On a Consistent Dynamic Finite-Strain Plate Theory and Its Linearization

Abstract

This paper develops a dynamic finite-strain plate theory consistent with three-dimensional Hamilton’s principle under general loadings with a fourth-order error. Starting from the three-dimensional field equations for a compressible hyperelastic material and by a series expansion about the bottom surface, we deduce a vector dynamic plate equation with three unknowns, which exhibits the local momentum-balance structure. Associated weak formulations are considered, in connection with various boundary conditions. Then, by linearization, we provide a novel linear plate theory for orthotropic materials, which takes into account both stretching and bending effects. For isotropic materials, it is further modified to a refined linear plate theory, leading to higher-order results under certain circumstances. To verify the present plate theory, an exhaustive study of the free vibration and static bending problems is carried out. Comparing with the available exact three-dimensional solutions for these problems, it is shown that the plate theory indeed provides correct asymptotic results for displacements and stresses distributions. The advantages of this linear plate theory include (i) its simplicity (as simple as the Kirchhoff-Love theory), (ii) high accuracy for frequencies and deflections, as well as capability of capturing the distributions of all concerned quantities, and (iii) applicability for general loadings.

Appendices

Appendix A: Recursion Relations and Moduli Tensors in the Linear Case

The component forms of (42) and (41)₁ read

$$ \begin{aligned} {x}_{i}^{(1)}&= \tilde{B}_{ij} \bigl( - {q}_{j}^{-} - { \mathscr{A}}_{03j\alpha k}^{1} {x}_{k,\alpha}^{(0)} + { \mathscr{A}}_{03jkk}^{1} \bigr) \\ &= - \tilde{B}_{ij} {q}_{j}^{-} - { \mathscr{A}}_{0\alpha k 3j}^{1} \tilde{B}_{ji} {x}_{k,\alpha}^{(0)} + {\mathscr{A}}_{0kk3j}^{1} \tilde{B}_{ji} \\ &= - \tilde{B}_{ij} {q}_{j}^{-} + C_{\alpha k i}^{(0)} {x}_{k,\alpha}^{(0)} - C_{k k i}^{(0)}, \\ S_{m i}^{(0)} &= {\mathscr{A}}_{0m i\beta j}^{1} {x}_{j,\beta}^{(0)} + {\mathscr{A}}_{0m i3 j}^{1} {x}_{j}^{(1)} - {\mathscr{A}}_{0m ikl}^{1} \delta_{kl} \\ &= {\mathscr{A}}_{0m i\beta j}^{1} {x}_{j,\beta}^{(0)} + {\mathscr{A}}_{0m i3 j}^{1} \bigl[- \tilde{B}_{jk} {q}_{k}^{-} + C_{\alpha k j}^{(0)} {x}_{k,\alpha}^{(0)} - C_{k k j}^{(0)} \bigr] - { \mathscr{A}}_{0m ikk}^{1} \\ &= D_{m i \alpha k}^{(0)} {x}_{k,\alpha}^{(0)} + C_{m i k}^{(0)} {q}_{k}^{-} -D_{m i kk}^{(0)}, \end{aligned} $$

(110)

where

$$ \begin{aligned} (\tilde{B}_{ij}) =& \bigl({ \mathscr{A}}_{03 i3 j}^{1}\bigr)^{-1}, \qquad C_{m i k}^{(0)}=- {\mathscr{A}}_{0m i3 j}^{1} \tilde{B}_{jk}, \\ D_{m i j k}^{(0)}=& {\mathscr{A}}_{0 m ij k}^{1}+ {\mathscr{A}}_{0mi 3 n}^{1} C_{jk n}^{(0)}= { \mathscr{A}}_{0 m ij k}^{1} - {\mathscr{A}}_{0mi 3 n}^{1} {\mathscr{A}}_{0jk 3 p}^{1} \tilde{B}_{pn}. \end{aligned} $$

(111)

It is easy to verify the symmetry and the special values

$$ \begin{aligned} C_{m i k}^{(0)} &= C_{imk}^{(0)}, \qquad C_{3 i k}^{(0)}=-\delta_{ik} , \\ D_{m i j k}^{(0)} &= D_{im j k}^{(0)} =D_{m i kj}^{(0)} =D_{j k m i}^{(0)}, \qquad D_{3 i j k}^{(0)} = 0, \end{aligned} $$

(112)

which imply that $S_{m i}^{(0)}$ in (110) does not involve ${x}_{3}^{(0)}$ and $S_{m 3}^{(0)}$ only involves $q_{k}^{-}$. Furthermore, for isotropic or orthotropic materials, $\tilde{\mathbb{B}}$ is always a diagonal matrix, and as a result

$$ \begin{aligned} {\mathscr{A}}_{0\alpha\beta\gamma3}^{1} =0, \qquad C_{\alpha\beta \gamma}^{(0)} =0, \qquad C_{\alpha33}^{(0)} =0, \qquad C_{ 33\alpha}^{(0)} =0, \end{aligned} $$

(113)

which imply that ${x}_{\gamma}^{(1)}$ in (110) only involves derivatives of ${x}_{3}^{(0)}$, while ${x}_{3}^{(1)}$ only involves the derivatives of ${x}_{\alpha}^{(0)}$.

Now we analyze

$$\begin{aligned} {x}_{i}^{(2)} =& \tilde{B}_{ij}\rho \ddot{x}_{j}^{(0)} - \tilde{B}_{ij} \bigl( {\mathscr{A}}_{03j\alpha k}^{1} {x}_{k,\alpha}^{(1)} + S_{\alpha j,\alpha }^{(0)} \bigr) \\ = &\tilde{B}_{ij}\rho \ddot{x}_{j}^{(0)} + C_{\alpha k i}^{(0)} {x}_{k,\alpha}^{(1)} - \tilde{B}_{ij} S_{\alpha j,\alpha}^{(0)}. \end{aligned}$$

(114)

For isotropic or orthotropic cases, ${x}_{\gamma}^{(2)}$ only involves $\ddot{x}_{\alpha}^{(0)}$ and spatial derivatives of ${x}_{3}^{(1)}$ and $S_{\alpha\beta}^{(0)}$, and thus derivatives of ${x}_{\alpha}^{(0)}$; on the other hand, ${x}_{3}^{(2)}$ only involves $\ddot{x}_{3}^{(0)}$ and spatial derivatives of ${x}_{\gamma}^{(1)}$ ($S_{\alpha3}^{(0)}$ becomes $-q_{\alpha}^{-}$), and thus derivatives of ${x}_{3}^{(0)}$. Then, we can calculate

$$\begin{aligned} S_{m i}^{(1)} =& {\mathscr{A}}_{0m i\alpha k}^{1} {x}_{k,\alpha}^{(1)} + { \mathscr{A}}_{0m i3 j}^{1} {x}_{j}^{(2)} \\ =& {\mathscr{A}}_{0m i\alpha k}^{1} {x}_{k,\alpha}^{(1)} + {\mathscr{A}}_{0m i3 j}^{1} \bigl( \tilde{B}_{j k} \rho \ddot{x}_{k}^{(0)} + C_{\alpha k j}^{(0)} {x}_{k,\alpha}^{(1)} - \tilde{B}_{jk} S_{\alpha k,\alpha}^{(0)} \bigr) \\ =&- C_{m i k}^{(0)} \rho\ddot{x}_{k}^{(0)} + D_{m i \alpha k}^{(0)} {x}_{k,\alpha}^{(1)} + C_{m i k}^{(0)} S_{\alpha k,\alpha}^{(0)}. \end{aligned}$$

(115)

For isotropic or orthotropic cases, $S_{\alpha\beta}^{(1)}$ only involves $\ddot{x}_{3}^{(0)}$ and spatial derivatives of ${x}_{\gamma}^{(1)}$, thus only derivatives of ${x}_{3}^{(0)}$; while $S_{\alpha 3}^{(1)}$ only involves $\ddot{x}_{\gamma}^{(0)}$ and spatial derivatives of ${S}_{\beta\gamma}^{(0)}$, thus only derivatives of ${x}_{\gamma}^{(0)}$; and $S_{3 3}^{(1)}$ only involves $\ddot{x}_{3}^{(0)}$.

We might also express ${x}_{i}^{(2)}$ and $S_{m i}^{(1)}$ in terms of ${x}^{(0)}$ by combining the above recursion relations

$$\begin{aligned} {x}_{i}^{(2)} = & \tilde{B}_{ij}\rho \ddot{x}_{j}^{(0)} + C_{\alpha k i}^{(0)} {x}_{k,\alpha}^{(1)} - \tilde{B}_{ij} S_{\alpha j,\alpha}^{(0)} \\ =&\tilde{B}_{ij}\rho \ddot{x}_{j}^{(0)} + C_{\alpha k i}^{(0)} \bigl( - \tilde{B}_{kn} {q}_{n,\alpha}^{-} + C_{\beta n k}^{(0)} {x}_{n,\alpha\beta}^{(0)} \bigr) \\ &{}- \tilde{B}_{ij} \bigl( D_{\alpha j \beta n}^{(0)} {x}_{n,\alpha\beta }^{(0)} + C_{\alpha j n}^{(0)} {q}_{n,\alpha}^{-} \bigr) \\ =&\tilde{B}_{ij}\rho \ddot{x}_{j}^{(0)} + C_{i\alpha n} {q}_{n,\alpha}^{-} + C^{(1)}_{\alpha i \beta n} {x}_{n,\alpha\beta}^{(0)}, \end{aligned}$$

(116)

where (changing $\alpha\rightarrow m$, $\beta\rightarrow k$)

$$ \begin{aligned} C_{im n}&= - C_{m k i}^{(0)} \tilde{B}_{kn} - C_{m j n}^{(0)} \tilde {B}_{ij}, \\ C^{(1)}_{m i kn}& = C_{m j i}^{(0)} C_{k n j}^{(0)} -\tilde{B}_{ij} D_{m j k n}^{(0)}, \end{aligned} $$

(117)

with the properties

$$ \begin{aligned} C_{im n}= C_{nmi},\qquad C_{i3 n}= 2 \tilde{B}_{in},\qquad C^{(1)}_{m i k n} = C^{(1)}_{m i n k}. \end{aligned} $$

(118)

And

$$\begin{aligned} S_{m i}^{(1)} =& -C_{m i k}^{(0)} \rho\ddot{x}_{k}^{(0)} + D_{m i \alpha k}^{(0)} {x}_{k,\alpha}^{(1)} + C_{m i j}^{(0)} S_{\alpha j,\alpha}^{(0)} \\ =&- C_{m i k}^{(0)} \rho\ddot{x}_{k}^{(0)} + D_{\alpha k m i }^{(0)} \bigl( - \tilde{B}_{kn}{q}_{n,\alpha}^{-} + C_{\beta n k}^{(0)}{x}_{n,\alpha\beta}^{(0)} \bigr) \\ &{}+ C_{m i j}^{(0)} \bigl( D_{\alpha j \beta n}^{(0)}{x}_{n,\alpha\beta}^{(0)} + C_{\alpha j n}^{(0)}{q}_{n,\alpha}^{-} \bigr) \\ =& - C_{m i k}^{(0)} \rho\ddot{x}_{k}^{(0)} + C_{\alpha n m i}^{(1)} {q}_{n,\alpha}^{-} + D_{m i \alpha\beta n }^{(1)} {x}_{n,\alpha\beta}^{(0)}, \end{aligned}$$

(119)

where (changing $\alpha\rightarrow p$, $\beta\rightarrow j$)

$$ \begin{aligned} D_{m i p j n }^{(1)}= D_{p k m i}^{(0)} C_{j n k}^{(0)} + D_{p k j n}^{(0)} C_{mi k }^{(0)}, \end{aligned} $$

(120)

with symmetry properties

$$ \begin{aligned} D_{m i p j n }^{(1)}= D_{im p j n }^{(1)} =D_{m i p n j }^{(1)} = D_{j n p m i }^{(1)}. \end{aligned} $$

(121)

Appendix B: Recursion Relations, Stress Coefficients and Functions for Isotropic Case

The high-order coefficients of current position $\mathbf{x}$ are

$$ \begin{aligned} &x_{1}^{(1)}=\hat{u}^{(1)}= -w_{X} - \frac{2 (\nu+1) }{E} q_{1}^{-}, \\ &x_{2}^{(1)}=\hat{v}^{(1)}= -w_{Y} - \frac{2 (\nu+1)}{E} q_{2}^{-}, \\ &x_{3}^{(1)}=1+ \hat{w}^{(1)} = 1+ \frac{\nu}{\nu-1} (u_{X} +v_{Y}) - \frac{ (2 \nu^{2}+\nu-1 )}{E (\nu-1)} q_{3}^{-}, \end{aligned} $$

(122)

and

$$ \begin{aligned} & x_{1}^{(2)}= \hat{u}^{(2)}= - \Delta u +\frac{1}{(\nu-1)} (u_{X} +v_{Y})_{X} +\frac{2(\nu+ 1)}{E} \rho\ddot{u} - \frac{1 + \nu}{E (\nu-1)} q_{3X}^{-}, \\ &x_{2}^{(2)}=\hat{v}^{(2)}= - \Delta v + \frac{1}{(\nu-1)} (u_{X} +v_{Y})_{Y} + \frac{2(\nu+ 1)}{E} \rho\ddot{v} - \frac{1 + \nu}{E (\nu-1)} q_{3Y}^{-}, \\ &x_{3}^{(2)}=\hat{w}^{(2)} = -\frac{\nu}{\nu-1} \Delta w + \frac{ (2 \nu^{2}+\nu-1 )}{E (\nu-1)} \rho\ddot{w} -\frac{1+ \nu}{E (\nu -1)} \bigl(q_{1X}^{-} + q_{2Y}^{-}\bigr). \end{aligned} $$

(123)

The subsequent coefficients of $\mathbf{x}$ are given by (for $i\ge1$)

$$ \begin{aligned} & x_{1}^{(i+2)}= \frac{2 (\nu+1) }{E} \rho\ddot {x}_{1}^{(i)}-x_{1YY}^{(i)} + \frac{1}{2 \nu-1}\bigl[x_{3X}^{(i+1)} +x_{2XY}^{(i)} \bigr]-\frac{2 (\nu-1)}{2 \nu-1} x_{1XX}^{(i)}, \\ & x_{2}^{(i+2)}= \frac{2 (\nu+1) }{E} \rho\ddot {x}_{2}^{(i)}-x_{2XX}^{(i)} + \frac{1}{2 \nu-1}\bigl[x_{3Y}^{(i+1)} +x_{1XY}^{(i)} \bigr]-\frac{2 (\nu-1)}{2 \nu-1} x_{2YY}^{(i)}, \\ & x_{3}^{(i+2)}= \frac{ (2 \nu^{2}+\nu-1 ) }{E (\nu -1)} \rho \ddot{x}_{3}^{(i)} + \frac{(1-2 \nu) }{2 (\nu-1)} \Delta x_{3}^{(i)} +\frac{1}{2 (\nu -1)} \bigl[x_{1X}^{(i+1)} + x_{2Y}^{(i+1)}\bigr]. \end{aligned} $$

(124)

The stress coefficients are given by

$$ \begin{aligned} &S_{11}^{(0)} = \frac{E }{1-\nu^{2}} (\nu v_{Y}+u_{X} ) + \frac{\nu }{\nu-1} q_{3}^{-}, \\ &S_{12}^{(0)} = \frac{E (u_{Y}+v_{X} )}{2 (\nu+1)}, \\ &S_{22}^{(0)} = \frac{E }{1- \nu^{2}} (v_{Y}+\nu u_{X} ) + \frac{\nu}{\nu-1} q_{3}^{-}, \\ &S_{13}^{(0)} = -q_{1}^{-},\qquad S_{23}^{(0)} = -q_{2}^{-},\qquad S_{33}^{(0)} = -q_{3}^{-}, \end{aligned} $$

(125)

and

$$ \begin{aligned} &S_{11}^{(1)} = \frac{E}{\nu^{2}-1}(\nu w_{YY} + w_{XX}) - \frac {\nu}{\nu -1} \rho\ddot{w} + \frac{1}{\nu-1} \bigl[(2 - \nu)q_{1X}^{-} + \nu q_{2Y}^{-} \bigr], \\ &S_{12}^{(1)} = -\frac{E w_{XY}}{\nu+1} - \bigl(q_{1Y}^{-} + q_{2X}^{-}\bigr), \\ &S_{22}^{(1)} = \frac{E}{\nu^{2}-1}(\nu w_{XX} + w_{YY}) - \frac {\nu}{\nu -1} \rho\ddot{w} + \frac{1}{\nu-1} \bigl[(2 -\nu)q_{2Y}^{-} + \nu q_{1X}^{-} \bigr], \\ &S_{13}^{(1)} = \frac{E}{2 (\nu^{2}-1 )} \bigl[(1-\nu) \Delta u+ (1+\nu ) (u_{X}+v_{Y})_{X} \bigr] + \rho \ddot{u} - \frac{\nu}{\nu-1} q_{3X}^{-}, \\ &S_{23}^{(1)} = \frac{E}{2 (\nu^{2}-1 )} \bigl[(1-\nu) \Delta v+ (1+\nu ) (u_{X}+v_{Y})_{Y} \bigr] + \rho \ddot{v} - \frac{\nu}{\nu-1} q_{3Y}^{-}, \\ &S_{33}^{(1)} =\rho\ddot{w} +q_{1X}^{-} + q_{2Y}^{-}. \end{aligned} $$

(126)

The components of $\mathbb{S}^{(i)}$ ($i\ge2$) can be obtained by using (39) and the above formulas of $\mathbf{x}^{(k)}$ together with the explicit expressions $\mathscr{A}_{0}^{1}$ for isotropic case.

The functions $f_{i}(\mathbf{q}^{-})$ in the isotropic linear system (62) are given by

$$\begin{aligned} f_{1} =& \frac{\nu q_{3X}^{-}}{\nu-1} - \frac{ h}{\nu-1} \bigl[(\nu-1) \Delta q_{1}^{-}- q_{\phi X}^{-} \bigr] + \frac{2 h (\nu+1) \rho \ddot{q}_{1}^{-} }{E} , \\ f_{2} =& \frac{\nu q_{3Y}^{-}}{\nu-1} - \frac{ h}{\nu-1} \bigl[(\nu-1) \Delta q_{2}^{-}- q_{\phi Y}^{-} \bigr] + \frac{2 h (\nu+1) \rho \ddot{q}_{2}^{-}}{E}, \\ f_{3} =& -q_{\phi}^{-} - \frac{\nu h}{\nu-1} \Delta q_{3}^{-} +\frac{2 (\nu-2) h^{2}}{3 (\nu-1)} \Delta q_{\phi}^{-} +\frac{h (\nu+1) \rho ((6 \nu-3) \ddot {q}_{3}^{-} -4 h (\nu-1) \ddot{q}_{\phi}^{-} )}{3 E (\nu-1)}, \end{aligned}$$

(127)

where $q_{\phi}^{-}$ are given in (64). And the functions in (65) are given by

$$ \begin{aligned} f_{\phi}\bigl(\mathbf{q}^{-} \bigr) &= f_{1X}+ f_{2Y} = \frac{\nu}{\nu-1} \Delta q_{3}^{-} - \frac{ (\nu-2) h}{\nu-1} \Delta q_{\phi}^{-} + \frac{2 h (\nu+1) \rho \ddot{q}_{\phi}^{-} }{E}, \\ f_{\varphi}\bigl(\mathbf{q}^{-}\bigr) &= f_{2X}- f_{1Y} =-h \Delta q_{\varphi}^{-} + \frac{2 h (\nu+1) \rho \ddot{q}_{\varphi}^{-} }{E}. \end{aligned} $$

(128)

The functions $F_{i}(\mathbf{q}^{\pm})$ in the refined dynamic system (81) are given by

$$\begin{aligned} F_{1} = & \bar{q}_{1} + \frac{q_{3X}^{-}-(\nu-1) q_{3X}^{+} }{\nu-1} + \biggl[\frac {2 (\nu+1) \rho (2 \ddot{q}_{1}^{-}- \ddot{q}_{1}^{+} )}{3 E}+\frac{(4 \nu-1) q_{\phi X}^{-} }{3 (\nu-1)} - \frac{2}{3} q_{\phi X}^{+} \\ &{} + \frac{1}{3} \Delta \bigl( q_{1}^{+} -2 q_{1}^{-} \bigr)\biggr]h + \biggl(\frac{2 (-6 \nu^{3}+5 \nu^{2}+7 \nu-4 ) \rho \ddot{q}_{3X}^{-} }{3 E (\nu-1)^{2}} + \frac{2}{3} \Delta{q}_{3X}^{-} \biggr)h^{2} \\ &{} + \biggl(-\frac{8 (\nu+1)^{2} \rho^{2} \ddddot{q}_{1}^{-} }{3 E^{2}} - \frac { (\nu^{2}-\nu-2 ) \rho\ddot{q}_{\phi X}^{-} }{3 E(\nu-1)^{2}} + \frac{2 (\nu+1) \rho\Delta\ddot{q}_{1}^{-} }{E}- \frac{1}{3} \Delta ^{2} {q}_{1}^{-} \biggr)h^{3}, \\ F_{2} = & \bar{q}_{2} +\frac{q_{3Y}^{-}-(\nu-1) q_{3Y}^{+} }{\nu-1} + \biggl[ \frac{2 (\nu+1) \rho (2 \ddot{q}_{2}^{-}- \ddot{q}_{2}^{+} )}{3 E}+\frac{(4 \nu-1) q_{\phi Y}^{-} }{3 (\nu-1)} -\frac{2}{3} q_{\phi Y}^{+} \\ &{}+ \frac{1}{3} \Delta \bigl( q_{2}^{+} -2 q_{2}^{-} \bigr)\biggr]h + \biggl(\frac{2 (-6 \nu^{3}+5 \nu^{2}+7 \nu-4 ) \rho \ddot{q}_{3Y}^{-} }{3 E (\nu-1)^{2}} + \frac{2}{3} \Delta{q}_{3Y}^{-} \biggr)h^{2} \\ &{}+ \biggl(-\frac{8 (\nu+1)^{2} \rho^{2} \ddddot{q}_{2}^{-} }{3 E^{2}} - \frac { (\nu^{2}-\nu-2 ) \rho\ddot{q}_{\phi Y}^{-} }{3 E(\nu-1)^{2}} + \frac{2 (\nu+1) \rho\Delta\ddot{q}_{2}^{-} }{E}- \frac{1}{3} \Delta ^{2} {q}_{2}^{-} \biggr)h^{3}, \\ F_{3} =& \bar{q}_{3} +\frac{1}{2} \bigl(q_{\phi}^{+}-q_{\phi}^{-}\bigr) + \biggl(\frac{ (2 \nu^{2}+\nu-1 ) \rho \ddot{q}_{3}^{-} }{E (\nu-1)}-\frac{2}{5} \Delta\bigl(q_{3}^{+} + q_{3}^{-}\bigr) \biggr) h \\ &{}+ \biggl(\frac{2 \nu (\nu+1) \rho\ddot{q}_{\phi}^{-}}{3 E (\nu-1)} +\frac{(6 \nu-1) \Delta q_{\phi}^{-} }{15 (\nu-1)} - \frac{1}{15} \Delta q_{\phi}^{+} \biggr)h^{2} \\ &{}+ \biggl(\frac{1}{3} \Delta^{2} q_{3}^{-} - \frac{ (14 \nu^{3}-22 \nu ^{2}-19 \nu+17 ) \rho\Delta\ddot{q}_{3}^{-}}{15 E (\nu-1)^{2}} \biggr) h^{3} \\ &{}+ \biggl(-\frac{8 (\nu+1)^{2} \rho^{2} \ddddot{q}_{\phi}^{-}}{15 E^{2}} + \frac{ (10 \nu^{3}-3 \nu^{2}-7 \nu+6 ) \rho \Delta\ddot {q}_{\phi}^{-} }{15 E (\nu-1)^{2}} - \frac{1}{5} \Delta^{2} q_{\phi}^{-} \biggr) h^{4}. \end{aligned}$$

(129)

Appendix C: The Replacement Procedure in Sect. 6.2

The replacement procedure takes the following steps in sequence

• 1. Replace $\ddddot{u}$ in (80)₁ by utilizing (62)₁, i.e., by taking twice time derivative of

  $$\frac{1}{2} \bar{E} \bigl[(1-\nu) \Delta u+ (1+\nu)\phi_{X} \bigr] + \bar{q}_{1} + f_{1} =\rho \ddot{u}, $$

  and similarly replace $\ddddot{v}$ in (80)₂ by utilizing (62)₂.

• 2. Replace $\Delta^{2} u$ in (80)₁ by also utilizing (62)₁, i.e., by taking a $\Delta$ on the equation

  $$\frac{1}{2} \bar{E} \bigl[(1-\nu) \Delta u+ (1+\nu)\phi_{X} \bigr] - \bar{E} h \Delta w_{X} +\bar{q}_{1} + f_{1} =\rho \ddot{u}, $$

  and similarly replace $\Delta^{2} v$ in (80)₂ by utilizing (62)₂.

• 3. Replace $\Delta\phi_{X}$ and $\Delta\phi_{Y}$ in (80)_1,2 by using (65)₁, i.e., by taking proper derivative of

  $$\bar{E} \Delta\phi - \bar{E} h \Delta^{2} w + \bar{q}_{\phi}+ f_{\phi}=\rho \ddot{\phi}, $$

  and replace $\Delta^{2} \phi$ in (80)₃ by conducting a $\Delta$ on the above equation without the term $\rho \ddot{\phi}$.

• 4. Replace $\Delta^{2} w_{X}$ and $\Delta^{2} w_{Y}$ in (80)_1,2 by using (66), i.e., taking proper derivative of

  $$-\frac{1 }{3} \bar{E} h^{2} \Delta^{2} w + \bar{q}_{3} + f_{3} + h(\bar{q}_{\phi}+ f_{\phi})= \rho \ddot{w} + \frac{\rho\nu h}{\nu-1} \ddot{\phi}, $$

  and replace $\Delta^{3} w$ in (80)₃ by conducting a $\Delta$ on the above equation without the term $\ddot{\phi}$.

• 5. Eliminate the term $\Delta\phi$ in the resulting (80)₃ by utilizing the first two equations (like the derivation of (66)), and neglect the resulting $\rho h^{3} \Delta\ddot{\phi }$ term on the right hand side.