Application scope studies of analytical methods based on the equivalent single-layer and layerwise theories for vibration analysis of symmetric sandwich plates

Abstract

This paper proposes two analytical methods to predict free and forced vibrations of symmetric sandwich plates in vacuum and water. The application scopes of the two methods are investigated. Displacement fields of the plates are described by the first-order shear deformation theory. The two methods are, respectively, built via the equivalent single-layer (ESL) theory and layerwise (LW) theory. The equations of motion are derived through Hamilton’s principle. The Helmholtz and Euler’s equations are utilized to represent the acoustic pressure in water domain and the continuities on acoustic-structure coupling faces. Through assuming the submerged sandwich as a thin plate, the formulas of transverse vibration in the two analytical models are reconstructed. The Navier method is employed to solve the new equations of the plates with simply supported boundaries at four sides. The obtained analytical nature frequencies and forced vibration responses are compared with the existing ones in the literature, and numerical results from the finite element method (FEM) and coupling finite element and boundary element method (FEM/BEM). Accuracies of the ESL and LW methods are studied for the hard or soft core case. The application scope of the ESL method is further analyzed by taking the fundamental frequency of the plates in vacuum as the evaluation object. Through five-level discussions which consider effects of the Young’s modulus ratio of core to panel, the thickness ratio of core to panel, the length–thickness ratio and the length–width ratio of the sandwich plate, a general quantitative critical dimensionless parameter is designed to determine the application scope of the ESL method and distinguish the hard or soft core.

Appendices

Appendix A

The coefficients in Eq. (24) are

$$\left\{ \begin{gathered} R_{11} = - D_{11} \alpha^{2} - D_{66} \beta^{2} - C_{55} + \overline{\rho }_{2} \omega^{2} , \, R_{12} = - (D_{12} + D_{66} )\alpha \beta , \, R_{13} = - C_{55} \alpha , \, \\ R_{21} = - (D_{12} + D_{66} )\alpha \beta , \, R_{22} = - D_{66} \alpha^{2} - D_{22} \beta^{2} - C_{44} + \overline{\rho }_{2} \omega^{2} , \, R_{23} = - C_{44} \beta , \, \\ R_{31} = - \alpha C_{55} \gamma_{mn \, mn} , \, R_{32} = - \beta C_{44} \gamma_{mn \, mn} , \\ R_{33} = ( - C_{55} \alpha^{2} - C_{44} \beta^{2} + \overline{\rho }_{0} \omega^{2} )\gamma_{mn \, mn} + \rho_{0} \omega^{2} g(m,m,L_{x} )g(n,n,L_{y} ) \\ \end{gathered} \right.$$

(A1)

where

$$f(n_{1} ,n_{2} ,l_{0} ) = \int_{0}^{{l_{0} }} {\cos \frac{{n_{1} \pi \xi }}{{l_{0} }}\cos \frac{{n_{2} \pi \xi }}{{l_{0} }}} {\rm{d}}\xi = \left\{ {\begin{array}{*{20}c} {0\;\;,\; \, n_{1} \ne n_{2} } \\ {{{l_{0} } \mathord{\left/ {\vphantom {{l_{0} } {2,}}} \right. \kern-\nulldelimiterspace} {2,}}\;n_{1} = n_{2} } \\ \end{array} } \right.$$

(A2)

$$g(n_{1} ,n_{2} ,l_{0} ) = \int_{0}^{{l_{0} }} {\sin \frac{{n_{1} \pi \xi }}{{l_{0} }}\sin \frac{{n_{2} \pi \xi }}{{l_{0} }}} {\rm{d}}\xi = \left\{ {\begin{array}{*{20}c} {0\;\;,\; \, \;n_{1} \ne n_{2} } \\ {{{l_{0} } \mathord{\left/ {\vphantom {{l_{0} } 2}} \right. \kern-\nulldelimiterspace} 2},\;n_{1} = n_{2} } \\ \end{array} } \right.$$

(A3)

$$\begin{gathered} \gamma_{pq \, mn} = \int_{{S_{p} }} {\int_{{S_{p} }} {\sin \frac{m\pi x}{{L_{x} }}\sin \frac{n\pi y}{{L_{y} }}\sin \frac{p\pi x}{{L_{x} }}\sin \frac{q\pi y}{{L_{y} }}\left. {\frac{{\partial^{2} G(M,M_{0} )}}{{\partial z_{M} \partial z_{0} }}} \right|_{{z_{M} ,z_{0} = 0}} } } {\rm{d}}x{\rm{d}}y{\rm{d}}x_{0} {\rm{d}}y_{0} \hfill \\ \qquad = 2jL_{x}^{2} L_{y}^{2} \pi^{2} \int_{0}^{\infty } {\int_{0}^{\infty } {\frac{{pqmn\sqrt {k_{0}^{2} - k_{x}^{2} - k_{y}^{2} } (1 - ( - 1)^{p} \cos k_{x} L_{x} )(1 - ( - 1)^{q} \cos k_{y} L_{y} )}}{{(L_{x}^{2} k_{x}^{2} - p^{2} \pi^{2} )(L_{y}^{2} k_{y}^{2} - q^{2} \pi^{2} )(L_{x}^{2} k_{x}^{2} - m^{2} \pi^{2} )(L_{y}^{2} k_{y}^{2} - n^{2} \pi^{2} )}}{\rm{d}}k_{x} {\rm{d}}k_{y} } } \hfill \\ \end{gathered}$$

(A4)

Appendix B

The coefficients in Eq. (26) are

$$O_{pq} = - \sum\limits_{m = 1}^{M} {\sum\limits_{n = 1}^{N} {q_{zmn} \gamma_{pq \, mn} } }$$

(B1)

$$\left\{ \begin{gathered} T11(...) = R_{11} f(p,m,L_{x} )g(q,n,L_{y} ) \\ T12(...) = R_{12} f(p,m,L_{x} )g(q,n,L_{y} ) \\ T13(...) = R_{13} f(p,m,L_{x} )g(q,n,L_{y} ) \\ \end{gathered} \right. \, \left\{ \begin{gathered} T21(...) = R_{21} g(p,m,L_{x} )f(q,n,L_{y} ) \\ T22(...) = R_{22} g(p,m,L_{x} )f(q,n,L_{y} ) \\ T23(...) = \, R_{23} g(p,m,L_{x} )f(q,n,L_{y} ) \\ \end{gathered} \right.$$

(B2)

$$\left\{ \begin{gathered} T31(...) = - \alpha C_{55} \gamma_{pq \, mn} , \, T32(...) = - \beta C_{44} \gamma_{pq \, mn} \\ T33(...) = ( - C_{55} \alpha^{2} - C_{44} \beta^{2} + \overline{\rho }_{0} \omega^{2} )\gamma_{pq \, mn} { + }\rho_{0} \omega^{2} g(p,m,L_{x} )g(q,n,L_{y} ) \\ \end{gathered} \right.$$

(B3)

in which $Tij\left( {...} \right)$ denotes $Tij\left( {pN - N + q,mN - N + n} \right)$.

Appendix C

The coefficients in Eq. (47) are

$$\left\{ \begin{gathered} \overline{R}_{11} = \sum\limits_{k = 1}^{3} {( - C_{55}^{\left( k \right)} \alpha^{2} - C_{44}^{\left( k \right)} \beta^{2} + \rho_{0}^{\left( k \right)} \omega^{2} )} \gamma_{mn \, mn} + \rho_{0} \omega^{2} g(m,m,L_{x} )g(n,n,L_{y} ), \, \hfill \\ \left\{ {\overline{R}_{12} , \, \overline{R}_{14} , \, \overline{R}_{16} } \right\} = - \alpha \left\{ {C_{55}^{\left( 1 \right)} , \, C_{55}^{\left( 2 \right)} , \, C_{55}^{\left( 3 \right)} } \right\}\gamma_{mn \, mn} , \hfill \\ \left\{ {\overline{R}_{13} , \, \overline{R}_{15} , \, \overline{R}_{17} } \right\} = - \beta \left\{ {C_{44}^{\left( 1 \right)} , \, C_{44}^{\left( 2 \right)} , \, C_{44}^{\left( 3 \right)} } \right\}\gamma_{mn \, mn} \hfill \\ \end{gathered} \right.$$

(C1)

$$\left\{ \begin{gathered} \overline{R}_{21} = - C_{55}^{\left( 1 \right)} \alpha , \hfill \\ \overline{R}_{22} = - (h_{1}^{2} A_{11}^{\left( 1 \right)} {/4} + D_{11}^{\left( 1 \right)} )\alpha^{2} - (h_{1}^{2} A_{66}^{\left( 1 \right)} {/4} + D_{66}^{\left( 1 \right)} )\beta^{2} - C_{55}^{\left( 1 \right)} + (h_{1}^{2} \rho_{0}^{\left( 1 \right)} {/4} + \rho_{2}^{\left( 1 \right)} )\omega^{2} \hfill \\ \overline{R}_{23} = - [h_{1}^{2} (A_{12}^{\left( 1 \right)} + A_{66}^{\left( 1 \right)} ){/4} + D_{12}^{\left( 1 \right)} + D_{66}^{\left( 1 \right)} ]\alpha \beta , \hfill \\ \overline{R}_{24} = h_{1} h_{2} ( - A_{11}^{\left( 1 \right)} \alpha^{2} - A_{66}^{\left( 1 \right)} \beta^{2} + \rho_{0}^{\left( 1 \right)} \omega^{2} ){/4}, \hfill \\ \overline{R}_{25} = - h_{1} h_{2} (A_{12}^{\left( 1 \right)} + A_{66}^{\left( 1 \right)} )\alpha \beta {/4, }\overline{R}_{26} = 0{, }\overline{R}_{27} = 0 \hfill \\ \end{gathered} \right.$$

(C2)

$$\left\{ \begin{gathered} \overline{R}_{31} = - C_{44}^{\left( 1 \right)} \beta , \hfill \\ \overline{R}_{32} = - [h_{1}^{2} (A_{12}^{\left( 1 \right)} + A_{66}^{\left( 1 \right)} ){/4} + D_{12}^{\left( 1 \right)} + D_{66}^{\left( 1 \right)} ]\alpha \beta , \hfill \\ \overline{R}_{33} = - (h_{1}^{2} A_{22}^{\left( 1 \right)} {/4} + D_{22}^{\left( 1 \right)} )\beta^{2} - (h_{1}^{2} A_{66}^{\left( 1 \right)} {/4} + D_{66}^{\left( 1 \right)} )\alpha^{2} - C_{44}^{\left( 1 \right)} + (h_{1}^{2} \rho_{0}^{\left( 1 \right)} {/4} + \rho_{2}^{\left( 1 \right)} )\omega^{2} , \hfill \\ \overline{R}_{34} = - h_{1} h_{2} (A_{12}^{\left( 1 \right)} + A_{66}^{\left( 1 \right)} )\alpha \beta {/4}, \hfill \\ \overline{R}_{35} = h_{1} h_{2} ( - A_{22}^{\left( 1 \right)} \beta^{2} - A_{66}^{\left( 1 \right)} \alpha^{2} + \rho_{0}^{\left( 1 \right)} \omega^{2} ){/4, }\overline{R}_{36} = 0{, }\overline{R}_{37} = 0 \hfill \\ \end{gathered} \right.$$

(C3)

$$\left\{ \begin{gathered} \overline{R}_{41} = - C_{55}^{\left( 2 \right)} \alpha {,} \hfill \\ \overline{R}_{42} = h_{1} h_{2} [ - A_{11}^{\left( 1 \right)} \alpha^{2} - A_{66}^{\left( 1 \right)} \beta^{2} + \rho_{0}^{\left( 1 \right)} \omega^{2} ]{/}4{, }\overline{R}_{43} = - h_{1} h_{2} (A_{12}^{\left( 1 \right)} + A_{66}^{\left( 1 \right)} )\alpha \beta {/}4{, } \hfill \\ \overline{R}_{44} = - [h_{2}^{2} (A_{11}^{\left( 1 \right)} + A_{11}^{\left( 3 \right)} ){/}4 + D_{11}^{\left( 2 \right)} ]\alpha^{2} - [h_{2}^{2} (A_{66}^{\left( 1 \right)} + A_{66}^{\left( 3 \right)} ){/}4 + D_{66}^{\left( 2 \right)} ]\beta^{2} \hfill \\ \, - C_{55}^{\left( 2 \right)} + [h_{2}^{2} (\rho_{0}^{\left( 1 \right)} + \rho_{0}^{\left( 3 \right)} ){/}4 + \rho_{2}^{\left( 2 \right)} ]\omega^{2} , \hfill \\ \overline{R}_{45} = - [h_{2}^{2} (A_{12}^{\left( 1 \right)} + A_{12}^{\left( 3 \right)} + A_{66}^{\left( 1 \right)} + A_{66}^{\left( 3 \right)} ){/4} + D_{12}^{\left( 2 \right)} + D_{66}^{\left( 2 \right)} ]\alpha \beta {,} \hfill \\ \overline{R}_{46} = h_{2} h_{3} ( - A_{11}^{\left( 3 \right)} \alpha^{2} - A_{66}^{\left( 3 \right)} \beta^{2} + \rho_{0}^{\left( 3 \right)} \omega^{2} ){/4, }\overline{R}_{47} = - h_{2} h_{3} (A_{12}^{\left( 3 \right)} + A_{66}^{\left( 3 \right)} )\alpha \beta {/}4 \hfill \\ \end{gathered} \right.$$

(C4)

$$\left\{ \begin{gathered} \overline{R}_{51} = - C_{44}^{\left( 2 \right)} \beta {,} \hfill \\ \overline{R}_{52} = - h_{1} h_{2} (A_{12}^{\left( 1 \right)} + A_{66}^{\left( 1 \right)} )\alpha \beta {/}4{, }\overline{R}_{53} = h_{1} h_{2} [ - A_{22}^{\left( 1 \right)} \beta^{2} - A_{66}^{\left( 1 \right)} \alpha^{2} + \rho_{0}^{\left( 1 \right)} \omega^{2} ]{/}4{, } \hfill \\ \overline{R}_{54} = - [h_{2}^{2} (A_{12}^{\left( 1 \right)} + A_{12}^{\left( 3 \right)} + A_{66}^{\left( 1 \right)} + A_{66}^{\left( 3 \right)} ){/4} + D_{12}^{\left( 2 \right)} + D_{66}^{\left( 2 \right)} ]\alpha \beta , \hfill \\ \overline{R}_{55} = - [h_{2}^{2} (A_{22}^{\left( 1 \right)} + A_{22}^{\left( 3 \right)} ){/}4 + D_{22}^{\left( 2 \right)} ]\beta^{2} - [h_{2}^{2} (A_{66}^{\left( 1 \right)} + A_{66}^{\left( 3 \right)} ){/}4 + D_{66}^{\left( 2 \right)} ]\alpha^{2} \hfill \\ \, - C_{44}^{\left( 2 \right)} + [h_{2}^{2} (\rho_{0}^{\left( 1 \right)} + \rho_{0}^{\left( 3 \right)} ){/}4 + \rho_{2}^{\left( 2 \right)} ]\omega^{2} {,} \hfill \\ \overline{R}_{56} = - h_{2} h_{3} (A_{12}^{\left( 3 \right)} + A_{66}^{\left( 3 \right)} )\alpha \beta {/}4{, }\overline{R}_{57} = h_{2} h_{3} ( - A_{22}^{\left( 3 \right)} \beta^{2} - A_{66}^{\left( 3 \right)} \alpha^{2} + \rho_{0}^{\left( 3 \right)} \omega^{2} ){/4} \hfill \\ \end{gathered} \right.$$

(C5)

$$\left\{ \begin{gathered} \overline{R}_{61} = - C_{55}^{\left( 3 \right)} \alpha {, }\overline{R}_{62} = 0{, }\overline{R}_{63} = 0{, } \hfill \\ \overline{R}_{64} = h_{2} h_{3} ( - A_{11}^{\left( 3 \right)} \alpha^{2} - A_{66}^{\left( 3 \right)} \beta^{2} + \rho_{0}^{\left( 3 \right)} \omega^{2} ){/4, }\overline{R}_{65} = - h_{2} h_{3} (A_{12}^{\left( 3 \right)} + A_{66}^{\left( 3 \right)} )\alpha \beta {/4,} \hfill \\ \overline{R}_{66} = - (h_{3}^{2} A_{11}^{\left( 3 \right)} {/4} + D_{11}^{\left( 3 \right)} )\alpha^{2} - \left( {h_{3}^{2} A_{66}^{\left( 3 \right)} {/4} + D_{66}^{\left( 3 \right)} } \right)\beta^{2} - C_{55}^{\left( 3 \right)} + \left( {h_{3}^{2} \rho_{0}^{\left( 3 \right)} {/4} + \rho_{2}^{\left( 3 \right)} } \right)\omega^{2} {,} \hfill \\ \overline{R}_{67} = - [h_{3}^{2} (A_{12}^{\left( 3 \right)} + A_{66}^{\left( 3 \right)} ) + D_{12}^{\left( 3 \right)} + D_{66}^{\left( 3 \right)} ]\alpha \beta \hfill \\ \end{gathered} \right.$$

(C6)

$$\left\{ \begin{gathered} \overline{R}_{71} = - C_{44}^{\left( 3 \right)} \beta {, }\overline{R}_{72} = 0{, }\overline{R}_{73} = 0{, } \hfill \\ \overline{R}_{74} = - h_{2} h_{3} (A_{12}^{\left( 3 \right)} + A_{66}^{\left( 3 \right)} )\alpha \beta {/4, }\overline{R}_{75} = h_{2} h_{3} ( - A_{22}^{\left( 3 \right)} \beta^{2} - A_{66}^{\left( 3 \right)} \alpha^{2} + \rho_{0}^{\left( 3 \right)} \omega^{2} ){/4,} \hfill \\ \overline{R}_{76} = - [h_{3}^{2} (A_{12}^{\left( 3 \right)} + A_{66}^{\left( 3 \right)} ) + D_{12}^{\left( 3 \right)} + D_{66}^{\left( 3 \right)} ]\alpha \beta {, } \hfill \\ \overline{R}_{77} = - (h_{3}^{2} A_{22}^{\left( 3 \right)} {/4} + D_{22}^{\left( 3 \right)} )\beta^{2} - \left( {h_{3}^{2} A_{66}^{\left( 3 \right)} {/4} + D_{66}^{\left( 3 \right)} } \right)\alpha^{2} - C_{44}^{\left( 3 \right)} + \left( {h_{3}^{2} \rho_{0}^{\left( 3 \right)} {/4} + \rho_{2}^{\left( 3 \right)} } \right)\omega^{2} \hfill \\ \end{gathered} \right.$$

(C7)

Appendix D

The coefficients in Eq. (49) are

$$\left\{ \begin{gathered} \overline{T}11(...) = \sum\limits_{k = 1}^{3} {\left( { - C_{55}^{\left( k \right)} \alpha^{2} - C_{44}^{\left( k \right)} \beta^{2} + \rho_{0}^{\left( k \right)} \omega^{2} } \right)} \gamma_{pq \, mn} { + }\rho_{0} \omega^{2} g(p,m,L_{x} )g(q,n,L_{y} ) \hfill \\ \left( {\overline{T}12(...), \, \overline{T}14(...), \, \overline{T}16(...)} \right) = - \alpha \left( {C_{55}^{\left( 1 \right)} , \, C_{55}^{\left( 2 \right)} , \, C_{55}^{\left( 3 \right)} } \right)\gamma_{pq \, mn} \hfill \\ \left( {\overline{T}13(...), \, \overline{T}15(...), \, \overline{T}17(...)} \right) = - \beta \left( {C_{44}^{\left( 1 \right)} , \, C_{44}^{\left( 2 \right)} , \, C_{44}^{\left( 3 \right)} } \right)\gamma_{pq \, mn} \hfill \\ \end{gathered} \right.$$

(D1)

$$\left\{ \begin{gathered} \overline{T}2j(...) = \overline{R}_{2j} f(p,m,L_{x} )g(q,n,L_{y} ) \, \overline{T}3j(...) = \overline{R}_{3j} g(p,m,L_{x} )f(q,n,L_{y} ) \hfill \\ \overline{T}4j(...) = \overline{R}_{4j} f(p,m,L_{x} )g(q,n,L_{y} ) \, \overline{T}5j(...) = \overline{R}_{5j} g(p,m,L_{x} )f(q,n,L_{y} ) \hfill \\ \overline{T}6j(...) = \overline{R}_{6j} f(p,m,L_{x} )g(q,n,L_{y} ) \, \overline{T}7j(...) = \overline{R}_{7j} g(p,m,L_{x} )f(q,n,L_{y} ) \hfill \\ \end{gathered} \right.$$

(D2)

in which $Tij\left( {...} \right)$ denotes $Tij\left( {pN - N + q,mN - N + n} \right)$.