This is a survey of the interesting phenomenology and the prominent regimes of granular flow, followed by a unified mathematical synthesis of continuum modeling. The unification is achieved by means of "parametric" viscoelasticity and hypoplasticity based on elastic and inelastic potentials. Fully nonlinear, anisotropic viscoelastoplastic models are achieved by expressing potentials as functions of the joint isotropic invariants of kinematic and structural tensors. These take on the role of evolutionary parameters or "internal variables," whose evolution equations are derived from the internal balance of generalized forces. The resulting continuum models encompass most of the mechanical constitutive equations currently employed for granular media. Moreover, these models are readily modified to include Cosserat and other multipolar effects. Several outstanding questions are identified as to the contribution of parameter evolution to dissipation; the distinction between quasielastic and inelastic models of material instability; and the role of multipolar effects in material instability, dense rapid flow, and particle migration phenomena. Copyright © 2014 by ASME.

This paper deals with the long-standing conflict between interpretations of thermoelectricity based on the original reversible thermodynamics of Thomson and the later irreversible thermodynamics of Onsager. It is shown that, by a slight modification of the Maxwellian relaxation treated in a previous paper [J. Goddard and K. Kamrin, “Dissipation potentials from elastic collapse,” Proc. R. Soc. A 475, 20190144 (2019)], Onsager's symmetry is simply a reflection of the underlying symmetry of equilibrium thermodynamics. It is also shown that a modern interpretation of Thomson's thermodynamics, as given recently by the present author, reveals thermoelectricity to be the analog of a fluid-mechanical transport process with the limit of thermodynamic equilibrium corresponding to the convection-dominated regime of large Péclet number.

This work considers strongly dissipative reaction-diffusion systems with constitutive equations given by Edelen's dissipation potentials, whose existence is tantamount to nonlinear Onsager symmetry. As the main goal of this work, it is shown that the associated extremum principle yields the relevant steady-state species balances and concentration fields for well-mixed reactors as well as for spatially inhomogeneous reactions accompanied by diffusion. The above principle is contrasted briefly with the popular notion of maximum entropy production. Potential applications of the theory to a wide range of reaction-diffusion systems are discussed, and questions are raised as to the possible breakdown of strong dissipation and nonlinear Onsager symmetry arising from reversible chemical or chemomechanical couplings.

Following is an elaboration on D. G. B. Edelen's (1972-1973) nonlinear generalization of the classical Rayleigh-Onsager dissipation potentials and the implications for the models of viscoplasticity. A brief derivation is given via standard vector calculus of Edelen's potentials and the associated non-dissipative or "gyroscopic" forces and fluxes. It is also shown that certain extensions of Edelen's formulae can be obtained by means of a recently proposed source-flux relation or "inverse divergence," a generalization of the classical Gauss-Maxwell construct. The Legendre-Fenchel duality of Edelen's potentials is explored, with important consequences for rate-independent friction or plasticity. The use of dissipation potentials serves to facilitate the development of viscoplastic constitutive equations, a point illustrated here by the special cases of Stokesian fluid-particle suspensions and granular media. In particular, we consider inhomogeneous systems with particle migration coupled to gradients in particle concentration, strain rate, and fabric. Employing a mixture-theoretic treatment of Stokesian suspensions, one is able to identify particle stress as the work conjugate of the global deformation of the particle phase. However, in contrast to past treatments, this stress is not assumed to be a privileged driving force for particle migration. A comparison is made with models based on extremal dissipation or entropy production. It is shown that such models yield the correct dissipative components of force or flux but generally fail to capture certain non-dissipative, but mechanically relevant components. The significance of Edelen's gyroscopic forces and their relation to reactive constraints or other reversible couplings is touched upon. When gyroscopic terms are absent, one obtains a class of strongly dissipative or hyperdissipative materials whose quasi-static mechanics are governed by variational principles based on dissipation potential. This provides an interesting analog to elastostatic variational principles based on strain energy for hyperelastic materials and to the associated material instabilities arising from loss of convexity. © 2014 Springer-Verlag Wien.

The following note shows that the symmetry of various resistance formulae, often based on Lorentz reciprocity for linearly viscous fluids, applies to a wide class of nonlinear viscoplastic fluids. This follows from Edelen's nonlinear generalization of the Onsager relation for the special case of strongly dissipative rheology, where constitutive equations are derivable from his dissipation potential. For flow domains with strong dissipation in the interior and on a portion of the boundary, this implies strong dissipation on the remaining portion of the boundary, with strongly dissipative traction-velocity response given by a dissipation potential. This leads to a nonlinear generalization of Stokes resistance formulae for a wide class of viscoplastic fluid problems. We consider the application to nonlinear Darcy flow and to the effective slip for viscoplastic flow over textured surfaces.

The following article deals with the role of compressibility in regularizing the well-known μ(I) model, i.e., eliminating the short-wavelength (Hadamard) instability revealed by Barker et al. [“Well-posed and ill-posed behaviour of the μ(I)-rheology for granular flow,” J. Fluid Mech. 779, 794–818 (2015)]. In particular, we discuss the compressible-flow models proposed in the recent papers by Heyman et al. [“Compressibility regularizes the μ(I)-rheology for dense granular flows,” J. Fluid Mech. 830, 553–568 (2017)] and Barker et al. [“Well-posed continuum equations for granular flow with compressibility and μ(I)-rheology,” Proc. R. Soc. A 473(2201), 20160846 (2017)]. In addition to a critique of certain aspects of their proposed constitutive models, we show that the main effect of their regularizations is to add viscous effects to the shear response in a way that appears unfortunately to eliminate quasi-static yield stress. Another goal of the present work is to show how the development and analysis of visco-plastic constitutive relations are facilitated by dissipation potentials and the dissipative analog of elastic potentials. We illustrate their utility in Sec. IV of this article, where it is shown that a constant non-zero yield stress leads to loss of convexity that can only be restored by substituting viscous effects or else by adding spatial-gradient effects proposed previously by the present authors [Goddard, J. and Lee, J., “On the stability of the μ(I) rheology for granular flow,” J. Fluid Mech. 833, 302–331 (2017)].

This article deals with the Hadamard instability of the so-called$\unicode[STIX]{x1D707}(I)$model of dense rapidly sheared granular flow, as reported recently by Barkeret al.(J. Fluid Mech., vol. 779, 2015, pp. 794–818). The present paper presents a more comprehensive study of the linear stability of planar simple shearing and pure shearing flows, with account taken of convective Kelvin wavevector stretching by the base flow. We provide a closed-form solution for the linear-stability problem and show that wavevector stretching leads to asymptotic stabilization of the non-convective instability found by Barkeret al.(J. Fluid Mech., vol. 779, 2015, pp. 794–818). We also explore the stabilizing effects of higher velocity gradients achieved by an enhanced-continuum model based on a dissipative analogue of the van der Waals–Cahn–Hilliard equation of equilibrium thermodynamics. This model involves a dissipative hyperstress, as the analogue of a special Korteweg stress, with surface viscosity representing the counterpart of elastic surface tension. Based on the enhanced-continuum model, we also present a model of steady shear bands and their nonlinear stability against parallel shearing. Finally, we propose a theoretical connection between the non-convective instability of Barkeret al.(J. Fluid Mech., vol. 779, 2015, pp. 794–818) and the loss of generalized ellipticity in the quasi-static field equations. Apart from the theoretical interest, the present work may suggest stratagems for the numerical simulation of continuum field equations involving the$\unicode[STIX]{x1D707}(I)$rheology and variants thereof.

As a summary and extension of previous work on arching in granular heaps, an analysis is given of statically admissible stress distributions in infinite planar wedges and axisymmetric cones composed of an Isotropie linear-elastic material subject to a noncohesive Mohr-Coulomb yield criterion. The treatment is based on a combination of analytical solutions for elastic and simple plastic states, together with numerical integration of the (Sokolovskii-Kötter) ordinary differential equations appropriate to more complex plastic states. For wedges, we obtain a one-parameter family of continuous elastoplastic solutions, with only three isolated symmetric solutions for symmetric wedges, one of which has a central pressure dip or 'arch'. For axisymmetric cones subject to a well-known closure for plastic hoop stress, only one continuous elastoplastic state is found, and it exhibits an arch. In addition to the above continuous solutions, a class of discontinuous plasticlimit states is considered, which exhibit a central pressure dip associated with the discontinuous transition from active to passive states proposed by Savage. The only solutions of this type found for symmetric wedges and cones involve central pressure dips. A brief discussion is given of the relation of this work to an extensive recent literature on the central pressure dip observed in certain experiments on granular heaps. © 2000 The Royal Society.