· 5 . what do we need ? • numerically discretize newton’s second law • provide contact laws...

Mechanics of Materials with Discrete Element Simulations

Christophe L. Martin

[email protected]

David Jauffres

Zilin Yan, Denis Roussel

mailto:[email protected]

Introduction to a ‘new’ method to treat mechanics of materials

Give enough elements to decide if Discrete Element simulations can be useful to you

Give application examples that relate to materials science

Aims of this lecture

2

What are Discrete Element simulations ?

A bit of history

Main principles of Discrete Element simulations

Contact kinematics

Force models (elasticity – adhesion – sintering- bonds)

Application examples

Contents

3

4

DEM = Discrete Element Method (sometime Distinct Element Method)

What a DEM code is supposed to do ?

model the motion of an assembly of discrete bodies


≠

‘continuum mechanics’

Finite Element Method, FEM

bodies interact through their

contacts

‘discrete element mechanics’ Ft

Fn

DEM: contact forces (and sometimes rolling resistances) are transmitted

5

What do we need ?

• Numerically discretize Newton’s second law

• Provide contact laws to model inter-bodies forces

normal forces

tangential forces

rolling resistance (optional)


exti

jI →= +∑i j iθ Γ Γ

, mass, inertia matrix

, translation acceleration, angular acceleration on force exerted by on

external force (typically gravity) on moment exerted by on gravity cen

exti

m I

ij i

i→

→ →

=

==

==

i i

j i

j i j i

x θF

FΓ F

ter of

moment exerted by on gravity center of ext exti i

i

i=Γ F

overlap or indentation, δn Ft

Fn

δn

exti

jm →= +∑i j ix F F

6


Several approaches for modeling the motion of an assembly of bodies

• molecular dynamics (potential instead of forces)

LAAMPS, ESPResSo, …

• DEM (Molecular dynamics like)

LIGGGHTS (based on LAMMPS), Yade, dp3D

PFC, EDEM

easy to code

large computation times

numerical scheme is conditionally stable (explicit numerical scheme)

contacts between bodies are smooth

7


Several approaches for modeling the motion of an assembly of bodies

Contact dynamics: contacts between bodies are non-smooth

LMGC90 (Montpellier)

Perfectly rigid bodies

not easy to code (and to understand)

numerical scheme is unconditionally stable (implicit numerical scheme)

Event-driven method

specific to granular gases

no enduring contacts, binary collisions

A bit of history

8

First MD simulation of a simplified biological folding process [1975]

Geomaterial community uses MD like simulations [1979]

Application to powder metallurgy by Prof. Shima [1994]

Application to sintering by Veringa [1994] and Parhami and McMeeking [1998]

Application to fragmentation by Kun and Herrmann [1999] Levitt, M. & Warshel, A. Computer simulation of protein folding. Nature 253, 694–698 (1975).

P A Cundall, O D L Strack. A discrete numerical model for granular assemblies. Géotechnique 29, 47-65 (1979).

J Lian, S Shima. Powder assembly simulation by particle dynamics method. International Journal for Numerical Methods in Engineering 37, 763-775 (1994).

W J Soppe, G J M Janssen, B C Bonekamp, L A Correeia and H J Veringa. A computer-simulation method for sintering in 3-dimensional powder compacts. J. Mater. Sci. 29, 754 - 761 (1994).

F Parhami, R M McMeeking. A network model for initial stage sintering. Mech. Mater. 27, 111-124 (1998).

Kun, F. & Herrmann, H. Transition from damage to fragmentation in collision of solids. Phys. Rev. E 59, 2623–2632 (1999).

A bit of history

9

main evolutions of DEM

2D 3D

Larger number of particles

More physics at the contact length scale

Using DEM to model continuum mechanics

Heavy parallelization of codes (MPI, openmp)

Main principles of DEM

10

Each particle is represented by a sphere (for simplicity, not necessity)

The motion of each particle is dictated by the 2nd law of Newton

The physics of the problem is embedded in the contact law

New contacts appear and contacts break naturally


11

• Contact detection

• calculation of contact forces (Normal + Tangential)

• calculation of the total force applied to each particle

• calculation of accelerations due to applied forces

same procedure for rotation and moments

• new positions, and velocities of particles

F

FN FT


12

Contact detection N particles ⇒ operations : Huge computational effort !

Verlet-Neighbor list: list of ‘potential contacts’ , to be updated

keep a list of potential contacts (to update) still O(N²) operation !

2

4

3

5

6

1

rc

Particles 2, 3 and 4: ‘potentially’ in contact with particle 1

• Put particles in boxes (box size > particle size )

• Visit only the particle’s box and adjacent boxes

Much less efficient if large distribution of sizes more involved methods necessary

operation !

Linked cell method


13

Numerical integration

ext toti i

jm →= + =∑i j ix F F F

Verlet, Velocity Verlet, leap-frog, gear predictor-corrector … algorithms

Velocity Verlet more commonly used

( ) ( ) ( ) ( ) 212

toti

i i ii

F tx t t x t x t t t

m+ ∆ = + ∆ + ∆

time stept∆ =

( ) ( ) ( ) ( )12

tot toti i

i ii

F t F t tx t t x t t

m+ + ∆

+ ∆ = + ∆

need to store positions, velocities and accelerations of particles


14

Time step consider the Kelvin-Voigt model (viscous damping)

Ri

a

Rj

, n nδ δ

kn

ηn kn=normal stiffness (N.m-1)

ηn= normal damping coefficient (N.m-1.s), not a viscosity

two frequencies characterize this harmonic model

n n n nk δ η δ= − −nF n n

0

2 2 ic

n

mtk

π πω

= =

typical duration of a contact (most rapid frequency):


15

Time step

Ri

a

Rj

, n nδ δ

kn

ηn

typical duration of a contact:

convergence one contact lasts several time steps

ic

n

mtk

π πω

= =

Thornton, C. & Antony, S. J. Quasi-static deformation of particulate media. Proc. Roy. Soc. Lond. A A 356, 2763–2782 (1998).

i

n

mt fk

∆ ≈

t∆

f of the order of 0.1 – 0.01

for quasi-static simulations, we can renormalize masses (increase by several orders of magnitude), or decrease stiffness

need to check if equilibrium is reached


16

Damping (dissipate energy within the system)

consider the Kelvin-Voigt model (viscous damping)

Ri

a

Rj

, n nδ δ

kn

ηn kn=normal stiffness (N.m-1)

ηn= normal damping coefficient (N.m-1.s), not a viscosity

two frequencies characterize this harmonic model

n n n nk δ η δ= − −nF n n

2n i nm kη <

0 0 , 2

n n

i i

km m

ηω µ= =

dictates ‘correct’ damping

dictates the time step


17

Damping other methods to damp the system:

• apply a force that opposes the motion of each particle (Stokes’s drag)

0

jm η→= −∑i j i ix F x

j→∑ j iF

1j j j

η→ → →= −∑ ∑ ∑j i j i j i iF F F x

• apply a global damping proportional to :

Useful to dissipate energy within the system, but to be used with care !

18

Main principles of DEM Dimensional analysis

Which category your process/material falls in ?

for a confinement pressure P and strain-rate

the inertial number (for a 3D system):

m particle mass of size d

mIPd

ε=

ε

dynamics I > 0.1

I 0 : quasi-static limit (equilibrium state at any time)

quasi-static I < 0.001 (beware with renormalization of mass)

Agnolin et al., 2007

19

Main principles of DEM Boundary conditions

Rigid walls

specific structure close to walls

Periodic conditions

extend system boundaries to ∞

Periodic cell face

Image particle

L

simple to code, for distance xij:

if(xij>0.5L) xij=xij-L

if(xij<-0.5L) xij=xij+L

simple way to get rid of boundary effects

i j

20

Main principles of DEM Imposing strains (with periodic conditions)

L0 z0

z0+∆zaff L1

affine displacement of particles

zz z tε∆ = ∆

L1

z0+∆zaff+∆zrearr

displacement of particles to reach force equilibrium (verlet)

21

Main principles of DEM Stress calculations

• when objects bounds the part:

object

1i j

jF n

S= ∑ijΣ

Fn Ft

( )contacts

1n tF F

V= +∑ij i i jΣ n t l

Weber, J. Recherches concernant les contraintes intergranulaires dans les milieux pulvérulents. Bull. liaison des Ponts Chaussées 1–20 (1966).

• when average stress is computed over volume V (ok for equilibrated configurations) Love Formula:

contact kinematics

, ,c j j j c j= + ∧v v ω r , , ,c rel c j c i= −v v v, ,c i i i c i= + ∧v v ω r

,c iriω

= position vector of a particle

= translational velocity

= angular velocity of a particle

= position vector from a particle center to the contact point

= velocity of particle at the contact point

= velocity of particle j relative to particle i at the contact point

= normal unit vector (from i to j)

= tangential unit vector (pointing in the same direction as )

ir

iv

,c relv

nt

defined in a global frame of reference:

,c iv

,c relv

cross product (x)

‘produit vectoriel’

x

z

y O

Ri

Rj x

x

n

jviω

t

,c jr

iv

jωjr

ir,c ir

22

( ) ( ), , ,. .c rel c rel c rel= +v v n n v t t

( )( )

, ,

, ,

.

.c rel c rel

c rel c rel

−=

−

v v n nt

v v n nj i

j i

−=

−

r rn

r r

contact kinematics

( )n j i i jR Rδ δ= = − − +r rif contact, δ <0, but may be used in the following

overlap or indentation:

δx

z

y O

Ri

Rj x

x

nt

,c ir,c jr

jrir

23

( )( )

, ,

, ,

.

.c rel c rel

c rel c rel

−=

−

v v n nt

v v n nj j

j j

−=

−

r rn

r r

tc : initial contact time

effective radius will be encountered everywhere * i j

i j

R RR

R R=

+

used for force calculation , , ,t tvδ δ δ , .t c relv = v t dc

t

t ttv tδ = ∫

Particles interact through their

contacts

Ft

Fn

Discrete Element Method, DEM: contact forces (and sometimes resisting moments) are transmitted

24

soft particle force model in DEM: particles are authorized to overlap each other (≠ contact dynamics) the overlap contact force determination

analytical force models in the literature for spheres particles remain spherical (or indented spheres)

overlap or indentation, δn

Ft

Fn

δn

Forces

soft-particle DEM is limited to small deformations/overlaps! :

not true but be careful about the validity limit of the model

in general, the contact force expression Fn(i,j) does not depend on the existence of contact j,k (pair interaction)

contact forces resolved into a normal and a tangential component

normal force does not depend (in general) on the tangential force

tangential depends (in general) on the normal force

overlap or indentation, δn

Ft

Fn(i,j) i

j

k

Fn(j,k)

25

Forces

force models weight no rotation

m=mass

g = gravity acceleration

fluid forces (Stokes force, …)

R=particle radius

v=particle velocity

not developed in the following

repulsive contact forces (elasticity, plasticity, …)

cohesive contact forces (adhesion, bonding, capillarity, …)

tangential contact forces (friction, stick-slip, …)

26

m=gF g

6 Rπ= −dF v

force models

elasticity (Hertzian contact)

elasticity + adhesion

linear spring

plasticity (perfect plasticity + hardening)

tangential forces

sintering

bonding (+ fracture)

capillarity

27

elasticity (Hertzian contact)

* i j

i j

R RR

R R=

+

22

*

111 ji

i jE E Eνν −−

= +

Ei=Young’s modulus, particle i

νi=Poisson’s ratio, particle i

[Johnson, 1987]

δn

Fn

stress

"On the contact of elastic solids" ("Ueber die Berührung fester elastischer Körper") by Heinrich Hertz, 1882

28

32

31 22** * *2*

4 43 3nF E R E R

Rδ

δ

= =

adhesion between elastic spheres

adhesion (or cohesion if same material) due to competing forces of attraction and repulsion between individual atoms or molecules at surfaces

becomes non-negligible for small particles

force to separate two bodies ?

difficult to measure forces directly:

measure the work necessary to separate the two surfaces

29

new free surface created work required to separate the two surfaces

w= work of adhesion per unit area (J.m-2)

γ= surface energy (J.m-2)

γ1,γ2= intrinsic surface energies (J.m-2)

γ1,2= interface energy (J.m-2)

( )1 2 122w γ γ γ γ= = + −

1

2

1

2

adhesion between elastic spheres JKR: Johnson, Kendall, Roberts

Cambridge

DMT: Derjaguin, Muller,Toporov

Russian Academy of Sciences

DMT derivation assumes that attractive stresses do not bring about deformation

JKRc

aa a=

pull-off force

30

‘large soft particles’ ‘small hard particles’

2w γ=w= work of adhesion

γ= surface energy

pull-off d 0

d

JKRnFa

= →

[Johnson, 1971; Derjaguin, 1975; Tabor, 1977; Maugis, 1992]

contact size still given by:

2 *na R δ=

* 3* 3

*

4 2 23

JKRn

E aF wE aR

π= −* 3

**

4 23

DMTn

E aF wRR

π= −

*32

JKRcF wRπ= − *2DMT

cF wRπ= −

13*2

*

98

JKRc

wRaE

π =

0DMT

ca =

adhesion and size effect

2w γ=w= work of adhesion

γ= surface energy

x

z

y O

Ri

Rj

nt

,c ir,c jr

jrir

[Johnson, 1971; Derjaguin, 1975] 31

DMT: Derjaguin, Muller,Toporov * 3

**

4 23

DMTn

E aF wRR

π= −

( )y ycl = −j ir r

xcF = x component of the force for contact c

= y component of the branch vector for contact c

,1 x y

xy n c ccontacts

F lV

σ = ∑

31 R2R RHertz

no size effect

,1 x y

xy n c ccontacts

F lV

σ = ∑

31 R R RDMT

(or JKR) size effect, fracture stress is 1 R∝

average stress

1 x yxy c c

contactsF l

Vσ = ∑

2 *na R δ=

Tangential Force models (friction)

δ

Fn

T

T

Ft

T

k nFµs nFµ

often µs=µk

in the models

stick slip

stick : small relative motion

slip : large relative motion

32

friction forces: resist the relative motion of the two surfaces

Coulomb Force widely used

µs=static friction coefficient

µk=kinetic friction coefficient

s n

tk n s n

T T FF

F T Fµ

µ µ<

= ≥

[Johnson, 1985; Vu-Quoc, 1999; Silbert, 2002; Gilabert, 2007]

Tangential Force models

accumulated tangential relative displacement

33

or in incremental form:

( )

d

t t n

n t n

k t F

F F

µ

µ µ

= − < ← ≥

t c,rel

tt

t

dF v FFF FF

t t n

n t n

k k F

F k F

µ

µ µ

− <= − ≥

t t

t tt

t

Δ ΔF Δ Δ

Δd

c

t

tt= ∫t c,relΔ v

tangential force opposes the accumulated relative tangential elastic displacement

back to zero when contact vanishes

only the repulsive part of Fn should be used (not the adhesive part for example)

kt can be related to elastic properties (a contact radius)

tΔ

*8tk G a=

Application: Towards Random Close Packing

34

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65Packing fraction, φ

Γ/Γ 0

10-0

10-1

10-2

10-3

10-4

10-5

10-6

10-71.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00


Γ/Γ

0

10-0

10-1

10-2

10-3

10-4

10-5

10-6

10-7

No friction, no adhesion Elastic interactions All particles of the same size

Densification at a small constant pressure (P/E=10-7)

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00


Γ/Γ 0

10-0

10-1

10-2

10-3

10-4

10-5

10-6

10-7

Adhesion + friction, w=1.0 J.m-2, 2R=1µm Packing fraction D

Nor

mal

ized

str

ain-

rate

Friction µ=0.2

Random close pack

D0=0.635 Z0 ≈ 6

D0 ≈ 0.3 Z0 =0

contact laws: Hertz + DMT adhesion + Coulomb friction

35

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00


Γ/Γ 0

10-0

10-1

10-2

10-3

10-4

10-5

10-6

10-7

Packing fraction D

Nor

mal

ized

stra

in-r

ate

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0 0.1 0.2 0.3 0.4 0.5aggregate density

rela

tive

dens

ity

P = 0.05 MPa

Pack

ing

frac

tion

dendritic Porous aggregate HCP

0!iZ =

contact laws: bonds (with resisting moments) + Coulomb friction

Application: Towards Random Close Packing

specific interactions at high temperature (T > 0.5Tf)

decrease of the total surface energy of the particulate system driving force for sintering

thermally activated mechanism time dependent (viscous effects)

Sintering

[Petzow , 1976; Bouvard, 1996, Martin, 2006]

Planar array of copper spheres at 950°C

as

δn

Rj

Ri

36

Sintering of two spheres

37

24

89dd

bbs

s s

h Rat a

∆γ

∆= − σ

Major mechanisms : grain boundary (limiting) and surface diffusion

• Bouvard & McMeeking (1996) γs = surface energy

as = contact radius Sintering term

Additional stress term

b b bDkT

δΩ∆ =

D Bouvard, R M McMeeking. J. Am. Ceram. Soc. 79, 666-672 (1996).

4 2

8d 4 1 co 8s sind 2 2

bs s

b

sst a ah R a∆ ψ ψ = − + γ −

∆

σ

• Parhami & McMeeking (1998)

Sintering term Additional stress term

F Parhami, R M McMeeking. Mech. Mater. 27, 111-124 (1998).

aeq

h

R

R

σ

ψ = dihedral angle

sin2eqa R ψ

=

ψ


38

24

89dd

bbs

s s

h Rat a

∆γ

∆= − σ



Sintering term


b b bDkT

δΩ∆ =

as

h

R

R

σ

as = contact radius

Sintering term

4 d 9 8 d 8s

ss

b

a hN Rt

π π γ= −∆

Normal viscous term


39

24

89dd

bbs

s s

h Rat a

∆γ

∆= − σ



Sintering term


b b bDkT

δΩ∆ =

as

h

R

R

σ

Sintering term

4 d 9 8 d 8s

ss

b

a hN Rt

π π γ= −∆

Normal viscous term

as = contact radius

2 2 d8 d

ss

b

a R uTt

πη= −∆

Tangential viscous parameter η

+

R Raj, M F Ashby. Metall. Trans. 2, 1113-1123 (1971).


40

24

89dd

bbs

s s

h Rat a

∆γ

∆= − σ


• Bouvard & McMeeking (1996)

• Neck growth: Coble’s model:

γs = surface energy

Sintering term


b b bDkT

δΩ∆ =

as = contact radius

d dd d

s

s

a R ht a t

=

R L Coble. J. Am. Ceram. Soc. 41, 55-62 (1958).

as

h

R2

R1

sNsT

1 2

1 2

2 2 R RR RR R

∗⇔ =+

F Parhami, R M McMeeking, A C F Cocks, Z Suo. Mech. Mater. 31, 43-61 (1999)

Sintering of Multi-Layer Ceramic Capacitors

41

before heterogeneities

after 2µm

2µm

nickel BaTiO3 (BTO)

after

(#2) E1

E2

D1

D2

E3

D3

D4

E4

D-dielectric E-electrode

Z Yan, O Guillon, S Wang, C L Martin, C-S Lee, D Bouvard. Appl. Phys. Lett. 100, 263107 (2012).

• few particles in layer thickness

• composite system

• constrained sintering

Ex-situ X-ray –nCT at Argone

0 50 100 150 200 250

-0.020

-0.016

-0.012

-0.008

-0.004

0.000

0.004

Dens

ificat

ion

rate

(%/m

in)

Sintering time (min)

Ni BaTiO3

(b)0 50 100 150 200 250

0

200

400

600

800

1000

1200

Tem

pera

ture

(oC)

Ni constrained by non sintering BTO for T < 1100°C

Ø 10µm

1.2µm


z r

Ni

(d)

42

free sintering

2 µm

~25000 particles

Discrete simulations: only Ni sintering from 700 to 1150°C discontinuities

constrained

BaTiO3

1 µm

~25000 particles

2.4µm 1.2µm


z r

Ni

43

constrained

Discrete simulations: only Ni sintering from 700 to 1150°C discontinuities

2 µm

1 µm

BaTiO3

~25000 particles

5 µm

44


0 min 20 min 40 min

60 min 100 min 80 min

Discontinuities start from low green density regions

Contact loss

heterogeneities

45


Relative density heterogeneities discontinuities

Ex-situ X-ray –nCT at Argone

Z Yan, O Guillon, C L Martin, S Wang, C-S Lee, D Bouvard. Appl. Phys. Lett. 102, 223107 (2013)

46


• Increase thickness not an option for the MLCC industry !

• Possible remedies ?

0.6 µm 1 µm 1.2 µm

2 %

0.8 µm

24 % 8 % 4 %

microstructure before BTO sintering:

47


1k/min

14 %

5k/min

6 %

10k/min

4 %

15k/min

2 %

30k/min

1.7%

50k/min

1.1%

• Possible remedies ?

• Increase heating rate decrease sintering driving force for Ni layer

microstructure before BTO sintering:

aim of the model: to mimic a solid neck (bond) between two particles

allows building clusters of particles a simple way to model non-spherical particles

the bond may have formed by various physical mechanisms (diffusion, precipitation, cementation, …), or may not represent any true process

Bonding model

[Jefferson, 2002; Potyondy, 2004; Jauffres, 2012] 48

a b

R j

t n

R i

nu

Γt Γn

assumption: the bond size (ab) is unvarying

in the literature, also called, cement,

bonded-particle model

transmit normal, tangential forces and

resisting moments

comes often with a fracture criterion

Bonding model

[Jefferson, 2002; Potyondy, 2004; Jauffres, 2012]

a b

R j

t n

R i

nu

normal bonding force

depends on relative normal displacement (not actual overlap δ)

recall Hertz law

n j i j i init

u = − − −r r r r31 22**4

3nF E R δ=

12

1 12 2**

2 221 2 1

nelast

F E R EE R aδδδ ν ν

∂ = = = ∂ − −

stiffness of the Hertzian law is proportional to contact size

Γt Γn

49

for bonds 2 ( , )1

bn n b n

aEF g a uR

νν

= −−

Bonding model


a b

R j

t n

R i

nu

Γt Γn

50

2 ( , )1

bn n b n

aEF g a uR

νν

= −− for bonds

n j i j i initu = − − −r r r r

( , )bn

agR

ν = departure from the small ab limit for ‘large’ bonds and constant ab

obtained by FEM simulations

( , ) 2bn

agR

ν ≈ for and 0.5baR

= 0.2ν =

51

Bonding model

n j i j i initu = − − −r r r r

( , )bn

agR

ν = departure from the small ab limit for ‘large’ bonds and constant ab

obtained by FEM simulations

( , ) 2bn

agR


= 0.2ν =

2 ( , )1

bn n b n

aEF g a uR

νν

= −− for bonds

lateral strain ?

no ‘natural’ Poisson effect (comes only with random particle packing)

Bonding model

[Jefferson, 2002; Potyondy, 2004; Jauffres, 2012] 52

bond interactions

)],()[,(' iiNb

Nb NguRa

faEN θν +=

Additional term accounting for the elastic deformation of the two spheres

normal force

ab

Rp

TN

MT

MN

Rq

Rp+ Rq- h- uNN1

N2

θ1

θ2

Finite element calculation

Bonding model


a b

R j

t n

R i

tδ

tangential bonding force

depends on relative tangential accumulated displacement

recall Hertz-Mindlin law

( )( )* 28

1 2t

elast elastt

F EG a aδ ν ν

∂= − = −

∂ + −

for bonds

( , )bt

agR

ν = departure from the small ab limit for ‘large’ bonds and constant ab , obtained by FEM

( , ) 1.3bt

agR


= 0.2ν =

*8t tF G aδ= −

Γt Γn

53

( )( )2 ( , )

1 2b

t t b taEF g aR

ν δν ν

= −+ −

Bonding model


a b

R j

t n

R i

nθ

resisting moments to oppose relative rotations

depends on relative accumulated relative rotations

and are the accumulated relative rotations between the two particles

tθ

nθ tθ

Γt Γn

54

212n t b nk a θΓ = −

214t n b tk a θΓ = −

( )( )3( , )

2 1n t b b nE g a aν θ

ν νΓ = −

− +

( )3

2( , )

4 1t n b b tE g a aν θ

νΓ = −

−

ab

solid bond between two particles bond fracture

Potyondy model (PFC type) : solid bond = cylindrical beam

maximum tensile stress on the periphery of the cylinder:

maximum shear stress on the cylinder:

need to model what happens after fracture: bond disappears or keeps some stiffness in compression ?

Bonding model: fracture

[Potyondy, 2004; Pizette, 2013]

R

t n

Γt

Γn

n tensionσ σ>

t shearσ σ>

55

2 3

4 tn

nb b

Fa a

σπ π

Γ= +

2 3

2 t n

tb b

Fa a

σπ π

Γ= +

solid bond between two particles bond fracture

solid bond: stress singularity use Linear Elastic Fracture Mechanics

need to model what happens after fracture: bond disappears or keeps some stiffness in compression ?

fracture: need to take into account the history of the contact

memory access in the code

Bonding model: fracture

[Jauffrès, 2012]

σb > σc ab

Fracture energy Γ = 2γs

Stress singularity

(mode I)

σb > σc

(mode II)

56

221c n

b

E ga

σν π

Γ=

−

( )( )2 22

2 1c tb

E ga

σπ ν ν π

Γ=

− +

57

Applications : porous electrodes, bone scaffolds, filters, catalyst, thermal insulation …

Microstructural optimization of porous ceramics

≠

‘continuum mechanics’

Finite Element Method, FEM Discrete Element Method, DEM Grenoble INP dp3D code

Particles interact through their

contacts

‘discrete element mechanics’

Inherent contradiction between functionalities and strength

Modeling strategies for partially sintered ceramics

5 µm

Application: porous ceramics

Porous electrodes aerogels

T N


58

Initial Compact

Numerical microstructures

[C.L. Martin and R. Bordia, 2008, Phys Rev E 77, 031307] [C.L. Martin and R. Bordia, 2009, Acta Mater 57, 549-558] [X. Liu et al., 2011, J. Power Sources, 196, 2046- 2054]

Sintered Process parameters: green density, particle size, pore formers …


59

Initial Compact


[C.L. Martin and R. Bordia, 2008, Phys Rev E 77, 031307] [C.L. Martin and R. Bordia, 2009, Acta Mater 57, 549-558] [X. Liu et al., 2011, J. Power Sources, 196, 2046- 2054]

Sintered

Sintering with pore formers

NiO(Ni)+YSZ

T. Suzuki, et al., Science 325 (2009) 852–855.

Process parameters: green density, particle size, pore formers …

Macroscopic elastic properties

60

0

20

40

60

80

100

120

140

160

180

0 0.1 0.2 0.3 0.4 0.5 0.6Porosity, ε

You

ng's

mod

ulus

(GPa

)Pihlatie et al.Radovic et al.Wang et al.Selçuk et al.

NiO/YSZ

NiO/YSZ-pf

0.32

0.25

0.16

YSZ NiO

ENiO= EYSZ= 220 GPa

Homogeneous microstructure

microstructure with pore formers

61

D0 = 0.5 ; D = 0.72

Crack tip: number of broken bonds per particle

Sintered sample with a pre-crack (D = 0.68)

7

6

5

4

3

2

8

1

0

7

6

5

4

3

2

8

1

0

Macropore

density = 0.8


Homogeneous microstructure

microstructure with pore formers

Toughness

Toughness

62

Fracture of a pre-cracked sample toughness

7

6

5

4

3

2

8

1

0

7

6

5

4

3

2

8

1

0

number of broken bonds per particle

2a

w/ macropores ; D = 0.68

Effect of pore formers

63

7.14.1

∝

RaZK b

IC

4.13.2~

∝

Ra

ZE b

is less sensitive to bond number than

5 numerical samples/ microstructure: dispersion

ab

Z=3

E0= 400GPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.5 0.6 0.7 0.8 0.9Relative density D

D060w/ pore formers

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.5 0.6 0.7 0.8 0.9

Relative density D

KIC

[Mpa

.m0.

5 ]

D060w/ pore formers

Z = number of bonds per particle

EICK

Scaling law

Toward optimized microstructures

64

Master curve /IcK E Z∝

Homogeneous

Normal distribution

Bimodal distribution

Pore formers Aggregated

aggregates

pore formers

homogeneous

Toward optimized microstructures

65

0

0.5

1

1.5

2

2.5

3

3.5

4

0.5 0.6 0.7 0.8 0.9Relative density D

D060D051-pfD050-agg

Aggregates

homogeneous Pore formers

IcKE

Performance index (imposed displacement)

aggregates

pore formers

homogeneous

• DEM simulations from X-ray tomography images of freeze-cast microstructures

Coupling DEM and X-ray tomography

66

A 3D image of the macropores

Hierarchical microstructure « carved » within a homogeneous sample

100 µm

Freezing direction

67

simple compression test

Coupling DEM and X-ray tomography

68

Coupling DEM and X-ray tomography Porous electrodes for SOFCs

FC-47-Fine FC-47-Coarse

FC-31-Fine

Homogeneous

PF-29

50 µm ~ 400 000 particles

Kel (LSM) 200 S/cm² Kio (8YSZ) 0.1 S/cm²

Kec 1E-5 S/cm²

Resistance Network [ ]

nbottomV

topV

ninn VK

=0...0

x

Current collector (Vtop)

Electrolyte (Vbottom)

Vi

V1

Vn

69

Tb Nb

MT MN

Linear spring, normal + tangent + resisting moments

fracture?

Particles do not represent anymore discrete bodies just a mean to “mesh” the structure

Bonding model for continuum


[Jefferson, 2002; Potyondy, 2004; Estrada, 2011; Jauffres, 2012]

a b

R j

t n

Γt

R i

nθ

bonding model in incremental form

kn, kt, kR,n, kR,t, can be used to model continuum

tθ

Γn

resisting moments

normal and tangential forces

2,R n n bk k a∝

70

, ,R n R n nk θ∆Γ = − ∆

, ,R t R t tk θ∆Γ = − ∆

t t tF k δ∆ = − ∆

n n nF k u∆ = − ∆

[Jefferson, 2002; Potyondy, 2004; Estrada, 2011; Jauffres, 2012]

what is the lateral deformation of this simple packing under simple compression ?

71

summary of bonding model in incremental form

kn, kt, kR,n, kR,t, can be used to model continuum


2,R n n bk k a∝

, ,R n R n nk θ∆Γ = − ∆

, ,R t R t tk θ∆Γ = − ∆

t t tF k δ∆ = − ∆

n n nF k u∆ = − ∆

72

0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,080

2

4

6

8

10

stre

ss (M

Pa)

Strain

DEM EXP

fitting of:

• normal and tangential stiffness (2 parameters)

• fracture stress (1 parameter)

Starch plate with a notch and a hole


[L. Hedjazi,, 2012]

73


elastic instability with DEM ?

Euler equation:

Rate effects !

Euler

simulation

Buckling of a cylindrical beam

74

• 3D alveolar structure

X-ray microtomography of an alveolar food (Miel pops, INRA Nantes) Large simulation (large strain and ∼ 7 millions bonds)

need for parallel computing

Coupling DEM and X-ray tomography: towards structures

[L. Hedjazi, 2014]

3D discrete microstructure

• size of debris

• mastication force

75

• 3D alveolar structure

X-ray microtomography of an alveolar food (Miel pops, INRA Nantes) Large simulation (large strain and ∼ 7 millions bonds)

need for parallel computing

Coupling DEM and X-ray tomography: towards structures

[L. Hedjazi, 2014]

3D discrete microstructure

• size of debris

• mastication force

Conclusions

76

DEM is relatively simple to code

The physics of the problem is embedded in the contact law

Very well adapted for fracture (contact loss is simple to code)

New contacts between surfaces are accounted naturally (change of topology)

DEM is CPU intensive

need for parallelization (or GPU ?)

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P. Pizette, C. L. Martin, G. Delette, F. Sans, and T. Geneves, “Green strength of binder-free ceramics,” J. Eur. Ceram. Soc., vol. 33, pp. 975–984, 2013.

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· 5 . what do we need ? • numerically discretize newton’s second law • provide contact laws...

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