This research develops a nonlocal couple stress theory to investigate static stability and free vibration characteristics of functionally graded (FG) nanobeams. The theory introduces two parameters based on nonlocal elasticity theory and modified couple stress theory to capture the size effects much accurately. Therefore, a nonlocal stress field parameter and a material length scale parameter are used to involve both stiffness-softening and stiffness-hardening effects on responses of FG nanobeams. The FG nanobeam is modeled via a higher order refined beam theory in which shear deformation effect is verified needless of shear correction factor. A power-law distribution is used to describe the graded material properties. The governing equations and the related boundary conditions are derived by Hamilton’s principle and they are solved applying Chebyshev–Ritz method which satisfies various boundary conditions. A comparison study is performed to verify the present formulation with the provided data in the literature and a good agreement is observed. The parametric study covered in this paper includes several parameters such as nonlocal and length scale parameters, power-law exponent, slenderness ratio, shear deformation and various boundary conditions on natural frequencies and buckling loads of FG nanobeams in detail.

Application of Chebyshev–Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures

The aim of this work is to establish a two dimensional (2D) and quasi three dimensional (quasi-3D) shear deformation theories, which can model the free vibration of FG plates resting on elastic foundations using a new shear strain shape function. The proposed theories have a novel displacement field which includes undetermined integral terms and contains fewer unknowns with taking into account the effects of both transverse shear and thickness stretching. The mechanical properties of the plates are assumed to vary through the thickness according to a power law distribution in terms of the volume fractions of the constituents. The elastic foundation parameters are introduced in the present formulation by following the Pasternak (two-parameters) mathematical model. Hamilton's principle is employed to determine the equations of motion. The closed form solutions are derived by using Navier's method and then fundamental frequencies are obtained by solving the results of eigenvalue problems. The efficiency of the proposed theory is ascertained by comparing the results of numerical examples with the different 2D, 3D and quasi-3D solutions found in literature.

New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations

Abstract A modified functionally graded beam theory based on the neutral axis is exploited to investigate natural frequencies of macro/nanobeams. The location of neutral axis is determined, where material properties of FG nanobeams are assumed to vary through the thickness according to a power law. The FG nanobeam is investigated on the basis of the nonlocal Eringen continuum model. The kinematic assumption of nanobeam is proposed according to Euler–Bernoulli beam theory. The finite element method is used to discretize the model and obtain a numerical approximation of the equation of motion. The verification of the model is obtained by comparing the current results with previously published work. Numerical results are presented to show the significance of neutral axis position, the material distribution profile, and nonlocal effect on the fundamental frequencies of nanobeams.

Determination of neutral axis position and its effect on natural frequenciesof functionally graded macro/nanobeams

The hygro-thermo-mechanical bending behavior of sigmoid functionally graded material (S-FGM) plate resting on variable two-parameter elastic foundations is discussed using a four-variable refined plate theory. The material characteristics are distributed within the thickness direction according to the two power law variation in terms of volume fractions of the constituents of the material. By employing a four variable refined plate model, both a trigonometric distribution of the transverse shear strains within the thickness and the zero traction boundary conditions on the top and bottom surfaces of the plate are respected without utilizing shear correction factors. The number of independent variables of the current formulation is four, as against five in other shear deformation models. The governing equations are deduced based on the four-variable refined plate theory incorporating the external load and hygro-thermal influences. The results of this work are compared with those of other shear deformation models. Various numerical examples introducing the influence of power-law index, plate aspect ratio, temperature difference, elastic foundation parameters, and side-to-thickness ratio on the static behavior of S-FGM plates are investigated.

Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory

In the present work, a simple first-order shear deformation theory is developed and validated for a variety of numerical examples of the thermal buckling response of functionally graded sandwich plates with various boundary conditions. Contrary to the conventional first-order shear deformation theory, the present first-order shear deformation theory involves only four unknowns and has strong similarities with the classical plate theory in many aspects such as governing equations of motion, and stress resultant expressions. Material properties and thermal expansion coefficient of the sandwich plate faces are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic material. The thermal loads are considered as uniform, linear and non-linear temperature rises within the thickness direction. The results reveal that the volume fraction index, loading type and functionally graded layers thickness have significant influence on the thermal buckling of functionally graded sandwich plates. Moreover, numerical results prove that the present simple first-order shear deformation theory can achieve the same accuracy of the existing conventional first-order shear deformation theory which has more number of unknowns.

Thermal stability of functionally graded sandwich plates using a simple shear deformation theory

This study aimed to presents a simple analytical approach to investigate the thermal buckling behavior of thick functionally graded sandwich by employing both the sinusoidal shear deformation theory and stress function. The material properties of the sandwich plate faces are continuously varied within the plate thickness according to a simple power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic material. The thermal loads are considered as uniform, linear and non-linear temperature rises across the thickness direction. Numerical examples are presented to prove the effect of power law index, loading type and functionally graded layers thickness on the thermal buckling response of thick functionally graded sandwich.

A simple analytical approach for thermal buckling of thick functionally graded sandwich plates

In the present work the dynamic analysis of the functionally graded rectangular nanoplates is studied. The theory of nonlocal elasticity based on the quasi 3D high shear deformation theory (quasi 3D HSDT) has been employed to determine the natural frequencies of the nanosize FG plate. In HSDT a cubic function is employed in terms of thickness coordinate to introduce the influence of transverse shear deformation and stretching thickness. The theory of nonlocal elasticity is utilized to examine the impact of the small scale on the natural frequency of the FG rectangular nanoplate. The equations of motion are deduced by implementing Hamilton\'s principle. To demonstrate the accuracy of the proposed method, the calculated results in specific cases are compared and examined with available results in the literature and a good agreement is observed. Finally, the influence of the various parameters such as the nonlocal coefficient, the material indexes, the aspect ratio, and the thickness to length ratio on the dynamic properties of the FG nanoplates is illustrated and discussed in detail.

Dynamic analysis of nanosize FG rectangular plates based on simple nonlocal quasi 3D HSDT

In this article, an efficient and simple refined theory is proposed for buckling analysis of functionally graded plates by using a new displacement field which includes undetermined integral variables. This theory contains only four unknowns, with is even less than the first shear deformation theory (FSDT). Governing equations are obtained from the principle of virtual works. The closed-form solutions of rectangular plates are determined. Comparison studies are carried out to check the validity of obtained results. The influences of loading conditions and variations of power of functionally graded material, modulus ratio, aspect ratio, and thickness ratio on the critical buckling load of functionally graded plates are examined and discussed.

An efficient and simple four variable refined plate theory for buckling analysis of functionally graded plates

Postbuckling of thick plates made of functionally graded material (FGM) subjected to in-plane compressive, thermal and thermomechanical loads is investigated in this work. It is assumed that the plate is in contact with a Pasternak-type elastic foundation during deformation. Thermomechanical nonhomogeneous properties are considered to be temperature independent, and graded smoothly by the distribution of power law across the thickness in the thickness in terms of the volume fractions of constituents. By employing the higher order shear deformation plate theory together the non-linear von-
Karman strain-displacement relations, the equilibrium and compatibility equations of imperfect FGM plates are derived. The Galerkin technique is used to determine the buckling loads and postbuckling equilibrium paths for simply supported plates. Numerical examples are presented to show the influences of power law index, foundation stiffness and imperfection on the buckling and postbuckling loading capacity of the plates.

Thermo-mechanical post-buckling behavior of thick functionally graded plates resting on elastic foundations

In this work, an analytical formulation based on both hyperbolic shear deformation theory and stress function, is presented to study the nonlinear post-buckling response of symmetric functionally graded plates supported by elastic foundations and subjected to in-plane compressive, thermal and thermo-mechanical loads. Elastic properties of material are based on sigmoid power law and varying across the thickness of the plate (S-FGM). In the present formulation, Von Karman nonlinearity and initial geometrical imperfection of plate are also taken into account. By utilizing Galerkin procedure, closed-form expressions of buckling loads and post-buckling equilibrium paths for simply supported plates are obtained. The effects of different parameters such as material and geometrical characteristics, temperature, boundary conditions, foundation stiffness and imperfection on the mechanical and thermal buckling and post-buckling loading capacity of the S-FGM plates are investigated.

Ahmed Bakora

Papers

A simple analytical approach for thermal buckling of thick functionally graded sandwich plates

Dynamic analysis of nanosize FG rectangular plates based on simple nonlocal quasi 3D HSDT

An efficient and simple four variable refined plate theory for buckling analysis of functionally graded plates

Thermo-mechanical post-buckling behavior of thick functionally graded plates resting on elastic foundations

Thermo-mechanical postbuckling of symmetric S-FGM plates resting on Pasternak elastic foundations using hyperbolic shear deformation theory