Static and dynamic analyses of two-phase/multi-phase carbon nanotube-reinforced functionally graded composite beams via warping-included mixed finite element method

Abstract

This study enhances the application of cross-sectional warping considered mixed finite element (W-MFE) formulation to accurately determine natural vibration, static displacement response, and shear and normal stress evaluation with very close to the precision of solid finite elements (FEs) in two-phase/multi-phase functionally graded (FG) laminated composite beams strength using carbon nanotubes (CNTs). The principles of three dimensional (3D) elasticity theory are used to derive constitutive equations. The mixed finite element (MFE) method is improved by accounting for warping effects by displacement-based FEs within the cross-sectional domain. The MFE with two nodes has a total of 24 degrees of freedom. The two-phase material consists of a polymer matrix reinforced with aligned CNTs that are FG throughout the beam thickness. The multi-phase FG beam is modeled as a three-component composite material, consisting of CNTs, a polymer matrix, and fibers. The polymer matrix is reinforced by longitudinally aligned fibers and randomly dispersed CNT particles. The fiber volume fractions are considered to change gradually through the thickness of the beam following a power-law variation. The W-MFE achieves satisfactory results with fewer degrees of freedom than 3D solid FEs. Benchmark examples examine the effects of ply orientation, configuration, and fiber gradation on FG beam behavior.

Graphical abstract

Keywords

stress analysis / natural vibration analysis / CNT-reinforced composites / mixed finite element method / functionally graded material

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