Materials Physics and Mechanics 44 (2020) 125-136
Received: July 29, 2019
CRYSTALLOGRAPHIC ORIENTATION, DELAY TIME AND
MECHANICAL CONSTANTS INFLUENCE ON THERMAL FATIGUE
STRENGTH OF SINGLE - CRYSTAL NICKEL SUPERALLOYS
A.V. Savikovskii1*, A.S. Semenov1, L.B. Getsov2
1
Peter the Great St. Petersburg Polytechnic University, Polytechnicheskaya 29, St. Petersburg, 195251, Russia
2
Joint-Stock Company "I.I. Polzunov Scientific and Development Association on Research and Design of Power
Equipment", Polytechnicheskaya 24, St. Petersburg, 194021, Russia
*e-mail: savikovskii.artem@yandex.ru
Abstract. The influence of a delay time at the maximum temperature on the number of cycles
for the macrocrack initiation for two thermal loading programs was investigated for two
single-crystal nickel-based superalloys VIN3 and ZhS32. An analytic approximation of a
delay time influence was proposed. Comparison of computational results and analytic formula
on the base of constitutive equations with the experimental data was performed for considered
single-crystal superalloys and showed a good accuracy. Influence of several mechanical
constants of nickel alloy on thermal fatigue strength is presented and discussed. The influence
of crystallographic orientation of the corset sample on the thermal fatigue durability with
delay times for various thermal loading programs and different single-crystal nickel
superalloys was investigated.
Keywords: thermal fatigue, single-crystal nickel-based superalloy, deformation criterion,
corset sample, crystallographic orientation, finite element modeling, analytic approximation
1. Introduction
Single-crystal nickel-based superalloys [1] are used for production of gas turbine engines
(GTE) [2]. These materials have a pronounced anisotropy and temperature dependence of
properties. Cracking in the turbine blades is caused often by thermal fatigue [3,4]. For
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 [3] on the installation developed in NPO CKTI [5] (see Fig. 1). Fixed
in axial direction by means of two bolts with a massive foundation the corset sample (see
Fig. 2) 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 FE simulation
is required for the computation of inhomogeneous stress and inelastic strain fields.
The aims of the study are: (I) to study numerically a stress-strain state of the sample
during cyclic heating and cooling due to its clamping, (II) to study systematically the effect of
delay at maximum temperature on the thermal fatigue durability on the base of the four-term
deformation criterion [6-8] of thermal-fatigue failure for single crystal superalloys using the
results of finite element (FE) simulation of full-scale experiments and results of analytical
formulae and (III) to study systematically the effect of crystallographic orientation on the
thermal fatigue durability. The results of simulation and their verification are obtained for
different single-crystal nickel-based superalloys: VIN3 and ZhS32.
http://dx.doi.org/10.18720/MPM.4412020_15
© 2020, Peter the Great St. Petersburg Polytechnic University
© 2020, Institute of Problems of Mechanical Engineering RAS
126
A.V. Savikovskii, A.S. Semenov, L.B. Getsov
Fig. 1. Testing setup for thermal fatigue experiments
Fig. 2. Geometrical parameters of the corset sample
2. Results of thermo-elasto-visco-plastic analysis
The axial fixing of the corset specimen under heating leads to the high axial stress and
inelastic strain appearance. The numerical simulation is required for the computation of
inhomogeneous stress and inelastic strain fields in sample. Modeling and simulation of
inelastic cyclic deformation of corset samples has been performed by means of the
FE programs ANSYS and PANTOCRATOR [9], which allows to apply the micromechanical
(microstructural, crystallographic, physical) models of plasticity and creep for single crystals
[10-14]. The micromechanical plasticity model accounting 12 octahedral slip systems with
lateral and nonlinear kinematic hardening [10,15] is used in the FE computation for the
simulation of single crystal superalloy behavior under cyclic loading. The Norton power-type
law is used to describe creep properties.
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 nonstationary temperature
field, mechanical contacts between bolt and the specimen, between specimen and foundation,
friction between the contact surfaces, temperature expansion in the specimen. The viscous
properties are taken into account because of high temperature despite of a quick time of
heating and cooling of the corset samples.
The two FE formulations for the thermo-mechanical problem have been considered:
• FE model with taking into account equipment;
• FE model without taking into account equipment (simplified formulation [16] for the
sample only).
Using of 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. It is
very actual for the numerous multivariant computations for different regimes of loading and
the crystallographic orientations. The validity of the simplified formulation is based on the
comparison with the results of full-scale formulation (with taking into account equipment), as
Crystallographic orientation, delay time and mechanical constants influence on thermal fatigue strength...
127
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 (Fig. 3a) 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. 3a) can be introduced. Equipment and bolts are modeled by linear elastic material (steel),
and for the sample elasto-visco–plastic model of material is used. The problem is solved in a
three-dimensional quasi-static formulation. As boundary conditions the symmetry conditions
are 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
are set. On the bolt cap the pressure of 100 MPa has been applied that is equivalent to the
tightening force of the bolt. The temperature boundary conditions are set from the
experimental data at maximum and minimum temperature with linear interpolation in time.
The results of finite element heat conduction simulations [17] consistent with experimental
temperature field distributions. The mechanical properties for the alloy VIN3 are taken from
the paper [18] and for the alloy ZHS32 from [19] are summarized in Tables 1 and 2. The
mechanical properties of bolts are taken for pearlitic steel [20].
Table 1. Mechanical properties of VIN3 used in simulations [18]
500
700
20
900
°C
T
E 001
MPa
104000
126000
110000
89000
0.39
0.41
0.42
0.42
ν
−5
−5
−5
1/K
α
1.21 ⋅ 10
1.33 ⋅ 10
1.5 ⋅ 10−5
1.4 ⋅ 10
σ y001
MPa
555
930
800
910
n
3
3
3
3
−27
−15
− n −1
−17
A
1 ⋅ 10
2.3 ⋅ 10
6.5 ⋅ 10−14
MPa s
8 ⋅ 10
Table 2. Mechanical properties of ZHS32 used in simulations [19]
500
700
900
20
°C
T
E 001
MPa
137000
110000
105000
99800
0.395
0.4248
0.4284
0.4317
ν
−5
−5
−5
1/K
α
1.81 ⋅ 10−5
1.24 ⋅ 10
1.6 ⋅ 10
1.7 ⋅ 10
σ y001
MPa
901
895
919
904
n
8
8
8
8
−42
−30
− n −1
−31
A MPa s
1 ⋅ 10
8.5 ⋅ 10
2 ⋅ 10−28
2.5 ⋅ 10
1000
80000
0.425
1.57 ⋅ 10−5
645
1050
75000
0.428
1.6 ⋅ 10−5
540
3
3.5 ⋅ 10−13
3
8 ⋅ 10−13
1000
94800
0.4347
2.22 ⋅ 10−5
670
8
6 ⋅ 10−27
1050
92300
0.4361
2.42 ⋅ 10−5
580
8
7 ⋅ 10−26
In simplified formulation of the problem (see Fig. 3b) we consider only the sample
without equipment, in which zero displacements on the symmetry planes xz and yz are 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 are also fixed in the
direction of the y and z axes.
128
a)
A.V. Savikovskii, A.S. Semenov, L.B. Getsov
a
b)
sample
sample
y
x
z
o
equipment
bolts
Fig. 3. FE models of the corset sample for thermo-elasto-visco-plastic problem solution:
a) model (¼ due to symmetry) with taking into account of equipment,
b) simplified model (¼ due to symmetry) without taking into account of equipment
Figure 4 shows distributions of plastic strain intensity field for two nickel superalloys
and three different temperature modes after 7 cycles (for VIN3 the effective length of the
sample is 42 mm, for ZHS32 is 50 mm) [17] obtained with using the FE model (¼ due to
symmetry) without taking into account equipment (Fig. 3b).
a)
b)
c)
d)
Fig. 4. Plastic strain intensity field distributions in the corset sample after 7 cycles for:
a) superalloy ZhS32, loading regime 150÷900°C;
b) superalloy ZhS32, loading regime 500÷1050°C;
c) superalloy ZhS32, loading regime 700÷1050°C;
d) superalloy VIN3, loading regime 500÷1050°C
Table 3 shows the equivalent (effective) length of the sample for the simplified
formulation for different alloys, which has been found by the comparison with full model
using the condition of equality of the inelastic strain ranges. 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.
Crystallographic orientation, delay time and mechanical constants influence on thermal fatigue strength...
129
Table 3. The equivalent length of the corset sample for different alloys
VIN3
ZhS32
38 − 46 mm
40 − 52 mm
3. Influence of the delay on the thermal fatigue durability
FE computations are carried out for a part of a corset sample (simplified FE model with
effective length of sample equal 40 mm, see Fig. 3b). The temperature fields are set from the
experimental data at maximum and minimum temperature cycle phase with using linear
interpolation in time.
The influence of the delay at maximum temperature 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 with:
• maximum temperature of 1050°C and minimum temperature of 700°C;
• maximum temperature of 1050°C and minimum temperature of 500°C;
• maximum temperature of 1000°C and minimum temperature of 500°C;
• maximum temperature of 900°C and minimum temperature of 150°C.
The heating time in the cycle is 10s, the cooling time is 16s for VIN3. The heating times
in the cycle are 15s, 15s, 10s, 25s, the cooling times are 15s, 15s, 14s, 75s for ZhS32. The
mechanical properties for the alloy VIN3 were taken from the paper [18] and for the alloy
ZhS32 from [19] (see also Tables 1 and 2).
The problem is solved in a quasi-static 3-dimensional formulation. The FE model is
shown in Fig. 3b. The boundary conditions are zero displacements in the direction of the xaxis on two side faces of the sample with the normal along the x-axis. To exclude rigid body
motions, a number of points on these faces in the direction of the y and z axes are also fixed.
Temperature evolutions in central point of sample with and without delay for loading
regimes 700÷1050°C, 500÷1050°C, 250÷1000°C and 150÷900°C are presented schematically
in Fig. 5.
a)
b)
c)
d)
Fig. 5. Schematic presentations of temperature evolutions in central point of sample for
loading regimes with and without delay time:
a) 700÷1050°C; b) 500÷1050°C; c) 500÷1000°C and d) 150÷900°C
130
A.V. Savikovskii, A.S. Semenov, L.B. Getsov
Damage calculation and estimation of the number of cycles for the macrocrack
initiation are made on the basis of four-term deformation criterion [6-8,14]:
N ( De p ) k
N ( De c ) m
e eqp
e eqc
eqi
eqi
D =∑
+∑
+ max p
+ max c
,
(1)
0 ≤t ≤tmax e (T )
0 ≤t ≤tmax e (T )
C1 (T )
i 1 C2 (T )
=i 1 =
r
r
where the first term takes into account the range of plastic strain within the cycle, the second
term deals with 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 to initiate macrocrack N is determined from the condition D = 1.
The equivalent strain for single crystal is defined by maximum shear strain in the slip system
with normal to the slip plane n{111} and the slip direction l 011 :
e=
n{111} ⋅ ε ⋅ l 011 .
eq
(2)
k
m
5
Usually it takes in (1) the values of constants: k = 2 , m = , C1 = (ε rp ) , C2 = ( 43 ε rc ) ,
4
p
c
where ε r and ε r are ultimate strains of plasticity and creep under uniaxial tension. In the
p
c
=
=
18% are used the same for all
FE computations the values of ultimate strains εε
r
r
considered alloys. Improvement of the prediction accuracy of the delay time influence on
durability can be achieved by the refinement of the constant strains 𝜀𝜀𝑟𝑟𝑝𝑝 on the basis of data
without delay.
Analytic approximation [21] is offered to enter for describing of delay time influence on
thermal fatigue durability. The additive strain decomposition [22] is used for the small strain
case under uniaxial relaxation at constant temperature and total strain:
e = ee + e p + ec + et = e0 ,
(3)
where ε is the total strain, e e =
σ
E
is the elastic strain, ε p is the plastic strain, ε c is the creep
strain and ε t is the thermal strain, ε 0 is constant. Differentiating (3) and using ε p =
σ
H
(where H is the hardening modulus), Norton law εc = Aσ n with taking into account notation
=
ET
(E
−1
+ H −1 ) for the tangent modulus it can be obtained equation:
−1
σ − nσ = − AET .
(4)
Using result of integration of differential equation (4) from t0 to t for σ (t ) in the
relation εc = Aσ n allow us to introduce differential equation for creep strain:
n
=
εc A σ 01− n + (n − 1) AET (t − t0 ) 1− n .
Result of integration of (5) from t0 to t has the form:
1
1
1
,
∆ε c =
−
−
1
1
ET 1− n
1− n 1− n
1
−
n
(σ 0 )
σ 0 + (n − 1) AET (t − t0 )
which can be rewritten as following:
1
−
1− n
σ0
(n − 1) ET
n −1
∆ε=
Aσ 0 (t − t0 )
1 − 1 +
.
c
ET
σ0
(5)
(6)
(7)
Crystallographic orientation, delay time and mechanical constants influence on thermal fatigue strength...
131
Using simplified two-term deformation criterion with taking into account creep strain
terms:
m
c
∆
εε
accumul
1,
+N c =
(8)
εε
r
r
where ε r is the ultimate strain of creep under uniaxial tension, N is the number of cycles of
macrocrack initiation we obtain:
c
er
e accumul
(9)
1
N =
⋅
−
.
1
−
e
r
σ 0 (n − 1) ET
1− n
Aσ 0 n −1tdelay
1 − 1 +
σ0
ET
In the simulations we use: σ=0 (a 20−Tmax ⋅ Tmax − a 20−Tmin ·Tmin ) ⋅ ET ⋅ 0.9 , a 20−Tmax and α 20−Tmin
m
are
the
coefficients
of
linear
thermal
expansion,
ET =
9.48 ⋅104 / 9.98 ⋅104 MPa ,
28
A=
2 ⋅10-/ 6 ⋅10 27 MPa -n s -1 , ε r = 0.18 for alloy ZhS32, A= 8 ⋅10-13 MPa -n s -1 , ε r = 0.18 for
alloy VIN3. Multiplier 1 −
c
ε accumul
is picking up to correlate one point with experiment.
εr
Comparison of results of FE simulations, experiments and analytical approximation (9)
concerning the effect of the delay time at the maximum temperature on the thermal fatigue
durability for single-crystal superalloys VIN3 and ZhS32 for four temperature modes is given
in Fig. 6.
a)
b)
d)
c)
Fig. 6. Comparison of results of FE simulations, analytical approximation and experimental
data for alloys: a) ZhS32, loading regime 150÷900°C, heating time is 25 s, cooling time is
75 s; b) ZhS32, loading regime 500÷1000°C, heating time is 10 s, cooling time is 14 s;
c) ZhS32, loading regime 700÷1050°C, heating time is 15 s, cooling time is 15 s;
d)VIN3, loading regime 500÷1050°C, heating time is 10 s, cooling time is 16 s
132
A.V. Savikovskii, A.S. Semenov, L.B. Getsov
Figure 7 shows creep strain intensity field distributions for nickel superalloys ZhS32
and VIN3 in case without delay time and with delay time 5 minutes after 10 cycles.
a)
c)
b)
d)
Fig. 7. Creep strain distributions:
a) ZhS32, loading regime 150÷900°C, without delay time; b) VIN3, loading regime
500÷1050°C, without delay time; c) ZhS32, loading regime 150÷900°C, with delay time
5 minutes; d) VIN3, loading regime 150÷900°C, with delay time 5 minutes
Several material parameters appearing in the analytic formula (9) and calculations such
as A, n and ε r are more complicated to obtain and to find in open sources. Influence of these
constants on thermal fatigue durability for nickel alloys ZhS32 and VIN3 and two
temperature loading regimes is investigated. The results of parametrical analysis are presented
in Fig. 8 with varied values of A, n and ε r .
Numerical simulations show that parameters n and ε r influence stronger, then
parameter A on thermal fatigue durability.
In gas turbine blades direction of mechanical, thermal and other loadings may not the
same as crystallographic orientation (CGO) of single-crystal blade. CGO of gas turbine blades
and the samples have an effect on creep rate and thermal fatigue durability. Influence of CGO
on thermal fatigue strength is important to predict behavior and damage of single-crystal
alloys. Influence of crystallographic orientation (CGO) on thermal fatigue strength for
superalloys ZhS32 and VIN3 for four temperature modes is presented in Fig. 9.
Crystallographic orientation, delay time and mechanical constants influence on thermal fatigue strength...
a)
c)
133
b)
d)
f)
e)
Fig. 8. Influence of material parameters of nickel-based alloys on thermal fatigue durability:
a) Influence of parameter A, ZhS32, loading regime 500÷1000°C, ε r = 0.18 , n = 8 ;
b) Influence of parameter A, VIN3, loading regime 500÷1050°C, ε r = 0.18 , n = 3 ;
c) Influence of parameter n, ZhS32,loading regime 500÷1050°, ε r = 0.18, A= 6 ⋅ 10-27 MPa -ns-1
d) Influence of parameter n,VIN3, loading regime 500÷1050,
ε r = 0.18, A= 8 ⋅ 10-13 MPa -ns-1;
e) Influence of parameter ε r , ZhS32, loading regime 500÷1050°C, n = 8 ,
A= 6 ⋅ 10-27 MPa -ns-1;
f) Influence of parameter ε r , VIN3, loading regime 500÷1050°C, n = 3 , A= 8 ⋅ 10-13 MPa -ns-1
134
A.V. Savikovskii, A.S. Semenov, L.B. Getsov
a)
b)
d)
c)
Fig. 9. Influence of crystallographic orientation on
thermal fatigue durability for single-crystal nickel-based superalloys:
a) ZhS32, T = 150÷900ºC, heating time is 25s, cooling time is 75s, ε r = 0.18 ;
b) ZhS32, T = 500÷1000ºC, heating time is 10s, cooling time is 14s, ε r = 0.18 ;
c) ZhS32, T = 700÷1050ºC, heating time is 15s, cooling time is 15s, ε r = 0.18 ;
d) VIN3, T = 500÷1050ºC, heating time is 10s, cooling time is 16s, ε r = 0.18
The reason of the superiority of thermal fatigue durability for samples with CGO <001>
over CGO <011> and <111> is associated with lower values of Young's modulus for CGO
=
=
E 111 / E 011 1.4 at 1000ºC).
<001>
( E 111 / E 001 2.4,
4. Conclusions
Computational results of thermal fatigue durability showed a good agreement with the
experiments, which suggests that finite-element modeling and analytical approximation (9) in
combination with deformation criterion (1) can be used to predict thermal-fatigue strength of
single-crystal nickel-based superalloys.
Investigation of material parameters influence show that creep exponent n and tensile
rupture strain 𝜀𝜀𝑟𝑟 affect more stronger that creep parameter A on thermal fatigue durability of
single-crystal nickel based superalloys. Constants n and 𝜀𝜀𝑟𝑟 should be more accurately
obtained from experimental data.
The thermal fatigue durability of corset samples from superalloys ZhS32 and VIN3 with
CGO <001> exceeds the thermal fatigue durabilities of CGO <011> and <111> (Fig. 9) for all
considered loading programs and alloys.
Comparison analysis of superalloys ZhS32 and VIN3 showed that superalloy ZhS32 has
thermal fatigue strength higher than superalloy VIN3 for the same considered loading
program.
Crystallographic orientation, delay time and mechanical constants influence on thermal fatigue strength...
135
Acknowledgments. The reported study was supported by RFBR according to the research
project №19-08-01252.
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