Primary and secondary resonance analysis of FG/lipid nanoplate with considering porosity distribution based on a nonlinear elastic medium

Abstract

In this study, the nonlinear vibration analysis of the new generation nanostructures is investigated. The composite nanoplate is fabricated from the functional-graded (FG) core and two lipid layers on top and bottom of the FG core as face sheets. The nonlinear vibration analysis is studied in the presence of the external harmonic excitation force. The porosity effect on the free and force vibration analysis of the composite nanoplate is investigated. The nonlocal elasticity theory is utilized to obtain the nonlinear differential governing equation. The Kelvin–Voigt model is used to model the viscoelastic effect of the lipid layers. The Hamilton's principle is utilized to obtain the differential governing equation. The Galerkin's method is used to discrete the nonlinear partial differential governing equation to a nonlinear ordinary differential equation. The multiple scale method is used to solve the ordinary differential equation. The numerical results are compared with the reported results in the literature. A comparison between the presented numerical results and the Runge–Kutta results is done and good agreement is obtained. In the presence and absence of the porosity, the system vibration behavior is studied in the primary and secondary resonance cases. The results show that the porosity distribution types play an important role in the mechanical behavior of the composite nanoplate. Also, the numerical results show that the nonlinear frequency of the system decreases by passing time. This study can be useful to product the sensors and devices at the nanoscale with high biocompatibility.

Keywords:

• Primary and secondary resonance
• porosity effect
• lipid layer
• composite nanoplate
• viscoelastic effect
• nonlinear vibration

Nomenclature

Symbols	=	Description
${\sigma }_{xx}^c$, ${\sigma }_{yy}^c$, ${\tau }_{xy}^c$	=	The normal and shear stress components of FG core.
${\epsilon }_{xx}$, ${\epsilon }_{yy}$, ${\gamma }_{xy}$	=	Strains of the sandwich nanoplate.
$Q_{11},Q_{22},Q_{12},Q_{21},Q_{66}$	=	Stiffness of the FG core.
$E(z)$	=	Young's modulus of the FG core.
$\overline{E}$	=	Young's modulus of the lipid layer.
${\upsilon }_c$	=	Poison ratio of the FG core.
${\upsilon }_f$	=	Poison ratio of the lipid layer.
k	=	Power index of material.
$P_{\text{top}}$, $P_{\text{bottom}}$	=	The top and bottom material properties of the FG core.
${\sigma }_{xx}^f$, ${\sigma }_{yy}^f$, ${\tau }_{xy}^f$	=	The normal and shear stress components of the lipid layer.
${\overline{Q}}_{11},{\overline{Q}}_{22},{\overline{Q}}_{12},{\overline{Q}}_{21},{\overline{Q}}_{66}$	=	The stiffness of lipid face sheet of the sandwich nanoplate.
${\rho }_c(z)$	=	Density of the FG core.
${\rho }_f$	=	Density of the lipid face.
$g$	=	Damping structural of the lipid layer.
$u$, $v$, $w$	=	The displacements of the sandwich nanoplate in directions x, y and z.
Nxx, Nxy, Nyy,	=	Stress resultants.
Mxx, Mxy, Myy	=
$a_i$, $b_i(i=1,2,3)$	=	The upper and lower bound of the integrals.
$h_c$	=	Thicknesses of FG core.
$h_f$	=	Thicknesses of the lipid face sheet.
$K_e$	=	Kinetic energy.
$W_f$	=	The work done by the external forces.
$U$	=	The total strain energy.
$q_f(x,y,t)$	=	Reaction force of the foundation
$q_l(x,y,t)$	=	Harmonic point load
$\overline{f}$	=	Amplitude of the harmonic point load
$\Omega$	=	Excitation frequency
$A_{ij},{\mathrm{B}}_{ij},{\mathrm{D}}_{ij}$$(i,j=1,2,6)$	=	Stretching and bending stiffness of the sandwich nanoplate
$\Theta$	=	Airy's function
$k_{nl}$, $k_l$, $k_s$	=	Elastic constants
$\overline{u}$, $\overline{v}$, $W$	=	The non-dimensional form of the displacement in x, y and z direction.
$\overline{\Theta }$	=	The non-dimensional form of the Airy's function
$K_{\text{nl}}$, $K_l$, $K_s$	=	The non-dimensional form of the elastic constants
$\widehat{t}$	=	The non-dimensional form of the time
$A_{ij}^{\ast },{\overline{B}}_{ij},{\overline{B}}_{ij}^{*},{\overline{D}}_{ij},{\overline{D}}_{ij}^{*}$	=	The non-dimensional forms of the FG and lipid layers stiffness $(\mathrm{i},\mathrm{j}=1,2,6)$.
${\omega }_n$	=	The natural vibration frequency
${\omega }_{nl}$	=	The nonlinear natural vibration frequency
$\epsilon$	=	Perturbation parameter
$\sigma$	=	Detuning parameter

Nomenclature

Symbols	=	Description
${\sigma }_{xx}^c$, ${\sigma }_{yy}^c$, ${\tau }_{xy}^c$	=	The normal and shear stress components of FG core.
${\epsilon }_{xx}$, ${\epsilon }_{yy}$, ${\gamma }_{xy}$	=	Strains of the sandwich nanoplate.
$Q_{11},Q_{22},Q_{12},Q_{21},Q_{66}$	=	Stiffness of the FG core.
$E(z)$	=	Young's modulus of the FG core.
$\overline{E}$	=	Young's modulus of the lipid layer.
${\upsilon }_c$	=	Poison ratio of the FG core.
${\upsilon }_f$	=	Poison ratio of the lipid layer.
k	=	Power index of material.
$P_{\text{top}}$, $P_{\text{bottom}}$	=	The top and bottom material properties of the FG core.
${\sigma }_{xx}^f$, ${\sigma }_{yy}^f$, ${\tau }_{xy}^f$	=	The normal and shear stress components of the lipid layer.
${\overline{Q}}_{11},{\overline{Q}}_{22},{\overline{Q}}_{12},{\overline{Q}}_{21},{\overline{Q}}_{66}$	=	The stiffness of lipid face sheet of the sandwich nanoplate.
${\rho }_c(z)$	=	Density of the FG core.
${\rho }_f$	=	Density of the lipid face.
$g$	=	Damping structural of the lipid layer.
$u$, $v$, $w$	=	The displacements of the sandwich nanoplate in directions x, y and z.
Nxx, Nxy, Nyy,	=	Stress resultants.
Mxx, Mxy, Myy	=
$a_i$, $b_i(i=1,2,3)$	=	The upper and lower bound of the integrals.
$h_c$	=	Thicknesses of FG core.
$h_f$	=	Thicknesses of the lipid face sheet.
$K_e$	=	Kinetic energy.
$W_f$	=	The work done by the external forces.
$U$	=	The total strain energy.
$q_f(x,y,t)$	=	Reaction force of the foundation
$q_l(x,y,t)$	=	Harmonic point load
$\overline{f}$	=	Amplitude of the harmonic point load
$\Omega$	=	Excitation frequency
$A_{ij},{\mathrm{B}}_{ij},{\mathrm{D}}_{ij}$$(i,j=1,2,6)$	=	Stretching and bending stiffness of the sandwich nanoplate
$\Theta$	=	Airy's function
$k_{nl}$, $k_l$, $k_s$	=	Elastic constants
$\overline{u}$, $\overline{v}$, $W$	=	The non-dimensional form of the displacement in x, y and z direction.
$\overline{\Theta }$	=	The non-dimensional form of the Airy's function
$K_{\text{nl}}$, $K_l$, $K_s$	=	The non-dimensional form of the elastic constants
$\widehat{t}$	=	The non-dimensional form of the time
$A_{ij}^{\ast },{\overline{B}}_{ij},{\overline{B}}_{ij}^{*},{\overline{D}}_{ij},{\overline{D}}_{ij}^{*}$	=	The non-dimensional forms of the FG and lipid layers stiffness $(\mathrm{i},\mathrm{j}=1,2,6)$.
${\omega }_n$	=	The natural vibration frequency
${\omega }_{nl}$	=	The nonlinear natural vibration frequency
$\epsilon$	=	Perturbation parameter
$\sigma$	=	Detuning parameter

Disclosure statement

No potential conflict of interest was reported by the authors.