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Prediction of intergranular micro-cracks initiation induced by the impingement of persistent slip bands on
grain boundaries J. Hazan, M. Sauzay
To cite this version:
J. Hazan, M. Sauzay. Prediction of intergranular micro-cracks initiation induced by the impingement of persistent slip bands on grain boundaries. Fatigue 2018, May 2018, Poitiers, France. �cea-02400189�
PREDICTION OF INTERGRANULAR
MICRO-CRACK INITIATION INDUCED
BY THE IMPINGEMENT OF
PERSISTENT SLIP BANDS ON
GRAIN BOUNDARIES
FATIGUE 2018 - MAY 29TH 2018
Jérôme HAZAN, Maxime SAUZAY
CONTENTS
• CONTEXT & GOALS
• METHODS • RESULTS
• CONCLUSIONS & WORK IN PROGRESS
CONTEXT
Cyclic deformation induces localization of plastic slip within Persistent Slip Bands (PSBs) for grains oriented for
single slip
Formation of a two phase microstructure (if 𝜏𝑝 = 𝜏𝑃𝑆𝐵) : - Elastic matrix - Elastic-plastic bands (PSBs) | 3 [Man et al. 2002] [Mughrabi 1979] Poor channels(~1013m-2) rich walls(~1015m-2) :
Ladder like structure
[Weidner et al. 2010] Copper single crystal 316L SS polycrystal Nickel polycrystal Fatigue 2018 | May 29th 2018
CONTEXT
The extrusion of PSBs is due to the production of vacancies during
cyclic deformation 18 MAI 2018 | 4 Extr u si o n h ei g h t, h [n m] Number ofcycles, N*103 [Man et al. 2003] Almost constant growth rate of PSBs
during cycling [Polák & Sauzay 2009]
Type A
Type B
Aim of the present work: Predicting micro-cracks initiation due to the impingement of PSBs on grain boundaries
Prediction of GB extrusions and GB stress fields
Transgranular initiation treated by [Liu & Sauzay 2014]
CONTEXT
Slip localization in PSBs leads to microcrack initiation (in LCF):
| 5
Transgranular (type B):
[Polák et al. 2009] [Zhang et al. 1999]
Grain Boundaries (type A):
5 µm
CONTEXT: EXISTING MODELS
Reference models mostly based on dislocations pile-up at GB : - Tanaka & Mura 1981
- Sangid et al. 2011 (Molecular Dynamics input for dislocation absorption and nucleation at GB)
| 6
𝑁𝑖 = 8 µ 𝛾𝑒𝑓𝑓
2 𝜋 1 − ν 𝑎 (∆𝜏 − 2𝜏𝑓)²
Our aim: Possibility of predicting GB micro-crack initiation through PSB cyclic plasticity and vacancy production ?
Cyclic deformation causes ratcheting of dislocations at grain boundaries Homogeneous slip within PSBs PSB containing Pile-up h = N.b t t [Weidner et al 2006] h = γp.t
[Sauzay & Ould Moussa 2013]
CONTENTS
• CONTEXT & GOAL
• METHODS
• Elastic-plastic model and behavior • Cohesive zone model
• RESULTS
• CONCLUSIONS & WORK IN PROGRESS
METHOD : ELASTIC-PLASTIC MODEL AND BEHAVIOR
| PAGE 8
Finite Element Calculations based on:
Elastic-Plastic behavior for PSBs
Armstrong-Fredericks non-linear kinematic hardening
Plateau behavior 2
Mixture rule of Winter: 𝛾𝑐𝑟𝑦𝑠𝑡𝑝 = 𝛾𝑀𝑎𝑡𝑝 𝑓𝑣𝑀𝑎𝑡 + 𝛾𝑃𝑆𝐵𝑝 𝑓𝑣𝑃𝑆𝐵 𝛾𝑀𝑎𝑡𝑝 ≈ ൗ1 100 𝛾𝑃𝑆𝐵𝑝 𝛾𝑐𝑟𝑦𝑠𝑡𝑝 = 𝛾𝑃𝑆𝐵𝑝 𝑓𝑣𝑃𝑆𝐵 Localisation of slip in PSBs 1 0 10 20 30 -0,01 -0,005 0 0,005 0,01 Sh ear str ess (M Pa ) Shear strain
Experiment - Single crystal behavior Experiment - PSB behavior
Fit Armstrong-Fredericks
[Mughrabi 1978]
METHOD
Production of vacancies inducing the free dilatation of persistent slip bands
Resistivity measurements on copper single crystals cycled at 4K [Polák 1987]
Determination of the vacancy production term
[Polák & Sauzay 2009]
p = 3.1 10-7 for Copper
| 9
Finite Element Calculations based on: 3 𝜀𝑣𝑎𝑐𝑎𝑛𝑐𝑦∗ = 𝑁𝑐𝑦𝑐𝑙𝑒𝑠 ∗ 𝑝 Grain size, L Slip band thickness, t 𝒃 Grain boundary impinged by the slip band
Elastic matrix 𝒃 : Burgers vector Grain 1, orientation 1 Grain 2, orientation 2 PSB Matrix cv x Diffusion of vacancies toward matrix Fatigue 2018 | May 29th 2018
Assumptions for the calculations:
Elastic-Plastic behavior of PSBs & production, annihilation of vacancies within PSBs
and diffusion toward the matrix
Elastic-Plastic behavior of neighbor grain Elastic matrix 10 20 30 40 50 60 S h e a r s tr e s s ( M P a )
Plastic shear strain amplitude
Experiment [001] Gong et al 1997 Experiment [011] Li et al 1998 Experiment [-111] Lepistö et al 1986 Experiment [-149] Mughrabi 1978 10-5 10-4 10-3 10-2 10-1 METHOD
Elastic-plastic behavior of neighbor grain impinged by the PSB
| 10
Finite Element Calculations based on: 4
Copper single crystals
CONTENTS
• CONTEXT & GOAL
• METHODS
• Elastic-plastic model and behavior • Cohesive zone model
• RESULTS
• CONCLUSIONS & WORK IN PROGRESS
Evaluation of three parameters:
- GB Young modulus EGB
- Fracture energy of the interface 𝛾𝑓𝑟𝑎𝑐𝑡 = 2𝛾𝑠 − 𝛾𝐺𝐵
- Critical stress
Output: Opening of the interface, δ => δ/δc = 1 : Complete separation
METHOD : COHESIVE ZONE MODELING
GB obeys a non linear cohesive law: 2 criteria for crack initiation :
- Critical stress (UBER model) [Rice & Wang, 1989] [Rose et al. 1981] - Critical energy released by the
cracked interface [Griffith]
18 MAI 2018 | PAGE 12 GB PSB 𝜎𝑐𝑟𝑖𝑡 = 𝐸 𝑑Τ 𝐺𝐵 ∗ 𝛾𝑓𝑟𝑎𝑐𝑡 𝑒 Grain 1 Grain 2 0 5 10 15 0 1 2 3 N o rm al s tr e ss (GPa ) Opening displacement, δ (Å) σcrit γGriffith δ = δc See [Barbé et al 2018] Fatigue 2018 | May 29th 2018
CONTENTS
• CONTEXT & GOAL • METHODS
• RESULTS
• GB extrusions and stress-fields • Prediction of crack initiation
• CONCLUSIONS & WORK IN PROGRESS
0 2 4 6 8 0 20 40 60 80 100 S h e a r s tr e s s ( G P a )
Distance to grain boudary / slip band interface (nm), r 1500 cycles 10500 cycles 20500 cycles 30000 cycles 0 2 4 6 8 0 20 40 60 80 100 N o rm a l s tr e s s ( G P a )
Distance to grain boudary / slip band interface (nm), r
1500 cycles 10500 cycles 20500 cycles 30000 cycles
RESULTS : PREDICTION OF MECHANICAL FIELDS
Copper | PSB thickness = 1 µm | Grain size = 50 µm | 30 000 cycles
| 14
GB
b
σnn
τnm Height of grain boundary extruded by the PSB impingement 0 20 40 60 80 100 0 5 10 15 20 G ra in b o u n d a ry e x tr u s io n ( n m )
Position along grain boundary (µm)
1500 cycles 10500 cycles 20500 cycles 30000 cycles Isotropic elastic grain r + 30000 cycles Fatigue 2018 | May 29th 2018
RESULTS: PREDICTION OF MECHANICAL FIELDS
Influence of the persistent slip band characteristic lengths:
=> grain boundary extrusion
| 15
Influence of slip band thickness, t Influence of grain size, L
Φ = 10 µm
Combined effect of the slip band thickness and grain size on the grain boundary extrusion : Higher damage expected
y = 1,7182x0,2348 y = 5,1967x0,2357 0 2 4 6 8 10 12 14 16 18 20 0 50 100 150 200 250 Gr ai n b o u n d ar y extr u si o n h ei g h t (n m) Grain size (µm) 10 000 cycles 30 000 cycles 𝒉 (𝑵) = 𝟎, 𝟓𝟔 . 𝒑. 𝑵. 𝒕𝟎,𝟕𝟕 ∗ 𝑳(𝟏−𝟎,𝟕𝟕) y = 0,0149x0,7652 y = 0,0455x0,7643 0 1 2 3 4 5 6 7 8 9 10 0 200 400 600 800 1000 Gr ai n b o u n d ar y extr u si o n h ei g h t (n m)
Slip band thickness (nm)
10 000 cycles 30 000 cycles
RESULTS: PREDICTION OF MECHANICAL FIELDS
Influence of neighbor grain crystallographic orientation ?
| 16
High increase in the grain boundary extrusion due to plasticity in the neighbor grain
[Li et al 2010]
8 active systems 4 active systems 6 active systems 1 active system 0 2 4 6 8 10 0 200 400 600 800 1000 G ra in b o u n d a ry e x tr u s io n ( n m )
Slip band thickness (nm)
SSO - 5 degree rotation SSO - 10 degree rotation SSO - 15 degree rotation SSO + 5 degree rotation SSO + 10 degree rotation SSO + 15 degree rotation Isotropic elastic 0 2 4 6 8 10 12 14 16 35 36 37 38 39 40 G ra in b o u n d a ry e x tr u s io n (n m )
Position along grain boundary (µm)
[100] [110] [111]
SSO + 15 degree rotation Isotropic Elastic
Grain size = 10µm
Grain size = 200µm
Slip band thickness = 1µm Neighbor grain :
Neighbor grain :
SSO : Single slip orientation
10000 cycles
RESULTS: PREDICTION OF MECHANICAL FIELDS
Influence of neighbor grain crystallographic orientation ?
| 17
Expected decrease of GB stress fields due to plasticity in the neighbor grain
[Li et al 2010]
8 active systems 4 active systems 6 active systems 1 active system 0 0,5 1 1,5 0 20 40 60 80 100 No rma l s tres s (GPa )
Distance to grain boundary / slip band interface (nm), r [100]
[110] [111]
SSO + 15 degree rotation Isotropic Elastic -0,5 0 0,5 1 1,5 0 20 40 60 80 100 S he a r s tre s s (GP a )
Distance to grain boundary / slip band interface (nm), r [100]
[110] [111]
SSO + 15 degree rotation Isotropic Elastic
Grain size = 200µm
Slip band thickness = 1µm Neighbor grain :
Neighbor grain : Grain size = 10µm
Slip band thickness = 1µm
RESULTS: PREDICTION OF MECHANICAL FIELDS
| 18
Comparison of the prediction to experimental data from literature : Impact of PSB thickness on GB extrusion height ?
[Zhang & Wang]
[Weidner et al. 2010]
Exponents adjusted using experimental data in reasonable agreement with the exponents provided by
the FE computations ~0.7 y = 0,2309x0,6845 R² = 0,6506 0 0,5 1 1,5 2 2,5 0 10 20 30 G ra in b o u n d a ry e x tr u s io n ( µ m )
Slip band thickness (µm) y = 0,3175x0,973 R² = 0,7208 0 0,2 0,4 0,6 0,8 1 0 1 2 3 G ra in b o u n d a ry e x tr u s io n ( µ m )
Slip band thickness (µm)
CONTENTS
• CONTEXT & GOAL • METHODS
• RESULTS
• GB extrusions and stress-fields • Prediction of crack initiation
• CONCLUSIONS & WORK IN PROGRESS
Prediction: Stronger influence of PSB thickness than grain size Order of magnitude in agreement with experimental observation ?
RESULTS: PREDICTION OF MICRO-CRACK INITIATION
Preliminary results: influence of PSBs characteristic lengths on GB micro-crack initiation : Copper, elastic neighbor
| 20 t (nm) L (µm) Ni (cycles) 50 10 60k 100 10 45k 200 10 25k 500 10 15k 1000 10 13k 1000 20 13k 1000 50 11k 1000 100 10k 1000 200 9k 0 0,2 0,4 0,6 0,8 1 100 1000 10000 100000 δ /δ c Number of cycles t = 50 nm | L = 10µm t = 100 nm | L = 10µm t = 200 nm | L = 10µm t = 500 nm | L = 10µm t = 1000 nm | L = 20µm t = 1000 nm | L = 50µm t = 1000 nm | L = 100µm t = 1000 nm | L = 200µm Fatigue 2018 | May 29th 2018
RESULTS: PREDICTION OF MICRO-CRACK INITIATION
| 21
Observations of intergranular initiation published in literature (RT, air environment):
Material Δεp/2 t (µm) L (µm) Ni Reference 316L -polycristal 10 -3 ≈0,45 µm <30> <5000 (20%Nr) [Mineur et al. 2000] 316L -polycristal 10 -3 ≈0,45 µm <47> <10000 (20%Nr) [Blochwitz et Richter 1999] 316L -polycristal 2.5 10 -4 ≈0,45 µm <47> <40000 (20%Nr) [Blochwitz et Richter 1999] Ni -polycristal 2.5 10 -4 ≈1 µm <24> <145000 (66% Nr) [Morrison et Moosbrugger 1997] Ni -polycristal 2.5 10 -3 ≈1 µm <290> <200 (4% Nr) [Morrison et Moosbrugger 1997] Ni -polycristal 2.5 10 -4 ≈1 µm <290> <17000 [Morrison et Moosbrugger 1997] Ni -polycristal 2.5 10 -3 ≈1 µm <24> <1200 (19% Nr) [Morrison et Moosbrugger 1997] Ni – polycristal 3 10
-4 ≈1 µm <150> 18000 [Dorr & Blochwitz 1987] Cu
-polycristal 10
-3 ≈1 µm <100> <10000
(25%Nr) [Liu et al 1992]
Cu - bicrystal 2 10-3 ≈1 µm ~5 mm <10000 [Zhang et Wang 2002]
Cu - bicrystal 5 10-4 ≈1 µm ~5 mm <20000 [Zhang Wang Hu 1999]
Fatigue 2018 | May 29th 2018
First prediction are in the good order of magnitude
CONTENTS
• CONTEXT & GOAL • METHODS
• RESULTS
• CONCLUSIONS & WORK IN PROGRESS
CONCLUSIONS & WORK IN PROGRESS
Conclusions :
• Simulation based on basic physical assumptions
• Prediction of stress-fields induced by impingement of PSBs at GB for different crystallographic orientations of the neighbor grains
• Analytical formulas of GB extrusion heights (elastic neighbor) Work in progress :
• Prediction of intergranular initiation introducing atomistic
considerations in the calculations as M.Sangid did [Sangid et al. 2011]
• GB type, environment, material, etc.
• Interrupted fatigue test carried out on two 316L SS containing (i) very large grains and (ii) small grains
• EBSD scan + SEM observations of crack-initiations
Comparison of predictions to real cases microstructures
• Analytical model for the prediction of intergranular, transgranular and twin boundaries microcrack initiation Fatigue 2018 | May 29th 2018| 23
WORK IN PROGRESS
Interrupted fatigue test :
316L SS, 2000 cycles, <200 µm>, Δεp/2 = 10-3
18 MAI 2018 | 24
=> Microstructure informed FE computations based on EBSD data and post-FIB cutting (3D information)
In ter granul ar i ni tiatio n Tw in bo un dary i ni tiatio n Fatigue 2018 | May 29th 2018
Nuclear Energy Division Nuclear Material Department
Section for Applied Metallurgy Research Materials Mechanical Analysis Laboratory
French Alternative Energies and Atomic Energy Commission Saclay Center| 91191 Gif-sur-Yvette Cedex
T. +33 (0)1 69 08 17 05 jerome.hazan@cea.fr | PAGE 25
Many thanks for your attention
Any question ?
WORK IN PROGRESS
=> Case of HCF & VHCF
Same methodology applicable for twin boundary cracking
18 MAI 2018 | 26
[Li et al. 2014]
[Man et al. 2012]
TRANSGRANULAR OR INTERGRANULAR INITIATION ?
18 MAI 2018 | SLIDE 27
[Obrtlik et al, 1997]
Cylindrical Copper single crystal cyclically deformed : - Each peak is offset by 180°
=> Preferential direction of slip
NUMBER OF GRAINS CONTAINING PSBS
18 MAI 2018 | SLIDE 28
[Sauzay 2006]
VALIDITY DOMAIN ?
18 MAI 2018 | 29
Element size = 3,33 nm
Compression test on Ni micropillar [Legros 2014]
1013 m-2< ρ
Matrix < 1015 m-2
Ndislocation, max = 3.33*10-10 x 3.33*10-10 x 1015 = 1.1*10-4 ≈ 0 dislocation
Why matrix is supposed to be elastic ?
ENVIRONMENT EFFECT ON MICRO-CRACK INITIATION
18 MAI 2018 | 30
Air, RT, Δεp/2 = 2.10-3 Vacuum, RT, Δε
p/2 = 2.10-3
Mostly transgranular initiation Mostly intergranular initiation
[Mineur et al, 2000]
METHOD : COHESIVE ZONE MODEL 18 MAI 2018 | PAGE 31 GB , CZM PSB Grain 1 Grain 2 Copper Nickel 316 SS Young modulus (Pa) 1.3 10 11 1.8 1011 1.9 1011 EGB min (Pa) 5.9 1010 8.1 1010 8.6 1010 EGB max (Pa) 1.3 1011 1.8 1011 1.9 1011 dGB min (Å) 3 3 3 dGB max (Å) 7 7 7 γsurf min (J/m²) 1.9 2 2 γsurf max (J/m²) 2.2 2.4 3 γGB min (J/m²) 0.3 0.45 0.48 γGB max (J/m²) 1 1.4 1.6 γfrac(J/m²) min 2.8 2.6 2.4 γfrac (J/m²) max 4.1 4.4 5.5 σcmin (GPa) 5.6 6.4 6.3 σcmax (GPa) 15 19 22
References : [Latapie & Farkas 2003] [Shen et al 1994], [Barbé et al 2018] [Tschopp & McDowell 2007] [Vitos 1998], [Holm et al 2010]
𝜎𝑐𝑟𝑖𝑡 = 𝐸 𝑑Τ 𝐺𝐵 ∗ 𝛾𝑓𝑟𝑎𝑐𝑡
𝑒
METHOD : COHESIVE ZONE MODEL
Many experimental and simulation efforts have been carried out to better understand grain boundaries role :
| 32 Influence of the Coincident Site Lattice Grains misorientation and GB characteristics GB energy, GB thickness Influence of Grain Size Influence of environment
[Lim & Raj 1984] [Shenyang group]
[Liu et al 1992]
[Tschopp & McDowell 2007]
[Yamakov et al 2006] [Morrison & Moosbrugger
1997]
[Taira et al 1979]
[Mineur et al 2000]