多孔功能梯度梁的热-力耦合屈曲行为

采用经典欧拉梁理论和高阶三角剪切变形理论,研究了多孔功能梯度梁的热-力耦合屈曲行为。分析中考虑了材料物性与温度的相关性,采用含孔隙率的Voight混合模型描述了多孔功能梯度的材料属性。利用迭代算法求解结构在均匀、线性和非线性温升(考虑热传导效应)下的热-力耦合屈曲临界温度,讨论了材料非均匀参数、孔隙率和长细比等参数对屈曲临界温度的影响。ABAQUS数值模拟结果和文献对比结果验证了理论的可靠性,同时表明高阶剪切变形理论较经典欧拉梁理论精确。结果表明,功能梯度材料梁的热屈曲分析必须考虑物性与温度的相关性,否则可能高估热屈曲临界温度10%~30%;随着孔隙率增大,材料的等效弹性模量减少,即结构刚度有所弱化,但屈曲临界温度反而大大增高。     

Reduction of thermal stresses by composition optimization of two-dimensional functionally graded materials

Reduction of the thermal stresses in machine elements that are subjected to severe thermal loadings was achieved by developing two-dimensional functionally graded materials, 2D-FGM. In the current investigation, composition optimization for ZrO₂/6061-T6/Ti-6Al-4V 2D-FGM, under a severe thermal loading cycle that consists of heating followed by cooling, was carried out based on the minimization of temperatures and thermal and residual stresses to achieve better reduction of the thermal stresses. From the current investigation it was found that the optimum composition based on the minimum value of the maximum temperature for ZrO₂/6061-T6/Ti-6Al-4V 2D-FGM was achieved for m _x = 0.1 and m _y = 0.1, while the optimum composition based on the minimum value of the maximum normalized equivalent stresses for ZrO₂/6061-T6/Ti-6Al-4V 2D-FGM was achieved for m _x = 0.1 and m _y = 5, where m _x and m _y are the composition variation parameters in x- and y-directions, respectively. Also, the obtained optimum composition of ZrO₂/6061-T6/Ti-6Al-4V 2D-FGM can stand well with the adopted severe thermal loading without any plastic deformation or residual stresses, where the maximum value of the normalized equivalent stresses during the heating stage was 0.8 and the maximum value of the normalized equivalent stresses during the cooling stage was 0.24.     

湿-热-机耦合作用下多孔功能梯度梁的振动及屈曲特性

采用一种拓展的n阶广义梁理论（GBT）,研究了轴向机械载荷作用下多孔功能梯度材料（FGM）梁在湿热环境中的振动及屈曲特性。考虑了材料的物性随温度变化,湿-热沿梁厚按三种不同类型分布,采用含孔隙率的修正Voigt混合率模型描述多孔功能梯度梁的材料属性,在宏-细观力学模型框架下应用Hamilton原理统一建立了系统的自由振动及屈曲方程,采用Navier法求解FGM简支梁的静动态响应。通过算例验证并讨论了GBT阶数n的理想取值,可用于丰富梁理论。探讨了湿热效应、湿-热-机耦合、孔隙率、材料梯度指标、跨厚比对FGM梁振动及屈曲特性的影响。结果表明:湿-热加剧降低了FGM梁的频率和临界载荷,且不同类型的湿热分布对其减小程度有显著差异;随着孔隙率增大,梁结构的整体刚度虽有所弱化,但在湿热环境中频率反而增大,稳定性增强;湿-热效应对多孔FGM细长梁频率和稳定性影响十分显著,但对短粗梁的影响比较有限。      

功能梯度材料梁的非线性随机振动响应

基于大变形理论建立功能梯度材料(FGM)梁运动方程,将梁的横向位移假定为时间函数和梁线性模态乘积之和,利用伽辽金方法离散为非线性常微分方程组;然后,运用等效线性化方法求得随机激励作用下简支约束的功能梯度材料梁均方位移,与NewMark法和蒙特卡罗方法获得的结果对比,验证该等效线性化方法的可靠性.最后讨论材料梯度指数、激励强度和梁长细比对功能梯度材料梁振动响应的影响.     

Free vibration of functionally graded spatial curved beams

A formulation for the free vibration analysis of functionally graded (FG) spatial curved beams is presented by taking into account the effects of thickness-curvature. The governing equation is based on the first-order shear deformation theory (FSDT) and Ritz method is employed to obtain the natural frequencies. The curved beams presented are in the form of the cylindrical helical spring. The material distribution is in the direction of the curvature of the curved beam. The results for isotropic planar curved beams are validated with the known data in the literature. The effects of helix pitch angle, number of turns and boundary conditions on frequency parameters of spatial curved beams are investigated.     

Thermal buckling analysis of functionally graded material beams

Buckling of beams made of functionally graded material under various types of thermal loading is considered. The derivation of equations is based on the Euler–Bernoulli beam theory. It is assumed that the mechanical and thermal nonhomogeneous properties of beam vary smoothly by distribution of power law across the thickness of beam. Using the nonlinear strain–displacement relations, equilibrium equations and stability equations of beam are derived. The beam is assumed under three types of thermal loading, namely; uniform temperature rise, nonlinear, and linear temperature distribution through the thickness. Various types of boundary conditions are assumed for the beam with combination of roller, clamped and simply-supported edges. In each case of boundary conditions and loading, a closed form solution for the critical buckling temperature for the beam is presented. The formulations are compared using reduction of results for the functionally graded beams to those of isotropic homogeneous beams given in the literature.     

Nonlinear electro-mechanical responses of functionally graded piezoelectric beams

This study presents analyses of the nonlinear electro-mechanical responses of functionally graded piezoelectric beams undergoing small deformation gradients. The studied functionally graded beams comprise of electro-active and inactive constituents with gradual compositions varying through the thickness of the beams. Two types nonlinear electro-mechanical responses are considered for the active constituents, which are nonlinear electro-mechanical behaviors for the polarized piezoelectric constituent under electric fields smaller than the coercive limit, and polarization switching responses due to cyclic electric fields with high amplitude. The inactive constituent is modeled with uncoupled linear electro-elastic response. The functionally graded beam is discretized into several graded layers through its thickness. Each layer is comprised of different compositions of the active (piezoelectric) inclusions and conductive matrix. A particle-unit-cell micromechanical model is used to obtain the nonlinear electro-mechanical responses in each layer and is integrated within the laminate theory in order to obtain the overall nonlinear electro-mechanical responses of the functionally graded piezoelectric beams. The numerical predictions are compared with experimental data available in literature. Parametric studies are then performed in order to examine the effects of the thickness of the beam, of the concentration of the constituent, and the frequency of the cyclic electric field on the overall electro-mechanical response of the functionally graded piezoelectric beams.     

Analytical solution for functionally graded magneto-electro-elastic plane beams

This paper investigates the plane stress problem of generally anisotropic magneto-electro-elastic beams with the coefficients of elastic compliance, piezoelectricity, dielectric impermeability, piezomagnetism, magnetoelectricity, and magnetic permeability being arbitrary functions of the thickness coordinate. Firstly, partial differential equations governing stress function, electric displacement function and magnetic induction function are derived for plane problems of anisotropic functionally graded magneto-electro-elastic materials. Secondly, these functions are assumed in forms of polynomials in the longitudinal coordinate and can be acquired through a successive integral approach. The analytical expressions of axial force, bending moment, shear force, average electric displacement, average magnetic induction, displacements, electric potential and magnetic potential are then deduced. Thirdly, problems of functionally graded magneto-electro-elastic plane beams are considered, with integral constants being completely determinable from boundary conditions. A series of analytical solutions are thus obtained, including the solutions for beams under tension and pure bending, for cantilever beams subjected to shear force applied at the free end, and for cantilever beams subjected to uniform load. These solutions can be easily degenerated into the solutions for homogenous anisotropic magneto-electro-elastic beams. Finally, a numerical example is presented to show the application of the proposed method.     

Topology optimization of functionally graded cellular materials

Design of functionally graded material (FGM), in which the mechanical property varies along one direction, is the focus of this study. It is assumed that the microstructure of the FGM is composed of a series of base cells in the variation direction and self-repeated in other directions. Bi-directional evolutionary structural optimization technique in the form of inverse homogenization is used for the design of the FGM for specified variation in bulk or shear modulus. Instead of designing a series of base cells simultaneously, the base cells are optimized progressively by considering three base cells at each stage. Thus, the proper connections between adjacent base cells can be achieved with high computational efficiency. Numerical examples demonstrate the effectiveness of the proposed method for designing microstructures of 2D and 3D FGMs with specified variation in bulk or shear modulus. The proposed algorithm can also be easily extended to design FGMs with other functional properties.     

Nonlinear forced vibration analysis of clamped functionally graded beams

Multiple time scale solutions are presented to study the nonlinear forced vibration of a beam made of symmetric functionally graded (FG) materials based on Euler?CBernoulli beam theory and von Kármán geometric nonlinearity. It is assumed that material properties follow either exponential or power law distributions through the thickness direction. A Galerkin procedure is used to obtain a second-order nonlinear ordinary equation with cubic nonlinear term. The natural frequencies are obtained for the nonlinear problem. The effects of material property distribution and end supports on the nonlinear dynamic behavior of FG beams are discussed. Also, forced vibrations of the system in primary and secondary resonances have been studied, and the effects of different parameters on the frequency-response have been investigated.     

Thermo-electrical buckling of piezoelectric functionally graded material Timoshenko beams

In this article, buckling analysis of functionally graded material (FGM) beams with or without surface-bonded piezoelectric layers subjected to both thermal loading and constant voltage is studied. Thermal and mechanical properties of FGM layer is assumed to follow the power law distribution in thickness direction, except Poisson’s ratio which is considered constant. The Timoshenko beam theory and nonlinear strain-displacement relations are used to obtain the governing equations of piezoelectric FGM beam. Beam is assumed under three types of thermal loading and various types of boundary conditions. For each case of boundary conditions, existence of bifurcation-type buckling is examined and for each case of thermal loading and boundary conditions, closed-form solutions are obtained which are easily usable for engineers and designers. The effects of the applied actuator voltage, beam geometry, boundary conditions, and power law index of FGM beam on critical buckling temperature difference are examined.     

Hybrid finite element for analysis of functionally graded beams

A hybrid finite element model is presented, where stiffness and mass distributions over a beam with functionally graded material (FGM) are accurately modeled for both elastic and inelastic material responses. Von Mises and Drucker-Prager plasticity models are implemented for metallic and ceramic parts of FGM, respectively. Three-dimensional stress-strain relations are solved by a general closest point projection algorithm, and then condensed to the dimensions of the beam element. Numerical examples and verification studies on a proposed element demonstrate accuracy and robustness under inelastic material response as well as capturing fundamental, higher, and mix modes of vibration frequencies and shapes.     

On the unified equations of functionally graded piezoceramic beams

In this work (part I), we establish the 1D unified equations of a functionally graded piezoceramic beam from the 3D equations of piezoelectricity in both differential and variational forms. The equations of the beam, including a theorem of uniqueness, are obtained using a unified variational principle together with a kinematic-based product method of reduction. In part II, the free vibrations of the beam are considered and the basic properties of eigenvalues are examined. In part III, the equations are derived for the beam under mechanical and electrical bias. Furthermore, a solution for a piezoceramic torsion problem is given.     

功能梯度梁在热冲击下的动态响应

基于Timoshenko梁理论研究了功能梯度材料(FGM)梁在一维热冲击载荷作用下的瞬态动力响应.采用Laplace变换将功能梯度材料中的一维热传导方程转化为拉氏域中的常微分方程进行求解,再进行反变换得到温度场.然后采用微分求积法(DQM)对位移形式的动力学方程及初边值条件进行DQ离散,数值求解离散后的动力学方程,得到了梁在热冲击下的动态位移和应力响应.分析了材料组份指数和几何参数对梁的动力响应的影响,并考察了DQM法对此类问题的有效性.     

On deformation of functionally graded narrow beams under transverse loads

Solution is obtained for functionally graded (FG) narrow beams under plane stress condition of elasticity by using the mixed semi analytical model developed by Kant et al. (Int J Comput Methods Eng Sci Mech 8(3): 165–177, 2007a). The mathematical model consists in defining a two-point boundary value problem (BVP) governed by a set of coupled first-order ordinary differential equations (ODEs) in the beam thickness direction. Analytical solutions based on two dimensional (2D) elasticity, one dimensional (1D) first order shear deformation theory (FOST) and a new 1D higher order shear-normal deformation theory (HOSNT) are also established to show the accuracy, simplicity and effectiveness of the developed mixed semi analytical model. It is observed from the numerical investigation that the present mixed semi analytical model predicts structural response as good as the one given by the elasticity analytical solution which in turn proves the robustness of the present development.     

Deformation analysis of functionally graded beams by the direct approach

In this paper we employ the direct approach to the theory of rods and beams, which is based on the deformable curve model with a triad of rotating directors attached to each point. We show that this model (also called directed curve) is an efficient approach for analyzing the deformation of elastic beams with a complex material structure. Thus, we consider non-homogeneous, composite and functionally graded beams made of isotropic or orthotropic materials and we determine the effective stiffness properties in terms of the three-dimensional elasticity constants. We present general analytical expressions of the effective stiffness coefficients, valid for beams of arbitrary cross-section shape. Finally, we apply this method for FGM beams made of metal foams and compare our analytical results with the numerical results obtained by a finite element analysis.     

Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams

This paper investigates the nonlinear free vibration of functionally graded nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on Timoshenko beam theory and von Kármán geometric nonlinearity. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction and estimated though the rule of mixture. The Ritz method is employed to derive the governing eigenvalue equation which is then solved by a direct iterative method to obtain the nonlinear vibration frequencies of FG-CNTRC beams with different end supports. A detailed parametric study is conducted to study the influences of nanotube volume fraction, vibration amplitude, slenderness ratio and end supports on the nonlinear free vibration characteristics of FG-CNTRC beams. The results for uniformly distributed carbon nanotube-reinforced composite (UD-CNTRC) beams are also provided for comparison. Numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fraction, vibration amplitude, slenderness ratio, end supports and CNT distribution on the nonlinear free vibration characteristics of FG-CNTRC beams.     

Modelling and process optimization for functionally graded materials

We optimize continuous quench process parameters to produce functionally graded aluminium alloy extrudates. To perform this task, an optimization problem is defined and solved using a standard non‐linear programming algorithm. Ingredients of this algorithm include (1) the process parameters to be optimized, (2) a cost function: the weighted average of the precipitate number density distribution, (3) constraint functions to limit the temperature gradient (and hence distortion and residual stress) and exit temperature, and (4) their sensitivities with respect to the process parameters. The cost and constraint functions are dependent on the temperature and precipitate size which are obtained by balancing energy to determine the temperature distribution and by using a reaction‐rate theory to determine the precipitate particle sizes and their distributions. Both the temperature and the precipitate models are solved via the discontinuous Galerkin finite element method. The energy balance incorporates non‐linear boundary conditions and material properties. The temperature field is then used in the reaction rate model which has as many as 10⁵ degrees‐of‐freedom per finite element node. After computing the temperature and precipitate size distributions we must compute their sensitivities. This seemingly intractable computational task is resolved thanks to the discontinuous Galerkin finite element formulation and the direct differentiation sensitivity method. A three‐dimension example is provided to demonstrate the algorithm. Copyright © 2004 John Wiley & Sons, Ltd.     

含裂纹功能梯度Euler-Bernoulli梁和Timoshenko梁的屈曲载荷计算与分析

含裂纹构件的屈曲载荷是结构是否安全的判定准则之一, 其计算与分析也是结构健康监测和安全评价中关注的重要问题。基于Euler-Bernoulli梁理论和Timoshenko梁理论, 建立了一种求解含裂纹功能梯度材料梁的屈曲载荷计算方法。首先裂纹导致的构件截面转角不连续性由转动弹簧模型进行模拟, 再根据功能梯度材料Euler-Bernoulli梁和Timoshenko梁的屈曲控制方程及其闭合解, 由传递矩阵法建立了求解含裂纹功能梯度材料梁在多种边界条件下屈曲载荷的循环递推公式和特征行列式, 使问题通过降阶的方法得到快速准确的解答。数值算例研究了剪切变形、 裂纹的不同数目及位置、 材料参数变化、 长细比和不同边界约束条件等对含裂纹功能梯度材料梁屈曲载荷的影响。结果表明该方法可以简单、 方便和准确地计算不同数目裂纹和任意边界条件下功能梯度材料梁的屈曲问题。      

Hierarchical theories for the free vibration analysis of functionally graded beams

This work addresses a free vibration analysis of functionally graded beams via several axiomatic refined theories. The material properties of the beam are assumed to vary continuously on the cross-section according to a power law distribution in terms of the volume fraction of the material constituents. Young’s modulus, Poisson’s ratio and density can vary along one or two dimensions all together or independently. The three-dimensional kinematic field is derived in a compact form as a generic N-order polynomial approximation. The governing differential equations and the boundary conditions are derived by variationally imposing the equilibrium via the Principle of Virtual Displacements. They are written in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier-type, closed form solution is adopted. Higher-order displacements-based theories that account for non-classical effects are formulated. Classical beam models, such as Euler–Bernoulli’s and Timoshenko’s, are obtained as particular cases. Bending, torsion and axial modes are investigated. Slender as well as short beams are considered. Numerical results highlight the effect of different material distributions on natural frequencies and mode shapes and the accuracy of the proposed models.