Appendix 1: Translational Formulations
The coefficients and rotational angles are translated as follows:
$$ \begin{aligned} & w_{,x} = w_{,r} ;\quad w_{,xx} = w_{,rr} ; \\ & w_{,y} = \frac{1}{r}w_{,\phi } ;\quad w_{,yy} = \frac{1}{r}w_{,r} + \frac{1}{{r^{2} }}w_{,\phi \phi } ;\quad w_{,xy} = \frac{1}{r}w_{,r\phi } - \frac{1}{{r^{2} }}w_{,\phi }\,; \cdots \\ \end{aligned} $$
(7.73)
$$ \begin{aligned} & \theta_{x} = \theta_{r} ;\quad \theta_{y} = \frac{1}{r}\theta_{\phi } ;\quad \theta_{x,x} = \theta_{r,r} ;\quad \theta_{x,y} = \frac{1}{r}\theta_{r,\phi } - \frac{1}{{r^{2} }}\theta_{\phi } ;\;\quad \theta_{x,xx} = \theta_{r,rr} ; \\ & \theta_{x,xy} = \frac{1}{r}\theta_{r,r\phi } - \frac{1}{r}\left( {\frac{1}{r}\theta_{\phi } } \right)_{,r} - \frac{1}{{r^{2} }}\theta_{r,\phi } + \frac{1}{{r^{3} }}\theta_{\phi } ;\quad \theta_{x,xxx} = \theta_{r,rrr} ; \\ & \theta_{y,x} = \left( {\frac{1}{r}\theta_{\phi } } \right);\quad \theta_{y,y} = \frac{1}{r}\theta_{r} + \frac{1}{{r^{2} }}\theta_{\phi ,\phi } ; \\ & \theta_{y,yy} = \frac{1}{r}\left( {\frac{1}{r}\theta_{\phi } } \right)_{,r} + \frac{2}{{r^{2} }}\theta_{r,\phi } + \frac{1}{{r^{3} }}\theta_{\phi ,\phi \phi } - \frac{1}{{r^{3} }}\theta_{\phi } ; \\ & \theta_{y,xy} = \frac{1}{r}\left[ {\theta_{r,r} + \left( {\frac{1}{r}\theta_{\phi } } \right)_{,r\phi } } \right] - \frac{1}{{r^{2} }}\left( {\theta_{r} + \frac{1}{r}\theta_{\phi ,\phi } } \right); \\ & \theta_{x,yy} = \frac{1}{r}\theta_{r,r} + \frac{1}{{r^{2} }}\theta_{r,\phi \phi } - \frac{1}{{r^{2} }}\theta_{r} - \frac{2}{{r^{3} }}\theta_{\phi ,\phi } ;\quad \theta_{y,xx} = \left( {\frac{1}{r}\theta_{\phi } } \right)_{,rr}; \cdots \\ \end{aligned} $$
(7.74)
Appendix 2: Coefficients and Load Terms
The coefficients in Eqs. (7.36)–(7.38) are defined as
$$ \begin{aligned} A_{{1mn\bar{m}\bar{n}}} & = \int\limits_{0}^{2\pi } {\int\limits_{0}^{{r_{0} }} {\left[ {Gh_{0} \kappa_{r} \alpha_{Gr,r} f_{{\bar{m}\bar{n},r}} + } \right.} } Gh_{0} \kappa_{r} \alpha_{Gr} f_{{\bar{m}\bar{n},rr}} \\ & \quad + Gh_{0} \kappa_{\phi } \alpha_{G\phi ,\phi } \frac{1}{{r^{2} }}f_{{\bar{m}\bar{n},\phi }} + \left. {Gh_{0} \kappa_{\phi } \alpha_{G\phi } \left( {\frac{1}{r}f_{{\bar{m}\bar{n},r}} + \frac{1}{{r^{2} }}f_{{\bar{m}\bar{n},\phi \phi }} } \right)} \right]f_{mn} \,\text{d}rr\,\text{d}\phi \\ \end{aligned} $$
(7.75)
$$ \begin{aligned} A_{{2mn\bar{m}\bar{n}}} & = \int\limits_{0}^{2\pi } {\int\limits_{0}^{{r_{0} }} {\bigg( {Gh_{0} \kappa_{r} \alpha_{Gr,r} g_{{r\bar{m}\bar{n}}} + Gh_{0} \kappa_{r} \alpha_{Gr} g_{{r\bar{m}\bar{n},r}} } } } \\ & \quad + { Gh_{0} \kappa_{\phi } \alpha_{G\phi } \frac{1}{r}g_{{r\bar{m}\bar{n}}} } \bigg)f_{mn} \,\text{d}rr\,\text{d}\phi \\ \end{aligned} $$
(7.76)
$$ A_{{3mn\bar{m}\bar{n}}} = \int\limits_{0}^{2\pi } {\int\limits_{0}^{{r_{0} }} {\left( {Gh_{0} \kappa_{\phi } \alpha_{G\phi ,\phi } \frac{1}{{r^{2} }}g_{{\phi \, \bar{m}\bar{n}}} + Gh_{0} \kappa_{\phi } \alpha_{G\phi } \frac{1}{{r^{2} }}g_{{\phi \, \bar{m}\bar{n},\phi }} } \right)} } f_{mn} \,\text{d}rr\,\text{d}\phi $$
(7.77)
$$ B_{{1mn\bar{m}\bar{n}}} = - \int\limits_{0}^{2\pi } {\int\limits_{0}^{{r_{0} }} {Gh_{0} \kappa_{r} \alpha_{Gr} f_{{\bar{m}\bar{n},r}} g_{rmn} \,\text{d}rr\,\text{d}\phi ,} } $$
(7.78)
$$ \begin{aligned} B_{{2mn\bar{m}\bar{n}}} & = \int\limits_{0}^{2\pi } {\int\limits_{0}^{{r_{0} }} {\left\{ {D_{0} \text{d}_{,r} g_{{r\bar{m}\bar{n},r}} + D_{0} \text{d}_{,r} \nu\frac{1}{r}g_{{r\bar{m}\bar{n}}} } \right.} } \\ & \quad + D_{0} \text{d}\left[ {g_{{r\bar{m}\bar{n},rr}} + \nu\,\left( {\frac{1}{r}g_{{r\bar{m}\bar{n},r}} - \frac{1}{{r^{2} }}g_{{r\bar{m}\bar{n}}} } \right)} \right] \\ &\quad+\, \frac{{\left( {1 - \nu} \right)}}{2}D_{0} \text{d}_{,\phi } \frac{1}{{r^{2} }}g_{{r\bar{m}\bar{n},\phi }} + \frac{{\left( {1 - \nu} \right)}}{2}D_{0} \text{d} \\ & \left.\quad \times \left( {\frac{2}{r}g_{{r\bar{m}\bar{n},r}} + \frac{1}{{r^{2} }}g_{{r\bar{m}\bar{n},\phi \phi }} - \frac{2}{{r^{2} }}g_{{r\bar{m}\bar{n}}} } \right) - {Gh_{0} \kappa_{r} \alpha_{Gr} g_{{r\bar{m}\bar{n}}} } \right\}g_{rmn} \,\text{d}rr\,\text{d}\phi \\ \end{aligned} $$
(7.79)
$$ \begin{aligned}B_{{3mn\bar{m}\bar{n}}} &= \int\limits_{0}^{2\pi } {\int\limits_{0}^{{r_{0} }} {\left[ {D_{0} \text{d}_{,r} \nu\frac{1}{{r^{2} }}g_{{r\bar{m}\bar{n},\phi }} + D_{0} \,\text{d}\nu} \right.} } \left( { - \frac{2}{{r^{3} }}g_{{\phi \bar{m}\bar{n},\phi }} + \frac{1}{{r^{2} }}g_{{\phi \bar{m}\bar{n},r\phi }} } \right) \\ &\quad + \frac{(1 - \nu)}{2}D_{0} \text{d}_{,\phi } \left( { - \frac{2}{{r^{3} }}g_{{\phi \bar{m}\bar{n}}} + \frac{1}{{r^{2} }}g_{{\phi \bar{m}\bar{n},r}} } \right) \\&\left. \quad + \frac{(1 - \nu)}{2}D_{0} \text{d}\left( {\frac{1}{{r^{2} }}g_{{\phi \bar{m}\bar{n},r\phi }} - \frac{4}{{r^{3} }}g_{{\phi \bar{m}\bar{n},\phi }} } \right) \right]g_{rmn} \,\text{d}rr\,\text{d}\phi \\ \end{aligned} $$
(7.80)
$$ C_{{1mn\bar{m}\bar{n}}} = - \int\limits_{0}^{2\pi } {\int\limits_{0}^{{r_{0} }} {Gh_{0} \kappa_{\phi } \alpha_{G\phi } \frac{1}{r}f_{{\bar{m}\bar{n},\phi }} g_{\phi mn}\,\text{d}rr\,\text{d}\phi } } $$
(7.81)
$$ \begin{aligned} C_{{2mn\bar{m}\bar{n}}} & = \int\limits_{0}^{2\pi } {\int\limits_{0}^{{r_{0} }} {\left\{ {D_{0} \text{d}_{,\phi } \left( {\frac{1}{{r^{2} }}g_{{r\bar{m}\bar{n}}} + \nu\frac{1}{r}g_{{r\bar{m}\bar{n},r}} } \right)} \right.} } \\ & \quad + D_{0} \text{d}\left[ {\frac{2}{{r^{2} }}g_{{r\bar{m}\bar{n},\phi }} + \nu\left( {\frac{1}{r}g_{{r\bar{m}\bar{n},r\phi }} - \frac{1}{{r^{2} }}g_{{r\bar{m}\bar{n},\phi }} } \right)} \right] + \frac{(1 - \nu)}{2}D_{0} \text{d}_{,r} \frac{1}{r}g_{{r\bar{m}\bar{n},\phi }} \\ & \quad \left. { + \frac{(1 - \nu)}{2}D_{0} \text{d} \times \left( {\frac{1}{r}g_{{r\bar{m}\bar{n},r\phi }} - \frac{1}{{r^{2} }}g_{{r\bar{m}\bar{n},\phi }} } \right)} \right\}g_{\phi mn}\,\text{d}rr\,\text{d}\phi , \\ \end{aligned} $$
(7.82)
$$ \begin{aligned} C_{{3mn\bar{m}\bar{n}}} & = \int\limits_{0}^{2\pi } {\int\limits_{0}^{{r_{0} }} {\left\{ {D_{0} \text{d}_{,\phi } \frac{1}{{r^{3} }}g_{{\phi \, \bar{m}\bar{n},\phi }} } \right.} } + D_{0} \text{d}\left[ - \frac{2}{{r^{3} }}g_{{\phi \, \bar{m}\bar{n}}} + \frac{1}{{r^{2} }}g_{{\phi \, \bar{m}\bar{n},r}} \right. \\ &\left. \quad+\, \frac{1}{{r^{3} }}g_{{\phi \, \bar{m}\bar{n},\phi \phi }} + \nu\left( {\frac{2}{{r^{3} }}g_{{\phi \, \bar{m}\bar{n}}} - \frac{1}{{r^{2} }}g_{{\phi \, \bar{m}\bar{n},r}} } \right) \right] \\ & \quad + \frac{(1 - \nu)}{2}D_{0} \text{d}_{,r} \left( { - \frac{2}{{r^{2} }}g_{{\phi \, \bar{m}\bar{n}}} + \frac{1}{r}g_{{\phi \, \bar{m}\bar{n},r}} } \right) \\ & \quad + \frac{(1 - \nu)}{2}D_{0} \text{d}\left( {\frac{4}{{r^{3} }}g_{{\phi \, \bar{m}\bar{n}}} - \frac{3}{{r^{2} }}g_{{\phi \, \bar{m}\bar{n},r}} + \frac{1}{r}g_{{\phi \, \bar{m}\bar{n},rr}} } \right) \\ & \quad \left. { - Gh_{0} \kappa_{\phi } \alpha_{G\phi } \frac{1}{r}g_{{\phi \, \bar{m}\bar{n}}} } \right\}g_{\phi \, mn} \,\text{d}rr\,\text{d}\phi \\ \end{aligned} $$
(7.83)
And the terms of external loads in Eq. (7.36) are
$$ P_{mn} = - \int\limits_{0}^{2\pi } {\int\limits_{0}^{{r_{0} }} {p\,f_{mn}\, \text{d}rr\,\text{d}\phi } } $$
(7.84)
They may be calculated easily by the method shown in Appendix 3.
Appendix 3: Calculation Including the Extend Dirac Function \( D(r - r_{i} ) \)
The integral calculation including the extended Dirac function \( D(r - r_{i} ) \) can be written as [4]
$$ \int\limits_{0}^{{r_{0} }} {D(r - r_{i} )} \,f(r)\,\text{d}r = \int\limits_{{r_{i} - (b_{ri,j} /2)}}^{{r_{i} + (b_{ri,j} /2)}} {f(\xi )} \,\text{d}\xi $$
(7.85)
in which \( \xi \) is a supplementary variable of r. The n-th derivatives of the extended Dirac functions can therefore be expressed as
$$ \int\limits_{0}^{{r_{0} }} {D^{(n)} (r - r_{i} )} \,f(r)\,\text{d}r = \int\limits_{{r_{i} - (b_{ri,j} /2)}}^{{r_{i} + (b_{ri,j} /2)}} {( - 1)^{n} f^{(n)} (\xi )} \,\text{d}\xi $$
(7.86)
in which the superscripts enclosed within parentheses indicate the differential order. For calculations including the extended Dirac function \( D(\phi - \phi_{j} ) \), similar expressions may be obtained.
When the conditions \( b_{ri,\,j} \ll r_{0} \) and \( b_{\phi \,i,\,j} \ll { 2}\pi \, r \) are satisfied, the specific functions \( D(r - r_{i} ) \) and \( D(\phi - \phi_{j} ) \) are approximately related to the Dirac delta functions, \( \delta \left( {r - r_{i} } \right) \) and \( \delta \left( {\phi - \phi_{j} } \right) \), by
$$ D(r - r_{i} ) \approx b_{ri,j\,} \delta (r - r_{i} ) $$
(7.87)
$$ D\,(\phi - \phi_{j} ) \approx b_{\phi i,j\,} \delta (\phi - \phi_{j} ) $$
(7.88)
Appendix 4: The Governing Equations for Solid Plates Without Voids Based on Mindlin-Reissner Hypothesis
In Sect. 7.2, we had stated that the governing equations of circular cellular plates based on Mindlin-Reissner hypothesis may easily transform into rectangular solid (normal) plates without voids based on the same Mindlin-Reissner hypothesis by changing \( \alpha_{h\,} \) and d to 1 and \( \kappa_{x} G_{x} \) and \( \kappa_{y} G_{y} \) to \( \kappa G \). We introduce the governing equations of circular solid plates based on Mindlin-Reissner hypothesis for practical use. From Eqs. (7.4) to (7.9), we have
$$ \begin{aligned} & \delta w:\quad m_{0} \alpha_{m} \ddot{w} - [\kappa Gh_{0} (w_{,r} + \theta_{r} )]_{,r} - (\kappa Gh_{0} )_{,\phi } \left( {\frac{1}{{r^{2} }}w_{,\phi } + \frac{1}{{r^{2} }}\theta_{\phi } } \right) \\ & \quad - \kappa Gh_{0} \left( {\frac{1}{r}w_{,r} + \frac{1}{{r^{2} }}w_{,\phi \phi } + \frac{1}{r}\theta_{r} + \frac{1}{{r^{2} }}\theta_{\phi ,\phi } } \right) - p + c\dot{w} = 0 \\ \end{aligned} $$
(7.89)
$$ \begin{aligned} & \delta \theta_{r} :\quad I_{p} \,\ddot{\theta }_{r} + \kappa Gh_{0} (w_{,r} + \theta_{r} ) - D_{0} \left[ {\theta_{r,rr} + \nu\left( {\frac{1}{r}\theta_{r,r} - \frac{2}{{r^{3} }}\theta_{\phi ,\phi } + \frac{1}{{r^{2} }}\theta_{\phi ,r\phi } - \frac{1}{{r^{2} }}\theta_{r} } \right)} \right] \\ & \quad - \frac{(1 - \nu)}{2}D_{0} \left( {\frac{2}{r}\theta_{r,r} + \frac{1}{{r^{2} }}\theta_{r,\phi \phi } + \frac{1}{{r^{2} }}\theta_{\phi ,r\phi } - \frac{1}{{r^{2} }}\theta_{r} - \frac{4}{{r^{3} }}\theta_{\phi ,\phi } } \right) = 0 \\ \end{aligned} $$
(7.90)
$$ \begin{aligned} & \delta \theta_{\phi } :\quad \frac{1}{r}I_{p} \,\ddot{\theta }_{\phi } + \kappa Gh_{0} \left( {\frac{1}{r}w_{,\phi } + \frac{1}{r}\theta_{\phi } } \right) \\ & \quad - D_{0} \left[ - \frac{2}{{r^{3} }}\theta_{\phi } + \frac{1}{{r^{2} }}\theta_{\phi ,r} + \frac{2}{{r^{2} }}\theta_{r,\phi } + \frac{1}{{r^{3} }}\theta_{\phi ,\phi \phi } \right.\\ &\left.\quad+\, \nu\left( {\frac{1}{r}\theta_{r,r\phi } + \frac{2}{{r^{3} }}\theta_{\phi } - \frac{1}{{r^{2} }}\theta_{\phi ,r} - \frac{1}{{r^{2} }}\theta_{r,\phi } } \right) \right] \\ & \quad - \frac{(1 - \nu)}{2}D_{0} \left( {\frac{1}{r}\theta_{r,r\phi } + \frac{4}{{r^{3} }}\theta_{\phi } - \frac{3}{{r^{2} }}\theta_{\phi ,r} - \frac{1}{{r^{2} }}\theta_{r,\phi } + \frac{1}{r}\theta_{\phi ,rr} } \right) = 0 \\ \end{aligned} $$
(7.91)
$$ w = w^{*} \quad \text{or}\quad \kappa Gh_{0} (w_{,r} + \theta_{r} ) = v_{r}^{*} $$
(7.92)
$$ \theta_{r} = \theta_{r}^{*} \quad \text{or}\quad D_{0} \left[ {\theta_{r,r} + \nu\left( {\frac{1}{r}\theta_{r} + \frac{1}{{r^{2} }}\theta_{\phi ,\phi } } \right)} \right] = m_{r}^{*} $$
(7.93)
$$ \theta_{\phi } = \theta_{\phi }^{*} \quad \text{or}\quad \frac{1 - \nu}{2}D_{0} \left( {\frac{1}{r}\theta_{r,\phi } - \frac{2}{{r^{2} }}\theta_{\phi } + \frac{1}{r}\theta_{\phi ,r} } \right) = m_{r\phi }^{*} $$
(7.94)
at \( r = r_{0} \), in which p are lateral loads; \( w^{*} \), \( \theta_{r}^{*} \), and \( \theta_{\phi }^{*} \) are prescribed on the geometrical boundary conditions; and \( v_{r}^{*} \), \( m_{r}^{*} \), and \( m_{r\phi }^{*} \) are prescribed on the mechanical boundary conditions at \( r = r_{0} \).
