physical foundations of meso-scale continua -...
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Physical foundations of meso-scale continua
S. Mesarovic
CISM course: MESOSCALE MODELS: FROM MICRO-PHYSICS TO MACRO-INTERPRETATION
Udine, May 22-26, 2017
Intro• Continuum theories are approximations. Loss of information.
• Almost never rigorous: Assumptions.
• Macroscale continua: phenomenological components are inevitable: separation of scales, complex microscale physics.
• Mesoscale continua: higher requirements. To bridge the gap between microscale and macroscale, its primary variables must be clearly connected to the microscale.
• How to formulate a continuum theory for (to keep it simple) an isothermal process?1. Choose primary variables (usually kinematic)
2. Express energy (rates) and dissipation. A. Define generalized forces (power-conjugates)
B. Limit the form of closure (constitutive law) to satisfiy the 2nd law
3. Formulate the weak form (principle of virtual power)
4. Derive the strong form
Outline
• Part 1: Kinematics
1.1 Mass continuum (fluids, disordered, mixing)
1.2 Lattice continuum (crystals, order + diffusion, creep)
1.3 Granular continuum (disordered, deformation mechanism)
• Part 2: (Thermo-) Dynamics
2.1 Phase field models for capillary flows
2.2 Diffusional creep in polycrystals
2.3 Size-dependent crystal plasticity
Part 1: Kinematics1.1 Kinematics of the mass continuum
S. Mesarovic
CISM course: MESOSCALE MODELS: FROM MICRO-PHYSICS TO MACRO-INTERPRETATION
Udine, May 22-26, 2017
Kinematics of the mass continuum
mass density ( , ) moves with ( , )
continuity equation: ( ) 0 (no mass creation, only transport)
material derivative: material field ( , )
transport thm: vol
x v x
v
x v
t t
t
DY YY t Y
Dt t
( ) ( )
ume ( ) follows material
V t V t
D DYV t YdV dV
Dt Dt
,dVx
Eulerian (spatial) point of viewMaterial moves with
instantaneous velocity
What is material?
Mathematically: material=mass
Physically: ?????
( , )v x t
Physical meaning of material in the mass continuum
1Mass density: (unit volume). What is ( , )?
barycentric (equivalent linear momentum) velocity
Why? Cauchy EOM: ( )
v x v v
σ b v
N i
iNm t m
D
Dt
Single component fluid,color the atoms inside the volume element.Simplest flow: Uniform continuum velocityMixing/diffusion always present
tv
Atomic linear momentum Continuum linear momentumDigressions: What about kinetic energy? Angular momentum?
Digression: kinetic energy and angular momentum
1 1
2 21
Atomic kinetic energy = continuum kinetic energy + thermal energy
( ) ( ) (thermal en. density)
based on separation of time (frequency) scales
v v v x v x
An assumption!!!!
Ni i
i
m dV dV
To simplify – we will consider only isothermal processes:Energy transformed into thermal: DISSIPATIONTo couple with heat transfer: DISSIPATION = HEAT SOURCE
Atomic ang. mom. = 0 (in volume element) Simple continuumContinuum ang. mom. = 0 Symmetry of the stress tensorAtomic ang. mom. ≠ 0 Higher order continua(polar, micromorphic, gradient,…)
(Note: This is not the only physical argument for higher order continua.)
Two-component fluid
v v vA B
A B
; ; A B A B
A B A BM M M M M
V V V V
average velocity of atoms .
Barycentric velocity equivalent lin. momentum :
vA B A B
/ /
mass fraction: ; ;
relative av. vel.
q q w w
w v v
A A
A BA B
A B A B
MD
MDt
2
1( ) Incompressibility cond. 0v q v
d
d
Full incompressibility prevents any diffusion.
Diffusion/mixingflux
Components A/B are incompressible (mass density is independent of pressure)Mass density of the mix is independent of pressure
( , , )d p T d dp dTp T
isothermal
Compositionally compressible (quasi-incompressible) flow.Lowengrub & Truskinovsky 1998, Dehsara et al 2017
Kinetic energy of a two-component fluid
Cauchy EOM: ( ) can still be written formally,
but the extra kinetic energy will have to be:
either neglected (sometimes justified, sometimes not)
or included in the definition of stres
σ b vD
Dt
s (cf. ) or body forcesvirial stress
( . .)Particle: lin. mom.
( . .)Continuum:
K E
K EdV
v
vv
2 2 2 21 1 1 1
2 2 2 2
A BA B
A B A Bv v v q
Flux kinetic energy
Summary
• In the mass continuum, material is not associated with physical entities (particles, atoms). Instead, it has a mathematical definition: mass density field
• Barycentric continuum velocity for a mix– guaranties linear momentum equivalence between
particles (atoms) and continuum, – but not kinetic energy equivalence,– It requires additional material vector field:
diffusion/mixing flux
• Incompressibility prevents mixing/diffusion
2
( , )Primary variables (rates): ( , ), ; are not independent:
1 1
Cannot impose incompressibility cond. 0
xv x
v q
v
D tt
Dt
d d D
d d Dt
References
Atkin, RJ & Crane, RE 1976 Continuum theories of mixtures: basic theory and historical development. Q. J. Mech. Appl. Math. 29, 209-244.
Joseph D.D. & Renardy, Y.Y. 1993 Fundamentals of two fluid mechanics. Springer. (Ch. X)Lowengrub, J. & Truskinovsky, L. 1998 Quasi-incompressible Cahn-Hilliard fluids and
topological transition. Proc. R. Soc. Lond. A 454, 2617-2654.Dehsara, M., Fu, H, Mesarovic, S.Dj., Sekulic, D.P. & Krivilyov M. 2017
(In)Compressibility and parameter identification in phase field models for capillary flows. To appear in Proc. R. Soc. Lond.
Part 1: Kinematics1.2 Kinematics of the lattice continuum
S. Mesarovic
CISM course: MESOSCALE MODELS: FROM MICRO-PHYSICS TO MACRO-INTERPRETATION
Udine, May 22-26, 2017
Diffusional creep in solids
Creep test: High T, Low (constant) stress,polycrystal
Mechanisms:Bulk diffusion (Nabarro-Herring)Grain boundary diffusion (Coble)Dislocation creep (various mechanisms)
Pure nickel, grain size 1 µm, work-hardened
Deformation-Mechanism Maps, The Plasticity and Creep of Metals and Ceramics, by Harold J Frost and Michael F Ashby
Diffusion
Vacancy flux
Atom flux
Atom-vacancy exchange mechanism
Larché & Cahn 1973, 1985 lattice constraintBerdichevsky et al 1997 note “absence of Lagrangean coordinates”Garikipati et al 2001 elements of lattice continuum, diffuse interface layerMesarovic 2016 full (sharp interface) formulation
Lattice grows
Latt
ice
dis
app
ear
s
Quasi-static (quasi-equilibrium) process
• Inertial mass becomes irrelevant
• Mass density is just another material field
• Lattice continuum: material=lattice density
• Key questions:
– Creep: lattice motion vs. boundary motion
– Lagrangean kinematics: reference configuration for newly created lattice
– Transport theorem
– Mass balance
Lattice continuumlattice, moves with velocity ( , )
material field ( , ) associated with lattice (not with mass)
material (lattice) derivative:
lattice density (# of lattice sites
v
v
L
material x t
Y x t
DY YY
Dt t
N
( ) ( )
per unit volume)
Continuity equation Local lattice site conservation:
(no lattice sources/sinks)
Transport theorem:
vLL
L L
V t V t
DNN
Dt
D DYN YdV N dV
Dt Dt
Each lattice site occcupied by one of the two species: atoms or vacancies
# of vacanciesLattice carries with ( , ) : , ( , )
( , )and deformation gradient ,
, lattice site referenc
v x x
x XF
X
X x
LL
t N c tN
t
e (????) and current position
New lattice and boundary
1
0
Velocity gradient: ;
New lattice is created with continuous: , ,
Vacancy flux: ( )
derive: ; det ;
vL F L F
x
F X F x
J v v
J F
L
v
L
L
D
Dt
N c d d
c
F
NDcF F
Dt N
: normal velocity of the boundary lattice velocity
lattice volume production rate per unit area:
Creep deformation rate: constant in each grain
1(
n v
C v
n
n
n
V
deformed
g V
V
)n nnd
nV
Motivation for the creep rate tensor
1 11
2 22
Volume average of strain rate from boundary velocities
1 1( , , ) ( )( , , )
For creep deformation rate:
(1) Net boundary growth ( )
(2) Only normal component exi
v n
i j j i i j j ii j j i
n
u u d u n u n du u
V
sts
1( )C v n nnnV d
(Mesarovic& Padbidri 2005)
Mass conservation
0
1Grain mass conservation: 0;
;
If grain boundarydiffusion is disallowed:
1Local mass conservation at a boundary: 0
1Creep rate:
n v n J
C
L n
n n
n
cmN g J d
F
g V J
cg J
F
FJ
1
piecewise constant (discontinuous) field
nnn dc
Note: For the moment we are assuming non-existent: - Dislocation climb (lattice site conservation)- Grain boundary diffusion- Grain boundary sliding
Lagrangean kinematics
heterotopologic isotopologic
F F Fec p
plasticelastic compositional
Multiplicative decomposition of deformation gradient into pseudo-gradients
0
Linearized kinematics
( ) ( )
Total e-c strain = elastic + compositional
( )
Simplest: volumetric, Vegard's law:
( ) ( )
F R U I Ω I ε I Ω ε
ε S :σ e
e I
ec
c
c c c
Summary• Kinematics of lattice continuum with lattice site conservation (no
dislocation climb) and bulk diffusion only:
– Material (lattice) derivative
– Transport theorem
– Deformation gradient for newly created lattice
• Multiplicative decomposition of deformation gradient: (elastic-compositional) and plastic
• Creep strain tensor is discontinuous (constant within a grain) and directly related to the normal boundary diffusion flux
( , )Primary variables (rates): lattice velocity ( , ) and vac. conc. rate
Boundary normal velocity:
1Local mass conservation at a boundary: ( ) 0
1Creep rate:
1
xv x J
n v n J
C nn
n
n
n
Dc tt F
Dt
V
cV
F
FJd
c
ReferencesBerdichevsky, V, Hazzledine, P. & Shoykhet, B. 1997 Micromechanics of
diffusional creep. Int. J. Engg. Sci. 35(10/11), 1003-1032.
Garikipati, K, Bassman, L. & Deal, M. 2001 A lattice-based micromechanical
continuum formulation for stress-driven mass transport in polycrystalline
solids. J. Mech. Phys. Solids 49, 1209-1237.
Larché, F.C. & Cahn, JW. 1973 A linear theory of thermochemical equilibrium of solids under stress. Acta Metall. 21, 1051-1063.
Larché, F.C. & Cahn, J.W. 1978 Thermochemical equilibrium of multiphase solids under stress. Acta Metall. 26, 1579-1589.
Larché, FC. & Cahn, J.W. 1985 Interaction of composition and stress in crystalline solids. Acta Metall. 33(3), 331-357.
Mesarovic, S.Dj. 2016 Lattice continuum and diffusional creep. Proc. R. Soc. A472, 20160039.
Mesarovic, S.Dj. & Padbidri J. 2005 Minimal kinematic boundary conditions for simulations of disordered microstructures. Phil. Mag. 85(1), 65-78.
Mishin, Y, Waren, J.A, Sekerka, R.F. & Boettinger, W.J. 2013 Irreversible thermodynamics of creep in crystalline solids. Phys. Rev. B 88, 184303.
Part 1: Kinematics1.3 Kinematics of granular matter
S. Mesarovic
CISM course: MESOSCALE MODELS: FROM MICRO-PHYSICS TO MACRO-INTERPRETATION
Udine, May 22-26, 2017
Granular matter: Fundamental questions
2 centuries after Coulomb, 1.5 after Mohr and Reynolds:
Much is known empirically, little is understood.
Models are phenomenological.
(1) Dilatancy
(2) Critical state
(3) Shear localization size
(4) Flow pattern
1. Dilatancy
Dilatancy(O.Reynolds, 1885)
Rowe, 1962
Zero sum?
(1) Why dilatancy persists?
(2) Why other materials don’t dilate? Atoms are nearly rigid (Pauli exclusion principle)
2. Critical state11 33
1 3
( 2 )p
q
1
33
constantp
q
1
1
V
lnv
lnp
dilation
compactionsolid
Vv
V
ln ln
ss
v A p
q Mp
Cambridge, 1960’sSchofield & Wroth (1968)
(1) Why does critical state depend on pressure?(2) What is ln(Mpa)?(3) Rate dependence?
3. Persistent shear bands
Alshlibli et al, 2003
Typical width of persistent shear bands: 10-20 particles
Particle rotations and localization
deformation
rotations
Kuhn & Bagi 2004 DEM simulations
4. Flow pattern
Abedi, Rechenmacher & Orlando 2012 Vortex formation and dissolution in sheared sands. Granular Matter
Mathematical frameworkEqual spheres:Voronoi TessellationPolyhedra with One particle each
Dual graphDelaunay cellsSIMPLEXES
Dirichlet tessellation Bagi (1996)
Materials cells (curved polyhedra)
Dual graphSpace cells (SIMPLEXES)
Convex particlesWhy not Voronoi?
contacts DelaunayS S
C0 continuum (analogous to CST-FE)
1 AB AB
ABV
σ f d
AB BA= f f
AB BA= d d
A
B
● Cauchy Stress (Christoffersen et al,1981)
● Piecewise constant stress field
● Delaunay cells are simplexes: triangles (2D), tetrahedra (3D) (Satake, 1993)● For non-spherical convex particles: Space cells (Bagi, 1996)● C0 continuum (analogous to CST-FE)
defined volume (fraction), velocity gradient tensor (deformation rate) piecewise constant
(Bagi, 1996, 2006, Satake, 2004)
0 0
0 0
( ) ( )
( ) ( ); 1 4;
12 12
cell cell
cell k cell k k
v x v L x x
v x v L x x
L D W
Deformation mechanism
Isotopologic deformation: deformed Delaunay graph (space cells) is topologically equivalent to the reference one. Strains of the order 104, (Roux & Combe, 2010)
Heterotopologic deformation: changes in Delaunay graph
2-2 flip
Generic mechanism: FLIPS (Edelsbrunner, 2000): 1 in 2D, 2 in 3D
Statistically: 0
Intermittent flips
Cells lost from 4 – 5 %
Cells lost from 9 – 10 %
Cells lost in both intervals
% Axial Strain
v
f t
% Axial Strain
(1) (2)
( ) volume fraction flipping at
2
V V
V
t t
f t t
ff
t t
(1) 3
(2) 3
0.5 10 ;
1.5 10
0.025
V
V
Between 1% and 5% axial strain:
Dilation vs. Compaction
• Dilating sample: In each flip
– particle leaves highly stressed configuration
– ends up in relaxed configuration
• Loose sample under disturbance compacts
– Until RDP: lower density than critical state! (Roscoe et al 1958)
• Flips are disturbance
– unstable configuration > dynamic relaxation
– Fast attenuation owing to friction, weak disturbance
• Critical state: compaction/dilation rate balance
Force chains
● Contact force network (Drescher & de
Josselin de Jong, 1972)
● Non-homogeneous force distribution: Flips upon unloading
● Need a measure of local fluctuations in elastic strain energy: intrinsic stress
1 AB AB
ABV
σ f d
AB BA= f f
AB BA= d d
A
B
● Stress tensor (Christoffersen et al,1981)
● Piecewise constant stress field
Why other materials don’t dilate?
• No friction between atoms
> Critical state = RDP (Peyneau & Roux, 2008)
• Atomic vibrations – effective disturbance
> homogeneous forces, no force chains
Dilatancy is observed at very high strain rates in metallic glasses
ReferencesAbedi, S, Rechenmacher, AL, and Orlando, AD 2012 Vortex Formation and Dissolution in Sheared Sands.
Granular Matter, 14(6): 695-705.Bagi, K. (1996). Stress and strain in granular materials. Mech. Mater. 22, 165–177.Christoffersen, J, Mehrabadi, M.M & Nemat-Nasser, S, 1981, A micromechanical description of granular
material behavior, ASME J. Appl. Mech., 48, 339-344.Edelsbrunner, H. 2000 Triangulations and meshes in computational geometry. Acta Numer. 9, 133–213.Kuhn MR & Bagi K. 2004, Contact Rolling and Deformation in Granular Media, Int. J. Solids Struct. 41,
5793–5820.Mesarovic, S.Dj., Padbidri, J.M, & Muhunthan, B. 2012 Micromechanics of dilatancy and critical state in
granular matter. Geotechnique Letters 2, 61-66.Peyneau, P-E & Roux, J-N 2008 Frictionless bead packs have macroscopic friction, but no dilatancy.
Phys. Rev. E 78(1), 011307. Roscoe, KH, Schofield, AN & Wroth, CP. 1958 On the yielding of soils. Geotechnique 8(1), 22–52. Roux, J.-N. & Combe, G. 2010 How granular materials deform in quasistatic conditions. CP1227, IUTAM-
ISIMM symp. on mathematical modeling and physical instances of granular flow (Goddard, J. Jenkins, JT & Giovine, P (eds)). College Park, MD: American Institute of Physics, 260–270.
Rowe, P. W. (1962). The stress–dilatancy relation for static equilibrium of an assembly of particles in contact. Proc. Roy. Soc. Lond. 269, No. 1339, 500–527.
Satake, M. (1993). New formulation of graph-theoretical approach in the mechanics of granular materials. Mech. Mater. 16, No. 1-2, 65–72.
Schofield, A. N. & Wroth, C. P. (1968). Critical state soil mechanics. London: McGraw-Hill, London.
2. Shear band width
Rotation propagation distance?
Shear bands are accompanied by massive rolling of particles (Kuhn & Bagi, 2004, Tordesillas, 2008, etc.)
…as opposed to sliding.
But rolling is constrained!
Setup
00.1
00.5
0
Rotation propagation
0
distance d
0.25
0.5
0.75
1.0
0.05 100
Transmission distance increases with:
-increasing friction
-increasing width of the size distribution
-(weakly) with decreasing pressure
Size distribution
Directional dependence
( )ˆˆ( )
i ii i
ii S R
b fR
s= T s b
( )ˆˆ
i ii i
ii S R
b ftr
s= T s b
ˆˆ( , ) i
i ii i
i
b fW
s
= s b
is
ib
R
Examples
Rotation transmission distance• Intrinsic length
• Directional
• Depends on force chains
• Depends on – particle size distribution
– friction
– pressure.
• …nevertheless – for typical experimental values ~ 5-10 particles (1/2 of a shear band?)