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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
Block #1week 16 Understand sound propagation (discrete and continuous)week 17 Prepare MD code – time-scales, sound, …
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?
Block #2weeks 20 - 22 Apply perturbation theory and stability analysis
When does my system become unstable? for which modes?
Block #3weeks 23 - 25 Apply static equilibrium, linear methods
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
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 ( )( )
1if f
j
jii j i
i jij
ffa a n n f n
m m mδ
=−
= − − ⋅ = − − ⋅ = − ⋅
� �
���� � � � ���
http://www2.msm.ctw.utwente.nl/sluding/PAPERS/coll2p.pdfhttp://www2.msm.ctw.utwente.nl/sluding/PAPERS/coll2p.pdf
i ijf m kδ δ γδ= − = +�� �
- really simple ☺☺☺☺
- linear, analytical
- very easy to implement
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 =
http://www2.msm.ctw.utwente.nl/sluding/PAPERS/coll2p.pdfhttp://www2.msm.ctw.utwente.nl/sluding/PAPERS/coll2p.pdf
i ijf m kδ δ γδ= − = +�� �
- really simple ☺☺☺☺
- linear, analytical
- very easy to implement
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
http://www2.msm.ctw.utwente.nl/sluding/PAPERS/coll2p.pdfhttp://www2.msm.ctw.utwente.nl/sluding/PAPERS/coll2p.pdf
Linear Contact model (mw=∞)
elastic freq.0
ij
km
ω =
eigen-freq.
visc. diss.
2 2
0ω ω η= −
2ij
m
γη =
contact duration ctπ
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= −
http://www2.msm.ctw.utwente.nl/sluding/PAPERS/coll2p.pdfhttp://www2.msm.ctw.utwente.nl/sluding/PAPERS/coll2p.pdf
Time-scales
contact duration ctπ
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
http://www2.msm.ctw.utwente.nl/sluding/PAPERS/coll2p.pdfhttp://www2.msm.ctw.utwente.nl/sluding/PAPERS/coll2p.pdf
Time-scales
contact duration ctπ
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
http://www2.msm.ctw.utwente.nl/sluding/PAPERS/coll2p.pdfhttp://www2.msm.ctw.utwente.nl/sluding/PAPERS/coll2p.pdf
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
Elastic hard spheres in gravity
• N particles
• Kinetic Energy
• What is the density profile ?
gravity
Elastic hard spheres in 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= =
Elastic hard spheres in gravity
• N particles
• Kinetic Energy
• What is the density profile ?
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)
Sheared systems – linear stability (2D)
Sheared systems – linear stability (2D)
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
• Pressure vs. density
• Global equation of state (crystallization)
• Shear stress (viscosity) divergence -> J
• Homogeneous and sheared
• But: which power law is it?
Summary
Which power law is it? … really -1?
Approach to jamming
Which power law is it? … really -1?
Otsuki, Hayakawa -> -3 !!!
Approach to jamming
Which power law is it? … really -1?
Otsuki, Hayakawa -> -3 !!!
Approach to jamming
Which power law is it? … really -1?
Otsuki, Hayakawa -> -3 !!!
Approach to jamming
Approach to jamming
• Which power law is it? … really -1?
• control parameter -> dim.less. dissip.rate
Approach to jamming
• Which power law is it? … really -1?
• control parameter -> dim.less. dissip.rate
Approach to jamming
• Which power law is it? … really -1?
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
• Pressure vs. density
• Global equation of state (crystallization)
• Shear stress (viscosity) divergence -> J
• Homogeneous and sheared
• 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
• Pressure vs. density
• Global equation of state (crystallization)
• Shear stress (viscosity) divergence -> J
• Homogeneous and sheared
• 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
How does sound propagation depend on
- structure?
Lattice+tiny disorder => enormous effect
- adhesion?
- friction?
- preparation history?
…
Goal
Block #1week 16 Understand sound propagation (discrete and continuous)week 17 Prepare MD code – time-scales, sound, …
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?
Block #2weeks 20 - 22 Apply perturbation theory and stability analysis
When does my system become unstable? for which modes?
Block #3weeks 23 - 25 Apply static equilibrium, linear methods
How are moduli related to eigen-modes? and sound-speed?
Question
How does sound propagation depend on
- structure?
Lattice+tiny disorder => enormous effect
- adhesion?
- friction?
Weak effect of adhesion and friction (strong for µ=0)
- preparation history?
Question
How does sound propagation depend on
- structure?
Lattice+tiny disorder => enormous effect
- adhesion?
- friction? wasn’t there something else?
Weak effect of adhesion and friction (strong for µ=0)
- preparation history?
+Tangential elasticityOptical branch? (kt/kn=2)
Dispersion relations with tangential elasticity
Partec2007 132
Rotation waves ?
Question
How does sound propagation depend on
- structure?
Lattice+tiny disorder => enormous effect
- adhesion?
- friction?
Weak effect of adhesion and friction (strong for µ=0)
- preparation history?
Question & Conclusion
How does sound propagation depend on
- structure?
Lattice+tiny disorder => enormous effect
- adhesion?
- friction?
Weak effect of adhesion and friction (strong for µ=0)
- preparation history?
… insensitive to pre-failure �
… much stronger post-peak damping
P-wave animation
Question
How does sound propagation depend on
- structure?
Lattice+tiny disorder => enormous effect
- adhesion?
- friction?
Weak effect of adhesion and friction (strong for µ=0)
- preparation history?
… insensitive to pre-failure �
… much stronger post-peak damping
Goal
Block #1week 16 Understand sound propagation (discrete and continuous)week 17 Prepare MD code – time-scales, sound, …
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?
Block #2weeks 20 - 22 Apply perturbation theory and stability analysis
When does my system become unstable? for which modes?
Block #3weeks 23 - 25 Apply static equilibrium, linear methods
How are moduli related to eigen-modes? and sound-speed?
Questions?