In our numerical approach, there is one single coordinate system. Thus, we may set
\(\epsilon =1\) and model the RVE in real physical dimensions. In general, the homothetic ratio is defined as:
$$\begin{aligned} \underbrace{\epsilon = \frac{l}{L} = \frac{\text {microscale length}}{\text {macroscale length}}}_{\text {Homothetic ratio}} \, , \ \underbrace{ y_j = \frac{1}{\epsilon }(X_j - \overset{c}{X}_j)}_{\text {Local coordinate}} \, , \end{aligned}$$
(31)
\(\longrightarrow y_{i,j} = \delta _{ij}/\epsilon \). Microscale displacement field for the RVE is then expanded with regard to
\(\epsilon \) as:
$$\begin{aligned} {\varvec{{\varvec{u}}}}^\text {m}({\varvec{X}}) = \underbrace{\overset{0}{{\varvec{{\varvec{u}}}}}({\varvec{X}},{\varvec{y}}) + \epsilon \overset{1}{{\varvec{{\varvec{u}}}}}({\varvec{X}},{\varvec{y}}) + \epsilon ^2 \overset{2}{{\varvec{u}}}({\varvec{X}},{\varvec{y}}) + \mathcal {O}(\epsilon ^3)}_{\text {Expanded microscale displacement field}} \end{aligned}$$
(32)
where
\(\overset{n}{{\varvec{u}}}({\varvec{X}},{\varvec{y}})\) is
\({\varvec{y}}\)-periodic. Furthermore, to ensure that temperature within RVE is assumed constant,
$$\begin{aligned} T^\text {m}({\varvec{X}}, {\varvec{y}}) = T^\text {M}({\varvec{X}}) , \end{aligned}$$
(33)
to ensure the
\({\varvec{y}}\)-periodicity of a scalar field. By using the chain rule, we obtain the first derivative of (microscale) displacement field,
$$\begin{aligned} \begin{aligned} u^\text {m}_{i,j} = \overset{0}{u}_{i,j} + \frac{1}{\epsilon }\frac{\partial \overset{0}{u}_i}{\partial y_j} + \epsilon \bigg ( \overset{1}{u}_{i,j} + \frac{1}{\epsilon } \frac{\partial \overset{1}{u}_i}{\partial y_j} \bigg ) + \epsilon ^2 \bigg ( \overset{2}{u}_{i,j} + \frac{1}{\epsilon }\frac{\partial \overset{2}{u}_i}{\partial y_j} \bigg ) + \mathcal {O}(\epsilon ^3) . \end{aligned}\end{aligned}$$
(34)
We utilize Eq. (
17), use chain rule, and insert Eq. (
34),
$$\begin{aligned}{} & {} \Bigg [ {C_{ijkl}^\text {m}}\bigg ( \overset{0}{u}_{k,l} + \frac{1}{\epsilon }\frac{\partial \overset{0}{u}_k}{\partial y_l} + \epsilon \overset{1}{u}_{k,l} + \frac{\partial \overset{1}{u}_k}{\partial y_l} + \epsilon ^2\overset{2}{u}_{k,l} + \epsilon \frac{\partial \overset{2}{u}_k}{\partial y_l} \bigg ) -\beta ^\text {m}_{ij}(T^\text {M}- T_\text {ref})\Bigg ]_{,j} \nonumber \\{} & {} \quad + \frac{1}{\epsilon }\frac{\partial }{\partial y_j}\Bigg [ {C_{ijkl}^\text {m}} \bigg ( \overset{0}{u}_{k,l} + \frac{1}{\epsilon }\frac{\partial \overset{0}{u}_k}{\partial y_l} + \epsilon \overset{1}{u}_{k,l} + \frac{\partial \overset{1}{u}_k}{\partial y_l} + \epsilon ^2\overset{2}{u}_{k,l} + \epsilon \frac{\partial \overset{2}{u}_k}{\partial y_l}\bigg ) -\beta ^\text {m}_{ij}(T^\text {M}- T_\text {ref}) \Bigg ] \nonumber \\{} & {} \quad +\rho ^\text {m}g_i = 0 . \end{aligned}$$
(35)
Comparing coefficients in Eq. (
35) of the same order of
\(\epsilon \) leads to:
-
\(\epsilon ^{-2}\)$$\begin{aligned} \begin{aligned} \frac{\partial }{\partial y_j}\bigg ( {C_{ijkl}^\text {m}} \frac{\partial \overset{0}{u}_k}{\partial y_l}\bigg ) = 0 . \end{aligned}\end{aligned}$$
(36)
-
\(\epsilon ^{-1}\)$$\begin{aligned}{} & {} \Big ({C_{ijkl}^\text {m}} \frac{\partial \overset{0}{u}_k}{\partial y_l}\Big )_{,j} + \frac{\partial }{\partial y_j}({C_{ijkl}^\text {m}} \overset{0}{u}_{k,l}) + \frac{\partial }{\partial y_j}\Big ({C_{ijkl}^\text {m}} \frac{\partial \overset{1}{u}_k}{\partial y_l}\Big )\nonumber \\{} & {} \quad - \frac{\partial }{\partial y_j}\Big (\beta ^\text {m}_{ij}(T^\text {M}- T_\text {ref})\Big )= 0 . \end{aligned}$$
(37)
-
\(\epsilon ^0\)$$\begin{aligned}{} & {} \big ({C_{ijkl}^\text {m}} \overset{0}{u}_{k,l} \big )_{,j} + \Big ({C_{ijkl}^\text {m}} \frac{\partial \overset{1}{u}_k}{\partial y_l}\Big )_{,j} + \frac{\partial }{\partial y_j}\Big ( {C_{ijkl}^\text {m}} \overset{1}{u}_{k,l} \Big ) + \frac{\partial }{\partial y_j}\Big ( {C_{ijkl}^\text {m}} \frac{\partial \overset{2}{u}_k}{\partial y_l} \Big )\nonumber \\{} & {} \quad - \big (\beta ^\text {m}_{ij}T^\text {M}\big )_{,j} + \rho ^\text {m}g_i = 0 . \end{aligned}$$
(38)
-
\(\epsilon ^1\)$$\begin{aligned} \begin{aligned} \bigg [ C^\text {m}_{ijkl} \Big ( \overset{1}{u}_{k,l} + \frac{\partial \overset{2}{u}_k }{\partial y_l} \Big ) \bigg ]_{,j} + \frac{\partial }{\partial y_j} \bigg ( C^\text {m}_{ijkl} \overset{2}{u}_{k,l} \bigg ) = 0 . \end{aligned}\end{aligned}$$
(39)
-
\(\epsilon ^2\)$$\begin{aligned} \begin{aligned} \big ( C^\text {m}_{ijkl} \overset{2}{u}_{k,l} \big )_{,j} = 0 . \end{aligned}\end{aligned}$$
(40)
Only possible solution for Eq. (
36) is to define
\(\overset{0}{{\varvec{u}}}\) as a function of
\({\varvec{X}}\) because
\({\varvec{C}}^\text {m}\) and
\({\varvec{\beta }}^\text {m}\) depends on local variable
\({\varvec{y}}\). This argumentation leads to,
$$\begin{aligned} \overset{0}{u}_i({\varvec{X}}) = {u}^\text {M}_i({\varvec{X}}) \, . \end{aligned}$$
(41)
We use a separation of variables, Bernoulli Ansatz to rewrite,
$$\begin{aligned} \begin{aligned} \overset{1}{u}_i({\varvec{X}}, {\varvec{y}}) =&u^\text {M}_{a,b}({\varvec{X}}) \varphi _{abi}({\varvec{y}}) - (T^\text {M}({\varvec{X}}) - T_\text {ref}) P_i({\varvec{y}}) \ , \\ \overset{2}{u}_i({\varvec{X}}, {\varvec{y}}) =&u^\text {M}_{a,bc}({\varvec{X}}) \psi _{abci}({\varvec{y}}), \end{aligned}\end{aligned}$$
(42)
where we introduce unknown tensors
\({\varvec{\varphi }}\),
\({\varvec{\psi }}\),
\({\varvec{P}}\) of one rank higher than associated displacement derivatives and temperature field, so that the formulation is general. We emphasize that the temperature is only expanded up to one order less than the displacement. By inserting these into Eq. (
37), we acquire
$$\begin{aligned} \begin{aligned}&\frac{\partial }{\partial y_j}({C_{ijkl}^\text {m}} u^\text {M}_{k,l}) + \frac{\partial }{\partial y_j}\Bigg [{C_{ijkl}^\text {m}} \frac{\partial }{\partial y_l} \Big (u^\text {M}_{a,b} \varphi _{abk} - (T^\text {M}- T_\text {ref}) P_k \Big )\Bigg ] \\ {}&\quad - \frac{\partial }{\partial y_j}\Big [\beta ^\text {m}_{ij}(T^\text {M}- T_\text {ref})\Big ]= 0 \ , \\&u^\text {M}_{a,b} \frac{\partial }{\partial y_j} \Bigg [ C_{ijkl}^\text {m}\Big ( \delta _{ak}\delta _{bl} + \frac{\partial \varphi _{abk}}{\partial y_l} \Big ) \Bigg ] - (T^\text {M}- T_\text {ref}) \frac{\partial }{\partial y_j} \Bigg [ C_{ijkl}^\text {m}\frac{\partial P_k}{\partial y_l} + \beta ^\text {m}_{ij} \Bigg ] = 0 . \end{aligned}\end{aligned}$$
(43)
To obtain general solution of Eq. (
43) we need to independently solve the following governing equations
$$\begin{aligned} \frac{\partial }{\partial y_j}\Bigg [\delta _{ak}\delta _{bl} + C^\text {m}_{ijkl}\bigg (\frac{\partial \varphi _{abk}}{\partial y_l} \bigg ) \Bigg ] = 0 . \end{aligned}$$
(44)
in order to obtain
\({\varvec{\varphi }}\) and
$$\begin{aligned} \frac{\partial }{\partial y_j}\Bigg ( C^\text {m}_{ijkl}\frac{\partial P_{k}}{\partial y_l} +\beta ^\text {m}_{ij} \Bigg ) = 0 \, . \end{aligned}$$
(45)
to acquire
\({\varvec{P}}\). Repeating the same procedure for Eq. (
38), we have
$$\begin{aligned} \begin{aligned}&\Bigg [ C_{ijkl}^\text {m}u^\text {M}_{a,b} \delta _{ak}\delta _{bl} + {C_{ijkl}^\text {m}} \frac{\partial }{\partial y_l} \Big ( u^\text {M}_{a,b} \varphi _{abk} - (T^\text {M}- T_\text {ref}) P_k \Big ) \Bigg ]_{,j} \\ {}&+ \frac{\partial }{\partial y_j} \Bigg [ C_{ijkl}^\text {m}\overset{1}{u}_{k,l} + C_{ijkl}^\text {m}\frac{\partial }{\partial y_l} \Big ( u^\text {M}_{a,bc} \psi _{abck} \Big ) \Bigg ] - \big (\beta ^\text {m}_{ij}T^\text {M}\big )_{,j} + \rho ^\text {m}g_i = 0 , \end{aligned}\end{aligned}$$
(46)
which is rewritten
$$\begin{aligned} \begin{aligned}&\Bigg [ u^\text {M}_{a,b} C_{ijkl}^\text {m}\Big ( \delta _{ak}\delta _{bl} + \frac{\partial \varphi _{abk}}{\partial y_l} \Big ) - C_{ijkl}^\text {m}(T^\text {M}- T_\text {ref}) \frac{\partial P_k}{\partial y_l} - \beta ^\text {m}_{ij}T^\text {M}\Bigg ]_{,j} \\ {}&+ \frac{\partial }{\partial y_j} \Bigg [ C_{ijkl}^\text {m}\bigg [ \Big ( u^\text {M}_{a,b} \varphi _{abk} - (T^\text {M}- T_\text {ref})P_k \Big )_{,l} + \frac{\partial }{\partial y_l} \Big ( u^\text {M}_{a,bc} \psi _{abck} \Big ) \bigg ] \Bigg ] + \rho ^\text {m}g_i = 0 , \\&u^\text {M}_{a,bj} C_{ijkl}^\text {m}\Big ( \delta _{ak}\delta _{bl} + \frac{\partial \varphi _{abk}}{\partial y_l} \Big ) - C_{ijkl}^\text {m}T^\text {M}_{,j} \frac{\partial P_k}{\partial y_l} - \beta ^\text {m}_{ij} T^\text {M}_{,j} \\ {}&+ \frac{\partial }{\partial y_j} \Bigg (C_{ijkl}^\text {m}\bigg ( u^\text {M}_{a,bl} \varphi _{abk} - T^\text {M}_{,l} P_k + u^\text {M}_{a,bc} \frac{\partial \psi _{abck}}{\partial y_l} \bigg ) \Bigg ] + \rho ^\text {m}g_i = 0 , \\&u^\text {M}_{a,bc} \Bigg [ \delta _{cj} C_{ijkl}^\text {m}\Big ( \delta _{ak}\delta _{bl} + \frac{\partial \varphi _{abk}}{\partial y_l} \Big ) + \frac{\partial }{\partial y_j} \bigg ( C_{ijkl}^\text {m}\Big ( \delta _{lc} \varphi _{abk} + \frac{\partial \psi _{abck}}{\partial y_l} \Big ) \bigg ) \Bigg ] \\&-T^\text {M}_{,a} \Bigg [ C_{iakl}^\text {m}\frac{\partial P_k}{\partial y_l} + \beta ^\text {m}_{ia} + \frac{\partial }{\partial y_j} \bigg ( C_{ijkl}^\text {m}\delta _{al} P_k \bigg ) \Bigg ] + \rho ^\text {m}g_i = 0 \ , \end{aligned}\end{aligned}$$
(47)
Furthermore, from Eq. (
22), by inserting
$$\begin{aligned} \begin{aligned} \rho ^\text {M}g_i = - C^\text {M}_{ijkl} u^\text {M}_{k,lj} +\beta ^\text {M}_{ij} T^\text {M}_{,j} - G^\text {M}_{ijklm} u^\text {M}_{k,lmj} + G^\text {M}_{lmijk} u^\text {M}_{l,mjk} \end{aligned}\end{aligned}$$
(48)
and using the same cut-off procedure for displacement third derivative and temperature second derivative, we obtain
$$\begin{aligned} \begin{aligned}&u^\text {M}_{a,bc} \Bigg [ \delta _{cj} C_{ijkl}^\text {m}\Big ( \delta _{ak}\delta _{bl} + \frac{\partial \varphi _{abk}}{\partial y_l} \Big ) + \frac{\partial }{\partial y_j} \Bigg ( C_{ijkl}^\text {m}\Big ( \delta _{lc} \varphi _{abk} + \frac{\partial \psi _{abck}}{\partial y_l} \Big ) \Bigg ) \Bigg ] \\&-T^\text {M}_{,a} \Bigg [ C_{iakl}^\text {m}\frac{\partial P_k}{\partial y_l} + \beta ^\text {m}_{ia} + \frac{\partial }{\partial y_j} \bigg ( C_{ijka}^\text {m}P_k \bigg ) \Bigg ] + \frac{\rho ^\text {m}}{\rho ^\text {M}} \Big ( - C^\text {M}_{ijkl} u^\text {M}_{k,lj} +\beta ^\text {M}_{ij} T^\text {M}_{,j} \Big ) = 0 , \end{aligned}\end{aligned}$$
(49)
Rearranging Eq.
49 by separation into the independent parts and enforcing
\(T^\text {M}_{,a}=0\) and
\(u^\text {M}_{a,bc}=0\) leads to the following equations, respectively.
$$\begin{aligned}{} & {} \frac{\partial }{\partial y_j}\Bigg [ C^\text {m}_{ijkl}\bigg (\frac{\partial \psi _{abck}}{\partial y_l} +\varphi _{abk}\delta _{cl} \bigg ) \Bigg ] + C^\text {m}_{ickl}\bigg (\frac{\partial \varphi _{abk}}{\partial y_l} +\delta _{ak}\delta _{lb} \bigg ) - \frac{\rho ^\text {m}}{\rho ^\text {M}}C^\text {M}_{icab} = 0 \ , \end{aligned}$$
(50)
$$\begin{aligned}{} & {} C_{iakl}^\text {m}\frac{\partial P_k}{\partial y_l} + \beta ^\text {m}_{ia} + \frac{\partial }{\partial y_j} \bigg ( C_{ijka}^\text {m}P_k \bigg ) - \frac{\rho ^\text {m}}{\rho ^\text {M}}\beta ^\text {M}_{ia} = 0 \ , \end{aligned}$$
(51)
Due to our initial assumption in ignoring higher-order temperature terms, Eq. (
50) is automatically satisfied. Therefore, we solve Eqs. (
44), (
45), (
50) to determine
\({\varvec{\varphi }}\),
\({\varvec{P}}\),
\({\varvec{\psi }}\), respectively, while honoring
\(T^\text {M}_{,a}=0\). By introducing following notations
$$\begin{aligned} \begin{aligned} L_{abij}&= \delta _{ia}\delta _{jb} + \frac{\partial \varphi _{abi}}{\partial y_j} \ , \\ N_{abcij}&= \varphi _{abi}\delta _{jc} + \frac{\partial \psi _{abci}}{\partial y_j} \ , \\ Z_{ij}&= \frac{\partial P_{i}}{\partial y_{j}} \ , \end{aligned}\end{aligned}$$
(52)
We need to fulfill
$$\begin{aligned} \begin{aligned} \varphi _{abi} \Leftarrow&\frac{\partial }{\partial y_j}\Bigg ( C^\text {m}_{ijkl} L_{abkl} \Bigg ) = 0 , \\ P_i \Leftarrow&\frac{\partial }{\partial y_j}\Bigg ( C^\text {m}_{ijkl} Z_{kl} +\beta ^\text {m}_{ij} \Bigg ) = 0 , \\ \psi _{abci} \Leftarrow&\frac{\partial }{\partial y_j}\Bigg ( C^\text {m}_{ijkl} N_{abckl} \Bigg ) + C^\text {m}_{ickl} L_{abkl} - \frac{\rho ^\text {m}}{\rho ^\text {M}}C^\text {M}_{icab} = 0 , \end{aligned}\end{aligned}$$
(53)
Equations (
42) and (
32) which leads to
$$\begin{aligned} \begin{aligned} u^\text {m}_{i}({\varvec{X}}, {\varvec{y}})&= u^\text {M}_{i}({\varvec{X}}) + \epsilon \Big [ \varphi _{abi}({\varvec{y}})u^\text {M}_{a,b}({\varvec{X}}) - P_{i}({\varvec{y}})( T ^\text {M}({\varvec{X}})- T_\text {ref}) \Big ] \\&+ \epsilon ^2 \Big ( \psi _{abci}({\varvec{y}})u^\text {M}_{a,bc}({\varvec{X}}) \Big ) . \\ u^\text {m}_{i,j}({\varvec{X}}, {\varvec{y}})&= u^\text {M}_{i,j} + \epsilon \Big ( \frac{\partial \varphi _{abi}}{\partial y_j} \frac{1}{\epsilon } u^\text {M}_{a,b} + \varphi _{abi} u^\text {M}_{a,bj} - \frac{\partial P_{i}}{\partial y_j} \frac{1}{\epsilon } ( T ^\text {M}- T_\text {ref}) - P_i T^\text {M}_{,j} \Big ) \\&+ \epsilon ^2 \Big ( \frac{\partial \psi _{abci}}{\partial y_j} \frac{1}{\epsilon } u^\text {M}_{a,bc} + \psi _{abci} u^\text {M}_{a,bcj} \Big ) . \end{aligned}\end{aligned}$$
(54)
To obtain the solution for the microscale free energy and determine the parameters we resort to Eq.
18, where the macroscale energy is given in Eq. (
30). We take advantage of the fact that
\(T^\text {M}_{,a}=0\) and that third derivative of displacement vanishes. Thus, we obtain the following expression:
$$\begin{aligned} \begin{aligned} u^\text {m}_{i,j} = \bigg (\delta _{ia}\delta _{jb} + \frac{\partial \varphi _{abi}}{\partial y_j} \bigg )u^\text {M}_{a,b} - \frac{\partial P_{i}}{\partial y_{j}}( T ^\text {M}- T_\text {ref}) + \epsilon u^\text {M}_{a,bc}\bigg ( \varphi _{abi}\delta _{jc} + \frac{\psi _{abci}}{\partial y_j}\bigg ) \, , \end{aligned}\end{aligned}$$
(55)
with the help of Eq. (
27), we acquire
$$\begin{aligned} \begin{aligned} u^\text {m}_{i,j} =&\bigg (\delta _{ia}\delta _{jb} + \frac{\partial \varphi _{abi}}{\partial y_j} \bigg )(\langle u_{a,b}^\text {M}\rangle + \epsilon y_{c}\langle u_{a,bc}\rangle ) - \frac{\partial P_{i}}{\partial y_{j}}(T ^\text {M}- T_\text {ref}) \\&+\epsilon \bigg ( \varphi _{abi}\delta _{jc} + \frac{\psi _{abci}}{\partial y_j}\bigg )\langle u^\text {M}_{a,bc}\rangle \, . \end{aligned} \end{aligned}$$
(56)
By introducing
$$\begin{aligned} \begin{aligned} M_{abcij} = y_c L_{abij} + N_{abcij} , \end{aligned}\end{aligned}$$
(57)
and using Eqs. (
52), (
56) can be rewritten as:
$$\begin{aligned} u^\text {m}_{i,j} = L_{abij}\langle u_{a,b}^\text {M}\rangle + \epsilon M_{abcij}\langle u_{a,bc}^\text {M}\rangle - Z_{ij}(T ^\text {M}- T_\text {ref}) \end{aligned}$$
(58)
Using the above equation microscale energy becomes
where
$$\begin{aligned} \begin{aligned} \bar{C}_{abcd}&=\frac{1}{V}\int _{\Omega }C_{ijkl}^\text {m}L_{abij} L_{cdkl} \, \hbox {d}V\\ \bar{G}_{abcde}&=\frac{2\epsilon }{V}\int _{\Omega }C_{ijkl}^\text {m}L_{abij}M_{cdekl} \, \hbox {d}V\\ \bar{\beta }_{ab}&=\frac{2}{V}\int _{\Omega }\Big [C_{ijkl}^\text {m}L_{abij}Z_{kl} - \beta _{ij}^\text {m}L_{abij}\Big ] \, \hbox {d}V\\ \bar{D}_{abcdef}&=\frac{\epsilon ^2}{V}\int _{\Omega }C_{ijkl}^\text {m}M_{abcij}M_{defkl} \, \hbox {d}V\\ \bar{\gamma }_{abc}&=\frac{2\epsilon }{V}\int _{\Omega }\Big [C_{ijkl}^\text {m}M_{abcij}Z_{kl} - \beta _{ij}^\text {m}M_{abcij}\Big ] \, \hbox {d}V\\ \bar{a}&=\frac{1}{V}\int _{\Omega }\Big [C_{ijkl}^\text {m}Z_{ij}Z_{kl} - 2\beta _{ij}^\text {m}Z_{ij}- a^\text {m}\Big ] \, \hbox {d}V\\ \bar{c}&= \frac{2}{V}\int _{\Omega }c^\text {m}\, \hbox {d}V \end{aligned}\end{aligned}$$
(60)
Comparing microscale energy in Eq.
59 to macroscale energy in Eq.
30, we obtain the homogenized values:
$$\begin{aligned} \begin{aligned} C_{ijkl}^\text {M}&= \bar{C}_{ijkl}\\ G_{ijklm}^\text {M}&= \frac{\bar{G}_{ijklm}}{2}\\ \beta _{ij}^\text {M}&= \frac{\bar{\beta }_{ij}}{2}\\ D_{ijklmn}^\text {M}&= \bar{D}_{ijklmn} - C_{ijlm}^\text {M}I_{kn}\\ \gamma _{ijk}^\text {M}&= -\frac{\bar{\gamma }_{ijk}}{2}\\ a^\text {M}&= - \bar{a}\\ c^\text {M}&= \frac{\bar{c}}{2} \end{aligned}\end{aligned}$$
(61)
It should be noted that in Eq. (
60), parameters such as thermoelastic interaction
\(\bar{\beta }_{ab}\) or coupling constant
\(\bar{\gamma }_{ab}\) show the contribution from both mechanical (stiffness matrix) and thermal (thermoelastic interaction) parameters.