Functionally Graded Cylindrical Shells Research Articles

Advanced Thick FGM Plates-Cylindrical Shells in Linear Unsteady Supersonic Flow

Introduction:: Thick functionally graded (FG) plates-cylindrical shells with third-order shear deformation theory (TSDT) of displacement models under thermal vibration in linear unsteady supersonic flow are investigated by using the generalized differential quadrature (GDQ) method. Method:: The nonlinear coefficient term of TSDT models and the advanced nonlinear shear correction coefficient expression are used to study the thermal vibration of thick FG plates and cylindrical shells in linear unsteady supersonic flow. At the intersection of FG plates-cylindrical shells, the displacements, stress resultants, and moment resultants could be reasonably assumed in the continuity condition for the differential volume in linear unsteady supersonic flow, respectively. Result:: The dynamic equilibrium differential equations of thick FG plates and cylindrical shells in linear unsteady supersonic flow, respectively in terms of partial derivatives of displacements and shear rotations subjected to partial derivatives of thermal loads, mechanical loads consist of the linear supersonic aerodynamic pressure and inertia terms can be presented. Two parametric effects including environment temperature and functionally graded material (FGM) power law index on the thermal stress and center displacement of thick FG plates-cylindrical shells in linear unsteady supersonic flow are investigated. Conclusion:: The linear unsteady supersonic flow over the outer surface of thick FG platescylindrical shells provides the additional aerodynamic effect on the values of displacements and stresses.

On the wave propagation characteristics of functionally graded porous shells

This paper concentrates on the wave propagation characteristics of functionally graded (FG) porous shells. The transversal shear deformation of the shell is taken into consideration by employing a simple higher-ordered shear deformation shell theory. The equations of motion are derived for the proposed model based on Hamilton’s principle. An eigenvalue problem that relates the wave propagation elements is formulated and solved to present the various dispersion relations of FG cylindrical and spherical shells. The effects of porosities, shells’ geometrical parameters, FG material exponent, and wavenumbers on the principal wave propagation frequency and the associated phase velocity are investigated in detail.

Exact solutions for the symmetric elastoplastic response of functionally graded pressure vessels

With the development of functionally graded (FG) composite material technology, mechanical analyses of FG pressure vessels have attracted attention. Due to the non-uniform stress distributions inside such vessels, irreversible plastic deformation first occurs in some zones of the structure under the action of external loads. It is necessary to perform an elastoplastic analysis of FG structures to determine their load-bearing capacities. In this paper, exact solutions are obtained for the purely elastic, partially plastic, and fully plastic deformation of FG cylindrical and spherical shell pressure vessels under mechanical loads. The material properties (elastic modulus and yield strength) are assumed to vary nonlinearly in the radial direction of the vessel and the Poisson's ratio is assumed to be constant due to Poisson's ratio slight variations in the engineering materials. The novelty of the current work lies in the presentation of complete exact elastoplastic solutions for FG thick cylindrical and spherical shell pressure vessels, taking into account all deformation zones. Tresca's yield criterion is used to formulate four different deformation zones for an ideal plastic material. All of these states of deformation are studied in detail. It is shown that the elastoplastic response of the FG pressure vessels is notably affected by the radial variation of the material properties. With an increase in the FG parameter β, the first yielding position of the FG pressure vessels changes from the inner surface to the outer surface. The conclusions of this work should help with developing designs of FG pressure vessels to prevent yielding under excessive circumferential stresses.

Damped response of porous functionally graded viscoelastic cylindrical shells

In this article, the behavior of viscoelastic porous functionally graded (FG) shells regarding the free and damped forced vibration analysis is investigated. The differential equations are derived by using higher order shear deformation theory. Using Hamilton’s principle and the energy method, the equations of motion are obtained and solved using Navier’s method. The damping effect is implemented into the analysis by means of Kelvin and linear standard viscoelastic models. With the correspondence principle, the transition from elastic material properties to viscoelastic material properties is achieved. First, a free vibration analysis is performed to verify the accuracy of the developed algorithm and obtained results are compared with the existing studies in the literature. Afterwards, a parametric study considering two different viscoelasticity models is performed. In the first parametric example, damped forced vibration analysis is performed for simply supported FG cylindrical shell using linear standard viscoelastic model as the viscoelastic model. Then, damped forced analysis is performed using Kelvin viscoelastic model for simply supported FG cylindrical shell. Analyses are performed in Laplace domain. The obtained results are transferred to time domain using Durbin’s inverse Laplace transform method. The displacement graphs are given for the damped forced vibration examples. In the performed parametric studies, the effects of various porosity coefficients, ratios of instantaneous value, retardation times of the relaxation function and damping ratios on the analysis are investigated.

Nonlinear oscillations of a cylindrical shell with a circumferentially discontinuous elastic base

This work investigates the linear and nonlinear vibrations of a functionally graded (FG) transversally excited cylindrical shell and in contact along its length with a circumferentially discontinuous elastic base, a configuration frequently found in practical applications. The elastic medium is modeled as a Winkler foundation. The adopted distribution of the FG material along the thickness is similar to that of a sandwich shell, with one material predominating in the core and the other on both faces. First a finite element analysis (FEA) is performed to obtain the natural frequencies and vibration modes of the FG cylindrical shell and a parametric analysis is conducted to study the influence of the contact region and foundation stiffness on natural frequencies and mode shapes. Even though the problem is a symmetric, the cylindrical shell displays a sequence of symmetric and antisymmetric vibrations modes and these modes may interact due to nonlinear coupling. For much geometry the lowest vibration modes have very close natural frequencies, leading to internal resonances, which influence strongly the nonlinear shell response. To obtain a reliable reduced order model (ROM) for the nonlinear dynamic analysis, Donnell’s nonlinear shallow shell theory is considered. To discretize the nonlinear equations of motion by the Galerkin method, a consistent modal solution is obtained considering nonlinear couplings and interactions. Here, first the displacements are consistently derived from a perturbation procedure proposed in previous papers. To start the perturbation procedure, a seed solution is necessary for the transversal displacement field. For this, first the linear vibration modes obtained from the finite element analysis are expanded in Fourier series in terms of the circumferential coordinate. Then, the essential modes to be used in the ROM are identified using a mathematical procedure that combines Fourier transform and Parseval theorem. The proposed modal solution for the transversal displacements is also used to obtain the modal expansions for the axial and circumferential displacements and their modal amplitudes are determined as a function of the modal amplitudes of the transversal displacements, thus reducing even further the number of unknows. These expansions are finally used to discretize the nonlinear equation of motion in the transversal direction and the resulting equations of motion are solved numerically using continuation software AUTO. A geometry displaying multiple internal resonances is selected and resonance curves, time-responses and phase-portraits are obtained to analyze the nonlinear dynamics of the shell. Depending on the excitation, different bifurcation scenarios are obtained, with important changes in stability and resonant peaks due to modal interaction. Competition between coexisting stable attractors is investigated using basins of attraction, which exhibit a complex topology, often with fractal boundaries.

Active control of free and forced vibration of a rotating FG cylindrical shell via FG piezoelectric patches

This article aims to study the active control of free and forced vibration of a rotating functionally graded (FG) cylindrical shell using FG piezoelectric patches. Accordingly, this study uses sensors and actuators embedded in the shell’s inner and outer surfaces. Considering classical theory and Love relations, ordinary differential equations of the system are extracted through the energy method and Lagrange equation. The validation part of the study involves comparing the results of this study with previous studies of rotating and nonrotating cylindrical shells. In addition, the convergence of vibration responses of the rotating cylindrical shell is investigated. Using the LQR method and considering the appropriate value for the weighting matrices, the system’s vibration is attenuated, and the required voltage of the actuators is also obtained. Lastly, the effects of parameters such as position and performance index of piezoelectric on the closed-loop response are studied. The results show that changing piezoelectric properties in the form of a power law lead to a quicker decrease in vibration amplitude.

Free vibration analysis of bi-directional functionally graded cylindrical shells with varying thickness

The article presents a semi-analytical solution for the free vibration problem in bi-directional functionally graded (FG) cylindrical shells with varying thickness, utilizing a modified shear deformation shell theory (MSDT). In this model, the mechanical properties of the cylindrical shells change along both the axial and thickness directions. Classical analytical methods for shell structures often face limitations in terms of boundary conditions, transverse shear deformation, and rotary inertia, making the use of MSDT and Galerkin procedure necessary for solving the problem of free vibration analysis in this study. The boundary conditions of the bi-directional FG cylindrical shells are chosen to be simply supported and clamped, and the presented method is compared to relevant results in the literature to validate its effectiveness. Moreover, a comprehensive parametric investigation is conducted to evaluate the influence of material and geometrical parameters on the natural frequencies. The results show that the natural frequency of the FG cylindrical shells is significantly affected by the material and geometrical parameters, including the power-law index, the thickness variation, the aspect ratio, and the boundary conditions.

Free vibration of non-Lévy-type functionally graded doubly curved shallow shells: New analytic solutions

• Symplectic superposition framework for free vibration of FG shells is presented. • New analytic solutions of non-Lévy-type FG doubly curved shallow shells are obtained. • Frequencies of FG shells with different boundaries are provided as benchmarks. • Rational and rigorous procedure, fast convergence, and high accuracy are validated. Analytic solutions are of great significance to serve as benchmarks for validating numerical methods and to guide efficient preliminary structural designs. In this paper, we concentrate on seeking new analytic free vibration solutions of non-Lévy-type functionally graded (FG) doubly curved shallow shells, which were not available in the literature to the best of our knowledge. The main challenge in gaining such analytic solutions comes from both the governing system of partial differential equations under complicated boundary conditions. Here, we extend an analytic symplectic superposition method (SSM), which we proposed originally for plate problems, to solve the current tough issue. The solution methodology is implemented via the following three steps: converting an original free vibration problem into two elementary problems and formulating the equations of each elementary problem in the Hamiltonian system; utilizing specific mathematical techniques in the symplectic space for the analytic solutions of the elementary problems; superposing the solutions of the elementary problems to yield the eventual analytic free vibration solutions. The primary advantage of the SSM is that it does not require any assumption of solution forms, but is conducted via rigorous derivation, which is realized by the exceptional mathematical treatments, such as the symplectic space-based separation of variables and the symplectic eigen expansion. Comprehensive new free vibration solutions are presented for non-Lévy-type FG doubly curved shallow shells, through which the solutions of FG cylindrical shell panels are also reported as extreme cases. All tabulated results are well validated by the finite element method via ABAQUS software, guaranteeing the accuracy of the present solutions for benchmark use. Quantitative analyses on the natural frequencies are also conducted to reveal the effects of boundary conditions, FG models, and aspect ratios.

Review of thermoelastic, thermal properties and creep analysis of functionally graded cylindrical shell

ABSTRACT Functionally graded (FG) materials have shown excellent thermoelastic and creep performance, being considered as excellent materials for cylindrical shells subjected to thermo-mechanical loads. This review provides a detailed account of thermoelastic and creep properties of FG cylindrical shells. A review of governing equations on properties of FG materials as well as thermoelastic response of materials is provided. In depth analysis of the works carried out on the thermoelastic and creep performance of FG cylindrical under different thermal and mechanical loads (such as internal/external pressure, axial loading, etc.) is then conducted. Results obtained so far indicate the superior creep performance of FG cylindrical shells. Although many studies have been carried on thermoelastic response of FG cylindrical shells, their creep properties under different loading is yet to be extended and is a subject of interest for the future researches.

Buckling analysis of FG cylindrical nano shell integrated with CNTRC patches

Buckling analysis of a functionally graded (FG) cylindrical nano shell is studied in this paper based on nonlocal elasticity theory and first-order shear deformation theory. The nano shell is made from a combination of ceramic and metal integrated with carbon-nanotube reinforced composite (CNTRC) sheets on outer radius. The reinforced composite nano shell is resting on Pasternak’s foundation. The governing equations of motion are derived using vitual work principle. The effective material properties of carbon-nanotube reinforced composite (CNTRC) sheets are evaluated using rule of mixture. A comparative study is performed before full presentation of numerical results for verification of the model and solution procedure. A large parametric analysis is performed to investigate effect of dimensions of the core as well as CNTRC sheets, CNT volume fraction, nonlocal parameter, number of CNTRC sheets, FG-index and foundation parameters on the buckling results.

New controller (NPDCVF) outcome of FG cylindrical shell structure

This paper presents a new active controller technique effect of functionally graded (FG) cylindrical shell structure model within joint harmonic and parametric excitations. The new active controller involves Nonlinear Proportional-Derivative (NPD) plus Negative Cubic Velocity Feedback (NCVF) as a new nonlinear control method (NPDCVF). The motivation of this study is the well-known observation that NPDCVF control method can offer a means to improve the performance of plant systems. The coupled nonlinear differential equations with two modes have been derived applying the von Kármán nonlinear theory, Galerkin’s process, and the static condensation technique. Static condensation is the process of decreasing the number of free displacements or degrees of freedom. We have added other three different controller methods over the structure to select the best controller. The three applied controller methods to the considered structure are: Integral Resonant Control (IRC), Positive Position Feedback (PPF), and Nonlinear Integral Positive Position Feedback (NIPPF) which are applied to the considered structure. We establish that the best one for dipping the high vibration amplitudes is the NPDCVF as a new controller. The perturbation technique is useful to solve the present model including quadratic and cubic nonlinearities subjected to mixed harmonic and parametric forces within the simultaneous primary resonance case (Ω=ω1,Ω=ω2) as the worst resonance case of the system. All numerical outcomes have been completed with the help of MATLAB Ra 18.0 program software over the investigated model. The obtained results have proved that this new controller is excellent for improving the structure by reducing the risky vibrations caused during primary resonances. Frequency response equations have helped in observing the stability examination of the obtained numerical solutions. The actions of several factors performed on the controlled model have been examined and described. Some comparisons have also been prepared with the available recent published papers of a similar model.

Vibration analysis of ribbed and ring-stiffened FG cylindrical shell under hydrostatic pressure

This paper proposed an energy method, based on the Donnell thin shell theory, to derive the vibration characteristic equations of underwater ribbed and ring-stiffened functionally graded (FG) cylindrical shells, and the characteristic equations are solved by MATLAB to obtain their natural frequencies. Through comparative analysis with the existing literature, the correctness and effectiveness of the calculation method are verified. By using numerical examples, the effects of the shell size, material composition, rib section length to width ratio, and rib arrangement on the natural frequency are analyzed, and it has been proved that the aforementioned parameters are the key parameters influencing natural frequency. It is found that the natural frequency of the FG cylindrical shell increases with the decrease in the length–diameter ratio of the shell, increases with the increase in the shell thickness–diameter ratio, and increases with the decrease in the rib spacing. When the cylindrical shell is short and thin, the addition of longitudinal ribs can significantly change the natural frequency of the cylindrical shell.

Asymmetric Large Deformation Superharmonic and Subharmonic Resonances of Spiral Stiffened Imperfect FG Cylindrical Shells Resting on Generalized Nonlinear Viscoelastic Foundations

This paper is devoted to superharmonic and subharmonic behavior investigation of imperfect functionally graded (FG) cylindrical shells with external FG spiral stiffeners under large amplitude excitations. The structure is embedded within a generalized nonlinear viscoelastic foundation, which is composed of a two-parameter Winkler–Pasternak foundation augmented by a Kelvin–Voigt viscoelastic model with a nonlinear cubic stiffness, to account for the vibration hardening/softening phenomena and damping considerations. The von Kármán strain-displacement kinematic nonlinearity is employed in the constitutive laws of the shell and stiffeners. The external spiral stiffeners of the cylindrical shell are modeled according to the smeared stiffener technique. The coupled governing equations are solved by using Galerkin’s method in conjunction with the stress function concept. The multiple scales method is utilized to detect the subharmonic and superharmonic resonances and the frequency–amplitude relations of the 1/3 and 1/2 subharmonic and 3/1 and 2/1 superharmonic resonances of the system. Finally, the influences of the stiffeners helical angles, foundation type, coefficient of the nonlinear elastic foundation, material distribution, and excitation amplitude on the system resonances are investigated comprehensively.

Active Vibration Control of a Fluid-Conveying Functionally Graded Cylindrical Shell using Piezoelectric Material

Active vibration control of a smart FG (functionally graded) cylindrical shell conveying fluid in thermal environment is studied theoretically by using a laminated piezoelectric actuator. Velocity feedback control law is implemented to activate the piezoelectric actuator. Considering the electric-thermo-fluid-structure interaction effect, a nonlinear dynamic model of the smart fluid-conveying FG cylindrical shell is developed based on Hamilton’s principle and von-Karman type geometrical nonlinear relationship. The inviscid, incompressible, isentropic and irrotational fluid is coupled into governing equations using the linearized potential theory. The Galerkin’s method is used to obtain the nonlinear governing equations of motion of the coupled system. The multiple time scales approach is applied to solve the resulting governing equations for analysing the nonlinear dynamic characteristics of the coupled system. The influence of fluid flow velocity, feedback control gains of piezoelectric voltage, external excitation and material properties of FGM on the frequency-response curves of system are investigated. The results indicate that the piezoelectric voltage is an effective controlling parameter for vibration control of the system, and the flow velocity can effect significantly the vibration amplitude and nonlinearity of the coupled system.

Mechanical Properties of Al 25 wt.% Cu Functionally Graded Material

Abstract The present work refers to describe the effects of Al2Cu variations on various properties of thick-walled functionally graded (FG) cylindrical shell. Al-25 wt.% Cu hypo-eutectic alloy ingot is melted and centrifugally casted to obtain high entropy FG composite. A series of microstructure examinations such as FESEM and EDX analysis were carried out to determine the distributions of constituent phases and elements. It is revealed that the maximum volume fraction of Al2Cu particle is reached near the inner surface with 35.7 Vol.% and then reduces gradually to 32.5 Vol.% at the outer surface of FG cylindrical shell. The effects of the variations Al2Cu along radial direction of FG tube are discussed through Vickers hardness, wear rate, coefficient of thermal expansion and compressive test measurements. The experimental results show that the wear and hardness are varied in graded manner which the highest wear resistance with wear rate of 9.1×10−5 g/mm 2 and hardness with 153HV are found towards Al2Cu enriched zone or inner periphery. Moreover, the studied FG cylindrical shell shows drop 2.5% in yield stress and 4.5% in elastic modulus from intermediate to inner layers due to Al2Cu particles clustering in metal matrix.

Parametric Instability of a Rotating Axially Loaded FG Cylindrical Thin Shell Under Both Axial Disturbances and Thermal Effects

Abstract Parametric instability of a rotating functionally graded (FG) cylindrical thin shell with axial compression under various boundary conditions is studied in this article. In particular, the shell is subjected to both axial periodic displacement disturbances and a thermal environment. The initial hoop tension and Coriolis effects due to rotation are also considered. The coupled dynamic equations of the shell under multiple conditions are formulated based on Love’s thin-shell theory. The instability boundaries of the shell with different boundary conditions considering thermal factors, axial disturbances, and other system parameters are obtained analytically under the case of primary and combination resonance; numerical illustrations are also given. It is found that high temperature weakens the stability of the system, while axial disturbances show stronger influence on the instability regions of the shell compared to other parameters such as thermal factors and the angular rotation velocity.

Decoupled stability equation for buckling analysis of FG and multilayered cylindrical shells based on the first-order shear deformation theory

Based on the first-order shear deformation and Donnell's shell theory with von Karman non-linearity, one decoupled stability equation for buckling analysis of functionally graded (FG) and multilayered cylindrical shells with transversely isotropic layers subjected to various cases of combined thermo-mechanical loadings is developed. To this end, the equilibrium equations are uncoupled in terms of the transverse deflection, the force function and a new potential function. Using the adjacent equilibrium method, one decoupled stability equation which is an eighth-order differential equation in terms of transverse deflection is obtained and conveniently solved to present analytical expressions for buckling loads of cylindrical shells under any type of loading. These analytical expressions can be used in design and as a benchmark in numerical studies. For numerical purpose, the formulation is applied to FG shells, a three-layer shell laminated of transversely isotropic layers, and sandwich shells with an isogrid lattice core and transversely isotropic face sheets. The results are validated with the existing ones in the literature. Finally, the effect of different parameters on the buckling loads in the presence of combined loadings is discussed in detail. Numerical results show that the existence of an initial hoop stress as a main type of initial imperfection has significant influence on the buckling behavior (including buckling values and mode shapes) of long cylindrical shells subjected to axial loading and this effect reduces for cylindrical shells with a lattice core. Furthermore, the critical values of torsion significantly reduce under combined loading.

Characterization of Al-Al2Cu functionally graded material produced by using horizontal centrifugal casting

Purpose The purpose of this paper is to investigate the characterizations of high energy thick-walled functionally graded (FG) cylinder containing Al-26%Cu fabricated by horizontal centrifugal casting technique. Design/methodology/approach Field emission scanning electron microscopy in conjunction with image analyser software and energy dispersion spectroscopy is applied to measure the variations of constituent phase’s content and elemental ratios along the radial direction, respectively. Distributions of the FG properties are measured through hardness, CTE, E and σy along the radial direction to investigate the mechanical and physical properties corresponding to the variations in microstructure. In addition, the variations of wear rate along the thickness are evaluated through a series of dry sliding wear tests using the pin-on-disk wear machine. Moreover, scanning electron microscopy is employed to characterize the worn-out surfaces and morphology of wear debris in order to clarify the dominant operative wear mechanism. Findings Results showed that Al2Cu content gradually decreases from the inner wall containing 33.3 vol.% to outer wall containing 26.4 vol.% in the FG cylindrical shell. The elastic modulus and yield strength measured through compression tests reveal that these mechanical properties are limited up to certain value of Al2Cu. The obtained optimum value of Al2Cu content for studied Al-Al2Cu FG is almost 31 vol.%. Originality/value The obtained optimum value of Al2Cu content for studied Al-Al2Cu FG was almost 31 vol.%.

Free vibration analysis of coupled functionally graded (FG) doubly-curved revolution shell structures with general boundary conditions

In this paper, free vibration behavior of coupled functionally graded (FG) doubly-curved revolution shell structures with general boundary conditions is studied by the using unified Jacobi-Ritz method. The first-order shear deformation theory in conjunction with a multilevel partition technique is adopted to establish the theoretical model. The substructure of the theoretical model mainly includes the FG elliptical, hyperbolical, paraboloidal and cylindrical shells and three kinds of coupled FG shell structures containing paraboloidal-cylindrical shells, elliptical-cylindrical shells and hyperbolical-cylindrical shells are also considered in actual calculation. To obtain the continuous conditions at the interface and satisfy the arbitrary boundary conditions, the boundary and coupling spring techniques are adopted in this paper. In despite of the shell components and the boundary conditions, a mix function which is with the Jacobi polynomials along the meridional direction and the standard Fourier series along the circumferential direction is used as the admissible displacements of each shell segment. The convergence and comparison studies for the FG doubly-curved revolution shell structures with different boundary conditions, coupling parameters and Jacobi parameters are carried out to verify the reliability and accuracy of the present solutions. To enhance the understanding of the titled problem, some mode shapes of coupled FG shell structure are provided. Through comparative analysis, including the experimental and numerical comparison, it is obvious that the current method has good stability and rapid convergence properties and the present results agree closely with those reference results no matter what the frequency parameters or mode shapes are. The influence of the geometric dimensions and material constants on the vibration behavior of coupled FG shell structure is also reported.

Parametric resonance of a FG cylindrical thin shell with periodic rotating angular speeds in thermal environment

• Researching parametric resonance of a FG cylindrical thin shell. • The shell considers both periodic rotating angular speeds and thermal environment. • Studying combined effects of rotation and thermal effects on parametric instability. • Analyzing primary and combination resonances of the shell systematically. • Discussing critical rotating angular speeds of rotating FG cylindrical thin shells. Parametric resonance of a functionally graded (FG) cylindrical thin shell with periodic rotating angular speeds subjected to thermal environment is studied in this paper. Taking account of the temperature-dependent properties of the shell, the dynamic equations of a rotating FG cylindrical thin shell based upon Love's thin shell theory are built by Hamilton's principle. The multiple scales method is utilized to obtain the instability boundaries of the problem with the consideration of time-varying rotating angular speeds. It is shown that only the combination instability regions exist for a rotating FG cylindrical thin shell. Moreover, some numerical examples are employed to systematically analyze the effects of constant rotating angular speed, material heterogeneity and thermal effects on vibration characteristics, instability regions and critical rotating speeds of the shell. Of great interest in the process is the combined effect of constant rotating angular speed and temperature on instability regions.