Functionally Graded Material Shells Research Articles

Three-dimensional nonlinear stress analysis of functionally graded shells subjected to displacement-dependent loads

The nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading, developed in the companion paper, is here extended to the stress analysis of functionally graded material (FGM) shells. The term GeX means that the middle surface is described by analytical functions, including spline functions, and, therefore, the coefficients of the first and second fundamental forms can be taken exactly at element nodes. The FGM solid-shell element formulation is based on the choice of N sampling surfaces (SaS) parallel to the middle surface and located at Chebyshev polynomial nodes within the shell as well as on the outer surfaces, to introduce the displacements of these surfaces as basic unknowns. The use of Lagrange polynomials of degree N–1 in the distributions of displacements, strains, stresses, and elastic coefficients through the thickness of the shell leads to an efficient 3N-parameter shell formulation. To develop the FGM four-node solid-shell element based on the hybrid-mixed method, in which strains and stresses are utilized as primary variables, the Hu-Washizu variational principle is applied. Due to the analytical integration used to evaluate the tangent stiffness matrix, the developed FGM solid-shell element subjected to displacement-dependent loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered benchmark problems. It has been established that the difference between the second Piola-Kirchhoff and Cauchy stress tensors in non-conservative problems for FGM shells undergoing moderately large strains can be significant.

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.

Adaptive stochastic isogeometric analysis for nonlinear bending of thin functionally graded shells with material uncertainties

This paper presents an adaptive stochastic isogeometric method to incorporate material uncertainties in the nonlinear bending analysis of thin functionally graded material (FGM) shells. The gradient index is modeled as a second-order random field to describe the spatial randomness of material properties. An adaptive, nested, and non-intrusive Chebyshev interpolation process based on Leja sequences is formulated to approximate the response surface concerning random inputs, in which the discrete responses are parallelly obtained and incrementally augmented to save computational effort. The response surface is taken as a surrogate model to promote the efficiency of the stochastic analysis. In particular, a new probabilistic post-processing technique is proposed to accelerate the evaluation of the probability characteristics of the response. Isogeometric analysis is applied to solve the discrete responses of FGM shells, eliminating geometric errors and simplifying the implementation. Through three examples with different computational costs, the effectiveness of the present method is demonstrated by comparing it with two forms of Monte Carlo simulation methods. The influence of the statistical characteristics of the gradient index field on the response surface analysis and the random response is investigated.

Distorted dynamic similitude of sandwich FGM plates and shells in high-temperature environments

Limit to the size of the experimental platform, scaled-down model test is a reliable mean to study the dynamic of sandwich functionally graded material (FGM) structures in high-temperature environments. However, due to the complexity of material composition and impact of high-temperature environments, relevant parameters cannot be scaled with traditional similitude methods, resulting in the similitude distortion. For the first time, the present work studies the distorted dynamic similitude of sandwich FGM plates and shells in high-temperature environments. Based on the Energy Similitude Correction Method proposed by the author in recent years, the similitude distortion derived from the non-scalability of material properties and thermal effects is solved. More importantly, the application object of this method is extended for the first time from plates to shells of various shapes, including cylindrical shells, conical shells and double curved shells. Meanwhile, heat conduction, thermal expansion and temperature-dependent material properties are taken into account for the impact of high-temperature environments. Various numerical examples are verified with different gradient coefficients, thicknesses, temperature conditions and shapes of shells. The results of high-precision similitude show that the prediction approach is correct and widely applicable.

Vibration Control of Fgm Thick Shells Using Higher Order Shear Deformation Theory

Analytical solutions of Functionally Graded Material (FGM) shells with embedded magnetostrictive layers are presented in this study. These magnetostrictive layers are used for the vibration suppression of the functionally graded shells. The higher order shear deformation theory (HSDT) is employed to study the vibration suppression characteristics. The exact solution for the FGM shell with simply supported boundary conditions is based on the Navier solution procedure. Negative velocity feedback control is used. The parametric effect of the location of the magnetostrictive layers, material properties, and control parameters on the suppression effect are investigated in detail. Higher order shear deformation theory has significant influence on prediction of vibration response of thick shells. Further, it is found that (i) the shortest vibration suppression time is achieved by placing the actuating layers farthest from the neutral plane (ii) the use of thinner smart material layers leads to better vibration attenuation characteristics, and (iii) the vibration suppression time is longer for a smaller value of the feedback control coefficient.

Improved mode acceleration-based vibroacoustic coupling analysis of functionally graded shell under random excitation

This paper focuses on the vibroacoustic response of functionally graded material (FGM) shell structures under stationary random excitations. An eight-node finite element model of FGM shells is developed, which is valid for both flat and curved shells of FGM. Then, the finite element formulation of the acoustic-structural coupled system for FGM shells under random excitation is established. It is shown that the commonly used combined method of the pseudo excitation method (PEM) and mode displacement method (MDM) in previous work is highly efficient for stochastic dynamics problems, but its accuracy is reduced by the modal truncation error. To remedy this, the combined method of PEM and the conventional mode acceleration method (MAM) is an effective alternative, but not applicable to the coupled acoustic-structural system due to its singular stiffness matrix. To circumvent this, an efficient and accurate method integrating the PEM and the improved MAM (IMAM) is proposed for the vibroacoustic coupling analysis of FGM shell structures under stationary random excitation. Numerical examples are given to verify the effectiveness of the proposed method and to explore the random vibroacoustic properties of the FGM shells with different power-law distributions.

Research on Dynamics Behaviors of Two Coaxial Circular Cylindrical Thin Shells Constructed with Functionally Graded Materials

The paper reports the dynamic characteristics of two coaxial circular cylindrical thin shells (CCTSs) containing internal or annular flowing fluid in which one shell is constructed with functionally graded materials (FGMs). A new set of shell equations are firstly used to analyse the vibration of fluid-conveying thin shells. Two cases, i.e. inner shell or outer shell is constructed with FGMs, are calculated by adopting the Galerkin’s method. The results show that as the volume fraction index increases, the natural frequencies don’t always decrease which depend on the positions of FGM shell and flowing fluid of two coaxial CCTSs. The paper indicates that the effects of FGM shell cannot be neglected for the design of FGM shells conveying fluid.

A semi-analytic model for vibro-acoustic analysis of functionally graded shells of revolution

A semi-analytic model with wider applications is developed for vibro-acoustics of submerged functionally graded material (FGM) shells of revolution, and independent or coupled FGM shells are specific cases. For structural model, the shell is divided to many narrow segments treated as conical shells. Materials vary continuously in thickness direction according to the four-parameter power-law distributions about volume fractions of two constituents. Based on the first-order shear deformation theory (FSDT), differential equations are solved via expanding displacements as power series and Fourier series in meridional and circumferential directions, respectively. Continuity and boundary conditions are assembled to vibration governing equation. For acoustic model, Helmholtz integral equation is simplified to one line integral after expanding velocity and acoustic pressure as Fourier series in circumferential direction, and the pressure is further expressed as displacements of the shell. Then, structural and acoustic models are coupled as adding the acoustic pressure to continuity conditions. Through comparing results of developed model with the ones of literature and coupled finite element–boundary element method for some FGM shells, rapid convergence and high accuracy are demonstrated. Furthermore, discrepancies between vibro-acoustic responses of FGM, homogeneous and layered shells are slight, and effects of heavy fluid become more obvious as the volume fraction of the material with large stiffness decreases.

Thermo-mechanical nonlinear transient dynamic and Dynamic-Buckling analysis of functionally graded material shell structures using an implicit conservative/decaying time integration scheme

In this paper, the nonlinear time history response and dynamic-buckling of functionally graded material (FGM) shell structures in thermal environments with temperature-dependent material properties is studied. An implicit conservative/decaying time integration scheme and a curved 8-node degenerated shell element are used for time and spatial discretization respectively. Numerical results show that the developed shell element formulation with the implicit time integration scheme is capable of solving nonlinear dynamic-buckling and large displacement dynamic problems of FGM shells in severe thermal environments. Th influence of material constituents, power-law indexes and thermo-mechanical loadings on the transient dynamic response and dynamic-buckling phenomena are considered.

Analysis of the modal frequency of a functionally graded cylindrical shell

The modal frequencies of a functionally graded material (FGM) cylindrical shell with different geometry dimensions and material constituents are studied under eight boundary conditions. The main goal is to reveal the influencing mechanism of different factors on the modal frequencies of the FGM shell. Firstly, the Voigt model is used to describe the properties of FGM. Secondly, the motion equation of the FGM shell is derived using Hamilton’s principle. Thirdly, the modal frequency equation of the FGM shell is obtained based on a trigonometric function for the eight boundary conditions. Lastly, the effects of geometry dimension, material constituents, and boundary conditions on the modal frequencies of the FGM shell are analyzed through case studies. Results indicate that the modal frequencies obtained are good agreement with those from previous literature and the finite element method. The geometry dimensions of the FGM shell, the material constituents of the FGM, and the boundary conditions are crucial to the modal frequencies. This work can serve as a reference for the design of FGM and optimization of the structure of FGM 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.

Buckling Analysis of Functionally Graded Shells under Mixed ‎Boundary ‎Conditions Subjected to Uniform Lateral Pressure

In this study, the buckling problem of shells consisting of functionally graded ‎materials (FGMs) under uniform compressive lateral pressure is solved at mixed ‎boundary conditions. After creating the FGM models, the basic differential equations ‎of FGM shells under compressive lateral pressure are derived within the scope of ‎classical shell theory (CST). The basic differential equations are solved with the help ‎of Galerkin method and the formula for the lateral buckling pressure is obtained. The ‎minimum values of the lateral buckling pressure are found numerically by minimizing ‎the obtained expression according to the numbers of transverse and longitudinal ‎waves. The accuracy is confirmed by comparing the numerical values for the lateral ‎buckling pressure of homogeneous and FGM shells with the results in the literature. ‎The influences of FGM profiles and shell characteristics on the magnitudes of lateral ‎buckling pressure are investigated in detail by performing specific numerical analyzes.

Vibro-acoustic characteristics of eccentrically stiffened functionally graded material sandwich cylindrical shell under external mean fluid

The main objective of this research work is focused on acoustic radiation analysis of functionally graded material (FGM) sandwich cylindrical shell structure reinforced by periodically eccentrically ring stiffeners. In the analysis, two common types of FGM sandwich cylindrical shell are considered, which include one with FGM face shell and homogeneous ceramic or metal core, and the other with FGM core and homogeneous face shell. Given the structure is immersed in a mean flowing fluid of velocity tangential to the accoustically deformed boundary and it is excited by a radial point force. The ring stiffeners are assumed to be identical and uniformly spaced, and the in-plane flexural and out-of-plane torsional motions of the ring stiffeners are employed to accurately describe the force moment coupling between the stiffeners and the shell. Fluid-structure coupling is considered by imposing velocity continuity condition at fluid-structure interfaces. By applying the Poisson summation formula and the Fourier transformation technique for periodic ring stiffeners FGM shell structures, the sound pressure level (SPL) about structural acoustic radiation performance index is expressed in a superposition form of space harmonics for a given wave number. The validity of the proposed theoretical model is verified by comparisons with previously published numerical results. Based on the developed theoretical mode, the influences of the FGM properties, gradient index, sandwich type, the size of ring stiffener, external mean flow and skin-core-skin ratios type on the structural acoustic radiation performance have been investigated. In addition, the results show that under the same skin-core-skin ratio and thickness, the acoustic radiation performances of type C seem better, and the larger the section size of ring stiffened, the greater the influence of structure acoustic radiation characteristics, whereas the magnitudes of the SPL curves increase with the increasing values of Mach number.

A unified semi-analytic method for vibration analysis of functionally graded shells of revolution

A unified semi-analytic method is developed for free vibration analysis of functionally graded material (FGM) shells of revolution subjected to arbitrary boundary conditions, and independent and coupled shells of revolution are specific cases. Material properties vary continuously in thickness direction according to the four-parameter power-low distributions in terms of volume fractions of two constituents. The shell is firstly divided to some narrow shell segments to be treated as conical shells. Differential equations of motion based on first-order shear deformation theory (FSDT) are solved by expanding displacements as power series and Fourier series in meridional and circumferential directions, so five displacements and five forces at cross-section are only expressed as 10 unknown coefficients. Continuity conditions and elastic boundary conditions are assembled to the final governing equation. Through comparing natural frequencies of present method with the ones of opened works for several typical FGM shells of revolution with uniform/stepped thicknesses and classic/elastic boundaries, rapid convergence, high accuracy and wide application of the developed method are demonstrated. Furthermore, influences of material and geometry parameters reveal that increasing the power-law exponent does not change mode shapes and circumferential mode numbers of fundamental frequencies, and mode shapes are slightly affected by the variation of thickness.

Free vibration of joined cylindrical–hemispherical FGM shells

Free vibration response of a joined shell system including cylindrical and spherical shells is analyzed in this research. It is assumed that the system of joined shell is made from a functionally graded material (FGM). Properties of the shells are assumed to be graded through the thickness. Both shells are unified in thickness. To capture the effects of through-the-thickness shear deformations and rotary inertias, first-order shear deformation theory of shells is used. The Donnell type of kinematic assumptions is adopted to establish the general equations of motion and the associated boundary and continuity conditions with the aid of Hamilton’s principle. The resulting system of equations is discretized using the semi-analytical generalized differential quadrature method. Considering the clamped and free boundary conditions for the end of the cylindrical shell and intersection continuity conditions, an eigenvalue problem is established to examine the vibration frequencies of the joined shell. After proving the efficiency and validity of the present method for the case of thin isotropic homogeneous joined shells, some parametric studies are carried out for the system of combined moderately thick cylindrical–spherical shell system. Novel results are provided for the case of FGM joined shells to explore the influence of power-law index and geometric properties.

Free vibration analysis of sandwich FGM shells using isogeometric B spline finite strip method

This paper presents a free vibration analysis of shell panels made of functionally graded material (FGM) in the form of the ordinary and sandwich FGM and laminated shells using the isogeometric B3-spline finite strip method (IG-SFSM). B3-spline and Lagrangian interpolation are employed along the longitudinal and transverse directions respectively in this type of finite strip. The introduced finite strip formulation is based on the degenerated shell method, which provides variable thickness, arbitrary geometries, and analysis of thin or thick shells. Validity of the obtained natural frequencies by IG-SFSM is checked by comparison with results extracted from references for similar cases in different examples. These examples incorporate several geometries, materials, boundary conditions, and continuous thickness variation. A comparison of these two kinds of results and their proximity showed that the introduced IG-SFSM is a reliable tool which can be used in analysis of shells with the aforementioned properties.

Dynamic buckling analysis of functionally graded material cylindrical shells under thermal shock

This study focuses on dynamic buckling of functionally graded material (FGM) cylindrical shells under thermal shock. The transient non-uniform temperature fields in the FGM shells subjected to dynamic thermal loading are determined using an analytic method. Based on the Hamiltonian principle, the dynamic thermal buckling problem of the FGM cylindrical shells is transformed into the symplectic space for solving. At the same time, the buckling thermal loads and buckling modes corresponding to generalized eigenvalues and eigen solutions of canonical equations can be calculated via the bifurcation conditions. The dynamic thermal buckling characteristics of the FGM cylindrical shells as well as the solving processes are given by the symplectic method. A complete dynamic buckling modes space is presented for the FGM cylindrical shells. The effects of the material gradient, parameters of structural geometry and thermal loadings on the dynamic buckling temperature are discussed.

Rotating sandwich cylindrical shells with an FGM core and two FGPM layers: free vibration analysis

The free vibration analysis of a rotating cylindrical shell with an analytical method is investigated. The shell is considered as a sandwich structure, where the middle layer is a functionally graded material (FGM) shell, and it is surrounded by two piezoelectric layers. Considering piezoelectric materials to be functionally graded (FG), the material properties vary along the thickness direction as one innovation of this study. Applying the first-order shear deformation theory (FSDT), the equations of motion of this electromechanical system are derived as the partial differential equations (PDEs) using Hamilton’s principle. Then, the Galerkin procedure is used to discretize the governing equations, and the present results are compared with the previously published results for both isotropic and FGM shells to verify the analytical method. Finally, the effects of FGM and functionally graded piezoelectric material (FGPM) properties as well as the thickness ratio and the axial and circumferential wave numbers on the natural frequencies are studied. Moreover, the Campbell diagram is plotted and discussed through the governing equations. The present results show that increasing the non-homogeneous index of the FGM decreases the natural frequencies on the contrary of the effect of non-homogeneous index of the FGPM.

3D shell model for the thermo-mechanical analysis of FGM structures via imposed and calculated temperature profiles

The exact three-dimensional (3D) shell model proposed in the present paper is able to perform the thermal stress analysis of simply-supported Functionally Graded Material (FGM) spherical and cylindrical shells, cylinders and plates. The model is based on the 3D equilibrium equations for spherical shells developed using an orthogonal mixed curvilinear coordinate system. The use of this reference system allows the investigation of cylindrical shells, cylinders and plates as particular cases of spherical shells by means of simple considerations on the radii of curvature. The 3D shell model uses a layer-wise approach and the exponential matrix method to calculate the general and the particular solutions through the thickness direction z. The system of second order differential equations in z is not homogeneous because of the thermal terms which are externally defined. The system is reduced to a group of first order differential equations in z simply redoubling the number of variables. The solution is in closed form in the in-plane directions α and β because of the hypotheses of simply-supported boundary conditions, harmonic forms for displacement and temperature fields, and isotropic behavior in the in-plane directions for functionally graded materials. In order to define the equivalent thermal load, the temperature profile through the thickness is separately defined by means of three possible ways. Using the hypothesis of temperature amplitudes imposed at the top and bottom external surfaces in steady-state conditions, the temperature profile can be: imposed as linear through the entire thickness direction, calculated by solving the 1D version of the Fourier heat conduction equation, or calculated by solving the 3D version of the Fourier heat conduction equation. The effects of different temperature profiles on the displacement and stress analyses of FGM plates and shells are here remarked. The first order differential equation system in z has not constant coefficients because of the presence of radii of curvature for shells and through-the-thickness variable elastic and thermal coefficients for the FGM layers. An appropriate number of mathematical layers is introduced to calculate the curvature influence for shells and the elastic and thermal material coefficients for FGM layers. Therefore, the system can be considered as differential equations with constant coefficients. The proposed results allow the evaluation of thickness ratio, geometry, lamination scheme, thickness material law and temperature profile effects in the related thermal stress analysis of single-layered and sandwich FGM plates, cylinders, spherical and cylindrical shells.

Axisymmetric nonlinear rapid heating of FGM cylindrical shells

Large amplitude thermally induced vibrations of cylindrical shells made of a through-the-thickness functionally graded material (FGM) are investigated in the current research. All of the thermo-mechanical properties of the FGM shell are assumed to be functions of temperature and thickness coordinate. Shell is subjected to rapid surface heating on the ceramic-rich surface while the other surface of the shell is kept at reference temperature. One dimensional heat conduction equation is constructed and solved by means of a hybrid finite difference-Crank–Nicolson algorithm. The constructed heat conduction equation is nonlinear since the thermal conductivity is temperature dependent. With the aid of first-order shear deformation shell theory under the axisymmetric Donnell kinematic assumptions and von Kármán type of strain-displacement relations, the total energy of the shell is established. Implementing the conventional Ritz method, a set of nonlinear coupled algebraic equations are obtained which govern the dynamics of the shell under thermal shock. These equations are solved in time domain using the Newmark time marching scheme and the simple Picard successive method. Parametric studies are given to explore the dynamics of an FGM cylindrical shell under thermal shock.