International Journal of Plasticity 54 (2014) 96–112
Contents lists available at ScienceDirect
International Journal of Plasticity
journal homepage: www.elsevier.com/locate/ijplas
A modelling approach to yield strength optimisation in a
nickel-base superalloy
D.M. Collins a,⇑, H.J. Stone b
a
b
Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK
Rolls-Royce UTC, Department of Materials Science & Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge CB3 0FS, UK
a r t i c l e
i n f o
Article history:
Received 14 February 2013
Received in final revised form 15 August
2013
Available online 25 August 2013
Keywords:
Nickel alloys
Heat treatment
Tensile
Precipitation
Plasticity
a b s t r a c t
A computational methodology combining models of precipitation and dispersion strengthening with grain growth and grain boundary hardening has been produced to provide a
predictive capability of the microstructure and yield strength of nickel-base superalloys
subjected to arbitrary thermal cycles. This methodology has been applied to optimise
the post-forging heat treatment of the advanced polycrystalline nickel-base superalloy,
RR1000, to provide an improved proof stress. The temperature dependent antiphase
boundary energies required were obtained using thermodynamic data and temperature
dependent lattice parameters obtained via in situ synchrotron X-ray diffraction. Optimal
yield strength properties between 600 and 700 °C were predicted with precipitates in
the range of 34–57 nm. The precipitation modelling software, PrecipiCalc was used to optimise the solution and ageing heat treatments to maximise the volume fraction of intragranular c0 precipitates within the target precipitate size range, whilst maintaining a
critical minimum volume fraction of primary c0 to give a grain size of 7 lm. The optimal
yield strength of the material was predicted following a heat treatment consisting of 4 h
at 1105 °C; cooling to ambient at 40 °C s1, and ageing for 16 h at 798 °C. Tensile testing
at 650 °C of samples subjected to this heat treatment showed a 125 MPa increase in yield
strength over RR1000 in the conventional microstructural condition. However, this was
accompanied by a significant loss of ductility.
Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Nickel-base superalloys are an important class of materials, capable of operating under high loads at elevated temperatures whilst maintaining good surface stability with inherent oxidation and hot corrosion resistance (Sims et al., 1987). Their
excellent high temperature strength principally arises from a dispersion of L12, c0 precipitates, with compositions selected to
have a low lattice misfit with the A1, c matrix (Mishima et al., 1985; Brückner et al., 1997). In the polycrystalline nickel-base
superalloys used for turbine disc applications, the microstructure typically contains three distinct c0 distributions, denoted as
primary, secondary and tertiary c0 . During manufacture of turbine disc components, heat treatments are applied that tailor
the size, distribution and morphology of these precipitates to provide mechanical properties suitable for the application.
The primary c0 forms at grain boundaries and serves to inhibit grain boundary migration by Zener pinning. These precipitates typically have diameters of 1–5 lm (Connor, 2009), with their size and volume fraction being controlled via solution
⇑ Corresponding author.
E-mail address: david.collins@materials.ox.ac.uk (D.M. Collins).
0749-6419/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijplas.2013.08.009
97
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
heat treatments close to the c0 solvus temperature. On cooling from the solution heat treatment temperature, sufficient undercooling of the supersaturated c matrix allows nucleation of the secondary c0 precipitates, which grow to an extent dependent on the time dependent diffusion fields surrounding each nuclei. With sufficiently high cooling rates, the lower
diffusional mobility of atomic species may lead to the formation of channels in the matrix which cannot interact with the
diffusional fields of the secondary precipitates (Radis et al., 2009). Upon further cooling, nucleation of a tertiary c0 population
may then occur within these channels (Wen et al., 2003). As the size, morphology and distribution of c0 precipitates critically
control mechanical properties, the multimodal c0 distributions produced in nickel-base superalloys have been extensively
studied using both experimental and modelling methods (Babu et al., 2001; Wen et al., 2003, 2006; Sarosi et al., 2007; Singh
et al., 2011). For the nickel-base superalloy, RR1000, the precipitate size distributions (PSDs) of the secondary and tertiary c0
precipitates typically lie in the ranges, 50–350 nm and 5–50 nm, respectively (Connor, 2009).
During the contemporary process of dual microstructure heat treatments (DMHT), the rim of the disc is subjected to
supersolvus temperatures, dissolving the primary c0 , allowing grain growth and an increased volume fraction of intragranular c0 , thereby improving creep properties. Conversely, the bore of the disc is subjected to a subsolvus heat treatment,
retaining the primary c0 , which inhibits grain growth and preserves the fine grain structure necessary for static strength
and good fatigue crack properties (Mitchell et al., 2008; Mourer and Williams, 2004). As each radial location will experience
a unique thermal cycle, as illustrated by Fig. 1, prediction of the location specific microstructure and properties is not trivial.
Unlike the location specific solution heat treatment experienced during DMHT processing, ageing heat treatments are
typically performed under isothermal conditions. This cannot compensate for any variations in secondary and tertiary c0
PSDs that arise as a consequence of the location specific thermal cycles experienced prior to ageing. The isothermal heat
treatment conditions selected will therefore be a compromise between the range of properties required across the disc
rather than optimised to meet the needs of a specific location. In principle, it may be possible to develop location specific
ageing heat treatments to optimise the PSDs and hence the properties across the disc. However, determination of the conditions required using experimental techniques is quite impractical, with each radial location requiring a unique heat treatment. It is therefore desirable to be able to accurately predict the precipitation behaviour for an arbitrary thermal cycle and,
using this, tailor a heat treatment for target precipitate sizes and subsequent properties.
In a study by Jackson and Reed (1999), the size of the secondary c0 in the nickel-base superalloy U720Li was found to be
strongly dependent on the cooling rate from the solution heat treatment. In addition, only coarsening of the tertiary c0 was
observed during ageing, with the secondary c0 remaining largely unchanged. Tensile testing of samples following various
heat treatments showed an approximately parabolic relationship between the proof stress and ageing time, with peak yield
strength properties being achieved with a precipitate size close to the transition from strong to weak dislocation coupling.
A later study by Kozar et al. (2009) assessed the strengthening mechanisms in the polycrystalline nickel-base superalloy,
IN100. Their investigation also incorporated models of weak and strong dislocation coupling, similar to the study by Jackson
and Reed (1999). Modifications of note included the assumption that all dislocations have mixed character and that precipitate sizes are described by distributions, rather than a mean radius alone. This is particularly important when the distribution extends across the transition from weak to strong dislocation coupling. In addition, the effects of temperature on pair
coupling was also incorporated by the addition of thermally activated deformation, as described by Kocks et al. (1975).
SOLUTION HEAT
TREATMENT (SHT)
Region of disc with a
supersolvus heat treatment
Temperature
’ solvus
temperature
Region of disc with a
subsolvus heat treatment
Cooling rate dependent
on position
radial position
AGEING HEAT
TREATMENT
Heating rate
dependent on
radial position
Radial Position
4 hrs at SHT
temperature
End of SHT
Time
Rim
{
Bore
16 hrs at 760 C
Fig. 1. Schematic illustration of the post-forging heat treatments experienced by a turbine disc during dual microstructure heat treatments.
98
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
These previous investigations each describe a simple and effective method of optimising the yield strength by tailoring
the size of the c0 precipitates. This study aims to extend this approach to optimise the yield strength of RR1000 by designing a
heat treatment using only data on the initial alloy condition (bulk composition, grain size and c0 particle size distributions).
The microstructural parameters produced after an arbitrary thermal cycle are calculated using; the commercial modelling
software, PrecipiCalc, to predict the PSDs and an expression to describe grain growth in the presence of grain boundary pinning primary c0 precipitates. The microstructural parameters predicted were then combined with values of the anti-phase
boundary energy, determined by thermodynamic calculations, and the measured lattice parameters of the c and c0 phases
to predict the critical resolved shear stress (CRSS). By identifying the peak CRSS as a function of intragranular c0 volume fraction and precipitate radius, heat treatments could be predicted that would provide the optimal yield strength. The effect of
solid solution strengthening has been omitted from these calculations as the variation in the composition of the c matrix
under the different heat treatment conditions examined, and hence also the extent of matrix solid solution strengthening,
was deemed insufficient to have a significant effect upon the overall strength of the alloy.
During the optimisation process, the following additional constraints were applied to obtain a suitable heat treatment: (i)
Sufficient primary c0 should be present in the material to retain a fine grain size. (ii) The heat treatment should consist of a
solution heat treatment, a controlled cool, and an ageing heat treatment. (iii) For economic reasons, the duration of the heat
treatments should not exceed those which are currently given during turbine disc processing.
2. Background
Reliable expressions are required for the effective prediction of the yield strength from microstructural parameters for
nickel-based superalloys. As such, a brief overview of the expressions used in this study are provided below.
2.1. Weak and strong dislocation coupling
The change in the critical resolved shear stress derived from the presence of coherent precipitates such as those found in
c=c0 nickel-base superalloys can be obtained from considering the effect on the passage of a single superpartial dislocation,
leaving an anti-phase boundary (APB) in its wake. This effect is shown schematically in Fig. 2. The force per unit length, F m ,
opposing the motion of the dislocation from the precipitate is given by
F m ¼ cAPB d
ð1Þ
(a)
(b)
Weak-Pair Coupling
Strong-Pair Coupling
leading superpartial
dislocation
anti-phase
boundary
trailing
superpartial dislocation
(c)
CRSS
optimal precipitate size
CRSS
Maximum
Weak-Pair Coupling
Strong-Pair Coupling
Observed behaviour
Precipitate size
Fig. 2. Weakly (a) and strongly (b) coupled dislocation pairs cutting ordered precipitates with the corresponding critical resolved shear stress relationship,
(c), for each of these mechanisms with respect to precipitate size.
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
99
where d is the average planar diameter of the dislocation in the glide plane within a c0 precipitate and cAPB is the APB energy
per unit area. For spherical precipitates, the maximum force on the dislocation, F 0m , will occur at d ¼ 2xr (Nembach et al.,
1988), where x is a parameter dependent on the size distribution of the precipitates, and is equal to p=4 when the precipitate size distribution is monodisperse. This parameter has been shown to be suitable to other superalloy systems, i.e.
(Nganbe and Heilmaier, 2009).
F 0m ¼
p
2
cAPB r
ð2Þ
A second trailing superpartial dislocation removes the APB, and restores the original c0 structure. Shearing of a precipitate
by the pair of superpartial dislocations will be governed by their respective forces, sbl1 & sbl2 , where b is the Burgers vector, s
is the applied shear stress and l1 & l2 are the dislocations lengths for the leading (1) and trailing (2) superpartial dislocations.
There is a repulsive force, F R , between the paired superpartials and an attractive force from the APB, denoted as cAPB d1 &
cAPB d2 , where d1 & d2 are the lengths of the dislocations lying within the precipitate for each superpartial. The force balance
for the pair of superpartials is then given by (Hüther and Reppich, 1978),
sbl1 þ F R cAPB d1 ¼ 0
sbl2 F R þ cAPB d2 ¼ 0
ð3Þ
Rearranging the above equations and eliminating F R gives
sAPB ¼
1 cAPB d1 d2
2 b
l1
l2
ð4Þ
Weak and strong coupling can be distinguished from the d1 =l1 and d2 =l2 ratios. For weak coupling, as the trailing superpartial is almost straight, as shown in Fig. 2(a), d2 =l2 ¼ /, where / is the volume fraction of the precipitates. The leading
superpartial has a bending component and is strongly dependent on the particle spacing. In this case, d1 ¼ 2r s and the length
l1 is defined by the Friedel condition:
l1 ¼ lF ¼ l
F 0m
2T
1=2
ð5Þ
where lF is the Friedel length and T is the line tension. The final expression to describe weak coupling can therefore be described by Eq. 6.
Ds ¼
1=2
1 cAPB 3=2 bds /
1 cAPB A
/
2 b
T
2 b
ð6Þ
where A is a geometric factor, equal to 0.72 for spherical precipitates, ds is the mean diameter of the precipitate and Ds is the
change in CRSS required to move coupled dislocation pairs. Assuming all of the dislocations are of screw character, the line
tension can be related to the shear modulus, G, giving (Brown and Ham, 1971),
2
T¼
Gb
2
ð7Þ
With supporting observations from microscopy, a theory was developed by Gleiter and Hornbogen (1965) and modified
by Reppich (1982) to describe strong coupling behaviour. Following the force balance given in Eq. (4), only the leading superpartial needs to be considered for weak coupling. However, for strongly coupled dislocations, the pair interaction is much
greater, as shown by Fig. 2(b), which requires a rigorous treatment of the trailing superpartial. For a full description of
the geometric considerations that give the dislocations lengths, the reader is referred to Hüther and Reppich (1978). Following this method, the change in the CRSS by strongly coupled dislocations is given by
Ds ¼
1=2
1 Gb 1=2
pds cAPB
/ 0:72w
1
2
2 ds
wGb
ð8Þ
where w is a parameter describing the repulsion between the pair of dislocations, and is found to be approximately 1 (Hüther
and Reppich, 1979). However, caution must be taken if the distance between partial dislocations becomes very small, due to
interactions between them (Püschl, 2001), in which case a Peierls model treatment is required for suitable corrections (Schoeck, 1994).
Increasing the precipitate diameter further yields another hardening mechanism. Above a critical precipitate size, Orowan
looping may occur, leading to a reduction in the CRSS. For turbine disc materials, it has been reported that Orowan loops are
not observed, with the largest precipitates being preferentially sheared by strongly coupled dislocation pairs instead (Reed,
2006).
The calculation of the weak and strong coupling mechanisms described above is critically dependent on the accurate prediction of APB energies. As such, the following section describes a suitable procedure for obtaining these values.
100
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
2.2. Calculation of APB energy
APB energies in L12 alloys may be calculated by modelling the atomic interactions with the nearest neighbours and higher
correlations. Early APB energy approximations assumed that only the first nearest atomic neighbours needed to be considered (Flinn, 1960). Using stoichiometric A3B compositions, and with the application to the Ni3Fe intermetallic, Inden et al.
(1986) demonstrated, using electron theory, that the energy parameters up to the fourth nearest atomic neighbours could
be calculated and used to obtain approximate APB energies for the {1 1 1} and {1 0 0} glide planes. A later analytical solution
demonstrated that the interaction energies could be described by a series of potentials, W i , describing interactions between
solutes in the ith nearest neighbour positions. Coefficients for these potentials were derived to provide a simple series that
can be used to approximate APB energies for the {1 1 1} and {1 0 0} planes (Khachaturyan and Jr., 1987). The expressions are
given by Eqs. (9) and (10)
2
3
cAPBf111g
16
h
7
¼ 2 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5ðW 1 3W 2 þ 4W 3 6W 4 þ Þ
a
2
2
2
ðh þ k þ l Þ
cAPBf100g
16
h
7
¼ 2 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5ðW 2 þ 4W 3 4W 4 Þ
a
2
2
2
ðh þ k þ l Þ
2
ð9Þ
3
ð10Þ
where a is the c0 lattice parameter and h; k and l are Miller indices. A simple algorithm, proposed by Miodownik and Saunders
(1995), has been developed to calculate APB energies from the thermodynamic databases typically used in the calculation of
phase equilibria with the CALPHAD method (Saunders et al., 2000). Specifically, relationships were found between ordering
enthalpies and interaction coefficients using the Bragg-Williams method (Bragg and Williams, 1934). Ordering enthalpies for
L12 phases were then obtained from thermodynamic databases to calculate the above interaction parameters for the first
three nearest neighbours.1,2 Their model has shown good agreement with simple Ni–Al binary systems and was extended
to higher order systems, including ternaries and commercial superalloys.
Accurate measurements of APB energies have been obtained in binary Ni–Al systems (Karnthaler et al., 1996), and boron
doped Ni3Al (Hemker and Mills, 1993) using a mixture of computer simulation and TEM measurements. APB energies have
also been calculated from experimental measurements in multicomponent alloy systems. Nembach et al. (1992) and Baither
et al. (2002) have also demonstrated that alternative techniques can be employed. With knowledge of the CRSS of an alloy, it
is possible to use the relations for weak and strong dislocation coupling to derive cAPB from Eq. (6) or (8). However, this requires extensive experimental work. If the material also displays the presence of Orowan loops, the APB energy can also be
calculated from the smallest observed loop size which can be sustained around the c0 precipitates (Nembach et al., 1992;
Baither et al., 2002). This latter method is limited to alloy systems that have been heat treated to provide sufficiently large
precipitates for Orowan looping to occur. It is unlikely, therefore, to be suitable for modern nickel-base superalloys used for
turbine disc applications, where precipitates are sufficiently small for the preferred dislocation interaction with c0 to be
shearing.
3. Modelling
3.1. Optimal precipitate size range
In this study, predictions were made of the critical resolved shear strength of the nickel-base superalloy, RR1000, the
nominal composition of which is given in Table 1. If it is taken that the maximum yield strength will be achieved at the transition from weak to strong pairwise cutting, where the maximum CRSS can be obtained from the point at which Eqs. (6) and
(8) intersect. These functions incorporate the precipitate volume fraction, but consider all of the precipitates to be of a single
c
c0
1
As ordering enthalpies obtained from thermodynamic databases (DHDB and DHDB ) include short range order effects, the required Bragg-Williams ordering
c
c0
enthalpies (DHBWG and DHBWG ) do not require this contribution, and thus a conversion is necessary. A parameterised expression to evaluate this is:
DHcBWG ¼
2
0
0
DHcDB
and DHcBWG ¼ DHcDB þ 0:091DHcDB :
1:1
The ratio between W 1 and W 3 is found to be approximately constant, giving an effective W 01 = W 1 + 2W 2 , and W 1 ; W 2 and W 3 are approximated as:
W 01 ¼
3A þ Bð1cÞ
c
;
24Rcð1 cÞ
W2 ¼
ðBð1 cÞ AcÞ
;
12Rc2 ð1 cÞ
0
W 1 ¼ 0:75W 01
and W 3 ¼ 0:125W 01 :
where A ¼ DHcBWG and B ¼ DHcBWG , R is the molar gas constant, c is the atomic fraction of a solute on the aluminium sublattice
in the L12 structure and a is the c0 lattice parameter.
101
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
Table 1
Nominal RR1000 chemical composition (wt.%) (Hessel et al., 2000).
Element
wt.%
Ni
Co
Cr
Mo
Ta
Ti
Al
B
C
Zr
Hf
bal
18.5
15
5
2
3.6
3
0.015
0.027
0.06
0.75
size, rather than a distribution, as would be measured experimentally. As a result, it is more useful to define a precipitate size
range, rather than a single precipitate size.
The majority of the variables in Eqs. (6) and (8) are temperature dependent physical parameters, specific to the alloy in
question. As such, it is appropriate that the yield strength properties are optimised for service-like temperatures. For this
reason, the target temperature range chosen in this study was 600–700 °C.
The APB energies required for calculating the weak and strong dislocation coupling using Eqs. (6) and (8) were calculated
using the algorithm proposed by Miodownik and Saunders (1995). The parameters used for these calculations are given as a
function of temperature in Fig. 3. This includes the temperature dependent lattice parameter of the c0 precipitates, shown in
Fig. 3(a), measured using in situ synchrotron X-ray diffraction data acquired from the I12 instrument at the Diamond Light
Source within the temperature range of interest in this study. The ordering enthalpy terms for both the c and c0 phases in
RR1000 were calculated and are shown in Fig. 3(b). The interaction energies between the 1st, 2nd and 3rd nearest atomic
neighbours are shown in Fig. 3(c).These values were obtained by simulating thermodynamic equilibrium for a range of temperatures, using the software ThermoCalc (ThermoCalc, 2012) with the TTNI8 thermodynamic database (ThermoTech, 2012).
For the purposes of the calculation of APB energies, the compositions have been modified by removing the hafnium content,
as this element is expected to preferentially form an oxide (Rolls-Royce, 2011), rather than providing solid solution strengthening, as would otherwise be predicted. A first simulation provided both the compositions and phase fractions of the c and c0
phases. The equilibrium calculation was run once again, using only the previously predicted c0 composition for the bulk
chemistry, and allowing only the formation of c0 . This provided the molar ordering enthalpy of c0 at the calculated temperature. This composition was simulated again, though now suppressing the formation of all phases except c, thereby enabling
the molar ordering enthalpy for the disordered c0 to also be obtained. With these data, the APB energies could then be calculated for the {1 1 1} and {1 0 0} planes using Eqs. (9) and (10). The calculated APB energies are shown in Fig. 4 as a function
of temperature and can be seen to decrease monotonically with increasing temperature for both of the calculated slip systems, demonstrating the temperature dependence is significant enough not to be neglected.
The Miodownik and Saunders model (Miodownik and Saunders, 1995) showed good correlation between the predicted
APB energies and those measured experimentally. During this investigation, the materials modelled in the work by Miodownik and Saunders (1995) were re-calculated using the TTNI8 database and provided very good agreement, validating the APB
energies predicted for RR1000 is this study.
The dislocation line tension could next be calculated using Eq. (7), in which the shear modulus was obtained from the
known Young’s Modulus,
pffiffiffi E, (224 GPa) and Poisson’s ratio, m, (0.3) of RR1000 (Rolls-Royce, 2011) and taking the Burgers vector, b, to be equal to 2a=2 for {1 1 1} slip. Assuming a homogeneous isotropic linear elastic relationship between the Young’s
modulus and shear modulus, the following expression was used:
G¼
E
2ð1 þ mÞ
ð11Þ
These variables permitted the CRSS to be calculated for both weak and strong coupling as a function of both precipitate
volume fraction and precipitate size, assuming slip occurs on the {1 1 1} planes. The volume fraction was varied from 0 to
0.38, which corresponds to the possible volume fraction range of secondary and tertiary c0 precipitates, after removing
(a)
(b)
(c)
Fig. 3. Input parameters used for the calculation of APB energies as proposed by Miodownik and Saunders (1995). In (a), the c0 lattice parameter and solute
concentration, c, on the aluminium sublattice of the L12 c0 structure are shown with respect to temperature. In (b) the Bragg–Williams ordering enthalpies
0
are plotted, given as DHcBWG for c and DHcBWG for c0 . The interaction energies also required for the APB calculation are shown in (c), denoted as W 1 ; W 2 and W 3
for the first, second and third nearest neighbours, and W 01 is an effective interaction energy, as used by the Miodownik method.
102
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
Fig. 4. Temperature dependence of calculated APB energies for the {1 1 1} and {1 0 0} slip planes.
1000
Weak Coupling: vf = 0
Weak Coupling: vf = 0.38
Strong Coupling: vf = 0
Strong Coupling: vf = 0.38
Maximum
for given vf
Critical resolved shear stress,
crss
/ M Pa
800
600
400
200
0
0
50
100
150
200
Precipitate size / nm
Fig. 5. Calculated values of the CRSS at 700 °C assuming strong and weak dislocation coupling. The ideal precipitate size to achieve maximum strength for a
given volume fraction are identified. The overlaid band denotes the final measured PSD following the optimised heat treatment.
the approximate volume fraction of primary c0 from the total equilibrium c0 volume fraction. To find the maximum CRSS, Eqs.
(6) and (8) were subtracted and solved using the bisection method with respect to precipitate size, adopting an iterative process until the root of the function was found. The results obtained at 700 °C are shown in Fig. 5, and illustrate the sensitivity
of the maximum critical resolved shear stress to the c0 volume fraction. Increasing the c0 volume fraction greatly increases
the maximum CRSS, and hence it is highly desirable to obtain the largest possible volume fraction of intragranular precipitates within the target precipitate size range.
The calculations of strong and weak dislocation coupling were repeated at 600 °C and once again solved to find the maximum CRSS. These values and the corresponding strongest particle sizes are shown as a function of volume fraction in Fig. 6.
The optimal precipitate size range was selected to be from 34 nm, corresponding to the strong to weak dislocation coupling
transition at 600 °C for low precipitate volume fractions and 57 nm, corresponding to the strong to weak dislocation coupling
transition at 700 °C for a precipitate volume fraction of 0.46 (the maximum volume fraction of c0 available in the material).
This allowed the optimal strengths to be calculated in the absence of any primary c0 .
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
103
Fig. 6. Calculated c0 precipitate diameter and corresponding maximum critical resolved shear stress at the strong to weak dislocation coupling transition as
a function of c0 volume fraction.
3.2. Modelling the microstructure
The properties of the material and their temporal evolution are critically dependent on the number density of precipitates
within a critical size range (Robson, 2004). Therefore, it has become common practice for models to calculate PSDs to accurately explicate precipitation behaviour. These distributions are typically described by distinct control volumes of precipitates within a specified size range and the temporal evolution of the material can be described by the flux of precipitates
into and out of these control volumes (Myhr and Grong, 2000). The commercial modelling software, PrecipiCalc is such a tool,
permitting the simulation of multimodal precipitate evolution, predicting PSDs as a function of user defined thermal cycles.
A full description of the equations used by PrecipiCalc including a model explanation has been given by Jou et al. (2004).
3.3. Simulation of heat treatments
With the target precipitate size range identified, heat treatments could next be designed to maximise the volume fraction
of precipitates within the desired range. First, an as-forged starting condition was selected, and the starting precipitate size
distribution was characterised using scanning electron microscopy. A matrix of solution heat treatments was simulated between 1080 °C and 1120 °C in steps of 1 °C. The length of each heat treatment was chosen to be 4 h, matching the solution
heat treatment time used during current turbine disc processing. To obtain an estimate of grain size, the equation
lim ¼ kðr=/Þ (Andersen and Grong, 1995) was used, with the Zener coefficient, k, taken to be 4/3 (Zener, 1948), where /
D
is the volume fraction of primary c0 and r is the mean radius of the precipitate size distribution. A grain growth study of
lim is a good approximation for the grain size when primary c0 is present (Collins et al., 2013). For imRR1000 showed that D
proved accuracy, the full primary c0 distribution could be considered, rather than a mean primary c0 precipitate size alone. In
addition, a refined value for k, specific to RR1000, may further improve the model predictions. The primary c0 volume frac lim , and the differential of D
lim , dðD
lim Þ=dT, has been plotted as a function of tempertion, mean radius of the distribution, D
lim deviates from approximate linearity,
ature in Fig. 7. The heat treatment temperature was chosen at the transition where D
lim Þ=dT. This corresponds to an estimated grain size of 7.2 lm, which is considered satas represented by the increase in dðD
isfactory for the bore region of a turbine disc. This condition corresponds to a primary c0 volume fraction of 0.096 and a mean
radius size is 0.54 lm. These values match well with the minimum values observed experimentally prior to rapid grain
growth in RR1000 (Collins et al., 2013). Using this approach, a solution heat treatment of 4 h at 1105 °C was selected.
An appropriate cooling rate from the solution heat treatment temperature was next required. During cooling, reprecipitation of secondary and tertiary c0 will occur, with faster cooling rates providing finer precipitates. A series of cooling rates
were modelled from the selected 1105 °C solution heat treatment, providing predictions of the PSDs that would be obtained
at room temperature. The cooling rates simulated ranged from 1 °C h1 to 1000 °C s1. A few examples of the PSDs predicted
in this way are shown in Fig. 8. The PSD data is shown as continuous data (with units m4), as used for modelling purposes,
though this can be readily integrated between an upper and lower radius to calculate a histogram of 3D number density (m3
units). For each of the PSDs shown, the target precipitate size range has been superimposed. For slower cooling rates, such as
shown in Fig. 8(a) at 1 °C sec1, 3 distinct c0 populations are predicted, where the primary c0 have the largest diameters of all
of the precipitate populations, are assumed to be present only at grain boundaries. For cooling rates slower than this, such as
1 °C hr1, the same observations were replicated. All precipitates of the intragranular secondary c0 can be seen to be above
the target optimal size range, and therefore would not confer best yield strength. Furthermore, tertiary c0 is significantly below the target size range. Cooling more rapidly, at 40 °C s1 for example, as shown in Fig. 8(b), predicts that a distinct sec-
104
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
lim and the derivative of D
lim with respect to
Fig. 7. Temperature dependence of the primary c0 volume fraction and mean radius; the grain size, D
temperature. A heat treatment temperature was selected that provided the minimum volume fraction and mean radius of c0 , prior to the onset of rapid grain
growth.
(a)
OPTIMAL PRECIPITATE RANGE
1°C/sec
1000°C/sec
40°C/sec
(b)
(c)
Fig. 8. PSDs predicted by PrecipiCalc for a series of cooling rates. The target precipitate size range is shown with the light grey band shown in each graph.
ondary c0 no longer has time to form, consistent with the observations of Wen et al. (2003). Instead, all intragranular c0 is
now below the target size range and can be coarsened into this range through a judiciously selected ageing heat treatment.
Cooling at even faster rates, such as 1000 °C s1 as shown in Fig. 8(c), simply provides finer intragranular c0 , which would be
unnecessarily small, necessitating longer ageing heat treatments to grow it to the desired size range. A cooling rate of
40 °C s1 was therefore chosen. It can also be seen that all of the PSDs predicted following cooling show a population doublet
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
105
Max vf
after 16
hours
Max vf
after 100
hours
0
Fig. 9. A contour plot of the volume fraction of c0 within the target size range plotted as a function of ageing temperature and time. The ageing temperatures
that provide the highest c0 precipitate volume fractions in this range after heat treatment for 16 h and 100 h are identified.
within the smallest precipitate population, suggesting a near pseudo-quadramodal c0 population may be found. To the
authors’ knowledge, this has not been previously reported.
The final stage of the heat treatment is the ageing cycle. Using the PSD predicted following the 40 °C s1 cool as a starting
condition for the PrecipiCalc model, a broad matrix of possible ageing heat treatments was simulated. The temperatures
modelled were between 700 °C and 900 °C in increments of 1 °C. Each temperature was modelled to monitor the changing
PSDs at a 5 s time resolution, up to a maximum time of 100 h. To identify the best heat treatments, the volume fraction of c0
within the desired precipitate size range was calculated. The results of this are shown in the contour plot in Fig. 9. At lower
temperatures, the coarsening of the distribution is limited, with only a small proportion of precipitates coarsening within the
critical size range after long exposures. Increasing the temperature allows the precipitates to coarsen more rapidly, with an
increased proportion of precipitates falling within the critical size range. A maximum c0 precipitate volume fraction of 0.278
is achieved within the critical range after 100 h at 747 °C. This ageing time is on the edge of the matrix tested, so clearly the
global maximum has not been found. Increasing the ageing time would find a higher volume fraction, though would require
a heat treatment that is impractically long. Ageing the material for a shorter time, but at a higher temperature will not
achieve such a high volume fraction within the desired size range, though does get close. Ageing the material instead at
798 °C for 16 h, a more commercially acceptable ageing time, achieves a c0 volume fraction in the critical range of 0.266. This
has therefore been selected as the optimised ageing heat treatment.
4. Validation of proposed heat treatment
A tensile test was conducted to compare RR1000 in the standard heat treatment condition to material subjected to the
optimised heat treatment. The specimens were sealed in a quartz glass tube filled with argon to limit any surface oxidation,
and solution heat treated in a calibrated box furnace at 1105 °C for 4 h, followed by an air cool. At this stage, the primary c0
should have reached the desired equilibrium volume fraction. To replicate the proposed solution heat treatment and cooling
cycle, producing the desired PSDs for the intragranular precipitates, a specially designed electro-thermal mechanical tester
(ETMT) was used. This apparatus can subject samples to arbitrary thermal cycles with resistive heating and can achieve controlled heating and cooling rates of up to 200 °C s1.
Tensile specimens were machined using electro-discharge machining (EDM) in a specially adapted geometry, accounting
for the parabolic temperature profile generated across a sample when using the ETMT. The temperature deviation was expected to be not more than ±5 °C across the central 1 mm of the test piece (Sokolov et al., 2002). For accurate control of the
thermal cycle, an R-type thermocouple was attached to the centre of the sample, the position reaching the highest temperature. A thermal cycle was applied using the ETMT, raising the temperature initially to the solution heat treatment temperature, holding for 15 min, followed by the 40 °C s1 cool. Conducting the entire solution heat treatment using the ETMT
would have been preferable. However, the limited oxidation protection offered by this instrument makes sustained high
temperature heat treatments impractical. The method used exposes the samples to a high temperature for a limited time
period only, initially dissolving the secondary and tertiary c0 formed from the previous cool, then allowing reprecipitation
of intragranular c0 at the desired cooling rate. The thermal cycle achieved deviated only from the programmed cycle at
low temperatures (<400 °C) and, as such, will not have influenced the precipitation behaviour. Following resolutioning
and controlled cooling at 40 °C s1, the samples were aged at 798 °C for 16 h using a calibrated box furnace.
106
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
To validate the modelling predictions, the PSDs of the primary c0 and intragranular c0 were compared with measurements
made by electron microscopy. Example micrographs are given in Fig. 10, showing (a) the primary c0 and grain structure, and
(b) all the remaining intragranular c0 within the grains and their associated PSDs, (c) and (d). The primary c0 remains present
at the grain boundaries, though the micrograph shows considerable growth of the grains. The mean grain size was determined to be around 7 lm, which is the desired size and suggests that the heat treatment was successful. The volume fraction
of primary c0 was measured to be 0.075 ± 0.01 and the mean precipitate size was measured to be 0.45 lm ± 0.06 lm. This
latter measurement includes an adjustment from 2D to a 3D, valid when assuming the precipitates can be fitted to a log-normal distribution. As the predicted volume fraction of 0.096 appears to be an overestimate, and the predicted mean precipitate size, 0.54 lm, appears to be an underestimate, the measured Dlim is almost identical to the desired value.
Inspection of the secondary c0 in Fig. 10(b) suggests that the majority of intragranular c0 is within a narrow size range,
exactly as was desired from the proposed heat treatment. Measurements of these fine precipitates give a mean radius of
30.3 ± 0.1 nm. The method used in preparing this material for microscopy involved electroetching the surface with an aqueous 10 vol.% phosphoric acid solution, which removes the c matrix leaving the c0 in relief. Though this method allows good
etching rate control by adjustment of the applied voltage, over etching still occurs when precipitates are very fine. This leads
to the measurement of excessively large volume fractions.
PrecipiCalc predicted that 0.266 (by volume fraction) of c0 would be present in the radius range of 17–28.5 nm. Assuming
the total available c0 volume fraction, including primary c0 is 0.46, this leaves a volume fraction of 0.364 c0 to reprecipitate
upon cooling. By proportion, 21% of c0 present is primary c0 whilst the remaining 79% is reprecipitated as intragranular c0
upon cooling. From the measured precipitate distribution following the optimised heat treatment, approximately 34% (by
proportion of all c0 ) is within the desirable size range. This corresponds to a volume fraction of 0.155. These results are very
encouraging, since the proposed precipitate size range was very narrow. It does appear, however, that the ageing heat treatment has underestimated the rate of coarsening.
To identify any improvement in yield strength, a comparison was made with data obtained by Connor (2009) on samples
of RR1000 removed from various positions in a disc after a DMHT and tensile tested at 650 °C. In the study by Connor, the
0.2% yield strengths were found to be 932, 1039 and 968 MPa at the rim, bore and transition regions, respectively. This present study has optimised the yield strength properties for 600–700 °C, making 650 °C a suitable test temperature. Prior to
conducting any tensile tests, any surface oxide from the prior heat treatments was removed using manual abrasion with
(a)
(c)
(b)
(d)
Fig. 10. SEM micrographs of RR1000 following the optimised heat treatment (a), at low magnification showing the grain structure and primary c0 , and (b) at
high magnification showing secondary and tertiary intragranular c0 . The associated precipitate size distributions of the primary c0 and intragranular c0 are
shown in (c) and (d), respectively.
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
107
1400
1200
Stress / MPa
1000
800
600
Optimised heat treatment
LDC, bore region
LDC, transition region
LDC, rim region
0.2% o set yield strength
400
200
0
0
5
10
15
Strain / %
20
25
Fig. 11. Tensile stress–strain curves at 650 °C of RR1000 in the optimised heat treatment condition and from RR1000 samples measured by Connor (2009).
SiC abrasive paper. As before, an R-type thermocouple was spot-welded to the centre of the specimen, along with Pt wires
2 mm either side, adjacent to the thermocouple, to measure changes in resistance and provide strain measurements (Roebuck et al., 2004).
A series of tensile tests were conducted at 650 °C under load control, increasing the load by 10 N s1 until the sample fractured. Stress–strain data obtained from the ETMT is shown in Fig. 11, referred to as the optimised heat treatment, along with
test data obtained by Connor (LDC) for RR1000 samples extracted from the rim, bore and transition regions of a DMHT disc.
The specimens tested in this study show a higher yield strength than all of the DMHT specimens, although they fail in a brittle manner, with an inferior ultimate tensile strength. The measured 0.2% yield strength of the optimised heat treated
RR1000 is 1164 MPa ± 10 MPa, which demonstrates an increase of 125 MPa over the best performing DMHT RR1000 specimen. The ultimate tensile stress was measured as 1172 MPa ± 10 MPa.
5. Discussion
5.1. Validity of the precipitate strengthening model
The Young’s modulus and Poisson’s ratio are temperature dependent material parameters, giving a shear modulus calculated in Eq. (11) that will also vary with temperature. The weak and strong dislocation coupling expressions used in this
study to predict the CRSS have different dependencies on the shear modulus. Consequently, the intersection of the two functions used to identify the maximum CRSS with respect to precipitate size will change as the temperature is increased. Without validated temperature dependent elastic constants available for RR1000, the methodology used in this study has
assumed that the temperature dependence of the shear modulus will not be significant when predicting a target precipitate
size range. For the polycrystalline nickel-base superalloy MAR-M200, at room temperature the Young’s modulus and shear
modulus are measured to be 230 GPa and 84 GPa, respectively, and at 650 °C, the target operating temperature proposed in
this study, the Young’s modulus and shear modulus decreases to 195 GPa and 70 GPa, respectively (Dandekar et al., 1981).
Assuming a homogenous linear elastic isotropic relationship from Eq. 11, the Poisson’s ratio at room temperature is 0.37
and at 650 °C is 0.39. This is notably a small change, and a similar increase has been reported in the CMSX-4 nickel-base
superalloy (Siebörger et al., 2001). This reduction in measured elastic modulus is comparable to measurements made on
other nickel-base superalloys, i.e. (Hicks and King, 1983; Kuhn and Sockel, 1988; Siebörger et al., 2001; Sawant et al.,
2008), and is therefore expected to be similar in the case of RR1000.
Using elastic constants from other superalloy materials, it is expected that the Young’s modulus may reduce by up to 15%
and the Poission’s ratio may increase by approximately 5%. Using this crude estimate, the shear modulus will drop from
86 GPa (from measured room temperature data) to 72 GPa. The corresponding target precipitate size range is calculated
to be 29–48 nm. compared to 34–57 nm when estimated using room temperature predictions. This therefore indicates that
room temperature elastic constants will overestimate the optimal size range, though further comprehensive measurements
will be required if this is to be suitably validated.
5.2. Material deformation
A tensile study by Sharghi-Moshtaghin and Asgari (2004) found a strong dependence of the ductility on the heat treatment used, observing an increased ductility with materials having larger precipitate sizes. However, this relationship was
shown to be strongly related to the test temperature. Good ductility would be expected in the presence of a homogenous
108
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
(a)
(b)
200 nm
1 m
Fig. 12. SEM micrographs of RR1000 following a conventional heat treatment showing (a) intragranular secondary and tertiary c0 precipitates and (b),
primary c0 precipitates present on the grain boundaries.
distribution of dislocations on the {1 1 1} slip planes, and reduced ductility may be explained by a local strain accumulation.
Other alloy systems have also shown reduced ductility as the c0 precipitates become smaller, and this was also reported to be
inversely correlated to any increase in yield strength (Xia et al., 2007; Chang and Liu, 2001).
The limited ductility observed in the tensile specimens following the optimised heat treatment will ultimately restrict its
feasibility. The significant drop in ductility has occurred with a microstructure having a narrow precipitate size range, quite
different from the polydisperse PSD with larger precipitates typically present in conventionally processed RR1000 (Fig. 12),
which does not display limited plasticity. This phenomena has additionally been observed in single crystal turbine blades,
where aged material provided greater ductility, due to homogenisation of deformation, and noting greater matrix deformation prior to precipitate shearing (Pessah-Simonetti et al., 1993). The accumulation of strain will lead eventually to sessile
dislocations, whilst at sufficiently high temperatures (pronounced at >700 °C in U720Li (Gopinath et al., 2008)), diffusion
controlled dynamic recovery enables dislocations to become mobile. From the lack of ductility observed in this study on
RR1000, it appears that the test temperature was too low for this mechanism to have an effect.
For the material to fail suddenly, dislocations become immobile once a critical strain is reached. Furthermore, the number
of obstacles must be sufficiently high, that the dislocations cannot take an alternative lower stress route. It is hypothesised
that the optimised heat treatment used in this study has produced a case where these factors have been satisfied. Though the
yield strength has been increased, the ultimate tensile strength is significantly lower than is typically observed in this material. A key difference between the microstructure produced by the optimised heat treatment in this study, compared to the
microstructure from conventionally processed RR1000, is the presence of a single, very fine intragranular size distribution
rather than the two distinct intragranular precipitate populations, secondary c0 and tertiary c0 . However, the volume fraction
of tertiary c0 is typically small compared to the secondary c0 also present. Failure will occur when the deformation creates a
localised strain concentration, corresponding to an inhomogeneous distribution of dislocations in the glide plane (Bettge
et al., 1995). The lack of ductility indicates the dislocations are piling up at the c=c0 interface, rather than shearing the precipitates, which would have alleviated the localised stress concentration whilst permitting homogenous strain accumulation,
and hence ductility.
Kozar et al. (2009) show clear evidence of dislocations cutting large c0 precipitates, demonstrating that cross slip pinning
is a strengthening mechanism. By creating a microstructure with only very fine precipitates, this strengthening effect has
been eliminated in this study. Furthermore, the high CRSS required to cut the precipitates, paired with a very small interprecipitate spacing, provides a greatly restricted area through which the dislocation is permitted to sweep when shearing
a precipitate. Thermally activated cross-slip pinning will now be severely restricted. For a microstructure with a range of
precipitate sizes, some strain will be accumulated as this effect is less significant when the precipitate size requires a lower
CRSS for it to be sheared by a dislocation. The effect observed in this study demonstrates that a small, narrow precipitate
population will heavily restrict strain accumulation, giving a very low ductility and a low ultimate tensile strength compared
to conventionally heat treated RR1000.
By the incorporation of an extended model to account for strain accumulation and yield behaviour, such as a crystal plasticity framework, i.e. (Shenoy et al., 2008), it is possible that heat treatments may be devised that offer improvements in the
yield strength of RR1000 whilst simultaneously ensuring adequate ductility is preserved. Furthermore, the selected heat
treatment should be commercially feasible; suitable in a manufacturing process, i.e. a cooling rate that can be achieved in
the thickest sections of a component, which will be significantly slower than the 40 °C s1 cooling rate selected from a theoretical optimisation. This study has also accounted only for deformation from the formation of APB partial dislocations, and
has not included the formation of stacking faults, microtwins (Reed, 2006), dislocation homogeneity/heterogeneity
109
Tertiary
DSC: PRIMARY ’ DISSOLUTION
Secondary
DSC: SECONDARY ’ DISSOLUTION
Primary
DSC: TERTIARY ’ DISSOLUTION
-1
/
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
DSC trace
Differential
/
Fig. 13. DSC trace of as-forged RR1000 and its first differential plotted alongside changes in precipitate volume fraction and radius predicted by PrecipiCalc
for the same thermal cycle.
(Gopinath et al., 2008) or solid solution strengthening (Kozar et al., 2009). Again, these factors may be accounted for in a
future detailed study of deformation modes in RR1000.
5.3. Precipitation modelling
The mean precipitate diameter measured on the sample subjected to the optimised heat treatment was 60.6 nm. This is
significantly larger than the 43 nm predicted to correspond to the transition from strong to weak dislocation coupling at the
equivalent c0 volume fraction of 0.155, as can be seen with reference to Fig. 6. This discrepancy may be attributed to the
assumptions made in the precipitate modelling scheme and their applicability to non conventional heat treatments, particularly those that include rapid cooling rates where highly non-equilibrium behaviour may be anticipated, such as those used
in this study.
PrecipiCalc was used by Yoon et al. (2007) in a Ni–Al–Cr–Re quaternary system exhibiting multimodal c0 populations similar to commercial turbine disc alloys. In their study, the assumption used in PrecipiCalc that all of the precipitates present are
at an equilibrium composition was shown to be inappropriate and a largely limiting factor of the model. It was also noted
that the success of modelling predictions are largely based on the accuracy of the thermodynamic databases used. In a different study by Booth-Morrison et al. (2009), a Ni–Al–Cr–Ta quaternary system was simulated using PrecipiCalc, modelling
the effect of reprecipitation of c0 at rapid cooling rates. Primary c0 predictions were shown to be in good agreement with
experimental measurements. However, the secondary c0 formed upon cooling showed little agreement.
The assessments made by Yoon et al. (2007) and Booth-Morrison et al. (2009) are consistent with the observations made
on RR1000 in this study. Using the assumption that all of the precipitates are at an equilibrium volume fraction is not valid
when a cooling rate from above a c0 distribution solvus temperature is rapid. In such circumstances, the rate of diffusion
110
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
would not be rapid enough for an equilibrium composition to be reached. This appears to be the case for the cooling rates
simulated in this study.
To further assess the predictive capability of PrecipiCalc for the alloy used in this study, RR1000, the dissolution temperatures of the c0 distributions were measured using differential scanning calorimetry (DSC), shown in Fig. 13. Using a sample
of RR1000 in an initially as-forged condition, the material was heated at a rate of 50 °C h1. The DSC technique is sensitive
enough to detect the dissolution of the primary, secondary and tertiary c0 , allowing the dissolution temperatures of each c0
distribution predicted by PrecipiCalc to be directly compared.
The predicted volume fractions and mean particle radii for the c0 distributions are shown as a function of temperature
with the DSC measurements in Fig. 13. As the temperature is increased, the DSC measurements show the tertiary c0 beginning to dissolve at 880 °C up to the distribution solvus temperature of 1000 °C. PrecipiCalc predictions, however, show a
drop in volume fraction from 700 °C, with a compete dissolution of the tertiary c0 by 800 °C. Disagreement for this distribution is significant. A similar disparity between the predicted and experimental results can be seen with the secondary c0 .
The DSC shows dissolution between a narrow temperature range of 1090 °C to 1115 °C, whereas PrecipiCalc shows the
distribution volume fraction decreases over 400 °C between 700 °C and 1105 °C. Whilst the most rapid reduction in the
mean radius does occur in the dissolution range identified by DSC, clearly at this temperature, the volume fraction is negligible. The simulated primary c0 dissolution again shows little agreement with the DSC measurements. Dissolution between
1145 °C and 1155 °C is measured, whereas PrecipiCalc predicts a drop in volume fraction between 1190 °C and 1200 °C.
Discrepancies arising between the measured and predicted dissolution temperatures are likely to be present due to a
number of the assumptions made by PrecipiCalc. The mean field assumption enables all precipitates to interact with each
other irrespective of proximity. However, this assumption will be invalid with very rapid cooling rates where the size of diffusion fields becomes increasingly important. If a secondary c0 may interact with any fine tertiary c0 , irrespective of distance,
clearly the rate of tertiary c0 coarsening or dissolution would be overestimated. In spite of the issues identified, the modelling
software has enabled a significant improvement in yield strength to be achieved in RR1000. With further developments and
incorporation of physically based behaviour, improved heat treatment optimisation could be achieved.
6. Summary and conclusions
The yield strength of the polycrystalline nickel-base superalloy, RR1000, has been improved by an optimisation of the
processing heat treatments after forging. The following conclusions can be drawn from this study:
Using calculated APB energies, measured lattice parameters and shear modulus, an optimal precipitate size range of 34–
57 nm between 600 °C and 700 °C was identified, to achieve the maximum CRSS in the alloy.
The modelling software PrecipiCalc was used to design a heat treatment to provide an optimised microstructure. By simulating an extensive matrix of heat treatments the optimal thermal cycle for maximum strength was determined to be; a
solution heat treatment of 1105 °C for 4 h, a controlled cooling rate of 40 °C s1, and finally an age of 16 h at 798 °C.
The proposed heat treatment was tested on as-forged RR1000 specimens to validate these predictions. Microscopy
revealed a satisfactory correlation between the simulated heat treatments and the measured precipitates. Tensile testing
at 650 °C displayed a 125 MPa improvement in yield strength over RR1000 following established dual microstructure
heat treatments, though this was achieved at the expense of ductility and ultimate tensile strength. This was attributed
to the narrow and small size distribution produced with regular interparticle spacings and the high homogeneous resistance to dislocation motion that results.
Validation of PrecipiCalc using DSC showed discrepancies between the observed and predicted c0 distribution solvus temperatures. The assumptions used in this model largely limit the accuracy of selecting an improved heat treatment, particularly with the high cooling rates examined in this study. However, further refinement of this software and
implementation of physically-based models may improve the predictive capability.
Acknowledgments
The authors would like to acknowledge EPSRC and Rolls-Royce plc. for their financial support. With thanks to H.-T. Pang
for help with DSC and ETMT materials testing. The assistance of H.-J. Jou (QuesTek Innovations LLC), R. Goetz (Rolls-Royce
Corp.) and M.C. Hardy (Rolls-Royce plc.) is gratefully acknowledged. Diamond Light Source are also gratefully acknowledged
for the allocation of beamtime on the I12 instrument (EE4791-1), in addition to the assistance of T. Connolley and L.D. Connor throughout the experiment.
References
Andersen, I., Grong, Ø, 1995. Analytical modelling of grain growth in metals and alloys in the presence of growing and dissolving precipitates – I. Normal
grain growth. Acta Metall. Mater. 43, 2673–2688.
Babu, S., Miller, M., Vitek, J., David, S., 2001. Characterization of the microstructure evolution in a nickel base superalloy during continuous cooling
conditions. Acta Mater. 49, 4149–4160.
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
111
Baither, D., Rentenberger, C., Karnthaler, H.P., Nembach, E., 2002. Three alternative experimental methods to determine the antiphase-boundary energies of
the c0 precipitates in superalloys. Philos. Mag. A 82, 1795–1805.
Bettge, D., Osterle, W., Ziebs, J., 1995. Temperature dependence of yield strength and elongation of the nickel-base superalloy IN738-LC and the
corresponding microstructural evolution. Z. Metallkd 86, 190–197.
Booth-Morrison, C., Noebe, R., Seidman, D., 2009. Effects of tantalum on the temporal evolution of a model Ni–Al–Cr superalloy during phase decomposition.
Acta Mater. 57, 909–920.
Bragg, W.L., Williams, E.J., 1934. The effect of thermal agitation on atomic arrangement in alloys. Proc. R. Soc. A 145, 699–730.
Brown, L.M., Ham, R., 1971. Strengthening Mechanisms in Crystals. Elsevier Publishing Company Ltd., London, UK.
Brückner, U., Epishin, A., Link, T., 1997. Local X-ray diffraction analysis of the structure of dendrites in single-crystal nickel-base superalloys. Acta Mater. 45,
5223–5231.
Chang, K.M., Liu, X., 2001. Effect of c0 content on the mechanical behavior of the WASPALOY alloy system. Mater. Sci. Eng. A 308, 1–8.
Collins, D., Conduit, B., Stone, H., Hardy, M., Conduit, G., Mitchell, R., 2013. Grain growth behaviour during near-c0 solvus thermal exposures in a
polycrystalline nickel-base superalloy. Acta Mater. 61, 3378.
L.D. Connor, 2009. The Development of a Dual Microstructure Heat Treated Ni-base Superalloy for Turbine Disc Applications. Ph.D. thesis, University of
Cambridge.
Dandekar, D., Martin, A., Kelley, J., 1981. High temperature elastic properties of polycrystalline MAR-M200 (a nickel base superalloy). Metall. Mater. Trans. A
12, 801–803.
Flinn, P., 1960. Theory of deformation in superlattices. Trans. AIME 218, 145–154.
Gleiter, H., Hornbogen, E., 1965. Beobachtung der wechselwirkung von versetzungen mit kohärenten geordneten Zonen (II). Phys. Status Solidi B 12, 251–
264.
Gopinath, K., Gogia, A.K., Kamat, S.V., Balamuralikrishnan, R., Ramamurty, U., 2008. Tensile properties of Ni-based superalloy 720Li: temperature and strain
rate effects. Metall. Mater. Trans. A 39, 2340–2350.
Hemker, K.J., Mills, M.J., 1993. Measurements of antiphase boundary and complex stacking fault energies in binary and B-doped Ni3Al using TEM. Philos.
Mag. A 68, 305–324.
Hessel, J., Voice, W., James, A., Blackham, S., Small, C., Winstone, M., 2000. Nickel alloy for turbine engine component. Patent Number EP0803585.
Hicks, M., King, J., 1983. Temperature effects on fatigue thresholds and structure sensitive crack growth in a nickel-base superalloy. Int. J. Fatigue 5, 67–74.
Hüther, W., Reppich, B., 1978. Interaction of dislocations with coherent, stress-free, ordered particles. Z. Metallkd 69, 628–634.
Hüther, W., Reppich, B., 1979. Order hardening of MgO by large precipitated volume fractions of spinel particles. Mater. Sci. Eng. 39, 247–259.
Inden, G., Bruns, S., Ackermann, H., 1986. Antiphase boundary energies in ordered f.c.c. alloys. Philos. Mag. A 53, 87–100.
Jackson, M., Reed, R., 1999. Heat treatment of UDIMET 720Li: the effect of microstructure on properties. Mater. Sci. Eng. A 259, 85–97.
Jou, H., Voorhees, P., Olsen, G., 2004. Computer simulations for the prediction of microstructure/property variation in aeroturbine disks. In: Green, K.,
Pollock, T., Harada, H., Howson, T., Reed, R., Schirra, J., Walston, S. (Eds.), Superalloys 2004. The Minerals, Metals & Materials Society, pp. 877–886.
Karnthaler, H., Mühlbacher, E., Rentenberger, C., 1996. The influence of the fault energies on the anomalous mechanical behaviour of Ni3Al alloys. Acta
Mater. 44, 547–560.
Khachaturyan, A., Jr., J.M., 1987. The interfacial tension of a sharp antiphase domain boundary. Philos. Mag. A 56, 517–532.
Kocks, U., Argon, A., Ashby, M., 1975. Thermodynamics and kinetics of slip. Prog. Mater. Sci. 19, 1–281.
Kozar, R., Suzuki, A., Milligan, W., Schirra, J., Savage, M., Pollock, T., 2009. Strengthening mechanisms in polycrystalline multimodal nickel-base superalloys.
Metall. Mater. Trans. A 40, 1588–1603.
Kuhn, H.A., Sockel, H.G., 1988. Comparison between experimental determination and calculation of elastic properties of nickel-base superalloys between 25
and 1200 °C. Phys. Status Solidi A 110, 449–458.
Miodownik, A., Saunders, N., 1995. The calculation of APB energies in L12 compounds. In: Nash, P., Sundman, B. (Eds.), Applications of Thermodynamics in
the Sythesis and Processing of Materials. The Minerals, Metals & Materials Society, pp. 91–104.
Mishima, Y., Ochiai, S., Suzuki, T., 1985. Lattice parameters of Ni(c), Ni3Al(c0 ) and Ni3Ga(c0 ) solid solutions with additions of transition and B-subgroup
elements. Acta Metall. 33, 1161–1169.
Mitchell, R.J., Lempsky, J.A., Ramanathan, R., Li, H.Y., Perkins, K.M., Connor, L.D., 2008. Process development & microstructure & mechanical property
evaluation of a dual microstructure heat treated advanced nickel disc alloy. In: Reed, R., Green, K., Caron, P., Gabb, T., Fahrmann, M., Huron, E., Woodard,
S. (Eds.), Superalloys 2008. TMS (The Minerals, Metals & Materials Society), pp. 347–356.
Mourer, D., Williams, J., 2004. Dual heat treat process development for advanced disk applications. In: Green, K., Pollock, T., Harada, H., Howson, T., Reed, R.,
Schirra, J., Walston, S. (Eds.), Superalloys 2004. TMS (The Minerals, Metals & Materials Society), pp. 401–407.
Myhr, O.R., Grong, Ø, 2000. Modelling of non-isothermal transformations in alloys containing a particle distribution. Acta Mater. 48, 1605–1615.
Nembach, E., Schänzer, S., Schröer, W., Trinckauf, K., 1988. Hardening of nickel-base superalloys by high volume fractions of c0 -precipitates. Acta Metall. 36,
1471–1479.
Nembach, E., Schänzer, S., Trinckauf, K., 1992. The antiphase boundary energy of c0 precipitates in nickel-based superalloys. Philos. Mag. A 66, 729–738.
Nganbe, M., Heilmaier, M., 2009. High temperature strength and failure of the Ni-base superalloy PM 3030. Int. J. Plast. 25, 822–837.
Pessah-Simonetti, M., Caron, P., Khan, T., 1993. Effect of a long-term prior aging on the tensile behaviour of a high-performance single crystal superalloy. J.
Phys. IV France 03, C7-347–C7-350.
Püschl, W., 2001. The activation energy of cross-slip in L12 ordered alloys. Mater. Sci. Eng. A 319–321, 266–269.
Radis, R., Schaffer, M., Albu, M., Kothleitner, G., Pölt, P., Kozeschnik, E., 2009. Multimodal size distributions of c0 precipitates during continuous cooling of
UDIMET 720 Li. Acta Mater. 57, 5739–5747.
Reed, R., 2006. The Superalloys - Fundamentals and Applications. Cambridge University Press, Cambridge, UK.
Reppich, B., 1982. Some new aspects concerning particle hardening mechanisms in c0 precipitating Ni-base alloys – I. Theoretical concept. Acta Metall. 30,
87–94.
Robson, J., 2004. Modelling the evolution of particle size distribution during nucleation, growth and coarsening. Mater. Sci. Technol. 20, 441–448.
Roebuck, B., Cox, D., Reed, R., 2004. An innovative device for the mechanical testing of miniature specimens of superalloys. In: Green, K., Pollock, T., Harada,
H., Howson, T., Reed, R., Schirra, J., Walston, S. (Eds.), Superalloys 2004. The Minerals, Metals & Materials Society, pp. 523–528.
Rolls-Royce, 2011. Personal Communication with M.C. Hardy.
Sarosi, P., Wang, B., Simmons, J., Wang, Y., Mills, M., 2007. Formation of multimodal size distributions of c0 in a nickel-base superalloy during interrupted
continuous cooling. Scr. Mater. 57, 767–770.
Saunders, N., Fahrmann, M., Small, C., 2000. The application of CALPHAD calculations to Ni-based superalloys. In: Pollock, T., Kissinger, R., Bowman, R.,
Green, K., Mclean, M., Olson, S., Schirra, J. (Eds.), Superalloys 2000. The Minerals, Metals & Minerals Society, pp. 803–811.
Sawant, A., Tin, S., Zhao, J.C., 2008. High temperature nanoindentation of Ni-base superalloys. In: Reed, R., Green, K., Caron, P., Gabb, T., Fahrmann, M., Huron,
E., Woodard, S. (Eds.), Superalloys 2008. TMS (The Minerals, Metals & Materials Society), pp. 863–871.
Schoeck, G., 1994. The generalized Peierls–Nabarro model. Philos. Mag. A 69, 1085–1095.
Sharghi-Moshtaghin, R., Asgari, S., 2004. The influence of thermal exposure on the c0 precipitates characteristics and tensile behavior of superalloy IN738LC. J. Mater. Process Technol. 147, 343–350.
Shenoy, M., Tjiptowidjojo, Y., McDowell, D., 2008. Microstructure-sensitive modeling of polycrystalline IN100. Int. J. Plast. 24, 1694–1730.
Siebörger, D., Knake, H., Glatzel, U., 2001. Temperature dependence of the elastic moduli of the nickel-base superalloy CMSX-4 and its isolated phases.
Mater. Sci. Eng. A 298, 26–33.
Sims, C., Stoloff, N., Hagel, W. (Eds.), 1987. Superalloys II – High Temperature Materials for Aerospace and Industrial Power. John Wiley & Sons, Inc..
112
D.M. Collins, H.J. Stone / International Journal of Plasticity 54 (2014) 96–112
Singh, A., Nag, S., Hwang, J., Viswanathan, G., Tiley, J., Srinivasan, R., Fraser, H., Banerjee, R., 2011. Influence of cooling rate on the development of multiple
generations of c0 precipitates in a commercial nickel base superalloy. Mater. Charact. 62, 878–886.
Sokolov, M., Landes, J., Lucas, G. (Eds.), 2002. Small Specimen Test Techniques, fourth ed. ASTM International, Chealsea, MA.
ThermoCalc. <http://www.thermocalc.com/> (accessed 24.10.12).
ThermoTech. http://www.thermocalc.com/res/pdfdbd/dbd_ttni8-2.pdf (accessed 24.10.12).
Wen, Y., Simmons, J., Shen, C., Woodward, C., Wang, Y., 2003. Phase-field modeling of bimodal particle size distributions during continuous cooling. Acta
Mater. 51, 1123–1132.
Wen, Y., Wang, B., Simmons, J., Wang, Y., 2006. A phase-field model for heat treatment applications in Ni-based alloys. Acta Mater. 54, 2087–2099.
Xia, P., Yu, J., Sun, X., Guan, H., Hu, Z., 2007. Influence of thermal exposure on c0 precipitation and tensile properties of DZ951 alloy. Mater. Charact. 58, 645–
651.
Yoon, K., Noebe, R., Seidman, D., 2007. Effects of rhenium addition on the temporal evolution of the nanostructure and chemistry of a model Ni–Cr–Al
superalloy. II: analysis of the coarsening behavior. Acta Mater. 55, 1159–1169.
Zener, C., 1948 (Quoted by C.S. Smith). Trans. Am. Inst. Min. Metall. Pet. Eng. 15, 15.