Appendix A
Stress resultants and couple stress resultants in terms of u,v,w,\(\phi _{x}\), \(\phi _{y}\), \(\gamma \), and \(\zeta \):
$$\begin{aligned} N_{xx}= & {} h\left[ {C_{11} u_{,x} +C_{12} v_{,y} } \right. +k_{s} C_{14} \left( {w_{,y} -\phi _{y} } \right) +k_{s} C_{15} \left( {w_{,x} -\phi _{x} } \right) +C_{16} \left( {u_{,y} +v_{,x} } \right) \nonumber \\&+\frac{1}{2}B_{11} \left( {w_{,xy} +\phi _{y,x} } \right) -\frac{1}{2}B_{12} \left( {w_{,xy} +\phi _{x,y} } \right) +\frac{1}{2}B_{13} \left( {\phi _{x,y} -\phi _{y,x} } \right) +\frac{1}{2}B_{14} \left( {v_{,xy} -u_{,yy} } \right) \nonumber \\&+\frac{1}{2}B_{15} \left( {v_{,xx} -u_{,xy} } \right) \left. {+\frac{1}{2}B_{16} \left( {w_{,yy} +\phi _{y,y} -w_{,xx} -\phi _{x,x} } \right) } \right] -\frac{2h}{\pi }e_{11} \gamma _{,x} -\frac{2h}{\pi }e_{21} \gamma _{,y} \nonumber \\&+2e_{31} \gamma _{0} -\frac{2h}{\pi }q_{11} \zeta _{,x} -\frac{2h}{\pi }q_{21} \zeta _{,y} +2q_{31} \zeta _{0} , \end{aligned}$$
(A.1)
$$\begin{aligned} N_{yy}= & {} h\left[ {C_{12} u_{,x} +C_{22} v_{,y} +k_{s} C_{24} \left( {w_{,y} -\phi _{y} } \right) } \right. +k_{s} C_{25} \left( {w_{,x} -\phi _{x} } \right) +C_{26} \left( {u_{,y} +v_{,x} } \right) \nonumber \\&+\frac{1}{2}B_{21} \left( {w_{,xy} +\phi _{y,x} } \right) -\frac{1}{2}B_{22} \left( {w_{,xy} +\phi _{x,y} } \right) +\frac{1}{2}B_{23} \left( {\phi _{x,y} -\phi _{y,x} } \right) +\frac{1}{2}B_{24} \left( {v_{,xy} -u_{,yy} } \right) \nonumber \\&+\frac{1}{2}B_{25} \left( {v_{,xx} -u_{,xy} } \right) +\left. {\frac{1}{2}B_{26} \left( {w_{,yy} +\phi _{y,y} -w_{,xx} -\phi _{x,x} } \right) } \right] -\frac{2h}{\pi }e_{12} \gamma _{,x} -\frac{2h}{\pi }e_{22} \gamma _{,y} \nonumber \\&+2e_{32} \gamma _{0} -\frac{2h}{\pi }q_{12} \zeta _{,x} -\frac{2h}{\pi }q_{22} \zeta _{,y} +2q_{32} \zeta _{0} , \end{aligned}$$
(A.2)
$$\begin{aligned} N_{zy}= & {} h\left[ {k_{s} C_{14} u_{,x} +k_{s} C_{24} v_{,y} } \right. +k_{s}^{2} C_{44} \left( {w_{,y} -\phi _{y} } \right) +k_{s}^{2} C_{45} \left( {w_{,x} -\phi _{x} } \right) +k_{s} C_{46} \left( {u_{,y} +v_{,x} } \right) \nonumber \\&+\frac{1}{2}k_{s} B_{41} \left( {w_{,xy} +\phi _{y,x} } \right) -\frac{1}{2}k_{s} B_{42} \left( {w_{,xy} +\phi _{x,y} } \right) +\frac{1}{2}k_{s} B_{43} \left( {\phi _{x,y} -\phi _{y,x} } \right) \nonumber \\&+\frac{1}{2}k_{s} B_{44} \left( {v_{,xy} -u_{,yy} } \right) +\frac{1}{2}k_{s} B_{45} \left( {v_{,xx} -u_{,xy} } \right) +\left. {\frac{1}{2}k_{s} B_{46} \left( {w_{,yy} +\phi _{y,y} -w_{,xx} -\phi _{x,x} } \right) } \right] \nonumber \\&-\frac{2h}{\pi }k_{s} e_{14} \gamma _{,x} -\frac{2h}{\pi }k_{s} e_{24} \gamma _{,y} +2k_{s} e_{34} \gamma _{0} -\frac{2h}{\pi }k_{s} q_{14} \zeta _{,x} -\frac{2h}{\pi }k_{s} q_{24} \zeta _{,y} +2k_{s} q_{34} \zeta _{0} , \end{aligned}$$
(A.3)
$$\begin{aligned} N_{zx}= & {} h\left[ {k_{s} C_{15} u_{,x} } \right. +k_{s} C_{25} v_{,y} +k_{s}^{2} C_{45} \left( {w_{,y} -\phi _{y} } \right) +k_{s}^{2} C_{55} \left( {w_{,x} -\phi _{x} } \right) +k_{s} C_{56} \left( {u_{,y} +v_{,x} } \right) \nonumber \\&+\frac{1}{2}k_{s} B_{51} \left( {w_{,xy} +\phi _{y,x} } \right) -\frac{1}{2}k_{s} B_{52} \left( {w_{,xy} +\phi _{x,y} } \right) +\frac{1}{2}k_{s} B_{53} \left( {\phi _{x,y} -\phi _{y,x} } \right) \nonumber \\&+\frac{1}{2}k_{s} B_{54} \left( {v_{,xy} -u_{,yy} } \right) +\frac{1}{2}k_{s} B_{55} \left( {v_{,xx} -u_{,xy} } \right) +\left. {\frac{1}{2}k_{s} B_{56} \left( {w_{,yy} +\phi _{y,y} -w_{,xx} -\phi _{x,x} } \right) } \right] \nonumber \\&-\frac{2h}{\pi }k_{s} e_{15} \gamma _{,x} -\frac{2h}{\pi }k_{s} e_{25} \gamma _{,y} +2k_{s} e_{35} \gamma _{0} -\frac{2h}{\pi }k_{s} q_{15} \zeta _{,x} -\frac{2h}{\pi }k_{s} q_{25} \zeta _{,y} +2k_{s} q_{35} \zeta _{0} , \end{aligned}$$
(A.4)
$$\begin{aligned} N_{xy}= & {} h\left[ {C_{16} u_{,x} +C_{26} v_{,y} } \right. +k_{s} C_{46} \left( {w_{,y} -\phi _{y} } \right) +k_{s} C_{56} \left( {w_{,x} -\phi _{x} } \right) +C_{66} \left( {u_{,y} +v_{,x} } \right) \nonumber \\&+\frac{1}{2}B_{61} \left( {w_{,xy} +\phi _{y,x} } \right) -\frac{1}{2}B_{62} \left( {w_{,xy} +\phi _{x,y} } \right) +\frac{1}{2}B_{63} \left( {\phi _{x,y} -\phi _{y,x} } \right) +\frac{1}{2}B_{64} \left( {v_{,xy} -u_{,yy} } \right) \nonumber \\&+\frac{1}{2}B_{65} \left( {v_{,xx} -u_{,xy} } \right) +\left. {\frac{1}{2}B_{66} \left( {w_{,yy} +\phi _{y,y} -w_{,xx} -\phi _{x,x} } \right) } \right] -\frac{2h}{\pi }e_{16} \gamma _{,x} -\frac{2h}{\pi }e_{26} \gamma _{,y} \nonumber \\&+2e_{36} \gamma _{0} -\frac{2h}{\pi }q_{16} \zeta _{,x} -\frac{2h}{\pi }q_{26} \zeta _{,y} +2q_{36} \zeta _{0} , \end{aligned}$$
(A.5)
$$\begin{aligned} M_{xx}= & {} -\frac{h^{3}}{12}\left[ {C_{11} \phi _{x,x} +C_{12} \phi _{y,y} } \right. +C_{16} \left( {\phi _{x,y} +\phi _{y,x} } \right) +\frac{1}{2}B_{14} \left( {\phi _{y,xy} -\phi _{x,yy} } \right) \nonumber \\&+\left. {\frac{1}{2}B_{15} \left( {\phi _{y,xx} -\phi _{x,xy} } \right) } \right] +\frac{2h}{\pi }e_{31} \gamma +\frac{2h}{\pi }q_{31} \zeta , \end{aligned}$$
(A.6)
$$\begin{aligned} M_{yy}= & {} -\frac{h^{3}}{12}\left[ {C_{12} \phi _{x,x} +C_{22} \phi _{y,y} } \right. +C_{26} \left( {\phi _{x,y} +\phi _{y,x} } \right) +\frac{1}{2}B_{24} \left( {\phi _{y,xy} -\phi _{x,yy} } \right) \nonumber \\&+\left. {\frac{1}{2}B_{25} \left( {\phi _{y,xx} -\phi _{x,xy} } \right) } \right] +\frac{2h}{\pi }e_{32} \gamma +\frac{2h}{\pi }q_{32} \zeta , \end{aligned}$$
(A.7)
$$\begin{aligned} M_{xy}= & {} -\frac{h^{3}}{12}\left[ {C_{16} \phi _{x,x} +C_{26} \phi _{y,y} } \right. +C_{66} \left( {\phi _{x,y} +\phi _{y,x} } \right) +\frac{1}{2}B_{64} \left( {\phi _{y,xy} -\phi _{x,yy} } \right) \nonumber \\&+\left. {\frac{1}{2}B_{65} \left( {\phi _{y,xx} -\phi _{x,xy} } \right) } \right] +\frac{2h}{\pi }e_{36} \gamma +\frac{2h}{\pi }q_{36} \zeta , \end{aligned}$$
(A.8)
$$\begin{aligned} Y_{xx}= & {} h\left[ {\frac{1}{2}A_{11} \left( {w_{,xy} +\phi _{y,x} } \right) } \right. -\frac{1}{2}A_{12} \left( {w_{,xy} +\phi _{x,y} } \right) +\frac{1}{2}A_{13} \left( {\phi _{x,y} -\phi _{y,x} } \right) +\frac{1}{2}A_{14} \left( {v_{,xy} -u_{,yy} } \right) \nonumber \\&+\frac{1}{2}A_{15} \left( {v_{,xx} -u_{,xy} } \right) +\frac{1}{2}A_{16} \left( {w_{,yy} +\phi _{y,y} -w_{,xx} -\phi _{x,x} } \right) +B_{11} u_{,x} +B_{21} v_{,y} \nonumber \\&+k_{s} B_{41} \left( {w_{,y} -\phi _{y} } \right) +k_{s} B_{51} \left( {w_{,x} -\phi _{x} } \right) +\left. {B_{61} \left( {u_{,y} +v_{,x} } \right) } \right] , \end{aligned}$$
(A.9)
$$\begin{aligned} Y_{yy}= & {} h\left[ {\frac{1}{2}A_{12} \left( {w_{,xy} +\phi _{y,x} } \right) } \right. -\frac{1}{2}A_{22} \left( {w_{,xy} +\phi _{x,y} } \right) +\frac{1}{2}A_{23} \left( {\phi _{x,y} -\phi _{y,x} } \right) +\frac{1}{2}A_{24} \left( {v_{,xy} -u_{,yy} } \right) \nonumber \\&+\frac{1}{2}A_{25} \left( {v_{,xx} -u_{,xy} } \right) +\frac{1}{2}A_{26} \left( {w_{,yy} +\phi _{y,y} -w_{,xx} -\phi _{x,x} } \right) +B_{12} u_{,x} +B_{22} v_{,y} \nonumber \\&+k_{s} B_{42} \left( {w_{,y} -\phi _{y} } \right) +k_{s} B_{52} \left( {w_{,x} -\phi _{x} } \right) +\left. {B_{62} \left( {u_{,y} +v_{,x} } \right) } \right] , \end{aligned}$$
(A.10)
$$\begin{aligned} Y_{zz}= & {} h\left[ {\frac{1}{2}A_{13} \left( {w_{,xy} +\phi _{y,x} } \right) } \right. -\frac{1}{2}A_{23} \left( {w_{,xy} +\phi _{x,y} } \right) +\frac{1}{2}A_{33} \left( {\phi _{x,y} -\phi _{y,x} } \right) +\frac{1}{2}A_{34} \left( {v_{,xy} -u_{,yy} } \right) \nonumber \\&+\frac{1}{2}A_{35} \left( {v_{,xx} -u_{,xy} } \right) +\frac{1}{2}A_{36} \left( {w_{,yy} +\phi _{y,y} -w_{,xx} -\phi _{x,x} } \right) +B_{13} u_{,x} +B_{23} v_{,y} \nonumber \\&\quad +k_{s} B_{43} \left( {w_{,y} -\phi _{y} } \right) +k_{s} B_{53} \left( {w_{,x} -\phi _{x} } \right) +\left. {B_{63} \left( {u_{,y} +v_{,x} } \right) } \right] , \end{aligned}$$
(A.11)
$$\begin{aligned} Y_{zy}= & {} h\left[ {\frac{1}{2}A_{14} \left( {w_{,xy} +\phi _{y,x} } \right) } \right. -\frac{1}{2}A_{24} \left( {w_{,xy} +\phi _{x,y} } \right) +\frac{1}{2}A_{34} \left( {\phi _{x,y} -\phi _{y,x} } \right) +\frac{1}{2}A_{44} \left( {v_{,xy} -u_{,yy} } \right) \nonumber \\&+\frac{1}{2}A_{45} \left( {v_{,xx} -u_{,xy} } \right) +\frac{1}{2}A_{46} \left( {w_{,yy} +\phi _{y,y} -w_{,xx} -\phi _{x,x} } \right) +B_{14} u_{,x} +B_{24} v_{,y} \nonumber \\&+k_{s} B_{44} \left( {w_{,y} -\phi _{y} } \right) +k_{s} B_{54} \left( {w_{,x} -\phi _{x} } \right) +\left. {B_{64} \left( {u_{,y} +v_{,x} } \right) } \right] , \end{aligned}$$
(A.12)
$$\begin{aligned} Y_{zx}= & {} h\left[ {\frac{1}{2}A_{15} \left( {w_{,xy} +\phi _{y,x} } \right) } \right. -\frac{1}{2}A_{25} \left( {w_{,xy} +\phi _{x,y} } \right) +\frac{1}{2}A_{35} \left( {\phi _{x,y} -\phi _{y,x} } \right) +\frac{1}{2}A_{45} \left( {v_{,xy} -u_{,yy} } \right) \nonumber \\&+\frac{1}{2}A_{55} \left( {v_{,xx} -u_{,xy} } \right) +\frac{1}{2}A_{56} \left( {w_{,yy} +\phi _{y,y} -w_{,xx} -\phi _{x,x} } \right) +B_{15} u_{,x} +B_{25} v_{,y} \nonumber \\&+k_{s} B_{45} \left( {w_{,y} -\phi _{y} } \right) +k_{s} B_{55} \left( {w_{,x} -\phi _{x} } \right) +\left. {B_{65} \left( {u_{,y} +v_{,x} } \right) } \right] , \end{aligned}$$
(A.13)
$$\begin{aligned} Y_{xy}= & {} h\left[ {\frac{1}{2}A_{16} \left( {w_{,xy} +\phi _{y,x} } \right) } \right. -\frac{1}{2}A_{26} \left( {w_{,xy} +\phi _{x,y} } \right) +\frac{1}{2}A_{36} \left( {\phi _{x,y} -\phi _{y,x} } \right) +\frac{1}{2}A_{46} \left( {v_{,xy} -u_{,yy} } \right) \nonumber \\&+\frac{1}{2}A_{56} \left( {v_{,xx} -u_{,xy} } \right) +\frac{1}{2}A_{66} \left( {w_{,yy} +\phi _{y,y} -w_{,xx} -\phi _{x,x} } \right) +B_{16} u_{,x} +B_{26} v_{,y} \nonumber \\&+k_{s} B_{46} \left( {w_{,y} -\phi _{y} } \right) +k_{s} B_{56} \left( {w_{,x} -\phi _{x} } \right) +\left. {B_{66} \left( {u_{,y} +v_{,x} } \right) } \right] , \end{aligned}$$
(A.14)
$$\begin{aligned} H_{yz}= & {} \frac{h^{3}}{12}\left[ {\frac{1}{2}A_{44} \left( {-\phi _{y,xy} +\phi _{x,yy} } \right) } \right. +\frac{1}{2}A_{45} \left( {-\phi _{y,xx} +\phi _{x,xy} } \right) -B_{14} \phi _{x,x} -B_{24} \phi _{y,y} \nonumber \\&-\left. {B_{64} \left( {\phi _{x,y} +\phi _{y,x} } \right) } \right] , \end{aligned}$$
(A.15)
$$\begin{aligned} H_{zx}= & {} \frac{h^{3}}{12}\left[ {\frac{1}{2}A_{45} \left( {-\phi _{y,xy} +\phi _{x,yy} } \right) } \right. +\frac{1}{2}A_{55} \left( {-\phi _{y,xx} +\phi _{x,xy} } \right) -B_{15} \phi _{x,x} -B_{25} \phi _{y,y} \nonumber \\&-\left. {B_{65} \left( {\phi _{x,y} +\phi _{y,x} } \right) } \right] , \end{aligned}$$
(A.16)
$$\begin{aligned} \Xi _{x}= & {} \frac{h}{2}\epsilon _{11} \gamma _{,x} +\frac{h}{2}\epsilon _{12} \gamma _{,y} -\frac{4}{\pi }\epsilon _{13} \gamma _{0} +\frac{h}{2}d_{11} \zeta _{,x} +\frac{h}{2}d_{12} \zeta _{,y} -\frac{4}{\pi }d_{13} \zeta _{0} +\frac{2h}{\pi }e_{11} u_{,x} \nonumber \\&+\frac{2h}{\pi }e_{12} v_{,y} +\frac{2h}{\pi }k_{s} e_{14} \left( {w_{,y} -\phi _{y} } \right) +\frac{2h}{\pi }k_{s} e_{15} \left( {w_{,x} -\phi _{x} } \right) +\frac{2h}{\pi }e_{16} \left( {u_{,y} +v_{,x} } \right) , \end{aligned}$$
(A.17)
$$\begin{aligned} \Xi _{y}= & {} \frac{h}{2}\epsilon _{12} \gamma _{,x} +\frac{h}{2}\epsilon _{22} \gamma _{,y} -\frac{4}{\pi }\epsilon _{23} \gamma _{0} +\frac{h}{2}d_{21} \zeta _{,x} +\frac{h}{2}d_{22} \zeta _{,y} -\frac{4}{\pi }d_{23} \zeta _{0} +\frac{2h}{\pi }e_{21} u_{,x} \nonumber \\&+\frac{2h}{\pi }e_{22} v_{,y} +\frac{2h}{\pi }k_{s} e_{24} \left( {w_{,y} -\phi _{y} } \right) +\frac{2h}{\pi }k_{s} e_{25} \left( {w_{,x} -\phi _{x} } \right) +\frac{2h}{\pi }e_{26} \left( {u_{,y} +v_{,x} } \right) , \end{aligned}$$
(A.18)
$$\begin{aligned} \Xi _{z}= & {} -\frac{\pi ^{2}}{2h}\epsilon _{33} \gamma -\frac{\pi ^{2}}{2h}d_{33} \zeta -\frac{2h}{\pi }e_{31} \phi _{x,x} -\frac{2h}{\pi }e_{32} \phi _{y,y} -\frac{2h}{\pi }e_{36} \left( {\phi _{x,y} +\phi _{y,x} } \right) , \end{aligned}$$
(A.19)
$$\begin{aligned} \Sigma _{x}= & {} \frac{h}{2}\mu _{11} \zeta _{,x} +\frac{h}{2}\mu _{12} \zeta _{,y} -\frac{4}{\pi }\mu _{13} \zeta _{0} +\frac{h}{2}d_{11} \gamma _{,x} +\frac{h}{2}d_{12} \gamma _{,y} -\frac{4}{\pi }d_{13} \gamma _{0} +\frac{2h}{\pi }q_{11} u_{,x} \nonumber \\&+\frac{2h}{\pi }q_{12} v_{,y} +\frac{2h}{\pi }k_{s} q_{14} \left( {w_{,y} -\phi _{y} } \right) +\frac{2h}{\pi }k_{s} q_{15} \left( {w_{,x} -\phi _{x} } \right) +\frac{2h}{\pi }q_{16} \left( {u_{,y} +v_{,x} } \right) , \end{aligned}$$
(A.20)
$$\begin{aligned} \Sigma _{y}= & {} \frac{h}{2}\mu _{12} \zeta _{,x} +\frac{h}{2}\mu _{22} \zeta _{,y} -\frac{4}{\pi }\mu _{23} \zeta _{0} +\frac{h}{2}d_{21} \gamma _{,x} +\frac{h}{2}d_{22} \gamma _{,y} -\frac{4}{\pi }d_{23} \gamma _{0} +\frac{2h}{\pi }q_{21} u_{,x} \nonumber \\&+\frac{2h}{\pi }q_{22} v_{,y} +\frac{2h}{\pi }k_{s} q_{24} \left( {w_{,y} -\phi _{y} } \right) +\frac{2h}{\pi }k_{s} q_{25} \left( {w_{,x} -\phi _{x} } \right) +\frac{2h}{\pi }q_{26} \left( {u_{,y} +v_{,x} } \right) , \end{aligned}$$
(A.21)
$$\begin{aligned} \Sigma _{z}= & {} -\frac{\pi ^{2}}{2h}\mu _{33} \zeta -\frac{\pi ^{2}}{2h}d_{33} \gamma -\frac{2h}{\pi }q_{31} \phi _{x,x} -\frac{2h}{\pi }q_{32} \phi _{y,y} -\frac{2h}{\pi }q_{36} \left( {\phi _{x,y} +\phi _{y,x} } \right) \end{aligned}$$
(A.22)
where \(k_{s} \) is a shape correction factor introduced to account for the non-uniformity of the shear strain components \(\varepsilon _{xz} \) and \(\varepsilon _{yz} \) over the plate thickness such that \(\varepsilon _{xz} \rightarrow k_{s} \varepsilon _{xz} \) and \(\varepsilon _{yz} \rightarrow k_{s} \varepsilon _{yz} \) (e.g., [30, 31, 50]).
Appendix B
In this Appendix, it is shown that the new anisotropic magneto-electro-elastic Mindlin plate model developed in Sect. 3 includes three models for orthotropic, transversely isotropic, and isotropic plates as special cases.
1.1 Orthotropic magneto-electro-elastic Mindlin plate model incorporating the microstructure effects
For an orthotropic magneto-electro-elastic material, the constitutive equations in Eqs. (8.1–4) reduce to (e.g., [39, 42, 52])
$$\begin{aligned}&\left\{ {{\begin{array}{*{20}c} {\sigma _{xx} } \\ {\sigma _{yy} } \\ {\sigma _{zz} } \\ {\begin{array}{l} \sigma _{yz} \\ \sigma _{zx} \\ \sigma _{xy} \\ \end{array}} \\ \end{array} }} \right\} =\left[ {{\begin{array}{*{20}c} {C_{11} } &{} {C_{12} } &{} {C_{13} } &{} 0 &{} 0 &{} 0 \\ {C_{12} } &{} {C_{22} } &{} {C_{23} } &{} 0 &{} 0 &{} 0 \\ {C_{13} } &{} {C_{23} } &{} {C_{33} } &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} {C_{44} } &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} {C_{55} } &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {C_{66} } \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {\varepsilon _{xx} } \\ {\varepsilon _{yy} } \\ {\varepsilon _{zz} } \\ {\begin{array}{l} 2\varepsilon _{yz} \\ 2\varepsilon _{zx} \\ 2\varepsilon _{xy} \\ \end{array}} \\ \end{array} }} \right\} -\left[ {{\begin{array}{*{20}c} 0 &{} 0 &{} {e_{31} } \\ 0 &{} 0 &{} {e_{32} } \\ 0 &{} 0 &{} {e_{33} } \\ 0 &{} {e_{24} } &{} 0 \\ {e_{15} } &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {E_{x} } \\ {E_{y} } \\ {E_{z} } \\ \end{array} }} \right\} -\left[ {{\begin{array}{*{20}c} 0 &{} 0 &{} {q_{31} } \\ 0 &{} 0 &{} {q_{32} } \\ 0 &{} 0 &{} {q_{33} } \\ 0 &{} {q_{24} } &{} 0 \\ {q_{15} } &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {H_{x} } \\ {H_{y} } \\ {H_{z} } \\ \end{array} }} \right\} , \end{aligned}$$
(B.1.1.1)
$$\begin{aligned}&\left\{ {{\begin{array}{*{20}c} {m_{xx} } \\ {m_{yy} } \\ {m_{zz} } \\ {\begin{array}{l} m_{yz} \\ m_{zx} \\ m_{xy} \\ \end{array}} \\ \end{array} }} \right\} =\left[ {{\begin{array}{*{20}c} {A_{11} } &{} {A_{12} } &{} {A_{13} } &{} 0 &{} 0 &{} 0 \\ {A_{12} } &{} {A_{22} } &{} {A_{23} } &{} 0 &{} 0 &{} 0 \\ {A_{13} } &{} {A_{23} } &{} {A_{33} } &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} {A_{44} } &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} {A_{55} } &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {A_{66} } \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {\chi _{xx} } \\ {\chi _{yy} } \\ {\chi _{zz} } \\ {\begin{array}{l} 2\chi _{yz} \\ 2\chi _{zx} \\ 2\chi _{xy} \\ \end{array}} \\ \end{array} }} \right\} , \end{aligned}$$
(B.1.1.2)
$$\begin{aligned}&\left\{ {{\begin{array}{*{20}c} {D_{x} } \\ {D_{y} } \\ {D_{z} } \\ \end{array} }} \right\} =\left[ {{\begin{array}{*{20}c} 0 &{} 0 &{} 0 &{} 0 &{} {e_{15} } &{} 0 \\ 0 &{} 0 &{} 0 &{} {e_{24} } &{} 0 &{} 0 \\ {e_{31} } &{} {e_{32} } &{} {e_{33} } &{} 0 &{} 0 &{} 0 \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {\varepsilon _{xx} } \\ {\varepsilon _{yy} } \\ {\varepsilon _{zz} } \\ {\begin{array}{l} 2\varepsilon _{yz} \\ 2\varepsilon _{zx} \\ 2\varepsilon _{xy} \\ \end{array}} \\ \end{array} }} \right\} +\left[ {{\begin{array}{*{20}c} {\epsilon _{11} } &{} 0 &{} 0 \\ 0 &{} {\epsilon _{22} } &{} 0 \\ 0 &{} 0 &{} {\epsilon _{33} } \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {E_{x} } \\ {E_{y} } \\ {E_{z} } \\ \end{array} }} \right\} +\left[ {{\begin{array}{*{20}c} {d_{11} } &{} 0 &{} 0 \\ 0 &{} {d_{22} } &{} 0 \\ 0 &{} 0 &{} {d_{33} } \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {H_{x} } \\ {H_{y} } \\ {H_{z} } \\ \end{array} }} \right\} , \end{aligned}$$
(B.1.1.3)
$$\begin{aligned}&\left\{ {{\begin{array}{*{20}c} {B_{x} } \\ {B_{y} } \\ {B_{z} } \\ \end{array} }} \right\} =\left[ {{\begin{array}{*{20}c} 0 &{} 0 &{} 0 &{} 0 &{} {q_{15} } &{} 0 \\ 0 &{} 0 &{} 0 &{} {q_{24} } &{} 0 &{} 0 \\ {q_{31} } &{} {q_{32} } &{} {q_{33} } &{} 0 &{} 0 &{} 0 \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {\varepsilon _{xx} } \\ {\varepsilon _{yy} } \\ {\varepsilon _{zz} } \\ {\begin{array}{l} 2\varepsilon _{yz} \\ 2\varepsilon _{zx} \\ 2\varepsilon _{xy} \\ \end{array}} \\ \end{array} }} \right\} +\left[ {{\begin{array}{*{20}c} {d_{11} } &{} 0 &{} 0 \\ 0 &{} {d_{22} } &{} 0 \\ 0 &{} 0 &{} {d_{33} } \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {E_{x} } \\ {E_{y} } \\ {E_{z} } \\ \end{array} }} \right\} +\left[ {{\begin{array}{*{20}c} {\mu _{11} } &{} 0 &{} 0 \\ 0 &{} {\mu _{22} } &{} 0 \\ 0 &{} 0 &{} {\mu _{33} } \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {H_{x} } \\ {H_{y} } \\ {H_{z} } \\ \end{array} }} \right\} . \end{aligned}$$
(B.1.1.4)
For an orthotropic elastic Mindlin plate with \(C_{\mathrm{11}}\), \(C_{\mathrm{12}}\), \(C_{\mathrm{13}}\), \(C_{\mathrm{22}}\), \(C_{\mathrm{23}}\), \(C_{\mathrm{33}}\), \(C_{\mathrm{44}}\), \(C_{\mathrm{55}}\), \(C_{\mathrm{66}}\), \(A_{\mathrm{11}}\), \(A_{\mathrm{12}}\), \(A_{\mathrm{13}}\), \(A_{\mathrm{22}}\), \(A_{\mathrm{23}}\), \(A_{\mathrm{33}}\), \(A_{\mathrm{44}}\), \(A_{\mathrm{55}}\), \(A_{\mathrm{66}}\), \(e_{\mathrm{15}}\), \(e_{\mathrm{24}}\), \(e_{\mathrm{31}}\), \(e_{\mathrm{32}}\), \(e_{\mathrm{33}}\), \(q_{\mathrm{15}}\), \(q_{\mathrm{24}}\), \(q_{\mathrm{31}}\), \(q_{\mathrm{32}}\), \(q_{\mathrm{33}}\), \(\epsilon _{\mathrm{11}}\), \(\epsilon _{\mathrm{22}}\), \(\epsilon _{\mathrm{33}}\), \(d_{\mathrm{11}}\), \(d_{\mathrm{22}}\), \(d_{\mathrm{33}}\), \(\mu _{\mathrm{11}}\), \(\mu _{\mathrm{22 }}\), and \(\mu _{\mathrm{33}}\) as the nonzero material constants in Eqs. (A.1)–(A.22) the equations of motion can be expressed in terms of the u,v,w, \(\phi _{x}\), \(\phi _{y}\), \(\gamma \), and \(\zeta \) as, after substituting the resulting equations from Eqs. (A.1)–(A.22) into Eqs. (31.1–7),
$$\begin{aligned}&h\left( {C_{11} u_{,xx} +C_{12} v_{,xy} } \right) +hC_{66} \left( {u_{,yy} +v_{,xy} } \right) +\frac{hA_{55} }{4}\left( {-u_{,xxyy} +v_{,xxxy} } \right) +\frac{hA_{44} }{4}\left( {-u_{,yyyy} +v_{,xyyy} } \right) \nonumber \\&\quad +f_{x} +\frac{1}{2}c_{z,y} =m_{0} \ddot{{u}}, \end{aligned}$$
(B.1.2.1)
$$\begin{aligned}&h\left( {C_{12} u_{,xy} +C_{22} v_{,yy} } \right) +hC_{66} \left( {u_{,xy} +v_{,xx} } \right) +\frac{hA_{55} }{4}\left( {u_{,xxxy} -v_{,xxxx} } \right) +\frac{hA_{44} }{4}\left( {u_{,xyyy} -v_{,xxyy} } \right) \nonumber \\&\quad +f_{y} -\frac{1}{2}c_{z,x} =m_{0} \ddot{{v}}, \end{aligned}$$
(B.1.2.2)
$$\begin{aligned}&hk_{s}^{2} C_{55} \left( {w_{,xx} -\phi _{x,x} } \right) +hk_{s}^{2} C_{44} \left( {w_{,yy} -\phi _{y,y} } \right) -\frac{1}{4}h\left[ {\left( {A_{11} -2A_{12} +A_{22} } \right) w_{,xxyy} +(-A_{12} +A_{22} } \right. \nonumber \\&\quad +A_{13} -A_{23} )\phi _{x,xyy} +\left( {A_{11} -A_{12} -A_{13} +A_{23} } \right) \left. {\phi _{y,xxy} } \right] +\frac{1}{4}hA_{66} (-w_{,xxxx} +2w_{,xxyy} -w_{,yyyy} \nonumber \\&\quad -\phi _{x,xxx} +\phi _{x,xyy} +\phi _{y,xxy} -\phi _{y,yyy} )-\frac{2}{\pi }hk_{s} \left( {e_{15} \gamma _{,xx} +e_{24} \gamma _{,yy} } \right) -\frac{2}{\pi }hk_{s} \left( {q_{15} \zeta _{,xx} +q_{24} \zeta _{,yy} } \right) \nonumber \\&\quad +f_{z} -\frac{1}{2}\frac{\partial c_{x} }{\partial y} +\frac{1}{2}\frac{\partial c_{y} }{\partial x}=m_{0} \ddot{{w}}, \end{aligned}$$
(B.1.2.3)
$$\begin{aligned}&hk_{s}^{2} C_{55} \left( {w_{,x} -\phi _{x} } \right) +\frac{1}{12}h^{3}\left( {C_{11} \phi _{x,xx} +C_{12} \phi _{y,xy} } \right) +\frac{h^{3}C_{66} }{12}\left( {\phi _{x,yy} +\phi _{y,xy} } \right) +\frac{1}{4}hA_{66} (w_{,xxx} -w_{,xyy} \nonumber \\&\quad +\phi _{x,xx} -\phi _{y,xy} )+\frac{1}{4}h\left[ {\left( {-A_{12} +A_{22} +A_{13} -A_{23} } \right) } \right. w_{,xyy} +(A_{22} +A_{33} -2A_{23} )\phi _{x,yy} \nonumber \\&\quad +\left( {A_{13} +A_{23} -A_{12} -A_{\text{33 }} } \right) \left. {\phi _{y,xy} } \right] -\frac{h^{3}}{48}A_{55} \left( {\phi _{x,xxyy} -\phi _{y,xxxy} } \right) -\frac{h^{3}}{48}A_{44} \left( {\phi _{x,yyyy} -\phi _{y,xyyy} } \right) \nonumber \\&\quad -\frac{2h}{\pi }\left( {k_{s} e_{15} +e_{31} } \right) \gamma _{,x} -\frac{2h}{\pi }\left( {k_{s} q_{15} +q_{31} } \right) \zeta _{,x} -\frac{1}{2}c_{y} =m_{2} \ddot{{\phi }}_{x} , \end{aligned}$$
(B.1.2.4)
$$\begin{aligned}&hk_{s}^{2} C_{44} \left( {w_{,y} -\phi _{y} } \right) +\frac{1}{12}h^{3}\left( {C_{12} \phi _{x,xy} +C_{22} \phi _{y,yy} } \right) +\frac{h^{3}C_{66} }{12}\left( {\phi _{x,xy} +\phi _{y,xx} } \right) \text{+ }\frac{1}{4}h\left[ ( \right. A_{11} -A_{12} \nonumber \\&\quad -A_{13} +A_{23} )w_{,xxy} +(-A_{12} +A_{13} +A_{23} -A_{33} )\phi _{x,xy} +(A_{11} -2A_{13} +A_{33} \left. {)\phi _{y,xx} } \right] \nonumber \\&\quad +\frac{1}{4}hA_{66} \left( {w_{,yyy} +\phi _{y,yy} -w_{,xxy} -\phi _{x,xy} } \right) \,+\frac{h^{3}}{48}A_{55} \left( {\phi _{x,xxxy} -\phi _{y,xxxx} } \right) +\frac{h^{3}}{48}A_{44} \left( {\phi _{x,xyyy} -\phi _{y,xxyy} } \right) \nonumber \\&\quad -\frac{2h}{\pi }\left( {k_{s} e_{24} +e_{32} } \right) \gamma _{,y} -\frac{2h}{\pi }\left( {k_{s} q_{24} +q_{32} } \right) \zeta _{,y} +\frac{1}{2}c_{x} =m_{2} \ddot{{\phi }}_{y} , \end{aligned}$$
(B.1.2.5)
$$\begin{aligned}&\frac{4}{\pi }k_{s} e_{15} \left( {w_{,xx} -\phi _{xx} } \right) +\frac{4}{\pi }k_{s} e_{24} \left( {w_{,yy} -\phi _{yy} } \right) -\frac{4}{\pi }e_{31} \phi _{x,x} -\frac{4}{\pi }e_{32} \phi _{y,y} +\epsilon _{11} \gamma _{,xx} +\epsilon _{22} \gamma _{,yy} +d_{11} \zeta _{,xx} \nonumber \\&\quad +d_{22} \zeta _{,yy} -\frac{\pi ^{2}}{h^{2}}\epsilon _{33} \gamma -\frac{\pi ^{2}}{h^{2}}d_{33} \zeta =0, \end{aligned}$$
(B.1.2.6)
$$\begin{aligned}&\frac{4}{\pi }k_{s} q_{15} \left( {w_{,xx} -\phi _{xx} } \right) +\frac{4}{\pi }k_{s} q_{24} \left( {w_{,yy} -\phi _{yy} } \right) -\frac{4}{\pi }q_{31} \phi _{x,x} -\frac{4}{\pi }q_{32} \phi _{y,y} +\mu _{11} \zeta _{,xx} +\mu _{22} \zeta _{,yy} +d_{11} \gamma _{,xx} \nonumber \\&\quad +d_{22} \gamma _{,yy} -\frac{\pi ^{2}}{h^{2}}\mu _{33} \zeta -\frac{\pi ^{2}}{h^{2}}d_{33} \gamma =0. \end{aligned}$$
(B.1.2.7)
It is seen from Eqs. (B.1.2.1–7) that the in-plane displacements u and v are uncoupled with the out-of-plane displacement w, the rotation \(\phi _{x} \), and the rotation \(\phi _{y} \), and can therefore be obtained separately from solving Eqs. (B.1.2.1) and (B.1.2.2) subject to prescribed boundary conditions and suitable initial conditions.
1.2 Transversely isotropic magneto-electro-elastic Mindlin plate model incorporating the microstructure effects
For a transversely isotropic magneto-electro-elastic material (with \(B_{ijkl} \quad =\) 0), the constitutive equations in Eqs. (8.1–4) reduce to (e.g., [4])
$$\begin{aligned}&\left\{ {{\begin{array}{*{20}c} {\sigma _{xx} } \\ {\sigma _{yy} } \\ {\sigma _{zz} } \\ {\begin{array}{l} \sigma _{yz} \\ \sigma _{zx} \\ \sigma _{xy} \\ \end{array}} \\ \end{array} }} \right\} =\left[ {{\begin{array}{*{20}c} {C_{11} } &{} {C_{12} } &{} {C_{13} } &{} 0 &{} 0 &{} 0 \\ {C_{12} } &{} {C_{11} } &{} {C_{13} } &{} 0 &{} 0 &{} 0 \\ {C_{13} } &{} {C_{13} } &{} {C_{33} } &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} {C_{44} } &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} {C_{44} } &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {\frac{C_{11} -C_{12} }{2}} \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {\varepsilon _{xx} } \\ {\varepsilon _{yy} } \\ {\varepsilon _{zz} } \\ {\begin{array}{l} 2\varepsilon _{yz} \\ 2\varepsilon _{zx} \\ 2\varepsilon _{xy} \\ \end{array}} \\ \end{array} }} \right\} -\left[ {{\begin{array}{*{20}c} 0 &{} 0 &{} {e_{31} } \\ 0 &{} 0 &{} {e_{31} } \\ 0 &{} 0 &{} {e_{33} } \\ 0 &{} {e_{15} } &{} 0 \\ {e_{15} } &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {E_{x} } \\ {E_{y} } \\ {E_{z} } \\ \end{array} }} \right\} -\left[ {{\begin{array}{*{20}c} 0 &{} 0 &{} {q_{31} } \\ 0 &{} 0 &{} {q_{31} } \\ 0 &{} 0 &{} {q_{33} } \\ 0 &{} {q_{15} } &{} 0 \\ {q_{15} } &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {H_{x} } \\ {H_{y} } \\ {H_{z} } \\ \end{array} }} \right\} , \nonumber \\ \end{aligned}$$
(B.2.1.1)
$$\begin{aligned}&\left\{ {{\begin{array}{*{20}c} {m_{xx} } \\ {m_{yy} } \\ {m_{zz} } \\ {\begin{array}{l} m_{yz} \\ m_{zx} \\ m_{xy} \\ \end{array}} \\ \end{array} }} \right\} =\left[ {{\begin{array}{*{20}c} {A_{11} } &{} {A_{12} } &{} {A_{13} } &{} 0 &{} 0 &{} 0 \\ {A_{12} } &{} {A_{11} } &{} {A_{13} } &{} 0 &{} 0 &{} 0 \\ {A_{13} } &{} {A_{13} } &{} {A_{33} } &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} {A_{44} } &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} {A_{44} } &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {\frac{A_{{11}} -A_{{12}} }{2}} \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {\chi _{xx} } \\ {\chi _{yy} } \\ {\chi _{zz} } \\ {\begin{array}{l} 2\chi _{yz} \\ 2\chi _{zx} \\ 2\chi _{xy} \\ \end{array}} \\ \end{array} }} \right\} , \end{aligned}$$
(B.2.1.2)
$$\begin{aligned}&\left\{ {{\begin{array}{*{20}c} {D_{x} } \\ {D_{y} } \\ {D_{z} } \\ \end{array} }} \right\} =\left[ {{\begin{array}{*{20}c} 0 &{} 0 &{} 0 &{} 0 &{} {e_{15} } &{} 0 \\ 0 &{} 0 &{} 0 &{} {e_{15} } &{} 0 &{} 0 \\ {e_{31} } &{} {e_{31} } &{} {e_{33} } &{} 0 &{} 0 &{} 0 \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {\varepsilon _{xx} } \\ {\varepsilon _{yy} } \\ {\varepsilon _{zz} } \\ {\begin{array}{l} 2\varepsilon _{yz} \\ 2\varepsilon _{zx} \\ 2\varepsilon _{xy} \\ \end{array}} \\ \end{array} }} \right\} +\left[ {{\begin{array}{*{20}c} {\epsilon _{11} } &{} 0 &{} 0 \\ 0 &{} {\epsilon _{11} } &{} 0 \\ 0 &{} 0 &{} {\epsilon _{33} } \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {E_{x} } \\ {E_{y} } \\ {E_{z} } \\ \end{array} }} \right\} +\left[ {{\begin{array}{*{20}c} {d_{11} } &{} 0 &{} 0 \\ 0 &{} {d_{11} } &{} 0 \\ 0 &{} 0 &{} {d_{33} } \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {H_{x} } \\ {H_{y} } \\ {H_{z} } \\ \end{array} }} \right\} , \nonumber \\ \end{aligned}$$
(B.2.1.3)
$$\begin{aligned}&\left\{ {{\begin{array}{*{20}c} {B_{x} } \\ {B_{y} } \\ {B_{z} } \\ \end{array} }} \right\} =\left[ {{\begin{array}{*{20}c} 0 &{} 0 &{} 0 &{} 0 &{} {q_{15} } &{} 0 \\ 0 &{} 0 &{} 0 &{} {q_{15} } &{} 0 &{} 0 \\ {q_{31} } &{} {q_{31} } &{} {q_{33} } &{} 0 &{} 0 &{} 0 \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {\varepsilon _{xx} } \\ {\varepsilon _{yy} } \\ {\varepsilon _{zz} } \\ {\begin{array}{l} 2\varepsilon _{yz} \\ 2\varepsilon _{zx} \\ 2\varepsilon _{xy} \\ \end{array}} \\ \end{array} }} \right\} +\left[ {{\begin{array}{*{20}c} {d_{11} } &{} 0 &{} 0 \\ 0 &{} {d_{11} } &{} 0 \\ 0 &{} 0 &{} {d_{33} } \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {E_{x} } \\ {E_{y} } \\ {E_{z} } \\ \end{array} }} \right\} +\left[ {{\begin{array}{*{20}c} {\mu _{11} } &{} 0 &{} 0 \\ 0 &{} {\mu _{11} } &{} 0 \\ 0 &{} 0 &{} {\mu _{33} } \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {H_{x} } \\ {H_{y} } \\ {H_{z} } \\ \end{array} }} \right\} .\nonumber \\ \end{aligned}$$
(B.2.1.4)
When
$$\begin{aligned} \begin{array}{l} C_{11} =C_{22} , C_{13} =C_{23} ,\,\,\,\,C_{44} =C_{55} ,\,\,\,\,C_{66} =\left( {C_{11} -C_{12} } \right) /2, \\ A_{11} =A_{22} ,\,\,\,\,A_{13} =A_{23} ,\,\,\,\,A_{44} =A_{55} ,\,\,\,\,A_{66} =\left( {A_{11} -A_{12} } \right) /2, \\ e_{15} =e_{24} ,\,\,\,\,e_{31} =e_{32} ,\,\,\,\,q_{15} =q_{24} ,\,\,\,\,q_{31} =q_{32} ,\,\,\,\,\epsilon _{11} =\epsilon _{22} ,\,\,\,\,d_{11} =d_{22} ,\,\,\,\,\mu _{11} =\mu _{22} , \\ \end{array} \end{aligned}$$
(B.2.2)
Eqs. (B.1.2.1–7) reduce to the equations of motion of the transverse isotropic magneto-electro-elastic Mindlin plate in terms of u,v, w, \(\phi _{x}\), \(\phi _{y}\), \(\gamma \), and \(\zeta \) as
$$\begin{aligned}&h\left( {C_{11} u_{,xx} +C_{12} v_{,xy} } \right) +\frac{(C_{11} -C_{12} )h}{2}\left( {u_{,yy} +v_{,xy} } \right) +\frac{A_{44} h}{4}(-u_{,xxyy} -u_{,yyyy} +v_{,xxxy} +v_{,xyyy} ) \nonumber \\&\quad +f_{x} +\frac{1}{2}c_{z,y} =m_{0} \ddot{{u}}, \end{aligned}$$
(B.2.3.1)
$$\begin{aligned}&h\left( {C_{12} u_{,xy} +C_{11} v_{,yy} } \right) +\frac{\left( {C_{11} -C_{12} } \right) h}{2}\left( {u_{,xy} +v_{,xx} } \right) -\frac{A_{44} h}{4}\left( {-u_{,xyyy} -u_{,xxxy} +v_{,xxxx} +v_{,xxyy} } \right) \nonumber \\&\quad +f_{y} -\frac{1}{2}c_{z,x} =m_{0} \ddot{{v}}, \end{aligned}$$
(B.2.3.2)
$$\begin{aligned}&hk_{s}^{2} C_{44} \left( {w_{,xx} +w_{,yy} -\phi _{x,x} -\phi _{y,y} } \right) -\frac{h}{8}\left( {A_{11} -A_{12} } \right) (w_{,xxxx} +2w_{,xxyy} +w_{,yyyy} +\phi _{x,xxx} \nonumber \\&\quad +\phi _{x,xyy} +\phi _{y,xxy} +\phi _{y,yyy} )-\frac{2}{\pi }hk_{s} e_{15} \left( {\gamma _{,xx} +\gamma _{,yy} } \right) -\frac{2}{\pi }hk_{s} q_{15} \left( {\zeta _{,xx} +\zeta _{,yy} } \right) +f_{z} \nonumber \\&\quad -\frac{1}{2}\frac{\partial c_{x} }{\partial y}+\frac{1}{2}\frac{\partial c_{y} }{\partial x}=m_{0} \ddot{{w}}, \end{aligned}$$
(B.2.3.3)
$$\begin{aligned}&hk_{s}^{2} C_{44} \left( {w_{,x} -\phi _{x} } \right) +\frac{1}{12}h^{3}\left( {C_{11} \phi _{x,xx} +C_{12} \phi _{y,xy} } \right) +\frac{\left( {C_{11} -C_{12} } \right) h^{3}}{24}\left( {\phi _{x,yy} +\phi _{y,xy} } \right) \nonumber \\&\quad +\frac{h}{4}\left[ {\left( {A_{11} +A_{33} -2A_{\text{13 }} } \right) \phi _{x,yy} +\left( {2A_{13} -A_{12} -A_{\text{33 }} } \right) \phi _{y,xy} } \right] +\frac{\left( {A_{11} -A_{12} } \right) h}{8}(w_{,xxx} \nonumber \\&\quad +w_{,xyy} +\phi _{x,xx} -\phi _{y,xy} )-\frac{A_{44} h^{3}}{48}(\phi _{x,xxyy} +\phi _{x,yyyy} -\phi _{y,xxxy} -\phi _{y,xyyy} )-\frac{2h}{\pi }\left( {k_{s} e_{15} +e_{31} } \right) \gamma _{,x} \nonumber \\&\quad -\frac{2h}{\pi }\left( {k_{s} q_{15} +q_{31} } \right) \zeta _{,x} -\frac{1}{2}c_{y} =m_{2} \ddot{{\phi }}_{x} , \end{aligned}$$
(B.2.3.4)
$$\begin{aligned}&hk_{s}^{2} C_{44} \left( {w_{,y} -\phi _{y} } \right) \text{+ }\frac{1}{12}h^{3}\left( {C_{12} \phi _{x,xy} +C_{11} \phi _{y,yy} } \right) +\frac{\left( {C_{11} -C_{12} } \right) h^{3}}{24}\left( {\phi _{x,xy} +\phi _{y,xx} } \right) \nonumber \\&\quad +\frac{h}{4}\left[ {\left( {2A_{13} -A_{12} -A_{33} } \right) \phi _{x,xy} +\left( {A_{11} +A_{33} -2A_{13} } \right) \phi _{y,xx} } \right] +\frac{\left( {A_{11} -A_{12} } \right) h}{8}(w_{,xxy} \nonumber \\&\quad +w_{,yyy} -\phi _{x,xy} +\phi _{y,yy} )+\frac{A_{44} h^{3}}{48}(\phi _{x,xxxy} +\phi _{x,xyyy} -\phi _{y,xxxx} -\phi _{y,xxyy} )-\frac{2h}{\pi }\left( {k_{s} e_{15} +e_{31} } \right) \gamma _{,y} \nonumber \\&\quad -\frac{2h}{\pi }\left( {k_{s} q_{15} +q_{31} } \right) \zeta _{,y} +\frac{1}{2}c_{x} =m_{2} \ddot{{\phi }}_{y} , \end{aligned}$$
(B.2.3.5)
$$\begin{aligned}&\frac{4}{\pi }k_{s} e_{15} \left( {w_{,xx} +w_{,yy} -\phi _{x,x} -\phi _{y,y} } \right) -\frac{4}{\pi }e_{31} \left( {\phi _{x,x} +\phi _{y,y} } \right) \nonumber \\&\quad +\epsilon _{11} \left( {\gamma _{,xx} +\gamma _{,yy} } \right) +d_{11} \left( {\zeta _{,xx} +\zeta _{,yy} } \right) -\frac{\pi ^{2}}{h^{2}}\epsilon _{33} \gamma -\frac{\pi ^{2}}{h^{2}}d_{33} \zeta =0, \end{aligned}$$
(B.2.3.6)
$$\begin{aligned}&\frac{4}{\pi }k_{s} q_{15} \left( {w_{,xx} +w_{,yy} -\phi _{x,x} -\phi _{y,y} } \right) -\frac{4}{\pi }q_{31} \left( {\phi _{x,x} +\phi _{y,y} } \right) \nonumber \\&\quad \,\,\,\,\,+\mu _{11} \left( {\zeta _{,xx} +\zeta _{,yy} } \right) +d_{11} \left( {\gamma _{,xx} +\gamma _{,yy} } \right) -\frac{\pi ^{2}}{h^{2}}\mu _{33} \zeta -\frac{\pi ^{2}}{h^{2}}d_{33} \gamma =0. \end{aligned}$$
(B.2.3.7)
It is observed from Eqs. (B.2.3.1–7) that the in-plane displacements u and v are uncoupled with the out-of-plane displacement w, the rotation \(\phi _{x}\) and the rotation \(\phi _{y} \).
1.3 Isotropic elastic Mindlin plate model incorporating the microstructure effects
When
$$\begin{aligned} C_{11} =C_{33} =\lambda +2\mu , C_{12} =C_{13} =\lambda , C_{44} =\left( {C_{11} -C_{12} } \right) /2=\mu , \end{aligned}$$
(B.3.1)
Eq. (B.2.1.1) reduces to
$$\begin{aligned}&\left\{ {{\begin{array}{*{20}c} {\sigma _{xx} } \\ {\sigma _{yy} } \\ {\sigma _{zz} } \\ {\begin{array}{l} \sigma _{yz} \\ \sigma _{zx} \\ \sigma _{xy} \\ \end{array}} \\ \end{array} }} \right\} =\left[ {{\begin{array}{*{20}c} {\lambda +2\mu } &{} \lambda &{} \lambda &{} 0 &{} 0 &{} 0 \\ \lambda &{} {\lambda +2\mu } &{} \lambda &{} 0 &{} 0 &{} 0 \\ \lambda &{} \lambda &{} {\lambda +2\mu } &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \mu &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \mu &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \mu \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {\varepsilon _{xx} } \\ {\varepsilon _{yy} } \\ {\varepsilon _{zz} } \\ {\begin{array}{l} 2\varepsilon _{yz} \\ 2\varepsilon _{zx} \\ 2\varepsilon _{xy} \\ \end{array}} \\ \end{array} }} \right\} , \end{aligned}$$
(B.3.2)
which is the constitutive equation for an isotropic linear elastic material.
Similarly, when \(A_{11} =A_{33} , A_{12} =A_{13} , A_{44} =\left( {A_{11} -A_{12} } \right) /2\), Eq. (B.2.1.2) becomes
$$\begin{aligned} \left\{ {{\begin{array}{*{20}c} {m_{xx} } \\ {m_{yy} } \\ {m_{zz} } \\ {\begin{array}{l} m_{yz} \\ m_{zx} \\ m_{xy} \\ \end{array}} \\ \end{array} }} \right\} =\left[ {{\begin{array}{*{20}c} {A_{11} } &{} {A_{12} } &{} {A_{12} } &{} 0 &{} 0 &{} 0 \\ {A_{12} } &{} {A_{11} } &{} {A_{12} } &{} 0 &{} 0 &{} 0 \\ {A_{12} } &{} {A_{12} } &{} {A_{11} } &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} {\frac{A_{{11}} -A_{{12}} }{2}} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} {\frac{A_{{11}} -A_{{12}} }{2}} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {\frac{A_{{11}} -A_{{12}} }{2}} \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {\chi _{xx} } \\ {\chi _{yy} } \\ {\chi _{zz} } \\ {\begin{array}{l} 2\chi _{yz} \\ 2\chi _{zx} \\ 2\chi _{xy} \\ \end{array}} \\ \end{array} }} \right\} . \end{aligned}$$
(B.3.3)
Using Eq. (23) in Eq. (B.3.3) then yields
$$\begin{aligned} m_{xx} =\left( {A_{11} -A_{12} } \right) \chi _{xx} ,\,\,\,m_{yy} =\left( {A_{11} -A_{12} } \right) \chi _{yy} ,\,\,\,m_{zz} =\left( {A_{11} -A_{12} } \right) \chi _{zz} . \end{aligned}$$
(B.3.4)
Hence, the constitutive equation for an isotropic linear elastic material can be written as
$$\begin{aligned} m_{ij} =\left( {A_{11} -A_{12} } \right) {\chi }_{\textit{ij}} . \end{aligned}$$
(B.3.5)
Note that when
$$\begin{aligned} \left( {A_{11} -A_{12} } \right) =2\,l^{2}\mu , \end{aligned}$$
(B.3.6)
Eq. (B.3.5) is the same as the constitutive relation in the modified couple stress theory [38, 48].
For the isotropic elastic Mindlin plate with
$$\begin{aligned} \begin{array}{l} C_{11} =C_{33} =\lambda +2\mu ,\,\,\,\,C_{12} =C_{13} =\lambda ,\,\,\,\,C_{44} =\left( {C_{11} -C_{12} } \right) /2=\mu , \\ A_{11} =A_{33} ,\,\,\,\,A_{12} =A_{13} ,\,\,\,\,A_{44} =\left( {A_{11} -A_{12} } \right) /2=\,l^{2}\mu ,\,\,\,\,\gamma =0,\,\,\,\,\zeta =0, \\ \end{array} \end{aligned}$$
(B.3.7)
Eqs. (B.2.3.1–7) reduce to
$$\begin{aligned}&(\lambda +2\mu )hu_{,xx} +\mu hu_{,yy} +(\lambda +\mu )hv_{,xy} +\frac{1}{4}l^{2}\mu h(-u_{,xxyy} -u_{,yyyy} +v_{,xxxy} +v_{,xyyy} )\nonumber \\&\quad +f_{x} +\frac{1}{2}c_{z,y} =m_{0} \ddot{{u}}, \end{aligned}$$
(B.3.8.1)
$$\begin{aligned}&(\lambda +2\mu )hv_{,yy} +\mu hv_{,xx} +(\lambda +\mu )hu_{,xy} +\frac{1}{4}l^{2}\mu h(-v_{,xxyy} -v_{,xxxx} +u_{,xxxy} +u_{,xyyy} ) \nonumber \\&\quad +f_{y} -\frac{1}{2}c_{z,x} =m_{0} \ddot{{v}}, \end{aligned}$$
(B.3.8.2)
$$\begin{aligned}&k_{s}^{2} \mu h(w_{,xx} +w_{,yy} -\phi _{x,x} -\phi _{y,y}) -\frac{1}{4}l^{2}\mu h(w_{,xxxx} +2w_{,xxyy} +w_{,yyyy} +\phi _{x,xxx} +\phi _{x,xyy} +\phi _{y,xxy} \nonumber \\&\quad +\phi _{y,yyy} )+f_{z} -\frac{1}{2}c_{x,y} +\frac{1}{2}c_{y,x} =m_{0} \ddot{{w}}, \end{aligned}$$
(B.3.8.3)
$$\begin{aligned}&\quad -\frac{h^{3}}{12}(\lambda +2\mu )\phi _{x,xx} -\frac{h^{3}}{12}\mu \phi _{x,yy} -\frac{h^{3}}{12}(\lambda +\mu )\phi _{y,xy} -k_{s}^{2} \mu h(w_{,x} -\phi _{x} )-\frac{l^{2}\mu h^{3}}{48}(-\phi _{x,xxyy} -\phi _{x,yyyy} \nonumber \\&\quad +\phi _{y,xxxy} +\phi _{y,xyyy} )+\frac{l^{2}\mu h}{4}(-w_{,xxx} -w_{,xyy} -\phi _{x,xx} -4\phi _{x,yy} +3\phi _{y,xy} )+\frac{1}{2}c_{y} =-m_{2} \ddot{{\phi }}_{x} , \nonumber \\ \end{aligned}$$
(B.3.8.4)
$$\begin{aligned}&\quad -\frac{h^{3}}{12}(\lambda +2\mu )\phi _{y,yy} -\frac{h^{3}}{12}\mu \phi _{y,xx} -\frac{h^{3}}{12}(\lambda +\mu )\phi _{x,xy} -k_{s}^{2} \mu h(w_{,y} -\phi _{y} )-\frac{l^{2}\mu h^{3}}{48}(\phi _{x,xxxy} +\phi _{x,xyyy} \nonumber \\&\quad -\phi _{y,xxxx} -\phi _{y,xxyy} )+\frac{l^{2}\mu h}{4}(-w_{,xxy} -w_{,yyy} +3\phi _{x,xy} -4\phi _{y,xx} -\phi _{y,yy} )-\frac{1}{2}c_{x} =-m_{2} \ddot{{\phi }}_{y} . \end{aligned}$$
(B.3.8.5)
These equations are identical to those derived in Gao and Zhang [12] using the same modified couple stress theory. Note that Eqs. (B.3.8.1–5) can be further simplified to the Mindlin plate based on classical elasticity (e.g., [27]) by suppressing the microstructure effect by setting \(l = 0\).
