• Advances in nano research
• /
• v.8 no.4
• /
• pp.283-292
• /
• 2020

In the present research, differential quadrature (DQ) method has been utilized for investigating free vibrations of porous functionally graded (FG) micro/nano beams in thermal environments. The exact location of neutral axis in FG material has been assumed where the material properties are described via porosity-dependent power-law functions. A scale factor related to couple stresses has been employed for describing size effect. The formulation of scale-dependent beam has been presented based upon a refined beam theory needless of shear correction factors. The governing equations and the associated boundary conditions have been established via Hamilton's rule and then they are solved implementing DQ method. Several graphs are provided which emphasis on the role of porosity dispersion type, porosity volume, temperature variation, scale factor and FG material index on free vibrational behavior of small scale beams.

• Structural Engineering and Mechanics
• /
• v.64 no.5
• /
• pp.591-610
• /
• 2017

This paper is concerned with the static analysis of variable thickness of two directional functionally graded porous materials (FGPM) circular plate resting on a gradient hybrid foundation (Horvath-Colasanti type) with friction force and subjected to compound mechanical loads (e.g., transverse, in-plane shear traction and concentrated force at the center of the plate).The governing state equations are derived in terms of displacements based on the 3D theory of elasticity, assuming the elastic coefficients of the plate material except the Poisson's ratio varying continuously throughout the thickness and radial directions according to an exponential function. These equations are solved semi-analytically by employing the state space method (SSM) and one-dimensional differential quadrature (DQ) rule to obtain the displacements and stress components of the FGPM plate. The effect of concentrated force at the center of the plate is approximated with the shear force, uniformly distributed over the inner boundary of a FGPM annular plate. In addition to verification study and convergence analysis, numerical results are displayed to show the effect of material heterogeneity indices, foundation stiffness coefficients, foundation gradient indices, loads ratio, thickness to radius ratio, compressibility, porosity and friction coefficient of the foundation on the static behavior of the plate. Finally, the responses of FG and FG porous material circular plates to compound mechanical loads are compared.

• Wind and Structures
• /
• v.28 no.1
• /
• pp.19-30
• /
• 2019

This article present the free vibration analysis of simply supported perfect and imperfect (porous) FG beams using a high order trigonometric deformation theory. It is assumed that the material properties of the porous beam vary across the thickness. Unlike other theories, the number of unknown is only three. This theory has a parabolic shear deformation distribution across the thickness. So it is useless to use the shear correction factors. The Hamilton's principle will be used herein to determine the equations of motion. Since, the beams are simply supported the Navier's procedure will be retained. To show the precision of this model, several comparisons have been made between the present results and those of existing theories in the literature.

• Steel and Composite Structures
• /
• v.45 no.3
• /
• pp.305-330
• /
• 2022

This article aims to investigate the static deflection and stress analysis of bi-directional functionally graded porous plate (BDFGPP) modeled by unified higher order kinematic theories to include the shear stress effects, which not be considered before. Different shear functions are described according to higher order models that satisfy the zero-shear influence at the top and bottom surfaces, and hence refrain from the need of shear correction factor. The material properties are graded through two spatial directions (i.e., thickness and length directions) according to the power law distribution. The porosities and voids inside the material constituent are described by different cosine functions. Hamilton's principle is implemented to derive the governing equilibrium equation of bi-directional FG porous plate structures. An efficient numerical differential integral quadrature method (DIQM) is exploited to solve the coupled variable coefficients partial differential equations of equilibrium. Problem validation and verification have been proven with previous prestigious work. Numerical results are illustrated to present the significant impacts of kinematic shear relations, gradation indices through thickness and length, porosity type, and boundary conditions on the static deflection and stress distribution of BDFGP plate. The proposed model is efficient in design and analysis of many applications used in nuclear, mechanical, aerospace, naval, dental, and medical fields.

• Steel and Composite Structures
• /
• v.49 no.4
• /
• pp.431-440
• /
• 2023

The functionally graded (FG) porous plates are usually characterized by the non-symmetric elastic modulus distribution through the thickness so that the plate neutral surface does not coincide with the mid-surface. Nevertheless, the conventional analysis models were mostly based on the plate mid-surface so that the accuracy of resulting numerical results is questionable. In this context, this paper presents the neutral surface-based static and free vibration analysis of FG porous plates and investigates the differences between the mid- and neutral surface-based analysis models. The neutral surface-based numerical method is formulated using the (3,3,2) hierarchical model and approximated by the last introduced natural element method (NEM). The volume fractions of metal and ceramic are expressed by the power-law function and the cosine-type porosity distributions are considered. The proposed numerical method is demonstrated through the benchmark experiment, and the differences between two analysis models are parametrically investigated with respect to the thickness-wise material and porosity distributions. It is found from the numerical results that the difference cannot be negligible when the material and porosity distributions are remarkably biased in the thickness direction.

• Structural Engineering and Mechanics
• /
• v.73 no.2
• /
• pp.191-207
• /
• 2020

This study aims at investigating the size-dependent free vibration of porous nanoplates when exposed to hygrothermal environment and rested on Kerr foundation. Based on the modified power-law model, material properties of porous functionally graded (FG) nanoplates are supposed to change continuously along the thickness direction. The generalized nonlocal strain gradient elasticity theory incorporating three scale factors (i.e. lower- and higher-order nonlocal parameters, strain gradient length scale parameter), is employed to expand the assumption of second shear deformation theory (SSDT) for considering the small size effect on plates. The governing equations are obtained based on Hamilton's principle and then the equations are solved using an analytical method. The elastic Kerr foundation, as a highly effected foundation type, is adopted to capture the foundation effects. Three different patterns of porosity (namely, even, uneven and logarithmic-uneven porosities) are also considered to fill some gaps of porosity impact. A comparative study is given by using various structural models to show the effect of material composition, porosity distribution, temperature and moisture differences, size dependency and elastic Kerr foundation on the size-dependent free vibration of porous nanoplates. Results show a significant change in higher-order frequencies due to small scale parameters, which could be due to the size effect mechanisms. Furthermore, Porosities inside of the material properties often present a stiffness softening effect on the vibration frequency of FG nanoplates.

• Steel and Composite Structures
• /
• v.46 no.2
• /
• pp.263-277
• /
• 2023

Free vibration analysis of multi-directional porous functionally graded (FG) sandwich plate has been performed for two cases namely: FG skin with homogeneous core and FG core with homogeneous skin. Hamilton's principle was employed and the solution was obtained using Navier's technique. This theory imposes traction-free boundary conditions on the surfaces and does not require shear correction factors. The results obtained are validated with those available in the literature. The composition of metal-ceramic-based functionally graded material (FGM) changes in longitudinal and transverse directions according to the power law. Imperfections in the functionally graded material introduced during the fabrication process were modeled with different porosity laws such as evenly, unevenly distributed, and logarithmic uneven distributions. The effect of porosity laws and geometry parameters on the natural frequency was investigated. On comparing the natural frequency of two cases for perfect and imperfect sandwich plates a reverse trend in natural frequency result was seen. The finding shows a multidirectional functionally graded structures perform better compared to uni-directional gradation. Hence, critical grading parameters and imperfection types have been identified which will guide experimentalists and researchers in selecting fabrication routes for improving the performance of such structures.

• Geomechanics and Engineering
• /
• v.35 no.4
• /
• pp.367-383
• /
• 2023

The present paper examines the natural frequency responses of the bi-directional (n_x-n_y, n_y-n_z and n_z-n_x) and multidirectional (n_x-n_y-n_z) functionally graded (FG) plate and curved structures with and without porosity. The even and uneven kind of porosity pattern are considered to observe the influence of porosity type and porosity index. The numerical findings have been obtained using a higher order shear deformation theory (HSDT) based isometric finite element (FE) approach generated in a MATLAB platform. According to the convergence and validation investigation, the proposed HSDT based FE model is adequate to predict free vibrational responses of multidirectional porous FG plates and curved structures. Further a parametric analysis is carried out by taking various design parameters into account. The free vibrational behavior of bidirectional (2D) and multidirectional (3D) perfect-imperfect FGM structure is examined against various power law index, support conditions, aspect, and thickness ratio, and for the curvature of curved structures. The results indicate that the maximum non-dimensional fundamental frequency (NFF) value is observed in perfect FGM plates and curved structures compared to porous FGM plates and curved structures and it is maximum for FGM plates and curved structures with uneven kind of porosity than even porosity.

• Steel and Composite Structures
• /
• v.49 no.3
• /
• pp.293-306
• /
• 2023

This article focuses on the study of the buckling behavior of two-dimensional functionally graded (2D-FG) nanosize tubes, including porosity, based on the first shear deformation and higher-order theory of the tube. The nano-scale tube is simulated using the nonlocal gradient strain theory, and the general equations and boundary conditions are derived using Hamilton's principle for the Zhang-Fu's tube model (as a higher-order theory) and Timoshenko beam theory. Finally, the derived equations are solved using a numerical method for both simply-supported and clamped boundary conditions. A parametric study is performed to investigate the effects of different parameters, such as axial and radial FG power indices, porosity parameter, and nonlocal gradient strain parameters, on the buckling behavior of the bi-dimensional functionally graded porous tube. Keywords: Nonlocal strain gradient theory; buckling; Zhang-Fu's tube model; Timoshenko theory; Two-dimensional functionally graded materials; Nanotubes; Higher-order theory.

• Steel and Composite Structures
• /
• v.33 no.5
• /
• pp.699-716
• /
• 2019

In this paper, wave propagations in sigmoid functionally graded (S-FG) plates are studied using new Higher Shear Deformation Theory (HSDT) based on two-dimensional (2D) elasticity theory. The current higher order theory has only four unknowns, which mean that few numbers of unknowns, compared with first shear deformations and others higher shear deformations theories and without needing shear corrector. The material properties of sigmoid functionally graded are assumed to vary through thickness according sigmoid model. The S-FG plates are supposed to be imperfect, which means that they have a porous distribution (even and uneven) through the thickness of these plates. The governing equations of S-FG plates are derived employed Hamilton's principle. Using technique of Navier, differential equations of S-FG in terms displacements are solved. Extensive results are presented to check the efficient of present methods to predict wave dispersion and velocity wave in S-FG plates.