Perturbation Method for Thermal Post-buckling Analysis of Shear Deformable FG-CNTRC Beams with Different Boundary Conditions

This research aims to analyze the thermal buckling and post-buckling of carbon nanotube (CNT) reinforced composite beams. It is assumed that the beam is rested on a nonlinear elastic foundation which contains the Winkler spring, shear layer, and nonlinear spring. Distribution of CNTs across the thickness may be non-uniform which results in a functionally graded media. The elastic properties of the beam are evaluated using the refined rule of mixtures which contains efficiency parameters. Temperature dependency of the constituents is also taken into account. Using three different beam models, namely, first-order, third-order, and sinusoidal theories, the governing equations for the composite beam are established. Three different types of edge supports are considered which are pinned–pinned, clamped–clamped, and clamped–roller. With the aid of the two-step perturbation technique, closed-form expressions are extracted to obtain the elevated temperature as a function of the post-buckling deflection in the beam. Results of this study are compared with the available data in the literature. After that, new results are given to discuss the effects of important factors such as foundation parameters, geometrical characteristics, boundary conditions, the CNT volume fraction, and CNT pattern. It is shown that the critical buckling temperature of pinned–pinned and clamped–roller beams is the same while their post-buckling responses are totally different.

Heat conduction in one-dimensional functionally graded media based on the dual-phase-lag theory

In this article, heat conduction in one-dimensional functionally graded media is investigated based on the dual-phase-lag theory to consider the microstructural interactions in the fast transient process of heat conduction. All material properties of the media are assumed to vary continuously according to a power-law formulation with arbitrary non-homogeneity indices except the phase lags which are taken constant for simplicity. The one-dimensional heat conduction equations based on the dual-phase-lag theory are derived in a unified form which can be used for Cartesian, cylindrical, and spherical coordinates. A semi-analytical solution for temperature and heat flux is presented using the Laplace transform to eliminate the time dependency of the problem. The results in the time domain are then given by employing a numerical Laplace inversion technique. The semi-analytical solution procedure leads to exact expressions for the thermal wave speed in one-dimensional functionally graded media with different geometries based on the dual-phase-lag and hyperbolic heat conduction theories. The transient temperature distributions have been found for various types of dynamic thermal loading. The numerical results are shown to reveal the effects of phase lags, non-homogeneity indices, and thermal boundary conditions on the thermal responses for different temporal disturbances. The results are verified with those reported in the literature for hyperbolic heat conduction in cylindrical and spherical coordinates.

FUNCTIONALLY GRADED MEDIA

The notions of uniformity and homogeneity of elastic materials are reviewed in terms of Lie groupoids and frame bundles. This framework is also extended to consider the case of Functionally Graded Media, which allows us to obtain some homogeneity conditions.