Linearly viscous behavior

In the present study, we will make use of the “variational” (Ponte Casta˜neda, 1991) and the “second-order” (Ponte Casta˜neda, 2002a) nonlinear homogenization methods to estimate the effective behavior of viscoplastic porous media. Due to the fact that these nonlinear methods make use of available results for linearly viscous and linearly thermo-viscous porous media, it is useful here to recall briefly certain relations for linear composites. Thus, this section deals with the determination of the effective behavior of linearly viscous, two-phase composites, which are specialized later to linearly viscous porous media. The study of linearly thermo-viscous composites is made in the following section. It is worth noting here that the linearly viscous and linearly thermo-viscous materials are mathematically analogous to the linear elastic and thermo-elastic materials.

More specifically, we consider a linearly viscous composite consisting of a matrix phase identified with the label 1, and an inclusion phase identified with label 2. Now, let these phases be described by quadratic stress potentials of the form

U^(r)(σ) = 1

2σ(x) · M^(r)σ(x), r= 1,2, ∀x ∈ Ω^(r), (2.51)

whereM^(r)are fourth-order, positive-definite, tensors that possess both major and minor symmetries.

The corresponding stress–strain-rate relation of such materials is linear and reads D^(r)=∂U^(r)(σ)

∂σ =M^(r)σ^(r), r= 1,2. (2.52)

This relation can be inverted to give

σ^(r)=L^(r)D^(r), r= 1,2. (2.53)

withL^(r)= (M^(r))⁻¹ denoting the viscous modulus tensor of the phases, which has both major and minor symmetries. These last relations specify completely the local behavior of the phases in the two-phase medium, whereas they are in accordance with the definitions (2.1) and (2.2).

It follows from the linearity of the problem, that the correspondinginstantaneous effective stress potentialUe of the two-phase linear composite is also of a quadratic form and can be written as

Ue(σ) =1

2σ · fMσ, (2.54)

with fM being a fourth-order symmetric (both minor and major symmetries) tensor denoting the effective viscous compliance tensor of the composite. Following definition (2.6), the resulting relation between the average stress and strain-rate is given by

D=∂Ue(σ)

∂σ =Mfσ. (2.55)

This relation can be inverted to yield

σ=LeD, (2.56)

with Le = fM⁻¹ denoting the effective viscous modulus tensor of the composite. For the case of two-phaseparticulate composites, where the inclusions and their distribution function have the same ellipsoidal shape and orientation, as discussed in subsection 2.2.1, the effective viscous compliance and modulus tensors,Mf andL, are given by (Willis, 1978; Ponte Casta˜e neda and Willis, 1995)

fM=M⁽¹⁾+c⁽²⁾

·

c⁽¹⁾Q+³

M⁽²⁾−M⁽¹⁾´₋₁¸₋₁

, (2.57)

and

Le =L⁽¹⁾+c⁽²⁾

·

c⁽¹⁾P+

³

L⁽²⁾−L⁽¹⁾

´₋₁¸₋₁

. (2.58)

In these expressions,c^(r)denote the volume fractions of the phases (r= 1 for the matrix andr= 2 for the inclusions). In addition, the fourth-order microstructural tensors Q and P are related to the Eshelby (1957) and Hill (1963) polarization tensor and contain information about the shape and orientation of the inclusions and their distribution function, given by (Willis, 1978)

Q= 1 4πdet(Z)

Z

|ζ|=1

H(ζ)b |Z⁻¹ζ|⁻³dS, with Hb =L⁽¹⁾−L⁽¹⁾H L⁽¹⁾, (2.59) and

P= 1 4πdet(Z)

Z

|ζ|=1

H(ζ)|Z⁻¹ζ|⁻³dS. (2.60)

Here, the tensor Z is given by relation (2.19) serving to characterize the instantaneous shape and orientation of the inclusions and their distribution function in this context of particulate microstruc-tures. In addition, H(ij)(kl) = (L⁽¹⁾_iakbζaζb)⁻¹ζjζl

¯¯

¯(ij)(kl), where the brackets denote symmetrization with respect to the corresponding indices, whileζ is a unit vector. Then, it follows from (2.59) and (2.60) that theQis related to thePtensor through

Q=L⁽¹⁾−L⁽¹⁾P L⁽¹⁾, since Z

|ζ|=1

|Z⁻¹ζ|⁻³dS= 4πdet(Z). (2.61) At this point, it is worth mentioning that both expressions (2.57) and (2.58) for Mf and L,e respectively, are equivalent and either of them can be used for the estimation of the instantaneous effective behavior of the linear two-phase medium (Sab, 1992). Moreover, it should be emphasized that the above Willis estimates for Mf (or L) lead to uniform fields in the inclusion phase (Willis,e 1978), which is consistent with the work of Eshelby (1957) in the dilute case. In fact, the Willis estimates are exact for dilute composites. On the other hand, for non-dilute media, the fields within the inclusions are, in general, non-uniform, but this “non-uniformity” is negligible (Bornert et al., 1996) provided that the inclusions are not in close proximity to each other, i.e., their volume fraction is not so large compared to the one of the matrix phase. This is an important observation that we should bear in mind when the application of these homogenization techniques is done for composites consisting of high concentrations of particles or voids. Nonetheless, the focus on this work is on porous media with low to moderate concentrations of voids, and hence the Willis procedure is expected to be sufficiently accurate in this case.

The above analysis provides a description of the instantaneous effective behavior of linearly viscous two-phase materials in terms of macroscopic measures, namely the effective stress potentialUe, and the macroscopic stress σ and strain-rateD. However, the homogenization theory is also capable of generating estimates for other stress and strain-rate quantities such as the first and second moments of the phase fields. In this work, the interest is mainly on the first moments of the phase fields, or equivalently, the average stressσ^(r)=hσi^(r), the average strain-rateD^(r)=hDi^(r)and the average spinΩ^(r) =hΩi^(r) in each phase. It should be noted that the phase average strain-rate D^(r) and spinΩ^(r)tensors are the symmetric and skew-symmetric parts of the phase average velocity gradient.

However, in addition to the first moments, expressions can also be derived for the second-moments of the stress and strain-rate fields, which will be presented in the sequel.

In this regard, in the case of linear, two-phase materials, the estimation of the average stress and strain-rate fields is given in terms of stress and strain-rate concentration tensors by (Hill, 1963; Laws, 1973; Willis, 1981)

σ^(r)=B^(r)σ, D^(r)=A^(r)D, (2.62)

wherer= 1,2. In this expression,B^(r)andA^(r)are fourth-order tensors that exhibit minor symmetry (but not necessarily major symmetry). It is important to note that the phase average stresses and strain-rates are related to the macroscopic stress and strain-rate tensor by

σ= X2

r=1

c^(r)σ^(r), D= X2

r=1

c^(r)D^(r). (2.63)

These last relations suggest that the stress and strain-rate concentration tensorsB^(r)andA^(r)should be consistent with the identities

X2

In addition, the phase average stressσ^(r) and strain-rates D^(r) can also be obtained by averaging the local constitutive relations (2.52) and (2.53), to obtain the following phase constitutive relations:

D^(r)=M^(r)σ^(r), σ^(r)=L^(r)D^(r). (2.65) By combining relations (2.62) and (2.65), the following relations for the macroscopic strain-rate and stress can be deduced:

Now, looking at this last relation together with expressions (2.55) and (2.56), it is straightforward that the effective viscous compliance and modulus tensorsMfandLe are directly related to the stress and strain-rate concentration tensorsB^(r) andA^(r), respectively, through

Mf= By making use of the identity (2.64), the concentration tensorsB^(r) and A^(r) can be expressed in terms offMandL, respectively, by relationse

c⁽²⁾B⁽²⁾=h It is further noted that by combining relations (2.62) and (2.65), the stress and strain-rate concen-tration tensors are related by

A^(r)=M^(r)B^(r)L,e or B^(r)=L^(r)A^(r)M,f r= 1,2. (2.69) In order to complete the set of relations for the estimation of the phase average fields, correspond-ing expressions have been introduced by Ponte Casta˜neda (1997) and Kailasam and Ponte Casta˜neda (1998) for the evaluation of the average spin tensorsΩ^(r) in each phase, such that

Ω^(r)=Ω−C^(r)D, for r= 1,2. (2.70)

In this expression,C_[ij](kl)^(r) are fourth-order tensors that are skew-symmetric in the first two indices and symmetric in the last two^†, and Ωis the macroscopic spin tensor which is applied externally in the problem.

Following expressions (2.63), the phase average spin tensors Ω^(r) are related to the macroscopic spin tensorΩthrough

which implies for the spin concentration tensorsC^(r)that X2

r=1

c^(r)C^(r)=0. (2.72)

Thus using this last relation together with results provided by Ponte Casta˜neda (1997), it is pos-sible to write the spin concentration tensors in terms of the strain-rate (or equivalently the stress) concentration tensorsA^(r), via

C⁽²⁾=c⁽¹⁾Π

³

L⁽²⁾−L⁽¹⁾

´

A⁽²⁾, c⁽¹⁾C⁽¹⁾ =−c⁽²⁾C⁽²⁾, (2.73) whereΠ is a microstructural tensor related to theQtensor (see (2.59)), given by

Π= 1 4πdet(Z)

Z

|ζ^|=1

H(ζ)ˇ |Z⁻¹ ·ζ|⁻³dS, Hˇijkl= (L⁽¹⁾_iakbζaζb)⁻¹ζjζl

¯¯

¯[ij](kl). (2.74) The square brackets denote the skew-symmetric part of the first two indices, whereas the simple brackets define the symmetric part of the last two indices. The second-order tensor Z serves to characterize theinstantaneous shape and orientation of the inclusions and is given by relation (2.19).

In addition to the phase average fields, it is also expedient to discuss here the determination of the second moments of the stress and the strain-rate fields. In order to make this calculation straightforward, we recall relations (2.4) and (2.54), to obtain the following expression:

1

2σ·Mfσ= X2

r=1

c^(r)h1

2σ(x)· M^(r)σ(x)i^(r), ∀x∈ Ω^(r), (2.75) where use of definition (2.51) has been made for the phase stress potential U^(r). Following Ponte Casta˜neda and Suquet (1998) (but see also Idiart and Ponte Casta˜neda (2007)), the second moments of the stress fields in the linear material can be evaluated by considering the partial derivative with respect to the compliance tensorsM^(r), so that

hσ(x)⊗σ(x)i^(r)= 1

c^(r)σ ∂Mf

∂M^(r)σ, ∀x∈ Ω^(r). (2.76)

It is worth noting that if the fields in any of the phases are uniform, i.e., σ(x) =σ^(r) for allx in Ω^(r), thenhσ⊗σi^(r)=σ^(r)⊗σ^(r).

As already pointed out previously, the Willis (1978) and Ponte Casta˜neda and Willis (1995) estimates for particulate microstructures result in uniform fields in the inclusion phase. Based on this observation, the fluctuations in the inclusion phases are zero, i.e.,

hσ⊗σi⁽²⁾−σ⁽²⁾⊗σ⁽²⁾=0 or σ(x) =σ⁽²⁾ ∀x ∈ Ω⁽²⁾. (2.77) Similar expressions can also be derived for the strain-rate fluctuations by employing duality (in terms of the Legendre-Fenchel transform) and by noting that

1

2D·LeD= X2

r=1

c^(r)h1

2D(x)· L^(r)D(x)i^(r), ∀x∈ Ω^(r). (2.78)

Following Ponte Casta˜neda and Suquet (1998), the second moments of the strain-rate fields in the linear material can be evaluated by considering the partial derivative with respect to the modulus tensorsL^(r), so that

hD(x)⊗D(x)i^(r)= 1

c^(r)D ∂Le

∂L^(r)D, ∀x∈ Ω^(r). (2.79)

As previously mentioned, the fields in the inclusion phase are uniform, as predicted by Willis (1978) and Ponte Casta˜neda and Willis (1995) estimates for particulate microstructures, and thus the cor-responding strain-rate fluctuations in the inclusions are zero, i.e.,

hD⊗Di⁽²⁾−D⁽²⁾⊗D⁽²⁾=0 or D(x) =D⁽²⁾ ∀x ∈ Ω⁽²⁾. (2.80) For the same reason, the average spinΩ⁽²⁾in the inclusion phase is also uniform, which implies that Ω(x) =Ω⁽²⁾ ∀x ∈ Ω⁽²⁾. (2.81) In the following subsection, we specialize the previous results for the particular case of linearly viscous porous media.

2.3.1 Linearly viscous porous media

More specifically, we consider a linear composite consisting of a matrix phase identified with the label 1, and a vacuous phase identified with label 2. The behavior of the matrix phase is described by a quadratic stress potentialU⁽¹⁾≡U of the form

U(σ) = 1

2σ · Mσ, ∀x ∈ Ω⁽¹⁾, (2.82)

where M and L = M⁻¹ are the viscous compliance and modulus tensors of the matrix phase. In contrast, the porous phase is described by a modulus tensorL⁽²⁾ with zero eigenvalues, and a stress potential U⁽²⁾ = 0. In this connection, the instantaneous effective stress potential of the porous material is given by relation (2.54), which is recalled here for completeness to be

Ue(σ) =1

2σ·Mfσ, (2.83)

where Mf is the effective viscous compliance tensor of the porous medium and has both minor and major symmetries. In addition, the corresponding constitutive macroscopic law for the linearly viscous porous medium is given by (2.55) and (2.56), such that

D=Mfσ, σ=LeD, (2.84)

withLe =Mf⁻¹ denoting the effective viscous modulus tensor of the linear porous medium. For the determination of the effective tensorsMfor Le of the porous material, it is necessary to setL⁽²⁾ =0 in relation (2.57) and (2.58), respectively. The resulting expressions read

Mf=M+ f

1−f Q⁻¹, (2.85)

and

Le=L+f [(1−f)P−M]⁻¹, (2.86)

where use of the notation f =c⁽²⁾ is made for the volume fraction of the voids or equivalently the porosity. This implies that the volume fraction of the matrix phase is simply c⁽¹⁾ = 1−f. The microstructural tensors Q and P, serving to characterize the instantaneous shape and orientation of the voids and their distribution function, are given by relations (2.59) and (2.60), respectively, whereL⁽¹⁾ is the viscous modulus of the matrix phase and for consistency with the notation of this subsection, it should be replaced byLin those expressions.

The corresponding stress and strain-rate concentration tensorsB^(r)andA^(r), respectively, given by relation (2.68) simplify dramatically to

(1−f)B⁽¹⁾=I, B⁽²⁾=0, (1−f)A⁽¹⁾ =ML,e fA⁽²⁾=I−ML.e (2.87) It follows from this result and relation (2.62) that

(1−f)σ⁽¹⁾=σ, σ⁽²⁾ =0. (2.88)

This last result is consistent with the fact that the stress in the voids is zero for all x in Ω⁽²⁾. In turn, it follows from relations (2.62) and (2.87) that the phase average strain-rate in the matrix and the vacuous phase are given simply by

D⁽¹⁾ =A⁽¹⁾D= 1

1−fMLD,e D⁽²⁾ =A⁽²⁾D= 1

f h

I−MLe i

D. (2.89)

These expressions relate the phase average strain-rates with the macroscopic strain-rate in the linear porous medium. On the other hand, those expressions can be manipulated so that we may write the phase average strain-rates in terms of the macroscopic stress tensorσ. As we will see in the sections to follow, this later representation is more convenient. Thus, by making use of definitions (2.84), (2.85) and (2.87), the phase average strain-rates take the form

D⁽¹⁾= 1 1−fMσ, D⁽²⁾= 1

f (fM−M)σ= 1

1−fQ⁻¹σ. (2.90)

Finally, the phase average spin in the inclusion phase is obtained by relation (2.70), which is recalled here for completeness to be

Ω⁽²⁾ =Ω−C⁽²⁾D, (2.91)

withC⁽²⁾ given by (2.73) after settingL⁽²⁾ =0resulting in C⁽²⁾= (f−1)Π L A⁽²⁾=1−f

f Π ³ Le−L´

. (2.92)

The fourth-order microstructural tensorΠis defined in expression (2.74) and is skew-symmetric with respect to the first two indices, and symmetric with respect to the last two ones. Note that by substituting (2.92) in (2.91), we obtain

Ω⁽²⁾=Ω+ (1−f)Π LD⁽²⁾=Ω+Π L Q⁻¹σ, (2.93) where use of relations (2.84) and (2.85) has also been made.

2.3.2 Brief summary

In this section, we presented certain constitutive relations for two-phase, linearly viscous particulate media. We have defined the instantaneous effective behavior of the material and it has been shown that both the macroscopic properties as well as the phase average fields can be completely defined in terms of the effective viscous compliance tensorfM(or equivalently the effective viscous modulus ten-sorL) of the composite material. These general results for two-phase materials have been specializede to porous media, which is the main subject of this work. It is worth mentioning at this point that in the sequel use will be made of nonlinear homogenization techniques to predict the instantaneous effective behavior of nonlinear porous media. These nonlinear methods make use of results for linear composites and hence the above results will be very helpful in the sections to follow.

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