Granular Solid Hydrodynamics

Granular Matter (2009) 11:139–156 DOI 10.1007/s10035-009-0137-3

Granular solid hydrodynamics

Yimin Jiang · Mario Liu

Received: 21 October 2008 / Published online: 21 April 2009 © The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract A complete continuum mechanical theory for 5 Elastic and plastic strain ...... 146 granular media, including explicit expressions for the energy 6 The granular free energy ...... 146 current and the entropy production, is derived and explained. 6.1 The elastic energy ...... 147 6.2 Density dependence of B ...... 148 Its underlying notion is: granular media are elastic when at 6.3 Higher-order strain terms ...... 148 rest, but turn transiently elastic when the grains are agitated— 6.4 Pressure contribution from agitated grains ...... 149 such as by tapping or shearing. The theory includes the true 6.5 The Edwards entropy ...... 150 temperature as a variable, and employs in addition a gran- 7 Granular hydrodynamic theory ...... 150 ular temperature to quantify the extent of agitation. A free 7.1 Derivation ...... 150 energy expression is provided that contains the full jamming 7.2 Results ...... 152 8 The hypoplastic regime ...... 152 phase diagram, in the space spanned by pressure, shear stress, 9 Conclusion ...... 153 density and granular temperature. We refer to the theory as GSH, for granular solid hydrodynamics. In the static limit, it reduces to granular elasticity, shown previously to yield realistic static stress distributions. For steady-state defor- 1 Introduction mations, it is equivalent to hypoplasticity, a state-of-the-art engineering model. Widespread interests in granular media were aroused among physicists a decade ago, stimulated in large part by review articles revealing the intriguing fact that something as Contents familiar as sand is still rather poorly understood [1–4]. The resultant collective efforts since have greatly enhanced our 1 Introduction ...... 139 understanding, though the majority of theoretic consider- 2 Sand: a transiently elastic medium ...... 142 ations are focused either on the limit of highly excited 3 Granular equilibria and jamming ...... 143 3.1 Liquid equilibrium ...... 143 gaseous state [5–10], or that of the fluid-like flow [11–13]. 3.2 Solid equilibrium ...... 143 Except in some noteworthy and insightful simulations 3.3 Granular equilibria ...... 143 [14–16], the quasi-static, elasto-plastic motion of dense gran- 4 Granular temperature Tg ...... 144 ular media—of technical relevance and hence a reign of 4.1 The equilibrium condition for T ...... 144 g engineers—received less attention among physicists. This 4.2 The equation of motion for sg ...... 144 4.3 Two fluctuation–dissipation theorems ...... 145 choice is due (at least in part) to the confusing state of engineering theories, where innumerable continuum mechanical Y. Jiang · M. Liu (B) models compete, employing strikingly different expressions. Theoretische Physik, Universität Tübingen, Although the better ones achieve considerable realism when 72076 Tübingen, Germany confined to the effects they were constructed for, these differ- e-mail: [email protected] ential equations are more a rendition of complex empirical Y. Jiang data, less a reflection of the underlying physics. In a forth- Central South University, 410083 Changsha, China coming book on soil mechanics by Gudehus, phrases such 123 140 Y. Jiang, M. Liu as morass of equations and jungle of data were used as met- itly, its full functional dependence. Hydrodynamic theories aphors. may well be derived without the second step, and the result Most engineering theories are elasto-plastic [17–19], is referred to as the structure of the theory. The form of all though there are also hypoplastic ones, which manage to fluxes, including especially the stress σij, are in this case retain the realism while being simpler and more explicit given in terms of the energy’s variables and conjugate vari- [20,21]. Both are continuum mechanical models that, starting ables. (The conjugate variables are the energy derivatives. If from momentum conservation, focus on the total stress σij. w is a function of entropy s, density ρ and strain εij,theyare Because an explicit expression appears impossible, incre- Temperature T ≡ ∂w/∂s, chemical potential µ ≡ ∂w/∂ρ, mental relations are constructed, expressing ∂t σij in terms of and elastic stress πij ≡−∂w/∂εij). σij, velocity gradient ∇ j vi and mass density ρ. (Any such The hydrodynamic approach [28,29] has been success- expression, either for σij or ∂t σij, is referred to as a constitu- fully employed to account for many condensed systems, tive relation). Typically, neither elasto-plastic nor hypoplastic including liquid crystals [30–36], superfluid 3He [37–42], theory considers energy conservation. superconductors [43–45], macroscopic electro-magnetism The continuum mechanical formalism laid down by [46–49] and ferrofluids [50–58]. Transiently elastic media Truesdell and others [22–24] spells out a number of con- such as polymers are under active consideration at pres- straints, such as material objectivity, material symmetry, and ent [59–62]. Constructing a granular hydrodynamic theory, Clausius–Duhem (or entropy) inequality, which any phys- we believe, is both useful and possible: useful, because it ically sound constitutive relation must satisfy. The remain- should help to illuminate and order the complex macroscopic ing, still considerable discretion is deemed necessary to behavior of granular solid; possible, because total energy is account for the variation between different materials, say conserved in granular media, as it is in any other system. simple fluids and elastic solids. This is in contrast to the When comparing agitated sand to molecular gas, it is fre- hydrodynamic theory, a powerful approach to macroscopic quently emphasized that the kinetic energy, although con- field theories pioneered by Landau and Lifshitz [25,26] and served in the latter system, is not in the former, because the Khalatnikov [27], in the context of superfluid helium. By grains collide inelastically. This is undoubtedly true, but it considering energy and momentum conservation simulta- does not rule out the conservation of total energy, which neously, and combining both with thermodynamic consider- includes the entropy that accounts for the heat in the grains, ations, this approach cogently deduces the proper constitutive and in the air (or liquid) between them. To construct granu- relation—for a given energy expression. In other words, if the lar hydrodynamics, we need to start from some assumptions energy expression is known, the inclusion of energy conser- about the essence of granular physics, in order to specify vation adds so much constraints for the constitutive relation the energy w, its variables and the functional dependence. that it becomes unique. The modeling discretion is hence re- Our choice is given below, and argued for throughout this duced to a scalar energy expression, which, however, is quite manuscript. Aiming to be both general and specific, we first sufficient to account for material-specific differences, such specify a simple yet fairly realistic energy density, then derive as between simple fluids and elastic solids. Energy conserva- the structure of granular hydrodynamics that remains valid tion is also considered in continuum mechanics, see, e.g. [24], for more complex energies. The full theory is presented here, but the discussion does not go far enough to achieve a similar to prod the community to respond, and to serve as a reference degree of cogency. for our future works, in which we shall zoom into varying The total energy w depends, in addition to the relevant aspects and details of the theory, comparing them with exper- macroscopic variables such as the density ρ and the strain εij, iments. Only then will we be sure whether our assumptions always on the entropy density s: There are different though are appropriate, whether the presented set of partial differen- equivalent ways to understand s. The appropriate one here is tial equations is indeed the proper granular hydrodynamics. to take it as the summary variable for all implicit, microscopic Next a rough sketch of what we believe is the basic physics degrees of freedom. So the energy change associated with s, of granular behavior: granular motion may be divided into written as (∂w/∂s)ds ≡ T ds, is the increase of energy con- two parts, the macroscopic one arising from the large-scaled, tained in these degrees of freedom—what we usually refer smooth velocity of the medium, and the mesoscopic one from to as heat increase. When the macroscopic energy (such as the small-scaled, stochastic movements of the grains. The the elastic contribution) dissipates, it is being transferred into first is as usual accounted for by the hydrodynamic variable the microscopic degrees of freedom. The associated change of velocity, the second we shall account for by a scalar, the in entropy is such that the increase in heat is equal to the granular temperature Tg—although the analogy to molecu- loss of macroscopic energy, with the total energy w being lar motion is quite imperfect, the grains do not typically have conserved. velocities with a Gaussian distribution, and equipartition is Two steps are involved in specifying the energy w:first, usually violated. All this, as we shall see, is irrelevant in the the identification of all its variables; next and more explic- present context. Tg may be created by external perturbations 123 Granular solid hydrodynamics 141 such as tapping, or internally, by nonuniform macroscopic Then we proceed, in Sect. 3, to discuss jamming,aword motion such as shear—as a result of both the grains will jig- coined to describe a system confined to a single state, and gle and slide. Then the grains will loose contact with one prevented from exploring the phase space. Although this idea another briefly, during which their individual deformation has proven rather useful [69], one must not forget that it will partially relax. When the deformation is being dimin- is a partial view, based on a truncated mesoscopic model, ished, so will the associated stress be. This is the reason and inappropriate for the present purpose. In this section, granular media can sustain a shear stress only when at rest, jamming is generalized and embedded in the concept of but looses it gradually when being tapped or sheared. And constrained equilibria. The point is, individual grains are our assumption is, this happens similarly no matter how the unlike atoms already macroscopic. They contain innumera- grains jiggle and slide, and we may therefore parameterize ble internal degrees of freedom that are neglected in mes- this stochastic motion as a scalar Tg. Our conclusion is: gran- oscopic models [5–9]. For instance, phonons contained in ular media are transiently elastic; the elastic stress relaxes individual grains do explore the phase space and arrive at a −1 toward zero, with a rate τ that grows with Tg, most simply distribution appropriate for the ambient temperature. Since −1 as τ ∼ Tg. jamming fixes only a tiny portion of the existing degrees of In granular statics, the grains are at rest, Tg ≡ 0. With freedom, the fact that grains are jammed is comparable to τ ∼ −1 Tg diverging, granular stress persists forever, display- the following textbook example: Two chambers of different ing in essence elastic behavior, see [63–67]. When granular pressure, separated by a jammed piston and prevented from media are being sheared, because Tg = 0, the stress relaxes going to the lowest-energy state of equal pressure. Such a irreversibly. This is a qualitative change from the elastic, system is in equilibrium and amenable to thermodynam- reversible behavior of ideal solids. We maintain that it is ics, albeit under the constraint of two constant subvolumes. this irreversible relaxation that is perceived as plastic gran- Similarly, a jammed granular system at Tg = 0isalsoin ular flows. If true, this insight would greatly simplify our equilibrium, not in a single state, and amenable to thermo- understanding of granular media: stress relaxation is an ele- dynamics, although under the local constraint of a given mentary process, while plastic flows are infamous for their packaging—or approximately a given density field ρ(r)— complexity. In a recent Letter [68], some simplified equations that cannot change even when nonuniform. Exploring this were derived based on the above physics. For steady-state analogy, Sect. 3 arrives at a number of equilibrium condi- shear maintaining a stationary Tg, these equations reproduce tions, useful both for describing granular statics and setting the basic structure of hypoplasticity [20], a rate-independent up granular dynamics. soil-mechanical model, and yields an account of granular In Sect. 4, the physics of the granular temperature Tg is plastic flow that is strikingly realistic. As this agreement is specified and developed. As mentioned, the energy change a result of fitting merely four numbers, we may with some dw from all microscopic, implicit variables is usually sub- confidence take it as an indication that transient elasticity is sumed as T ds, with s the entropy and T ≡ ∂w/∂s its indeed a sound starting point. More work and exploration is conjugate variable. From this, we divide out the mesoscop- needed for further validation, and especially cyclic loading, ic, intergranular degrees of freedom (such as the kinetic and critical state, shear banding and tapping need to be consid- elastic energy of random, small-scaled granular motion), ered. We reserve the study of these phenomena for the future. denoting them summarily as the granular entropy sg, with In this paper, we confine ourselves to deriving a consistent, Tg ≡ ∂w/∂sg. This is necessary, because these are fre- hydrodynamic framework starting from transient elasticity, quently more strongly agitated than the truly microscopic and call it GSH, for granular solid hydrodynamics. ones, Tg T . (Note, however, that we are also interested in The paper is organized as follows. In Sect. 2, we discuss the regime Tg ≈ T .) In Sect. 4, the equilibrium condition for to what extent granular media are elastic—or more precisely, sg and its equation of motion are derived, without any pre- permanently elastic. It is well known that, although the pro- conception about how “thermal” the associated mesoscopic cess leading to a given granular state is typically predom- degrees of freedom are. We only assume a two-step irrevers- inantly plastic, the excess stress field induced by a small ibility, w → sg → s, that the energy only goes from the mac- external force in a pre-stressed, static state can be described roscopic degrees of freedom to the mesoscopic, intergranular by the equations of elasticity. We explain why, for Tg = 0, ones summarized in sg, and from there to the microscopic, in- elasticity extends well beyond this limit, and how to employ nergranular ones of s, never backwards. The final subsection granular elasticity to calculate all static stresses, not only of 4 refutes the misconception that Onsager relation is not incremental ones. The basic reason is, without a finite Tg, valid in granular media, because the fluctuation–dissipation there is no stress relaxation and plastic flow. Similarly, if an theorem (FDT)—in terms of Tg—is frequently violated. The incremental strain is small enough, producing insufficient Tg, point is, the validity of FDT in terms of the true temperature there is too little plastic flow to mar the elasticity of a stress is never in question, and this is what the Onsager relation increment. depends on. 123 142 Y. Jiang, M. Liu

≡− 2 ≡ 0 0 0 ≡ − 1 δ In Sect. 5, the equation of motion for the elastic strain where u , us uijuij, uij uij 3 u ij. Three is elucidated, and shown to fully determine the evolution classical cases: silos, sand piles and granular sheets under a of the plastic strain as well. In Sect. 6, an explicit expres- point load were solved employing these equations, produc- sion for the free energy f is presented. This is necessary, ing rather satisfactory agreement with experiments [66,67]. because the energy w, or equivalently the free energy f , The elastic coefficient B, a measure of overall rigidity, is a are (as discussed above) material-dependent quantities. As function of the density ρ. Assuming a uniform ρ (hence a such, the free energy must be found either by careful obser- spatially constant B), the stress at the bottom of a sand pile vation of experimental data, an exercise in trial and error, is (as one would expect) maximal at the center. But a stress or more systematically, through simulation and microscopic dip appears if an appropriate nonuniform density is assumed. consideration. We proceed along the first line, making use Because the difference in the two density fields are plausibly mainly of the jamming transition that occurs as a function caused by how sand is poured to form the piles, this presents of ρ,Tg, uij, to find this expression. Section 7 presents the a natural resolution for the dip’s history dependence, long formal derivation of the hydrodynamic structure. The result- considered mystifying. ing equations are then applied to reproduce the hypoplastic Moreover, the energy w1 is convex only for model in Sect. 8. Finally, Sect. 9 gives a brief summary. us/ ≤ 2ξ, or πs/P ≤ 2/ξ, (3) ≡ 1 π π 2 ≡ π 0 π 0 π 0 ≡ π − 1 π δ (where P 3 , s ij ij, ij ij 3 ij)implying no elastic solution is stable outside this region. Identifying 2 Sand: a transiently elastic medium ◦ its boundary with the friction angle of 28 gives [66,67] Granular media possess different phases that, depending on ξ ≈ 5/3(4) the grain’s ratio of elastic to kinetic energy, may loosely be for sand. Because the plastic strain pij is clearly irrelevant for referred to as gaseous, liquid and solid. Moving fast and being the static stress, one may justifiably consider granular media free most of the time, the grains in the gaseous phase have at rest, say a sand pile, as elastic. much kinetic, but next to none elastic, energy [5–9]. In the If this sand pile is perturbed by periodic tapping at its denser liquid phase, say in chute flows, there is less kinetic base, circumstances change qualitatively: shear stresses are energy, more durable deformation, and a rich rheology that no longer maintained, and the conic form degrades until the has been scrutinized recently [11–13]. In granular statics, surface becomes flat. This is because part of the grains in the with the grains deformed but stationary, the energy is all pile lose contact with one another temporarily, during which elastic. This state is legitimately referred to as solid because their individual deformation decreases, implying a dimin- static shear stresses are sustained. If a granular solid is slowly ishing elastic strain uij, and correspondingly, smaller elastic sheared, the predominant part of the energy remains elastic. energy w1(uij) and stress πij(uij). The system is now elastic For simplicity, we shall continue to refer to it (though tran- only for a transient period of time. The typical example for siently elastic) as solid. transient elasticity is of course polymer, and the reason for When a granular solid is being compressed and sheared, its elasticity being transient is the appreciable time it takes the deformation of individual grains leads to reversible en- to disentangle polymer strands. Although the microscopic ergy storage that sustains a static, elastic stress. But they also mechanisms are different, tapped granular media display jiggle and slide, heating up the system irreversibly. There- similar macroscopic behavior, and share the same hydrody- fore, the macroscopic granular strain field ε = u + p ij ij ij namic structure. has two contributions, an elastic one u for deforming the ij When being slowly sheared, or otherwise deformed, gran- grains, and a plastic one p for the rest. The elastic energy ij ular media behaves similarly to being tapped, and turn tran- w (u ) is a function of u , not ε , and the elastic contribu- 1 ij ij ij siently elastic. This is because in addition to moving with tion to the stress σij is given as πij(uij) ≡−∂w1/∂uij. With the large-scale shear velocity vi , the grains also slip and jig- the total and elastic stress being equal in statics, σ = π , ij ij gle, in deviation of it. Again, this allows temporary, partial stress balance ∇ j σij = 0 may be closed with πij = πij(uij), unjamming, and leads to a relaxing uij. and uniquely determined employing appropriate boundary One does not have to assume that this deviatory motion is conditions. Our choice [63–65] for the elastic energy w = 1 completely random, satisfying equipartition and resembling w (u ) is 1 ij molecular motion in a gas. It suffices that the elasticity turns transient the same way, no matter what kind of deviatory √ √ 2 w = 2B2 + A 2 ≡ B 22 + us , motion is present. In either cases, it is sensible to quantify this 1 us (1) 5 5 ξ motion with a scalar. Referring to it as the granular entropy or ∂w √ u2 temperature is suggestive and helpful. The granular entropy π ≡− 1 = (Bδ − A 0 ) + A √s δ , ij ij 2 uij ij (2) ∂uij 2 sg thus introduced is an independent variable of GSH, with 123 Granular solid hydrodynamics 143

2 an equation of motion that accounts for the generation of Tg and altering the chemical one to ∂t vi +∇i (µ − v /2) = 0. by shear flows, and how the energy contained in Tg leaks into We focus on Eqs. (9) here. heat. Only when Tg is large enough, of course, is granular Including gravitation, the energy is w¯ 0 = w0 + φ, with elasticity noticeably transient. Gk =−∇i φ the gravitational constant pointing downwards. The generalized chemical potential is µ(ρ)¯ ≡ ∂w¯ /∂ρ = µ + φ, (11) 3 Granular equilibria and jamming 0 while chemical equilibrium, ∇i µ¯ = 0, now reads Solid and liquid equilibria are first described, then shown ∇ µ = . to correspond, respectively, to the jammed and unjammed i Gi (12) equilibria of granular media. This implies that a nonuniform density now represents the optimal mass distribution minimizing the energy (or maxi- 3.1 Liquid equilibrium mizing the entropy). With the pressure given as PT =−w0 + TS+µρ, see Appendix 9, the condition for mechanical equi- In liquid, the conserved energy density w(s,ρ,gi ) depends librium, on the densities of entropy s,massρ, and momentum gi = ∇ P = s∇ T + ρ∇ µ = ρG (13) ρvi . The dependence on gi is universal, given simply by i T i i i w( ,ρ, ) = w ( ,ρ)+ 2/( ρ), is a combination of the thermal and chemical ones. s gi 0 s gi 2 (5) w leaving the rest-frame energy 0 to contain the material 3.2 Solid equilibrium dependent part. Its infinitesimal change, dw0 =(∂w0/∂s)ds+ (∂w /∂ρ) ρ 0 d , is conventionally written as In solids, if the subtle effect of mass defects is neglected, density is not an independent variable and varies with the dw0 = T ds + µdρ, (6) strain (for small strains) as by defining dρ/ρ =−du . (14) T ≡ ∂w0/∂s|ρ,µ≡ ∂w0/∂ρ|s. (7) Defining πij ≡−∂w0/∂uij|s, we write the change of the ∂w/∂ρ| , ≡ It is useful to note that given Eq. (5), the relation s gi energy as µ − v2/2 holds, hence dw (s, u ) = T ds − π du . (15) 2 0 ij ij ij dw = T ds + (µ − v /2)dρ + vi dgi . (8) Maximal entropy, with the displacement vanishing at the sys- = 3 Consider a closed system, of given volume V d r, tem’s surface, implies the following thermal and mechanical w 3 ρ 3 energy d r, and mass d r. Whatever the initial equilibrium conditions (see Appendix 9), conditions, it will eventually arrive at equilibrium, in which the entropy sd3r is maximal, or equivalently, at minimal ∇i T = 0, ∇ j πij = 0. (16) energy for given entropy, mass and volume. To obtain the So force balance is simply an expression of maximal mathematical expression for this final state, one varies wd3r entropy—quite analogous to uniformity of temperature. It for given sd3r and ρ d3r, arriving at the following equi- implies the dominance of phonon distribution that satisfies librium conditions, force balance, and the rarity of phonon fluctuations that vio- w¯ ( , ) = ∇i T = 0, ∇i µ = 0. (9) late it. Including gravitation, the energy is d 0 s uij T ds −¯πijduij, with π¯ij = πij + ρφ, and mechanical equi- Being expressions for optimal distribution of entropy and librium becomes mass, these two conditions may, respectively, be referred to as the thermal and chemical one. ∇ j πij = ρGi (17) In mathematics, Eqs. (9) are referred to as the Euler– Lagrange equations of the calculus of variation. The calcu- 3.3 Granular equilibria lation is given in Appendix 9. More details may be found in [70], in which three additional conserved quantities: In granular media, the density is an independent variable, 3 3 momentum gi d r, angular momentum (r × g)i d r, and because the grains may be differently packaged, leading to a 3 booster (ρri − gi t)d r were also considered, adding a density variation of between 10 and 20% at vanishing defor- motional condition, mation. So the energy depends on all three variables, vij ≡ (∇i v j +∇j vi )/2 = 0, (10) dw0(s,ρ,uij) = T ds + µdρ − πijduij. (18) 123 144 Y. Jiang, M. Liu

This is a combination of Eqs. (6) and (15), and depending excited, Tg T . The second expression, algebraically on whether Tg is zero or finite, sand flip-flops between the identical, takes w as a function of s and sg, with T ds be- liquid and solid behavior. If Tg is finite, the elastic stress ing the total heat if all degrees were at T , and (Tg − T )dsg πij relaxes until it vanishes. The equilibrium conditions are the increase in energy when some of the degrees are at Tg. therefore, including gravitation, If unperturbed, a stable system will always return to equilibrium, at which the second pool is empty, sg = 0. This ∇i T = 0, ∇i PT = ρ Gi ,πij = 0, (19) implies the free energy f ≡ w0 − Ts has a minimum at = similar to that of a liquid, with ∇i PT = ρ Gi (or ∇i µ = Gi ) sg 0. Assuming analyticity, we expand the free energy ( , ) = enforcing an appropriate density field, and πij = 0 forbid- f T sg around sg 0, arriving at ding any free surface other than horizontal. = ( ) + 2/( ρ), f f0 T sg 2b (24) For vanishing Tg, sand is jammed, implying two points: π first, ij no longer relaxes; second, without slipping and jig- where b is a positive material parameter, a function of ρ and gling, the packaging density cannot change, and the density uij. (The factor ρ will turn out later to be convenient.) With ρ/ρ =− is again a dependent variable, d du . The suitable d f =−sdT + (Tg − T )dsg we have equilibrium conditions, as derived in Appendix 9,are ¯ Tg ≡ Tg − T ≡ ∂ f/∂sg|T = sg/(bρ), (25) ∇i T = 0, ∇ j (PT δij + πij) = ρGi , (20) a quantity that vanishes in equilibrium which allow static shear stresses and tilted free surfaces. So, ¯ although jammed states are prevented from arriving at the Tg ≡ Tg − T = 0. (26) liquid-like conditions of Eqs. (19), they do possess reach- able thermal and mechanical equilibria. We shall employ the Legendre transformed potential, ˜( , ¯ ) ≡ ( , ) − ¯ If the energy (as given in Sect. 4) depends in addition on f T Tg f T sg Tgsg, below (that has a maximum rather than a minimum at T = T ), the granular entropy, dw = T ds + Tgdsg +···, the pressure g contribution PT (see Sect. 7.1)is ˜( , ¯ ) = ( ) − ρ ¯ 2/ . f T Tg f0 T b Tg 2 (27) ˜ PT =−w0 + Ts + Tgsg + µρ =−f + µρ , (21) Because a rather high Tg is implied by any random motion ∇ = ∇ + ∇ + ρ∇ µ. with i PT s i T sg i Tg i (22) of the grains, neglecting T in comparison to Tg or taking ¯ Tg ≈ Tg is frequently a good approximation, though not ¯ close to Tg = 0. So it is prudent not to implement it while 4 Granular temperature Tg deriving the equations.

Granular temperature is not a new concept. Haff, along with 4.2 The equation of motion for sg Jenkins and Savage [5–9], introduced it in the context of ∼ w granular gas, taking (in an analogy to ideal gas) Tg kin, First of all, s must obey a relaxation equation, −∂ s = w g t g where kin is the kinetic energy density of the grains in a γ∂f/∂s = γ T¯ . Since it is macroscopically slow, s also ≡ ∂w /∂ ∼ ∂ /∂ g g g quiescent granular gas. With Tg kin sg Tg sg, displays characteristics of a quasi-conserved quantity, and ∼ the granular entropy is sg ln Tg. As discussed above, gran- removal of local accumulations is accounted for by a con- ular temperature is also a crucial variable in granular solids. vective and a diffusive term, But one must not expect this gas-like behavior to extend to ¯ ¯ the vicinity of Tg = T : as the system, if left alone, always − ∂t sg =∇i [sgvi − κg∇i Tg]+γ Tg = 2 2 returns to Tg T , the energy has a minimum there. And =∇i (sgvi ) + (1 − χ ∇ )sg/τg, (28) w ∼ 2 ∼ ( − )2 something like sg Tg T would be needed. where τg ≡ bρ/γ is the relaxation time, while χ ≡ κg/γ 4.1 The equilibrium condition for T is the characteristic length associated with the diffusion. (The g ¯ second line of Eq. (28) assumes κg,γ = constant.) If Tg is = The energy change dw from all microscopic, implicit vari- held at T0 at the boundary x 0, and allowed to relax for > ∼ ¯ ( ) ( − χ 2∇2) = ables is generally subsumed as T ds, with s the entropy and x 0, the field sg Tg x obeys 1 sg 0inthe ∂ ,v = T ≡ ∂w0/∂s its conjugate variable. From this, we divide out stationary limit t sg i 0, and decays as the energy of granular random motion, denoting it as Tgdsg, Tg(x) = T + T0 exp(−x/χ). (29) dw = T d(s − s ) + T ds = T ds + (T − T )ds . (23) 0 g g g g g Equation (28) is not complete. To see this, compare it with The first expression distinguishes between two heat pools: the true entropy s. In liquid, s is governed by a balance equa- s − sg and sg, with the latter sometimes more strongly tion with a positive source term R that is fed by shear and 123 Granular solid hydrodynamics 145 compressional flows, and by temperature gradients [25,26], A direct consequence for the stationary uniform case, ¯ Rg = 0 and ∇i Tg = 0, is ∂t s +∇i (svi − κ∇i T ) = R/T, (30) 0 0 2 2 R = ηv v + ζv + κ(∇ T ) , (31) γ ¯ 2 = η v0 v0 + ζ v2 , ij ij i Tg g ij ij g (35) 0 1 where v is the traceless part of vij ≡ (∇i v j +∇j vi ) and ij 2 which quantifies how much T¯ is excited by shear or com- v its trace; η, ζ > 0 are, respectively, the shear and com- g pressional viscosity, and κ>0 the heat diffusion coefficient. pressional flows. η ,ζ Entropy production R must vanish in equilibrium and be In dry sand, the viscosities g g probably dominate, as η, ζ are possibly insignificant—though these should be quite positive definite off it. The thermodynamic forces ∇i T and a bit larger in sand saturated with water: a macroscopic shear vij do vanish in equilibrium [see Eqs. (9, 10)]; off it, they may be taken to quantify the “distance from equilibrium.” flow of water implies much stronger microscopic ones in the The entropy production R increases with this distance and fluid layers between the grains, and the energy dissipated there contributes to R, instead of to Rg first. may be expanded in ∇i T and vij. The given terms are the lowest order, positive ones that are compatible with isotropy. In granular media, equilibrium conditions are more numer- 4.3 Two fluctuation–dissipation theorems ous than in liquid. As discussed in Sect. 3.3,theseare,in ¯ addition, the vanishing of πij, ∇ j πij, and Tg, hence we have There are many in the granular community who dispute the validity of the Onsager reciprocity relation in granular media, R = ηv0 v0 + ζv2 + κ(∇ T )2 + γ T¯ 2 ij ij i g enlisting any of the following reasons: (1) The FDT does not + β(π0 )2 + β π 2 + β P (∇ π )2. ij 1 j ij (32) hold. (2) The microscopic dynamics is not reversible. (3) Sand is too far off equilibrium. Three additional points: (1) Being an expansion in the ther- Careful scrutiny shows that none of these arguments holds modynamic forces, the transport coefficients η, ζ, κ, κg,γ, P water. First, with F denoting the free energy, fluctuations say β,β1,β may still depend on the variables of the energy, , ¯ ,ρ π π 2 ≡ π 0 π 0 of the volume are always given as T Tg , and s ij ij, but not on the forces them- selves, such as ∇i T or vij. (2) More terms are conceivable in − − V 2 =T (∂2 F/∂V 2) 1 = T (−∂ P/∂V ) 1, (36) Eq. (32), say α1∇i T ∇ j πij or κ1πij∇i T ∇ j T . These may be included when necessary. (3) The above reasoning leaves the which holds for unjammed sand, jammed sand, as well as question open why ∇i µ does not contribute to R, not even in liquid—or more precisely, why the coefficient preceding a copper block. The only difference between jammed and 2 unjammed sand is that there is a contribution to F from Tg, (∇i µ) always vanishes. The answer is given in [70], see also [71] and references therein. see Sect. 6. Exploring the analogy between T and Tg,an natural question is, whether The granular entropy sg should obey a balance equation with the same structure, 2 −1 V =Tg(−∂ P/∂V ) (37) ¯ ¯ ∂t sg +∇i (sgvi − κg∇i Tg) = Rg/Tg, (33) holds for unjammed sand. Although the answer is frequently though the source term R has positive as well as negative g “no”, the crucial point is, the validity of the Onsager relation contributions: two positive ones from shear and compres- depends on Eq. (36), not Eq. (37). sional flows, and the negative relaxation term discussed in Second, the dynamics typically employed in granular sim- Eq. (28), ulations is indeed irreversible, but only as a result of a model- = η v0 v0 + ζ v2 + κ (∇ ¯ )2 − γ ¯ 2. dependent approximation that treats grains as elementary Rg g ij ij g g i Tg Tg (34) constituent entities. The true microscopic dynamics that ¯ 2 γ The fact that the coefficient preceding Tg is both in resolves the atomic building blocks of the grains remains Eqs. (32) and (34) derives from energy conservation: taking reversible. And this is the basis for the Onsager relation. ¯ the system to be uniform, we have ∂t w = T ∂t s + Tg∂t sg = Third, “too far off equilibrium” is not a convincing argu- + ¯ (−γ ¯ ) ∂ w = = γ ¯ 2 R Tg Tg .So t 0 implies R Tg .It ment, as turbulent fluids, truly far off equilibrium, are known expresses the fact that the same amount of heat leaving sg to obey the Onsager relation. Some argue further that sand, must arrive at s. (Note it is possible to have ∂t sg +∇i [sgvi − whether jammed or in motion, are always far from equilib- (κ1 + κ2)∇ ¯ ) = / ¯ = ···+κ1(∇ ¯ )2 g g i Tg Rg Tg, with Rg g i Tg and rium. Yet as the careful discussion in Sect. 3 shows, granular =···+κ2(∇ ¯ )2 R g i Tg , a complication that we shall not con- media are certainly not always far from equilibrium, they sider here, but must keep in mind should experimental finding just have different ones to go to—solid-like if jammed and require it). liquid-like if unjammed. 123 146 Y. Jiang, M. Liu

5 Elastic and plastic strain The equation of motion for the elastic strain uij is in fact more complicated [cf. the derivation leading to Eq. (76)], and As discussed in Sect. 2, the elastic strain uij accounts for given as the deformation of individual grains, while their rolling and sliding is, on average, described by the plastic strain pij. dt uij − (1 − α)vij + uij/τ Together, they form the total strain ε = u + p . The elas- ij ij ij =−[(uik∇ j vk +∇i y j /2) + (i ↔ j)], (40) tic energy w(uij) is by definition a function of uij, not of ε , and the elastic stress is given as π (u ) ≡−∂w/∂u . ij ij ij ij where dt ≡ ∂t +vk∇k, and (i ↔ j) signifies the same expres- When Tg is finite, the elastic strain relaxes, sions as in the preceding bracket, only with the indices i and j exchanged. In this equation, the term (u ∇ v )+(i ↔ j), ∂ − v =− /τ. ik j k t uij ij uij (38) important for large strain field and frequently negligible for hard grains, is of geometric origin, see [59–62] for expla- implying a diminishing elastic strain uij, and correspond- nations. The dissipative flux y ∼∇π will be derived in w( ) π ( ) i j ij ingly, smaller elastic energy uij and stress ij uij . Sect. 7.1. It is quite similar to the diffusive heat current κ∇ T , ∂ ε = i Because the total strain is a purely kinematic quantity, t ij which aims to reduce temperature gradients and establish vij, the evolution of the plastic strain pij is given as ∂t pij = ∇i T = 0. We can take yi to be a current that aims to reduce vij − ∂t uij. ∇ j πij and establish the equilibrium condition, ∇ j πij = 0, of It is the relaxation term −uij/τ that gives rise to plasticity. 0 Eq. (20). The term αvij is a simplification of α0v +α1v δij, To see how it works, take a constant τ and consider the fol- ij assuming 1 α = α . lowing scenario. If a transiently elastic medium is deformed 3 0 1 quickly enough by an external force, leaving little time for relaxation, (uij/τ) dt ≈ 0, we have uij ≈ εij = vijdt, pij = 0 right after the deformation. The built-up in elastic 6 The granular free energy energy and stress πij is maximal. If released at this point, the system would snap back toward its initial state, as pre- As explained in Sect. 1, the structure of the hydrodynamic scribed by momentum conservation, ∂t (ρvi ) +∇j πij = 0, theory is determined by general principles, especially energy displaying a reversible and elastic behavior. But if we hold the and momentum conservation, but the explicit form of the system still for long enough, vij = 0, hence ∂t εij = 0, the energy w is not. Although there are general requirements elastic part uij will relax, ∂t uij =−uij/τ, while the plastic that w must always satisfy, most of its functional dependence part grows accordingly, ∂t pij =−∂t uij. When uij vanishes, reflects the specific behavior of the material. To arrive at an elastic energy w(uij) and the stress πij are also gone, imply- expression for the energy of granular media, there are two ing ∂t (ρvi ) = 0. The system now stays where it is when obvious methods, either a microscopic derivation, possibly released, and no longer returns to its original position. This via simulation, or more pragmatically, examining constraints is what we mean by a plastic deformation. from key experiments, opting for simplicity whenever pos- τ ∼ −1 Next take Tg . As discussed in the introduction, sible, as we do here. this should be appropriate for granular media. Assuming Because we are interested in the limit of small Tg and uij, 2 = (for simplicity) a stationary granular temperature, or Tg see Eqs. (1) and (24), and because the dependence on the true 2 (ηg/γ )vijvij ≡ (ηg/γ )||vs|| ,seeEq.(35), we obtain from temperature is usually less directly relevant, the difficult part Eq. (38) the equation, is the density dependence of the energy. Fortunately, quite a number of known features may be used as input. First, ∂t uij − vij ∼||vs||(−uij) ηg/γ , (39) there are two characteristic granular densities, the minimal and maximal ones, ρ p and ρcp, respectively, referred to as the rate-independent structure of which closely resembles random loosest and closest packing. In the first case, the the hypoplastic one [20]. As a result, both the elastic strain grains necessarily loose contact with one another when the uij and the stress σij will display incremental nonlinear- density is further decreased; in the second, the density can no ity, i.e., behave differently depending whether the load is longer be increased without compression, at which point the being increased (vij > 0, ||vs|| > 0) or decreased (vij < system is orders of magnitude stiffer [17–19,72]. Then there 0, ||vs|| > 0). Not surprisingly, this equation leads to plastic is the jamming transition of sand, especially the so-called vir- flows very similar to the hypoplastic results. However, under gin consolidation line, which we believe is the limit beyond cyclic loading of small amplitudes, because Tg never has which no stable elastic solutions are possible, see Fig. 1a. time to grow to its stationary value, the plastic term uij/τ ∼ These in conjunction with the density dependence of sound Tguij remains small, and the system’s behavior is rather more velocity and the pressure exerted by agitated grains contain elastic. sufficient information to fix the expression for the energy. 123 Granular solid hydrodynamics 147

( ,ρ)= ( )/ ρ, (a) (b) f0 T F1 T m (44)

where F1 is the free energy of a single grain, m its mass, and F1(T )/m the free energy per unit mass, averaged over a number of grains. It is important to realize that the equilibrium stress is given, once one knows what the free energy density f˜ = F/V is (see Appendix 9), ∂( f˜/ρ) ∂ f˜ σij = PT δij + πij =− δij − . (45) ∂(1/ρ) ∂uij The first term is the local expression for the more familiar one, (c) ∂ F ∂( fV˜ /M) ∂( f˜/ρ) PT ≡− =− =− ∂V ∂(V/M) ∂(1/ρ) M ˜ ˜ ¯ = ρ∂ f /∂ρ − f = ρµ + Ts + Tgsg − w. (46) In liquids, only this term exists, since f˜ does not depend on uij; in ideal crystals, only the second term exists, because the density is not an independent variable, see the discussion in Sect. 3. In granular media, both terms coexist. Given the free ˜ energy f = fi of Eq. (41), each term yields the pressure Fig. 1 Granular yield surface, or jamming phase diagram, for T = 0, g contribution, as a function of the pressure P, shear stress σs ,andvoidratioe ≡ ρ /ρ − G 1. All thick solid lines are calculated using Eqs. (42, 50, 53). ≡ ρ(∂ /∂ρ) − , a Maximal void ratio e versus pressure P,orthevirgin consolida- Pi fi fi (47) tion line. The dotted line is an empirical formula, e = 0.679 − ≡ ≡ ρ∂ /∂ρ − = 0.097 ln(P/0.5),withP in MPa. The thin line (designated as simple with PT Pi and P0 f0 f0 0. model) renders Eq. (52). The circle at the top is the random loosest packing value for e. b The straight Coulomb yield line bends over 6.1 The elastic energy depending on e, a behavior usually accounted for by the cap model in elasto-plastic theories. c The 3D combination of a and b. Values for B = , ρ∗ = . ρ ρ = . ρ The elastic part of the free energy, Eq. (42), has previously the calculation are: 0 7 000 MPa, p 0 445 G , cp 0 645 G , −4 −5 3 been successfully tested under varying circumstances, cf. the 1 = 10 ,andk1 = 10 m /kg, k2 = 1,000, k3 = 0.01 discussion in Sect. 2, below Eq. (2). It is not analytic in the elastic strain, but does contain the lowest order terms. As it ˜ ¯ Instead of the energy, we consider the potential f (T, Tg, takes some deliberation to arrive at its density dependence ¯ u ρ,uij) ≡ w0 − Ts− Tgsg,seeEq.(27). Referring to it (for and the terms of higher order in ij, we consider them in two simplicity) still as the free energy density, we write separate sections below. First, a conceptual point. We take any yield surface as ˜ ¯ the divide between two regions: one in which stable elastic f = f0(T,ρ)+ f1(ρ, uij) + f2(ρ, Tg), (41) √ solutions are possible, the other in which they are not—so a ≡ w = B ( 2/ + 2/ξ), f1 1 2 5 us (42) system under stress must flow and cannot come to rest here. = ρ ( − ρ/ρ )a(− ¯ 2/ ), < , f2 b0 1 cp Tg 2 0 a 1 (43) Accepting this, the natural approach is to have a convex elastic energy turn concave at the yield surface. The idea behind where f0(T,ρ)is the free energy at vanishing granular tem- it is, the energy is an extremum if the equilibrium conditions ¯ perature and elastic deformation, Tg, uij = 0, while w1(uij) of Sect. 3, including especially Eq. (20), are met. Convexity ¯ and f2(Tg) are the respective lowest order term. (It is a sim- implies the energy is at a minimum there, and concavity that plifying assumption that the temperature T enters the free it is at a maximum. Where w1 is concave, any elastic solution energy only via f0, and not w1, f2. This neglects effects such satisfying Eq. (20) has maximal energy, and is eager to get as thermal expansion that, however, are easily added when rid of it. It is not stable because infinitesimal perturbations necessary). suffice to destroy it. B,ξ = w Being cohesionless, the grains possess no interaction en- As discussed√ in Sect. 2,for constant, 1 is convex ergy, f0(T,ρ)is therefore the sum of the free energy in each for πs/P ≤ 2/ξ and concave otherwise, and already pos- of the grains, sesses the right form to account for the Coulomb yield line, 123 148 Y. Jiang, M. Liu see Fig. 1b. Our task now is to appropriately generalize it such that the density ρ is included as a third variable. Instead of ρ, the void ratio, e ≡ ρG /ρ − 1, is frequently employed. It remains constant at elastic compressions and accounts for granular packaging only. (ρG is the bulk density of granular material at the same pressure, typically around 2700 kg/m3 for sand).

6.2 Density dependence of B

We shall take B as density dependent, but not ξ: since the Coulomb yield line is approximately independent of the den- Fig. 2 Equation (48), obtained by employing the Hardin–Richart sity, so must the coefﬁcient ξ be, see Eq. (4). Granular sound relation directly, violates the stability condition Eq. (49), because − / velocity was measured by Hardin and Richart [73], who ∂2B 2 3/∂ρ2 > 0 for all density values. Although numerically sim- found it linear in the void ratio, c ∼ 2.17−e.GivenEq.(42), ilar, see insert, the expression from Eq. (50) suitably becomes concave √ at ρ c, and satisﬁes the stability condition between ρ c and ρcp.The the velocity of sound is c ∼ B/ρ, implying ρ∗ = . ρ ρ = . ρ plots are calculated with c 0 445 G , pc 0 645 G (implying ρ = . ρ B = , B = B ( . − ρ /ρ)2(ρ/ρ ). p 0 555 G ), and 0 7 000 MPa, appropriate for Ham River 0 3 17 G G (48) sand [74] Since this expression properly accounts for the measured

[74] density dependence of the compliance tensor Mijk ,the convex, between ρ and a density larger than ρ . We there- B ρ cp p dependence of on seems settled [75]. It is not, because the fore propose resultant w1 is concave in the variables ρ and , and could ρ − ρ∗ 0.15 not possibly sustain any static solution. Inserting Eq. (48)into p B = B0 × C, for ρ>ρ p; (50) (42), we find the energy violating the stability condition, ρcp − ρ 2 −2/3 2 ∂ B /∂ρ ≤ 0, (49) B = 0, for ρ ≤ ρ p. (51) ≡ (∂2w /∂ρ2) ρ∗ <ρ obtained from inserting Eq. (42) with us 0into 1 With an appropriate p p, this expression renders the 2 2 2 2 (∂ w1/∂ ) ≥ (∂ w1/∂ρ∂) . Clearly, the widely em- energy divergent at ρcp, stable and convex up to ρ p, and ployed Hardin–Richart relation, c ∼ 2.17−e, is not accurate approximates the Hardin–Richart relation between them, see enough for a direct input into the energy. It works fine as long Fig. 2.(TakeC = 1 for now, until it is specified otherwise in as the sand is jammed, Tg = 0, and ρ is only a given param- the next section). eter, not a free variable—such as in the experiments of [74], or when determining static stress distributions. But if a finite 6.3 Higher-order strain terms Tg frees the density to become a variable, this instability will wreck havoc with the hydrodynamic theory. So our task is Next, we consider compaction by pressure, the fact that to reconstruct the density dependence of B, such that the denser sand can sustain more compression before getting energy w1 unjammed, before elastic solutions become unstable: see the dotted line of Fig. 1a, depicting a well-known empirical for- 1. vanishes for densities smaller than the random loosest mula from soil mechanics, e = e0 − ln P,see[17–19]. packing value (around the void ratio of e p ≈ 0.8for Referred to as the virgin (or primary) consolidation line,it sand of uniform grain size), or ρ ≤ ρ p; represents the boundary that sand (at rest) will not cross when 2. (as a simplification) diverges at ρ = ρcp, the random compressed. Instead, it will collapse, becoming more com- closest packing value (around ecp ≈ 0.55); pact, with a smaller e, close to or at the curve, but not beyond. 3. is convex and reproduces the Hardin–Richart relation (Note the dotted line does not appear to cut the e-axis, as it between ρ p and ρcp. should at ρ p—this is where sand becomes instable for any pressure. The discrepancy may derive from difficulties of Alas, these points are more easily stated than combined making reliable measurements close to ρ p.) This behavior in an energy expression, and no continuous B seems feasi- is a natural consequence of higher-order strain terms such as B ρ − ρ ble: if analytic, would be proportional to p close − (ζ 3 + ζ 2), α 1 2 us (52) to ρ p. More generally, we may take B ∼ (ρ − ρ p) , with α 2.5 α positive. But the resulting energy, w ∼ (ρ − ρ p) , which are to be added to w1,Eq.(42). Take ζ1,ζ2 > 0 and ρ 2 = remains concave. Only when including the divergence at cp consider pure compression, us 0. For small ,theterm α β 3 by taking B ∼ (ρ − ρ p) /(ρcp − ρ) does the energy turn −ζ1 is negligible, and w1 remains convex. But if is large 123 Granular solid hydrodynamics 149 enough, its negative second derivative will turn w1 concave, Stability is given only if the energy w1 is convex with ρ,, 0 making any elastic solution impossible. The value of at respect to its seven variables, uij. As linear transforma- which this happens, grows with B—a larger third-order term tions do not alter the convexity property of any function,√ we is needed for a larger B.Now,B is smallest at ρ = ρ p,grows may take√ the energy as√w7(ρ, , x1−5) where x1 √≡ 2uxy, monotonically with ρ, and diverges at ρcp. As a result, the x2 ≡ 2uxz, x3 ≡ √2u yz, x4 ≡ (uxx − uzz)/ 2, x5 ≡ instability line cuts the e-axis at ρ p, veers toward larger (uxx − 2u yy + uzz)/ 6. The characteristic polynomial N7 ρ w = (λ − −1∂w /∂ )4 (or larger P) at higher density , and heads for infinity at cp, of the Hessian matrix of 7 is N7 us 1 us N3, see the thin line depicted as “simple model” in Fig. 1a, drawn with N3 the characteristic polynomial of w1(, us,ρ). Since ζ = −1∂w /∂ with a constant 1 24500 MPa. (It is of course possible, us 1 us is always positive, it is sufficient to consider employing a density-dependent ζ1, to improve the agreement w1(, us,ρ). Requiring N3 to have only positive eigen- to the dotted line.) In Fig. 1b, the point of maximal pressure values defines the stable region in the strain space, spanned for a given void ratio e is located at where the P-axis is being by ,us, e. Using Eqs. (54, 55), we may convert this into one ∼ 2 ,σ , cut by the associated curve. If the term us did not exist, in the stress space, spanned by P s e. The result, obtained ∼ 2 these curves would be vertical lines. The presence of us numerically, is the yield surface plotted in Fig. 1. reduces the value of (or P) for growing us (or σs), bending the lines to the left. Although qualitative figures of these curves—frequently 6.4 Pressure contribution from agitated grains referred to as “caps”—abound in textbooks [17–19], we did not find enough quantitative data, especially not a generally Agitated grains are known to exert a pressure in granular accepted empirical expression, that we could have compared liquid. Using the model of ideal gas (better: non-interacting our results to. Presumably, it is not easy to observe caps in atoms with excluded volumes), with w2 ∼ ρTg denoting dry sand. Given this lack of reliable data, we decided against the energy density of agitated grains, the pressure expres- the expansion of Eq. (52), and opted for a flexible “cap func- sion, tion,” C of Eq. (50), capable of accounting for any possible cap-like unjamming transition, PT (ρ, Tg) ∼ w2/(1 − ρ/ρcp), (56)

2C = 1 + tanh[(0 − )/1], where (53) = ρ − 2 − = /( + ) − 2 − . was employed and found to account realistically for the 0 k1 k2us k3 k1 e 1 k2us k3 behavior of granular liquid sandwiched between two cylin- With C ≈ 1for 0, and C ≈ 0for 0,the ders rotating at different velocities [76–79]. cap function is constructed to be relevant only in a narrow In ideal gas, both the energy density w and pressure P are −4 neighborhood around 0,for| − 0| 1 ≈ 10 , such proportional to the temperature T . As a consequence, the en- that the energy’s convexity is destroyed around 0. Taking tropy is s∼ ln T , and diverges for T → 0. (The free energy k1, k2, k3 as constant, 0 grows with the density and falls has a contribution ∼T ln T that vanishes for T → 0.) As 2 with us , giving rise to the typical appearance reproduced in quantum effects become important long before T vanishes, Fig. 1. the unphysical feature of a diverging entropy is inconse- Together, Eqs. (42, 50, 53) give the energy density w1, quential for ideal gases. Yet this would be a highly rele- appropriate for cohesionless granular materials at Tg = 0. vant defect for granular solids, for which important phys- ¯ There are two contributions to the pressure, P = P1 + P, ics occurs at or around Tg = 0. This is the reason ideal where P1 ≡ ρ(∂w1/∂ρ) − w1 from Eq. (47), and πij = gas is not an appropriate model for granular solids. The −∂w /∂ ≡ δ − σ 0 / w , ∼ ¯ 2 1 uij P ij suij us. Because we still take to considerations of Sect. 4 show that 2 f2 Tg close to 2.5 ¯ be a small quantity, P1 ∼ may be neglected. (Similarly, Tg =0— implying a pressure contribution, P2 =ρ(∂f2/∂ρ)− π ∼ 2.5 ∼ ¯ 2 ∼ w terms such as iku jk from Eq. (65) below are also f2 Tg ,seeEq.(47). Note ﬁrst that P2 2 is re- π = ≈ negligible). So the stress is simply ik, with pressure and tained, and second that because P0 0, P1 0, we have shear stress given as PT ≡ Pi ≈ P2. √ Unfortunately, the density dependence of Eq. (56)also 3 2 ∗ P = B ( + u /) − w C / , (54) = ρ ( − √ 10 s 1 1 poses a problem, as it implies a free energy f2 b0 ln 1 ∗ 2 π = 6 B − w C / , ρ/ρcp)(−T /2) and a granular entropy, sg =−∂ f2/∂Tg = s 5 us 2k2us 1 1 (55) g b0ρ ln(1−ρ/ρcp) Tg, both diverging for ρ → ρcp. We there- ∗ ∗ where C ≡ 1 − tanh[(0 − )/1], hence C → 0away fore take f2 to be given as in Eq. (43), with a positive but from the cap. (The terms of higher order in are kept in small a. The resulting entropy is physically acceptable, and ∗ C , because 1 is small. This is how we make C a function the pressure is easily rendered numerically indistinguishable relevant for ≈ 0, not → 0). from Eq. (56), 123 150 Y. Jiang, M. Liu

Taking the entropy S(W, V ) as a function of the energy W and volume V ,ordS = (1/T )dW + (P/T )dV , the authors of [81,82] argue that a mechanically stable agglomerate of infinitely rigid grains at rest has, irrespective of its volume, vanishing energy, W ≡ 0, dW = 0. The physics is clear: however, we package these rigid grains that neither attract nor repel each other, the energy remains zero. Therefore, the basic expression of GSM, dS = (P/T )dV ,ordV = (T/P)dS ≡ XdS hold, with X a relevant quantity charac- terizing granular media at rest. The entropy S is obtained ,σ ∼ 2 Fig. 3 Jamming transition as a function of e s and PT Tg ,for by counting the number of possibilities to package grains P = . 0 4 MPa. Values of model parameters are the same as those in eS Fig. 1 for a given volume, and taking it to be . Because a stable agglomerate is stuck in one single configuration, some tapping (or a similar disturbance) is taken to be needed to enable ρ ¯ 2 ρ a b0Tg the system to explore the phase space. PT = P = , (57) 2 2ρ (1 − ρ/ρ )1−a In GSH, the present theory, grains are neither infinitely cp cp ∂ f ρ a rigid, nor always at rest. An elastic and a Tg-dependent energy s =− 2 = ρ b T¯ 1 − . (58) g ¯ 0 g ρ contribution, denoted, respectively, as f1 and f2,seeEq.(41), ∂Tg cp account for these effects. GSH also possesses a Tg-switch As the total pressure is now P = PT + P,cf.Eq.(54), the that determines whether the system’s behavior is solid- or jamming transition discussed above is modified. For instance, liquid-like, with phase space exploration enabled in the sec- the yield condition of Eq. (3), with ξ = 5/3, now reads ond case. That grains neither attract nor repel each other

is accounted for by the stress vanishing if Tg and uij do. πs πs 6 = ≤ , (59) Then f1, f2 = 0, with only f0 ∼ ρ finite, implying πij = P P − PT 5 ∂( f0/ρ)/∂(1/ρ)δij = 0. implying a smaller maximal πs for given P. On the other Given this comparison, it is natural to ask whether GSM hand, the maximal value for the void ratio e is larger when is a legitimate limit of GSH. The answer is probably no, PT is present: any given e has a maximal elastic compres- as both appear conceptually at odds—in two points, the first sion that will not sustain a larger e.ButifP is fixed and more direct, the second quite fundamental: (1) Because of the Tg is finite, the elastic compression will be appropriately Hertz-like contact between grains, very little material is being smaller to sustain a larger e. This behavior is depicted in deformed at first contact, and the compressibility diverges at Fig. 3. vanishing compression. This is a geometric fact independent The jamming transition, from elastic solid to liquid, is of of how rigid the bulk material is. Infinite rigidity is therefore course no longer completely sharp at a finite Tg, because Tg not a realistic limit for sand. (2) In considering the entropy, turns the elastic body into a transiently elastic one for all one must not forget that the number of possibilities to pack- values of stress and density. Nevertheless, there is a huge age grains for a given volume is vastly overwhelmed by the quantitative difference between catastrophic unjamming and much more numerous configurations of the inner granular the gradual process of stress relaxation. A sand pile may degrees of freedom. Maximal entropy S for given energy slowly degrade, relaxing toward the flat surface. But when therefore realistically implies minimal macroscopic energy, turning on Tg violates Eq. (59), sudden events such as lique- such that a maximally possible amount of energy is in S faction happen. (PT may be substituted by the pore pressure (or heat), equally distributed among the numerous inner gran- to account for a similar collapse, if the soil is filled with ular degrees of freedom. Maximal number of possibilities water.) The frequently reported phenomenon of a primary to package grains for a given volume is a very different earthquake emitting elastic waves that trigger earthquakes criterion. elsewhere [80], may well be connected to Eq. (59): Tg as given by Eq. (35) accompanies elastic waves. It may be suf- ficiently large to violate Eq. (59) if stability was precarious. 7 Granular hydrodynamic theory

6.5 The Edwards entropy 7.1 Derivation

It is useful, with the free energy obtained in this chapter We take the conserved energy w(s, sg,ρ,gi , uij) of granular in mind, to revisit the starting points of granular statistical media to depend on entropy s, granular entropy sg, density mechanics (GSM), especially the Edwards entropy [81,82]. ρ, momentum density gi , and the elastic strain uij. Deﬁning 123 Granular solid hydrodynamics 151

¯ the conjugate variables as T ≡ ∂w/∂s, Tg ≡ Tg − T ≡ vanish independently, ∂w/∂ µ − v2/ ≡ ∂w/∂ρ sg [see Eq. (23)], 2 [see Eq. (8)], ¯ Qi = Tfi + Tg Fi + µρvi + v j σij − y j πij, (67) vi ≡ ∂w/∂gi = gi /ρ [see Eq. (5)], πij ≡−∂w/∂uij,we = D∇ + σ Dv + ∇ π + π + γ ¯ 2, write R fi i T ij ij yi j ij Xij ij Tg (68) = Dv + D∇ ¯ − γ ¯ 2. Rg ij ij Fi i Tg Tg (69) ¯ 2 dw = T ds + Tgdsg + (µ − v /2)dρ + vi dgi − πijduij. Comparing R, Rg with Eqs. (32, 34), the currents are found (60) as D = κ∇ ,σD = ζv δ + ηv0 + απ , The equations of motion for the energy and its variables are fi i T ij ij ij ij D = κ ∇ ¯ ,D = ζ v δ + η v0, Fi g i Tg ij g ij g ij (70) ∂ w +∇ Q = 0,∂ρ +∇(ρv ) = 0, (61) t i i t i i = β P ∇ π , = βπ0 + β δ π − αv . yi j ij Xij ij 1 ij ij ∂t gi +∇j (σij + gi v j ) = 0, (62) D∇ ¯ ∂t s +∇i fi = R/T,∂t sg +∇i Fi = Rg/TG , (63) (It is an assumption to take Fi i Tg as part of Rg rather than R.) The two terms preceded by α contribute ±απijvij to dt uij − vij − Xij R, respectively, hence cancel each other and are compatible =−[( ∇ v +∇ / ) + ( ↔ )]. uik j k i y j 2 i j (64) with Eq. (32). (More such pairs of terms, mutually canceling or contributing in equal parts, are possible. They have been The first three equations are conservation laws, with the excluded as a simplification. In the language of the Onsag- σ fluxes Qi and ij as yet unknown, to be determined in this er force–flux relation, the above fluxes possess only diago- section. The next two are the balance equation for the two nal elements, with the exception of the reactive, off-diagonal entropies, the form of which are already given, in Eqs. (30, terms ∼ α). Defining two relaxation times, 32, 33, 34). Nevertheless, to see that they indeed fit the con- 1 √ 1 √ Au2 straints required by energy and momentum conservation, we ≡ 2βA , ≡ 3β B + s . (71) = v − D = v − D τ τ 1 2 designate the currents as fi s i fi , Fi sg i Fi , 1 2 D, D, , leaving fi Fi R Rg unspecified. The last is the equation the last of Eqs. (70) may be written as of motion for the elastic strain field, as discussed in Sect. 5, = δ /τ − 0 /τ − αv . with yi , Xij the unknown fluxes to be determined here. Next, Xij ij 1 uij ij (72) σ D + D we introduce ij ij,as To ensure permanent elasticity in granular statics, we must in addition require σ ≡ (− ˜ + µρ)δ − σ D + D ij f ij ij ij Xij → 0forTg → 0. (73) + π − π u − π u , (65) ij ik jk jk ik D, D,σD, This completes the derivation of GSH. Given fi Fi ij ˜ ¯ D, y , X , the structure of all currents in the set of equa- where f ≡ w0 − Ts− Tgsg,asinEqs.(27, 47). This is sim- ij i ij σ D + D tion, Eqs. (61–64), are known. The question that remains is ply a definition of ij ij, which transfer our task from whether these expressions are unique. For simpler hydrody- determining σij to finding the new quantity. This simplifies namic theories, such as for isotropic liquid, nematic liquid our task, notationally, of finding the form of σij, it does not in anyway prejudice it. crystal, or elastic solid, this procedure (frequently referred ¯ to as the standard procedure) is easily shown to be unique, Differentiating the energy, ∂t w = T ∂t s + Tg∂t sg + (µ − 2 because one can convince oneself that as long as the energy v /2)∂t ρ + vi ∂t gi − πij∂t uij,seeEq.(60), then inserting ¯ w remains unspecified, there is only one way to write the Eqs. (61–64) into it, employing relations such as Tg∂t sg = ¯ ¯ ¯ time derivative of the energy ∂ w as the sum of a divergence Tg Rg/TG + vksg∇k Tg −∇k(Tgsgvk), we obtain t and a series of expressions that vanish in equilibrium. In the ¯ present case, with two levels of entropy productions, one of ∇i Qi =∇i (Tfi + Tg Fi + µρvi + v j σij − y j πij) which controls the switch between permanent and transient − R + f D∇ T + σ Dv + y ∇ π + X π i i ij ij i j ij ij ij elasticity, the hydrodynamic theory is singularly intricate, +γ ¯ 2 − + Dv + D∇ ¯ − γ ¯ 2 Tg Rg ij ij Fi i Tg Tg (66) and peripheral ambiguities remain. Nevertheless, displaying energy and momentum conservation explicitly, and reduc- This is a useful result, which shows one can rewrite ∂t w as the ing to liquid and solid hydrodynamics in the proper limits, divergence of something (first line), plus something (second the given set of equations is certainly a viable and consistent and third line) that vanishes in equilibrium—see Sect. 3.3 theory. why ∇i T,vij,πij, ∇ j πij and TG vanish. We take the first A more formal way of obtaining the fluxes of Eqs. (70)is = ( D, ,σD, ) line to yield the energy flux, Qi , and the next two lines to to define the flux and force vectors as Z fi yi ij Xij , 123 152 Y. Jiang, M. Liu

= ( D,D) = (∇ , ∇ π ,v ,π ) = Zg Fi ij , Y i T j ij ij ij , Yg which, as we shall see next, gives rise to the same dynamic ¯ (∇i Tg,vij). And because R = Z · Y , Rg = Zg · Yg,the structure as hypoplasticity. The density dependence is more Onsager force–flux relations are given as subtle, hence harder and less urgent to determine. However, it seems plausible that λ, λ1 should decrease for growing =ˆ· , =ˆ · . Z c Y Zg cg Yg (74) density, and the compressional relaxation should stop being operative at the random close packing density ρcp. To account The transport matrices, cˆ and cˆg, have positive diagonal elements and off-diagonal ones that satisfy the Onsager for this, the simplest dependence would be reciprocity relation. Our example above has only diagonal λ1 ∼ (ρ − ρcp). (81) elements, with the single exception of the reactive, off-diag- ∼α The coefficient α needs to vanish in the elastic limit, for onal terms . ¯ ¯ Tg → 0, and be constant in the hypoplastic one, when Tg is σ = π 7.2 Results moderately large: We have ij ij in the elastic regime, and σij = (1 − α)πij +··· with 1 − α ≈ 0.2 in the hypo- Collecting the terms derived above, in Sect. 7.1, the equations plastic one, implying sand is much softer here—same strain, yet stress is smaller by a factor of about five. The behavior of of GSH, with σij valid to lowest order in strain, are α is probably the result of granular agitation disrupting force ∂t ρ +∇i (ρvi ) = 0, (75) chains. They are all intact in the elastic limit, making the = ( − α)v − 0 /τ − δ /τ system comparatively stiff. A finite T¯ breaks up the chains, dt uij 1 ij uij u ij 1 g and when most of chains are destroyed, the remaining ones − ∇ v +∇[β P ∇ π / ] − ( ↔ ), uik j k i k jk 2 i j (76) become essential in the sense that their disruption leads to

σij = (1 − α)πij − πiku jk − π jkuik local collapse, which in turn immediately repair the chains ˜ 0 by some rearrangement. This is why α saturates and becomes +(µρ − f )δij − (ζ + ζg)v δij − (η + ηg)v , (77) ij constant. [∂ +∇( v − κ ∇ ¯ )]= Tg t sg i sg i g i Tg Rg Finally, as long as Eq. (39) holds, the rate independence = ζ v2 + η v0 v0 + κ (∇ ¯ )2 − γ ¯ 2, g g ij ij g i Tg Tg (78) it entails would prevent the propagation of sound and elas- 2 0 0 ¯ 2 tic waves: because both the elastic and the plastic part are T [∂t s +∇i (svi − κ∇i T )]=ζv + ηv v + γ T ij ij g linear in the velocity, and of the same order in the wave vec- +κ(∇ )2 + β P (∇ π )2 + βπ0 π 0 + β π 2 . i T j ij ij ij 1 (79) tor q, sound damping is comparable to sound velocity, and ρ wave propagation could at most persist for a few periods. We Given in terms of the variables: (s, sg, , gi , uij), conju- ¯ ¯ therefore expand γ,ηg in Tg,as gate variables (T , Tg, µ, vi , πij), and 11 transport coeffi- α τ τ ζ ζ η η γ β P κ κ ¯ ¯ cients, ( , , 1, , g, , g, , , , g), these equations γ = γ0 + γ1Tg,ηg = η1Tg, (82) are valid irrespective of the functional form of the energy η w and the transport coefficients. Therefore, they only pro- assuming g lacks a constant term, because viscous dissipa- η ¯ → vide the hydrodynamic structure, a framework into which tion occurs directly via for Tg 0, see Eq. (70). Insert- different concrete theories fit. This circumstance, though also ing these expression into Eq. (35) for a quick, qualitative ¯ ∼ v v ≡ v2 γ γ ¯ true for Newtonian fluids, is not as relevant there, because estimate, we find Tg ij ij s for 0 1Tg, and ¯ ∼ v γ γ ¯ static susceptibilities (such as the compressibility or specific Tg s for 0 1Tg. The first regime is essentially elas- /τ ∼ ¯ ∼ v2 heat) and transport coefficients may frequently be approxi- tic, because the relaxation term, uij uijTg uij s ,is mated as constant. So the structure alone already possesses of second order and small. This ensures the propagation of /τ ∼ considerable predicting power. This is not true for granu- sound modes. In the second regime, the same term, uij ¯ ∼ v lar media, which typically possess more involved functional uijTg uij s, is of first order and rather more prominent, ¯ giving rise to the hypoplastic behavior discussed in the next dependence—especially concerning the Tg → 0 limit, which does not have a counter part in other systems. This is one of section. the less recognized reasons, we believe, underlying the complexity of granular systems. In Sect. 6, a free energy was presented that we are con- 8 The hypoplastic regime fident is fairly realistic. The situation with respect to the 11 transport coefficients is less settled, and in need of much Hypoplasticity [20], a modern, well-verified, yet compara- future work, though a few limits are clear from the onset: tively simple theory of soil mechanics, models solid dynam- first, a simple, analytic way to assure the elastic limit for ics as realistically as the best of the elasto-plastic theories. Its T¯ = starting point is the rate-independent constitutive relation, g 0 and satisfy the requirement of Eq. (73) is given by /τ = λ ¯ , /τ = λ ¯ , ∂ σ = v + v0 v0 + v2 , 1 Tg 1 1 1Tg (80) t ij Hijk k ij ij ij (83) 123 Granular solid hydrodynamics 153 where the coefficients Hijk ,ij, are functions of σij,ρ, are: ζg/ηg = 0.33 implies shear flows are three times as specified using experimental data mainly from triaxial appa- effective in creating Tg as compressional flows. τ/τ1 = 0.09 ratus. (Rate-independence means ∂t σij is linearly propor- means, plausibly, that the relaxation rate of shear stress is tional to the magnitude of the velocity, such that the change ten times higher than that of pressure. The factor (1 − α)2 in stress remains the same for given displacement irrespec- accounts for an overall softening of the static compliance λ tive how fast the change is applied, a well verified observation tensor Mij k. Finally,√ controls the stress relaxation rate η /γ in both the elastic and hypoplastic regime.) Great efforts are for given Tg, and 1 1 how well shear flow excites Tg. invested in finding accurate expressions for the coefficients, Together, λ ηg/γ = 114 determines the relative weight of of which a recent set [20]is = 1/3, plastic versus reactive response: the term in Eq. (83) preceded

by H is the reversible, elastic response, the second term = 2δ δ + 2σ σ /σ 2 ijk Hijk f F ik j a ij k nn ,(84) preceded by ij comes from stress relaxation, is dissipative, , → = σ + σ 0 /σ , irreversible and plastic. For small strain, us 0, the elas- ij affd F ij ij nn (85) tic part is dominant, |Hijk ||ij|.But|ij|/|Hijk |∼ = . = , = . = . | 0 |· /( − α) | | −3 where a 2 76, hs 1 600 MPa, ed 0 44ei , ec 0 85ei , uk 114 1 is of order unity for uij around 10 . −1 = (σ / )0.19 ei exp hs , and Although the functions of Eqs. (87, 88) appear rather different from that of Eqs. (84, 85), the stress–strain increments − 0.25 . ( + ) σ 0.81 e ed 8 7hs 1 ei are quite similar, as the comparison in [68] shows. Moreover, fd = , f =− , e − e 3 (σ /σ + 1) e h c d s s the residual discrepancies may be eliminated by discarding 3σ 2 2σ 2σ − 3σ 4/σ 3 σ the simplifying assumption of constant transport coefficients, F = s + s s − s . 2 2 0 0 0 σ independent of the stress. This agreement provides valuable 8σ 2σ σ − 6σ σ σ 8 s ij j i insights into the physics of Hypoplasticity, showing why it GSH, as derived above, reduces to Eq. (83)forasta- works, what its range of validity is, and how it may be gen- tionary Tg, with Hijlk,ij, given in terms of Mijk ≡ eralized. And it conversely also verifies GSH. 2 −∂ w/∂uij∂uk and four scalars that are combinations of transport coefficients. We assume uniformity and stationa- ¯ rity, with especially ∇i Tg, ∇ j πij,∂t vi = 0, and only include 9 Conclusion the lowest order terms in the strain uij.Wealsotakeα, ηg,ζg as constants, and neglect PT from Eq. (57), the pressure rel- The success of granular elasticity, the theory we employ evant in granular liquid, assuming Tg is too small for the to account for static stress distribution in granular media, given velocity. It is then quite easy to evaluate ∂t σij employ- is mainly due to the fact that the information on the plas- ing Eqs. (76, 77), tic strain is quite irrelevant there. This is no longer true in granular dynamics, when the system is being deformed— ∂t σij = (1 − α)∂t πij = (1 − α)Mijk ∂t uk sheared, compressed or tapped. Starting from the working = ( − α)M [( − α)v − u0 /τ − δ u /τ ]. 1 ijk 1 k k k mm 1 hypothesis that granular media are transiently elastic while (86) being deformed, we aim to understand the notoriously com- Clearly, given Eqs. (35, 80), this expression already has the plex plastic motion by combining two simple and transparent structure of Eq. (83) that Hypoplasticity postulates. And the elements, elasticity and stress relaxation. In a recently pub- coefficients are lished Letter [68], we proposed a model for granular solids based on this hypothesis. In the present manuscript, we give 2 H = ( − α) M ,= ζ /η , ijk 1 ijk g g (87) this model a consistent, hydrodynamic framework, compat- = ( − α) [(τ/τ )δ − 0 ]λ η /γ . ij 1 Mijk 1 k uk g (88) ible with conservation laws and thermodynamics. The framework is valid for any healthy energy, but is + + H , , HPM has 43 free parameters (36 6 1for ijk ij ), essentially devoid of predictive power if the energy is left all functions of the stress and density. Expressed as here, unspecified. Therefore, an explicit expression for the total, the stress and density dependence are essentially determined conserved energy is given. Encapsulating the key features of M by ijk that is a known quantity [66,67]. For the four free static granular media: stress distribution, incremental stress– constants, we take strain relation, minimal and maximal density, the virgin con- 1 − α = 0.22,τ/τ1 = 0.09, solidation line, the Coulomb yield line and the cap model, (89) this expression should prove realistic enough for rendering ζ η η 114 g = . , g = 1 = , the specific hydrodynamic theory useful. Much future work is η 0 33 γ γ λ g 1 needed to see whether further agreement between theory and to be realistic choices, as these numbers yield satis- experiments may be achieved, especially concerning cyclic factory agreement with Hypoplasticity. Their significance loading, tapping and shear band. 123 154 Y. Jiang, M. Liu

Open Access This article is distributed under the terms of the Creative by changing the volume, or the length in the one-dimen- Commons Attribution Noncommercial License which permits any sional case. Holding δUi ≡ 0 at the moving surface will then noncommercial use, distribution, and reproduction in any medium, allow Eq. (A4) to hold. Denoting the energy, entropy and provided the original author(s) and source are credited. volume per unit mass, respectively, as e ≡ w/ρ, σ ≡ s/ρ, v ≡ V/M = 1/ρ, and f ≡ w − Ts, the equivalent expression, Appendix A: Equilibrium conditions de = T dσ − PT dv − (πij/ρ)d∇ j Ui , (A5) First, noting π du = π d∇ U because π is symmetric, ij ij ij j i ij PT ≡−w + Ts + µρ =−f + µρ , (A6) we write the energy density per unit volume (dropping the holds. Now we have E = edM, S = σ dM, V = vdM, subscript of w0 in this section) as where dM = ρdV is the integrating mass element. Varying dw = T ds + µdρ − πijd∇ j Ui . (A1) the energy for given entropy, volume and requiring it to vanish, δE − S − V = 0, we find Next, we vary the energy wdV for given entropy sdV , 1 2 mass ρdV , and for fixed displacement at the medium’s sur- [(T − 1) δσ + (PT − 2) δv] dM = ∇ j πij δUi dV. face, δUi = 0. Taking 1, 2 as constant Lagrange parameters and denoting the surface element as dAi , we require the implying ∇i T = 0, ∇i PT = 0, and ∇ j πij = 0. These are the variation of the energy to be extremal, same conditions as Eq. (A3), because ∇ P = s∇ T +ρ∇ µ. i T i i But if Eq. (A4) is implemented, turning Eq. (A5)to δ (w − 1s − 2ρ) dV = 0. (A2) = σ − ( + π / ) v − (π 0 /ρ) ∇ , de T d PT 3 d ij d j Ui (A7) − Inserting Eq. (A1)into(A2), we find = T dσ − ρ 1(P δ + π ) d∇ U , (A8) T ij ij j i T δs + µδρ + πijδ∇ j Ui − 1δs − 2δρ dV the equilibrium conditions are altered to become ∇ = , ∇ ( + π / ) = , ∇ π 0 = . i T 0 i PT 3 0 j ij 0 (A9) = (T − 1) δs + (µ − 2) δρ − ∇ j πij δUi dV Clearly, the Cauchy, or total, stress in equilibrium is given as + π δ = , ij Ui dAi 0 σij = PT δij + πij, with ∇ j σij = 0. (A10) ρφ w −∇ φ = where the last term vanishes because δUi ≡ 0 at the surface. Including the gravitational energy in , with i If δs, δρ and δU j vary independently, all three brackets must Gi , the gravitational constant pointing downward, we have vanish. And because 1, 2 are constant, T,µalso need to be. dw = T ds + (µ + φ)dρ − πijd∇ j Ui + ρ dφ, (A11) So the conditions for the energy (or entropy) being extremal are and find (via the same calculation as above) that µ+φ is now a constant, implying an alteration of the second of Eqs. (A3) ∇ = , ∇ µ = , ∇ π = . i T 0 i 0 j ij 0 (A3) to ¯ = In granular media for Tg 0, density and compression ∇i µ = Gi , (A12) are coupled as or equivalently, ∇i PT = s∇i T + ρ∇i µ = ρGi .IfEq.(A4) =− ρ/ρ = ρ v, du d d (A4) holds, ∇ j σij = 0 is analogously changed to and do not vary independently. Simply inserting this rela- ∇ j σij =∇j (PT δij + πij) = ρGi . (A13) tion into the above calculation, we find ∇i (µ + π /3ρ) = 0 to replace the last two conditions of Eq. (A3). This is not the correct result, because we have been varying the References energy and its variables above,keeping the volume unchanged throughout, with δUi ≡ 0 at the surface. But then u is 1. Jaeger, H.M., Nagel, S.R., Behringer, R.P.: Granular solids, liquids, fixed and cannot change with the density ρ: Consider a one- and gases. Rev. Mod. Phys. 68(4), 1259 (1996) = = 2. Liu, A.J., Nagel, S.R.: Jamming is not just cool any more. dimensional medium between x 0 and x x0, with Nature 396, 21 (1998) ( ), ( ) ∇ π ∼ ∂2[ ( ) − ( )]= Ux 0 Ux x0 given. Since j ij x Ux x0 Ux 0 3. de Gennes, P.G.: Granular matter: a tentative view. Rev. Mod. Phys. 71(2), 347 (1999) 0, the one-dimensional strain is u = ∂x Ux = (Ux (x0) − U (0))/x and cannot change. 4. Kadanoff, L.P.: Built upon sand: theoretical ideas inspired by the x 0 flow of granular materials. Rev. Mod. Phys. 71(1), 435 (1999) To find the proper expression, we may take mass M rather 5. Haff, P.K.: Grain flow as a fluid-mechanical phenomenon. J. Fluid than volume V as the constant quantity, and vary the density Mech. 134, 401 (1983) 123 Granular solid hydrodynamics 155

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