UDK 539.42:669.018	ISSN 1580-2949
Original scientific article/Izvirni znanstveni članek	MTAEC9, 48(2)255(2014)
THERMOCYCLIC- AND STATIC-FAILURE CRITERIA FOR SINGLE-CRYSTAL SUPERALLOYS OF GAS-TURBINE BLADES
TERMOCIKLIČNA IN STATIČNA MERILA ZA PORUŠITVE LOPATIC PLINSKIH TURBIN IZ MONOKRISTALNIH SUPERZLITIN
Leonid B. Getsov1, Artem S. Semenov2, Elena A. Tikhomirova3, Alexander I. Rybnikov1
1NPO ZKTI, Russia 2St. Petersburg State Polytechnical University, Russia
3Klimov Company, Russia guetsov@yahoo.com, semenov.artem@googlemail.com
Prejem rokopisa - received: 2013-05-17; sprejem za objavo - accepted for publication: 2013-06-28
The problem of the thermo-mechanical fatigue (TMF) of single-crystal turbine blades has not been fully investigated theoretically or experimentally. In the present work TMF tests were performed for two single-crystal nickel-based alloys (ZhS36 and ZhS32) with various crystallographic orientations ((001), (011), (111)) under different temperatures and cycle durations. The dependence of the failure modes (crystallographic or non-crystallographic) on the loading regimes was analyzed. The non-linear viscoelastic, elastoplastic and viscoelastoplastic material models with non-linear kinematic hardening were used to predict the cyclic stress-strain state, ratcheting and creep of the samples. The deformation criterion of damage accumulation was introduced to the lifetime prediction. A stress analysis of the single-crystal samples, with concentrators (in the form of circular holes) and without them, was carried out using physical models of the plasticity and creep. These material models take into account that an inelastic deformation occurs due to a slip mechanism and it is determined with the crystallographic orientation. The proposed failure model using the deformation criterion allows qualitative and quantitative predictions of the TMF fracture process in single crystals.
Keywords: gas-turbine blade, single crystal, thermo-mechanical fatigue, damage, crystallographic and non-crystallographic failure modes
Problem termo-mehanske utrujenosti (TMF) monokristalnih turbinskih lopatic še ni v celoti raziskan niti teoretično niti eksperimentalno. V tem delu so bili izvršeni TMF-preizkusi na dveh monokristalnih zlitinah na osnovi niklja (ZhS36 in ZhS32) z različno kristalografsko orientacijo ((001), (011), (111)) pri različnih temperaturah in trajanju ciklov. Analizirana je bila odvisnost načina porušitve (kristalografska ali nekristalografska) od vrste obremenjevanja. Modeli nelinearne viskoelastičnosti, elastoplastičnosti in viskoelastoplastičnosti z nelinearnim kinematičnim utrjevanjem so bili uporabljeni za napovedovanje cikličnega stanja napetost - raztezek, nazobčanja in lezenja vzorcev. V napovedovanje dobe trajanja je bilo vpeljano deformacijsko merilo akumuliranja poškodb. Izvršena je bila analiza napetosti v monokristalnem vzorcu s koncentratorji (v obliki okroglih lukenj) in brez njih, z uporabo fizikalnega modela plastičnosti in lezenja. Ti materialni modeli upoštevajo, da se pojavi neelastična deformacija z mehanizmom lezenja in je določena s kristalografsko orientacijo. Predlagani model porušitve z uporabo deformacijskih meril omogoča kvalitativno in kvantitativno napovedovanje TMF- procesa preloma monokristala.
Ključne besede: lopatica plinske turbine, monokristal, termo-mehanska utrujenost, poškodba, kristalografski in nekristalografski način porušitve
1	INTRODUCTION	carbon amounts and were designed to expand the infor-
mation given in1,2. The tests of the mechanical properties, The use of single-crystal alloys for the manufacturing creep and thermal-fatigue resistance at different tempe-of the blades of a gas-turbine engine allows a significant ratures were carried out.
increase in the gas temperature before a turbine and sets The creep tests were conducted on an installation of a number of tasks that should help increase the reliability ATS (Applied Test Systems, Inc.) determining the kine-of a stress and strength analysis. In the present investi- tics of the accumulated inelastic deformation during the gation, the results of the experimental studies of single- first stage and in the steady state of the creep. The test crystal nickel-based alloys, as well as the approaches to methods for TMF are described in detail in3,4. For the the computation of the stress-strain state and strength of tests, rigidly clamped samples with polished surfaces the structural parts are considered and discussed.	were used, as shown in Figure 1. The tests were con-
ducted in a vacuum that allowed us, during the testing, to
2	MATERIALS AND METHODS OF RESEARCH observe the formation of slip bands and crack initiation,
and to determine the rate of the crack propagation on a Numerous experimental studies were performed on polished surface using the magnification of 250-times. two single-crystal alloys, ZhS32 and ZhS36 (Table 1) The tests were conducted with various values for the with different alloying and, most importantly, different maximum (r^ax = 900-1100 °C) and minimum (Tmin =
Table 1: Chemical compositions of the ZhS32 and ZhS36 single-crystal alloys, w/% Tabela 1: Kemijska sestava monokristalnih zlitin ZhS32 in ZhS36, w/%
Alloy	C	Cr	Co	Mo	W	Ta	Re	Nb	Al	Ti	Ni
ZhS32	0.12-0.18	4.3-5.6	8.0-10.0	0.8-1.4	7.7-9.5	3.5-4.5	3.5-4.5	1.4-1.8	5.6-6.3	-	Base
ZhS36	0.03	4.03	8.48	1.41	11.50	-	1.94	1.07	5.70	0.91	Base
Figure 1: a) Specimen for the thermal-fatigue test, b) with typical cyclic-temperature changes over time in the central point Slika 1: a) Vzorec za preizkus toplotnega utrujanja, b) z zna~ilnim nihanjem temperature v srednjem delu
150-700 °C) cycle temperatures. The tests with the retardation times of 2 min and 5 min at Tmax were carried out for some parts of the samples. Some samples had the stress concentrator in the form of a central hole with the diameter of 0.5 mm. The test specimens had different crystallographic orientations: (001), (011) or (111). To determine the crystallographic orientation for each sample Laue's diffraction patterns were obtained and three Euler angles and Schmid factors were computed.
On the basis of the results of a crystallographic analysis, possible directions (angles) of the slip bands on the specimen surfaces were calculated with the aim to compare them with the angles, at which the samples
were destroyed. The location of the fracture nucleus was determined with the results of fractographic studies using a TESCAN microscope. The comparison of the experimental data on the orientation of the fracture surfaces with the results of the crystallographic analysis and with the results of the finite-element analysis of the specimens allowed us to define the dependence of the failure mechanism (crystallographic or non-crystallo-graphic) on the parameters of the thermal cycle.
3 RESULTS OF EXPERIMENTAL STUDIES
The tests of mechanical properties show that single-crystal alloys do not have a high plasticity at all the temperatures (see, for example, Table 2). Low values of plasticity ö are observed for the carbon-free ZhS36 alloy (as opposed to the carbon ZhS32 alloy) at 500 °C (as opposed to the cases of high temperatures where T > 900 °C).
Table 2: Mechanical properties of single-crystal alloys with orientation (001) at 500 °C
Tabela 2: Mehanske lastnosti monokristalne zlitine z orientacijo (001) pri 500 °C
Alloy		Äp0.2 MPa	Rm MPa	A %	Z %
ZhS36		964 967	982 1000	1.3 2.3	5.0 6.9
ZhS32	Schedule t/t 1	850	880	19.5	35.5
	Schedule t/t 2	810	1110	13.0	11.7
Figure 2 shows the short-term creep curves of alloy ZhS32. The curves obtained under the stress of 550 MPa at 850 °C significantly differ for various samples. The results of the creep tests for alloy ZhS36 are given in5.
Figure 2: Creep curves of alloy ZhS32 at (850, 975 and 1050) °C Slika 2: Krivulje lezenja zlitine ZhS32 pri (850, 975 in 1050) °C
Figure 4: Ratcheting of alloys ZhS32 and ZhS36 under thermal cyclic loading
Slika 4: Zob~enje pri zlitinah ZhS32 in ZhS36 pri toplotnem cikli-~nem obremenjevanju
such a possibility on the basis of two (stress and deformation) failure criteria.
The effect of stress on deformation capacity £* is determined with the formulas of Mahutov N. A. or Hancock J. W. and Mackenzie A. C.7:
£* = £ pr 1.7 exp
£* =
£ pr Ke q 2 1 q mean
(1)
(2)
Figure 3: Map of the fracture mechanisms for alloy ZhS32 under thermal cyclic loading
Slika 3: Prikaz podro~ij mehanizmov poru{itve za zlitino ZhS32 pri toplotnem cikli~nem obremenjevanju
The tests of the ZhS36 alloy show that the conditions for failure under the thermal cyclic loading depend on the crystallographic orientation of the single-crystal alloy and also on the temperatures of the cycle. Unfortunately, the experiments conducted on the ZhS36 alloy with orientations (001), (011) and (111) were not numerous and the obtained results reflect only a trend. However, a formulation of the failure criterion depends on the failure mode (crystallographic or non-crystallographic).6 In this research, we obtained the conditions (a range of thermal-cycle parameters) (Figure 3) for the fracture modes of the ZhS32 alloy with the orientations close to (001). An accumulation of unilateral irreversible deformation (ratcheting) was observed in all the tests (Figure 4) for both alloys.
4 CRITERIA OF FAILURE UNDER STATIC LOADING
Single-crystal superalloys, as a rule, are plastic materials and the possibility of a brittle fracture under static loading of gas-turbine blades is remote. However, this issue requires a special investigation. We considered
where £ pr is the maximum deformation, determined from the experiments under short-term tension, and Ke is the characteristic of the material state (in a brittle state Ke = 1, in a viscous state Ke = 1.2).
We need to consider the effect of stress on the fracture conditions in terms of the power criterion. Let us consider the general case of stress: 02 = A1CT1, 0-3 = A2CT1, where A1 and A2 can vary from to 1. Depending on the relations between 01, 02, 03 and on the ratio of Opr/00,2, there is an area of brittle damage, in which the ultimate tensile stress is used as the limiting strength characteristic 0pr for the local stresses.
The above relation can also be written in a different form. Let us consider the case where 01/0; > 1. Using an approximation of the stress-strain curve in the form of 0 i = 0 0,2 + A£ f and the fracture condition according to the first theory of strength, 01 = 0pr. q = 01/0;, we obtain q(0 „ ^ + A£ p ) = 0pr, where the plastic deformation is defined with the equation:
1
r 0 pr/ q - 0 0 2 lm
£ f =
A
(3)
For k = Opr/Oo,2, using:
1
r 0 pr k / q - 0 0 2 lm
£ f =
A
r 0 pr(k / q -1) Im
A
with k/q > 1, a brittle fracture takes place as k/q = 1.
An analysis of crack initiation in the blades under static loading (centrifugal force and bending) on the basis of the stress-failure criterion should include: 1. A thermal finite-element analysis of the steady-state temperature field in a blade;
An elastic finite-element analysis of the stress fields with the subsequent definitions of q = 01/02 and k = 0pr/00.2 at the corresponding temperatures for all the elements of cooled blades;
According to the first strength theory we assume that 0pr = 0separation ~ 0f/ (1 - ^) and Verify the absence of equality for q = k at all the points. For the remaining cases, we calculate the values of the plastic strain using equation (3);
An estimation of the strength by comparing £ip (3) with the limiting plasticity £* (1) or (2).
2.
3.
4.
An analysis of the crack initiation in the blades under
static loading (centrifugal force and bending) on the
basis of the strain-failure criterion should include:
1.	A thermal finite-element analysis of the steady-state temperature field in a blade;
2.	An elastoplastic finite-element analysis with a definition of the zones of plastic deformation and maximum values £ fmax for all the elements of cooled blades;
3.	A comparison of the obtained value for £ Pmax with the
limiting characteristic £* (1) or (2) at the corresponding temperatures, also taking into account a decrease under the effect of aging at the operating temperatures and during long exposures;
4.	A viscoelastic finite-element analysis of the stressrelaxation processes with the initial conditions obtained with the elastoplastic analysis (see 2);
5.	A calculation of the equivalent stress. If the value of (CTq - ares)IE is lower than, or approximately equal to, the maximum ductility under creep conditions at a suitable temperature, determined with the formulas:
D3 = ^ max £ pq £
p* = £ c -1.7 exp
p* =
^ K a2
3a 1a m
(4)
(5)
D, = ^ 1 (A£ pqi )k
C1 i=1
the cyclic creep strain:
1 n
D2 = 1 (A£ cqi )m
C 2 i=1
the unilaterally accumulated plastic strain:
(7)
(8)
and the unilaterally accumulated creep strain:
D4 =-
1
-max £ eq
(1Q)
C1, C2, k, m, £ T, £ c are the material parameters depending on the temperature and on the crystallographic orientation. Usually the relations k = 2, m = 5I4, C1 = (£p)k, C2 = (It£c)m are used.
Different norms of the strain tensor are considered as an equivalent strain for (6); among them there are: the maximum shear strain in the slip system with normal n{iii} to the slip plane and slip direction 1(q11):
£eq = n{111! ■ e ■ 1(Q11>	(11)
the maximum tensile strain (the maximum eigenvalue of the strain tensor):
£eq = £j = max arg {det (e - H) = Q} the von Mises strain intensity: £eq ^ ^/^deve^-^deve =
then a brittle fracture under the conditions of stress relaxation is possible.
An acceptance of the assumption that fine micro-cracks are formed in the zone of inelastic deformation. Determination of the size of the zone of inelastic deformation (plastic and creep) and a comparison with the limit values corresponding to the beginning of the accelerated crack growth in non-linear fracture mechanics.
5 CRITERIA OF FAILURE UNDER THERMAL CYCLIC LOADING
For a prediction of a TMF failure of single-crystal materials, it is reasonable to use the deformation criterion proposed in6. The crack-initiation criterion is the condition for achieving the critical value of the total damage initiated by different mechanisms:
D1 (A£ pq ) + D2 ( A£ cq ) + D3 (£ pq ) + D4 (£ cq ) =1 (6)
The criterion (6) is based on the linear summation of the damages caused by the cyclic plastic strain:
= L(£x - £ y ) + (£ y - £z ) + (£z - £x ) + -2(Y2y + Y yz + Yzx.
and the maximum shear strain:
£ eq = [£1 - £ 2 ]= ^ [maxarg{det(e - ll) = Q}--max arg {det( e - llj=Q}]
(12)
(13)
(14)
Equivalent strain (11) corresponds to the crystallo-graphic failure mode, while equivalent strains (12)-(14) correspond to the non-crystallographic failure mode.
6 RESULTS OF THE FINITE-ELEMENT SIMULATION
The stress-strain analysis of single-crystal samples (Figure 1), with a concentrator (in the form of a central
circular hole) or without a concentrator, was carried out
on the basis of the finite-element program PANTO-CRATOR8 with an application of physical models of plasticity. These material models take into account that
an inelastic deformation occurs in accordance with the crystal-slip systems due to a slip mechanism and, therefore, the deformation is strongly sensitive to the crystal-lographic orientation. The elastoplastic and viscoelasto-plastic material models9,10 with non-linear kinematic and isotropic hardening, also accounting for the self-hardening of each system and the latent hardening11 between the slip systems, are used in the finite-element simulations. The application of viscoelastic models leads to unrealistically overestimated levels of the stress.
The obtained results for the inhomogeneous stress, strain and damage fields allow us to find the location of the specimen critical point. The damage field is obtained with criterion (6) on the basis of the analysis of the
Figure 5: Damage-field distribution after the first cycle for sample 5-1 with the (001) orientation
Slika 5: [irjenje podro~ja po{kodbe po prvem ciklu pri vzorcu 5-1 z orientacijo (001)
Figure 6: Influence of the crystal orientation on the cyclic stressstrain curve. Results of the finite-element simulations of the thermal cycles with rmax = 900 °C and Tmin = 150 °C (the central point of the specimen).
Slika 6: Vpliv orientacije kristala na cikli~no krivuljo napetost - raz-tezek. Rezultati simulacije s kon~nimi elementi za toplotne cikle s Tmax = 900 °C in T^^^ = 150 °C (sredinski del vzorca).
Figure 7: Influence of the temperature program on the cyclic stressstrain curve. Results of the finite-element simulations of the specimens with the (001) orientation (the central point of a specimen). Slika 7: Vpliv temperaturnega programa na krivuljo napetost - raz-tezek. Rezultati simulacije s kon~nimi elementi za vzorce z orientacijo (001) (sredinski del vzorca).
strain-field evolution using the experimental data on the creep and elastoplastic deformation-resistance curves. The typical damage-field distribution after the first thermal cycle (20 °C ^ Tmax = 900 °C ^ Tmin = 150 °C) is presented in Figure 5 for sample 5-1 of ZhS36 with orientation (001).
The results of the finite-element simulations show that the crystallographic orientation has a significant influence on the stress-strain state of the samples (Figure 6), as confirmed also by the experiments.12 The width of the hysteresis loops and the unilaterally accumulated strain are also very sensitive to the thermal cyclic regime (Figure 7).
The numbers of the cycles to crack initiation, calculated on the basis of criterion (6) using different equivalent strains, (11)-(14), are given in Table 3. A satisfactory correlation is observed between the criterion predictions and the results of the experiments (without a sufficient statistical representation).
7 CONCLUSIONS
1.	In the investigations of I. L. Svetlov, E. R. Golubov-sky and other researchers a series of tests were conducted regarding the definition of the thermal fatigue resistance and short-term creep of the single-crystal ZhS32 alloy, determining the temperature range causing the changes in the failure modes.
2.	The failure criteria for the single-crystal alloys under static and thermal cyclic loadings are proposed and discussed. A satisfactory correlation is observed between deformation criterion (6) and the obtained experimental results. All the considered variants of equivalent strains (11)-(14) give practically the same results. The criterion using von Mises strain intensity (13) offers the most conservative estimation.
3.	The finite-element simulations of single-crystal specimens under thermal cyclic loading were performed using different material models. The obtained results indicate an applicability of the proposed deformation criteria of failure for the single-crystal ZhS32 and ZhS36 alloys for the temperatures up to 1100 °C.
Table 3: Deformation-criterion (6) predictions vs. experimental results for the crack initiation life
Tabela 3: Napovedi merila deformacij (6) v odvisnosti od eksperimentalnih rezultatov za za~etek nastanka razpoke
	Orientation	Tmax/°C	Tmin/°C	Number of cycles to crack initiation				
				eeq = enl (11)	eeq = £1 (12)	eeq = ei (13)	eeq = ymax (14)	Experiment
Sample 5-1		900	150	338	275	195	280	435
Sample 5-3		1000	500	218	196	150	172	305
Sample 1-2		900	150	85	73	59	75	190
Sample 2-1		900	150	15	9	10	15	17
Sample 1-1*		900	150	5	4	3	4	10
Sample 2-6*		1000	500	61	44	56	57	10-130
Sample 4-1*		900	150	6	4	4	5	10
♦Specimen with a concentrator (the radius of the central hole is 0.25 mm)
Acknowledgements
The study was supported by the Russian Fundamental Research Program, Project No.12-08-00943. The authors are also grateful to S. M. Odabai-Fard for helping prepare the paper.
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