The nodes of both sublattices must be included in this cycle which is why the diffusion mobility of elements in the crystal lattice of theγ phase can be decreased essentially although th
Trang 16.6 Composition of the γ Phase and Atomic Vibrations 103 Tab 6.5 Compositions ofB3A (γ ) phases and the mean-square
atomic displacements
A,pm 2
723 K 873 K 1023 K 723 K 873 K 1023 K
(Al0.51Ti0.31Nb0.07Mo0.08Cr0.03)
Ni2.83Fe0.04Cr0.08W0.03Mo0.02 230 340 450 30 60 80 (Al0.43Ti0.28W0.08Mo0.03Cr0.10)
Ni2.76Co0.09Fe0.01Cr0.06Mo0.02 260 340 580 20 40 50 (Al0.79W0.08Mo0.11Cr0.03)
the diffusion mobility in the crystal lattice The probability of an atom transfer
to an adjacent node of the crystal lattice also increases The smaller the am-plitudes of the atomic vibrations, the greater the high-temperature strength
of theγ phase A decrease in amplitudes results in an increase in the value
of the activation energy∆U in Eq (6.13) and the creep rate decreases.
In the first phase, practically pureNi3Al (Table 6.5), the atomic displace-ments of the lighter atoms of aluminum (nodes A) appreciably exceed the dis-placement of the nickel atoms (nodes B) at all tested temperatures,u2
A> u2
B The replacement of approximately one sixth of the aluminum atoms with titanium atoms decreases the amplitudes of the heat vibrations in the alu-minum sublattice, so that nowu2
A < u2
B The effect of titanium dissolution
in the sublattice A on decreasing amplitudes becomes more evident if the temperature rises The effect of adding molybdenum and tungsten to theγ
phase is even greater One can see from Table 6.5 that the valuesu2
Aare 10–16 times less at 1023K in phases containing both these elements
The vibration amplitudes decrease less in the sublattice B The decrease
of u2
Ais not proportional to the growth of the mean atomic mass because
of Mo and W dissolution, but exceeds it essentially Actually, the effective mass of atoms A in the investigated strengthening phases increases only
by 64–78% in comparison with the effective mass of aluminum atoms The mass of atoms dissolved at the B nodes is close to that of nickel This fact is evidence of the growth of interatomic bonds in the crystal lattice of the alloyed
γ phase Comparing the values of the mean-square atomic displacements in
the sublattice A for phases of different composition we can conclude that they are related, mainly, to the sum of the atomic parts of molybdenum and tungsten In Table 6.5 the phases are ordered by increasing this sum while decreasingu2
A
Trang 2An energetically efficient mechanism of an elementary diffusion event in theγ phase consists of a certain cycle of a vacancy shift The nodes of both
sublattices must be included in this cycle which is why the diffusion mobility
of elements in the crystal lattice of theγ phase can be decreased essentially
although the mean-square amplitudes are decreased only significantly in the aluminum sublattice
We have compared the mean-square atomic amplitudes in strengthening phases of superalloys with the rate of the steady-state creep ˙ε of the same
alloys The superalloy EI437B withu2
A = 460 pm2 at 1023K has the least high-temperature strength of the three alloys listed in Table 2.1 whereas the superalloy EI867 withu2
A= 50 pm2, has the greatest strength
6.7
Influence of the Particle Size and Concentration
It is the cut of the precipitates that is the rate-controlling process when the av-erage distance between particles is smaller than a critical value However, the interaction of dislocations with particles is dependent on the size of the parti-cles and on the distance between them The Orowan mechanism of obstaparti-cles bowing can also be rate-controlling
The mean particle size and the average distance between particles are de-pendent on one another Manonukul et al [37], by assuming that the precipi-tates of theγ phase are arranged in a three-dimensional array, have proposed
to a simple, first approximation, that
λp=
4π
3c
1/3
− 2
whereλpis the average distance between particles,c is the volume fraction of
theγ phase,2r is the mean particle size, as before The results of calculations
according to Eq (6.20) are presented in Table 6.6
Particle sizes change during ageing at a constant temperature Both the particle size and the volume fraction increase This process is called
coarsen-Tab 6.6 The mean particles distanceλp, nm as a function of the average particle size and of theγ phase concentration
Trang 36.7 Influence of the Particle Size and Concentration 105
ing and has an appreciable effect on creep rate This is reflected in Eq (6.13),
in which the average particle size,2¯r, appears within the exp term.
The dependence of precipitate size on time can be represented by the em-pirical equation given in [37]
2r = 2r0+ 2.96 × 105t 1/3 exp(0.012T ) (6.21) where the timet is in hours In Fig 6.19 the corresponding dependence is
shown for two temperatures
There is a critical valueλpc as well as a critical particle size Ifλp ≤ λpc the rate-controlling mechanism is the cutting ofγ -particles by slipping
dis-locations Ifλp > λpc the rate-controlling mechanism becomes dislocation bowing
The C263 superalloy is strengthened by titanium, aluminum, chromium, molybdenum and cobalt The volume fraction ofγ phase is equal to 0.095.
In Fig 6.20 the creep curves for this superalloy are shown At 1073K and stress 160MPa the critical particle size was found to be 85 nm Therefore the dislocations overcome obstacles with thermally activated cutting Coarsening
of particles takes place, however, and after approximately1.7×106s (472 h) the critical particle size, 85 nm is achieved, and the rate-controlling mechanism becomes dislocation bowing This is indicated in Fig 6.20 where the transfer from steady-state to tertiary creep is clearly seen
It is reasonable, for completeness, to consider the case in which the pre-cipitate dissolves out such thatc = 0 The slip of dislocation is controlled by
dislocation networks The length of the activated dislocation segmentl equals
the distance between pinning points
Fig 6.19 The increase in the average particle size with time
B, 973K; C, 1073K
Trang 4Fig 6.20 Creep curves in C236 superalloy at 1073K B, stress
160MPa; C, stress 200MPa Data from Ref [37]
6.8
The Prediction of Properties on the Basis of Integrated Databases
Comprehensive tests and investigations are needed in order to design new superalloys Variation of the chemical composition of superalloys is known to
be the main method for optimization of alloy properties, however, there are
an astronomical number of combinations Therefore another approach has been developed, which consists in the prediction of properties of new alloys The idea is as follows [30] The known data (from the literature and indus-trial sources) on dependences of certain characteristics (such as the rupture life, the yield stress, the lattices parameters) on the element concentration have to be gathered and put into a computer database A special program is used to predict the properties of a new superalloy
This approach does not take into account the physical processes The struc-ture of alloys, solution and precipitation of phases, the influence of elements
on interatomic bonds, the mechanism of deformation are not considered Moreover, the additive action of alloying elements on properties is also ig-nored Nevertheless, such an approach is substantiated, because our knowl-edge about the extremely complex processes in superalloys is too limited to
be used to predict the properties The investigators try to evaluate the errors
of prediction and to compare the results with experimental data
Tancret et al [30] write: “Because the influence of the composition and processing parameters on the material properties is extremely complex and multivariate, designing an alloy “to measure” is not feasible using experi-ence alone Modern alloys contain many chemical elements added to achieve particular properties The influence of individual alloying elements on
Trang 5me-6.8 The Prediction of Properties 107
chanical properties can be measured and understood in isolated cases; simple interactions between two or three elements can be formulated, but describing all the interactions as a whole is generally impossible”
The Gaussian processes of the modelling of mechanical properties have been described A stochastic process is called a Gaussian process if all the distributions are subjected to the corresponding distribution
Suppose that N alloys have been studied Each tested ith alloy is
asso-ciated with an input vector x i The obtained data D are considered as N
input vectors { x1, x2, x N } = [X N ] There are N corresponding outputs
or targets{t1, t2, , t N } = t N, each target being a measurement The joint
probability distribution, in anN-dimensional space, of the target vector t N
is denoted by an expression P ( t N |[X N]) The known data are denoted by
D = { t N , [X N ]} One may consider the data D as an input matrix, where
every column represents one superalloy and every row contains the values of the same characteristic throughout all alloys
The aim is to predict the output value,t N+1, corresponding to a new in-put vector, x N+1 (i.e a new alloy or test conditions) This means to
calcu-late the one-dimensional probability distribution over the predicted point
P (t N+1 |x N+1 , D), given a knowledge of the corresponding input vector,
x N+1, and the dataD The new point is given by the following relationship
P (t N+1 |x N+1 , D) = P (t N+1 ,t N |x N+1 , [X N])
P (t N |[X N]) (6.22) The model assumes that the joint probability distribution of anyN output
values is a multivariate Gaussian
Equation (6.22) can be reduced to a univariate Gaussian of the form
P (t N+1 |x N+1 , D) = √ 1
2πσ texp
− (t N+1 − tm)2
2σ2
t
(6.23)
wheretmis the mean value,σ tis its standard deviation.
We refer the reader for details to Ref [30] and to references therein The authors have collected the data concerning mechanical properties and, espe-cially, creep rupture stress
In Fig 6.21 the predicted and actual effects of stress on rupture life for IN939 superalloy are presented This figure shows that the error of the method
is about±15–20%.
Using the described technique the authors try to estimate the influence
of alloying elements upon the properties of superalloys Titanium and alu-minum, when used as theγ formers, increase creep rupture stress However,
the respective influence of Al and Ti atoms seems to be inverted Since tita-nium atoms are bigger than aluminum atoms (by 4%) they induce an increase
Trang 6Fig 6.21 Predicted and actual relation between creep
rupture stress for IN939 superalloy and its rupture life,tr
Temperature of tests 1143K Reprinted from Ref [30] with
permission from Maney Publishing
in theγ parameter andγ /γ mismatch and thus influence the strain fields Ti
also increases the anti-phase boundary energy of theγ phase, which makes
the cutting of particles by dislocations more difficult Titanium is thus ex-pected to increase the yield strength more than aluminum and this has been recognized by the Gaussian process On the other hand, it has been shown that an excessive increase in theγ /γ misfit reduces the creep strength Thus,
adding titanium, although it increases creep strength by promoting γ
pre-cipitation, is less effective than adding aluminum because of a higher lattice misfit, which is also predicted by the model
Fig 6.22 Predicted influence of Co, Mo and W content on the
creep rupture stress at 1023K of the Ni–20Cr–1Al–1Ti–10Co
superalloy Reprinted from [30] with permission from Maney
Publishing
Trang 76.9 Summary 109
The authors’ predicted influence of Co, Mo and W atoms is presented in Fig 6.22 All these elements increase the creep resistance of nickel-based superalloys, in agreement with known results It is obvious from this figure that the error of prediction of about 50% and more shown by the bars is too great
The value of the method seems to lie in the possibility to cast out “bad” solutions even though it fails to find “good” solutions
6.9
Summary
Superalloys have been developed for aircraft and industrial gas turbines, rocket engines and space vehicles
The structure of nickel-based superalloys consists of theγ matrix and the
γ phase precipitations This phase is a solid solution of various elements in
the intermetallic compoundNi3Al The γ phase has remarkable properties;
in particular it hardens with increasing temperature
Dislocation sub-boundaries form in the matrix of superalloys under the influence of the applied stress The loading results in an increase in the subgrain misorientations and a decrease in subgrain size in the matrix The moving dislocations cut the particles during the steady-state stage of creep The most probable mechanism of the cutting is a diffusion-controlled motion
of partial dislocations with the anti-phase boundary in the ordered phase At the stage of the tertiary creep the shape of the particles becomes irregular as the coarsening of precipitates takes place
The experimental dependences between the logarithm of the minimum strain rate and the applied stress for superalloys are linear Hence the min-imum creep rate of superalloys depends exponentially on stress The com-puted lengths of the thermally activated dislocation segments are one order less than the average particle sizes
The minimum strain rate of superalloys is expressed as
˙ε = bf(c)N 0.24 · 2r νb2 exp
− ∆U kT
exp
0.12τb2· 2r kT
(6.24)
wheref(c) is a decreasing function of the concentration of the γ phase,N
is the dislocation density,2¯r is the average particle size, ∆U is an activation
energy,τ is the shear stress in the slip plane.
The activation energies,∆U, for the high-temperature steady-state creep
of superalloys are estimated to vary from4.18 × 10 −19to5.22 × 10 −19Jat.−1
when the dislocation density is assumed to be of the order of1012m−2.
Trang 8In a phase of typeB3A (Ni3Al) Ti atoms occupy places in sublattice A Cr,
Fe and Co atoms are located mostly in sublattice B Atoms of Mo, Nb and
W partition between the two sublattices but are mainly located in A Under the effect of solution of W, Mo and Ti atoms in theNi3Al phase the mean-square vibration amplitudes of A-atoms decrease from 810 to 50–80pm2at 1023K The smaller the mean-square amplitudes of the atom vibrations in the hardening phase the higher the creep strength of a superalloy
Methods of prediction of the properties of new superalloys are being devel-oped on the basis of integrated databases of superalloy composition and test results The error of prediction is from 15 to 50%
Trang 9High Temperature Strain of Metals and Alloys, Valim Levitin (Author)
Copyright c 2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim
ISBN: 3-527-313389-9
111
7
Single Crystals of Superalloys
7.1
Effect of Orientation on Properties
Single crystals of the nickel-based superalloys were developed for gas tur-bine applications in the 1970s and 1980s Single-grained castings replaced polycrystalline moldings for some engines with a view to avoiding slip and fracture along grain boundaries The single-crystal superalloys have superior creep, fatigue and thermal properties compared to conventional cast alloys because the grain boundaries have been eliminated Their components are able to operate at temperatures 200–300K higher, up to 1373K The rupture life is considered to be at least 20–40% greater than that of polycrystalline superalloys
Cast nickel-based single-crystal superalloys are utilized for blades in aircraft and in stationary gas turbines The blades of very complex geometry are often constructed with channels so that cooler air can be forced to flow within the blades during operation
Single crystals are manufactured by means of the directional crystallization technique The preferred crystal growth direction is the< 001 >, i.e along the
cube side, as compared with< 011 > and < 111 > The Young’s modulus has
a minimal value in the< 001 > direction, so that the thermal stresses in this
direction are minimal The anisotropy coefficient of the Young’s modulus is equal to 2.2, whereas the mechanical properties have the anisotropy coefficient
of 1.4 at high temperatures
Many manufacturers recommend to produce the single crystal blades with
< 001 > orientation, however, in practice, there is a misalignment from exact
< 001 > orientation.
The creep behavior of single crystal superalloys is highly anisotropic Under otherwise equal conditions the orientation of a superalloy single crystal is the factor which contributes to the overall creep strength
Trang 10The shear stress in a crystalline plane is known to be determined by the Schmid law
whereτ is the shear stress in the slip plane, σ is the applied stress, ϕ is the
angle between the applied stress and the perpendicular to the slip plane,ψ is
the angle between the applied stress and the slip direction
The values of the Schmid factors are shown in Table 7.1 The< 001 >
ori-entation is a multiple slip oriori-entation All the {111} slip planes are equivalent When an orientation moves from< 001 > the Schmid factor increases since
the anglesϕ and ψ decrease One can see that the Schmid factors are less
for the< 111 > tensile direction than for < 100 > (0.27 in comparison with
0.41) A< 111 > oriented single crystal has to have a higher creep strength.
Some dislocation splittings are also shown in Table 7.1 For crystals oriented near the< 001 > direction (the tensile axis is also directed along the < 001 >)
the dominant crystalline shear system is {111} < 110 > This is the
so-called octahedral slip The system{111} < 110 > operates most frequently
at temperatures in the vicinity of 1023K The dislocations with the Burgers
Tab 7.1 The Schmid factors in the cubic face-centeredγ, γ
crystal lattices and examples of the dislocation splitting (hkl),
indexes of the slip plane; [uvw], indexes of the slip direction
APB, anti-phase sub-boundary; SISF, the superlattice intrinsic
stacking fault
[001] (111) [¯101] 0.41 a/2[¯101] → a/6[¯112] + a/6[¯2¯11]+APB
[0¯11] 0.41
(¯111) [0¯11] 0.41 a/2[0¯11] → a/3[1¯12] + a/6[¯2¯1¯1]+SISF
[10¯1] 0.41
[101] 0.41
[¯101] 0.41
[¯111] (111) [¯110] 0.27 a/2[¯110] → a/6[¯121] + a/6[¯21¯1]+APB
[¯101] 0.27
[111] (¯111) [110] 0.27 a/2[110] → a/3[211] + a/6[¯11¯2]+SISF
[101] 0.27