Effect of continuously grading fiber orientation face sheets on vibration of sandwich panels with FGM core

The main purpose of this paper is to investigate effect of continuously grading fiber orientation face sheets on free vibration of sandwich panels with functionally graded core using generalized power-law distribution. The benefit of using generalized power-law distribution is to illustrate and present useful results arising from symmetric, asymmetric and classic profiles. The fiber-reinforced composite material studied in the present work consists of silicon carbide fibers embedded in titanium matrix with the fiber volume fraction and fiber orientation graded in the radial direction. The face sheets have variation of the fiber orientation while the core has variation of fiber volume fraction. Generalized differential quadrature (GDQ) method is used to yield natural frequencies of the simply supported functionally graded sandwich (FGSW) panels on the basis of the 2-D, linear and small strain elasticity theory. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. In this research work, a detailed parametric study is carried out to highlight the influences of continuously grading fiber orientation face sheets and different profile of fiber volume fraction and fiber orientation on the vibration characteristics of the FGSW panels.     

Mechanical properties of fibrous core sandwich panels

Fibrous core sandwich panels are thin, lightweight structures with face sheets separated by an irregular arrangement of independent fibres; the fibres have a random angle of fibre inclination and a range of initial curvatures. These panels have small thickness so that they can be pressed into 3D curvature in a forming operation. This paper analyses the effects of core morphology on the through-thickness elastic moduli, compressive strength and the through-thickness shear strength of the fibrous core. For a specific panel construction, analytical results are compared with both Finite Element analysis and experiment. A new approach to measure the through-thickness shear modulus of fibrous core is described.     

FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations

In this paper, the generalized differential quadrature (GDQ) method is applied to study the dynamic behavior of functionally graded materials (FGMs) and laminated doubly curved shells and panels of revolution with a free-form meridian. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature a generalization of the Reissner-Mindlin theory, proposed by Toorani and Lakis, is adopted. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. Firstly, the differential quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Secondly, the discretization of the system by means of the GDQ technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the Reissner-Mindlin and the Toorani-Lakis theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed. Finally, different lamination schemes are considered to expand the combination of the two functionally graded four-parameter power-law distributions adopted. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the lamina thickness direction. A two-constituent functionally graded lamina consists of ceramic and metal those are graded through the lamina thickness. A parametric study is performed to illustrate the influence of the parameters on the mechanical behavior of shell and panel structures considered.     

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.     

梯度材料球罐热弹性应力分析

对梯度材料球罐承压条件下的热弹性响应进行分析,通过求解应力控制方程分别考察不同弹性梯度因子和温差梯度因子对球罐中应力分布的影响。分析结果表明,即使不考虑梯度温差应力的影响,梯度材料球罐的最大环向应力也不总在球罐的最内层,而是取决于材料的弹性梯度分布;并且,不同球罐壁厚下,弹性梯度因子对球罐应力分布的影响作用也明显不同。随着温差梯度因子的增加,球罐中径向应力将从完全受压状态转变为部分受拉状态,而球罐内外壁的环向应力差值也随之增大。     

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.     

Interface crack of two dissimilar bonded functionally graded strips with arbitrary distributed properties under plane deformations

In this paper the plane elasticity problem of two bonded dissimilar functionally graded strips containing an interface crack with material properties varying arbitrarily is studied. The governing equation in terms of Airy stress function is formulated and exact solutions are obtained for several special variations of material properties in Fourier transformation domain. A multi-layered model is employed to model arbitrary variations of material properties based on two linear-distributed material compliance parameters. The mixed boundary problem is reduced to a system of singular integral equations that are solved numerically. Some numerical examples are given to demonstrate the accuracy, efficiency and versatility of the model. Numerical results show that fracture behavior of materials can be greatly affected by graded variation of elastic modulus and the influence of the specific form of elastic modulus on the fracture behavior of FGM is limited.     

Effect of material inhomogeneity on the rotating functionally of a graded orthotropic hollow cylinder

The effect of the material inhomogeneity on the stress field of a rotating orthotropic infinite hollow cylinder made of functionally graded materials is investigated. The original functionally graded hollow cylinder is approximated by the laminate model, of which the solution will gradually approach the exact one with the increase in number of fictitious layers. The analysis is performed by means of the state space method. The validity of the solution is verified by utilizing the exact solution of a special non-homogeneous rotating hollow cylinder and earlier results. The distributions of the radial and tangential stresses of the internally pressurized rotating functionally graded hollow cylinder for various material inhomogeneity parameters are depicted graphically. The degree of the orthotropy on the stress fields is also discussed.     

Free vibration analysis of multi-directional functionally graded circular and annular plates

This paper addresses the free vibration of multi-directional functionally graded circular and annular plates using a semianalytical/ numerical method, called state space-based differential quadrature method. Three-dimensional elasticity equations are derived for multi-directional functionally graded plates and a solution is given by the semi-analytical/numerical method. This method gives an analytical solution along the thickness direction, using a state space method and a numerical solution using differential quadrature method. Some numerical examples are presented to show the accuracy and convergence of the method. The most of simulations of the present study have been validated by the existing literature. The non-dimensional frequencies and corresponding displacements mode shapes are obtained. Then the influences of thickness ratio and graded indexes are demonstrated on the non-dimensional natural frequencies.     

Functionally graded piezoelectric strip with eccentric crack under anti-plane shear

In this paper, we examine the singular stresses and electric fields in a functionally graded piezoelectric ceramic strip containing an eccentric crack off the center line under anti-plane shear loading with the theory of linear piezoelectricity. It is assumed that the properties of the functionally graded piezoelectric ceramic strip vary continuously along the thickness. Fourier transforms are used to reduce the problem to the solution of two pairs of dual integral equations, which are then expressed to a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and the energy release rate are obtained.     

A new beam finite element for the analysis of functionally graded materials

A new beam element is developed to study the thermoelastic behavior of functionally graded beam structures. The element is based on the first-order shear deformation theory and it accounts for varying elastic and thermal properties along its thickness. The exact solution of static part of the governing differential equations is used to construct interpolating polynomials for the element formulation. Consequently, the stiffness matrix has super-convergent property and the element is free of shear locking. Both exponential and power-law variations of material property distribution are used to examine different stress variations. Static, free vibration and wave propagation problems are considered to highlight the behavioral difference of functionally graded material beam with pure metal or pure ceramic beams.     

Eshelby-Mori-Tanaka approach for vibrational behavior of functionally graded carbon nanotube-reinforced plate resting on elastic foundation

In this study, based on the three-dimensional theory of elasticity, free vibration characteristics of functionally graded (FG) nanocomposite plates reinforced by randomly-oriented straight single-walled carbon nanotubes (SWCNTs) resting on an elastic foundation are considered. Material properties are graded in the thickness direction of the plate according to the volume fraction power law distribution. An embedded carbon nanotube (CNT) in a polymer matrix and its surrounding inter-phase which is perfectly bonded to surrounding resin is replaced with an equivalent fiber to predict the mechanical properties of the carbon nanotube/polymer composite. The Mori-Tanaka approach is employed to calculate the effective elastic moduli of the plate. The natural frequencies of the plate are obtained by means of the generalized differential quadrature (GDQ) method. Detailed parametric studies have been carried out to investigate the influences of the CNT volume fraction, Winkler foundation modulus, shear elastic foundation modulus and various geometrical parameters on the vibration behavior of the functionally graded carbon nanotube-reinforced (FG-CNTR) plates.     

On the plane contact problem of a functionally graded elastic layer loaded by a frictional sliding flat punch

This article is concerned with the contact mechanics of a functionally graded layer loaded by a frictional sliding flat punch. The coefficient of friction is assumed to be constant and the lower side of the graded layer is firmly attached to a rigid foundation. The graded, nonhomogeneous property of the medium is represented in terms of an exponential variation of the shear modulus, while Poisson’s ratio is taken to be constant. Based on the use of plane elasticity equations and the Fourier integral transform technique, the formulation of the current contact mechanics problem lends itself to a Cauchy-type singular integral equation of the second kind for the unknown contact pressure, which is solved numerically. As a result, the effects of several parameters, i.e., the material nonhomogeneity, the friction coefficient, the punch width, and Poisson’s ratio, on the distributions of the contact pressure and the in-plane surface stress component are presented.     

Dynamic characteristics of an eccentric crack in a functionally graded piezoelectric ceramic strip

The dynamic response of an eccentric Griffith crack in functionally graded piezoelectric ceramic strip under anti-plane shear impact loading is analysed using integral transform method. Laplace transform and Fourier transform are used to reduce the problem to two pairs of dual integral equations, which are then expressed to Fredholm integral equations of the second kind. We assume that the properties of the functionally graded piezoelectric material vary continuously along the thickness. The impermeable crack boundary condition is adopted. Numerical values on the dynamic stress intensity factors are presented for the functionally graded piezoelectric material to show the dependence of the gradient of material properties and electric loadings.     

Dynamic propagation of a finite eccentric crack in a functionally graded piezoelectric ceramic strip

The dynamic propagation of an eccentric Griffith crack in a functionally graded piezoelectric ceramic strip under anti-plane shear is analyzed using the integral transform method. A constant velocity Yoffe-type moving crack is considered. Fourier transform is used to reduce the problem to a pair of dual integral equations, which is then expressed in a Fredholm integral equation of the second kind. We assume that the properties of the functionally graded piezoelectric material vary continuously along the thickness. The impermeable crack boundary condition is adopted. Numerical values on the dynamic stress intensity factors are presented for the functionally graded piezoelectric material to show the dependence of the gradient of material properties, crack moving velocity, and eccentricity. The dynamic stress intensity factors of a moving crack in functionally graded piezoelectric material increases when the crack moving velocity, eccentricity of crack location, material property gradient, and crack length increase. This paper was recommended for publication in revised form by Associate Editor Hyeon Gyu Beom Jeong Woo Shin received a B.S. and M.S. degree in Mechanical Engineering from Yonsei University in Seoul, Korea in 1998 and 2000, respectively. A major field of Mr. Shin is fracture mechanics. He is currently working on the KARI (Korea Aerospace Research Institute) as a senior researcher. He conducted load analysis of fixed wing aircraft and full scale airframe static test at the KARI. He is now developing landing gear in the KHP (Korea Helicopter Program) as a performance engineer.     

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.     

含有泡沫铝芯的复合板弯曲断裂行为的原位研究

对由泡沫金属铝芯和金属面板组成的三层和多层复合板四点弯曲条件下的变形和断裂行为进行原位观察。研究结果表明：在弯曲条件下,复合板有两种基本的破坏方式,一种是复合板表面凹陷(Indentation, ID),它是表面局部集中塑性变形的结果;另一种是泡沫铝内芯切断 (Core shear, CS),它是内芯在最大切应力作用下的破坏。对一个给定的三层复合板,当凹陷破坏的载荷极限FID大于内芯切断的载荷极限FCS时发生内芯切断式破坏,反之发生表面凹陷式破坏。对于多层复合板,破坏方式受金属面板制约,不能直接应用三层板的破坏判据。若三层板发生凹陷型破坏,具有与三层板相同金属面板厚度的多层复合板发生凹陷加内芯切断的混合型破坏。当三层板只发生内芯切断型破坏时,具有与三层板相同金属面板厚度的多层复合板完全发生内芯切断型破坏。     

Analysis of a moving crack in a functionally graded strip between two homogeneous layers

In this paper a finite crack with constant length (Yoffe-type crack) propagating in a functionally graded strip with spatially varying elastic properties between two dissimilar homogeneous layers under in-plane loading was studied. By utilizing the Fourier transformation technique, the mixed boundary problem is reduced to a system of singular integral equations that are solved numerically. The influences of the geometric parameters, the graded parameter, the crack length and speed on the stress intensity factors are investigated. The numerical results show that the graded parameters, the thicknesses of the functionally graded strip and the two homogeneous layers, the crack size and speed have significant effects on the dynamic fracture behavior.     

高速公路用防眩板在风荷载作用下的应力分析

高速公路用防眩板的抗风能力是其关键指标之一,对防眩板在风荷载作用时的应力分析具有重要意义。本文建立了防眩板在风荷载作用下的有限元模型,用相关试验数据对模型的有效性进行了验证,并利用有限元技术和力学抗风试验的结果比较,探讨了在风荷载作用下防眩板的破坏形式和应力集中点。结果表明,计算得到的防眩板的应力集中点和断裂位置与试验结果基本吻合,建立的模型可用于防眩板外形和结构设计研究。     

Investigation of constantly and transiently propagating cracks in functionally graded materials by dynamic photoelasticity

In this study, the stress intensity factors (SIFs) for steady and transient propagation of cracks in transparent homogeneous functionally graded materials were analyzed by using the photoelasticity technique. The fracture analysis was carried out for the cracks propagating from a region with high elasticity towards low elasticity, as well as the cracks propagating from a region with low elasticity towards high elasticity. The analysis includes cracks propagating (1) at an almost steady speed, and (2) with the rapid increase, followed by a decrease in speed. For cracks with almost constant velocity, the SIFs were greater when a crack started from a high elasticity region, as compared to the cracks which initiated from a low elasticity region. For cracks propagating with rapid acceleration and deceleration, when the strain energy accumulated in the material due to an increase in load or stress was released at the moment of crack propagation, the SIF was momentarily lowered by approximately 45 %–50 % of the static SIF(before crack initiation), which subsequently increases by approximately 30 % eventually, the crack acceleration approaches nearly zero; the SIF decreases and increases respectively as the crack propagates in a material with decreasing and increasing modulus of elasticity.