where ADiff and QDiff are effective parameters accounting for all point defects and obtained through the fitting to the experimental
data. Dg is the grain size. Note that the above equation is very similar to the power law creep model with a stress exponent of 1, except
here, the intrinsic material properties and microstructure are accounted for explicitly.
2.5. Dislocation density evolution
The evolution of dislocation density governs the transients in the creep rate via dislocation density-dependent glide shear rate and
climb velocity. The dislocation density law introduced in (Wen et al., 2017) is employed here. The following three different dynamic
processes are considered to evolve the dislocations within the cells: plastic shear-induced generation, dynamic recovery, and trapping
of dislocations at the preexisting boundaries. Accordingly, the evolution of the dislocation density within the cell is expressed as:
ρ̇sci = ρ̇s,+
ci,g −
s,+
ci,g ,
ρ̇s,−
ci,a −
(20)
ρ̇s,−
ci,trap
s,−
ci,a ,
s,−
ci,trap
Where ρ̇
ρ̇
and ρ̇
denotes the rate of change in dislocation content due to generation, dynamic recovery, and trapping at
the sub-grain boundaries. The dislocation generation rate is associated with the area swept by the moving dislocations, and it is written
as (Kitayama et al., 2013; Wen et al., 2016):
ρ̇s,+
ci,g =
k1 s
|γ̇ |
bλs
(21)
The dynamic recovery involves several mechanisms, such as cross-slip and climb, that allow the dislocation to move to another slip
plane and annihilate with dislocations with the opposite Burger vector. Estrin (1998) proposed a general expression of the dynamic
recovery rate:
ρ̇s,−
ci,a = k2
(ε̇ )n1
0
ε̇
0
(22)
s
ρsci |γ̇ |
6
International Journal of Plasticity 158 (2022) 103411
M.A. Kumar and L. Capolungo
Where ε̇0 is a reference strain rate. The parameter n0 should be associated with the dominant mechanism, and it should be between
3 and 5 for climb dominated process, or higher otherwise. The dislocation trapping rate at the subgrain boundaries is related to the
subgrain size λsg :
ρ̇s,−
ci,trap =
k3 s
|γ̇ |
λsg
(23)
The trapped dislocations will essentially become part of the cell wall structure. Meanwhile, the dislocations in the cell wall will also
annihilate. Thus, the rate of change of ρscw is written as:
(24)
s,−
ρ̇scw = ρ̇s,−
ci,trap − ρ̇cw,a
Dislocation annihilation in the sub-grain boundaries is complex, and its mechanism for the Fe-Cr-Mo alloy is not fully understood.
For the sake of simplicity, the annihilation rate is given as:
⃒ s ⃒( s )3/2
⃒ ⃒
ρ̇s,−
(25)
cw,a = k4 vc ρcw
Here vsc is the climb velocity that is given in Eq. (15). The parameters k1 , k2 , k3 and k4 are material constants and are fitted against
the experimental data.
3. Results and discussion
3.1. Model calibration and validation
The model parameters can be classified into three categories such as (i) microstructure descriptors; (ii) intrinsic material properties;
(iii) model-specific parameters. The initial microstructure descriptors for the calculations are obtained from experimental studies
which will be discussed later. The intrinsic material properties like mass density, diffusivity, and vacancy concentration are obtained
from the single crystal experiments or lower-length scale calculations. These values are listed in Table 1, along with appropriate
references. Other model-specific parameters are fitted to the experimental data. The calibrated values are listed in Table 1, and these
parameters are explicitly labeled as “fitted”.”
In this work, experimental data from the literature is used for model calibration and validation. As mentioned in Section 1, a nonnegligible variability exists in the reported experimental data for samples with similar chemical compositions. Thus, model calibration
will rely on the selection of a single or a few experimental sources that cover the creep conditions which are required to calibrate
different deformation mechanisms, and provides the microstructural details. To this end, creep data is extracted from numerous
sources. In total, a database containing 281 creep tests (59, 132, 69, and 21 creep tests for 550 ◦ C, 600 ◦ C, 650 ◦ C, and 700 ◦ C,
respectively) is generated from the literature. Note that the initial microstructure is not reported in the respective source articles for all
of these creep tests. To calibrate and validate the model parameters, a total of 29 experimental data points from two sources will be
used as they (i) provide an in-depth description of the microstructure and (ii) report creep rates consistent with one another. The rest of
the data, i.e., 252 tests, will be used to assess whether the polycrystal model can rationalize variability in performance on the basis of
differences in the initial microstructure.
Regarding the calibration data, the experimental observations of P91 alloy from Basirat et al. (2012), Shrestha et al. (2012) for
three different temperatures (600 ◦ C, 650 ◦ C, and 700 ◦ C) and varying stresses from 50 MPa to 200 MPa are considered for the model
calibration and validation. Apart from the macroscopic creep measurements, as mentioned before, these experimental studies also
characterize the initial microstructure of P91 alloy. This data helps seed the initial microstructure for the simulations, which is given in
detail later. Note, however, that the stress and temperature conditions considered in these studies are most suitable for calibrating the
parameters associated with the dislocation mediated plasticity but are insufficient to constrain the diffusional creep model parameters
(Frost and Ashby, 1982). To calibrate the diffusional creep model parameters (ADiff and QDiff ), creep responses at lower stresses and
moderate temperatures are required. Such experimental observations are not available for P91 alloy. Thus, only for diffusional creep
parameters, creep responses of T91 alloy at 550 ◦ C, and 600 ◦ C from Kimura et al. (2009) are used. The chemical composition of P91
and T91 alloys are the same, and they only differ in the prior thermo-mechanical processes. Diffusional creep is known to be governed
by the intrinsic diffusional properties of the material, which primarily depends on the chemistry and the grain size. Thus, in this work,
experimental results of T91 alloy are used only for diffusional creep model parameters calibration. These two sources (Basirat et al.,
2012; Kimura et al., 2009; Shrestha et al., 2012) provide experimental data for 29 creep conditions. Among them, the following seven
conditions were used for the model calibration: 600 ◦ C/70 MPa, 600 ◦ C/120 MPa, 600 ◦ C/150 MPa, 650 ◦ C/125 MPa, 650 ◦ C/150
MPa, 700 ◦ C/50 MPa, and 700 ◦ C/150 MPa. The first two cases from Kimura et al. (2009) were used to calibrate the diffusional creep
model parameters. In these cases, ~30% of the creep rate is due to diffusional creep; such a noticeable effect is sufficient to calibrate
the diffusional parameters. The other five cases are taken to get the dislocation plasticity model-related parameters. The remaining
creep conditions, a total of 22, are reserved for model validation against the experiment.
Central to this calibration exercise lies the description of the initial microstructure, which is detailed in what follows. The initial
material used for the creep tests in Basirat et al. (2012), Shrestha et al. (2012) is normalized at 1040 ◦ C, followed by air cooling and
tempered at 790 ◦ C. It results in tempered martensitic microstructure with almost equiaxed grains, which contains precipitates and cell
structures. Since the initial material is heat-treated, the initial texture of the material for the calculations is approximated with 50
uniformly distributed crystallographic orientations. In each body-centered cubic (BCC) VPSC grains of ferritic-martensitic Fe-Cr-Mo
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International Journal of Plasticity 158 (2022) 103411
M.A. Kumar and L. Capolungo
alloy, {110}〈111〉 and {112}〈111〉 slip modes were allowed to accommodate the dislocation mediated plasticity, and thus a total of 24
slip systems are considered.
Transmission electron microscopy-based microstructural characterization in Basirat et al. (2012), Shrestha et al. (2012) showed the
presence of both MX and M23C6 precipitates. The spherical-shaped MX precipitates are of ∼ 37 ± 15nm in size. These precipitates are
distributed within the grain and sub-grain structures. The work of Maruyama et al. (2001) with similar heat treatment reports that the
MX precipitates are distributed inside the grain with a mean spacing of 320 nm. Based on these details, the number density (NMX ) and
trapping coefficient (hMX ) for MX precipitates are selected as 3 × 10− 20 m− 3 and 1.0, respectively. Following Eq. (9), these precipitate
size and density values give an interspacing of 300.1 nm, which is close to the value given in Maruyama et al. (2001), and also other
works (Abe, 2008; Danielsen, 2016; Li and Chen, 2020). Interestingly, the grain size is not described in [30,31]. We will thus choose it
to be equal to 9.1 μm, consistently with that reported from the source (Kimura et al., 2009). Note that (Kimura et al., 2009) corresponds
to T91, which is used to calibrate the diffusional creep parameters. The actual grain size of the samples used in Basirat et al. (2012),
Shrestha et al. (2012) could be slightly different than the selected value. In any case, the precise choice of the grain size is not critical,
given that the mean free path of dislocations is driven by the sub-grain size of the grade-91 alloy. Following the work of Maruyama
et al. (2001), the sub-grain size is taken as 0.5 μm, similar to the values reported in Abe (2008), Golański et al. (2017a), Li and Chen
(2020), Panait et al. (2010b). The dislocation density in the cell interior (ρci ) and cell walls (ρcw ) are not known. However, consistently
the experimental literature reports that the total dislocation density in the grade-91 alloy is in the order of 1014 m− 2 and the dislocation
density in the cell walls is high compared to that of in the cell interior (Golański et al., 2017b; Li and Mitchell, 2013; Maruyama et al.,
2001; Panait et al., 2010b; Shrestha et al., 2012). Accordingly, ρci and ρcw are taken to be 4 × 1012 m− 2 and 10 × 1012 m− 2 ,
respectively, and this gives a total dislocation density of 3.36 × 1014 m− 2 once all 24 slip systems are considered. The selected initial
microstructural parameters are listed in Table 1. All other model parameters are either obtained from the literature or fitted against the
experimental data.
The predicted steady-state creep rates of grade-91 alloy for four different temperatures and varying stresses are shown in Fig. 1,
along with the experimental observations. The empty and solid symbols correspond to experiment and model predictions. The cases
considered for model calibration are marked by an ellipse. Note that the creep tests at 550 ◦ C for all stresses, and for stresses above 150
MPa for all temperatures are not considered for the model calibration. These cases are primarily outside the calibrated regime and used
for the model validation. Fig. 1 clearly shows that the developed model correctly captures the effect of stress and temperature on the
steady-state creep rate, including the extrapolation regime.
The predicted evolution of creep strain (top row) and strain rate (bottom row) are shown in Fig. 2, along with the experimental
Fig. 1. Calibration and validation of model parameters for grade-91 alloy. VPSC model predicted steady-state creep rate as a function of stress for
various temperatures along with experiments. Empty and solid symbols correspond to experiment and model predictions. A total of 7 cases were
used for the model calibration that are highlighted by ellipses, and the remaining 22 cases were used for model verification and validation.
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International Journal of Plasticity 158 (2022) 103411
M.A. Kumar and L. Capolungo
values. For the sake of compactness, the creep curves are shown only for two stresses for three different temperatures (600 ◦ C, 650 ◦ C,
and 700 ◦ C). The solid lines and symbols correspond to the model predictions and experimental values. The horizontal dotted lines in
the creep strain-rate plots represent the experimentally reported steady-state creep rates. The experimental curves shown in Fig. 2
include the signature of tertiary creep, i.e., an increase in creep rate and an increase in the creep strain curve’s slope after reaching the
true steady state. This tertiary creep is mainly associated with the damage nucleation and growth. As the current model does not
account for the damage phenomena, the model predictions can only be compared until the secondary creep. Except for the tertiary
creep regime, the developed model correctly captures the experimentally measured creep curves for all stress and temperature cases.
Particularly, the model captures well the primary creep rate and primary-to-secondary transients. Furthermore, note that these creep
curves are not used for the model calibration; only the reported steady state creep rates are used. The steady-state creep rate definition
in those experimental studies is unclear and sometimes not consistent. In some studies, the minimum creep rate is reported as a steady
state creep rate, or in others, the effective steady-state creep rate is derived from the creep curves. In turn, in some cases like 700 ◦ C/80
MPa and 700 ◦ C/100 MPa, the model calculated creep rate curves are slightly off from the experimental values. Nevertheless, Figs. 1
and 2 suggest that the developed mechanistic model is capable of capturing the stress and temperature dependence on the transients in
addition to the steady-state creep rates.
Further, to understand the deformation mechanisms that contribute to the precited creep responses, the relative contribution of
dislocation glide and diffusional creep is plotted in Fig. 3. The relative contribution of dislocation glide (f glide ) is defined as the ratio
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ )
(
glide
d
2ϵ̇d
between the von-Mises strain rate accommodated by the dislocation glide ϵ̇vM =
and von Mises total plastic strain rate
3 ij : ϵ̇ij
(
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ )
p
2 p
Diff
ϵ̇vM =
= 1 − f glide . The main focus of this
3ϵ̇ij : ϵ̇ij . Following Eq. (1), the relative contribution of the diffusion process is f
analysis is to understand the stress and temperature dependence on the activation of different deformation mechanisms. Thus, in Fig. 3,
the evolution of relative activities is shown only for a few representative cases. Accordingly, two different temperatures (600 ◦ C and
700 ◦ C) are selected to capture the temperature effect, and for each temperature, two different stresses are considered to understand
the stress effect. At 600 ◦ C and 70 MPa, both dislocation glide and diffusional creep contribute to the predicted creep rates. As ex­
pected, the diffusional creep contribution decreases significantly with an increase in stress from 70 MPa to 150 MPa, and dislocation
glide predominantly accommodates the deformation. At 700 ◦ C, for both low and high stresses, dislocation glide dominates with
almost negligible contribution from the diffusional creep. Overall, the dislocation glide dominates at high stress and temperature
conditions, whereas, at low stress and moderate temperature, diffusional creep also contributes equally.
3.2. Effect of initial microstructure on creep rates
In this section, the effect of initial microstructure on the creep rates is investigated to mainly test out whether the variability in the
experimental creep data could be due to the differences in the initial microstructure. Commonly, engineering components are shaped
Fig. 2. Model predicted creep responses along with the experimental values for different stresses at (a)-(b) 600 ◦ C; (c)-(d) 650 ◦ C; (e)-(f) 700 ◦ C. The
top and bottom rows show the creep strain and creep strain rate evolution as a function of time, respectively. Solid lines and symbols correspond to
model predictions and experimental measurements, respectively. The dotted lines in the bottom row represent the experimentally measured steadystate creep rates.
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International Journal of Plasticity 158 (2022) 103411
M.A. Kumar and L. Capolungo
Fig. 3. Relative contribution of dislocation glide and diffusional creep mechanisms during the creep deformation at (a) 600 ◦ C and (b) 700 ◦ C for
two different stresses.
via a metal forming process such as hot/cold rolling, forging, etc., followed by controlled heat treatment. The metal forming process
induces significant plastic work hardening to the material and thus develops a complex microstructure with sub-grain boundaries and
higher dislocation content. The rolling process can increase the dislocation density in steels to the order of 1015 m− 2 (Shakhova et al.,
2016). Further heat treatment, e.g., annealing, aging, and normalizing/tempering, relaxes the prior plastic work hardening developed
dislocation content and alters the precipitate content and grain structures. For example, the MX precipitate size in grade-91 alloy
increased from 24 nm to 50 nm upon normalizing at 1050 ◦ C (Yoshino et al., 2008). Tempering of the 9Cr-Fe alloy at 720 ◦ C increases
the sub-grain lath size from 250 nm to 300 nm and decreases the dislocation density from 7 × 1014 m− 2 to 6 × 1014 m− 2 (Dudko et al.,
2012). However, tempering of the same material in the temperature range from 400 ◦ C to 625 ◦ C does not result in a noticeable change
in the dislocation density or sub-grain size. Furthermore, the experimental studies report that based on the prior thermal processes, the
grain size in grade 91 alloys can vary from ~5.0μm to ∼ 20.0μm (Maruyama et al., 2020; Zavaleta Gutiérrez et al., 2014). Thus, the
evolution of microstructure is strongly correlated with the heat treatment temperature and duration (Dudko et al., 2012; Li and Chen,
2020; Maruyama et al., 2020; Yoshino et al., 2008; Zavaleta Gutiérrez et al., 2014).
Based on these findings, a range for the grain size, dislocation density, and precipitate content is selected, see Table 2. The grain size
is varied from 5.0 to 15.0μm. The dislocation density in the cell interior varies from 1 × 1012 m− 2 to 8 × 1012 m− 2 . The dislocation
content in the cell wall is represented in terms of the cell interior dislocation content and the ratio ρcw /ρci is varied from 1 to 10. Note
that the ratio ρcw /ρci cannot be less than one, which means that the dislocation cell wall structure is a dominant feature in all the
considered microstructures, consistent with the experiments (Abe, 2008; Golański et al., 2017a; Li and Mitchell, 2013; Li and Chen,
2020; Maruyama et al., 2001; Murchú et al., 2017; Panait et al., 2010a, 2010b; Prasad et al., 2012). This combination of ρci and ρcw
leads to a total dislocation density range from 4.8 × 1013 m− 2 to 2.1 × 1015 m− 2 . The lower and upper limits can be associated with the
over-aged microstructure and plastically work-hardened material. The most commonly employed heat treatment processes will be
bounded between these two limits. Following the work of Yoshino et al. (2008), the MX precipitate size is varied from 25 nm to 50 nm.
The number density of precipitates is varied from 1 × 1020 m− 3 to 6 × 1020 m− 3 , which is similar to the ranges given in Prasad et al.
(2012). The considered ranges for the dislocation density and precipitate content lead to a total of 300 different initial microstructures
for each grain size.
Thermal creep simulations for all the considered initial microstructures are performed at 550 ◦ C, 600 ◦ C, 650 ◦ C, and 700 ◦ C for
various stresses. The imposed stress ranges from 2 MPa to 220 MPa for 550 ◦ C and 600 ◦ C, 2 MPa to 200 MPa for 650 ◦ C, and 2 MPa to
Table 2
Considered ranges for the initial cell-interior and cell-wall dislocation densities, precipitate size
and number density, and grain size. These ranges lead to a total of 900 different microstructures.
Parameter
Values
12
− 2
Dislocation density in the cell interior, ρci (10 m
Dislocation density at the cell walls: ρcw (m− 2 )
Precipitate number density, NMX (1020 m− 3 )
Precipitate size, dMX (10− 9 m)
Grain size, Dg (10− 6 m)
10
)
[1, 2, 4, 6, 8]
[1, 2.5, 5, 10]. ρci
[1, 2, 3, 4, 6]
[25, 37, 50]
[5.0, 9.2, 15.0]
International Journal of Plasticity 158 (2022) 103411
M.A. Kumar and L. Capolungo
Fig. 4. Model predicted steady-state creep rates as a function of stress for different microstructures at (a) 550 ◦ C, (b) 600 ◦ C, (c) 650 ◦ C, and (d)
700 ◦ C, along with the experimental values. A total of 300 different microstructures listed in Table 2 are considered for each grain size. The shaded
region shows the spread in the creep rates. The discrepancy in the available experimental data is likely to be associated with the differences in the
initial microstructure.
170 MPa for 700 ◦ C. Accordingly, a total of 23 stresses were considered for 550 ◦ C and 600 ◦ C, whereas a total of 22 and 18 stresses
were considered for 650 ◦ C and 700 ◦ C cases, respectively. The model predicted steady-state creep rates along with the experimental
data is shown in Fig. 4. As mentioned earlier, the grain size is directly controlling only the diffusional creep, the model predictions are
shown for each grain size separately. The shaded region in Fig. 4 shows the spread in the predicted creep rates for every stress due to
differences in the initial dislocation density and precipitate content for each grain size. The symbols correspond to the creep rates from
the experimental literature (Basirat et al., 2012; Guguloth and Roy, 2017; Haney et al., 2009; Kimura et al., 2009; Kloc and Sklenička,
1997; Sklenička et al., 2003) including the NIMS database (Sawada et al., 2014). The results shown in Fig. 4 reveal the following key
findings. The effect of stress on the secondary creep rates is not linear, and also the dependency varies with temperature. Commonly,
the stress effect on creep rates for every temperature is represented independently by two stress exponents using the
Bird-Mukherjee-Dorn (BMD) equation (Bird et al., 1969). The predictions shown in Fig. 4 reveal that the analysis based on the BMD
equation will not satisfactorily capture the evolution of the power-law exponent with stress.
Importantly, the effect of grain sizes is critical in the low-stress regime, where the diffusional creep mechanism is dominant. This
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International Journal of Plasticity 158 (2022) 103411
M.A. Kumar and L. Capolungo
can be assessed by comparing the creep rates in Fig. 4(a) and (b), where a change in grain size of ~5μm can lead to orders of changes in
the steady-state creep rate. For example, at 550 ◦ C and 10 MPa, the average steady-state creep rate is 1.6 × 10− 12 s− 1 and 1.0 ×
10− 11 s− 1 for the grain sizes of 9.1μm and 5.0μm, respectively. Interestingly, one sees that at a constant grain size, the creep rate for the
low-stress regimes also varies noticeably with the changes in the initial dislocation density and precipitate content. For example, the
variation in the creep rates is significant even for stresses lower than 50 MPa at 600 ◦ C, 650 ◦ C and 700 ◦ C. This is due to the fact that
both diffusional creep and dislocation mediated plasticity are equally contributing to the creep rates in those regimes. As expected, the
variation in the creep rates decreases with the grain size in the high-stress regime but remains important. This non-negligible effect of
grain size on the creep rates in high-stress regimes is directly related to the relative contribution of diffusional creep to the overall creep
rate. This grain size effect in the high stress-regime decreases with the temperature, and interestingly, at 700 ◦ C, there is no effect of
grain size beyond 120 MPa. Overall, the effect of grain size on the steady-state creep rate is proportional to the relative contribution of
diffusional creep.
The sensitivity of the creep rates on initial dislocation density and precipitate content increases with stress and temperature.
Remarkably, this trend is supported by the experimental data from the NIMS database. This evolving effect of microstructure is also
directly related to the relative contribution of different deformation mechanisms. As shown in Fig. 3, with an increase in stress and
temperature, the relative contribution of dislocation-mediated plasticity increases. In the employed mechanistic model, the variation
Fig. 5. Model predicted time to reach 1% creep strain as a function of stress for different microstructures at (a) 550 ◦ C, (b) 600 ◦ C, (c) 650 ◦ C, and
(d) 700 ◦ C, along with the experimental values (Guguloth and Roy, 2017; Kimura et al., 2009; Sklenička et al., 2003; Yadav et al., 2018). A total of
300 different microstructures listed in Table 2 are considered for each grain size and each stress level. The shaded region shows the variation in t1%ϵ
c
due to the changes in the initial microstructure.
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International Journal of Plasticity 158 (2022) 103411
M.A. Kumar and L. Capolungo
in the dislocation density and precipitate content only affects the motion of dislocation motion. Thus, the effect of changes in the initial
microstructure on creep rates is significant in the dislocation process dominated high stress and temperature regime. Finally, these
simulations strongly suggest that the variability in the steady-state creep rates from the experimental literature is most likely due to the
differences in the initial microstructures of the samples used for the experiments. The possible uncertainty associated with the
experimental procedure and data processing is unlikely to be the key reason for the observed discrepancies.
3.3. Effect of initial microstructure on transients via creep strain accumulation
In this section, the effect of initial microstructure on the transients during creep deformation is investigated. Here, the “time taken
to reach 1% creep strain” (tc1%ϵ ) is considered as a measure to capture the effect of initial microstructure on both the primary creep rate
and the kinetics of primary-to-secondary creep. Also, this time for 1% strain is often used as a failure or creep lifetime criteria in the
creep-fatigue community (Boiler, 1998). Note that the calculated strain does not include the elastic contribution. The elastic strain
accumulated during the loading stage and primary creep is expected to be minimal compared to the plastic creep strain. Thus, the
calculated time to reach 1% strain can reasonably be compared with experimental values. The predicted time to reach 1% strain is
shown in Fig. 5 for four different temperatures as a function of the imposed stress along with the experimental measurements
(Guguloth and Roy, 2017; Kimura et al., 2009; Sklenička et al., 2003; Yadav et al., 2018). Here also, the color shade provides the
variation in the tc1%ϵ due to differences in the initial dislocation density and precipitate content for each grain size. The experimental
data points are extracted from the experimentally reported creep strain curves.
First, as shown in Fig. 5(b), the envelope of predicted time to 1% strain encloses the experimental values. Note that these values are
not considered in the model calibration. Second, similar to the steady-state creep rates shown in Fig. 4, the effect of initial micro­
structure on tc1%ϵ increases with an increase in stress and temperature. The underlying reason for this variation is the same as the one
presented in the previous section, i.e., the role of initial microstructure is significant in the dislocation plasticity-dominated high stress
and temperature regime, not in the diffusion-dominated regime. At the same time, in contrast to the steady-state creep rate profiles
shown in Fig. 4, the differences in initial microstructure induced variation in tc1%ϵ suddenly reduces beyond a particular imposed stress
for high-temperature cases, see Fig. 5(c) and (d). A detailed analysis of these cases found that the stress effect on primary creep rate
changes tc1%ϵ significantly. The creep strain curves for an imposed stress of 140 MPa and 150 MPa at 700 ◦ C are plotted in Fig. 6. These
two stresses are selected to understand the abrupt changes in the tc1%ϵ spread observed in Fig. 5(d). The creep curves shown in Fig. 6
correspond to the initial microstructure of Dg = 9.2μm, ρci = 1 × 1012 m− 2 , ρcw /ρci = 10, NMX = 1 × 1020 m− 3 and dMX = 25 × 10− 9 ,
which is the microstructure gives higher tc1%ϵ for 140 MPa loading condition. The inset in Fig. 6 shows the evolution of creep rate in the
early stages of creep simulation. Fig. 6 clearly shows that the significant increase in the primary creep rate with stress (ϵ̇P = 1.3 ×10− 4
at 140 MPa and 2.4 × 10− 4 at 150 MPa) and the short duration of primary creep (<0.5h) significantly changes the time required to
accumulate the creep strain. At 150 MPa, 1% strain is accumulated within 0.17h, whereas for the same strain level, the 140 MPa case
requires around 30 h. Note that the critical stress at which one sees such a change in the initial microstructure effect is highly sensitive
to the selected strain value for the analysis. Furthermore, a thorough analysis of all the cases reveals that the observed changes in the
tc1%ϵ spread is mainly associated with the lower precipitate content microstructures.
Fig. 6. Model predicted creep strain profiles for the imposed stresses of 140 and 150 MPa at 700C. The figure inset shows the creep rate in the initial
stages of the creep simulation. The initial microstructure for this particular case is: Dg = 9.2μm, ρci = 1 × 1012 m− 2 , ρρcw = 10, NMX = 1 ×
1020 m−
3
ci
and dMX = 25 × 10− 9 m.
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International Journal of Plasticity 158 (2022) 103411
M.A. Kumar and L. Capolungo
4. Conclusions
In this work, a recently developed advanced mechanistic model is employed to investigate the effect of initial microstructure on the
creep response of grade-91 alloy. First, the model parameters were calibrated using a few experimental creep data from the literature.
Then, the model is tested against a wide set of experimental data for validation purposes -hundreds of creep tests covering a wide range
of stress and temperature were used for that purpose-. The proposed model is found to quantitatively reproduce the experimentally
observed stress and temperature effects on (i) steady-state creep rates and (ii) primary-to-secondary transients. Further, the poly­
crystalline framework is shown to correctly predict the creep responses outside the stress and temperature regimes used for calibration.
Critically the model was then used to quantify the effects of most relevant microstructural features (dislocation density, precipitate
content, and grain size) on the creep responses and also reports its dependency on the stress and temperature. The effect of initial
dislocation density and precipitate content on the creep responses is found to be significant in the high stress and temperature regime
where dislocation plasticity is dominant, compared to the low stress and temperature regime where the diffusional creep is dominant.
On the other hand, in this low-stress and temperature regime, grain size plays a critical role.
Upon comparing model predictions against hundreds of experimental tests, it is found that experimentally reported variability in
creep responses is likely due to differences in the initial microstructure. Thus, using this mechanistic modeling approach, one could
potentially identify and optimize the key microstructural features to obtain the desired performance for a given operating condition.
Further, it is found that by sampling through the microstructural fingerprint, one can get an absolute upper bound for the secondary
creep rate as a function of stress and temperature. These limiting cases are likely to enable a microstructure-based risk assessment in
using grade-91 under creep loading scenarios.
Data availability statement
Data available on request from the authors: The data that support the findings of this study are available from the corresponding
author upon reasonable request.
CRediT authorship contribution statement
Mariyappan Arul Kumar: Conceptualization, Methodology, Investigation, Software, Visualization, Writing – original draft.
Laurent Capolungo: Conceptualization, Funding acquisition, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Data availability
Data will be made available on request.
Acknowledgments
This work is funded by the US Department of Energy, FE program XMAT and NE program NEAMS.
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