From particles to continuum – Micro-Macro Methods · Micro-Macro Methods Stefan Luding V. Magnanimo, A. R. Thornton, S. Srivastava Multi ScaleMechanics, TS, CTW, UTwente, POBox217,

From particles to continuum –Micro-Macro Methods

Stefan Luding

V. Magnanimo, A. R. Thornton, S. Srivastava

Multi Scale Mechanics, TS, CTW, UTwente,

POBox 217, 7500AE Enschede, NL --- [email protected]

Overview

Block #1week 16 Introduction: Sound propagation (discrete and continuous)week 17 – free time for practice and assignmentsweek 18 Particle Methods (contact and efficient detection) and

MD for fluids and solids with examples and programmingweek 19 - free week

Block #2weeks 20 - 22 Perturbation theory and stability analysis

Applied to flowing systems – practice and applications

Block #3weeks 23 - 25 Static equilibrium mechanical systems

(discrete and continuous)

Goal

Block #1week 16 Understand sound propagation (discrete and continuous)week 17 Prepare MD code – time-scales, sound, …

week 18 Implement contact model and efficient contact detectionweek 19 - free week

Block #2weeks 20 - 22 Apply perturbation theory and stability analysis

Block #3weeks 23 - 25 Apply static equilibrium, linear methods

Goal


What are the time-scales? what is sound speed? in ‘my’ system?week 18 Implement contact model and efficient contact detection

How many particles can I simulate with ‘my’ code? which time?


When does my system become unstable? for which modes?


How are moduli related to eigen-modes? and sound-speed?

• Introduction

• Contact Models

• DEM/MD simulations

• Application/Example Sound

• Outlook

Single

particle

Contacts

Many

particle

simulation

Continuum Theory

Content

Molecular Dynamics – soft/hard

1. Specify interactions

between bodies (for example

two colloids/atoms)

2. Compute all forces

3. Integrate the equations

of motion for all particles

i j i

j i

m →≠

=∑x f��

j i→f

i ijf m kδ δ γδ= − = +��

- really simple ☺☺☺☺

- linear, analytical

- very easy to implement

Linear Contact modeli

f

δ

overlap ( ) ( )1

2i j i jd d r r nδ = + − − ⋅

� � �

rel. velocity ( )i j nδ = − − ⋅� � �

� v v

acceleration ( )δ = − − ⋅� � �

��i ja a n

http://www2.msm.ctw.utwente.nl/sluding/PAPERS/coll2p.pdfhttp://www2.msm.ctw.utwente.nl/sluding/PAPERS/coll2p.pdf





Linear Contact modeli

f

δ

overlap ( ) ( )1

2i j i jd d r r nδ = + − − ⋅

� � �

rel. velocity ( )i j nδ = − − ⋅� � �

� v v

acceleration ( )( )

1if f

j

jii j i

i jij

ffa a n n f n

m m mδ

=−

= − − ⋅ = − − ⋅ = − ⋅

� �

��






Linear Contact model

elastic freq.0

ij

km

ω =

eigen-freq.

visc. diss.

0ijk mδ γδ δ+ + =� ��

2 02ij ij

k

m m

γδ δ δ+ + =� ��

2

0 2 0ω δ ηδ δ+ + =� ��

2 2

0ω ω η= −

2ij

m

γη =






Linear Contact model

elastic freq.0

ij

km

ω =

eigen-freq.

visc. diss.

( ) exp( )sin( )t t tδ η ωω

= −0v

0ijk mδ γδ δ+ + =� ��

2 02ij ij

k

m m

γδ δ δ+ + =� ��

2

0 2 0ω δ ηδ δ+ + =� ��

2 2

0ω ω η= −

2ij

m

γη =

[]

( ) exp( ) sin( )

cos( )

t t t

t

δ η η ωω

ω ω

= − −

+

� 0v

contact duration ctπ

ω=

restitution coefficient( )

exp( )

c

c

tr

tη

= −

= −0

v

v


Linear Contact model (mw=∞)

elastic freq.0

ij

km

ω =

eigen-freq.

visc. diss.

2 2

0ω ω η= −

2ij

m

γη =


ω=

restitution coeff. exp( )c

r tη= −

particle-particle particle-wall

00

2

wall

i

km

ωω = =

2 2

0 2 4wallω ω η= −

2 2

wall

im

γ ηη = =

wallwallc c

t tπω

= >

exp( )wall wall wall

cr tη= −


Time-scales


ω= wallwallc c

t tπω

= >

time-step50

ct

t∆ <=

time between contacts

n ct t<

n ct t>

sound propagation ... with number of layers L c L

N t N

experiment T


Time-scales


ω= argl e small

c ct t>

time-step50

ct

t∆ <=

different sized particlesn c

t t<

n ct t>

sound propagation ... with number of layers L c L

N t N

experiment T

time between contacts


Molecular Dynamics – soft/hard

1. Specify interactions

between bodies (for example

two colloids/atoms)

2. Compute all forces

3. Integrate the equations

of motion for all particles

i j i

j i

m →≠

=∑x f��

j i→f

Continuum theory – hard …

• Pressure P

• Deviator Stress

• Energy Dissipation Rate I

( ) 0i

i

ut x

ρ ρ∂ ∂

+ =∂ ∂

( ) ( ) dev

i i i

i j

ijk

k

u u u P gt x x x

ρ ρ ρσ∂ ∂ ∂ ∂

+ = − + +∂ ∂ ∂ ∂

( )2 2 2 21 1 1 12 2 2 2k

k

u u uP

t xρ ρ ρ

ρ

∂ ∂+ = − + +

∂ ∂ v v

mass conservation:

momentum conservation:

energy balance:

dev

ijσ

( )212

dev

iki i i

k

u K gx

Iuρσ ρ∂

− − + −∂

v

Elastic hard spheres

• Pressure P

• Deviator Stress

• Energy Dissipation Rate I=0

0t

∂=

∂

0ixP

∂= −

∂

elastic steady state:

mass & energy conservation – OK

momentum balance:

dev 0ijσ =

0i

u I= =

0i

g =

First example … pressure EQS

( ) ( )( )21 1 ν ν= + +

arP

Eg

V

( )2?ν =

ag

Pressure (Equation of State – 2D)

fluid, disordered

solid, ordered

phase transitionat critical density

PV/E-1=2ννννg(νννν)

S. Luding, Nonlinearity, Dec. 2009

Elastic hard spheres in gravity

• N particles

• Kinetic Energy

• What is the density profile ?

gravity


• N particles

• Kinetic Energy


gravity


• Pressure P = global equation of state

• Deviator Stress

• Energy Dissipation Rate I=0

0t

∂=

∂

0i

i

P gx

ρ∂

= − +∂

elastic steady state:

mass & energy conservation – OK

momentum balance:

dev 0ijσ =

0i

u I= =


• N particles

• Kinetic Energy


gravity

Hard sphere gas in gravity – which EQS?

fluid, disorderedexponential tail

solid, ordered

phase transitionat critical density

Structure formation under shear

Low density -> linear velocity profile

High density -> shear localization

Sheared systems – linear stability (2D)




Shukla, Alam, Luding, 2008

Shear (first normal stress difference)

Shear (first normal stress difference)

Structure formation under shear

Low density -> linear velocity profile

High density -> shear localization

shear “viscosity” (2D)

S. Luding, Nonlinearity, Dec. 2009

Global equations of state (2D)

Shear (viscosity at high density)

homogeneous

inhomogeneous=> dilatancy

critical density

( )1

133

E

η

η ην ν

= +

−

R. Garcia-Rojo, S. Luding, J. J. Brey, PRE 2006

Shear viscosity divergence: power -1

( )1

133

E

η

η ην ν

= +

−

• Pressure vs. density

• Global equation of state (crystallization)

• Shear stress (viscosity) divergence -> J

• Homogeneous and sheared …

Summary




• Homogeneous and sheared

• But: which power law is it?

Summary

Which power law is it? … really -1?

Approach to jamming


Otsuki, Hayakawa -> -3 !!!

Approach to jamming



Approach to jamming



Approach to jamming

Approach to jamming

• Which power law is it? … really -1?

• control parameter -> dim.less. dissip.rate

Approach to jamming


• control parameter -> dim.less. dissip.rate

Approach to jamming


M. Otsuki, H. Hayakawa, S. Luding, JTP, 2010

Approach to jamming

• Which time-scales?

1. shear rate (inverse)

2. contact duration tc

3. dissipation time

ττττw=nT/S=1./3.

M. Otsuki, H. Hayakawa, S. Luding, JTP, 2010





• Which power law is it? Hard vs. Soft

• Hard/soft jamming

• Almost elastic vs. dissipative

• Hard/rigid vs. soft

• Kinetic theory vs. multi-particle contacts

Summary





• Which power law is it? Hard vs. Soft

• Open issues:

• Dense (and inhomogeneous) systems

• Anisotropy, micro-structure, …

• Experimental validation, …

Summary

Application/Example - SOUND

An estimate for sound propagation speed …

Application/Example - SOUND

Soil investigation

- Seismology

- Oil exploration

- …

Compressive (P) and Shear (S) waves

Sound

Model system

P-wave animation

Influence of “micro” propertiesCompressive (P)-wave

Elastic Visco-Elastic

How relevant is the damping coefficient in our model ?

2

1 1

1

2

C Cc c c c t c c c c

p V c cV

aC k n n n n k n t n tαβγφ α β γ φ α β γ φ

∈ = =

= +

∑ ∑ ∑

Wave speed from the stiffness tensor

zzzzpz

CV

ρ=

Modes

• P-waves

• S-waves

• R-waves

• …

Structure

• Mono-disperse

• Poly-disperse

• Disordered

• …

Micro-Parameters

• Damping

• Friction (Rotations)

• Adhesion

• Contact laws …

• …

Towards complexity

Velocities

with rotation+friction

Weak polydispersity

δδδδ = a/1000

∆∆∆∆a = δδδδ/2, δδδδ and 2δδδδa

- The system is practically unchanged at the structure level

- Wide distribution of weak and strong contacts

and most important opening of contacts

P-wave animation

P-wave animation

Velocities

with rotation+friction frictionless+weak disorder

Dispersion relations

∆∆∆∆a = δ/2δ/2δ/2δ/2space-time-FFT

from eigenvalue calc.

Density of states

ordered

disordered

Eigenmodes

Eigenmodes

Question

How does sound propagation depend on

- structure?

Lattice+tiny disorder => enormous effect

Question


- structure?


- adhesion?

- friction?

- preparation history?

…

Goal








Question


- structure?


- adhesion?

- friction?

Weak effect of adhesion and friction (strong for µ=0)


Question


- structure?


- adhesion?

- friction? wasn’t there something else?



+Tangential elasticityOptical branch? (kt/kn=2)

Dispersion relations with tangential elasticity

Partec2007 132

Rotation waves ?

Question


- structure?


- adhesion?

- friction?



Question & Conclusion


- structure?


- adhesion?

- friction?



… insensitive to pre-failure �

… much stronger post-peak damping

P-wave animation

Question


- structure?


- adhesion?

- friction?



… insensitive to pre-failure �

… much stronger post-peak damping

Goal








Questions?

From particles to continuum – Micro-Macro Methods · Micro-Macro Methods Stefan Luding V. Magnanimo, A. R. Thornton, S. Srivastava Multi ScaleMechanics, TS, CTW, UTwente, POBox217,

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