A new semi-analytical approach for obtaining crack-tip stress distributions under variable-amplitude loading

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A new semi-analytical approach for obtaining crack-tip stress A new semi-analytical approach for obtaining crack-tip stress distributions under PCF variable-amplitude loading XV Portuguese Conference on Fracture, 2016, 10-12 February 2016, Paço de Arcos, Portugal distributions under variable-amplitude loading a A.N. Savkinaa, D.S. Denisevich *, K.A. Badikovaa, A.A. Sedovaa blade of an Thermo-mechanical modeling of turbine a a high pressure A.N. Savkin , D.S. Denisevich *, K.A. Badikov , A.A. Sedov Volgograd State Technical University, Lenin Avenue 28, Volgograd 400005, Russian Federation airplane gas turbine engine Volgograd State Technical University, Lenin Avenue 28, Volgograd 400005, Russian Federation a a

P. Brandãoa, V. Infanteb, A.M. Deusc* AbstractaDepartment of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal Abstract b Department of Mechanical Instituto Superior Técnico, Universidade de Lisboa, Av.the Rovisco Pais, 1, NearIDMEC, threshold fatigue crack responseEngineering, is extremely sensitive to load history. It is associated with influence of1049-001 near-tip Lisboa, stress Portugal Near threshold fatigue crack response extremely to load history. is associated conditions. with the influence of near-tip on cinstantaneous resistance of crackis tip surfacesensitive layers to failure underIt atmospheric The paper deals stress with CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, on instantaneous resistance of and crack tip distributions surface layers toPortugal failuretip.under atmospheric paper deals stress with computational aspects of stress strain at the crack A new approach isconditions. proposed toThe obtain near-tip computational of loading. stress andThe strain distributions crack tip. A approach is proposed to obtain near-tip(Linear stress under variable aspects amplitude method considersat athe combination of new standard rules for determining local strains under variable amplitude loading.etc.) Theand method considers a combination of standard rulesreturn for determining local strains (Linear rule, Neuber rule, ESED method, well-known in Finite Element Analysis (FEA) mapping algorithm in context of rule, Neuber rule,hardening ESED method, etc.) and well-known in Finitebased Element Analysis (FEA) return algorithm in context of nonlinear mixed material model. Iterative procedure on Newton-Raphson methodmapping is proposed to calculate plastic Abstract nonlinear and mixed hardening Iterative Newton-Raphson method proposedwith to calculate plastic variables near-tip stressmaterial at eachmodel. load step. The procedure calculated based resultsonobtained by new model areisvalidated the traditional variables and near-tip stressmodern at alloy eachaircraft loadconstant step. The calculated results obtained by model are validated with the traditional During in their operation, engine components areafter subjected to new increasingly demanding operating conditions, approach case of Al2024-T3 for amplitude loading tensile/comprehensive overloads. especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent approach in case of Al2024-T3 alloy for constant amplitude loading after tensile/comprehensive overloads. one of Published which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict © degradation, 2018 The Authors. by Elsevier B.V. © 2019 The Authors. Published by Elsevier B.V. the creep behaviour of HPT blades. Flight data records for a specific aircraft, provided by a commercial aviation © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license(FDR) (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an openwere accessused article thethermal CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) company, to under obtain mechanical data for responsibility three differentofflight cycles. In organizers. order to create the 3D model This is an and openpeer-review access article under the CCand BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection under responsibility of Peer-review under 2018 Selection responsibility Peer-review underscanned, responsibility SICE the 2018SICE organizers. neededand forpeer-review the FEM under analysis, a HPT of blade was and of itsthe chemical and material properties were Selection and peer-review under responsibility ofscrap Peer-review under responsibility of thecomposition SICE 2018 organizers. obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a loading, simplified 3D Keywords: crack growth rate, crack closure, cyclic plastic zone, stress intensity factor, threshold stress intensity, variable-amplitude rectangular shape, in hardening; order to better the stress model, and then with the realstress 3D mesh obtained from the blade scrap. The plasticity, back-stress, kinematic Keywords: crackblock growth rate, crack closure, cyclicestablish plastic zone, intensity factor, threshold intensity, variable-amplitude loading, overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a plasticity, back-stress, kinematic hardening; model can be useful in the goal of predicting turbine blade life, given a set of FDR data. © 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.: +7927-535-1068. E-mail address:author. [email protected] * Corresponding Tel.: +7927-535-1068. E-mail address: [email protected] 2452-3216 © 2018 The Authors. Published by Elsevier B.V. This is an © open article under theby CCElsevier BY-NC-ND 2452-3216 2018access The Authors. Published B.V. license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

Selection under responsibility of Peer-review responsibility of the SICE 2018 organizers. This is an and openpeer-review access article under the CC BY-NC-ND licenseunder (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under * Corresponding author. Tel.: +351responsibility 218419991. of Peer-review under responsibility of the SICE 2018 organizers. 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  2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of Peer-review under responsibility of the SICE 2018 organizers. 10.1016/j.prostr.2019.05.085

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A.N. Savkin et al. / Procedia Structural Integrity 14 (2019) 684–687 Author name / Structural Integrity Procedia 00 (2018) 000–000

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1. Introduction Near threshold fatigue crack response is extremely sensitive to load sequence effects. It is associated with the influence of near-tip stress on instantaneous resistance of crack tip surface layers to failure under atmospheric conditions (Brittle Metal Fracture model Sunder (2005)). As a result threshold stress intensity can be considered as not a material constant, but a variable which changes its value cycle by cycle depends on near-tip stress state response Sunder (2007), Sunder (2012). Therefore, an effective algorithm for stress calculation at the crack tip becomes significant problem in fatigue application. Traditional approaches which are well-known in notch fatigue analysis can be implemented in context of local inelastic response of crack tip. These methods involve solving Neuber equation or another rule for local inelasticity in combination with Ramberg–Osgood equation after accounting for material memory through Massing hypothesis. Using direct expressions for stress-strain relationship brings some limitations in material cyclic response, creep effects or cyclic unstable materials cannot be considered. Another disadvantage of such approaches is that they can be effectively used only for simple load sequences. The paper mainly deals with computational aspects of stress and strain distributions at the crack tip. A new approach is proposed to obtain near-tip stress under variable amplitude loading. 2. Proposed algorithm 2.1. Calculation strategy As LSS approaches attend to get direct analytical expression for local stress increment during cyclic loading and then apply material memory rules, we propose alternative computational procedure:  Calculation of stress intensity range K n 1 on current half cycle;  Determining of local strain increment at the crack tip ε *n 1 ;  Calculation of local stress σ *n 1 according to the accepted plasticity model. This algorithm is similar to the same used in finite element analysis (FEA) except strain calculation which is obtained from some approximation rule instead of variational formulation in finite element approach. In this paper we accept only linear rule in determining strain increment:

ε *n 1 

K n  1 2r *E

,

(1)

r * — physically possible minimum distance from the crack tip for stress calculation Sunder, E — Young modulus. Other strain approximation techniques also can be implemented (ESED method, etc.). Main advantage of proposed scheme is that all applicable in FEA material models can be considered. Fig. 1 shows graphical presentation of proposed algorithm in case of Al2024-T3 alloy for constant amplitude loading after tensile/comprehensive overloads.

2.2. Material model implementation Calculation of stress increment σ *n 1 is closely connected with material response during cyclic loading. A lot of mechanical effects (Bauschinger effect, ratcheting, cyclic hardening/softening, etc.) should be taken into account to provide adequate simulation results. Therefore appropriate hardening plasticity model need to be accepted. Further, we consider additive decomposition of total strain into elastic and plastic parts:

A.N. name Savkin/ Structural et al. / Procedia Structural 14 000–000 (2019) 684–687 Author Integrity ProcediaIntegrity 00 (2018)

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ε *  ε *e  ε *p ,

3

(2)

* ε *е — elastic strain increment; ε — plastic strain increment. p

In this paper we accept von Mises yield criterion in combination with isotropic-kinematic hardening rule:

f  σ *  α  σ y ε *   0,

(3)

p

Translation of yield surface is based on Chaboche summing rule, every individual translation follows Frederic– Armstrong kinematic hardening model: α

α i

i

α i  C i  γ i α i ψ ε * ,

(4)

p

α — sum of translations; α i — individual translation; С i , γ i — material constants; ψ — loading parameter («-1» — compression, «+1» — tension). Limit surface radius increasing/decreasing follows simple saturation model:

σ y ε *p   σ 0y  σ y  σ 0y 1  exp bε *p ,

(5)

σ 0y — initial yield stress; σ y — asymptotic yield stress; b — isotropic hardening constant. All needed material constants can be determined during standard multistep experimental procedure. Proposed integration algorithm of incremental constitutive expressions (3) can be interpreted as return mapping (closest point projection) scheme in one-dimensional implementation which is widely used in FEA. The first step of algorithm is called elastic predictor. After determining total strain increment it is assumed that its purely elastic. The stress and hardening parameters are predicted elastically as: tr

σ *  σ *n  E  ε *n 1 ,

tr

α  αn ,

tr

(6)

ε *p  ε *p n .

If trial stress is within the elastic domain (on the stress curve), f n 1 

tr

σ *  tr α  σ y n 1  0, then stress and other

internal variables stay unchanged:

σ *n 1  tr σ * , α n 1  tr α,

ε  * p

n 1

 tr ε *p

If trial stress is outside the elastic domain f n 1 

(7) tr

σ *  tr α  σ y n 1  0, then material response becomes plastic

and plastic correction and internal variables updating is needed (mapping): σ *n 1  tr σ *  E  ε *p n 1 , α n 1  tr α 

 C i

i

 γ i α i ψ   ε *p n 1 , ε *p n 1  ε *p n  ε *p n 1 .

(8)

Actual stress and plastic variables can be determined from yield condition which is a nonlinear equation in terms of plastic strain increment:

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A.N. Savkin et al. / Procedia Structural Integrity 14 (2019) 684–687 Author name / Structural Integrity Procedia 00 (2018) 000–000

f n 1 

tr

  σ  tr α   E   C i  γ i α i ψ   ε *p n  i   0  0  σ y  σ y  σ y  1  exp  b  ε *p n







  0

687

(9)

Equation (9) can be solved iteratively using Newton–Raphson method until convergence condition is achieved.

Fig. 1. Graphical interpretation of the proposed procedure for determining near-tip stresses under various loading sequences: underload – overload (a); overload – underload (b)

3. Results and conclusion The calculated results obtained by new model are qualitatively and quantitatively similar to the same obtained with the traditional approach based on Neuber rule and Ramberg–Osgood stress strain relation. Further algorithm enhancement will be towards the research of more realistic rule for obtaining strain increment and extending to the elevated temperature for creep resistant superalloys, including consideration of more difficult stress states.

Acknowledgements This paper was financially supported by the RFBR grant 17-08-01648 A, № 17-08-01742 A and the President of the Russian Federation grant MK-943.2017.8.

References Sunder, R. 2005, "Fatigue as a process of cyclic brittle microfracture", Fatigue and Fracture of Engineering Materials and Structures, vol. 28, no. 3, pp. 289-300. Sunder, R. 2012, "Unraveling the science of variable amplitude fatigue", ASTM Special Technical Publication, pp. 20. Sunder, R. 2007, "A unified model of fatigue kinetics based on crack driving force and material resistance", International Journal of Fatigue, vol. 29, no. 9-11, pp. 1681-1696.