Estimating the orientation of crack initiation planes under constant and ...

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

Estimating the orientation of crack initiation planes under constant XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal and variable amplitude multiaxial fatigue loading Thermo-mechanical modeling of turbine blade of an a, a high pressure b Yingyu Wanga,*, Luca Susmelb airplane gas turbine engine Key Key Laboratory Laboratory of of Fundamental Fundamental Science Science for for National National Defense-Advanced Defense-Advanced Design Design Technology Technology of of Flight Flight Vehicle, Vehicle, Nanjing Nanjing University University of of a a

bb

Aeronautics Aeronautics and and Astronautics, Astronautics, Nanjing, Nanjing, 210016, 210016, China China a b c Department Department of of Civil Civil and and Structural Structural Engineering, Engineering, the the University University of of Sheffield, Sheffield, Sheffield Sheffield S1 S1 3JD, 3JD, UK UK

P. Brandão , V. Infante , A.M. Deus *

a

Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Abstract Abstract Portugal c CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, This paper investigates the accuracy of the shear strain Maximum Variance This paper investigates the accuracy of the shear strain Maximum Variance Method Method (γ−MVM) (γ−MVM) and and Maximum Maximum Damage Damage Method Method Portugal b

(MDM) (MDM) in in predicting predicting the the orientation orientation of of crack crack initiation initiation planes planes under under both both constant constant and and variable variable amplitude amplitude multiaxial multiaxial fatigue fatigue loading. loading. The The γ−MVM γ−MVM defines defines the the critical critical plane plane as as the the plane plane on on which which the the variance variance of of the the resolved resolved shear shear strain strain reaches reaches its its maximum Abstractvalue. maximum value. In In contrast, contrast, aa specific specific multiaxial multiaxial fatigue fatigue criterion criterion is is needed needed to to be be used used along along with with the the MDM MDM to to predict predict the the orientation orientation of of the the critical critical plane plane under under multiaxial multiaxial fatigue fatigue loading. loading. As As far far as as variable variable amplitude amplitude multiaxial multiaxial loading loading is is concerned, concerned, the the MDM have to be used by applying with a certain fatigue criterion, a cycle counting method and a cumulative damage In During their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, MDM have to be used by applying with a certain fatigue criterion, a cycle counting method and a cumulative damage rule. rule. In this this especially the high pressure turbine (HPT) blades.criterion, Such conditions cause these parts to undergo types of time-dependent paper, the is with Fatemi & Bannantine & cycle counting method Palmgren-Miner’s paper, the MDM MDM is applied applied with Fatemi & Socie’s Socie’s criterion, Bannantine & Socie’s Socie’s cycle countingdifferent method and and Palmgren-Miner’s degradation, of which is creep. A critical model using finite element method the (FEM) was developed, in order to beExperimental able to predict linear rule. MDM assumes that plane is plane experiencing maximum accumulated damage. linear rule. The Theone MDM assumes that the the critical plane the is the the plane experiencing the maximum accumulated damage. Experimental the creep behaviour of HPT blades. Flight data records (FDR) for a specific aircraft, provided by a commercial aviation data for several metals tested under constant and variable amplitude multiaxial fatigue loading taken from literature to data for several metals tested under constant and variable amplitude multiaxial fatigue loading taken from literature are are used used to company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model assess the accuracy of these two methodologies. The results show that the predictions made by both the γ−MVM and MDM have assess the accuracy of these two methodologies. The results show that the predictions made by both the γ−MVM and MDM have needed for the FEMinvestigated analysis, a HPT blade scrap was scanned, and its chemical compositionmade and material properties were good good accuracy accuracy for for the the investigated materials materials and and investigated investigated load load histories: histories: 90% 90% of of the the predictions predictions made by by the the γ−MVM γ−MVM and and 80% 80% obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D of the the predictions predictions made made by by the the MDM MDM fall fall within within aa scatter scatter band band of of 20%. 20%. of block shape, in order better B.V. establish the model, and then with the real 3D mesh obtained from the blade scrap. The © rectangular 2016 The The Authors. Authors. Published by to Elsevier © 2016 Published by Elsevier B.V. by Elsevier overall expected behaviour in terms of displacement was observed, in particular © 2016, PROSTR (Procedia Structural Integrity) Hosting rights reserved. at the trailing edge of the blade. Therefore such a Peer-review under under responsibility responsibility of of the the Scientific Scientific Committee CommitteeLtd. of All ECF21. Peer-review of ECF21. Peer-review under the Scientific Committee of PCF 2016. model can be responsibility useful in theofgoal of predicting turbine blade life, given a set of FDR data. Keywords: multiaxial fatigue; variable amplitude; critical plane; crack initiation plane; nonproportional loading Keywords: fatigue; variable by amplitude; © 2016 multiaxial The Authors. Published Elseviercritical B.V. plane; crack initiation plane; nonproportional loading

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. author. Tel.: Tel.: +86-258-489-6297; +86-258-489-6297; fax: fax: +86-258-489-1422. +86-258-489-1422. ** Corresponding E-mail address: address: [email protected] [email protected] E-mail 2452-3216 2452-3216 © © 2016 2016 The The Authors. Authors. Published Published by by Elsevier Elsevier B.V. B.V.

* Corresponding Tel.: +351of Peer-review under responsibility the Peer-review underauthor. responsibility of218419991. the Scientific Scientific Committee Committee of of ECF21. ECF21. E-mail address: [email protected]

2452-3216 © 2016 The Authors. Published by Elsevier B.V.

Peer-review under responsibility of the Scientific Committee of PCF 2016. 2452-3216 © 2016, PROSTR (Procedia Structural Integrity) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of the Scientific Committee of PCF 2016. 10.1016/j.prostr.2016.06.403

Yingyu Wang et al. / Procedia Structural Integrity 2 (2016) 3233–3239 Author name / Structural Integgrity Procedia 00 0 (2016) 000–0000

3234 2

1. Introduction C Critical plane approaches taake as a startinng point the idea i that, at th he critical locaations, fatiguee cracks initiatte and proppagate on cerrtain planes. As A to definitiion of the crittical plane itsself, there aree different waays. Findley (1956) ( deteermined the critical c plane by b maximizinng a linear com mbination of the shear streess amplitude and the max ximum valuue of the norrmal stress. Brown B and Miller M (1973) defined the critical planee as the planee experiencin ng the maxximum shear strain amplituude. Fatemi annd Socie (19888) assumed th hat the critical plane is the plane of max ximum sheaar strain ampllitude when thhe crack initiattion process iss Mode II gov verned. As farr as variable am mplitude multtiaxial loadd histories aree concerned, defining d the orientation o of the critical pllane itself beccomes very coomplicate and d time connsuming. To address a this inntractable probblem, several approaches have h been prooposed which include: the MDM M firstt proposed byy Bannantine and Socie (19991), the MV VM proposed by Macha (19989), Susmel (2010), and Wang W andd Susmel (20155, 2016), and the weight fuunction methodd. T There always exists an arguument on thatt the predictions of the critiical plane appproach to the ccracking orien ntation is nnot coincide with w the physiccal concept which w it is baseed on. And, a limited workk has been donne in the evaluation of thhe capability of the critical plane approacch to predict the t crack direcctions (Jiang, 2007). IIn this scenariio, this paper aims to invesstigate the accuracy in pred dicting the orieentation of Staage I fatigue cracks c of tthe γ−MVM proposed p by Wang and Suusmel (2015, 2016) and th he MDM appplied with thee shear strain based critiical plane appproach proposeed by Fatemi and Socie (19988) (FS). 2. T The shear straain Maximum m Variance Method M (γ-MV VM) B Based on the research r by Macha M (1989),, Susmel (2010) formulated d the shear strress-Maximum m Variance Method M (τ−M MVM) to preedict, in the high-cycle faatigue regime, the orientattion of the crritical plane uunder constan nt and variiable amplitudde multiaxial fatigue loadinng. Then, by extending e thiss idea to in terrms of strain, Wang and Su usmel (2015, 2016) prooposed the sheear strain-Maaximum Variaance Method (γ−MVM) ( whhich is suitablle to be used in the w/medium-cycle fatigue regiime. low T The body as shhown in Fig. 1a is subjectedd to a complex system of fo orces, which results r in a tri--axial time-vaariable streess and strain states at the critical c locatioon, point O. The T stress and d strain statess at point O ccan be expresssed as follows:

Fig. 1 (a) body b subjected too an external systeem of forces, (b) definition of a geeneric material pllane, Δ, and coorddinate system

Yingyu Wang et al. / Procedia Structural Integrity 2 (2016) 3233–3239 Author name / Structural Integrity Procedia 00 (2016) 000–000

⎡σ x (t ) τ xy (t ) τ xz (t ) ⎤ [σ (t )] = ⎢⎢τ xy (t ) σ y (t ) τ yz (t )⎥⎥ ⎢τ xz (t ) τ yz (t ) σ z (t ) ⎥ ⎣ ⎦

⎡ ⎢ ε x (t ) ⎢ γ (t ) [ε (t )] = ⎢⎢ xy 2 ⎢ γ (t ) xz ⎢ ⎢⎣ 2

γ xy (t ) 2

ε y (t ) γ yz (t ) 2

3235 3

(1)

γ xz (t ) ⎤

⎥ 2 ⎥ γ yz (t ) ⎥ 2 ⎥ ⎥ ε z (t ) ⎥ ⎥⎦

(2)

Then the shear strain resolved along of a generic direction on a generic plane, γq(t), can be obtained by tensor rotation of the above strain tensor to this direction. Alternatively, γq(t) can also be expressed by the following scalar product: γ q (t ) 2

= e (t ) • d

(3)

where, d and e can be expressed as follows:

[ [

] ⎤⎥ ⎥ ]⎥⎥

⎡1 2 ⎢ 2 sin(θ ) sin(2φ ) cos(α ) + sin(α ) sin(2θ ) cos(φ ) d ⎡ 1⎤ ⎢ ⎢d ⎥ ⎢ 1 − sin(θ ) sin(2φ ) cos(α ) + sin(α ) sin(2θ ) sin(φ ) 2 ⎢ 2⎥ ⎢2 ⎢d ⎥ ⎢ 1 − sin(α ) sin(2θ ) d = ⎢ 3⎥ = ⎢ 2 ⎢d 4 ⎥ ⎢ ⎢ d ⎥ ⎢ 1 sin(α ) sin(2φ ) sin(2θ ) − cos(α ) cos(2φ ) sin(θ ) ⎢ 5⎥ ⎢ 2 ⎣⎢d 6 ⎦⎥ ⎢ sin(α ) cos(φ ) cos(2θ ) + cos(α ) sin(φ ) cos(θ ) ⎢ sin(α ) sin(φ ) cos(2θ ) − cos(α ) cos(φ ) cos(θ ) ⎣

γ xy (t ) ⎡ e(t ) = ⎢ε x (t ) ε y (t ) ε z (t ) 2 ⎣

γ xz (t ) γ yz (t ) ⎤

(4)

(5)

⎥ 2 ⎦

2

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Therefore, the variance of the shear strain resolved along a generic direction q can be expressed in the following form: ⎡ γ q (t ) ⎤ ⎡ ⎤ Var ⎢ ⎥ = Var ⎢∑ d k ek (t )⎥ = ∑ ∑ d i d j Cov ei (t ), e j (t ) 2 k ⎣ ⎦ i j ⎣ ⎦

[

]

(6)

All the candidate critical planes can be obtained by determining the global maxima of the variance of the resolved shear strain, as discussed by Susmel (2010) and Wang and Susmel (2016).

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3. The Maximum Damage Method (MDM) The stress and strain states summarized via Eq. (1) and Eq. (2) are also used in the MDM, and the stress and strain components on a generic direction of a generic plane are obtained by tensor rotation. The MDM always needs to be applied with a specific multiaxial fatigue criterion to predict the orientation of the critical plane. In the present paper, the FS criterion is used along with the MDM. As far as variable amplitude multiaxial loading is concerned, the cycle counting method and the fatigue damage cumulative rule are also required. Bannantine & Socie’s cycle counting method (Bannantine and Socie, 1991) and Palmgren-Miner’s linear damage rule (Palmgren, 1924; Miner, 1945) are used in this paper. In more detail, the resolved shear stain and the normal stress can be obtained by projecting the loading history on a generic direction of a generic plane. The resolved shear strain cycles and the maximum normal stresses during the shear strain cycles are identified by using Bannantine & Socie’s cycle counting method. The i-th loading damage is calculated by using the FS criterion, and the accumulated damage is calculated according to Palmgen-Miner’s linear rule. This calculation should be done in every direction on every plane at the critical location. The shear-strain based multiaxial fatigue criterion proposed by Fatemi and Socie (1988) can be expressed as follows: Δγ 2

⎛ σ n , max ⎜ ⎜⎜ 1 + k σ y ⎝

⎞ τ′ ⎟ f ⎟⎟ = G 2 N ⎠

(

f

)b0

+ γ ′f ⎛⎜ 2 N f ⎞⎟ ⎝ ⎠

c0

(7)

where Δγ/2 is the shear stain amplitude relative to the critical direction on the certain plane, σn,max is the maximum normal stress occurring during the same cycle of Δγ/2 on this plane, k is a material constant, and σy is the material yield strength. The total damage is calculated according to Palmgren-Miner’s linear rule as follows: ni i =1 N f ,i j

Dtot = ∑

(8)

where ni is the number of loading cycles, Nf ,i is the total cycles to failure under the i-th loading, and Dtot is the total value of the damage sum. 4. Experimental valuations In order to check the accuracy of the γ-MVM and MDM in predicting the orientation of the critical plane under multiaxial loading the experimental research for S45C steel by Kim et al. (1999) under short variable amplitude multiaxial loading, the observation of cracking behavior for S460N by Jiang et al. (2007) under multiaxial constant loading and the observed cracking behavior of 1050QT steel and 304L steel under discriminating strain paths by Shamsaei et al. (2011) were taken into consideration. According to the research by Forsyth (1961), Socie and Marquis (2000), Marciniak et al. (2014) and Susmel et al. (2014), the propagation process of micro/meso-crack can be described by two stages: Stage I is crack initiation, and Stage II is crack propagation. Usually, for elasto-plastic metallic materials, Stage I cracks initiate on those plane of maximum shear. Stage II cracks tend to propagate perpendicular to the normal stress. Therefore, in the current paper, if the length of the observed crack in the original literature is of the order of millimeters, the observed angle is deemed to be the orientation of the crack initiation plane. If the crack length is in the centimeter scale, with


3237 5

reference to the method used by Susmel et al. (2014) and Marciniak et al. (2014), then an angle of ±45° was added to the observed crack angle in the original literature to obtain the orientation of the crack initiation plane. The static and fatigue properties of the investigated materials are listed in the Table 1. The investigated loading paths are shown in Fig. 2. Figs 3 and 4 show the comparison of the predicted orientation of crack initiation plane and the experimental observation on the cracking behavior. The ranges of the crack shown in these two figures are from 0° to 260°. Here, the crack plane at angle 0° is the same plane as the one which have orientation angle equal to 180°. The definition of the angle of crack plane is shown down-right in Figs 3 and 4. 80 data taken from literature are used in this evaluation. However, some data are hidden from view because of some other data in front of them. The results show that both the γ-MVM and MDM can predict the orientation of the crack initiation plane with a good level of accuracy. 90% of the predictions made by the γ-MVM fall within the 20% scatter band, and 95% of the predictions fall within the 30% scatter band. The MDM provides good predictions. 80% of the estimates made by the MDM fall within the 20% scatter band, and 90% of the estimates fall within the 30% scatter band. Table 1: Static and fatigue properties of the investigated materials. b0

c0

0.198

࣎ᇱ܎ (MPa) 685

-0.12

-0.36

3.48

777

-0.062

-0.725

0.15

0.211

743

-0.145

-0.394

1

0.487

559.8

-0.086

-0.493

Material

Ref.

E (MPa)

G (MPa)

S45C

Kim et al. (1999)

186,000

70,600

496

1

1050 QT Steel

Shamsaei et al. (2011)

203,000

81,000

1009

0.6

σy (MPa)

304L steel

Shamsaei et al. (2011)

195,000

77,000

208

S460N

Jiang et al. (2007)

208,500

80,200

500

γ/√3

γ/√3

ε

A

ε

T

γ/√3

γ/√3

E1

γ/√3

M5

N5

O1

ε

R02

ξ

ε Δξ

ε Δξ

FR

Fig. 2. Investigated strain paths

O2

γ/√3

γ/√3 ξ

R01

N1

ε

γ/√3

ε

ε

γ/√3

N4

γ/√3

ε

M4

ε

N3

γ/√3

ε

M3

ε

E2

γ/√3

ε

γ/√3

N2

ε

AT

ε

γ/√3

ε

γ/√3

M2

γ/√3

ε

P

γ/√3

ε

M1

γ/√3

઻ᇱ܎

k in FS criterion

PI

3238 6


Fig. 3. Comparison of experimental and calculated crack initiaton plane according to the γ-MVM

Fig. 4. Comparison of experimental and calculated crack initiaton plane according to the MDM


3239 7

5. Conclusions 1.

For the investigated materials under investigated loading conditions, both the γ-MVM and the MDM can predict the orientation of the crack initiation plane with a good level of accuracy.

2.

90% of the predictions made by the γ-MVM fall within the 20% scatter band, and 95% of the predictions fall within the 30% scatter band. The MDM provides a good prediction. 80% of the data estimated by the MDM fall within the 20% scatter band, and 90% of the estimates fall within the 30% scatter band.

Acknowledgements This work was partly supported by the Fundamental Research Funds for the Central Universities, within the research project No. NS2015003. References Bannantine, J.A., Socie, D.F., 1991.A variable amplitude multiaxial life prediction method. In: Kussmaul, K., McDiarmid, D. Socie, D.F. (Eds). Fatigue under Biaxial and Multiaxial Loading. ESIS 10. Lodon: Mechanical Engineering Publications, pp. 35. Brown, M.W., Miller, K.J., 1973. A theory for fatigue under multiaxial stress-strain conditions. In: Proc. Institution of Mechanical Engineering 187, 745-56. Fatemi, A., Socie, D.F., 1988. A critical plane approach to multiaxial fatigue damage including out-of-phase loading. Fatigue & Fracture of Engineering Material and Structure 11, 149-165. Findley, W.N., 1956. Modified theory of fatigue failure under combined stress, In: Proc. of the society of experimental stress analysis 14, 35-46. Forsyth, P.J.E., 1961. A two-stage fatigue fracture mechanisms. In: Proceeding of the crack propagation symposium, Vol. 1, Cranfield, UK, 7694. Jiang, Y., Hertel, O., Vormwald, M., 2007. An experimental evaluation of three critical plane multiaxial fatigue criteria. International Journal of Fatigue 29, 1490-1502. Kim, K.S., Park, J.C., Lee, J.W., 1999. Multiaxial fatigue under variable amplitude loads, Trans ASME Journal of Engineering Materials and Technology 121, 286-93. Macha, E., 1989. Simulation investigations of the position of fatigue fracture plane in materials with biaxial loads. Materialwiss Werkstofftech 20, 132-6. Marciniak, Z., Rozumek, D., Macha, E., 2014. Verification of fatigue critical plane position according to variance and damage accumulation methods under multiaxial loading. International Journal of Fatigue 58, 84-93. Miner, M.A., 1945. Cumulative damage in fatigue. Journal of Applied Mechanics 67, AI59–64. Palmgren, A., 1924. Die Lebensdauer von Kugellagern. vol. 68. Verfahrenstechnik, Berlin, pp. 339–41. Shamsaei, N., Fatemi, A., Socie, D.F., 2011. Multiaxial fatigue evaluation using discriminating strain paths. International Journal of Fatigue 33, 597-609. Socie, D.F., Marquis, G.B., 2000. Multiaxial Fatigue. Warrendale, PA: SAE. Susmel, L., 2010. A simple and efficient numerical algorithm to determine the orientation of the critical plane in multiaxial fatigue problems. International Journal of Fatigue 32, 1875-1883. Susmel, L., Tovo, R., Socie, D.F., 2014. Estimating the orientation of stage I crack paths through the direction of maximum variance of the resolved shear stress. International Journal of Fatigue 58, 94-101. Wang, Y., Susmel, L., 2015. Critical plane approach to multiaxial variable amplitude fatigue loading. Fracture and Structural Integrity 33, 345356. Wang, Y., Susmel, L., 2016. The modified Manson-Coffin Curve Method to estimate fatigue lifetime under complex constant and variable amplitude multiaxial fatigue loading. International Journal of Fatigue 83, 135-149.