Mixed-interpolated finite elements for functionally graded cylindrical shells

In this paper cylindrical shells made of functionally graded materials (FGMs) are studied. A two-constituent material distribution through the thickness is considered, varying with a simple power rule of mixture. The equations governing the FGMs shells are determined using a variational formulation arising from the Naghdi theory. Moreover a strategy to achieve an improved transverse shear factor is investigated by energy equivalence. To approximate the problem a family of mixed-interpolated finite elements is used. It is based on a suitable reduction of the shear and membrane energy. Several numerical simulations are carried out in order to show the capability of the proposed elements to capture the properties of shells of various gradings, subjected to thermo-mechanical loads.     

A quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates

This paper presents an original hyperbolic sine shear deformation theory for the bending and free vibration analysis of functionally graded plates. The theory accounts for through-the-thickness deformations.     

Stress intensity factors for functionally graded solid cylinders

This paper presents the mode I stress intensity factors for functionally graded solid cylinders with an embedded penny-shaped crack or an external circumferential crack. The solid cylinders are assumed under remote uniform tension. The multiple isoparametric finite element method is used. Various types of functionally graded materials and different gradient compositions for each type are investigated. The results show that the material property distribution has a quite considerable influence on the stress intensity factors. The influence for embedded cracks is quite different from that for external cracks.     

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.     

Finite element analysis of thermoelastic contact problem in functionally graded axisymmetric brake disks

An analysis of thermoelastic contact problem of functionally graded (FG) rotating brake disk with heat source due to contact friction is presented. Finite element method (FEM) is used. The material properties of disk are assumed to be represented by power-law distributions in the radial direction. The inner and outer surfaces considered are metal and ceramic, respectively. Pure material is considered for the brake pad. Coulomb contact friction is assumed as the heat source. It is divided into two equal parts between pad and brake disk which leads to thermal stresses. Mechanical response of FG disks are compared and verified with the known results from the literatures. The results show that the maximum value of radial displacement in mounted FG brake disk is not at outer surface. It is found that the all areas between pad and brake disk is in full-contact status when the ratio of pad thickness to brake disk thickness is 0.66. It is observed that the total strain due to thermomechanical load is negative for some parts of the disks, whereas, the thermal strains are always positive. It can be concluded that gradation index of the metal-ceramic has significant effect in the thermomechanical response of FG disks.     

Three-dimensional thermomechanical buckling analysis for functionally graded composite plates

Three-dimensional thermomechanical buckling analysis is investigated for functionally graded composite structures that composed of ceramic, functionally graded material (FGM), and metal layers. Material properties are assumed to be temperature dependent, and in FGM layer, they are varied continuously in the thickness direction according to a simple power law distribution in terms of the ceramic and metal volume fractions. The finite element model is adopted by using an 18-node solid element to analyze more accurately the variation of material properties and temperature field in the thickness direction. Temperature at each node is obtained by solving the thermomechanical equations. For a time discretization, Crank–Nicolson method is used. In numerical results, the thermal buckling behavior of FGM composite structures due to FGM thickness ratios, volume fraction distributions, and system geometric parameters are analyzed.     

A review of meshless methods for laminated and functionally graded plates and shells

This review focuses mainly on the developments of element-free or meshless methods and their applications in the analysis of composite structures. This review is organized as follows: a brief introduction to shear deformation plate and shell theories for composite structures, covering the first-order and higher-order theories, is given in Section 2. A review of meshless methods is provided in Section 3, with main emphasis on the element-free Galerkin method and reproducing kernel particle method. The applications of meshless methods in the analysis of composite structures are discussed in Section 4, including static and dynamic analysis, free vibration, buckling, and non-linear analysis. Finally, the problems and difficulties in meshless methods and possible future research directions are addressed in Section 5.     

Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates

In this paper, nonlinear static and free vibration analysis of functionally graded piezoelectric plates has been carried out using finite element method under different sets of mechanical and electrical loadings. The plate with functionally graded piezoelectric material (FGPM) is assumed to be graded through the thickness by a simple power law distribution in terms of the volume fractions of the constituents. Only the geometrical nonlinearity has been taken into account and electric potential is assumed to be quadratic across the FGPM plate thickness. The governing equations are obtained using potential energy and Hamilton’s principle that includes elastic and piezoelectric effects. The finite element model is derived based on constitutive equation of piezoelectric material accounting for coupling between elasticity and electric effect using higher order plate elements. The present finite element is modeled with displacement components and electric potential as nodal degrees of freedom. Results are presented for two constituent FGPM plate under different mechanical boundary conditions. Numerical results for PZT-4/PZT-5H plate are given in dimensionless graphical forms. Effects of material composition and boundary conditions on nonlinear response are also studied. The numerical results obtained by the present model are in good agreement with the available solutions reported in the literature.     

Thermoelastic analysis of a functionally graded rotating disk

A semi-analytical thermoelasticity solution for hollow and solid rotating axisymmetric disks made of functionally graded materials is presented. The radial domain is divided into some virtual sub-domains in which the power-law distribution is used for the thermomechanical properties of the constituent components. Imposing the necessary continuity conditions between adjacent sub-domains, together with the global boundary conditions, a set of linear algebraic equations are obtained. Solution of the linear algebraic equations yields the thermoelastic responses for each sub-domain as exponential functions of the radial coordinate. Some results for the stress, strain and displacement components along the radius are presented due to centrifugal force and thermal loading. Results obtained within this solution are compared with those of a finite element analysis in the literature. Based on the results, it is shown that the property gradation correlates with thermomechanical responses of FG disks.     

Vibration analysis of CNT-reinforced functionally graded rotating cylindrical panels using the element-free kp-Ritz method

In this paper, analysis of free vibration of carbon nanotube (CNT) reinforced functionally graded rotating cylindrical panels is presented. The analysis is performed by using the element-free kernel particle Ritz method or in short the kp-Ritz method. The rotating cylindrical panels are reinforced by single-walled carbon nanotubes (SWCNTs) with different types of distributions along thickness direction of the panels. Extended rule of mixture is selected to estimate the effective material properties of the resulting nanocomposite rotating panels. Two-dimensional displacement fields of the plates are approximated by a set of mesh-free kernel particle functions. The discretized governing eigen-equations are developed via the Ritz procedure. This kp-Ritz method enforces essential boundary conditions through the full transformation method. Detailed parametric studies have been carried out to reveal the influences of volume fraction of carbon nanotubes, edge-to-radius ratio and rotation speed on the frequency characteristics, with mode shape visualization provided. In addition, effects of different boundary conditions and types of distributions of carbon nanotubes are examined in detail.     

Free vibration analysis of functionally graded conical shell panels by a meshless method

A free vibration analysis of metal and ceramic functionally graded conical shell panels is presented using the element-free kp-Ritz method. The first-order shear deformation shell theory is used to account for the transverse shear strains and rotary inertia, and mesh-free kernel particle functions are employed to approximate the two-dimensional displacement fields. The material properties of the conical shell panels are assumed to vary continuously through their thickness in accordance with a power-law distribution of the volume fractions of their constituents. Convergence studies are performed in terms of the number of nodes, and comparisons of the current solutions and those reported in literature are provided to verify the accuracy of the proposed method. Two types of functionally graded conical shell panels, including Al/ZrO₂ and Ti–6Al–4V/aluminum oxide, are chosen in the study, and the effects of the volume fraction, boundary condition, semi-vertex angle, and length-to-thickness ratio on their frequency characteristics are discussed in detail.     

Finite element simulations of crack propagation in functionally graded materials under flexural loading

Cracks in stepped and continuously graded material specimens under flexural loading were investigated via finite element analysis. Calculation of mechanical energy release rates and propagation angles with crack-opening displacement correlation and the local symmetry (K_II = 0) criterion, respectively, provided results most efficiently and accurately, as compared with compliance and J-integral approaches and other deflection criteria. A routine was developed for automatic crack extension and remeshing, enabling simulation of incremental crack propagation. Effects of gradient profile and crack geometry on crack-tip stresses and crack propagation path are examined, and implications of these for optimal design of graded components against failure by fast fracture are discussed.     

3D crack analysis in functionally graded materials

Elastostatic crack analysis in three-dimensional, continuously non-homogeneous, isotropic and linear elastic functionally graded materials and structures is presented in this paper. A boundary-domain-integral equation formulation is applied for this purpose, which uses the elastostatic fundamental solutions for homogeneous, isotropic and linear elastic materials and involves a domain-integral due to the material’s non-homogeneity. To avoid displacement gradients in the domain-integral, normalized displacements are introduced. The domain-integral is transformed into boundary-integrals over the global boundary of the cracked solids by using the radial integration method. A meshless scheme is developed, which requires only the conventional boundary discretization and additional interior nodes instead of interior cells or meshes. Numerical examples for three-dimensional crack problems in continuously non-homogeneous, isotropic and linear elastic FGMs are presented and discussed, to show the effects of the material gradation on the crack-opening-displacements and the stress intensity factors.     

Study on postbuckling of axial compressed elastoplastic functionally graded cylindrical shells

Postbuckling of plates and shells is an important and cumbersome problem in the structural stability field. Presently, postbuckling behaviors of elastoplastic functionally graded cylindrical shells are investigated by a numerical simulation. The elastoplastic material properties are assumed to be of a multilinear hardening type, according to the constituent distributions, and are modeled using the laminate method. The Riks algorithm is used to obtain the equilibrium path. The postbuckling deformation and stain history of elastoplastic functionally graded cylindrical shells are investigated and various effects of the shell thickness and the constituent distributions are discussed. The results show material unloading effects in the postbuckling state.     

The effects of residual stresses and degradation on the response of viscoplastic functionally graded materials

Functionally graded materials (FGMs), having ceramic and metallic constituents, are commonly used for extreme temperature applications. Under extreme temperature changes, the mismatches in the thermo-mechanical properties of the ceramics and metallic constituents could cause pronounced thermal stresses and could lead to degradation in the properties of the constituents. High stresses in the metallic constituent lead to plastic deformations and high tensile stresses in the ceramic constituent induce cracking. An elastic–viscoplastic micromechanical model is formulated for analyzing residual stresses and strains and degradation in ceramic–metal FGMs undergoing temperature changes due to conduction of heat. A degradation parameter that depends on the temperature and strain is introduced in order to determine the level of material degradation in the ceramics and metallic constituents. The Perzyna viscoplastic model is considered for the metallic constituent while the ceramic constituent is assumed linear elastic. The material parameters in these constituents change with the degradation. The problem leads to time-dependent coupled thermal, deformation, and degradation behaviors. The micromechanical model is implemented in a displacement based finite element (FE), which is used to determine the performance of the viscoplastic functionally graded structures subject to external thermo-mechanical stimuli.     

Three-dimensional free vibration analysis of functionally graded annular sector plates with general boundary conditions

The main purpose of this paper is to investigate free vibration behaviors of functionally graded sector plates with general boundary conditions in the context of three-dimensional theory of elasticity. Generally, the material properties of functionally graded sector plates are assumed to vary continuously and smoothly in thickness direction. However, the changes in the material properties may occur in the other directions, such as radial direction. Therefore, two types of functionally graded annular sector plates are considered in the paper. In this work, both the Voigt model and Mori-Tanaka scheme are adopted to evaluate the effective material properties. Each of displacements of annular sector plate, regardless of boundary conditions, is expressed as modified Fourier series which consists of three-dimensional Fourier cosine series plus several auxiliary functions introduced to overcome the discontinuity problems of the displacement and its derivatives at edges. To ensure the validity and accuracy of the method, numerous examples for isotropic and functionally graded sector plates with various boundary conditions are presented. Furthermore, new results for functionally graded sector plates with elastic restraints are given. The effects of the material profiles and boundary conditions on the free vibration of the functionally sector plates are also studied.     

Free vibration analysis of rotating functionally graded cylindrical shells in thermal environment

The free vibration analysis of rotating functionally graded (FG) cylindrical shells subjected to thermal environment is investigated based on the first order shear deformation theory (FSDT) of shells. The formulation includes the centrifugal and Coriolis forces due to rotation of the shell. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The initial thermo-mechanical stresses are obtained by solving the thermoelastic equilibrium equations. The equations of motion and the related boundary conditions are derived using Hamilton’s principle. The differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to discretize the thermoelastic equilibrium equations and the equations of motion. The convergence behavior of the method is demonstrated and comparison studies with the available solutions in the literature are performed. Finally, the effects of angular velocity, Coriolis acceleration, temperature dependence of material properties, material property graded index and geometrical parameters on the frequency parameters of the FG cylindrical shells with different boundary conditions are investigated.     

On the size-dependent behavior of functionally graded micro-beams

In this paper, the size-dependent static and vibration behavior of micro-beams made of functionally graded materials (FGMs) are analytically investigated on the basis of the modified couple stress theory in the elastic range. Functionally graded beams can be considered as inhomogeneous composite structures, with continuously compositional variation from usually a ceramic at the bottom to a metal at the top. The governing equations of motion and boundary conditions are derived on the basis of Hamilton principle. Closed-form solutions for the normalized static deflection and natural frequencies are obtained as a function of the ratio of the beam characteristic size to the internal material length scale parameter and FGM distribution functions of properties. The results show that the static deflection and natural frequencies developed by the modified couple stress theory have a significant difference with those obtained by the classical beam theory when the ratio of the beam characteristic size to the internal material length scale parameter is small.     

Active control of functionally graded laminated cylindrical shells

An analytical method on active vibration control of smart FG laminated cylindrical shells with thin piezoelectric layers is presented based on Hamilton’s principle. The thin piezoelectric layers embedded on inner and outer surfaces of the smart FG laminated cylindrical shell act as distributed sensor and actuator, which are used to control vibration of the smart FG laminated cylindrical shell under thermal and mechanical loads. Here, the modal analysis technique and Newmark’s integration method are used to calculate the dynamic response of the smart FG laminated cylindrical shell with thin piezoelectric layers. Constant-gain negative velocity feedback approach is used for active vibration control with the structures subjected to impact, step and harmonic excitations. The influences of different piezoelectric materials (PZT-4, BaTiO₃ and PZT-5A) and various loading forms on the active vibration control are described in the numerical results.     

Comment on ‘Thermoelsatic analysis of the functionally graded rotating disk’

In their paper, Hosseini Kordkheili and Naghdabadi (2007) [1] have obtained the Navier thermoelastic equation, along with some coefficients, incorrectly. This has resulted in considerable error in their numerical results. Other authors in this field frequently use the work of Kordkheili and Naghdabadi in their own calculations. Asghari and Ghafoori (2010) [2] is an example of another work that has used the wrong Navier thermoelastic equation directly in their work. It is clear there is a need for corrected formulations, which will be presented here.