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We assume that the suspended particles are homogeneous and made of incompressible, isotropic solids. We consider particles with viscoelastic behavior and make use of a generalized Kelvin-Voigt (KV) model to describe their constitutive response. This model consists of a hyperelastic spring and a dashpot connected in parallel. We will also consider suspensions of purely elastic particles, which are one limiting case of the KV particles. For the incompressible KV material, the Cauchy stressσ

can be written as (Joseph, 1990)

σ=−p′I+τ, τ =τ_e+τ_v, (4.1)

where p′ _{is an arbitrary hydrostatic pressure associated with the incompressibility constraint, and}

τ_eandτ_vare the elastic and viscous parts of the total “extra” stress tensorτ in the particle (which

need not be deviatoric in general, tr(τ)6= 0.) Note that the actual hydrostatic pressure pis given

byp=p′_−trτ.

The elastic stress may be described in terms of a stored-energy functionψ, which, on account of frame invariance, is a function ofC, via

τ_e= 2F∂ψ(C)

∂C F

T_{, det(F) = 1. (4.2)}

In addition, elastic isotropy (and incompressibility) implies thatψ depends on C through its first two invariants. For simplicity, in this chapter, we will consider generalized neo-Hookean behavior such that ψ(C) =g(I), where I= tr(C) and g is a generally nonlinear function ofI satisfying the requirements thatg(3) = 0, g′_{(3) =µ/2, where µ is the ground state shear modulus of the elastic}

particle. Then, the elastic extra stress tensorτ_ein (4.2) can be expressed as

τ_e= 2g′(I)B−µI, (4.3)

where the term promotional toµarises from the linearization requirements at the ground state (i.e.,

τ_e(I) =0).

Making use of the fact that (Joseph, 1990)

∇

B=DB

Dt −LB−BL

T _{=0, (4.4)}

a rate (hypo-elastic) form of equation for the elastic constitutive relation (4.3) may be obtained in terms of the upper-convected (or Truesdell) time derivative such that

∇

τ_e= ˙τ_e−Lτ_e−τ_eLT = 4g′′tr (D B)B+ 2µD, (4.5)

where ˙τ_e= (∂τ_e/∂t) +v· ∇∇∇τ_e denotes the material time derivative of the tensorτ_e.

The simplest possible choice for the elastic behavior of the particles is of course the neo-Hookean model withg(I) = µ₂(I−3). However, this model is unrealistic at large stretches for most materials, including elastomers, as it ignores the significant stiffening that such materials exhibit at large stretches. For this reason, in the applications to be considered below, we will make use of theGent

model (Gent, 1996), characterized by the choice

g(I) =−Jm₂µln 1−I_J−3 m , (4.6)

to the limiting value of I−3 at which the elastomer locks up (and the argument of the logarithm vanishes). Note that the Gent model (4.6) reduces to the neo-Hookean model in the limit asJm→ ∞.

The corresponding specialization of the constitutive relation (4.3) can then be written as

τ_e=µ " 1−I_J−3 m −1 B−I # , (4.7)

which in turn leads to the following evolution equation for the elastic extra stress tensor

∇

τ_e= 2µD+ 2

µ Jm

tr [D(τ_e+µI)] (τ_e+µI), (4.8)

where use has been made of (4.7) to express Bin terms ofτ_e. Note that these expressions reduce

to the well-known neo-Hookean expressions (Joseph, 1990) in the limit asJm→ ∞, namely,

τ_e=µ(B−I), and

∇

τ_e= 2µD. (4.9)

Going back to the general expressions for the KV material, the viscous part of the extra stress can likewise be described in terms of a dissipation potentialφ, which, on account of incompressibility and frame invariance, is a function of the last two invariants ofD, via

τ_v=∂φ(D)

∂D , tr(D) = 0. (4.10)

Although more general nonlinear forms could be considered, in this chapter, again for simplicity, we will focus our attention on linearly viscous behavior, such that

τ_v= 2η(2)D, (4.11)

whereη(2) _{describes the constant viscosity of the particle material.}

It should be noted that the set of constitutive relations for KV particles reduce to those for purely elastic particles by taking the limit as the viscosity η(2) goes to zero. In this limit, the KV model simplifies to a hyperelastic model characterized by the stored-energy functionψ(F). In other words, in this limit, the viscous part of the stress vanishes (τ_v = 0) and the elastic part

coincides with the total stress (τ_e=τ). Therefore, for the case of (incompressible) Gent particles,

the constitutive relation for the extra stress tensor (τ) and its evolution (

∇

τ) are given by (4.7) and

(4.8), respectively, withτ_e being replaced byτ. These relations have been shown to provide good

agreement with experimental data for rubber-like materials (Ogden et al., 2004). Similarly, for the case of neo-Hookean particles, the corresponding relations are given by (4.9) withτ_ebeing replaced

byτ.

The suspending fluid (or the matrix) will be assumed here to be an incompressible Newtonian fluid, with constitutive relation given by expression (4.1) with

In other words, the constitutive behavior of the matrix will be taken to be purely viscous with linear response and constant viscosityη(1)_.