FGM Shells Research Articles

Fluid-structure coupling of concentric double FGM shells with different lengths

The aim of this study is to develop a semi-analytical method to investigate fluid-structure coupling of concentric double shells with different lengths and elastic behaviours. Co-axial shells constitute a cylindrical circular container and a baffle submerged inside the stored fluid. The container shell is made of functionally graded materials with mechanical properties changing through its thickness continuously. The baffle made of steel is fixed along its top edge and submerged inside fluid such that its lower edge freely moves. The developed approach is verified using a commercial finite element computer code. Although the model is presented for a specific case in the present work, it can be generalized to investigate coupling of shellplate structures via fluid. It is shown that the coupling between concentric shells occurs only when they vibrate in a same circumferential mode number, n. It is also revealed that the normalized vibration amplitude of the inner shell is about the same as that of the outer shell, for narrower radial gaps. Moreover, the natural frequencies of the fluid-coupled system gradually decrease and converge to the certain values as the gradient index increases.

A new approach for nonlinear dynamic buckling of S-FGM toroidal shell segments with axial and circumferential stiffeners

This paper presents an analytical approach to study the nonlinear dynamic buckling of sigmoid functionally graded material (S-FGM) toroidal shell segments surrounded by elastic foundations subjected to axial compression. Shells are reinforced by closely spaced rings and stringers. The nonlinear dynamic buckling analysis is based on the classical shell theory with von Kármán–Donnell-type of geometrical nonlinearity. By using the Galerkin method, the closed-form expression of the deflection–time curves is obtained. Numerical solutions are presented in tabular and graphical forms to investigate the deflection–time curves of FGM shells surrounded by elastic foundations. Finally, the influences of geometrical parameters, volume fraction index, elastic foundations, and the effectiveness of stiffeners on the nonlinear dynamic buckling of the S-FGM toroidal shell segments are discussed. The present results are compared with the available data for a special case.

The stability of FGM truncated conical shells under combined axial and external mechanical loads in the framework of the shear deformation theory

Abstract The major goal of this research was to obtain a closed form of the solution for critical combined loads (combined effects of the axial load and l lateral pressure or the axial load and hydrostatic pressure) of functionally graded (FG) truncated conical shell in the framework of the shear deformation theory (SDT). The basic equations of FG truncated conical shell shells subjected to the combined loads are derived in the framework of the SDT. By using the Galerkin method to basic equations are obtained the expressions for critical combined loads of FG truncated conical shell in the framework of the SDT. In particular, similar expressions in the framework of the classical shell theory (CST) are obtained, also. Our numerical experiments reveal that the proposed solution may offer accurate critical combined loads for the FGM shells as compared with reference solutions available in the literature. Finally, the calculation and presentation of the effects of many parameters included in the analysis conclude the goals to be reached in the study.

Fonksiyonel Değişimli Çekirdeğe Sahip Üç Tabakalı Silindirik Kabukların Parametrik Titreşimi

In this study, the parametric vibration of three-layered cylindrical shells with functionally graded (FG) core is investigated. Firstly, analytical models of an FG core, which uses the intermediate member and outer sheath layers of the cylindrical shell formed from metal-rich and ceramic-rich represented as ternary systems. The governing equations of three-layered cylindrical shells with an FG core are derived and are reduced Mathieu-Hill equation by using Galerkin method. The expressions for non-dimensional frequency, critical axial load and boundaries of instability regions of three-layered cylindrical shell with an FG core are found. The expressions for non-dimensional frequency parameter, critical axial load and boundaries of instability regions are obtained for the monolayer FGM, pure metal and pure ceramic cylindrical shells, as a special case. Finally, the effects of variations of volume fractions, shell characteristics and variations of the thickness of a core on the values of critical parameters of three-layered cylindrical shells made of different types of FG core are studied numerically. Comparisons are made with the available studies in the open literature to validate this study

Nonlinear thermal stability of eccentrically stiffened functionally graded truncated conical shells surrounded on elastic foundations

Abstract This paper studies the thermal stability of an eccentrically stiffened functionally graded truncated conical shells in thermal environment and surrounded on elastic foundations. Both of the FGM shell as well as the stiffeners are deformed under temperature. The formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation. By applying Galerkin method, the closed-form expression for determining the thermal buckling load is obtained. The numerical results show that the critical thermal load in the case of the uniform temperature rise is smaller than one of the linear temperature distribution through the thickness of the shell, and the critical thermal load increases when increasing the coefficient of stiffeners and vice versa. The paper also analyzes and discussed the significant effects of material and geometrical properties, elastic foundations on the thermal buckling capacity of the eccentrically stiffened FGM truncated conical shell in thermal environment. The obtained results are validated by comparing with those in the literature.

Transient responses of functionally graded double curved shallow shells with temperature-dependent material properties in thermal environment

Abstract An analytical approach is presented to investigate the nonlinear dynamic response and vibration of imperfect eccentrically stiffened FGM double curved thin shallow shells on elastic foundation using a simple power-law distribution (P-FGM) in thermal environment. The formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation. By applying Galerkin method and using stress function, explicit relations of thermal load-deflection curves for simply supported curved eccentrically stiffened FGM shells are determined. Effects of material and geometrical properties, temperature, elastic foundation and eccentrically stiffeners on the dynamic response and vibration of the imperfect eccentrically stiffened P-FGM double curved shallow shells in thermal environments are analyzed and discussed. The numerical results in this paper are compared with results reported in other publications.

Nonlinear vibration of imperfect eccentrically stiffened functionally graded double curved shallow shells resting on elastic foundation using the first order shear deformation theory

Abstract This paper presents an analytical approach to investigate the nonlinear dynamic response and vibration of imperfect eccentrically stiffened FGM thick double curved shallow shells on elastic foundation using both the first order shear deformation theory and stress function with full motion equations (not using Volmir's assumptions). The FGM shells are assumed to rest on elastic foundation and subjected to mechanical and damping loads. Numerical results for dynamic response of the FGM shells are obtained by Runge–Kutta method. The results show the influences of geometrical parameters, the material properties, imperfections, the elastic foundations, eccentrically stiffeners and mechanical loads on the nonlinear dynamic response and nonlinear vibration of functionally graded double curved shallow shells. The numerical results in this paper are compared with results reported in other publications.

A semi-analytical three-dimensional free vibration analysis of functionally graded curved panels integrated with piezoelectric layers

Abstract Free vibration analysis of functionally graded (FG) curved panels integrated with piezoelectric layers under various boundary conditions is studied. A panel with two opposite edges is simply supported, and arbitrary boundary conditions at the other edges are considered. Two different models of material property variations based on the power law distribution in terms of the volume fractions of the constituents and the exponential law distribution of the material properties through the thickness are considered. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. For the simply supported boundary conditions, closed-form solution is given by making use of the Fourier series expansion, and applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free-end conditions. Natural frequencies of the hybrid curved panels are presented by solving the eigenfrequency equation, which can be obtained by using edges boundary conditions in this state equation. The results obtained for only FGM shell is verified by comparing the natural frequencies with the results obtained in the literature.

Nonlinear axisymmetric response of FGM shallow spherical shells on elastic foundations under uniform external pressure and temperature

Abstract Based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation, the nonlinear axisymmetric response of shallow spherical FGM shells under mechanical, thermal loads and different boundary conditions is considered in this paper. Using the Bubnov–Galerkin method and stress function, obtained results show effects of elastic foundations, external pressure, temperature, material and geometrical properties on the nonlinear buckling and postbuckling of the shells. The snap-through behaviors of the FGM spherical shallow shells on elastic foundations also are analyzed carefully in this paper. Some results were compared with the ones of other authors.

Nonlinear postbuckling of imperfect doubly curved thin shallow FGM shells resting on elastic foundations and subjected to mechanical loads

The nonlinear response of buckling and posbuckling of imperfect thin functionally graded doubly curved thin shallow shells resting on elastic foundations and subjected to some mechanical loads is investigated analytically. The elastic moduli of materials, Young’s modulus, and Poisson ratio are all graded in the shell thickness direction according to a simple power-law in terms of volume fractions of constituents. All formulations are based on the classical theory of shells with account of geometrical nonlinearity, an initial geometrical imperfection, and a Pasternak-type elastic foundation. By employing the Galerkin method, explicit relations for the load–deflection curves of simply supported doubly curved shallow FGM shells are determined. The effects of material and geometrical properties, foundation stiffness, and imperfection of shells on the buckling and postbuckling loadcarrying capacity of spherical and cylindrical shallow FGM shells are analyzed and discussed.

Nonlinear postbuckling of imperfect eccentrically stiffened P-FGM double curved thin shallow shells on elastic foundations in thermal environments

This paper first time presents an analytical investigation on the nonlinear postbuckling for imperfect eccentrically stiffened FGM double curved thin shallow shells on elastic foundation using a simple power-law distribution (P-FGM) in thermal environments. The formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation. By applying Galerkin method and using stress function, explicit relations of thermal load–deflection curves for simply supported curved eccentrically stiffened FGM shells are determined. Effects of material and geometrical properties, temperature, elastic foundation and eccentrically stiffeners on the buckling and postbuckling loading capacity of the imperfect eccentrically stiffened P-FGM double curved shallow shells in thermal environments are analyzed and discussed.

Thermal Buckling of Imperfect Deep Functionally Graded Spherical Shell

Thermal buckling analysis of deep imperfect functionally graded (FGM) spherical shell is considered in this paper. A mixture of ceramic and metal is considered for the FGM shell and the material properties, such as the modulus of elasticity and coefficient of thermal expansion, vary by a power law function through the thickness. Employing the Sanders non-linear kinematic relations, total potential energy function is derived and the equilibrium and stability equations are obtained for the imperfect shell. Approximate solutions satisfying the simply supported boundary condition are assumed and using the Galerkin method the error due to the approximation is minimized. The geometrically imperfect shell is considered and three types of thermal loadings, such as the uniform temperature rise (UTR), linear temperature rise through the thickness (LTR), and non-linear temperature rise through the thickness (NLTR) are considered and their associated buckling temperatures are obtained. The effects of different temperature functions and the magnitude of initial geometric imperfection are examined on the thermal buckling loads of the shell.

The Higher Order Asymptotic Crack-Tip Field for Reissner’s Linear Functionally Graded Shell

The eigen-problem of a crack in functionally graded Reissner’s spherical shell is analyzed. By adopting the asymptotic expansion method, the higher order crack tip asymptotic fields which are similar to the Williams’ solutions of plane crack problems in homogenous materials are obtained. The grade direction is assumed to be parallel to the crack. The results can be widely adopted in numerical analysis, experimental investigation and the engineering application of FGM shell structure.

Finite Element Modeling and Analysis of Functionally Graded (FG) Composite Shell Structures

This article deals with the finite element modeling and analysis of functionally graded (FG) shell structures under different loading such as thermal and mechanical. Free vibration analysis of functionally graded (FG) spherical shell structure has also been presented. In order to study the influences of important parameters on the responses of FG shell structures, different types of shells have been considered. The responses obtained for FG shells are compared with the homogeneous shells of pure ceramic (Al2O3) and pure metal (steel) shells and it has been observed that the responses of the FGM shells are in between the responses of the homogeneous shells. Based on the analysis, some important results are presented and discussed for thick as well as thin shells.

Nonlinear vibration of shear deformable FGM cylindrical shells surrounded by an elastic medium

This paper investigates the large amplitude vibration behavior of a shear deformable FGM cylindrical shell of finite length embedded in a large outer elastic medium and in thermal environments. The surrounding elastic medium is modeled as a Pasternak foundation. Two kinds of micromechanics models, namely, Voigt model and Mori–Tanaka model, are considered. The motion equations are based on a higher order shear deformation shell theory that includes shell-foundation interaction. The thermal effects are also included and the material properties of FGMs are assumed to be temperature-dependent. The equations of motion are solved by a two step perturbation technique to determine the nonlinear frequencies of the FGM shells. Numerical results demonstrate that in most cases the natural frequencies of the FGM shells are increased but the nonlinear to linear frequency ratios of the FGM shells are decreased with increase in foundation stiffness. The results confirm that in most cases Voigt model and Mori–Tanaka model have the same accuracy for predicting the vibration characteristics of FGM shells.

Thermal buckling of FGM shells resting on a two-parameter elastic foundation

This paper focuses on the thermal buckling analysis of FGM shells resting on the two-parameter elastic foundation. Material properties of the constituents are graded in the thickness direction according to the power-law distribution. The surrounding elastic medium is modeled as an elastic foundation of the Pasternak-type. After giving the fundamental relations, the stability and compatibility equations of an FGM truncated conical shell subjected to thermal load and resting on a two-parameter elastic foundation have been derived. Critical temperature differences of FGM truncated conical shells with or without elastic foundations subjected to non-linearly distributed temperature across the thickness of the shells are obtained by solving eigenvalue problems. The appropriate formulas for FGM cylindrical shells with or without elastic foundations are found as a special case. In order to assure the accuracy of the present study, convergence properties of the critical temperature are examined in detail.

Nonlinear vibrations of functionally graded doubly curved shallow shells

Nonlinear forced vibrations of FGM doubly curved shallow shells with a rectangular base are investigated. Donnell’s nonlinear shallow-shell theory is used and the shell is assumed to be simply supported with movable edges. The equations of motion are reduced using the Galerkin method to a system of infinite nonlinear ordinary differential equations with quadratic and cubic nonlinearities. Using the multiple scales method, primary and subharmonic resonance responses of FGM shells are fully discussed and the effect of volume fraction exponent on the internal resonance conditions, softening/hardening behavior and bifurcations of the shallow shell when the excitation frequency is (i) near the fundamental frequency and (ii) near two times the fundamental frequency is shown. Moreover, using a code based on arclength continuation method, a bifurcation analysis is carried out for a special case with two-to-one internal resonance between the first and second doubly symmetric modes with respect to the panel’s center ( ω 13≈2 ω 11). Bifurcation diagrams and Poincaré maps are obtained through direct time integration of the equations of motion and chaotic regions are shown by calculating Lyapunov exponents and Lyapunov dimension.

Dynamic buckling of FGM truncated conical shells subjected to non-uniform normal impact load

Abstract Dynamic buckling of functionally graded materials truncated conical shells subjected to normal impact loads is discussed in this paper. In the analysis, the material properties of functionally graded materials shells are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Geometrically nonlinear large deformation and the initial imperfections are taken into account. Galerkin procedure and Runge–Kutta integration scheme are used to solve nonlinear governing equations numerically. From the characteristics of dynamic response obtain critical loads of the shell according to B-R criterion. From the research results it can be found that gradient properties of the materials have significant effects on the critical buckling loads of FGM shells.

Geometrically Nonlinear Analysis of Functionally Graded Plates and Shells

In this paper we perform the geometric nonlinear analysis of functionally graded (FGM) plates and shells. We use the Marguerre shell element, modified to incorporate the graded properties across the thickness. We describe the basic element formulation and then present and discuss some FGM plate and shell linear and nonlinear examples.

The buckling of FGM truncated conical shells subjected to combined axial tension and hydrostatic pressure

Abstract The purpose of this paper is to investigate the elastic buckling of FGM truncated thin conical shells under combined axial tension and hydrostatic pressure. Here axial tensions are separately applied to small and large bases of the truncated conical shell, respectively. It is assumed that the cone is a mixture of metal and ceramic, and that its properties changes as the power and exponential functions of the shell thickness. After giving the fundamental relations, the stability and compatibility equations of an FGM truncated conical shell, subject to combined axial tension and hydrostatic pressure, have been derived. Applying Galerkin’s method general formulas have been obtained for the critical combined and separate loads of FGM conical shells. The appropriate formulas for homogenous and FGM cylindrical shells are found as a special case. Effects of changing shell characteristics, material composition and volume fraction of constituent materials on the critical combined and separate loads of FGM shells with simply supported edges are also investigated. The results obtained for homogeneous cases are compared with their counterparts in the literature.