C. Keller, L. Duchêne, M. Afteni, E. Hug, A-M Habraken 2-4 June 2010 ICACM Paris France

Dimension reduction ? Ni micropillars, Ø=1 µm (Dimiduk 2005) “ smaller is stronger “ Role played by dislocation sources on surface Classical mechanical behavior Polycrystal or large single crystal Introduction

Meso scale -> microsystems scale Characteristics of small parts: dimensions lower than 500 µm metallic alloy with complex microstructure (second phase, precipitates…) Geiger et al. CRIP 2001Vollertsen et al. JMPT 2004 Introduction 0.5 mm

Forming processes and industrial use may be problematic Forming process: Know-how for bulk parts cannot be used; turn/cast necessary; low production rates/high costs. Reliability: reduced reliability; unexpected fracture; can lead to security problem. Small axis  18 step process Geiger et al. CRIP 2001 Airbag sensor: inflate start without accident Problem linked to our weak knowledge of the mechanical properties Introduction

 well known mechanical properties  simple microstructure  used in Micro-Electro-Mechanical systems (MEMS) Fundamentals aspects Application to microforming Multi-scale analysis Experimental/Numerical study of miniaturization Mechanical behavior of nickel Why nickel?

Experimental study PhD thesis C. Keller, Supervisor E. Hug, CRISMAT Lab, Caen/France Thickness between 10 µm and 3.2 mm and constant grain size 100 µm t Strong mechanical behavior modification due to the decrease of t/d ratio Tensile tests for Ni sheets

Experimental study Three kinds of behavior depending on grain size and thickness Keller et al., Int. J. Plasticity. Submitted PhD thesis C. Keller, Supervisor E. Hug, CRISMAT Lab, Caen/France Tensile tests for Ni sheets

Experimental study Statistical TEM analysis of dislocation cells  stress gradient t/d=2,5; ε =0,1 core, Φ=1,25 µm50 µm below surface: Φ=1,58 µm Keller et al., Mechanics of Materials, 2010 Tensile tests for Ni sheets PhD thesis C. Keller, Supervisor E. Hug, CRISMAT Lab, Caen/France

Experimental study Synthesis Surface effects enhanced by a decrease of polycrystalline character What are the characteristics of the surface effects (deep of stress gradient…)? What is the role played by dislocations (escape through free surfaces…) ? How to model the mechanical behavior of thin samples (prediction of the behavior) ? multiscale modeling with strain gradient crystal plasticity is needed

Numerical modeling “Non-local crystal plasticity model with intrinsic SSD and GND effects”, Evers, L.P., Brekelmans, W.A.M., Geers, M.G.D.: J Mech. Phys. Solids 52 (2004)  Modified L. Duchêne + C. Keller Features of the model:  Based on dislocation glide on slip systems  Accounts for dislocation densities  Distinction between SSD (statistically stored dislocations), GND (geometrically necessary dislocations)  Visco-plastic slip law including a back-stress accounting for internal stresses due to GND Strain gradient crystal plasticity model

slip rate on slip system α slip resistance for slip system α SSD density rate of slip system α mean free path of dislocation on slip system α Numerical modeling Strain gradient crystal plasticity model Classic equations for crystalline plasticity

Numerical modeling Strain gradient crystal plasticity model Specific equations for GND and backstress GND density rate of slip system α, f depends on the screw or edge dislocation character Formulation of the backstress involved by GND. g function depends on the screw or edge character of the dislocations Size effects reproduced by the model, 36 parameters to indentify

Numerical modeling Strain gradient crystal plasticity model F.E. implementation Starting equations of the strong form:  Equilibrium  GND densities evolution laws  3D coupled element with 20 nodes and 8 IP  Nodal DOF: - Displacements (3) - GND densities (18)

Numerical modeling Identification for nickel Most of parameter values are obtained from literature  Nickel crystallograpical characteristics (µ, b, elastic tensor…)  handbook;  Dislocation interaction matrix  work of B. Devincre with DDD; Other parameters identified by simulations of single crystal tensile curves Three different orientations Orientation A [001] (X.Feaugas) Orientation B [111] (A.W.Thompson,1976) Orientation C S-G (P.Haasen,1956 )

Numerical modeling Identification acceptable but not perfect. Many reasons:  Experimental orientations given +/- 2°  strong influence on simulations  Old experimental tests  Difference of environment for single glide orientations (test realized in air, simulations correspond to vacuum) Identification for nickel Feaugas 2009 Thompson 1976 Haasen 1956

Numerical modeling Application to single crystals Preliminary surface effect study Tensile test simulation for different thickness single crystals Single glide orientation Effects similar to those observed experimentally by Mughrabi (Phys. Stat. Sol. 1971) and Fourie (Phil. Mag. 1967) on Cu single crystals Stage II delayed if thickness decreases  surface effects

Numerical modeling SSD distribution into the median cross section profile along the slip direction Dislocations can emerge through free surfaces Single glide orientation, stage I Core region thickness decrease  reduction of core regions Softening effect of free surfaces deep of gradient depends on dislocation mean free path Keller et al., J. Mech. Phys. Sol. To be published Application to single crystals

dislocations are blocked in case of hard layer Numerical modeling Effect of surface hard layer 001 orientation Free surfaces Hard layer Strengthening effect of free surfaces Slip directions profile along the vertical slip direction Application to single crystals

Numerical modeling 12 elements / grain, 300 µm edge grain, grain orientations  EBSD Effect of t/d ratio correctly reproduced by the model Application to polycrystals

Numerical modeling 12 elements / grain, 300 µm edge grain, grain orientations  EBSD t/d=2, median cross section Strong stress gradients, surface grain affected on 2/3 grain size Keller et al., Metal Forming 2010 Application to polycrystals profile along the line

Numerical modeling New strategy of modeling for metal forming 2/3 surface grains affected composite modeling for metal forming: 2 elastoplastic constitutive laws Surface constitutive law applied for distance ≈ 2/3 equivalent grain size below free surface

Numerical modeling New strategy of modeling for metal forming Application to tensile tests Keller et al., Numiform 2010  surface constitutive law identified from experimental tensile tests of thin samples (t/d<1)  core constitutive law identified from experimental tensile tests of bulk samples (t/d=27) Simulations with elastoplastic laws

Numerical modeling Application to micro deep drawing F.E. modeling strategies A. Modeling with 2 constitutive laws (composite model) B. Analytical Mixture modeling: C. Classical bulk modeling (1 constitutive law )

Numerical modeling Application to micro deep drawing, t=250 µm, punch radius: 2.5 mm “surface effect” approach mixture approach Stress distribution modified Keller et al., Numiform 2010 Application to micro deep drawing

Numerical modeling New strategy of modeling for metal forming Application to micro deep drawing Prediction of damage, Cockroft-latham criterion Damage distribution and maximal value modified mixture approach “surface effect” approach

Numerical modeling New strategy of modeling for metal forming Application to dome test Thickness: 0.1; 0.2; 0.3 and 0.4 mm, punch radius: 4.8 mm Force prediction depends on strategy, need experimental validation Keller et al., Metal forming 2010

Conclusions  Miniaturization effects governed for meso-scale by free surfaces  Strong stress gradients appear and must be taken into account  Composite approach of modeling is pertinent and better reproduce stress and damage distribution Perspectives  Surface effects must be investigated for multi-axial loading (decrease of dislocation mean free path)  Experimental validation of composite approach: dome test (Singapour/SIMTECH) and deep drawing (Galati/Romania)