Effects Of Porosity Coefficient Research Articles

Static analysis of functionally graded porous beam-column frames by the complementary functions method

In this paper, an efficient and accurate numerical approach is implemented to examine the static behavior of Functionally Graded Porous (FGP) planar frames for the first time. The symmetric material constitutive relationship (SMCR), monotonic material constitutive relationships (MMCR), and uniform material constitutive relationship (UMCR) models are used to define the gradation of porosity along the thickness direction. The effects of porosity coefficient and material porosity distribution type on the static behavior of the FGP frames are investigated in detail. A direct relation between porosity coefficient and nodal deformations is observed.The effect of shear deformation is considered in the governing equations based on the Timoshenko Beam Theory (TBT). Within the scope of the presented procedure, fifth-order Runge-Kutta (RK5) algorithm is applied in the solution of the initial value problems. Imposing the rigidity matrix-based Complementary Functions Method (CFM) on the static analysis of the FGP planar frames is the novelty of this research. The results of the suggested approach are verified with the those of the finite element method (FEM), and those of the available literature. It has been carried out that type of the porosity distribution and porosity gradient index have an important impact on the response of FGP planar frames. By increasing the porosity coefficient, the displacement values of the frames made of UMCR and MMCR porosity distribution are affected more than those of the UMCR distributions. It can be noted that the SMCR can be preferred for designing the FGP beam-column structure as the minimum deformation values are obtained.

Functionally graded porous plate reinforced by carbon nanotubes subjected to low-velocity impact: an analytical and numerical analysis

PurposeThe purpose of this study is a numerical simulation and an analytical analysis about the low-velocity impact on a functionally graded porous plate with porosity distribution in the thickness direction. In this article, polymethyl methacrylate is used for matrix, and single-walled carbon nanotube (CNTs) (10,10) with consideration agglomeration sizes and lumping of CNT inside the agglomerations is applied for reinforcement.Design/methodology/approachIn analytical formulation, the non-linear Hertz contact law is applied for interaction between projectile and plate surface. High-order shear deformation plate theory is developed, and energy of the system for impactor and plate is written. The governing equations are derived using Ritz method and Lagrange equations and are solved using the fourth-order Runge–Kutta method. Also, ABAQUS finite element model of functionally graded porous plate with all edges simply supported and reinforced by CNT under low-velocity impact is simulated and is compared with those is achieved in the present analytical approach.FindingsIn parametric studies, the influence of porosity distribution patterns include uniform, non-uniform symmetric and non-uniform asymmetric on the histories of contact force and impactor displacement of simply supported plate reinforced by CNT are presented. Eventually, the effects of porosity coefficient, impactor initial velocity, impactor radius and CNTs lumping inside agglomerations for non-uniform symmetric distribution patterns are discussed in impact event in detail.Originality/valueIn this paper, the effect of combination of polymethyl methacrylate and CNTs with consideration agglomeration sizes and lumping of CNTs inside the agglomerations in the form of a functionally graded porous plate is studied in the problem of low-velocity impact analysis.

Free Vibration Analysis of Spinning Sandwich Annular Plates with Functionally Graded Graphene Nanoplatelet Reinforced Porous Core.

This paper conducted the free vibration analysis of a sandwich annular thin plate with whirl motion. The upper and lower faces of the annular plate are made of uniform solid metal, while its core is porous foamed metal reinforced by graphene nanoplatelets (GPLs). Both uniform and non-uniform distributions of GPLs and porosity along the direction of plate thickness which leads to a functionally graded (FG) core are taken into account. The effective material properties including Young’s modulus, Poisson’s ratio and mass density are calculated by employing the Halpin–Tsai model and the rule of mixture, respectively. Based on the Kirchhoff plate theory, the differential equations of motion are derived by applying the Lagrange’s equation. Then, the assumed mode method is utilized to obtain free vibration behaviors of the sandwich annular plate. The finite element method is adopted to verify the present model and vibration analysis. The effects of porosity coefficient, porosity distribution, graphene nanoplatelet (GPL) distribution, graphene nanoplatelet (GPL) weight fraction, graphene nanoplatelet length-to-thickness ratio (GPL-LTR), graphene nanoplatelet length-to-width ratio (GPL-LWR), spinning speed, outer radius-to-thickness ratio and inner radius-to-thickness ratio of the plate, are examined in detail.

Linear and nonlinear static bending of sandwich beams with functionally graded porous core under different distributed loads

In this investigation, linear and nonlinear bending analyses of sandwich beams with functionally graded cores are determined under different types of distributed loads. These sandwich beams are composed of two isotropic faces and a porous core with different gradients of internal pores. The governing formulation used to describe the beam’s linear and nonlinear behavior is constructed from Reddy's third-order shear deformation theory and nonlinear strain–displacement relations of von Kármán. The Gram-Schmidt orthogonalization procedure is adopted to generate numerically stable functions for the displacement field to solve the beam problems with various boundary conditions. Then, the Ritz method is utilized to find out linear and nonlinear bending results in conjunction with the iterative technique. The accuracy of our solutions is validated, and our numerical results agree well with some cases available in the literature. New results of the sandwich beams based on several effects of porosity coefficient, slenderness ratio, loading types, porous distributions of the core, etc., are presented in graphical and tabular forms, serving as a benchmark solution for future studies.

Limit load analysis and imperfection sensitivity of porous FG micro-tubes surrounded by a nonlinear softening elastic medium

An imperfection sensitivity analysis for the nonlinear post-buckling behavior of functionally graded (FG) porous micro-tubes is performed in this research. The case of geometrically imperfect micro-tubes surrounded by a nonlinear elastic medium under axial compressive load is analyzed. Properties of the micro-tube with uniform distributed porosity are FG across the radius of the cross-section. Two types of boundary conditions as simply-supported and clamped are considered. The high-order shear deformation theory of tubes is utilized to approximate the displacement field. Differential equations governing the equilibrium position of the micro-tube are extracted using the virtual displacement principle. These nonlinear equations are analytically solved by means of the two-step perturbation technique and Galerkin procedure. It is shown that when the imperfect micro-tube is in contact with a sufficiently soft foundation, the post-buckling path of the structure is unstable, and therefore the structure is imperfection sensitive. Since the imperfection sensitivity of micro-tubes is not reported in literature, results of this study are compared with buckling responses of perfect FGM tubes. The effects of porosity coefficient, power law index, length scale parameter, and geometrical parameters upon the limit buckling load of imperfect micro-tubes are investigated.

Nonlinear bending analysis of arbitrary-shaped porous nanocomposite plates using a novel numerical approach

In this paper, an efficient numerical strategy is used to study the geometrically nonlinear static bending of functionally graded graphene platelet-reinforced composite (FG-GPLRC) porous plates with arbitrary shape. Porous nanocomposite plates including cutout with various shapes can be modeled by the present approach. Four types of porous distribution scheme and four GPL dispersion patterns are selected, and the material properties are calculated based on the closed-cell Gaussian random field scheme, the Halpin–Tsai micromechanical model together with the rule of mixture. First, the variational statement of governing equations based on the virtual work principle and higher-order shear deformation theory (HSDT) is derived and presented in vector–matrix form for computational aims. Then, using the ideas of variational differential quadrature and finite element methods (VDQ and FEM), a numerical approach called as VDQ-FEM is used to address the considered problem. In VDQ-FEM, the domain of problem is first transformed into a number of finite elements. In the next step, the VDQ discretization technique is implemented within each element. Then, the assemblage procedure is performed to obtain the set of Studying effects of porosity and GPL distributions and porosity coefficient matricized governing equations which is finally solved by means of the pseudo arc-length continuation algorithm. One of the main novelties of the present work in implementing VDQ-FEM is proposing an efficient way based on mixed-formulation to guarantee the continuity condition of first-order derivatives in entire domain for the used HSDT model. A detailed parametric study is conducted to investigate the nonlinear bending of FG-GPLRC porous plates with different shapes. In the numerical results, the effects of porosity coefficient, porosity distribution pattern, GPL distribution pattern and boundary conditions on the nonlinear bending response of plates are analyzed.

Chebyshev collocation technique for vibration analysis of sandwich cylindrical shells with metal foam core

AbstractIn this paper, we present vibration analysis of a metal foam core sandwich cylindrical thin shell via the Chebyshev collocation method. Three types of metal foam core are taken into account. The governing equations are derived using Hamilton's principle and the Kirchhoff‐Love shell theory. Then, the Chebyshev collocation method is developed to evaluate natural frequencies of the metal foam core sandwich cylindrical shell under various boundary conditions. Results show that the Chebyshev collocation method possesses high accuracy. Moreover, the Sym‐1 metal foam core leads to the largest stiffness of the shell while the Sym‐2 metal foam core leads to the smallest stiffness. The effects of porosity coefficient and core thickness are also highlighted on the vibration characteristics of the metal foam core sandwich cylindrical shell.

Analysis of a functionally graded nanocomposite sandwich beam considering porosity distribution on variable elastic foundation using DQM: Buckling and vibration behaviors

In the present study, according to the important of porosity in low specific weight in comparison of high stiffness of carbon nanotubes reinforced composite, buckling and free vibration analysis of sandwich composite beam in two configurations, of laminates using differential quadrature method (DQM) is studied. Also, the effects of porosity coefficient and three types of porosity distribution on critical buckling load and natural frequency are discussed. It is shown the buckling loads and natural frequencies of laminate 1 are significantly larger than the results of laminate 2. When configuration 2 (the core is made of FRC) and laminate 1 ([0/90/0/45/90]s) are used, the first natural frequency rises noticeably. It is also demonstrated that the influence of the core height in the case of lower carbon volume fractions is negligible. Even though, when volume fraction of fiber increases, the critical buckling load enhances smoothly. It should be noticed the amount of decline has inverse relationship with the beam aspect ratio. Investigating three porosity patterns, beam with the distribution of porosity Type 2 has the maximum critical buckling load and first natural frequency. Among three elastic foundations (constant, linear and parabolic), buckling load and natural frequency in linear variation has the least amount. For all kind of elastic foundations, when the porosity coefficient increases, critical buckling load and natural frequency decline significantly.

Aeroelastic Flutter Analysis of Thick Porous Plates in Supersonic Flow

The aeroelastic flutter analysis of thick porous plates surrounded with piezoelectric layers in supersonic flow is studied. In order to aeroelastic analysis of the thick porous-cellular plate, Reddy’s third-order shear deformation plate theory and first-order piston theory are used. Furthermore, the plate is composed of two face piezoelectric layers and three functionally graded porous distributions core. Applying the extended Hamilton’s principle and Maxwell’s equation, the governing equations of motion are obtained. The partial differential governing equations are transformed into a set of ordinary differential equations by applying Galerkin’s approach. The effects of porosity coefficient, porosity distributions, piezoelectric layers, geometric dimensions, electrical and mechanical boundary conditions on the flutter aerodynamic pressure and natural frequencies of porous-cellular plates are investigated.

Structural failure performance of the encased functionally graded porous cylinder consolidated by graphene platelet under uniform radial loading

Abstract This paper investigates the structural failure mechanism of the functionally graded porous (FGP) cylinder consolidated by graphene platelet (GPL) encased in the rigid medium. Three porous distributions and three GPL dispersion patterns are discussed. The pressurized FGM-GPL cylinder may deform in the shapes characterized by “single-lobe” or “multi-lobe” due to the constraint of the medium. The nonlinear equilibrium equations are developed by the differential of the total potential energy function of the cylinder. The critical buckling pressure of the confined cylinder is obtained theoretically and increases with a higher value of the lobe number. The verification of the present analytical solution is conducted by establishing a two-dimensional (2D) finite element model (FEM). The pressure-displacement equilibrium paths are traced, from which, the maximum pressure (buckling pressure) is determined. Both the analytical and numerical results indicate that the confined FGP-GPL cylinder sustains its highest pressure capacity for the case that the porosity and the GPL are distributed symmetrically to the mid-surface, but non-uniformly through the cross-section. Finally, a full parametric study is mainly focused on the effects of porosity coefficient, and the weight fraction of the GPL on the buckling pressure, the hoop force, the bending moment, the stress and strain distribution on the crown position with the maximum deformation when the critical buckling occurs.

Analytical consideration and numerical verification of the confined functionally graded porous ring with graphene platelet reinforcement

The present work is concerned with the stability behavior of the functionally graded porous (FGP) ring with graphene platelet (GPL) reinforcement. The ring is surrounded by an elastic medium and a point load is introduced at the crown position. Assuming no interface friction between the ring and medium, the buckling load of the ring with a “single-lobe” deformation is obtained analytically. Afterward, the numerical examination is developed by comparing the critical buckling load from the numerical and analytical results. The good comparison indicates the analytical solutions are applicable to the nine cases by combining three porous distributions and three GPL dispersion patterns, respectively. It is explored that the combination of symmetrical pores and symmetrical GPL to the mid-surface can resist the highest external load of the confined FGP–GPL ring, and this case is successfully compared with other closed-form expressions for the homogeneous ring without pores and GPL. Finally, the effects of porosity coefficient, and weight fraction of the GPL are examined and discussed in terms of the load capacity, the hoop force, the bending moment, the hoop stress and strain, and the geometric shapes of the GPL.

Buckling and postbuckling of porous cylindrical shells with functionally graded composite coating under torsion in thermal environment

This paper investigates the nonlinear buckling and postbuckling of functionally graded porous circular cylindrical shells reinforced by orthogonal stiffeners resting on Pasternak elastic foundations in thermal environment and under torsional load by an analytical approach. Shells are reinforced by closely spaced stringers and rings in which material properties of the shell and the stiffeners are assumed to be continuously graded in the thickness direction. Basing on the classical shell theory with von Karman geometrical nonlinearity and smeared stiffeners technique, the governing equations are derived. Using the Galerkin method with the three-term solution of deflection, the closed form to find critical load and post-buckling response are obtained. The effects of porosity coefficient, material, temperature, dimensional parameters, stiffener and foundation are analyzed.

Free vibration analysis of porous plates with porosity distributions in the thickness and in-plane directions using isogeometric approach

The main purpose of this paper is to study the free vibration of porous square plate, circular plate, and rectangle plate with a central circular hole in the framework of isogeometric analysis (IGA). Generally, the porosity distributions of plates are assumed to happen in the thickness direction. However, the graded distributions of porosity may occur through the in-plane direction of plates. Therefore, porosity distributions along both the thickness direction and in-plane direction are considered in this study. The displacement fields are described by the first order shear deformation theory (FSDT) and the exact geometric models are formulated wholly by non-uniform rational B-spline (NURBS) basis functions, which bear high-order continuity inherently. To ensure the versatility of IGA-FSDT, several numerical examples for isotropic and porous plates with different boundary conditions and various types of porosity distributions are presented. Moreover, some innovative results are presented and discussed, which can be the benchmark data for other algorithm researches. The effects of porosity coefficient, boundary conditions and geometric parameters on the free vibration of porous plates are investigated comprehensively.

Size-dependent vibration analysis of a three-layered porous rectangular nano plate with piezo-electromagnetic face sheets subjected to pre loads based on SSDT

ABSTRACTIn the present research vibration of a porous rectangular plate which is located between two piezo-electromagnetic layers based on two variables sinusoidal shear deformation plate theory and according to nonlocal theory is investigated. The plate is resting on Winkler–Pasternak foundation and was subjected to pre loads. The motion equations have been obtained using Hamilton principle and are solved using analytical Navier's solution method. The effects of porosity coefficient, pores distribution, nonlocal parameter, pre load values, foundation constants and geometric size of the plate have been discussed in details. The results can be used to design more efficient sensors and actuators.

Axisymmetric post-buckling behavior of saturated porous circular plates

This study aimed to investigate axisymmetric post-buckling behavior of a circular plate made of porous material under uniformly distributed radial compression with simply supported and clamped boundary conditions. Pores are saturated with fluid and plate properties vary continuously in the thickness direction. Governing equations are obtained based on classical plate theory and applying Sanders nonlinear strain-displacement relation. Shooting numerical method is used to solve the governing equations of problem. Effects of porosity coefficient, pore distribution, pore fluid compressibility, thickness change and boundary conditions on the post-buckling behavior of the plate are investigated. The results obtained for post-bucking of homogeneous/isotropic plates and critical buckling load of porous plates are compared with the results of other researchers.