Haar Wavelet Method for Nonlinear Vibration of Functionally Graded CNT-Reinforced Composite Beams Resting on Nonlinear Elastic Foundations in Thermal Environment

Jianyu Fan,Jin Huang

doi:10.1155/2018/9597541

Abstract

This paper presents a simple and effective approach based on the Haar wavelet discretization method (HWDM) for the nonlinear vibration analysis of carbon nanotube-reinforced composite (CNTRC) beams resting on a nonlinear elastic foundation in a thermal environment. Material properties are assumed to be functionally graded (FG) in the thickness direction and temperature-dependent and are evaluated through the extended rule of mixture. Based on the first-order shear deformation beam theory in conjunction with the von Kármán nonlinearity, the nonlinear governing equations of CNTRC beams on nonlinear elastic foundations are derived as well as the related boundary conditions. Moreover, the initial thermal stress due to uniform temperature rise is considered in this study. To evaluate the nonlinear natural frequencies, the obtained equations are discretized into a set of nonlinear algebraic equations through HWDM and then the direct iteration technique is employed to solve the resulting algebraic equations. The convergence and comparison studies are carried out, and the results indicate that the numerical rate of convergence of the proposed method is in agreement with the convergence theorem, and good accuracy of the present results is observed. Studies on the effects of different parameters such as CNT volume fraction, distribution type of CNTs, foundation stiffness coefficients, boundary condition, slenderness ratio, temperature rise, and initial thermal stress on the linear frequencies and the nonlinear frequency ratios are also reported. This study offers an insight into the vibration behaviors of CNTRC beams resting on nonlinear elastic foundation subjected to the uniform temperature rise.

Highlights

Owing to their extraordinary mechanical, thermal, and electrical properties, carbon nanotubes (CNTs) have been considered to be an ideal reinforcement material for high performance composites
En, parametric studies are conducted to show the influences of the CNT volume fraction, CNT distribution type, slenderness ratio, boundary condition, and elastic foundation coefficients on the nonlinear vibration behavior of the functionally graded (FG)-carbon nanotube-reinforced composite (CNTRC) beams resting on a nonlinear elastic foundation
Otherwise specified, the CNTRC beam considered here is made of the PMMA matrix and (10, 10) armchair SWCNT reinforcements. e material properties of the PMMA matrix are assumed to be temperaturedependent and are given by

Summary

Introduction

Owing to their extraordinary mechanical, thermal, and electrical properties, carbon nanotubes (CNTs) have been considered to be an ideal reinforcement material for high performance composites. Structural applications of carbon nanotubereinforced composites (CNTRCs) may include aviation, automotive, military, and space-related parts, in which the material with high strength, stiffness, and light weight is required. Ey found that the strength and stiffness of the resulting CNT/epoxy composite can be remarkably enhanced. Omidi et al [3] showed that only a 3 wt.% addition of multiwalled CNTs increases Young’s modulus and the tensile strength of epoxy-based composites up to 43.1% and 55.2%, respectively. E above researches have proved that the addition of CNTs into the matrix leads to significant improvements in the strength and stiffness of the composite Omidi et al [3] showed that only a 3 wt.% addition of multiwalled CNTs increases Young’s modulus and the tensile strength of epoxy-based composites up to 43.1% and 55.2%, respectively. e above researches have proved that the addition of CNTs into the matrix leads to significant improvements in the strength and stiffness of the composite

Methods

Results

Conclusion