We apply the asymptotic homogenization assumptions (
57) and (
58) to Eqs. (
38–
53) to obtain, accounting for periodicity, the following multiscale system of PDEs
$$\begin{aligned} \nabla _y \cdot {{\mathsf {T}}_{\scriptscriptstyle \mathrm {I}}^{\epsilon }}+\epsilon \nabla _x \cdot {{\mathsf {T}}_{\scriptscriptstyle \mathrm {I}}^{\epsilon }}=0&\quad \text{ in } \quad \varOmega _{\scriptscriptstyle \mathrm {I}}, \end{aligned}$$
(60)
$$\begin{aligned} \nabla _y \cdot {{\mathsf {T}}_{\scriptscriptstyle \mathrm {II}}^{\epsilon }}+\epsilon \nabla _x \cdot {{\mathsf {T}}_{\scriptscriptstyle \mathrm {II}}^{\epsilon }}=0&\quad \text{ in } \quad \varOmega _{\scriptscriptstyle \mathrm {II}}, \end{aligned}$$
(61)
$$\begin{aligned} \nabla _y \cdot {\mathsf {T_f}^{\epsilon }}+\epsilon \nabla _x \cdot {\mathsf {T_f}^{\epsilon }}=0&\quad \text{ in } \quad \varOmega _\mathrm {f}, \end{aligned}$$
(62)
$$\begin{aligned} \nabla _y \cdot \mathbf{v^{\epsilon }}+\epsilon \nabla _x \cdot \mathbf{v^{\epsilon }}=0&\quad \text{ in } \quad \varOmega _\mathrm {f}, \end{aligned}$$
(63)
$$\begin{aligned} \dot{\mathbf{u}}_{\scriptscriptstyle \mathrm {I}}^{\epsilon }= \mathbf{v^{\epsilon }}&\quad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {I}}, \end{aligned}$$
(64)
$$\begin{aligned} \dot{\mathbf{u}}_{\scriptscriptstyle \mathrm {II}}^{\epsilon }= \mathbf{v^{\epsilon }}&\quad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {II}}, \end{aligned}$$
(65)
$$\begin{aligned} \mathsf {T_f}^{\epsilon }\mathbf{n}_{\scriptscriptstyle \mathrm {I}}={\mathsf {T}}_{\scriptscriptstyle \mathrm {I}}^{\epsilon }\mathbf{n}_{\scriptscriptstyle \mathrm {I}}&\quad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {I}}, \end{aligned}$$
(66)
$$\begin{aligned} \mathsf {T_f}^{\epsilon }\mathbf{n}_{\scriptscriptstyle \mathrm {II}}={\mathsf {T}}_{\scriptscriptstyle \mathrm {II}}^{\epsilon }\mathbf{n}_{\scriptscriptstyle \mathrm {II}}&\quad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {II}}, \end{aligned}$$
(67)
$$\begin{aligned} {\mathsf {T}}_{\scriptscriptstyle \mathrm {I}}^{\epsilon }\mathbf{n}_{\scriptscriptstyle \mathrm {III}}={\mathsf {T}}_{\scriptscriptstyle \mathrm {II}}^{\epsilon }\mathbf{n}_{\scriptscriptstyle \mathrm {III}}&\quad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {III}}, \end{aligned}$$
(68)
$$\begin{aligned} \mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{\epsilon }=\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{\epsilon }&\quad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {III}}, \end{aligned}$$
(69)
equipped with multiscale constitutive equations for the fluid and solid stress tensors
\(\mathsf {T_f}^{\epsilon }\),
\({\mathsf {T}}_{\scriptscriptstyle \mathrm {I}}^{\epsilon }\),
\({\mathsf {T}}_{\scriptscriptstyle \mathrm {II}}^{\epsilon }\), given by
$$\begin{aligned}&\mathsf {T_f}^{\epsilon }=-p^{\epsilon }{} \mathbf{I}+\epsilon (\nabla _y\mathbf{v^{\epsilon }}+(\nabla _y\mathbf{v^{\epsilon }})^{\mathrm T})+\epsilon ^2(\nabla _x\mathbf{v^{\epsilon }}+(\nabla _x\mathbf{v^{\epsilon }})^{\mathrm T}),&\end{aligned}$$
(70)
$$\begin{aligned}&\epsilon {\mathsf {T}}_{\scriptscriptstyle \mathrm {I}}^{\epsilon }={\mathbb {C}}_{\scriptscriptstyle \mathrm {I}}\xi _y(\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{\epsilon })+\epsilon {\mathbb {C}}_{\scriptscriptstyle \mathrm {I}}\xi _x(\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{\epsilon }),&\end{aligned}$$
(71)
$$\begin{aligned}&\epsilon {\mathsf {T}}_{\scriptscriptstyle \mathrm {II}}^{\epsilon }={\mathbb {C}}_{\scriptscriptstyle \mathrm {II}}\xi _y(\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{\epsilon })+\epsilon {\mathbb {C}}_{\scriptscriptstyle \mathrm {II}}\xi _x(\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{\epsilon }),&\end{aligned}$$
(72)
while the balance equations in terms of the elastic displacement, fluid velocity, and pressure
\(\mathbf{u}^{\epsilon }_{\scriptscriptstyle {\mathrm {I}}}\),
\(\mathbf{u}^{\epsilon }_{\mathrm {I}}\),
\(\mathbf{v}^{\epsilon }\),
\(p^{\epsilon }\) read
$$\begin{aligned}&\nabla _y \cdot ({\mathbb {C}}_{\scriptscriptstyle \mathrm {I}} \xi _y( \mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{\epsilon }))+\epsilon \nabla _y \cdot ({\mathbb {C}}_{\scriptscriptstyle \mathrm {I}} \xi _x (\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{\epsilon }))~\nonumber \\&\qquad +\epsilon \nabla _x \cdot ({\mathbb {C}}_{\scriptscriptstyle \mathrm {I}} \xi _y (\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{\epsilon }))+ \epsilon ^2\nabla _x \cdot ({\mathbb {C}}_{\scriptscriptstyle \mathrm {I}} \xi _x (\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{\epsilon }))=0\quad \text{ in } \quad \varOmega _{\scriptscriptstyle \mathrm {I}}, \end{aligned}$$
(73)
$$\begin{aligned}&\nabla _y \cdot ({\mathbb {C}}_{\scriptscriptstyle {\scriptscriptstyle \mathrm {II}}} \xi _y (\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{\epsilon }))+\epsilon \nabla _y \cdot ({\mathbb {C}}_{\scriptscriptstyle \mathrm {II}} \xi _x (\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{\epsilon }))~\nonumber \\&\qquad +\epsilon \nabla _x \cdot ({\mathbb {C}}_{\scriptscriptstyle \mathrm {II}} \xi _y (\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{\epsilon }))+ \epsilon ^2\nabla _x \cdot ({\mathbb {C}}_{\scriptscriptstyle \mathrm {II}} \xi _x (\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{\epsilon }))=0\quad \text{ in } \quad \varOmega _{\scriptscriptstyle \mathrm {II}}, \end{aligned}$$
(74)
$$\begin{aligned}&\epsilon ^3 \nabla _x^2\mathbf{v}^{\epsilon }+\epsilon ^2 \nabla _x \cdot (\nabla _y \mathbf{v}^{\epsilon })+\epsilon ^2\nabla _y \cdot (\nabla _x \mathbf{v}^{\epsilon })+\epsilon \nabla _y^2\mathbf{v}^{\epsilon }\nonumber \\&\quad =\nabla _y p^{\epsilon }+\epsilon \nabla _x p^{\epsilon } \quad \text{ in } \quad \varOmega _\mathrm {f}. \end{aligned}$$
(75)
We can now substitute power series of type (
58) into the relevant fields in (
60–
75). Then by equating the coefficients of
\(\epsilon ^l\) for
\(l=0,1,\ldots \), we derive the macroscale model for the material in terms of the relevant leading (zeroth)-order fields. Whenever a component in the asymptotic expansion retains a dependence on the microscale, we can take the integral average, which we define as
$$\begin{aligned} \langle \varphi \rangle _i=\frac{1}{|\varOmega |}\int _{\varOmega _i}\varphi (\mathbf{x,y},t){\mathrm{d}}{} \mathbf{y} \quad i=\mathrm {f}, {\mathrm {I}}, {\mathrm {II}}, \end{aligned}$$
(76)
where
\(\varphi \) is a field, and also where the integral average can be performed over one representative cell due to
\(\mathbf{y}\)-periodicity and
\(|\varOmega |\) is the volume of the domain and the integration is performed over the microscale. We note that
\(|\varOmega |=|\varOmega _{\mathrm{f}}|+|\varOmega _{\scriptscriptstyle \mathrm {I}}|+|\varOmega _{\scriptscriptstyle \mathrm {II}}|\). Due to the assumption of
\(\mathbf{y}\)-periodicity, the integral average can be performed over one representative cell. Therefore, (
76) represents a cell average. For the sake of brevity, we also introduce the notation
$$\begin{aligned} \langle \varphi _{\scriptscriptstyle \mathrm {I}}+\varphi _{\scriptscriptstyle \mathrm {II}}\rangle _s=\frac{1}{|\varOmega |}\left( \int _{\varOmega _{\scriptscriptstyle \mathrm {I}}}\varphi _{\scriptscriptstyle \mathrm {I}}(\mathbf{x,y},t)\text{ d }\mathbf{y}+\int _{\varOmega _{\scriptscriptstyle \mathrm {II}}}\varphi _{\scriptscriptstyle \mathrm {II}}(\mathbf{x,y},t)\text{ d }{} \mathbf{y}\right) , \end{aligned}$$
(77)
for fields
\(\varphi \) with components
\(\varphi _{\scriptscriptstyle \mathrm {I}}\) and
\(\varphi _{\scriptscriptstyle \mathrm {II}}\) defined in the solid cell portions
\(\varOmega _{\scriptscriptstyle \mathrm {I}}\) or
\(\varOmega _{\scriptscriptstyle \mathrm {II}}\), respectively.
Equating coefficients of
\(\epsilon ^0\) in (
60–
69) we obtain
$$\begin{aligned} \nabla _y \cdot {{\mathsf {T}}}_{\scriptscriptstyle \mathrm {I}}^{(0)}=0&\qquad \qquad \text{ in } \quad \varOmega _{\scriptscriptstyle \mathrm {I}}, \end{aligned}$$
(78)
$$\begin{aligned} \nabla _y \cdot {{\mathsf {T}}}_{\scriptscriptstyle \mathrm {II}}^{(0)}=0&\qquad \qquad \text{ in } \quad \varOmega _{\scriptscriptstyle \mathrm {II}}, \end{aligned}$$
(79)
$$\begin{aligned} \nabla _y \cdot {\mathsf {T_f}}^{(0)}=0&\qquad \qquad \text{ in } \quad \varOmega _{\mathrm{f}}, \end{aligned}$$
(80)
$$\begin{aligned} \nabla _y \cdot \mathbf{v}^{(0)}=0&\qquad \qquad \text{ in }\quad \varOmega _{\mathrm{f}}, \end{aligned}$$
(81)
$$\begin{aligned} \dot{\mathbf{u}}_{\scriptscriptstyle \mathrm {I}}^{(0)}= \mathbf{v}^{(0)}&\qquad \qquad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {I}}, \end{aligned}$$
(82)
$$\begin{aligned} \dot{\mathbf{u}}_{\scriptscriptstyle \mathrm {II}}^{(0)}= \mathbf{v}^{(0)}&\qquad \qquad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {II}}, \end{aligned}$$
(83)
$$\begin{aligned} \mathsf {T_f}^{(0)}\mathbf{n}_{\scriptscriptstyle \mathrm {I}}={\mathsf {T}}_{\scriptscriptstyle \mathrm {I}}^{(0)}\mathbf{n}_{\scriptscriptstyle \mathrm {I}}&\qquad \qquad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {I}}, \end{aligned}$$
(84)
$$\begin{aligned} \mathsf {T_f}^{(0)}\mathbf{n}_{\scriptscriptstyle \mathrm {II}}={\mathsf {T}}_{\scriptscriptstyle \mathrm {II}}^{(0)}\mathbf{n}_{\scriptscriptstyle \mathrm {II}}&\qquad \qquad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {II}}, \end{aligned}$$
(85)
$$\begin{aligned} {\mathsf {T}}_{\scriptscriptstyle \mathrm {I}}^{(0)}\mathbf{n}_{\scriptscriptstyle \mathrm {III}}={\mathsf {T}}_{\scriptscriptstyle \mathrm {II}}^{(0)}\mathbf{n}_{\scriptscriptstyle \mathrm {III}}&\qquad \qquad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {III}}, \end{aligned}$$
(86)
$$\begin{aligned} \mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{(0)}=\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{(0)}&\qquad \qquad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {III}}, \end{aligned}$$
(87)
and the constitutive equations (
70–
72) for
\(\mathsf {T_f}^{\epsilon }\),
\({\mathsf {T}}_{\scriptscriptstyle \mathrm {I}}^{\epsilon }\),
\({\mathsf {T}}_{\scriptscriptstyle \mathrm {II}}^{\epsilon }\) have coefficients of
\(\epsilon ^0\)$$\begin{aligned} {\mathsf {T_f}}^{(0)}=-p^{(0)}{} \mathbf{I}&\qquad \qquad \text{ in } \quad \varOmega _{\mathrm{f}}, \end{aligned}$$
(88)
$$\begin{aligned} {\mathbb {C}}_{\scriptscriptstyle \mathrm {I}}\xi _y(\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{(0)})=0&\qquad \qquad \text{ in } \quad \varOmega _{\scriptscriptstyle \mathrm {I}}, \end{aligned}$$
(89)
$$\begin{aligned} {\mathbb {C}}_{\scriptscriptstyle \mathrm {II}}\xi _y(\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{(0)})=0&\qquad \qquad \text{ in } \quad \varOmega _{\scriptscriptstyle \mathrm {II}}, \end{aligned}$$
(90)
and the balance equations (
73–
75) have coefficients of
\(\epsilon ^0\)$$\begin{aligned} \nabla _y \cdot ({\mathbb {C}}_{\scriptscriptstyle \mathrm {I}} \xi _y (\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{(0)}))=0&\qquad \qquad \text{ in } \quad \varOmega _{\scriptscriptstyle \mathrm {I}}, \end{aligned}$$
(91)
$$\begin{aligned} \nabla _y \cdot ({\mathbb {C}}_{\scriptscriptstyle \mathrm {II}} \xi _y( \mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{(0)}))=0&\qquad \qquad \text{ in } \quad \varOmega _{\scriptscriptstyle \mathrm {II}}, \end{aligned}$$
(92)
$$\begin{aligned} \nabla _y p^{(0)}=0&\qquad \qquad \text{ in }\quad \varOmega _{\mathrm{f}}. \end{aligned}$$
(93)
Similarly, we now wish to equate the coefficients of
\(\epsilon ^1\) in Eqs. (
60–
69) which gives
$$\begin{aligned} \nabla _y \cdot {{\mathsf {T}}}_{\scriptscriptstyle \mathrm {I}}^{(1)}+\nabla _x \cdot {{\mathsf {T}}}_{\scriptscriptstyle \mathrm {I}}^{(0)}=0&\quad \text{ in } \quad \varOmega _{\scriptscriptstyle \mathrm {I}}, \end{aligned}$$
(94)
$$\begin{aligned} \nabla _y \cdot {{\mathsf {T}}}_{\scriptscriptstyle \mathrm {II}}^{(1)}+\nabla _x \cdot {{\mathsf {T}}}_{\scriptscriptstyle \mathrm {II}}^{(0)}=0&\quad \text{ in } \quad \varOmega _{\scriptscriptstyle \mathrm {II}}, \end{aligned}$$
(95)
$$\begin{aligned} \nabla _y \cdot {\mathsf {T_f}}^{(1)}+\nabla _x \cdot {\mathsf {T_f}}^{(0)}=0&\quad \text{ in } \quad \varOmega _{\mathrm{f}}, \end{aligned}$$
(96)
$$\begin{aligned} \nabla _y \cdot \mathbf{v}^{(1)}+\nabla _x \cdot \mathbf{v}^{(0)}=0&\quad \text{ in } \quad \varOmega _{\mathrm{f}}, \end{aligned}$$
(97)
$$\begin{aligned} \dot{\mathbf{u}}_{\scriptscriptstyle \mathrm {I}}^{(1)}= \mathbf{v}^{(1)}&\quad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {I}}, \end{aligned}$$
(98)
$$\begin{aligned} \dot{\mathbf{u}}_{\scriptscriptstyle \mathrm {II}}^{(1)}= \mathbf{v}^{(1)}&\quad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {II}}, \end{aligned}$$
(99)
$$\begin{aligned} \mathsf {T_f}^{(1)}\mathbf{n}_{\scriptscriptstyle \mathrm {I}}={\mathsf {T}}_{\scriptscriptstyle \mathrm {I}}^{(1)}\mathbf{n}_{\scriptscriptstyle \mathrm {I}}&\quad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {I}}, \end{aligned}$$
(100)
$$\begin{aligned} \mathsf {T_f}^{(1)}\mathbf{n}_{\scriptscriptstyle \mathrm {II}}={\mathsf {T}}_{\scriptscriptstyle \mathrm {II}}^{(1)}\mathbf{n}_{\scriptscriptstyle \mathrm {II}}&\quad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {II}}, \end{aligned}$$
(101)
$$\begin{aligned} {\mathsf {T}}_{\scriptscriptstyle \mathrm {I}}^{(1)}\mathbf{n}_{\scriptscriptstyle \mathrm {III}}={\mathsf {T}}_{\scriptscriptstyle \mathrm {II}}^{(1)}\mathbf{n}_{\scriptscriptstyle \mathrm {III}}&\quad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {III}}, \end{aligned}$$
(102)
$$\begin{aligned} \mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{(1)}=\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{(1)}&\quad \text{ on } \quad \varGamma _{\scriptscriptstyle \mathrm {III}}, \end{aligned}$$
(103)
and the constitutive equations (
70–
72) for
\(\mathsf {T_f}^{\epsilon }\),
\({\mathsf {T}}_{\scriptscriptstyle \mathrm {I}}^{\epsilon }\),
\({\mathsf {T}}_{\scriptscriptstyle \mathrm {II}}^{\epsilon }\) have coefficients of
\(\epsilon ^1\)$$\begin{aligned} {\mathsf {T_f}}^{(1)}=-p^{(1)}{} \mathbf{I}+(\nabla _y\mathbf{v}^{(0)}+(\nabla _y\mathbf{v}^{(0)})^{\mathrm T})&\quad \text{ in } \quad \varOmega _{\mathrm{f}}, \end{aligned}$$
(104)
$$\begin{aligned} {\mathsf {T}}_{\scriptscriptstyle \mathrm {I}}^{(0)}={\mathbb {C}}_{\scriptscriptstyle \mathrm {I}}\xi _y(\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{(1)})+{\mathbb {C}}_{\scriptscriptstyle \mathrm {I}}\xi _x(\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{(0)})&\quad \text{ in } \quad \varOmega _{\scriptscriptstyle \mathrm {I}}, \end{aligned}$$
(105)
$$\begin{aligned} {\mathsf {T}}_{\scriptscriptstyle \mathrm {II}}^{(0)}={\mathbb {C}}_{\scriptscriptstyle \mathrm {II}}\xi _y(\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{(1)})+{\mathbb {C}}_{\scriptscriptstyle \mathrm {II}}\xi _x(\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{(0)})&\quad \text{ in } \quad \varOmega _{\scriptscriptstyle \mathrm {II}}, \end{aligned}$$
(106)
and the balance equations (
73–
75) have coefficients of
\(\epsilon ^1\)$$\begin{aligned} \nabla _y \cdot ({\mathbb {C}}_{\scriptscriptstyle \mathrm {I}} \xi _y (\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{(1)}))+\nabla _y \cdot ({\mathbb {C}}_{\scriptscriptstyle \mathrm {I}} \xi _x (\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{(0)}))+\nabla _x \cdot ({\mathbb {C}}_{\scriptscriptstyle \mathrm {I}} \xi _y (\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{(0)}))=0&~~ \text{ in } ~~ \varOmega _{\scriptscriptstyle \mathrm {I}}, \end{aligned}$$
(107)
$$\begin{aligned} \nabla _y \cdot ({\mathbb {C}}_{\scriptscriptstyle \mathrm {II}} \xi _y (\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{(1)}))+\nabla _y \cdot ({\mathbb {C}}_{\scriptscriptstyle \mathrm {II}} \xi _x (\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{(0)}))+\nabla _x \cdot ({\mathbb {C}}_{\scriptscriptstyle \mathrm {II}} \xi _y (\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{(0)}))=0&~~ \text{ in } ~~ \varOmega _{\scriptscriptstyle \mathrm {II}}, \end{aligned}$$
(108)
$$\begin{aligned} \nabla _y^2\mathbf{v}^{(0)}=\nabla _y p^{(1)}+\nabla _x p^{(0)}&~~ \text{ in } ~~ \varOmega _{\mathrm{f}}, \end{aligned}$$
(109)
We can now see from (
80) and (
88) that the leading order pressure
\(p^{(0)}\) does not depend on the microscale
\(\mathbf{y}\). That is
$$\begin{aligned} p^{(0)}=p^{(0)}(\mathbf{x},t). \end{aligned}$$
(110)
We also have from (
89) and (
90) that
\(\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{(0)}\) and
\(\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{(0)}\), which are the leading order solid displacements, are rigid body motions and therefore, by
\(\mathbf{y}\)-periodicity, do not depend on the microscale
\(\mathbf{y}\). That is
$$\begin{aligned} \mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{(0)}=\,&\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{(0)}(\mathbf{x},t) \end{aligned}$$
(111)
$$\begin{aligned} \mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{(0)}=\,&\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{(0)}(\mathbf{x},t). \end{aligned}$$
(112)
Since we have the boundary condition
\(\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{(0)}=\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{(0)}\) on
\(\varGamma _{\scriptscriptstyle \mathrm {III}}\) given by (
87) we can define
$$\begin{aligned} \mathbf{u}^{(0)}=\mathbf{u}_{\scriptscriptstyle \mathrm {I}}^{(0)}=\mathbf{u}_{\scriptscriptstyle \mathrm {II}}^{(0)}, \end{aligned}$$
(113)
which we will use throughout the following sections.