Types Of Porosity Distributions Research Articles

Size-dependent finite element buckling analysis of porous cylindrical micro-shells reinforced by graphene platelets

This study introduces a spatial shear deformable rectangular element formulation to investigate the size-dependent buckling behavior of porous graphene platelets (GPL) reinforced micro-shells within the framework of the modified couple stress theory (MCST) for the first time in the worldwide term. The rectangular element, characterized by 4 nodes with 19 degrees of freedom each and possessing weak continuity of order C 1 , incorporates a length scale parameter enabling size-dependent analyses of shells. The research considers three distinct functions for the distribution of porosity combined with three different patterns of the GPL distribution (called by patterns A to C) along the micro-shell thickness. The elastic modulus of the nanocomposite in the absence of porosity is obtained using the Halpin-Tsai micromechanics model. Subsequently, the study meticulously validates the modeling accuracy and then investigates variations in the critical buckling load with respect to porosity distribution functions, distribution patterns of GPL, porosity coefficient, weight percentage of GPL, geometric parameters of the cylinder, and the length-scale parameter. The results reveal that the combination of the GPL pattern A with the first symmetric type porosity distribution yields the highest critical buckling loads. On the contrary, if the pattern B is combined with the second symmetric type of porosity distribution, the lowest critical buckling load will be observed. Given the accuracy of the results obtained and the versatility of the finite element method (FEM) in handling complex geometries together with diverse boundary conditions, the developed element holds promise for analyzing shells of various shapes under different boundary conditions.

Nonlinear thermal-mechanical coupled isogeometric analysis for GPLs reinforced functionally graded porous plates

This work presents an isogeometric model for thermal-mechanical coupled analysis of geometrically nonlinear responses of graphene platelets reinforced functionally graded porous (GPLs-FGP) plates. The closed-cell Gaussian random field (GRF) scheme, Halpin-Tsai micromechanics model and rule of mixture are utilized to determine the material properties of GPLs-FGP plates with three types of porosity distribution and GPLs dispersion pattern. Unlike previous studies, the influences of GPLs and porosity on the thermal conductivity of GPLs-FGP plates are considered simultaneously. A geometrically nonlinear isogeometric analysis (IGA) model is developed base on a four-variable refined plate theory (RPT), using the von Kármán nonlinear theory, Hamilton’s principle and Total Lagrangian (TL) incremental scheme. After ensuring the accuracy and reliability of the present model through comparison with several examples, the thermal conductivity, temperature distribution, geometrically nonlinear static bending and transient responses of the plate, taking into account the temperature effects of porosity and graphene under thermal-mechanical loads are investigated. Numerical results show that porosity exerts influences on the thermal conductivity and temperature distribution of GPLs-FGP plates, which subsequently impact their geometrically nonlinear thermal-mechanical coupled responses. The results of this study are helpful for designing and manufacturing GPLs-FGP structures with improved performance.

Modeling and static analysis of porous functionally graded and FG-sandwich plates

The present work is emphasized towards the development of models for the porosity occurring in the functionally graded (FG) structures. The porosities are modeled across the thickness for the FG plate as well as for the FG core of the sandwich plate considering three different types of porosity distribution i.e. symmetric center enhanced, top enhanced, and bottom enhanced. The influence of the porosities on the material properties is examined and the properties are evaluated in the presence of porosities. Further, the influence of the porosities on the static behavior of FG and FG-sandwich plates is investigated quantitatively wherein the considered structures are modeled in the framework of an inverse hyperbolic shear deformation theory. The governing equations are obtained through principle of virtual work considering linear structural kinematics and generalized Hooke’s law. These equations are solved in the closed form for the simply supported boundary conditions and the response is obtained. The results are verified with the existing literature wherever possible and are in good agreement. The effect of various parameters such as porosity distribution, porosity parameter, core-thickness distribution, span-thickness ratio, power-law index, and loading conditions on the static response is examined and, on those basis, noteworthy conclusions are made. The results indicate that the plate with top enhanced porosity distribution possess lowest deflection while the plate with bottom enhanced distribution possess the highest deflection.

Bending, free vibration and buckling finite element analysis of porous functionally graded plates with various porosity distributions using an improved FSDT

Functionally graded materials (FGMs) are advanced composite materials with spatially varying properties, and their porosity distribution further enhances their complexity. The distribution pattern of porosity within a porous material plays a crucial role in determining the mechanical response of these structures. Therefore, the main objective of this study is to analyze the bending, free vibration, and buckling characteristics of porous FG plates by considering different porosity distributions and their effects on the overall behavior. To achieve this goal, a new finite element model is developed in the framework of an improved first-order shear deformation theory (IFSDT). In contrast to the conventional Mindlin–Reissner theory, the present IFSDT incorporates an improved mathematical formulation and provides a more realistic parabolic depiction of shear strain throughout the plate’s thickness without using any shear correction factors. In the present study, five types of porosity distribution functions are considered for the analysis. The material characteristics of the FGM porous plate change gradually in the thickness direction based on a power-law function. The governing equations are derived here using Hamilton’s principle, and a finite element method is employed for numerical analysis. Comparative analyses with previously published literature underscore the precision and simplicity of our developed finite element model. Moreover, the effects of various types of loads, porosity parameters, power-law index, side-to-thickness ratio, aspect ratio, porosity distributions and boundary conditions on the deflections, natural frequencies, and critical buckling loads are thoroughly analyzed in detail. Finally, the findings of this research contribute to the understanding of the mechanical behavior of FGMs and pave the way for designing and optimizing novel porous functionally graded structures.

Vibration analysis of graphene-reinforced porous aluminum-based variable-walled thickness sandwich joined conical-conical panel with elastic boundary conditions using differential quadrature method

In this paper, a unified solution is proposed to investigate the vibration characteristics of the variable-walled thickness graphene-reinforced porous aluminum-based (GRPA) sandwich joined conical-conical panel (JCCP) with the arbitrary elastic support boundary conditions by using the differential quadrature method (DQM). The two surfaces of the sandwich JCCP are made of metallic aluminum and the central core layer is the GRPA. The core thickness of each conical plate varies linearly with its generatrix. Three types of the graphene distributions and two types of porosity distributions are considered along the core thickness direction. Based on the first-order shear deformation theory (FSDT), von-Karman strain displacement relationship, constitutive relationship and Hamiltonian principle, the partial differential governing equations of motion are obtained for the variable wall thickness GRPA sandwich JCCP. Using the DQM, the dynamic equation is discretized into the ordinary differential equation. The matrix of the characteristic equation is analyzed to solve the frequencies and mode shapes of the variable wall thickness GRPA sandwich JCCP. The effects of the spring stiffness, boundary conditions, graphene distributions, porosity distributions and geometric parameters on the vibration properties are studied for the variable wall thickness GRPA sandwich JCCP with two interesting elastic supported boundary conditions. At the same time, this article provides a useful approach for studying the arbitrary boundary coupled plate and shell structures with the variable wall thickness.

Study of the dynamic behavior of plates in gradient properties with logarithmic and exponential porosity

In this paper, we are interested in the dynamic behavior of porous plates in property gradient, the property gradient material is a new generation of composite material, which is characterized by a gradual and continuous variation of physical properties, in order to attenuate the major singularity of laminar materials, which is the direct transition from one material to another. The proposed theory is based on assumption that the in-plane and transverse displacements consist of bending and shear components, in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments.In order to model our plate as well as possible, we will use a high-order theory with four variables only, by injecting the exact position of the neutral surface. The originality of this work lies in the consideration of two types of porosity distribution “logarithmic and exponential” in the calculation of the position of the neutral surface.The plate is subjected to a vertical load of sinusoidal distribution expressed in double series of Fourier. The equation of motion for FG rectangular plates is obtained through Hamilton’s principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigen value problems.The object of this work is to study the influence of the variation of the volume fraction, the power index and The dimension ratio on the response of the frequency , and ,the constraints , and and the dimensionless deflection and the dimensionless displacement obtained in the presence of porosity with logarithmic and exponential distribution compared to those uniformly distributed.

Free vibration analysis of hybrid CNT/GPL-reinforced Porous composite plates under fluid-loading

The present study aims to explore free vibration analysis of hybrid reinforced porous composite plates subjected to fluid loading. The hybrid composite comprises a porous core layer covered by two reinforced porous face sheets. Graphene Platelets (GPLs) and Carbon Nanotubes (CNTs) are considered as reinforcement materials. The linear three-dimensional piezo-elasticity theory is employed to model the structural part of the system, while the classical Rayleigh integral is used to obtain the fluid interaction part. The accuracy and reliability of the proposed model are validated by comparing the calculated natural frequencies of the plate in vacuum and water-loaded conditions with those reported in the literature. For the first time, several parametric studies are performed to investigate the effects of dimension ratios, porosity distributions, reinforcement dispersion pattern, porosity coefficient, and weight fraction of nanofiller on the natural frequency of hybrid reinforced porous composites. The results indicate that adjusting the porosity and weight fraction of the face sheets can enhance stiffness without changing the mass, emphasizing the importance of hybrid composites for achieving optimal stiffness while maintaining a constant mass. Furthermore, the influence of core layer porosity distributions on the composite behavior differs from that of porosity distributions in the face layers. Additionally, the effectiveness of the reinforcement pattern depends on the type of porosity distribution in the face layers.

Free Vibration Analysis of Porous Functionally Graded Material Plates with Variable Thickness on an Elastic Foundation Using the R-Functions Method

Free vibrations of porous functionally graded material (FGM) plates with complex shapes are analyzed by using the R-functions method. The thickness of the plate is variable in the direction of one of the axes. Two types of porosity distributions through the thickness are considered: uniform (even) and non-uniform (uneven). The elastic foundation is defined by two parameters (Winkler and Pasternak). To obtain the mathematical model of the problem, the first-order shear deformation theory of the plate (FSDT) is used. The effective material properties in the thickness direction are modeled by means of a power law. Variational Ritz’s method joined with the R-functions theory is used for obtaining a semi-analytical solution of the problem. The approach is applied to a number of case studies and validated by means of comparative analyses carried out on rectangular plates with a traditional finite element approach. The proof of the efficiency of the approach and its capability to handle actual engineering problems is fulfilled for FGM plates having complex shapes and various boundary conditions. The effect of different parameters, such as porosity distribution, volume fraction index, elastic foundation, FGM types, and boundary conditions, on the vibrations is studied.

Nonlinear Phase Angle and Amplitude Analysis of Porous Functionally Graded Material Rotating Shaft Under Electromagnetic Loads

In this research, the nonlinear response including amplitude and phase angle of a functionally graded inextensional rotating shaft is studied under the electromagnetic load. Three types of porosity distributions through the radial direction are considered in this work. For the rotating shaft with nonlinear curvature and inertia, the governing equations and corresponding boundary conditions are derived. The Galerkin and the multiple scales methods are used to obtain the modulation equations. The effect of the power law index for a functionally graded shaft fabricated from the mixture of ceramic and metal is presented in the frequency response diagrams for primary resonance under electromagnetic force. Numerical simulations include the effects of the power law index of functionally graded material (FGM), porosity distribution, air-gap length, electromagnetic load, and vibration modes of a rotor on the phase angle and amplitude of steady state responses.

Analytical and mesh-free approaches to dynamic analysis and active control of smart FGP-GPLRC beam

This paper aims to perform the dynamic analysis and active control of a functionally graded porous beam reinforced with graphene platelets (GPLs) and embedded in piezoelectric layers (smart FGP-GPLRC beam). Two types of porosity distribution combined with three dispersions of GPL pattern for the FGP-GPLRC beam are considered. The Halpin–Tsai model is used for estimating the effective elastic modulus of the FGP-GPLRC beam. The linear electric potential field through the thickness of each piezoelectric layer is assumed. Energy principles are applied to establish the equations of motion, which are then solved by two approaches. The first one is an analytical method with the Navier technique and the second one is a mesh-free method based on the polynomial basis in conjunction with the C1 Hermite interpolation technique. Newmark's integration algorithm with constant average acceleration is employed to determine the time-dependent response. The velocity feedback control algorithm is utilized to actively control the vibration response of the smart FGP-GPLRC beam. The convergence of mesh-free analysis is performed through a numerical test. Various comparative examples are carried out rigorously to verify the accuracy of the current study. Influences of porosity and GPL parameters, open- and closed-circuit states on the natural frequency, and forced vibration of the smart beam are investigated via numerical examples. In addition, the study also examines the effects of velocity feedback control gain on vibration suppression and the prevention of the beat and resonance problems occurring in the smart beam. Finally, the solution of using a collocated piezoelectric sensor/actuator pair on the same side of the FGP-GPLRC beam to guarantee the stability of the vibration control with the stretching-bending coupling effect is illustrated.

Simple first-order shear deformation theory for free vibration of FGP-GPLRC spherical shell segments

This study presents a free vibration analysis of spherical shell segments using simple first-order shear deformation shell theory (S-FSDT) for the first time. The shell structure is made of functionally graded porous graphene platelet reinforced composite (FGP-GPLRC) – one kind of porous material strengthened by graphene platelets (GPLs). Effective material properties of the FGP-GPLRC are determined by the modified Halpin-Tsai micromechanical model and the mixture rule. Four types of porosity distributions and GPL dispersions are considered fully in this study. The governing equations of the shell are derived based on the S-FSDT and classical shell theory, then solved by the well-known Rayleigh-Ritz method and the artificial spring technique. The results show that the S-FSDT can capture well the behaviors of the spherical shell segments. Besides, the best profile of novel FGP-GPLRC is not fixed; the physical parameters, geometric parameters, and material characteristics of the shells need to be investigated fully.

Nonlinear fully-coupled thermoelastic analysis of bidirectional porous functionally graded doubly-curved shell panels with optimum material distribution

This work focuses on the nonlinear fully-coupled thermoelastic solutions of bidirectional porous functionally graded doubly-curved panels under uniform-pressure and heat-flux. The material properties are assumed to vary in x-y directions and achieved using multivariable power-law based Voigt’s scheme. Three types of porosity distribution (even, uneven, and cosine) are considered for the analysis. The nonlinear model is developed, considering cubic-polynomial based temperature-dependent material properties and higher-order Green-Lagrange kinematics. The fully-coupled model is developed using the extended potential-energy principle and 2D-isoparametric finite-element approximations via Picard’s iterative scheme. Two-way coupling of deflection and temperature responses is adopted to estimate the thermomechanical behavior. Here, the effects of curvature ratio, power-law, and porosity indices on the longitudinal/transverse/out-of-plane deflections, and temperature-profile are demonstrated. Additionally, the optimum material-distributions are obtained using response surface method by minimizing the thermoelastic responses. Mixed hardening/softening nonlinearity behaviors of bidirectional FGM curved panels is observed for the coupled thermoelastic responses.

Nonlinear Finite Element Model for Bending Analysis of Functionally-Graded Porous Circular/Annular Micro-Plates under Thermomechanical Loads Using Quasi-3D Reddy Third-Order Plate Theory.

A nonlinear finite element model for axisymmetric bending of micro circular/annular plates under thermal and mechanical loading was developed using quasi-3D Reddy third-order shear deformation theory. The developed finite element model accounts for a variation of material constituents utilizing a power-law distribution of a two-constituent material, three different porosity distributions through plate thickness, and geometrical nonlinearity. The modified couple stress theory was utilized to account for the strain gradient effects using a single material length scale parameter. Three different types of porosity distributions that have the same overall volume fraction but different enhanced areas were considered as a form of cosine functions. The effects of the material and porosity distribution, microstructure-dependency, the geometric nonlinearity, and various boundary conditions on the bending response of functionally-graded porous axisymmetric microplates under thermomechanical loads were studied using the developed nonlinear finite element model.

Vibration analysis of the porous metal cylindrical curved panel by using the differential quadrature method

In this paper, the natural vibration characteristic of the porous metal cylindrical curved panel under the spring supported boundary conditions is studied. The material of cylindrical curved plate is porous metal, and in thickness direction, three types of porosity distribution are considered. Theoretical formulations for the free vibration of the cylindrical curved panel are established using the first-order shear deformation theory (FSDT) and Hamilton’s principle, and a novel dynamics system for porous cylindrical curved panel in physical space is presented. Then, by employing differential quadrature method (DQM), the governing equations of natural vibration in form of partial differential equation and equations describing boundary conditions are discretized into algebraic equations. Unified solution for free vibration of curved panel with arbitrary boundary conditions is obtained. The system’s mode shapes and natural frequencies are identified. Moreover, the influence of geometric dimensions, spring stiffness, porosity and types of distribution on the natural vibration of the cylindrical curved panel are studied in detail. The results show that the approach proposed here can efficiently extract analytical expressions of vibration for cylindrical curved plate with arbitrary boundary conditions in physical space

Modeling and Free Vibration Analysis of Variable Stiffness System for Sandwich Conical Shell Structures with Variable Thickness

This paper presents modeling and free vibration analysis of variable stiffness system for the truncated sandwich conical shell made of porous aluminum foam core with variable thickness and carbon fiber face sheets under the simply supported boundary condition. The thickness of the core layer varies along the longitudinal direction. Five different types of porosity distribution of the aluminum foam core, which contains Type-X, Type-O, Type-U, Type-V and Type-[Formula: see text] along the direction of thickness, are considered. Considering the effect of thermal environment, we derive the nonlinear dynamic equations based on first-order shear deformation theory and Hamilton’s principle, and obtain the natural frequencies of the system by employing the Galerkin method. The comparison and validation are conducted by contrast with the determined results of the literature. The influences of porosity distribution pattern, porosity coefficient, the total number of layers, temperature increment, semi-vertex angle, the exponent of thickness function, the minimum radius-thickness and length-thickness ratio of the core layer on the natural frequencies, modal and mode shapes are studied comprehensively.

Free vibration analysis of FGM plates and shallow shells by the R-functions method

Research of free vibration of porous functionally graded material (FGM) plates with complex shape is carried out. It is supposed that thickness of the plate changes in direction of one of the axes. Two types of porosity distributions through the thickness even and uneven are considered. Elastic foundation is two parameters (Winkler and Pasternak). To obtain the mathematical model of the given problem the first order shear deformation theory of the plate (FSDT) is used. The effective material properties in the thickness direction is modelled by the power law. Variational Ritz’s method joined with the R-functions theory is used for solution of the formulated problem. The developed approach is tested on a lot of problems. Comparative analysis of the obtained results for rectangular plates confirms their verification. Demonstration of the efficiency of the proposed approach is fulfilled for FGM plate with complex shape and various boundary conditions. Effect of the different parameters on vibration characteristics such as porosity, volume exponent, elastic foundation, types of FGM, boundary conditions is studied.

Thermal nonlinear stability of functionally graded porous material nearly cylindrical shells with surrounding elastic media and tangentially restrained edges

This paper aims to investigate the influences of surrounding elastic media and tangential edge constraints on the buckling and postbucking behaviors of nearly cylindrical shell (NCS) made of functionally graded porous material (FGPM) and subjected to uniform temperature rise. Three different types of porosity distributions are considered and assumed to be in forms of cosine functions of thickness variable. The volume fractions of material constituents are assumed to vary through the shell thickness according to power‐law functions. Temperature‐dependent material properties and elasticity of tangential constraints of boundary edges are included. Governing equations are established within the framework of classical shell theory taking into account von Kármán–Donnell nonlinearity and interactive pressure from surrounding elastic media. Multiterm analytical solutions for simply supported NCSs are assumed and Galerkin procedure is utilized to derive expressions of buckling load and nonlinear load–deflection relation. An iteration process is adopted to determine critical thermal loads and postbuckling temperature–deflection paths. Numerous parametric studies are carried out to assess many different effects on thermal buckling resistance and postbuckling load carrying capabilities. The study reveals that porosity and tangential edge constraint have beneficial and deteriorative influences on the thermal stability of the NCSs, respectively. Furthermore, the obtained results demonstrate that surrounding elastic media have significantly beneficial influences on the thermal nonlinear stability of FGPM NCSs.

Free vibration analysis of functionally graded porous cylindrical panels and shells with porosity distributions along the thickness and length directions

This paper mainly investigates the free vibration behaviors of functionally graded porous (FGP) cylindrical panels and shells. According to the direction and type of porosity distribution, besides three typical thickness-direction distribution kinds: symmetric, non-symmetric and uniform, two new distribution types along the length direction: symmetric and non-symmetric are considered for the first time. By utilizing the first order shear deformation theory (FSDT) in conjunction with Hamilton’s principle, the weak form for free vibration of FGP shells is derived. Then the isogeometric analysis (IGA) method is applied to solve this problem, in which the non-uniform rational B-spline (NURBS) is employed to both exactly build geometric model and approximately describe displacement fields. The accuracy of IGA-FSDT is examined via comparing the current results with published data. Several numerical examples for FGP cylindrical panels and shells are presented and discussed in terms of porosity coefficient, boundary conditions, geometric parameters and types of porosity distributions. Some innovative results may provide benchmark solutions for other algorithm research and some guidelines for designing FGP structures.

The effects of the elastic foundations on the frequency of the viscoelastic FG porous doubly-curved macro/nanosize shell subjected to hygro-thermo-magnetic fields using a novel quasi-3D shell theory

This article investigates the size-dependent dynamic behavior of the viscoelastic functionally graded (FG) porous doubly-curved macro/nanosize shells rested on various foundations based on the nonlocal strain gradient theory (NSGT). Also, a parametric study was conducted to describe the impact of small-scale parameters, magneto-hygro-thermal environments, as well as Winkler, Pasternak, and Kerr foundations on the vibration characteristics of the system. A novel quasi-3D shell model is developed to scrutinize the dynamics of the structure by dividing the transverse displacement into deflection and thickness stretching components. For porosity-dependent dynamic analysis, two different types of porosity distributions are considered. A novel higher-order shear deformation theory (HSDT) is employed to reflect the shear effects along the system thickness. Hamilton’s principle and Kelvin–Voigt damping effects are utilized to obtain the governing equations of motion. The Navier solution approach is applied to acquire the dynamic response of the panel with simply-supported boundary conditions. It is indicated that the vibration frequency of the considered system is substantially affected by the porosity volume fraction, FG index, porosity distribution, foundation parameters, feedback control gain, viscoelastic structural damping, and length-to-thickness ratio of the structure.

Free Vibration Analysis of a Graphene-Platelet-Reinforced, Porous, Two-Cylindrical-Panel System

In this study, a novel, dynamic model of a graphene-platelet-reinforced, porous (GPLRP) double-cylindrical-panel system is proposed. The material properties of a graphene-platelet-reinforced, porous, double-cylindrical-panel system were determined by the Halpin–Tsai micromechanics model and the typical mechanical properties of open-cell metal foams. Different types of porosity distribution and graphene platelet (GPL) distribution patterns were considered. Love’s shell theory was utilized to derive the theoretical formulation, and the Rayleigh–Ritz method was used to calculate the natural frequencies of the system. The proposed model was validated by several comparison studies with the natural frequencies in the existing literature. Finally, the effects of stiffness of Winkler springs, boundary condition, porosity coefficient, porosity distribution, GPL distribution pattern, and GPL weight fraction on the free vibration characteristics of the system were evaluated.