Joe Goddard, jgoddard@ucsd.edu
Continuum modeling of granular media
This talk summarizes a recent survey [1] of the interesting phenomenology and the prominent régimes of
granular flow which also offers a unified mathematical synthesis of continuum modeling. The unification is
based on “parametric” viscoelasticity and hypoplasticity involving elastic and inelastic potentials. Fully nonlinear, anisotropic visco-elastoplastic models are achieved by expressing the potentials as functions of the joint
isotropic invariants of kinematic and structural tensors. These take on the role of evolutionary parameters or
internal variables, whose evolution equations are derived from the internal balance of generalized forces. The
resulting continuum models encompass most of the mechanical constitutive equations currently employed for
granular media. Moreover, these models are readily modified to include Cosserat and other multipolar effects.
Several outstanding questions are identified as to the contribution of parameter evolution to dissipation, the
distinction between quasi-elastic and inelastic models of material instability, and the role of multipolar effects
in material instability, dense rapid flow and particle migration phenomena.
References
[1] J. Goddard, Appl. Mech. Rev. 64:5, 2014.
Background
Continuum Model
Continuum Modeling of Granular Media
Joe Goddard1
Department of Mechanical & Aerospace Engineering
University of California, San Diego
La Jolla, California, USA
Continuum Models of Disrete Systems - 13
Salt Lake City, Utah, USA
21-25 July 2014
1
”JDG” in references to follow
Examples, etc.
Background
Continuum Model
Examples, etc.
Typical granular media (d & 10 µM)
Real granular media
DEM simulation of a granular medium & force chains (”fabric”)
Background
Continuum Model
Examples, etc.
Outline
• Premise: Granular media, representing disordered,
mechanically non-linear, polydisperse, and athermal systems,
pose a challenge to traditional methods of homogenization.
• Present Contribution2 :
• Propose a constitutive framework based loosely on
representative microscopic parameters and broad enough
to encompass existing elasto-viscoplastic models, as
potential target of homogenization.
• Show how this can be simplified rigorously by the use of
elastic and dissipation potentials.
• Scope of this talk: Brief review of basic granular mechanics
and flow regimes, followed by a broad outline of the
continuum modeling with examples.
• ”Loose ends” and conclusions.
2
Based on JDG Appl. Mech. Rev. 66(5):050801-18, 2014. Items in
green font are not covered in that review.
Background
Continuum Model
Examples, etc.
Characteristic Parameters, Variables and Non-dimensional Groups
• Grain properties:
ρs , µ s
d, Gs
ec , ps
φ, ηf
A, Z =tr(A)
γ, γ̇ ∼ |D|
grain density and intergranular friction coefficient
representative grain diameter and elastic modulus
collisional restitution coeff. and confining pressure
granular volume fraction pore fluid viscosity
granular ”fabric” tensor and coordination number
characteristic strain from rest and strain rate
• Characteristic elastic, inertial, viscous, & Knudsen numbers:
E = Gs /ps , I = γ̇λ, H = ηf γ̇/ps , K = d/L,
p
where λ = d ρf /ps is an inertial/frictional relaxation time,
and L is a macro-length scale or inverse spatial gradient.
• NB: E ∼ 105 −106 for geomaterials at ∼ atmospheric confining
pressures, and the transition from quasi-static to grain-inertial
flow takes place over the range I ∼ 0.1−1.
• This talk focuses largely on dry granular media H = 0 and
”simple-material” models with K << 1.
Background
Continuum Model
Examples, etc.
Characteristic Parameters, Variables and Non-dimensional Groups
• Grain properties:
ρs , µ s
d, Gs
ec , ps
φ, ηf
A, Z =tr(A)
γ, γ̇ ∼ |D|
grain density and intergranular friction coefficient
representative grain diameter and elastic modulus
collisional restitution coeff. and confining pressure
granular volume fraction pore fluid viscosity
granular ”fabric” tensor and coordination number
characteristic strain from rest and strain rate
• Characteristic elastic, inertial, viscous, & Knudsen numbers:
E = Gs /ps , I = γ̇λ, H = ηf γ̇/ps , K = d/L,
p
where λ = d ρf /ps is an inertial/frictional relaxation time,
and L is a macro-length scale or inverse spatial gradient.
• NB: E ∼ 105 −106 for geomaterials at ∼ atmospheric confining
pressures, and the transition from quasi-static to grain-inertial
flow takes place over the range I ∼ 0.1−1.
• This talk focuses largely on dry granular media H = 0 and
”simple-material” models with K << 1.
Background
Continuum Model
Examples, etc.
Regimes of Granular Flow
τ/ps
b
μC
a
I
II
III
X ∼ µC Eγ/(Eγ + µC ) + I2
I.
a.
b.
II.
III.
3
Regime
Quasi-static:(Hertz-Coulomb)
elastoplastic “solid”
(Hertz) elastic
(Coulomb) elastoplastic
Dense-rapid: viscoplastic “liquid”3
Rarified-rapid: (Bagnold )
viscous “granular gas”
X
τ scaling
Gs γ
µC
ps f (I)
ρs d 2 γ̇ 2 ∼ ηB γ̇
Numerical simulations of Campbell (2002) and Chialvo et al. (2012)
indicate a dependence of f on E for soft particles with E ∼ 1.
Background
Continuum Model
Examples, etc.
Conceptual Model
Maxwellian spring-dashpot composed of Herztian spring,
Reynolds-Rowe dilatant slide block plus Bagnold dashpot:
ps
μC
~ ~
μ
E ~ 0
ηB
ps
T = [τij ] = T0 − ps I = TE = TP
µE = [µE ijkl ]
µC = [µC ijkl ]
η B = [ηB ijkl ]
η P = ps µC /|DP | + η B
DP = D − DE
ηP
μE
T
solid confining pressure
stress tensor
incremental elastic moduli
Coulomb modulus
”Bagnold” viscosity
Plastic viscosity
Viscoplastic deformation rate
where various moduli depend on T or DP .
[Pa]
[Pa]
[Pa]
[ ]
[Pa-s]
[Pa-s]
[s−1 ],
Background
Continuum Model
Examples, etc.
Viscoplasticity and Hypoplasticity
• Ansatz D = sym(∇v) = DP + DE , together with inverse
DP (T) of
T = η P : DP , with η P = ps µC /|DP | + η B (DP )
leads to Lagrangian ODE (LODE) governing stress evolution:
◦
2
−1
T= µE (T) : DE = µE (T) : (D − DP (T)), with µE = (∂T ϕE ) ,
◦
where ( ) is Jaumann (or other objective) rate, and
(ˆ) = ( )/|( )| is the director of a second-rank tensor. D serves
as control variable, and ”stiff” limit E → ∞ yields strictly
dissipative viscoplasticity D = DP (T).
• In quasi-static regime I → 0, reduce to standard incremental,
rate-independent plasticity:
◦
T= µE (T) : D − N(T, D̂)|D| (homogeneous degree 1 in D)
Added dependence of N on D̂ necessary to describe elastic
unloading from a yield surface. Otherwise a special case of
hypoplasticity with symmetric µE .
Background
Continuum Model
Examples, etc.
Viscoplasticity and Hypoplasticity
• Ansatz D = sym(∇v) = DP + DE , together with inverse
DP (T) of
T = η P : DP , with η P = ps µC /|DP | + η B (DP )
leads to Lagrangian ODE (LODE) governing stress evolution:
◦
2
−1
T= µE (T) : DE = µE (T) : (D − DP (T)), with µE = (∂T ϕE ) ,
◦
where ( ) is Jaumann (or other objective) rate, and
(ˆ) = ( )/|( )| is the director of a second-rank tensor. D serves
as control variable, and ”stiff” limit E → ∞ yields strictly
dissipative viscoplasticity D = DP (T).
• In quasi-static regime I → 0, reduce to standard incremental,
rate-independent plasticity:
◦
T= µE (T) : D − N(T, D̂)|D| (homogeneous degree 1 in D)
Added dependence of N on D̂ necessary to describe elastic
unloading from a yield surface. Otherwise a special case of
hypoplasticity with symmetric µE .
Background
Continuum Model
Examples, etc.
Bipotential Representation
• The above results follow readily from elastic free energy
ϕE (T), with elastic strain E = ∂T ϕE , and a viscoplastic
(Edelen) dissipation potential ϕD (T), with dissipative strain
rate DP = ∂T ϕP , so that:
◦
2
T= µE : (D − ∂T ϕD (T)), with µE = ∂T ϕE (T)
• Rate-independent plasticity implies DP = |DP |D̂P (T), with
indeterminate |DP | to be specified in terms of applied
deformation rate D. (Hypoplasticity takes |DP | = |D| vs.
conventional plasticity which employs yield surface to obtain
forms given above.)
• In every case, the bipotential structure
• guarantees thermodynamic admissibility
• reduces model to two scalar potentials (depending on
joint isotropic scalar invariants of the independent
variables), and
• describes material instability as loss of convexity of at
least one potential.
Background
Continuum Model
Examples, etc.
Bipotential Representation
• The above results follow readily from elastic free energy
ϕE (T), with elastic strain E = ∂T ϕE , and a viscoplastic
(Edelen) dissipation potential ϕD (T), with dissipative strain
rate DP = ∂T ϕP , so that:
◦
2
T= µE : (D − ∂T ϕD (T)), with µE = ∂T ϕE (T)
• Rate-independent plasticity implies DP = |DP |D̂P (T), with
indeterminate |DP | to be specified in terms of applied
deformation rate D. (Hypoplasticity takes |DP | = |D| vs.
conventional plasticity which employs yield surface to obtain
forms given above.)
• In every case, the bipotential structure
• guarantees thermodynamic admissibility
• reduces model to two scalar potentials (depending on
joint isotropic scalar invariants of the independent
variables), and
• describes material instability as loss of convexity of at
least one potential.
Background
Continuum Model
Examples, etc.
Bipotential Representation
• The above results follow readily from elastic free energy
ϕE (T), with elastic strain E = ∂T ϕE , and a viscoplastic
(Edelen) dissipation potential ϕD (T), with dissipative strain
rate DP = ∂T ϕP , so that:
◦
2
T= µE : (D − ∂T ϕD (T)), with µE = ∂T ϕE (T)
• Rate-independent plasticity implies DP = |DP |D̂P (T), with
indeterminate |DP | to be specified in terms of applied
deformation rate D. (Hypoplasticity takes |DP | = |D| vs.
conventional plasticity which employs yield surface to obtain
forms given above.)
• In every case, the bipotential structure
• guarantees thermodynamic admissibility
• reduces model to two scalar potentials (depending on
joint isotropic scalar invariants of the independent
variables), and
• describes material instability as loss of convexity of at
least one potential.
Background
Continuum Model
Examples, etc.
Evolutionary Parameters
ps
X(t)
μC
~
μ
E
ηP
ηB
μE
T
• Soil mechanics models and recent DEM simulations
(Kolymbas, Bauer, & Tejchman et al., Radjai et al., Sun et
al., Luding et al., ...) indicate need for additional evolutionary
parameters or ”internal variables” X = [X α ] = {φ, A, ...} to
define shear-induced changes in microstructure, including
compacity φ, fabric A = [Aij ], ...
• Enlarged set Y = {T, X } requires evolution equation (LODE):
◦
Y= f(Y, D)
Background
Continuum Model
Examples, etc.
Evolutionary Parameters
ps
X(t)
μC
~
μ
E
ηP
ηB
μE
T
• Soil mechanics models and recent DEM simulations
(Kolymbas, Bauer, & Tejchman et al., Radjai et al., Sun et
al., Luding et al., ...) indicate need for additional evolutionary
parameters or ”internal variables” X = [X α ] = {φ, A, ...} to
define shear-induced changes in microstructure, including
compacity φ, fabric A = [Aij ], ...
• Enlarged set Y = {T, X } requires evolution equation (LODE):
◦
Y= f(Y, D)
Background
Continuum Model
Examples, etc.
Bipotential (”GSM”) Model
• Constitutive model is reduced once again to two scalar
potentials, depending now on scalar invariants of variables and
parameters. This represents the ”generalized standard
material” (GSM) of Halphen and Nguyen (1975), based on a
phenomenological inelastic or dissipation potential ϕD (whose
existence is guaranteed by the work of Edelen, 1970-3).
• In contrast to arbitrary phenomenological forms for parameter
evolution (Kolymbas, Bauer, Tejchman et al., JDG), GSM
assigns dissipative and elastic forces to the parameters, with:
◦
◦
2
U =X , U E = ∂ F
ϕE · F , U P = ∂F ϕD ,
with potentials ϕE (T ), ϕD (T ), where T = [Tα ] = {T, F } is the
force conjugate to velocity V = [V α ] = {D, U },
• Previous Ansatz V = V E + V P , and absence of external forces
or inertia, yields balance of internal forces F = F P = F E and
evolution equations. (Examples below.)
Background
Continuum Model
Examples, etc.
Bipotential (”GSM”) Model
• Constitutive model is reduced once again to two scalar
potentials, depending now on scalar invariants of variables and
parameters. This represents the ”generalized standard
material” (GSM) of Halphen and Nguyen (1975), based on a
phenomenological inelastic or dissipation potential ϕD (whose
existence is guaranteed by the work of Edelen, 1970-3).
• In contrast to arbitrary phenomenological forms for parameter
evolution (Kolymbas, Bauer, Tejchman et al., JDG), GSM
assigns dissipative and elastic forces to the parameters, with:
◦
◦
2
U =X , U E = ∂ F
ϕE · F , U P = ∂F ϕD ,
with potentials ϕE (T ), ϕD (T ), where T = [Tα ] = {T, F } is the
force conjugate to velocity V = [V α ] = {D, U },
• Previous Ansatz V = V E + V P , and absence of external forces
or inertia, yields balance of internal forces F = F P = F E and
evolution equations. (Examples below.)
Background
Continuum Model
Examples, etc.
Examples
• Neglecting elastic velocity U E of parameters, one obtains
◦
X =
˙ U P = ∂F ϕD (T, F ), with F = ∂X ψE (T, X ),
where ψE is the (mixed) Legendre dual to ϕE , giving
◦
◦
X = g(T, X ), and ∴ Y= f(Y, D), where Y = {T, X }
• When elastic effects are negligible, it is convenient to employ
the Legendre-Fenchel dual (convex conjugate)
ψD (V) = ψD (V; X ) of ϕD (T ), where V = {D, U } and
◦
T = {T, F } = ∂V ψD , with F = 0 = ∂U ψD (D, X ; X )
◦
Invertibility of last equation gives X and T in terms of
{D, X }. This is method proposed5 for fabric evolution in
viscoplastic fluid-particle systems with H > 0.
5
JDG, Acta Mech. 2014, involving invariants of the form tr(Dn Am )
Background
Continuum Model
Examples, etc.
Examples
• Neglecting elastic velocity U E of parameters, one obtains
◦
X =
˙ U P = ∂F ϕD (T, F ), with F = ∂X ψE (T, X ),
where ψE is the (mixed) Legendre dual to ϕE , giving
◦
◦
X = g(T, X ), and ∴ Y= f(Y, D), where Y = {T, X }
• When elastic effects are negligible, it is convenient to employ
the Legendre-Fenchel dual (convex conjugate)
ψD (V) = ψD (V; X ) of ϕD (T ), where V = {D, U } and
◦
T = {T, F } = ∂V ψD , with F = 0 = ∂U ψD (D, X ; X )
◦
Invertibility of last equation gives X and T in terms of
{D, X }. This is method proposed5 for fabric evolution in
viscoplastic fluid-particle systems with H > 0.
5
JDG, Acta Mech. 2014, involving invariants of the form tr(Dn Am )
Background
Continuum Model
Examples, etc.
Examples and ”Loose Ends”
• Interesting special case of (Legendre-Fenchel) dual dissipation
potentials:
ψ(D) + ϕ(T) = T : D, with T = ∂D ψ, D = ∂T ϕ,
arises for ψ, ϕ homogeneous of degree p > 1, q > 1, resp.
• Requires (q − 1) = (p − 1)−1 , with rate-independent limit
p → 1 giving q → ∞ and 6-dimensional ”step-function”
associated with any limit surface |T| = r0 (T̂):
^ ^
^
D = s0 (T)D (T)
^
^
s0 (T)
0
^
r0(T)
T
^ ^
T = r0 (T) T
D=0
with D normal to that surface, i.e. normal plastic flow rule is
delivered ”free of charge”. But granular materials generally
do not obey this flow rule.
Background
Continuum Model
Examples, etc.
”Loose Ends” (cont’d)
• Edelen’s theory allows for additive ”powerless” or
”gyroscopic” stresses or deformation rates6 which may serve
to capture above failure of ”normality” and the
non-associative plasticity of granular media. Do these effects
have microscopic origins in Reynolds dilatancy?
• Material instability: In hypoplastic models, this seems to arise
from “quasi-elastic” (Hadamard) instability associated with
non-positivity of incremental modulus µE and associated
(Hill) “second-order work” criterion:
◦
T: D = D : µE : D ≤ 0
pursued extensively by Chambon, Nicot, Darve et al. (2005-).
In contrast, one may envisage strictly dissipative viscoplastic
instabilities arising from loss of convexity of the dissipation
potential (cf. complex fluids).
6
breakdown of non-linear Onsager symmetry (JDG, Acta Mech. 2014)
Background
Continuum Model
Examples, etc.
”Loose Ends” (cont’d)
• Edelen’s theory allows for additive ”powerless” or
”gyroscopic” stresses or deformation rates6 which may serve
to capture above failure of ”normality” and the
non-associative plasticity of granular media. Do these effects
have microscopic origins in Reynolds dilatancy?
• Material instability: In hypoplastic models, this seems to arise
from “quasi-elastic” (Hadamard) instability associated with
non-positivity of incremental modulus µE and associated
(Hill) “second-order work” criterion:
◦
T: D = D : µE : D ≤ 0
pursued extensively by Chambon, Nicot, Darve et al. (2005-).
In contrast, one may envisage strictly dissipative viscoplastic
instabilities arising from loss of convexity of the dissipation
potential (cf. complex fluids).
6
breakdown of non-linear Onsager symmetry (JDG, Acta Mech. 2014)
Background
Continuum Model
Examples, etc.
Examples (cont’d)
• Non-locality and gradient effects (K not << 1): Needed,
particularly in the quasi-static regime, to regularize the field
equations, e.g. to impart a length scale to shear bands arising
from material instability or slip at solid boundaries. Examples
are:
• ”Weakly” nonlocal models: Cosserat plasticity is treated
by Lippmann (1995), Mohan et al. (2002), and others
based on associated plastic potential, and Tejchman and
coworkers (2008-), employ empirical hypoplastic Cosserat
models in geomechanics. One simply enlarges set of
kinematic tensors and conjugate (hyper)stresses, to yield
LODEs discussed above. Boundary conditions are
required as part of the constitutive theory as in ”gradient
plasticity” models.
• Fully nonlocal models involve functionals connecting
spatial fields of kinematics and/or stress and are not
covered by the above models.
Background
Continuum Model
Examples, etc.
Examples (cont’d)
• Non-locality and gradient effects (K not << 1): Needed,
particularly in the quasi-static regime, to regularize the field
equations, e.g. to impart a length scale to shear bands arising
from material instability or slip at solid boundaries. Examples
are:
• ”Weakly” nonlocal models: Cosserat plasticity is treated
by Lippmann (1995), Mohan et al. (2002), and others
based on associated plastic potential, and Tejchman and
coworkers (2008-), employ empirical hypoplastic Cosserat
models in geomechanics. One simply enlarges set of
kinematic tensors and conjugate (hyper)stresses, to yield
LODEs discussed above. Boundary conditions are
required as part of the constitutive theory as in ”gradient
plasticity” models.
• Fully nonlocal models involve functionals connecting
spatial fields of kinematics and/or stress and are not
covered by the above models.
Background
Continuum Model
Examples, etc.
Example: The Kamrin-Koval (2012-) Model
• Slight generalization of the K-K ”granular fluidity” model:
D0 = g S, where S = T0 /ps , ps = −tr(T), g = |D0 |/|S| = I/λ|S|,
p
with λ = d ps /ρs . ”Fluidity” g is functional g = G[S], of
entire field S(x), given by the solution to the PDE
ξ 2 ∇2 g + g = g0 , with ξ = ξ(S),
where g0 is local form of the relation I(S) pioneered by
Pouliquen et al. (2002-), and ξ is a length scale that diverges
as |S| approaches its yield value from above.
• Simple form of the above model gives remarkable agreement
with various steady simple shearing (”viscometric flow”)
experiments (Kamrin et al. 2002- ). How is it related to the
above models?
• Moreau’s theorem (J-J. Moreau,C. R. Acad. Sci., 255, 238-40,
1962) suggests the existence of a dissipation potential
represented by convex functional ϕD = F[S] such that
g = G = δF/δS(x). Does the K-K PDE imply such convexity?
Can other non-local models be represented by convex
functionals?
Background
Continuum Model
Examples, etc.
Example: The Kamrin-Koval (2012-) Model
• Slight generalization of the K-K ”granular fluidity” model:
D0 = g S, where S = T0 /ps , ps = −tr(T), g = |D0 |/|S| = I/λ|S|,
p
with λ = d ps /ρs . ”Fluidity” g is functional g = G[S], of
entire field S(x), given by the solution to the PDE
ξ 2 ∇2 g + g = g0 , with ξ = ξ(S),
where g0 is local form of the relation I(S) pioneered by
Pouliquen et al. (2002-), and ξ is a length scale that diverges
as |S| approaches its yield value from above.
• Simple form of the above model gives remarkable agreement
with various steady simple shearing (”viscometric flow”)
experiments (Kamrin et al. 2002- ). How is it related to the
above models?
• Moreau’s theorem (J-J. Moreau,C. R. Acad. Sci., 255, 238-40,
1962) suggests the existence of a dissipation potential
represented by convex functional ϕD = F[S] such that
g = G = δF/δS(x). Does the K-K PDE imply such convexity?
Can other non-local models be represented by convex
functionals?
Background
Continuum Model
Examples, etc.
Conclusions
• Evolutionary elasto-viscoplastic models offer a convenient
framework for the continuum modeling of granular mechanics.
• Formulation is facilitated by a ”bipotential” structure,
involving elastic and inelastic potentials anticipated by the
long-standing ”GSM” model. Shouldn’t homogenization be
based on such potentials?
• Dependence of the potentials on parameters or ”internal
variables”, such as granular compacity and fabric, can give
internal forces balance governing parameter evolution.
• Models are readily generalized to include gradient effects in
”weakly non-local” models.
• Can they be extended to fully non-local models via potential
functionals? This could lend variational a structure that may
be useful in homogenization.