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Dynamic responses of saturated functionally graded porous plates resting on elastic foundation and subjected to a moving mass using pb2-Ritz method

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Abstract

This work analyzes the dynamic responses of plates with an elastic foundation under a concentrated moving mass. The plate is made of fluid-saturated functionally graded porous material (FFGPM). The material properties of the FFGPM plate are assumed to vary smoothly across the thickness, with symmetric, asymmetric, and uniform patterns for porosity distribution. The governing equations of the FFGPM plate are developed using Reddy’s third-order shear deformation plate theory (Reddy’s TSDT), Biot’s poroelasticity theory, pb2-Ritz formulation, and the Lagrange equation. In the numerical results, the Newmark scheme is used to obtain the deflection response of the FFGPM plate with the traveling mass. Two approaches are considered and discussed, including moving load and moving mass. Moreover, the influences of the porosity coefficient, distribution patterns, different boundary conditions, elastic foundations, geometry parameters, Skempton coefficient, moving mass velocity, and the mass of moving mass on the dynamic characteristics of the FFGPM plate are studied.

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Acknowledgements

This research is funded by Ministry of Education and Training under grand number B2023-XDA-10.

Funding

Funding for this research was provided by: Ministry of Education and Training (B2023-XDA-10).

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Authors and Affiliations

  1. Department of Strength of Materials, Faculty of Building and Industrial Construction, Hanoi University of Civil Engineering, 55 Giai Phong Road, Hanoi, Vietnam

    Van-Long Nguyen, Van-Loi Nguyen & Minh-Tu Tran

  2. Department of Civil Engineering, Vinh University, 182 Le Duan, Vinh City, Vietnam

    Tuan-Anh Nguyen

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Appendix

Appendix

Matrices \({\varvec{K}}\), \({\varvec{M}}\), \({\varvec{F}}\), \({\varvec{R}}\), \(\overline{\user2{H}}\), and \({\varvec{S}}\) in Eq. (36) are given as follows:

$$\begin{aligned} & {\varvec{K}} = \left[ {\begin{array}{*{20}c} {{\varvec{K}}^{uu} } & {{\varvec{K}}^{uv} } & {{\varvec{K}}^{uw} } & {{\varvec{K}}^{ux} } & {{\varvec{K}}^{uy} } \\ {{\varvec{K}}^{vu} } & {{\varvec{K}}^{vv} } & {{\varvec{K}}^{vw} } & {{\varvec{K}}^{vx} } & {{\varvec{K}}^{vy} } \\ {{\varvec{K}}^{wu} } & {{\varvec{K}}^{wv} } & {{\varvec{K}}^{ww} } & {{\varvec{K}}^{wx} } & {{\varvec{K}}^{wy} } \\ {{\varvec{K}}^{xu} } & {{\varvec{K}}^{xv} } & {{\varvec{K}}^{xw} } & {{\varvec{K}}^{xx} } & {{\varvec{K}}^{xy} } \\ {{\varvec{K}}^{yu} } & {{\varvec{K}}^{yv} } & {{\varvec{K}}^{yw} } & {{\varvec{K}}^{yx} } & {{\varvec{K}}^{yy} } \\ \end{array} } \right];\;{\varvec{M}} = \left[ {\begin{array}{*{20}c} {{\varvec{M}}^{uu} } & {\varvec{0}} & {{\varvec{M}}^{uw} } & {{\varvec{M}}^{ux} } & {\varvec{0}} \\ {\varvec{0}} & {{\varvec{M}}^{vv} } & {{\varvec{M}}^{vw} } & {\varvec{0}} & {{\varvec{M}}^{vy} } \\ {{\varvec{M}}^{wu} } & {{\varvec{M}}^{wv} } & {{\varvec{M}}^{ww} } & {{\varvec{M}}^{wx} } & {{\varvec{M}}^{wy} } \\ {{\varvec{M}}^{xu} } & {\varvec{0}} & {{\varvec{M}}^{xw} } & {{\varvec{M}}^{xx} } & {\varvec{0}} \\ {\varvec{0}} & {{\varvec{M}}^{yv} } & {{\varvec{M}}^{yw} } & {\varvec{0}} & {{\varvec{M}}^{yy} } \\ \end{array} } \right]; \\ & {\varvec{F}} = \left\{ {\begin{array}{*{20}c} {\varvec{0}} \\ {\varvec{0}} \\ {{\varvec{F}}^{w} } \\ {\varvec{0}} \\ {\varvec{0}} \\ \end{array} } \right\}; \\ \end{aligned}$$
$$\begin{aligned} & {\varvec{R}} = \left[ {\begin{array}{*{20}c} {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {{\varvec{R}}^{ww} } & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ \end{array} } \right];\;\overline{\user2{H}} = \left[ {\begin{array}{*{20}c} {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {\overline{\user2{H}}^{ww} } & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ \end{array} } \right]; \\ & {\varvec{S}} = \left[ {\begin{array}{*{20}c} {{\varvec{S}}^{uu} } & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {{\varvec{S}}^{vv} } & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {{\varvec{S}}^{ww} } & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ \end{array} } \right]; \\ \end{aligned}$$

where entire elements of matrices \({\varvec{K}}\), \({\varvec{M}}\), \({\varvec{R}}\), \(\overline{\user2{H}}\), \({\varvec{S}}\) and \({\varvec{F}}\) are determined by:

$${\varvec{K}}^{uu} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {A_{11} \frac{4}{{a^{2} }}\frac{{\partial {\varvec{U}}}}{\partial \xi }\frac{{\partial {\varvec{U}}^{T} }}{\partial \xi } + A_{66} \frac{4}{{b^{2} }}\frac{{\partial {\varvec{U}}}}{\partial \eta }\frac{{\partial {\varvec{U}}^{T} }}{\partial \eta }} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{vv} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {A_{66} \frac{4}{{a^{2} }}\frac{{\partial {\varvec{V}}}}{\partial \xi }\frac{{\partial {\varvec{V}}^{T} }}{\partial \xi } + A_{22} \frac{4}{{b^{2} }}\frac{{\partial {\varvec{V}}}}{\partial \eta }\frac{{\partial {\varvec{V}}^{T} }}{\partial \eta }} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{ww} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ \begin{gathered} \kappa^{2} G_{11} \frac{16}{{a^{4} }}\frac{{\partial^{2} {\varvec{W}}}}{{\partial \xi^{2} }}\frac{{\partial^{2} {\varvec{W}}^{T} }}{{\partial \xi^{2} }} + 2\kappa^{2} G_{12} \frac{16}{{a^{2} b^{2} }}\frac{{\partial^{2} {\varvec{W}}}}{{\partial \xi^{2} }}\frac{{\partial^{2} {\varvec{W}}^{T} }}{{\partial \eta^{2} }} \hfill \\ + \kappa^{2} G_{22} \frac{16}{{b^{4} }}\frac{{\partial^{2} {\varvec{W}}}}{{\partial \eta^{2} }}\frac{{\partial^{2} {\varvec{W}}^{T} }}{{\partial \eta^{2} }} + 4\kappa^{2} G_{66} \frac{16}{{a^{2} b^{2} }}\frac{{\partial^{2} {\varvec{W}}}}{\partial \xi \partial \eta }\frac{{\partial^{2} {\varvec{W}}^{T} }}{\partial \xi \partial \eta } \hfill \\ + A^{s} \left( {\frac{4}{{a^{2} }}\frac{{\partial {\varvec{W}}}}{\partial \xi }\frac{{\partial {\varvec{W}}^{T} }}{\partial \xi } + \frac{4}{{b^{2} }}\frac{{\partial {\varvec{W}}}}{\partial \eta }\frac{{\partial {\varvec{W}}^{T} }}{\partial \eta }} \right) \hfill \\ + k_{w} {\varvec{WW}}^{T} + k_{sx} \frac{4}{{a^{2} }}\frac{{\partial {\varvec{W}}}}{\partial \xi }\frac{{\partial {\varvec{W}}^{T} }}{\partial \xi } + k_{sy} \frac{4}{{b^{2} }}\frac{{\partial {\varvec{W}}}}{\partial \eta }\frac{{\partial {\varvec{W}}^{T} }}{\partial \eta } \hfill \\ \end{gathered} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{xx} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {C_{11}^{*} \frac{4}{{a^{2} }}\frac{{\partial \user2{\varphi }}}{\partial \xi }\frac{{\partial \user2{\varphi }^{T} }}{\partial \xi } + C_{66}^{*} \frac{4}{{b^{2} }}\frac{{\partial \user2{\varphi }}}{\partial \eta }\frac{{\partial \user2{\varphi }^{T} }}{\partial \eta } + A^{s} \user2{\varphi \varphi }^{T} } \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{yy} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {C_{66}^{*} \frac{4}{{a^{2} }}\frac{{\partial {\varvec{\psi}}}}{\partial \xi }\frac{{\partial {\varvec{\psi}}^{T} }}{\partial \xi } + C_{22}^{*} \frac{4}{{b^{2} }}\frac{{\partial {\varvec{\psi}}}}{\partial \eta }\frac{{\partial {\varvec{\psi}}^{T} }}{\partial \eta } + A^{s} \user2{\psi \psi }^{T} } \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{uv} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {A_{12} \frac{4}{ab}\frac{{\partial {\varvec{U}}}}{\partial \xi }\frac{{\partial {\varvec{V}}^{T} }}{\partial \eta } + A_{66} \frac{4}{ab}\frac{{\partial {\varvec{U}}}}{\partial \eta }\frac{{\partial {\varvec{V}}^{T} }}{\partial \xi }} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{vu} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {A_{12} \frac{4}{ab}\frac{{\partial {\varvec{V}}}}{\partial \eta }\frac{{\partial {\varvec{U}}^{T} }}{\partial \xi } + A_{66} \frac{4}{ab}\frac{{\partial {\varvec{V}}}}{\partial \xi }\frac{{\partial {\varvec{U}}^{T} }}{\partial \eta }} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{uw} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ \begin{gathered} - \kappa D_{11} \frac{8}{{a^{3} }}\frac{{\partial {\varvec{U}}}}{\partial \xi }\frac{{\partial^{2} {\varvec{W}}^{T} }}{{\partial \xi^{2} }} - \kappa D_{12} \frac{8}{{ab^{2} }}\frac{{\partial {\varvec{U}}}}{\partial \xi }\frac{{\partial^{2} {\varvec{W}}^{T} }}{{\partial \eta^{2} }} \hfill \\ - 2\kappa D_{66} \frac{8}{{ab^{2} }}\frac{{\partial {\varvec{U}}}}{\partial \eta }\frac{{\partial^{2} {\varvec{W}}^{T} }}{\partial \xi \partial \eta } \hfill \\ \end{gathered} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{wu} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ \begin{gathered} - \kappa D_{11} \frac{8}{{a^{3} }}\frac{{\partial^{2} {\varvec{W}}}}{{\partial \xi^{2} }}\frac{{\partial {\varvec{U}}^{T} }}{\partial \xi } - \kappa D_{12} \frac{8}{{ab^{2} }}\frac{{\partial^{2} {\varvec{W}}}}{{\partial \eta^{2} }}\frac{{\partial {\varvec{U}}^{T} }}{\partial \xi } \hfill \\ - 2\kappa D_{66} \frac{8}{{ab^{2} }}\frac{{\partial^{2} {\varvec{W}}}}{\partial \xi \partial \eta }\frac{{\partial {\varvec{U}}^{T} }}{\partial \eta } \hfill \\ \end{gathered} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{vw} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ \begin{gathered} - \kappa D_{12} \frac{8}{{a^{2} b}}\frac{{\partial {\varvec{V}}}}{\partial \eta }\frac{{\partial^{2} {\varvec{W}}^{T} }}{{\partial \xi^{2} }} - \kappa D_{22} \frac{8}{{b^{3} }}\frac{{\partial {\varvec{V}}}}{\partial \eta }\frac{{\partial^{2} {\varvec{W}}^{T} }}{{\partial \eta^{2} }} \hfill \\ - 2\kappa D_{66} \frac{8}{{a^{2} b}}\frac{{\partial {\varvec{V}}}}{\partial \xi }\frac{{\partial^{2} {\varvec{W}}^{T} }}{\partial \xi \partial \eta } \hfill \\ \end{gathered} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{wv} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ \begin{gathered} - \kappa D_{12} \frac{8}{{a^{2} b}}\frac{{\partial^{2} {\varvec{W}}}}{{\partial \xi^{2} }}\frac{{\partial {\varvec{V}}^{T} }}{\partial \eta } - \kappa D_{22} \frac{8}{{b^{3} }}\frac{{\partial^{2} {\varvec{W}}}}{{\partial \eta^{2} }}\frac{{\partial {\varvec{V}}^{T} }}{\partial \eta } \hfill \\ - 2\kappa D_{66} \frac{8}{{a^{2} b}}\frac{{\partial^{2} {\varvec{W}}}}{\partial \xi \partial \eta }\frac{{\partial {\varvec{V}}^{T} }}{\partial \xi } \hfill \\ \end{gathered} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{ux} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {B_{11}^{*} \frac{4}{{a^{2} }}\frac{{\partial {\varvec{U}}}}{\partial \xi }\frac{{\partial \user2{\varphi }^{T} }}{\partial \xi } + B_{66}^{*} \frac{4}{{b^{2} }}\frac{{\partial {\varvec{U}}}}{\partial \eta }\frac{{\partial \user2{\varphi }^{T} }}{\partial \eta }} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{xu} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {B_{11}^{*} \frac{4}{{a^{2} }}\frac{{\partial \user2{\varphi }}}{\partial \xi }\frac{{\partial {\varvec{U}}^{T} }}{\partial \xi } + B_{66}^{*} \frac{4}{{b^{2} }}\frac{{\partial \user2{\varphi }}}{\partial \eta }\frac{{\partial {\varvec{U}}^{T} }}{\partial \eta }} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{uy} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {B_{12}^{*} \frac{4}{ab}\frac{{\partial {\varvec{U}}}}{\partial \xi }\frac{{\partial {\varvec{\psi}}^{T} }}{\partial \eta } + B_{66}^{*} \frac{4}{ab}\frac{{\partial {\varvec{U}}}}{\partial \eta }\frac{{\partial {\varvec{\psi}}^{T} }}{\partial \xi }} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{yu} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {B_{12}^{*} \frac{4}{ab}\frac{{\partial {\varvec{\psi}}}}{\partial \eta }\frac{{\partial {\varvec{U}}^{T} }}{\partial \xi } + B_{66}^{*} \frac{4}{ab}\frac{{\partial {\varvec{\psi}}}}{\partial \xi }\frac{{\partial {\varvec{U}}^{T} }}{\partial \eta }} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{vx} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {B_{12}^{*} \frac{4}{ab}\frac{{\partial {\varvec{V}}}}{\partial \eta }\frac{{\partial \user2{\varphi }^{T} }}{\partial \xi } + B_{66}^{*} \frac{4}{ab}\frac{{\partial {\varvec{V}}}}{\partial \xi }\frac{{\partial \user2{\varphi }^{T} }}{\partial \eta }} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{xv} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {B_{12}^{*} \frac{4}{ab}\frac{{\partial \user2{\varphi }}}{\partial \xi }\frac{{\partial {\varvec{V}}^{T} }}{\partial \eta } + B_{66}^{*} \frac{4}{ab}\frac{{\partial \user2{\varphi }}}{\partial \eta }\frac{{\partial {\varvec{V}}^{T} }}{\partial \xi }} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{vy} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {B_{22}^{*} \frac{4}{{b^{2} }}\frac{{\partial {\varvec{V}}}}{\partial \eta }\frac{{\partial {\varvec{\psi}}^{T} }}{\partial \eta } + B_{66}^{*} \frac{4}{{a^{2} }}\frac{{\partial {\varvec{V}}}}{\partial \xi }\frac{{\partial {\varvec{\psi}}^{T} }}{\partial \xi }} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{yv} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {B_{22}^{*} \frac{4}{{b^{2} }}\frac{{\partial {\varvec{\psi}}}}{\partial \eta }\frac{{\partial {\varvec{V}}^{T} }}{\partial \eta } + B_{66}^{*} \frac{4}{{a^{2} }}\frac{{\partial {\varvec{\psi}}}}{\partial \xi }\frac{{\partial {\varvec{V}}^{T} }}{\partial \xi }} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{wx} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ \begin{gathered} E_{11}^{*} \frac{8}{{a^{3} }}\frac{{\partial^{2} {\varvec{W}}}}{{\partial \xi^{2} }}\frac{{\partial \user2{\varphi }^{T} }}{\partial \xi } + E_{12}^{*} \frac{8}{{ab^{2} }}\frac{{\partial^{2} {\varvec{W}}}}{{\partial \eta^{2} }}\frac{{\partial \user2{\varphi }^{T} }}{\partial \xi } \hfill \\ + 2E_{66}^{*} \frac{8}{{ab^{2} }}\frac{{\partial^{2} {\varvec{W}}}}{\partial \xi \partial \eta }\frac{{\partial \user2{\varphi }^{T} }}{\partial \eta } + A^{s} \frac{2}{a}\frac{{\partial {\varvec{W}}}}{\partial \xi }\user2{\varphi }^{T} \hfill \\ \end{gathered} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{xw} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ \begin{gathered} E_{11}^{*} \frac{8}{{a^{3} }}\frac{{\partial \user2{\varphi }}}{\partial \xi }\frac{{\partial^{2} {\varvec{W}}^{T} }}{{\partial \xi^{2} }} + E_{12}^{*} \frac{8}{{ab^{2} }}\frac{{\partial \user2{\varphi }}}{\partial \xi }\frac{{\partial^{2} {\varvec{W}}^{T} }}{{\partial \eta^{2} }} \hfill \\ + 2E_{66}^{*} \frac{8}{{ab^{2} }}\frac{{\partial \user2{\varphi }}}{\partial \eta }\frac{{\partial^{2} {\varvec{W}}^{T} }}{\partial \xi \partial \eta } + A^{s} \frac{2}{a}\user2{\varphi }\frac{{\partial {\varvec{W}}^{T} }}{\partial \xi } \hfill \\ \end{gathered} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{wy} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ \begin{gathered} E_{12}^{*} \frac{8}{{a^{2} b}}\frac{{\partial^{2} {\varvec{W}}}}{{\partial \xi^{2} }}\frac{{\partial {\varvec{\psi}}^{T} }}{\partial \eta } + E_{22}^{*} \frac{8}{{b^{3} }}\frac{{\partial^{2} {\varvec{W}}}}{{\partial \eta^{2} }}\frac{{\partial {\varvec{\psi}}^{T} }}{\partial \eta } \hfill \\ + 2E_{66}^{*} \frac{8}{{a^{2} b}}\frac{{\partial^{2} {\varvec{W}}}}{\partial \xi \partial \eta }\frac{{\partial {\varvec{\psi}}^{T} }}{\partial \xi } + A^{s} \frac{2}{b}\frac{{\partial {\varvec{W}}}}{\partial \eta }{\varvec{\psi}}^{T} \hfill \\ \end{gathered} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{yw} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ \begin{gathered} E_{12}^{*} \frac{8}{{a^{2} b}}\frac{{\partial {\varvec{\psi}}}}{\partial \eta }\frac{{\partial^{2} {\varvec{W}}^{T} }}{{\partial \xi^{2} }} + E_{22}^{*} \frac{8}{{b^{3} }}\frac{{\partial {\varvec{\psi}}}}{\partial \eta }\frac{{\partial^{2} {\varvec{W}}^{T} }}{{\partial \eta^{2} }} \hfill \\ + 2E_{66}^{*} \frac{8}{{a^{2} b}}\frac{{\partial {\varvec{\psi}}}}{\partial \xi }\frac{{\partial^{2} {\varvec{W}}^{T} }}{\partial \xi \partial \eta } + A^{s} \frac{2}{b}{\varvec{\psi}}\frac{{\partial {\varvec{W}}^{T} }}{\partial \eta } \hfill \\ \end{gathered} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{xy} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {C_{12}^{*} \frac{4}{ab}\frac{{\partial \user2{\varphi }}}{\partial \xi }\frac{{\partial {\varvec{\psi}}^{T} }}{\partial \eta } + C_{66}^{*} \frac{4}{ab}\frac{{\partial \user2{\varphi }}}{\partial \eta }\frac{{\partial {\varvec{\psi}}^{T} }}{\partial \xi }} \right]d\xi d\eta } } ;$$
$${\varvec{K}}^{yx} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {C_{12}^{*} \frac{4}{ab}\frac{{\partial {\varvec{\psi}}}}{\partial \eta }\frac{{\partial \user2{\varphi }^{T} }}{\partial \xi } + C_{66}^{*} \frac{4}{ab}\frac{{\partial {\varvec{\psi}}}}{\partial \xi }\frac{{\partial \user2{\varphi }^{T} }}{\partial \eta }} \right]d\xi d\eta } } ;$$
$${\varvec{M}}^{uu} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {I_{0} {\varvec{UU}}^{T} } \right]d\xi d\eta } } ;\;M^{vv} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {I_{0} {\varvec{VV}}^{T} } \right]d\xi d\eta } } ;$$
$${\varvec{M}}^{ww} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {I_{0} {\varvec{WW}}^{T} + \kappa^{2} I_{6} \left( {\frac{4}{{a^{2} }}\frac{{\partial {\varvec{W}}}}{\partial \xi }\frac{{\partial {\varvec{W}}^{T} }}{\partial \xi } + \frac{4}{{b^{2} }}\frac{{\partial {\varvec{W}}}}{\partial \eta }\frac{{\partial {\varvec{W}}^{T} }}{\partial \eta }} \right)} \right]d\xi d\eta } } ;$$
$${\varvec{M}}^{xx} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {K_{2} \user2{\varphi \varphi }^{T} } \right]d\xi d\eta } } ;\;{\varvec{M}}^{yy} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {K_{2} \user2{\psi \psi }^{T} } \right]d\xi d\eta } } ;$$
$${\varvec{M}}^{uw} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ { - \kappa I_{3} \frac{2}{a}{\varvec{U}}\frac{{\partial {\varvec{W}}^{T} }}{\partial \xi }} \right]d\xi d\eta } } ;\;{\varvec{M}}^{wu} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ { - \kappa I_{3} \frac{2}{a}\frac{{\partial {\varvec{W}}}}{\partial \xi }{\varvec{U}}^{T} } \right]d\xi d\eta } } ;$$
$${\varvec{M}}^{vw} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ { - \kappa I_{3} \frac{2}{b}{\varvec{V}}\frac{{\partial {\varvec{W}}^{T} }}{\partial \eta }} \right]d\xi d\eta } } ;\;{\varvec{M}}^{wv} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ { - \kappa I_{3} \frac{2}{b}\frac{{\partial {\varvec{W}}}}{\partial \eta }{\varvec{V}}^{T} } \right]d\xi d\eta } } ;$$
$${\varvec{M}}^{ux} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {J_{1} \user2{U\varphi }^{T} } \right]d\xi d\eta } } ;\;{\varvec{M}}^{xu} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {J_{1} \user2{\varphi U}^{T} } \right]d\xi d\eta } } ;$$
$${\varvec{M}}^{vy} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {J_{1} \user2{V\psi }^{T} } \right]d\xi d\eta } } ;\;{\varvec{M}}^{yv} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ {J_{1} \user2{\psi V}^{T} } \right]d\xi d\eta } } ;$$
$${\varvec{M}}^{wx} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ { - \kappa J_{4} \frac{2}{a}\frac{{\partial {\varvec{W}}}}{\partial \xi }\user2{\varphi }^{T} } \right]d\xi d\eta } } ;\;{\varvec{M}}^{xw} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ { - \kappa J_{4} \frac{2}{a}\user2{\varphi }\frac{{\partial {\varvec{W}}^{T} }}{\partial \xi }} \right]d\xi d\eta } } ;$$
$${\varvec{M}}^{wy} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ { - \kappa J_{4} \frac{2}{b}\frac{{\partial {\varvec{W}}}}{\partial \eta }{\varvec{\psi}}^{T} } \right]d\xi d\eta } } ;\;{\varvec{M}}^{yw} = \frac{ab}{4}\int\limits_{ - 1}^{1} {\int\limits_{ - 1}^{1} {\left[ { - \kappa J_{4} \frac{2}{b}{\varvec{\psi}}\frac{{\partial {\varvec{W}}^{T} }}{\partial \eta }} \right]d\xi d\eta } } ;$$
$${\varvec{F}}^{w} = - c(t)Mg\left. {\varvec{W}} \right|_{{\xi = \xi_{M} ,\eta = \eta_{M} }} ;$$
$${\varvec{R}}^{ww} = c(t)\left[ {v_{x} v_{y} \frac{4}{ab}\frac{{\partial {\varvec{W}}}}{\partial \xi }\frac{{\partial {\varvec{W}}^{T} }}{\partial \eta } + v_{x}^{2} \frac{4}{{a^{2} }}\frac{{\partial {\varvec{W}}}}{\partial \xi }\frac{{\partial {\varvec{W}}^{T} }}{\partial \xi } + v_{y}^{2} \frac{4}{{b^{2} }}\frac{{\partial {\varvec{W}}}}{\partial \eta }\frac{{\partial W^{T} }}{\partial \eta }} \right]_{{\xi = \xi_{M} ,\eta = \eta_{M} }} ;$$
$${\varvec{S}}^{uu} = \left. {c(t){\varvec{UU}}^{T} } \right|_{{\xi = \xi_{M} ,\eta = \eta_{M} }} ;\;{\varvec{S}}^{vv} = \left. {c(t){\varvec{VV}}^{T} } \right|_{{\xi = \xi_{M} ,\eta = \eta_{M} }} ;\;{\varvec{S}}^{ww} = \left. {c(t){\varvec{WW}}^{T} } \right|_{{\xi = \xi_{M} ,\eta = \eta_{M} }} ;$$
$$\overline{\user2{H}}^{ww} = c(t)\left[ {v_{x} \frac{2}{a}{\varvec{W}}\frac{{\partial {\varvec{W}}^{T} }}{\partial \xi } + v_{y} \frac{2}{b}{\varvec{W}}\frac{{\partial {\varvec{W}}^{T} }}{\partial \eta }} \right]_{{\xi = \xi_{M} ,\eta = \eta_{M} }} ;$$

with \(\xi_{M} = \frac{{2x_{0} }}{a} - 1;\eta_{M} = \frac{{2y_{0} }}{b} - 1.\)

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Nguyen, VL., Nguyen, VL., Nguyen, TA. et al. Dynamic responses of saturated functionally graded porous plates resting on elastic foundation and subjected to a moving mass using pb2-Ritz method. Acta Mech (2024). https://doi.org/10.1007/s00707-024-03978-z

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