Vibration and damping characteristics of CNTR viscoelastic skewed shell structures under the influence of hygrothermal conditions

Abstract

The vibration and damping characteristics of carbon nanotubes reinforced (CNTR) skewed shell structure under a hygrothermal environment have been investigated using the finite element method. CNT as reinforcing phase and polymer as matrix phase are considered for the nanocomposites (NCs) based viscoelastic skewed shell structure. Dynamic mechanical analysis is used to conduct the creep test for NCs samples which were fabricated, as per ASTM-D4065 standard, and obtained the viscoelastic properties in the frequency domain under different hygrothermal conditions. The shell geometry is defined by considering an arbitrary coordinate system for the skewed shell structure. Finite element modelling has been done with Serendipity element with five degrees of freedom in all eight nodes. The present formulation is based on Koiter’s shell theory and first-order shear deformation theory is considered to incorporate the transverse shear effect based on Mindlin’s hypothesis. The frequency dependant viscoelastic properties are directly used to obtain the frequency responses of the skewed shell panel using fast Fourier transform (FFT) whereas the transient responses are determined using inverse fast Fourier transform (IFFT). An in-house MATLAB code is developed for the numerical simulation and the accuracy of the proposed formulation is validated with available results in literatures and using ANSYS software. A parametric study has been carried out for the skewing angle and CNT volume fraction on the vibration behaviour of different thin and thick NC skewed shell structures under various hygrothermal conditions.

Appendix

Shape functions (${N}_{i}$) used for the shell element are mentioned below:

$$\begin{gathered} N_{1} = \frac{1}{4}(1 - \xi )(1 - \eta )( - 1 - \xi - \eta ) \hfill \\ N_{2} = \frac{1}{2}(1 - \xi^{2} )(1 - \eta ) \hfill \\ N_{3} = \frac{1}{4}(1 + \xi )(1 - \eta )( - 1 + \xi - \eta ) \hfill \\ N_{4} = \frac{1}{2}(1 + \xi )(1 - \eta^{2} ) \hfill \\ N_{5} = \frac{1}{4}(1 + \xi )(1 + \eta )( - 1 + \xi + \eta ) \hfill \\ N_{6} = \frac{1}{2}(1 - \xi^{2} )(1 + \eta ) \hfill \\ N_{7} = \frac{1}{4}(1 - \xi )(1 + \eta )( - 1 - \xi + \eta ) \hfill \\ N_{8} = \frac{1}{2}(1 - \xi )(1 - \eta^{2} ). \hfill \\ \end{gathered}$$

(67)

Strain–displacement relation based on the Koiter shell theory

$$\left\{ \varepsilon \right\} = \sum\limits_{i = 1}^{8} {\left[ {\begin{array}{*{20}l} { \, \frac{1}{{A_{1} }}\frac{{\partial N_{i} }}{{\partial \alpha_{1} }}} \hfill & { \, \frac{{N_{i} }}{{A_{1} A_{2} }}\frac{{\partial A_{1} }}{{\partial \alpha_{2} }} \, } \hfill & {\frac{{N_{i} }}{{R_{1} }}} \hfill & { 0} \hfill & { \, 0} \hfill \\ { \, \frac{{N_{i} }}{{A_{1} A_{2} }}\frac{{\partial A_{2} }}{{\partial \alpha_{1} }}} \hfill & { \, \frac{1}{{A_{2} }}\frac{{\partial N_{i} }}{{\partial \alpha_{2} }}} \hfill & {\frac{{N_{i} }}{{R_{2} }}} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ { \, \frac{1}{{A_{2} }}\frac{{\partial N_{i} }}{{\partial \alpha_{2} }} - \frac{{N_{i} }}{{A_{1} A_{2} }}\frac{{\partial A_{1} }}{{\partial \alpha_{2} }}} \hfill & { \, \frac{1}{{A_{1} }}\frac{{\partial N_{i} }}{{\partial \alpha_{1} }} - \frac{{N_{i} }}{{A_{1} A_{2} }}\frac{{\partial A_{2} }}{{\partial \alpha_{1} }}} \hfill & {\frac{{2N_{i} }}{{R_{12} }}} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ { - \frac{1}{2}\frac{1}{{R_{12} }}\left( {\frac{1}{{A_{2} }}\frac{{\partial N_{i} }}{{\partial \alpha_{2} }} + \frac{{N_{i} }}{{A_{1} A_{2} }}\frac{{\partial A_{1} }}{{\partial \alpha_{2} }}} \right)} \hfill & { \, \frac{1}{2}\frac{1}{{R_{12} }}\left( {\frac{1}{{A_{1} }}\frac{{\partial N_{i} }}{{\partial \alpha_{1} }} + \frac{{N_{i} }}{{A_{1} A_{2} }}\frac{{\partial A_{2} }}{{\partial \alpha_{1} }}} \right)} \hfill & 0 \hfill & { \, \frac{1}{{A_{1} }}\frac{{\partial N_{i} }}{{\partial \alpha_{1} }}} \hfill & { \, \frac{{N_{i} }}{{A_{1} A_{2} }}\frac{{\partial A_{1} }}{{\partial \alpha_{2} }}} \hfill \\ { \, \frac{1}{2}\frac{1}{{R_{12} }}\left( {\frac{1}{{A_{2} }}\frac{{\partial N_{i} }}{{\partial \alpha_{2} }} + \frac{{N_{i} }}{{A_{1} A_{2} }}\frac{{\partial A_{1} }}{{\partial \alpha_{2} }}} \right)} \hfill & { - \frac{1}{2}\frac{1}{{R_{12} }}\left( {\frac{1}{{A_{1} }}\frac{{\partial N_{i} }}{{\partial \alpha_{1} }} + \frac{{N_{i} }}{{A_{1} A_{2} }}\frac{{\partial A_{2} }}{{\partial \alpha_{1} }}} \right)} \hfill & 0 \hfill & { \, \frac{{N_{i} }}{{A_{1} A_{2} }}\frac{{\partial A_{2} }}{{\partial \alpha_{1} }}} \hfill & { \, \frac{1}{{A_{2} }}\frac{{\partial N_{i} }}{{\partial \alpha_{2} }}} \hfill \\ { \, C_{0} \left( {\frac{1}{{A_{2} }}\frac{{\partial N_{i} }}{{\partial \alpha_{2} }} + \frac{{N_{i} }}{{A_{1} A_{2} }}\frac{{\partial A_{1} }}{{\partial \alpha_{2} }}} \right)} \hfill & { \, - C_{0} \left( {\frac{1}{{A_{1} }}\frac{{\partial N_{i} }}{{\partial \alpha_{1} }} + \frac{{N_{i} }}{{A_{1} A_{2} }}\frac{{\partial A_{2} }}{{\partial \alpha_{1} }}} \right)} \hfill & 0 \hfill & {\frac{1}{{A_{2} }}\frac{{\partial N_{i} }}{{\partial \alpha_{2} }} - \frac{{N_{i} }}{{A_{1} A_{2} }}\frac{{\partial A_{1} }}{{\partial \alpha_{2} }}} \hfill & {\frac{1}{{A_{1} }}\frac{{\partial N_{i} }}{{\partial \alpha_{1} }} - \frac{{N_{i} }}{{A_{1} A_{2} }}\frac{{\partial A_{2} }}{{\partial \alpha_{1} }}} \hfill \\ \end{array} } \right]} \left\{ {\begin{array}{*{20}l} {u_{0i} } \hfill \\ {v_{0i} } \hfill \\ {w_{i} } \hfill \\ {\theta_{1i} } \hfill \\ {\theta_{2i} } \hfill \\ \end{array} } \right\},$$

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where, $C_{0}=\frac{1}{2}\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$.

The $\rho$ matrix (symmetric) for the Eq. 53 is mentioned as:

$$\left[ \rho \right] = \left[ {\begin{array}{*{20}c} \rho & 0 & 0 & \rho & 0 \\ {} & \rho & 0 & 0 & \rho \\ {} & {} & \rho & 0 & 0 \\ {} & {} & {} & \rho & 0 \\ {} & {} & {} & {} & \rho \\ \end{array} } \right]$$

(69)

$$\left[N\right]= \left[ {\begin{array}{*{20}c} {N_{1} } & 0 & 0 & 0 & 0 \\ 0 & {N_{1} } & 0 & 0 & 0 \\ 0 & 0 & {N_{1} } & 0 & 0 \\ 0 & 0 & 0 & {N_{1} } & 0 \\ 0 & 0 & 0 & 0 & {N_{1} } \\ \end{array} .......\begin{array}{*{20}c} {N_{n} } & 0 & 0 & 0 & 0 \\ 0 & {N_{n} } & 0 & 0 & 0 \\ 0 & 0 & {N_{n} } & 0 & 0 \\ 0 & 0 & 0 & {N_{n} } & 0 \\ 0 & 0 & 0 & 0 & {N_{n} } \\ \end{array} } \right] .$$

(70)

The coefficient of thermal expansion and coefficient of moisture expansion terms for the Eq. 45 are mentioned as:

$$\alpha = \frac{{E_{f} \alpha_{f} v_{f} + E_{m} \alpha_{m} v_{m} }}{{E_{f} v_{f} + E_{m} v_{m} }},$$

(71)

$$\beta = \frac{{E_{m} \beta_{m} v_{m} }}{{E_{f} v_{f} + E_{m} v_{m} }}.$$

(72)