A Quasi-3D Higher-Order Theory for Bending of FG Nanoplates Embedded in an Elastic Medium in a Thermal Environment

This paper presents the effects of temperature and the nonlocal coefficient on the bending response of functionally graded (FG) nanoplates embedded in an elastic foundation in a thermal environment. The effects of transverse normal strain, as well as transverse shear strains, are considered where the variation of the material properties of the FG nanoplate are considered only in thickness direction. Unlike other shear and deformations theories in which the number of unknown functions is five and more, the present work uses shear and deformations theory with only four unknown functions. The four-unknown normal and shear deformations model, associated with Eringen nonlocal elasticity theory, is used to derive the equations of equilibrium utilizing the principle of virtual displacements. The effects due to nonlocal coefficient, side-to-thickness ratio, aspect ratio, normal and shear deformations, thermal load and elastic foundation parameters, as well as the gradation in FG nanoplate bending, are investigated. In addition, for validation, the results obtained from the present work are compared to ones available in the literature.

Vibrations of an Incompressible Linearly Elastic Plate Using Discontinuous Finite Element Basis Functions for Pressure

We numerically analyze, with the finite element method, free vibrations of incompressible rectangular plates under different boundary conditions with a third-order shear and normal deformable theory (TSNDT) derived by Batra. The displacements are taken as unknowns at the nodes of a 9-node quadrilateral element and the hydrostatic pressure at four interior nodes. The plate theory satisfies the incompressibility condition, and the basis functions satisfy the Babuska-Brezzi condition. Because of the singular mass matrix, Moler's QZ algorithm (also known as the generalized Schur decomposition) is used to solve the resulting eigenvalue problem. Computed results for simply supported, clamped, and clamped-free rectangular isotropic plates agree well with the corresponding analytical frequencies of simply supported plates and with those found using the commercial software, abaqus, for other edge conditions. In-plane modes of vibrations are clearly discerned from mode shapes of square plates of aspect ratio 1/8 for all three boundary conditions. The magnitude of the transverse normal strain at a point is found to equal the sum of the two axial strains implying that higher-order plate theories that assume null transverse normal strain will very likely not provide good solutions for plates made of rubberlike materials that are generally taken to be incompressible. We have also compared the presently computed through-the-thickness distributions of stresses and the hydrostatic pressure with those found using abaqus.

A refined sinusoidal model for functionally graded plates subjected to thermomechanical loading

A refined sinusoidal model considering transverse normal strain has been developed for thermoelastic analysis of functionally graded material plate. Although transverse normal strain has been considered, the additional displacement parameters are not increased as transverse normal strain only includes the thermal expansion coefficient and thermal loading. Moreover, the merit of the previous sinusoidal model satisfying tangential stress-free boundary conditions on the surfaces can be maintained. It is important that the effects of transverse normal thermal deformation are incorporated in the in-plane displacement field, which can actively influence the accuracy of in-plane stresses. To assess the performance of the proposed model, the thermoelastic behaviors of functionally graded material plates with various configurations have been analyzed. Without increase of displacement variables, accuracy of the proposed model can be significantly improved by comparing to the previous sinusoidal model. Agreement between the present results and quasi-dimensional solutions are very good, and the proposed model only includes the five displacement variables which can illustrate the accuracy and effectiveness of the present model. In addition, new results using several models considered in this paper have been presented, which can serve as a reference for future investigations.

Effects of transverse normal strain on bending of laminated composite beams

Effect of transverse normal strain on bending of laminated composite beams is proposed in this paper. A Quasi-3D beam theory which accounts for a higher-order variation of both axial and transverse displacements is used to consider the effects of both transverse shear and normal strains on bending behaviours of laminated composite beams. Ritz method is used to solve characteristic equations in which trigonometric shape functions are proposed. Numerical results for different boundary conditions are presented to compare with those from earlier works, and to investigate the effects of thickness stretching, fibre angles, span-to-height ratio and material anisotropy on the displacement and stresses of laminated composite beams.

BENDING OF FGM PLATES BY A SIMPLIFIED FOUR-UNKNOWN SHEAR AND NORMAL DEFORMATIONS THEORY

The bending response of FGM plates is presented based upon a simplified shear and normal deformations theory. The present simplified theory is accounted for an adequate distribution of transverse shear strains through the plate thickness and tangential stress-free on the plate surfaces. The effect of transverse normal strain is also included. The number of unknown functions involved here is only four as against six in case of other shear and normal deformations theories. The principle of virtual work is employed to derive the governing equations. A comparison with the corresponding results is made to check the accuracy and efficiency of the present theory. Additional results for all stresses are investigated through-the-thickness of the FGM plate.

STRESS ANALYSIS OF THICK LAMINATED PLATES USING TRIGONOMETRIC SHEAR DEFORMATION THEORY

A trigonometric shear deformation theory (TSDT) taking into account transverse shear deformation effect as well as transverse normal strain effect is presented. The inplane displacement field uses sinusoidal function in terms of thickness coordinate to include the shear deformation effect. The cosine function in thickness coordinates is used in transverse displacement to include the effect of transverse normal strain. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. The results of displacements and stresses for static flexure of simply supported symmetric and anti-symmetric cross-ply laminated square plates subjected to parabolic load and line load are obtained. The results obtained by present theory are compared with those of classical, first-order and higher-order plate theories.