The Hill–Mandel condition for viscous suspensions reads
$$\begin{aligned} \langle W_\mathsf{{V}} \rangle = \langle \varvec{\sigma }_\mathsf{{V}}\cdot \varvec{D}\rangle = \langle \varvec{\sigma }_\mathsf{{V}}\rangle \cdot \langle \varvec{D}\rangle , \end{aligned}$$
(11)
or equivalently
\({\langle \hat{\varvec{\sigma }}_\mathsf{{V}}\cdot \hat{\varvec{D}}\rangle =0}\). In order to derive Eq. (
11), the local viscous dissipation
\(W_\mathsf{{V}}\) given in Eq. (
4) and its volume average
\(\langle W_\mathsf{{V}} \rangle \) are considered
$$\begin{aligned} W_\mathsf{{V}}&= \varvec{\sigma }_\mathsf{{V}}\cdot \varvec{D},&\langle W_\mathsf{{V}} \rangle&= \langle \varvec{\sigma }_\mathsf{{V}}\cdot \varvec{D}\rangle . \end{aligned}$$
(12)
The decomposition regarding the local fields
\(\varvec{\sigma }_\mathsf{{V}}(\varvec{x}),\varvec{D}(\varvec{x})\) into the volume averages
\(\langle \varvec{\sigma }_\mathsf{{V}}\rangle ,\langle \varvec{D}\rangle \) and the local fluctuations
\(\hat{\varvec{\sigma }}_\mathsf{{V}}(\varvec{x}),\hat{\varvec{D}}(\varvec{x})\)$$\begin{aligned} \varvec{\sigma }_\mathsf{{V}}&= \langle \varvec{\sigma }_\mathsf{{V}}\rangle + \hat{\varvec{\sigma }}_\mathsf{{V}},&\varvec{D}&= \langle \varvec{D}\rangle + \hat{\varvec{D}}, \end{aligned}$$
(13)
leads to the following volume-averaged viscous dissipation by using
\({\langle \hat{\varvec{\sigma }}_\mathsf{{V}}\rangle =\mathbf {0}}\) and
\({\langle \hat{\varvec{D}}\rangle =\mathbf {0}}\):
$$\begin{aligned} \langle W_\mathsf{{V}} \rangle = \langle \varvec{\sigma }_\mathsf{{V}}\rangle \cdot \langle \varvec{D}\rangle + \langle \hat{\varvec{\sigma }}_\mathsf{{V}}\cdot \hat{\varvec{D}}\rangle . \end{aligned}$$
(14)
The volume average of the fluctuation terms
\(\langle \hat{\varvec{\sigma }}_\mathsf{{V}}\cdot \hat{\varvec{D}}\rangle \) can be reformulated as follows by applying the product rule, the symmetry of
\(\varvec{\sigma }_\mathsf{{V}}\) and
\({\varvec{D}=\mathrm {sym\,grad}(\varvec{v})}\) with the velocity field
\(\varvec{v}\)$$\begin{aligned} \langle \hat{\varvec{\sigma }}_\mathsf{{V}}\cdot \hat{\varvec{D}}\rangle&= \langle \hat{\varvec{\sigma }}_\mathsf{{V}}\cdot \mathrm {grad}(\hat{\varvec{v}})\rangle = \langle \mathrm {div}({\hat{\varvec{\sigma }}_\mathsf{{V}}}^\mathsf{T}\hat{\varvec{v}})\rangle - \langle \hat{\varvec{v}}\cdot \mathrm {div}(\hat{\varvec{\sigma }}_\mathsf{{V}})\rangle . \end{aligned}$$
(15)
To further simplify Eq. (
15), the balance of linear momentum for viscous suspensions without body forces is considered in the common form without dimensions [
103]
$$\begin{aligned} \displaystyle \frac{\partial \varvec{v}^*}{\partial t^*} + \mathrm {grad^*}(\varvec{v}^*)\varvec{v}^* = -\mathrm {grad^*}(p^*) + \frac{1}{\mathrm {Re}} \mathrm {div^*}(\varvec{\sigma }_\mathsf{{V}}^*). \end{aligned}$$
(16)
In Eq. (
16), fields and operations without dimensions are denoted by
\((\cdot )^*\), and
\(\mathrm {Re}\) refers to the Reynolds number. Similar to previous studies [
4,
11,
92,
95], Stokes flow (
\({\mathrm {Re}\ll 1}\)) is considered in the context of this work for which the pressure and the viscous forces are in equilibrium. As a consequence, local accelerations due to changes in time and convective accelerations are neglected. Note that for the latter, also small spatial gradients of the flow are assumed leading to the simplified linear momentum balance
\({\mathrm {grad}(p)=\mathrm {div}(\varvec{\sigma }_\mathsf{{V}})}\). In this context, the parallels of linear elastic solids and linear viscous suspensions are addressed in the literature [
38,
95] as long as the kinematic constraints are the same for both suspensions and composites. Next, the splittings
\({p=\langle p\rangle +\hat{p}}\) and
\({\varvec{\sigma }_\mathsf{{V}}=\langle \varvec{\sigma }_\mathsf{{V}}\rangle +\hat{\varvec{\sigma }}_\mathsf{{V}}}\) are used in combination with the linearity of
\(\mathrm {grad}(\cdot )\) and
\(\mathrm {div}(\cdot )\) leading to the linear momentum balance in terms of averages and fluctuations
$$\begin{aligned} \mathrm {grad}(\langle p\rangle ) + \mathrm {grad}(\hat{p})&= \mathrm {div}(\langle \varvec{\sigma }_\mathsf{{V}}\rangle ) + \mathrm {div}(\hat{\varvec{\sigma }}_\mathsf{{V}}). \end{aligned}$$
(17)
Since
\(\langle p\rangle \) and
\(\langle \varvec{\sigma }_\mathsf{{V}}\rangle \) are constant, Eq. (
17) reduces in a trivial way to the linear momentum balance for the fluctuations
$$\begin{aligned} \mathrm {grad}(\hat{p}) = \mathrm {div}(\hat{\varvec{\sigma }}_\mathsf{{V}}). \end{aligned}$$
(18)
By inserting Eq. (
18) into Eq. (
15), the volume average of the fluctuation terms read after applying the product rule again
$$\begin{aligned} \langle \hat{\varvec{\sigma }}_\mathsf{{V}}\cdot \hat{\varvec{D}}\rangle&= \langle \mathrm {div}({\hat{\varvec{\sigma }}_\mathsf{{V}}}^\mathsf{T}\hat{\varvec{v}})\rangle - \langle \hat{\varvec{v}}\cdot \mathrm {grad}(\hat{p})\rangle = \langle \mathrm {div}({\hat{\varvec{\sigma }}_\mathsf{{V}}}^\mathsf{T}\hat{\varvec{v}})\rangle - \langle \mathrm {div}(\hat{\varvec{v}}\hat{p})\rangle + \langle \hat{p}\,\mathrm {div}(\hat{\varvec{v}})\rangle . \end{aligned}$$
(19)
For the special case of incompressibility in terms of velocity fluctuations
\({\mathrm {div}(\hat{\varvec{v}})=0}\), Eq. (
19) can be expressed as follows
$$\begin{aligned} \langle \hat{\varvec{\sigma }}_\mathsf{{V}}\cdot \hat{\varvec{D}}\rangle&= \langle \mathrm {div}({\hat{\varvec{\sigma }}_\mathsf{{V}}}^\mathsf{T}\hat{\varvec{v}})\rangle - \langle \mathrm {div}(\hat{\varvec{v}}\hat{p})\rangle = \frac{1}{V} \int _V\mathrm {div}({\hat{\varvec{\sigma }}_\mathsf{{V}}}^\mathsf{T}\hat{\varvec{v}})\mathrm {d}V - \frac{1}{V} \int _V\mathrm {div}(\hat{\varvec{v}}\hat{p})\mathrm {d}V. \end{aligned}$$
(20)
In the next step, the Gaussian integral theorem with singular surface
\(\Gamma \) [
96] is applied to transform the volume integrals into surface integrals
$$\begin{aligned} \langle \hat{\varvec{\sigma }}_\mathsf{{V}}\cdot \hat{\varvec{D}}\rangle&= \frac{1}{V} \int _{\partial V \backslash \Gamma }({\hat{\varvec{\sigma }}_\mathsf{{V}}}^\mathsf{T}\hat{\varvec{v}})\cdot \varvec{n}\mathrm {d}A - \frac{1}{V} \int _{\Gamma }\llbracket {\hat{\varvec{\sigma }}_\mathsf{{V}}}^\mathsf{T}\hat{\varvec{v}}\rrbracket \cdot \varvec{n}\mathrm {d}A - \frac{1}{V} \int _{\partial V \backslash \Gamma }(\hat{\varvec{v}}\hat{p})\cdot \varvec{n}\mathrm {d}A + \frac{1}{V} \int _{\Gamma }\llbracket \hat{\varvec{v}}\hat{p}\rrbracket \cdot \varvec{n}\mathrm {d}A. \end{aligned}$$
(21)
To achieve a convenient notation being compatible with the solid mechanics considerations, the stress vectors
\({\varvec{t}_\mathsf{{V}}=\varvec{\sigma }_\mathsf{{V}}\varvec{n}}\) and
\({\varvec{t}_\mathsf{{p}}=-p\varvec{n}}\) representing viscous and pressure forces are introduced leading to
$$\begin{aligned} \langle \hat{\varvec{\sigma }}_\mathsf{{V}}\cdot \hat{\varvec{D}}\rangle = \frac{1}{V} \int _{\partial V \backslash \Gamma }(\hat{\varvec{t}}_\mathsf{{p}}+\hat{\varvec{t}}_\mathsf{{V}})\cdot \hat{\varvec{v}}\mathrm {d}A - \frac{1}{V} \int _{\Gamma }\llbracket (\hat{\varvec{t}}_\mathsf{{p}}+\hat{\varvec{t}}_\mathsf{{V}})\cdot \hat{\varvec{v}}\rrbracket \mathrm {d}A = \frac{1}{V} \int _{\partial V \backslash \Gamma }\hat{\varvec{t}}\cdot \hat{\varvec{v}}\mathrm {d}A - \frac{1}{V} \int _{\Gamma }\llbracket \hat{\varvec{t}}\cdot \hat{\varvec{v}}\rrbracket \mathrm {d}A. \end{aligned}$$
(22)
In Eq. (
22), the addition of the force vectors
\({\varvec{t}=\varvec{\sigma }\varvec{n}=(-p\varvec{I}+\varvec{\sigma }_\mathsf{{V}})\varvec{n}=\varvec{t}_\mathsf{{p}} + \varvec{t}_\mathsf{{V}}}\) is used to obtain a structure like the one commonly used in solid mechanics. The fluctuation term
\(\langle \hat{\varvec{\sigma }}_\mathsf{{V}}\cdot \hat{\varvec{D}}\rangle \) in Eq. (
22) depends on the stress vector fluctuations
\(\hat{\varvec{t}}\) and the velocity fluctuations
\(\hat{\varvec{v}}\) on
\(\partial V \backslash \Gamma \), if cracks and voids are excluded. Therefore,
\(\varvec{t}\) and
\(\varvec{v}\) are continuous over
\(\Gamma \) leading to a vanishing integral over
\(\Gamma \), since
\({\llbracket \varvec{t}\rrbracket =\mathbf {0}}\) and
\({\llbracket \varvec{v}\rrbracket =\mathbf {0}}\) hold. The remaining integral over
\(\partial V \backslash \Gamma \) vanishes and, as a consequence, Eq. (
11) holds if any of the following four cases hold (implied by the list in Sect.
3.3.1):
-
Homogeneous velocity on the boundary \(\partial V \backslash \Gamma \): \({\hat{\varvec{v}}=\mathbf {0}}\)
-
Homogeneous stress on the boundary \(\partial V \backslash \Gamma \): \({\hat{\varvec{t}}=\mathbf {0}}\)
-
Periodic boundary conditions
-
Ergodic media in the limit \({V\rightarrow \infty }\): \(\langle W_\mathsf{{V}}\rangle \rightarrow \langle \varvec{\sigma }_\mathsf{{V}}\rangle \cdot \langle \varvec{D}\rangle \)
Similar to the previous Sect.
3.3.1 and motivated by the Hill–Mandel condition and the accessibility of
\(\langle \varvec{\sigma }_\mathsf{{V}}\rangle \) and
\(\langle \varvec{D}\rangle \) over
\(\partial V \backslash \Gamma \), the effective fields are equal to the volume averages as used by Bertóti [
12]
$$\begin{aligned} \langle \varvec{\sigma }_\mathsf{{V}}\rangle&= \bar{\varvec{\sigma }}_\mathsf{{V}},&\langle \varvec{D}\rangle&= \bar{\varvec{D}},&\langle W_\mathsf{{V}}\rangle&= \bar{W}_\mathsf{{V}}. \end{aligned}$$
(23)
The derivation does not imply that the viscous stress and the strain rate are constitutively coupled. Equation (
11) corresponds to the Hill–Mandel condition given in Traxl et al. [
17] and refers to the macroscopic viscous dissipation addressed by Bertóti et al. [
4].