### 2.1. A micromechanical based continuum approach

Despite the fact that granular materials are discontinuous media, their behavior is commonly described by a continuum approach. Continuum mechanics theories solve the conservation equations of the whole medium, that is, the balance of mass, momentum, and when necessary, energy. Although the balance laws are easily deducible, the big challenge is the definition of the constitutive relations, that is, the rheology. The latter captures the macroscopic behavior of the system, incorporating the microscale grain‐grain interaction dynamics.

A granular flow can undergo different behaviors depending on both properties at the particle level and the macroscopic characteristic of the flow (i.e., velocity and concentration). At the microscopic level, each particle is characterized by its shape, dimension, material, and contact properties. For the sake of simplicity, in this chapter an assembly of identical spheres, of diameter *d*, density *ρp,* and equivalent linear contact stiffness *kn* is considered. The density of the continuum medium can be computed as the product of the particle density and the volume fraction, *ν*, defined as the fractional, local volume occupied by the spheres: *ρ = ρp ν.* Given that each grain *i* moves with velocity *vi*, the macroscopic velocity of *N*‐particles flow in a volume *V* can be defined as the average u=1V∑i=1Nvi. Similarly, we can introduce the strain‐rate tensor, calculated as the symmetric part of the velocity gradient. Its off‐diagonal components describe the shear rate between two Cartesian directions and are often used as control parameters to describe flow problems. In particular, considering a granular system with mean flow in the *x*‐direction only and sheared along the *y*‐direction, we introduce the shear rate as γ˙=2ε˙xy=∂ux/∂y. Finally, in continuum mechanics, the stress tensor, **σ**, represents the manner in which force is internally transmitted. Each component of the stress tensor, *σij* represents the force in the *i*‐direction on a surface with inward pointing normal unit vector in the *j*‐direction. The isotropic part of the stress tensor is the hydrostatic stress or pressure *p*, while the shear stress *τ* is proportional to the second invariant of the stress tensor. A detailed description of how to calculate strain rate and stress tensors in the case of granular assemblies will be provided in Section 2.2.

In the framework of continuum mechanics, dimensionless numbers are often introduced in order to describe the material behavior. These dimensionless numbers are defined as the ratio of different time scales or forces, thus signifying the relative dominance of one phenomenon over another.

In the case of granular flows, the macroscopic time scale associated with the shear rate parallel to the flow plays an important role. Then, it is convenient to scale all the quantities using the particle diameter, particle density, and shear rate γ˙, so that the dimensionless pressure and stiffness are given as p/(ρpd2γ˙2)and kn/(ρpd3γ˙2), respectively. On the other hand, when particle deformability becomes relevant, quantities are usually made dimensionless using the particle stiffness; pressure and shear rate are then expressed as p d/knand γ˙(ρpd3/kn). In the following sections, we will see how these dimensionless numbers are used to characterize granular flows in their different regimes, namely fluid‐like and solid‐like.

### 2.2. Continuum models

In the early modeling attempts, granular flow is envisaged as existing in either dense solid‐like or loose gas‐like regimes. Early works using shear cell experiments observed these regimes by varying the shear rate and allowing the bed to dilate or compact. Granular materials exhibit solid‐like behavior if the particles are packed densely enough and a network of persistent contacts develops within the medium, resulting in a jammed mechanically stable structure of the particles. On the other hand, when the grains are widely spaced and free to move in any direction, interacting only through collisions, the medium is unjammed and behaves like a fluid [7].

In the fluid‐like limit, the system is very dilute and the grains interact mainly through binary, instantaneous, uncorrelated collisions. One of the first rheological models for granular flows in this regime was proposed in 1954 by Bagnold [8]. This empirical model, derived from experiments in two‐dimensional plane shear flows, basically states that the stresses are proportional to the square of the strain rate. This simple law, now known as “Bagnold scaling,” has been the first to understand the physics of granular dynamics at large deformations and has been verified for dry grains in a number of experimental and numerical studies [9–12]. In the fluid‐like regime, the generalization of kinetic theory of granular gases provides a meaningful hydrodynamic description.

On the other hand, when the system is very dense, its response is governed by the enduring contacts among grains, which are involved in force chains; the deformations are extremely slow because the entire network of contacts has to be continuously rearranged (jammed structure). In these conditions, the granular material behaves like a solid, showing an elastic response in which stresses are rate independent. The corresponding flow regime is usually referred to as quasi‐static. Slowly deforming quasi‐static dense granular material has been mainly investigated in the framework of geo‐mechanics. There, the majority of the constitutive models are based on the theories of elasto‐plasticity and visco‐plasticity [13–16], and many of them have been conceived by starting from the well‐known critical state theory [17, 18].

In the transition phase, the grains interact via both force chains and collisions. None of the models cited above is able to deal with this phase‐transition of granular materials from a solid‐like to a fluid‐like state and vice‐versa. Intensive studies of the granular rheology at the phase transition have been conducted in the last decades, for example, by Campbell [19], Ji and Shen [20, 21], and Chialvo et al. [22] using 3D simulations of soft frictional spheres at imposed volume fractions. In these works, the authors derived a flow‐map of the various flow regimes and analyzed the transition areas. In particular, they found that, for a collection of particles, the solid‐fluid transition occurs in the limit of zero confining pressure at the critical volume fraction *νc*. Then the solid‐like regime, in which stresses are independent of shear rate, occurs for volume fractions *ν > νc*, whereas, at volume fractions *ν < νc* the system shows a fluid‐like behavior with stresses scaling with the square of the shear rate. In the proximity of the critical volume fraction, a continuous transition between the two extreme regimes takes place, for which the rheological behavior is still not fully understood.

More recently, new theories have been developed to model the phase transition. The French research group GDR‐MiDi [23] has suggested that dense granular materials obey a local, phenomenological rheology, known as μ(I)‐rheology, that can be expressed in terms of relations between three nondimensional quantities: volume fraction, shear to normal stress ratio, usually called *μ*, and inertial parameter *I*. The latter is defined as the ratio of the time scales associated with the motion perpendicular and parallel to the flow: I= γ˙dρp/p[24, 25]. The inertial number provides an estimate of the local rapidity of the flow, with respect to pressure, and is of significance in dynamic/inertial flows, as shown in Ref. [26]. In dense, quasi‐static flows, particles interact by enduring contacts and inertial effects are negligible, that is *I* goes to zero. Two main assumptions on the basis of the μ(I)‐rheology are: (i) perfectly rigid (i.e., nondeformable) particles and (ii) homogeneous flow. Various constitutive relations, based on the GDR‐MiDi rheology, have been developed [9, 27–29] in order to extend the validity of the model. In particular, the influence of particle deformability has been accounted for in the soft μ(I)‐rheology proposed in Refs. [30–32].

Below we present a summary of the two continuum theories that well describe the flow behavior in the limits and their extension to the intermediate regime. Kinetic theory in its standard form (SKT) provides a meaningful hydrodynamic description for frictionless particles in the very dilute regime, while μ(I)‐rheology holds for both frictionless and frictional particles for dense flows. It is important to mention that both theories work only for ideal systems, made of rigid, perfectly elastic, monodisperse particles. Finally, the extension of μ(I)‐rheology to deal with soft and deformable particles is also introduced.

#### 2.2.1. Standard kinetic theory (SKT)

This section is largely based on the notable works of Brilliantov et al. [33], Garzo et al. [34, 35], Goldhirsch [6, 36], and Pöschel et al. [37].

The term “granular gas” is used in analogy with a (classical) molecular gas, where the molecules are widely separated and are free to move in all directions, interacting only through instantaneous, uncorrelated collisions. The main differences between molecular and granular gases are that in the latter case part of the energy is irreversibly lost whenever particles interact and the absence of strong scale separation. These facts have numerous consequences on the rheology of granular gases, one of which being the sizeable normal stress differences [38].

Analogous to the molecular gases (or liquids), the macroscopic fields velocity and mass density are defined for granular systems [6]. An additional variable of the system, the granular temperature, *T*, is introduced as the mean square of the velocity fluctuations of the grains, in analogy with molecular gases, quantitatively describing the degree of agitation of the system.

Following the statistical mechanics approach, the kinetic theory of granular gases rigorously derives the set of partial differential equations given by the conservation laws of mass, momentum, and energy (the latter describing the time development of the granular temperature) for the dilute gas of inelastically colliding particles.

In this section, we summarize the standard kinetic theory (SKT) for the case of steady and homogeneous flows for a collection of ideal particles, that is, they are rigid, monodisperse, frictionless with diameter, *d*, and density, *ρp*. In this case, the mass balance is automatically satisfied, the momentum balance trivially asserts that the pressure, *p*, and the shear stress, *τ*, are homogeneous and the flow is totally governed by the balance of energy, which reduces to

where *Γ* is the rate of energy dissipation due to collisions and *γ* is the shear rate. The constitutive relations for *p*, *τ*, and *Γ* are given as [39]

where, *f1*, *f2*, and *f3*, are explicit functions of the volume fraction *ν* and the coefficient of restitution, *en*, (ratio of precollisional to postcollisional relative velocity between colliding particles in the normal impact direction), and are listed in Table 1.

f1=4νGFf2=8J5π1/2νGf3=12π1/2(1−en2)νGG= ν(2−ν)2(1−ν)3F= (1+en)2+14GJ= (1−en)2+π32[5+2(1+en)(3en−1)G][5+4(1+en)G][24−6(1+en)2−5(1−en2)]G2 |

### Table 1.

List of coefficients as introduced in the constitutive relations of SKT (standard kinetic theory).

Further, by substituting the constitutive relations for *τ* and *Γ* into the energy balance, the granular temperature drops out, so that the pressure becomes proportional to the square of the strain rate (Bagnold scaling [8])

SKT was rigorously derived under very restrictive assumptions. In particular, the granular system is assumed to be monodisperse and composed of spherical, frictionless, and rigid particles, interacting only through binary and uncorrelated collisions [7, 40, 41]. Several modifications to the SKT have been introduced in the literature accounting for different effects: interparticle friction [4, 7, 42–44], nonsphericity [45], or polydispersity [46]. As one example, Jenkins [47, 48] extended the kinetic theory to account for the existence of correlated motion among particles at high concentration.

#### 2.2.2. Traditional µ(I) rheology

A convincing, yet simple phenomenological model that predicts the flow behavior in moderate‐to‐dense regime is the µ(I) rheology. Once again, this rheological law is based on the assumption of homogeneous flow of idealized rigid, monodisperse particles, though the extra constraint of frictionless particles can be dropped. According to this empirical model, only three dimensionless variables are relevant for steady shear flows of granular materials: the volume fraction *ν*, the shear stress to normal stress ratio *µ* = *τ*/*p*, and the inertial number *I* [9, 23, 28]. The collaborative study GDR‐Midi showed the data collapse for various shear geometries such as inclined plane, rotating drum, and annular shear when analyzed in terms of the inertial number. µ(I) rheology in the standard form is given by

with *µ0*, *µ∞*, and *I0* being dimensionless, material parameters which are affected by the micromechanical properties of the grains [49].

To account for the polydispersity of particles, the generalized inertial number taking into account the average diameters of the particles was introduced by [50]. Traditional µ(I) rheology had been successful in describing the flow behavior of homogeneous flows (both dense and fast). But it has failed to capture the slow and nonhomogeneous flow, where a shear rate gradient is present. Researchers have made significant efforts into developing nonlocal models for granular flows [51].

#### 2.2.3. Soft µ(I) rheology

When particles are not perfectly rigid, instead they have a finite stiffness (or softness), the binary collision time is nonzero and hence presents an additional timescale, which is ignored in the standard inertial number phenomenology. A dimensionless number signifying the finite softness of the particles is the dimensionless pressure p*= pd/kn, which is needed to describe the flow behavior, as proposed recently in Refs. [30–32].

with the dimensionless pressure *p** being the characteristic pressure at which this correction becomes considerable.

The other dimensionless number needed for the full flow characterization is the volume fraction *ν*. In case of rigid particles under shear, the packing will dilate and hence *ν* depends only on the inertial number *I*. On the other hand, a packing made up of soft particles will dilate due to shear, at the same time pressure will lead the compression of the particles. Hence *ν* depends on both *I* and *p** as

where *Ic* and *pc** are material dependent dimensionless quantities [49, 52] and *νc* is the critical volume fraction, governing the fluid‐solid transitions introduced in the previous section. Its dependence on the polydispersity of the system will be discussed in Section 4.