MATEC Web of Conferences 245, 10006 (2018)
EECE-2018
https://doi.org/10.1051/matecconf/201824510006
Analysis
of
crystallographic
orientation
influence on thermal fatigue with delay of the
single-crystal corset sample by means of
thermo-elasto-visco-plastic
finite-element
modeling
Artem Savikovskii1, *, Artem Semenov1, and Leonid Getsov2
1Peter
the Great Saint-Petersburg Polytechnic University, 194064, Polytechnicheskaya 29, Russia
and production association for research and design of power equipment I. I. Polzunova,
194021, Polytechnicheskaya 24, Russia
2Research
Abstract. The influence of a delay time at the maximum temperature on the
number of cycles before the macrocrack initiation for two thermal loading
programs was investigated for single-crystal nickel-based superalloy
VZhM4. 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
various single-crystal nickel-based superalloys 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 loading programs and single-crystal nickel-based superalloys.
1 Introduction
Single-crystal nickel based superalloys [1, 2] are promising used for production of gas turbine
engines (GTE) [3]. These materials have a pronounced anisotropy and temperature
dependence of properties. Cracking in the turbine blades is caused often by thermal fatigue
[4, 5]. 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 [4] on the installation developed in NPO CKTI [6] (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 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.
*
Corresponding author: temachess@yandex.ru
© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons
Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).
MATEC Web of Conferences 245, 10006 (2018)
EECE-2018
https://doi.org/10.1051/matecconf/201824510006
Fig. 1. Setup for thermal fatigue experimental
investigations.
Fig. 2. Geometry of corset sample for thermal
fatigue experiment.
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 criterion [7-11]
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 VZhM4.
2 Methods
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.
The two FE formulations for the thermomechanical problem have been considered:
•
with taking into account equipment;
•
without taking into account equipment (simplified formulation [12] 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 [13]. The results of finite element heat conduction
simulations [13,14] consistent with experimental temperature field distributions.
The mechanical properties for alloy VZHM4 were taken from the paper [15] are presented
in Table 1. The mechanical properties of bolts are taken for pearlitic steel [16]. Used material
properties consistent with considered in [17,18].
Table 1. Mechanical properties of VZHM4 used in simulations [15].
T
E001
𝜈
α
𝜎𝑌 001
⁰C
MPa
1/K
MPa
20
700
800
900
1000
1050
130000
101000
96000
91000
86000
82000
0.39
0.42
0.422
0.425
0.428
0.43
1.11·10−5 1.68·10−5 1.74·10−5 1.87·10−5 2.1·10−5 2.3·10−5
846
950
820
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MATEC Web of Conferences 245, 10006 (2018)
EECE-2018
n
A
MPa−𝑛 𝑠 −1
8
1·10−42
https://doi.org/10.1051/matecconf/201824510006
8
3·10−31
8
1·10−29
8
1·10−28
8
2·10−27
8
1·10−26
In simplified formulation (see Fig. 3) 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.
sample
b)
a)
y
z
equipment
bolt
x
Fig. 3. 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.4 shows distributions of plastic strain intensity for two different temperature loading
programs after 7 cycles (for VZHM4 the length of the sample is 42 mm).
b)
a)
Fig. 4. Distributions of plastic strain intensity for a) superalloy VZhM4, T = 700÷1050 °C; b) VZhM4,
T = 500÷1050 °C after 7 cycles.
The full effective length for superalloy VZhM4 for several temperature modes was
42 mm [13]. In the FE simulations the full length of the specimen for all alloys was taken to
be 40 mm.
Simulation of inelastic cyclic deformation of corset samples were performed with using
of the FE program PANTOCRATOR [19], which allows to apply the micromechanical
(physical) models of plasticity and creep for single crystals [20-22]. The micromechanical
plasticity model accounting 12 octahedral slip systems with lateral and nonlinear kinematic
hardening [20] 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.
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MATEC Web of Conferences 245, 10006 (2018)
EECE-2018
https://doi.org/10.1051/matecconf/201824510006
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. 9b) with:
•
maximum temperature of 1050 °C and a temperature range of 350 °C;
•
maximum temperature of 1050 °C and a temperature range of 550 °C;
The heating times in the cycle were 24s and 7s, the cooling time was 15 s for VZhM4.
The mechanical properties for the alloy VZhM4 were taken from the paper [15]. 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 (fig. 5).
z
y
x
Fig. 5. Finite element model of sample (simplified formulation) for analysis of delay influence
Temperature evolutions in central point of sample with and without delay for thermal
loading program T = 700÷1050 °C and T = 500 ÷1050 °C are presented in fig. 6.
Fig. 6. Temperature evolutions in central point of sample with and without delay for T = 700÷1050 °C
and T = 500 ÷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 [7-11]:
N
D
i 1
p
eqi
k
С1 T
N
i 1
с
eqi
m
С2 T
max
0t tmax
eqp
rp T
max
0t tmax
eqc
rc T
,
(1)
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
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MATEC Web of Conferences 245, 10006 (2018)
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https://doi.org/10.1051/matecconf/201824510006
(ratcheting), the fourth term is accumulated creep strain. The number of cycles before the
formation of macrocrack N is determined from the condition D = 1. Usually it takes the values
5
k = 2, m = ,
4
, C
C1 rp
k
3
4
2
c m
r
rp
, where
and
rc are ultimate strains of plasticity
𝑝
and creep under uniaxial tension. In the FE computations the values of ultimate strain 𝜀𝑟 =
𝜀𝑟𝑐 = 𝜀𝑟 = 0.17 for VZhM4 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 in case of uniaxial
loading:
Ɛ = Ɛ𝑒 + Ɛ𝑝 +Ɛ𝑐 + Ɛ𝑡 = Ɛ0 ,
(2)
𝜎
where Ɛ is the full initial strain, Ɛ𝑒 = is the elastic strain, Ɛ𝑝 is the plastic strain, Ɛ𝑐 is the
𝐸
𝜎̇
creep strain and Ɛ𝑡 is the temperature strain. Differentiation (2), using Ɛ𝑝̇ = , where H is the
𝐻
hardening modulus [23], Norton law Ɛ𝑐̇ = A𝜎 𝑛 , taking into account E+H=𝐸𝑇 is the tangent
modulus [24] and dividing the equation by 𝜎 𝑛 we put:
𝜎 −𝑛 𝜎̇ = -A𝐸𝑇
(3)
Splitting variables, integrating from 𝑡0 to time t and using Ɛ𝑐̇ = A𝜎 𝑛 we put:
𝑛
Ɛ𝑐̇ = A (𝜎01−𝑛 + (𝑛 − 1)𝐴𝐸𝑇 (𝑡 − 𝑡0 ))1−𝑛
(4)
Using variables changing τ = 𝜎01−𝑛 + (𝑛 − 1)𝐴𝐸𝑇 (𝑡 − 𝑡0 ) and integrating from 𝑡0 to time t
we obtain:
∆Ɛ𝑐 = -
1
𝐸𝑇
(
1
1
(𝜎01−𝑛 + (n−1)𝐴𝐸𝑇 (𝑡−𝑡0 ))1−𝑛
1
-
1
(𝜎01−𝑛 )1−𝑛
)=
𝜎0
𝐸𝑇
(1 - (1 +
(n−1)𝐸𝑇
𝜎0
1
𝐴𝜎0𝑛 (𝑡 − 𝑡0 ))1−𝑛 )
that leads to:
∆Ɛ𝑐 =
𝜎0
𝐸𝑇
(1 - (1 +
(n−1)𝐸𝑇
𝜎0
1
𝐴𝜎0𝑛 (𝑡 − 𝑡0 ))1−𝑛 )
(5)
Using simplified deformation criterion with taking into account creep deformation terms:
Ɛ𝑎𝑐𝑐𝑢𝑚𝑢𝑙
𝑐
Ɛ𝑟
+N(
∆Ɛ𝑐 𝑚
)
Ɛ𝑟
= 1,
(6)
where Ɛ𝑟 is the ultimate strain of creep under uniaxial tension, N is the number of cycles of
macrocrack initiation we obtain:
N=(𝜎
Ɛ𝑟
1
0 (1 − (1+ (n−1)𝐸𝑇 𝐴𝜎 𝑛 (𝑡
0 𝑑𝑒𝑙𝑎𝑦 ))1−𝑛 )
𝐸𝑇
𝜎0
)𝑚 *(1-
Ɛ𝑎𝑐𝑐𝑢𝑚𝑢𝑙
𝑐
Ɛ𝑟
),
(7)
In the simulations we use 𝐸𝑇 = 8.2·104 MPa, 𝜎0 = (𝛼20−Tmax *Tmax-𝛼20−Tmin *Tmin)* 𝐸𝑇 ∗ 0.9,
𝛼20−Tmax and 𝛼20−Tmin are the coefficients of linear thermal expansion, A = 1026
MPa−𝑛 𝑠 −1 , Ɛ𝑟 =0.17, multiplier (1 -
Ɛ𝑎𝑐𝑐𝑢𝑚𝑢𝑙
𝑐
Ɛ𝑟
) picking up to correlate one point with
experiment.
3 Results and discussion
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 and is given in Fig. 7.
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MATEC Web of Conferences 245, 10006 (2018)
EECE-2018
a)
https://doi.org/10.1051/matecconf/201824510006
b)
c)
Fig. 7. Comparison of results of FE simulation and experimental data for alloy VZhM4: a)
T = 700÷1050 ⁰C, heating time is 7s, cooling time is 15s, 𝜀𝑟 = 0.17,b) T = 500÷1050 ⁰C, heating time is
7s, cooling time is 15s, 𝜀𝑟 = 0.17, c) T = 500÷1050 ⁰C, heating time is 24s, cooling time is 15s, 𝜀𝑟 = 0.17.
Comparison of results of experiment and analytical approximation concerning the effect
of the delay time at the maximum temperature on the thermal fatigue durability for singlecrystal superalloy VZhM4 is given in Fig. 8.
Fig. 8. Comparison of results of experiment and analytical approximation for alloy VZhM4: a)
T = 700÷1050 ⁰C, heating time is 7s, cooling time is 15s, 𝜀𝑟 = 0.17, b) T = 500÷1050 ⁰C, heating time
is 7s, cooling time is 15s, 𝜀𝑟 = 0.17,c) T = 500÷1050 ⁰C, heating time is 24s, cooling time is 15s,
𝜀𝑟 =0.17.
Note that the additive experimental verification is required for the near to horizontal
branches of curves in fig. 7 and 8 corresponding to remarkable delays.
Influence of crystallographic orientation (CGO) on thermal fatigue strength for
superalloys VZhM4 for two temperature modes is presented in fig. 9.
Fig. 9. Influence of crystallographic orientation on thermal fatigue strength for superalloy alloy
VZhM4: a) T = 700÷1050 ⁰C, heating time is 7s, cooling time is 15s, 𝜀𝑟 = 0.17, T = 500÷1050 ⁰C,
heating time is 7s, cooling time is 15s, 𝜀𝑟 = 0.17, c) T = 500÷1050 ⁰C, heating time is 24s, cooling
time is 15s, 𝜀𝑟 = 0.17.
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MATEC Web of Conferences 245, 10006 (2018)
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https://doi.org/10.1051/matecconf/201824510006
The thermal fatigue durability of samples from superalloy VZhM4 with CGO <001>
exceeds the thermal fatigue durabilities of CGO <011> and <111> (fig. 9) for all considered
loading programs.
Further improvement of the accuracy of thermal fatigue durability calculations with
delays can be achieved by considering more complex creep models [25, 26] and taking into
account the rafting process [27] at high temperatures.
4 Conclusions
The results of the computations and the analytical approximations of delay-time influence on
thermal fatigue durability show a good agreement with the experiment, which suggests that
the finite-element and analytical computations in combination with application of
deformation criterion (7) can be used to predict the thermal-fatigue strength of various singlecrystal superalloy samples with different delays. 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.
The research is supported by RFBR grant No. 16-08-00845.
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