In the next step, we establish the weak representation of the balance equations (
3), (
4) and (
9) using the notation for the generalized variational format presented in [
16]. Hence, we seek solutions in the trial spaces
\(\mathbb {U}_M\) and
\(\mathbb {P}_\Box =\mathbb {P}_M\times \mathbb {P}_{F}\) of admissible displacements and fluid pressure fields
3 that are sufficiently regular in
\(\varOmega _M\) and
\(\partial F\). Furthermore, we introduce the corresponding trial spaces of self-equilibrated fluxes
\(\mathbb {T}_M\),
\(\mathbb {Q}_M\) and
\(\mathbb {Q}_{F}\) that are sufficiently regular on
\(\partial \varOmega _M^+\) and
\(\partial \partial F^+\). We write the equations for finding
\(\mathbf{u},\,p,\,\mathbf{t},\,q_M,\,q_{F}\,\in \,\mathbb {U}_M\times \mathbb {P}_\Box \times \mathbb {T}_M\times \mathbb {Q}_M\times \mathbb {Q}_{F}\) as
$$\begin{aligned}&\mathfrak {a}^u_M(\mathbf{u},\delta \mathbf{u}) + \mathfrak {b}^u_M(p,\delta \mathbf{u}) -\, \mathfrak {c}^u_M(\mathbf{t},\delta \mathbf{u}) - \mathfrak {d}^u_M(p,\delta \mathbf{u}) = 0,\nonumber \\\end{aligned}$$
(23)
$$\begin{aligned}&-\, \mathfrak {a}^{p}_M(p,\delta p) + \mathfrak {b}^{p}_M(\dot{\mathbf{u}},\delta p) + \mathfrak {m}^{p}_M(\dot{p},\delta p)\nonumber \\&+ \,\mathfrak {c}^{p}_M(q_M,\delta p) + \mathfrak {d}^{p}_M(p,\delta p) =0, \end{aligned}$$
(24)
$$\begin{aligned}&- \,\mathfrak {a}^p_F(p,\delta p) + \mathfrak {b}^p_F(\dot{\mathbf{u}},\delta p) + \mathfrak {m}^p_F(\dot{p},\delta p)\nonumber \\&+ \, \mathfrak {c}^p_F(q_F,\delta p)+ \mathfrak {d}^p_F(p,\delta p) + \sum \limits _{k=1}^n\mathfrak {e}^p_{F,k}(\hat{q}_k,\delta p) = 0, \end{aligned}$$
(25)
for
\(k=1,2,\dots ,n\), under
$$\begin{aligned} - \mathfrak {c}^u_M(\delta \mathbf{t},\mathbf{u})= & {} - \mathfrak {c}^u_M(\delta \mathbf{t},\bar{{\varvec{\varepsilon }}}\cdot \mathbf{x}), \end{aligned}$$
(26)
$$\begin{aligned} \mathfrak {c}^{p}_M(\delta q_M,p)= & {} 0, \end{aligned}$$
(27)
$$\begin{aligned} \mathfrak {c}^{p}_F(\delta q_F,p)= & {} 0, \end{aligned}$$
(28)
which hold for any admissible test functions
\(\delta \mathbf{u}\),
\(\delta p\),
\(\delta \mathbf{t}\),
\(\delta q_M\),
\(\delta q_{F}\)\(\in \,\mathbb {U}_M\times \mathbb {P}_\Box \times \mathbb {T}_M\times \mathbb {Q}_M\times \mathbb {Q}_{F}\). Here, we use the abbreviations
$$\begin{aligned} \mathfrak {a}^u_M(\mathbf{u},\delta \mathbf{u})= & {} \left\langle {\varvec{\sigma }}^\mathrm{eff}({\varvec{\varepsilon }}(\mathbf{u})):(\delta \mathbf{u}\otimes {\varvec{\nabla }})\right\rangle _M, \end{aligned}$$
(29)
$$\begin{aligned} \mathfrak {b}^u_M(p,\delta \mathbf{u})= & {} \left\langle {\varvec{\sigma }}^p(p):(\delta \mathbf{u}\otimes {\varvec{\nabla }})\right\rangle _M, \end{aligned}$$
(30)
$$\begin{aligned} \mathfrak {c}^u_M(\mathbf{t},\mathbf{u})= & {} \frac{1}{V_\Box } \int \limits _{\partial \varOmega _M^+} \mathbf{t}\cdot \llbracket {\mathbf{u}}\rrbracket _\Box \,\hbox {d}a, \end{aligned}$$
(31)
$$\begin{aligned} \mathfrak {c}^u_M(\mathbf{t},\bar{{\varvec{\varepsilon }}}\cdot \mathbf{x})= & {} \frac{1}{V_\Box } \int \limits _{\partial \varOmega _M^+} \mathbf{t}\otimes \llbracket {\mathbf{x}}\rrbracket _\Box \,\hbox {d}a:\bar{{\varvec{\varepsilon }}}, \end{aligned}$$
(32)
$$\begin{aligned} \mathfrak {d}^u_M(p,\delta \mathbf{u})= & {} -\sum \limits _{k=1}^n \frac{1}{V_\Box } \int \limits _{\partial \varOmega _M^{i,k}} p\,\delta \mathbf{u}\cdot \mathbf{n}\,\hbox {d}a, \end{aligned}$$
(33)
and
$$\begin{aligned} \mathfrak {a}^p_M(p,\delta p)= & {} \left\langle \mathbf{q}_M({\varvec{\nabla }}p)\cdot {\varvec{\nabla }}\delta p\right\rangle _M, \end{aligned}$$
(34)
$$\begin{aligned} \mathfrak {b}^p_M(\dot{\mathbf{u}},\delta p)= & {} \left\langle \alpha \,\dot{e}\,\delta p\right\rangle _M, \end{aligned}$$
(35)
$$\begin{aligned} \mathfrak {m}^p_M(\dot{p},\delta p)= & {} \left\langle \frac{\dot{p}}{M}\,\delta p\right\rangle _M,\end{aligned}$$
(36)
$$\begin{aligned} \mathfrak {c}^p_M(q_M,p)= & {} \frac{1}{V_\Box }\int \limits _{\partial \varOmega _M^+} q_M\,\llbracket {p}\rrbracket _\Box \,\hbox {d}a,\end{aligned}$$
(37)
$$\begin{aligned} \mathfrak {d}^p_M(p,\delta p)= & {} -\sum \limits _{k=1}^n \frac{1}{V_\Box }\int \limits _{\partial \varOmega _M^{i,k}}q_M\,\delta p\,\hbox {d}a, \end{aligned}$$
(38)
and
$$\begin{aligned} \mathfrak {a}^p_F(p,\delta p)= & {} \sum \limits _{k=1}^n \left\langle \mathbf{q}_F({\varvec{\nabla }}^F p)\cdot {\varvec{\nabla }}^F\delta p\,\tau _{0}\right\rangle _{\partial F_k}, \end{aligned}$$
(39)
$$\begin{aligned} \mathfrak {b}^p_F(\dot{\mathbf{u}},\delta p)= & {} \sum \limits _{k=1}^n \left\langle \dot{e}_{F,k}\,\delta p\,\tau _{0}\right\rangle _{\partial F_k},\end{aligned}$$
(40)
$$\begin{aligned} \mathfrak {m}^p_F(\dot{p},\delta p)= & {} \sum \limits _{k=1}^n \left\langle \frac{\dot{p}}{K^f}\,\delta p\,\tau _{0}\right\rangle _{\partial F_k}\end{aligned}$$
(41)
$$\begin{aligned} \mathfrak {c}^p_F(q_F,p)= & {} \sum \limits _{k=1}^n \frac{1}{V_\Box }\int \limits _{\partial \partial F^+_k}q_F\,\llbracket {p}\rrbracket _\Box \,\tau _{0}\,\hbox {d}s,\end{aligned}$$
(42)
$$\begin{aligned} \mathfrak {d}^p_F(p,\delta p)= & {} \sum \limits _{k=1}^n \left\langle q_{L}(p)\,\delta p\right\rangle _{\partial F_k}, \end{aligned}$$
(43)
$$\begin{aligned} \mathfrak {e}^p_{F,k}(\hat{q}_k,\delta p)= & {} -\left\langle \hat{q}_k\,\delta p\right\rangle _{\partial F_k}. \end{aligned}$$
(44)
Using (
12) and the constraint
\(\hat{q}_{kl}=-\hat{q}_{lk}\), we see that
\(\sum _{k=1}^n\mathfrak {e}^p_{F,k}(\hat{q}_k,\delta p)=0\). Moreover, using (
15)
\(_2\), we conclude that
\(\mathfrak {d}^p_M(p,\delta p)+\mathfrak {d}^p_F(p,\delta p)=0\). In order to abbreviate notation we introduce
\(q=(q_M,q_F)\) and the corresponding set
\(\mathbb {Q}_\Box =\mathbb {Q}_M\times \mathbb {Q}_{F}\). For a unified description of the fully coupled pressure diffusion process on SVE level we add (
24) and (
25), and we write the SVE continuity equation solving for
\(p,\,q\,\in \,\mathbb {P}_\Box \times \mathbb {Q}_\Box \) as
$$\begin{aligned}&\, -\mathfrak {a}^p_\Box (p,\delta p) + \mathfrak {b}^p_\Box (\dot{\mathbf{u}},\delta p) + \mathfrak {m}^p_\Box (\dot{p},\delta p) \nonumber \\&\quad +\, \mathfrak {c}^p_\Box (q,\delta p) = 0, \end{aligned}$$
(45)
$$\begin{aligned}&\mathfrak {c}^p_\Box (\delta q,p) = 0, \end{aligned}$$
(46)
which holds for any admissible test functions
\(\delta p,\,\delta q\,\in \,\mathbb {P}_\Box \times \mathbb {Q}_\Box \). In (
45) we introduce the SVE forms
$$\begin{aligned} \mathfrak {a}^p_\Box (p,\delta p)= & {} \mathfrak {a}^p_M(p,\delta p) + \mathfrak {a}^p_F(p,\delta p), \end{aligned}$$
(47)
$$\begin{aligned} \mathfrak {b}^p_\Box (\dot{\mathbf{u}},\delta p)= & {} \mathfrak {b}^p_M(\dot{\mathbf{u}},\delta p) + \mathfrak {b}^p_F(\dot{\mathbf{u}},\delta p),\end{aligned}$$
(48)
$$\begin{aligned} \mathfrak {m}^p_\Box (\dot{p},\delta p)= & {} \mathfrak {m}^p_M(\dot{p},\delta p) + \mathfrak {m}^p_F(\dot{p},\delta p),\end{aligned}$$
(49)
$$\begin{aligned} \mathfrak {c}^p_\Box (q,p)= & {} \mathfrak {c}^p_M(q_M,\,p) + \mathfrak {c}^p_F(q_F,p). \end{aligned}$$
(50)
For the sake of completeness, we derive Hill’s principle of macro-homogeneity for the given problem. We, therefore, choose
\(\delta \mathbf{u}\rightarrow \dot{\mathbf{u}}\) and
\(\delta p\rightarrow p\). We add (
23) and (
45) and write in expanded form
$$\begin{aligned}&\left\langle {\varvec{\sigma }}:\dot{{\varvec{\varepsilon }}}\right\rangle _M - \left\langle ({\varvec{\nabla }}\cdot \mathbf{q}_M)\,p\right\rangle _M - \left\langle \mathbf{q}_M\cdot {\varvec{\nabla }}^M p\right\rangle _M \nonumber \\&\qquad - \sum \limits _{k=1}^n\left[ \left\langle p\,\dot{e}_{F}\,\tau _{0}\right\rangle _{\partial F_k} + \left\langle ({\varvec{\nabla }}^F\cdot \mathbf{q}_F)\,p\,\tau _{0}\right\rangle _{\partial F_k}\right. \nonumber \\&\qquad \left. + \left\langle \mathbf{q}_F\cdot {\varvec{\nabla }}^F p\,\tau _{0}\right\rangle _{\partial F_k} \right] \nonumber \\&\quad = \underbrace{\frac{1}{V_\Box }\int \limits _{\partial \varOmega _M^+} (\mathbf{t}\otimes \llbracket {\mathbf{x}}\rrbracket _\Box )^\mathrm{sym}\,\hbox {d}a}_{=:\bar{{\varvec{\sigma }}}} : \dot{\bar{{\varvec{\varepsilon }}}}. \end{aligned}$$
(51)
Hereby, we employ the periodic boundary conditions in the form of (
20), (
21) and (
22). On the right-hand side of the macro-homogeneity condition (
51) we can see that, under the chosen periodic boundary conditions, the macroscopic substitute medium is represented by a monophasic Cauchy continuum model. We use the divergence theorem and rewrite the macroscopic stress tensor
\(\bar{{\varvec{\sigma }}}\) as
$$\begin{aligned} \bar{{\varvec{\sigma }}}:= & {} \frac{1}{V_\Box }\int \limits _{\partial \varOmega _M^+}(\mathbf{t}\otimes \llbracket {\mathbf{x}}\rrbracket _\Box )^\mathrm{sym}\,\hbox {d}a\nonumber \\= & {} \left\langle {\varvec{\sigma }}\right\rangle _M - \sum \limits _{k=1}^n\frac{1}{V_\Box }\int \limits _{\partial \varOmega _M^{i,k}}(\mathbf{t}\otimes \mathbf{x})^\mathrm{sym}\,\hbox {d}a\nonumber \\= & {} \left\langle {\varvec{\sigma }}\right\rangle _M - \sum \limits _{k=1}^n \left\langle p\,\tau _{0}\right\rangle _{\partial F_k}\,\mathbf{I}. \end{aligned}$$
(52)