A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates

An exact closed-form frequency equation is presented for free vibration analysis of circular and annular moderately thick FG plates based on the Mindlin's first-order shear deformation plate theory. The edges of plate may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. The material properties change continuously through the thickness of the plate, which can vary according to a power-law distribution of the volume fraction of the constituents, whereas Poisson's ratio is set to be constant. The equilibrium equations which govern the dynamic stability of plate and its natural boundary conditions are derived by the Hamilton's principle. Several comparison studies with analytical and numerical techniques reported in literature and the finite element analysis are carried out to establish the high accuracy and superiority of the presented method. Also, these comparisons prove the numerical accuracy of solutions to calculate the in-plane and out-of-plane modes. The influences of the material property, graded index, thickness to outer radius ratios and boundary conditions on the in-plane and out-of-plane frequency parameters are also studied for different functionally graded circular and annular plates.     

Thermoelastic and vibration analysis of functionally graded cylindrical shells

The static response and free vibration of metal and ceramic functionally graded shells are analyzed using the element-free kp-Ritz method. The material properties are assumed to vary continuously along the depth direction. The displacement field is expressed in terms of a set of mesh-free kernel particle functions according to Sander's first-order shear deformation shell theory. The effects of the volume fraction, material property, boundary condition, and length-to-thickness ratio on the shell deflection, axial stress, and natural frequency are examined in detail. Convergence studies of node numbers are performed to verify the effectiveness of the proposed method. Comparisons reveal that the numerical results obtained from the proposed method agree well with those from the classical and finite element methods.     

Geometrically nonlinear analysis of functionally graded shells

The nonlinear response of functionally graded ceramic-metal shell panels under mechanical and thermal loading is studied. The nonlinear formulation is based on a modified version of Sander's nonlinear shell theory, in which the geometric nonlinearity takes the form of von Kármán strains. It is assumed that the material properties vary through the thickness according to a power-law distribution of the volume fraction of the constituents. The displacement field is expressed in terms of a set of mesh-free kernel particle functions. The bending stiffness is evaluated using a stabilized conforming nodal integration technique, and the shear and membrane terms are computed using a direct nodal integration to eliminate shear and membrane locking. The arc-length method, combined with the modified Newton-Raphson approach, is employed to trace the full load-displacement path. The characteristic of the displacement and the axial stress in panels under thermal and mechanical loading is investigated, and the effects of the volume fraction exponent, boundary conditions, and material properties on the nonlinear response of shell panels are also examined.     

A new exact analytical approach for free vibration of Reissner-Mindlin functionally graded rectangular plates

An exact closed-form procedure is presented for free vibration analysis of moderately thick rectangular plates having two opposite edges simply supported (i.e. Lévy-type rectangular plates) based on the Reissner-Mindlin plate theory. The material properties change continuously through the thickness of the plate, which can vary according to a power law distribution of the volume fraction of the constituents. By introducing some new potential and auxiliary functions, the displacement fields are analytically obtained for this plate configuration. Several comparison studies with analytical and numerical techniques reported in literature are carried out to establish the high accuracy and reliability of the solutions. Comprehensive benchmark results for natural frequencies of the functionally graded (FG) rectangular plates with six different combinations of boundary conditions (i.e. SSSS-SSSC-SCSC-SCSF-SSSF-SFSF) are tabulated in dimensionless form for various values of aspect ratios, thickness to length ratios and the power law index. Due to the inherent features of the present exact closed-form solution, the present results will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.     

An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression

In this research, mechanical buckling of circular plates composed of functionally graded materials (FGMs) is considered. Equilibrium and stability equations of a FGM circular plate under uniform radial compression are derived, based on the higher order shear deformation plate theory (HSDT). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations are established. A buckling analysis of a functionally graded circular plate (FGCP) under uniform radial compression is carried out and the results are given in closed-form solutions. The results are compared with the buckling loads of plates obtained for FGCP based on the first order shear deformation plate theory (FSDT) and classical plate theory (CPT) given in the literature. The study concludes that HSDT accurately predicts the behavior of FGCP, whereas the FSDT and CPT overestimates buckling loads.     

Thermal buckling of functionally graded rectangular plates subjected to partial heating

We present the thermal buckling analysis of functionally graded rectangular plates subjected to partial heating in a plane and uniform temperature rise through its thickness. The plate is simply supported for out-of-plane deformation and perfectly clamped for in-plane deformation. It is assumed that the functionally graded material properties such as the coefficient of linear thermal expansion and Young's modulus are changed individually in the thickness direction of the plate with the power law, while Poisson's ratio is assumed to be constant. Analytical developments consist of two stages. First, the nonuniform in-plane resultant forces are determined by solving a plane thermoelastic problem. Then the critical buckling temperatures of the plates with the predetermined resultant forces are calculated as the generalized eigenvalue problem which is constructed by using the Galerkin method. Finally, the effects of material inhomogeneity, aspect ratio, and heated region on the critical buckling temperatures are examined.     

Stress concentration factor due to a circular hole in functionally graded panels under uniaxial tension

The effect of the material property inhomogeneity on the stress concentration factor (SCF) due to a circular hole in functionally graded panels is numerically investigated. The multiple isoparametric finite element formulation is used to simulate the elastostatic boundary value problem. A parametric study is performed by varying the functional form and the direction of the material property gradation. The material property inhomogeneity is characterized by the intrinsic inhomogeneity length scale, modulus ratio and the power-law index. The results from our parametric study showed that the SCF is reduced when Young's modulus progressively increased away from the hole. The angular position of the maximum tensile stress on the surface of the hole remains unaffected by the material property inhomogeneity. The SCF is seen to be most influenced by the power-law index, followed by the variation of the inhomogeneity length scale. The SCF is least affected by the modulus ratio.     

A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate

A new hyperbolic shear deformation theory taking into account transverse shear deformation effects is presented for the buckling and free vibration analysis of thick functionally graded sandwich plates. Unlike any other theory, the theory presented gives rise to only four governing equations. Number of unknown functions involved is only four, as against five in case of simple shear deformation theories of Mindlin and Reissner (first shear deformation theory). The plate properties are assumed to be varied through the thickness following a simple power law distribution in terms of volume fraction of material constituents. The theory presented is variationally consistent, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Equations of motion are derived from Hamilton's principle. The closed-form solutions of functionally graded sandwich plates are obtained using the Navier solution. The results obtained for plate with various thickness ratios using the theory are not only substantially more accurate than those obtained using the classical plate theory, but are almost comparable to those obtained using higher order theories with more number of unknown functions.     

一种改进的网格模型功能梯度材料体素化算法

针对目前面向快速成形的三维CAD功能梯度材料表达方法存在的问题,分析了基于深度缓存三维模型体素法与基于八叉树结构的体素化算法,并提出一种改进的基于最小欧氏距离测度的网格模型体素化算法,以最小欧氏距离测度的网格模型体素化算法为基础,通过在体数据压缩结构RLE上增加体素逻辑坐标的方法,从而在执行效率、模型精度和修改方面达到很好的效果,所得体素模型为零件模型的功能梯度材料表达提供了良好的数据模型.     

Adaptive slicing of functionally graded material objects for rapid prototyping

Slicing is a fundamental process planning task and an important procedure in rapid prototyping. However, most research currently focuses on the slicing of homogeneous objects and few approaches for slicing of heterogeneous objects have been reported in the literature. In this article, we present an approach for adaptive slicing of functionally graded material objects. Unlike homogeneous objects, functionally graded material objects contain both geometry and material information. The layer thickness is computed by considering not only geometry but also material variation along the build direction. The continuous material distribution in each layer is discretised into step-wise gradings by subdividing the slice into sub-regions, which can be regarded as homogeneous material regions. An algorithm that summarised the slicing procedure is described and an example is also presented.     

Nonlinear cylindrical bending analysis of shear deformable functionally graded plates under different loadings using analytical methods

An exact solution is presented for the nonlinear cylindrical bending and postbuckling of shear deformable functionally graded plates in this paper. A simple power law function and the Mori–Tanaka scheme are used to model the through-the-thickness continuous gradual variation of the material properties. The von Karman nonlinear strains are used and then the nonlinear equilibrium equations and the relevant boundary conditions are obtained using Hamilton's principle. The Navier equations are reduced to a linear ordinary differential equation for transverse deflection with nonlinear boundary conditions, which can be solved by exact methods. Finally, by solving some numeral examples for simply supported plates, the effects of volume fraction index and length-to-thickness ratio are studied. It is shown that there is no bifurcation point for simply supported functionally graded plates under compression. The behavior of near-boundary areas predicted by the shear deformation theory and the classical theory is remarkably different.     

基于材料特征的功能梯度材料CAD模型分层算法

为了利用快速成形技术制造功能梯度材料零件,在分析以往基于设计的功能梯度材料零件计算机辅助设计模型的基础上,提出一种基于制造的功能梯度材料零件计算机辅助设计模型.采用PIC格式存储层片轮廓,存储层片材料分布向量的数据结构为四又树和线性表,材料分布向量精度可调且存储空间可以优化.基于该模型,提出一种新的材料分布向量计算方法,实例证明该方法计算材料分布向量简单.采朋不符合要求的采样点占总采样点百分比λ作为判定依据,比以往判定依据更灵活.基于此削据提出一种基于材料特征的层厚计算方法,实例证明该算法正确、有效.     

Thermomechanical postbuckling analysis of moderately thick functionally graded plates and shallow shells

In this paper, an analytical solution is provided for the postbuckling behaviour of moderately thick plates and shallow shells made of functionally graded materials (FGMs) 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 moderately thick rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection and high-order shear deformation theory for moderately thick plates. The solution is obtained in terms of mixed Fourier series and the obtained results are compared with those of the Reissner–Mindlin's theory for moderately thick plates and the classical theory ignoring transverse shear deformation. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behaviour and the associated 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.     

Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams

An improved third order shear deformation theory is employed to investigate thermal buckling and vibration of the functionally graded beams. A power law distribution is used to describe the variation of volume fraction of material compositions. The functionally graded material properties are assumed to vary smoothly and continuously across the thickness of the beams. The Ritz method is adopted to solve the eigenvalue problems that are associated with thermal buckling and vibration in various types of immovable boundary conditions. The parametric study covered in this paper includes the effects of material composition, temperature-dependent material properties, and slenderness ratio.     

Three-dimensional solution for transient thermal stresses of functionally graded rectangular plate due to nonuniform heat supply

This paper is concerned with the theoretical treatment of transient thermoelastic problem involving a functionally graded rectangular plate due to nonuniform heat supply. The thermal and thermoelastic constants of the rectangular plate are assumed to vary exponentially in the thickness direction. The transient three-dimensional temperature is analyzed by the methods of Laplace and finite cosine transformations. We obtain the three-dimensional solution for the simple supported rectangular plate. Some numerical results for the temperature change, the displacement and the stress distributions are shown in figures. Furthermore, the influence of the nonhomogeneity of the material is investigated.     

Finite element analysis of thermal stresses in functionally gradient layered composites

A new composite material has been introduced which has a great potential satisfying successfully the desired functions under severe thermal circumstances. Because of the sharp material discontinuity at the interfaces between different material layers in classical layered composites, thermal and mechanical stress concentration may exist at such interfaces, and which results in undesired failure. The functionally gradient material (FGM), a new concept for future-oriented composite material, has a continuously varying material variation through the thickness of layered composites. And, hence, it can eliminate the defect occurred in classical layered composites. The purpose of this study is to develop a technique for finite element analysis of the thermal characteristics of FGMs, and to investigate the effects of significant governing parameters, a variation function of material composition and a relative thickness of FGM layer inserted between metal and ceramic layers. Through numerous numerical simulations carried out with the developed FEM program, we investigated the thermal characteristics for different concerning parameters. Considerable improvement and parametric dependence on temperature and thermal stress distributions are obtained.     

The free vibration analysis and optimal design of an adhesively bonded functionally graded single lap joint

In this study, three-dimensional free vibration and stress analyses of an adhesively bonded functionally graded single lap joint were carried. The effects of the adhesive material properties, such as modulus of elasticity, Poisson's ratio and density were found to be negligible on the first ten natural frequencies and mode shapes of the adhesive joint. Both the finite element method and the back-propagation artificial neural network (ANN) method were used to investigate the effects of the geometrical parameters, such as overlap length, plate thickness and adhesive thickness; and the material composition variation through the plate thickness on the natural frequencies, mode shapes and modal strain energy of the adhesive joint. The suitable ANN models were trained successfully using a series of free vibration and stress analyses for various random geometrical parameters and compositional gradient exponents. The ANN models showed that the support length, the plate thickness and the compositional gradient exponent played important role on the natural frequencies, mode shapes and modal strain energies of the adhesive joint whereas the adhesive thickness had a minor effect. In addition, the optimal joint dimensions and compositional gradient exponent were determined using genetic algorithm and ANN models so that the maximum natural frequency and the minimum modal strain energy conditions are satisfied for each natural frequency of the adhesively bonded functionally graded single lap joint.     

Vibration of functionally graded cylindrical shells

Functionally gradient materials (FGMs) have attracted much attention as advanced structural materials because of their heat-resistance properties. In this paper, a study on the vibration of cylindrical shells made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented. The objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of configurations of the constituent materials on the frequencies. The properties are graded in the thickness direction according to a volume fraction power-law distribution. The results show that the frequency characteristics are similar to that observed for homogeneous isotropic cylindrical shells and the frequencies are affected by the constituent volume fractions and the configurations of the constituent materials. The analysis is carried out with strains–displacement relations from Love’s shell theory and the eigenvalue governing equation is obtained using Rayleigh–Ritz method. The present analysis is validated by comparing results with those in the literature.     

Frictionally excited thermoelastic instability in a thin layer of functionally graded material sliding between two half-planes

Frictionally excited thermoelastic instability (TEI) is investigated in the system of a layer of finite thickness which is composed of functionally graded material (FGM), sliding against two homogeneous frictional materials of half-planes at speed V. Results show that ceramic-based FGM, which is composed of ceramic at the sliding interface and steel at the middle of the layer, reduces the susceptibility towards TEI. The effect of nonhomogeneous parameters of FGM on critical speed is investigated for the several combinations of FGM such as steel- or ceramic-based FGM with positive and negative nonhomogeneous parameters, respectively. The material sensitivity to the critical speed for the nonhomogeneous parameter of FGM shows that the nonhomogeneous parameters of the thermal expansion coefficient are strong influential factors in ceramic- or steel-based FGM. The transient evolution of temperature perturbation is also obtained to determine the susceptibility towards hot spotting, showing that the ceramic-based FGM with negative nonhomogeneous parameters has the lowest temperature perturbation amplitude during engagement.     

Free vibration analysis of functionally graded plates with multiple circular and non-circular cutouts

Cutouts are inevitable in structures due to practical consideration.In order to investigate the free vibration of functionally graded plates with multiple circular and non-circular cutouts,finite eleme...