Functionally Graded Porous Research Articles

The vibration study of a sandwich conical shell with a saturated FGP core

This paper is provided to analyze the free vibration of a sandwich truncated conical shell with a saturated functionally graded porous (FGP) core and two same homogenous isotropic face sheets. The mechanical behavior of the saturated FGP is assumed based on Biot’s theory, the shell is modeled via the first-order shear deformation theory (FSDT), and the governing equations and boundary conditions are derived utilizing Hamilton’s principle. Three different porosity distribution patterns are studied including one homogenous uniform distribution pattern and two non-homogenous symmetric ones. The porosity parameters in mentioned distribution patterns are regulated to make them the same in the shell’s mass. The equations of motion are solved exactly in the circumferential direction via proper sinusoidal and cosinusoidal functions, and a numerical solution is provided in the meridional direction utilizing the differential quadrature method (DQM). The precision of the model is approved and the influences of several parameters such as circumferential wave number, the thickness of the FGP core, porosity parameter, porosity distribution pattern, the compressibility of the pore fluid, and boundary conditions on the shell’s natural frequencies are investigated. It is shown that the highest natural frequencies usually can be achieved when the larger pores are located close to the shell’s middle surface and in each vibrational mode, there is a special value of the porosity parameter which leads to the lowest natural frequencies. It is deduced that in most cases, natural frequencies decrease by increasing the thickness of the FGP core. In addition, reducing the compressibility of the porefluid a small growth in the natural frequencies can be seen.

Effect of porosity on free vibration of functionally graded porous beam based on simple beam theory

In this paper, the natural frequencies of functionally graded porous (FGP) beam are determined associated with two kinds of porosity. The simple beam theory is applied in this study with various boundary conditions. The results obtained in this paper are presented and compared with other results in the references to verify the correctness in implementing the formula and writing the Matlab code. Last but not least, this article can help researchers to have an overview of the free vibration characteristics of the functionally graded porous beams.

Nonlinear Large-Amplitude Oscillations of PFG Composite Rectangular Microplates Based Upon the Modified Strain Gradient Elasticity Theory

In this research work, the nonlinear large-amplitude free vibration characteristics of composite microplates made of a porous functionally graded (PFG) material are addressed numerically in the presence of different size-dependent strain gradient tensors as microscale. Accordingly, for the first time, the effect of each microstructural tensor is analyzed separately on the nonlinear free oscillations of PFG microplates with and without a central cutout. In order to fulfill this goal, the isogeometric computation approach is engaged to integrate the finite element approach into the nonuniform B-spline-based computer aided design tool. Accordingly, the geometry of the microplate with a central cutout is modeled smoothly to verify C−1 continuity based upon a refined higher-order plate formulations. In this regard, the microstructural-dependent frequency responses associated with the nonlinear free oscillations of microplates are traced. In both the cases of simply supported and clamped boundary conditions, it was revealed that the fundamental frequency is enhanced about 1.20% by considering only the symmetric rotation gradient tensor, about 3.27% by taking only the dilatation gradient tensor, and 9.43% by considering only the deviatoric stretch gradient tensor. On the other hand, the anisotropic character of PFG composite microplates results in an unsymmetrical frequency response curve, as the nonlinear frequencies associated with negative oscillation amplitudes are a bit higher than those of positive ones.

A unified Jacobi–Ritz approach for vibration analysis of functionally graded porous rectangular plate with arbitrary boundary conditions based on a higher-order shear deformation theory

In this paper, based on the higher-order shear deformation theory (HSDT) variational equation, a unified solution for the transient state vibration problem of functionally graded porous (FGP) rectangular plates are proposed using the combination of Jacobi–Ritz method and multi-segment strategy. Using the artificial spring technique makes a FGP rectangular plate can be separated into several segments, which is proved to be helpful to improve the calculation accuracy of natural frequency. Some common boundary conditions are applied to the mode including classical boundary conditions and elastic boundary conditions. Three common types of porous distributions are considered which contain uneven symmetrical, uneven asymmetric and evenly distributions. Through the study of truncated numbers and continuous conditional parameters, it is found that the results have evident convergence. Moreover, the accuracy of this method is proved by comparison. On this premise, the effect of different parameters on the free vibration and pulsating forced vibration of FGP rectangular plates are briefly studied.

Sound Transmission Loss of a Honeycomb Sandwich Cylindrical Shell with Functionally Graded Porous Layers

To examine the acousto-structural behavior of a sandwich cylindrical shell benefiting from hexagonal honeycomb structures in its core and functionally graded porous (FGP) layers on its outer and inner surfaces, a comprehensive study based on an analytical model which also considers the effect of an external flow is conducted. A homogenous orthotropic model is used for the honeycomb core while its corresponding material features are found from the modified Gibson’s equation. The distribution pattern of FGP parts is either even or logarithmic-uneven, and a special rule-of-mixture relation governs their properties. Based on the first-order shear deformation theory (FSDT), Hamilton’s principle is exploited to derive the final coupled vibro-acoustic equations, which are then solved analytically to allow us to calculate the amount of sound transmission loss (STL) through the whole structure. This acoustic property is further investigated in the frequency domain by changing a set of parameters, i.e., Mach number, wave approach angle, structure’s radius, volume fraction, index of functionally graded material (FGM), and different honeycomb properties. Overall, good agreement is observed between the result of the present study and previous findings.

Linear and nonlinear free vibration analysis of functionally graded porous nanobeam using stress-driven nonlocal integral model

Linear and nonlinear free vibration analysis of functionally graded porous (FGP) nanobeam with four different porous distribution patterns is performed on the basis of stress-driven two-phase local/nonlocal integral model. The strain is defined according to von Karman nonlinearity, and the differential governing equations as well as standard boundary conditions are derived through the Hamilton’s principle. The integral relation between nonlinear strain and nonlocal stress is transformed equivalently into differential form with constitutive boundary conditions. Neglecting the nonlinear terms, the bending deflection and moment are derived and expressed explicitly through the Laplace transformation with six unknown constants for linear free vibration, and the nonlinear characteristic equation for vibration frequency and linear vibration mode shape can be calculated through standard and constitutive boundary conditions. Two analytic methods including Ritz–Galerkin and perturbation methods based on both local and nonlocal linear vibration mode shape are applied to study the linear and nonlinear free vibration of nonlocal FGP nanobeams under different boundary conditions Meanwhile, direct numerical method based on general differential quadrature method and Newton’s iterative process are utilized to obtain the numerical solutions of nonlinear vibration frequencies. The influence of nonlocal parameters and vibration amplitude on the free vibration frequencies is investigated numerically. The numerical results show that influence of nonlocal parameters on linear vibration mode shape is not significant. Local linear vibration mode shape based Ritz–Galerkin​ and perturbation methods would overestimate and underestimate the linear free vibration of nonlocal simply-supported and clamped–clamped nanobeams. Nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods could provide more accurate prediction on nonlinear vibration frequencies than local mode shape based methods when W(0.5) is small. However, the relative error increases quickly with the increase of W(0.5). Based on the results obtained in this study, local and nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods cannot provide accurate prediction on nonlinear vibration frequencies when the vibration magnitude is big, since linear vibration mode shape assumption is not suitable for strong nonlinear vibration.

Free vibration analysis of functionally graded porous elliptical microshells using modified couple stress theory

In this study, the free vibration analysis of functionally graded porous (FGP) elliptical microshells was studied for the first time. The microshell was modeled based on Donnell’s shell theory along with modified coupled stress theory (MCST). The set of the governing equations were derived using Hamilton’s principle and were solved via the Galerkin's method. The accuracy of the presented model was confirmed and the effects on the natural frequencies of the microshell are investigated such as porosity parameter, porosity distribution pattern, length scale parameter, geometrical parameters of the microshell, and the circumferential wave number (n). Numerical results show that for n = 1, the length scale parameter has no considerable effect on the natural frequencies, but for n > 1, the natural frequencies grow by increasing the length scale parameter. Also, it is shown that the effect of length scale parameter on the natural frequencies grows the higher values of elliptical ratio (large-to-small radii ratio).

Orthogonal Polynomials-Ritz Method for Dynamic Response of Functionally Graded Porous Plates Using FSDT

This work focuses on the dynamic response of the functionally graded porous (FGP) plates. To construct the FGP plate model, the material-distributed profiles of the FGP plates are given first. In addition, the relations between the stress and strain are given by means of the first-order shear deformation theory (FSDT). Furthermore, the boundary conditions are defined based on the virtual spring technique. Moreover, the third kind of Chebyshev orthogonal polynomials are employed to construct the displacement components of the FGP plate, and the energy expressions of the FGP plate are constructed and then changed into the Lagrange equation form. Ultimately, the FGP plate energy expressions are solved by means of the Rayleigh–Ritz method. On this basis, the verification of the proposed method is conducted, and then, a series of investigations are given with respect to diverse influence factors.

Effect of graphene platelets reinforcement on vibration behavior of functionally graded porous arches under thermal environment

Functionally graded porous (FGP) composites reinforced with graphene platelets (GPL) are a new class of advanced nanocomposite that has surfaced recently in the past years in which the microstructure (mainly porosity) of the single-phase material (i.e., metal or polymer) varies continuously in the preferred directions to meet various functional requirements. Key advantages like low density, light-weightiness, high stiffness-to-weight ratios, excellent energy absorption, and thermal management properties make them the most promising candidates for structural and functional engineering applications. The application of these nanocomposites into the basic structural elements like arches, plates, and shells which are widely used in aerospace industries, requires a thorough and accurate analysis of these components which are made of these nanocomposite materials considering various environmental effects like temperature, moisture, etc.Therefore, this paper conducts the free vibration analysis of functionally graded (FG) porous arches reinforced with graphene platelets (GPLs) which are subjected to thermal loading conditions. A C0 finite element model based on higher-order shear deformation theory has been utilized to predict the structural response of the FG-GPL reinforced porous arches. Convergence and validation have also been performed to ascertain the accuracy of the present finite element formulation. Effect of various influencing parameters such as volume content of GPLs, porosity index, and temperature difference on frequency parameter has been shown, and various mode shapes for different curvature angles and boundary conditions have also been plotted. It is shown that graphene platelets (GPLs) as reinforcement can increase the stiffness of FG porous arches significantly under thermal environment and the mode shapes of FG-GPL reinforced arches are mainly the function of the curvature angle and boundary conditions.

BENDING ANALYSIS OF FG POROUS BEAM BASED ON SIMPLE TIMOSHENKO BEAM ELEMENTS

In this document, the static bending of functionally graded porous (FGP) beam is studied under various boundary conditions and uniform load. Besides, finite element procedure is based on the simple Timoshenko beam theory. The effects of material properties like two types of porosity on bending behaviors are also considered. The solutions achieved in this document are given and evaluated with other solutions in the cited papers to check the feasibility in deployment the formula and creating the Matlab program. Furthermore, this document helps researchers to get some information about the bending behaviors of the proposed structures.

Nonlinear Stability Characteristics of Porous Graded Composite Microplates Including Various Microstructural-Dependent Strain Gradient Tensors

In this study, the nonlinear buckling and postbuckling characteristics of composite microplates made of a porous functionally graded (PFG) material are scrutinized in the presence of different size-dependent strain gradient tensors as microscale. Accordingly, for the first time, the effect of each microstructural tensor is analyzed separately on the nonlinear stability of PFG microplates with and without a central cutout. In order to fulfill this goal, the isogeometric computation approach is engaged to integrate the finite element approach into the nonuniform B-spline-based computer aided design tool. Accordingly, the geometry of the microplate with a central cutout is modeled smoothly to verify [Formula: see text] continuity based upon a refined higher-order plate formulations. In this regard, the microstructural-dependent load-deflection paths associated with the nonlinear stability of axially compressed microplates are traced.

A new two-variable shear deformation theory for bending, free vibration and buckling analysis of functionally graded porous beams

This paper presents a simple two-variable shear deformation theory for bucking, bending, and vibration behaviours of functionally graded porous (FGP) beams. The displacement field of beams is developed from the separation of variables. Three typical porosity distribution types are considered. Mass density and elastic modulus of FGP beams are assumed to be graded in the beam’s thickness. Governing equations are derived from Lagrange’s principle. The exponential approximation functions are developed for various boundary conditions to predict frequency, buckling load, deflection, and stress of beams. The effects of boundary condition, span-to-height ratio, porous distribution pattern, porosity parameter, and shear deformation on the critical buckling load, frequency, deflection, and stress of beams are investigated in detail.

A Semianalytic Method for Vibration Analysis of a Sandwich FGP Doubly Curved Shell with Arbitrary Boundary Conditions

Due to the excellent mechanical properties of doubly curved structure and functionally graded porous (FGP) material, the study of their vibration characteristics has attracted wide attention. The main aim of this research is to establish a formulation for free and forced vibration analysis of a new Sandwich FGP doubly curved structure. Four models of Sandwich materials are considered. The potential energy and kinetic energy functions are obtained on the foundation of the first-order shear deformation theory (FSDT). The idea of domain energy decomposition is applied to the theoretical modeling, where the structure is segmented along the generatrix direction. The continuity conditions for the interfaces between adjacent segments are balanced by the weighted parameters. For each segment, the displacement functions are selected as the Jacobi orthogonal polynomials and trigonometric series. The boundary conditions of the structure are obtained by the boundary spring simulation technique. The solution is obtained by the variational operation of the structural functional. The convergence performance and correctness of the theoretical model are examined by several numerical examples. Finally, some novel results are given, where free and forced vibration characteristics of Sandwich FGP doubly curved structures are examined in detail.

Nonlinear in-plane instability of the confined FGP arches with nanocomposites reinforcement under radially-directed uniform pressure

The nonlinear buckling of the functionally graded porous (FGP) arches with nanocomposites reinforcement is studied in this paper. Both the pores and the nanocomposites are distributed symmetrically to the mid-axis of the cross-section. When the arch is encased in the elastic medium, the deformed shape of the arch can be described as a “single-lobe”, which can be expressed precisely by a cosine displacement function. By employing the nonlinear thin-walled shell theory and the principle of minimum potential energy, an analytical solution of the loading capacity is obtained, which is examined by developing a three-dimensional (3D) simulated model. After introducing the geometric nonlinearities and the Riks technique, the maximum load (loading capacity) is obtained numerically. Both the analytical and numerical results are successfully compared and verified by other closed-form expressions. It is found that the pores reduce the loading capacity, while a small addition of the nanocomposites increases prominently the loading capacity of the FGP arches. The confinement effect is examined by comparing the loading capacity between the confined and unconfined arches. The results indicate that the confinement effect increases with the increase of the central angle. Finally, parametric evaluations are taken to examine the effects of geometric shapes of the nanocomposites and interface friction on the loading capacity of the FGP-GPLs arches.

Static and buckling characteristics of the porous ring reinforced by graphene nanofillers

The present work focuses on the static and buckling characteristics of the confined functionally graded porous (FGP) ring consolidated by graphene nanofillers (GNF). The ring is under a temperature rise field when a point load is applied on the crown position. The Halpin-Tsai micromechanics theory is used to evaluate the distribution of Young’s modulus, and the Gaussian random field theory is used to evaluate the distribution of the porosity and mass density, respectively. The influence of the thermal field on the buckling load is quantized by applying the energy function and nonlinear thin-walled shell theory. The analytical buckling load is examined numerically, and good agreement is obtained. Furthermore, the present analytical and numerical results are verified effectively by other closed-form predictions when the FGP-GNF ring reduces to a homogeneous ring. Finally, comprehensive parametric study is taken by analyzing the effect of parameters on the buckling load. These parameters include the temperature rise, porosity coefficient, weight fraction, geometric parameters of GNF, friction coefficient and Young’s modulus of the medium.

Lateral-torsional buckling of functionally graded porous arches with graphene platelets reinforcements under an arbitrary radial concentrated load

This paper investigates the lateral-torsional buckling of functionally graded porous (FGP) circular arches with an I-section made of graphene platelets reinforced composites (GPLRC) under an arbitrary radial concentrated load, which has not been reported in open literature so far. The effective elastic modulus of the FGP- GPLRC arches is estimated by the Halpin-Tsai micromechanics model. After pre-buckling internal forces caused by the concentrated load have been obtained, the principle of stationary potential energy and Rayleigh-Ritz method are employed to derive theoretical solutions for the lateral-torsional buckling loads of FGP-GPLRC arches based on Timoshenko beam theory. The present theoretical solutions for the buckling loads are validated against the finite element results. The analytical expression of the effective warping stiffness and torsional stiffness of FGP-GPLRC arches are then obtained by an integral method. Finally, the influences of pore distribution pattern, porosity coefficient, weight fraction of graphene platelets (GPLs), load position, include angle and slenderness ratio of the arch on the lateral-torsional buckling load are analyzed in detail. Results show that a higher porosity coefficient leads to lower buckling loads but an increase in GPL weight fraction can effectively enhance the buckling resistance of the arch. Compared with their counterparts with uniform or asymmetric porosity distribution, FGP-GPLRC arches with symmetric non-uniform porosity distribution have higher lateral-torsional buckling loads.

Nonlinear bending analysis of fgp plates under various boundary conditions using an analytical approach

In this paper, the nonlinear bending behavior of functionally graded porous (FGP) plates under uniformly distributed transverse loads is studied based on the neutral surface concept within the framework of first-order shear deformation theory (FSDT) including geometrical nonlinearity. The FGP materials with three porosity distribution patterns namely uniform, non-uniform symmetric and non-uniform asymmetric are considered. Using the stress function and Galerkin method, the analytical solutions are obtained to examine the load vs deflection and load vs bending moment curves for various edges boundary conditions. The present results are compared with reference solutions to show the accuracy and the effectiveness of the proposed approach. The effects of the geometric, material parameters, elastic foundations and in-plane constraints on the nonlinear bending behavior of FGP plates are studied in detail through numerical investigations.

A unified modeling method for dynamic analysis of GPLs-FGP sandwich shallow shell embedded SMA wires with general boundary conditions under hygrothermal loading

In this research, the dynamic analysis of graphene platelets (GPLs) reinforced functionally graded porous (FGP) sandwich shallow shell embedded shape memory alloy (SMA) wires with general boundary conditions under hygrothermal loading is conducted. Four different porosity distributions of FGP reinforced with four various GPLs dispersion patterns are considered. Based on the Brinson formulation, the constitutive equation of the SMA is established. The energy functional of the sandwich shallow shell is constructed by employing the Hooke’s law and the first-order shear deformation theory (FSDT). Ultimately, the dynamic characteristics of the sandwich shallow shell are obtained by solving the energy functional with the Rayleigh-Ritz method. After the verification of accuracy and reliability, parametric studies on the effects of material properties, geometrical parameters, and boundary conditions are investigated.

A Unified Modeling Method for Dynamic Analyses of FGP Annular and Circular Plates with General Boundary Conditions

The porous material is an emerging lightweight material, which is able to reduce structural weight and also keeps the superiority of the structure. Therefore, this work is devoted to the investigation of the functionally graded porous (FGP) annular and circular plates with general boundary conditions. The unified modeling method is proposed by combining the first-order shear deformation theory, the virtual spring technology, the multi-segment partition method, and the semi-analysis Rayleigh–Ritz approach. Afterwards, the convergency and correctness of the proposed method are verified, respectively. The frequency parameters, modal shapes, and forced vibration responses are uniformly calculated based on the proposed method. Finally, the dynamic analyses of the FGP annular and circular plates with different parameters, such as the porosity distribution types, porosity ratios, boundary condition types, geometry parameters, and load types, are conducted in detail. It is found that the reasonable porous design is able to keep the dynamic stability of the structure under different parameter variations.

Analyses of dynamic characteristics of functionally graded porous (FGP) sandwich plates with viscoelastic materials-filled square‐celled core

The dynamic characteristics of functionally graded porous (FGP) sandwich plates with viscoelastic material-filled square‐celled core (VSC) are investigated based on the constructed sandwich plate model. The dynamic model is built by combining the first-order shear deformation theory, the equivalent theory of material mechanics, the virtual spring technology, the modified Fourier-Ritz method, and the Newmark-Beta approach. Furthermore, the model validation is conducted, including the convergence study and the numerical validation under different geometrical parameters, material parameters, and boundary conditions. With the validated model, the parameter analysis is conducted to investigate the influence of the porous types, geometrical parameters, structure types, boundary conditions, and load types on the dynamic performance of the FGP-VSC sandwich plate.