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A refined quasi-3D model for buckling and free vibration of functionally graded saturated porous plate resting on elastic foundation

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Abstract

This study investigates the buckling and free vibration behavior of functionally graded saturated porous (FGSP) using a refined quasi-3D theory that ensures zero transverse shear stress at the top and bottom surfaces of the plate. The material properties depend on the porosity coefficient according to three patterns. Hamilton's principle and Biot's poroelasticity theory are employed to derive the equations of motion, which are then solved using Navier's technique. After examining the accuracy of the suggested approach, the effect of fluid compressibility on natural frequency and critical buckling load is investigated in the undrained condition. Also, the effect of porosity, geometrical parameters, and elastic foundation on the vibration and buckling response of FGSP plates are examined. The study reveals that saturating the pores with fluid leads to increased plate stiffness. This translates to higher critical buckling loads and fundamental frequencies.

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Acknowledgements

This research is funded by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) under grant number: 107.02-2021.16.

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Vu Thi Thu Trang: Methodology, Supervision, Writing – original draft. Nguyen Van Long: Methodology, Project administration, Software, Writing – original draft, Writing – review & editing. Tran Minh Tu: Formal analysis, Software, Supervision, Writing – review & editing. Le Thanh Hai: Data curation, Investigation, Validation, Writing – review & editing.

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Correspondence to Nguyen Van Long.

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Appendix: The global linear stiffness matrix [S], and global mass matrix [M] Coefficients of matrix [S]:

Appendix: The global linear stiffness matrix [S], and global mass matrix [M] Coefficients of matrix [S]:

$$s_{11} = A_{1} \alpha^{2} + A_{3} \beta^{2} ;s_{12} = \left( {A_{2} + A_{3} } \right)\alpha \beta ;s_{13} = - \frac{{D_{1} }}{3}\kappa \alpha^{3} - \left( {\frac{{D_{2} }}{3} + \frac{{2D_{3} }}{3}} \right)\kappa \alpha \beta^{2} ;$$
$$s_{14} = \left( {B_{1} - \frac{{D_{1} \kappa }}{3}} \right)\alpha^{2} + \left( {B_{3} - \frac{{D_{3} \kappa }}{3}} \right)\beta^{2} ;s_{15} = \left( {B_{2} + B_{3} - \frac{{D_{2} \kappa }}{3} - \frac{{D_{3} \kappa }}{3}} \right)\alpha \beta ;$$
$$s_{16} = s_{61} = - \frac{{C_{1} }}{2}\alpha^{3} - A_{2} \alpha - \left( {\frac{{C_{2} }}{2} + C_{3} } \right)\alpha \beta^{2} ;s_{17} = - \frac{{D_{1} }}{3}\alpha^{3} - 2B_{2} \alpha - \left( {\frac{{D_{2} }}{3} + \frac{{2D_{3} }}{3}} \right)\alpha \beta^{2} ;$$
$$s_{22} = A_{3} \alpha^{2} + A_{1} \beta^{2} ;s_{23} = - \left( {\frac{{D_{2} }}{3} + \frac{{2D_{3} }}{3}} \right)\kappa \alpha^{2} \beta - \frac{{D_{1} }}{3}\kappa \beta^{3} ;$$
$$s_{24} = \left( {B_{2} + B_{3} - \frac{{D_{2} \kappa }}{3} - \frac{{D_{3} \kappa }}{3}} \right)\alpha \beta ;s_{25} = \left( {B_{3} - \frac{{D_{3} \kappa }}{3}} \right)\alpha^{2} + \left( {B_{1} - \frac{{D_{1} \kappa }}{3}} \right)\beta^{2} ;$$
$$s_{26} = - \left( {\frac{{C_{2} }}{2} + C_{3} } \right)\alpha^{2} \beta - A_{2} \beta - \frac{{C_{1} }}{2}\beta^{3} ;s_{27} = - \left( {\frac{{D_{2} }}{3} + \frac{{2D_{3} }}{3}} \right)\alpha^{2} \beta - 2B_{2} \beta - \frac{{D_{1} }}{3}\beta^{3} ;$$
$$s_{33} = \frac{{G_{1} \kappa^{2} }}{9}\alpha^{4} + A^{s} \alpha^{2} + \frac{{2\left( {G_{2} + 2G_{3} } \right)\kappa^{2} }}{9}\alpha^{2} \beta^{2} + A^{s} \beta^{2} + \frac{{G_{1} \kappa^{2} }}{9}\beta^{4} + \xi ;$$
$$s_{{34}} = \left( {\frac{{G_{1} \kappa ^{2} }}{9} - \frac{{E_{1} \kappa }}{3}} \right)\alpha ^{3} + A^{s} \alpha - \left( {\frac{{E_{2} \kappa }}{3} + \frac{{2E_{3} \kappa }}{3} - \frac{{G_{2} \kappa ^{2} }}{9} - \frac{{2G_{3} \kappa ^{2} }}{9}} \right)\alpha \beta ^{2} ;$$
$$s_{{35}} = s_{{53}} = - \left( {\frac{{E_{2} \kappa }}{3} + \frac{{2E_{3} \kappa }}{3} - \frac{{G_{2} \kappa ^{2} }}{9} - \frac{{2G_{3} \kappa ^{2} }}{9}} \right)\alpha ^{2} \beta + A^{s} \beta + \left( {\frac{{G_{1} \kappa ^{2} }}{9} - \frac{{E_{1} \kappa }}{3}} \right)\beta ^{3} ;$$
$$s_{{36}} = \frac{{F_{1} \kappa }}{6}\alpha ^{4} + \frac{{D_{2} \kappa }}{3}\alpha ^{2} + \left( {\frac{{F_{2} }}{3} - \frac{{2F_{3} }}{3}} \right)\kappa \alpha ^{2} \beta ^{2} + \frac{{D_{2} \kappa }}{3}\beta ^{2} + \frac{{F_{1} \kappa }}{6}\beta ^{4} - \frac{h}{2}\xi ;$$
$$s_{37} = \frac{{G_{1} \kappa }}{9}\alpha^{4} + \frac{{2E_{2} \kappa }}{3}\alpha^{2} + \left( {\frac{{2G_{2} }}{9} + \frac{{4G_{3} }}{9}} \right)\kappa \alpha^{2} \beta^{2} + \frac{{2E_{2} \kappa }}{3}\beta^{2} + \frac{{G_{1} \kappa }}{9}\beta^{4} + \frac{{h^{2} }}{4}\xi ;$$
$$s_{44} = \left( {\frac{{G_{1} \kappa^{2} }}{9} - \frac{{2E_{1} \kappa }}{3} + C_{1} } \right)\alpha^{2} + \left( {\frac{{G_{3} \kappa^{2} }}{9} - \frac{{2E_{3} \kappa }}{3} + C_{3} } \right)\beta^{2} + A^{s} ;$$
$$s_{45} = s_{54} = \left( {C_{2} + C_{3} + \frac{{G_{2} \kappa^{2} }}{9} + \frac{{G_{3} \kappa^{2} }}{9} - \frac{{2E_{2} \kappa }}{3} - \frac{{2E_{3} \kappa }}{3}} \right)\alpha \beta ;$$
$$s_{46} = - \left( {\frac{{D_{1} }}{2} - \frac{{F_{1} \kappa }}{6}} \right)\alpha^{3} + \left( {\frac{{D_{2} \kappa }}{3} - B_{2} } \right)\alpha - \left( {\frac{{D_{2} }}{2} + D_{3} - \frac{{F_{2} \kappa }}{6} - \frac{{F_{3} \kappa }}{3}} \right)\alpha \beta^{2} ;$$
$$s_{47} = - \left( {\frac{{E_{1} }}{3} - \frac{{G_{1} \kappa }}{9}} \right)\alpha^{3} + \left( {\frac{{2E_{2} \kappa }}{3} - 2C_{2} } \right)\alpha - \left( {\frac{{E_{2} }}{3} + \frac{{2E_{3} }}{3} - \frac{{G_{2} \kappa }}{9} - \frac{{2G_{3} \kappa }}{9}} \right)\alpha \beta^{2} ;$$
$$s_{55} = \left( {\frac{{G_{3} \kappa^{2} }}{9} - \frac{{2E_{3} \kappa }}{3} + C_{3} } \right)\alpha^{2} + \left( {\frac{{G_{1} c_{0}^{2} }}{9} - \frac{{2E_{1} \kappa }}{3} + C_{1} } \right)\beta^{2} + A^{s} ;$$
$$s_{56} = - \left( {\frac{{D_{2} }}{2} + D_{3} - \frac{{F_{2} \kappa }}{6} - \frac{{F_{3} \kappa }}{3}} \right)\alpha^{2} \beta + \left( {\frac{{D_{2} \kappa }}{3} - B_{2} } \right)\beta - \left( {\frac{{D_{1} }}{2} - \frac{{F_{1} \kappa }}{6}} \right)\beta^{3} ;$$
$$s_{57} = - \left( {\frac{{E_{2} }}{3} + \frac{{2E_{3} }}{3} - \frac{{G_{2} \kappa }}{9} - \frac{{2G_{3} \kappa }}{9}} \right)\alpha^{2} \beta + \left( {\frac{{2E_{2} \kappa }}{3} - 2C_{2} } \right)\beta - \left( {\frac{{E_{1} }}{3} - \frac{{G_{1} \kappa }}{9}} \right)\beta^{3} ;$$
$$s_{66} = \frac{{E_{1} }}{4}\alpha^{4} + C_{2} \alpha^{2} + \left( {\frac{{E_{2} }}{2} + E_{3} } \right)\alpha^{2} \beta^{2} + C_{2} \beta^{2} + \frac{{E_{1} }}{4}\beta^{4} + A_{1} + \frac{{h^{2} }}{4}\xi ;$$
$$s_{67} = \frac{{F_{1} }}{6}\alpha^{4} + 2D_{2} \alpha^{2} + \left( {\frac{{F_{2} }}{3} + \frac{{2F_{3} }}{3}} \right)\alpha^{2} \beta^{2} + 2D_{2} \beta^{2} + \frac{{F_{1} }}{6}\beta^{4} + 2B_{1} - \frac{{h^{3} }}{8}\xi ;$$
$$s_{77} = \frac{{G_{1} }}{9}\alpha^{4} + \frac{{4E_{2} }}{3}\alpha^{2} + \left( {\frac{{2G_{2} }}{9} + \frac{{4G_{3} }}{9}} \right)\alpha^{2} \beta^{2} + \frac{{4E_{2} }}{3}\beta^{2} + \frac{{G_{1} }}{9}\beta^{4} + 4C_{1} + \frac{{h^{4} }}{16}\xi ;$$

with \(\xi = k_{w} + k_{sx} \alpha^{2} + k_{sy} \beta^{2} .\)

1.1 Coefficients of matrix [M]:

$$\left[ M \right] = \left[ {\begin{array}{*{20}l} {m_{11} } \hfill & 0 \hfill & {m_{13} } \hfill & {m_{14} } \hfill & 0 \hfill & {m_{16} } \hfill & {m_{17} } \hfill \\ 0 \hfill & {m_{22} } \hfill & {m_{23} } \hfill & 0 \hfill & {m_{25} } \hfill & {m_{26} } \hfill & {m_{27} } \hfill \\ {m_{13} } \hfill & {m_{23} } \hfill & {m_{33} } \hfill & {m_{34} } \hfill & {m_{35} } \hfill & {m_{36} } \hfill & {m_{37} } \hfill \\ {m_{14} } \hfill & 0 \hfill & {m_{34} } \hfill & {m_{44} } \hfill & 0 \hfill & {m_{46} } \hfill & {m_{47} } \hfill \\ 0 \hfill & {m_{25} } \hfill & {m_{35} } \hfill & 0 \hfill & {m_{55} } \hfill & {m_{56} } \hfill & {m_{57} } \hfill \\ {m_{16} } \hfill & {m_{26} } \hfill & {m_{36} } \hfill & {m_{46} } \hfill & {m_{56} } \hfill & {m_{66} } \hfill & {m_{67} } \hfill \\ {m_{17} } \hfill & {m_{27} } \hfill & {m_{37} } \hfill & {m_{47} } \hfill & {m_{57} } \hfill & {m_{67} } \hfill & {m_{77} } \hfill \\ \end{array} } \right];$$
$$m_{11} = m_{22} = I_{0} ;m_{13} = - \frac{\alpha \kappa }{3}I_{3} ;m_{14} = J_{1} ;m_{16} = - \frac{\alpha }{2}I_{2} ;m_{17} = - \frac{\alpha }{3}I_{3} ;$$
$$m_{23} = - \frac{\beta \kappa }{3}I_{3} ;m_{25} = J_{1} ;m_{26} = - \frac{\beta }{2}I_{2} ;m_{27} = - \frac{\beta }{3}I_{3} ;$$
$$m_{33} = I_{0} + \frac{{\left( {\alpha^{2} + \beta^{2} } \right)\kappa^{2} }}{9}I_{6} ;m_{34} = - \frac{\alpha \kappa }{3}J_{4} ;m_{35} = - \frac{\beta \kappa }{3}J_{4} ;$$
$$m_{36} = I_{1} + \frac{{\left( {\alpha^{2} + \beta^{2} } \right)\kappa }}{6}I_{5} ;m_{37} = I_{2} + \frac{{\left( {\alpha^{2} + \beta^{2} } \right)\kappa }}{9}I_{6} ;m_{44} = m_{55} = K_{2} ;$$
$$m_{46} = - \frac{\alpha }{2}J_{3} ;m_{47} = - \frac{\alpha }{3}J_{4} ;m_{56} = - \frac{\beta }{2}J_{3} ;m_{57} = - \frac{\beta }{3}J_{4} ;$$
$$m_{66} = I_{2} + \frac{{\left( {\alpha^{2} + \beta^{2} } \right)}}{4}I_{4} ;m_{67} = I_{3} + \frac{{\left( {\alpha^{2} + \beta^{2} } \right)}}{6}I_{5} ;m_{77} = I_{4} + \frac{{\left( {\alpha^{2} + \beta^{2} } \right)}}{9}I_{6} .$$

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Trang, V.T.T., Van Long, N., Tu, T.M. et al. A refined quasi-3D model for buckling and free vibration of functionally graded saturated porous plate resting on elastic foundation. Arch Appl Mech 94, 1703–1721 (2024). https://doi.org/10.1007/s00419-024-02613-6

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