PSI - Issue 2_B

Volume 2 • 201 6 B

ISSN 2452-3216

ELSEVIER

21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy

Guest Editors: Francesco I acoviello L uca Susmel

D onato Firrao Giuse pp e Ferro

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XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal Thermo-mechanical modeling of a high pressure turbine blade of an airplane gas turbine engine P. Brandão a , V. Infante b , A.M. Deus c * a Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal b IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal c CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal Abstract During their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict the creep behaviour of HPT blades. Flight data records (FDR) for a specific aircraft, provided by a commercial aviation company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a model can be useful in the goal of predicting turbine blade life, given a set of FDR data. 21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy General effects of pulse electric breakdown of dielectric gaps and dynamic failure of continuous media Yuri Petrov and Ivan Smirnov* Saint Petersburg University, Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia Abstract In this paper we consider some general effects observed at pulse electric breakdown of dielectric gaps and dynamic failure of continuous media. The effect of time and strain rate dependence of limiting characteristics, the substitution effect of maximal strength, as well as failure and breakdown with delay are considered. Despite the different physical nature of mechanical failure and electrical breakdown, these effects can be modeled based on a common approach of the incubation time criterion. It is discussed that the strain/stress rate dependence of strength and the volt-time characteristic of a dielectric medium cannot be used as a universal characteristic of the material’s mechanical and dielectric strength and should be determined for each specific case. It is shown that the time parameter, which is invariant to the action history, is more appropriate as a characteristic of the dynamic strength. With the incubation time criterion one can construct a unified time dependence of the mechanical or electrical strength consisting of quasi-static and dynamic regimes of action. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. Keywords: dynamic failure; pulse electric breakdown; the incubation time criterion; strain rate dependence; volt-time characteristic; strength substitution effect; failure delay. 1. Introduction Research results on dynamic failure of continuous media and pulsed electrical breakdown of dielectric gaps exhibit a number of effects, which ar common to these seemingly quite different physical processes. The effects relate to a fundamental difference between the behavior of medium under dynamic and quasi-static actions. For dynamic failure of co P e -re lity o the ntif tee c breakdown; the incubation time criterion; strain rate dependence; volt-time characteristic; strength ctrical breakdown of dielectric gaps Copyright © 2016 The Authors. ublishe by Elsevier B.V. This is an open access articl under the CC BY-NC-ND license (http://creativecommons. rg/lice ses/by-nc- /4.0/). Peer-review under responsibility of the Scientific Committee of ECF21. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016. Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation.

* Corresponding author. E-mail address: i.v.smirnov@spbu.ru

* Corresponding author. Tel.: +351 218419991. E-mail address: amd@tecnico.ulisboa.pt 2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. bility of the Scientific Committee of ECF21.

2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016. Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ). Peer review under responsibility of the Scientific Committee of ECF21. 10.1016/j.prostr.2016.06.056

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example, one of the main problems of determining the dynamic strength properties is associated with the functional dependence of limiting characteristics on the history and method of applying a load. Whereas a limiting characteristic is a constant for a material in the static case, limiting characteristics in dynamics are strongly unstable and, as a result, their behavior becomes unpredictable. In the case of the mechanical rupture or compression, the dynamic strength is usually expressed by the experimentally measured strain rate dependence of limiting stresses, see e.g. Antoun et al. (2003) and Freund (1990). In the case of the electric breakdown, the dynamic electric strength is usually expressed by the experimentally measured voltage–time characteristic, see e.g. Vavilov and Mesyats (1970) and Mesyats et al. (1972). Other typical effects of the behavior of medium under dynamic actions are the change of maximum strength of two materials (Petrov et al. 2013, Vorob’ev 1998) and a delay of failure (breakdown) (Antoun et al. 2003, Kuznetsov et al. 2011). The change of maximum strength of two materials or the substitution effect is that one material can have a greater quasi-static strength than the other material, but the second material can withstand more high dynamic loads than the first. The delay of failure (breakdown) corresponds to failure (breakdown) at the time of a reduction of stress in the material (electric field between gaps). In this paper, we analyze examples illustrating typical dynamic effects inherent in the processes of mechanical failure and electrical breakdown. We propose a unified interpretation for the failure of continuous media and electrical breakdown of dielectric gaps using the structural-time approach (Petrov and Morozov 1994) based on the concept of the failure incubation time criterion (Petrov 2004). 2. Calculation of limiting characteristics Under slow action, there is a phenomenological approach for evaluation of the limiting fields, which proves to be a reasonably efficient tool of modeling and prediction of the electric and mechanical strength: ( ) c F t F  (1), where F ( t ) is the intensity of a local force field causing the failure of the medium; F c is the limit intensity of the local force filed, which can depend on many material and geometrical factors; t is the time. The basic cause of difficulties in modeling the dynamic effects of mechanical or electrical strength is the absence of an adequate limiting condition that determines the instant of rupture or breakdown. This problem can be solved by using both the structural macro mechanics of failure and the concept of the failure incubation time, which represents the kinetic processes of macroscopic breaks formation (Morozov and Petrov 2000). The dynamic effects become essential for actions whose periods are comparable with the scale determined by the failure incubation time associated with preparatory processes of developing micro defects in the material structure. The criterion of the failure incubation time makes it possible to calculate effects of the unstable behavior of dynamic-strength characteristics. This criterion can be generalized in the form of the condition (Petrov 2004)

*

1 ( ) t F t



1

dt



(2),

* F    t

c

where F ( t ) is the intensity of a local force field causing the failure of the medium; F c is the static critical intensity of the local force filed; τ is the incubation time associated with the dynamics of a process preparing the break. The time of failure or breakdown t * is defined as the time at which the condition (2) becomes an equality. Depending on the physics of the process, the local force field can correspond to the stress in the place of failure or the electric field between the electrodes. The static critical intensity of the field is determined by standard experiments. If the local force field and the time of the break can be registered in an experiment, then the condition (2) has only one unknown - the incubation time. This parameter can be defined by fitting the condition (2) to the experimental points.

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According to the terminology of the approach of the failure incubation time, the static critical intensity of the local force filed characterizes the material strength under slow actions, and the incubation time characterizes the material strength under dynamic actions. Thus, knowing only two parameters ( F c , τ ), one can find different dependencies and parameters, such as current–voltage characteristics, limiting stress at different strain rates, breakdown time, etc. 3. Effects of dynamic actions 3.1. Time dependence of limiting characteristics A typical example illustrating the complicated behavior of the dynamic mechanical strength of medium is the time dependence of strength observed at spall fracture of solids (Zlatin et al. 1974) and cavitation of liquids (Besov et al. 2001), see Fig. 1. This dependence of the fracture time t * on the critical pulse amplitude F * for different pulse durations shows that the dynamic strength is not a material constant but depends on the time to fracture (i.e., sample “life time”). The criterion of critical action (1) describes well long-term quasi-static failure/breakdown caused by long-duration wave pulses. However, in the case of short-duration pulses, the fracture time weakly depends on the threshold pulse amplitude, and this dependence has a certain asymptote. This effect is called the phenomenon of the dynamical branch of the strength time dependence. Neither the conventional theory of strength nor the known time criteria explains this phenomenon. The total time dependence of strength can be obtained on the basis of the incubation time criterion (2).

Fig. 1. Time dependence of limiting characteristics calculated by Petrov (2004). (a) Logarithm of the fracture-process duration t * vs. the threshold amplitude F * of a stress pulse that causes spall fracture in aluminum samples (Zlatin et al. 1974); (b) Mechanical strength of water P * as a function of the pulse duration T (Besov et al. 2001). The schemes for the application of the criterion (2) to spall problems are given by Petrov et al. (2010). An example of a calculation using the criterion (2) for the time dependence of the spall strength of aluminium for triangular pulses realized in the experiments reported by Zlatin et al. (1974) is represented in Fig. 1 by the solid curve. The calculated parameters of the material are τ = 0.45 μs and F c = 103 MPa. The experiments of Besov et al. (2001) show that the cavitation strength of liquids increases nonlinearly with decrease of loading-pulse duration. Using the incubation time criterion (2) makes it possible to calculate the experimentally observed increase in the cavitation threshold P* with decreasing the pulse duration T (Fig.1b). The calculation was made for the static critical pressure P c = 1 atm and the incubation time τ = 19 μs. The above effect is also observed in pulsed electrical breakdown of dielectric gaps. The typical feature of pulsed breakdown is an increase in the breakdown voltage with reducing pulse duration. As an example, the breakdown electric field E * measured by Khaneft (2000) for ammonium perchlorate single crystals is presented in Fig. 2 as a

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function of the duration T of the leading edge of the pulse. This dependence also characterizes the electrical strength as a function of the voltage growth rate in a sample and can be called the time dependence of strength by analogy with the above examples of spall fracture and cavitation. The curves in Fig. 2 are the time dependences for the electrical strength of ammonium perchlorate calculated by Petrov (2004) according to the criterion (2) with the incubation time τ = 0.33 μs, the static electrical strength E c = 0.52×10 6 V/cm and E c = 0.2×10 6 V/cm for different material thicknesses h = 0.01 and 0.03 cm, respectively. The onset time of increasing the breakdown field in the dependences plotted in Fig. 2 is entirely determined by the τ value. As was shown by Khaneft (2000), this time was virtually independent of the interelectrode distance. This indicates that the incubation time in the case under discussion may be considered as a material characteristic.

Fig. 2. Electrical strength E * of ammonium perchlorate vs. the duration T of the leading edge of an electrical pulse for the interelectrode gaps h = 0.1 cm - (1) and 0.03 cm – (2) received by Khaneft (2000) and calculated by Petrov (2004).

Note that Fig. 1 and 2 reveal two branches of the time dependence belonging to the slow quasi-static and fast dynamic input of energy. The quasi-static branch depends mainly on the parameter F c , whereas the dynamic branch is caused by approaching the values of characteristic times of applied loads to the duration of the failure incubation period τ . Thus, τ can be considered as the parameter integrally describing the dynamic strength of a material. 3.2. Substitution effect of maximal strength A construction material is selected on the basis of its ability to withstand a certain stress (as one of the defining parameters). There is a set of test standards governing determination of the ultimate strength of a material under quasi-static tension, compression, bending, etc. However, tests under dynamic loading conditions show essential differences of dynamic strength characteristics in comparison with those of quasi-static tests. Under dynamic loading the critical stresses are characterized by very strong instabilities and cannot consider as material parameters. Moreover, the dynamic loads may lead to an unexpected substitution effect of maximal strength. A material, which has a lower strength compared to another material in quasi-static tests, can have greater strength under dynamic loading. Fig. 3 shows the results of split tests of the fibre reinforced concrete (CARDIFRC) and gabbro-diabase under quasi-static and high strain rates on a semi-logarithmic scale. The tests were carried out using the modification of Kolsky method for dynamic splitting (the Brazil test). Detailed schemes of tests and results are presented in work of Bragov et al. (2003) for gabbro-diabase and in work of Bragov et al. (2012) for CARDIFRC. The curves in Fig. 3

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correspond to the calculation by the criterion (2) with the following parameter values: F c = 23 MPa and τ = 15 μs for concrete and F c = 18 MPa and τ = 70 μs for gabbro-diabase. It is clear from the results that carrying capacity of both materials increases with the growth of loading rate. However, although CARDIFRC has a higher quasi-static split strength than that of gabbro-diabase, its dynamic carrying capacity in splitting is lower at high stress rates (>10 2.5 ).

10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 15 20 25 30 35 40 45 50 55 60 CARDIFRC gabbro-diabase

Maximum Stress (MPa)

Stress Rate (GPa/s)

Fig. 3. The split tests of CARDIFRC and gabbro-diabase. Black squares are the experimental values for CARDIFRC (Bragov et al. 2012); black line is predictions of Eq. (2) for CARDIFRC (Petrov et al. 2013); red triangles are the experimental values for gabro-diabase (Bragov et al. 2003); and red dashed is predictions of Eq. (2) for gabro-diabase (Petrov et al. 2013). The similar effect can be observed at the development of the breakdown channel in liquid or solid dielectrics in dependence on the steepness of the voltage pulse front. For example, if the electric strength of solid dielectrics often exceeds the strength of liquid dielectric media at a slow quasi-static input of energy, the electric strength of liquids can occur higher than the strength of solid dielectric materials including rocks at a fast pulse voltage (Vorob’ev et al. 1998).

Fig. 4. Dependencies of electric strength on the time of breakdown for various media calculated on the basis of criterion (2) (Petrov 2014) and experimental ones (Vorob’ev et al. 1971).

Using the criterion (2) for analysis of particular experiments of Vorob'ev et al. (1971) for the interelectrode gap filled with liquid or solid dielectric, it is possible to obtain the voltage-time characteristics corresponding to various

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parameters of breakdown media. Fig. 4 shows the experimental and the calculated time dependencies of the electric strength for various media. The curves were calculated by Petrov (2014) with the following parameters: τ = 0.65×10 -6 s and E c = 2.8×10 4 V/cm for water; τ = 0.2×10 -6 s and E c = 8.5×10 4 V/cm for marble; τ = 0.08×10 -6 s and E c = 1.8×10 5 V/cm for quartz. It can be seen that with increasing the steepness of the voltage pulse front and correspondingly decreasing the breakdown time t * , the ratio between the breakdown voltages for different media can be changed to the opposite. In particular, water having substantially lower static strength can be broken down at appreciably higher electric field intensities than rock in the case of fast input of energy. In this case, it is possible to assert that the dynamic strengths of the compared media are arranged in inverse order as compared with their quasi-static strengths E c expressed in terms of the incubation time τ . 3.3. Failure and breakdown with delay Since the critical stresses at dynamic loads are unstable, it is usual to plot diagrams of strain/stress rate dependence of the strength. In this case, each loading or strain rate corresponds to one's critical stress. These diagrams are taken as a material property. So dynamic strength is related to the strain rate without regard for the load time and shape. However, as the analysis shows (see e.g. Petrov and Utkiv 2015), the action time and the applied pulse shape and amplitude equally determine the critical fracture characteristics. The shape and parameters of action on medium are often determined by the characteristics of the facility used for tests (e.g., the flyer plate thickness, the capacity and the inductance of an electric charging unit, the laser power). At the same time, one of the specific features of failure or breakdown during dynamic loading is the possibility of application of an action that is highly than the critical action, which is required to break the material. Let the action shape (it can be an isosceles or right-angled triangle) and action time T be specified. Let a pulse action of a given shape that results in failure (or breakdown) be the minimum breaking pulse if its decrease is due to a decrease in the amplitude or the time does not cause failure (or breakdown). If the applied pulse is higher than the minimum required pulse, we can speak about failure (or breakdown) with overloading. If the failure (or electrical breakdown) occurs after the passage of the peak of the local stress (or an electric field), then we say that the failure (or electrical breakdown) is delayed. The time elapsed from the peak of the pulse until the moment of break characterizes the delay. Thus, the delay duration depends on the shape and overload of an action pulse. Therefore, the time of the break and the critical value of the local force field are determined by the pulse shape and the magnitude of the overload.

a

b

* 2 >F

* 1

t *

* 1

F

2 < t

F *

* 2 >F

* 1

F

F

3 >F

F * 3

 = const

* 3 < t

* 2 < t

* 1

t

g = dF/dt = const

F * 2

F * 2

F * 1

F * 1

t * 2

t * 1

t * 1

t * 3

* 2

t



t



t

i is the critical local force field; t *

i is the time of break. (a) pulses

Fig. 5. Some possible variants of failure or electrical breakdown with a delay. F *

with the same action rate; (b) pulses with the different action rate.

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Fig. 5 shows some variants of failure or electrical breakdown with the delay. Pulses with the same action rate can lead to the break both the leading edge of the pulse and the trailing edge of the pulse, Fig.5a. Increasing an action rate at a constant duration of the action leads to the pulse overload and reduction of the delay, Fig.5b. Experimental studies of the delay were carried out, for example, by Zlatin et al. (1974) for spall fracture and Kuznetsov et al. (2011) for electrical breakdown. Since failure and breakdown can occur at the trailing edge of the action pulse, it becomes clear that break depends primarily on the evolution of the process over a certain preceding time interval rather than on the instantaneous value of the effective force field. Integral basis of criteria (2) allows describing this effect. The studies of temporal effects of failure and electric breakdown in terms of the incubation time criterion were carried out by Petrov et al. (2010) and Petrov et al. (2015). It was shown that the diagrams of the strain rate dependence of medium strength (see e.g. Fig. 3) and the volt-time characteristics (see e.g. Fig. 2) of medium cannot be considered as properties of this medium. A result of any dynamic actions should be assessed separately on the basis of simple and clear engineering principles. 4. Conclusion The effect of time and strain rate dependence of limiting characteristics, the substitution effect of maximal strength, as well as failure and breakdown with delay characterize the nature of medium behavior under dynamic actions. These effects show the fundamental importance of investigating incubation processes preparing abrupt structural changes (failure and phase transitions) in continua medium under intense pulsed actions. The parameter with dimensions of time can be a universal basic characteristic of the dynamic strength and should become one of the main material parameters to be experimentally determined. The considered results show that the structural-time approach based on the incubation time criterion is fundamental and makes it possible to adequately represent the dynamics of both the failure of continuous media and electrical breakdown of dielectric gaps. The presence of such effective criteria predicting the mechanical and electrical strength of a medium in simple engineering terms is vital for application in practice, and it can help eliminate the need to conduct laborious research of material behavior in a wide range of action conditions. Acknowledgements This work was supported by St. Petersburg State University (grant no. 6.38.243.2014). References Antoun, T., Seaman, L., Curran, D.R., Kanel, G.I., Razorenov, S.V., Utkin, A.V., 2003. Spall Fracture. Springer, 417 p. Besov, A.S., Kedrinskii, V.K., Morozov, N.F., Petrov, Yu.V., Utkin, A.A., 2001. On the Similarity of the Initial Stage of Failure of Solids and Liquids under Impulse Loading. Doklady Physics 46(5), 363-365. Bragov, A.M., Bolshakov, A.P., Gerdyukov, N.N., Lomunov, A.K., Novikov, S.A., Sergeichev, I.V., 2003. Research of Dynamic Properties of Some Rocks. In: International Conference "V Kharitonov thematic scientific reading", VNIIEF, Sarov, p. 43. Bragov, A.M., Karihaloo, B.L., Petrov, Yu.V., Konstantinov, A.Yu., Lamzin, D.A., Lomunov, A.K., Smirnov, I.V. 2012. High-Rate Deformation and Fracture of Fiber Reinforced Concrete. J. of Applied Mechanics and Technical Physics, 53(6), 926. Freund, L.B., 1998. Dynamic Fracture Mechanics. Cambridge University Press, 584 p. Khaneft, I.G., Khaneft, A.V., 2000. Effect of the Duration of the Front Edge of the Voltage Pulse on the Electric Breakdown of Ammonium Perchlorate. Technical Physics 45(4), 423-426. Kuznetsov, Yu.I., Vazhov, V.F., Zhurkov, M.Yu., 2011, Electrical Breakdown of Solid Dielectrics and Rocks on the Trailing Edge of a Voltage Pulse. Russian Physics Journal 54, 410. Mesyats, G.A., Bychkov, Yu.I., Kremnev, V.V., 1972. Pulsed Nanosecond Electric Discharges in Gases. Sov. Phys. Usp. 15, 282–297. Morozov, N. and Petrov, Y., 2000. Dynamics of Fracture. Springer, 98 p. Petrov, Y., Morozov, N., 1994. On the Modeling of Fracture of Brittle Solids. ASME J Appl Mech 61, 710-712. Petrov, Y.V., 2004. Incubation Time Criterion and the Pulsed Strength of Continua: Fracture, Cavitation, and Electrical Breakdown. Doklady Physics 49(4), 246–249. Petrov, Yu.V., Smirnov, I.V., and Utkin, A.A., 2010. Effects of Strain-Rate Strength Dependence in Nanosecond Load Duration Range. Mechanics of Solids 45(3), 476-484.

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Petrov, Y., Smirnov, I., Evstifeev, A., Selyutina, N., 2013. Temporal Peculiarities of Brittle Fracture of Rocks and Concrete. Frattura ed Integrità Strutturale 24, 112-118. Petrov, Y., 2014. Structural-Time Criterion of Pulsed Electric Strength. Doklady Physics 59(1), 56–58. Petrov, Y.V., and Utkin. A.A., 2015. Time Dependence of the Spall Strength under Nanosecond Loading. Technical Physics 60(8), 1162–1166. Petrov, Yu.V., Morozov, V.A., Smirnov, I.V., Lukin, A.A., 2015. Electrical Breakdown of a Dielectric on the Voltage Pulse Trailing Edge: Investigation in Terms of the Incubation Time Concept. Technical Physics 60(12), 1733-1737. Vavilov, S.P., Mesyats, G. A., 1970. Current Growth During Pulsed Breakdown of Millimeter Gaps in Vacuum. Soviet Physics Journal 13(8), 1058-1061. Vorob’ev, A.A., Vorob’ev, G. A., Zavadovskaya, E.K., et al., 1971. Pulse Breakdown and Fracture of Dielectrics and Rocks. Tomsk University, Tomsk, 228 p. (in Russian) Vorob’ev, A.A., Vorob’ev, G. A., Chepikov, A.T., 1998. Regularity of Breakdown in Solid–Liquid Dielectric Interface under a Pulse of Voltage. Inventor’s Certificate No. A-122. Zlatin, N.A., Mochalov, S.M., Pugachev, G.S., and Bragov, A.M., 1974. Temporal Features of Fracture in Metals under Pulsed Intense Actions. Sov. Phys. Solid State 16, 1137-1140.

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XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal Thermo-mechanical modeling of a high pressure turbine blade of an airplane gas turbine engine P. Brandão a , V. Infante b , A.M. Deus c * a Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal b IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal c CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal Abstract During their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict the creep behaviour of HPT blades. Flight data records (FDR) for a specific aircraft, provided by a commercial aviation company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a model can be useful in the goal of predicting turbine blade life, given a set of FDR data. 21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy General estimation equation of transient C ( t ) under load and displacement control Han-Sang Lee a , Dong-Jun Kim a , Jin-Ho Je a , Yun-Jae Kim a *, Robert A Ainsworth b , Peter J Budd n c a Korea University, 5Ka Anam-Dong, Sungbuk-Gu, Seoul 136-701, Republic of Korea b The University f Mancherste, Manchester 13 9PL, UK c Assessment Technology Group, EDF Energy, Barnwood Gloucester GL4 3RS, UK Abstract This paper propose estimation equations of transient C ( t )-integrals for general material properties where plastic and creep stress exponent are different under load and displacement control. The new equations are made by modifying the plasticity correction term in the existing equations. The modified plasticity corrections term is expressed in terms of initial elastic-plastic and steady state creep stress fields. For validation, elastic-plastic-creep finite element analysis are performed. FE results are compared with predicted C ( t ) results using proposed equations. Good agreement with FE results is found even when plastic and creep stress exponents are different. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. Keywords: Transient C ( t )-integrals; Elastic-plastic-creep; Crack-tip stress fields; Load and displacement control 1. Introduction Creep crack growth is important fa tor into life assessment of components operating at high temperature. Creep crack growth rate can be quantified by the C ( t )-integral which characterizes the singular stress and strain fields at the crack tip (Riedel, 1987). Note that the notation C * is used for the value of C ( t ) at the steady state creep conditions. Copyright © 2016 The Authors. ublished by E sevier B.V. This is an open access rticl under the CC BY-NC-ND license (http://creativecommons. rg/licenses/by-nc- d/4.0/). Peer-review under responsibility of the Scientific Committee of ECF21. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016. Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation.

* Corresponding author. Tel.: +82-2-3290-3372; fax: +82-2-929-1718. E-mail address: kimy0308@korea.ac.kr

* Corresponding author. Tel.: +351 218419991. E-mail address: amd@tecnico.ulisboa.pt 2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21.

2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016. Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ). Peer review under responsibility of the Scientific Committee of ECF21. 10.1016/j.prostr.2016.06.105

Han-Sang Lee et al. / Procedia Structural Integrity 2 (2016) 817–824 Han-Sang LEE et al. / Structural Integrity Procedia 00 (2016) 000–000

818 2

Nomenclature a

crack length

A , B

material constants (plasticity and creep) C-integrals (transient and steady state creep)

C ( t ), C *

normalized opening stress at t=0 (initial conditions), HRR fields

D E F L r m

Young’s modulus

normalized opening stress at t→∞ (steady state creep conditions), RR fields

J (0)

J-integrals for initial (t=0) conditions parameter related to plastic yielding

strain hardening exponent

creep exponent

n

M , M L

applied load and plastic limit load polar coordinates at the crack tip time and redistribution time

r , θ

t , t red

x , y

Cartesian coordinates elastic follow-up factor

Z

ε , ε e , ε p

strain, elastic strain and plastic strain

Poisson’s ratio

ν τ σ

normalized time, = t / t red

stress

yield strength reference stress

σ o

σ ref

plasticity correction factor under load control plasticity correction factor under displacement control parameter related to elastic follow-up, = Z /( Z -1)

φ

γ

Ф

Thus estimations of C ( t ) and C * are needed to assess creep crack growth in conjunction with creep crack growth rate data determined in terms of C ( t ) and C * from specimen tests. For elastic-power law creep problem, Ehlers and Riedel (1981) proposed a C ( t )/ C * relaxation curve. A slightly different equation was developed by Ainsworth and Budden (1990). However, the approach can invalidate under widespread plasticity. For widespread plasticity, Joch and Ainsworth (1992) presented the effect of initial plasticity on the magnitude of C ( t )-integral during the transient creep. Based on the approach of Ainsworth and co-workers, Lei (2005) proposed equation of C ( t )/ C * relaxation curve for secondary loading cases. Note that above equations (Joch and Ainsworth, 1992; Lei, 2005) which take account of initial plasticity are valid only for equal power law stress exponents, i.e., the plastic hardening exponent ( m ) and creep exponent ( n ) are the same. Generally, materials have unequal stress exponents for plasticity and creep. Therefore, a more general equation is needed to apply for general stress exponent cases. The present work presents estimation equation of transient C ( t ) for general elastic-plastic-creep conditions where the plastic and creep exponents are different under load and displacement control. The new equation is made by modifying the plasticity correction term in the existing equations. The proposed equations are validated against elastic-plastic-creep finite element (FE) analysis results for plane strain single-edge-cracked bend (SE(B)) specimen. 2. Finite element analysis 2.1. Geometry One typical geometry with high crack-tip constraint levels was considered in this paper: plane strain single-edge cracked bend (SE(B)) specimen, as depicted in Fig. 1. The specimen width, W , was taken to be W =50mm with the relative crack depth a / W =0.5 In Fig.1, r and θ denote polar coordinated at the crack tip; y denotes crack opening direction.

Han-Sang Lee et al. / Procedia Structural Integrity 2 (2016) 817–824 Han-Sang LEE et al. / Structu al Integrity Procedia 00 (2016) 000–000

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Fig. 1. Specimen conceded in this paper, schematics: SE(B).

2.2. Material properties For elastic-plastic analyses, an isotropic material was assumed to follow the Ramberg-Osgood relationship:

m

o        

0.002

E 

E



e p       

o

with

and











(1)

o

o

E



o

 

m

A

 

E

where ε , ε e , ε p denote total, elastic, plastic strain, respectively; σ is stress (MPa); A and m are material constants. For elastic properties, Young’s modulus E =200GPa and Poisson’s ratio ν =0.3 were used. For plastic properties, the yield strength σ o was assumed to be 300MPa with two values of the strain hardening exponent, m =5 and 10. For creep analyses, the material was assumed to follow power-law behavior, characterized by: (2) where c   denote creep strain rate; B and n are material constants. Two values of the creep exponents n were considered, n =5 and 10. The following values were assumed, B =3.2x10 -15 (MPa) - n h -1 for n =5 and B =3.2x10 -25 for n =10. However, the values of constant B don not affect the results as these are presented in a normalized manner. 2.3. Finite element analysis Elastic-plastic-creep Fe analyses of SE(B) specimen were performed using ABAQUS (2013).To avoid problems associated with incompressibility, eight-noded plane strain element with reduced integration were used. A small geometry change continuum FE model was assumed. Figure 2 depicts the FE mesh for SE(B) specimen. The crack tip was designed with collapsed elements, and a ring of wedge-shaped elements was used in the crack-tip region. The number of elements and nodes in the FE meshes were 4543 and 14055. To apply pure bending loading conditions, the multi-point constraint (MPC) option within ABAQUS was used. To quantify the applied loading magnitude, a parameter related to plastic yielding, L r , is used: c n B    



(3)

ref

r L M M

 



L o

1.261

(4)

 2



for SE(B)

M b W a  



L

o

2 3

Han-Sang Lee et al. / Procedia Structural Integrity 2 (2016) 817–824 Han-Sang LEE et al. / Structural Integrity Procedia 00 (2016) 000–000

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4

Fig. 2. FE mesh for SE(B) specimen.

where M and M L denote the applied load and the plastic limit load based on the von Mises yield condition (Webster and Ainsworth, 1987); b is the specimen thickness; and σ ref is the reference stress. In this work, three different values of L r , L r =0.5, 0.8 and 1.0, were considered. Elastic-plastic-creep FE analyses were performed as follows. For load controlled cases, constant loading was applied in the first step (at time t =0). The load was then held constant for t >0. For displacement controlled cases, constant displacement which related to L r within load control was applied in the first step (at time t =0). The displacement was then held constant for t >0 and subsequent time-dependent creep calculations were performed. For time-dependent creep calculations, an implicit method was selected within ABAQUS. 3. Transient C ( t ) estimation 3.1. Existing transient C(t) estimation equation For elastic-creep conditions under load control, Ehlers and Riedel (1981) proposed a relaxation curve for C ( t ) in terms of time t and steady-state C *:

* ( ) 1 1  

C t

t



(5)

1  









1 n t 

1

n



C

red

where τ denotes the normalized time which is given in terms of redistribution time t red

  * 0

red t

C t

(6)

  

t

J

J (0) denotes the initial (at time t =0) FE value of J -integrals. For elastic-plastic-creep conditions under load control, Joch and Ainsworth (1992) proposed another equation that the effect of initial plasticity on C ( t ) could be incorporated using a factor φ :

1

n







*

1





( )

C t

AC

(7)

with

1  





  0

*

1

n



BJ





C

1

  



The material constant for plasticity and creep, A and B , are given in Eqs. (1) and (2), respectively.

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5

For elastic-plastic-creep conditions under displacement control, Lei (2005) proposed equation of C ( t )/ C * relaxation curve based on the approach of Ainsworth and co-workers:

1

1

n

n





   

       

   

 

 

ref

ref

(8)

o

o

o

   

   



( )

C t

Z

  

  

ref

ref

ref o 

With

and

1  





 

*

1

n

1



Z



    

    

C

E

   

   

ref

 

ref o ref

1  





where Z denotes elastic follow-up factor. The following values were assumed, Z =2.0 for n =5 and Z =2.5 for n =10. An important point to note is that Eqs. (7) and (8) were derived based on the assumption of equal stress exponents for plasticity and creep ( m = n ). When the stress exponent are different ( m ≠ n ), Eqs. (7) and (8) cannot be applied. 3.2. Proposed transient C(t) estimation equation A new estimation equation is made by changing plasticity correction term φ , γ in terms of the crack-tip stress fields at the initial and steady state creep conditions. At initial conditions (time t =0), the crack-tip stress field should follow the Hutchinson-Rice-Rosengren field (1968), and is denoted as D :

1

  0

   

   



  

  

J

1

m



(9)

yy

( , ) m D 

yy   



1

m





m o I A r 

o

=0

t

where r and θ denote polar coordinate at the crack-tip. At t >0, the crack-tip stress under creep conditions is given by:

1

  n n o C t I B r   1



   



1

n



(10)

yy

( , ) n    yy

 



 

o

where I m (or I n ) is an constant that depend on stress exponent. At long times under steady-state creep conditions, the crack-tip stress field follow the RR field (Riedel and Rice, 1980), and is denotes as F :

1

*

   

   



  

  

1

n



C

(11)

yy

( , )    yy

n F 



1

n





n o I B r 

o

t



For load controlled cases, using Eqs. (7) and (11), Eq. (10) can be re-written as

(12)







1

F





yy o



1



 1    n   

  



1

n



1

 

Equation (12) is crack-tip stress field at transient creep condition under load control. By matching Eq. (9) and Eq. (12) at time t =0, we can obtain that

1

n



1

n







1





( )

C t

F D      

(13)

 If



 

 

 

with

1  

0, then

0







*

1

n







C

    

1

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6

Equation (13) is the proposed C ( t ) estimation equation under load control. Eq. (13) has the same form as Eq. (7), but the plasticity-correction factor is different. In the cases of m = n , the equations are the same. For displacement controlled cases, using Eqs. (8) and (11), Eq. (10) can be re-written as

   

   

   

ref ref o o ref ref

F

(14)



yy o



1



    

    

1

n



    

    

1

n



   

   

 

ref o ref

1  





Equation (14) is crack-tip stress field at transient creep condition under displacement control. By matching Eq. (9) and Eq. (14) at time t =0, we can obtain that

1

1

n

n





   

       

   

 

 

ref

ref

(15)

o

o

1

n



( )

C t

F D       

ref

ref

with ' 



*

1

n



    

    

C

   

   

 

ref o ref

1 ' 



 

Equation (15) is the proposed C ( t ) estimation equation under displacement control. Eq. (15) has the same form as Eq. (8), but the plasticity-correction factor is different. In the cases of m = n , the equations are the same. 3.3. Validation Elastic-plastic values of J -integral at t =0, J (0), and elastic-plastic-creep values of C -integral at steady state creep, C *, are determined from FE analysis. Determined values of J (0) and C * are presented in Table 1. Using determined J (0) and C *, values of factor φ ’ in Eq. (13) and factor γ ’ in Eq. (15) are calculated. The proposed C ( t ) estimation equations are compared with the FE results in Fig. 3 (for load control) and Fig. 4 (for displacement control). Although the prediction is slightly non-conservative for the case of m =10, n =5 with L r =1.0 in Fig. 3, overall C ( t )/ C * relaxation curves using the new equation agree well with FE results.

Table 1. Values of J (0) and C * from FE analysis. J (0) (MPa ∙ mm)

C * (MPa ∙ mm/h)

L r

L r

0.5

0.8

1.0

0.5

0.8

1.0

m = n =5 m = n =10

6.03 5.67 6.03 5.67

20.20 17.92 20.20 17.92

42.40 42.39 42.40 42.39

1.06 6.84 6.84 1.06

17.85 1203 1203 17.85

68.07 14001 14001 68.07

m =5, n =10 m =10, n =5

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(a)

(b)

(c)

(d)

Fig. 3. Variations of C ( t )/ C * for load controlled cases: (a) m=n=5, (b) m=n=10, (c) m=5, n=10, and (d) m=10, n=5.

4. Conclusions In this work, estimation equation for transient C ( t ) are proposed for general material where the plastic and creep stress exponents are different under load and displacement control. The new equations are expressed in terms of crack-tip stress fields at initial elastic-plastic and steady-stated creep conditions. These can be calculated from analytical HRR and RR field expression. To validate the proposed equations, the predicted C ( t ) values are compared with elastic-plastic-creep FE results for plane strain single-edge-crack bend specimen. It is found that the proposed equations provide good agreement with the FE results, even when plastic and creep stress exponent are different. The present results can provide insight on the estimation of transient C ( t ) under load and displacement control. Acknowledgements This research was supported by the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2013M2A7A1076396, NRF-2013M2B2B1075733).