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Variational-asymptotic homogenization of thermoelastic periodic materials with
thermal relaxation

Deison Préve, Andrea Bacigalupo, Marco Paggi

##  Section A Variational-asymptotic homogenization scheme: additional
mathematical details

###  A.1. Asymptotic expansions of the microscopic field equations on the
Laplace domain

In order to land on the homogenized field equations set in Sec. 2, let us
consider the formulas coming from the chain rule of differentiation

$\displaystyle\dfrac{D}{Dx_{k}}$
$\displaystyle\hat{\boldsymbol{u}}\left(\textbf{x},\boldsymbol{\xi}=\dfrac{\textbf{x}}{\varepsilon}\right)=\left.\left[\dfrac{\partial\hat{u}_{h}(\textbf{x},\boldsymbol{\xi})}{\partial
x_{k}}+\dfrac{\partial\hat{u}_{h}(\textbf{x},\boldsymbol{\xi})}{\partial\xi_{k}}\dfrac{\partial\xi_{k}}{\partial
x_{k}}\right]\right|_{\boldsymbol{\xi}=\dfrac{\textbf{x}}{\varepsilon}}=\left.\left[\dfrac{\partial}{\partial
x_{k}}\hat{u}_{h}(\textbf{x},\boldsymbol{\xi})+\dfrac{1}{\varepsilon}\hat{u}_{h,k}\right]\right|_{\boldsymbol{\xi}=\dfrac{\textbf{x}}{\varepsilon}},$
(1bza) $\displaystyle\dfrac{D}{Dx_{j}}$
$\displaystyle\hat{\theta}\left(\textbf{x},\boldsymbol{\xi}=\dfrac{\textbf{x}}{\varepsilon}\right)=\left.\left[\dfrac{\partial\hat{\theta}(\textbf{x},\boldsymbol{\xi})}{\partial
x_{j}}+\dfrac{\partial\hat{\theta}(\textbf{x},\boldsymbol{\xi})}{\partial\xi_{j}}\dfrac{\partial\xi_{j}}{\partial
x_{j}}\right]\right|_{\boldsymbol{\xi}=\dfrac{\textbf{x}}{\varepsilon}}=\left.\left[\dfrac{\partial}{\partial
x_{j}}\hat{\theta}(\textbf{x},\boldsymbol{\xi})+\dfrac{1}{\varepsilon}\hat{\theta}_{,j}\right]\right|_{\boldsymbol{\xi}=\dfrac{\textbf{x}}{\varepsilon}},$
(1bzb)

which introduces the macroscopic derivatives $\partial\hat{u}_{h}/\partial
x_{k},$ $\partial\hat{\theta}/\partial x_{j},$ and the microscopic derivatives
$\hat{u}_{h,k},$ $\hat{\theta}_{,j}$ on the transformed Laplace domain.
Applying them to the asymptotic expansions $\leavevmode\nobreak\
\eqref{eqr:ulaplace}$ and $\leavevmode\nobreak\ \eqref{eqr:thetalaplace},$ one
leads to

$\displaystyle\dfrac{D}{Dx_{k}}\hat{\boldsymbol{u}}\left(\textbf{x},\boldsymbol{\xi}=\dfrac{\textbf{x}}{\varepsilon}\right)=\left.\left[\left(\dfrac{\partial\hat{u}^{(0)}_{h}}{\partial
x_{k}}+\varepsilon\dfrac{\partial\hat{u}^{(1)}_{h}}{\partial
x_{k}}+\varepsilon^{2}\dfrac{\partial\hat{u}^{(2)}_{h}}{\partial
x_{k}}+\cdots\right)+\dfrac{1}{\varepsilon}\left(\hat{u}^{0}_{h,k}+\varepsilon\hat{u}^{(1)}_{h,k}+\varepsilon^{2}\hat{u}^{(2)}_{h,k}+\cdots\right)\right]\right|_{\boldsymbol{\xi}=\dfrac{\textbf{x}}{\varepsilon}},$
(1caa)
$\displaystyle\dfrac{D}{Dx_{j}}\hat{\theta}\left(\textbf{x},\boldsymbol{\xi}=\dfrac{\textbf{x}}{\varepsilon}\right)=\left.\left[\left(\dfrac{\partial\hat{\theta}^{(0)}}{\partial
x_{j}}+\varepsilon\dfrac{\partial\hat{\theta}^{(1)}}{\partial
x_{j}}+\varepsilon^{2}\dfrac{\partial\hat{\theta}^{(2)}}{\partial
x_{j}}+\cdots\right)+\dfrac{1}{\varepsilon}\left(\hat{\theta}^{0}_{,j}+\varepsilon\hat{\theta}^{(1)}_{,j}+\varepsilon^{2}\hat{\theta}^{(2)}_{,j}+\cdots\right)\right]\right|_{\boldsymbol{\xi}=\dfrac{\textbf{x}}{\varepsilon}}.$
(1cab)

As mentioned in Sec. 2, rearranging properly the terms with equal powers of
$\varepsilon,$ and taking advantage of the derivative equations
$\leavevmode\nobreak\ \eqref{eq:gradu}$ and $\leavevmode\nobreak\
\eqref{eq:gradtheta},$ yield to the following asymptotic field equations

$\displaystyle\begin{split}&\left(\varepsilon^{-2}\left(C^{m}_{ijhk}\hat{u}^{(0)}_{h,k}\right)_{,j}+\varepsilon^{-1}\left[\left(C^{m}_{ijhk}\left(\dfrac{\partial\hat{u}^{(0)}_{h}}{\partial
x_{k}}+\hat{u}^{(1)}_{h,k}\right)\right)_{,j}+\dfrac{\partial}{\partial
x_{j}}\left(C^{m}_{ijhk}\hat{u}^{(0)}_{h,k}\right)-\left(\alpha^{m}_{ij}\hat{\theta}^{(0)}\right)_{,j}\right]+\right.\\\
&+\varepsilon^{0}\left[\left(C^{m}_{ijhk}\left(\dfrac{\partial\hat{u}^{(1)}_{h}}{\partial
x_{k}}+\hat{u}^{(2)}_{h,k}\right)\right)_{,j}+\dfrac{\partial}{\partial
x_{j}}\left(C^{m}_{ijhk}\left(\dfrac{\partial\hat{u}^{(0)}_{h}}{\partial
x_{k}}+\hat{u}^{(1)}_{h,k}\right)\right)-\left(\alpha^{m}_{ij}\hat{\theta}^{(1)}\right)_{,j}\right.+\\\
&-\left.\dfrac{\partial}{\partial
x_{j}}\left({\alpha}^{m}_{ij}\hat{\theta}^{(0)}\right)+\hat{b}_{i}-\rho^{m}s^{2}u^{(0)}_{h}\right]+\varepsilon\left[\left(C^{m}_{ijhk}\left(\dfrac{\partial\hat{u}^{(2)}_{h}}{\partial
x_{k}}+\hat{u}^{(3)}_{h,k}\right)\right)_{,j}+\right.\\\
&\left.\left.+\dfrac{\partial}{\partial
x_{j}}\left(C^{m}_{ijhk}\left(\dfrac{\partial\hat{u}^{(1)}_{h}}{\partial
x_{k}}+\hat{u}^{(2)}_{h,k}\right)\right)-\left.\left({\alpha}^{m}_{ij}\hat{\theta}^{(2)}\right)_{,j}-\dfrac{\partial}{\partial
x_{j}}\left({\alpha}^{m}_{ij}\hat{\theta}^{(1)}\right)-\rho^{m}s^{2}\hat{u}^{(1)}_{h}\right]+\mathcal{O}(\varepsilon^{2})\right)\right|_{\boldsymbol{\xi}=\frac{\textbf{x}}{\varepsilon}}=0,\end{split}$
(1cba)
$\displaystyle\begin{split}&\left(\varepsilon^{-2}\left({K}^{m}_{ij}\hat{\theta}^{(0)}_{,j}\right)_{,i}+\varepsilon^{-1}\left[\left({K}^{m}_{ij}\left(\dfrac{\partial\hat{\theta}^{(0)}}{\partial
x_{j}}+\hat{\theta}^{(1)}_{,j}\right)\right)_{,i}+\dfrac{\partial}{\partial
x_{i}}\left({K}^{m}_{ij}\hat{\theta}^{(0)}_{,j}\right)-\left(\alpha^{m}_{ij}s\hat{u}^{(0)}_{i,j}\right)\right]+\right.\\\
&+\varepsilon^{0}\left[\left({K}^{m}_{ij}\left(\dfrac{\partial\hat{\theta}^{(1)}}{\partial
x_{j}}+\hat{\theta}^{(2)}_{,j}\right)\right)_{,i}+\dfrac{\partial}{\partial
x_{i}}\left({K}^{m}_{ij}\left(\dfrac{\partial\hat{\theta}^{(0)}}{\partial
x_{j}}+\hat{\theta}^{(1)}_{,j}\right)\right)-\alpha^{m}_{ij}s\left(\dfrac{\partial\hat{u}^{(0)}_{i}}{\partial
x_{j}}+\hat{u}^{(1)}_{i,j}\right)\right.+\\\
&=\left.\hat{r}-p^{m}s\hat{\theta}^{(0)}\right]+\varepsilon\left[\left({K}^{m}_{ij}\left(\dfrac{\partial\hat{\theta}^{(2)}}{\partial
x_{j}}+\hat{\theta}^{(3)}_{,j}\right)\right)_{,i}+\dfrac{\partial}{\partial
x_{i}}\left({K}^{m}_{ij}\left(\dfrac{\partial\hat{\theta}^{(1)}}{\partial
x_{j}}+\hat{\theta}^{(2)}_{,j}\right)\right)+\right.\\\
&\left.\left.\left.-\alpha^{m}_{ij}s\left(\dfrac{\partial\hat{u}^{(1)}_{i}}{\partial
x_{j}}+\hat{u}^{(2)}_{i,j}\right)-p^{m}s\hat{\theta}^{(1)}\right]+\left.\mathcal{O}(\varepsilon^{2})\right]\right)\right|_{\boldsymbol{\xi}=\frac{\textbf{x}}{\varepsilon}}=0,\end{split}$
(1cbb)

where the equation $\leavevmode\nobreak\ \eqref{eqn:uhomogenized}$ is the
homogenized micro-displacement field, and the equation $\leavevmode\nobreak\
\eqref{eqn:thetahomogenized}$ is the homogenized micro-temperature field.

Recalling the interface conditions
$(\ref{eq:InterfaceConditionMicroDisplacement})-(\ref{eq:InterfaceConditionMicrotemperature1}),$
they may be rewritten in terms of the components, thus

$\displaystyle\left.\left[\left[\hat{u}_{h}\right]\right]\right|_{\mathbf{x}\in\Sigma}=0,$
(1cca)
$\displaystyle\left.\left[\left[\left(C^{m}_{ijhk}\dfrac{D\hat{u}_{h}}{Dx_{k}}-\alpha^{m}_{ij}\hat{\theta}\right)n_{j}\right]\right]\right|_{\mathbf{x}\in\Sigma}=0,$
(1ccb)
$\displaystyle\left.\left[\left[\hat{\theta}\right]\right]\right|_{\mathbf{x}\in\Sigma}=0,$
(1ccc)
$\displaystyle\left.\left[\left[-K^{m}_{ij}(s)\dfrac{D\hat{\theta}}{Dx_{j}}n_{i}\right]\right]\right|_{\mathbf{x}\in\Sigma}=0.$
(1ccd)

Also, once the micro fields
$\hat{\boldsymbol{u}}(\mathbf{x},\boldsymbol{\xi},s),$ and
$\hat{\theta}(\mathbf{x},\boldsymbol{\xi},s)$ are supposed to be
$\mathcal{Q}$-periodic regular functions of the variable $\mathbf{x},$ it is
then possible to write the interface conditions
$(\ref{eq:InterfaceConditionMicroDisplacementComponents})$ to
$(\ref{eq:InterfaceConditionMicrotemperatureComponents1})$ in terms of the
fast variable $\boldsymbol{\xi}.$ [28]. Therefore the asymptotic expansions
$(\ref{eqn:u})$ and $(\ref{eqn:theta}),$ among the derivative formulae
$\leavevmode\nobreak\ \eqref{eq:gradu}$ and $\leavevmode\nobreak\
\eqref{eq:gradtheta},$ and the interface conditions
$(\ref{eq:InterfaceConditionMicroDisplacementComponents})$ and
$(\ref{eq:InterfaceConditionMicroDisplacementComponents1})$ are rewritten in
terms of $\varepsilon$ as

$\displaystyle\begin{split}&\left.\left[\left[\hat{u}_{h}^{(0)}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}+\varepsilon\left.\left[\left[\hat{u}_{h}^{(1)}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}+\varepsilon^{2}\left.\left[\left[\hat{u}_{h}^{(2)}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}+\mathcal{O}(\varepsilon^{3})=0,\\\
&\dfrac{1}{\varepsilon}\left.\left[\left[\left(C_{ijhk}^{m}\,\hat{u}_{h,k}^{(0)}\right)n_{j}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}+\varepsilon^{0}\left.\left[\left[\left(C_{ijhk}^{m}\left(\dfrac{\partial\hat{u}_{h}^{(0)}}{\partial
x_{k}}+\hat{u}_{h,k}^{(1)}\right)-\alpha_{ij}^{m}\,\hat{\theta}^{(0)}\right)n_{j}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}+\\\
&+\varepsilon\left.\left[\left[\left(C_{ijhk}^{m}\left(\dfrac{\partial\hat{u}_{h}^{(1)}}{\partial
x_{k}}+\hat{u}_{h,k}^{(2)}\right)-\alpha_{ij}^{m}\,\hat{\theta}^{(1)}\right)n_{j}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}+\\\
&+\varepsilon^{2}\left.\left[\left[\left(C_{ijhk}^{m}\left(\dfrac{\partial\hat{u}_{h}^{(2)}}{\partial
x_{k}}+\hat{u}_{h,k}^{(3)}\right)-\alpha_{ij}^{m}\,\hat{\theta}^{(2)}\right)n_{j}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}+\mathcal{O}(\varepsilon^{3})=0,\end{split}$
(1cd)

and the interface conditions
$(\ref{eq:InterfaceConditionMicrotemperatureComponents})$ and
$(\ref{eq:InterfaceConditionMicrotemperatureComponents1})$ become

$\displaystyle\begin{split}&\left.\left[\left[\hat{\theta}^{(0)}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}+\varepsilon\left.\left[\left[\hat{\theta}^{(1)}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}+\varepsilon^{2}\left.\left[\left[\hat{\theta}^{(2)}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}+\mathcal{O}(\varepsilon^{3})=0,\\\
&\dfrac{1}{\varepsilon}\left.\left[\left[\left(K_{ij}^{m}\,\hat{\theta}_{,j}^{(0)}\right)n_{i}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}+\varepsilon^{0}\left.\left[\left[\left(K_{ij}^{m}\left(\dfrac{\partial\hat{\theta}^{(0)}}{\partial
x_{j}}+\hat{\theta}_{,j}^{(1)}\right)\right)n_{i}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}+\\\
&+\varepsilon\left.\left[\left[\left(K_{ij}^{m}\left(\dfrac{\partial\hat{\theta}^{(1)}}{\partial
x_{j}}+\hat{\theta}_{,j}^{(2)}\right)\right)n_{i}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}+\varepsilon^{2}\left.\left[\left[\left(K_{ij}^{m}\left(\dfrac{\partial\hat{\theta}^{(2)}}{\partial
x_{j}}+\hat{\theta}_{,j}^{(3)}\right)\right)n_{i}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}+\mathcal{O}(\varepsilon^{3})=0,\end{split}$
(1ce)

where $\Sigma_{1}$ denotes the interface between two different phases in the
unit cell $\mathcal{Q}$.

###  A.2. Recursive differential problems and their solutions

After some algebraic rearrangements on the asymptotic field Eqs.
$\eqref{eqn:uhomogenized}$ and $\eqref{eqn:thetahomogenized},$ a hierarchical
set of recursive partial differential problems, in terms of the sensitivities
${\hat{u}}^{(l)}_{h},$ and $\hat{\theta}^{(l)},$ has arisen. For this matter,
at the order $\varepsilon^{-2}$, the differential problems coming from Eqs.
$\eqref{eqn:uhomogenized}$ and $\leavevmode\nobreak\
\eqref{eqn:thetahomogenized}$ are, respectively,

$\displaystyle\left(C^{m}_{ijhk}\boldsymbol{\hat{u}}^{(0)}_{h,k}\right)_{,j}=f^{(0)}_{i}(\textbf{x}),$
(1cfa)
$\displaystyle\left(K^{m}_{ij}\hat{\theta}^{(0)}_{,j}\right)_{,i}=g^{(0)}(\textbf{x}),$
(1cfb)

with interface conditions

$\displaystyle\left.\left[\left[\hat{u}_{h}^{(0)}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}=0,$
(1cga)
$\displaystyle\left.\left[\left[\left(C_{ijhk}^{m}\,\hat{u}_{h,k}^{(0)}\right)n_{j}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}=0,$
(1cgb)
$\displaystyle\left.\left[\left[\hat{\theta}^{(0)}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}=0,$
(1cgc)
$\displaystyle\left.\left[\left[\left(K_{ij}^{m}\hat{\theta}_{,j}^{(0)}\right)n_{i}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}=0.$
(1cgd)

As matter of fact, since we are searching for solutions in the class of
$\mathcal{Q}$-periodic solutions $\hat{u}^{(0)}_{h}$ and $\hat{\theta}^{(0)},$
it can be checked [28, 29] that there exists a unique solution of the
equations $\leavevmode\nobreak\ \eqref{eq:epsilon-2f}$ and
$\leavevmode\nobreak\ \eqref{eq:epsilon-2g},$ up to a constant, more
specifically, since they are elliptic differential equations in the divergence
forms, with vanishing mean values of the source terms over the unit cell
$\mathcal{Q},$ implies the existence of a $\mathcal{Q}$-periodic regular
solution. Such result gives rise to the also known as solvability condition
for these differential problems, implying that the source terms are
$f^{(0)}_{i}(\textbf{x})=0$ and $g^{(0)}(\textbf{x})=0,$ therefore the
differential problems $\leavevmode\nobreak\ \eqref{eq:epsilon-2f}$ and
$\leavevmode\nobreak\ \eqref{eq:epsilon-2g},$ take the form

$\displaystyle\left(C^{m}_{ijhk}\hat{u}^{(0)}_{h,k}\right)_{,j}=0,$ (1cha)
$\displaystyle\left({K}^{m}_{ij}\hat{\theta}^{(0)}_{,j}\right)_{,i}=0.$ (1chb)

Hence, the solution of the first term of the macroscopic displacement
expansion field transformed is given by

$\hat{u}^{(0)}_{h}(\textbf{x},\boldsymbol{\xi},s)=\hat{U}^{M}_{h}(\textbf{x},s),$
(1ci)

and the solution for the first term of the macroscopic temperature expansion
field is given by

$\hat{\theta}^{(0)}(\textbf{x},\boldsymbol{\xi},s)=\hat{\Theta}^{M}(\textbf{x},s).$
(1cj)

It is important to notice here that both solutions
$\hat{U}^{M}_{h}(\textbf{x},s)$ and $\hat{\Theta}^{M}(\textbf{x},s)$ are no
longer dependent on the fast variable.

Proceeding with the succeeding terms related to $\varepsilon^{-1}$ in Eqs.,
$\leavevmode\nobreak\ \eqref{eqn:uhomogenized}$ and $\leavevmode\nobreak\
\eqref{eqn:thetahomogenized},$ and using recursively the two solutions above
obtained, it follows that

$\displaystyle\left(C^{m}_{ijhk}\hat{u}^{(1)}_{h,k}\right)_{,j}+C^{m}_{ijhk,j}\dfrac{\partial\hat{U}^{M}_{h}}{\partial
x_{k}}-{\alpha}^{m}_{ij,j}\hat{\Theta}^{M}=f^{(1)}_{i}(\textbf{x}),$ (1cka)
$\displaystyle\left({K}^{m}_{ij}\hat{\theta}^{(1)}_{,j}\right)_{,i}+{K}^{m}_{ij,i}\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{j}}=g^{(1)}(\textbf{x}),$ (1ckb)

since $\hat{U}^{M}_{h,k}=0$ and $\hat{\Theta}^{M}_{,j}=0.$ The interface
conditions are, respectively,

$\displaystyle\left.\left[\left[\hat{u}_{h}^{(1)}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}=0,$
(1cla)
$\displaystyle\left.\left[\left[\left(C_{ijhk}^{m}\left(\dfrac{\partial\hat{U}_{h}^{M}}{\partial
x_{k}}+\hat{u}_{h,k}^{(1)}\right)-\alpha_{ij}^{m}\,\hat{\Theta}^{M}\right)n_{j}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}=0,$
(1clb)
$\displaystyle\left.\left[\left[\hat{\theta}^{(1)}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}=0,$
(1clc)
$\displaystyle\left.\left[\left[\left(K_{ij}^{m}\left(\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{j}}+\hat{\theta}_{,j}^{(1)}\right)\right)n_{i}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}=0.$
(1cld)

Likewise, the solvability condition on the class of $\mathcal{Q}$-periodic
functions guarantees that

$\displaystyle f^{(1)}_{i}(\textbf{x})=$ $\displaystyle\langle
f^{(1)}_{i}(\textbf{x})\rangle=\langle
C^{m}_{ijhk,j}\rangle\dfrac{\partial\hat{U}^{M}_{h}}{\partial
x_{k}}-\langle{\alpha}^{m}_{ij,j}\rangle\hat{\Theta}^{M},$ (1cma)
$\displaystyle g^{(1)}(\textbf{x})=$ $\displaystyle\langle
g^{(1)}(\textbf{x})\rangle=\langle{K}^{m}_{ij,i}\rangle\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{j}},$ (1cmb)

where
$\langle(\cdot)\rangle=\frac{1}{|\mathcal{Q}|}\int_{\mathcal{Q}}(\cdot)d\boldsymbol{\xi}$
and $|\mathcal{Q}|=\delta\gamma$ gives the mean value over the unit cell
$\mathcal{Q}.$ Moreover, the $\mathcal{Q}$-periodicity of the components
$C^{m}_{ijhk},$ $\alpha^{m}_{ij}$ and ${K}^{m}_{ij},$ along with the
divergence theorem, entail both $f^{(1)}_{i}(\textbf{x})=0$ and
$g^{(1)}(\textbf{x})=0,$ thus the differential problems

$\displaystyle\left(C^{m}_{ijhk}\hat{u}^{(1)}_{h,k}\right)_{,j}+C^{m}_{ijhk,j}\dfrac{\partial\hat{U}^{M}_{h}}{\partial
x_{k}}-{\alpha}^{m}_{ij,j}\hat{\Theta}^{M}=0,\quad\forall\dfrac{\partial\hat{U}^{M}_{h}}{\partial
x_{k}},\hat{\Theta}^{M}$ (1cna)
$\displaystyle\left({K}^{m}_{ij}\hat{\theta}^{(1)}_{,j}\right)_{,i}+{K}^{m}_{ij,i}\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{j}}=0,\quad\forall\dfrac{\partial\hat{\Theta}^{M}}{\partial x_{j}}$ (1cnb)

have the following solutions, respectively

$\displaystyle\hat{u}^{(1)}_{h}(\textbf{x},\boldsymbol{\xi},s)=$
$\displaystyle
N^{(1,0)}_{hpq_{1}}(\boldsymbol{\xi})\dfrac{\partial\hat{U}^{M}_{p}}{\partial
x_{q_{1}}}+\tilde{N}^{(1,0)}_{h}(\boldsymbol{\xi})\hat{\Theta}^{M},$ (1coa)
$\displaystyle\hat{\theta}^{(1)}(\textbf{x},\boldsymbol{\xi},s)=$
$\displaystyle
M^{(1,0)}_{q_{1}}(\boldsymbol{\xi},s)\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{q_{1}}},$ (1cob)

where $N^{(1,0)}_{hpq_{1}},$ $\tilde{N}^{(1,0)}_{h}$ and $M^{(1,0)}_{q_{1}},$
are the perturbation functions, which each of them depends on the fast
variable $\boldsymbol{\xi}.$ On Sec. $\ref{Sec::CellsProblems},$ in order to
impose uniqueness of the homogenized solutions $\hat{u}_{h}$ and
$\hat{\theta},$ the perturbation functions must be supposed to have zero mean
value over the unit cell $\mathcal{Q}$ and so $N^{(1,0)}_{hpq_{1}},$
$\tilde{N}^{(1,0)}_{h}$ and $M^{(1,0)}_{q_{1}}$ comply with the normalization
condition, which means that

$\displaystyle\langle
N^{(1,0)}_{hpq_{1}}\rangle=\dfrac{1}{|\mathcal{Q}|}\int_{\mathcal{Q}}N^{(1,0)}_{hpq_{1}}(\boldsymbol{\xi})d\boldsymbol{\xi}=0,$
(1cpa)
$\displaystyle\langle\tilde{N}^{(1,0)}_{h}\rangle=\dfrac{1}{|\mathcal{Q}|}\int_{\mathcal{Q}}\tilde{N}^{(1,0)}_{h}(\boldsymbol{\xi})d\boldsymbol{\xi}=0,$
(1cpb) $\displaystyle\langle
M^{(1,0)}_{q_{1}}\rangle=\dfrac{1}{|\mathcal{Q}|}\int_{\mathcal{Q}}M^{(1,0)}_{q_{1}}(\boldsymbol{\xi},s)d\boldsymbol{\xi}=0.$
(1cpc)

At order $\varepsilon^{0},$ the differential problems are

$\displaystyle\begin{split}&\left(C^{m}_{ijhk}\left(\dfrac{\partial\hat{u}^{(1)}_{h}}{\partial
x_{k}}+\hat{u}^{(2)}_{h,k}\right)\right)_{,j}+\dfrac{\partial}{\partial
x_{j}}\left(C^{m}_{ijhk}\left(\dfrac{\partial\hat{u}^{(0)}_{h}}{\partial
x_{k}}+\hat{u}^{(1)}_{h,k}\right)\right)-\left(\alpha^{m}_{ij}\hat{\theta}^{(1)}\right)_{,j}-\dfrac{\partial}{\partial
x_{j}}\left(\alpha^{m}_{ij}\hat{\theta}^{(0)}\right)+\\\
&-\rho^{m}s^{2}\hat{u}^{(0)}_{i}=f^{(2)}_{i}(\textbf{x}),\end{split}$ (1cqa)
$\displaystyle\begin{split}&\left({K}^{m}_{ij}\left(\dfrac{\partial\hat{\theta}^{(1)}}{\partial
x_{j}}+\hat{\theta}^{(2)}_{,j}\right)\right)_{,i}+\dfrac{\partial}{\partial
x_{i}}\left({K}^{m}_{ij}\left(\dfrac{\partial\hat{\theta}^{(0)}}{\partial
x_{j}}+\hat{\theta}^{(1)}_{,j}\right)\right)-\alpha^{m}_{ij}\left(\dfrac{\partial\hat{u}^{(0)}_{i}}{\partial
x_{j}}+\hat{u}^{(1)}_{i,j}\right)+\\\
&-\rho^{m}s\hat{\theta}^{(0)}=g^{(2)}(\textbf{x}).\end{split}$ (1cqb)

Considering the solutions $\leavevmode\nobreak\
\eqref{eqn:solutiondisplac-2},$ $\leavevmode\nobreak\
\eqref{eqn:solutiondisplac-1}$ at order $\varepsilon^{-2}$ and
$\varepsilon^{-1},$ respectively, from the displacement expansion, and the
solutions $\leavevmode\nobreak\ \eqref{eqn:solutiontemp-2},$
$\leavevmode\nobreak\ \eqref{eqn:solutiontemp-1},$ at order $\varepsilon^{-2}$
and $\varepsilon^{-1}$, respectively, from the temperature expansion, the
differential problems $\leavevmode\nobreak\ \eqref{eq:epsilon0f}$ and
$\leavevmode\nobreak\ \eqref{eq:epsilon0g}$ are turned into

$\displaystyle\begin{split}&\left(C^{m}_{ijhk}\hat{u}^{(2)}_{h,k}\right)_{,j}+\left(\left(C^{m}_{ijhk}N^{(1,0)}_{hpq_{1}}\right)_{,j}+C^{m}_{iq_{1}pk}+\left(C^{m}_{ikhj}N^{(1,0)}_{hpq_{1},j}\right)\right)\dfrac{\partial^{2}\hat{U}^{M}_{p}}{\partial
x_{q_{1}}\partial x_{k}}+\\\
&+\left(\left(C^{m}_{ijhk}\tilde{N}^{(1,0)}_{h}\right)_{,j}+C^{m}_{ikhj}\tilde{N}^{(1,0)}_{h,j}-\left(\alpha^{m}_{ij}M^{(1,0)}_{k}\right)_{,j}-\alpha^{m}_{ik}\right)\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{k}}-\rho^{m}s^{2}\hat{U}^{M}_{i}=f^{(2)}_{i}(\textbf{x}),\\\ \end{split}$
(1cra)
$\displaystyle\begin{split}&\left({K}^{m}_{ij}\hat{\theta}^{(2)}_{,j}\right)_{,i}+\left(\left({K}^{m}_{ij}M^{(1,0)}_{q_{1}}\right)_{,i}+{K}^{m}_{q_{1}j}+\left({K}^{m}_{ji}M^{(1,0)}_{q_{1},i}\right)\right)\dfrac{\partial^{2}\hat{\Theta}^{M}}{\partial
x_{q_{1}}\partial x_{j}}+\\\
&-\left(\alpha^{m}_{ij}N^{(1,0)}_{ipq_{1},j}+\alpha^{m}_{pq_{1}}\right)s\dfrac{\partial\hat{U}^{M}_{p}}{\partial
x_{q_{1}}}-\left(\alpha^{m}_{ij}\tilde{N}^{(1,0)}_{i,j}+p^{m}\right)s\hat{\Theta}^{M}=g^{(2)}(\textbf{x}),\end{split}$
(1crb)

with interface conditions

$\displaystyle\begin{split}&\left.\left[\left[\hat{u}_{h}^{(2)}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}=0,\\\
&\left.\left[\left[\left(C_{ijhk}^{m}\left(N^{(1,0)}_{hpq_{1},j}\dfrac{\partial^{2}\hat{U}^{M}_{p}}{\partial
x_{q_{1}}\partial
x_{k}}+\tilde{N}^{(1,0)}_{h}\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{k}}+\hat{u}_{h,k}^{(2)}\right)-\alpha_{ij}^{m}M^{(1,0)}_{q_{1}}\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{q_{1}}}\right)n_{j}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}=0,\end{split}$
(1csa)
$\displaystyle\begin{split}&\left.\left[\left[\hat{\theta}^{(2)}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}=0,\\\
&\left.\left[\left[\left(K_{ij}^{m}\left(M^{(1,0)}_{q_{1}}\dfrac{\partial^{2}\hat{\Theta}^{M}}{\partial
x_{q_{1}}\partial
x_{j}}+\hat{\theta}_{,j}^{(2)}\right)\right)n_{i}\right]\right]\right|_{\boldsymbol{\xi}\in\Sigma_{1}}=0.\end{split}$
(1csb)

Simetrizing the terms with the derivative of second order on the two
differential problems $\leavevmode\nobreak\ \eqref{eq:epsilon0fsubst}$ and
$\leavevmode\nobreak\ \eqref{eq:epsilon0gsubst},$ and once again, from the
solvability condition of differential problem on the class of
$\mathcal{Q}$-periodic functions and the divergence theorem lead to,
respectively,

$\displaystyle\begin{split}f^{(2)}_{i}(\textbf{x})=&\dfrac{1}{2}\left\langle
C^{m}_{iq_{1}pk}+C^{m}_{ikhj}N^{(1,0)}_{hpq_{1},j}+C^{m}_{ikpq_{1}}+C^{m}_{iq_{1}hj}N^{(1,0)}_{hpk,j}\right\rangle\dfrac{\partial^{2}\hat{U}^{M}_{p}}{\partial
x_{q_{1}}\partial x_{h}}+\\\ +&\left\langle
C^{m}_{ikhj}\tilde{N}^{(1,0)}_{h,j}-\alpha^{m}_{ik}\right\rangle\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{k}}-\Big{\langle}\rho^{m}\Big{\rangle}s^{2}\hat{U}^{M}_{i},\end{split}$
(1cta)
$\displaystyle\begin{split}g^{(2)}(\textbf{x})=&\dfrac{1}{2}\left\langle{K}^{m}_{q_{1}j}+{K}^{m}_{ji}M^{(1,0)}_{q_{1},i}+{K}^{m}_{jq_{1}}+{K}^{m}_{q_{1}i}M^{(1,0)}_{j,i}\right\rangle\dfrac{\partial^{2}\hat{\Theta}^{M}}{\partial
x_{q_{1}}\partial x_{j}}-\\\
+&\left\langle\alpha^{m}_{ij}N^{(1,0)}_{ipq_{1},j}+\alpha^{m}_{pq_{1}}\right\rangle
s\dfrac{\partial\hat{U}^{M}_{p}}{\partial
x_{q1}}-\left\langle\alpha^{m}_{ij}\tilde{N}^{(1,0)}_{i,j}+p^{m}\right\rangle
s\hat{\Theta}^{M}.\end{split}$ (1ctb)

Note that, even though the interface conditions
$(\ref{eq:InterfaceConditionsDisplacOrder0})$ and
$(\ref{eq:InterfaceConditionsTermicaOrder0})$ are simetrized as well, they are
kept hidden on this regard.

Consequently, the solution for each $\hat{U}^{M}_{p},$ $\hat{\Theta}^{M}$ and
their derivatives of the differential problems above, at the order
$\varepsilon^{0},$ are

$\hat{u}^{(2)}_{h}(\textbf{x},\boldsymbol{\xi},s)=N^{(2,0)}_{hpq_{1}q_{2}}(\boldsymbol{\xi})\dfrac{\partial^{2}\hat{U}^{M}_{p}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}+\tilde{N}^{(2,1)}_{hq_{1}}(\boldsymbol{\xi})\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{q_{1}}}+s^{2}N^{(2,2)}_{hq_{1}}(\boldsymbol{\xi})\hat{U}^{M}_{p},$ (1cua)
$\hat{\theta}^{(2)}(\textbf{x},\boldsymbol{\xi},s)=M^{(2,0)}_{q_{1}q_{2}}(\boldsymbol{\xi})\dfrac{\partial^{2}\hat{\Theta}^{M}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}+s\tilde{M}^{(2,1)}_{pq_{1}}(\boldsymbol{\xi})\dfrac{\partial\hat{U}^{M}_{p}}{\partial
x_{q_{1}}}+sM^{(2,1)}(\boldsymbol{\xi})\hat{\Theta}^{M},$ (1cub)

where $N^{(2,0)}_{hpq_{1}q_{2}},$ $\tilde{N}^{(2,1)}_{hq_{1}},$
$N^{(2,2)}_{hq_{1}},$ $M^{(2,0)}_{q_{1}q_{2}},$ $\tilde{M}^{(2,1)}_{pq_{1}}$
and $M^{(2,1)}$ are the perturbation functions depending on the parameters
$\boldsymbol{\xi}$ and $s.$

###  A.3. Average field equation of infinite order

Recalling the structure of the field equations $(\ref{fsourceterms})$ and
$(\ref{gsourceterms})$ and substituting each $f_{i}^{(l)}(\mathbf{x})$ and
$g^{(l)}(\mathbf{x}),$ obtained from each relative recursive differential
problem into them, one derives the average field equations of infinite order

$\displaystyle
n^{(2,0)}_{ipq_{1}q_{2}}\dfrac{\partial^{2}\hat{U}_{h}^{M}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}+\tilde{n}^{(2,0)}_{iq_{1}}\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{q_{1}}}-n_{ip}^{(2,2)}s^{2}\hat{U}_{h}^{M}+\mathcal{O}\left({\bf{\varepsilon}}\right)+\hat{b}_{i}=0,$
(1cva) $\displaystyle
w^{(2,0)}_{q_{1}q_{2}}\dfrac{\partial^{2}\hat{\Theta}^{M}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}-\tilde{w}^{(2,1)}_{pq_{1}}s\dfrac{\partial\hat{U}^{M}_{h}}{\partial
x_{q_{1}}}-w^{(2,1)}s\hat{\Theta}^{M}+\mathcal{O}\left({\bf{\varepsilon}}\right)+\hat{r}=0,$
(1cvb)

where the constant global constitutive tensors factors are

$\displaystyle n^{(2,0)}_{ipq_{1}q_{2}}=\dfrac{1}{2}\left\langle
C_{iq_{1}pq_{2}}^{m}+C_{iq_{2}hj}^{m}N^{(1,0)}_{hpq_{1},j}+C_{iq_{2}pq_{1}}^{m}+C_{iq_{1}hj}^{m}N^{(1,0)}_{hpq_{2},j}\right\rangle,$
(1cwa) $\displaystyle\tilde{n}^{(2,0)}_{iq_{1}}=\left\langle
C_{iq_{1}hj}^{m}\tilde{N}^{(1,0)}_{h,j}-\alpha^{m}_{iq_{1}}\right\rangle,$
(1cwb) $\displaystyle
n_{ip}^{(2,2)}=\left\langle\rho^{m}\right\rangle\delta_{ip},$ (1cwc)
$\displaystyle w^{(2,0)}_{q_{1}q_{2}}=\dfrac{1}{2}\left\langle
K^{m}_{q_{1}q_{2}}+K^{m}_{q_{2}i}M^{(1,0)}_{q_{1},i}+K^{m}_{q_{2}q_{1}}+K^{m}_{q_{1}i}M^{(1,0)}_{q_{2},i}\right\rangle,$
(1cwd)
$\displaystyle\tilde{w}^{(2,1)}_{pq_{1}}=\left\langle\alpha_{ij}^{m}N^{(1,0)}_{ipq_{1},j}+\alpha^{m}_{pq_{1}}\right\rangle,$
(1cwe) $\displaystyle
w^{(2,1)}=\left\langle\alpha_{ij}^{m}\tilde{N}^{(1,0)}_{i,j}-p^{m}_{iq_{1}}\right\rangle.$
(1cwf)

In the interest of deriving explicit solutions of the equations
$(\ref{eq:averagefieldu})$ and $(\ref{eq:averagefieldtheta}),$ the macroscopic
variables $\hat{U}^{M}_{h}$ and $\hat{\Theta}^{M}$ are asymptotically expanded
as follows

$\displaystyle\hat{U}^{M}_{h}\left({{\mathbf{x}},s}\right)=\sum\limits_{j=0}^{+\infty}{{\varepsilon^{j}}}{\hat{U}_{h}^{(j)}}\left({{\mathbf{x}},s}\right),$
(1cxa)
$\displaystyle\hat{\Theta}^{M}\left({{\mathbf{x}},s}\right)=\sum\limits_{j=0}^{+\infty}{{\varepsilon^{j}}}{\hat{\Theta}^{(j)}}\left({{\mathbf{x}},s}\right).$
(1cxb)

By plugging the solutions $(\ref{seriesU})$ and $(\ref{seriesTheta}),$ into
the equations $(\ref{eq:averagefieldu})$ and $(\ref{eq:averagefieldtheta}),$
the asymptotic expansions of the average field equations of infinite order
become

$\displaystyle\begin{split}&n^{(2,0)}_{ipq_{1}q_{2}}\left(\dfrac{\partial^{2}\hat{U}_{h}^{(0)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}+\varepsilon\dfrac{\partial^{2}\hat{U}_{h}^{(1)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}+\varepsilon^{2}\dfrac{\partial^{2}\hat{U}_{h}^{(2)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}+\cdots\right)+\tilde{n}^{(2,0)}_{iq_{1}}\left(\dfrac{\partial\hat{\Theta}^{(0)}}{\partial
x_{q_{1}}}+\varepsilon\dfrac{\partial\hat{\Theta}^{(1)}}{\partial
x_{q_{1}}}+\varepsilon^{2}\dfrac{\partial\hat{\Theta}^{(2)}}{\partial
x_{q_{1}}}+\cdots\right)+\\\
&-n_{ip}^{(2,2)}s^{2}\left(\hat{U}_{h}^{(0)}+\varepsilon\hat{U}_{h}^{(1)}+\varepsilon^{2}\hat{U}_{h}^{(2)}\right)+\cdots+\hat{b}_{i}=0,\end{split}$
(1cya)
$\displaystyle\begin{split}&w^{(2,0)}_{q_{1}q_{2}}\left(\dfrac{\partial^{2}\hat{\Theta}^{(0)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}+\varepsilon\dfrac{\partial^{2}\hat{\Theta}^{(1)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}+\varepsilon^{2}\dfrac{\partial^{2}\hat{\Theta}^{(2)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}+\cdots\right)-\tilde{w}^{(2,1)}_{pq_{1}}s\left(\dfrac{\partial\hat{U}_{h}^{(0)}}{\partial
x_{q_{1}}}+\varepsilon\dfrac{\partial\hat{U}_{h}^{(1)}}{\partial
x_{q_{1}}}+\varepsilon^{2}\dfrac{\partial\hat{U}_{h}^{(2)}}{\partial
x_{q_{1}}}+\cdots\right)+\\\
&-w^{(2,1)}s\left(\hat{\Theta}^{(0)}+\varepsilon\hat{\Theta}^{(1)}+\varepsilon^{2}\hat{\Theta}^{(2)}\right)+\cdots+\hat{r}=0.\end{split}$
(1cyb)

Rearranging the terms at the same order of $\varepsilon$ in
$(\ref{eq:averagefieldassymptoticelastic})$ and
$(\ref{eq:averagefieldassymptoticthermal}),$ an infinite set of macroscopic
hierarchical differential problems expressed in terms of the sensitivities
$\hat{U}^{(j)}_{h}$ and $\hat{\Theta}^{(j)}$ of both macroscopic displacement
$\hat{U}_{h}^{M}$ and macroscopic temperature $\hat{\Theta}^{M}$ fields can be
determined. Thus,

$\displaystyle\begin{split}&\varepsilon^{0}\left(n^{(2,0)}_{ipq_{1}q_{2}}\dfrac{\partial^{2}\hat{U}_{h}^{(0)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}+\tilde{n}^{(2,0)}_{iq_{1}}\dfrac{\partial\hat{\Theta}^{(0)}}{\partial
x_{q_{1}}}-n_{ip}^{(2,2)}s^{2}\hat{U}_{h}^{(0)}+\hat{b}_{i}\right)+\\\
&+\varepsilon\left(n^{(2,0)}_{ipq_{1}q_{2}}\dfrac{\partial^{2}\hat{U}_{h}^{(1)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}+\tilde{n}^{(2,0)}_{iq_{1}}\dfrac{\partial\hat{\Theta}^{(1)}}{\partial
x_{q_{1}}}-n_{ip}^{(2,2)}s^{2}\hat{U}_{h}^{(1)}+s_{i}^{(1)}(\textbf{x},s)\right)+\\\
&+\varepsilon^{2}\left(n^{(2,0)}_{ipq_{1}q_{2}}\dfrac{\partial^{2}\hat{U}_{h}^{(2)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}+\tilde{n}^{(2,0)}_{iq_{1}}\dfrac{\partial\hat{\Theta}^{(2)}}{\partial
x_{q_{1}}}-n_{ip}^{(2,2)}s^{2}\hat{U}_{h}^{(2)}+s_{i}^{(2)}(\textbf{x},s)\right)+\mathcal{O}(\varepsilon^{3})=0,\end{split}$
(1cza)
$\displaystyle\begin{split}&\varepsilon^{0}\left(w^{(2,0)}_{q_{1}q_{2}}\dfrac{\partial^{2}\hat{\Theta}^{(0)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}-\tilde{w}^{(2,1)}_{pq_{1}}s\dfrac{\partial\hat{U}_{h}^{(0)}}{\partial
x_{q_{1}}}-w^{(2,1)}s\hat{\Theta}^{(0)}+\hat{r}\right)+\\\
&+\varepsilon\left(w^{(2,0)}_{q_{1}q_{2}}\dfrac{\partial^{2}\hat{\Theta}^{(1)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}-\tilde{w}^{(2,1)}_{pq_{1}}s\dfrac{\partial\hat{U}_{h}^{(1)}}{\partial
x_{q_{1}}}-w^{(2,1)}s\hat{\Theta}^{(1)}+\upsilon^{(1)}(\textbf{x},s)\right)+\\\
&+\varepsilon^{2}\left(w^{(2,0)}_{q_{1}q_{2}}\dfrac{\partial^{2}\hat{\Theta}^{(2)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}-\tilde{w}^{(2,1)}_{pq_{1}}s\dfrac{\partial\hat{U}_{h}^{(2)}}{\partial
x_{q_{1}}}-w^{(2,1)}s\hat{\Theta}^{(2)}+\upsilon^{(2)}(\textbf{x},s)\right)+\mathcal{O}(\varepsilon^{3})=0,\end{split}$
(1czb)

Therefore, the recursive problem at the macroscopic scale at order
$\varepsilon^{0}$ reads

$\displaystyle
n^{(2,0)}_{ipq_{1}q_{2}}\dfrac{\partial^{2}\hat{U}_{h}^{(0)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}+\tilde{n}^{(2,0)}_{iq_{1}}\dfrac{\partial\hat{\Theta}^{(0)}}{\partial
x_{q_{1}}}-n_{ip}^{(2,2)}s^{2}\hat{U}_{h}^{(0)}+\hat{b}_{i}=0,$ (1daa)
$\displaystyle
w^{(2,0)}_{q_{1}q_{2}}\dfrac{\partial^{2}\hat{\Theta}^{(0)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}-\tilde{w}^{(2,1)}_{pq_{2}}s\dfrac{\partial\hat{U}_{h}^{(0)}}{\partial
x_{q_{1}}}-w^{(2,1)}s\hat{\Theta}^{(0)}+\hat{r}=0,$ (1dab)

while the generic recursive problem at the macroscopic scale of order
$\varepsilon^{r}$ with $l\in\mathbb{Z},$ $l\geq 1$ is found as

$\displaystyle
n^{(2,0)}_{ipq_{1}q_{2}}\dfrac{\partial^{2}\hat{U}_{h}^{(l)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}+\tilde{n}^{(2,0)}_{iq_{1}}\dfrac{\partial\hat{\Theta}^{(l)}}{\partial
x_{q_{1}}}-n_{ip}^{(2,2)}s^{2}\hat{U}_{h}^{(l)}+s_{i}^{(l)}(\mathbf{x},s)=0,$
(1dba) $\displaystyle
w^{(2,0)}_{q_{1}q_{2}}\dfrac{\partial^{2}\hat{\Theta}^{(l)}}{\partial
x_{q_{1}}\partial
x_{q_{2}}}-\tilde{w}^{(2,1)}_{pq_{2}}s\dfrac{\partial\hat{U}_{h}^{(l)}}{\partial
x_{q_{1}}}-w^{(2,1)}s\hat{\Theta}^{(l)}-\upsilon^{(l)}(\mathbf{x},s)=0,$
(1dbb)

where the functions $s_{i}^{(l)}(\textbf{x},s)$ and
$\upsilon^{(l)}(\textbf{x},s)$ are the source terms $\mathcal{Q}$-periodic
fields, which depend both on the higher order constant tensors that appear in
the terms at orders equal or higher than $\varepsilon^{m}$ of the equations
$(\ref{eq:averagefieldu})$ and $(\ref{eq:averagefieldtheta}),$ and on the
sensitivities $\hat{U}_{h}^{(j)}$ and $\hat{\Theta}^{(j)}$ from macroscopic
hierarchical differential problems of an order lower than $\varepsilon^{m}.$

###  A.4. Approximation of the power-like functional through truncation of its
asymptotic expansion

Thus, plugging the down-scaling relations truncated at the first order (1aia),
(1aib) and their gradients (1aj), (1ak), into the transformed power-like
functional (3), it results

$\displaystyle\begin{split}\hat{\Lambda}_{m}(\hat{U}^{M}_{h},\hat{\Theta}^{M})=&\int_{\mathfrak{L}}\langle\hat{\lambda}_{m}(\textbf{x},\boldsymbol{\xi})\rangle
d\textbf{x}=\\\
=&s\left[\dfrac{1}{2}s^{2}\langle\rho^{m}\rangle\int_{\mathfrak{L}}\hat{U}^{M}_{h}\hat{U}^{M}_{h}d\textbf{x}+\left\langle\dfrac{1}{2}B^{(1,0)}_{ijlr_{1}}C^{m}_{ijhk}B^{(1,0)}_{hkpq_{1}}\right\rangle\int_{\mathfrak{L}}\dfrac{\partial\hat{U}^{M}_{l}}{\partial
x_{r_{1}}}\dfrac{\partial\hat{U}^{M}_{p}}{\partial
x_{q_{1}}}d\textbf{x}\quad+\right.\\\
+&\left\langle\dfrac{1}{2}B^{(1,0)}_{ijlr_{1}}C^{m}_{ijhk}\tilde{B}^{(1,0)}_{hk}\right\rangle\int_{\mathfrak{L}}\dfrac{\partial\hat{U}^{M}_{l}}{\partial
x_{r_{1}}}\hat{\Theta}^{M}d\textbf{x}+\left\langle\dfrac{1}{2}\tilde{B}^{(1,0)}_{ij}C^{m}_{ijhk}B^{(1,0)}_{hkpq_{1}}\right\rangle\int_{\mathfrak{L}}\hat{\Theta}^{M}\dfrac{\partial\hat{U}^{M}_{l}}{\partial
x_{q_{1}}}d\textbf{x}\quad+\\\
+&\left\langle\dfrac{1}{2}\tilde{B}^{(1,0)}_{ij}C^{m}_{ijhk}\tilde{B}^{(1,0)}_{hk}\right\rangle\int_{\mathfrak{L}}\hat{\Theta}^{M}\hat{\Theta}^{M}d\textbf{x}-\left\langle
B^{(1,0)}_{ijlr_{1}}\alpha^{m}_{ij}\right\rangle\int_{\mathfrak{L}}\dfrac{\partial\hat{U}^{M}_{l}}{\partial
x_{r_{1}}}\hat{\Theta}^{M}d\textbf{x}\quad+\\\
-&\left\langle\tilde{B}^{(1,0)}_{ij}\alpha^{m}_{ij}\right\rangle\int_{\mathfrak{L}}\hat{\Theta}^{M}\hat{\Theta}^{M}d\textbf{x}-\left.\int_{\mathfrak{L}}\hat{U}^{M}_{h}\hat{b}_{h}d\textbf{x}\right]\quad+\\\
-&\left\langle\dfrac{1}{2}A^{(1,0)}_{ir_{1}}K^{m}_{ij}A^{(1,0)}_{jq_{1}}\right\rangle\int_{\mathfrak{L}}\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{r_{1}}}\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{q_{1}}}d\textbf{x}-\dfrac{1}{2}s\langle
p^{m}\rangle\int_{\mathfrak{L}}\hat{\Theta}^{M}\hat{\Theta}^{M}d\textbf{x}\quad+\\\
+&\int_{\mathfrak{L}}\hat{r}\hat{\Theta}^{M}d\textbf{x}+\mathcal{O}(\varepsilon^{2}).\end{split}$
(1dc)

Using the concept of the variation of a functional to use the necessary
condition for a functional to have an extremum, the stability condition of the
transformed power-like functional $\hat{\Lambda}_{m}$ is given by the
governing equation of a non-local homogeneous continuum, in other words, one
arises to the first variation of the average transformed power-like functional
$\delta\hat{\Lambda}_{m}(\hat{U}^{M}_{h},\delta\hat{U}^{M}_{h},\hat{\Theta}^{M},\delta\hat{\Theta}^{M}),$
thus

$\displaystyle\begin{split}\delta\hat{\Lambda}_{m}=&\displaystyle\int_{\mathfrak{L}}s\left[s^{2}\langle\rho^{m}\rangle\hat{U}^{M}_{l}-\left\langle
B^{(1,0)}_{ijlr_{1}}C^{m}_{ijhk}B^{(1,0)}_{hkpq_{1}}\right\rangle\dfrac{\partial^{2}\hat{U}^{M}_{p}}{\partial
x_{q_{1}}\partial
x_{r_{1}}}-\left\langle\dfrac{1}{2}B^{(1,0)}_{ijlr_{1}}C^{m}_{ijhk}\tilde{B}^{(1,0)}_{hk}\right\rangle\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{r_{1}}}\quad+\right.\\\
-&\left.\left\langle\dfrac{1}{2}\tilde{B}^{(1,0)}_{ij}C^{m}_{ijhk}B^{(1,0)}_{hklq_{1}}\right\rangle\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{q_{1}}}+\left\langle
B^{(1,0)}_{ijlr_{1}}\alpha^{m}_{ij}\right\rangle\dfrac{\partial\hat{\Theta}^{M}}{\partial
x_{r_{1}}}-\hat{b}_{l}\right]\delta\hat{U}^{M}_{l}d\textbf{x}\quad+\\\
+&\displaystyle\int_{\mathfrak{L}}\left[s\left(\left\langle\dfrac{1}{2}B^{(1,0)}_{ijlr_{1}}C^{m}_{ijhk}\tilde{B}^{(1,0)}_{hk}\right\rangle\dfrac{\partial\hat{U}^{M}_{l}}{\partial
x_{r_{1}}}+\left\langle\dfrac{1}{2}\tilde{B}^{(1,0)}_{ij}C^{m}_{ijhk}B^{(1,0)}_{hkpq_{1}}\right\rangle\dfrac{\partial\hat{U}^{M}_{p}}{\partial
x_{q_{1}}}\quad+\right.\right.\\\
-&\left.\left\langle\tilde{B}^{(1,0)}_{ij}C^{m}_{ijhk}\tilde{B}^{(1,0)}_{hk}\right\rangle\hat{\Theta}^{M}-\left\langle
B^{(1,0)}_{ijlr_{1}}\alpha^{m}_{ij}\right\rangle\dfrac{\partial\hat{U}^{M}_{l}}{\partial
x_{r_{1}}}-2\left\langle\tilde{B}^{(1,0)}_{ij}\alpha^{m}_{ij}\hat{\Theta}^{M}\right\rangle\right)\quad+\\\
+&\left.\left\langle
A^{(1,0)}_{ir_{1}}K^{m}_{ij}A^{(1,0)}_{jq_{1}}\right\rangle\dfrac{\partial^{2}\hat{\Theta}^{M}}{\partial
x_{r_{1}}x_{q_{1}}}-s\langle
p^{m}\rangle\hat{\Theta}^{M}+\hat{r}\right]\delta\hat{\Theta}^{M}d\textbf{x},\end{split}$
(1dd)

where the divergence theorem has been applied in the first variation of the
transformed power-like functional (1dd).

###  A.5. Euler-Lagrange equation via power-like functional at micro-scale

Regarding the obtained governing field equations at the macro-scale, namely
(1aoa) and (1aob), and the overall constitutive tensors (1ana) to (1and) in
Sec. 3, the procedure only holds true in case that the variational approach at
the micro-scale also holds. Having said that, this Section it will provide an
equivalence between the governing field equations at the micro-scale
$(\ref{eq:Gov1T}),$ $(\ref{eq:Gov2T})$ and the power-like functional (3)
through the first variation of it, a similar procedure may be seen also in
[84].

If the functional $\hat{\Lambda}$ in equation (3) attains a local minimum at
$(\hat{\boldsymbol{u}},\hat{\theta}),$ $\delta$ is an arbitrary functions that
has at least one derivative and vanishes at the boundary of $\mathfrak{L},$
and then defining here $\eta$ as any number close to $0,$ yields

$\displaystyle\begin{split}\delta\hat{\Lambda}_{m}(\hat{\boldsymbol{u}},\delta\hat{\boldsymbol{u}},\hat{\theta},\delta\hat{\theta})&=\dfrac{d}{d\eta}\left[\int_{\mathfrak{L}}s\left(\dfrac{1}{2}\rho^{m}s^{2}(\hat{\boldsymbol{u}}+\eta\delta\hat{\boldsymbol{u}})\cdot(\hat{\boldsymbol{u}}+\eta\delta\hat{\boldsymbol{u}})\right.\right.\quad+\\\
&+\dfrac{1}{2}\nabla(\hat{\boldsymbol{u}}+\eta\delta\hat{\boldsymbol{u}}):(\mathfrak{C}^{m}\nabla(\hat{\boldsymbol{u}}+\eta\delta\hat{\boldsymbol{u}}))\quad+\\\
&-\left.\dfrac{1}{2}\nabla(\hat{\boldsymbol{u}}+\eta\delta\hat{\boldsymbol{u}}):(\boldsymbol{\alpha}^{m}(\hat{\theta}+\eta\delta\hat{\theta}))-(\hat{\boldsymbol{u}}+\eta\delta\hat{\boldsymbol{u}})\cdot\hat{\boldsymbol{b}}\right)d\mathbf{x}\quad+\\\
&-\int_{\mathfrak{L}}\left(\dfrac{1}{2}\nabla(\hat{\theta}+\eta\delta\hat{\theta})\cdot\left(\mathbf{K}^{m}\nabla(\hat{\theta}+\eta\delta\hat{\theta})\right)\right.\quad+\\\
&+\dfrac{1}{2}s(\hat{\theta}+\eta\delta\hat{\theta})\left(\boldsymbol{\alpha}^{m}:\nabla(\hat{\boldsymbol{u}}+\eta\delta\hat{\boldsymbol{u}})\right)\quad+\\\
&+\left.\left.\left.\dfrac{1}{2}s(\hat{\theta}+\eta\delta\hat{\theta})(p^{m}(\hat{\theta}+\eta\delta\hat{\theta}))-(\hat{\theta}+\eta\delta\hat{\theta})\hat{r}\right)d\mathbf{x}\right]\right|_{\eta=0}\,,\end{split}$
(1de)

where it must be highlighted that $\eta$ as defined in this Section does not
stand for thickness.

Reckon that the divergence theorem, the symmetry of the tensors
$\mathfrak{C}^{m},$ $\mathbf{K}^{m},$ $\boldsymbol{\alpha}^{m}$ and evaluating
the equation (1de) on $\eta=0,$ the first variation of the power-like
functional lying on the Laplace transform space at the micro-scale takes the
form

$\displaystyle\begin{split}\delta\hat{\Lambda}_{m}&=\int_{\mathfrak{L}}\left[s\left(\rho^{m}s^{2}\hat{\boldsymbol{u}}-\nabla\cdot(\mathfrak{C}^{m}\nabla\hat{\boldsymbol{u}})+\nabla\cdot(\boldsymbol{\alpha}^{m}\hat{\theta})-\hat{\boldsymbol{b}}\right)\delta\hat{\boldsymbol{u}}\right]d\mathbf{x}\quad+\\\
&+\int_{\mathfrak{L}}\left[\left(\nabla\cdot\left(\mathbf{K}^{m}\nabla\hat{\theta}\right)-s\left(\boldsymbol{\alpha}^{m}\nabla\hat{\boldsymbol{u}}\right)-sp^{m}\hat{\theta}+\hat{r}\right)\delta\hat{\theta}\right]d\mathbf{x}\,,\end{split}$
(1df)

and since
$\delta\hat{\Lambda}_{m}(\hat{\boldsymbol{u}},\delta\hat{\boldsymbol{u}},\hat{\theta},\delta\hat{\theta})=0,$
the Euler-Lagrange equations corresponding to the power-like functional at the
micro-scale are

$\displaystyle\rho^{m}s^{2}\hat{\boldsymbol{u}}-\nabla\cdot(\mathfrak{C}^{m}\nabla\hat{\boldsymbol{u}})+\nabla\cdot(\boldsymbol{\alpha}^{m}\hat{\theta})-\hat{\boldsymbol{b}}=\boldsymbol{0},$
(1dga)
$\displaystyle\nabla\cdot\left(\mathbf{K}^{m}\nabla\hat{\theta}\right)-s\left(\boldsymbol{\alpha}^{m}\nabla\hat{\boldsymbol{u}}\right)-sp^{m}\hat{\theta}+\hat{r}=0,$
(1dgb)

hence the variational approach at the micro-scale gives exactly the field
equations defined over the Laplace transform space seen in equations
$(\ref{eq:Gov1T})$ and $(\ref{eq:Gov2T}).$ By consequence, the
thermoelasticity governing field equations (1oa) and (1ob) over the time space
$t$ emerge once again after applying the inverse Laplace transform.

##  Section B Thermoelastic wave propagation modelling

###  B.1. Wave propagation in heterogeneous periodic material

On one hand, Sec. 4 studied the wave propagation an homogenized continuum,
taking advantage of the thermoelastic field equations on the complex frequency
space $s$ at the macro-scale, provided by the asymptotic homogenization
process seen previously in the present work. Now, on the other hand, this
Section is focused on investigate the heterogeneous material to determine the
frequency spectrum via Floquet-Bloch theory [74, 75, 76], from the
thermoelastic field equations at the micro-scale over Laplace transformed
space.

To this aim, firstly we recall the thermoelastic field equations (1dga) and
(1dgb), without the source terms $\hat{\boldsymbol{b}}$ and $\hat{r},$ i.e.,

$\displaystyle\nabla\cdot(\mathfrak{C}^{m}\nabla\hat{\boldsymbol{u}})-\nabla\cdot(\boldsymbol{\alpha}^{m}\hat{\theta})=\rho^{m}s^{2}\hat{\boldsymbol{u}},$
(1dha)
$\displaystyle\nabla\cdot\left(\mathbf{K}^{m}\nabla\hat{\theta}\right)-s\left(\boldsymbol{\alpha}^{m}\nabla\hat{\boldsymbol{u}}\right)=sp^{m}\hat{\theta}.$
(1dhb)

Secondly, from Floquet-Bloch theory, the decompositions for the thermoelastic
medium are given by

$\displaystyle\hat{\boldsymbol{u}}\left({\mathbf{x}},s\right)={\boldsymbol{\hat{U}}}^{B}({\mathbf{x}},s){\mathrm{e}^{\mathrm{i}\left({{\boldsymbol{k}}\cdot{\mathbf{x}}}\right)}},$
(1dia)
$\displaystyle\hat{\theta}\left({\mathbf{x}},s\right)=\hat{\Theta}^{B}({\mathbf{x}},s){\mathrm{e}^{\mathrm{i}\left({{\boldsymbol{k}}\cdot{\mathbf{x}}}\right)}},$
(1dib)

where ${\boldsymbol{\hat{U}}}^{B}({\mathbf{x}},s)$ and
$\hat{\Theta}^{B}({\mathbf{x}},s)$ are $\mathcal{A}$-periodic Bloch amplitudes
of the displacement field and temperature field, on the Laplace transformed
space, respectively. Equations (1dia) and (1dib) satisfy the Floquet-Bloch
periodicity boundary conditions, in which they arose by the
$\mathcal{Q}$-periodicity of the medium. Let $\boldsymbol{k}\in\mathbb{C}^{3}$
be the wave vector, the Floquet-Bloch boundary conditions read

$\displaystyle\hat{\boldsymbol{u}}(\mathbf{x}+\boldsymbol{v}_{p},s)$
$\displaystyle={\mathrm{e}^{\mathrm{i}\left({{\boldsymbol{k}}\cdot{{\boldsymbol{v}}_{p}}}\right)}}\hat{\boldsymbol{u}}\left({\mathbf{x}},s\right),$
(1dja) $\displaystyle\hat{\theta}(\mathbf{x}+\boldsymbol{v}_{p},s)$
$\displaystyle={\mathrm{e}^{\mathrm{i}\left({{\boldsymbol{k}}\cdot{{\boldsymbol{v}}_{p}}}\right)}}\hat{\theta}\left({\mathbf{x}},s\right),$
(1djb)

where $\boldsymbol{v}_{p}$ is the periodicity vector $(p=1,2,3).$

Finally, coupling the Floquet-Bloch decompositions above into the field
equations (1dha) and (1dhb), and making some simplifications, the tensorial
form yields

$\displaystyle\nabla^{B}\cdot(\mathfrak{C}^{m}\nabla^{B}{\boldsymbol{\hat{U}}}^{B})-\nabla^{B}\cdot(\boldsymbol{\alpha}^{m}\hat{\Theta}^{B})-\rho^{m}s^{2}{\boldsymbol{\hat{U}}}^{B}=\boldsymbol{0},$
(1dka)
$\displaystyle\nabla^{B}\cdot\left(\mathbf{K}^{m}\nabla^{B}\hat{\Theta}^{B}\right)-s\left(\boldsymbol{\alpha}^{m}\nabla^{B}{\boldsymbol{\hat{U}}}^{B}\right)-sp^{m}\hat{\Theta}^{B}=0,$
(1dkb)

where the differential operator $\nabla^{B}$ was defined as

$\displaystyle\nabla\hat{\boldsymbol{u}}=\Big{(}\nabla\boldsymbol{\hat{U}}^{B}+\mathrm{i}\boldsymbol{k}\otimes\boldsymbol{\hat{U}}^{B}\Big{)}{\mathrm{e}^{\mathrm{i}\left({{\boldsymbol{k}}\cdot{\mathbf{x}}}\right)}}=\nabla^{B}\Big{(}{\boldsymbol{\hat{U}}}^{B}\Big{)}{\mathrm{e}^{\mathrm{i}\left({{\boldsymbol{k}}\cdot{\mathbf{x}}}\right)}},$
(1dla)
$\displaystyle\nabla\hat{\theta}=\Big{(}\nabla\hat{\Theta}^{B}+\mathrm{i}\boldsymbol{k}\otimes\hat{\Theta}^{B}\Big{)}{\mathrm{e}^{\mathrm{i}\left({{\boldsymbol{k}}\cdot{\mathbf{x}}}\right)}}=\nabla^{B}\Big{(}\hat{\Theta}^{B}\Big{)}{\mathrm{e}^{\mathrm{i}\left({{\boldsymbol{k}}\cdot{\mathbf{x}}}\right)}}.$
(1dlb)

Particularly, we may also represent the field equations in terms of the
components

$\displaystyle\begin{split}&\left(C^{m}_{ijhk}\hat{U}^{B}_{h,k}\right)_{,j}+\mathrm{i}k_{j}\left[\Big{(}C^{m}_{ijhk}+C^{m}_{ikhj}\Big{)}\hat{U}^{B}_{h,k}+C^{m}_{ikhj,k}\hat{U}^{B}_{h}-\alpha^{m}_{ij}\hat{\Theta}^{B}\right]+\\\
&-\Big{(}k_{k}k_{j}C^{m}_{ijhk}+\rho^{m}s^{2}\delta_{ih}\Big{)}\hat{U}^{B}_{h}-\left(\alpha^{m}_{ij}\hat{\Theta}^{B}\right)_{,j}=0,\end{split}$
(1dma)
$\displaystyle\begin{split}&\left(K^{m}_{ij}\hat{\Theta}^{B}_{,j}\right)_{,i}+\mathrm{i}k_{j}\left[\Big{(}K^{m}_{ij}+K^{m}_{ji}\Big{)}\hat{\Theta}^{B}_{,i}+K^{m}_{ij,i}\hat{\Theta}^{B}-s\alpha^{m}_{ij}\hat{U}^{B}_{i}\right]+\\\
&-\Big{(}k_{i}k_{j}K^{m}_{ij}+sp^{m}\Big{)}\hat{\Theta}^{B}-s\alpha^{m}_{ij}\hat{U}^{B}_{i,j}=0,\end{split}$
(1dmb)

although the derivative $(\cdot)_{,j}$ was once defined as the microscopic
derivative, for convenience here it represents the partial derivative in
$x_{j},$ i.e. $\partial/\partial x_{j}=(\cdot)_{,j}$

Recalling the spatial damping method in the Sec. 4, let the conditions of an
homogeneous wave be plugged into the field equations (1dma) and (1dmb), so
given a direction $\boldsymbol{n}$ of the wave vector
$\boldsymbol{k}=\kappa\boldsymbol{n}$ one obtains an eigenvector-eigenvalue
problem where $\kappa$ is the eigenvalue and the $\mathcal{Q}$-periodic Bloch
amplitudes $\boldsymbol{\hat{U}}^{B},$ $\hat{\Theta}^{B}$ are the
eigenfunctions associated to it, while the angular frequency $\omega$ might be
a fixed parameter. Consequently to the study of harmonic waves over a layered
media in the next Sec. B.2., it arrives to the frequency spectrum
$\kappa{\omega}$ (1bh).

Similarly through the time damping path seen in Sec. 4 together with the
results from the Sec. B.2., it is possible to find an eigenvector-eigenvalue
problem, where once resolved in this scenario it allows to determine the
eigenvalues corresponding to the dispersion surfaces $s(\boldsymbol{k}),$ that
is writing the complex angular frequency $s$ as functions of the wave vector
$\boldsymbol{k}$ and also the eigenfunctions corresponding to the wave
polarization, with the Bloch amplitudes being its components [85].

###  B.2. Frequency-band structure for periodic heterogeneous thermoelastic
layered material

In the present Section is outlined the procedure followed to obtain the
frequency spectrum corresponding to the spatial damping over a periodic
heterogeneous thermoelastic orthotropic layered material, which has led us to
evaluate the benchmark test detailed in Sec. 5.2, where the corresponding
method drives into the dispersion relation (1be). Specifically, this procedure
is based on the transfer matrix method.

Initially, let us consider a body made of a given number $n$ of overlapped
layers bonded at their interfaces and stacked normal to the axis
$\boldsymbol{e}_{2}$ of the plane $(\boldsymbol{e}_{1},\boldsymbol{e}_{2})$
($\boldsymbol{\xi}=\xi_{1}\boldsymbol{e}_{1}+\xi_{2}\boldsymbol{e}_{2}$). By
assumption, the line boundary of each layer must be parallel to the
$\boldsymbol{e}_{1}$ vector which is chosen to coincide whit the exact half of
the layered plate. We also assign for each layer the number $j$ (with
$j=1,2,\dots,n$) a local coordinate $\mathfrak{s}_{i}^{{}_{(j)}}$ such that
its origin is located in the barycentre of the layer with
$\mathfrak{s}_{2}^{{}_{(j)}}$ normal to it. Thus layer $j$ occupies the region
$-d^{{}_{(j)}}/2\leq\mathfrak{s}_{3}^{{}_{(j)}}\leq d^{{}_{(j)}}/2,$ where
$d^{{}_{(j)}}$ is its thickness, and hence $d,$ the sum of the thickness of
all individual layers
$d_{2}=d^{{}_{(1)}}+\cdots+d^{{}_{(j)}}+\cdots+d^{{}_{(n)}}$ must be equal to
the total height of the layered body, where the body occupies the region
$-d_{2}/2\leq\xi_{2}\leq d_{2}/2.$

Let us assume the same hypothesis taken in the Sec. 5.1 for the heterogeneous
thermoelastic problem, this means an orthotropic bi-phase layered material
with the orthotropy axes perpendicular to the layering direction
$\boldsymbol{e_{2}}$ and the wave vector as
$\boldsymbol{k}=(k_{1},k_{2})^{T}=(0,k_{2})^{T},$ which make the field
equations (1dma) and (1dmb) only dependent on the variable $\xi_{2},$
therefore for each layer $j$ we have a set of following governing equations

$\displaystyle\begin{split}&C^{{}_{(j)}}_{1212}\hat{U}^{{B}}_{1,22}+\mathrm{i}2k_{2}C^{{}_{(j)}}_{1212}\hat{U}^{{B}}_{1,2}-\Big{(}k_{2}^{2}C^{{}_{(j)}}_{1212}+\rho^{m}s^{2}\Big{)}\hat{U}^{{B}}_{1}=0,\end{split}$
(1dna)
$\displaystyle\begin{split}&C^{{}_{(j)}}_{2222}\hat{U}^{{B}}_{2,22}+\mathrm{i}2k_{2}\left(C^{{}_{(j)}}_{2222}\hat{U}^{{B}}_{2,2}-\alpha^{{}_{(j)}}_{22}\hat{\Theta}^{{B}}\right)-\Big{(}k_{2}^{2}C^{{}_{(j)}}_{2222}+\rho^{{}_{(j)}}s^{2}\Big{)}\hat{U}^{{B}}_{2}-\alpha^{{}_{(j)}}_{22}\hat{\Theta}^{{B}}_{,2}=0,\end{split}$
(1dnb)
$\displaystyle\begin{split}&K^{{}_{(j)}}_{22}\hat{\Theta}^{{B}}_{,22}+\mathrm{i}2k_{2}\left(K^{{}_{(j)}}_{22}\hat{\Theta}^{{B}}_{,2}-s\alpha^{{}_{(j)}}_{22}\hat{U}^{{B}}_{2}\right)-\Big{(}k_{2}^{2}K^{{}_{(j)}}_{22}+sp^{{}_{(j)}}\Big{)}\hat{\Theta}^{{B}}-s\alpha^{{}_{(j)}}_{22}\hat{U}^{{B}}_{2,2}=0.\end{split}$
(1dnc)

In order to arrive at the transfer matrix, the solutions in the Floquet-Bloch
form (1dia) and (1dib) of the specialized governing equations (1dna), (1dnb)
and (1dnc) for each layer $j$ are obtained and are evaluated for both the
upper $(+)$ and lower $(-)$ boundary surfaces of layer $j.$ Among them, from
the constitutive relations (LABEL:sigma) and (1a) transformed in the Floquet-
Bloch form as well, the components $\hat{\sigma}^{{B}}_{12},$
$\hat{\sigma}^{{B}}_{22}$ and $\hat{q}^{{B}}_{2}$ are derived, which are also
evaluated for the upper $(+)$ and lower $(-)$ boundary surfaces of layer $j.$
Proceeding with some algebraic manipulations over them, the transformed
components specialized for the upper $(+)$ boundary of layer $j$ can be
written in terms of the same components but specialized for the lower $(-)$
boundary of layer $j.$

In the following, by applying the above procedure to a single layer, one
obtains

$[\boldsymbol{P}]_{j}^{+}=[\boldsymbol{A}]_{j}[\boldsymbol{P}]_{j}^{-},\quad
j=1,2,\dots,n,$ (1do)

where the vector
$[\boldsymbol{P}]_{j}^{\pm}=\left(\left[\hat{U}^{B}_{1},\hat{U}^{B}_{2},\hat{\Theta}^{B},\hat{\sigma}^{{B}}_{12},\hat{\sigma}^{{B}}_{22},\hat{q}^{{B}}_{2}\right]_{j}^{\pm}\right)^{T}$
($T$ here is for transpose) defines the column vectors of the displacement
$\boldsymbol{\hat{U}}^{B},$ temperature $\boldsymbol{\hat{\Theta}}^{B},$
stress $\boldsymbol{\hat{\sigma}}^{B}$ and heat flux
$\boldsymbol{\hat{q}}^{B},$ specialized to the upper $(+)$ and lower $(-)$
boundary surfaces of layer $j,$ and the matrix $[\boldsymbol{A}]_{j}$
constitutes the local transfer matrix for layer $j.$

Finding the equation (1do) to each layer, followed by the individual matrix
multiplication
$[\boldsymbol{A}]=[\boldsymbol{A}]_{n}\dots[\boldsymbol{A}]_{2}[\boldsymbol{A}]_{1}$
and reminding the continuity of the thermoelastic solutions and constitutive
tensors at the layer interfaces namely,
$[\boldsymbol{P}]_{j+1}^{-}=[\boldsymbol{P}]_{j}^{+},$ one relates the
solutions and tensors at the upper boundary, to those at its lower boundary,
this results in $[\boldsymbol{P}]^{+}=[\boldsymbol{A}][\boldsymbol{P}]^{-},$
where $[\boldsymbol{P}]^{-}$ and $[\boldsymbol{P}]^{+}$ are now the
displacement, temperature and tensor column vectors specialized to the upper
and lower faces of the total plate, respectively and $[\boldsymbol{A}]$ is the
global transfer matrix of the total cell. Finally, imposing the Floquet-Bloch
periodic boundary conditions namely, equations (1dja) and (1djb), one gives
$[\boldsymbol{P}]^{+}={\mathrm{e}^{\mathrm{i}\left({{\boldsymbol{k}}\cdot{\mathbf{x}}}\right)}}[\boldsymbol{P}]^{-},$
and therefore one arises the linear problem

${\mathrm{e}^{\mathrm{i}\left({{\boldsymbol{k}}\cdot{\mathbf{x}}}\right)}}[\boldsymbol{P}]^{-}=[\boldsymbol{A}][\boldsymbol{P}]^{-}.$
(1dp)

At this point, even though both the local and global transfer matrices show
themselves with several properties, which are discussed and listed in [86], we
only exploit them computationally herein. Such properties classify the
transfer matrix as being a symplectic matrix, and the characteristic
polynomial attached to its respective eigenvector-eigenvalue problem i.e.
$([\boldsymbol{A}]-{\mathrm{e}^{\mathrm{i}\left({{\boldsymbol{k}}\cdot{\mathbf{x}}}\right)}}[\boldsymbol{I}])[\boldsymbol{P}]^{-}=([\boldsymbol{A}]-\lambda[\boldsymbol{I}])[\boldsymbol{P}]^{-}=[\boldsymbol{0}],$
is a palindromic polynomial [87].

The palindromic polynomial
$\mathscr{P}(\lambda)=\text{det}\left([\boldsymbol{A}]-\lambda[\boldsymbol{I}]\right)=0$
to our bi-layered material problem given by the field equations (1dna), (1dnb)
and (1dnc), wrote in terms of the invariants $I_{k}$ of $[\boldsymbol{A}]$ as

$\mathscr{P}(\lambda)=I_{6}+I_{5}\lambda+I_{4}\lambda^{2}+I_{3}\lambda^{3}+I_{2}\lambda^{4}+I_{1}\lambda^{5}+I_{0}\lambda^{6},$
(1dq)

it has the following symmetric relations

$\quad I_{0}=I_{6}=1,\quad I_{1}=I_{5},\quad I_{2}=I_{4}.$ (1dr)

In addition, once recognized that
${e^{i\left({{\boldsymbol{k}}\cdot{\mathbf{x}}}\right)}}$ is an eigenvalue of
the transfer matrix $[\boldsymbol{A}],$ via the transformation
$\mathfrak{z}=\lambda+1/\lambda$ its palindromic polynomial (1dq) can still be
formally rephrased as

$\mathscr{P}(\mathfrak{z})=\mathfrak{z}^{3}+I_{1}\mathfrak{z}^{2}+(I_{2}-3)\mathfrak{z}+(I_{3}-2I_{1}).$
(1ds)

Last but not least, it is possible to invoke the transfer matrix structure and
rewrite the invariants $I_{1},$ $I_{2},$ and $I_{3}$ in terms of the trace of
$[\boldsymbol{A}]$ as stabilised in [88], so

$\displaystyle I_{1}$ $\displaystyle=-\textrm{tr}([\boldsymbol{A}]),$ (1dta)
$\displaystyle I_{2}$
$\displaystyle=-\dfrac{1}{2}\textrm{tr}([\boldsymbol{A}]^{2})+\dfrac{1}{2}(\textrm{tr}([\boldsymbol{A}]))^{2},$
(1dtb) $\displaystyle I_{3}$
$\displaystyle=-\dfrac{1}{3}\textrm{tr}([\boldsymbol{A}]^{3})+\dfrac{1}{2}\textrm{tr}([\boldsymbol{A}]^{2})\textrm{tr}([\boldsymbol{A}])-\dfrac{1}{6}(\textrm{tr}([\boldsymbol{A}]))^{3}.$
(1dtc)

Still at the same assumptions made in the beginning of this Section, it is
also feasible factorize the polynomial (1dq) into two minor factors, namely
one of a second degree polynomial associated to the equation (1dna) and
another of a fourth degree polynomial associated to the equations (1dnb) and
(1dnc). Each sub-polynomial will also be a palindromic polynomial and
therefore governed by invariants of its sub-matrices. Furthermore, choosing
the right changing of variables, after a similar process as made before, one
will have factorized the third degree polynomial (1ds) into two sub-
polynomials a linear one and a quadratic one.

###  B.3. Invariants of the dispersive wave propagation in the periodic
heterogeneous thermoelastic layered material

Throughout this Section it is presented the invariants that characterize the
wave propagation along the heterogeneous material. So, for the uncoupled
hypothesis of the thermoelastic problem taken in Sec. 5.2 for the comparative
analysis and among the latter theories written in the Secs. B.1. and B.2., the
invariants may be obtained with the following analytical expressions

$\displaystyle\begin{split}\tilde{I}_{1}^{{}_{1}}(s)=&-\dfrac{\sqrt{A_{1212}}B_{1212}+2A_{1212}}{4A_{1212}}\left(\mathrm{e}^{-s\left(\sum\limits^{2}_{r=1}\varpi_{1212}^{{}_{(r)}}\right)}+\mathrm{e}^{s\left(\sum\limits^{2}_{r=1}\varpi_{1212}^{{}_{(r)}}\right)}\right)+\\\
&+\dfrac{\sqrt{A_{1212}}B_{1212}-2A_{1212}}{4A_{1212}}\left(\mathrm{e}^{-s\left(\sum\limits^{2}_{r=1}(-1)^{r}\varpi_{1212}^{{}_{(r)}}\right)}+\mathrm{e}^{s\left(\sum\limits^{2}_{r=1}(-1)^{r}\varpi_{1212}^{{}_{(r)}}\right)}\right),\end{split}$
(1du)

where
$A_{1212}=C_{1212}^{{}_{(2)}}\rho^{{}_{(2)}}C_{1212}^{{}_{(1)}}\rho^{{}_{(1)}},$
$B_{1212}=C_{1212}^{{}_{(1)}}\rho^{{}_{(1)}}+C_{1212}^{{}_{(2)}}\rho^{{}_{(2)}},$
$\varpi_{1212}^{{}_{(j)}}=\mathfrak{s}_{j}\sqrt{C_{1212}^{{}_{(j)}}\rho^{{}_{(j)}}}/C_{1212}^{{}_{(j)}}$
with $j={1,2}.$ Analogously, for the compressional wave the associate
dispersion relation provides the wave number $k_{2}$ as an explicit function
of the complex frequency $s,$

$\displaystyle\begin{split}\tilde{I}_{1}^{{}_{2}}(s)=&-\dfrac{\sqrt{A_{2222}}B_{2222}+2A_{2222}}{4A_{2222}}\left(\mathrm{e}^{-s\left(\sum\limits^{2}_{r=1}\varpi_{1212}^{{}_{(r)}}\right)}+\mathrm{e}^{s\left(\sum\limits^{2}_{r=1}\varpi_{1212}^{{}_{(r)}}\right)}\right)+\\\
&+\dfrac{\sqrt{A_{2222}}B_{2222}-2A_{2222}}{4A_{2222}}\left(\mathrm{e}^{-s\left(\sum\limits^{2}_{r=1}(-1)^{r}\varpi_{1212}^{{}_{(r)}}\right)}+\mathrm{e}^{s\left(\sum\limits^{2}_{r=1}(-1)^{r}\varpi_{1212}^{{}_{(r)}}\right)}\right),\end{split}$
(1dv)

where
$A_{2222}=C_{2222}^{{}_{(2)}}\rho^{{}_{(2)}}C_{2222}^{{}_{(1)}}\rho^{{}_{(1)}},$
$B_{2222}=C_{2222}^{{}_{(1)}}\rho^{{}_{(1)}}+C_{2222}^{{}_{(2)}}\rho^{{}_{(2)}},$
$\varpi_{2222}^{{}_{(j)}}=\mathfrak{s}_{j}\sqrt{C_{2222}^{{}_{(j)}}\rho^{{}_{(j)}}}/C_{2222}^{{}_{(j)}}$
with $j={1,2}.$ Lastly, for the thermal wave propagation follows that

$\displaystyle\begin{split}\tilde{I}_{1}^{{}_{3}}(s)=&-\left(\dfrac{|s|\sum\limits^{2}_{r=1}\left(s\tau^{{}_{(r)}}+1\right)\mu^{{}_{(r)}}_{22}}{4s\sqrt{\left(\prod^{2}_{r=1}\left(s\tau^{{}_{(r)}}+1\right)\mu^{{}_{(r)}}_{22}\right)}}+\dfrac{1}{2}\right)\left(\mathrm{e}^{-\sum\limits^{2}_{r=1}\left(s\tau^{{}_{(r)}}+1\right)\varphi^{{}_{(r)}}_{22}}+\mathrm{e}^{\sum\limits^{2}_{r=1}\left(s\tau^{{}_{(r)}}+1\right)\varphi^{{}_{(r)}}_{22}}\right)+\\\
&+\left(\dfrac{|s|\sum\limits^{2}_{r=1}\left(s\tau^{{}_{(r)}}+1\right)\mu^{{}_{(r)}}_{22}}{4s\sqrt{\left(\prod^{2}_{r=1}\left(s\tau^{{}_{(r)}}+1\right)\mu^{{}_{(r)}}_{22}\right)}}-\dfrac{1}{2}\right)\left(\mathrm{e}^{-\sum\limits^{2}_{r=1}(-1)^{r}\left(s\tau^{{}_{(r)}}+1\right)\varphi^{{}_{(r)}}_{22}}+\mathrm{e}^{\sum\limits^{2}_{r=1}(-1)^{r}\left(s\tau^{{}_{(r)}}+1\right)\varphi^{{}_{(r)}}_{22}}\right),\end{split}$
(1dw)

where $\mu^{{}_{(r)}}_{22}=1/\bar{K}^{{}_{(r)}}_{22}p^{{}_{(r)}},$
$\varphi^{{}_{(r)}}_{22}=T_{0}\mathfrak{s}_{r}\sqrt{\bar{K}^{{}_{(r)}}_{22}p^{{}_{(r)}}}/\bar{K}^{{}_{(r)}}_{22}.$