A continuum based macroscopic unified low- and high cycle fatigue - PowerPoint PPT Presentation

A continuum based macroscopic unified low- and high cycle fatigue model Tero Frondelius 1 , 2 , Sami Holopainen 3 , Reijo Kouhia 3 , Niels Saabye Ottosen 4 , Matti Ristinmaa 4 , Joona Vaara 1 1 W¨ artsil¨ a Finland Oy, J¨ arvikatu 2-4, FI-65100 Vaasa, Finland 2 Oulu University, Materials and Mech. Eng., P. Kaiteran katu 1, FI-90014 Oulu, Finland 3 Tampere University, Structural Mechanics, P.O. Box 600, FI-33014 Tampere University 4 Lund University, Solid Mechanics, P.O. Box 117, SE-22100 Lund, Sweden ICMFF12 - 12th International Conference on Multiaxial Fatigue and Fracture Bordeaux, France, June 24-26, 2019

Introduction - fatigue models Problems in fatigue analyses: ◮ low-cycle- and high-cycle -fatigue regimes are treated separately, ◮ mostly based on well defined cycles, ◮ multiaxiality. A more fundamental approach for HCF based on evolution equations proposed by Ottosen, Stenstr¨ om and Ristinmaa in IJF 2008. https://doi.org/10.1016/j.ijfatigue.2007.08.009 In this study this idea is combined with a plasticity model to obtain a unified model.

Evolution equation based HCF model Key ingredients are: B σ 1 Endurance surface β > 0 d s β ( σ , { α } ; parameters ) = 0 β < 0 evolution equations for the fatigue damage D d α α D = g ( β, D ) ˙ ˙ β and the internal variables { α } σ 2 σ 3 A α } = { G } ( σ , { α } ) ˙ { ˙ β

Conditions for evolution σ 1 σ 1 d s β > 0 β > 0 d s ˙ ˙ β ≥ 0 β < 0 α ̸ = 0 ˙ α = 0 ˙ s s ˙ ˙ D ≥ 0 d α D = 0 α α σ 2 σ 3 σ 2 σ 3 (a) (b)

Original formulation for HCF Endurance surface: 1 �� � 3 ¯ β = J 2 + AI 1 − σ − 1 = 0 σ − 1 where ¯ J 2 = 1 2 tr ( s − α ) 2 , I 1 = tr σ , A = σ − 1 /σ 0 − 1 and σ a σ − 1 = σ af ,R = − 1 A σ 0 = σ af ,R =0 σ − 1 σ m Evolution equations: α = C ( s − α ) ˙ D = K exp( Lβ ) ˙ ˙ ˙ β, β

LCF-HCF approach Couples with the plasticity model, Chaboche type model adopted: σ 1 d s β > 0 � ˙ 3 β ≥ 0 f ( σ , X , R ) = 2 ( s − X ) : ( s − X ) − ( σ y + R ) = 0 α ̸ = 0 ˙ s ˙ D ≥ 0 d α λ ∂f � ε p = ˙ ε p : ˙ ε p 2 ε p ˙ ∂ σ , ˙ eff = 3 ˙ α X � ˙ ε p R = R i , R i = γR ∞ ,i (1 − R i /R ∞ ,i ) ˙ eff ε p − γ i ˙ � ε p X i = 2 ˙ X = X i , 3 X ∞ ,i ˙ eff X i σ 2 σ 3 (a)

LCF-HCF approach - damage evolution d D d t = g ( β )d β d t + M d d t (exp( Qβ ) ε p eff ) where the high cycle part is modified to � � 1 + 1 − exp( − ˜ L ( β − b )) g ( β ) = K ≈ K exp Lβ when β � 1 a + exp( − ˜ L ( β − b )) Parameters M and Q from two standard cyclic tests with different amplitudes: d N ≈ M exp( Qβ )d ε p d D d N = 4 M exp( Qβ ) ε p eff a β ( N 1 ) − β ( N 2 ) ln N 2 ε p 1 1 a2 Q = , M = N 1 ε p 4 N i exp( Qβ ( N i )) ε p a1 a i

LCF-parameters Using the Coffin-Manson and Ramberg-Osgood relations: ε p f (2 N ) − c , c ( ε p a ) n c a = ε ′ σ a = σ ′ log( ε a ) LCF HCF 1 − c � N 2 � ε ′ Q = β ( N 1 ) − β ( N 2 ) ln f N 1 1 M = σ ′ 4 N i exp( Qβ ( N i )) ε ′ f (2 N i ) − c f E 1 f ) − c (2 N i ) − cn c − σ − 1 � (1 + A ) σ ′ c ( ε ′ � β ( N i ) = ∆ ε p > ∆ ε e ∆ ε p < ∆ ε e σ − 1 log(2 N )

bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc Preliminary results, S-N curve for AISI 4340 900 σ − 1 = 315 MPa , A = 0 . 225 , C = 1 . 2 , ˜ 800 K = 2 . 5 · 10 − 6 , L = 14 . 5 , a = 0 . 005 , b = 0 . 5 , M = 10 − 11 , Q = 16 . 700 σ a [MPa] σ y = 331 MPa , X ∞ , 1 = 35921 MPa , 600 bc bc X ∞ , 2 = 6972 MPa , X ∞ , 3 = 4222 MPa , 500 γ 1 = 651 , γ 2 = 53 . 3 , γ 3 = 5 . 7 , no isotr. hardening, 400 300 Chaboche model data from Y. Gorash, D. MacKenzie, Open Engineering , 7 , 126 (2017) 200 10 2 10 3 10 4 10 5 10 6 10 7 https://doi.org/10.1515/eng-2017-0019 N Present model fit with blue solid line. Dashed red line fit by Gorash, MacKenzie. Experimental results (black dots) from N.E. Dowling: Mean stress effects in stress-life and strain-life fatigue. SAE Technical Paper 1 (2004), 1-14.

ut ut ut ut ut ut ut ut ut ut ut Two-level test 1 . 4 1 . 2 Two-level loading 735 → 810 MPa (blue), 810 → 735 MPa (red). 1 . 0 0 . 8 n 2 /N 2 Experimental data shown by triangles from W.H. Erickson, C.E. Work, A study of the accumulation of fatigue 0 . 6 damage in steel, 64th Annual Meeting of ASTM , 704-718 (1961). 0 . 4 0 . 2 Present model predictions by solid lines. 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 n 1 / ( n 1 + n 2 )

Concluding remarks and future work ◮ Continuum based Unified LCF-HCF model. ◮ Multiaxial, applicable to arbitrary loading history. ◮ Applicable for post-processing. ◮ Can be easily extended to include anisotropic, gradient and stochastic effects. ◮ Parameter estimation. ◮ Micromechanical motivation of the evolution equations. Human fatigue illustrated by Akseli Gallen-Kallela 1894 Acknowledgements : The work was partially funded by TEKES - The National Technology Foundation of Finland (Business Finland from January 1, 2018), project MaNuMiES. Thank you for your attention!