Appendix A: Asymptotic expansions
The terms of orders \(\varepsilon ^0\), \(\varepsilon ^1\), \(\varepsilon ^2\) in the asymptotic expansions of the thermomechanical fields are given below. In the expansion of \({{\varvec{q}}}^{\varepsilon }\) given in Eq. (16), the terms \({{\varvec{q}}}^{(p)}\), \(p = 0, 1, 2\) are:
$$\begin{aligned} \displaystyle q^{(-1)}_{j}=-k\frac{\partial {T^{(0)}}}{\partial {y_{j}}} ; \ q^{(0)}_{j}=-k \Big ( \frac{\partial {T^{(0)}}}{\partial {x_{j}}}+\frac{\partial {T^{(1)}}}{\partial {y_{j}}} \Big ) ; \ q^{(1)}_{j}=-k \Big ( \frac{\partial {T^{(1)}}}{\partial {x_{j}}}+\frac{\partial {T^{(2)}}}{\partial {y_{j}}} \Big ) \end{aligned}$$
(62)
and the corresponding terms for \({\varvec{\sigma }}^{\varepsilon }\) in development (17) are:
$$\begin{aligned}&\sigma ^{(-1)}_{ij} = a_{ijkl} e_{ykl}(\mathbf{u}^{(0)}) \end{aligned}$$
(63)
$$\begin{aligned}&\sigma ^{(0)}_{ij}= a_{ijkl} \big ( e_{xkl}(\mathbf{u}^{(0)})+e_{ykl}(\mathbf{u}^{(1)}) \big ) - b_{ij}(T^{(0)} - T_0) \end{aligned}$$
(64)
$$\begin{aligned}&\sigma ^{(0)}_{ij}= a_{ijkl} \big ( e_{xkl}(\mathbf{u}^{(1)})+e_{ykl}(\mathbf{u}^{(2)}) \big ) - b_{ij}T^{(1)} \end{aligned}$$
(65)
For the expressions of different order terms \(D^{(i)}\) in asymptotic expansion (26) of the crack-tip heat flow \(D^{\varepsilon }\) one obtains:
$$\begin{aligned}&\displaystyle D^{(0)} =\lim _{r \rightarrow 0}\int _{\varGamma Y_{r}}-k\frac{\partial T^{(0)}}{\partial y_j}n_j \mathrm{d}s_{y} ; \ D^{(1)} =\lim _{r \rightarrow 0}\int _{\varGamma Y_{r}}-k(\frac{\partial T^{(0)}}{\partial x_j}+\frac{\partial T^{(1)}}{\partial y_j}) n_j \mathrm{d}s_{y} ; \nonumber \\&\displaystyle D^{(2)} =\lim _{r \rightarrow 0}\int _{\varGamma Y_{r}}-k(\frac{\partial T^{(1)}}{\partial x_j}+\frac{\partial T^{(2)}}{\partial y_j}) n_j \mathrm{d}s_{y} \end{aligned}$$
(66)
The terms of different orders in asymptotic expansion (27) of the energy-release rate \({{\mathcal {G}}}^{d \varepsilon }\) are:
$$\begin{aligned} \displaystyle {{\mathcal {G}}}^{(-1)}&=\lim _{r \rightarrow 0}\int _{\varGamma Y_{r}}\left( \frac{1}{2}a_{ijkl} e_{ykl}(\mathbf{u}^{(0)}) e_{yij}(\mathbf{u}^{(0)})n_1 - a_{ijkl} e_{ykl}(\mathbf{u}^{(0)})n_j \frac{\partial u^{(0)}_i}{\partial y_1}\right) \mathrm{d}s_y \end{aligned}$$
(67)
$$\begin{aligned} \displaystyle {{\mathcal {G}}}^{(0)}&= \lim _{r \rightarrow 0}\int _{\varGamma Y_{r}} \Bigg (\Big ( a_{ijkl} e_{ykl}(\mathbf{u}^{(0)})\big (e_{xij}(\mathbf{u}^{(0)})+e_{yij}(\mathbf{u}^{(1)})\big ) + b_{ij}T_0 e_{yij}(\mathbf{u}^{(0)})\Big )n_1 \nonumber \\&\displaystyle \quad - a_{ijkl}\big ( e_{xkl}(\mathbf{u}^{(0)})+e_{ykl}(\mathbf{u}^{(1)})\big )n_j\frac{\partial u^{(0)}_i}{\partial y_1} - a_{ijkl} e_{ykl}(\mathbf{u}^{(0)})n_j\bigg (\frac{\partial u^{(0)}_i}{\partial x_1}+\frac{\partial u^{(1)}_i}{\partial y_1}\bigg ) \Bigg ) \mathrm{d}s_y \end{aligned}$$
(68)
$$\begin{aligned} \displaystyle {{\mathcal {G}}}^{(1)}&= \lim _{r \rightarrow 0}\int _{\varGamma Y_{r}} \Bigg (\bigg (\frac{1}{2}a_{ijkl} \Big (2e_{ykl}(\mathbf{u}^{(0)})\big (e_{xij}(\mathbf{u}^{(0)}) + e_{yij}(\mathbf{u}^{(1)})\big ) \nonumber \\&\quad \displaystyle + \big (e_{xij}(\mathbf{u}^{(0)})+e_{yij}(\mathbf{u}^{(1)})\big )\big (e_{xkl}(\mathbf{u}^{(0)})+e_{ykl}(\mathbf{u}^{(1)})\big )\Big ) \nonumber \\&\quad \displaystyle + b_{ij}T_0 \big (e_{xij}(\mathbf{u}^{(0)})+e_{yij}(\mathbf{u}^{(1)})\big )+cT^{(0)} +\frac{\rho }{8}\frac{\partial u^{(0)}_i}{\partial y_1}\frac{\partial u^{(0)}_i}{\partial y_1}\Big (\frac{\mathrm{d}d}{\mathrm{d}t}\Big )^2 \bigg )n_1 \nonumber \\&\quad \displaystyle - \bigg (a_{ijkl}e_{ykl}(\mathbf{u}^{(0)})n_j\Big (\frac{\partial u^{(1)}_i}{\partial x_1}+\frac{\partial u^{(2)}_i}{\partial y_1}\Big ) \nonumber \\&\quad \displaystyle +\Big (a_{ijkl}( e_{xkl}(\mathbf{u}^{(0)})+e_{ykl}(\mathbf{u}^{(1)})) - b_{ij}(T^{(0)}-T_0)\Big )n_j\Big (\frac{\partial u^{(0)}_i}{\partial x_1}+\frac{\partial u^{(1)}_i}{\partial y_1}\Big ) \nonumber \\&\quad \displaystyle +\Big (a_{ijkl} ( e_{xkl}(\mathbf{u}^{(1)}) + e_{ykl}(\mathbf{u}^{(2)})) - b_{ij}T^{(1)}\Big )n_j\frac{\partial u^{(0)}_i}{\partial y_1} \bigg ) \Bigg ) \ \mathrm{d}s_y \end{aligned}$$
(69)
For the velocity field the local relation \( \frac{\partial u^{\varepsilon }_i}{\partial t} \simeq - \frac{\mathrm{d} (l/2)}{\mathrm{d}t} \frac{\partial u^{\varepsilon }_i}{\partial x_1}= - \frac{\varepsilon {L_\mathrm{c}}}{2}\frac{\mathrm{d}d}{\mathrm{d}t} \frac{\partial u^{\varepsilon }_i}{\partial x_1}\) in the vicinity of the crack tips [7, 23] has been used.
From Eqs. (18), (21), (22), (23) and (28), we deduce at the order \(\varepsilon ^{-2}\) the problems for \(\mathbf{u}^{(0)}\) and \(T^{(0)}\) as:
$$\begin{aligned}&\displaystyle \frac{\partial }{\partial y_j}\Big (a_{ijkl} e_{ykl}(\mathbf{u}^{(0)})\Big ) = 0 \qquad \text {in}\qquad {Y} \end{aligned}$$
(70)
$$\begin{aligned}&\displaystyle \big [a_{ijkl} e_{ykl}(\mathbf{u}^{(0)}) N_j\big ] = 0 ; \quad \big [u^{(0)}_i N_i\big ] = 0 ; \quad N_i a_{ijkl} e_{ykl}(\mathbf{u}^{(0)}) N_j < 0 \end{aligned}$$
(71)
$$\begin{aligned}&\displaystyle \mid T_i a_{ijkl} e_{ykl}(\mathbf{u}^{(0)}) N_j \mid = - {\mu }_\mathrm{f} N_i a_{ijkl} e_{ykl}(\mathbf{u}^{(0)}) N_j \end{aligned}$$
(72)
and
$$\begin{aligned}&\displaystyle -k\Big ( \frac{\partial ^2 T^{(0)}}{\partial y_1^2}+\frac{\partial ^2 T^{(0)}}{\partial y_2^2} \Big ) =0 \qquad \text {in}\qquad {Y} \end{aligned}$$
(73)
$$\begin{aligned}&\displaystyle \big [T^{(0)}\big ] = 0; \quad \Big [-k \frac{\partial T^{(0)}}{\partial y_{j}} N_j \Big ] = {\mu }_\mathrm{f} N_i a_{ijkl} e_{ykl}(\mathbf{u}^{(0)})\Big [\frac{\partial u^{(0)}_m}{\partial t} T_m\Big ] N_j \end{aligned}$$
(74)
$$\begin{aligned}&\displaystyle \lim _{r \rightarrow 0}\int _{\varGamma Y_{r}}-k\frac{\partial T^{(0)}}{\partial y_j}n_j \mathrm{d}s_{y} =\frac{1}{2}\frac{\mathrm{d}d}{\mathrm{d}t} {L_\mathrm{c}} {{\mathcal {G}}}^{(-1)} \ \ \text {at} \ \ {(y_{1},y_{2})= \left( \pm \frac{d L_\mathrm{c}}{2} ,0\right) } \end{aligned}$$
(75)
The solution is obtained by choosing the macroscopic displacements and temperature fields \(\mathbf{u}^{(0)}(\mathbf{x}, t )\) and \(T^{(0)}(\mathbf{x}, t )\), independent of the microscopic variables \(y_i\), which verify systems (70–72) and (73–75).
Since \(\mathbf{u}^{(0)}\) is not depending on \(\mathbf{y}\), expressions (67–68) show that \({{\mathcal {G}}}^{(-1)}\) and \({{\mathcal {G}}}^{(0)}\) are vanishing. As remarked in [16], the kinetic energy does not contribute to \({{\mathcal {G}}}^{(-1)}\) unless higher-order correctors are considered for the thermomechanical fields.
Appendix B: Computation of the effective coefficients
We provide here the details concerning the computation of the homogenized coefficients appearing in the formulation of the macroscopic elastodamage problem.
The cell problem for \({{\varvec{\xi }}}^{pq}\) is obtained in the form:
$$\begin{aligned}&\displaystyle \frac{\partial }{\partial y_j}\big (a_{ijkl} e_{ykl}({{\varvec{\xi }}}^{pq})\big )=0 \qquad \text {in}\quad Y \end{aligned}$$
(76)
$$\begin{aligned}&\displaystyle \Big [a_{ijkl} e_{ykl}({{\varvec{\xi }}}^{pq}) N_j\Big ] = - \Big [a_{ijkl} E^{pq}_{kl} N_j\Big ] \end{aligned}$$
(77)
$$\begin{aligned}&\displaystyle N_i a_{ijkl} \big (e_{ykl}({{\varvec{\xi }}}^{pq}) + E^{pq}_{kl} \big ) N_j < 0 \end{aligned}$$
(78)
$$\begin{aligned}&\displaystyle \big \vert T_i a_{ijkl} \big (e_{ykl}({{\varvec{\xi }}}^{pq}) + E^{pq}_{kl} \big ) N_j \big \vert = - {\mu }_\mathrm{f} N_i a_{ijkl} \big (e_{ykl}({{\varvec{\xi }}}^{pq}) + E^{pq}_{kl} \big ) N_j \end{aligned}$$
(79)
$$\begin{aligned}&\displaystyle \big [{\xi }^{pq}_i N_i\big ] = 0 \end{aligned}$$
(80)
The function \({\varvec{\phi }}\) is the solution of the boundary-value problem:
$$\begin{aligned}&\displaystyle \frac{\partial }{\partial y_j}\big (a_{ijkl} e_{ykl}({{\varvec{\phi }}})\big )=0 \qquad \text {in}\quad Y \end{aligned}$$
(81)
$$\begin{aligned}&\displaystyle \Big [a_{ijkl} e_{ykl}({{\varvec{\phi }}}) N_j\Big ] = \Big [b_{ij} N_j\Big ] \end{aligned}$$
(82)
$$\begin{aligned}&\displaystyle N_i \big ( a_{ijkl} e_{ykl}({{\varvec{\phi }}}) - b_{ij} \big ) N_j < 0 \end{aligned}$$
(83)
$$\begin{aligned}&\displaystyle \big \vert T_i \big ( a_{ijkl} e_{ykl}({{\varvec{\phi }}}) - b_{ij} \big ) N_j \big \vert = - {\mu }_\mathrm{f} N_i \big ( a_{ijkl} e_{ykl}({{\varvec{\phi }}}) - b_{ij} \big ) N_j \end{aligned}$$
(84)
$$\begin{aligned}&\displaystyle \big [{\phi }_i N_i\big ] = 0 \end{aligned}$$
(85)
And for the function \(\theta \) we obtain the cell problem:
$$\begin{aligned}&\displaystyle -k \Big (\frac{\partial ^2 \theta }{\partial y_1^2} + \frac{\partial ^2 \theta }{\partial y_2^2} \Big )=0 \qquad \text {in}\quad Y \end{aligned}$$
(86)
$$\begin{aligned}&\displaystyle \Big [\frac{\partial \theta }{\partial y_2}\Big ] = 0 ; \qquad [\theta ] = 0 \quad \text {on}\quad CY \end{aligned}$$
(87)
$$\begin{aligned}&\displaystyle \lim _{r \rightarrow 0}\int _{\varGamma Y_{r}}-k \Big (\frac{\partial \theta }{\partial y_1}n_1 + \frac{\partial \theta }{\partial y_2}n_2 \Big ) \mathrm{d}s_{y} = 0 \quad \text {at} \ {(y_{1},y_{2})= \left( \pm \frac{\mathrm{d} L_\mathrm{c}}{2},0\right) } \end{aligned}$$
(88)
As concerns the expressions of the homogenized coefficients, we obtain a thermal modulus \( S^{*} \) given by the formula:
$$\begin{aligned} {S}^{*}=\frac{1}{L_\mathrm{c}^2}\int _{Y} b_{ij} e_{yij} ({{\varvec{\phi }}})\mathrm{d}y \end{aligned}$$
(89)
The effective elastic moduli \(C^{\varrho }_{ijpq}\) are computed with the cell solutions as:
$$\begin{aligned} C^{\varrho }_{ijpq}=\frac{1}{L_\mathrm{c}^2}\int _{Y} a_{ijkl} \big ( E^{pq}_{kl} + e_{ykl}({{\varvec{\xi }}}^{pq})\big ) \mathrm{d}y \end{aligned}$$
(90)
and the homogenized mass density:
$$\begin{aligned} {\rho }^\mathrm{eff}=\frac{1}{\vert Y \vert }\int _{Y} \rho \mathrm{d}y \end{aligned}$$
(91)
For the effective thermoelastic coupling coefficients we obtain:
$$\begin{aligned} {\beta }^{\varrho }_{pq}= & {} \frac{1}{L_\mathrm{c}^2}\int _{Y} b_{ij}(E^{pq}_{ij} + e_{yij} ({{\varvec{\xi }}}^{pq}))\mathrm{d}y \end{aligned}$$
(92)
$$\begin{aligned} {\vartheta }_{ij}= & {} \frac{1}{L_\mathrm{c}^2}\int _{Y} \big ( b_{ij}-a_{ijkl}e_{ykl}({{\varvec{\phi }}}) \big )\mathrm{d}y \end{aligned}$$
(93)
The homogenized thermal conduction components are:
$$\begin{aligned}&\displaystyle k^\mathrm{eff}_{11}=\frac{1}{\vert Y \vert }\int _{Y} k \mathrm{d}y ; \quad k^\mathrm{eff}_{12}=\frac{1}{\vert Y \vert }\int _{Y} k \frac{\partial \theta }{\partial y_1} \mathrm{d}y ; \nonumber \\&\displaystyle k^\mathrm{eff}_{22}=\frac{1}{\vert Y \vert }\int _{Y} k \Big ( 1+\frac{\partial \theta }{\partial y_2} \Big ) \mathrm{d}y \end{aligned}$$
(94)
As explained in [53], a particular system of generators for the cell problem is necessary to respect the frictional contact conditions on the crack lips. The construction is specific to the present orientation of microcracks. The macroscopic strains \(e_{xij}(\mathbf{u}^{(0)})\) are linear combinations of the elements of the system of macroscopic strains generators \(\mathbf{E}^{pq}\):
$$\begin{aligned} e_{xij}(\mathbf{u}^{(0)}) = {\varrho }_{pq}(\mathbf{u}^{(0)}) E^{pq}_{ij} \end{aligned}$$
(95)
where \(\mathbf{E}^{pq}\) has the form:
$$\begin{aligned} \displaystyle \mathbf{E}^{11} = \begin{pmatrix} \displaystyle -\frac{1}{\gamma } &{} &{} &{} 0 \\ \\ 0 &{} &{} &{} 0 \end{pmatrix} ;\quad&\displaystyle \mathbf{E}^{12} = \begin{pmatrix} \displaystyle 0 &{} &{} \displaystyle -\frac{1}{\gamma } \\ \\ \displaystyle -\frac{1}{\gamma } &{} &{} \displaystyle -\frac{1}{2 \gamma } \end{pmatrix} ;\quad&\displaystyle \mathbf{E}^{22} = \begin{pmatrix} \displaystyle 0 &{} &{} \displaystyle -\frac{1}{\gamma } \\ \\ \displaystyle -\frac{1}{\gamma } &{} &{} \displaystyle -\frac{1}{\gamma } \end{pmatrix} \qquad \end{aligned}$$
(96)
and \({\varrho }_{pq}(\mathbf{u}^{(0)})\) are defined as:
$$\begin{aligned} {\varrho }_{11}(\mathbf{u}^{(0)})= & {} - \gamma e_{x11}(\mathbf{u}^{(0)}); \quad {\varrho }_{12}(\mathbf{u}^{(0)}) = \gamma \big ( e_{x22}(\mathbf{u}^{(0)}) - e_{x12}(\mathbf{u}^{(0)}) \big ); \qquad \nonumber \\ {\varrho }_{22}(\mathbf{u}^{(0)})= & {} \gamma \big ( e_{x12}(\mathbf{u}^{(0)}) - 2 e_{x22}(\mathbf{u}^{(0)}) \big ) \end{aligned}$$
(97)
In (46) the coefficients \(\beta _{ij}\) are obtained as linear combination of \(\beta ^{\varrho }_{ij}\) using Eq. (97):
$$\begin{aligned} \displaystyle \beta _{11} = -\gamma \beta ^{\varrho }_{11}; \quad \beta _{12} = \frac{\gamma }{2} (\beta ^{\varrho }_{22} - 2 \beta ^{\varrho }_{12}); \quad \beta _{22} = 2 \gamma (\beta ^{\varrho }_{12} - \beta ^{\varrho }_{22}) \end{aligned}$$
(98)
Using the equality \(b_{ij} = \alpha a_{ijkl} \delta _{kl} \), we can write:
$$\begin{aligned} {\beta }^{\varrho }_{pq}= \alpha C^{\varrho }_{ijpq} \delta _{ij} \end{aligned}$$
(99)
This gives:
$$\begin{aligned} \displaystyle \beta ^{\varrho }_{11}= & {} \alpha \big (C^{\varrho }_{1111} + C^{\varrho }_{2211} \big ) ; \quad \beta ^{\varrho }_{12} = \alpha \big (C^{\varrho }_{1112} + C^{\varrho }_{2212} \big ) ; \nonumber \\ \displaystyle \beta ^{\varrho }_{22}= & {} \alpha \big (C^{\varrho }_{1122} + C^{\varrho }_{2222} \big ) \end{aligned}$$
(100)
Finally we obtain \(\beta _{ij}\) as linear combination of \(C^{\varrho }_{ijpq}\):
$$\begin{aligned} \displaystyle \beta _{11}= & {} -\gamma \alpha \big (C^{\varrho }_{1111} + C^{\varrho }_{2211} \big ); \nonumber \\ \displaystyle \beta _{12}= & {} \frac{\gamma }{2} \alpha \big (C^{\varrho }_{1122} + C^{\varrho }_{2222} - 2 (C^{\varrho }_{1112} + C^{\varrho }_{2212} ) \big ); \nonumber \\ \displaystyle \beta _{22}= & {} 2 \gamma \alpha \big ( C^{\varrho }_{1112} + C^{\varrho }_{2212} - C^{\varrho }_{1122} -C^{\varrho }_{2222} \big ) \end{aligned}$$
(101)
The coefficients \(C_{ijpq}\) in Eq. (47) are linear combination of \(C^{\varrho }_{ijpq}\):
$$\begin{aligned} \displaystyle C_{ij11}= & {} -\gamma C^{\varrho }_{ij11}; \quad C_{ij12} = \frac{\gamma }{2} (C^{\varrho }_{ij22} - 2 C^{\varrho }_{ij12}); \nonumber \\ \displaystyle C_{ij22}= & {} 2 \gamma (C^{\varrho }_{ij12} - C^{\varrho }_{ij22}) \end{aligned}$$
(102)
Appendix C: Construction of the damage law
In this appendix we give details about the procedure used to obtain the damage law presented in Sect. 4.
We first prove energy balance relation (50). Starting from similar cell problems as the one in Sect. 3 for the microscopic corrector \(\mathbf{u}^{(1)}\), in [15, 53] the following energy balance has been established:
$$\begin{aligned}&\dfrac{\mathrm{d}}{\mathrm{d}t}\int _{Y}\frac{1}{2}a_{ijkl} e_{ykl}(\mathbf{u}^{(1)}) e_{yij}(\mathbf{u}^{(1)})\mathrm{d}y + \frac{\mathrm{d}d}{\mathrm{d}t} {L_\mathrm{c}} {{\mathcal {G}}}^{(1)} \nonumber \\&\quad = \int _{CY} a_{ijkl} e_{ykl}(\mathbf{u}^{(1)})N_j \left[ {\dot{u}}_i^{(1)}\right] \mathrm{d}s_y \end{aligned}$$
(103)
The equivalent form of Eq. (70):
$$\begin{aligned} \frac{\partial }{\partial y_j}\Big ( a_{ijkl}e_{xkl}(\mathbf{u}^{(0)}) - b_{ij}(T^{(0)} - T_0) \Big ) = 0 \end{aligned}$$
(104)
multiplied by \({{\dot{u}}}^{(1)}_i\) and integrated over Y leads to:
$$\begin{aligned}&\displaystyle \int _{CY} \big ( a_{ijkl}e_{xkl}(\mathbf{u}^{(0)}) - b_{ij}(T^{(0)} - T_0) \big ) N_j \Big [{{\dot{u}}}^{(1)}_i\Big ] \mathrm{d}s_y \nonumber \\&=\displaystyle \int _{Y} \big ( a_{ijkl}e_{xkl}(\mathbf{u}^{(0)}) - b_{ij}(T^{(0)} - T_0) \big ) e_{yij}({\dot{\mathbf{u}}}^{(1)}) \mathrm{d}y \end{aligned}$$
(105)
Combination of relations (103) and (105) gives:
$$\begin{aligned}&\int _{CY} \Big (a_{ijkl} \big ( e_{xkl}(\mathbf{u}^{(0)}) + e_{ykl}(\mathbf{u}^{(1)}) \big ) - b_{ij}(T^{(0)} -T_0) \Big ) N_j \left[ {\dot{u}}_i^{(1)}\right] \mathrm{d}s_y \nonumber \\&\quad = \frac{\mathrm{d}d}{\mathrm{d}t} {L_\mathrm{c}} {{\mathcal {G}}}^{(1)} + \dfrac{\mathrm{d}}{\mathrm{d}t}\int _{Y} \frac{1}{2}a_{ijkl} e_{ykl}(\mathbf{u}^{(1)}) e_{yij}(\mathbf{u}^{(1)})\mathrm{d}y \qquad \nonumber \\&\qquad + \int _{Y} \big ( a_{ijkl} e_{xkl}(\mathbf{u}^{(0)}) - b_{ij}(T^{(0)} -T_0) \big )e_{yij}({\dot{\mathbf{u}}}^{(1)}) \mathrm{d}y \end{aligned}$$
(106)
Using the Reynolds transport theorem [52] together with the singularity of the \(\mathbf{u}^{(1)}\) field, for the last integral in (106) we get:
$$\begin{aligned}&\int _{Y} \big ( a_{ijkl} e_{xkl}(\mathbf{u}^{(0)}) - b_{ij}(T^{(0)} -T_0) \big )e_{yij}({\dot{\mathbf{u}}}^{(1)}) \mathrm{d}y \nonumber \\&\quad = \dfrac{\mathrm{d}}{\mathrm{d}t}\int _{Y} \big ( a_{ijkl} e_{xkl}(\mathbf{u}^{(0)}) - b_{ij}(T^{(0)} -T_0) \big )e_{yij}(\mathbf{u}^{(1)}) \mathrm{d}y \quad \nonumber \\&\qquad - \int _{Y} \big ( a_{ijkl} e_{xkl}({\dot{\mathbf{u}}}^{(0)}) - b_{ij} {\dot{T}}^{(0)} \big )e_{yij}(\mathbf{u}^{(1)}) \mathrm{d}y \end{aligned}$$
(107)
such that (106) becomes:
$$\begin{aligned}&\int _{CY} \Big (a_{ijkl} \big ( e_{xkl}(\mathbf{u}^{(0)}) + e_{ykl}(\mathbf{u}^{(1)}) \big ) - b_{ij}(T^{(0)} -T_0) \Big ) N_j \left[ {\dot{u}}_i^{(1)}\right] \mathrm{d}s_y - \frac{\mathrm{d}d}{\mathrm{d}t} {L_\mathrm{c}} {{\mathcal {G}}}^{(1)} \nonumber \\&\qquad -\frac{1}{2} \dfrac{\mathrm{d}}{\mathrm{d}t}\int _{Y} \Big ( a_{ijkl} \big (e_{xkl}(\mathbf{u}^{(0)}) + e_{ykl}(\mathbf{u}^{(1)}) \big ) - b_{ij}(T^{(0)} -T_0) \Big )e_{yij}(\mathbf{u}^{(1)}) \mathrm{d}y \nonumber \\&\quad = \frac{1}{2} \dfrac{\mathrm{d}}{\mathrm{d}t}\int _{Y} \big ( a_{ijkl} e_{xkl}(\mathbf{u}^{(0)}) - b_{ij}(T^{(0)} -T_0) \big )e_{yij}(\mathbf{u}^{(1)}) \mathrm{d}y \nonumber \\&\qquad - \int _{Y} \big ( a_{ijkl} e_{xkl}({\dot{\mathbf{u}}}^{(0)}) - b_{ij} {\dot{T}}^{(0)} \big )e_{yij}(\mathbf{u}^{(1)}) \mathrm{d}y \end{aligned}$$
(108)
On the other hand, we can rewrite Eq. (29) as:
$$\begin{aligned} \frac{\partial }{\partial y_j}\Big ( a_{ijkl} \big (e_{xkl}(\mathbf{u}^{(0)}) + e_{ykl}(\mathbf{u}^{(1)}) \big ) - b_{ij}(T^{(0)} - T_0) \Big ) = 0 \end{aligned}$$
(109)
Multiplication by \(u_i^{(1)}\) and integration on Y give:
$$\begin{aligned}&\int _{Y} \Big ( a_{ijkl} \big (e_{xkl}(\mathbf{u}^{(0)}) + e_{ykl}(\mathbf{u}^{(1)})\big ) - b_{ij}(T^{(0)} - T_0) \Big ) e_{yij}(\mathbf{u}^{(1)}) \mathrm{d}y \nonumber \\&\quad = \int _{CY} \Big ( a_{ijkl} \big (e_{xkl}(\mathbf{u}^{(0)}) + e_{ykl}(\mathbf{u}^{(1)})\big ) - b_{ij}(T^{(0)} - T_0) \Big ) N_j \Big [{u}^{(1)}_i\Big ] \mathrm{d}s_y \end{aligned}$$
(110)
This identity can be used in (108) to obtain energy relation (50).
Let us now prove damage relations (51–53). From the expression of \(\mathbf{u}^{(1)}\) in (38), we deduce the following relations:
$$\begin{aligned} \int _Y a_{ijkl}e_{yij}(\mathbf{u}^{(1)})\mathrm{d}y= & {} \int _Y a_{ijkl}e_{yij}({{\varvec{\xi }}}^{pq})\mathrm{d}y \varrho _{pq}(\mathbf{u}^{(0)}) \nonumber \\&+ \int _Y a_{ijkl}e_{yij}({\varvec{\phi }})\mathrm{d}y(T^{(0)} - T_0) \end{aligned}$$
(111)
$$\begin{aligned} \int _Y b_{ij}e_{yij}(\mathbf{u}^{(1)})\mathrm{d}y= & {} \int _Y b_{ij}e_{yij}({{\varvec{\xi }}}^{pq})\mathrm{d}y \varrho _{pq}(\mathbf{u}^{(0)}) \nonumber \\&+ \int _Y b_{ij}e_{yij}({\varvec{\phi }})\mathrm{d}y(T^{(0)} - T_0) \end{aligned}$$
(112)
And using the expressions of effective coefficients (90), (92), (93) and (89),we deduce:
$$\begin{aligned} \int _Y a_{ijkl}e_{yij}({{\varvec{\xi }}}^{pq})\mathrm{d}y= & {} L_\mathrm{c}^2 C^{\varrho }_{klpq} - a_{klij} E^{pq}_{ij} \end{aligned}$$
(113)
$$\begin{aligned} \int _Y b_{ij}e_{yij}({{\varvec{\xi }}}^{pq})\mathrm{d}y= & {} \alpha \delta _{kl} \big ( L_\mathrm{c}^2 C^{\varrho }_{klpq} - a_{klij} E^{pq}_{ij} \big ) \end{aligned}$$
(114)
$$\begin{aligned} \int _Y a_{ijkl}e_{yij}({{\varvec{\phi }}})\mathrm{d}y= & {} b_{kl} - L_\mathrm{c}^2 \vartheta _{kl} \end{aligned}$$
(115)
$$\begin{aligned} \int _Y b_{ij}e_{yij}({{\varvec{\phi }}})\mathrm{d}y= & {} L_\mathrm{c}^2 S^* \end{aligned}$$
(116)
Replacing Eqs. (113–116) in (111–112) gives:
$$\begin{aligned}&\displaystyle \int _Y a_{ijkl}e_{yij}(\mathbf{u}^{(1)})\mathrm{d}y = \big ( L_\mathrm{c}^2 C^{\varrho }_{klpq} - a_{klij} E^{pq}_{ij} \big ) \varrho _{pq}(\mathbf{u}^{(0)}) \nonumber \\&\quad \displaystyle + \big ( b_{kl} - L_\mathrm{c}^2 \vartheta _{kl} \big )(T^{(0)} - T_0) \end{aligned}$$
(117)
$$\begin{aligned}&\int _Y b_{ij}e_{yij}(\mathbf{u}^{(1)})\mathrm{d}y = \alpha \delta _{kl} \big ( L_\mathrm{c}^2 C^{\varrho }_{klpq} - a_{klij} E^{pq}_{ij} \big ) \varrho _{pq}(\mathbf{u}^{(0)}) \nonumber \\&\quad + L_\mathrm{c}^2 S^* (T^{(0)} - T_0) \end{aligned}$$
(118)
With relations (117) and (118), we can calculate the first term in the right side of (50) to obtain:
$$\begin{aligned}&\displaystyle \dfrac{1}{2}\dfrac{\mathrm{d}}{\mathrm{d}t}\int _{Y} \big ( a_{ijkl} e_{xkl}(\mathbf{u}^{(0)}) - b_{ij}(T^{(0)} -T_0) \big )e_{yij}(\mathbf{u}^{(1)}) d_y \nonumber \\&\quad =\displaystyle \dfrac{1}{2} L_\mathrm{c}^2 \dfrac{\mathrm{d}C^{\varrho }_{klpq}}{\mathrm{d}d} \dfrac{\mathrm{d}d}{\mathrm{d}t} \varrho _{pq}(\mathbf{u}^{(0)}) \varrho _{mn}(\mathbf{u}^{(0)}) E^{mn}_{kl} \nonumber \\&\qquad \displaystyle - \dfrac{1}{2} L_\mathrm{c}^2 \dfrac{\mathrm{d} {\vartheta }_{kl}}{\mathrm{d}d} \dfrac{\mathrm{d}d}{\mathrm{d}t} (T^{(0)} - T_0) \varrho _{mn}(\mathbf{u}^{(0)}) E^{mn}_{kl} \nonumber \\&\qquad \displaystyle - \dfrac{1}{2} L_\mathrm{c}^2 \dfrac{\mathrm{d}C^{\varrho }_{klpq}}{\mathrm{d}d} \dfrac{\mathrm{d}d}{\mathrm{d}t} (T^{(0)} - T_0) \alpha \delta _{kl} \varrho _{pq}(\mathbf{u}^{(0)}) \nonumber \\&\qquad \displaystyle - \dfrac{1}{2} L_\mathrm{c}^2 \dfrac{\mathrm{d} S^*}{\mathrm{d}d} \dfrac{\mathrm{d}d}{\mathrm{d}t} (T^{(0)} - T_0)^2 \nonumber \\&\qquad \displaystyle + \dfrac{1}{2} \big ( L_\mathrm{c}^2 C^{\varrho }_{klpq} - a_{klij} E^{pq}_{ij} \big ) \big ( \varrho _{pq}(\dot{\mathbf{u}}^{(0)}) \varrho _{mn}(\mathbf{u}^{(0)}) - \varrho _{pq}(\mathbf{u}^{(0)}) \varrho _{mn}(\dot{\mathbf{u}}^{(0)}) \big ) E^{mn}_{kl} \nonumber \\&\qquad \displaystyle + \dfrac{1}{2} \big ( b_{kl} - L_\mathrm{c}^2 {\vartheta }_{kl} \big ) \big ( (T^{(0)} - T_0) \varrho _{mn}(\dot{\mathbf{u}}^{(0)}) + \varrho _{mn}(\mathbf{u}^{(0)}) {\dot{T}}^{(0)} \big ) E^{mn}_{kl} \nonumber \\&\qquad \displaystyle - \dfrac{1}{2} \alpha \delta _{kl} \big ( L_\mathrm{c}^2 C^{\varrho }_{klpq} - a_{klij} E^{pq}_{ij} \big ) \big ( (T^{(0)} - T_0) \varrho _{pq}(\dot{\mathbf{u}}^{(0)}) + \varrho _{pq}(\mathbf{u}^{(0)}) {\dot{T}}^{(0)}\big ) \nonumber \\&\qquad \displaystyle - L_\mathrm{c}^2 S^* {\dot{T}}^{(0)} (T^{(0)} - T_0) \; \end{aligned}$$
(119)
In the same way, we obtain for the second term in the right side of (50):
$$\begin{aligned}&\displaystyle \int _{Y} \big ( a_{ijkl} e_{xkl}({\dot{\mathbf{u}}}^{(0)}) - b_{ij} {\dot{T}}^{(0)} \big )e_{yij}(\mathbf{u}^{(1)}) d_y \nonumber \\&\quad =\displaystyle \big ( L_\mathrm{c}^2 C^{\varrho }_{klpq} - a_{klij} E^{pq}_{ij} \big ) \varrho _{pq}(\mathbf{u}^{(0)}) \varrho _{mn}(\dot{\mathbf{u}}^{(0)}) E^{mn}_{kl} \nonumber \\&\qquad \displaystyle + \big ( b_{kl} - L_\mathrm{c}^2 {\vartheta }_{kl} \big ) (T^{(0)} - T_0) \varrho _{mn}(\dot{\mathbf{u}}^{(0)}) E^{mn}_{kl} \nonumber \\&\qquad \displaystyle - \alpha \delta _{kl} \big ( L_\mathrm{c}^2 C^{\varrho }_{klpq} - a_{klij} E^{pq}_{ij} \big ) \varrho _{pq}(\mathbf{u}^{(0)}) {\dot{T}}^{(0)} - L_\mathrm{c}^2 S^* {\dot{T}}^{(0)} (T^{(0)} - T_0) \end{aligned}$$
(120)
Replacing (119, 120) in (50) we obtain:
$$\begin{aligned}&\displaystyle \dfrac{1}{L_\mathrm{c}^2} \int _{CY} \sigma ^{(0)}_{ij} N_j \left[ {\dot{u}}_i^{(1)}\right] \mathrm{d}s_y - \dfrac{1}{2L_\mathrm{c}^2} \dfrac{\mathrm{d}}{\mathrm{d}t} \int _{CY} \sigma ^{(0)}_{ij} N_j \Big [{u}^{(1)}_i\Big ] \mathrm{d}s_y - \dfrac{1}{L_\mathrm{c}} \frac{\mathrm{d}d}{\mathrm{d}t} {{\mathcal {G}}}^{(1)} \nonumber \\&\quad =\displaystyle \dfrac{\mathrm{d}d}{\mathrm{d}t} \bigg ( \dfrac{1}{2} \dfrac{\mathrm{d}C^{\varrho }_{klpq}}{\mathrm{d}d} \varrho _{pq}(\mathbf{u}^{(0)}) \varrho _{mn}(\mathbf{u}^{(0)}) E^{mn}_{kl} - \dfrac{1}{2} \dfrac{\mathrm{d} {\vartheta }_{kl}}{\mathrm{d}d} (T^{(0)} - T_0) \varrho _{mn}(\mathbf{u}^{(0)}) E^{mn}_{kl} \nonumber \\&\qquad - \dfrac{1}{2} \dfrac{\mathrm{d}C^{\varrho }_{klpq}}{\mathrm{d}d} (T^{(0)} - T_0) \alpha \delta _{kl} \varrho _{pq}(\mathbf{u}^{(0)}) - \dfrac{1}{2} \dfrac{\mathrm{d} S^*}{\mathrm{d}d} (T^{(0)} - T_0)^2 \bigg ) \nonumber \\&\qquad + \displaystyle \dfrac{1}{2 L_\mathrm{c}^2}\int _{CY} a_{ijkl}E^{mn}_{kl}N_j \big [ \xi ^{pq}_i \big ] \mathrm{d}s_y \big ( \varrho _{pq}(\dot{\mathbf{u}}^{(0)}) \varrho _{mn}(\mathbf{u}^{(0)}) - \varrho _{pq}(\mathbf{u}^{(0)}) \varrho _{mn}(\dot{\mathbf{u}}^{(0)}) \big ) \nonumber \\&\qquad + \displaystyle \dfrac{1}{2 L_\mathrm{c}^2}\int _{CY} a_{ijkl}E^{mn}_{kl}N_j \big [ \phi _i \big ] \mathrm{d}s_y \big ( \varrho _{mn}(\mathbf{u}^{(0)}) {\dot{T}}^{(0)} - (T^{(0)} - T_0) \varrho _{mn}(\dot{\mathbf{u}}^{(0)}) \big ) \nonumber \\&\qquad + \displaystyle \dfrac{1}{2 L_\mathrm{c}^2}\int _{CY} b_{ij} N_j \big [ \xi ^{pq}_i \big ] \mathrm{d}s_y \big ( \varrho _{pq}(\mathbf{u}^{(0)}) {\dot{T}}^{(0)} - (T^{(0)} - T_0) \varrho _{pq}(\dot{\mathbf{u}}^{(0)}) \big ) \end{aligned}$$
(121)
With the expressions of \(\mathbf{u}^{(1)}\) and \(\sigma _{ij}^{(0)}\) we calculate the second term in the left side of (121) as:
$$\begin{aligned}&\displaystyle \dfrac{1}{2L_\mathrm{c}^2} \dfrac{\mathrm{d}}{\mathrm{d}t} \int _{CY} \sigma ^{(0)}_{ij} N_j \left[ u_i^{(1)}\right] \mathrm{d}s_y = \displaystyle \bigg ( \dfrac{1}{2} \dfrac{\mathrm{d}{{\mathcal {Z}}}^{\varrho 1}_{mnpq}}{\mathrm{d}d} \varrho _{mn}(\mathbf{u}^{(0}) \varrho _{pq}(\mathbf{u}^{(0})\nonumber \\&\quad \displaystyle + \dfrac{1}{2} \Big ( \dfrac{\mathrm{d}{{\mathcal {Z}}}^{\varrho 2}_{mn}}{\mathrm{d}d} + \dfrac{\mathrm{d}{{\mathcal {Z}}}^{\varrho 3}_{mn}}{\mathrm{d}d} \Big ) \varrho _{mn}(\mathbf{u}^{(0}) (T^{(0)} - T_0) + \dfrac{1}{2} \dfrac{\mathrm{d}{{\mathcal {Z}}}^4}{\mathrm{d}d} (T^{(0)} - T_0)^2 \bigg ) \dfrac{\mathrm{d}d}{\mathrm{d}t} \nonumber \\&\quad \displaystyle + \dfrac{1}{2} {{\mathcal {Z}}}^{\varrho 1}_{mnpq} \big ( \varrho _{mn}(\dot{\mathbf{u}}^{(0}) \varrho _{pq}(\mathbf{u}^{(0}) + \varrho _{mn}(\mathbf{u}^{(0}) \varrho _{pq}(\dot{\mathbf{u}}^{(0}) \big ) \nonumber \\&\quad \displaystyle + \dfrac{1}{2} \big ( {{\mathcal {Z}}}^{\varrho 2}_{mn} + {{\mathcal {Z}}}^{\varrho 3}_{mn} \big ) \big ( \varrho _{mn}(\dot{\mathbf{u}}^{(0}) (T^{(0)} - T_0) + \varrho _{mn}(\mathbf{u}^{(0}) {\dot{T}}^{(0)} \big ) \nonumber \\&\quad \displaystyle + {{\mathcal {Z}}}^4 {\dot{T}}^{(0)} (T^{(0)} - T_0) \end{aligned}$$
(122)
where
$$\begin{aligned} \displaystyle {{\mathcal {Z}}}_{mnpq}^{\varrho 1}= & {} \dfrac{1}{L_\mathrm{c}^2} \int _{CY} a_{ijkl} \big ( E^{mn}_{kl} + e_{ykl} ({{\varvec{\xi }}}^{mn}) \big ) N_j \left[ \xi _i^{pq}\right] \mathrm{d}s_y \end{aligned}$$
(123)
$$\begin{aligned} \displaystyle {{\mathcal {Z}}}_{mn}^{\varrho 2}= & {} \dfrac{1}{L_\mathrm{c}^2} \int _{CY} a_{ijkl} \big ( E^{mn}_{kl} + e_{ykl} ({{\varvec{\xi }}}^{mn}) \big ) N_j \left[ \phi _i \right] \mathrm{d}s_y \end{aligned}$$
(124)
$$\begin{aligned} \displaystyle {{\mathcal {Z}}}_{mn}^{\varrho 3}= & {} -\dfrac{1}{L_\mathrm{c}^2}\int _{CY} \big ( b_{ij} - a_{ijkl} e_{ykl} ({{\varvec{\phi }}}) \big ) N_j \big [ \xi ^{mn}_i \big ] \mathrm{d}s_y \end{aligned}$$
(125)
$$\begin{aligned} \displaystyle {{\mathcal {Z}}}^4= & {} - \dfrac{1}{L_\mathrm{c}^2} \int _{CY} \big ( b_{ij} - a_{ijkl} e_{ykl} ({{\varvec{\phi }}}) \big ) N_j \left[ \phi _i\right] \mathrm{d}s_y \end{aligned}$$
(126)
When the time derivative of \(\mathbf{u}^{(1)}\):
$$\begin{aligned} \displaystyle \dot{\mathbf{u}}^{(1)}&= \dot{{{\varvec{\xi }}}}^{pq}(\mathbf{y}) {\varrho }_{pq}(\mathbf{u}^{(0)})(\mathbf{x},t) + {{\varvec{\xi }}}^{pq}(\mathbf{y}) {\varrho }_{pq}(\dot{\mathbf{u}}^{(0)})(\mathbf{x},t) \nonumber \\&\quad + \dot{{{\varvec{\phi }}}}(\mathbf{y})(T^{(0)}{(\mathbf{x}}, t)-T_0) + {{\varvec{\phi }}}(\mathbf{y}){\dot{T}}^{(0)}{(\mathbf{x}}, t) \end{aligned}$$
(127)
is used in the first integral in the left member of (121) one obtains:
$$\begin{aligned}&\displaystyle \dfrac{1}{L_\mathrm{c}^2} \int _{CY} \sigma ^{(0)}_{ij} N_j \left[ {\dot{u}}_i^{(1)}\right] \mathrm{d}s_y = \Big ( I^{\varrho 2}_{mnpq} \varrho _{mn}(\mathbf{u}^{(0}) \varrho _{pq}(\mathbf{u}^{(0}) \nonumber \\&\quad \displaystyle + J^{\varrho 2}_{mn} \varrho _{mn}(\mathbf{u}^{(0}) (T^{(0)} - T_0) - P^2 (T^{(0)} - T_0)^2 \Big ) \dfrac{\mathrm{d}d}{\mathrm{d}t} \nonumber \\&\quad \displaystyle + {{\mathcal {Z}}}^{\varrho 1}_{mnpq} \varrho _{mn}(\mathbf{u}^{(0}) \varrho _{pq}(\dot{\mathbf{u}}^{(0}) + {{\mathcal {Z}}}^{\varrho 2}_{mn} \varrho _{mn}(\mathbf{u}^{(0}) {\dot{T}}^{(0)} \nonumber \\&\quad \displaystyle + {{\mathcal {Z}}}^{\varrho 3}_{mn} \varrho _{mn}(\dot{\mathbf{u}}^{(0}) (T^{(0)} - T_0) + {{\mathcal {Z}}}^4 {\dot{T}}^{(0)} (T^{(0)} - T_0) \end{aligned}$$
(128)
with
$$\begin{aligned} \displaystyle I^{\varrho 2}_{mnpq}&= \dfrac{1}{L_\mathrm{c}^2} \int _{CY} a_{ijkl} \big ( E^{mn}_{kl} + e_{ykl} ({{\varvec{\xi }}}^{mn}) \big ) N_j \left[ \dfrac{\mathrm{d} \xi _i^{pq}}{\mathrm{d}d} \right] \mathrm{d}s_y \end{aligned}$$
(129)
$$\begin{aligned} \displaystyle J^{\varrho 2}_{mn}&= \dfrac{1}{L_\mathrm{c}^2} \int _{CY} a_{ijkl} \big ( E^{mn}_{kl} + e_{ykl} ({{\varvec{\xi }}}^{mn}) \big ) N_j \left[ \dfrac{\mathrm{d} \phi _i}{\mathrm{d}d} \right] \mathrm{d}s_y \end{aligned}$$
(130)
$$\begin{aligned}&\quad \displaystyle - \dfrac{1}{L_\mathrm{c}^2} \int _{CY} \big ( b_{ij} - a_{ijkl} e_{ykl} ({{\varvec{\phi }}}) \big ) N_j \left[ \dfrac{\mathrm{d} \xi _i^{mn}}{\mathrm{d}d}\right] \mathrm{d}s_y \nonumber \\ \displaystyle P^2&= \dfrac{1}{2L_\mathrm{c}^2} \dfrac{\mathrm{d}}{\mathrm{d}d} \int _{CY} \big ( b_{ij} - a_{ijkl} e_{ykl} ({{\varvec{\phi }}}) \big ) N_j \left[ \dfrac{\mathrm{d} \phi _i}{\mathrm{d}d} \right] \mathrm{d}s_y \end{aligned}$$
(131)
Using cell problems (76), (81) one can prove that the integral \( \int _{CY} a_{ijkl} e_{ykl} ({{\varvec{\xi }}}^{mn}) N_j \left[ \xi _i^{pq}\right] \mathrm{d}s_y\) is symmetrical in respect of the pairs mn and pq and we have
$$\begin{aligned} \displaystyle \int _{CY} a_{ijkl} e_{ykl} ({{\varvec{\xi }}}^{mn}) N_j \left[ \phi _i \right] \mathrm{d}s_y = \displaystyle \int _{CY} a_{ijkl} e_{ykl} ({{\varvec{\phi }}}) N_j \left[ \xi _i^{pq}\right] \mathrm{d}s_y \end{aligned}$$
Combination with (122) and (128) and substitution in (121) allow us to obtain the damage relation:
$$\begin{aligned}&\displaystyle \dfrac{\mathrm{d}d}{\mathrm{d}t} \bigg ( \displaystyle \dfrac{{{\mathcal {G}}}^{(1)}}{L_\mathrm{c}} + \dfrac{1}{2} \dfrac{\mathrm{d}C^{\varrho }_{klpq}}{\mathrm{d}d} \varrho _{pq}(\mathbf{u}^{(0)}) \varrho _{mn}(\mathbf{u}^{(0)}) E^{mn}_{kl} \nonumber \\&\quad \displaystyle - \dfrac{1}{2} \dfrac{\mathrm{d} {\vartheta }_{kl}}{\mathrm{d}d} (T^{(0)} - T_0) \varrho _{mn}(\mathbf{u}^{(0)}) E^{mn}_{kl} \nonumber \\&\quad \displaystyle - \dfrac{1}{2} \dfrac{\mathrm{d}C^{\varrho }_{klpq}}{\mathrm{d}d} (T^{(0)} - T_0) \alpha \delta _{kl} \varrho _{pq}(\mathbf{u}^{(0)}) - \dfrac{1}{2} \dfrac{\mathrm{d} S^*}{\mathrm{d}d} (T^{(0)} - T_0)^2 \nonumber \\&\quad \displaystyle + I^{\varrho }_{mnpq} \varrho _{mn}(\mathbf{u}^{(0}) \varrho _{pq}(\mathbf{u}^{(0}) \nonumber \\&\quad + J^{\varrho }_{mn} \varrho _{mn}(\mathbf{u}^{(0}) (T^{(0)} - T_0) - P^{\varrho } (T^{(0)} - T_0)^2 \bigg ) = 0 \end{aligned}$$
(132)
with the effective coefficients
$$\begin{aligned} \displaystyle I^{\varrho }_{mnpq}&= \dfrac{1}{2L_\mathrm{c}^2} \dfrac{\mathrm{d}}{\mathrm{d}d} \int _{CY} a_{ijkl} \big ( E^{mn}_{kl} + e_{ykl} ({{\varvec{\xi }}}^{mn}) \big ) N_j \left[ \xi _i^{pq}\right] \mathrm{d}s_y \nonumber \\&\quad \displaystyle - \dfrac{1}{L_\mathrm{c}^2} \int _{CY} a_{ijkl} \big ( E^{mn}_{kl} + e_{ykl} ({{\varvec{\xi }}}^{mn}) \big ) N_j \left[ \dfrac{\mathrm{d} \xi _i^{pq}}{\mathrm{d}d} \right] \mathrm{d}s_y \end{aligned}$$
(133)
$$\begin{aligned} \displaystyle J^{\varrho }_{mn}&= \dfrac{1}{2L_\mathrm{c}^2} \dfrac{\mathrm{d}}{\mathrm{d}d} \int _{CY} a_{ijkl} \big ( E^{mn}_{kl} + e_{ykl} ({{\varvec{\xi }}}^{mn}) \big ) N_j \left[ \phi _i \right] \mathrm{d}s_y \nonumber \\&\quad - \displaystyle \dfrac{1}{2L_\mathrm{c}^2} \dfrac{\mathrm{d}}{\mathrm{d}d} \int _{CY} \big ( b_{ij} - a_{ijkl} e_{ykl} ({{\varvec{\phi }}}) \big ) N_j \left[ \xi _i^{mn}\right] \mathrm{d}s_y \nonumber \\&\quad \displaystyle - \dfrac{1}{L_\mathrm{c}^2} \int _{CY} a_{ijkl} \big ( E^{mn}_{kl} + e_{ykl} ({{\varvec{\xi }}}^{mn}) \big ) N_j \left[ \dfrac{\mathrm{d} \phi _i}{\mathrm{d}d} \right] \mathrm{d}s_y \nonumber \\&\quad \displaystyle + \int _{CY} \big ( b_{ij} - a_{ijkl} e_{ykl} ({{\varvec{\phi }}}) \big ) N_j \left[ \dfrac{\mathrm{d} \xi _i^{mn}}{\mathrm{d}d}\right] \mathrm{d}s_y \end{aligned}$$
(134)
$$\begin{aligned} \displaystyle P&= \dfrac{1}{2L_\mathrm{c}^2} \dfrac{\mathrm{d}}{\mathrm{d}d} \int _{CY} \big ( b_{ij} - a_{ijkl} e_{ykl} ({{\varvec{\phi }}}) \big ) N_j \left[ \phi _i \right] \mathrm{d}s_y \nonumber \\&\quad \displaystyle - \dfrac{1}{L_\mathrm{c}^2} \int _{CY} \big ( b_{ij} - a_{ijkl} e_{ykl} ({{\varvec{\phi }}}) \big ) N_j \left[ \dfrac{\mathrm{d} \phi _i}{\mathrm{d}d}\right] \mathrm{d}s_y \end{aligned}$$
(135)
In Eq. (53) the coefficients \(I_{mnpq}\) and \(J_{mn}\) are linear combinations of \(I^{\varrho }_{mnpq}\) and, respectively, \(J^{\varrho }_{mn}\). For \(I_{mnpq}\) we get:
$$\begin{aligned} \displaystyle I_{1212}= & {} \dfrac{\gamma ^2}{4} \big ( I^{\varrho }_{2222} - 2 I^{\varrho }_{1222} - 2 I^{\varrho }_{2212} + 4 I^{\varrho }_{1212} \big ) \nonumber \\ \displaystyle I_{2222}= & {} 4 \gamma ^2 \big ( I^{\varrho }_{2222} - I^{\varrho }_{1222} - I^{\varrho }_{2212} + I^{\varrho }_{1212} \big ) \nonumber \\ \displaystyle I_{1222} + I_{2212}= & {} \gamma ^2 \big ( -2 I^{\varrho }_{2222} + 3 I^{\varrho }_{1222} + 3 I^{\varrho }_{2212} - 4 I^{\varrho }_{1212} \big ) \end{aligned}$$
(136)
while the other components \(I_{1111}, \; I_{1112}, \; I_{1122}, \; I_{2211}, \; I_{1211}\) are vanishing.
Similar combinations are obtained for \(I^{2}_{mnpq}\) of Eq. (61).
$$\begin{aligned} \displaystyle I^2_{1212}= & {} \dfrac{\gamma ^2}{4} \big ( I^{\varrho 2}_{2222} - 2 I^{\varrho 2}_{1222} - 2 I^{\varrho 2}_{2212} + 4 I^{\varrho 2}_{1212} \big ) \nonumber \\ \displaystyle I^2_{2222}= & {} 4 \gamma ^2 \big ( I^{\varrho 2}_{2222} - I^{\varrho 2}_{1222} - I^{\varrho 2}_{2212} + I^{\varrho 2}_{1212} \big ) \nonumber \\ \displaystyle I^2_{1222} + I^2_{2212}= & {} \gamma ^2 \big ( -2 I^{\varrho 2}_{2222} + 3 I^{\varrho 2}_{1222} + 3 I^{\varrho 2}_{2212} - 4 I^{\varrho 2}_{1212} \big ) \nonumber \\ I^2_{1111}= & {} I^2_{1112} = I^2_{1122} = I^2_{2211} = I^2_{1211} = 0 \end{aligned}$$
(137)
Expressions of \(J_{mn}\) and \(J^2_{mn}\) in definition (53) of \({{\mathcal {Y}}}_\mathrm{f}\) and, respectively, (61) are:
$$\begin{aligned}&\displaystyle J_{11}= -\gamma J^{\varrho }_{11} \, ;\quad J_{12}= \dfrac{\gamma }{2} \big ( J^{\varrho }_{22} - 2 J^{\varrho }_{12}\big ) \, ;\quad J_{22}= 2 \gamma \big ( J^{\varrho }_{12} - J^{\varrho }_{22}\big ) \end{aligned}$$
(138)
$$\begin{aligned}&\displaystyle J^2_{11}= -\gamma J^{\varrho 2}_{11} \, ;\quad J^2_{12}= \dfrac{\gamma }{2} \big ( J^{\varrho 2}_{22} - 2 J^{\varrho 2}_{12}\big ) \, ;\quad J^2_{22}= 2 \gamma \big ( J^{\varrho 2}_{12} - J^{\varrho 2}_{22}\big ) \end{aligned}$$
(139)
Finally, the coefficients \({{\mathcal {Z}}}^1_{mnpq}\), \({{\mathcal {Z}}}_{mn}^{2}\) and \({{\mathcal {Z}}}_{mn}^{3}\) in (61) are linear combinations of \({{\mathcal {Z}}}^{\varrho 1}_{mnpq}\), \({{\mathcal {Z}}}_{mn}^{\varrho 2}\) and \({{\mathcal {Z}}}_{mn}^{\varrho 3}\) correspondingly:
$$\begin{aligned} \displaystyle {{\mathcal {Z}}}^1_{1212}= & {} \dfrac{\gamma ^2}{4} \big ( {{\mathcal {Z}}}^{\varrho 1}_{2222} - 2 {{\mathcal {Z}}}^{\varrho 1}_{1222} - 2 {{\mathcal {Z}}}^{\varrho 1}_{2212} + 4 {{\mathcal {Z}}}^{\varrho 1}_{1212} \big ) \nonumber \\ \displaystyle {{\mathcal {Z}}}^1_{2222}= & {} 4 \gamma ^2 \big ( {{\mathcal {Z}}}^{\varrho 1}_{2222} - {{\mathcal {Z}}}^{\varrho 1}_{1222} - {{\mathcal {Z}}}^{\varrho 1}_{2212} + {{\mathcal {Z}}}^{\varrho 1}_{1212} \big ) \nonumber \\ \displaystyle {{\mathcal {Z}}}^1_{2212}= & {} \gamma ^2 \big ( - {{\mathcal {Z}}}^{\varrho 1}_{2222} + {{\mathcal {Z}}}^{\varrho 1}_{1222} + 2 {{\mathcal {Z}}}^{\varrho 1}_{2212} - 2 {{\mathcal {Z}}}^{\varrho 1}_{1212} \big ) \nonumber \\ \displaystyle {{\mathcal {Z}}}^1_{1222}= & {} \gamma ^2 \big ( - {{\mathcal {Z}}}^{\varrho 1}_{2222} + 2 {{\mathcal {Z}}}^{\varrho 1}_{1222} + {{\mathcal {Z}}}^{\varrho 1}_{2212} - 2 {{\mathcal {Z}}}^{\varrho 1}_{1212} \big ) \nonumber \\ {{\mathcal {Z}}}^1_{1111}= & {} {{\mathcal {Z}}}^1_{1112} = {{\mathcal {Z}}}^1_{1122} = {{\mathcal {Z}}}^1_{2211} = {{\mathcal {Z}}}^1_{1211} = 0. \end{aligned}$$
(140)
$$\begin{aligned} \displaystyle {{\mathcal {Z}}}^2_{11}= & {} -\gamma {{\mathcal {Z}}}^{\varrho 2}_{11} \, ;\quad {{\mathcal {Z}}}^2_{12}= \dfrac{\gamma }{2} \big ( {{\mathcal {Z}}}^{\varrho 2}_{22} - 2 {{\mathcal {Z}}}^{\varrho 2}_{12}\big ) \, ;\quad {{\mathcal {Z}}}^2_{22}= 2 \gamma \big ( {{\mathcal {Z}}}^{\varrho 2}_{12} - {{\mathcal {Z}}}^{\varrho 2}_{22}\big ) \end{aligned}$$
(141)
$$\begin{aligned} \displaystyle {{\mathcal {Z}}}^3_{11}= & {} -\gamma {{\mathcal {Z}}}^{\varrho 3}_{11} \, ;\quad {{\mathcal {Z}}}^3_{12}= \dfrac{\gamma }{2} \big ( {{\mathcal {Z}}}^{\varrho 3}_{22} - 2 {{\mathcal {Z}}}^{\varrho 3}_{12}\big ) \, ;\quad {{\mathcal {Z}}}^3_{22}= 2 \gamma \big ( {{\mathcal {Z}}}^{\varrho 3}_{12} - {{\mathcal {Z}}}^{\varrho 3}_{22}\big ) \end{aligned}$$
(142)
