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University of Illinois at Urbana-Champaign Department of Mechanical Science and Engineering Recent Advances in Modeling Fatigue in Metals H. Sehitoglu, M. Sangid,T. Ezaz, H.J.Maier University of Illinois, USA, University of Paderborn, Germany Symposium in honor of C. Tome TMS, San Diego, February 28, 2011 Work Supported by Rolls Royce, National Science Foundation, DMR, Metals 1 www.mechse.uiuc.edu Outline Modeling of Fatigue • Analysis of Grain Boundaries- Energy Barriers for G.B. Slip Transmission And G.B. Slip Nucleation • Energy Formulation for Crack Initiation via Persistent Slip Bands, Life results Fatigue Crack Initiation Experimentally observed mechanism: transgranular facets forming from persistent slip bands (PSBs) across GBs Sangid MD, Maier HJ, Sehitoglu H, “A physically-based model for prediction of crack initiation from persistent slip bands in polycrystals,” Acta Materialia 59 328-341 (2011). Atomic Simulations Use Molecular Dynamics Code, LAMMPs • Construct GBs from crystal lattices (axis/angle pairs) • Ni with Foiles-Hoyt EAM Potential: FCC Structure • 3D Periodic Boundary Conditions • Atoms ‘relax’ to determine grain boundary energy GB N FCC E Perfect M Area GB ECSL Atomic Simulations of Prevalent Tilt & Twist GBs Grain Boundary Energy (mJ/m2) 1600 <110> Tilt <111> Twist <001> Tilt 1400 Σ5 1200 1000 Σ19 Σ5 Σ9 <001> Σ17 θ( 800 600 Σ13 Σ7 Σ11 Σ21 400 200 Perfect FCC 0 0 60 Σ3 90 <110> Σ7 Σ21 Σ21 Σ3 30 Σ13 Σ13 Σ7 Perfect FCC Σ3 120 Rotation Angle (degrees) 150 θ 180 ( <111> Mechanical Behavior of GBs: Slip Transmission A Σ3 (Twin) GB a/2 [011] t (100)t a/6 [211](111) Y [111] X [211] Z [011] a a a 211 011T 100 6 2 3 Incident Lomer Stair Rod Sangid MD, Ezaz T, Sehitoglu H, Robertson IM, “Energy of slip transmission and nucleation at grain boundaries,” Acta Materialia 59 283-296 (2011). Contour Plot of Energy Barriers GB Energy Barriers to Slip • Monitor the energy of the atoms within a control box at the GB n E • E i load Estatic i volum e Obtain the energy barrier for slip to penetrate the GB Validated values of slip in a perfect FCC lattice by comparing with GSFE curve • 400 Static GSFE Curve Control Box Method DFT Data from Siegel, 2005 Experimental Data Stacking Fault Energy (mJ/m2) 350 300 250 200 150 100 Error ~ 6% 50 0 0 0.5 1 1.5 Reaction Coordinate: Uz/<112>/6 2 Measured Energy Barrier for Slip Transmission for various GBs Validation of MD Results of Slip Transmission Criterion for Slip Transmission • Lee, Robertson, and Birnbaum, 1989 Geometrical condition – Minimizes angle between lines of intersection of slip planes with GB, maximize M: • Resolved shear stress condition – After predicting active slip plane from GC, choose direction based on max resolved shear stress • Residual grain boundary dislocation condition – Minimize Burgers vector of residual dislocation (difference in b of incoming and outgoing ┴): Energy Barriers for Slip Transmission through GB Energy Barrier for Dislocation - GB Interaction (mJ/m3) 2.5E+12 Σ3 2.0E+12 E Transmissi on Barrier 1.5E+12 Σ7 Σ11 1.0E+12 Σ13 Σ21 Σ17 Σ5 5.0E+11 Σ9 Perfect FCC Σ19 0.0E+00 0 200 400 600 800 2.8 10 E 13 1000 Static GB Energy (mJ/m2) 1200 1400 0.6 GB Static Twin boundary has inherently high energy barriers for slip transmission Coherent twin boundary • Stable defect structure • Low static GB energy • High energy barrier for slip transmission Note: • Applied loading is normal to GB • No applied shear stress on GB • General twin-slip interaction energies please refer to: Ezaz T, Sangid MD, Sehitoglu H, “Energy barriers associated with sliptwin interactions,” Philosophical Magazine, (2011). DOI: 10.1080/14786435.2010.541166 Slip is initially impeded by the twin resulting in a dislocation pile-up. As the applied load increases, slip transmits past the twin boundary. Measured Energy Barrier for Slip Nucleation from various GBs Energy Barriers for Slip Nucleation from GB Energy to Nucleate a Dislocation (mJ/m3) 2.0E+12 Σ7 1.8E+12 Σ21 Σ13 1.6E+12 Please note: the Σ1,3,&11 GBs have a simple dislocation structure and stable configurations. Hence dislocations were nucleated in the matrix material during the simulation, preventing the energy barrier to be measured Σ17 1.4E+12 1.2E+12 1.0E+12 8.0E+11 Σ9 Σ5 6.0E+11 Σ19 4.0E+11 2.0E+11 0.0E+00 0 200 400 600 800 1000 Static GB Energy (mJ/m2) 1200 1400 Fatigue Crack Initiation Experimentally observed mechanism: transgranular facets forming from persistent slip bands (PSBs) across GBs Sangid MD, Maier HJ, Sehitoglu H, “A physically-based model for prediction of crack initiation from persistent slip bands in polycrystals,” Acta Materialia 59 328-341 (2011). PSB Modeling and Energy Contributions • • Model the energy of a persistent slip band through a physics based approach Create an energy balance evolving with increasing loading cycles, which addresses all the PSB energy contributions: – Stress-field which must be overcome to have slip within the PSB • Applied Strain • Dislocation-dislocation interaction within the PSB (work-hardening) • Internal stress-field from dislocations – PSB interaction with GBs, particularly dislocations: • Piling-up at GBs • Nucleating from GBs and agglomerating within the PSB • Transmission through the GB – Formation of the PSB, dislocations shearing the γ matrix and γ’ precipitates • Crack initiates when PSB reaches minimum energy wrt to plastic deformation, i.e. dislocation motion Energy Formulation for a PSB Applied Work Extrusion Formation at GBs Dislocation Nucleation at GBs Shearing of γ’ Precipitates Exp functions σ ≡ Applied stress h ≡ Width of PSB d ≡ Dislocation spacing ρ ≡ PSB dislocation density N ≡ Number of cycles Microstructure Inputs where: Output Dislocation Pile-ups disl slipGB Enuc (m,,h,L,L',N) Eextrusion (m,,h,L,L',N) EAPB (L,dist ,N) E SF (L,dist ,N) Continuum Work Hardening in Bands MD disl Energy Eapp (,m,L,N) Ehard (,L,N) E pileup (h,d,L,N) Monitor a PSB and when it reaches a stable point, the material fails! m ≡ Schmid factor L ≡ Grain size Σ ≡ Characteristic of GB γ’ ≡ Distribution of precipitate L’ ≡ Grain size of neighboring grain Each term is expressed in terms of a slip increment, ∂X Sangid MD, Maier HJ, Sehitoglu H, “A physically-based model for prediction of crack initiation from persistent slip bands in polycrystals,” Acta Materialia 59 328-341 (2011). Shearing of γ Matrix Energy associated with overcoming stress field within the PSB for motion of glissile dislocations r layers E bLn X where the total stress is given by: dis Total stressPile-up of dislocations A Applied stress h where: b = Burgers vector L = grain size nlayers = number of dislocation layers in the PSB related to width of PSB, normalized by annihilation distance Lattice resistance ∂X = increment of slip Sangid MD, Maier HJ, Sehitoglu H, “A physically-based model for prediction of crack initiation from persistent slip bands in polycrystals,” Acta Materialia 59 328-341 (2011). PSB-GB Interaction Energy Atomistic Based Formulation: • Dislocations nucleate from the GB and localize in slip bands disl nuc E Xi EMD nucGB r 2 o bhL i – The number of dislocations that are emitted from the GB and aggregate within the PSB is given by: o hL • PSBs form extrusions at grain boundaries in polycrystalline material slipGB extrusion E Xi EMD slipGB r n bh pen dis i • Leverage energy barriers for slip transmission and nucleation at a GB from atomic simulations (previously shown) Sangid MD, Maier HJ, Sehitoglu H, “The role of grain boundaries in fatigue crack initiation – an energy approach,” Accepted to International Journal of Plasticity (In Press - 2011). Energy due to shearing γ matrix and γ’ precipitates Stacking Fault Energy + Anti-Phase Boundary Energy Atomistic Based Formulation: layers EAPB E SF f APB dL 1 f SF dL neff X d d o o where f is the area fraction of γ’, fU720 ~ 0.20 Shearing of γ’ precipitates O M Sangid MD, Maier HJ, Sehitoglu H, “A physically-based model for prediction of crack initiation from persistent slip bands in polycrystals,” Acta Materialia 59 328-341 (2011). R M’ O’ PSB Energy Balance Sangid MD, Ezaz T, Sehitoglu H, Robertson IM, “Energy of slip transmission and nucleation at grain boundaries,” Acta Materialia 59 283296 (2011). Sangid MD, Maier HJ, Sehitoglu H, “A physically-based model for prediction of crack initiation from persistent slip bands in polycrystals,” Acta Materialia 59 328-341 (2011). Criterion for Crack Initiation • Create an expression for energy (as energy components were previously shown) • Increment the number of loading cycles and update the energy expression as variables evolve: pen layers h, d , A , , ndis , nlayers , neff • Criterion for crack initiation – Energy of PSB reaches a stable value – Minimize energy to check for stability of PSB: E 0 X i where Xi represents the position of the glissile (mobile) dislocations in our energy expression Sangid MD, Maier HJ, Sehitoglu H, “A physically-based model for prediction of crack initiation from persistent slip bands in polycrystals,” Acta Materialia 59 328-341 (2011). More on PSB-GB Interaction θ > 15˚ θ < 15˚ PSBs transmit through low-angle GBs, θ < 15˚ High-angle GBs, θ > 15˚, impede dislocations, resulting in pile-ups, stress concentration, local increase in energy, and ultimately crack initiation TEM pictures from: Zhang & Wang, Acta Mat 51 (2003). Clustering of grains Failure occurs due to PSB-GB interaction or Grain cluster (multiple grains connected by LAGBs) Single large grain Normalized Applied Strain Range, % 1.2 Fatigue Scatter Results Model - Simulated Specimens Model - Average U720 Experimental Data U720 Data - Average 1.1 1 1000 simulated specimens vs. 84 experimental results 0.9 Each simulated specimen takes <30 seconds to construct its microstructure and predict fatigue life for a series of strain ranges 0.8 0.7 0.6 100 1000 10000 100000 1000000 Cycles to Initiation Sangid MD, Maier HJ, Sehitoglu H, “An energy-based microstructure model to account for fatigue scatter in polycrystals,” Journal of the Mechanics and Physics of Solids (In Press - 2011). Conclusions • Atomistic Simulations – Quantified the strengthening mechanisms of slip transmission and nucleation from GBs – Inverse correlation between energy barrier and static interfacial energy • Lower GB energy results in a stronger barrier. • Fatigue modeling – Introduced methodology to model persistent slip bands energetics, in order to predict fatigue life. • Prediction of fatigue life – Accurately predict scatter in a deterministic framework