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Free vibration of advanced composite plates resting on elastic foundations based on refined non-polynomial theory

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Abstract

This paper presents a free vibration analysis of functionally graded plates resting on elastic foundation by using a generalized quasi-3D hybrid-type higher order shear deformation theory (HSDT). The displacement field is modeled based on a hybrid-type (hyperbolic and sinusoidal) quasi-3D HSDT with six unknowns in which the stretching effect is taken into account. The elastic foundation follows the Pasternak (two-parameter) mathematical model. The governing equations are obtained through the Hamilton’s principle. These equations are then solved via Navier-type, closed form solutions. The fundamental frequencies are found by solving the eigenvalue problem. The performance of the present theory is demonstrated by comparing results with the 3D exact solution and other closed form solutions available in literature.

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Acknowledgments

The author would like to dedicate this job to his song, Italo.

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Correspondence to J. L. Mantari.

Appendices

Appendix 1: Constitutive equations

$$ \begin{aligned} {\text{N}}_{i} &= {\text{A}}_{ij} \varepsilon_{j}^{0} + B{}_{ij}\varepsilon_{j}^{1} + C{}_{ij}\varepsilon_{j}^{2} + D{}_{ij}\varepsilon_{j}^{3} + E{}_{ij}\varepsilon_{j}^{4} + F{}_{ij}\varepsilon_{j}^{5} , \hfill \\ &\quad {({\text{i}} = 1,2,4,5,6)} \hfill \\ {\text{M}}_{i} &= {\text{B}}_{ij} \varepsilon_{j}^{0} + G{}_{ij}\varepsilon_{j}^{1} + H{}_{ij}\varepsilon_{j}^{2} + I{}_{ij}\varepsilon_{j}^{3} + J{}_{ij}\varepsilon_{j}^{4} + K^{\prime}_{ij} \varepsilon_{j}^{5} , \hfill \\ &\quad {({\text{i}} = 1,2,6)} \hfill \\ {\text{P}}_{\text{i}} &= C_{\text{ij}} \varepsilon_{j}^{0} + H{}_{ij}\varepsilon_{j}^{1} + L{}_{ij}\varepsilon_{j}^{2} + M^{\prime}{}_{ij}\varepsilon_{j}^{3} + N^{\prime}{}_{ij}\varepsilon_{j}^{4} + O{}_{ij}\varepsilon_{j}^{5} , \hfill \\ &\quad {({\text{i}} = 1, 2, 6)} \hfill \\ {\text{Q}}_{i} &= {\text{D}}_{ij} \varepsilon_{j}^{0} + I{}_{ij}\varepsilon_{j}^{1} + M_{ij} \varepsilon_{j}^{2} + P^{\prime}{}_{ij}\varepsilon_{j}^{3} + Q^{\prime}{}_{ij}\varepsilon_{j}^{4} + R^{\prime}{}_{ij}\varepsilon_{j}^{5} , \hfill \\ &\quad {({\text{i}} = 4, 5)} \hfill \\ {\text{K}}_{i} &= {\text{E}}_{ij} \varepsilon_{j}^{0} + J{}_{ij}\varepsilon_{j}^{1} + N^{\prime}{}_{ij}\varepsilon_{j}^{2} + Q^{\prime}{}_{ij}\varepsilon_{j}^{3} + S{}_{ij}\varepsilon_{j}^{4} + T{}_{ij}\varepsilon_{j}^{5} , \hfill \\ &\quad {({\text{i}} = 4, 5)} \hfill \\ {\text{R}}_{i} &= {\text{F}}_{ij} \varepsilon_{j}^{0} + K^{\prime}{}_{ij}\varepsilon_{j}^{1} + O{}_{ij}\varepsilon_{j}^{2} + R^{\prime}{}_{ij}\varepsilon_{j}^{3} + T{}_{ij}\varepsilon_{j}^{4} + U{}_{ij}\varepsilon_{j}^{5} , \hfill \\ &\quad {({\text{i}} = 3)} \hfill \\ \end{aligned} $$
(31a{-}f)

where

$$ \begin{aligned} & (A{}_{ij},B_{ij} ,C_{ij} ,D_{ij} ,E_{ij} ,F_{ij} ) \\ & \quad = \int\nolimits_{ - h/2}^{h/2} {Q_{ij(z)}^{(k)} (1,z,f(z),g(z),f^{\prime}(z),g^{\prime}(z))dz} \\ & (G_{ij} ,H_{ij} ,I_{ij} ,J_{ij} ,K^{\prime}_{ij} )\\ &\quad = \int\nolimits_{ - h/2}^{h/2} {Q_{ij(z)}^{(k)} (z^{2} ,zf(z),zg(z),zf^{\prime}(z),zg^{\prime}(z))dz} \\ &(L_{ij} ,M_{ij}^{{\prime }} ,N_{ij}^{{\prime }} ,O_{ij} )\\ &\quad = \int\nolimits_{ - h/2}^{h/2} {Q_{ij(z)}^{(k)} (f^{2} (z),f(z)g(z),f(z)f^{\prime}(z),f(z)g^{\prime}(z))dz} \\ &(P_{ij}^{{\prime }} ,Q_{ij}^{{\prime }} ,R_{ij}^{{\prime }} )\\ &\quad= \int\nolimits_{ - h/2}^{h/2} {Q_{ij(z)}^{(k)} (g^{2} (z),g(z)f^{\prime}(z),g(z)g^{\prime}(z))dz} \\ &(S_{ij} ,T_{ij} ) = \int\nolimits_{ - h/2}^{h/2} {Q_{ij(z)}^{(k)} (f^{{{\prime }2}} (z),f^{\prime}(z)g^{\prime}(z))dz} \\ &U_{ij} = \int\nolimits_{ - h/2}^{h/2} {Q_{ij(z)}^{(k)} g^{{{\prime }2}} (z)dz} \\ \end{aligned} $$
(32a{-}f)

Appendix 2: Definition of constants in Eq. (19)

As mentioned before, these matrices are associated with the expressions of the plate governing Eq. (17a–f) they used to calculate the Kij element matrices. This method is perhaps more convenient and simple than the others [51]. The advantage of the present technique is that infinite shear deformation theories can be created and calculated by using the same following matrices, only ‘‘y* and q*’’ should be changed.

1.1 Calculation of Kij

$$ \begin{aligned} & \frac{{\partial^{2} (N_{i} ,M_{i} )}}{{\partial x^{2} }} \\ & \quad = ({\text{A}}_{\text{ij}} ,{\text{B}}_{\text{ij}} )\left[ {\begin{array}{*{20}c} {\alpha^{3} } & 0 & 0 & 0 & 0 & 0 \\ 0 & {\alpha^{2} \beta } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - y^{*} \alpha^{2} } & { - q^{*} \alpha^{2} \beta } \\ 0 & 0 & 0 & { - y^{*} \alpha^{2} } & 0 & { - q^{*} \alpha^{3} } \\ { - \alpha^{2} \beta } & { - \alpha^{3} } & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ & \quad \quad \times \left[ {\begin{array}{*{20}c} {U_{{rs^{\prime}}} } \\ {V_{{rs^{\prime}}} } \\ {W_{{rs^{\prime}}} } \\ {\varTheta_{{rs^{\prime}}}^{1} } \\ \begin{aligned} \varTheta_{{rs^{\prime}}}^{2} \hfill \\ \varTheta_{{rs^{\prime}}}^{3} \hfill \\ \end{aligned} \\ \end{array} } \right]^{T} \left\{ {\begin{array}{*{20}c} {SS} \\ {SS} \\ {SS} \\ {SC} \\ {CS} \\ {CC} \\ \end{array} } \right\} \\ & \quad \quad + ({\text{B}}_{\text{ij}} ,{\text{G}}_{\text{ij}} )\left[ {\begin{array}{*{20}c} 0 & 0 & { - \alpha^{4} } & {y^{*} \alpha^{3} } & 0 & {q^{*} \alpha^{4} } \\ 0 & 0 & { - \alpha^{2} \beta^{2} } & 0 & {y^{*} \alpha^{2} \beta } & {q^{*} \alpha^{2} \beta^{2} } \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {2\alpha^{3} \beta } & { - y^{*} \alpha^{2} \beta } & { - y^{*} \alpha^{3} } & { - 2q^{*} \alpha^{3} \beta } \\ \end{array} } \right] \\ & \quad \quad \times \left[ {\begin{array}{*{20}c} {U_{{rs^{\prime}}} } \\ {V_{rs'} } \\ {W_{{rs^{\prime}}} } \\ {\varTheta_{{rs^{\prime}}}^{1} } \\ \begin{aligned} \varTheta_{{rs^{\prime}}}^{2} \hfill \\ \varTheta_{{rs^{\prime}}}^{3} \hfill \\ \end{aligned} \\ \end{array} } \right]^{T} \left\{ {\begin{array}{*{20}c} {SS} \\ {SS} \\ {SS} \\ {SC} \\ {CS} \\ {CC} \\ \end{array} } \right\} \\ & \quad \quad + ({\text{C}}_{\text{ij}} ,{\text{H}}_{\text{ij}} )\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & {\alpha^{3} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {\alpha^{2} \beta } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - \alpha^{2} \beta } & { - \alpha^{3} } & 0 \\ \end{array} } \right] \\ & \quad \quad \times \left[ {\begin{array}{*{20}c} {U_{{rs^{\prime}}} } \\ {V_{{rs^{\prime}}} } \\ {W_{{rs^{\prime}}} } \\ {\varTheta_{{rs^{\prime}}}^{1} } \\ \begin{aligned} \varTheta_{{rs^{\prime}}}^{2} \hfill \\ \varTheta_{{rs^{\prime}}}^{3} \hfill \\ \end{aligned} \\ \end{array} } \right]^{T} \left\{ {\begin{array}{*{20}c} {SS} \\ {SS} \\ {SS} \\ {SC} \\ {CS} \\ {CC} \\ \end{array} } \right\} \\ \end{aligned} $$
$$ \begin{aligned} & \quad \quad + ({\text{D}}_{\text{ij}} ,{\text{I}}_{\text{ij}} )\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - \alpha^{2} \beta } \\ 0 & 0 & 0 & 0 & 0 & { - \alpha^{3} } \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ & \quad \quad \times \left[ {\begin{array}{*{20}c} {U_{{rs^{\prime}}} } \\ {V_{{rs^{\prime}}} } \\ {W_{{rs^{\prime}}} } \\ {\varTheta_{{rs^{\prime}}}^{1} } \\ \begin{aligned} \varTheta_{{rs^{\prime}}}^{2} \hfill \\ \varTheta_{{rs^{\prime}}}^{3} \hfill \\ \end{aligned} \\ \end{array} } \right]^{T} \left\{ {\begin{array}{*{20}l} {SS} \hfill \\ {SS} \hfill \\ {SS} \hfill \\ {SC} \hfill \\ {CS} \hfill \\ {CC} \hfill \\ \end{array} } \right\} \\ & \quad \quad + ({\text{E}}_{\text{ij}} ,{\text{J}}_{\text{ij}} )\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - \alpha^{2} } & 0 \\ 0 & 0 & 0 & { - \alpha^{2} } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ & \quad \quad \times \left[ {\begin{array}{*{20}c} {U_{{rs^{\prime}}} } \\ {V_{{rs^{\prime}}} } \\ {W_{{rs^{\prime}}} } \\ {\varTheta_{{rs^{\prime}}}^{1} } \\ \begin{aligned} \varTheta_{{rs^{\prime}}}^{2} \hfill \\ \varTheta_{{rs^{\prime}}}^{3} \hfill \\ \end{aligned} \\ \end{array} } \right]^{T} \left\{ {\begin{array}{*{20}c} {SS} \\ {SS} \\ {SS} \\ {SC} \\ {CS} \\ {CC} \\ \end{array} } \right\} \\ & \quad \quad + ({\text{F}}_{\text{ij}} ,{\text{K}}_{\text{ij}}^{{\prime }} )\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - \alpha^{2} } \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ & \quad \quad \times \left[ {\begin{array}{*{20}c} {U_{{rs^{\prime}}} } \\ {V_{{rs^{\prime}}} } \\ {W_{{rs^{\prime}}} } \\ {\varTheta_{{rs^{\prime}}}^{1} } \\ \begin{aligned} \varTheta_{{rs^{\prime}}}^{2} \hfill \\ \varTheta_{{rs^{\prime}}}^{3} \hfill \\ \end{aligned} \\ \end{array} } \right]^{T} \left\{ {\begin{array}{*{20}c} {SS} \\ {SS} \\ {SS} \\ {SC} \\ {CS} \\ {CC} \\ \end{array} } \right\}. \\ \end{aligned} $$
(33)

where \( {\text{SS}} = \sin (\alpha x)\sin (\beta y) \), and the elements of the 6 × 6 matrices are the coefficients obtained after taking the second derivation of the strains expression in the Eq. (31a–f). As is known, the strains are expressed as a function of the 6DOF (6 unknowns), described in Eq. (2a–c). These unknowns are expressed as shown in the Eqs. (38a–e) in order to satisfy the simply supported boundary conditions.

The 6 × 6 matrices associated with \( \frac{{\partial^{2} M_{i} }}{{\partial x^{2} }} \) in Equations B1, is called \( \bar{M}_{x}^{2,b} \)(b = 0,…,5). The symbols used in \( \overline{M}_{v}^{a,b} \) are as follow: the first upper and lower (a,v) indicates the derivative (second derivative with respect to x, in the example), and the second upper character, b, indicates that the derivative is associates with the strain ε b j (b = 0,…,5). Therefore, the expression \( \frac{{\partial^{2} (N_{i} ,M_{i} )}}{{\partial x^{2} }} \), can be expressed as:

$$ \begin{aligned} \frac{{\partial^{2} (N_{i} ,M_{i} )}}{{\partial x^{2} }} &= \left(({\text{A}}_{\text{ij}} ,{\text{B}}_{\text{ij}} )\bar{M}_{x}^{2,0} + ({\text{B}}_{\text{ij}} ,{\text{G}}_{\text{ij}} )\bar{M}_{x}^{2,1}\right. \\ & \left. \quad+\, ({\text{C}}_{\text{ij}} ,{\text{H}}_{\text{ij}} )\bar{M}_{x}^{2,2} + ({\text{D}}_{\text{ij}} ,{\text{I}}_{\text{ij}} )\bar{M}_{x}^{2,3} \right. \\ & \left. \quad +\, { ({\text{E}}_{\text{ij}} ,{\text{J}}_{\text{ij}} )\bar{M}_{x}^{2,4} + ({\text{F}}_{\text{ij}} ,{\text{K}}_{\text{ij}}^{{\prime }} )\bar{M}_{x}^{2,5} } \right) \times {\text{sin}}(\alpha x)\sin (\beta y) \\ \end{aligned} $$
(34)

where, for example, \( \bar{M}_{x}^{2,0} \) is:

$$ \bar{M}_{x}^{2,0} = \left[ {\begin{array}{*{20}c} {\alpha^{3} } & 0 & 0 & 0 & 0 & 0 \\ 0 & {\alpha^{2} \beta } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - y^{*} \alpha^{2} } & { - q^{*} \alpha^{2} \beta } \\ 0 & 0 & 0 & { - y^{*} \alpha^{2} } & 0 & { - q^{*} \alpha^{3} } \\ { - \alpha^{2} \beta } & { - \alpha^{3} } & 0 & 0 & 0 & 0 \\ \end{array} } \right] $$
(35)

All matrices of type, \( \bar{M}_{v}^{a,b} \), associated with the expressions of the plate governing Eqs. (17a–f), for example \( \frac{{\partial^{2} M_{i} }}{\partial x\partial y} \) or Qi, are given in what follows.

$$ \begin{aligned} \bar{M}^{0,0} & = \left[ {\begin{array}{*{20}c} { - \alpha } & 0 & 0 & 0 & 0 & 0 \\ 0 & { - \beta } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {y^{*} } & {q^{*} \beta } \\ 0 & 0 & 0 & {y^{*} } & 0 & {q^{*} \alpha } \\ \beta & \alpha & 0 & 0 & 0 & 0 \\ \end{array} } \right],\quad \bar{M}^{0,1} = \left[ {\begin{array}{*{20}c} 0 & 0 & {\alpha^{2} } & { - y^{*} \alpha } & 0 & { - q^{*} \alpha^{2} } \\ 0 & 0 & {\beta^{2} } & 0 & { - y^{*} \beta } & { - q^{*} \beta^{2} } \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & { - 2\alpha \beta } & {y^{*} \beta } & {y^{*} \alpha } & {2q^{*} \alpha \beta } \\ \end{array} } \right] \\ \bar{M}^{0,2} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & { - \alpha } & 0 & 0 \\ 0 & 0 & 0 & 0 & { - \beta } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \beta & \alpha & 0 \\ \end{array} } \right],\quad \bar{M}^{0,3} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \beta \\ 0 & 0 & 0 & 0 & 0 & \alpha \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ \bar{M}^{0,4} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],\quad \bar{M}^{0,5} = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right] \\ \bar{M}_{x}^{1,0} & = \left[ {\begin{array}{*{20}c} { - \alpha^{2} } & 0 & 0 & 0 & 0 & 0 \\ 0 & { - \alpha \beta } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {y^{*} \alpha } & {q^{*} \alpha \beta } \\ 0 & 0 & 0 & { - y^{*} \alpha } & 0 & { - q^{*} \alpha^{2} } \\ { - \alpha \beta } & { - \alpha^{2} } & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ \bar{M}_{x}^{1,2} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & { - \alpha^{2} } & 0 & 0 \\ 0 & 0 & 0 & 0 & { - \alpha \beta } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - \alpha \beta } & { - \alpha^{2} } & 0 \\ \end{array} } \right],\quad \bar{M}_{x}^{1,3} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\alpha \beta } \\ 0 & 0 & 0 & 0 & 0 & { - \alpha^{2} } \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ \bar{M}_{x}^{1,4} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & { - \alpha } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],\quad \bar{M}_{x}^{1,5} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \alpha \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ \bar{M}_{y}^{1,0} & = \left[ {\begin{array}{*{20}c} { - \alpha \beta } & 0 & 0 & 0 & 0 & 0 \\ 0 & { - \beta^{2} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - y^{*} \beta } & { - q^{*} \beta^{2} } \\ 0 & 0 & 0 & {y^{*} \beta } & 0 & {q^{*} \alpha \beta } \\ { - \beta^{2} } & { - \alpha \beta } & 0 & 0 & 0 & 0 \\ \end{array} } \right],\quad \bar{M}_{y}^{1,1} = \left[ {\begin{array}{*{20}c} 0 & 0 & {\alpha^{2} \beta } & { - y^{*} \alpha \beta } & 0 & { - q^{*} \alpha^{2} \beta } \\ 0 & 0 & {\beta^{3} } & 0 & { - y^{*} \beta^{2} } & { - q^{*} \beta^{3} } \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {2\alpha \beta^{2} } & { - y^{*} \beta^{2} } & { - y^{*} \alpha \beta } & { - 2q^{*} \alpha \beta^{2} } \\ \end{array} } \right] \\ \bar{M}_{y}^{1,2} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & { - \alpha \beta } & 0 & 0 \\ 0 & 0 & 0 & 0 & { - \beta^{2} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - \beta^{2} } & { - \alpha \beta } & 0 \\ \end{array} } \right],\quad \bar{M}_{y}^{1,3} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - \beta^{2} } \\ 0 & 0 & 0 & 0 & 0 & {\alpha \beta } \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ \bar{M}_{y}^{1,4} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - \beta } & 0 \\ 0 & 0 & 0 & \beta & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],\quad \bar{M}_{y}^{1,5} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \beta \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ \end{aligned} $$
$$ \begin{aligned} \bar{M}_{y}^{2,0} & = \left[ {\begin{array}{*{20}c} {\alpha \beta^{2} } & 0 & 0 & 0 & 0 & 0 \\ 0 & {\beta^{3} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - y^{*} \beta^{2} } & { - q^{*} \beta^{3} } \\ 0 & 0 & 0 & { - y^{*} \beta^{2} } & 0 & { - q^{*} \alpha \beta^{2} } \\ { - \beta^{3} } & { - \alpha \beta^{2} } & 0 & 0 & 0 & 0 \\ \end{array} } \right], \\ \bar{M}_{y}^{2,1} & = \left[ {\begin{array}{*{20}c} 0 & 0 & { - \alpha^{2} \beta^{2} } & {y^{*} \alpha \beta^{2} } & 0 & {q^{*} \alpha^{2} \beta^{2} } \\ 0 & 0 & { - \beta^{4} } & 0 & {y^{*} \beta^{3} } & {q^{*} \beta^{4} } \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {2\alpha \beta^{3} } & { - y^{*} \beta^{3} } & { - y^{*} \alpha \beta^{2} } & { - 2q^{*} \alpha \beta^{3} } \\ \end{array} } \right] \\ \bar{M}_{y}^{2,2} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - \beta^{2} } & 0 \\ 0 & 0 & 0 & { - \beta^{2} } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right], \\ \bar{M}_{y}^{2,3} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - \beta^{3} } \\ 0 & 0 & 0 & 0 & 0 & { - \alpha \beta^{2} } \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ \end{aligned} $$
$$ \begin{aligned} \bar{M}_{y}^{2,4} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - \beta^{2} } & 0 \\ 0 & 0 & 0 & { - \beta^{2} } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right], \\ \bar{M}_{y}^{2,5} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - \beta^{2} } \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ \bar{M}_{xy}^{2,0} & = \left[ {\begin{array}{*{20}c} { - \alpha^{2} \beta } & 0 & 0 & 0 & 0 & 0 \\ 0 & { - \alpha \beta^{2} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - y^{*} \alpha \beta } & { - q^{*} \alpha \beta^{2} } \\ 0 & 0 & 0 & { - y^{*} \alpha \beta } & 0 & { - q^{*} \alpha^{2} \beta } \\ {\alpha \beta^{2} } & {\alpha^{2} \beta } & 0 & 0 & 0 & 0 \\ \end{array} } \right], \\ \bar{M}_{xy}^{2,1} & = \left[ {\begin{array}{*{20}c} 0 & 0 & {\alpha^{3} \beta } & { - y^{*} \alpha^{2} \beta } & 0 & { - q^{*} \alpha^{3} \beta } \\ 0 & 0 & {\alpha \beta^{3} } & 0 & { - y^{*} \alpha \beta^{2} } & { - q^{*} \alpha \beta^{3} } \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & { - 2\alpha^{2} \beta^{2} } & {y^{*} \alpha \beta^{2} } & {y^{*} \alpha^{2} \beta } & {2q^{*} \alpha^{2} \beta^{2} } \\ \end{array} } \right] \\ \bar{M}_{xy}^{2,2} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & { - \alpha^{2} \beta } & 0 & 0 \\ 0 & 0 & 0 & 0 & { - \alpha \beta^{2} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\alpha \beta^{2} } & {\alpha^{2} \beta } & 0 \\ \end{array} } \right], \\ \bar{M}_{xy}^{2,3} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - \alpha \beta^{2} } \\ 0 & 0 & 0 & 0 & 0 & { - \alpha^{2} \beta } \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ \bar{M}_{xy}^{2,4} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - \alpha \beta } & 0 \\ 0 & 0 & 0 & { - \alpha \beta } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right], \\ \bar{M}_{xy}^{2,5} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\alpha \beta } \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ \end{aligned} $$
(36)

With the help of the previous matrices it is possible to obtain a matrix from the operations of the strain energy “U” called “K U ”, where for example,\( K_{U} (1,j) = \frac{{\partial N_{1} }}{\partial x} + \frac{{\partial N_{6} }}{\partial y} \) (j = 1,2,…,6). Then, considering the matrix derived from the elastic potential energy of the elastic foundation “V e ” called “K Ve ”, the following matrix is obtained:

$$ \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & { - k_{0} - k_{1} (\alpha^{2} + \beta^{2} )} & 0 & 0 & {q^{*} k_{0} + q^{*} k_{1} (\alpha^{2} + \beta^{2} )} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {q^{*} k_{0} + q^{*} k_{1} (\alpha^{2} + \beta^{2} )} & 0 & 0 & { - q^{*2} k_{0} - q^{*2} k_{1} (\alpha^{2} + \beta^{2} )} \\ \end{array} } \right] $$
(37)

Finally, the matrix [K] = [K U ] + [K Ve ] used in Eq. (19) is obtained.

Appendix 3: Solution Fourier series

$$ \begin{aligned} u(x,y) &= \sum\limits_{r = 1}^{\infty } {\sum\limits_{s = 1}^{\infty } {U_{rs} \cos (\alpha x)} } \sin (\beta y)e^{i\omega t} , \hfill \\ & \quad {0 \le x \le a;\;0 \le y \le b} \hfill \\ v(x,y) &= \sum\limits_{r = 1}^{\infty } {\sum\limits_{s = 1}^{\infty } {V_{rs} \sin (\alpha x)} } \cos (\beta y)e^{i\omega t} , \hfill \\ & \quad {0 \le x \le a;\;0 \le y \le b} \hfill \\ w(x,y) &= \sum\limits_{r = 1}^{\infty } {\sum\limits_{s = 1}^{\infty } {W_{rs} \sin (\alpha x)} } \sin (\beta y)e^{i\omega t} , \hfill \\ & \quad {0 \le x \le a;\;0 \le y \le b} \hfill \\ \theta_{1} (x,y) &= \sum\limits_{r = 1}^{\infty } {\sum\limits_{s = 1}^{\infty } {\varTheta_{rs}^{1} \cos (\alpha x)} } \sin (\beta y)e^{i\omega t} , \hfill \\ & \quad {0 \le x \le a;\;0 \le y \le b} \hfill \\ \theta_{2} (x,y) &= \sum\limits_{r = 1}^{\infty } {\sum\limits_{s = 1}^{\infty } {\varTheta_{rs}^{2} \sin (\alpha x)} } \cos (\beta y)e^{i\omega t} , \hfill \\ &\quad {0 \le x \le a;\;0 \le y \le b} \hfill \\ \theta_{3} (x,y) &= \sum\limits_{r = 1}^{\infty } {\sum\limits_{s = 1}^{\infty } {\varTheta_{rs}^{3} \sin (\alpha x)} } \sin (\beta y)e^{i\omega t} , \hfill \\ & \quad {0 \le x \le a;\;0 \le y \le b} \hfill \\ \end{aligned} $$
(38a{-}e)

where

$$ \alpha = \frac{r\pi }{a}, \quad \beta = \frac{s\pi }{b}. $$
(39)

Appendix 4: Calculation of M ij

$$ \left[ {\begin{array}{*{20}c} { - I_{1} } & 0 & {I_{2} \alpha } & { - (y^{*} I_{2} + I_{4} )} & 0 & { - q^{*} I_{2} \alpha } \\ {} & { - I_{1} } & {I_{2} \beta } & 0 & { - (y^{*} I_{2} + I_{4} )} & { - q^{*} I_{2} \beta } \\ {} & {} & { - I_{3} (\alpha^{2} + \beta^{2} ) - I_{1} } & {(y^{*} I_{3} + I_{5} )\alpha } & {(y^{*} I_{3} + I_{5} )\beta } & {q^{*} I_{3} (\alpha^{2} + \beta^{2} ) - I_{7} } \\ {} & {} & {} & { - (y^{*2} I_{3} + 2y^{*} I_{5} + I_{6} )} & 0 & { - q^{*} (y^{*} I_{3} + I_{5} )\alpha } \\ {} & {} & {} & {} & { - (y^{*2} I_{3} + 2y^{*} I_{5} + I_{6} )} & { - q^{*} (y^{*} I_{3} + I_{5} )\beta } \\ {symm} & {} & {} & {} & {} & { - q^{*2} I_{3} (\alpha^{2} + \beta^{2} ) - I_{8} } \\ \end{array} } \right] $$
(40)

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Mantari, J.L. Free vibration of advanced composite plates resting on elastic foundations based on refined non-polynomial theory. Meccanica 50, 2369–2390 (2015). https://doi.org/10.1007/s11012-015-0160-x

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