Another way in which we can potentially tackle the coupling of the two scales, is to propose a decomposition of the pre-stress into two parts, one depending on the macroscale and the other depending on only the microscale. We make the following assumptions: firstly, the elasticity tensor depends only on the microscale
$$\begin{aligned} \mathbb {C}^{\scriptscriptstyle \textrm{c}}=\mathbb {C}^{\scriptscriptstyle \textrm{c}}(\textbf{y}) \quad \text{ and } \quad \mathbb {C}^{\scriptscriptstyle \textrm{p}}=\mathbb {C}^{\scriptscriptstyle \textrm{p}}(\textbf{y}), \end{aligned}$$
(80)
and secondly, the leading-order pre-stresses have the following decomposition
$$\begin{aligned}&(\varvec{\sigma }_{\scriptscriptstyle \textrm{0c}}^{(0)})_{ij}=(\varvec{\sigma }_{\scriptscriptstyle \textrm{0c}}^{L(0)}(\textbf{y}))_{ijkl}(\varvec{\sigma }_{\scriptscriptstyle \textrm{0c}}^{G(0)}(\textbf{x}))_{kl} \end{aligned}$$
(81a)
$$\begin{aligned}&(\varvec{\sigma }_{\scriptscriptstyle \textrm{0p}}^{(0)})_{ij}=(\varvec{\sigma }_{\scriptscriptstyle \textrm{0p}}^{L(0)}(\textbf{y}))_{ijkl}(\varvec{\sigma }_{\scriptscriptstyle \textrm{0p}}^{G(0)}(\textbf{x}))_{kl}, \end{aligned}$$
(81b)
where the new superscripts
L and
G represent
Local (microscale) and
Global (macroscale), respectively. Here we have decomposed in such a way that the second-rank pre-stress arises from the double contraction of a fourth-rank microscale component with a second-rank macroscale component. We need to take this decomposition into account in the ansatz that we propose for elastic problem (
54a)–(
54f). The new solution that we propose is given as
$$\begin{aligned} \textbf{u}_{\scriptscriptstyle \textrm{c}}^{(1)}(\textbf{x}, \textbf{y})&={\mathcal {A}}^{\scriptscriptstyle \textrm{c}}(\textbf{y}) \xi _x (\textbf{u}^{(0)}(\textbf{x}))+\textbf{a}^{\scriptscriptstyle \textrm{c}}(\textbf{y})p^{(0)}(\textbf{x})+{\mathcal {B}}^{\scriptscriptstyle \textrm{c}}(\textbf{y}) \varvec{\sigma }_{\scriptscriptstyle \textrm{0c}}^{G(0)}(\textbf{x})+{\mathcal {D}}^{\scriptscriptstyle \textrm{c}}(\textbf{y}) \varvec{\sigma }_{\scriptscriptstyle \textrm{0p}}^{G(0)}(\textbf{x}) \end{aligned}$$
(82a)
$$\begin{aligned} \textbf{u}_{\scriptscriptstyle \textrm{p}}^{(1)}(\textbf{x}, \textbf{y})&={\mathcal {A}}^{\scriptscriptstyle \textrm{p}}(\textbf{y}) \xi _x (\textbf{u}^{(0)}(\textbf{x}))+\textbf{a}^{\scriptscriptstyle \textrm{p}}(\textbf{y})p^{(0)}(\textbf{x})+{\mathcal {B}}^{\scriptscriptstyle \textrm{p}}(\textbf{y}) \varvec{\sigma }_{\scriptscriptstyle \textrm{0p}}^{G(0)}(\textbf{x}) +{\mathcal {D}}^{\scriptscriptstyle \textrm{p}}(\textbf{y}) \varvec{\sigma }_{\scriptscriptstyle \textrm{0c}}^{G(0)}(\textbf{x}), \end{aligned}$$
(82b)
where
\({\mathcal {A}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})\),
\({\mathcal {A}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})\),
\({\mathcal {B}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})\),
\({\mathcal {B}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})\),
\({\mathcal {D}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})\) and
\({\mathcal {D}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})\) are third-rank tensors with a dependence on only the microscale variable
\(\textbf{y}\) and
\(\textbf{a}^{\scriptscriptstyle \textrm{c}}(\textbf{y})\) and
\(\textbf{a}^{\scriptscriptstyle \textrm{p}}(\textbf{y})\) are vectors depending on the microscale variable
\(\textbf{y}\). There are now two choices that can be made that will result in different cell problems. If we assume that the second-rank tensors
\(\varvec{\sigma }_{\scriptscriptstyle \textrm{0c}}^{G(0)}(\textbf{x})\ne \varvec{\sigma }_{\scriptscriptstyle \textrm{0p}}^{G(0)}(\textbf{x})\), then we obtain the following cell problems that do not retain any macroscale dependency
$$\begin{aligned} \nabla _\textbf{y} \cdot (\mathbb {C}^{\scriptscriptstyle \textrm{c}} \xi _y ({\mathcal {A}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})))+\nabla _\textbf{y} \cdot \mathbb {C}^{\scriptscriptstyle \textrm{c}}=0&\quad \quad \text{ in } \quad \Omega _{\scriptscriptstyle \textrm{c}} \end{aligned}$$
(83a)
$$\begin{aligned} \nabla _\textbf{y} \cdot (\mathbb {C}^{\scriptscriptstyle \textrm{p}} \xi _y ({\mathcal {A}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})))+\nabla _\textbf{y} \cdot \mathbb {C}^{\scriptscriptstyle \textrm{p}}=0&\quad \quad \text{ in } \quad \Omega _{\scriptscriptstyle \textrm{p}} \end{aligned}$$
(83b)
$$\begin{aligned} \mathbb {C}^{\scriptscriptstyle \textrm{c}} \xi _y ({\mathcal {A}}^{\scriptscriptstyle \textrm{c}}(\textbf{y}))\textbf{n}_{\scriptscriptstyle \textrm{cp}}-\mathbb {C}^{\scriptscriptstyle \textrm{p}} \xi _y ({\mathcal {A}}^{\scriptscriptstyle \textrm{p}}(\textbf{y}))\textbf{n}_{\scriptscriptstyle \textrm{cp}}=(\mathbb {C}^{\scriptscriptstyle \textrm{p}}-\mathbb {C}^{\scriptscriptstyle \textrm{c}})\textbf{n}_{\scriptscriptstyle \textrm{cp}}&\quad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{cp}} \end{aligned}$$
(83c)
$$\begin{aligned} {\mathcal {A}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})={\mathcal {A}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})&\quad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{cp}} \end{aligned}$$
(83d)
$$\begin{aligned} (\mathbb {C}^{\scriptscriptstyle \textrm{c}} \xi _y ({\mathcal {A}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})))\textbf{n}_{\scriptscriptstyle \textrm{c}}+ \mathbb {C}^{\scriptscriptstyle \textrm{c}}{} \textbf{n}_{\scriptscriptstyle \textrm{c}}=0&\quad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{c}} \end{aligned}$$
(83e)
$$\begin{aligned} (\mathbb {C}^{\scriptscriptstyle \textrm{p}} \xi _y({\mathcal {A}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})))\textbf{n}_{\scriptscriptstyle \textrm{p}}+ \mathbb {C}^{\scriptscriptstyle \textrm{p}}\textbf{n}_{\scriptscriptstyle \textrm{p}}=0&\quad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{p}} \end{aligned}$$
(83f)
and
$$\begin{aligned} \nabla _\textbf{y} \cdot (\mathbb {C}^{\scriptscriptstyle \textrm{c}} \xi _y (\textbf{a}^{\scriptscriptstyle \textrm{c}}(\textbf{y})))=\textbf{0}&\qquad \quad \text{ in } \quad \Omega _{\scriptscriptstyle \textrm{c}} \end{aligned}$$
(84a)
$$\begin{aligned} \nabla _\textbf{y} \cdot (\mathbb {C}^{\scriptscriptstyle \textrm{p}} \xi _y (\textbf{a}^{\scriptscriptstyle \textrm{p}}(\textbf{y})))=\textbf{0}&\qquad \quad \text{ in } \quad \Omega _{\scriptscriptstyle \textrm{p}} \end{aligned}$$
(84b)
$$\begin{aligned} (\mathbb {C}^{\scriptscriptstyle \textrm{c}} \xi _y (\textbf{a}^{\scriptscriptstyle \textrm{c}}(\textbf{y})))\textbf{n}_{\scriptscriptstyle \textrm{cp}}=(\mathbb {C}^{\scriptscriptstyle \textrm{p}} \xi _y( \textbf{a}^{\scriptscriptstyle \textrm{p}}(\textbf{y})))\textbf{n}_{\scriptscriptstyle \textrm{cp}}&\qquad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{cp}} \end{aligned}$$
(84c)
$$\begin{aligned} \textbf{a}^{\scriptscriptstyle \textrm{c}}(\textbf{y})=\textbf{a}^{\scriptscriptstyle \textrm{p}}(\textbf{y})&\qquad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{cp}} \end{aligned}$$
(84d)
$$\begin{aligned} (\mathbb {C}^{\scriptscriptstyle \textrm{c}} \xi _y (\textbf{a}^{\scriptscriptstyle \textrm{c}}(\textbf{y})))\textbf{n}_{\scriptscriptstyle \textrm{c}}+ \textbf{n}_{\scriptscriptstyle \textrm{c}}=\textbf{0}&\qquad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{c}} \end{aligned}$$
(84e)
$$\begin{aligned} (\mathbb {C}^{\scriptscriptstyle \textrm{p}} \xi _y (\textbf{a}^{\scriptscriptstyle \textrm{p}}(\textbf{y})))\textbf{n}_{\scriptscriptstyle \textrm{p}}+ \textbf{n}_{\scriptscriptstyle \textrm{p}}=\textbf{0}&\qquad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{p}} \end{aligned}$$
(84f)
which are the cell problems (
57a)–(
57f) and (
58a)–(
58f) which are now
\(\textbf{y}\)-dependent only since we make assumption (
80). We have the change in the cell problems which include the pre-stresses which we have decomposed and these are
$$\begin{aligned} \nabla _\textbf{y} \cdot (\mathbb {C}^{\scriptscriptstyle \textrm{c}} \xi _y ({\mathcal {B}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})))+\nabla _\textbf{y} \cdot \varvec{\sigma }_{\scriptscriptstyle \textrm{0c}}^{L(0)}(\textbf{y})={ 0}&\qquad \quad \text{ in } \quad \Omega _{\scriptscriptstyle \textrm{c}} \end{aligned}$$
(85a)
$$\begin{aligned} \nabla _\textbf{y} \cdot (\mathbb {C}^{\scriptscriptstyle \textrm{p}} \xi _y ({\mathcal {D}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})))={ 0}&\qquad \quad \text{ in } \quad \Omega _{\scriptscriptstyle \textrm{p}} \end{aligned}$$
(85b)
$$\begin{aligned} (\mathbb {C}^{\scriptscriptstyle \textrm{c}} \xi _y ({\mathcal {B}}^{\scriptscriptstyle \textrm{c}}(\textbf{y}))-\mathbb {C}^{\scriptscriptstyle \textrm{p}} \xi _y ({\mathcal {D}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})))\textbf{n}_{\scriptscriptstyle \textrm{cp}}=-\varvec{\sigma }_{\scriptscriptstyle \textrm{0c}}^{L(0)}(\textbf{y})\textbf{n}_{\scriptscriptstyle \textrm{cp}}&\qquad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{cp}} \end{aligned}$$
(85c)
$$\begin{aligned} {\mathcal {B}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})={\mathcal {D}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})&\qquad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{cp}} \end{aligned}$$
(85d)
$$\begin{aligned} (\mathbb {C}^{\scriptscriptstyle \textrm{c}} \xi _y ({\mathcal {B}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})))\textbf{n}_{\scriptscriptstyle \textrm{c}}+ \varvec{\sigma }_{\scriptscriptstyle \textrm{0c}}^{L(0)}(\textbf{y})\textbf{n}_{\scriptscriptstyle \textrm{c}}={0}&\qquad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{c}} \end{aligned}$$
(85e)
$$\begin{aligned} (\mathbb {C}^{\scriptscriptstyle \textrm{p}} \xi _y ({\mathcal {D}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})))\textbf{n}_{\scriptscriptstyle \textrm{p}}={ 0}&\qquad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{p}} \end{aligned}$$
(85f)
and
$$\begin{aligned} \nabla _\textbf{y} \cdot (\mathbb {C}^{\scriptscriptstyle \textrm{c}} \xi _y ({\mathcal {D}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})))={ 0}&\qquad \quad \text{ in } \quad \Omega _{\scriptscriptstyle \textrm{c}} \end{aligned}$$
(86a)
$$\begin{aligned} \nabla _\textbf{y} \cdot (\mathbb {C}^{\scriptscriptstyle \textrm{p}} \xi _y ({\mathcal {B}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})))+\nabla _\textbf{y} \cdot \varvec{\sigma }_{\scriptscriptstyle \textrm{0p}}^{L(0)}(\textbf{y})={0}&\qquad \quad \text{ in } \quad \Omega _{\scriptscriptstyle \textrm{p}} \end{aligned}$$
(86b)
$$\begin{aligned} \mathbb {C}^{\scriptscriptstyle \textrm{c}} \xi _y ({\mathcal {D}}^{\scriptscriptstyle \textrm{c}}(\textbf{y}))\textbf{n}_{\scriptscriptstyle \textrm{cp}}-\mathbb {C}^{\scriptscriptstyle \textrm{p}} \xi _y ({\mathcal {B}}^{\scriptscriptstyle \textrm{p}}(\textbf{y}))\textbf{n}_{\scriptscriptstyle \textrm{cp}}=\varvec{\sigma }_{\scriptscriptstyle \textrm{0p}}^{L(0)}(\textbf{y})\textbf{n}_{\scriptscriptstyle \textrm{cp}}&\qquad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{cp}} \end{aligned}$$
(86c)
$$\begin{aligned} {\mathcal {D}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})={\mathcal {B}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})&\qquad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{cp}} \end{aligned}$$
(86d)
$$\begin{aligned} (\mathbb {C}^{\scriptscriptstyle \textrm{c}} \xi _y ({\mathcal {D}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})))\textbf{n}_{\scriptscriptstyle \textrm{c}}={0}&\qquad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{c}} \end{aligned}$$
(86e)
$$\begin{aligned} (\mathbb {C}^{\scriptscriptstyle \textrm{p}} \xi _y({\mathcal {B}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})))\textbf{n}_{\scriptscriptstyle \textrm{p}}+ \varvec{\sigma }_{\scriptscriptstyle \textrm{0p}}^{L(0)}(\textbf{y})\textbf{n}_{\scriptscriptstyle \textrm{p}}={0}&\qquad \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{p}}. \end{aligned}$$
(86f)
These cell problems can be solved completely on the microscale.
If we instead make the assumption that the second-rank tensors
$$\begin{aligned} \varvec{\sigma }_{\scriptscriptstyle \textrm{0c}}^{G(0)}(\textbf{x})=\varvec{\sigma }_{\scriptscriptstyle \textrm{0p}}^{G(0)}(\textbf{x})=\varvec{\sigma }_{\scriptscriptstyle \textrm{0}}^{G(0)}(\textbf{x}), \end{aligned}$$
(87)
then we can rewrite the ansatz (
82a)–(
82b) as
$$\begin{aligned}&\textbf{u}_{\scriptscriptstyle \textrm{c}}^{(1)}(\textbf{x}, \textbf{y})={\mathcal {A}}^{\scriptscriptstyle \textrm{c}}(\textbf{y}) \xi _x (\textbf{u}^{(0)}(\textbf{x}))+\textbf{a}^{\scriptscriptstyle \textrm{c}}(\textbf{y})p^{(0)}(\textbf{x})+{\mathcal {B}}^{\scriptscriptstyle \textrm{c}}(\textbf{y}) \varvec{\sigma }_{\scriptscriptstyle \textrm{0}}^{G(0)}(\textbf{x}) \end{aligned}$$
(88a)
$$\begin{aligned}&\textbf{u}_{\scriptscriptstyle \textrm{p}}^{(1)}(\textbf{x}, \textbf{y})={\mathcal {A}}^{\scriptscriptstyle \textrm{p}}(\textbf{y}) \xi _x (\textbf{u}^{(0)}(\textbf{x}))+\textbf{a}^{\scriptscriptstyle \textrm{p}}(\textbf{y})p^{(0)}(\textbf{x})+{\mathcal {B}}^{\scriptscriptstyle \textrm{p}}(\textbf{y}) \varvec{\sigma }_{\scriptscriptstyle \textrm{0}}^{G(0)}(\textbf{x}), \end{aligned}$$
(88b)
where again
\({\mathcal {A}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})\),
\({\mathcal {A}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})\),
\({\mathcal {B}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})\) and
\({\mathcal {B}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})\) are third-rank tensors with a dependence on only the microscale variable
\(\textbf{y}\) and
\(\textbf{a}^{\scriptscriptstyle \textrm{c}}(\textbf{y})\) and
\(\textbf{a}^{\scriptscriptstyle \textrm{p}}(\textbf{y})\) are vectors that depend only on the microscale. This allows us to obtain the following cell problems that do not retain any macroscale dependency. We have exactly (
83a)–(
83f) and (
84a)–(
84f) and the following different microscale cell problem
$$\begin{aligned}&\nabla _\textbf{y} \cdot (\mathbb {C}^{\scriptscriptstyle \textrm{c}} \xi _y ({\mathcal {B}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})))+\nabla _\textbf{y} \cdot \varvec{\sigma }_{\scriptscriptstyle \textrm{0c}}^{L(0)}(\textbf{y})=0 \quad \text{ in } \quad \Omega _{\scriptscriptstyle \textrm{c}} \end{aligned}$$
(89)
$$\begin{aligned}&\nabla _\textbf{y} \cdot (\mathbb {C}^{\scriptscriptstyle \textrm{p}} \xi _y ({\mathcal {B}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})))+\nabla _\textbf{y} \cdot \varvec{\sigma }_{\scriptscriptstyle \textrm{0p}}^{L(0)}(\textbf{y})=0 \quad \text{ in } \quad \Omega _{\scriptscriptstyle \textrm{p}} \end{aligned}$$
(90)
$$\begin{aligned}&\mathbb {C}^{\scriptscriptstyle \textrm{c}} \xi _y ({\mathcal {B}}^{\scriptscriptstyle \textrm{c}}(\textbf{y}))\textbf{n}_{\scriptscriptstyle \textrm{cp}}-\mathbb {C}^{\scriptscriptstyle \textrm{p}} \xi _y ({\mathcal {B}}^{\scriptscriptstyle \textrm{p}}(\textbf{y}))\textbf{n}_{\scriptscriptstyle \textrm{cp}}=(\varvec{\sigma }_{\scriptscriptstyle \textrm{0p}}^{L(0)}(\textbf{y})\nonumber \\&\quad -\varvec{\sigma }_{\scriptscriptstyle \textrm{0c}}^{L(0)}(\textbf{y}))\textbf{n}_{\scriptscriptstyle \textrm{cp}} \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{cp}} \end{aligned}$$
(91)
$$\begin{aligned}&{\mathcal {B}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})={\mathcal {B}}^{\scriptscriptstyle \textrm{p}}(\textbf{y}) \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{cp}} \end{aligned}$$
(92)
$$\begin{aligned}&(\mathbb {C}^{\scriptscriptstyle \textrm{c}} \xi _y ({\mathcal {B}}^{\scriptscriptstyle \textrm{c}}(\textbf{y})))\textbf{n}_{\scriptscriptstyle \textrm{c}}+ \varvec{\sigma }_{\scriptscriptstyle \textrm{0c}}^{L(0)}(\textbf{y})\textbf{n}_{\scriptscriptstyle \textrm{c}}=0 \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{c}} \end{aligned}$$
(93)
$$\begin{aligned}&(\mathbb {C}^{\scriptscriptstyle \textrm{p}} \xi _y({\mathcal {B}}^{\scriptscriptstyle \textrm{p}}(\textbf{y})))\textbf{n}_{\scriptscriptstyle \textrm{p}}+ \varvec{\sigma }_{\scriptscriptstyle \textrm{0p}}^{L(0)}(\textbf{y})\textbf{n}_{\scriptscriptstyle \textrm{p}}=0 \quad \text{ on } \quad \Gamma _{\scriptscriptstyle \textrm{p}}. \end{aligned}$$
(94)
These different decompositions and assumptions lead to us having well-posed cell problems that when supplemented with the periodic conditions on the cell boundary, an additional condition for uniqueness can be easily solved to determine the coefficients of the macroscale model. Here we have introduced a more complicated notation that emphasises the scale that the quantity has a dependence on and shows the decomposition of the pre-stress into the local and global components. Despite this increased complexity in the notation, the resulting model can actually be solved at a reduced computational cost than when assuming that the pre-stresses depend on both the microscale and macroscale in full. We actually find that the decomposition of the pre-stresses leads to a complete decoupling between the macroscale and the microscale problems eliminating the issues discussed in the previous bullet point of this subsection. Since we have this decoupling, we then solve the six elastic-type cell problems (
83a)–(
83f), the vector problems (
84a)–(
84f) and then either vector problems (
85a)–(
85f) and (
86a)–(
86f) or (
89)–(
94).