Planificación de tareas y coaliciones: problemas ST-MR-TE

We consider arepresentative volume element (RVE) of the suspension, which occupies a volume Ω with boundary∂Ω. The fluid and particle phases are in turn assumed to occupy volumes Ω(1) _and

Ω(2)_{, respectively, such that Ω = Ω}(1)_{+ Ω}(2)_{. It is assumed that the RVE satisfies the separation} of length scales hypothesis implying that the typical size of the neutrally buoyant particles is much smaller than the size of the RVE, as well as the Stoke’s condition in the fluid phase, such that

Re= ρ

(1)_{γ d˙} 2 p

η(1) →0, (4.16)

whereρ(1)_{is the density of the fluid, ˙γis a measure of the macroscopic strain rate andd}

pis a measure

of the particle diameter. Noting that the microstructure of the RVE isstatisticallyuniform, a uniform

macroscopic stress field will be generated in the RVE when anaffine velocity boundary condition is applied on the boundary of the RVE (∂Ω). Thus, the suspension is subjected to the boundary condition

v(x) = ¯Lx, on ∂Ω, (4.17) where ¯L(tr ¯L= 0) is the macroscopic, or average velocity gradient, defined by1

¯ L= 1 Vol(Ω) Z Ω LdV. (4.18)

1_{Themean value theorem for the strain rate(e.g., Ponte Casta˜neda and Suquet, 1998) states that the (volume)}

average of the local strain-rate tensorLover the RVE under the affine velocity boundary condition (4.17) is precisely

¯

Similarly, the average or macroscopic Cauchy stress is defined as ¯ σ= 1 Vol(Ω) Z Ω σdV, (4.19)

and the instantaneous macroscopic constitutive response is determined by the relation between ¯σ

and ¯L.

For future reference, we also define the phase averages of the strain-rate field over phase r

(r= 1,2) via ¯ D(r)= 1 Vol(Ω(r)₎ Z Ω(r) DdV, (4.20) such that ¯ D=c(1)D¯(1)+c(2)D¯(2), (4.21) with thec(1) _andc(2) _{denoting the volume fractions of fluid and particle phases, respectively. Sim-}

ilarly, defining ¯τ(1) and ¯τ(2) as the averages of the extra stress in the fluid and particle phases,

respectively, the macroscopic stress, as defined by (4.19), can be rewritten (on account of the in- compressibility of the phases) as

¯

σ=−p¯′I+c(1)τ¯(1)+c(2)τ¯(2), (4.22)

where ¯p′ _{is an indeterminate hydrostatic pressure associated with the overall incompressibility of}

the suspension.

Now, taking advantage of the special form of the constitutive relations for the fluid matrix and solid particle phases, as described in Section 4.2.1,modifieddissipation potentials are introduced

W(r)(D) =η(r)D·D+τ_e·D, tr(D) =0, (4.23)

such that the local constitutive relation of the phases can be written as

σσσ=−p′I+τ, where τ = ∂W

(r)

∂D = 2η

(r)_D+τ

e, (4.24)

where p′ _{is an indeterminate hydrostatic pressure. It should also be emphasized that the elastic}

strainsτ_eare considered to be fixed in taking the derivative with respect toD. Thus, it can be seen

that the addition of the linear term in the strain-rate tensorDto the dissipation functionφ, defined by equation (4.10), allows the inclusion of the elastic stressτ_e, assuming that it is known at the given

instant. More specifically, labeling the quantities associated with the matrix and particle phases by the superscripts (1) and (2), respectively, the local constitutive relation (4.24) can be used to recover the constitutive relations for the elastic particles and fluid matrix phases, as given by (4.1) to (4.12), provided that we let η(1) _{and η}(2) _{be the viscosities of the fluid and elastic particles, respectively,}

and that we letτ_e= 0 in the fluid phase andτ_e=τ(2)_e , as characterized by the evolution equation

(4.8), in the particle phase. In addition, in this last expression, we use µ and Jm to describe the

been dropped from µand Jm for convenience (since only the particle phase has elastic properties,

thus eliminating the risk of confusion).

We define next, for compactness, the local modified dissipation potential

W(x,D) =χ(1)(x)W(1)(D) +χ(2)(x)W(2)(D), (4.25) where the χ(r)_{(x) (r= 1,2) are the characteristic functions of the two phases, such that they are}

equal to one if the position vectorxis in phaser (i.e.,x∈Ω(r)_{) and zero otherwise. Then, we can}

state the principle of minimum dissipation via min

D∈K

Z

Ω

W(x,D) dV, (4.26) whereK denotes the set of kinematically admissible strain rates:

K={D|there isvsuch thatD= (∇v+ (∇v)T)/2,divv= 0 in Ω,andv= ¯Lx on∂Ω}. (4.27) It is noted (see Ekeland and T´emam, 1999, for the purely viscous problem) that the Euler-Lagrange equations of this variational principle are precisely the Stoke’s equations for the fluid phase

2η(1)∇2v− ∇p=0, ∇ ·v= 0, (4.28) together with the equilibrium equations for the solid particles, which in Eulerian form become

2η(2)_∇2_{v− ∇p}′_{+∇ ·τ}(2)

e =0, ∇ ·v= 0, (4.29)

where once again the elastic extra stressτ(2)_e in the particles is assumed to be known at the present

instant. Note that the variational principle also ensures continuity of the velocity v and traction components of the total stressσ across the particle-fluid boundaries, as well as satisfaction of the

affine boundary condition (4.17).

Finally, it is noted that the dissipation functional in equation (4.26), evaluated at the minimum, defines a function of the macroscopic strain-rate ¯D, as given by the symmetric part of the average velocity gradient ¯L. When normalized by the volume of the RVE Ω, it can be shown (see, for example, Ponte Casta˜neda and Suquet, 1998) that it provides a modified dissipation potential for the macroscopic constitutive relation, in the sense that

¯ σ σσ=−p¯′I+ ¯τ, where τ¯ =∂ f W ∂D¯ , (4.30)

and where ¯p′_{is the Lagrange multiplier associated with the macroscopic incompressibility constraint}

and f W( ¯D) = min D∈K 1 Vol(Ω) Z Ω W(x,D) dV. (4.31) The homogenization problem defined by equations (4.30) and (4.31) for the instantaneous re-

sponse of the viscoelastic composite characterized by (4.23)–(4.25) is mathematically analogous to the corresponding problem for an incompressible thermoelasticcomposite with elastic moduli η(r)

and thermal stresses τ_e (provided that the strain rate and velocity fields are identified with the

strain and displacement fields, respectively). For the specific problem of interest here, the viscosi- ties (moduli) η(r) _{areuniform-per-phase, and while the elastic stress (thermal stress) in the matrix}

phase is zero, the corresponding elastic stress (thermal stress) in the particle phase is not only non- vanishing, but in fact also non-uniform. More general situations, including the case of nonuniform thermal stresses in the matrix phase has been considered recently by Lahellec et al. (2011).

To estimate the effective dissipation function fW( ¯D), we make use of the Hashin-Shtrikman- Willis (HSW) variational method, which was originally developed for isotropic elastic composites by Hashin and Shtrikman (1963), and extended later for generally anisotropic elastic composites by Willis (1977, 1981). For the particulate material systems of interest in this chapter consisting of random distributions of ellipsoidal inclusions in a given matrix, more specific estimates have been given by Ponte Casta˜neda and Willis (1995) still making use of the HSW variational method. Applied to the above-described viscous systems, the key feature of this method is the use of a “polarization field” relative to a homogeneous “comparison fluid” (with viscosityη0_{). In this way, it}

is possible to make use of simple, constant-per-phase trial fields for the polarization to obtain bounds and estimates for the effective response of the composite system. The application of the method of Ponte Casta˜neda & Willis (PCW) to the class of suspensions of thermoelastic particles of interest in this chapter was given by Ponte Casta˜neda (2005). For completeness, the adaptation of these results for viscoelastic particles is given in Appendix C.1. In this section, we will only provide the final results for the macroscopic response in terms of the average local fields in the particle phase.

Thus, the resulting variational estimate for the effective dissipation function fW( ¯D) can be ex- pressed as f W( ¯D) =η(1)D¯ ·D¯ +cη(2)−η(1)D¯(2)·D¯ +c 2τ¯ (2) e · ¯ D(2)+ ¯D, (4.32) where ¯ D(2)=nI₋2 (1−c)η(1)−η(2)Po−1nD¯ −(1−c)Pτ¯(2) e o (4.33) is the corresponding estimate for the average strain-rate over the particles. In these expressions, it is recalled from Appendix C.1 thatc =c(2) _{is the volume fraction of the particle phase, ¯τ}(2)

e is

the average elastic stress in the particles, andPis a microstructural (Eshelby-type) tensor given by (C.11)1.

For later reference, we note that the procedure also provides an estimate for the average vorticity tensor in the particle phase (see Appendix C.1), which is given by

¯

W(2) = ¯W+ (1−c)Rh2η(1)−η(2)D¯(2)−τ¯(2)

e

i

, (4.34) whereRis the microstructural tensor defined by (C.11)2.

Finally, the instantaneous macroscopic constitutive relation for the suspension of viscoelastic particles can be obtained from the estimate (4.32) forfW( ¯D) by means of equation (4.30). However, given ¯D(2) _{and ¯τ}(2)

in the matrix phase as ¯ τ(1)= 2η(1)D¯(1)= 2η(1)(1−c)−1 ¯ D−cD¯(2). (4.35) Then, substituting this expression into expression (4.22) for the macroscopic stress, we arrive at

¯ σ=−p¯′I+ 2η(1)D¯ +c ¯ τ(2)−2η(1)D¯(2) , (4.36) which, using (4.24)2to express the average extra stress over the inclusion phase in terms of the extra