Thermal buckling and elastic vibration analysis of functionally graded beams and plates using improved third-order shear deformation theory

Functionally graded materials (FGMs) have been developed for general purpose structural components such as rocket engine components or turbine blades where the components are exposed to extreme temperatures. The earliest FGMs were introduced by Japanese scientists in the mid-1980s as ultra-high temperature-resistant materials for aerospace applications. Recently, these materials have found other uses in electrical devices, energy transformation, biomedical engineering, optics, etc. FGMs are microscopically inhomogeneous spatial composite materials, typically composed of a ceramic-metal or ceramic-polymer pair of materials. Therefore, it is important to investigate the behaviors of engineering structures such as beams and plates made from FGMs when they are subjected to thermal and dynamic loads for appropriate design. The material property profiles of FGMs vary across the graded direction. Therefore, using an improved third order shear deformation theory (TSDT) based on more rigorous kinetics of displacements to predict the behaviors of functionally graded beams and plates is expected to be more suitable than using other theories. Thus, in this research, the improved TSDT is used to investigate thermal buckling and elastic vibration response of functionally graded beams and plates. For the first time in this research temperature dependent material property solutions, are adopted to investigate thermal buckling results of functionally graded beams and plates. Additionally, the research includes natural frequency and forced vibration analysis of functionally graded plates subjected to a uniformly distributed dynamic load acting over the plate domain. To obtain the solutions, the Ritz method using polynomial and trigonometric functions for defining admissible displacements and rotations is applied to solve the governing equations. The numerical results are validated by published and experimental results. To clearly understand functionally graded materials beam specimens were manufactured from alumina-epoxy using a multi-step sequential infiltration technique. These beams were then subject to microscopic analysis to determine the profiles of the constituents. Finally experiments were conducted to determine the vibration characteristics and the results were compared to analysis using the improved TSDT. To compute theoretical parts in this research, the material compositions of the functionally graded beams and plates are assumed to vary smoothly and continuously throughout the thickness according to the power law distribution. Several significant aspects such as thickness and aspect ratios, materials, temperature, added mass etc. which affect analytical results are taken into account and discussed in detail. The original work in this thesis includes the application of the improved TSDT to thermal buckling and elastic vibration problems of functionally graded beams and plates. New critical buckling temperature results for the case of temperature dependent material properties have been solved by an iterative calculation technique. The results reveal that the effect of temperature dependent material on reduced buckling temperatures is more profound for a thicker beam and plate than a thinner one. The relationship between the critical temperatures and natural frequencies of the beam and plate structures are also presented and discussed

A study on dynamic response of functionally graded sandwich beams under different dynamic loadings

In this research, free and forced vibration of functionally graded sandwich beams is considered using Timoshenko beam theory which takes into account the significant effects of transverse shear deformation and rotary inertia. The governing equations of motion are formulated from Lagrange's equations and they are solved by using The Ritz and Newmark methods. The results are presented in both tabular and graphical forms to show the effects of layer thickness ratios, boundary conditions, length to height ratios, etc. on natural frequencies and dynamic deflections of the beams. According to the numerical results, all parametric studies considered in this research have significant impact on free and forced behaviour of the beams; for example, the frequency is low and the dynamic deflection is large for the beams which are hinged at both ends

Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories

In this paper, various higher-order shear deformation beam theories for bending and free vibration of functionally graded beams are developed. The developed theories account for higher-order variation of transverse shear strain through the depth of the beam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the beam. A shear correction factor, therefore, is not required. In addition, these theories have strong similarities with Euler–Bernoulli beam theory in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the functionally graded beam are assumed to vary according to power law distribution of the volume fraction of the constituents. Equations of motion and boundary conditions are derived from Hamilton's principle. Analytical solutions are presented, and the obtained results are compared with the existing solutions to verify the validity of the developed theories. Finally, the influences of power law index and shear deformation on the bending and free vibration responses of functionally graded beams are investigated

Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory

The first-order shear deformation beam theory for static and free vibration of axially loaded rectangular functionally graded beams is developed. In this theory, the improved transverse shear stiffness is derived from the in-plane stress and equilibrium equation and thus, associated shear correction factor is then obtained analytically. Equations of motion are derived from the Hamilton’s principle. Analytical solutions are presented for simply-supported functionally graded beams. The obtained results are compared with the existing solutions to verify the validity of the developed theory. Effects of the power-law index, material contrast and Poisson’s ratio on the displacements, natural frequencies, buckling loads and load–frequency curves as well as corresponding mode shapes are investigated

A finite element model for the thermo-elastic analysis of functionally graded porous nanobeams

In this study, for the first time, a nonlocal finite element model is proposed to analyse thermo-elastic behaviour of imperfect functionally graded porous nanobeams (P-FG) on the basis of nonlocal elasticity theory and employing a double-parameter elastic foundation. Temperature-dependent material properties are considered for the P-FG nanobeam, which are assumed to change continuously through the thickness based on the power-law form. The size effects are incorporated in the framework of the nonlocal elasticity theory of Eringen. The equations of motion are achieved based on first-order shear deformation beam theory through Hamilton's principle. Based on the obtained numerical results, it is observed that the proposed beam element can provide accurate buckling and frequency results for the P-FG nanobeams as compared with some benchmark results in the literature. The detailed variational and finite element procedure are presented and numerical examinations are performed. A parametric study is performed to investigate the influence of several parameters such as porosity volume fraction, porosity distribution, thermal loading, material graduation, nonlocal parameter, slenderness ratio and elastic foundation stiffness on the critical buckling temperature and the nondimensional fundamental frequencies of the P-FG nanobeams. Based on the results of this study, a porous FG nanobeam has a higher thermal buckling resistance and natural frequency compared to a perfect FG nanobeam. Also, uniform distributions of porosity result in greater critical buckling temperatures and vibration frequencies, in comparison with functional distributions of porosities

Vibration analysis of viscoelastic single-walled carbon nanotubes resting on a viscoelastic foundation

Vibration responses were investigated for a viscoelastic Single-walled carbon nanotube (visco-SWCNT) resting on a viscoelastic foundation. Based on the nonlocal Euler-Bernoulli beam model, velocity-dependent external damping and Kelvin viscoelastic foundation model, the governing equations were derived. The Transfer function method (TFM) was then used to compute the natural frequencies for general boundary conditions and foundations. In particular, the exact analytical expressions of both complex natural frequencies and critical viscoelastic parameters were obtained for the Kelvin-Voigt visco-SWCNTs with full foundations and certain boundary conditions, and several physically intuitive special cases were discussed. Substantial nonlocal effects, the influence of geometric and physical parameters of the SWCNT and the viscoelastic foundation were observed for the natural frequencies of the supported SWCNTs. The study demonstrates the efficiency and robustness of the developed model for the vibration of the visco-SWCNT-viscoelastic foundation coupling system

Tensile behaviours of single-walled carbon nanotubes

This research is carried out to investigate linear and non-linear mechanical properties of the single-walled carbon nanotube. The approach of the analysis is carbon nanotube treated as a space-frame structure where the carbon nanotube atoms are treated as nodes and the covalent bond between atoms are treated as beam elements. The 3 types of carbon nanotubes i. e. the armchair, chiral and zigzag single-walled carbon nanotubes have been modelled using the Nanotube Modeler software. These geometric models will then be exported to the ANSYS Parametric Design Language software where the linear and non-linear analyses on the carbon nanotube structure have been conducted. The linear analysis is to determine the Young's modulus of the carbon nanotube structure. The non-linear analysis is to determine the ultimate tensile strength of the carbon nanotube structure. The material property (stress-strain data) of the individual beam is estimated using the Morse potential equation between carbon atoms. This material property is applied to the ANSYS's software where the carbon nanotube structure is assumed to behave in isotropic hardening manner. The Young's modulus and the tensile strength of the carbon nanotube structure are found to be close to past value taken from literature