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Extended constraints, arches and soft modes in granular materials
Journal of Non-Crystalline Solids 352 (2006) 4505–4508 www.elsevier.com/locate/jnoncrysol
Extended constraints, arches and soft modes in granular materials N. Rivier
*
IPCMS, Universite´ Louis Pasteur, 67084 Strasbourg cedex, France Available online 22 August 2006
Abstract A granular material is a network of fixed length edges linking two grains in contact, and flexible hinges if the grains can rotate without sliding on each other. The network is then dynamically unfrustrated: it can deform freely under shear and behaves like a dry fluid. A sufficient condition for non-frustration in 3D is that all circuits of grains in contact are even. Frustration is concentrated on odd circuits, and the ‘odd vorticity’ forms loops in 3D. Thus, some constraints in the network cannot be accounted for by local mean-field analysis. These extended constraints and the means of accounting for them are discussed. A granular packing can be described as a graph, with a connection carried by the edges. If all circuits of grains in contact are even, the connection is pure gauge, independent of the path between grains. When shear stress is applied over a large scale, it can break the frustrating odd circuits by disconnecting selected grains in contact, leaving all other contacts and even circuits unaffected. The granular material deforms freely like dry quicksand, as occurs in silent earthquakes, soil liquefaction, wet sand under foot (dilatancy) and, to some extent, in geological faults. Ó 2006 Elsevier B.V. All rights reserved. PACS: 83.80.Fg; 45.70.n; 64.70.Pf; 02.10.Ox; 46.25.y Keywords: Glass transition; Jamming; Stability; Granular materials; Intermediate phase
1. Introduction It is convenient to describe dry, hard granular matter by a graph. Two grains in hard, repulsive contact are connected by an edge. There are no attractive, cohesive forces between grains and the packing is held together by external forces (gravity or boundaries). The arches, chains of grains in contact, are prevented to buckle by the presence of odd circuits of grains in contact, which, in 3D, are sufficient to insure stability [1]. This role of parity explains why there can be no simple continuous description of granular matter. A granular material that contains only even circuits of grains in contact, makes up a novel state of matter, a dry fluid. It is a non-viscous fluid that does not resist shear, whose constituents roll without slip on each other or on any boundary or inclusion (the point of contact between any two grains has zero velocity, even though it changes *
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all the time). It only feels hydrostatic pressure, i.e. forces normal to any plane inside the material on any scale larger than a few grains; a solid inclusion is subject to Archimedean pull only; arches can buckle freely and make ripples on free boundaries. The packing is bichromatic: every red grain has only blue neighbors, consistently throughout. Examples of dry fluids are aerated sand [2,3], and numerical space-filling bearings [4]. An edge of the graph represents two hard grains (hard spheres, say) in contact. The contact force is scalar, repulsive and infinite. The edges are struts that carry forces with a sign constraint. Thus, the edges are geometrical and boolean (they exist or not) objects of fixed length. The extension of structural rigidity concepts to granular contact networks has been discussed first by Guyon et al. [5], and recently, by Moukarzel [6,7], who argued that a granular packing, represented as a finite network of struts with a sign constraint, is exactly isostatic (minimally rigid, exactly at the rigidity percolation threshold) in the limit of large stiffness-to-load ratio, thus for a hard granular material subject to moderate external forces. We will qualify this argument, and show
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N. Rivier / Journal of Non-Crystalline Solids 352 (2006) 4505–4508
the existence, below the percolation threshold, of rigid, underconstrained networks stabilized by odd circuits. Hinges (two edges incident on the same vertex) are free of constraints if and only if the grains in contact can roll on each other without slip. Then, arches and circuits are unstable and can buckle. A circuit of the graph is a skew, s-sided polygon that remains closed by definition. It is unstable and the seat of soft rotation modes if s is even. The condition is only sufficient in 3D, but necessary and sufficient in 2D. Thus, odd circuits are all stable because the grains in contact cannot roll on each other without slip, and the hinges are blocked. Frustration is geometric but it affects the dynamics or stability of the circuit, not its statics. By contrast, only 3-sided circuits are stable according to the classic, mean-field constraint-counting argument of Maxwell–Phillips–Thorpe (recent review: [8]). Hard granular materials differ in two ways from the structural rigidity analysis of elastic networks, from floppy to overconstrained, through isostatic at the rigidity percolation threshold. (a) Stresses carried by edges have a sign constraint. It follows that (repulsive) self-stresses cannot be sustained in an overconstrained network: the two grains connected by a redundant edge pull apart, and the network becomes isostatic. On the other hand, a floppy network will rearrange and add edges, if it can, to become isostatic ([7], my italics). (b) But hinges in a floppy network can only buckle through grains rolling on each other, a motion blocked by odd circuits. Thus, as a function of the number of edges, a hard granular material ranges from a dry fluid (isostatic network without odd circuits) to the unrestricted, isostatic network of rigidity percolation, through an intermediate phase [9,10] of underconstrained, but dynamically rigid networks with odd circuits. 2. Granular materials as graphs 2.1. Vertices, edges, circuits A granular packing can be represented as a graph with the grains as vertices, and straight edges linking (spherical) grains through their point of contact. For hard, incompressible grains, the edges have a fixed length, otherwise they do not exist: grains lose contact as soon as they move apart. An applied stress may rotate an edge, or it destroys it by pulling apart the two grains. When two grains roll on each other without slip, the edge carries a connection a that relates the rotations Rixi of the two grain. The hinge between two edges incident on the same vertex is flexible as the points of contact change upon rotation of the grains. Edges constitute arches and circuits. The arches are stable under compression, but may buckle under rotation of the hard grains. Stability is the absence of soft buckling mode(s) of zero energy or zero frequency. A stable granular packing is called dynamically frustrated. It is a stronger condition than the static stability of grain heaps, as one of the villains in the film Witness discovered to his cost. Dry fluids are dynamically unfrustrated. In a dry granular
material, the non-slip rotation of grains is blocked by odd circuits [1]. The edge linking grains i and i + 1 is represented by the vector Ri,i+1 = (Ri + Ri+1)ti,i+1, with fixed length Ri + Ri+1 and unitary directional vector ti,i+1. With time t, it can rotate at a rate /i,i+1 around the axis ki,i+1, thus dRi;iþ1 =dt ¼ /i;iþ1 ðRi þ Riþ1 Þni;iþ1 ; thereby defining a local orthonormal frame (t,n,k) for each edge (i,i + 1), with (k^t) = n, etc. Thus, dt=dt ¼ /ðk^tÞ ¼ /n;
dn=dt ¼ /ðk^nÞ ¼ /t:
(The local frame is not Frenet’s because it is defined through time derivative on a discrete polygonal curve, rather than as derivative along the curve.) 2.2. Closure relations A circuit of s edges is the skew polygonal curve in the t’s, RRi;iþ1 ¼ RðRi þ Riþ1 Þti;iþ1 ¼ 0 ðR from i ¼ 1 to sðs þ 1 1ÞÞ: Also; RdRi;iþ1 =dt ¼ R/i;iþ1 ðRi þ Riþ1 Þni:iþ1 ¼ 0 is an orthogonal, skew polygon in the n’s. Higher time derivatives contain combinations of t and n. Replacing vertices and edges by spherical grains in contact, we will obtain a third polygon in the k’s (Eq. (4) below), Rð1Þi /i;iþ1 ðRi þ Riþ1 Þki;iþ1 ¼ 0 (for s even). 2.3. Rolling without slip The condition for rolling without slip is that the two grains have the same velocity at the point of contact v1 + x1^(R1t12) = v2 + x2^(R2t12), with the velocities of the centers of the two grains related by v2 = v1 + dR12/dt. The non-slip condition is a relation between the angular rotation velocity vectors x of the two spheres in contact: (R1x1 + R2x2)^t12 = dR12/dt. With R1 x1 þ R2 x2 ¼ a12 t12 b12 n12 c12 k12 ; one has b = 0, c12 = /12(R1 + R2), but a remains an arbitrary coefficient of connection between x1 and x2. Consider first the situation of a bearing, where the centers of the grains are at rest (/ = 0). In 2D (plane polygons of cogwheels), R1x1 = R2x2 (a = 0), the axes of rotation are collinear, the angular velocities have opposite signs (different colors) and a necessary and sufficient condition for non-slip rotation is that all circuits are even. In 3D, where neither are the centers of the grains coplanar nor the axes of rotation collinear, the same condition holds, but it is only sufficient [1]. The non-slip condition, R1x1 + R2x2 = a12t12 defines a connection a between two spheres in contact that gives R2x2 in terms of R1x1, then R3x3 in terms of R2x2, etc. Around a circuit with s edges, s + 1 1, R1 x1 þ ð1Þs R1 x1 ¼ Rð1Þi ai;iþ1 ti;iþ1 :
N. Rivier / Journal of Non-Crystalline Solids 352 (2006) 4505–4508
If s is even, it yields the identity, i
Rð1Þ ai;iþ1 ti;iþ1 ¼ 0:
ð1Þ
The connection a is carried from one sphere to the next, and, around a circuit, back to the initial sphere. The connection, i
ai;iþ1 ¼ cð1Þ ðRi þ Riþ1 Þ
ð2Þ
is consistent (‘pure gauge’) for a circuit, regardless of the starting sphere, for all circuits involving the same sphere, and for all contact paths between any two spheres. It reduces identity (1) to closure of the polygonal circuit. c is a constant for the whole packing (c = 0 implies that the axes of rotation of all the grains are collinear). If grain centers move, two grains in non-slip contact are connected by the relation R1 x1 þ R2 x2 ¼ a12 t12 þ /12 ðR1 þ R2 Þk12 :
4507
loops, such a surface is the locus of all the critical contacts to be broken. It is therefore sufficient, and most economical, to separate those contacts located on the (minimal) soap film(s) attached to the R-loops of the material. The separation of the critical contacts requires a little space: the unfrustrated packing has less contacts than the original. In practice, this is done by aerating gently the granular material before the start of the experiment [2], by injecting water in soils, as in silent earthquakes and soil liquefaction [13], or by applying shear from an extensive boundary, as in the granular fluids of [3] (here, shear is applied throughout the experiment), in geological faults with sliding tectonic plates [14] or in the phenomenon of dilatancy [15,16]. 4. Contact (adjacency) matrices 4.1. From one circuit
The n-components R1 x1 n12 ¼ R2 x2 n12
ð3Þ
have opposite signs (different colors). The consistency relation is now s
i
R1 x1 þ ð1Þ R1 x1 ¼ Rð1Þ ai;iþ1 ti;iþ1 i
þ Rð1Þ /i;iþ1 ðRi þ Riþ1 Þki;iþ1 : For any even circuit, it is the identity,
M N ¼ 0 ¼ N M:
i
Rð1Þ /i;iþ1 ðRi þ Riþ1 Þki:iþ1 ¼ 0 ðs evenÞ;
ð4Þ
together with the pure gauge connection (2). 2.4. Chains of grains In a granular material with even circuits only, a circuit can be interrupted at j, and the chain from 1 to j depends only on the end points, independent of the return path. The chain i
Rð1Þ /i;iþ1 ðRi þ Riþ1 Þki;iþ1 ¼ R1 x1 þ ð1Þ
For an odd circuit, Eq. (1) is only a definition of R1x1 in terms of the connections a, given by inverting the matrix M = 1 + W, where Wij = di,i+1 is the elementary circulant, and Ws = 1. The non-slip condition is expressed as M(Rx) = (at). Since det M = 1 + (1)s, M is not invertible for s even, and the identity (1) is simply N(at) = 0, where the matrix N is the kernel of M,
j1
Rj xj cRij
depends only on the end points i and j (R goes now from i = 1 to j 1). 3. Odd circuits Even circuits are always dynamically unfrustrated. Odd circuits are frustrated in general. A packing subjected to shear will, if it can, break its frustrating odd circuits, with often spectacular consequences (soil liquefaction, sliding tectonic plates, dilatancy, etc. See [11] for further discussion). This liquefaction by breaking frustrating odd circuits is done by pulling slightly apart the minimal number of neighboring grains that close all the odd circuits. In a graph, odd circuits do not occur in isolation, but form the pearls of necklaces, ‘odd vorticity’ loops, called R-loops (or lines terminating at the boundary of the material) [12]. Since all odd circuits cut across once a surface bounded by the set of R-
M is a positive matrix describing a given circuit. Its Perron eigenvector (1,1,1,. . .)t is not a non-slip mode (it yields maximal edge-bending energy). If s is even, the alternating eigenvector (1,1,1,1,. . .)t is a non-slip mode, with zero bending energy. If s is odd, there are almost alternating eigenvectors with low but finite bending energy. Now, M + Mt = D + A is the dynamical matrix for the circuit, where D is a diagonal matrix whose diagonal entry Dii = zi (= 2 here) is the connectivity (valency) of the vertex i, and A is the adjacency matrix (Aij = 1 if vertices i and j are connected by an edge – if grains i and j are in contact , = 0 otherwise). 4.2. To the whole graph The analysis can be extended to the whole graph (connected, with n vertices and adjacency matrix A), through its dynamical matrix K = D + A = 2D Q, where Q = D A is the usual dynamical matrix for a network of unit masses connected by springs of unit stiffness (see [17] for an application to proteins). [Two grains in contact have opposite stress-free rotations (hi + hj) = 0, but equal stress-free translations (xi xj) = 0 if they are connected by a spring]. From the Perron–Frobenius theorem (see, e.g. [18], p. 135), K, a positive matrix, has a non-degenerate highest eigenvalue k0, and the corresponding Perron eigenvector (p1, p2, p3,. . .) with all components pi > 0, such that k0 ¼ zi þ R0 pj =pi for all i (R0 sums over the zi grains in contact with i).
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N. Rivier / Journal of Non-Crystalline Solids 352 (2006) 4505–4508
What interests us is the lowest eigenvalue(s) k1 of K that measures the frustration. Indeed, for a bichromatic graph, without odd circuits, k1 = 0 (det K = 0, since the adjacency matrix A connects vertices of opposite colors). Consider first a bichromatic graph without odd circuits. Let (1)i be the color of vertex i. The adjacency matrix has a non-zero entry Aij = 1 only if i and j have opposite colors. The unitary transformation Oij = (1)idij changes the sign of odd rows and columns, A0ij ¼ ð1Þi Aij ð1Þj and trans0 forms the dynamical matrix K = D + A into K = OKO1 = D A = Q that is the dynamical matrix for an elastic network. Now, the lowest eigenvalue of Q is zero. It is non-degenerate, with the uniform, positive eigenvector j = (1,1,1,. . .)t, because Ri Qij ¼ 0 ¼ Rj Qij . ([19], §6). [Let z0 be the highest connectivity in the graph, and consider the positive matrix z01 Q. It has a non-degenerate, highest eigenvalue z0, with corresponding Perron eigenvector j]. Transforming back, the lowest eigenvalue of the dynamical matrix K of the bichromatic packing is zero, a soft mode with an alternating eigenvector a = (1,1,1,1,. . .)t = Oj. Any graph containing odd circuits can be systematically bi-colored, simply by removing one selected edge from every odd circuit (separating those contacts located on the (minimal) soap film(s) attached to the R-loops of the material): For each odd circuit, set Aab ¼ 0 between the two vertices a and b with the same color (1)a = (1)b to be separated. The original graph is thus partitioned A = A0 + A* into a bichromatic, vertex-spanning, edge subgraph described by the adjacency matrix A0, and the few, removed edges Aab ¼ 1 between same color vertices. The valency (connectivity) of all vertices is zi ¼ z0i , except for the vertices that have been separated z0a ¼ z0a þ 1. Under the unitary transformation O, A0 ¼ A0 changes sign, whereas A* 0 = A* remains unchanged because it connects vertices with the same color. Thus, K = (D0 + A0) + 0 (1* + A*), is transformed into K = (D0 A0) + (1* + 0 A*) = Q + J*, where (1* + A*) = J* is a very sparse matrix with non-zero entries 1 only for the pair of separated vertices a,b with the same color in every odd circuit. J* is the sum over all odd circuits of the matrix whose entries Jij = 1 if i,j = a,b, and zero elsewhere. The lowest eigenvalue of Q0 is zero, with eigenvector j. Thus, the lowest eigenvalue of K is lower, but almost equal to 4c/n, k1 6 ðj; K0 jÞ=ðj; jÞ ¼ ða; KaÞ=ða; aÞ ¼ ðj; J jÞ=ðj; jÞ ¼ Rodd circuits ð4=nÞ ¼ 4c=n;
ð5Þ
where c is the number of odd circuits and n the number of vertices in the graph. Thus, k1 4c/n is a measure of the frustration generated by the odd circuits, and diluted into
the whole connected network. The granular material is rigid, but fragile (in the mechanical sense of the word). The corresponding eigenvector is alternating, nearly homogeneously. 5. Conclusions The dynamical stability of granular materials is caused by the frustration of odd circuits. They form extended, linear defects (R-loops) that do not move. The shear stress, blocked by the odd circuits, is extended within the material, but it is fully released by the disconnection of a few contacts localized in a critical region stretched on the R-loop. (This is similar to shear bands.) The material with only even circuits of grains in contact, constitutes a novel state of matter, a dry fluid that does not resist shear. As in network glasses [9,10], there is in granular materials an intermediate phase between liquid and isostatic solid, that is rigid but fragile. References [1] R. Mahmoodi-Baram, H.J. Herrmann, N. Rivier, Phys. Rev. Lett. 93 (2004) 044301. [2] D. Lohse, R. Rauhe´, R. Bergmann, D. van der Meer, Nature 432 (2004) 689. [3] D.A. Huerta, V. Sosa, M.C. Vargas, J.C. Ruiz-Sua´rez, Phys. Rev. E. 72 (2005) 031307. [4] R. Mahmoodi-Baram, H.J. Herrmann, Fractals 12 (2004) 293. [5] E. Guyon, S. Roux, A. Hansen, D. Bideau, J.-P. Troadec, H. Crapo, Rep. Progr. Phys. 53 (1990) 373. [6] C.F. Moukarzel, Phys. Rev. Lett. 81 (1998) 1634. [7] C.F. Moukarzel, in: M.F. Thorpe, P. Duxbury (Eds.), Rigidity Theory and Applications, Plenum Press, 1999. [8] P. Boolchand, G. Lucovsky, J.C. Phillips, M.F. Thorpe, Phil. Mag. 85 (2005) 3823. [9] P. Boolchand, X. Feng, W.J. Bresser, J. Non-cryst. Solids 293–295 (2001) 348. [10] P. Boolchand, D.G. Georgiev, B. Goodman, J. Optoelectr. Adv. Mat. 3 (2001) 703. [11] N. Rivier, in: R. Garcı´a-Rojo, H.J. Herrmann, S. McNamara (Eds.), Powder and Grains (2005), Balkema, Leiden, 2005, 29. [12] N. Rivier, Phil. Mag. 40 (1979) 859. [13] J. Johansson, Soil liquefaction. 2000. Available from:. [14] W. McCann, S. Nishenko, L. Sykes, J. Krause, Pure Appl. Geophys. 117 (1979) 1082. [15] O. Reynolds, Phil. Mag. 20 (1885) 223. [16] T. Aste, D. Weaire, The Pursuit of Perfect Packing, Institute of Physics Publication, Bristol, 2000, Chapter 3.5. [17] J.F. Sadoc, Eur. Phys. J E 18 (2005) 321. [18] J.K. Percus, Combinatorial Methods, Springer, 1971. [19] N. Biggs, Algebraic Graph Theory, Cambridge University Press, 1974.
Extended constraints, arches and soft modes in granular materials N. Rivier
*
IPCMS, Universite´ Louis Pasteur, 67084 Strasbourg cedex, France Available online 22 August 2006
Abstract A granular material is a network of fixed length edges linking two grains in contact, and flexible hinges if the grains can rotate without sliding on each other. The network is then dynamically unfrustrated: it can deform freely under shear and behaves like a dry fluid. A sufficient condition for non-frustration in 3D is that all circuits of grains in contact are even. Frustration is concentrated on odd circuits, and the ‘odd vorticity’ forms loops in 3D. Thus, some constraints in the network cannot be accounted for by local mean-field analysis. These extended constraints and the means of accounting for them are discussed. A granular packing can be described as a graph, with a connection carried by the edges. If all circuits of grains in contact are even, the connection is pure gauge, independent of the path between grains. When shear stress is applied over a large scale, it can break the frustrating odd circuits by disconnecting selected grains in contact, leaving all other contacts and even circuits unaffected. The granular material deforms freely like dry quicksand, as occurs in silent earthquakes, soil liquefaction, wet sand under foot (dilatancy) and, to some extent, in geological faults. Ó 2006 Elsevier B.V. All rights reserved. PACS: 83.80.Fg; 45.70.n; 64.70.Pf; 02.10.Ox; 46.25.y Keywords: Glass transition; Jamming; Stability; Granular materials; Intermediate phase
1. Introduction It is convenient to describe dry, hard granular matter by a graph. Two grains in hard, repulsive contact are connected by an edge. There are no attractive, cohesive forces between grains and the packing is held together by external forces (gravity or boundaries). The arches, chains of grains in contact, are prevented to buckle by the presence of odd circuits of grains in contact, which, in 3D, are sufficient to insure stability [1]. This role of parity explains why there can be no simple continuous description of granular matter. A granular material that contains only even circuits of grains in contact, makes up a novel state of matter, a dry fluid. It is a non-viscous fluid that does not resist shear, whose constituents roll without slip on each other or on any boundary or inclusion (the point of contact between any two grains has zero velocity, even though it changes *
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0022-3093/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.02.158
all the time). It only feels hydrostatic pressure, i.e. forces normal to any plane inside the material on any scale larger than a few grains; a solid inclusion is subject to Archimedean pull only; arches can buckle freely and make ripples on free boundaries. The packing is bichromatic: every red grain has only blue neighbors, consistently throughout. Examples of dry fluids are aerated sand [2,3], and numerical space-filling bearings [4]. An edge of the graph represents two hard grains (hard spheres, say) in contact. The contact force is scalar, repulsive and infinite. The edges are struts that carry forces with a sign constraint. Thus, the edges are geometrical and boolean (they exist or not) objects of fixed length. The extension of structural rigidity concepts to granular contact networks has been discussed first by Guyon et al. [5], and recently, by Moukarzel [6,7], who argued that a granular packing, represented as a finite network of struts with a sign constraint, is exactly isostatic (minimally rigid, exactly at the rigidity percolation threshold) in the limit of large stiffness-to-load ratio, thus for a hard granular material subject to moderate external forces. We will qualify this argument, and show
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N. Rivier / Journal of Non-Crystalline Solids 352 (2006) 4505–4508
the existence, below the percolation threshold, of rigid, underconstrained networks stabilized by odd circuits. Hinges (two edges incident on the same vertex) are free of constraints if and only if the grains in contact can roll on each other without slip. Then, arches and circuits are unstable and can buckle. A circuit of the graph is a skew, s-sided polygon that remains closed by definition. It is unstable and the seat of soft rotation modes if s is even. The condition is only sufficient in 3D, but necessary and sufficient in 2D. Thus, odd circuits are all stable because the grains in contact cannot roll on each other without slip, and the hinges are blocked. Frustration is geometric but it affects the dynamics or stability of the circuit, not its statics. By contrast, only 3-sided circuits are stable according to the classic, mean-field constraint-counting argument of Maxwell–Phillips–Thorpe (recent review: [8]). Hard granular materials differ in two ways from the structural rigidity analysis of elastic networks, from floppy to overconstrained, through isostatic at the rigidity percolation threshold. (a) Stresses carried by edges have a sign constraint. It follows that (repulsive) self-stresses cannot be sustained in an overconstrained network: the two grains connected by a redundant edge pull apart, and the network becomes isostatic. On the other hand, a floppy network will rearrange and add edges, if it can, to become isostatic ([7], my italics). (b) But hinges in a floppy network can only buckle through grains rolling on each other, a motion blocked by odd circuits. Thus, as a function of the number of edges, a hard granular material ranges from a dry fluid (isostatic network without odd circuits) to the unrestricted, isostatic network of rigidity percolation, through an intermediate phase [9,10] of underconstrained, but dynamically rigid networks with odd circuits. 2. Granular materials as graphs 2.1. Vertices, edges, circuits A granular packing can be represented as a graph with the grains as vertices, and straight edges linking (spherical) grains through their point of contact. For hard, incompressible grains, the edges have a fixed length, otherwise they do not exist: grains lose contact as soon as they move apart. An applied stress may rotate an edge, or it destroys it by pulling apart the two grains. When two grains roll on each other without slip, the edge carries a connection a that relates the rotations Rixi of the two grain. The hinge between two edges incident on the same vertex is flexible as the points of contact change upon rotation of the grains. Edges constitute arches and circuits. The arches are stable under compression, but may buckle under rotation of the hard grains. Stability is the absence of soft buckling mode(s) of zero energy or zero frequency. A stable granular packing is called dynamically frustrated. It is a stronger condition than the static stability of grain heaps, as one of the villains in the film Witness discovered to his cost. Dry fluids are dynamically unfrustrated. In a dry granular
material, the non-slip rotation of grains is blocked by odd circuits [1]. The edge linking grains i and i + 1 is represented by the vector Ri,i+1 = (Ri + Ri+1)ti,i+1, with fixed length Ri + Ri+1 and unitary directional vector ti,i+1. With time t, it can rotate at a rate /i,i+1 around the axis ki,i+1, thus dRi;iþ1 =dt ¼ /i;iþ1 ðRi þ Riþ1 Þni;iþ1 ; thereby defining a local orthonormal frame (t,n,k) for each edge (i,i + 1), with (k^t) = n, etc. Thus, dt=dt ¼ /ðk^tÞ ¼ /n;
dn=dt ¼ /ðk^nÞ ¼ /t:
(The local frame is not Frenet’s because it is defined through time derivative on a discrete polygonal curve, rather than as derivative along the curve.) 2.2. Closure relations A circuit of s edges is the skew polygonal curve in the t’s, RRi;iþ1 ¼ RðRi þ Riþ1 Þti;iþ1 ¼ 0 ðR from i ¼ 1 to sðs þ 1 1ÞÞ: Also; RdRi;iþ1 =dt ¼ R/i;iþ1 ðRi þ Riþ1 Þni:iþ1 ¼ 0 is an orthogonal, skew polygon in the n’s. Higher time derivatives contain combinations of t and n. Replacing vertices and edges by spherical grains in contact, we will obtain a third polygon in the k’s (Eq. (4) below), Rð1Þi /i;iþ1 ðRi þ Riþ1 Þki;iþ1 ¼ 0 (for s even). 2.3. Rolling without slip The condition for rolling without slip is that the two grains have the same velocity at the point of contact v1 + x1^(R1t12) = v2 + x2^(R2t12), with the velocities of the centers of the two grains related by v2 = v1 + dR12/dt. The non-slip condition is a relation between the angular rotation velocity vectors x of the two spheres in contact: (R1x1 + R2x2)^t12 = dR12/dt. With R1 x1 þ R2 x2 ¼ a12 t12 b12 n12 c12 k12 ; one has b = 0, c12 = /12(R1 + R2), but a remains an arbitrary coefficient of connection between x1 and x2. Consider first the situation of a bearing, where the centers of the grains are at rest (/ = 0). In 2D (plane polygons of cogwheels), R1x1 = R2x2 (a = 0), the axes of rotation are collinear, the angular velocities have opposite signs (different colors) and a necessary and sufficient condition for non-slip rotation is that all circuits are even. In 3D, where neither are the centers of the grains coplanar nor the axes of rotation collinear, the same condition holds, but it is only sufficient [1]. The non-slip condition, R1x1 + R2x2 = a12t12 defines a connection a between two spheres in contact that gives R2x2 in terms of R1x1, then R3x3 in terms of R2x2, etc. Around a circuit with s edges, s + 1 1, R1 x1 þ ð1Þs R1 x1 ¼ Rð1Þi ai;iþ1 ti;iþ1 :
N. Rivier / Journal of Non-Crystalline Solids 352 (2006) 4505–4508
If s is even, it yields the identity, i
Rð1Þ ai;iþ1 ti;iþ1 ¼ 0:
ð1Þ
The connection a is carried from one sphere to the next, and, around a circuit, back to the initial sphere. The connection, i
ai;iþ1 ¼ cð1Þ ðRi þ Riþ1 Þ
ð2Þ
is consistent (‘pure gauge’) for a circuit, regardless of the starting sphere, for all circuits involving the same sphere, and for all contact paths between any two spheres. It reduces identity (1) to closure of the polygonal circuit. c is a constant for the whole packing (c = 0 implies that the axes of rotation of all the grains are collinear). If grain centers move, two grains in non-slip contact are connected by the relation R1 x1 þ R2 x2 ¼ a12 t12 þ /12 ðR1 þ R2 Þk12 :
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loops, such a surface is the locus of all the critical contacts to be broken. It is therefore sufficient, and most economical, to separate those contacts located on the (minimal) soap film(s) attached to the R-loops of the material. The separation of the critical contacts requires a little space: the unfrustrated packing has less contacts than the original. In practice, this is done by aerating gently the granular material before the start of the experiment [2], by injecting water in soils, as in silent earthquakes and soil liquefaction [13], or by applying shear from an extensive boundary, as in the granular fluids of [3] (here, shear is applied throughout the experiment), in geological faults with sliding tectonic plates [14] or in the phenomenon of dilatancy [15,16]. 4. Contact (adjacency) matrices 4.1. From one circuit
The n-components R1 x1 n12 ¼ R2 x2 n12
ð3Þ
have opposite signs (different colors). The consistency relation is now s
i
R1 x1 þ ð1Þ R1 x1 ¼ Rð1Þ ai;iþ1 ti;iþ1 i
þ Rð1Þ /i;iþ1 ðRi þ Riþ1 Þki;iþ1 : For any even circuit, it is the identity,
M N ¼ 0 ¼ N M:
i
Rð1Þ /i;iþ1 ðRi þ Riþ1 Þki:iþ1 ¼ 0 ðs evenÞ;
ð4Þ
together with the pure gauge connection (2). 2.4. Chains of grains In a granular material with even circuits only, a circuit can be interrupted at j, and the chain from 1 to j depends only on the end points, independent of the return path. The chain i
Rð1Þ /i;iþ1 ðRi þ Riþ1 Þki;iþ1 ¼ R1 x1 þ ð1Þ
For an odd circuit, Eq. (1) is only a definition of R1x1 in terms of the connections a, given by inverting the matrix M = 1 + W, where Wij = di,i+1 is the elementary circulant, and Ws = 1. The non-slip condition is expressed as M(Rx) = (at). Since det M = 1 + (1)s, M is not invertible for s even, and the identity (1) is simply N(at) = 0, where the matrix N is the kernel of M,
j1
Rj xj cRij
depends only on the end points i and j (R goes now from i = 1 to j 1). 3. Odd circuits Even circuits are always dynamically unfrustrated. Odd circuits are frustrated in general. A packing subjected to shear will, if it can, break its frustrating odd circuits, with often spectacular consequences (soil liquefaction, sliding tectonic plates, dilatancy, etc. See [11] for further discussion). This liquefaction by breaking frustrating odd circuits is done by pulling slightly apart the minimal number of neighboring grains that close all the odd circuits. In a graph, odd circuits do not occur in isolation, but form the pearls of necklaces, ‘odd vorticity’ loops, called R-loops (or lines terminating at the boundary of the material) [12]. Since all odd circuits cut across once a surface bounded by the set of R-
M is a positive matrix describing a given circuit. Its Perron eigenvector (1,1,1,. . .)t is not a non-slip mode (it yields maximal edge-bending energy). If s is even, the alternating eigenvector (1,1,1,1,. . .)t is a non-slip mode, with zero bending energy. If s is odd, there are almost alternating eigenvectors with low but finite bending energy. Now, M + Mt = D + A is the dynamical matrix for the circuit, where D is a diagonal matrix whose diagonal entry Dii = zi (= 2 here) is the connectivity (valency) of the vertex i, and A is the adjacency matrix (Aij = 1 if vertices i and j are connected by an edge – if grains i and j are in contact , = 0 otherwise). 4.2. To the whole graph The analysis can be extended to the whole graph (connected, with n vertices and adjacency matrix A), through its dynamical matrix K = D + A = 2D Q, where Q = D A is the usual dynamical matrix for a network of unit masses connected by springs of unit stiffness (see [17] for an application to proteins). [Two grains in contact have opposite stress-free rotations (hi + hj) = 0, but equal stress-free translations (xi xj) = 0 if they are connected by a spring]. From the Perron–Frobenius theorem (see, e.g. [18], p. 135), K, a positive matrix, has a non-degenerate highest eigenvalue k0, and the corresponding Perron eigenvector (p1, p2, p3,. . .) with all components pi > 0, such that k0 ¼ zi þ R0 pj =pi for all i (R0 sums over the zi grains in contact with i).
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What interests us is the lowest eigenvalue(s) k1 of K that measures the frustration. Indeed, for a bichromatic graph, without odd circuits, k1 = 0 (det K = 0, since the adjacency matrix A connects vertices of opposite colors). Consider first a bichromatic graph without odd circuits. Let (1)i be the color of vertex i. The adjacency matrix has a non-zero entry Aij = 1 only if i and j have opposite colors. The unitary transformation Oij = (1)idij changes the sign of odd rows and columns, A0ij ¼ ð1Þi Aij ð1Þj and trans0 forms the dynamical matrix K = D + A into K = OKO1 = D A = Q that is the dynamical matrix for an elastic network. Now, the lowest eigenvalue of Q is zero. It is non-degenerate, with the uniform, positive eigenvector j = (1,1,1,. . .)t, because Ri Qij ¼ 0 ¼ Rj Qij . ([19], §6). [Let z0 be the highest connectivity in the graph, and consider the positive matrix z01 Q. It has a non-degenerate, highest eigenvalue z0, with corresponding Perron eigenvector j]. Transforming back, the lowest eigenvalue of the dynamical matrix K of the bichromatic packing is zero, a soft mode with an alternating eigenvector a = (1,1,1,1,. . .)t = Oj. Any graph containing odd circuits can be systematically bi-colored, simply by removing one selected edge from every odd circuit (separating those contacts located on the (minimal) soap film(s) attached to the R-loops of the material): For each odd circuit, set Aab ¼ 0 between the two vertices a and b with the same color (1)a = (1)b to be separated. The original graph is thus partitioned A = A0 + A* into a bichromatic, vertex-spanning, edge subgraph described by the adjacency matrix A0, and the few, removed edges Aab ¼ 1 between same color vertices. The valency (connectivity) of all vertices is zi ¼ z0i , except for the vertices that have been separated z0a ¼ z0a þ 1. Under the unitary transformation O, A0 ¼ A0 changes sign, whereas A* 0 = A* remains unchanged because it connects vertices with the same color. Thus, K = (D0 + A0) + 0 (1* + A*), is transformed into K = (D0 A0) + (1* + 0 A*) = Q + J*, where (1* + A*) = J* is a very sparse matrix with non-zero entries 1 only for the pair of separated vertices a,b with the same color in every odd circuit. J* is the sum over all odd circuits of the matrix whose entries Jij = 1 if i,j = a,b, and zero elsewhere. The lowest eigenvalue of Q0 is zero, with eigenvector j. Thus, the lowest eigenvalue of K is lower, but almost equal to 4c/n, k1 6 ðj; K0 jÞ=ðj; jÞ ¼ ða; KaÞ=ða; aÞ ¼ ðj; J jÞ=ðj; jÞ ¼ Rodd circuits ð4=nÞ ¼ 4c=n;
ð5Þ
where c is the number of odd circuits and n the number of vertices in the graph. Thus, k1 4c/n is a measure of the frustration generated by the odd circuits, and diluted into
the whole connected network. The granular material is rigid, but fragile (in the mechanical sense of the word). The corresponding eigenvector is alternating, nearly homogeneously. 5. Conclusions The dynamical stability of granular materials is caused by the frustration of odd circuits. They form extended, linear defects (R-loops) that do not move. The shear stress, blocked by the odd circuits, is extended within the material, but it is fully released by the disconnection of a few contacts localized in a critical region stretched on the R-loop. (This is similar to shear bands.) The material with only even circuits of grains in contact, constitutes a novel state of matter, a dry fluid that does not resist shear. As in network glasses [9,10], there is in granular materials an intermediate phase between liquid and isostatic solid, that is rigid but fragile. References [1] R. Mahmoodi-Baram, H.J. Herrmann, N. Rivier, Phys. Rev. Lett. 93 (2004) 044301. [2] D. Lohse, R. Rauhe´, R. Bergmann, D. van der Meer, Nature 432 (2004) 689. [3] D.A. Huerta, V. Sosa, M.C. Vargas, J.C. Ruiz-Sua´rez, Phys. Rev. E. 72 (2005) 031307. [4] R. Mahmoodi-Baram, H.J. Herrmann, Fractals 12 (2004) 293. [5] E. Guyon, S. Roux, A. Hansen, D. Bideau, J.-P. Troadec, H. Crapo, Rep. Progr. Phys. 53 (1990) 373. [6] C.F. Moukarzel, Phys. Rev. Lett. 81 (1998) 1634. [7] C.F. Moukarzel, in: M.F. Thorpe, P. Duxbury (Eds.), Rigidity Theory and Applications, Plenum Press, 1999. [8] P. Boolchand, G. Lucovsky, J.C. Phillips, M.F. Thorpe, Phil. Mag. 85 (2005) 3823. [9] P. Boolchand, X. Feng, W.J. Bresser, J. Non-cryst. Solids 293–295 (2001) 348. [10] P. Boolchand, D.G. Georgiev, B. Goodman, J. Optoelectr. Adv. Mat. 3 (2001) 703. [11] N. Rivier, in: R. Garcı´a-Rojo, H.J. Herrmann, S. McNamara (Eds.), Powder and Grains (2005), Balkema, Leiden, 2005, 29. [12] N. Rivier, Phil. Mag. 40 (1979) 859. [13] J. Johansson, Soil liquefaction. 2000. Available from: