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Analysis of nanoindentation size effect based on Crystal Plasticity and Dislocation Dynamics

Analysis of nanoindentation size effect based on Crystal Plasticity and Dislocation Dynamics. Hyung–Jun CHANG 9 th October, 2009 Supervisors Marc Fivel, Marc Verdier (Grenoble INP) Laurent Tabourot (Univ. de Savoie) Heung Nam Han, Kyu Hwan Oh (Seoul Natl’ Univ.)

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Analysis of nanoindentation size effect based on Crystal Plasticity and Dislocation Dynamics

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  1. Analysis of nanoindentation size effectbased on Crystal Plasticity and Dislocation Dynamics Hyung–Jun CHANG 9th October, 2009 Supervisors Marc Fivel, Marc Verdier (Grenoble INP) Laurent Tabourot (Univ. de Savoie) Heung Nam Han, Kyu Hwan Oh (Seoul Natl’ Univ.) Samuel Forest (Centre des matériaux)

  2. Macro scale Micro scale P Indenter Pile up Sink in h Specimen ∞ ∞ hf ∞ ∞ Principle of indentation Loading state Measurement 1. Continuous measure : Force, Depth, Stiffness Unloading state 2. Post measurement : surface imprint (contact area) Objective : Obtain material property using the measurements

  3. H P S σ E h 0 σy 0 0 ε εy 0 h Work done by L. Charleux (ideal case) For - Isotropic material : amorphous or poly crystal with fine grain - Self symmetric indenter : conical or pyramidal Model analysis (2D FEM) Indentation test Mechanical parameters Mechanical parameters Indentation curve : sy and E : Hardness Surface imprint Obtain - two mechanical parameters - stable hardness L. Charleux, ph.D thesis, Grenoble INP (2006)

  4. Indentation with single crystal metal With Self symmetric indenter : conical or pyramidal Surface imprint Mesured hardness : Anisotropic slip line : Indentation size effect Need 3D modeling for anisotropic behavior and indentation size effect Work done by M. Verdier (2007) Y.Y. Lim and M. M. Chaudhri, Philosophical Magazine A (1999)

  5. Simulation model for indentation Isotropic model (2D, axisymmetric) : Only for ideal (isotropic) indentation anisotropic model (3D,continuum) anisotropic model (3D,discrete) : Constitutive law for anisotropy  crystal plasticity model : Dislocation dynamics coupled with FEM

  6. sphere Berkovich Cone Vickers Scale of indentation Scale mm nm Å mm Indentation Micro-Indentation Nano-Indentation Nano-Indentation Experiment Single crystal (Copper) Single crystal Poly crystal Specimen Indenters Sphere Cone Pyramidal, cube Sphere Cone Simulation (analysis tool) Theoretical analysis (ideal, FEM) Reproducing experiment (single crystal, CPFEM) Reproducing experiment (single crystal, CPFEM) FEM (isotropy) FEM (anisotropy) FEM (size effect) DD MD FEM (anisotropy) Theoretical analysis (size effect, DD&CPFEM) Theoretical analysis (initiation+evolution, DD) FEM (size effect) Theoretical analysis (initiation, MD) Multiscale method ( DD base ) MD

  7. Contents • Part 0 : Introduction • Part I : macro scale : experiments and FEM • Experiments • Crystal plasticity modeling • Finite element simulations • Part II : micro scale : DD simulation • Simulation method • DD simulation of spherical indentation • DD simulation of conical indentation • Discussion & prospective • FEM modeling (size effect) • DD modeling (Size effect, local rotation)

  8. Contents • Part 0 : Introduction • Part I : macro scale : experiments and FEM • Experiments • Crystal plasticity modeling • Finite element simulations • Part II : micro scale : DD simulation • Simulation method • DD simulation of spherical indentation • DD simulation of conical indentation • Discussion & prospective • FEM modeling (size effect) • DD modeling (Size effect, local rotation) Continuum modeling

  9. FEM Experiment Sapphire sphero-conical tip Rigid body tip (identical) Tip angle = 71.2o Tip radius = 3.3 mm Indentation depth 1400 nm hsphere=200 nm Crystal plasticity theory 4 Cu single crystals 5mm × 5mm 5mm × 5mm Velocity control Strain rate control - Experiment Nanoindentation procedure (EXP) Nanoindentation procedure • Objective : To check • Orientation effect • Dislocation density effect • Mean • Force, Stiffness, Contact area and Hardness • Surface displacement (symmetry, pile-up height) Tip 4 Specimens Loading condition

  10. - Experiment Sample preparation for experiment (110), (001), (111) surface orientation Cut by spark erosion from bulk single crystal (high rini) (110) (111) (001) (123), (111low) surface orientation Grown from high purity Cu using Bridgman technique (low rini) (12-3) (-541) (111 low) - 4 surface orientations - 2 initial dislocation densities on (111) orientation

  11. S - Experiment Quantitative results (EXP) Strong effect of initial dislocation density & Weak effect of orientation

  12. 5mm 5mm 5mm 5mm 5mm - Experiment Surface morphologies (AFM) (001) Surface (110) Surface (123) Surface (111) Surface (highrini) (111) Surface (lowrini) Surface morphology strongly affected by Surface orientation To check by FEM modelling

  13. FEM FEM Experiment Sapphire conical tip Rigid body tip (identical) Rigid body tip (identical) Tip angle = 71.2o Tip radius = 3.3 mm Indentation depth Indentation depth 1400 nm 1400 nm hsphere=200 nm Crystal plasticity theory Crystal plasticity theory 4 Cu single crystals 5mm × 5mm 5mm × 5mm 5mm × 5mm Velocity control Velocity control Strain rate control - CP model Nanoindentation procedure Nanoindentation procedure (FEM) • Ingredient : Physical constitutive equations • Dislocation density based model of crystal plasticity • Procedure : • Inverse method to find the best set of parameters Tip 4 Specimens Loading condition

  14. - CP model Crystal plasticity theory Plasticity = activation of many slip systems 12 slip systems for Copper single crystal

  15. - CP model Constitutive law for crystal plasticity Equations Key variable CE Elastic Flow law rini , asp asp, Kg Hardening law + Surface orientation Evolution law L. Tabourot, M. Fivel, E. Rauch, Mater. Sci. Eng. A (1997)

  16. Elastic properties for From literatures Initial dislocation density and Surface orientation From X-ray results Hardening parameters From DD theory - CP model Obtaining variables Unknown M. Fivel, PhD These, (1997) L. Kubin, B. Devincre and T. Hoc, Acta Mater., (2008)

  17. - CP model Implementation in ABAQUS (UMAT / VUMAT) & Validation (Taylor / normal)  Well know three stage curve reproduced

  18. - CP FEM Application to nanoindentation Effect of orientation (Hetero / Same) (001) Surface (110) Surface 3% 5mm 5mm (111) Surface (123) Surface (rini= 1.2×1013/m2) Weak effect on loading curve 5mm 5mm Strong effect on surface displacement

  19. (111) Surface (high rini) 5mm (111) Surface (low rini) 5mm - CP FEM Effect of dislocation densityfor (111) orientation (Hetero / Same) Strong effect on loading curve Optimized initial dislocation density rini= 6.0×1012/m2 for (111low) rini= 1.2×1014/m2 for (111high) What about the other orientations ? Strong effect on surface shape

  20. - CP FEM Comparison of quantitative results (hetero / same) rini= 1.56×1014/m2 (110) rini= 1.20×1014/m2 (111) rini= 1.20×1014/m2 (001) rini= 3.00×1013/m2 (123) rini= 6.00×1012/m2 (111low) Optimized initial densities

  21. EXP 5mm 5mm 5mm 5mm 5mm 5mm 5mm - CP FEM Effect of hardening parametersfor (111) orientation Hetero Taylor Normal Same Weak effect on loading curve Obtained hardening parameter High K Hetero + Same Need to check for other orientation Influence surface morphology

  22. - CP FEM Comparison of Surface morphology (hetero / same) FEM FEM EXP EXP (001) Surface (111) Surface (high rini) 5mm 5mm 5mm 5mm (111) Surface (low rini) (110) Surface 5mm 5mm 5mm 5mm (123) Surface Confirmed hardening parameter 5mm 5mm

  23. q 123 110 001 - CP FEM Comparison of pile up morphology (hetero / same) 111 (lowrini) 111 (highrini) The hardening parameter (hetero/Same) is confirmed quantitatively

  24. - CP FEM Conclusion : best set of parameters Elastic Initial dislocation density and Surface orientation Hardening L. Kubin, B. Devincre and T. Hoc, Acta Mater., (2008)

  25. Summary of EXP and FEM studies • Effect of key variables • Orientation • Weak effect to Force displacement curve • Strong effect on surface morphology (imprint symmetry) • Initial dislocation density • Strong effect on Force displacement curve • Effect to pile up height • Hardening parameters • Weak effect to Force displacement curve • Effect to details of surface morphology and pile up height • Comparison between EXP and FEM with best parameter set  Reasonable both qualitatively and quantitatively • Limits • No size effect Dislocation dynamics

  26. Contents • Part 0 : Introduction • Part I : macro scale : experiments and FEM • Experiments • Crystal plasticity modeling • Finite element simulations • Part II : micro scale : DD simulation • Simulation method • DD simulation of spherical indentation • DD simulation of conical indentation • Discussion & prospective • FEM modeling (size effect) • DD modeling (Size effect, local rotation) Discrete modeling

  27. FS FS -FS Virtual indenter DE =DTOT DDD Virtual indenter DE =DTOT-DDD DDD DTOT DTOT FDD + FDD = + = FFEM SFEM SDD FFEM SFEM SDD FTOT= FFEM+ FDD FTOT= FFEM+ FDD Bulk Infinite region Bulk Bulk Infinite region Bulk DTOT : Total indenting depth DDD : depth from dislocation field DE : depth from Elastic field SFEM : Elastic stress field DTOT : Total indenting depth DDD : depth from dislocation field DE : depth from Elastic field SFEM : Elastic stress field FTOT : Total applied load FDD : Load from dislocation field FFEM : Load from Elastic field FS : Surface load from dislocation field FTOT : Total applied load FDD : Load from dislocation field FFEM : Load from Elastic field FS : Surface load from dislocation field 100nm 100nm 100nm - Simulation method Coupling (Superposition) method Coupled FEM DD E. van-der-Giessen, A. Needleman, Mater. Sci. Eng. , (1995)

  28. START CALCULATION Finite Element Method (CASTEM) Dislocation dynamics (TRIDIS) Get dDD, dFEM, FS,DD from previous step Update dDD, FS,DD Impose B.C. at top Surface Solve an elastic FEM problem Remove over constraints Over constraint ? Yes Nucleate ? No Yes Nucleation (it could be failed) No Equilibrium ? Dislocation Dynamics Steps Recalculate FEM ? Nucleate ? No Yes Yes Yes No No Update dDD, FS,DD END CALCULATION - Simulation method Algorithm of the coupled simulation Dislocation dynamics  • dislocation multiplication • dislocation nucleation ?  Need a criterion ! What (shape) ? When (condition) ?  MD simulations

  29. 2.5 First Generation of Dislocation loops Ni1 Ni2 Ni3 Ni4 2.0 Ni6 Ni8 1.5 F [mN] 1.0 0.5 0.0 2.0 0.0 0.5 1.0 1.5 depth [nm] - Simulation method Nucleation criterion #1 : MD global criterion Shape of Nucleation (MD,111) Master curve (MD,111) What : 3 Prismatic loops When : Global criterion (Master curve) Good : criterion without any experimental results Weak : MD cannot tell the force for deeper indentation depth Need a nucleation criterion without master curve

  30. START CALCULATION Finite Element Method (CASTEM) xind xDD,Nucle Dislocation dynamics (TRIDIS) Get dDD, dFEM, FS,DD from previous step Nucleation (it could be failed) Over constraint ? Impose B.C. at top Surface Solve an elastic FEM problem Remove over constraints Yes Equilibrium ? No Dislocation Dynamics Steps No Yes Update dDD, FS,DD END CALCULATION - Simulation method Nucleation criterion #2 : GND criterion Flow Chart Deformation and GND Nucleation imposed by indentation depth Good : without any experimental results Good : without any master curves W. D. Nix, H. Gao, J. Mech. Phys. Solids, (1998)

  31. Copper single crystal (111 surface) Sphere(r=150nm) Cone(angle = 71.2o) 50nm 50nm - Simulation method Calculation Parameters Specimen Tip geometries 1. Global criterion : Force controlled Nucleation 2. GND criterion : Depth controlled Nucleation Nucleation method Master curve from MD (sphere) or Exp (cone) Cross-slip probability : For indentation ?

  32. - Spherical indentation (111) Contents • Part 0 : Introduction • Part I : macro scale : experiments and FEM • Experiments • Crystal plasticity modeling • Finite element simulations • Part II : micro scale : DD simulation • Simulation method • DD simulation of spherical indentation • DD simulation of conical indentation • Discussion & prospective • FEM modeling (size effect) • DD modeling (Size effect, local rotation) Discrete modeling

  33. (1 0- 1) (1 0- 1) (1 0- 1) (1-2 1) (1-2 1) (1-2 1) (1 1 1) (1 1 1) (1 1 1) 150nm 150nm 230nm (1-2 1) (1-2 1) (1-2 1) - Spherical indentation (111) Dislocation evolution (MD global crit. + no cross-slip) 5nm depth 10nm depth 60nm depth s > line tension  F-R source Contact area ↑  loop length ↑ Loop Length ↑ Line tension ↓ No space to nucleation Nucleation only (similar to MD) Nucleation and Frank-Read source Frank-Read source only

  34. - Spherical indentation (111) Force displacement curve MD global crit. + no cross-slip GND crit. + no cross-slip MD global crit. Nucleation allow Nucleation forbidden GND crit. MD master curve MD master curve MD global crit. + Cross-slip condition GND crit.  closer to MD prediction MD master curve Cross-slip  Deviation delayed

  35. 100nm 100nm 100nm (1 0 -1) (1 1 1) (1 1 1) (1 0 -1) (1 0 -1) (1 1 1) 150nm 150nm 150nm (1-2 1) (1-2 1) (1-2 1) (1-2 1) (1-2 1) (1-2 1) - Spherical indentation (111) Cross-slip effect (30nm depth) Easy cross-slip No cross-slip Hard cross-slip Cross-slip↑ def. homogeneous ↑, Glissile loops ↓, Plastic zone size ↑

  36. No cross-slip Hard cross-slip Easy cross-slip 230nm (1 0- 1) (1 0- 1) (1 0- 1) (1-2 1) (1-2 1) (1-2 1) 230nm 230nm 230nm 230nm 230nm - Spherical indentation (111) Cross-slip effect after unloading Before unloading (60nm depth) After unloading (60nm depth) Cross-slip↑ more irreversible micro structure

  37. - Conical indentation (111) Contents • Part 0 : Introduction • Part I : macro scale : experiments and FEM • Experiments • Crystal plasticity modeling • Finite element simulations • Part II : micro scale : DD simulation • Simulation method • DD simulation of spherical indentation • DD simulation of conical indentation • Discussion & prospective • FEM modeling (size effect) • DD modeling (Size effect, local rotation) Discrete modeling

  38. - Conical indentation (111) Nucleation criterion for conical indentation Hardness Total Force MD master curve, sphere Exp. master curve, Berkovich MD global crit. GND crit. • Two phase behavior ( linear to parabolic) Hardness : Drop down  Stable Exp. global crit.  single behavior (  linear response) Exp. global crit.  long range decreasing

  39. (1 0- 1) (1 0- 1) (1 0- 1) (1-2 1) (1-2 1) (1-2 1) (1 1 1) (1 1 1) (1 1 1) 230nm 230nm 230nm 230nm 230nm 230nm (1-2 1) (1-2 1) (1-2 1) - Conical indentation (111) Dislocation Structure (60nm depth) GND crit. MD global crit. Exp. global crit. Prismatic loops & Glissile loops Prismatic loops & Junctions

  40. b1 - b2 b3 (1 1 2) (1 -1 0) - Conical indentation (111) Coplanar Junction : unexpected behavior !! Coplanar Junctions b3=(1 -1 0) b2=(1 0 1) b1=(0 1 1) -Activate dislocations perpendicular to loading direction -Mobile dislocations  Immobile dislocations 230nm

  41. Summary of DD results • Dislocation evolution source : Size effect • Nucleation in size effect phase • Frank-Read source in no size effect phase • Cross slip • Coplanar junction (>50nm, Conical indenter with Exp. global crit.) • Dislocation structures • Prismatic loops • Increase by nucleation • Glissile loops • Increase by Frank Read source • Decrease by Cross slip, Coplanar junction, Unloading

  42. Contents • Part 0 : Introduction • Part I : macro scale : experiments and FEM • Experiments • Crystal plasticity modeling • Finite element simulations • Part II : micro scale : DD simulation • Simulation method • DD simulation of spherical indentation • DD simulation of conical indentation • Discussion & prospective • FEM modeling (size effect) • DD modeling (Size effect, local rotation)

  43. FEM (rini) FEM (lower rini) EXP - FEM modeling (Size effect) Problem of CPFEM model :Size effect Force curve of (111) Surface Problem !!! : Different rini for deeper indentation depth Solution : Fit the curvature How ? : Using Generalized plasticity theory (Ex. Strain gradient theory) Access to size effect Curvature of EXP < Curvature of FEM

  44. - FEM modeling (Size effect) Simplified strain gradient theory dislocation theory strain gradient theory Simplified theory (proposed) = = : SSD : GND PROBLEM !! Very high calculation cost Usual internal stress Extra stress  sextra↓ if indentation depth ↑ - No extra parameter and extra degree of freedom - Easy to implement in FEM

  45. - FEM modeling (Size effect) First FEM results of simplified model (001) Surface orientation • reproduce indentation size effect, qualitatively • Need detailed verification

  46. (1 1 1) (1 1 1) 150nm 230nm (1-2 1) (1-2 1) - DD modeling (Size effect) Origin of Size effect (Microscale) Prismatic loops dominant Glissile loops dominant • Dislocation structures beneath indenter • Size effect phase : Prismatic loops dominant • Stable hardening phase : Glissile loops dominant Dislocation dynamics can help understanding size effect

  47. - DD modeling (local rotation) Link between size effect and local rotation • Local rotation with scale (Simulation) • Microscale DD : • Size effect theory : • Crystal plasticity : • Local rotation with scale (Experimental, EBSD) κ : lattice torsion curvature ANR CATSIZE project with Samuel Forest (CMR) : Connenction bet. Cosserat & DD M. Rester, C. Motz, R. Pippan, Acta Mater., (2008)

  48. Thank you for your attention

  49. Analysis of nanoindentation size effect based on Crystal Plasticity and Dislocation Dynamics Contents • Part 0 : Introduction • Part I : macro scale : experiments and FEM • Experiments • Crystal plasticity modeling • Finite element simulations • Part II : micro scale : DD simulation • Simulation method • DD simulation of spherical indentation • DD simulation of conical indentation • Discussion & prospective • FEM modeling (size effect) • DD modeling (Size effect, local rotation)

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