An isogeometric approach to static and transient analysis of fluid-infiltrated porous metal foam piezoelectric nanoplates with flexoelectric effects and variable nonlocal parameters

Abstract

In this work, a novel refined higher-order shear deformation plate theory is integrated with nonlocal elasticity theory for analyzing the free vibration, bending, and transient behaviors of fluid-infiltrated porous metal foam piezoelectric nanoplates resting on Pasternak elastic foundation with flexoelectric effects. Isogeometric analysis (IGA) and the Navier solution are applied to the problem. The innovation in the present study is that the influence of the in-plane variation of the nonlocal parameter on the free and forced vibration of the piezoelectric nanoplates is investigated for the first time. The nonlocal parameter and material characteristics are assumed to be material-dependent and vary gradually over the thickness of structures. Based on Hamilton’s principle, equations of motion are built, then the IGA approach combined with the Navier solution is used to analyze the static and dynamic response of the nanoplate. Lastly, we investigate the effects of the porosity coefficients, flexoelectric parameters, elastic stiffness, thickness, and variation of the nonlocal parameters on the mechanical behaviors of the rectangular and elliptical piezoelectric nanoplates.

Graphical abstract

Keywords

isogeometric analysis / fluid-infiltrated porous / piezoelectric nanoplates / static and transient analysis / flexoelectric effect / variable nonlocal parameters

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Competing interests

The authors declare that they have no competing interests.

Appendix A

The coefficients of the material stiffness matrix

\left[\begin{array}{ccc}{\mathcal{A}}_{11}& {\mathcal{A}}_{12}& 0\\ {\mathcal{A}}_{12}& {\mathcal{A}}_{22}& 0\\ 0& 0& {\mathcal{A}}_{66}\end{array}\right]={\int }_{-\frac{h}{2}}^{\frac{h}{2}}{\mathit{D}}_{\mathrm{b}}^0\mathrm{d}\mathit{\text{z}},

\left[\begin{array}{ccc}{\mathcal{B}}_{11}& {\mathcal{B}}_{12}& 0\\ {\mathcal{B}}_{12}& {\mathcal{B}}_{22}& 0\\ 0& 0& {\mathcal{B}}_{66}\end{array}\right]={\int }_{-\frac{h}{2}}^{\frac{h}{2}}{\mathit{D}}_{\mathrm{b}}^{\mathrm{z}}\mathrm{d}\mathit{\text{z}},

\left[\begin{array}{ccc}{\mathcal{B}}_{11}^{\mathrm{s}}& {\mathcal{B}}_{12}^{\mathrm{s}}& 0\\ {\mathcal{B}}_{12}^{\mathrm{s}}& {\mathcal{B}}_{22}^{\mathrm{s}}& 0\\ 0& 0& {\mathcal{B}}_{66}^{\mathrm{s}}\end{array}\right]={\int }_{-\frac{h}{2}}^{\frac{h}{2}}{\mathit{D}}_{\mathrm{b}}^{\mathrm{f}}\mathrm{d}\mathit{\text{z}};

\left[\begin{array}{ccc}{\overline{\mathcal{A}}}_{11}& {\overline{\mathcal{A}}}_{12}& 0\\ {\overline{\mathcal{A}}}_{12}& {\overline{\mathcal{A}}}_{22}& 0\\ 0& 0& {\overline{\mathcal{A}}}_{66}\end{array}\right]={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\widetilde{\mathit{\text{z}}}{\mathit{D}}_{\mathrm{b}}^0\mathrm{d}\mathit{\text{z}},

\left[\begin{array}{ccc}{\mathcal{F}}_{11}& {\mathcal{F}}_{12}& 0\\ {\mathcal{F}}_{12}& {\mathcal{F}}_{22}& 0\\ 0& 0& {\mathcal{F}}_{66}\end{array}\right]={\int }_{-\frac{h}{2}}^{\frac{h}{2}}{\widetilde{\mathit{\text{z}}}\mathit{D}}_{\mathrm{b}}^{\mathrm{z}}\mathrm{d}\mathit{\text{z}},

\left[\begin{array}{ccc}{\mathcal{F}}_{11}^{\mathrm{s}}& {\mathcal{F}}_{12}^{\mathrm{s}}& 0\\ {\mathcal{F}}_{12}^{\mathrm{s}}& {\mathcal{F}}_{22}^{\mathrm{s}}& 0\\ 0& 0& {\mathcal{F}}_{66}^{\mathrm{s}}\end{array}\right]={\int }_{-\frac{h}{2}}^{\frac{h}{2}}{\widetilde{\mathit{\text{z}}}\mathit{D}}_{\mathrm{b}}^{\mathrm{f}}\mathrm{d}\mathit{\text{z}},

\left[\begin{array}{ccc}{\overline{\mathcal{B}}}_{11}^{\mathrm{s}}& {\overline{\mathcal{B}}}_{12}^{\mathrm{s}}& 0\\ {\overline{\mathcal{B}}}_{12}^{\mathrm{s}}& {\overline{\mathcal{B}}}_{22}^{\mathrm{s}}& 0\\ 0& 0& {\overline{\mathcal{B}}}_{66}^{\mathrm{s}}\end{array}\right]={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\tilde{F}{\mathit{D}}_{\mathrm{b}}^0\mathrm{d}\mathit{\text{z}},

\left[\begin{array}{ccc}{\overline{\mathcal{F}}}_{11}^{\mathrm{s}}& {\overline{\mathcal{F}}}_{12}^{\mathrm{s}}& 0\\ {\overline{\mathcal{F}}}_{12}^{\mathrm{s}}& {\overline{\mathcal{F}}}_{22}^{\mathrm{s}}& 0\\ 0& 0& {\overline{\mathcal{F}}}_{66}^{\mathrm{s}}\end{array}\right]={\int }_{-\frac{h}{2}}^{\frac{h}{2}}{\tilde{F}\mathit{D}}_{\mathrm{b}}^{\mathrm{z}}\mathrm{d}\mathit{\text{z}},

\left[\begin{array}{ccc}{\mathcal{H}}_{11}& {\mathcal{H}}_{12}& 0\\ {\mathcal{H}}_{12}& {\mathcal{H}}_{22}& 0\\ 0& 0& {\mathcal{H}}_{66}\end{array}\right]={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\tilde{F}{\mathit{D}}_{\mathrm{b}}^{\mathrm{f}}\mathrm{d}\mathit{\text{z}},

\left[\begin{array}{cc}{\mathcal{A}}_{44}^{\mathrm{s}}& 0\\ 0& {\mathcal{A}}_{55}^{\mathrm{s}}\end{array}\right]={\int }_{-\frac{h}{2}}^{\frac{h}{2}}{\tilde{\mathcal{G}}}^2{\mathit{D}}_{\mathrm{s}}^0\mathrm{d}\mathit{\text{z}},

\left[\begin{array}{ccc}{\mathcal{C}}_{11}& {\mathcal{C}}_{12}& {\mathcal{C}}_{13}\\ {\mathcal{D}}_{11}& {\mathcal{D}}_{12}& {\mathcal{D}}_{13}\end{array}\right]={\int }_{-\frac{h}{2}}^{\frac{h}{2}}{\mathit{D}}_{\mathrm{s}}^1\mathrm{d}\mathit{\text{z}},

where

\begin{array}{rl}& {\mathit{D}}_{\mathrm{b}}^0=\left[\begin{array}{ccc}{C}_{11}& C_{12}& 0\\ C_{12}& C_{22}& 0\\ 0& 0& C_{66}\end{array}\right]+\frac{e_{31}^2}{{\kappa }_{33}}\left[\begin{array}{ccc}1& 1& 0\\ 1& 1& 0\\ 0& 0& 0\end{array}\right],\\ & {\mathit{D}}_{\mathrm{s}}^0=\left[\begin{array}{cc}{C}_{44}& 0\\ 0& C_{55}\end{array}\right],\end{array}

{\mathit{D}}_{\mathrm{b}}^{\mathrm{z}}=\widetilde{\mathit{\text{z}}}\left[\begin{array}{ccc}{C}_{11}& C_{12}& 0\\ C_{12}& C_{22}& 0\\ 0& 0& C_{66}\end{array}\right]+\left(\frac{e_{31}^2\widetilde{\mathit{\text{z}}}}{{\kappa }_{33}}+\frac{e_{31}{f}_{14}}{{\kappa }_{33}}\right)\left[\begin{array}{ccc}1& 1& 0\\ 1& 1& 0\\ 0& 0& 0\end{array}\right],

\begin{array}{rl}{\mathit{D}}_{\mathrm{b}}^{\mathrm{f}}=& \tilde{F}\left[\begin{array}{ccc}{C}_{11}& C_{12}& 0\\ C_{12}& C_{22}& 0\\ 0& 0& C_{66}\end{array}\right]\\ & +\left(\frac{e_{31}^2\tilde{F}}{{\kappa }_{33}}+\frac{e_{31}{f}_{14}{\tilde{F}}_{,\mathit{\text{z}}}}{{\kappa }_{33}}\frac{e_{31}^2}{{\kappa }_{33}}\right)\left[\begin{array}{ccc}1& 1& 0\\ 1& 1& 0\\ 0& 0& 0\end{array}\right],\end{array}

{\mathit{D}}_{\mathrm{s}}^1=\left[\begin{array}{ccc}{\displaystyle \frac{e_{31}{f}_{14}}{{\kappa }_{33}}}& -\left({\displaystyle \frac{e_{31}{f}_{14}}{{\kappa }_{33}}}\widetilde{\mathit{\text{z}}}+{\displaystyle \frac{f_{14}^2}{{\kappa }_{33}}}\right)& -\left({\displaystyle \frac{e_{31}{f}_{14}}{{\kappa }_{33}}}\tilde{F}+{\displaystyle \frac{f_{14}^2}{{\kappa }_{33}}}{\tilde{F}}_{,\mathit{\text{z}}}\right)\\ {\displaystyle \frac{e_{31}{f}_{14}}{{\kappa }_{33}}}{\tilde{F}}_{,\mathit{\text{z}}}& -\left({\displaystyle \frac{e_{31}{f}_{14}}{{\kappa }_{33}}}\widetilde{\mathit{\text{z}}}+{\displaystyle \frac{f_{14}^2}{{\kappa }_{33}}}\right){\tilde{F}}_{,\mathit{\text{z}}}& -\left({\displaystyle \frac{e_{31}{f}_{14}}{{\kappa }_{33}}}\tilde{F}+{\displaystyle \frac{f_{14}^2}{{\kappa }_{33}}}{\tilde{F}}_{,\mathit{\text{z}}}\right){\tilde{F}}_{,\mathit{\text{z}}}\end{array}\right].

Appendix B

1) Stiffness matrix

\begin{array}{rl}& k_{\text{11}}={\mathcal{A}}_{11}{\theta }^2+{\mathcal{A}}_{66}{\phi }^2,\\ & k_{\text{12}}=k_{\text{21}}=\theta \phi \left({\mathcal{A}}_{12}+{\mathcal{A}}_{66}\right),\\ & k_{\text{22}}={\mathcal{A}}_{22}{\phi }^2+{\mathcal{A}}_{66}{\theta }^2,\end{array}

\begin{array}{rl}& k_{\text{13}}=-{\mathcal{B}}_{11}{\theta }^3-\left({\mathcal{B}}_{12}+2{\mathcal{B}}_{66}\right){\phi }^2\theta ,\\ & k_{\text{14}}=-{\mathcal{B}}_{11}^{\mathrm{s}}{\theta }^3-\left({\mathcal{B}}_{12}^{\mathrm{s}}+2{\mathcal{B}}_{66}^{\mathrm{s}}\right){\phi }^2\theta ,\end{array}

\begin{array}{rl}& k_{\text{23}}=-{\mathcal{B}}_{22}{\phi }^3-\phi {\theta }^2\left({\mathcal{B}}_{12}+2{\mathcal{B}}_{66}\right),\\ & k_{\text{24}}=-{\mathcal{B}}_{22}^{\mathrm{s}}{\phi }^3-\phi {\theta }^2\left({\mathcal{B}}_{12}^{\mathrm{s}}+2{\mathcal{B}}_{66}^{\mathrm{s}}\right),\end{array}

k_{\text{31}}=-\left({\overline{\mathcal{A}}}_{11}-{\mathcal{C}}_{11}\right){\theta }^3-\left({\overline{\mathcal{A}}}_{12}+2{\overline{\mathcal{A}}}_{66}-{\mathcal{C}}_{11}\right){\phi }^2\theta ,

k_{\text{32}}=-\left({\overline{\mathcal{A}}}_{22}-{\mathcal{C}}_{11}\right){\phi }^3-\left({\overline{\mathcal{A}}}_{12}+2{\overline{\mathcal{A}}}_{66}-{\mathcal{C}}_{11}\right)\phi {\theta }^2,

k_{33}=2{\mathcal{F}}_{12}{\theta }^2{\phi }^2+4{\mathcal{F}}_{66}{\theta }^2{\phi }^2+\left({\mathcal{F}}_{22}-{\mathcal{C}}_{12}\right){\phi }^4+\left({\mathcal{F}}_{11}-{\mathcal{C}}_{12}\right){\theta }^4+L,

k_{34}=2{\mathcal{F}}_{11}^{\mathrm{s}}{\theta }^2{\phi }^2+4{\mathcal{F}}_{66}^{\mathrm{s}}{\theta }^2{\phi }^2+\left({\mathcal{F}}_{22}^{\mathrm{s}}-{\mathcal{C}}_{12}\right){\phi }^4+\left({\mathcal{F}}_{11}^{\mathrm{s}}-{\mathcal{C}}_{12}\right){\theta }^4+L,

k_{\text{41}}=-\left({\overline{\mathcal{B}}}_{11}^{\mathrm{s}}-{\mathcal{D}}_{11}\right){\theta }^3-\left({\overline{\mathcal{B}}}_{12}^{\mathrm{s}}+2{\overline{\mathcal{B}}}_{66}^{\mathrm{s}}+{\mathcal{D}}_{11}\right){\phi }^2\theta ,

k_{\text{42}}=-\left({\overline{\mathcal{B}}}_{22}^{\mathrm{s}}-{\mathcal{D}}_{11}\right){\phi }^3-\phi {\theta }^2\left({\overline{\mathcal{B}}}_{12}^{\mathrm{s}}+2{\overline{\mathcal{B}}}_{66}^{\mathrm{s}}+{\mathcal{D}}_{11}\right),

k_{43}=2{\overline{\mathcal{F}}}_{11}^{\mathrm{s}}{\theta }^2{\phi }^2+4{\overline{\mathcal{F}}}_{66}^{\mathrm{s}}{\theta }^2{\phi }^2+\left({\overline{\mathcal{F}}}_{22}^{\mathrm{s}}-{\mathcal{D}}_{12}\right){\phi }^4+\left({\overline{\mathcal{F}}}_{11}^{\mathrm{s}}-{\mathcal{D}}_{12}\right){\theta }^4+L,

\begin{array}{rl}{k}_{44}=& {\mathcal{A}}_{44}^{\mathrm{s}}{\phi }^2+\left({\mathcal{H}}_{11}-{\mathcal{D}}_{13}\right){\theta }^4+\left({\mathcal{H}}_{22}-{\mathcal{D}}_{13}\right){\phi }^4\\ & +{\mathcal{A}}_{55}^{\mathrm{s}}{\theta }^2+4{\mathcal{H}}_{66}{\theta }^2{\phi }^2+2{\mathcal{H}}_{12}{\theta }^2{\mathit{\text{β}}}^{\text{2}}+L,\end{array}

\begin{array}{rl}L=& (k_{\mathrm{s}}{\theta }^4+k_{\mathrm{s}}{\phi }^4+2{\theta }^2{\phi }^2{k}_{\mathrm{s}}+{\theta }^2{k}_{\mathrm{w}}\\ & +{\phi }^2{k}_{\mathrm{w}}){\mu }^2\left(-h/2\right)+({\theta }^2+{\phi }^2{)k}_{\mathrm{s}}+k_{\mathrm{w}}.\end{array}

2) Mass matrix coefficients m_{ij}:

m_{11}=I_0+I_{0\text{μ}}\left({\theta }^2+{\phi }^2\right),m_{12}=0,m_{13}=-I_1\theta -I_{1\text{μ}}\theta \left({\theta }^2+{\phi }^2\right),

m_{14}=-J_1\theta -J_{1\text{μ}}\theta \left({\theta }^2+{\phi }^2\right),m_{22}=I_0+I_{0\text{μ}}\left({\theta }^2+{\phi }^2\right),

m_{23}=-I_1\phi -I_{1\text{μ}}\phi \left({\theta }^2+{\phi }^2\right);m_{24}=-J_1\phi -J_{1\text{μ}}\phi \left({\theta }^2+{\phi }^2\right),

m_{33}=\left[I_0+I_2\left({\theta }^2+{\phi }^2\right)\right]+\left[I_{0\text{μ}}+I_{2\text{μ}}\left({\theta }^2+{\phi }^2\right)\right]\left({\theta }^2+{\phi }^2\right),

m_{34}=\left[I_0+J_2\left({\theta }^2+{\phi }^2\right)\right]+\left[I_{0\text{μ}}+J_{2\text{μ}}\left({\theta }^2+{\phi }^2\right)\right]\left({\theta }^2+{\phi }^2\right),

m_{44}=\left[I_0+K_2\left({\theta }^2+{\phi }^2\right)\right]+\left[I_{0\text{μ}}+K_{2\text{μ}}\left({\theta }^2+{\phi }^2\right)\right]\left({\theta }^2+{\phi }^2\right).

3) Force vectors:

\tilde{Q}=Q_{\mathrm{m}\mathrm{n}}\left[1+{\mu }^2\left(h/2\right)\left({\theta }^2+{\phi }^2\right)\right].

Appendix C

{\mathit{B}}_I^1=\left[\begin{array}{cccc}{R}_{I,x}& 0& 0& 0\\ 0& R_{I,y}& 0& 0\\ R_{I,y}& R_{I,x}& 0& 0\end{array}\right],{\mathit{B}}_I^2=\left[\begin{array}{cccc}0& 0& {-R}_{I,xx}& 0\\ 0& 0& {-R}_{I,yy}& 0\\ 0& 0& {-2R}_{I,xy}& 0\end{array}\right],

{\mathit{B}}_I^3=\left[\begin{array}{cccc}0& 0& 0& {-R}_{I,xx}\\ 0& 0& 0& {-R}_{I,yy}\\ 0& 0& 0& {-2R}_{I,xy}\end{array}\right],{\mathit{B}}_I^4=\left[\begin{array}{cccc}0& 0& 0& R_{I,x}\\ 0& 0& 0& R_{I,y}\end{array}\right],

{\mathit{B}}_I^5=\left[\begin{array}{cccc}0& 0& -R_{I,xx}& 0\\ 0& 0& 0& -R_{I,xx}\end{array}\right],{\mathit{B}}_I^6=\left[\begin{array}{cccc}0& 0& -R_{I,yy}& 0\\ 0& 0& 0& -R_{I,yy}\end{array}\right],

{\mathit{B}}_I^7=\left[\begin{array}{cccc}{-R}_{I,x}& {-R}_{I,y}& 0& 0\\ 0& 0& R_{I,xx}& R_{I,xx}\\ 0& 0& R_{I,yy}& R_{I,yy}\end{array}\right].

\begin{array}{rl}& {\mathit{N}}_I^1=\left[\begin{array}{cccc}{R}_I& 0& 0& 0\\ 0& 0& R_{I,x}& 0\\ 0& 0& 0& R_{I,x}\end{array}\right],{\mathit{N}}_I^2=\left[\begin{array}{cccc}0& R_I& 0& 0\\ 0& 0& R_{I,y}& 0\\ 0& 0& 0& R_{I,y}\end{array}\right],\\ & {\mathit{N}}_I^3=\left[\begin{array}{cccc}0& 0& R_I& 0\\ 0& 0& 0& R_I\\ 0& 0& 0& 0\end{array}\right].\end{array}

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