Vibration analysis of multi-scale hybrid nanocomposite shells by considering nanofillers’ aggregation

Abstract

Free oscillation analysis of shells consisted of multi-scale hybrid nanocomposites is carried out with regard to the destroying effect of the nanofillers’ agglomeration on the system’s dynamics. The equivalent material properties of the hybrid nanocomposite are obtained in the framework of a bi-level micromechanical procedure. The influence of existence of agglomerated carbon nanotubes (CNTs) on the stiffness of the nanocomposite is included with the aid of the Eshelby-Mori-Tanaka method. Thereafter, the first-order shear deformation theory (FSDT) of the shells will be combined with the dynamic form of the principle of virtual work to reach the Euler-Lagrange equations of the problem. Next, the governing equations will be extracted from the Euler-Lagrange ones by using the constitutive equations of the nanocomposite. To obtain the natural frequencies of the system for shells with clamped and simply supported ends, the Galerkin’s method is hired. The validity check reveals that the utilized methodology is powerful enough to predict the frequency behaviors of nanocomposite shells. According to the results, it can be inferred that the hybrid nanocomposite shells may be involved in resonance phenomenon in low-range frequencies if the impact of the CNTs’ aggregation is neglected. Obviously, the volume fraction of the CNTs inside the inclusions plays a magnificent role in the determination of the system’s dynamic behavior.

KEYWORDS:

• Vibration analysis
• agglomeration phenomenon
• multi-scale hybrid nanocomposites
• Galerkin's method

Disclosure statement

No potential conflict of interest was reported by the author(s).

Appendix

The components of stiffness matrix can be written as (A1) $\begin{array}{rl}{k}_{11}& =A_{11}{\int }_0^L\frac{d^3{X}_m(x)}{d{x}^3}\frac{d{X}_m(x)}{dx}dx-n^2\frac{A_{66}}{R^2}{\int }_0^L\frac{d{X}_m(x)}{dx}\frac{d{X}_m(x)}{dx}dx,\\ k_{12}& =-n\frac{A_{12}+A_{66}}{R}{\int }_0^L\frac{d^2{X}_m(x)}{d{x}^2}{X}_m(x)dx,k_{13}=-\frac{A_{12}}{R}{\int }_0^L\frac{d^2{X}_m(x)}{d{x}^2}{X}_m(x)dx,\\ k_{14}& =B_{11}{\int }_0^L\frac{d^3{X}_m(x)}{d{x}^3}\frac{d{X}_m(x)}{dx}dx-n^2\frac{B_{66}}{R^2}{\int }_0^L\frac{d{X}_m(x)}{dx}\frac{d{X}_m(x)}{dx}dx,\\ k_{15}& =-n\frac{B_{12}+B_{66}}{R}{\int }_0^L\frac{d^2{X}_m(x)}{d{x}^2}{X}_m(x)dx,\\ k_{22}& =A_{66}{\int }_0^L\frac{d^2{X}_m(x)}{d{x}^2}\frac{d{X}_m(x)}{dx}dx-\left[n^2\frac{A_{11}}{R^2}+\frac{A_{55}^s}{R^2}\right]{\int }_0^L{X}_m(x)X_m(x)dx,\\ k_{23}& =-n\frac{A_{11}+A_{55}^s}{R^2}{\int }_0^L{X}_m(x)X_m(x)dx,k_{24}=n\frac{B_{12}+B_{66}}{R}{\int }_0^L\frac{d{X}_m(x)}{dx}\frac{d{X}_m(x)}{dx}dx,\\ k_{25}& =B_{66}{\int }_0^L\frac{d^2{X}_m(x)}{d{x}^2}{X}_m(x)dx+\left[\frac{A_{55}^s}{R}-n^2\frac{B_{11}}{R^2}\right]{\int }_0^L{X}_m(x)X_m(x)dx,\\ k_{33}& =A_{55}^s{\int }_0^L\frac{d^2{X}_m(x)}{d{x}^2}{X}_m(x)dx-\left[\frac{A_{11}}{R^2}+n^2\frac{A_{55}^s}{R^2}\right]{\int }_0^L{X}_m(x)X_m(x)dx,\\ k_{34}& =\left(\frac{B_{12}}{R}-A_{55}^s\right){\int }_0^L\frac{d{X}_m(x)}{dx}\frac{d{X}_m(x)}{dx}dx,k_{35}=n\left(\frac{A_{55}^s}{R}-\frac{B_{11}}{R^2}\right){\int }_0^L{X}_m(x)X_m(x)dx,\\ k_{44}& =D_{11}{\int }_0^L\frac{d^3{X}_m(x)}{d{x}^3}\frac{d{X}_m(x)}{dx}dx-n^2\frac{D_{66}}{R^2}{\int }_0^L\frac{d{X}_m(x)}{dx}\frac{d{X}_m(x)}{dx}dx\\ & -A_{55}^s{\int }_0^L\frac{d{X}_m(x)}{dx}\frac{d{X}_m(x)}{dx}dx,\\ k_{45}& =-n\frac{D_{12}+D_{66}}{R}{\int }_0^L\frac{d^2{X}_m(x)}{d{x}^2}{X}_m(x)dx,\\ k_{55}& =D_{66}{\int }_0^L\frac{d^2{X}_m(x)}{d{x}^2}{X}_m(x)dx-\left[n^2\frac{D_{11}}{R^2}+A_{55}^s\right]{\int }_0^L{X}_m(x)X_m(x)dx\end{array}$(A1)

Also, the nonzero arrays of mass matrix are in the following form: (A2) $\begin{array}{rl}{m}_{11}& =I_0{\int }_0^L\frac{d{X}_m(x)}{dx}\frac{d{X}_m(x)}{dx}dx,m_{22}=m_{33}=m_{55}=I_0{\int }_0^L{X}_m(x)X_m(x)dx,\\ m_{25}& =I_1{\int }_0^L{X}_m(x)X_m(x)dx,m_{41}=I_1{\int }_0^L\frac{d{X}_m(x)}{dx}\frac{d{X}_m(x)}{dx}dx,\\ m_{44}& =I_2{\int }_0^L\frac{d{X}_m(x)}{dx}\frac{d{X}_m(x)}{dx}dx\end{array}$(A2)