Functionally Graded Material Cylindrical Shell Research Articles

Advanced frequency of thick FGM cylindrical shells with fully homogeneous equation

The effects of shear deformation theory and improved shear correction factors on the advanced computation of frequencies by using the fully homogeneous equation for thick functionally graded material (FGM) circular cylindrical shells are studied. It is quite reasonable to consider the extra advanced effect of third-order shear deformation theory (TSDT) of displacements on the varied shear correction coefficient. The values of advanced nonlinear shear correction coefficient are usually functions of the nonlinear coefficient term in TSDT, power-law exponent parameter, and environment temperature. The main achievements in the nonlinear case of displacements and with the varied value of shear correction coefficients, the non-dimensional fundamental frequencies are estimated, investigated, and compared with the values in the linear case. This paper aims to study the fundamental frequencies of very thick FGM cylindrical shells under the advanced effects of shear deformation theory and improved shear correction factors.

Assessment of Axisymmetric Dynamic Snap-Through and Thermally Induced Vibrations in FGM Cylindrical Shells Under Instantaneous Heating

This paper addresses the investigation of axisymmetric thermally induced vibrations in Functionally Graded Material (FGM) cylindrical shells. It considers temperature-dependent (TD) properties and geometric non-linearity (the von Karman effect). The study systematically solves a transient heat conduction equation using finite differences method and the Crank–Nicolson method. During the heating stages, the evaluation of thermal forces and moments takes place. Equations of motion are derived through the application of Hamilton’s principle. Spatial dependencies are discretized using the generalized Ritz method, while temporal dependencies are approximated using the [Formula: see text]-Newmark method with Newton–Raphson linearization. A comparative analysis validates the procedure’s efficiency and precision. Parametric studies explore the influence of parameters, including the temperature-dependency material properties, geometric nonlinearity, and shell’s power-law index, providing valuable insights into FGM shell behavior under thermal shock.

Advanced Frequency Study of Thick FGM Cylindrical Shells by Using TSDT and Nonlinear Shear

For the advanced frequency study of thick functionally graded material (FGM) circular cylindrical shells, it is interesting to consider the extra effects of nonlinear coefficient term in third-order shear deformation theory (TSDT) of displacements on the calculation of varied shear correction coefficient. The formulation for the advanced nonlinear shear correction coefficient are based on the energy equivalence principle. The values of nonlinear shear correction coefficient are usually functions of nonlinear coefficient term of TSDT, power-law exponent parameter and environment temperature. The free vibration frequencies of thick FGM circular cylindrical shells are investigated with the simply homogeneous equation by considering that simultaneous effects of the TSDT, the nonlinear shear correction coefficient of transverse shear force and the two direction of mode shapes. The novelty is more important and reasonable for the thick FGM circular cylindrical shells, especially for the ratio of length to thickness is five by considering the effects of advanced nonlinear shear correction coefficient and nonlinear coefficient term of TSDT on the advanced calculation of fundamental first natural frequencies.

Magneto-thermo-elastic coupled free vibration and nonlinear frequency analytical solutions of FGM cylindrical shell

The magneto-thermo-elastic coupled free vibration of functionally graded materials (FGM) cylindrical shell is investigated. Based on the physical neutral surface theory and Kirchhoff-Love theory, the expressions of internal force and internal torque are obtained by considering the constitutive relationship that includes thermal stress. According to the electromagnetic elasticity theory, the model of Lorentz forces of the shell in magnetic field are derived. The expressions of potential energy, kinetic energy and its variational operators are given by introducing geometric nonlinearity, respectively. Based on Hamilton principle, the magneto-thermal-elastic coupled vibration equation of FGM cylindrical shell in multi-physical field is established. The displacement functions under simply supported boundary conditions are solved, which are combined with Galerkin method for derivation of nonlinear ordinary differential equations. The second-order approximate analytical solution of natural frequency is obtained by applying the multi-scale method. The characteristic curves of natural frequency with different parameters are drawn through numerical examples. The results show that reducing the volume fraction index and increasing the initial amplitude can lead to the increase of the natural frequency under transient conditions. With the increase of temperature and magnetic induction intensity, the natural frequency decreases. In addition, the natural frequency is also influenced by the shell size and volume fraction index.

Research on prediction of critical pressure of FGM cylindrical shells under hydrostatic pressure based on homogenization transformation method

This study uses the classical Flügge shell theory and homogenization transformation method to predict the critical pressure of functionally graded material (FGM) cylindrical shells under hydrostatic pressure. The method considers the similarity in physical properties and mechanical behavior between FGM and uniform materials. The vibration equation of the coupled system of an underwater FGM cylindrical shells is derived using the wave method, taking into account the influence of fluids. The natural frequency of the FGM cylindrical shell under hydrostatic pressure is obtained using the Newton iteration method. Based on the linear correlation between critical load and load level with zero natural frequency, the fitting curve method and the formula method after homogenization transformation are used to predict and analyze the critical pressure of FGM cylindrical shells under hydrostatic pressure. The effects of various parameters on the critical pressure are discussed. The results show that the material elastic modulus E, geometric dimensions h/R and L/R of FGM cylindrical shells, and different boundary conditions have a significant impact on the critical pressure. Through multiple comparative analyses of examples, the correctness and effectiveness of the proposed method are verified. The method has high prediction accuracy and low computational cost, providing a new exploration path for the analysis of non-uniform structural mechanical behavior.

Nonlinear primary resonance of axially moving functionally graded cylindrical shells in thermal environment

Based on Love’s thin shell theory, an effort is made to investigate the nonlinear primary resonance response of axially moving functionally graded materials (FGM) cylindrical shells subjected to external excitations in thermal environment. Two kinds of temperature fields, namely, uniform temperature rise and heat conduction, are considered. The effective material properties of the FGM cylindrical shells are temperature-dependent. The equations of motion are derived using Euler-Lagrange principle and solved by multiple scale method, in which the cylindrical shell with four clamped edges is considered. Finally, the effects of different influencing factors on the nonlinear resonance response are discussed.

Advanced dynamic thermal vibration of thick FGM plates-cylindrical shells

The third-order shear deformation theory (TSDT) effects on the thick functionally graded material (FGM) plates-cylindrical shells under thermal vibration are investigated by using the generalized differential quadrature (GDQ) method. The nonlinear coefficient c1 term of TSDT displacement field is used to obtain the equations of motion and the advanced nonlinear shear correction coefficient expression for thermal vibration of thick FGM plates and cylindrical shells, respectively. At the intersection of FGM plates-cylindrical shells, the displacements, stress resultants and moment resultants could be considered and reasonably assumed in the continuity condition for the differential volume in the plates and cylindrical shells, respectively. The effects of advanced nonlinear shear correction, FGM power law index and environment temperature on the thermal stress and center displacement of FGM plates-cylindrical shells are investigated.

On the torsional buckling moment of cylindrical shells consisting of functionally graded materials resting on the Pasternak-type soil

In this study, the buckling analysis of cylindrical shells made of functionally graded materials (FGMs) under the torsional moment resting on the Pasternak-type soil is performed. After establishing the linear constitutive relations of FGM cylindrical shells within the framework of the modified Donnell type shell theory, the governing equations of FGM cylindrical shells under the torsional moment are derived considering the influence of Pasternak-type soil. Analytical formula for the torsional moment is obtained by choosing the approximation functions that satisfies the boundary conditions in an integral sense. From the obtained formula, the formulas for the critical torsional moment in the presence of Winkler soil and absence of soils are obtained as a special case. Variations of critical torsional moment for different soil coefficients, volume fraction ratio and shell characteristics are investigated in detail. Keywords: functionally graded materials; cylindrical shell; buckling; critical torsional moment; Pasternak-type soil.

Thermo-torsional buckling and postbuckling of thin FGM cylindrical shells with porosities and tangentially restrained edges

This paper aims to analyze the effects of porosities, tangential constraints of boundary edges and elevated temperature on the buckling and postbuckling behaviors of thin functionally graded cylindrical shells subjected to uniform torsion. The porosities are assumed to exist within functionally graded material (FGM) according to even and uneven distributions. The properties of constituents are assumed to be temperature dependent and effective properties of porous FGM are determined using a modified rule of mixture. Mathematical formulations are established within the framework of classical shell theory taking into account von Kármán–Donnell nonlinearity and elasticity of tangential edge constraints. Multi-term analytical solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is used to determine buckling torsional load and load–deflection relation for postbuckling analysis. Numerical results show that porosities have remarkably deteriorative influences on the torsional buckling resistance and postbuckling load capacities of FGM cylindrical shells. This study also reveals that tangential edge constraints have pronouncedly beneficial and detrimental effects on the postbuckling strength of torsion-loaded porous FGM cylindrical shells at room and elevated temperatures, respectively.

A hybrid algorithm for modeling and studying of the effect of material and mechanical uncertainties on stability of sandwich FGM materials under thermal shock

Functionally graded material (FGM) structures are commonly used to withstand thermal stresses. These structures are very likely to experience thermal shocks. Given their nature, cylindrical shells are often subjected to a variety of dynamic loads so that the risk of instability (particularly dynamic instability) is very anticipated if they are subjected to a sudden thermal stress and a dynamic load, simultaneously. In fact, the possibility of uncertainty is very high in the fundamental parameters which determine thermal stresses. There is also uncertainty in the other properties of the structure including mechanical and thermal properties. Therefore, the present study addresses the dynamic stability of FGM cylindrical shells subjected to thermal and mechanical stresses based on Budiansky and Ruth’s criterion. The point estimation method and the artificial neural network-based algorithm were used to simulate the sources of uncertainty and to extract the statistical properties including the mean value, standard deviation, probability density functions (PDF), and cumulative distribution functions (CDF). The higher-order sheardeformationtheory (HSDT) was used to obtain the equilibrium equations, while the differential quadrature (DQ) method was used for their discretization. Functionally graded materials were assumed to change gradually along the thickness. The thermal shock distributions were based on a laboratory model (aerodynamic heating conditions) and two mathematical functions. The primary conditions, the type of thermal shock distribution, and the properties of the FGM core had the highest influence on the different stochastic properties of sandwich FGM cylindrical shells.

Natural Frequencies Optimization of Thin-Walled Circular Cylindrical Shells Using Axially Functionally Graded Materials.

One method to avoid vibration resonance is shifting natural frequencies far away from excitation frequencies. This study investigates optimizing the natural frequencies of circular cylindrical shells using axially functionally graded materials. The constituents of functionally graded materials (FGMs) vary continuously in the longitudinal direction based on a trigonometric law or using interpolation of volume fractions at control points. The spatial change of material properties alters structural stiffness and mass, which then affects the structure’s natural frequencies. The local material properties at any place in the structure are obtained using Voigt model. First-order shear deformation theory and finite element method are used for estimating natural frequencies, and a genetic algorithm is used for optimizing material volume fractions. To demonstrate the proposed method, two optimization problems are presented. The goal of the first one is to maximize the fundamental frequency of an FGM cylindrical shell by optimizing the material volume fractions. In the second problem, we attempt to find the optimal material distribution that maximizes the distance between two adjoining natural frequencies. The optimization examples show that building cylindrical shells using axially FGM is a useful technique for optimizing their natural frequencies.

3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions

This paper presents the steady-state stress analysis of single-layered and multilayered plates and shells embedding Functionally Graded Material (FGM) layers under moisture conditions. This solution relies on an exact layer-wise approach; the formulation is unique despite the geometry. It studies spherical and cylindrical shells, cylinders, and plates in an orthogonal mixed curvilinear coordinate system (α, β, z). The moisture conditions are defined at the external surfaces and evaluated in the thickness direction under steady-state conditions following three procedures. This solution handles the 3D Fick diffusion equation, the 1D Fick diffusion equation, and the a priori assumed linear profile. The paper discusses their assumptions and the different results they deliver. Once defined, the moisture content acts as an external load; this leads to a system of three non-homogeneous second-order differential equilibrium equations. The 3D problem is reduced to a system of partial differential equations in the thickness coordinate, solved via the exponential matrix method. It returns the displacements and their z-derivatives as a direct result. The paper validates the model by comparing the results with 3D analytical models proposed in the literature and numerical models. Then, new results are presented for one-layered and multilayered FGM plates, cylinders, and cylindrical and spherical shells, considering different moisture contents, thickness ratios, and material laws.

Performance of FGM bilayered cylindrical shell placed on cantilever edge

Functionally graded materials (FGMs) are designed for specific purpose and applications. Functionally graded materials for bi-layered cylindrical shell was discussed for different boundary conditions. Functionally graded materials (FGMs) are that kind of material in which function and formation may deviate continuously. Cylindrical shells are mainly significant in various fields of science as well as advanced technology of engineering like aerospace engineering, mechanical engineering and civil engineering. Wide applications of cylindrical shell in different fields like aircraft, aerospace and pressure vessels etc. Bi-layered cylindrical shells consist of two layers and in this work, one layer is of FGM material whose constituents are nickel (Ni) and zirconia (Zr) and other is of isotropic material whose constituent is stainless steel. In this work, effect of trigonometric volume fraction law on cantilever FGM bi-layered cylindrical shell with internal pressure has analyzed by using Rayleigh-Ritz technique and Love's shell theory. Present results of FGM bi-layered cylindrical shell are compared with FGM cylindrical shell. Validity of present technique has verified by way of comparisons with current conclusions and those obtained in the past studies.

Thermomechanical Nonlinear Buckling of Pressurized Shear Deformable FGM Cylindrical Shells Including Porosities and Elastically Restrained Edges

This work proposes an effective analytical solution to study the nonlinear buckling of functionally graded material (FGM) porous cylindrical shells subjected to external pressure and elevated temperature. Volume fractions of constituents are varied through the thickness direction according to power functional rules, and effective properties are estimated according to a modified rule of mixture. The porosities are evenly or unevenly distributed within the shell. Basic equations of a simply supported shell are based on first-order shear deformation theory, including von Kármán-Donnell nonlinearity, and solved using analytical solutions and the Galerkin method. To account for practical situations, temperature-dependent properties, together with the elastic constraint of edges, are included. The suggested form of two-term deflection overcomes mathematical difficulties and is suitable for shear deformation cylindrical shells. Numerical results indicate that porosities detrimentally influence the buckling resistance of pressurized shells. Furthermore, tangential edge constraints have slight and significant effects on critical buckling loads and the pressure bearing capacity of porous shells at normal and high temperatures, respectively. Besides being an innovative approach, the presented study aims to address practical aspects in engineering applications of pressurized porous shells.

An analytical method for free vibrations of functionally graded cylindrical shells with arbitrary intermediate ring supports

An analytical method is presented to study free vibrations of functionally graded material (FGM) cylindrical shells with arbitrary intermediate ring supports. Material properties continuously vary in thickness direction in accordance with the four-parameter power-law distributions in terms of volume fractions of constituents. To establish the final governing equation, the shell is firstly divided into several shell segments according to locations of ring supports. Based on the first-order shear deformation theory (FSDT), differential equations about displacements are solved by expanding displacements as exponential functions in axial direction and Fourier series in circumferential direction. Correspondingly, five displacements and five forces at any cross section of segments are further expressed as 10 unknown coefficients for every circumferential mode number. The artificial spring technique is adopted to restrain displacements at boundaries and ring supports, and all segments are assembled to the overall shell through continuity conditions. To test the validity of the developed model, natural frequencies and mode shapes of several homogenous and FGM cylindrical shells with/without ring supports are compared with appropriate ones in the literature or calculated by finite element method (FEM), which demonstrates high accuracy and wide application of the proposed method. Furthermore, effects of ring supports and material parameters are analyzed. The results reveal that restraining radial and/or circumferential displacements at ring supports significantly affects natural frequencies, but mode shapes are mainly affected by radial displacements. In addition, natural frequencies of the FGM shell obviously vary with the power-law exponent, whereas the corresponding mode shapes almost keep unchanged.

Thermal Nonlinear Buckling of Shear Deformable Functionally Graded Cylindrical Shells with Porosities

In this paper, for the first time, two-term deflection is suggested for nonlinear analysis of shear deformable cylindrical shells and, to the authors’ knowledge, a first investigation on thermal nonlinear buckling behavior of functionally graded material (FGM) cylindrical shells including porosities is presented. Material properties are assumed to be temperature dependent, volume fractions of constituents are graded in the thickness direction according to a power law, and effective properties of porous FGM are estimated according to a modified rule of mixture. Porosities are evenly or unevenly distributed within the shell. Mathematical formulations are based on a first-order shear deformation theory taking into account geometrical nonlinearity and elasticity of in-plane restraints of edges. Two-term deflection and a multiterm stress function are assumed to satisfy simply supported boundary conditions. The Galerkin method and an iteration procedure are employed to evaluate critical buckling temperatures and trace temperature-deflection paths in the postbuckling region. The two-term solution of deflection is a significant suggestion for nonlinear buckling analysis of shear deformable cylindrical shells in general and thermal postbuckling analysis of porous FGM cylindrical shells in particular. This two-term solution includes inherent uniform deflection at the prebuckling state and overcomes mathematically tremendous difficulty in using three-term deflection for shear deformable cylindrical shells. Besides this innovative approach, the obtained results are of great interest for many engineering applications. Unlike the case of mechanical loads, porosities have beneficial influences on the stability of thermally loaded FGM cylindrical shells.

Natural frequencies analysis of functionally graded circular cylindrical shells

In the present work, a study on natural frequencies of functionally graded materials (FGM) circular cylindrical shells is presented. TheFGM is considered to be a mixture of two materials. The volumetric fractions are considered to vary in the radial direction (i.e., through the thickness) in compliance with a conventional power-law distribution. The equivalent material properties are estimated based on the Voigt model. The analysis of the FGM cylindrical shells is performed using the third-order shear deformation shell theory and the principle of virtual displacements. Moreover, the third-order shear deformation shell theory coupled with Carrera’s unified formulation is applied for the derivation of the governing equations associated with the free vibration of circular cylindrical shells. The accuracy of this method is examined by comparing the obtained numerical results with other previously published results. Additionally, parametric studies are performed for FGM cylindrical shells with several boundary conditions in order to show the effect of several design variables on the natural frequencies such as the power-law exponent, the circumferential wave number, the length to radius ratio and the thickness to radius ratio.

Resonance response of clamped functionally graded cylindrical shells with initial imperfection in thermal environments

In this paper, the resonance behavior of initial imperfect functionally graded material (FGM) cylindrical shells with the clamped–clamped boundary condition at two ends is investigated. The material properties considered here are graded in the thickness direction according to a simple power law in terms of the volume fractions and are affected by temperature. The classical deformation theory, von-Karman type nonlinear geometric relations and Hamilton’s principle are employed to derive nonlinear partial differential equation of the clamped circular cylindrical shell at two ends. The partial differential governing equations are truncated by the Galerkin technique. Under the effect of transverse external excitation and in-plane force, averaged equations in term of polar coordinate are obtained by using multiple scales method. It is assumed that the dynamic system is in the case of primary resonance and 1:2 internal resonance. The effects of the volume fraction, temperature, damping, imperfection, transverse external excitation and in-plane force on amplitude frequency response of FGM cylindrical shell with initial imperfection are studied in detail by the application of numerical continuation method. With varying the tuning parameter of the excitation frequency, saddle node bifurcation, hopf bifurcation, saddle node bifurcation and hopf bifurcation are detected.

STATIC ANALYSIS OF FGM CYLINDRICAL SHELLS UNDER LOCAL LOAD USING QUASI-3D HIGHER-ORDER SHEAR DEFORMATION THEORY

The paper presents static analysis results of functionally graded material (FGM) cylindrical shells under concentrated load by an analytical approach. The basic equations are formulated based on a higher-order shear strain theory, taking into account the effects of transverse shear strain and stress. The one-dimensional material distribution in the thickness direction follows Voigt’s power law. In this article influences of several geometric parameters and material distribution coefficients on the stress state of FGM cylindrical shells are considered. Based on the analysis results at the clamped boundary zone, the local load concentration is found, and the transverse normal stress is demonstrated to be considerable at this zone.

Wave based method for free vibration characteristics of functionally graded cylindrical shells with arbitrary boundary conditions

Abstract The wave based method (WBM) is used to analyze the free vibration characteristics of functionally graded material (FGM) cylindrical shell with arbitrary boundary conditions. The motion relationship is described by the first-order shear deformation shell theory (FSDST). The displacement components and transverse rotations are expressed as wave function expansions. In accordance with the dynamic relationship, the final governing equation and global matrix are assembled by incorporating the boundary matrices. The natural frequency of the system is obtained by solving the determinant of the global matrix. By comparing the results with those in the literature, the validity of the proposed method is verified. In addition, the influences of power-law exponents and boundary conditions on natural frequencies are analyzed. The effects of geometric parameters including the ratio of thickness to radius and the ratio of length to the radius on natural frequencies are discussed. The purpose of this paper is to demonstrate the ease of application of the WBM for the free vibration of FGM cylindrical shells with arbitrary boundary conditions. Furthermore, the advantage of the WBM are: (1) the global matrix is easy to construct; (2) different boundary conditions can be conveniently adjusted; (3) it is with high computational efficiency and precision.