Open problems of granular solid hydrodynamics: principled determination of transport coefficients and energy closures, the Edwards-statistics limit, and the range of validity

Granular solid hydrodynamics (GSH), the nonequilibrium-thermodynamic continuum theory of Jiang and Liu for dense granular media, is pinned at the level of structure but not at the level of content: the functional dependence of its transport coefficients and relaxation parameters on the state variables is tentative and fitted rather than derived or systematically measured; the universality of its elastic-energy closure and of the coefficient b(rho) is unestablished; whether the Edwards volume ensemble of jammed matter emerges as a limit of the theory is explicitly left open by the authors; the theory's validity range in inertial, collisional, and fluid-saturated regimes is uncharacterized; and broad quantitative experimental validation is lacking.

Background

A dense assembly of grains (sand, soil, powder) is a many-body system whose macroscopic behavior eludes simple continuum description: it supports shear stress like a solid, flows like a fluid, and dissipates energy through inter-granular collisions and friction. Jiang and Liu developed granular solid hydrodynamics (GSH), a continuum theory built by the methods of nonequilibrium thermodynamics that were originally devised for superfluid helium, applied here to what they call the 'granular gas of statics'. The central idea is that besides the true thermodynamic temperature, the state of a dense granular medium includes a granular temperature, denoted $\bar T_g$, an intensive variable conjugate to an internal entropy $S_g$ that accounts for the chaotic jiggling and rearrangement of grains - as distinct from the thermal entropy carried by the vibrational modes inside each grain. The independent state variables are the mass density $\rho$, the elastic strain tensor $u_{ij}$ (decomposing the total deformation into an elastic part and plastic rearrangements), and $\bar T_g$; the elastic stress $\pi_{ij}$ is derived from a stored elastic energy, and a relaxation term drives $u_{ij}$ toward its yield surface, which is how plasticity emerges thermodynamically rather than being postulated as a separate yield criterion. The energy is stored in the deformed grain contacts, and the theory takes the grains to be perfectly rigid except at their contacts.

The thermodynamic structure fixes the form of the equations but not their content. Jiang and Liu formulate the theory as a two-step hydrodynamic program: the first step specifies the theory's structure - the differential equations are given in terms of the energy density $w$, its variables and conjugate variables, while the form of $w$ and the functional dependence of the transport coefficients are left unspecified; the second step makes the energy and the transport coefficients specific: the elastic energy density $w$ (including its $\bar T_g$-dependent terms in the entropy production), the viscosities and thermal conductivities, the relaxation coefficients $\lambda, \lambda_1$ that set the plastic relaxation rate of $u_{ij}$, and the various coefficients appearing in the production and relaxation functions $R$ and $R_g$ of $\bar T_g$, all as functions of the state variables $(\bar T_g, \rho, \pi_{ij})$ and the true temperature. In their review 'The Physics of Granular Mechanics' (arXiv:1206.2835), they give explicit choices for these functions but state that they remain tentative: the chosen elastic energy $w_1$ and the density-dependent coefficient $b(\rho) = b_0(1-\rho/\rho_{cp})^a$ with exponent $a \approx 0.1$ reproduce many observed features but rest on empirical parametrizations ($\mathcal{B}_1, \mathcal{C}, \Delta_0, \Delta_1$) whose universality is not established.

The theory also stands in an unresolved relation to granular statistical mechanics. Edwards and Oakeshott proposed that jammed granular media admit a statistical ensemble in which all blocked configurations of a given volume and grain number are equally weighted, with a conjugate temperature-like variable, the compactivity, defined by $\partial S/\partial V = 1/X$. Whether this construction is a legitimate limit of GSH - whether, in the limit of perfectly rigid and purely static grains, the granular entropy $S_g$ reduces to the Edwards entropy - is stated by Jiang and Liu as an open question; they consider a positive answer unlikely, for two reasons: Hertzian contact mechanics (the elastic theory of curved bodies in contact, under which contact stiffness vanishes as the load goes to zero) makes the compressibility diverge as the compression vanishes, so the rigid limit is singular, and the number of internal vibrational degrees of freedom of the grains overwhelmingly exceeds the number of inter-granular configurations, so the granular entropy cannot be identified with a configurational entropy alone. The Edwards school, for its part, develops the ensemble program independently (compactivity measurements, ergodicity tests, ensemble equivalence) without deriving it as any hydrodynamic limit.

Neighboring continuum frameworks frame the stakes. The $\mu(I)$ rheology empirically captures dense inertial flow via a friction coefficient depending on the inertial number $I$ (a dimensionless ratio of the timescale of plastic rearrangement to the timescale set by the confining pressure, which orders dense-flow regimes); critical-state soil mechanics and hypoplasticity accumulate plasticity postulates in abundance; Onsager-symmetric constitutive laws for the inertial regime have recently been constructed from a yield condition plus discrete-element (DEM) steady-state data. GSH is the framework that claims to unify static, elasto-plastic, and dense-flow behavior from thermodynamic principles alone - but only if its second step, the principled determination of the coefficient functions and the validation of the closures, is completed.

Problem Statement

Complete the second step of the granular solid hydrodynamics program of Jiang and Liu, as left open in 'The Physics of Granular Mechanics' (arXiv:1206.2835). The single overarching objective is to make GSH's thermodynamic structure quantitative, principled, and validated, by resolving five jointly linked issues. Each of the five issues stands alone as an independently attackable and refutable sub-problem: settling any one of them is a result in its own right, judged on its own terms. For each issue, resolution means a definite outcome for the relevant quantity or relation - a derivation, a systematic determination, or a settled argument for the stated alternative - and not a further tentative parametrization:

1. Determine, systematically, the functional dependence of the transport coefficients and kinetic parameters on the state variables $(\bar T_g, \rho, \pi_{ij})$ and the true temperature: the viscosities, thermal conductivities, the plastic relaxation coefficients $\lambda, \lambda_1$, and the coefficients of the granular-temperature production/relaxation functions $R$ and $R_g$, replacing the present fitted tentative forms.

2. Establish the universality of the elastic-energy closure (the form $w_1$, its revision, and the coefficient $b(\rho) = b_0(1-\rho/\rho_{cp})^a$ with $a \approx 0.1$, and the empirical constants $\mathcal{B}_1, \mathcal{C}, \Delta_0, \Delta_1$) across materials, preparations, and loading paths.

3. Settle the relation between GSH and granular statistical mechanics: whether the Edwards volume ensemble and its compactivity emerge as a (possibly singular) limit of the granular entropy $S_g$, or whether the two frameworks are irreconcilable, with a clear statement of which alternative holds and why.

4. Characterize the limits of the present closures when inertial, collisional, and fluid-mediated effects become dominant - including when the neglected viscosity-type terms $\eta, \zeta$ must be retained, as in water-saturated sand - and extend the theory accordingly.

5. Provide systematic, quantitative validation of the resulting theory against a broad set of experiments spanning grain shapes, packing procedures, boundary conditions, and loading/frequency/amplitude ranges, beyond the preliminary tests reported to date. The solver specifies and justifies this set, naming the experiments or datasets and the quantitative comparison criteria, and states why the chosen set is adequate.

References

1. Yimin Jiang and Mario Liu, 'Granular solid hydrodynamics', Granular Matter 11, 139 (2009). DOI: 10.1007/s10035-009-0137-3, https://doi.org/10.1007/s10035-009-0137-3
2. Yimin Jiang and Mario Liu, 'The Physics of Granular Mechanics', arXiv:1206.2835 (2012; also in Mechanics of Natural Solids, Springer 2009, pp. 27-46). arXiv: 1206.2835, https://arxiv.org/abs/1206.2835
3. Yimin Jiang, Hepeng Zheng, Zheng Peng, Liping Fu, Shixiong Song, Qicheng Sun, Michael Mayer, and Mario Liu, 'An Expression for the Granular Elastic Energy', arXiv:1206.2831 (2012). arXiv: 1206.2831, https://arxiv.org/abs/1206.2831
4. Yimin Jiang and Mario Liu, 'Granular Solid Hydrodynamics (GSH): from Quasi-Static Motion to Rapid Dense Flow', arXiv:1207.1284 (2012). arXiv: 1207.1284, https://arxiv.org/abs/1207.1284
5. Yimin Jiang and Mario Liu, 'Applying GSH to a Wide Range of Experiments in Granular Media', arXiv:1407.7219 (2014). arXiv: 1407.7219, https://arxiv.org/abs/1407.7219
6. Yimin Jiang and Mario Liu, 'Why Granular Media Are Thermal, and Quite Normal, After All', The European Physical Journal E 40, 10 (2017); preprint arXiv:1604.01534 (2016, v3 2022). DOI: 10.1140/epje/i2017-11497-4, https://doi.org/10.1140/epje/i2017-11497-4; arXiv: 1604.01534, https://arxiv.org/abs/1604.01534
7. Mario Liu, 'Temperatures in Grains and Plasma', arXiv:2208.09904 (2022). arXiv: 2208.09904, https://arxiv.org/abs/2208.09904
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