2
APM Proceedings
Crystallographic orientation and delay time
influence on thermal fatigue strength of
single-crystal nickel superalloys
Artem V. Savikovskii, Artem S. Semenov, Leonid B. Getsov
temachess@yandex.ru
Abstract
The influence of a delay time at the maximum temperature on the number
of cycles before the macrocrack initiation for three thermal loading programs
was investigated for single-crystal nickel-based superalloy ZhS32. An analytic approximation of a delay time influence was proposed. Comparison of
the computational results and analytic formula on the basis of constitutive
equations with the experimental data was performed for single-crystal nickelbased superalloy ZhS32 and showed a good accuracy. The influence of crystallographic orientation of the corset sample on the thermal fatigue durability
with delay times was investigated for various thermal loadings.
1
Introduction
Single-crystal nickel based superalloys [1] are used for production of gas turbine
engines (GTE). These materials have a pronounced anisotropy and temperature
dependence of properties. Cracking in the turbine blades is caused often by thermal
fatigue [2]. For the investigation of thermal fatigue durability under a wide range
of temperatures with and without delay times the experiments are carried out on
different types of samples, including corset (plane) specimen [2] on the installation
developed in NPO CKTI [3] (see Fig. 1 a). Fixed in axial direction by means of
two bolts with a massive foundation the corset sample (see Fig. 1 b) is heated
periodically by passing electric current through it. The fixing of sample under
heating leads to the high stress level and inelastic strain appearance. The local
strain and stress concentration is observed in the central (working) part of sample.
The FE simulation is required for the computation of inhomogeneous stress and
inelastic strain fields.
The aim of the research is to study systematically the effect of delay at maximum temperature on the thermal fatigue durability on the base of the deformation
Crystallographic orientation and delay time influence on thermal fatigue strength
of single-crystal nickel superalloys
3
a)
Figure 1: a) Setup for thermal fatigue experimental investigations, b) Geometry of
corset sample for thermal fatigue experiment.
criterion [4, 5, 6] for single crystal superalloys using the results of finite element (FE)
simulation of full-scale experiments and results of analytical formulae and to study
systematically the effect of crystallographic orientation on the thermal fatigue durability. The results of simulation and their verification are obtained for single-crystal
nickel-based superalloy ZhS32.
2
Results of thermo-elasto-plastic analysis
Modeling of inelastic deformation in the corset samples has been performed with taking into account of the temperature dependence of all material properties, anisotropy
of mechanical properties of single crystal sample, inhomogeneous temperature field,
mechanical contacts between bolt and the specimen, between specimen and foundation, temperature expansion in the specimen.
Two FE formulations for the thermomechanical problem have been considered:
• with taking into account of equipment;
• without taking into account equipment (simplified formulation [7] for the sample only).
The validity of the simplified formulation is based on the comparison with the
results of full-scale formulation (with taking into account equipment), as well as
on the comparison with the displacements of two markers measured in experiments.
The problem was solved in a three-dimensional, quasi-static formulation. As boundary conditions the symmetry conditions were set: zero displacements on the y-axis
on the xz plane and zero displacements on the x-axis on the yz plane. On the lower
side of the equipment zero displacements along the x and z axes were set. Tightening force was applied on the bolt cap. The temperature field distributions were
set from the experimental data at maximum and minimum temperature with linear
interpolation in time. The results of finite element heat conduction simulations [8]
consistent with experimental temperature field distributions.
The mechanical properties for alloy ZhS32 were taken from [3] are presented in
Table 1. The mechanical properties of bolts are taken for pearlitic steel [9].
b)
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APM Proceedings
Table
T
E001
ν
α
σY 001
n
A
1. Mechanical properties of ZhS32 used in simulations
◦C
20
700
900
1000
MPa
137000
110000
99800
94800
0.395
0.425
0.43
0.435
−5
−5
−5
1/K
1.2·10
1.6·10
1.8·10
2.2·10−5
MPa
919
904
895
670
8
8
8
8
−n −1
−42
−31
−28
MPA c 1 · 10
2 · 10
2 · 10
6 · 10−27
[3]:
1050
92300
0.436
2.4·10−5
580
8
7 · 10−26
In simplified formulation (see Fig. 2 ) we consider only the sample without equipment, in which zero displacements on the symmetry planes xz and yz were set, the
outer face of the sample parallel to the symmetry plane xz was fixed in the direction
of the axis x. To exclude solid body motions, a number of points on this face were
also fixed in the direction of y and z axes.
Figure 2: Finite-element models in mechanical problem: a) with taking into account
equipment, (1/4 of structure due to symmetry) b) without taking into account
equipment (simplified formulation), (1/4 of structure due to symmetry).
Fig. 3 shows distributions of plastic strain intensity for two different temperature
loading programs after 7 cycles.
a)
b)
Figure 3: Distributions of plastic strain intensity for a) superalloy ZhS32, T =
150 ÷ 900 ◦ C; b) superalloy ZhS32, T = 700 ÷ 1050 ◦ C after 7 cycles.
The full effective length for superalloy ZhS32 for several temperature modes was
45 mm [8]. In the FE simulations the full length of the specimen for all alloys was
taken to be 40 mm.
Crystallographic orientation and delay time influence on thermal fatigue strength
of single-crystal nickel superalloys
5
3
Simulation and an analytical approximation for
delay influence on thermal fatigue strength. Influence of crystallographic orientation
Simulation of inelastic cyclic deformation of corset samples were performed with
using of the FE program PANTOCRATOR [10], which allows to apply the micromechanical (physical) models of plasticity and creep for single crystals [11]. The
micromechanical plasticity model accounting 12 octahedral slip systems with lateral
and nonlinear kinematic hardening was used in the FE computation for single crystal alloy. FE computations were carried out for a part of a corset sample (simplified
FE model with half-effective length of sample equal 20 mm, see Fig. 3b). The
temperature boundary conditions were set from the experimental data at maximum
and minimum temperature with linear interpolation in time.
The influence of the delay at maximum temperature and the influence of crystallographic orientation on the number of cycles to the formation of macrocrack is
analyzed in the range from 1 min to 1 hour for the cyclic loading regimes (see, for
example, Fig. 4 b) with:
• maximum temperature of 900 ◦ C and a temperature range of 750 ◦ C;
• maximum temperature of 1000 ◦ C and a temperature range of 500 ◦ C.
The heating times in the cycle were 10s and 25s, the cooling times were 14 s
and 75s for ZhS32. The mechanical properties for alloy ZhS32 were taken from [3].
The problem was solved in a quasi-static 3-dimensional formulation. The boundary
conditions were zero displacements in the direction of the x-axis on two side faces
of the sample with the normal along the x-axis. To exclude solid-state motions,
a number of points on these faces in the direction of the y and z axes were also
fixed. Finite element model in simplified formulation and temperature evolution in
central point of sample with and without delay for thermal loading program T =
150 ÷ 900 ◦ C are presented in Fig. 4.
a)
b)
Figure 4: a) Finite element model of sample (simplified formulation) for analysis of
delay influence, b) temperature evolution in central point of the sample with and
without delay for T = 150 ÷ 900 ◦ C.
Damage calculation and estimation of the number of cycles before the formation
of macrocracks were made on the basis of deformation four-member criterion [4, 5, 6]:
D=
N
N
X
(∆εpeqi )k X
(∆εceqi )m
∆εpeq
∆εceq
+
+ max p
+ max c
,
0≤t≥tmax εr (T )
0≤t≥tmax εr (T )
C
C
1 (T )
2 (T )
i=1
i=1
(1)
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APM Proceedings
where the first term takes into account the range of plastic strain within the
cycle, the second term is the range of creep strain within the cycle, the third term is
accumulated plastic strain (ratcheting), the fourth term is accumulated creep strain.
The number of cycles before the formation of the macrocrack N is determined from
the condition D = 1. Usually it takes the values k = 2, m = 5/4, C1 = (εpr )k ,
C2 = 3/4 · (εcr )m where εpr and εcr are ultimate strains of plasticity and creep under
uniaxial tension. In the FE computations the values of ultimate strain εpr = εcr =
εr = 0.13 and 0.18 for ZhS32 were used.
Analytic approximation is offer to enter for describing of delay time influence on
thermal fatigue strength. We consider the principle of deformation additivity [12]
in case of uniaxial loading:
(2)
ε = εe + εp + εc + εt = ε0 ,
where ε is the full initial strain, εe = σE is the elastic strain, εp is the plastic strain,
εc is the creep strain and εt is the temperature strain. Differentiating (2), using
ε˙p = Hσ̇ , where H is the hardening modulus, Norton law ε˙c = A · σn , taking into
account E + H = ET is the tangent modulus and dividing the equation by σn we put:
σ−n σ̇ = −AET
(3)
Splitting variables, integrating from t0 to time t and using ε˙c = A · σn , we put:
n
ε˙c = A(σ0 1−n + (n − 1)AET (t − t0 )) 1−n
(4)
Using variables changing τ = σ0 1−n + (n − 1)AET (t − t0 ) and integrating from t0 to
time t we obtain:
∆εc =
1
σ0
(1 − (1 + A(n − 1)ET σ0 n−1 (t − t0 )) 1−n )
ET
(5)
Using simplified deformation criterion with taking into account creep deformation
terms:
m
εc accumul
∆εc
= 1,
(6)
+N
εr
εr
where εr is the ultimate strain of creep under uniaxial tension, N is the number of
cycles of macrocrack initiation we obtain:
!m
εr
εc accumul
N= σ
· 1−
, (7)
1
0
εr
(1 − (1 + A(n − 1)ET σ0 n−1 (t − t0 )) 1−n )
ET
where we use σ0 = (α20−Tmax · Tmax − α20−Tmin · Tmin ) · ET · 0.9, α20−Tmax and
α20−Tmin are the coefficients of linear thermal expansion, ET = 9.48 · 104 MPa /
9.98 · 104 MPa, A = 2 · 10−28 / 6 · 10−27 MPa−n s−1 , εr = 0.13/0.18 for alloy ZhS32,
accumul
multiplier (1 − εc εr ) is picking up to correlate one point with experiment.
Crystallographic orientation and delay time influence on thermal fatigue strength
of single-crystal nickel superalloys
7
Comparison of results of FE simulations and experiments concerning the effect
of the delay time at the maximum temperature on the thermal fatigue durability
for single-crystal superalloy ZhS32 for two temperature modes is given in fig. 5.
a)
b)
Figure 5: Comparison of results of FE simulation and experimental data for alloy
ZhS32: a) T = 150 ÷ 900 ◦ C, heating time is 25s, cooling time is 75s, εr = 0.13, b)
T = 500 ÷ 1000 ◦ C, heating time is 10s, cooling time is 14s, εr = 0.18.
Influence of crystallographic orientation (CGO) on thermal fatigue strength for
superalloy ZhS32 for two temperature modes is presented in fig. 6.
a)
b)
Figure 6: Influence of crystallographic orientation on thermal fatigue strength for
superalloy ZhS32: a) T = 150 ÷ 900 ◦ C, heating time is 25s, cooling time is 75s,
εr = 0.13, b) T = 500 ÷ 1000 ◦ C, heating time is 10s, cooling time is 14s, εr = 0.18.
4
Conclusions
Computational results of thermal fatigue durability showed a good agreement with
the experiment, which suggests that the finite-element and analytical computations
in combination with deformation criterion (1) can be used to predict the thermalfatigue strength of single-crystal superalloys. Researching of CGO influence has
showed that thermal fatigue durability of specimens with crystallographic orientation <001> is the highest among all considered variants and specimens with crystallographic orientation <111> is the weakest among all variants of orientations.
Acknowledgements
The financial support of the Russian Science Foundation, Grant 18-19-00413, is
acknowledged.
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APM Proceedings
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Artem V. Savikovskii,SPbPU,Politechnicheskaya 29;195251,St.Petersburg,Russia
Artem S. Semenov,SPbPU,Politechnicheskaya 29;195251, St.Petersburg, Russia
Leonid B. Getsov,NPO CKTI,Politechnicheskaya 24;194021,St. Petersburg,Russia
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