THERMO-ELECTRO-MECHANICAL NUMERICAL MODELING WITH
AN ANALYTICAL APPROXIMATION OF COUPLED THERMAL
FATIGUE FAILURE PROCESS OF CORSET SAMPLES FROM SINGLECRYSTAL NICKEL-BASED SUPERALLOYS
A.V. Savikovskii1*, A.S. Semenov1, L.B. Getsov2
SPBPU, Polytechnicheskaya 29, Russia
NPO CKTI, Polytechnicheskaya 24, Russia
Abstract. The results of computations of the thermal and stress-strain state of single-crystal corset specimens
subjected to the action of periodic electric current, leading to variable inhomogeneous heating and subsequent
thermal fatigue failure, are presented. An analytic approximation was introduced for the process of nonstationary
heating of corset sample and a comparation of modeling results with experimental data and analytic model was
evaluated which showed a good correlation with experimental data. The influence of maximum value and range
of temperature and also influence of a delay time at the maximum temperature on the number of cycles before
the macrocrack formation were investigated. Also an analytic approximation was considered of a delay influence
and comparison of the computational results and analytic formulaes with the experimental data for various
single-crystal nickel-based superalloys showed a good accuracy.
Single-crystal nickel based superalloys  are used for production of gas turbine engines
(GTE). These materials have a pronounced anisotropy of properties and a dependence properties on
temperature and thermal fatigue strength of superalloys is not studied very well. 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  (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. During cycling the maximum and minimum
temperatures are automatically maintained constant.
Fig. 1. Installation for carrying out
experiments on thermal fatigue.
Fig. 2. Geometry corset sample for thermal
The aim of the study is to investigate numerically a process of heating and cooling of corset sample
and to request analytical approximation for this process, to study numerically the stress-strain state
of the sample during cyclic heating and cooling due to its clamping and 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 full-scale experiments and results of analytical formulaes. The
results of simulation and their verification are obtained for the different single-crystal nickel-based
superalloys: VZhM4, VIN3 and ZhS32.
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 was carried in the FE program ANSYS with taking into account the
temperature dependence of all material properties, thermoelectric contacts between the sample and
an equipment, 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. 3.
The problem was solved for different single-crystal nickel-based 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.
Fig. 3. Finite-element model in thermoelectric problem.
Modeling of heating and cooling processes of sample was carried out for five temperature
regimes (modes): 100÷800, 150÷900, 250÷1000, 500÷1050 and 700÷1050 °C. There are
nonstationary experimental data for four regimes: 100÷800 °C, a heating time is 19s, a cooling time
is 46s, 150÷900 °C, a heating time is 42s, a cooling time is 59,5s, 500÷1050 °C, a heating time is
14s, a cooling time is 10s, 700÷1050 °C, a heating time is 8s, a cooling time is 7s. In case of
temperature mode 250÷1000 time of heating is 80 s. The used in FE simulations material properties
for the single crystal nickel superalloy sample and for the steel equipment were taken from
literature , - (see also Tables 1-2). While specifying the properties of nickel alloy and steel
the implementation of the Wiedemann-Franz’s law was controlled: 𝜆 · 𝜌𝑒 = 𝐿𝑇, where 𝜆 is the
thermal conductivity, e is the specific electrical resistance, T is the temperature in K, L= 2.22 *10-8
W··K-2 is the Lorentz’s constant.
Table 1 Thermo-electric properties of nickel superalloy used in simulations:
Table 2 Thermo-electric properties of pearlitic steel used in simulations:
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:
𝑞𝑛 = ℎ(𝑇 − 𝑇0 ) ,
where n is the normal to body, qn is the heat flux density, ℎ = 20 𝑚2 𝐾 is the coefficient of
convective heat transfer, 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 − 𝑇04) ,
where ε = 0.8 is the black factor of the body, 𝜎𝑆𝐵 = 5.67 ∙ 10 𝑊𝑚 𝐾 is the coefficient of
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).
Fig. 4. The statement of simplify thermal problem
The aim of our analogy is to simplify a task of heating the corset sample to one-dimensional
problem with equivalent boundary conditions. The boundary conditions of a lack of heat flow were
set on surfaces S3, S4, S5, S6. On the surface S 2 was fixed the temperature, on the surface S1
boundary condition of convection with a convective heat transfer coefficient h is equal to 20 𝑚2𝐾 .
The sample in the thermal problem is heating by electric current that’s why boundary condition of
heat generation was 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
∆T - 𝜕𝜏 = - 𝜆 ,
where T is the temperature, ∆ 𝑖s the laplassian operator, τ is the slow time and τ = 𝐶 𝜌, Q is the heat
generation, 𝜆, 𝐶𝜌 , 𝜌 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
laplassian operator in Cartesian coordinates, we come to the equation
- 𝜕𝜏 = - 𝜆 ,
where x is the axial coordinate along the sample. Representing T as a sum of two functions T 1(x)
and T2(x, τ), we come to two equations. One of these equations has two variable, x and time:
Using Fourier method , we put two equations with variables X and 𝜴 respectively. The equation
with a variable X is
𝑋 ′′ + 𝛽𝑋 = 0,
where 𝛽 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′ + 𝛽Ω =0,
where 𝛽 is the arbitrary constant from equation (6). Finding a solution of equation (6) as a sum of
sinus and cosine with constants and substituting boundary conditions we put a transcendental
tgɣ𝑛 = - ℎ𝑙𝑛 ,
where ɣ𝑛 = √𝛽𝑛 l, h = 20 𝑚2 𝐾 ,l = 32.5 mm. General solution of equation (7) is Ω = C𝑒 −𝛽𝑛𝜏 ,
− 𝑛 𝜏
where 𝛽𝑛 is the eigenvalue and 𝛽𝑛 = 𝑙𝑛2 . In general, we put solution T2(x, τ) = C·X(x)· 𝑒 𝑙2 ,
where ɣ𝑛 𝑐𝑎𝑛 be found from an equation (8). We use simple approximation for temperature
changing in time for a heating and cooling as one exponential with exponent - 𝑙𝑛2 with constants.
Returning to usual time t, we can rewrite an analytical approximation for heating as:
ɣ 2 𝜆
− 𝑛2 ·
𝑇 = 𝐴 − 𝐵 · 𝑒 𝑙 𝐶𝜌 𝜌 ,
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, ɣ𝑛 𝑐𝑎𝑛 be found from a transcendental equation (8), l is the length of the sample. For a
process of the cooling of the sample similar analytical approximation was considered:
ɣ 2 𝜆
− 𝑛2 ·
𝑇 = 𝐶 + 𝐷 · 𝑒 𝑙 𝐶𝜌 𝜌 ,
Materials constants 𝜆, 𝐶𝜌 , 𝜌 were set to 20 𝑚2𝐾 , 550𝑘𝑔·𝐾 and 8400 𝑚3 respectively.Comparison of
experimental data, computational results and analytical approximation for temperature changing in
time are presented in fig. 5 for temperature modes 100÷800, 150÷900, 500÷1050 and 700÷1050 °C.
Fig. 5. Comparison of experimental data, simulation results and analytical approximation for
temperature modes: a) 100÷800, a heating time is 19s, a cooling time is 46s, b) 150÷900 °C, a
heating time is 42s, a cooling time is 59,5s, c) 500÷1050 °C, a heating time is 14s, a cooling time is
10s, d) 700÷1050 °C, a heating time is 8s, a cooling time is 7s.
Comparison of experimental data and computations for maximum temperature and temperature
distributions along the corset sample in different times for temperature modes 150÷900, 250÷1000,
500÷1050 and 700÷1050 °C are shown in fig. 6.
Fig.6. Comparison of experimental data and computations for maximum temperature and
temperature distributions along the corset sample in different times for temperature modes
a) 150÷900, a heating time is 42s, b) 250÷1000, a heating time is 80s c) 500÷1050, a heating
time is 14s d) 700÷1050 a heating time is 8s
Temperature plane distributions for maximum temperature for temperature modes 100÷800,
150÷900, 250÷1000 and 700÷1050 °C are presented in fig. 7.
Fig.7. Temperature plane distributions for maximum temperature for temperature modes
a) 100÷800, b) 150÷900, c) 250÷1000 d) 500÷1050 °C
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-elastovisco-plastic problem solution.
3. Results of thermo-elasto-visco-plastic analysis
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 and isotropic hardening, 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 were taking into account because of a quick time
of heating and cooling of the corset sample.
The two FE formulations for the thermomechanical problem have been considered:
• with taking into account equipment;
• without taking into account equipment (simplified formulation  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 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 due to anisotropy of mechanical
properties of single crystal sample. However in the important for practice case of 
crystallographic orientation of sample the symmetry in respect to planes xz and yz (see Fig. 8)
can be introduced. Equipment and bolts were modeled by linear elastic material (steel), and for
the sample the elasto-visco–plastic model of the material was used. 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 temperature boundary conditions were set from the experimental
data at maximum and minimum temperature with linear interpolation in time. The mechanical
properties for the alloys VZHM4 and VIN3 were taken from the papers [12, 13] and for ZHS32
from  are presented in Table 3, 4, 5. The mechanical properties of bolts are taken for pearlitic
Table 3. Mechanical properties of VZHM4 used in simulations :
𝜎𝑌 001 MPa
Table 4. Mechanical properties of VIN3 used in simulations :
𝜎𝑌 001 MPa
−𝑛 −1 1·10−42
Table 5. Mechanical properties of ZHS32 used in simulations :
𝜎𝑌 001 MPa
In simplified formulation (see Fig. 8) 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 zaxes.
Fig. 8. Finite-element models in mechanical problem: a) with taking into account equipment,
symmetric statement, b) with taking into account equipment, full statement, c) without taking into
account equipment (simplified formulation), symmetric statement, d) without taking into account
equipment (simplified formulation), full statement.
Fig.9 shows distributions of plastic strain intensity for three nickel superalloys and three
different temperature modes after 7 cycles (for VZHM4 and VIN3 the length of the sample is 42
mm, for ZHS32 is 50 mm).
Fig. 9 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. FE simulations showed that effective length doesn’t
depend on 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
In the FE simulations the length of the specimen for all alloys was taken to be 40 mm.
4. Influence of delay on the thermal fatigue durability and analytical approximation for
Simulation of inelastic cyclic deformation of corset samples were performed with using
of the FE program PANTOCRATOR , which allows to apply the micromechanical
(physical) models of plasticity and creep for single crystals , . The Norton power-type
law without hardening was used to describe creep properties. 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 effective length of sample equal 40 mm,
see Fig. 9a). 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 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. 9b) with:
• maximum temperature of 1050 °C and a temperature range of 350 °C and 550 °C;
• maximum temperature of 1100 °C and a temperature range of 900 °;
• maximum temperature of 1000 °C and a temperature range of 750 °;
• maximum temperature of 900 °C and a temperature range of 750 °C.
The heating times in the cycle were 24s, 7s, 18 s, 28s and 17s, the cooling time was 15s,
15s, 40s, 52s and 60s for VZhM4. The heating time in the cycle was 10s, the cooling time was
16s for VIN3. The heating times in the cycle was 35s, 25 s and 15s, the cooling time was 55,
75s and 15s for ZhS32. The mechanical properties for the alloys VZhM4 and VIN3 were taken
from the papers ,  and for ZhS32 from . The problem was solved in a quasi-static 3dimensional 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 solidstate motions, a number of points on these faces in the direction of the y and z axes were also
Fig. 10. Finite element model of sample (simplified formulation) for analysis of delay influence
Temperature evolutions in central point of sample with and without delay for temperature modes
700÷1050 °C, 500 ÷1050 °C, 250÷1000 ⁰C and 150 ÷ 900 °C are presented in fig. 11.
Fig. 11. Temperature evolutions in central point of sample with and without delay for
temperature modes 700÷1050 °C, 500 ÷1050 °C, 250÷1000 ⁰C and 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 [3-5]:
0 t tmax
0 t tmax
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. Usually it takes the values k=2, m = 4 ,
C1 rp , C2 34 rc , where rp and rc are ultimate strains of plasticity and creep under
uniaxial tension. In the FE computations the values of ultimate strain 𝜀𝑟 = 𝜀𝑟𝑐 = 𝜀𝑟 = 0.17 and
𝜀𝑟 = 𝜀𝑟𝑐 = 𝜀𝑟 = 0.13 for VZhM4, 𝜀𝑟 = 𝜀𝑟𝑐 = 𝜀𝑟 = 0.18 and 𝜀𝑟 = 𝜀𝑟𝑐 = 𝜀𝑟 = 0.13 for ZhS32, 𝜀𝑟 = 𝜀𝑟𝑐 =
𝜀𝑟 = 0.17 for VIN3 were used because of dependence ultimate strains from temperature.
Improvement of the accuracy of prediction of influence the delay time on durability can be
achieved by the refinement of the constant 𝜀𝑟 on the basis of data without delay.
In order to reduce computations, an analytical approximation of delay time influence in
thermal fatigue strength was performed. Dependence thermal fatigue strength on delay time was
N = Nmin + (N0 - Nmin)·𝑒 −𝑡/50 ,
where N is the number of cycles till the formation of macrocracks as function of delay time, N 0
is the the computational number of cycles till the formation of macrocracks in case without
delay, Nmin is the number of cycle in case delat time is equal to 1 hour, t is the delay time. 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. 12-13.
Fig. 12. Comparison of results of FE simulation and experimental data for the alloy VZhM4:
a) mode 150÷900 ⁰C, heating time is 28s, cooling time is 52s, 𝜀𝑟 = 0.13,
b) mode 150÷900 ⁰C, heating time is 17s, cooling time is 60s, 𝜀𝑟 = 0.13,
c) mode 250÷1000 ⁰C, heating time is 18s, cooling time is 40s, 𝜀𝑟 = 0.17,
d) mode 500÷1050 ⁰C, heating time is 24s, cooling time is 15s, 𝜀𝑟 = 0.17,
e) mode 500÷1050 ⁰C heating time is 7s, cooling time is 15s, 𝜀𝑟 = 0.17,
f) mode 700÷1050 ⁰C heating time is 7s, cooling time is 15s, 𝜀𝑟 = 0.17.
Fig. 13. Comparison of results of FE simulation and
experimental data for the alloy:
a) ZhS32, mode 150÷900, heating time is 35 s, cooling time is 55 s, 𝜀𝑟 = 0.13,
b) ZhS32, mode 200÷1100, heating time is 25 s, cooling time is 75 s, 𝜀𝑟 = 0.18,
c) ZhS32, mode 500÷1000, heating time is 10 s, cooling time is 14 s, 𝜀𝑟 = 0.18,
d) ZhS32, mode 500÷1050, heating time is 15 s, cooling time is 15 s, 𝜀𝑟 = 0.18,
e) ZhS32, mode 700÷1050, heating time is 15 s, cooling time is 15 s, 𝜀𝑟 = 0.18
f) VIN3, mode 500÷1050, heating time is 10s, cooling time is 16 s, 𝜀𝑟 = 0.17.
Comparison of results of FE simulation and analytical approximation 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. 14-15.
Fig. 14. Comparison of results of FE simulation and analytical approximation for the alloy VZhM4:
a) mode 150÷900 ⁰C, heating time is 28s, cooling time is 52s,
b) mode 150÷900 ⁰C, heating time is 17s, cooling time is 60s,
c) mode 250÷1000 ⁰C, heating time is 18s, cooling time is 40s,
d) mode 500÷1050 ⁰C, heating time is 24s, cooling time is 15s,
e) mode 500÷1050 ⁰C heating time is 7s, cooling time is 15s,
f) mode 700÷1050 ⁰C heating time is 7s, cooling time is 15s.
Fig. 15. Comparison of results of FE simulation and analytical approximation for the alloy:
a) ZhS32, mode 150÷900, heating time is 35 s, cooling time is 55 s,
b) ZhS32, mode 200÷1100, heating time is 25 s, cooling time is 75 s,
c) ZhS32, mode 500÷1000, heating time is 10 s, cooling time is 14 s,
d) ZhS32, mode 500÷1050, heating time is 15 s, cooling time is 15 s,
e) ZhS32, mode 700÷1050, heating time is 15 s, cooling time is 15 s,
f) VIN3, mode 500÷1050, heating time is 10s, cooling time is 16 s.
The results of the computations and the analytical approximations show a good agreement with
the experiment, which suggests that the finite-element computations in combination with
application of deformation criterion (3) can be used to predict the thermal-fatigue strength of
various single-crystal superalloy samples in wide range temperatures with different delay times.
The research is supported by the SIEMENS Scholarship Program, RFBR grant No. 16-0800845 and RSF grant No. 18-19-00413.
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