Materials Physics and Mechanics 42 (2019) 296-310
Received: September 1, 2018
COUPLED THERMO-ELECTRO-MECHANICAL MODELING OF
THERMAL FATIGUE OF SINGLE-CRYSTAL CORSET SAMPLES
A.V. Savikovskii1*, A.S. Semenov1, L.B. Getsov2
1
SPBPU, Polytechnicheskaya 29, Russia
2
NPO CKTI, Polytechnicheskaya 24, Russia
*e-mail: temachess@yandex.ru
Abstract. The possibilities of predicting thermal fatigue durability for single crystal on the
base of coupled thermo-electro-mechanical finite-element modeling with using of
deformational criterion and microstructural models of inelastic deformation are investigated.
Results of thermal and stress-strain state simulations of single-crystal corset specimens under
cyclic electric heating and cooling are presented and discussed. Comparison of computational
results with experimental data for various single-crystal nickel-based superalloys
demonstrates a good accuracy in the prediction of the number of cycles for the macrocrack
initiation. The influence of maximum / minimum values of temperature in cycle and delay
duration on the number of cycles for the macrocrack initiation are analyzed. The simplified
analytic approximation for thermal fatigue durability curves is proposed.
Keywords: thermal fatigue, single-crystal nickel based superalloy, deformation criterion,
corset sample, thermo-electric problem, finite element modeling
1. Introduction
Single-crystal nickel based superalloys [1] are used for manufacturing of blades of gas turbine
engines (GTE). These materials have a pronounced anisotropy of properties and sensitivity to
crystallographic orientation. The thermal fatigue strength of single crystal superalloys for
various crystallographic orientations is not studied very well. For investigation of thermal
fatigue durability under a wide range of temperatures with and without intermediate delays
the experiments are carried out on corset (plane) specimen on the installation developed in
NPO CKTI [2] (see Fig. 1). Fixed in axial direction by means of two bolts with a massive
foundation a corset sample (see Fig. 2) is heated cyclic by passing electric current through it.
During cycling the maximum and minimum temperatures are automatically maintained
constant.
The aims of the study are: (I) to investigate numerically a process of heating and
cooling of the corset sample and to obtain analytical approximation of this process, (II) to
study numerically a stress-strain state of the sample during cyclic heating and cooling due to
its clamping and (III) to study systematically the effect of delay at maximum temperature on
the thermal fatigue durability on the base of the deformation criterion [3-5] of thermal-fatigue
failure for single crystal superalloys using the results of finite element (FE) simulation of fullscale experiments. The results of simulation and their verification are obtained for the
different single-crystal nickel-based superalloys: VZhM4, VIN3 and ZhS32.
http://dx.doi.org/10.18720/MPM.4232019_5
© 2019, Peter the Great St. Petersburg Polytechnic University
© 2019, Institute of Problems of Mechanical Engineering RAS
Coupled thermo-electro-mechanical modeling of thermal fatigue of single-crystal corset samples
Fig. 1. Testing setup for thermal fatigue
experiments
297
Fig. 2. Geometrical parameters of the
corset sample
2. Results of numerical thermo-electric nonstationary analysis and analytical
approximation for temperature changing during time
Modeling of a heating process by an electric current and a cooling process without an electric
current of the corset sample has been performed with help of the FE program ANSYS with
taking into account a temperature dependence of all material properties, thermos-electric
contacts between the sample and an equipment, nonstationary Joule heating, convective heat
exchange and radiative heat transfer between the sample and the environment. The full-scale
FE model (¼ due to symmetry) of experimental object including discrete models of the
specimen and equipment is presented in Fig. 3.
The simulations have been performed for single-crystal superalloys VZhM4, VIN3 and
ZhS32. The properties of three alloys were accepted the same because of lack of information
about nickel alloys properties dependence on temperature.
sample
y
bolt
equipment
x
z
Fig. 3. Finite-element model of the corset sample with taking into account of equipment for
the solution of thermo-electric problem
Modeling of heating and cooling processes of sample is carried out for five loading
regimes (modes), which will be denoted further by indicating the minimum and maximum
temperature of the cycle:
• 100÷800°C, a heating time is 19 s, a cooling time is 46 s;
• 150÷900°C, a heating time is 42 s, a cooling time is 59,5 s;
• 250÷1000°C, a heating time is 80 s, a cooling time is 10 s;
• 500÷1050°C, a heating time is 14 s, a cooling time is 10 s;
• 700÷1050°C, a heating time is 8 s, a cooling time is 7 s.
298
A.V. Savikovskii, A.S. Semenov, L.B. Getsov
The material properties used in FE simulations for the single crystal nickel superalloy
sample and for the steel equipment were taken from literature [6-9] (see also Tables 1-2).
While specifying the properties of nickel alloy and steel the implementation of the
Wiedemann-Franz's law was controlled: 𝜆𝜆 · 𝜌𝜌𝑒𝑒 = 𝐿𝐿𝐿𝐿, where 𝜆𝜆 is the thermal conductivity, ρe
is the specific electrical resistance, T is the temperature in K, L= 2.22·10-8 W·Ω·K-2 is the
Lorentz's constant.
Table 1. Thermo-electric properties of nickel superalloy used in simulations
°С
Τ
20
200
400
800
1000
1150
3
Kg/m
8550
8500
8450
8350
8330
8310
ρ
Cp J/(kg∙K)
440
520
520
570
590
600
7.4
11.2
14.1
19.8
26.7
36.7
λ W/(m∙K)
-7
-7
-6
-6
-6
8.7·10
9.3·10
8.9·10-7
1·10
1.2·10
1·10
ρe
Ω∙m
Table 2. Thermo-electric properties of pearlitic steel used in simulations
°С
27
127
327
527
927
Τ
3
Kg/m
7778
7772
7767
7762
7754
ρ
Cp J/(kg∙K)
469
506
521
660
577
48
47
41
37
23
λ W/(m∙K)
-7
-7
-7
-7
2·10
2.6·10
4.2·10
6.4·10
1.16·10-6
ρe
Ω∙m
1127
7751
530
12
1.4·10-6
Ref.
[7]
[7]
[6]
[6]
Ref.
[8]
[8]
[8]
[9]
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 is considered
in simulations. Thermal and electric contacts between the sample and bolts, between the
sample and the foundation are taken into account. The initial temperature for the sample and
the equipment is set to 30°C. For the free surface of sample the boundary condition of
convective heat transfer is used:
(1)
𝑞𝑞𝑛𝑛 = ℎ(𝑇𝑇 − 𝑇𝑇0 ) ,
𝑊𝑊
where qn is the heat flux density, ℎ = 20 𝑚𝑚2 𝐾𝐾 is the coefficient of convective heat transfer,
𝑇𝑇0 is the ambient temperature.
The condition of radiative heat transfer is also set on the surfaces of central (high
temperature) part of the sample (10 mm length):
𝑞𝑞𝑛𝑛 = ε𝜎𝜎𝑆𝑆𝑆𝑆 (𝑇𝑇 4 − 𝑇𝑇04 ) ,
(2)
−8
−2 −4
where ε = 0.8 is the black factor of the body, 𝜎𝜎𝑆𝑆𝑆𝑆 = 5.67 ∙ 10 𝑊𝑊𝑚𝑚 𝐾𝐾 is the coefficient
of Stefan-Boltzmann.
In order to realize an analytical approximation for the curve of temperature change in
time, we consider the problem of mathematical physics of heating the sample with a constant
cross-section. For example, the sample has a length and a depth the same with the corset
sample 32.5 mm and 3 mm respectively, but the sample with a constant cross section has
width is equal to 10 mm (Fig. 4).
Coupled thermo-electro-mechanical modeling of thermal fatigue of single-crystal corset samples
299
3 mm
10 mm
32.5 mm
Fig. 4. The statement of simplified thermal problem
The aim of our analogy is to simplify a task of heating the corset sample to onedimensional problem with equivalent boundary conditions. The isolation boundary conditions
are set on surfaces S3, S4, S5, S6. On the surface S2 is fixed the temperature, on the surface S1
boundary condition of convection with a convective heat transfer coefficient h is equal to
𝑊𝑊
20 𝑚𝑚2 𝐾𝐾. The sample in the thermal problem is heating by electric current that’s why boundary
condition of heat generation is set on the sample is equal to some constant Q, which does not
depend on time. The equation of unsteady thermal conductivity can be represented as [10]:
𝜕𝜕𝜕𝜕
𝑄𝑄
∆T - 𝜕𝜕𝜕𝜕 = - 𝜆𝜆 ,
(3)
𝜆𝜆𝜆𝜆
where T is the temperature, ∆ is the Laplace operator, τ = 𝐶𝐶
𝜌𝜌 𝜌𝜌
is the slow time, Q is the heat
generation, 𝜆𝜆, 𝐶𝐶𝜌𝜌 , 𝜌𝜌 are the conductive coefficient, the specific heat and the density
respectively. Considering that boundary conditions in the axis y and z are a lack of heat flux
and overwriting the Laplacian in Cartesian coordinates, we come to the equation:
𝑑𝑑2 𝑇𝑇
𝜕𝜕𝜕𝜕
𝜕𝜕 2 𝑇𝑇2
𝜕𝜕𝑇𝑇
𝑄𝑄
- 𝜕𝜕𝜕𝜕 = - 𝜆𝜆 ,
(4)
where x is the axial coordinate along the sample. Representing T as a sum of two functions
T1(x) and T2(x, τ), we come to two equations. One of these equations has two variable,
coordinate x and time τ:
𝑑𝑑𝑥𝑥 2
- 𝜕𝜕𝜕𝜕2 = 0.
(5)
Using Fourier method [10], we put two equations with variables X(x) and Ω(𝜏𝜏)
respectively. The equation for the variable X is:
𝑋𝑋 ′′ + 𝛽𝛽𝛽𝛽 = 0,
(6)
where 𝛽𝛽 is the arbitrary constant. Boundary conditions for the equation (6) are a convective
boundary condition in the middle of the sample and temperature is equal to zero on the edges.
Also the equation with a variable Ω is:
Ω′ + 𝛽𝛽Ω =0.
(7)
Finding a solution of equation (6) as a sum of sinus and cosine with constants and
substituting boundary conditions we put a transcendental equation:
𝜆𝜆ɣ
tg ɣ𝑛𝑛 = - ℎ𝑙𝑙𝑛𝑛 ,
(8)
𝜕𝜕𝑥𝑥 2
𝑊𝑊
where ɣ𝑛𝑛 = �𝛽𝛽𝑛𝑛 l, h = 20 𝑚𝑚2 𝐾𝐾 , l = 32.5 mm. General solution of equation (7) is Ω = C𝑒𝑒 −𝛽𝛽𝑛𝑛𝜏𝜏 ,
ɣ 2
− 𝑛𝑛 𝜏𝜏
ɣ 2
where 𝛽𝛽𝑛𝑛 is the eigenvalue and 𝛽𝛽𝑛𝑛 = 𝑙𝑙𝑛𝑛2 . As a result, we get T2(x, τ) = C·X(x)· 𝑒𝑒 𝑙𝑙2 , where
ɣ𝑛𝑛 can be found from an equation (8). We use simple approximation for temperature changing
ɣ 2
in time for a heating and cooling as one exponential with exponent - 𝑙𝑙𝑛𝑛2 with constants.
Returning to usual time t, we can rewrite an analytical approximation for heating as:
𝑇𝑇 = 𝐴𝐴 − 𝐵𝐵 · e
ɣ 2 𝜆𝜆
− n2 ·
𝑡𝑡
𝑙𝑙
Cρ ρ
,
(9)
300
A.V. Savikovskii, A.S. Semenov, L.B. Getsov
where A and B are constants, which are selected from conditions of equality in the beginning
of the heating to minimum temperature in the cycle and in the end of the heating to maximum
temperature in the cycle, ɣ𝑛𝑛 can be found from a transcendental equation (8), l is the length of
the sample. For a process of the cooling of the sample the similar analytical approximation is
introduced:
ɣ 2 𝜆𝜆
− 𝑛𝑛2 ·
𝑡𝑡
𝑇𝑇 = 𝐶𝐶 + 𝐷𝐷 · 𝑒𝑒 𝑙𝑙 𝐶𝐶𝜌𝜌𝜌𝜌 .
(10)
Signs before constants B and D in (9) and (10) provide positive values of B and D.
𝑊𝑊
𝐽𝐽
𝐾𝐾𝐾𝐾
Material constants 𝜆𝜆, 𝐶𝐶𝜌𝜌 , 𝜌𝜌 are set to 20 𝑚𝑚𝐾𝐾 , 550𝑘𝑘𝑘𝑘·𝐾𝐾 and 8400 𝑚𝑚3 respectively.
Comparison of experimental data, computational results and analytical approximation for
temperature changing in time are presented in Fig. 5 for loading regimes 100÷800, 150÷900,
500÷1050 and 700÷1050 °C.
a)
b)
c)
d)
Fig. 5. Comparison of experimental data, simulation results and
analytical approximation for temperature modes:
a) 100÷800°C, a heating time is 19 s, a cooling time is 46 s;
b) 150÷900°C, a heating time is 42 s, a cooling time is 60 s;
c) 500÷1050°C, a heating time is 14 s, a cooling time is 10 s;
d) 700÷1050°C, a heating time is 8 s, a cooling time is 7 s
Comparison of experimental data and computational results for temperature distribution
along the corset sample at cycle phase with maximum temperature and also for the different times
for loading regimes 150÷900, 250÷1000, 500÷1050 and 700÷1050°C are shown in Fig. 6.
Coupled thermo-electro-mechanical modeling of thermal fatigue of single-crystal corset samples
301
a)
b)
c)
d)
Fig. 6. Temperature distribution along the corset sample at cycle phase with maximum
temperature (left) and also for the different times (right) for loading regimes:
a) 150÷900°C, a heating time is 42 s, b) 250÷1000°C, a heating time is 80 s;
c) 500÷1050°C, a heating time is 14 s, d) 700÷1050°C a heating time is 8 s
Temperature field distributions for cycle phase with maximum temperature for loading
modes 100÷800, 150÷900, 250÷1000 and 700÷1050°C are presented in Fig. 7.
302
A.V. Savikovskii, A.S. Semenov, L.B. Getsov
a)
b)
c)
d)
Fig. 7. Temperature field distributions for maximum
temperature for loading regimes:
a) 100÷800°C; b) 150÷900°C; c) 250÷1000°C; d) 500÷1050°C
The results of thermo-electric problem simulations in form spatial and temporal
distribution of the temperature field are the base for the strain and stress field computation
within the framework of thermo-elasto-visco-plastic problem.
3. 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 local strain and stress concentration is observed in the central
part of the corset sample. 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 [11], which allows to apply the
micromechanical (microstructural, crystallographic, physical) models of plasticity and creep
for single crystals [12,13]. The micromechanical plasticity model accounting 12 octahedral
slip systems with lateral and nonlinear kinematic hardening [12] is used in the FE
computation for the simulation 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 and foundation.
The viscous properties are taken into account because of high temperature despite 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 [14] 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. One of the aims of the investigations is the selection of the
equivalent (effective) length of the sample for the simplified formulation. The validity of the
Coupled thermo-electro-mechanical modeling of thermal fatigue of single-crystal corset samples
303
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 (Fig. 8b) 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. 8a) 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 mechanical properties for the alloys VZHM4 and VIN3 are taken from the papers [15,16]
and for ZHS32 from [17] are presented in Tables 3-5. The mechanical properties of bolts are
taken for pearlitic steel [9].
Table 3. Mechanical properties of VZHM4 used in simulations [15]
T
20
700
800
900
̊C
E001
MPa
130000
101000
96000
91000
0.39
0.42
0.422
0.425
𝜈𝜈
-5
-5
-5
α
1/K
1.11·10
1.68·10
1.74·10
1.87·10-5
MPa
846
950
σY 001
n
8
8
8
8
A
MPa-ns-1
1·10-42
3·10-31
1·10-29
1·10-28
Table 4. Mechanical properties of VIN3 used in simulations [16]
T
20
500
700
900
̊C
E001
MPa
126000
110000
104000
89000
0.39
0.41
0.42
0.42
𝜈𝜈
-5
-5
-5
1/K
1.21·10
1.33·10
1.4·10
1.5·10-5
α
MPa
555
800
930
910
𝜎𝜎𝑌𝑌 001
n
8
8
8
8
-n -1
-42
-34
-30
A
MPa s
1·10
4·10
1.5·10
5.8·10-27
Table 5. Mechanical properties of ZHS32 used in simulations [17]
T
20
700
800
900
⁰C
E001
MPa
137000
110000
105000
99800
0.395
0.4248
0.4284
0.4317
𝜈𝜈
-5
-5
-5
-5
1/K
1.24·10
1.6·10
1.7·10
1.81·10
α
MPa
919
904
901
895
𝜎𝜎𝑌𝑌 001
n
8
8
8
8
A
MPa-ns-1
1·10-42
2.5·10-31 8.5·10-30
2·10-28
1000
86000
0.428
2.1·10-5
8
2·10-27
1000
80000
0.425
1.57·10-5
645
8
3.5·10-25
1000
94800
0.4347
2.22·10-5
670
8
6·10-27
1050
82000
0.43
2.3·10-5
820
8
1·10-26
1050
75000
0.428
1.6·10-5
540
8
1.5·10-24
1050
92300
0.4361
2.42·10-5
580
8
7·10-26
304
A.V. Savikovskii, A.S. Semenov, L.B. Getsov
In simplified formulation (see Fig. 8c) 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.
a)
b)
sample
equipment
c)
bolts
equipment
d)
sample
z
sample
y
bolt
sample
x
Fig. 8. FE models of the corset sample for thermo-elasto-visco-plastic problem solution:
a) full model with taking into account equipment (no symmetry),
b) model (¼ due to symmetry) with taking into account equipment,
c) simplified model without taking into account equipment (no symmetry),
d) simplified model (¼ due to symmetry) without taking into account equipment
Figure 9 shows distributions of plastic strain intensity field for three nickel superalloys
and three different temperature modes after 7 cycles (for VZHM4 and VIN3 the effective
length of the sample is 42 mm, for ZHS32 is 50 mm) obtained with using the FE model
(¼ due to symmetry) with taking into account equipment (Fig. 8b).
b)
a)
c)
Fig. 9. Distributions of plastic strain intensity field after 7 cycles in the corset sample for:
a) superalloy VZhM4, loading regime 700÷1050°C;
b) superalloy VIN3, loading regime 500÷1050°C;
c) superalloy ZhS32, loading regime 150÷900°C
The Table 6 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. FE simulations show that the
Coupled thermo-electro-mechanical modeling of thermal fatigue of single-crystal corset samples
305
effective length doesn’t depend on the type of hardening (isotropic and kinematic) and
doesn’t depend on temperature mode. 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
34-42 mm
38-46 mm
ZHS32
40-52 mm
4. 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. 8d). 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 1100°C and a temperature range of 900°C;
• maximum temperature of 1050°C and a temperature range of 550°C;
• maximum temperature of 1050°C and a temperature range of 350°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 are 24s, 7s, 18 s, 28s, the cooling time are 15s, 15s, 40s,
52s for VZhM4. The heating time in the cycle is 10 s, the cooling time is 16s for VIN3. The
heating times in the cycle is 25 s, the cooling time is 75s for ZhS32.
The mechanical properties for the alloys VZhM4 and VIN3 were taken from the papers
[15], [16] and for ZhS32 from [17] (see also Tables 3-5).
The problem is solved in a quasi-static 3-dimensional formulation. The FE model is
shown in Fig. 8d. 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. 10.
Damage calculation and estimation of the number of cycles for the macrocrack
initiation are made on the basis of four-member deformation criterion [3-5]:
N
D =∑
( De )
p
eqi
k
N
( De )
с
eqi
m
e eqp
e eqc
(11)
,
+∑
+ max
+ max
С (T )
С2 (T ) 0≤t ≤tmax e rp (T ) 0≤t ≤tmax e rc (T )
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 :
=i 1 =
i 1
1
e eq = n {111} ⋅ ε ⋅ l
011
.
(12)
306
A.V. Savikovskii, A.S. Semenov, L.B. Getsov
a)
b)
c)
d)
Fig. 10. Schematic presentations of temperature evolutions in central point of sample for
loading regimes with and without delay:
a) 700÷1050°C; b) 150÷900°C; c) 500÷1050°C and d) 250÷1000°C
Usually it takes in (11) the values of constants: k = 2, m =
5
4
( )
, C1 = ε rp , C2 = ( 34 ε rc ) ,
k
m
where ε rp and ε rc are ultimate strains of plasticity and creep under uniaxial tension. In the
FE computations the values of ultimate strains 𝜀𝜀𝑝𝑝𝑟𝑟 = 𝜀𝜀𝑐𝑐𝑟𝑟 = 17 % are used the same for all
considered alloys. Improvement of the prediction accuracy of the delay time influence on
durability can be achieved by the refinement of the constant strains 𝜀𝜀𝑝𝑝𝑟𝑟 on the basis of data
without delay.
An analytical approximation of delay time influence in thermal fatigue strength has
been proposed in the form:
N = Nmin + (N0 - Nmin)·𝑒𝑒 −𝑡𝑡𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 /𝜏𝜏 ,
(13)
where N is the number of cycles to crack initiation as function of delay time 𝑡𝑡𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 , N0 is the
computational number of cycles in case without delay time, Nmin is the number of cycle in
case with delay time is equal to 1 hour, 𝜏𝜏 is a constant (50 s for all considered materials).
Comparison of results of FE simulations with experimental data for single-crystal superalloys
VZhM4, VIN3 and ZhS32 is given in Fig. 11. The good agreement between computational
and experimental results is observed.
Comparison of results of FE simulations and analytical approximation (13) 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. 12. Small deviations are
observed only in the vicinity of the region of maximum curvature.
Coupled thermo-electro-mechanical modeling of thermal fatigue of single-crystal corset samples
a)
c)
307
b)
d)
e)
f)
Fig. 11. Comparison of results of FE simulations and experimental data for alloys:
a) VZhM4, loading regime 150÷900°C, a heating time is 28 s, a cooling time is 52 s;
b) VZhM4, loading regime 250÷1000°C, a heating time is 18 s, a cooling time is 40 s;
c) VZhM4, loading regime 500÷1050°C, a heating time is 24 s, a cooling time is 15 s;
d) VZhM4, loading regime 700÷1050°C, a heating time is 7 s, a cooling time is 15 s;
e) ZhS32, loading regime 200÷1100°С, a heating time is 25 s, a cooling time is 75 s;
f) VIN3, loading regime 500÷1050°С, a heating time is 10 s, a cooling time is 16 s
308
A.V. Savikovskii, A.S. Semenov, L.B. Getsov
a)
b)
c)
d)
e)
f)
Fig. 12. Comparison of results of FE simulations and
analytical approximation (13) for alloys:
a) VZhM4, loading regime 150÷900°C, a heating time is 28 s, a cooling time is 52 s;
b) VZhM4, loading regime 250÷1000°C, a heating time is 18 s, a cooling time is 40 s;
c) VZhM4, loading regime 500÷1050°C, a heating time is 24 s, a cooling time is 15 s;
d) VZhM4,loading regime 700÷1050°C, a heating time is 7 s, cooling time is 15 s;
e) ZhS32, loading regime 200÷1100°С, a heating time is 25 s, a cooling time is 75 s;
f) VIN3, loading regime 500÷1050°С, a heating time is 10 s, a cooling time is 16 s
5. Conclusions
The results of thermal and stress-strain state simulations for single-crystal corset specimens
under cyclic electric heating show a good agreement with the experimental data for a wide
range of temperature alteration and different single-crystal nickel based superalloys. Obtained
results point out on the possibility of predicting thermal fatigue durability for single crystal by
Coupled thermo-electro-mechanical modeling of thermal fatigue of single-crystal corset samples
309
means of thermo-electro-mechanical finite-element simulation with using of four-member
deformational criterion of damage accumulation and microstructural models of inelastic
deformation.
A systematic numerical analysis of the delay effect at maximum temperature on thermal
fatigue durability was carried out for various single-crystal superalloy samples in wide range
temperatures. The simplified analytic approximation for durability curves are proposed on the
base of results of multivariate computational experiments.
Acknowledgments. Research was conducted under the financial support of the Russian
Science Foundation, Grant no. 18-19-00413.
References
[1] Shalin RE, Svetlov IL, Kachanov EB, Toloraiya VN, Gavrilin OS. Single crystals of
nickel heat-resistant alloys. Moscow: Mashinostroenie; 1997. (In Russian)
[2] Getsov LB. Materials and strength of gas turbine parts. Rybinsk: Gazoturbinnye
Tekhnologii; 2010. (In Russian)
[3] Getsov LB, Semenov AS. Criteria of fracture of polycrystalline and single crystal
materials under thermal cyclic loading. In: Proceedings of CKTI. Vol. 296. 2009. p.83-91.
[4] Semenov AS, Getsov LB. Thermal fatigue fracture criteria of single crystal heat-resistant
alloys and methods for identification of their parameters. Strength of Materials. 2014;46(1):
38-48.
[5] Getsov LB, Semenov AS, Staroselsky A. A failure criterion for single-crystal superalloys
during thermocyclic loading. Materials and technology. 2008;42(1): 3-12.
[6] Petrushin NO, Logunov AV, Kovalev AI, Zverev AF, Toropov VM, Fedotov NH.
Thermophysical properties of Ni3Al-Ni3Nb directly crystallized eutectic composition. High
temperature thermophysics. 1976;14(3): 649-652.
[7] Zinoviev VE. Thermo-physical properties of metals at high temperatures. Moscow:
Metallurgia; 1989. (In Russian)
[8] Chirkin VS. Thermophysical properties of nuclear materials. Moscow: Atomizdat; 1968.
(In Russian)
[9] Maslenkov SB, Maslenkova EA. Steels and alloys for high temperatures. Moscow:
Metallurgia; 1991. (In Russian)
[10] Courant R, Hilbert D. Methods of mathematical physics. Vol. 2. Leningrad: GTTI; 1945.
[11] Semenov AS. PANTOCRATOR – finite-element program specialized on the solution of
non-linear problems of solid body mechanics. In: Proc. of the V-th International. Conf.
"Scientific and engineering problems of reliability and service life of structures and methods
of their decision". Saint Petersburg: Izd-vo SPbGPU; 2003. p.466-480.
[12] Cailletaud GA. Micromechanical approach to inelastic behaviour of metals. Int. J. Plast.
1991;8(1): 55-73.
[13] Semenov AS. Identification of anisotropy parameters of phenomenological plasticity
criterion for single crystals on the basis of micromechanical model. Scientific and technical
sheets SPbGPU. Physical and mathematical Sciences. 2014;2(194): 15-29. (In Russian)
[14] May S, Semenov AS. Modeling of inelastic cyclic deformation of monocrystalline
specimens. In: Proc. of the XXXIX week of science of SPbPU. Vol. 5. 2010. p.73-74. (In
Russian)
[15] Kablov EN, Petrushin NO, Svetlov IL, Demonis IM. Nickel casting heat-resistant alloys
of the new generation. In: The jubilee nauch.-tech. sat. Aviation materials and technologies.
Moscow: Proceedings of VIAM; 2012. p.36-52. (In Russian)
[16] Semenov SG, Getsov LB, Semenov AS, Petrushin NV, Ospennikova OG,
Zhivushkin AA. The issue of enhancing resource capabilities of nozzle blades of gas turbine
310
A.V. Savikovskii, A.S. Semenov, L.B. Getsov
engines through the use of new single crystal alloy. Journal of Machinery Manufacture and
Reliability. 2016;4: 30-38.
[17] Getsov LB, Semenov AS, Tikhomirova EA, Rybnikov AI. Thermocyclic and static
failure criteria for single crystal superalloys of gas turbine blades. Materials and technology.
2014;48(2): 255-260.
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