3D EBSD analysis of mechanical size effects during indentation: EBSD tomography and crystal plasticity FEM

Transcription

1 3D EBSD analysis of mechanical size effects during indentation: EBSD tomography and crystal plasticity FEM E. Demir, N. Zaafarani, F. Weber, D. Ma, S. Zaefferer, F. Roters, D. Raabe Department of Microstructure Physics and Metal Forming Düsseldorf, Germany January 07 th, 2009; PLASTICITY 2009, St Thomas, Virgin Islands, USA

2 Overview Motivation and approach Sample preparation Deformation experiments Characterization Orientation gradients Dislocation analysis Results Other examples: Wire bending, pillar compression Conclusions 1

3 Overview Motivation and approach Sample preparation Deformation experiments Characterization Orientation gradients Dislocation analysis Results Other examples: Wire bending, pillar compression Conclusions 2

4 Overview Challenges in in size dependent mechanics Theoretical concept (GND, gradients, sources, hardening, localization, ) Experiments asking a specific question Homogeneity of of the microstructure Boundary conditions of of the experiment 3

5 Homogeneity and boundary conditions large scale Microstructure not nothomogeneous 3% 8% Boundary conditions essential 15% M. Sachtleber, Z. Zhao, D. Raabe: Mater. Sc. Engin. A 336 (2002) 81 D. Raabe, M. Sachtleber, Z. Zhao, F. Roters, S. Zaefferer: Acta Mater. 49 (2001)

6 Crystal plasticity FEM, grain scale mechanics (3D) orientation 5mm 5mm 8mm 21m m 1mm vm strain Z. Zhao, M. Ramesh, D. Raabe, A. Cuitino, R. Radovitzky: Intern. J. Plast. 24 (2008)

7 Homogeneity and boundary conditions small scale [111] [11-2] [-110] 6

8 Motivation and approach: specific Relationship between ISE and GND: hardness and GND in the same experiment Indents of different depths, Cu single crystal High resolution 3D EBSD Determine plastic volume beneath indents Calculate GNDs around indents from 3D orientation gradients and plastic volume Also: Wire bending; pillar compression 7

9 Overview Motivation and approach Sample preparation Deformation experiments Characterization Orientation gradients Dislocation analysis Results Other examples: Wire bending, pillar compression Conclusions 8

10 Nanoindentation 60 conical, tip radius 1μm, loading rate 1.82mN/s, loads of 4000μN, 6000μN, 8000μN and 10000μN [111] [11-2] [-110] Y. Wang, D. Raabe, C. Klüber, F. Roters: Acta Mater. 52 (2004)

11 3D electron microscopy, 3D EBSD, tomography J. Konrad, S. Zaefferer, D. Raabe, Acta Mater. 54 (2006) 1369 S. Zaefferer, S. I. Wright, D. Raabe, Metall. Mater. Trans. A 39A (2008)

12 Crystal Mechanics FEM (General) DFT, ultrasonics Statistical dislocation-based hardening law in CPFEM DDD, TEM, single crystal tests Crystallography, MD, Peierls analysis Kinematics in CPFEM EBSD, RVE,. D. Raabe, P. Klose, B. Engl, K.-P. Imlau, F. Friedel, F. Roters: Adv. Eng. Mater. 4 (2002)

13 Crystal Plasticity FEM (constitutive laws) Multiplicative Decomposition of the Deformation Gradient constitutive reduction of DOF: ~ L p 12 ~ = & γ αd α = 1 flow law α n ~ α & γ α ( τα, τ cα, θ ) Phenomenological & γ ( τ, ρ, θ ) Dislocation density laws α α F* : elastic and rotation deformation gradient F : total deformation gradient Fp : plastic deformation gradient Lp : plastic velocity gradient Le : elastic velocity gradient dγ dx b 1 & γ = = n = dt X Z dt : = F& F 1 L e e e 1 L ~ : p = F& pfp L Definitions: L : = F & F 1 e 12 p ρ bv m ~ : = F L F e 1 p

14 Physics-based constitutive laws: mean field theory 1 1. set internal variables dyadic flow law based on dislocation rate theory Taylor, Kocks, Mecking, Estrin, Kubin,... T T T T T T T T 2 2. set internal variables plastic gradients, size scale and orientation gradients (implicit) Nye, Ashby, Kröner, set internal variables grain boundaries activation concept: energy of formation upon slip penetration: conservation law Ma, Roters, Raabe: Acta Mater. 54 (2006) 2169; Ma, Roters, Raabe: Acta Mater. 54 (2006) 2181; Ma, Roters, Raabe: Intern. J Sol. Struct. 43 (2006)

15 Dislocation classes Mobile dislocations accommodate the external plastic deformation Immobile dislocations (cell interior, cell walls) work hardening, including locks and dipoles Geometrically necessary dislocations preserve the lattice continuity 14

16 Rate equations Rate equations are formulated for the immobile dislocation densities: & ρ & & + + SSD = ( ρi ) SSD+ ( ρii) SSD+ ( ρi ) SSD+ ( ρii) SSD & & Four mechanisms: mechanism 1 : lock formation mechanism 2 : dipole formation mechanism 3 : athermal annihilation mechanism 4 : thermally activated annihilation by climb A. Ma F. Roters D. Raabe: Acta Mater. 54 (2006)

17 Al Bicrystals, low angle g.b. [112] 7.4, v Mises strain experiment von Mises strain [1] SSD viscoplastic phenomen. model dislocationbased model; g.b. model A. Ma F. Roters D. Raabe: Acta Mater. 54 (2006) % 20% 30% 40% 50% 16

18 Overview Motivation and approach Sample preparation Deformation experiments Characterization Orientation gradients Dislocation analysis Results Other examples: Wire bending, pillar compression Conclusions 17

19 Crystal orientation distribution around nanoindents 20 0 Misorientation angle [111] [-110] [11-2] Zaafarani, Raabe, Singh, Roters, Zaefferer: Acta Mater. 54 (2006) 1707; Zaafarani, Raabe, Roters, Zaefferer: Acta Mater. 56 (2008) 31 18

20 Comparison, crystal rotations about [11-2] axis phenomenological CPFEM experiment 3D EBSD dislocation-based CPFEM scan 9 experiment simulation scan 8 scan 7 N. Zaafarani D. Raabe, R. N. Singh, F. Roters, S. Zaefferer: Acta Mater. 54 (2006) 1707 N. Zaafarani, D. Raabe, F. Roters and S. Zaefferer: Acta Mater. 56 (2008) [111] [11-2] [-110]

21 Simplify 20

22 Pattern quality, rotation field, indents of different size 21

23 Overview Motivation and approach Sample preparation Deformation experiments Characterization Orientation gradients Dislocation analysis Results Other examples: Wire bending, pillar compression Conclusions 22

24 Local misorientation analysis 23

25 From local misorientations to GNDs misorientation orientation difference orientation gradient (spacing d from EBSD scan) 24

26 From local misorientations to GNDs distortion (sym, a-sym) dislocation tensor (GND) J. F. Nye. Some geometrical relations in dislocated crystals. Acta Metall. 1:153, E. Demir, D. Raabe, N. Zaafarani, S. Zaefferer: Acta Mater. 57 (2009) E. Kröner. Kontinuumstheorie der Versetzungen und Eigenspannungen (in German). Springer, Berlin, E. Kröner. Physics of defects, chapter Continuum theory of defects, p.217. North-Holland Publishing, Amsterdam, Netherlands,

27 From local misorientations to GNDs Frank loop through area r DDT in terms of 18 b,t combinations DDT in terms of 9 b,t combinations E. Demir, D. Raabe, N. Zaafarani, S. Zaefferer: Acta Mater. 57 (2009)

28 Overview Motivation and approach Sample preparation Deformation experiments Characterization Orientation gradients Dislocation analysis Results Other examples: Wire bending, pillar compression Conclusions 27

29 Distribution of GNDs for different gradient resolutions 50 nm 100 nm center section, 2D analysis, color code: GND density (decadic log. (1/m 2 ) ) 200 nm 28

30 Identification of the plastic volume depth [μm] Reference volumes (red color) indicates the plastic volume using lower threshold of GNDs of /m 2 center section, 2D analysis, color code: GND density (decadic log. (1/m 2 ) ) E. Demir, D. Raabe, N. Zaafarani, S. Zaefferer: Acta Mater. 57 (2009)

31 Extract geometrically necessary dislocations E. Demir, D. Raabe, N. Zaafarani, S. Zaefferer: Acta Mater. 57 (2009)

32 Overview Motivation and approach Sample preparation Deformation experiments Characterization Orientation gradients Dislocation analysis Results Other examples: Wire bending, pillar compression Conclusions 31

33 Similar experiment for bending F. Weber, I. Schestakow, D. Raabe, F. Roters: Adv. Engin. Mater. 10 (2008)

34 Metallic micropillar deformation Parameter study: friction D. Raabe, D. Ma, F. Roters: Acta Mater. 55 (2007)

35 Metallic micropillar deformation Parameter study: individual slip activity and heterogeneity D. Raabe, D. Ma, F. Roters: Acta Mater. 55 (2007)

36 Metallic micropillar deformation corresponding experiments by : Robert Maaß, Steven Van Petegem, Julien Zimmermann, Daniel Grolimund, Helena Van Swygenhoven Simulation results 35

37 Overview Motivation and approach Sample preparation Deformation experiments Characterization Orientation gradients Dislocation analysis Results Other examples: Wire bending, pillar compression Conclusions 36

38 Conclusions (indents) The GNDs have an inhomogeneous distribution underneath the indents with very high local density values. No homogeneous distribution. The total GND density below the indents decreases with decreasing indentation depths. This observation does not conform to strain gradient theories attributing size-dependent material properties to GNDs. The amount of deformation imposed reduces with decreasing indentation depths. Therefore, SSDs that evolve through strain do not account for the increasing hardness values with decreasing indentation depth. Inhomogeneous distribution of GNDs plays a main role. Explaining size dependent material strengthening effects by using average density measures for both GNDs and SSDs is not sufficient to understand the indentation size effect. More critical experiments on physics of size effects Control and numerical analysis of boundary conditions 37