Axisymmetric Free Vibration of Soft Electroactive Circular Plates under Biasing Fields

Abstract

Soft electroactive materials (SEAMs) with large elastic deformation capacity as well as excellent electromechanical coupling characteristic have attracted increasing attention in the fields of mechanics and related engineering disciplines. Based on the nonlinear theory of electroelasticity and its linearized version for incremental fields, we derive the state-space formulations for small-amplitude free vibrations of an SEAM circular plate under large pre-deformation due to static biasing fields. An exact three-dimensional solution is then obtained by adopting the finite Hankel transform for the plate with an elastic simple support at the circular boundary. The exact solution for an isotropic linear elastic circular plate can be obtained as a particular and degenerated case. The model of generalized neo-Hookean compressible material is considered in numerical simulations. It is found that the natural frequency, while depending on the intrinsic parameters of the plate (e.g., initial thickness and electromechanical coupling coefficients), can be controlled effectively by the extrinsic factors (e.g., pre-stretch and biasing electric displacement). Results further indicate that Euler’s instability will occur under a certain combination of the biasing electric displacement and pre-stretch, which should be of practical importance when one intends to tune the dynamic characteristics of a plate by means of external loading.

Appendices

Appendix A: Effective Electroelastic Moduli

The nonzero coefficients appearing in Eq. (12) under a biasing field in Fig. 1 are given by [12]:

$$\begin{aligned} \mathscr {A}_{01111}= & {} \frac{{2}}{\lambda _{{3}} }\{\varOmega _{{1}} \text{+ }\varOmega _{{3}} \lambda _{{1}}^{{2}} \lambda _{{3}}^{{2}} +[\lambda _{{1}}^{{2}} (2\varOmega _{{11}} \text{+ }\varOmega _{{2}} )+4\varOmega _{{12}} \lambda _{{1}}^{{2}} \text{+2 }\varOmega _{{22}} \lambda _{{1}}^{{4}} ]\text{+ }\varOmega _{{2}} \lambda _{{3}}^{{2}}\nonumber \\&{+2}\lambda _{{1}}^{{2}} \lambda _{{3}}^{{2}} [2\varOmega _{{12}} \text{+2 }\left( {\varOmega _{{13}} {+}\varOmega _{{22}} } \right) \lambda _{{1}}^{{2}} +{2}\varOmega _{{23}} \lambda _{{1}}^{{4}} \text{+ }\varOmega _{{22}} \lambda _{{3}}^{{2}} \text{+2 }\varOmega _{{23}} \lambda _{{1}}^{{2}} \lambda _{{3}}^{{2}} \text{+ }\varOmega _{{33}} \lambda _{{1}}^{{4}} \lambda _{{3}}^{{2}}]\} \end{aligned}$$

(A.1)

$$\begin{aligned} \mathscr {A}_{01122}= & {} \frac{{2}\lambda _{{1}}^{{2}}}{\lambda _{{3}} }[2\varOmega _{{11}} \text{+2 }\varOmega _{{2}} +4\varOmega _{{12}} \lambda _{{1}}^{{2}} +2\varOmega _{{22}} \lambda _{{1}}^{{4}} +({4}\varOmega _{{12}} +{2}\varOmega _{3} )\lambda _{{3}}^{{2}} \nonumber \\&+2\varOmega _{{22}} \lambda _{3}^{{4}} +2\lambda _{{1}}^{{2}} \lambda _{{3}}^{{2}} (2\varOmega _{{13}} +2\varOmega _{{22}} +{2}\varOmega _{{23}} \lambda _{{1}}^{{2}} +{2}\varOmega _{{23}} \lambda _{{3}}^{{2}} +\varOmega _{{33}} \lambda _{{1}}^{2} \lambda _{{3}}^{{2}} )] \end{aligned}$$

(A.2)

$$\begin{aligned} \mathscr {A}_{01133}= & {} {2}\lambda _{{3}} [2\varOmega _{{11}} \text{+2 }\varOmega _{{2}} +(6\varOmega _{12} +2\varOmega _{{3}} )\lambda _{{1}}^{{2}} +2(\varOmega _{{13}} +2\varOmega _{{22}} )\lambda _{{1}}^{{4}} +2\varOmega _{{23}} \lambda _{{1}}^{6} \nonumber \\&+2\varOmega _{{12}} \lambda _{{3}}^{{2}} +2(\varOmega _{{13}} +2\varOmega _{{22}} )\lambda _{{1}}^{{2}} \lambda _{{3}}^{{2}}+6\varOmega _{{23}} \lambda _{{1}}^{{4}} \lambda _{{3}}^{{2}} +2\varOmega _{{33}} \lambda _{{1}}^{6} \lambda _{{3}}^{{2}}] \nonumber \\&+\frac{4D_{3}^{2} }{\lambda _{{1}}^{{2}} \lambda _{3} }\{\lambda _{1}^{8} \lambda _{3}^{4} (\varOmega _{35} +2\varOmega _{36} \lambda _{{3}}^{{2}} )+\lambda _{{1}}^{6} \lambda _{{3}}^{{2}} [\varOmega _{{15}} +2\varOmega _{{16}} \lambda _{{3}}^{{2}} \nonumber \\&+(\varOmega _{{25}} +2\varOmega _{{26}} \lambda _{{3}}^{{2}} )(\lambda _{{1}}^{{2}} +\lambda _{{3}}^{{2}} )]\} \end{aligned}$$

(A.3)

$$\begin{aligned} \mathscr {A}_{03333}= & {} \frac{2\lambda _{{3}} }{\lambda _{{1}}^{{2}} }\{\varOmega _{1} +2\varOmega _{2} \lambda _{{1}}^{{2}} +\varOmega _{3} \lambda _{1}^{4} +2\lambda _{3}^{{2}} [\varOmega _{{11}} +4\varOmega _{12} \lambda _{{1}}^{{2}} +\lambda _{{1}}^{{4}} (2\varOmega _{{13}} +4\varOmega _{{22}} \nonumber \\&+4\varOmega _{{23}} \lambda _{{1}}^{2} +\varOmega _{{33}} \lambda _{{1}}^{4} )]\}+\frac{2D_{3}^{2} }{\lambda _{{1}}^{{2}} \lambda _{3} }\{4\varOmega _{35} \lambda _{1}^{8} \lambda _{3}^{4} +\lambda _{3}^{{2}} [8\varOmega _{36} \lambda _{1}^{8} \lambda _{3}^{4} +\varOmega _{5} \lambda _{{1}}^{{4}} \nonumber \\&+2\lambda _{{1}}^{{4}} \lambda _{{3}}^{{2}} (2\varOmega _{{15}} +3\varOmega _{6} +4\varOmega _{{25}} \lambda _{{1}}^{2} +4\varOmega _{16} \lambda _{3}^{2} +8\varOmega _{26} \lambda _{{1}}^{{2}} \lambda _{3}^{{2}} )]\} \nonumber \\&+4D_{3}^{4} \lambda _{{1}}^{6} \lambda _{3}^{3} [\varOmega _{55} +4\lambda _{{3}}^{{2}} (\varOmega _{56} +\varOmega _{66} \lambda _{{3}}^{{2}} )] \end{aligned}$$

(A.4)

$$\begin{aligned} \mathscr {A}_{01212}= & {} \frac{2}{\lambda _{3} }(\varOmega _{1} +\varOmega _{2} \lambda _{3}^{{2}} ),{\, \, }\mathscr {A}_{01221} =-\frac{2\lambda _{{1}}^{{2}} }{\lambda _{3} }(\varOmega _{2} +\varOmega _{3} \lambda _{3}^{{2}} ) \end{aligned}$$

(A.5)

$$\begin{aligned} \mathscr {A}_{01313}= & {} \frac{2}{\lambda _{3} }(\varOmega _{1} +\varOmega _{2} \lambda _{1}^{{2}} +D_{3}^{2} \varOmega _{6} \lambda _{1}^{4} \lambda _{3}^{{2}} ),{\, \, }\mathscr {A}_{01331} =-2\lambda _{3} (\varOmega _{2} +\varOmega _{3} \lambda _{1}^{{2}} -D_{3}^{2} \varOmega _{6} \lambda _{{1}}^{{4}} ) \end{aligned}$$

(A.6)

$$\begin{aligned} \mathscr {A}_{03131}= & {} \frac{2\lambda _{{3}} }{\lambda _{{1}}^{{2}} }\{\varOmega _{1} +\varOmega _{2} \lambda _{1}^{{2}} +D_{3}^{2} \lambda _{1}^{4} [\varOmega _{5} +\varOmega _{6} (\lambda _{{1}}^{{2}} +2\lambda _{3}^{{2}} )]\} \end{aligned}$$

(A.7)

$$\begin{aligned} \varvec{\varGamma }_{01\text{13 }}= & {} \frac{{4}D_{3} \lambda _{1}^{{4}} }{\lambda _{{3}} }\{\varOmega _{{14}} +\varOmega _{{24}} (\lambda _{{1}}^{{2}} +\lambda _{3}^{{2}} )\text{+ }\lambda _{{3}}^{{2}} [\varOmega _{15} +\varOmega _{34} \lambda _{{1}}^{{2}} +\varOmega _{16} \lambda _{{3}}^{{2}} +(\varOmega _{26} +\varOmega _{35} )\lambda _{{1}}^{{2}} \lambda _{3}^{{2}}\nonumber \\&+\varOmega _{26}\lambda _{3}^{4} +\varOmega _{36} \lambda _{{1}}^{{2}} \lambda _{3}^{4} +\varOmega _{25} (\lambda _{{1}}^{{2}} +\lambda _{3}^{{2}} )]\} \end{aligned}$$

(A.8)

$$\begin{aligned} \varvec{\varGamma }_{0131}= & {} 2D_{3} \lambda _{1}^{{2}} \lambda _{{3}} [\varOmega _{5} +\varOmega _{6} (\lambda _{{1}}^{{2}} +\lambda _{3}^{{2}} )] \end{aligned}$$

(A.9)

$$\begin{aligned} \varvec{\varGamma }_{0333}= & {} 4D_{3} \lambda _{1}^{{2}} \lambda _{{3}} \{\varOmega _{14} +\varOmega _{5} +2\varOmega _{24} \lambda _{1}^{{2}} +\varOmega _{34} \lambda _{1}^{4} +(\varOmega _{15} +2\varOmega _{6} )\lambda _{3}^{{2}} +\varOmega _{16} \lambda _{3}^{4} \nonumber \\&+\lambda _{1}^{2}\lambda _{3}^{{2}} (2\varOmega _{25} +2\varOmega _{26} \lambda _{3}^{{2}} +\varOmega _{35} \lambda _{1}^{2} +\varOmega _{36} \lambda _{1}^{2} \lambda _{3}^{2} )+D_{3}^{2} \lambda _{1}^{4} [\varOmega _{45} \nonumber \\&+(2\varOmega _{46}+\varOmega _{55} )\lambda _{3}^{{2}} +3\varOmega _{56} \lambda _{3}^{4} +2\varOmega _{66} \lambda _{3}^{6} ]\} \end{aligned}$$

(A.10)

$$\begin{aligned} K_{011}= & {} 2\lambda _{{3}} (\varOmega _{4} +\varOmega _{5} \lambda _{1}^{2} +\varOmega _{6} \lambda _{1}^{2} ) \end{aligned}$$

(A.11)

$$\begin{aligned} K_{033}= & {} \frac{2\lambda _{1}^{2} }{\lambda _{{3}} }\{\varOmega _{4} +\lambda _{3}^{2} (\varOmega _{5} +\varOmega _{6} \lambda _{3}^{2} )+2D_{3}^{2} \lambda _{{1}}^{4} [\varOmega _{44} +2\varOmega _{45} \lambda _{3}^{{2}}\nonumber \\&+\lambda _{3}^{4} (2\varOmega _{46} +\varOmega _{55} +2\varOmega _{56} \lambda _{3}^{{2}} +\varOmega _{66} \lambda _{3}^{4} )]\} \end{aligned}$$

(A.12)

In addition, the following relation holds

$$\begin{aligned} \mathscr {A}_{01111} -\mathscr {A}_{01122} =\mathscr {A}_{01221} +\mathscr {A}_{01212} \end{aligned}$$

(A.13)

If adopting the generalized neo-Hookean model given in Eq. (48), the expressions for the corresponding effective electroelastic moduli are given by:

$$\begin{aligned} c_{11}= & {} K\lambda _{{1}}^{2} \lambda _{3} +\frac{7}{9}\mu \lambda _{{1}}^{{-4/3}} \lambda _{{3}}^{{-5/3}} +\frac{5}{9}\mu \lambda _{{1}}^{{-10/3}} \lambda _{{3}}^{{1/3}}\nonumber \\ c_{12}= & {} -\frac{D_{3}^{2} }{\varepsilon _{0} \lambda _{3}^{2} }\left( {\gamma _{1} +\lambda _{3}^{2} \gamma _{2} } \right) +K\left( {2\lambda _{1}^{2} \lambda _{3} -1} \right) +\frac{2}{9}\mu \lambda _{{1}}^{{-10/3}} \lambda _{{3}}^{{-5/3}} \left( {-4\lambda _{1}^{2} +\lambda _{3}^{2} } \right) \nonumber \\ c_{13}= & {} \frac{D_{3}^{2} \left( {-\gamma _{1} +\gamma _{2} \lambda _{3}^{2} } \right) }{\varepsilon _{0} \lambda _{3}^{2} }+K\left( {2\lambda _{1}^{2} \lambda _{3} -1} \right) -\frac{2}{9}\mu \lambda _{{1}}^{{-10/3}} \lambda _{{3}}^{{-5/3}} \left( {\lambda _{1}^{2} +2\lambda _{3}^{2} } \right) \nonumber \\ c_{33}= & {} -\frac{2D_{3}^{2} \gamma _{2} \left( {-3\gamma _{1} +\gamma _{2} \lambda _{3}^{2} } \right) }{\varepsilon _{0} \left( {\gamma _{1} +\gamma _{2} \lambda _{3}^{2} } \right) }+K\lambda _{1}^{2} \lambda _{3} +\frac{2}{9}\mu \lambda _{{1}}^{\mathrm{-10/3}} \lambda _{{3}}^{\mathrm{-5/3}} \left( {5\lambda _{1}^{2} +\lambda _{3}^{2} } \right) \nonumber \\ c_{58}= & {} -K\left( {\lambda _{1}^{2} \lambda _{3} -1} \right) +\frac{1}{3}\mu \lambda _{{1}}^{\mathrm{-10/3}} \lambda _{{3}}^{\mathrm{-5/3}} \left( {2\lambda _{1}^{2} +\lambda _{3}^{2} } \right) \nonumber \\&+\frac{D_{3}^{2} }{\varepsilon _{0} \left( {\gamma _{1} +\gamma _{2} \lambda _{1}^{2} } \right) \lambda _{3}^{2} }\left[ {\gamma _{1}^{2} -\gamma _{2}^{2} \lambda _{1}^{2} \lambda _{3}^{2} +\gamma _{1} \gamma _{2} \left( {\lambda _{1}^{2} +\lambda _{3}^{2} } \right) } \right] \nonumber \\ c_{55}= & {} -\frac{2D_{3}^{2} \gamma _{2}^{2} \lambda _{1}^{2} }{\varepsilon _{0} \left( {\gamma _{1} +\gamma _{2} \lambda _{1}^{2} } \right) }+\mu \lambda _{{1}}^{\mathrm{-4/3}} \lambda _{{3}}^{\mathrm{-5/3}} ,\quad c_{88} =\frac{2D_{3}^{2} \gamma _{1} \gamma _{2} }{\varepsilon _{0} \left( {\gamma _{1} +\gamma _{2} \lambda _{1}^{2} } \right) }+\mu \lambda _{{1}}^{\mathrm{-10/3}} \lambda _{{3}}^{\mathrm{1/3}} \nonumber \\ e_{15}= & {} -\frac{D_{3} \gamma _{2} \lambda _{1}^{2} }{\gamma _{1} +\gamma _{2} \lambda _{1}^{2} },\quad e_{31} =D_{3} , \quad e_{33} =-\frac{2D_{3} \gamma _{2} \lambda _{3}^{2} }{\gamma _{1} +\gamma _{2} \lambda _{3}^{2}}+D_{3} ,\nonumber \\ \varepsilon _{11}= & {} \frac{\varepsilon _{0} \lambda _{1}^{2} }{2(\gamma _{1} +\gamma _{2} \lambda _{1}^{2} )},\quad \varepsilon _{33} =\frac{\varepsilon _{0} \lambda _{3}^{2} }{2(\gamma _{1} +\gamma _{2} \lambda _{3}^{2} )} \end{aligned}$$

(A.14)

Appendix B: Coefficients in Eqs. (19), (20), (21), (25) and (26)

The coefficients in Eqs. (19), (20) and (21) are given by:

$$\begin{aligned} \beta _{1}= & {} 1/c_{88} , \quad \beta _{2} =c_{58} \beta _{1} , \quad \beta _{3} =e_{15} \beta _{1} , \quad \beta _{4} =\beta _{1} (c_{58} -c_{88} )+1 \end{aligned}$$

(B.1)

$$\begin{aligned} \beta _{5}= & {} \beta _{2} (c_{58} -c_{88} )-\left( {c_{55} -c_{58} } \right) , \quad \beta _{6} =\beta _{3} \left( {c_{58} -c_{88} } \right) , \quad \beta _{7} =e_{15} \beta _{1}\end{aligned}$$

(B.2)

$$\begin{aligned} \beta _{8}= & {} e_{15} \left( {\beta _{2} -1} \right) , \quad \beta _{9} =e_{15} \beta _{3} +\varepsilon _{11} , \quad \beta _{10} =\frac{e_{31} e_{33} +c_{13} \varepsilon _{33} }{e_{33}^{2}+c_{33} \varepsilon _{33} }\end{aligned}$$

(B.3)

$$\begin{aligned} \beta _{11}= & {} \frac{\varepsilon _{33} }{e_{33}^{2}+c_{33} \varepsilon _{33}},\quad \beta _{12} =\frac{e_{33} }{e_{33}^{2}+c_{33} \varepsilon _{33} }, \quad \beta _{13} =\frac{c_{13} e_{33} -c_{33} e_{31} }{e_{33}^{2}+c_{33} \varepsilon _{33} }\end{aligned}$$

(B.4)

$$\begin{aligned} \beta _{14}= & {} \frac{c_{33} }{e_{33}^{2}+c_{33} \varepsilon _{33} }, \quad \beta _{15} =-c_{11} +c_{13} \beta _{10} +e_{31} \beta _{13} , \quad \beta _{16} =c_{13} \beta _{11} +e_{31} \beta _{12}\end{aligned}$$

(B.5)

$$\begin{aligned} \beta _{17}= & {} c_{13} \beta _{12} -e_{31} \beta _{14} , \quad \beta _{18} =c_{11} -c_{13} \beta _{10} -e_{31} \beta _{13} , \quad \beta _{19} =c_{12} -c_{13} \beta _{10} -e_{31} \beta _{13} \end{aligned}$$

(B.6)

$$\begin{aligned} \beta _{20}= & {} 1/\beta _{4} , \quad \beta _{21} =\left( {c_{55} -c_{58} } \right) -\beta _{2} (c_{58} -c_{88} ), \quad \beta _{22} =(c_{58} -c_{88} )\beta _{3} \end{aligned}$$

(B.7)

In Eqs. (25) and (26), the coefficients are given by:

$$\begin{aligned} f_{1}= & {} c_{11} \beta _{1}, \quad f_{2} =t_{c} \beta _{2}, \quad f_{3} =t_{c} \beta _{3} \sqrt{\frac{c_{11} }{\varepsilon _{33} }}, \quad f_{4} =t_{c} \beta _{4} , \quad f_{5} =\frac{t_{c}^{2} \beta _{5} }{c_{11} } \end{aligned}$$

(B.8)

$$\begin{aligned} f_{6}= & {} \frac{t_{c}^{2} \beta _{6} }{\sqrt{c_{11} \varepsilon _{33}}}, \quad f_{7} =t_{c}^{2} \beta _{7} \sqrt{\frac{c_{11} }{\varepsilon _{33} }}, \quad f_{8} =\frac{t_{c}^{2} \beta _{8} }{\sqrt{c_{11} \varepsilon _{33}}} \quad f_{9} =\frac{t_{c}^{2} \beta _{9} }{\varepsilon _{33}} \end{aligned}$$

(B.9)

$$\begin{aligned} f_{10}= & {} \frac{t_{c}^{2} \beta _{15} }{c_{11}}, \quad f_{11} =t_{c} \beta _{16}, \quad f_{12} =t_{c} \beta _{17} \sqrt{\frac{\varepsilon _{33} }{c_{11}}}, \quad f_{13} =t_{c} \beta _{10} \end{aligned}$$

(B.10)

$$\begin{aligned} f_{14}= & {} c_{11} \beta _{11}, \quad f_{15} =\sqrt{c_{11} \varepsilon _{33} } \beta _{12}, \quad f_{16} =t_{c} \beta _{13} \sqrt{\frac{\varepsilon _{33} }{c_{11}}}, \quad f_{17} =\beta _{14} \varepsilon _{33} \end{aligned}$$

(B.11)

Appendix C: Free Vibration of an Isotropic Linear Elastic Circular Plate

Substituting $D_{3} =0$ and $\lambda _{1} =\lambda _{3} =1$ into Eq. (A.14) gives

$$\begin{aligned} \begin{array}{l} c_{11} =c_{33} =K+\frac{4}{3}\mu ,\quad c_{12} =c_{13} =K-\frac{2}{3}\mu ,\quad c_{58} =c_{55} =c_{88} =\mu , \\ e_{15} =e_{31} =e_{33} =0,\quad \varepsilon _{11} =\varepsilon _{33} =\varepsilon _{0} \\ \end{array} \end{aligned}$$

(C.1)

The material then becomes isotropic and linear elastic. In fact, Eq. (38) is accordingly decoupled into two equations under the same boundary conditions:

$$\begin{aligned} \frac{\text{ d }{\varvec{R}}_{1}}{\text{ d }\zeta }= & {} {\varvec{M}}^{*}{\varvec{R}}_{1} \end{aligned}$$

(C.2)

$$\begin{aligned} \frac{\text{ d }{\varvec{R}}_{2}}{\text{ d }\zeta }= & {} {\varvec{M}}^{**}{\varvec{R}}_{2} \end{aligned}$$

(C.3)

where

$$\begin{aligned} {\varvec{R}}_{1}= & {} \left[ {{\begin{array}{*{20}c} {U\left( \zeta \right) } &{}\quad {\sigma \left( \zeta \right) } &{}\quad {\tau \left( \zeta \right) } &{}\quad {W\left( \zeta \right) } \\ \end{array} }} \right] ^{\mathrm{T}},\quad {\varvec{R}}_{2} =\left[ {{\begin{array}{*{20}c} {D\left( \zeta \right) } &{}\quad {\Phi \left( \zeta \right) } \\ \end{array} }} \right] ^{\mathrm{T}} \end{aligned}$$

(C.4)

$$\begin{aligned} {\varvec{M}}^{*}= & {} \left[ {{\begin{array}{*{20}c} 0 &{}\quad {{\varvec{M}}_{1}^{*} } \\ {{\varvec{M}}_{2}^{*} } &{}\quad 0 \\ \end{array} }} \right] ,\quad {\varvec{M}}^{**}=\left[ {{\begin{array}{*{20}c} 0 &{}\quad {-f_{9} k^{2}} \\ {-f_{17} } &{}\quad 0 \\ \end{array} }} \right] \end{aligned}$$

(C.5)

$$\begin{aligned} {\varvec{M}}_{1}^{*}= & {} \left[ {{\begin{array}{*{20}c} {f_{1} } &{}\quad {f_{2} k} \\ {-f_{4} k} &{}\quad {-f_{5} k^{2}-\frac{\rho \omega ^{2}h^{2}}{c_{11} }} \\ \end{array} }} \right] ,\quad {\varvec{M}}_{2}^{*} =\left[ {{\begin{array}{*{20}c} {-\frac{\rho \omega ^{2}h^{2}}{c_{11} }-f_{10} k^{{2}}} &{}\quad {f_{11} k} \\ {-f_{13} k} &{}\quad {f_{14} } \\ \end{array} }} \right] \end{aligned}$$

(C.6)

The solutions of Eqs. (C.2) and (C.3), respectively, are:

$$\begin{aligned} {\varvec{R}}_{k} \left( \zeta \right) ={\varvec{T}}_{k} \left( \zeta \right) {\varvec{R}}_{k} \left( 0 \right) \quad (k=1,2) \end{aligned}$$

(C.7)

where ${\varvec{T}}_{1} \left( \zeta \right) =e^{{\varvec{M}}^{*}\zeta }, \quad {\varvec{T}}_{2} \left( \zeta \right) =e^{{\varvec{M}}^{**}\zeta }$.

Setting $\zeta ={1}$ in Eq. (C.7) gives:

$$\begin{aligned} {\varvec{R}}_{1} \left( 1 \right) ={\varvec{F}}^{*}{\varvec{R}}_{1} \left( 0 \right) \end{aligned}$$

(C.8)

where ${\varvec{F}}^{*}={\varvec{T}}_{1} \left( {{1}} \right)$.

The boundary conditions at the bottom and top surfaces of the plate are:

$$\begin{aligned} \dot{{T}}_{0zz} (h)=\dot{{T}}_{0rz} (h)=\dot{{T}}_{0zz} (0)=\dot{{T}}_{0rz} (0)=0 \end{aligned}$$

(C.9)

Substituting Eq. (C.9) into Eq. (C.8) gives:

$$\begin{aligned} \left[ {{\begin{array}{*{20}c} {F_{21}^{*} } &{}\quad {F_{24}^{*} } \\ {F_{31}^{*} } &{}\quad {F_{34}^{*} } \\ \end{array} }} \right] \left\{ {{\begin{array}{*{20}c} {U(0)} \\ {W(0)} \\ \end{array} }} \right\} =0 \end{aligned}$$

(C.10)

The determinant of the coefficient matrix must be zero for non-trivial solutions, giving rise to the corresponding characteristic equation. Then, the corresponding mode shape can also be obtained by the inverse Hankel transform. For more details, the reader can also refer to Ding et al. [33], which is for transversely isotropic laminated circular plates.

Appendix D: Vibration Mode Shapes

See Figs. 11, 12, 13 and 14.

Fig. 11

The mode shape of an SEAM circular plate with $\lambda _{{1}} =1.5$, $\lambda _{{3}} =1$, $D=0.1$, $t_{0} =0.1$, $\gamma _{1} =\gamma _{2} ={0.2}$ and the lowest dimensionless frequency $\bar{{\omega }}=2.953$ (Dotted lines: initial mesh; Solid lines: mode shape)

Fig. 12

The mode shape of an SEAM circular plate with $\lambda _{{1}} =1.5$, $\lambda _{{3}} =1$, $D=10$, $t_{0} =0.1$, $\gamma _{1} =\gamma _{2} ={0.2}$ and the lowest dimensionless frequency $\bar{{\omega }}=2.22{2}$ (Dotted lines: initial mesh; Solid lines: mode shape)

Fig. 13

The mode shape of an SEAM circular plate with $\lambda _{{1}} =1$, $\lambda _{{3}} =2$, $D=0.1$, $t_{0} =0.1$, $\gamma _{1} =\gamma _{2} ={0.2}$ and the lowest dimensionless frequency $\bar{{\omega }}=2.598$ (Dotted lines: initial mesh; Solid lines: mode shape)

Fig. 14

The mode shape of an SEAM circular plate with $\lambda _{{1}} =1$, $\lambda _{{3}} =2$, $D{=10}$, $t_{0} =0.1$, $\gamma _{1} =\gamma _{2} ={0.2}$ and the lowest dimensionless frequency $\bar{{\omega }}=2.63{0}$ (Dotted lines: initial mesh; Solid lines: mode shape)

Appendix E: Euler’s Instability

This type of instability is common in daily life and engineering, which also attracts a lot of interest in the study of elastomeric structures. The static criterion of Euler instability is as follows. A small disturbance is first applied to the elastomer under a biasing field. If the disturbance satisfies the incremental governing equations and the incremental boundary conditions, Euler instability then occurs and the biasing field is the critical biasing field. The following gives the detailed analysis according to the above criterion of Euler instability.

The constitutive relations for the incremental field are given in Eq. (15). The equilibrium differential equations and electrostatic equation are

$$\begin{aligned} \begin{array}{l} \frac{\partial \dot{{T}}_{0rr} }{\partial r}+\frac{\dot{{T}}_{0rr} -\dot{{T}}_{0\theta \theta } }{r}+\frac{\partial \dot{{T}}_{0zr} }{\partial z}=0 \\ \frac{\partial \dot{{T}}_{0rz} }{\partial r}+\frac{\dot{{T}}_{0rz} }{r}+\frac{\partial \dot{{T}}_{0zz} }{\partial z}=0 \\ \frac{\partial \dot{{D}}_{l0r} }{\partial r}+\frac{\dot{{D}}_{l0r} }{r}+\frac{\partial \dot{{D}}_{l0z} }{\partial z}=0 \\ \end{array} \end{aligned}$$

(E.1)

which is obtained from Eq. (17) by simply discarding the inertia terms.

The state equation is still in the form of Eq. (18), where, however, all inertia terms should be discarded. As in Sect. 3.1, we can derive:

$$\begin{aligned} \frac{\text{ d }{\varvec{R}}(\zeta )}{\text{ d }\zeta }=\varvec{\bar{{M}}R}(\zeta ) \end{aligned}$$

(E.2)

where $\varvec{\bar{{M}}}$ is the corresponding system matrix.

Under the three boundary conditions of Eqs. (41), (42) and (43), the characteristic equations for non-trivial solutions can be similarly obtained as:

$$\begin{aligned}&\left| {{\begin{array}{*{20}c} {\bar{{F}}_{21} } &{}\quad {\bar{{F}}_{25} } &{}\quad {\bar{{F}}_{26} } \\ {\bar{{F}}_{31} } &{}\quad {\bar{{F}}_{35} } &{}\quad {\bar{{F}}_{36} } \\ {\bar{{F}}_{41} } &{}\quad {\bar{{F}}_{45} } &{}\quad {\bar{{F}}_{46} } \\ \end{array} }} \right| =0 \end{aligned}$$

(E.3)

$$\begin{aligned}&\left| {{\begin{array}{*{20}c} {\bar{{F}}_{21} } &{}\quad {\bar{{F}}_{23} } &{}\quad {\bar{{F}}_{25} } \\ {\bar{{F}}_{41} } &{}\quad {\bar{{F}}_{43} } &{}\quad {\bar{{F}}_{45} } \\ {\bar{{F}}_{61} } &{}\quad {\bar{{F}}_{63} } &{}\quad {\bar{{F}}_{65} } \\ \end{array} }} \right| =0 \end{aligned}$$

(E.4)

$$\begin{aligned}&\left| {{\begin{array}{*{20}c} {\bar{{F}}_{21} } &{}\quad {\bar{{F}}_{23} } &{}\quad {\bar{{F}}_{25} } \\ {\bar{{F}}_{31} } &{}\quad {\bar{{F}}_{33} } &{}\quad {\bar{{F}}_{35} } \\ {\bar{{F}}_{41} } &{}\quad {\bar{{F}}_{43} } &{}\quad {\bar{{F}}_{45} } \\ \end{array} }} \right| =0 \end{aligned}$$

(E.5)

where $\varvec{\bar{{F}}}=\text{ e}^{\varvec{\bar{{M}}}}$.

When Eqs. (E.3), (E.4) or (E.5) is satisfied, instability occurs. The results for the critical points in Figs. 5b and 7b can be all determined from these characteristic equations.