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

A postbuckling analysis is presented for a functionally graded cylindrical shell with piezoelectric actuators subjected to lateral or hydrostatic pressure combined with electric loads in thermal environments. Heat conduction and temperature-dependent material properties are both taken into account. The temperature field considered is assumed to be a uniform distribution over the shell surface and varied in the thickness direction and the electric field considered only has non-zero-valued component E_Z. The material properties of functionally graded materials (FGMs) 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, and the material properties of both FGM and piezoelectric layers are assumed to be temperature-dependent. The governing equations are based on a higher order shear deformation theory with a von Kármán–Donnell-type of kinematic nonlinearity. A boundary layer theory of shell buckling is extended to the case of FGM hybrid laminated cylindrical shells of finite length. A singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of pressure-loaded, perfect and imperfect, FGM cylindrical shells with fully covered piezoelectric actuators under different sets of thermal and electric loading conditions. The results reveal that temperature dependency, temperature change and volume fraction distribution have a significant effect on the buckling pressure and postbuckling behavior of FGM hybrid cylindrical shells. In contrast, the control voltage only has a very small effect on the buckling pressure and postbuckling behavior of FGM hybrid cylindrical shells.     

Mechanical buckling of functionally graded material cylindrical shells surrounded by Pasternak elastic foundation

In this study, the mechanical buckling of functionally graded material cylindrical shell that is embedded in an outer elastic medium and subjected to combined axial and radial compressive loads is investigated. The material properties are assumed to vary smoothly through the shell thickness according to a power law distribution of the volume fraction of constituent materials. Theoretical formulations are presented based on a higher-order shear deformation shell theory (HSDT) considering the transverse shear strains. Using the nonlinear strain–displacement relations of FGMs cylindrical shells, the governing equations are derived. The elastic foundation is modelled by two parameters Pasternak model, which is obtained by adding a shear layer to the Winkler model. The boundary condition is considered to be simply-supported. The novelty of the present work is to achieve the closed-form solutions for the critical mechanical buckling loads of the FGM cylindrical shells surrounded by elastic medium. The effects of shell geometry, the volume fraction exponent, and the foundation parameters on the critical buckling load are investigated. The numerical results reveal that the elastic foundation has significant effect on the critical buckling load.     

Thermomechanical postbuckling analysis of functionally graded plates and shallow cylindrical shells

Summary. In this paper, an analytic solution is provided for the postbuckling behavior of plates and shallow cylindrical shells made of functionally graded materials under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell according to a power law distribution of the volume fraction of the constituents. The fundamental equations for thin rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection, and the solution is obtained in terms of mixed Fourier series. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behavior and stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.     

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

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.     

Dynamic characteristics of fluid-conveying functionally graded cylindrical shells under mechanical and thermal loads

Considering rotary, in-plane inertias, and fluid velocity potential, the dynamic characteristics of fluid-conveying functionally graded materials (FGMs) cylindrical shells subjected to dynamic mechanical and thermal loads are investigated, where material properties of FGM shells are considered as graded distribution across the shell thickness according to a power-law, and dynamic thermal loads applied on the shell is considered as non-linear distribution across the thickness of the shell. The linear response characteristics of fluid-conveying FGM cylindrical shells are obtained by using modal superposition and Newmark’s direct time integration method.     

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

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.     

The elastic response of functionally graded cylindrical shells to low-velocity impact

This paper deals with the problem of functionally graded (FG) cylindrical shells subjected to low-velocity impact by a solid striker. An analytic solution to predict the impact response of the FG cylindrical shells with one layer or multi-layers is presented. The solution includes both contact deformation and transverse shear deformation. The effective material properties of functionally graded materials (FGMs) for the cylindrical shells are assumed to vary continuously through the shell thickness and are graded in the shell thickness direction according to a volume fraction power law distribution. This is implemented in the governing equation of motion and thus included in the present solution. Four types of FG cylindrical shells composed of stainless steel and silicon nitride are configured and their transient responses to impact are computed using the present solution. The effects of the constituent volume fraction and the FGM configuration on the transient response of the laminated cylindrical shell induced by impact are examined.     

功能梯度板壳的力学研究进展

功能梯度材料(FGM)是一种材料组分或微观结构沿一个或几个方向连续变化的新型复合材料。FGM中基本消除了宏观界面,同时具有很好的可设计性,设计人员可以有目的地改变材料组成,以获得所期望的性能。由FGM构成的功能梯度板壳结构在许多工程领域中有着广阔的应用前景,评述了FGM板壳结构的弯曲、屈曲和后屈曲、振动和动力稳定性等力学问题研究的发展现状,阐述了常用方法和理论的优缺点,并提出了未来的发展趋势和需要研究的方向。     

Nonlinear bending and postbuckling of FGM cylindrical panels subjected to combined loadings and resting on elastic foundations in thermal environments

A nonlinear analysis is presented for FGM cylindrical panels resting on elastic foundations subjected to the combined actions of uniform lateral pressure and compressive edge loads in thermal environments. The two cases of postbuckling of initially pressurized FGM cylindrical panels and of nonlinear bending of initially compressed cylindrical panels are considered. Heat conduction and temperature-dependent material properties are both taken into account. Material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction based on Mori-Tanaka micromechanics model. The formulations are based on a higher order shear deformation theory and von Kármán strain displacement relationships. The panel-foundation interaction and thermal effects are also included. The governing equations are solved by a singular perturbation technique along with a two-step perturbation approach. The numerical illustrations concern the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels with two constituent materials resting on Pasternak elastic foundations. The effects of volume fraction index, temperature variation, foundation stiffness as well as initial stress on the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels are discussed in detail.     

Postbuckling analysis of axially-loaded functionally graded cylindrical shells in thermal environments

A postbuckling analysis is presented for a functionally graded cylindrical thin shell of finite length subjected to compressive axial loads and in thermal environments. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The governing equations are based on the classical shell theory with von Kármán–Donnell-type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of functionally graded cylindrical shells. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of axially-loaded, perfect and imperfect, cylindrical thin shells with two constituent materials and under different sets of thermal environments. The effects played by temperature rise, volume fraction distribution, shell geometric parameter, and initial geometric imperfections are studied.     

A mesh-free method for analysis of the thermal and mechanical buckling of functionally graded cylindrical shell panels

The buckling response of functionally graded ceramic-metal cylindrical shell panels under axial compression and thermal load is presented here. The formulation is based on the first-order shear deformation shell theory and element-free kp-Ritz method. The material properties of shell panels are assumed to vary through their thickness direction according to a power-law distribution of the volume fraction of constituents. Approximations of the displacement field are expressed in terms of a set of mesh-free kernel particle functions. A stabilized conforming nodal integration approach is employed to estimate the bending stiffness, and the shear and membrane terms are evaluated using a direct nodal integration technique to eliminate membrane and shear locking for very thin shells. The mechanical and thermal buckling responses of functionally graded shell panels are investigated, and the influences of the volume fraction exponent, boundary conditions, and temperature distribution on their buckling strengths are also examined.     

On the stability of FGM shells subjected to combined loads with different edge conditions and resting on elastic foundations

In this study, the stability analysis of functionally graded material (FGM) cylindrical, truncated and complete conical shells subjected to combined loads and resting on elastic foundations for two boundary conditions is investigated. The functionally graded material properties are assumed to vary continuously through the thickness of the conical shell. At first, the basic relations, the stability and compatibility equations of the FGM truncated conical shell on the Pasternak-type elastic foundation are obtained. By applying the Galerkin method to the foregoing equations, the critical combined loads of clamped–clamped and sliding–sliding FGM shells on the Pasternak-type elastic foundation are obtained. Finally, carrying out some computations, effects of the elastic foundation, boundary conditions, the variation of shell characteristics and material composition profiles on the values of critical combined loads have been studied.     

Creep buckling and post-buckling analyses for a hybrid laminated viscoelastic FGM cylindrical shell under in-plane loading

In this paper, creep buckling and post-buckling of a hybrid laminated viscoelastic functionally graded material (FGM) cylindrical shell under in-plane loading are investigated. Considering the high-order transverse shear deformation and geometric nonlinear theory, the von Karman geometric relation of the hybrid laminated viscoelastic FGM cylindrical shell with initial deflection is established. Based on the Donnell theory, elastic piezoelectric theory and Boltzmann superposition principle, nonlinear creep governing equations of the hybrid laminated viscoelastic FGM cylindrical shell under in-plane loading are derived. By means of the finite difference method and the Newton–Newmark method, the problem for creep buckling and post-buckling of the laminated shell’s structure is solved. Numerical results are presented to show effects of geometric parameters, power law index and loading on creep buckling and post-buckling of the hybrid laminated viscoelastic FGM cylindrical shell.     

Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments,Part I: Axially-loaded shells

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to axial compression in thermal environments. Two kinds of carbon nanotube-reinforced composite (CNTRC) shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The governing equations are based on a higher order shear deformation theory with a von Kármán-type of kinematic nonlinearity. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of axially-loaded, perfect and imperfect, FG-CNTRC cylindrical shells under different sets of thermal environmental conditions. The results for UD-CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell. The results show that the linear functionally graded reinforcements can increase the buckling load as well as postbuckling strength of the shell under axial compression. The results reveal that the CNT volume fraction has a significant effect on the buckling load and postbuckling behavior of CNTRC shells.     

Magnetothermoelastic Bending Analysis of a Functionally Graded Material Cylindrical Shell

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.     

Nonlinear dynamics of FGM circular cylindrical shell with clamped-clamped edges

An analysis on the nonlinear dynamics of a clamped-clamped FGM circular cylindrical shell subjected to an external excitation and uniform temperature change is presented in this paper. Material properties of the constituents are assumed to be temperature-independent and the effective properties of FGM cylindrical shell are graded in thickness direction according to a simple power law function in terms of the volume fractions. Based on the first-order shear deformation shell theory and von Karman type nonlinear strain-displacement relationship, the nonlinear governing equations of motion are derived by using Hamilton’s principle. Galerkin’s method is then 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. Numerical results including the bifurcations, waveform, phase plots and Poincare maps are presented for clamped-clamped FGM cylindrical shells showing the influences of material gradient index, the thickness and the external loading on the nonlinear dynamics.     

Nonlinear static and dynamic buckling analysis of functionally graded shallow spherical shells including temperature effects

This paper presents an analytical approach to investigate the nonlinear static and dynamic unsymmetrical responses of functionally graded shallow spherical shells under external pressure incorporating the effects of temperature. Governing equations for thin FGM spherical shells are derived by using the classical shell theory taking into account von Karman–Donnell geometrical nonlinearity. Approximate solutions are assumed and Galerkin procedure is applied to determine explicit expressions of static critical buckling loads of the shells. For the dynamical response, motion equations are numerically solved by using Runge–Kutta method and the criterion suggested by Budiansky–Roth. A detailed analysis is carried out to show the effects of material and geometrical parameters, boundary conditions and temperature on the stability and dynamical characteristics of FGM shallow spherical shells.     

Dynamic response of an FGM cylindrical shell under moving loads

This paper presents an analytical study on the dynamic behavior of the infinitely-long, FGM cylindrical shell subjected to combined action of the axial tension, internal compressive load and ring-shaped compressive pressure with constant velocity. It is assumed that the cylindrical shell is a mixture of metal and ceramic that its properties changes as a function of the shell thickness. The problem is studied on the basis of the theory of vibrations of cylindrical shells. Derived formulas for the maximum static and dynamic displacements, dynamic factors and critical velocity for the FGM cylindrical shell subjected to moving loads. Numerical calculations have been made for fully metal, fully ceramic and FGM (Si₃N₄/SUS304) cylindrical shells. A parametric study is conducted to demonstrate the effects of the material property gradient, the radius to thickness ratio and the velocity of the moving load on the dynamic displacements and dynamic factors of the inner and ring-shaped pressures for FGM cylindrical shells.     

复合载荷作用变刚度复合材料回转壳屈曲优化

通过曲线纤维轨迹设计,变刚度复合材料回转壳将拥有比常刚度（直线纤维）回转壳更好的抗屈曲稳定性,为此,研究了复合载荷作用下曲线纤维铺层形式和几何参数对变刚度复合材料回转壳屈曲性能的影响规律。首先根据回转壳横截面圆弧变化改进曲线纤维角度线性描述方法,建立了变刚度复合材料回转壳的参数化有限元模型;其次,结合序列二次响应面方法和回转壳屈曲优化模型,搭建了复合材料回转壳曲线纤维轨迹优化的设计流程;最后,以准各向同性铺层复合材料回转壳为比较基准,对弯扭载荷作用变刚度圆柱壳和轴压、弯矩和扭矩分别作用变刚度椭圆柱壳在不同铺层方式、不同几何参数下的屈曲性能进行了优化比较。结果表明:弯扭载荷作用下,变刚度圆柱壳的屈曲性能随弯矩载荷占比增加而提高,且均好于准各向同性圆柱壳,但扭矩载荷占优时,优化常刚度圆柱壳的屈曲性能更具有优势;不同载荷作用下,具有较小截面方向比的变刚度椭圆柱壳屈曲性能要明显好于对应的准各向同性椭圆柱壳,且横截面越接近圆形,曲线纤维对椭圆柱壳屈曲性能的改善越弱。      

Effects of uniform interior pressure distribution on vibration of FGM cylindrical shell with rings support based on first-order theory subjected to ten boundary conditions

This paper presents the study on natural frequency characteristics of a thin-walled functionally graded material (FGM) cylindrical shell with rings support under symmetric uniform interior pressure distribution. The FGM properties are graded along the thickness direction of the shell. The FGM shell equations with rings support and interior pressure are established based on first-order shear deformation theory. The governing equations of motion were employed, using energy functional and by applying the Ritz method. Ten boundary conditions represented by end conditions of the FGM shell are the following: simply supported-simply supported, clamped-clamped, free-free, clamped-free, clamped-simply supported, free-simply supported, sliding-sliding, sliding-simply supported, sliding-free and sliding-clamped. This problem was solved with computer programming using MAPLE package for numerical investigation. Comparison of the results is carried out to verify the validity of the proposed procedure with published works. The influence of interior pressure, ring support position and number of rings support, and effect of the ten boundary conditions on natural frequency characteristics are studied. The results presented can be used as an important benchmark for researchers to validate their numerical methods when studying natural frequencies of shells with ring and pressure.