Overcoming paradoxes of Drucker-Prager theory for unconsolidated granular matter. (Dated: January 23, 2014)

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1 Overcoming paradoxes of Drucker-Prager theory for unconsolidated granular matter (Dated: January 23, 2014) 1

2 Abstract Drucker-Prager model of granular matter contradicts to experiments in several aspects some of which are quite "annoying", like, for example, the steady shear paradox: in a steady simple shear ow the continuum extends inde nitely in the direction normal to the ow. In this paper we discuss modi cations of Drucker-Prager model, which eliminate most obvious paradoxes while the most essential feature of the model, its simple variational structure, is preserved. Keywords: granular material, Drucker-Prager, dissipation minimization, fast granular ow I. INTRODUCTION Granular matter exhibits the features of gases, liquids and solids. The development of a conceptually simple continuum model capturing most of these features remains a challenge. The e orts go from two opposite sides: some start from mechanics of a dilute dissipative system of rigid colliding particles and increase density to allow the system to get to the jamming state, others begin with models of elasto-plastic continua and complicate these models to mimic dissipative particle ows. Another source of ideas stem from an analogy between granular ows and ows of molecules in deformed glassy metals. Over the years several indepth reviews of the subject have been given (Savage 1984, Campbell 1990, Nedderman 1992, Mehta 1994, Herrmann 1995, Jaeger et al. 1996, de Gennes 1999, Kadano 1999, Goldhirsch 2003, GDR MiDi 2004, Forterre et al. 2008, Goddard 2012). This relieves us of the necessity to present here a detailed overview, and we will mention further only the papers that are most relevant to our work. Modeling of granular matter was put on the solid ground of continuum mechanics by D.C. Drucker and W. Prager (1952). They suggested to treat bearing capacity of soil masses by a simple model of rigid-plastic body. Drucker-Prager model is based on the associated rule which, in fact, means that the model can be formulated as a minimum principle for rateindependent dissipation. A remarkable feature of Drucker-Prager model is that it establishes a link between friction coe cient and dilatancy of sheared granular matter, the phenomenon rst noticed and studied by Reynolds (1885). Here, however, comes a contradiction to experiments: the observed values of dilatancy appear to be smaller than predictions of 2

3 Drucker-Prager model. We call this further the friction coe cient paradox. There are other inconsistencies with experimental observations as well, they are discussed in Section 3. The inconsistencies led to abandoning the associated rule, and most models developed further do not possess a variational structure (Abaqus Drucker-Prager-Cap model, Spencer 1964, Goddard 1984, 1986, 2010, Vardoulakis et al. 1991, Gudehus 1996, Anand et al. 2000, Bocquet et al. 2001, Aranson et al. 2002, Bazant 2006, Jop et al. 2006, Khain et al. 2006, Kamrin et al. 2007, 2007, Khain 2007, 2011, Kamrin 2010, Andrade et al. 2012). A noticeable exception is Cam clay model and the critical state theory (Wood 1999). Our goal is to change Drucker-Prager model to get rid of the most obvious contradictions to experiments and preserve the most essential feature of Drucker-Prager model, its variational structure. The equations governing slow motion of granular matter should follow from minimization of dissipation for a simple reason: Since inertia is negligible while particle impenetrability and friction determine micromotion, the particles move to minimize dissipation. For fast owing granular matter a possibility to obtain the governing equations by dissipation minimization is less obvious. However, as will be seen further, the relations obtained for this case previously, also follow from minimization of the dissipation after an appropriate modi cation. In the next Section we describe Drucker-Prager model and in Section 3 its paradoxes. Then we modify the dissipation of Drucker-Prager model (Section 4), derive the corresponding constitutive equations (Section 5), nd velocity eld in Couette ow (Section 6), discuss the resolution of paradoxes (Section 7). Finally, we incorporate fracture of grains in Section 8, introduce a simple model with smooth dissipation in Section 9 and discuss the modi cations of previous theories rendering their variational structure in Section 10. In the concluding Section 11 it is formulated the main outcome of the paper, a simple model of granular matter that are free of the paradoxes of Drucker-Prager theory. II. DRUCKER-PRAGER MODEL Drucker-Prager model is a model of rigid-plastic body. First, we brie y outline the theory of rigid-plastic bodies. Rigid-plastic bodies. The central concept of the theory of rigid-plastic bodies is that not 3

4 all values of stress tensor components ij are possible: in the six-dimensional space of ij the admissible stresses lie within some convex region described by an inequality f ( ij ) 6 0: (1) If f ( ij ) < 0; then continuum does not move. If ij are on the boundary, f ( ij ) = 0; (2) then motion may occur or may not occur. If it does occur, then the strain rate e ij e ij = j i (3) u i being velocity eld, is linked to the stress tensor by the associated law e ij : (4) Parameter is determined by the entire boundary value problem which includes the equilibrium j = 0 (5) and prescribing surface force or velocity at the boundary of the body. If only surface forces are known, remains undetermined and the process can go with any rate; prescribing boundary velocity on a part of the boundary speci es the process rate. In (5) and in what follows summation over repeated indices is implied. We denote the region f ( ij ) 6 0 by ; the reason for supplying with index will be come clear further. Formula (4) makes sense for smooth functions f ( ij ) : Another assumption was that the region is to convex. If is strictly convex, then, formula (4) establishes a one-to-one correspondence between stresses and strain rates up to a factor : If the conditions of strict convexity and smoothness are not satis ed, then (4) must be recti ed. This is done by Drucker postulate, the maximum principle for the dissipation density D = ij e ij : for given e ij the stress state is determined by the variational principle D (e ij ) = max ij 2 ije ij : (6) 4

5 For smooth functions f ( ij ) and convex (4) follows from (6). For non-smooth functions f ( ij ) and/or not strictly convex the one-to-one correspondence of ij and 1 e ij is lost. For a given e ij there might be di erent ij maximizing dissipation, and one maximizer may correspond to di erent e ij : Variational principle. The beauty of Drucker postulate is that the entire theory of rigidplastic bodies gets a variational form. To formulate the corresponding variational principles, it is convenient to write (6) introducing function 0 D ij 2 ( ij ) = +1 ij 2 : (7) Then D (e ij ) = max ij (e ij ij D ( ij )) ; (8) For convex ; function D ( ij ) is also convex, and the relation that is inverse to (8) is true D ( ij ) = max e ij ( ij e ij D (e ij )) : (9) We see that D is the Young-Fenchel transformation (usually marked by *) of D: If D (e ij ) is strictly convex, there is only one e ij corresponding to a given ij : Drucker-Prager model. Drucker-Prager model corresponds to the special choice of function f ( ij ) : f ( ij ) = 0 ij 0 ij Here 0 ij is the deviator of the stress tensor a ii a 1 : (10) ij = p ij + 0 ij; p = 1 3 ii; (11) ij the Kronecker delta, a and a 1 are positive constants, p pressure. Formula (1) with function f (10) is a generalization of Mohr-Coulomb s friction law for continuum: at each cut friction force does not exceed normal force times friction coe cient k. Parameter a 1 in (10) is responsible for the cohesion: shear stress required for simple slip must not be smaller some cohesion stress. Zero a 1 corresponds to unconsolidated granular matter. Further we focus on that case. Parameter a 1 can be also set to zero for consolidated granular materials if cohesion is negligible in comparison with pressure. The region for a 1 = 0 is a cone with the vertex at the origin ij = 0: This region is not strictly convex, therefore, there is no one-to-one correspondence between ij and e ij when the material moves. 5

6 The relation between the friction coe cient k and the coe cient a can be established by considering two-dimensional stress states ( 13 = 23 = 0) with the condition for 33 inherited from the incompressible elasticity 33 = p or 0 33 = 0: Then p = ( ) =2; while the condition that at every cut shear stress does not exceed k times normal stress yields 0 ij ij 6 p 2 sin p; (12) where friction angle is de ned in terms of the friction coe cient as k tan : (13) Comparing (12) and (10) one gets a = p 2 sin : (14) One can show that at every cut the admissible surface forces are compressive, as one would expect for unconsolidated granular matter, if a 6 3 p 6: According to (14), this feature holds for any : In particular, pressure p cannot be negative; this follows also directly from (12). Further we call by Drucker-Prager model a special case of the rigid-plastic body with admissible stresses selected by condition (12). The cone has the singularity at ij = 0: A remarkable consequence of that is that admissible strain rates also can be only inside some conjugated cone in the six dimensional strain rate space: it turns out that dissipation density D (e ij ) de ned by (6) is equal to +1 if e ij are outside of ; i.e. such e ij are not realizable. Indeed, calculation D (e ij ) for admissible stress (12) yields: Here e and e 0 ij respectively: p 0 e > a e 0 D (e ij ) = ij e 0 ij +1 e < a p : (15) e 0 ij e0 ij are the rst invariant of the strain rate tensor and strain rate deviator, e = e ii = div! u ; e 0 ij = e ij 1 3 e ij: Further for brevity we denote the shear rate, the square root of the second invariant of strain rate deviator, by e 0 : e 0 = q e 0 ij e0 ij : 6

7 In nite dissipation in (15) means that there is a one-side constraint: some strain rates are prohibited. Namely, if shear rate e 0 is not zero, the volume increase rate e cannot be too small: e > ae 0 : (16) First of all, continuum cannot decrease its volume, and can only expand. Second, any shear causes volume increase. Such a picture is quite consistent with the behavior of an ensemble of close-packed rigid particles. Dilation of a sheared set of close-packed particles was rst theoretically and experimentally established by Reynolds (1885). Strikingly, as Drucker- Prager model claims, the volume rate of a sheared ensemble of particles, which is a purely geometrical parameter, is linked by (16), (14) and (13) to the friction coe cient k. In Drucker-Prager model the dissipation in owing particle ensemble is always zero. If e > ae 0 ; i.e. volume grows faster than it must do due to shear, particles loose the contacts, and dissipation must be zero indeed. If e = ae 0 ; then zero dissipation means that the power of shear stresses, 0 ije 0 ij; is equal to the power of pressure, pe. Let the granular material be subject to surface force density f i at a part of material f ; and another part of the u have a prescribed velocity u (b) i : Then the true velocity eld minimizes the dissipation functional Z Z I(u) = D (e ij ) dv f i u i da: f Due to (15), it is enough to consider the velocity elds for which div! q u > a e 0 ij e0 ij (17) while the dissipation functional is equal to I(u) = f f i u i da: (18) The true velocity eld minimizes the functional (18) on the set of all velocity elds obeying the constraint (17) and the constraint u i = u (b) i u : (19) 7

8 This problem is set at each instant t. Apparently, if u (b) i 6= 0; the region V changes in the course of motion. This type of variational problems has been discussed in much detail by Mosolov and Miasnikov (1981). The minimum value in the variational problem (18)-(19) can be there is no a nite velocity eld obeying to all equations, i.e. applied load. 1: This means that material cannot carry the The problem (18)-(19) is not strictly convex and may have many solutions. To select a unique solution the problem must be regularized. One of possible regularizations is the inclusion of viscous dissipation, Z I = V e ij e ij dv f f i u i da: (20) A unique solution is selected by taking the limit solution that corresponds to vanishing viscosity,! 0: Formally, the equations of Drucker-Prager model can be obtained by including the constraint (17) by means of Lagrange multiplier p: Z q I(u) = p a e 0 ij e0 ij div! u Varying functional (21) one gets V dv f f i u i da: (21) e 0 ij ij = app e 0 mn e 0 mn p ij ; (22) equilibrium equations (5) and usual boundary conditions of continuum mechanics. Hence, p has the meaning of pressure. One can show that at the minimizer pressure can be only non-negative. III. PARADOXES OF DRUCKER-PRAGER MODEL Here we summarize the most annoying contradictions of Drucker-Prager model to experiments. Steady shear paradox. Consider a steady homogeneous shear with velocity eld u x = _y; _ = const: (23) 8

9 Then and from (16) Hence, for _ 6= 0; u z = 0; u y = u y (x; y) : e ; e0 xx = 1 3 e; e0 zz = 1 3 e; e0 yy = 2 3 e; e0 xy = > 3a s 1 2 _ > 0 : and material expands inde nitely in the direction normal to the ow direction. The dilation is expected at the beginning of the ow, which started from a close-packed state. However, after the initial dilation, the material ceases to dilate further, and a steady ow must settle with no further volume change. Boundary layer paradox. In Couette ow one observes boundary layers near the moving boundaries. The thickness of the boundary layers is of the order of several grain diameters. To exhibit such a boundary layer, a continuum theory should contain a material characteristic with the dimension of length. The only material characteristic in Drucker-Prager model, a, is dimensionless. This makes Drucker-Prager model incapable of describing the boundary layers. Friction coe cient paradox. According to (16) and (14), nonzero shear rate causes some volume rate, and e e 0 = p 2 sin : (25) Observable values of e=e 0 are smaller (Spencer 1964). This constitutes the friction coe cient paradox. There are less disturbing issues with Drucker-Prager model, like, e.g., the absence of hysteresis in loading/unloading, but we focus further on resolving only the three ones described. Though the above-mentioned paradoxes diminish the applied value of Drucker-Prager model, its fundamental role as one of the benchmarks of granular material phenomenology is undeniable. 9

10 IV. MODIFICATION OF DRUCKER-PRAGER MODEL A modi cation of Drucker-Prager model considered here is based on the following physical reasoning. If a granular matter moves, the volume concentration of grains c evolves and obeys the equation of conservation of grain volume Here d=dt is the material time derivative dc dt + c div! u = 0: (26) d dt dt + i i In random packing of grains, c takes values in some range c jam 6 c 6 c max 6 1: (27) Parameters c jam and c max are the material characteristics. The low limit c jam is de ned as such value of c that for c < c jam grains are not in contact with overwhelming probability. In other words, for c < c jam the granular matter is uidized. The existence of the upper limit c max in random packing was established experimentally (see review by Forterre et al. (2008)). Emphasize that the maximum concentration in random packing c max is smaller than the maximum possible concentration. For example, for random packing of rigid spheres c jam 0:55; c max 0:64; while the maximum possible concentration is 0:74; it is achieved on hcp and bcc periodic packing. We assume that the microstructure is isotropic, and accept that the only essential macroscopic parameters of microstructure are the grain volume concentration c and the averaged grain diameter d. If d was not included, then the de nition of c jam given above could be meaningless. Indeed, for a given c jam < c < c max ; one can take a close-packed con guration with a concentration c 0 > c: Let us take this con guration and reduce diameters of all particles without changing their positions. Then particles loose the contacts. The new radii can be chosen in such a way that the new concentration is c. So, we construct a grain microstructure with concentration c and particles that are not in contact. Including average grain diameter in the set of macroscopic characteristics eliminates such microstructures. It 10

11 FIG. 1: Symbolic plots of dissipation density in Drucker-Prager model. remains unclear how wide is the set of available grain con gurations for a given c and d and how massive is the subset with non-touching particles. We assume that such subset, if exists, does not a ect considerably the grain ow dynamics. The mechanisms of deformation in the loose-packed grains could be similar to that of glassy metals at low temperature (see Kamrin and Bouchbinder (2013)). Limiting the characteristics of microstructure by only two numbers, c and d, we ignore, for example, the e ects of grain microstructure "fabric", which is of tensorial nature. The goal here is to outline the logic structure of the approach; further complications of our model incorporating the anisotropic features of the ow can be done within the same framework. Consider dissipation in Drucker-Prager model. A symbolic plot of Drucker-Prager dissipation is shown in Fig.1. Emphasize that dissipation is zero for e=e 0 = a: D (e; e 0 ) = 0 if e = ae 0 : (28) For any e=e 0 < a; dissipation is +1: The rst point of departure from Drucker-Prager model is the assumption that parameter a is associated not with the friction coe cient but with Reynolds dilation: if e 0 6= 0; then the material must dilate with some rate e > ae 0 ; otherwise the grains fracture (incorporation of grain fracture is considered further in Section 9). Prohibiting the volume rates e < ae 0 corresponds to setting D = +1 for such rates. The dilation coe cient is a a pure kinematic characteristic of the granular microstructure. In principle, it is not related to the friction coe cient. The friction coe cient a ects the values of dissipation. 11

12 FIG. 2: Typical dependence of a on c: Our next deviation from Drucker-Prager model is the assumption that a may depend on c: e > a(c)e 0 : One can expect that a(c) decays with c, and vanishes at c = c jam (Fig. 2), i.e. at c = c jam for arbitrary small shear rates, the volume can increase with any rate, while the volume decrease is prohibited: e > 0: The general form of dissipation D for granular materials can be established from dimension reasoning. Apparently, the parameters on which D may depend on are: ; d; e; e 0 ; c; (29) being the grain mass density. We do not mention explicitly the friction coe cient, which is dimensionless. From theorem where D 0 (e=e 0 ; c) is a dimensionless function. D = d 2 e 03 D 0 e e 0 ; c Consider rst deformation of granular material in the close-packed range of concentrations, c jam 6 c 6 c max : It is natural to assume that deformation in this range is rate independent, i.e. the work done to deform granular mass does not depend on the rate of deformation. For rate independent deformations, dissipation must be a homogeneous function of the rst order. Obviously, function (30) can be such a function only if D 0 is equal to zero or in nity, as in the case of Drucker-Prager model. We consider rst the most simple possibility and set for c jam 6 c 6 c max D = (30) 0 a(c)e 0 e a(c)e 0 e > 0 : (31) 12

13 FIG. 3: Characteristic regions in (c; e=e 0 ) plane. For c < c jam the granular matter is uidized and behaves as a nonlinear viscous uid. The uid is nearly incompressible, therefore, the parameter e=e 0 is small. Expanding D 0 in Taylor series for small e=e 0 and keeping only the rst two terms we have D = 3 d2 e 03 1 e e 0 : (32) Parameters and may depend on c. Parameter is positive because the larger e the smaller dissipation must be (particles y away and the number of collisions drops). Parameter is, obviously, positive as well. Function (32) must be modi ed for nite e=e 0 to warrant positiveness of D. In such modi cation D should be decaying function of e=e 0 : We set D = 3 d2 e 03 e e=e0 : (33) Presumably, this relation holds in a small vicinity of the point c jam : c. c jam : The dissipation introduced is discontinuous. This is natural, because granular materials experience solid-like phase transitions. The lines of discontinuity are shown in Fig. 3. There are three characteristic regions in the (c; e=e 0 ) plane. In the transition from c < c jam to c > c jam uid-like behavior changes to solid- uid one. Similar is the transition across the line e=e 0 = a(c) for c > c jam : The essential di erence between the states with c < c jam and c > c jam is that for c > c jam dissipation due to collisions is practically absent. Dissipation discontinuities does not cause di culties in setting the variational problem. Regularization 13

14 can be done by means of vanishing linear viscosity. Further in Section 9 we discuss a way to smooth dissipation. V. CONSTITUTIVE EQUATIONS The governing equations of the model follow from minimization of dissipation functional, Z Z I = D (e 0 ; e; c) dv f i u i da; (34) f over velocity elds for each time instant. Regularization of this functional, if needed, can be done by vanishing viscosity as in Section 2. Typically the minimizing velocity eld is zero in some subregion V 0 of V; a rigid core, and nonzero outside of V 0 : A simple way to obtain the constitutive equations is to vary strain rates e ij and velocities ui that are subject to constraints (3). Denoting Lagrange multipliers for these constraints by ij, we obtain the functional Z V D (e 0 ; e; c) ij e ij j dv i f f i u i da: (35) In accordance with Lagrange multiplier method (see e.g. Berdichevsky 2009, Sect. 5.10), functional (35) must be minimized over e ij and u i and maximized over ij. Minimization over u i yields the equilibrium equations (5) and natural boundary conditions. Minimization over e ij is a local problem that should be done at each space point min e ij (D (e 0 ; e; c) ij e ij ) : (36) Split of stress tensor in its deviator and spherical parts renders a more convenient form of the variational problem (36) min D (e 0 ; e; c) 0 ije 0 e 0 ij ;e ij + pe : (37) It is easy to see that, for a known e 0 ; e 0 ij is determined from the equation e 0 ij e 0 q = 0 ij ; 0 0 = 0 ij 0 ij (38) 14

15 while e 0 and e should be found from the minimization problem min [D (e 0 ; e; c) 0 e 0 + pe] : (39) e;e 0 In case of the model described in the previous Section, (39) yields the following constitutive equations: if c jam 6 c 6 c max ; then, for each c, Drucker-Prager equations hold; if c < c max ; For small e=e 0 (40) simpli es to 0 = d 2 e e e 0 3 e 0 = 3 d2 e 02 e e=e0 : (40) 0 = d 2 e 02 p = 3 d2 e 02 : (41) These are the equations we will use in the next Section, because, as was mentioned, steady uidized granular motion is approximately incompressible. VI. COUETTE FLOW Consider a simple ow geometry shown in Fig. 4: granular matter is con ned between two walls; the bottom wall does not move, the upper wall moves with velocity U along x axis; ow is unbounded in x, z directions; upper wall can move in y direction and loaded with normal surface force density P, stresses in x, z directions, xx and zz ; are also equal to P. We are going to nd the ow using the model of Section 5. Stresses are constant over space coordinates xx = yy = zz = P; xy = const thus p = P: In steady regime only u x is nonzero, u x = u x (y) ; and e = 0; e 0 = ju x;y j p 2 : From (41) p = (c)(c) d 2 u 2 x;y = P; (42) 6 15

16 FIG. 4: Geometry of Couette ow. 0 = p 2 j xy j = (c)d 2 u 2 x;y. 2 = const: (43) In general, u x;y and c may depend on y: This dependence is determined by the history of the wall acceleration. We consider a simpler case when does not depend on c: Then, from (43), u x;y is a constant, and from (42), c is a constant as well. If the thickness of the ow region is h, then u x;y = U=h: In the rigid core, 0 = p 2 j xy j 6 a P: Assuming that motion started from a random close-packed state with concentration c 0 ; we obtain the value of xy ; Then ju x;y j = Consistency of (45) yields the equation for c: Then equation (42) determines the thickness of ow region j xy j = a (c 0) p 2 P: (44) s s 2a (c 0 ) P 6P = d 2 (c) d : (45) 2 (c) = 3 a (c 0 ) : (46) U 2 h = 6P 2 (c) d : (47) 2 The thickness of the boundary layer is of the order of grain size d; as expected. As follows from (42), (43), the friction coe cient k is k = j xyj p 16 = 3 p 2 :

17 Function (c) is expected to be a growing function of c, taking its maximum value at c = c jam : A solution of equation (46) exists as long as 3=a (c max ) < (c jam ) and c 0 is su ciently close to c max : If c 0 is close to c jam (the random packing is very loose), then solution of (46) does not exist for 3=a (c 0 ) > (c jam ) : In other words, in shear ow su ciently loose packing dissolve and do not form a rigid core. Absence of data on Reynolds dilation does not allow for quantitative analysis of this problem. VII. DISCUSSION OF PARADOXES As the solution of Couette problem shows, our model exhibits boundary layers with the thickness of the order of grain size as it must be. Apparently, the steady shear paradox is also resolved: volume increases until a becomes zero, and, accordingly, steady shear with no volume change becomes possible. The situation with the friction coe cient paradox is less clear. This can be reexamined after the model parameters are experimentally determined. VIII. INCORPORATION OF GRAIN FRACTURE Grain fracture was prohibited by the volume rate constraint e > ae 0 : If grains fracture, e can be less than the low limit ae 0 required by geometry of rigid grains. Accordingly, dissipation is nite for e < ae 0 : We mention here a simple model when dissipation becomes a linear function of e=e 0 in the entire region e 6 ae 0 : { (ae D (e 0 0 e) ae 0 ; e; c) = 0 ae 0 e > 0 e 6 0 : (48) Here { and a are some functions of c. The admissible stresses in such model are in the region 0 6 p 6 {; 0 6 ap: This region is shown in Fig. 5. If 0 < p < { and 0 < ap, then If p = { and 0 < ap; e = e 0 = 0: e < 0; e 0 = 0: 17

18 FIG. 5: Admissible stresses. If 0 < p < { and 0 = ap, e = ae 0 : If simultaneously p = { and 0 = ap; then vector fe; e 0 g is a linear combination with positive coe cients of two vectors f 1; 0g ; and f1; ag : This model is reminiscent of Abaqus Drucker-Prager-Cap model with one essential di erence: it has a variational structure and the associate rule is respected. IX. SMOOTHING THE DISSIPATION If fracture and non-elastic deformation of grains are possible, like in soils, then for c jam 6 c 6 c max dissipation can be nite and non-zero for both e < ae 0 and e > ae 0. A simple generalization of the model of Section 4 is to set for all c D = (c) 3 d2 e 03 e (c)(a(c) e=e0) : (49) Here, as before, a (c) characterizes Reynolds dilation; a (c) is extended by zero for c < c jam : Function (49) is a continuous function of strain rates. Solid- uid transition is associated with vanishing of a (c) at c = c jam and possible discontinuities of (c) and (c) at this point. The model of Section 4 is obtained by tending (c) to in nity for c > c jam : 18

19 X. ON FAST GRANULAR FLOWS Over last two decades considerable e orts were focused on developing equations of granular matter starting from gaseous phase and increasing grain concentration until jamming (Savage 1998, Losert et al. 2000, Bocquet et al. 2001, Garcia-Rojo et al. 2006, Khain et al. 2006, Khain 2007, 2011, Luding 2009, Otsuki et al. 2010). In this Section we show that some models suggested in this eld can be obtained from minimization of dissipation if some simplifying assumptions are made. The granular gas/liquid continuum is characterized, in addition, to grain concentration c and velocity, by granular temperature T (average squared velocity uctuation), and the closed system of equations consist of continuity equation (24), momentum equation energy equation constitutive equations for stress tensor and "heat ux" q i ; du i dt j ; (50) dt dt = ijv i;j q i;i T 3=2 (51) ij = p ij + 1 e ij + 2 ij ; p = p 0 T q i = T ;i : (52) The last term in (51) describes the energy loss due to non-elastic collisions of grains. From dimension reasoning, viscosities 1 ; 2 and "heat conduction" are proportional to T 1=2 : 1 = 1 d p T ; 2 = 2 d p T ; = 0 d p T (53) while = 0 =d; where 1 ; 2 ; 0 ; 0 are dimensionless functions of c. When c approaches c jam all these parameters diverge, presumably as (c jam c) 1 : These equations are rmly established in the limit c! 0 by an asymptotic analysis of Boltzmann equation incorporating inelasticity of particle collisions (Sela et al. 1996, 1998, Goldhirsch 2003). For c being closed to c jam these are pure phenomenological equations. 19

20 The equations do not have a variational structure even for quasi-static motion, when one neglects the left hand sides in (50) and (51). The situation changes if for quasi-static motion the "heat ux" in energy equation can be neglected, and motion is nearly incompressible (e 0) : Indeed, plugging (52) and (53) in (51) one nds temperature: T = function of c d 2 e 02 : Then viscosity becomes while 1 = function of c d 2 e 0 p = function of c d 2 e 02 (54) 0 ij = function of c d 2 e 02 e0 ij e 0 : (55) Equations (54), (55) can be obtained from minimization of dissipation D = 1 3 d2 e 03 e e 2 e + : (56) 0 e 0 Here the coe cients are positive functions of concentration. The sign minus in the second term is caused by decrease of dissipation for ying out grains (e > 0) : The last term is added to avoid negativeness of dissipation for large e, but the model makes physical sense only for small e. For small e=e 0 (54), (55) coincides with the constitutive equations of Section 5 (41). Note an apparent aw of using dissipation (56): pressure p becomes negative for su ciently large e=e 0 ; while an unconsolidated granular matter does not withstand negative pressure. Dissipation (33) is free from this defect. Equations (54), (55) can be presented in another form. As follows from (54) and (55) 0 ij p = e0 ij e 0 ; (57) being a function of c. Equation (54) can be written in terms of inertial number, I = e0 d p p= ; as 1 = function of c: (58) I2 20

21 Equation (58) can be solved with respect to c, and c becomes a function of inertial number. Plugging this dependence in (57) we arrive at Jop-Forterre-Pouliquen constitutive equation (Jop et al. 2006): 0 ij p = (I) e0 ij e 0 (59) where (I) is a prescribed function of I. Equation (59), was complemented by the incompressibility condition div! u = 0: (60) Together with equilibrium equations (5) these equations form a closed system of equations. Unfortunately, such setting has a aw: pressure is a required function of space coordinates, which is determined by the solution of a boundary value problem. In principle, pressure found from the boundary value problem might be negative. Besides, concentration is constant due to (60), and back path to (54), (55) is lost. Equations loose also the variational structure. The form of equations for liquid phase (54), (55) seem preferable. XI. CONCLUSION The main outcome of the paper is a simple model of granular matter formulated in Section 9. Here we present it in the form of a variational principle: Consider region V (t) occupied by a granular media. The values of velocity are given at the (t) of V (t) : u i = u (b) i (t): (61) Let the current volume concentration of particles of a granular media be c(t; x): Then at each instant t the true velocity eld minimizes the dissipation functional Z 1 3 (c)d2 e 03 e (c)(a(c) e=e0) dv where V (t) e q i ; e 0 = e 0 e0 ij ; e0 ij = i j 1 i 3 e ij while functions (c) ; (c) and a(c) are material characteristics, the features of which are described in the text; d 2 is a constant factor with the dimension mass density times length 21

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