On the role of DD in multiscale modeling: perspectives and new developments

Transcription

1 On the role of DD in multiscale modeling: perspectives and new developments Laurent Capolungo Cameron Sobie Nicolas Bertin Cedric Pradalier Saclay, December 1o 12 th 2014 International workshop on dislocation dynamics Acknowledgements simulations : EU Radinterfaces, ANR Magtwin 1

2 Plasticity in irradiated metals: a multiscale problem How to establish a link between irradiated microstructure and mechanical response? He Bubble Void Dislocation loops Li, 2010, Hernandez et al. 2010

3 Multiscale Perspective 1. Definition and parametrization of constitutive laws Connection with XRD, hexagonal systems Single defect populationfriedel Kroupa Hirsh vs Dispersed barrier model vs Bacon Kocks Scattergood Multiple defect: superposition approach 2. NEB with DDD: application to glide dislocation/prismatic loop interactions CP FEM DFT MS &MD Computation of activation barriers: NEB, FENEB, ART

4 DDD Cycle Calculate Forces Junctions, Remeshing Solve EOM N s t ( f k B α k V α ) j =1 j ( ) b δr k ds = 0 f pk = σ +Σ t Update Positions Time Integration <110> Drag Coefficient, B i, in 10 5 * N.s.m 2 B edge =5 B screw =500 4

5 Connection with CP: Wilkens approach ( ( )) N Scattering intensity r Restrictedly random distributions: I ( κ )= C exp i r κ R r j R r l The crystal can be subdivided in subvolumes of equal size r I ( s)= in exp which: ( iπs r n r ) C V dn3 dr 3 exp 2πigu r ( r + n r /2) u( r n r /2) -All have same dislocation densities. -Which have a null net dislocation polarity. -All Scattering dislocations intensity are is proportional infinitely long to the straight Fourrierand Transform parallel of -Within each volume the dislocation distribution is random A( n r )= 1 V dr 3 exp 2πigu r ( r + n r /2) u( r n r /2) j,l =2 ( ( )) ( ( )) Warren Averbach relation Wilkens A r n Asymmetric peaks in work of Groma et al. Effect of dislocation contrasts Ungar et al. ( > ) ( ) = exp 2π 2 n 2 g 2 2 < ε g,n 2 < ε g,n >= b 2π η = πρ f ( η) exp ( 1/4 )L R e R e 5

6 Connection with CP: Connection to XRD line profile analysis LPA vs DDD Compute elastic strain Complex Fourier Transform (CFT) Generate peaks using CFT Analyze peaks using the Stokes Wilson method and check accuracy line profile analysis (Wilkens model) Balogh et al, Acta Mat2012 6

7 Connection with CP: Latent hardening in HCP τ s = τ 0 s + μb s u a su ρ u Bertin et al, IJP

8 Multiscale Perspective 1. Definition and parametrization of constitutive laws Single defect populationfriedel Kroupa Hirsh vs Dispersed barrier model vs Bacon Kocks Scattergood Multiple defect: superposition approach 2. NEB with DDD: application to glide dislocation/prismatic loop interactions CP FEM MS &MD DFT

9 Irradiation Hardening Models Dispersed Barrier Hardening: based on mean spacing between defects. Δσ = αμb Nd Friedel Kroupa Hirsch: based on elastic interactions between SIA loops and straight dislocations. Δσ = α μb0 RN Bacon Kocks Scattergood: based on random array of spherical obstacles. Includes elastic self interaction. Δσ = α μb 2π L ln L b 1/2 ln d b /2 9

10 Irradiation Hardening Models Procedure Δσ = αμb Nd Randomly generate defect distribution for for many N,d Fit α, and compare quality of fit between models. 10

11 Perspectives Nogaret et al., 2008., Line tension based Mohles et al., 2002., DD Terentyev,

12 Uniform distribution of SIA loop size DBH Model Best fit: FKH model BKS Model Sobie et al, MET Trans, submitted

13 Effect of Size Distribution Can Δσ be predicted for a Gaussian size distribution by using the mean size? Δσ = αμb Nd Sobie et al, MET Trans, submitted

14 Uniform void size distribution DBH Model Best Fit: BKS Model Bacon et al (2009) 14

15 Spatial Distribution effect Does the same law apply if the defects are centered off the glide plane? Mean size also accurate for asymmetric distributions 15

16 Hardening due to both voids and SIA loops Loops and voids both present, how to use previous results? Δσ T n = Δσ 1 n + Δσ 2 n n=2.22 SIA loops, m 3 Voids, m 3 Typical values reported in literature 1, 2(KW), 1.5(Labusch), 2(HIratani), transition (Dong et al.) Dong et al. Met Trans

17 Multiscale Perspective 1. Definition and parametrization of constitutive laws Single defect populationfriedel Kroupa Hirsh vs Dispersed barrier model vs Bacon Kocks Scattergood Multiple defect: superposition approach 2. NEB with DDD: application to glide dislocation/prismatic loop interactions CP FEM DFT MS &MD Terentyev et al. 2013, Scripta Mater

18 NEB Method The images are evolved according to two forces: a spring force F s, which keeps images spaced apart, and the real (physical) force F R, moving the image to the minimum energy path. F i = F i s In 2D, the parallel and normal directions can be visualized. In higher dimensions, it is an (3)N dimensional vector. + F i R F R F s NEB used to find MEP between two states. Henkelman and Jónsson (2000) 18

19 NEB Method The images are evolved according to two forces: a spring force F s, which keeps images spaced apart, and the real (physical) force F R, moving the image to the minimum energy path. F i = F i s + F i R In 2D, the parallel and normal directions can be visualized. In higher dimensions, it is an 3N dimensional vector. τ i = R i R i 1 R i R i 1 + R i +1 R i R i +1 R i F i s F i R = (k[(r i +1 R i ) (R i R i 1 )] τ i )τ i = F ir (F i R τ i )τ i R i is a vector length 3N, the number of degrees of freedom in the system. R i = x r 1 y r 1 z r 1.. z r N 19

20 NEB in DDD: remeshing strategy Without remeshing, the relaxation can form kinks Points are interpolated using a cubic spline (red) and repositioned to be equally spaced, thus kinking cannot occur. Cannot change NDOF by removing/adding points. 20

21 NEB Method Terentyev et al 2008 Infinite dislocation impinging on a <100> SIA loop. 21

22 SIA Loop Size/Stress Effect For a range of sizes and applied stresses, a linear fit with a single slope and free y intercept was performed. The stress dependence is more complex, and best fit with a cubic function. 22

23 SIA Loop Interaction Geometry z +z Dislocations intersecting a <001> loop for varying vertical offset, equal applied stress. The activation energy for bypassing varies greatly with Burgers vector and offset position (z/r). Activation energy is highly sensitive to intersection offset distance. 23

24 Multiscale Perspective 1. Definition and parametrization of constitutive laws Single defect populationfriedel Kroupa Hirsh vs Dispersed barrier model vs Bacon Kocks Scattergood Multiple defect: superposition approach 2. NEB with DDD: application to glide dislocation/prismatic loop interactions CP FEM MS &MD DFT

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