Functionally Graded Porous Research Articles

Porosity and inhomogeneity effects on the buckling and vibration of double-FGM nanoplates via a quasi-3D refined theory

This study is devoted to illustrate the mechanical buckling and free vibration analyses of double-porous functionally graded (FG) nanoplates embedded in an elastic foundation. A new quasi-3D refined plate theory is presented to model the displacement field. This theory contains only five unknown functions and considers the shear strain as well as thickness stretching. Based on the modified Mooney-type exponential relation, a new exponential law is presented to govern the materials variation and porosities distribution through the thickness of the nanoplates. The two porous nanoplates are bonded together by a set of parallel elastic springs and surrounded by Pasternak medium. The nonlocal strain gradient theory containing the nonlocal parameter and gradient coefficient is utilized to study the size-dependent effects. Based on Hamilton’s principle, the equations of motion are drawn including the material parameters, elastic foundation reaction and biaxial compressive forces. An analytical approach for simply-supported and clamped bilayer porous FG nanoplates is implemented. The obtained results are compared with those available in the literature. Additional numerical calculations are introduced to show the influences of the material length scale parameters, inhomogeneity parameter, porosity factor and other parameters on the critical buckling and frequencies of the double-porous FG nanoplates.

Wave dispersion characteristics of heterogeneous nanoscale beams via a novel porosity-based homogenization scheme

The important effect of porosity on the mechanical behaviors of a continuum, makes it necessary to be accounted for while analyzing the structure. Motivated by this fact, a new porosity-dependent homogenization scheme is presented in this article to investigate the wave propagation responses of functionally graded (FG) nanobeams by considering the coupling effects between density and Young’s moduli in porous materials. In the introduced homogenization method, which is a modified form of the power-law model, dependency of effective Young’s modulus to the mass density is covered. Based on the Hamilton principle, the Navier equations are developed using the Euler-Bernoulli beam model. Thereafter, the constitutive equations are obtained employing the nonlocal elasticity theory of Eringen. Next, the governing equations are solved in order to reach the wave frequency. Once the validity of presented methodology is proved, a set of parametric studies are adopted to emphasize the role of each variant on the wave dispersion behaviors of porous FG nanobeams.

Finite element study of functionally graded porous femoral stems incorporating body-centered cubic structure.

The mismatch between stiffness of the femoral dense stem and host bone causes complications to patients, such as aseptic loosening and bone resorption. Three-dimensional finite-element models of homogeneous porous (HGP) and functionally graded porous (FGP) stems incorporating body-centered cubic (BCC) structures are proposed in this article as an alternative to the dense stems. The relationship between the porosity and strut thickness of the BCC structure was developed to construct the finite-element models. Three levels of porosities (20%, 50%, and 80%) were modeled in HGP and FGP stems. The porosity of the stems was decreased distally according to the sigmoid function (n=0.1, n=1 and n=10) with 3 grading exponents. The results showed that FGP stems transferred 120%-170% higher stresses to the femur (Gruen zone 7) as compared to the solid stem. Conversely, the stresses in HGP and FGP stems were 12%-34% lower than the dense stem. The highest micromotions (105-147µm) were observed for stems of 80% overall porosity, and the lowest (42-46µm) was for stems of 20% overall porosity. Finally, FGP stems with a grading exponent of n=10 resulted in an 11%-28% reduction in micromotions.

An isogeometric Bézier finite element analysis for piezoelectric FG porous plates reinforced by graphene platelets

In this study, we for the first time present an isogeometric Bézier finite element formulation for bending and transient analysis of functionally graded porous (FGP) plates reinforced by graphene platelets (GPLs) embedded in piezoelectric layers. We name it as PFGP-GPLs for short. The plates are constituted by a core layer, which contains the internal pores and GPLs dispersed in the metal matrix either uniformly or non-uniformly according to three different patterns, and two piezoelectric layers perfectly bonded on the top and bottom surfaces of host plate. The modified Halpin–Tsai micromechanical model is used to estimate the effective mechanical properties which vary continuously along thickness direction of the core layer. In addition, the electric potential is assumed to vary linearly through the thickness for each piezoelectric sublayer. A generalized C0-type higher-order shear deformation theory (C0-HSDT) in association with isogeometric analysis (IGA) based on Bézier extraction is investigated. Our approach allows performing all computations the same as in the conventional finite element method (FEM) yet the present formulation shows more advantages. The system of time-dependent equations is solved by the Newmark time integration scheme. The effects of weight fractions and dispersion patterns of GPLs, the coefficient and distribution types of porosity as well as external electrical voltages on structure’s behaviors are investigated through several numerical examples. These results, which have not been published before, can be considered as reference solutions for future works.

High-order shear theory for static analysis of functionally graded plates with porosities

The bending responses of porous functionally graded (FG) thick rectangular plates are investigated according to a high-order shear deformation theory. Both the effect of shear strain and normal deformation are included in the present theory and so it does not need any shear correction factor. The equilibrium equations according to the porous FG plates are derived. The solution to the problem is derived by using Navier's technique. Numerical results have been reported and compared with those available in the open literature for non-porous plates. The effects of the exponent graded and porosity factors are investigated.

Nonlinear behaviour and stability of functionally graded porous arches with graphene platelets reinforcements

This research presents an analytical approach for nonlinear static responses and stability analysis of functionally graded porous (FGP) arches with graphene platelets (GPLs) reinforcements (i.e., FGP-GPLRC arches). The constitutive material composition of the FGP-GPLRC arch varies along the radial direction of the cross section specifically, so that the mechanical performance of the arch such as buckling strength and weight can be well controlled for various engineering design purposes. The effective Young's modulus of the FGP-GPLRC arch is determined by the volume fraction distribution of materials. Based on the Euler-Bernoulli hypothesis, the structural responses of the arch considering the geometric nonlinearity are derived by using the virtual work method. Two boundary conditions are considered which are including the pinned-pinned and the fixed-fixed supports. The loading condition is defined as uniformly distributed load in the radial direction of the arch. Different buckling modes are discussed by the illustration of the equilibrium paths. By adopting the developed analytical solution, the relationship between the structural response, buckling load, self-weight, porosity level and the percentage of content of the GPLs can be investigated efficiently. The applicability and effectiveness of the proposed analytical approach for the geometric nonlinear analysis of FGP-GPLRC arch structures are demonstrated through numerical examples.

Analyzing post-buckling behavior of continuously graded FG nanobeams with geometrical imperfections

This research is concerned with post-buckling investigation of nano-scaled beams constructed from porous functionally graded (FG) materials taking into account geometrical imperfection shape. Hence, two types of nanobeams which are perfect and imperfect have been studied. Porous FG materials are classified based on even or uneven porosity distributions. A higher order nonlinear refined beam theory is used in the present research. Both perfect and imperfect nanobeams are formulated based on this refined theory. A detailed study is provided to understand the effects of geometric imperfection, pore distribution, material distribution and small scale effects on buckling of FG nanobeams.

Nonlinear bending analysis of porous FG thick annular/circular nanoplate based on modified couple stress and two-variable shear deformation theory using GDQM

This is the first attempt to consider the nonlinear bending analysis of porous functionally graded (FG) thick annular and circular nanoplates resting on Kerr foundation. The size effects are captured based on modified couple stress theory (MCST). The material properties of the porous FG nanostructure are assumed to vary smoothly through the thickness according to a power law distribution of the volume fraction of the constituent materials. The elastic medium is modeled by Kerr elastic foundation which consists of two spring layers and one shear layer. The governing equations are extracted based on Hamilton's principle and two variables refined plate theory. Utilizing generalized differential quadrature method (GDQM), the nonlinear static behavior of the nanostructure is obtained under different boundary conditions. The effects of various parameters such as material length scale parameter, boundary conditions, and geometrical parameters of the nanoplate, elastic medium constants, porosity and FG index are shown on the nonlinear deflection of the annular and circular nanoplates. The results indicate that with increasing the material length scale parameter, the nonlinear deflection is decreased. In addition, the dimensionless nonlinear deflection of the porous annular nanoplate is diminished with the increase of porosity parameter. It is hoped that the present work may provide a benchmark in the study of nonlinear static behavior of porous nanoplates.

Free vibrations of functionally graded porous rectangular plate with uniform elastic boundary conditions

In this paper, the free vibrations of functionally graded porous (FGP) rectangular plate with uniform elastic boundary conditions is investigated by means of an improved Fourier series method (IFSM). It is assumed that the distributions of porosity are uniform or non-uniformly along a certain direction and three types of the porosity distribution are considered, among which material property of two non-uniform porous distributions was expressed as the simple cosine. The size of the pore in a rectangular plate is determined by the porosity coefficients. Using the first-order shear deformation theory(FSDT), the energy expression of FGP rectangular plate is created. In order to obtain the admissible function of displacement for functionally graded porous rectangular plate, the IFSM is employed. Then, the Rayleigh-Ritz method is used to solve coefficients in the Fourier series which determine natural frequencies and modal shapes. Convergence and comparative research are performed to prove the convergence, reliability and accuracy of the current method. On this foundation, some new results covering the influence of the geometrical parameters subject to classical and elastic boundary condition are presented, and the parametric studies are also investigated in detail, which can provide a reference for future research by other researchers.

Bending and free vibration analysis of Levy-type porous functionally graded plate using state space approach

This paper aims to investigate the free vibration and bending characteristics of porous functionally graded (FG) plate with two opposite simply supported edges based on a four variable plate theory. Material properties of porous FG plate are defined by rule of the mixture with an additional term of porosity in the through-thickness direction. Equations of motion are derived using Hamilton principle. State-space approach is used to obtain the governing equation of porous FG plate. The effect of porosity parameter, slenderness ratio, and the power-law index is investigated on bending and vibration of porous FG plate. The results are compared with available results in the literature to verify this theory.

A modified series solution for free vibration analyses of moderately thick functionally graded porous (FGP) deep curved and straight beams

As a novel class of weight-efficient engineering materials, the functionally graded porous (FGP) beam structures have great potential value. However, the current research on it is relatively small. Based on this research status, the aim of this paper is establishing a unified analytical model to study the vibration behavior of moderately thick functionally graded porous deep curved and straight beam with general boundary conditions. The first-order beam theory which considering the influence of shear deformation, inertia rotary and deepness term are adopted in the formulation. The theoretical solution model is obtained by means of modified series solution which core soul is using the modified Fourier series including a standard cosine Fourier series with two auxiliary terms to expand the admissible function. This fact gives the opportunity to derive the exact solution for FGP beam with general boundary conditions by utilizing a reasonable spring stiffness value at both ends. A series of numerical examples show that the current model has superior convergence characteristics, computational accuracy and stability. On this basis, a series of innovative results are also highlighted in the text, which may be providing basic data for other algorithm research in the future.

Vibration analysis of functionally graded porous cylindrical shell with arbitrary boundary restraints by using a semi analytical method

The main purpose of this paper is to provide a new semi analytical method to analyze the free vibration of functionally graded porous (FGP) cylindrical shell with arbitrary boundary restraints. According to the distributions of porous along thickness direction of the structure, two typical types of symmetric and non-symmetric porosity distributions are performed in this paper. The formulations are established on the basis of energy method and first-order shear deformation theory (FSDT). The displacement functions are expressed by unified Jacobi polynomials and Fourier series. The arbitrary boundary restraints are realized by penalty method. The final solutions of FGP cylindrical shell structure are obtained by Rayleigh–Ritz method. To sufficient illustrate the effectiveness of proposed method, some numerical examples about spring stiffness, Jacobi parameters etc. are carried out. In addition, to verify the accuracy of this method, the results are compared with those obtained by FEM, experiment and published literature. The results show that the proposed method has ability to solve the free vibration behaviors of FGP cylindrical shell.

Porosity-dependent nonlinear transient responses of functionally graded nanoplates using isogeometric analysis

The study using numerical methods on porous functionally graded (FG) nanoplates is still somewhat limited. This paper focuses on porosity-dependent nonlinear transient analysis of FG nanoplates using isogeometric finite element approach. In order to capture the small size effects, the Eringen's nonlocal elasticity based on higher order shear deformation theory (HSDT) are used to model the porous FG nanoplates. Two distributions of porosities inside FG materials are incorporated and defined via a modified rule of mixture. The nonlinear transient nonlocal governing equations under transverse dynamic loads are formulated by using the von Kármán strains and are solved by Newmark time integration scheme to obtain geometrically nonlinear responses. It is indicated that nonlinear transient deflections of the porous FG nanoplate are significantly influenced by material composition, porosity, nonlocal parameters, volume fraction exponent, porosity distributions, geometrical parameters and dynamic load characteristics.

Transient response of porous FG nanoplates subjected to various pulse loads based on nonlocal stress-strain gradient theory

Based on nonlocal strain gradient theory (NSGT), transient behavior of a porous functionally graded (FG) nanoplate due to various impulse loads has been studied. The porous nanoplate has evenly and unevenly distributed pores inside its material structure. Impulse point loads are considered to be rectangular, triangular and sinusoidal types. These impulse loads lead to transient vibration of the nanoplate which is not studied before. NSGT introduces a nonlocal coefficient together with a strain gradient coefficient to characterize small size influences due to non-uniform stress and strain fields. Galerkin's approach has been performed to solve the governing equations and also inverse Laplace transform method is used to obtain transient response due to impulse loads. It is explained in this research that the transient response of a nanoplate is dependent on nonlocal coefficient, strain gradient parameter, pore dispersion, pore amount, type of impulse load and loading time.

A general vibration analysis of functionally graded porous structure elements of revolution with general elastic restraints

In this paper, a general vibration analysis of functionally graded porous (FGP) structure elements of revolution with general elastic restraints is performed for the first time. Under the basic framework of the first order shear theory, the moderately thick and thin FGP structure elements of revolution are considered. The specific FGP structure elements include the cylindrical shell, conical shell, spherical shell, cylindrical panel, conical panel and spherical panel. In order to simulate general elastic restraints, the artificial spring boundary technique is adopted this paper. And the modified series solution composed of a standard cosine Fourier series and two auxiliary terms is introduced in the Rayleigh-Ritz method to express the admissible function. A series of numerical examples are conducted, which show that the current model has superior convergence characteristics, computational accuracy and stability. On this basis, a series of innovative results are also given in the paper, which may provide basic data for other algorithm research in the future.

Analysis of Bending of Ceramic-Metal Functionally Graded Plates with Porosities Using of High Order Shear Theory

This work consists of the analysis of the bending responses of porous Ceramic-Metal functionally graded (FG) rectangular plates are investigated according to high order shear deformation theory. The proposed theory contains four unknowns unlike the other theories which contains five unknowns, but it checks the boundary conditions without constraints on the upper and lower plate surfaces. Both the effect of shear strain and normal deformation are included in the present theory and so it does not need any shear correction factor. The equilibrium equations according to the porous FG plates Ceramic-Metal are derived. The solution of the problem is derived by using Navier’s technique. Numerical results have been reported, and compared with those available in the open literature for non-porous plates. Effects of the exponent graded and porosity factors are investigated.

Three-dimensional exact solution for vibration analysis of thick functionally graded porous (FGP) rectangular plates with arbitrary boundary conditions

In this paper, a novel three-dimensional exact solution is performed for vibration analysis of thick functionally graded porous rectangular plates subject to arbitrary boundary conditions. The theoretical model is based on the three-dimensional elastic theory. Three kinds of porosity distributions including even, uneven and the logarithmic-uneven are performed. Non consideration boundary conditions, all displacements of the rectangular plate are unified expanded as a improve Fourier series, which consists of standard three-dimensional (3-D) Fourier cosine series supplemented with closed-form auxiliary functions introduced to eliminate all the relevant discontinuities with the displacements and its derivatives at the edges. As the displacement field is sufficiently smooth in the entire solution domain, the exact solution of the energy function of the plate is obtained through the Rayleigh Ritz process. The numerical examples are given to verify the accuracy and reliability of the present method, in which both the present results and those reported in the literature are provided. Besides, several numerous new results for thick FGP plates with elastic boundary conditions are also presented.

On vibrations of porous FG nanoshells

An analysis of forced vibration of a porous functionally graded (FG) nanoshell is conduced employing a two-parameter non-classical elasticity theory called nonlocal strain gradient theory. A transverse partial dynamic load with specific frequency of excitation is applied to the nanoshell. Even and uneven dispersion of pores due to material imperfections have been considered. The modified power-law modeling of FG materials is introduced to incorporate the pore content effects. The problem of forced vibration is solved by using a hybrid analytical-numerical method in order to derive deflection-frequency curves. According to the provided graphical results, it can be concluded that pore amount, type of pore distribution, two scale parameters, material gradation and geometrical parameters of applied dynamic load have important influences on deflection-frequency curves of nanoshells.

Dynamics analysis of functionally graded porous (FGP) circular, annular and sector plates with general elastic restraints

In this paper, the dynamics analysis of functionally graded porous (FGP) circular, annular and sector plates with general elastic restraints is performed in a unified form for the first time. The overall theoretical model is based on the first order shear deformation theory. The kinetic energy and potential energy function of the plates are unified representation of five kinds of displacement admissible function. Then, each of displacement admissible function is expanded as a modified Fourier series to obtain general elastic restraints. Lastly, the solutions are obtained by using the variational operation. The convergence and accuracy of the present modeling are validated by comparing its results with those available in the literature and FEM results. Based on that, a series of innovative results are also highlighted in the text, which may be as the basic data for other algorithm research in the future.

A unified solution for the vibration analysis of functionally graded porous (FGP) shallow shells with general boundary conditions

The main purpose of this paper is to illustrate the vibration characteristics of functionally graded porous (FGP) shallow shells with general boundary conditions for the first time. The general boundary condition of FGP shallow shells is realized by the virtual spring technique. The imposing procedures of the boundary conditions are simplified so that a certain kind of restraints can be easily achieved by merely setting different stiffness of the springs. It is assumed that the distributions of porosity are uniform or non-uniformly along a certain direction and three types of the porosity distribution are considered, among which material property of two non-uniform porous distributions are expressed as the simple cosine. The size of the pore in a shallow shell is determined by the porosity coefficients. Based on the first-order shear deformation theory (FSDT), all kinetic energy and potential energy of FGP shallow shells are expressed by displacement admissible function. On this basis, the author describes the displacement admissible function of the FGP shallow shells by using the modified Fourier series which increases the auxiliary function, so that the auxiliary function can be used to eliminate the discontinuity or jumping of the traditional Fourier series at the edges. Lastly, the natural frequencies as well as the associated mode shapes of FGP shallow shells are achieved by replacing the modified Fourier series into the above energy expression and using the variational operation for unknown expansion coefficients. Several numerical examples are carried out to demonstrate the validity and accuracy of the present solution by comparing with the results obtained by other researchers. In addition, a series of innovative results are also highlighted in the text, which may provide basic data for other algorithm research in the future.