Joe Goddard, Continuum modeling of granular media

Transcription

1 Joe Goddard, 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.

2 Continuum Modeling of Granular Media Joe Goddard 1 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 July JDG in references to follow

3 Typical granular media (d 10 µm) Real granular media DEM simulation of a granular medium & force chains ( fabric )

4 Outline Premise: Granular media, representing disordered, mechanically non-linear, polydisperse, and athermal systems, pose a challenge to traditional methods of homogenization. Present Contribution 2 : 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): , Items in green font are not covered in that review.

5 Characteristic Parameters, Variables and Non-dimensional Groups Grain properties: ρ s, µ s grain density and intergranular friction coefficient d, G s representative grain diameter and elastic modulus e c, p s collisional restitution coeff. and confining pressure φ, η f granular volume fraction pore fluid viscosity A, Z =tr(a) granular fabric tensor and coordination number γ, γ D characteristic strain from rest and strain rate Characteristic elastic, inertial, viscous, & Knudsen numbers: E = G s /p s, I = γλ, H = η f γ/p s, K = d/l, where λ = d ρ f /p s is an inertial/frictional relaxation time, and L is a macro-length scale or inverse spatial gradient. NB: E for geomaterials at atmospheric confining pressures, and the transition from quasi-static to grain-inertial flow takes place over the range I This talk focuses largely on dry granular media H = 0 and simple-material models with K << 1.

6 Characteristic Parameters, Variables and Non-dimensional Groups Grain properties: ρ s, µ s grain density and intergranular friction coefficient d, G s representative grain diameter and elastic modulus e c, p s collisional restitution coeff. and confining pressure φ, η f granular volume fraction pore fluid viscosity A, Z =tr(a) granular fabric tensor and coordination number γ, γ D characteristic strain from rest and strain rate Characteristic elastic, inertial, viscous, & Knudsen numbers: E = G s /p s, I = γλ, H = η f γ/p s, K = d/l, where λ = d ρ f /p s is an inertial/frictional relaxation time, and L is a macro-length scale or inverse spatial gradient. NB: E for geomaterials at atmospheric confining pressures, and the transition from quasi-static to grain-inertial flow takes place over the range I This talk focuses largely on dry granular media H = 0 and simple-material models with K << 1.

7 τ/p s Regimes of Granular Flow μ C b a I II III X µ C Eγ/(Eγ + µ C ) + I 2 Regime τ scaling I. Quasi-static:(Hertz-Coulomb) elastoplastic solid a. (Hertz) elastic G s γ b. (Coulomb) elastoplastic µ C II. Dense-rapid: viscoplastic liquid 3 p s f (I) III. Rarified-rapid: (Bagnold ) viscous granular gas ρ s d 2 γ 2 η B γ 3 Numerical simulations of Campbell (2002) and Chialvo et al. (2012) indicate a dependence of f on E for soft particles with E 1. X

8 Conceptual Model Maxwellian spring-dashpot composed of Herztian spring, Reynolds-Rowe dilatant slide block plus Bagnold dashpot: p s μ C ~ μ E ~ 0 η B η P μ E T p s solid confining pressure [Pa] T = [τ ij ] = T p s I = T E = T P stress tensor [Pa] µ E = [µ E ijkl ] incremental elastic moduli [Pa] µ C = [µ C ijkl ] Coulomb modulus [ ] η B = [η B ijkl ] Bagnold viscosity [Pa-s] η P = p s µ C / D P + η B Plastic viscosity [Pa-s] D P = D D E Viscoplastic deformation rate [s 1 ], where various moduli depend on T or D P.

9 Viscoplasticity and Hypoplasticity Ansatz D = sym( v) = D P + D E, together with inverse D P (T) of T = η P :D P, with η P = p s µ C / D P + η B (D P ) leads to Lagrangian ODE (LODE) governing stress evolution: T= µ E (T):D E = µ E (T):(D D P (T)), with µ E = ( 2 Tϕ E ) 1, 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 = D P (T). In quasi-static regime I 0, reduce to standard incremental, rate-independent plasticity: T= µ E (T):D N(T, ˆD) D (homogeneous degree 1 in D) Added dependence of N on ˆD necessary to describe elastic unloading from a yield surface. Otherwise a special case of hypoplasticity with symmetric µ E.

10 Viscoplasticity and Hypoplasticity Ansatz D = sym( v) = D P + D E, together with inverse D P (T) of T = η P :D P, with η P = p s µ C / D P + η B (D P ) leads to Lagrangian ODE (LODE) governing stress evolution: T= µ E (T):D E = µ E (T):(D D P (T)), with µ E = ( 2 Tϕ E ) 1, 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 = D P (T). In quasi-static regime I 0, reduce to standard incremental, rate-independent plasticity: T= µ E (T):D N(T, ˆD) D (homogeneous degree 1 in D) Added dependence of N on ˆD necessary to describe elastic unloading from a yield surface. Otherwise a special case of hypoplasticity with symmetric µ E.

11 Bipotential Representation The above results follow readily from elastic free energy ϕ E (T), with elastic strain E = T ϕ E, and a viscoplastic (Edelen) dissipation potential ϕ D (T), with dissipative strain rate D P = T ϕ P, so that: T= µ E :(D T ϕ D (T)), with µ E = 2 Tϕ E (T) Rate-independent plasticity implies D P = D P ˆD P (T), with indeterminate D P to be specified in terms of applied deformation rate D. (Hypoplasticity takes D P = 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.

12 Bipotential Representation The above results follow readily from elastic free energy ϕ E (T), with elastic strain E = T ϕ E, and a viscoplastic (Edelen) dissipation potential ϕ D (T), with dissipative strain rate D P = T ϕ P, so that: T= µ E :(D T ϕ D (T)), with µ E = 2 Tϕ E (T) Rate-independent plasticity implies D P = D P ˆD P (T), with indeterminate D P to be specified in terms of applied deformation rate D. (Hypoplasticity takes D P = 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.

13 Bipotential Representation The above results follow readily from elastic free energy ϕ E (T), with elastic strain E = T ϕ E, and a viscoplastic (Edelen) dissipation potential ϕ D (T), with dissipative strain rate D P = T ϕ P, so that: T= µ E :(D T ϕ D (T)), with µ E = 2 Tϕ E (T) Rate-independent plasticity implies D P = D P ˆD P (T), with indeterminate D P to be specified in terms of applied deformation rate D. (Hypoplasticity takes D P = 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.

14 Evolutionary Parameters p s X(t) μ C ~ μ E η B η P μ E 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 = [A ij ],... Enlarged set Y = {T, X } requires evolution equation (LODE): Y= f(y, D)

15 Evolutionary Parameters p s X(t) μ C ~ μ E η B η P μ E 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 = [A ij ],... Enlarged set Y = {T, X } requires evolution equation (LODE): Y= f(y, D)

16 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 ϕ D (whose existence is guaranteed by the work of Edelen, ). 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: U = X, U E = 2 Fϕ E F, U P = F ϕ D, with potentials ϕ E (T ), ϕ D (T ), where T = [T α ] = {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.)

17 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 ϕ D (whose existence is guaranteed by the work of Edelen, ). 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: U = X, U E = 2 Fϕ E F, U P = F ϕ D, with potentials ϕ E (T ), ϕ D (T ), where T = [T α ] = {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.)

18 Examples Neglecting elastic velocity U E of parameters, one obtains X = U P = F ϕ D (T, F), with F = X ψ E (T, X ), where ψ E is the (mixed) Legendre dual to ϕ E, 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) ψ D (V) = ψ D (V; X ) of ϕ D (T ), where V = {D, U} and T = {T, F} = V ψ D, with F = 0 = U ψ D (D, X ; X ) Invertibility of last equation gives X and T in terms of {D, X }. This is method proposed 5 for fabric evolution in viscoplastic fluid-particle systems with H > 0. 5 JDG, Acta Mech. 2014, involving invariants of the form tr(d n A m )

19 Examples Neglecting elastic velocity U E of parameters, one obtains X = U P = F ϕ D (T, F), with F = X ψ E (T, X ), where ψ E is the (mixed) Legendre dual to ϕ E, 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) ψ D (V) = ψ D (V; X ) of ϕ D (T ), where V = {D, U} and T = {T, F} = V ψ D, with F = 0 = U ψ D (D, X ; X ) Invertibility of last equation gives X and T in terms of {D, X }. This is method proposed 5 for fabric evolution in viscoplastic fluid-particle systems with H > 0. 5 JDG, Acta Mech. 2014, involving invariants of the form tr(d n A m )

20 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 = r 0 (ˆT): ^ ^ ^ D = s (T)D (T) 0 ^ s (T) 0 0 D = 0 ^ r(t) 0 ^ T ^ ^ T = r (T) 0 T 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.

21 Loose Ends (cont d) Edelen s theory allows for additive powerless or gyroscopic stresses or deformation rates 6 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 µ E and associated (Hill) second-order work criterion: T:D = D:µ E :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)

22 Loose Ends (cont d) Edelen s theory allows for additive powerless or gyroscopic stresses or deformation rates 6 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 µ E and associated (Hill) second-order work criterion: T:D = D:µ E :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)

23 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.

24 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.

25 Example: The Kamrin-Koval (2012-) Model Slight generalization of the K-K granular fluidity model: D = gs, where S = T /p s, p s = tr(t), g = D / S = I/λ S, with λ = d p s /ρ s. Fluidity g is functional g = G[S], of entire field S(x), given by the solution to the PDE ξ 2 2 g + g = g 0, with ξ = ξ(s), where g 0 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 ). How is it related to the above models? Moreau s theorem (J-J. Moreau,C. R. Acad. Sci., 255, , 1962) suggests the existence of a dissipation potential represented by convex functional ϕ D = 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?

26 Example: The Kamrin-Koval (2012-) Model Slight generalization of the K-K granular fluidity model: D = gs, where S = T /p s, p s = tr(t), g = D / S = I/λ S, with λ = d p s /ρ s. Fluidity g is functional g = G[S], of entire field S(x), given by the solution to the PDE ξ 2 2 g + g = g 0, with ξ = ξ(s), where g 0 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 ). How is it related to the above models? Moreau s theorem (J-J. Moreau,C. R. Acad. Sci., 255, , 1962) suggests the existence of a dissipation potential represented by convex functional ϕ D = 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?

27 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.