Performance of FGM bilayered cylindrical shell placed on cantilever edge

In most published papers, in order to obtain the analytical solution of the crack problems in functionally graded materials (FGMs), the thermomechanical properties of FGMs are usually assumed to be very particular functions and, hence, may not be physically realizable for many actual material combinations. Very few analytical methods can be used to solve the thermal shock crack problem of an FGM cylindrical shell or plate with general thermomechanical properties. In this article, a set of analytical methods is proposed for the thermal shock crack problem of an FGM plate or cylindrical shell with general thermomechanical properties. The crack problem of a cylindrical shell is modeled by a plate on an elastic foundation. Greatly different from previous studies, a set of analytical methods using both the perturbation method and a piecewise model are developed to obtain the transient temperature field and thermal stress intensity factor (TSIF). The perturbation method is applied to deal with the general thermal properties and the piecewise model is used to deal with the general mechanical properties. In the analytical procedure, integral transform, the residue theorem, and the theory of singular integral equation are used. Several representative examples are considered to check the capability of the present method. The transient thermal shock behavior of a ZrO2/Ti-6Al-4 V FGM plate with a surface crack and a Rene 41-Zirconia FGM cylindrical shell with a circumferential crack are analyzed.

The increased use of composites in aerospace and civil engineering has resulted in growing demand for engineers versed in the advanced design of composite structures. Composite structures are understood in the wide sense, including microstructured materials, composite structural shapes, composite members built up by subelements of various materials, laminate or sandwich members, small-scale composite structures, etc. Compositelike structural elements are increasingly used in traditional engineeringfields, such as civil engineering, mechanical engineering, or aeronautical engineering, and they have also found some new applications with some recent developments, such as biomedical engineering or biomechanics, nanotechnology, etc. This special issue is primarily based on the presentations given in the symposium on the stability of composite structures organized by the EngineeringMechanics Institute (EMI) Stability Committee during the ASCE EMI Annual Conference held in Boston on June 2–4, 2011. This symposium has provided a forum to discuss recent advances and address the future prospects in the area of the stability of composite structures. The symposium has covered a large number of topics related to the stability of composite structures including the following: • Buckling of composite members; • Plate buckling; • Buckling and vibration of thin-walled structures; • Shear and large displacement analysis; • Interactive buckling and nonlocal mechanics; • Shear effects for in-plane and out-of-plane analyses; • Inelastic buckling; • Anisotropic effects, microstructured materials, and stability problems; • Buckling of sandwich structures; • Stability of partially composite members and delamination effects; • Postbuckling; and • Dynamic buckling. After going through an external review process, the following papers have been accepted and are included in this special issue on the stability of composite structures. As indicated by the title of this special issue, most of the presented papers uniquely deal with the challenges in stability analysis brought by the complexity of composite materials and structures. Their specific advances by these contributions are briefly summarized as follows: 1. In “Local Buckling Analysis of Restrained Orthotropic Plates under Generic In-Plane Loading,” Qiao et al. investigated analytically and numerically the local buckling of composite orthotropic plates under generic in-plane loading and with rotationally restrained boundary conditions. They used both semianalytical and finite-element method (FEM) numerical approaches to characterize the stability behavior of laminated plates or panels of fiber-reinforced plastic structural shapes. The approximated semianalytical solution is obtained from the variationally based Rayleigh-Ritz method, and it is successively compared with available results published in the literature. 2. In “Modeling of Interactive Buckling in Sandwich Struts with Functionally Graded Cores,” Yiatros et al. studied the interactive buckling in sandwich struts with functionally graded cores. The postbuckling behavior is analyzed for this sandwich column composed of two Euler-Bernoulli extensible columns connected with a shear functionally graded core. Linearized and nonlinear analyses are presented for this nonlinear elastic structural problem, and the possibility of capturing secondary bifurcations for such nonlinear systems is numerically shown. This nonlinear model is compared successively to some FEM results based on two-dimensional analyses for the core behavior. 3. In “Buckling Loads of Two-Layer Composite Columns with Interlayer Slip and Stochastic Material Properties,” Schnabl et al. presented an efficient stochastic buckling model for studying the structural reliability of layered composite columns with interlayer slip between the layers and random material and loading parameters. The deterministic model is composed of two connected elastic columns with the shear effect neglected for each column. The nonderterministic approach is based on the exact buckling model, response surface method, and Monte Carlo simulations. The probability of failure for this stochastic buckling problem is computed, and it is shown to be very sensitive to loading and material distribution parameters. 4. In “Effect of Fiber Orientation on Buckling and First-Ply Failures of Cylindrical Shear-Deformable Laminates,” Cagdas and Adali investigated the effect of fiber orientation on buckling and first-ply failures of cylindrical shear-deformable laminates using an 8-node shell finite element (FE). The best ply angle is optimized for each stacking sequence to maximize the failure load. They identified that the rotational restraints at the curved edges have a pronounced effect on the failure load. 5. In “Thermal Postbuckling of Shear Deformable FGM Cylindrical Shells Surrounded by an Elastic Medium,” Shen incorporated the additional effect of an elastic medium on the thermal postbuckling of shear deformable functionally graded material (FGM) cylindrical shells. The surrounding elastic medium is modeled as a Pasternak foundation. The nonlinear governing equations are solved using a singular perturbation technique. The numerical results show that in some cases the FGM cylindrical shell with an intermediate volume fraction index does not have an intermediate buckling temperature and thermal postbuckling strength. In the case of heat conduction, the nonlinear equilibrium path for geometrically perfect FGM shells with simply supported boundary conditions is no longer of the bifurcation type. 6. In “Generalized Beam Theory to Analyze the Vibration of Open-Section Thin-Walled Composite Members,” Silvestre

An analysis on the nonlinear dynamics of a cantilever functionally graded materials (FGM) cylindrical shell subjected to the transversal and static in-plane pre-applied excitation is presented in a thermal environment. Material properties are assumed to be temperature-dependent. Based on Reddy's first-order shell theory, the nonlinear governing equations of motion for the FGM cylindrical shell are derived using Hamilton's principle. Galerkin's method is utilized to discretize the governing partial equations to a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms under combined external excitations. It is desirable to choose a suitable mode function to satisfy the first two modes of transverse nonlinear oscillations and the boundary conditions for the cantilever FGM cylindrical shell. The effects of external excitations on the internal resonance relationship and nonlinear dynamic response of FGM cylindrical shell are studied. The resonant case considered here is 1:2 internal resonance. A numerical method is used to find the nonlinear dynamic responses of the cantilever FGM cylindrical shell with different volume fraction index. It is found that the vibration amplitudes are not very sensitive to the value of the volume fraction index in the cases of 1:2 internal resonance. With the increasing of the volume fraction index, the value of and increased remarkably. There is a boundary value of the forcing amplitude, which divides the response into two parts for the case of different volume fraction index. One is the multiple periodic motion of the FGM cantilever cylindrical shell and the other is the quasi-period response. A larger value of volume fraction index leads to a larger region of quasi-period response.

In this paper, an analysis on nonlinear dynamics of a simply supported functionally graded material (FGM) cylindrical shell subjected to the different excitation in thermal environment. Material properties of cylindrical shell are assumed to be temperature-dependent. Based on the Reddy’s third-order plates and shells theory[1], the nonlinear governing partial differential equations of motion for the FGM cylindrical shell are derived by using Hamilton’s principle. Galerkin’s method is utilized to transform the partial differential equations into a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms under combined parametric and external excitation. The effects played by different excitation and system initial conditions on the nonlinear vibration of the cylindrical shell are studied. In addition, the Runge–Kutta method is used to find out the nonlinear dynamic responses of the FGM cylindrical shell.

M IRZAVAND and Eslami [1] have presented the thermal buckling analysis of simply supported functionally graded material (FGM) cylindrical shells that are integratedwith the surfacebonded piezoelectric actuators. Shen [2] reported the thermal postbuckling behavior of FGM cylindrical shells based on the thirdorder theory of shell via a singular perturbation method. Shen et al. [3,4] also made a study on the postbuckling response of an FGM cylindrical shell embedded in a Pasternak elastic medium and under mechanical load in thermal environments. The buckling analysis of FGM truncated conical shells subjected to combined axial extension loads and hydrostatics pressure and resting on the Pasternak-type elastic foundationwere studied analytically by Sofiyev [5]. Recently, the present authors reported the explicit expressions for mechanical buckling loads of thick FGM cylindrical shells in contact with a Pasternak-type elastic medium [6]. In the present Note, buckling of cylindrical shells made of FGM in contact with the Pasternak elastic foundation subjected to uniform temperature rise is investigated. The material properties of an FGM shell are assumed to be temperature-dependent and vary continuously as a power form through the thickness of a shell. The boundary conditions are assumed to be a fixed simply supported type. The equilibrium and stability equations are obtained, and stability equations are reduced to one equation. The investigation ends in a closed-form solution for the FGM cylindrical shells under uniform thermal load, and numerical results are presented.

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.

Sound radiation of a functionally graded material cylindrical shell in water by mobility method

Postbuckling of axially loaded FGM hybrid cylindrical shells in thermal environments

Postbuckling of pressure-loaded FGM hybrid cylindrical shells in thermal environments

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.

An analytic method for bending analysis of a cylindrical shell composed by a functionally graded material (FGM), placed in a uniform magnetic field, subjected to mechanical and thermal loads is presented. Based on the classical linear shell theory, the equations with the radial deflection and horizontal displacement are derived, and exact solution of the magnetothermoelastic bending problem is obtained. For the FGM cylindrical shell with fixed and simply supported boundary conditions, the effects of mechanical load, thermal load, magnetic intensity, volume exponent, and geometric parameters on the bending deformation of the FGM cylindrical shell are discussed.

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.

An analysis on nonlinear vibrations of a simply supported functionally graded materials (FGM) cylindrical shell at its two ends which subjected to the radial excitation in the uniform thermal environment is presented for the first time. Materials properties of the constituents are graded in the thickness direction according to a power-law distribution. In the framework of the First-order shear deformation shell theory, the nonlinear governing equations of motion for the functionally graded materials cylindrical shell is derived by using the Hamilton’s principle. The Galerkin’s method is utilized to discretize the governing equations of motion to a five-degree-of-freedom nonlinear system under combined thermal and external excitations. By using the numerical method, the five-degree-of-freedom nonlinear system is analyzed to find the nonlinear responses of the FGMs cylindrical shell.

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.

In this paper, a layer-wise finite element formulation is developed for the analysis of a functionally graded material (FGM) cylindrical shell with finite length under dynamic load. For this purpose, FGM cylinder is divided into many sub-layers and then the general layerwise laminate theory is formulated by introducing piecewise continuous approximations through the thickness for each state In this model the radial displacement is approximated linearly through each “mathematical” layer. The properties are controlled by volume fraction that is an exponential function of radius. The governing equations are derived from virtual work statement and solved by finite element method. The main contribution of the present study is to develop a discrete layerwise finite element for a 2-dimensional thick FGM cylindrical shell. Results are obtained for the time history of the displacement and stress components with different exponent “n” of functionally graded material. In addition, natural frequency and mean velocity of the radial wave propagation for different exponent “n” of functionally graded material (FGM) are studied and compared with similar ones currently obtained for FGM cylindrical shell of infinite length.