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 2001 Vollertsen 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

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 µm 50 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 studySynthesis

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

112SSDcSSD y

bLα α α

αρ ρ γ⎛ ⎞= −⎜ ⎟⎝ ⎠

&&

∑ ∑+=

ξ ξ

ξαξξαξ

α

ρρ GNDSSD HH

KL

slip rate on slip system α

SSDGNDsbAAα αξ ξ αξ ξ

ξ ξ

μ ρ ρ= +∑ ∑ slip resistance for slip system α

SSD density rate of slip system α

mean free path of dislocation on slip system α

Numerical modelingStrain gradient crystal plasticity model

Classic equations for crystalline plasticity

Numerical modelingStrain gradient crystal plasticity model

int2/esGNDµbRg α

ασ ρ⎛ ⎞= ∇⎜ ⎟

⎝ ⎠∑

Specific equations for GND and backstress

GND density rate of slip system α, f dependson 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 modelingStrain 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 modelingIdentification 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 modelingApplication 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 orientationFree surfaces

Hard layer

Strengthening effect of free surfaces

Slip directions

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 modelingNew 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 modelingNew 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 modelingApplication 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 )

.. surf coresurfcore

tottot

V VVV

σ σ σ= +

Numerical modeling

Application to micro deep drawing, t=250 µm, punch radius: 2.5 mm

“surface effect” approachmixture approach

Stress distribution modified

Keller et al., Numiform 2010

Application to micro deep drawing

Numerical modelingNew 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

0

f

eqeqdCε

σ ε =∫

“surface effect” approach

Numerical modelingNew 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)