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APM Proceedings
Thermo-electro-mechanical modeling of thermal
fatigue failure process of corset samples from
single-crystal nickel superalloys
Artem V. Savikovskii, Artem S. Semenov, Leonid B. Getsov
temachess@yandex.ru
Abstract
The results of computations of the thermal and stress-strain state of singlecrystal corset specimens subjected to the action of periodic electric current,
leading to variable inhomogeneous heating and subsequent thermal fatigue
failure, are presented. The influence of maximum value and range of temperature and also delay time at the maximum temperature on the number
of cycles before the macrocrack formation is investigated. Comparison of the
computational results with the experimental data for various single-crystal
nickel-based superalloys showed a good accuracy.
1
Introduction
Single-crystal nickel-based superalloys [1] are widely used for the manufacture of nozzle and working blades of gas turbine engines (GTE). The thermal-fatigue strength
of such materials with a pronounced anisotropy and a sensitivity of mechanical
properties to the temperature is currently not fully studied. For the investigation
of thermal fatigue durability under a wide range of temperatures with and without
intermediate delays the experiments are carried out on different types of samples,
including corset (plane) specimen on the installation developed in NPO CKTI [2]
(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. During cycling the maximum and minimum temperatures are
automatically maintained constant.
The objective of the study is to determinate numerically the thermal and stressstrain states of the corselet specimens under cyclic electric loading and to study
systematically the effect of delay at maximum temperature on the thermal fatigue
durability on the base of the deformation criterion [3, 4, 5] of thermal-fatigue failure
Thermo-electro-mechanical modeling of thermal fatigue failure process of corset
samples from single-crystal nickel superalloys
3
a)
Figure 1: a) Installation for carrying out experiments on thermal fatigue, b) Geometry corset sample for thermal fatigue experiment.
for single crystal superalloys using the results of finite element (FE) simulation of
full-scale experiments. The results of simulation and their verification are obtained
for the different single-crystal nickel-based superalloys: VZhM4, VIN3 and ZhS32.
2
Results of thermo-electric analysis
Modeling of heating process in the corset samples was carried in the FE program
ANSYS with taking into account the temperature dependence of all material properties, nonstationary Joule heating, the convective heat exchange and radiative heat
transfer between the sample and the environment. The full-scale FE model of experimentation object including discrete models of the specimen and the setup is
presented in Fig. 2.
Figure 2: Finite-element model for thermoelectric problem.
Modeling of heating processes and thermal fatigue fracture of sample was carried
out for four temperature regimes (modes): 150 ÷ 900, 250 ÷ 1000, 500 ÷ 1050 and
700 ÷ 1050 ◦ C. The used in FE simulations material properties for the single crystal
nickel superalloy sample and for the steel equipment were taken from literature [6],
[13, 14, 15] (see also Table 1). While specifying the properties of nickel alloy and
steel the implementation of the Wiedemann-Franzs law was controlled: λρe = LT,
b)
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APM Proceedings
where λ is the thermal conductivity,ρe is the specific electrical resistance, T is the
temperature in K, L = 2.22 · 10−8 WΩK−2 is the Lorentz, s constant.
Table 1.Thermo-electric properties of nickel superalloy used in FE simulations.
T ◦C
20
200
400
800
1000
1150
Ref.
kg
ρ
8550
8500
8450
8350
8330
8310
[13]
m3
J
Cρ kg·K
440
520
520
575
590
600
[13]
W
λ
7.4
11.2
14.1
19.8
26.7
36.7
[6]
m·K
−7
−7
−6
−6
−6
−7
ρe Ω · m 8.7·10
9.3·10
1.1·10
1.2·10
1 · 10
8.9·10
[6]
The coupled three-dimensional transient thermo-electrical analysis has been performed. Due to the symmetry in respect to the xz and yz planes, a quarter of the
structure was considered. The thermal and electric contacts between the sample
and bolts, between the sample and the foundation were taken into account. The
initial temperature for the sample and the equipment was set to 30 ◦C. For the free
surface of sample the boundary condition of convective heat transfer is used:
(1)
qn = h(T − T0 ),
where n is the normal to body, qn is the heat flux density, h = 20 mW2 K is the coefficient
of convective heattransfer, T0 is the ambient temperature. The condition of radiative
heat transfer was also set on the surfaces of central (high temperature) part of the
sample (10 mm length):
qn = εσSB (T 4 − T0 4 ),
(2)
where ε = 0.8 is the black factor of the body, σSB = 5.67 · 10−8 Wm−2 K−4 is the
coefficient of Stefan-Boltzmann.
The temperature field distribution in the VZhM4 sample is shown in Fig. 3a
for the loading regime with Tmax = 1050 ◦ C. Note that the solution of thermoelectric problem has been obtained for the complete FE model shown in Fig. 2 with
taking into account the equipment.The evolution of temperature spatial-distribution
is given in Fig. 3b (x is a distance from the sample center). The bell-form of the
curve is keeping during whole heating process.
a)
b)
Figure 3: a) Temperature field distributions in VZhM4 sample by heating for regime
with Tmax = 900 ◦ C, b) Evolution of temperature distribution along the VZhM4
sample axis for different heating times.
The comparison of FE results with experimental data for axial temperature
distribution demonstrates a good agreement for the all considered loading regimes
(see, for example, Fig. 4).
Thermo-electro-mechanical modeling of thermal fatigue failure process of corset
samples from single-crystal nickel superalloys
5
a)
b)
c)
Figure 4: Comparison of computational results with experimental data for axial
temperature distributions in VZhM4 sample for regimes with: a) Tmax = 900 ◦ C,b)
Tmax = 1000 ◦ C, c) Tmax = 1050 ◦ C
3
Results of thermo-elasto-visco-plastic analysis
The obtained in thermo-electric problem the spatial and temporal distribution of
the temperature field is the base for the strain and stress field computation within
the framework of thermo-elasto-visco-plastic problem solution.
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. 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, kinematic hardening, inhomogeneous nonstationary temperature
field, mechanical contacts bolt/specimen and specimen/foundation, friction between
the contact surfaces, temperature expansion in the specimen, bolt and foundation.
The two FE formulations for the thermo-mechanical problem have been considered:
•
with taking into account of equipment;
• without taking into account equipment (simplified formulation [7] for the sample only).
Using the second formulation provides significant saving computational time due
to reduction in the number of degrees of freedom and refusal to solve a contact
problem that is very actual for the numerous multivariant computations for different
regimes of loading and the crystallographic orientations. One of the aims of the
investigations was the selection of the equivalent (effective) length of the sample for
the simplified formulation. 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 relative displacements of two
markers measured in experiments.
In the general case there is no symmetry in the problem (see Fig.4 a) due to
anisotropy of mechanical properties of single crystal sample. However in the important for practice case of [001] crystallographic orientation of sample the symmetry
in respect to planes xz and yz (see Fig.4 b) can be introduced. Equipment and
bolts were modeled by linear elastic material (steel), and for the sample the elastoviscoplastic model of the material was used.
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APM Proceedings
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.
On the bolt cap the pressure of 100 MPa has been applied that is equivalent to the
tightening force of the bolt. The mechanical properties for the alloys VZhM4 and
VIN3 were taken from the papers [11, 12] and for ZhS32 from [2] (see Table 2 for
details). The mechanical properties of bolts are taken for pearlitic steel [13].
Table 2. Mechanical properties of VZhM4 used in simulations [11]:
T
◦C
20
700
900
1000
1050
E001 MPa
130000
96000
91000
86000
82000
ν
0.39
0.422
0.425
0.428
0.43
−5
−5
−5
−5
α
1/K
1.1·10
1.7·10
1.9·10
2.1·10
2.3·10−5
001
σY
MPa
846
950
820
n
8
8
8
8
8
−n −1
−42
−29
−28
−27
A
MPA c 1 · 10
1 · 10
1 · 10
2 · 10
1 · 10−26
In simplified formulation (see Fig. 5 d) 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 the y and z axes.
a)
b)
c)
d)
Figure 5: Finite-element models in mechanical problem: a) complete model
(sample/bolt/equipment) without symmetry account, b) complete model (sample/bolt/equipment) with symmetry account, c) simplified model (sample only)
without symmetry account, d) simplified model (sample only) with symmetry account.
Fig. 6 shows distributions of plastic strain intensity for nickel superalloys and
Thermo-electro-mechanical modeling of thermal fatigue failure process of corset
samples from single-crystal nickel superalloys
7
three temperature modes after 7 cycles for thermoplasticity problem in ANSYS (for
VZHM4 and VIN3 the length of the sample is 42 mm, for ZHS32 is 50 mm).
a)
b)
c)
Figure 6: Distributions of plastic strain intensity for a) superalloy VZhM4, mode
700 ÷ 1050 ◦ C; b) superalloy VIN3, mode 500 ÷ 1050 ◦ C; c) superalloy ZhS32, mode
150 ÷ 900 ◦ C after 7 cycles.
The Table 6 shows the equivalent (effective) length of the sample for the simplified formulation, which has been found from the condition of equality of the inelastic
strain ranges with complete model for different alloys. In the FE simulations with
acceptable engineering accuracy can be used the value 40 mm. Effective length takes
into account the compliance of equipment and its variation in considered range has
no appreciable on the results.
Table 6. The equivalent length of the corset sample for different alloys.
VZhM4
VIN3
ZhS32
34-42 mm
38-46 mm
40-52 mm
4
Influence of delay on the thermal fatigue durability
Simulations of inelastic cyclic deformation of corset samples were performed by
means of the FE program PANTOCRATOR [8], which allows to use the micromechanical (physical) models of plasticity and creep for single crystals [9, 10]. The
Norton power-type law without hardening was applied to describe creep properties.
The micromechanical plasticity model accounting 12 octahedral slip systems with
lateral and nonlinear kinematic hardening [9] 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 effective length of sample equal 40 mm, see Fig. 7a.
The influence of the delay at maximum temperature on the number of cycles to
the formation of macro cracks is analyzed in the range from 1 min to 1 hour for the
cyclic loading regimes (see, for example, Fig. 7b) with:
•
maximum temperature of 1000 ◦ C and a temperature range of 350 ◦ C and
550 ◦ C;
• maximum temperature of 1000 ◦ C and a temperature range of 750 ◦ C;
• maximum temperature of 900 ◦ C and a temperature range of 750 ◦ C.
The heating times in the cycle were 7s, 25s, 18 s and 28s, the cooling time was
15s, 17s, 40s and 52s for VZhM4. The heating time in the cycle was 25 s, the cooling
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APM Proceedings
time was 17s for VIN3. The heating times in the cycle was 25 s and 15s, the cooling
time was 15s and 75s for ZhS32. The mechanical properties for the alloys VZhM4
and VIN3 were taken from the papers [11, 12] and for ZhS32 from [2].
a)
b)
Figure 7: a) Finite element model of sample (simplified formulation) for analysis
of delay influence;b) temperature evolutions in central point of sample with and
without delay for temperature regime 700-1050 ◦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
[3],[4],[5]:
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
unilaterally accumulated plastic strain (ratcheting), the fourth term is unilaterally
accumulated creep strain. The number of cycles before the formation of macrocracks
N is determined from the condition D = 1. The maximum shear strain in the sliding
system with normal to the slip plane n and the sliding direction l is considered as
equivalent deformation. The values k=2, m = 5/4 , C1 = (εpr )k , C2 = 3/4 ∗ (εcr )m
are usually accepted, where εpr and εcr ultimate strains of plasticity and creep under
uniaxial tension.
In the FE computations the values of ultimate strain εpr = 0.40 for VZhM4, εpr
= 0.36 for ZhS32, εpr = 0.42 for VIN3 were used. Improvement of the accuracy of
prediction of influence the delay time on durability can be achieved by the refinement
of the constant εpr on the basis of data without delay.
The comparison of the 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 superalloys VZhM4, VIN3 and ZhS32 is given in Fig.
8,9,10.
Thermo-electro-mechanical modeling of thermal fatigue failure process of corset
samples from single-crystal nickel superalloys
9
a)
b)
c)
d)
Figure 8: Comparison of results of FE simulation and experimental data for the
alloy VZhM4: a)mode 150 ÷ 900 ◦ C, heating time is 28s, cooling time is 52s,b)mode
500 ÷ 1050 ◦ C, heating time is 7s, cooling time is 15s,c)mode 700 ÷ 1050 ◦ C heating
time is 25s, cooling time is 17s,d)mode 250 ÷ 1000 ◦ C heating time is 18s, cooling
time is 40s.
a)
b)
Figure 9: Comparison of results of FE simulation and experimental data for alloy
ZhS32: a) mode 150 ÷ 900 ◦ C, heating time is 25 s, cooling time is 75 s, b) mode
700 ÷ 1050 ◦ C, heating time is 15 s, cooling time is 15 s.
Figure 10: Comparison of calculation and experiment for alloy VIN3, mode 500-1050
◦C, heating time-25 s, cooling time-17s
10
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APM Proceedings
Conclusions
The results of the computations show a good agreement with the experiment,
which suggests that the finite-element computations in combination with application of deformational criterion can be used to predict the thermal-fatigue
strength of various single-crystal superalloy samples in wide range temperatures.
Acknowledgements
The study was performed under financial support of RFBR grant No. 16-08-00845
and scholarship program Siemens.
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Artem V. Savikovskii,SPbSPU,Politechnicheskaya 29;195251,St.Petersburg,Russia
Artem S. Semenov,SPbSPU,Politechnicheskaya 29;195251, St.Petersburg, Russia
Leonid B. Getsov,NPO CKTI,Politechnicheskaya 24;194021,St. Petersburg,Russia
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