Entropy and Material Instability in the Quasi-static Mechanics of Granular Media

Transcription

1 Entropy and Material Instability in the Quasi-static Mechanics of Granular Media J.D. Goddard Department of Aerospace & Mechanical Engineering University of California, San Diego, USA ABSTRACT Starting from a maximum-entropy model of granular statics, this brief note explores a possible material instability in the form of a stress localization anticipated in previous work (Goddard, 2002). After a brief review of the maximum-entropy model, it is shown that a special case allows for non-convex pressure-volume response of a kind that could lead to heterogeneous stress states in an isotropically compressed granular packing. 1 STATISTICS OF KINEMATICS AND STRESS Maximum-entropy estimates for the quasi-static mechanics of granular media date back at least to the pioneering work of Kanatani (1980) on the voidage of 2D granular packings. Recent works (Bagi, 1997; Kruyt & Rothenburg, 2002; Kruyt, 2003; Goddard, 2004a) treat both stress and infinitesimal strain, the latter pointing out the necessity of some specification of a priori probability in the relevant state space. This fact is already recognized in the work of Kanatani (1980), who refers to it as the density of states (a term also employed in quantum mechanics). 1.1 Delaunay triangulation, deformation and stress Following previous works, Kanatani (1980); Satake (1992); Bagi (1996), we assign to a granular assembly a (Satake) graph, which we associate with a Delaunay triangulation. The graph consists of a network of vertices or nodes, representing particle centroids, connected by edges or bonds, representing nearest-neighbor pairs. The latter correspond to real and latent mechanical contacts and define the edges of an in Exadaktylos, G.E. and Vardoulakis, I. G. (eds), Bifurcations, Instabilities,Degradation in Geomechanics (Proc. 7th International Workshop, Chania, Crete, 2005), Springer Verlag, 2007, pp

2 2 J.D. Goddard elementary space-filling volumes known as Delaunay simplices. In space dimension d the Delaunay simplex represents the minimal cluster of particles for which a d- volume can be assigned, with d + 1 vertices connected d(d + 1)/2 edges, defining triangles in 2D, tetrahedra (Fig. 1) in 3D, etc. The Delaunay triangulation in any dimension d suffices to define the global stress and kinematics of a granular assembly. g 3 g 2 g 1 Fig. 1. Delaunay simplex (tetrahedron) in 3D, with vertices representing grain centroids and edges representing contacts, actual (solid lines) and latent (dotted lines). Basis vectors g i, foreshortened for clarity, have length of Delaunay edges. Stress and deformation for an assembly of rigid grains involve involve contributions from the forces and relative motion between particles. We restrict attention to globally homogeneous deformations, where the difference between particle rotation and mean material rotation (Cosserat effect) is negligible. Hence, we can focus on the kinematics and stress associated with Delaunay edges. As in Goddard (2004b), the global or effective velocity gradient L =( v) T of a granular assembly is provisionally taken to be the volume-weighted average over simplexes s =1, 2,... L L s := 1 V s L s (1) V where V s is the volume of simplex s and s V = s V s (2) is that of the assembly. For statistically homogeneous assemblies, averages of the type (1) can be identified with ensemble averages based on an appropriate probability measure. The relation (1) written in terms of a given edge, say i =1in simplex s, is

3 Entropy & Material Instability 3 L 1,s = 1 V s u 1 a 1,s (3) where u 1 ġ 1 is the difference in velocity at the vertices connected by this edge, and a 1,s = V s g 1 = 1 2 g 2 g 3, V s = 1 2 g 1 g 2 g 3 (4) are, respectively, the area vector of the opposite face and the volume of the simplex. (a) (b) Fig. 2. Edge (contact) complexes in (a) 2D, (b) 3D. As illustrated in Fig 2, we identify the simplicial edge complex c(e) for a given edge e as the set of simplices having e as common edge. Replacing subscript 1 in (3) by e, we obtain the contribution of an arbitrary edge e to the global velocity gradient: L e = 1 V e s c(e) where a e = V s L e,s = 1 V e u e a e (5) s c(e)a e,s, l e a e = s ce V s = V e The corresponding volume-average velocity gradient is given by Stress and stress power L = 1 V e L e = 1 u e a e (6) V V e The conventional expression for granular (Cauchy) stress as dipolar force density can be expressed in terms of edge contributions as T = 1 V e e V e T e, where T e = f e l e V e, (7) where f e is interparticle force at particle contact represented by edge e. At this level, (5) and (7) provide the correct expression for stress-power:

4 4 J.D. Goddard T e :L e = f e u e V e, where A:B = tr(ab T )=A α β B α β (8) By contrast, it is generally recognized that the corresponding expression T : L, with T defined by (7) and L by (6), does not represent global stress power, owing essentially to micro-level heterogeniety. The latter requires additional work terms to account higher-order moment stresses and gradients (Goddard, 2006), reconsidered briefly below. 2 MAXIMUM ENTROPY & VIRTUAL THERMODYNAMICS Following Goddard (2004a), letting z Ω denote a representative point in the relevant state (or phase) space Ω, having probability measure P (z)dω(z), where dω(z) is an elemental state-space measure that remains to be specified. P (z) may depend on time, bu we restrict attention here to spatially homogeneous systems, such that P (z) and dω(z) are independent of spatial position. Statistical averages of arbitrary mechanical variables A(z) are then given by A = A(z)P (z)dω(z), (9) with relation between probability densities Ω P (z)dω(z) =ρ(z)dv (z), with dv (z) := dz 1 dz 2... dz n, dω(z) =J(z)dV (z), and ρ(z) =J(z)P (z) (10) connecting P and the probability density ρ in dv (z). Specification of J, the density of states (Kanatani, 1980), is essential to the determination of ρ(z). Several examples are considered elsewhere (Goddard, 2004a), where it is conjectured that J generally should represent a dynamically invariant measure determined by the micromechanics. The standard statistical-thermodynamical estimate for the unknown probability distribution P (z) is based on maximization of the entropy functional: S[P ]= log P = P (z) log P (z)dω(z) (11) subject to a discrete set of constraints of the form Λ(z) = const. For either constrained stress or constrained kinematics, the expressions (6) and (7) can be put into the common form Goddard (2004b) Ω A = n c M(x, y), with M(x, y) := x y, (12) with sums interpreted in terms of ensemble averages over contacts or edges, and with n c denoting contact-number density. With state space is now defined by z = x y, the maximization of (11) subject to stationarity of A yields the canonical distribution

5 Entropy & Material Instability 5 P (x, y) =Z 1 exp { βλ:m} = Z 1 exp { βx Λ y}, (13) where Z is (the partition function): Z(Λ) = exp { βx Λy} dω(x, y), (14) Ω a function of a tensor-valued Lagrange multiplier βλ. The arbitrary constant β, having units of inverse energy or power, plays the role of temperature in the usual thermodynamic formalism (Goddard, 2004b). This same formalism dictates that all macroscopic properties be derivable from Z, in particular A = Λ χ, with χ(β, Λ) = n c β 1 log Z(β, Λ), (15) with χ assuming the role of potential-energy function and Λ denoting partial derivative at constant β. Table 1. Constraints and variables constraint x y Λ A χ stress f l L T(β, L) ψ(β, L) deformation u a T L(β, T) ϕ(β, T) Table 1, from Goddard (2004b), summarizes the complementary relations between stress T and velocity or displacement gradient L. With a further restriction to infinitesimal deformation, one may interpret L as displacement gradient. The relation (15) represents a virtual thermomechanics, with ψ serving as analog of Helmholtz free energy and ϕ as its complementary energy. The standard thermodynamic interpretation of Table 1 would require that the measures dω(f, l) and dω(u, a) be such as to satisfy the (Legendre) relation: ψ(β, L)+ϕ(β, T) =T : L (16) As also pointed out by Goddard (2004b), the general validity of (15) appears to hinge on the possibility of capturing both elastic and frictional effects (conservative and non-conservative forces) via an elastic-plastic decomposition of the type employed in well-known incremental plasticity theories. However, such decompositions impose severe restrictions on particle-level statistical mechanics and need further micromechanical justification. 3 MATERIAL INSTABILITY As the main contribution of the present article, we consider the possible nonconvexity (Goddard, 2002, 2004b) of the functions ψ or ϕ as a condition for material

6 6 J.D. Goddard instability, with non-unique relation between stress and deformation. The analog of thermodynamic phase transition, this could allow for the well-known strain localization or its less familiar counterpart, stress localization. To illustrate the latter, we consider the simplest case of a static isotropic compression, with T = p1 as constraint and t = n c u a as local compressive volumetric strain, which we shall allow to take on values in [0, ). Then, by means of (15) and the second row of Table 1, the nondimensional pressure s = βp/n c is given as implicit function of global volumetric strain ɛ V by: ɛ V = 0 e st tj(t)dt 0 e st J(t)dt, (17) To make further progress, we assume a local density of states J(t) of the general form J(t) =t ν [J 0 + J 1 t J n t n ], ν> 1, n =0, 1,..., (18) which gives an (n + 1)th-degree polynomial for s: where P 0 s n+1 + P 1 s n P n+1 =0, (19) P k =(ν + k)... (ν + 1)(J k 1 J k ɛ V ),k=0..., n +1,J 1 = J n+1 =0 (20) By Descartes rule of sign, one can readily predict the number of real positive roots s = βp/n c, and this prediction is independent of the exponent ν in (18). For example, with n =2and J(t) > 0 for t 0, then J 0 > 0, and we obtain three positive roots for J 2 > 0,J 1 <J 2 ɛ V,J 0 >J 1 ɛ V. If J 1 > 0, this reduces to J k > 0, k =0, 1, 2, with J 1 /J 2 <ɛ V <J 0 /J 1, (21) a situation depicted schematically in Fig. 3 (Goddard, 2002). Presumably, the above multiplicity could lead to a two-phase granular structure such as that found in the numerical simulations of Radjai et al. (1998), consisting of an isotropic high-pressure network of force chains, supported laterally by an amorphous granular phase having much smaller contact forces and pressure. Although not pursued here, the simplicity of the above construct suggests a similar construct for multiple stress states involving shear stress with anisotropic twophase structures like those of Radjai et al. (1998). Of course, a more fundamental micromechanical analysis is needed to justify phenomenological models like (18) for the density of states. In the author s opinion, a particle-chain buckling instability with local densification represents a plausible mechanism. In closing, we note that there is a further interesting question as to length scales of the conjectured two-phase structures. While a micromechanical analysis is obviously needed to establish the implied spatial correlations, one can anticipate that these must signal the emergence of moment stresses T αβγ,... and conjugate kinematic quantities L αβγ,... (Goddard, 2006).

7 Entropy & Material Instability 7 p ε V Fig. 3. Schematic of pressure vs. volumetric compressive strain for conditions in (21). One can further anticipate that the maximum entropy principal would add terms T αβγ u α a β a γ,... or L αβγ f α l β l γ,... to the canonical exponential form (13). However, it is not clear how the corresponding constraints are to be imposed in a granular medium with nominally homogeneous stress or deformation. 4 CONCLUSIONS The preceding analysis suggests that maximum-entropy estimates of quasi-static granular stress and kinematics, together with a relatively simple phenomenological model for the density of states, can provide a model of material instability with twophase structure. Further micromechanical modeling is required to justify the purely phenomenological forms assumed above and to determine the relevant mesoscopic length scales. Acknowledgement This work is an outgrowth of research supported in part by National Aeronautics and Space Administration Grant NAG References K. Bagi (1996). Stress and strain in granular assemblies. Mechanics of Materials 22(3): K. Bagi (1997). Analysis of micro-variables through entropy principle. In R. Behringer & J. T. Jenkins (eds.), Powders and Grains, pp Balkema, Rotterdam.

8 8 J.D. Goddard J. D. Goddard (2002). Material instability with stress localization. In J. Labuz & A. Drescher (eds.), Bifurcations and Instabilities in Geomechanics, pp Balkema, Lisse. J. D. Goddard (2004a). On entropy estimates of contact forces in static granular assemblies. International Journal of Solids and Structures 41(21): J. D. Goddard (2004b). On maximum-entropy estimates for granular statics. In J. e. a. Vermeer (ed.), 2nd Intl. Symp. on Continuous and Discontinuous Modelling of Cohesive Frictional Materials, pp Balkema, Lisse, University of Stuttgart, Germany, September 27-28, J. D. Goddard (2006). From granular matter to generalized continuum. In P. Mariano, G. Capriz, & P. Giovine (eds.), Mathematical models of granular matter,, pp. 22pp., to appear. Springer, Berlin. K.-I. Kanatani (1980). An entropy model for shear deformation of granular materials. Letters in Applied Engineering Science (International Journal of Engineering Science) 18: N. P. Kruyt (2003). Contact forces in anisotropic frictional granular materials. International Journal of Solids and Structures 40(13-14): N. P. Kruyt & L. Rothenburg (2002). Probability density functions of contact forces for cohesionless frictional granular materials. International Journal of Solids and Structures 39(3): F. Radjai, D. E. Wolf, M. Jean, & J. J. Moreau (1998). Bimodal character of stress transmission in granular packings. Physical Review Letters 80(1):61 4. M. Satake (1992). Discrete-mechanical approach to granular materials. International Journal of Solids and Structures 30(10):