Material tailoring for functionally graded hollow cylinders and spheres

We present a technique to tailor materials for functionally graded (FG) linear elastic hollow cylinders and spheres to attain through-the-thickness either a constant hoop (or circumferential) stress or a constant in-plane shear stress. The volume fractions of two phases of a FG material (FGM) are assumed to vary only with the radius and the effective material properties are estimated by using either the rule of mixtures or the Mori-Tanaka scheme; the analysis is applicable to other homogenization methods. For a FG cylinder we find the required radial variation of the volume fractions of constituents to make a linear combination of the radial and the hoop stresses uniform throughout the thickness. The through-the-thickness uniformity of the hoop stress automatically eliminates the stress concentration near the inner surface of a very thick cylinder. The through-the-thickness variations of Young’s moduli obtained with and without considering the variation of Poisson’s ratio are very close to each other for a moderately thick hollow cylinder but are quite different in a very thick hollow cylinder. For an FG sphere the required radial variation of the volume fractions of the two phases to get a constant circumferential stress is similar to that in an FG cylinder. The material tailoring results presented here should help structural engineers and material scientists optimally design hollow cylinders and spheres comprised of inhomogeneous materials.     

复合材料层合板临界屈曲载荷分散性

基于随机场理论, 将纤维和基体性能以及纤维体积分数作为随机场变量, 利用局部平均法对随机场进行离散。结合MATLAB与ANSYS的PDS模块对复合材料层合板临界屈曲载荷进行Monte-Carlo模拟, 分析各类随机场变量、随机场的相关长度、对称性和边界条件对临界屈曲载荷分散性的影响。结果表明: 不同随机场变量对层合板屈曲载荷分散系数影响的程度不同, 纤维体积分数的影响最大, 其次为纤维性能与基体性能; 屈曲载荷的分散系数存在尺寸效应, 随着板尺寸的增加, 屈曲载荷分散系数逐渐减小; 减小相关长度可有效地减小屈曲载荷的分散系数; 纤维正对称铺设所引起的屈曲载荷分散系数稍大于反对称铺设情况, 而两对边固支板的屈曲载荷分散系数一般大于四边简支板的结果。     

Optimal laminations of thin underwater composite cylindrical vessels

This paper deals with the optimal design of deep submarine exploration housings and autonomous underwater vehicles. The structures under investigation are thin-walled laminated composite unstiffened vessels. Structural buckling failure due to the high external hydrostatic pressure is the dominant risk factor at exploitation conditions. The search of fiber orientations of the composite cylinders that maximize the stability limits is investigated. A genetic algorithm procedure coupled with an analytical model of shell buckling has been developed to determine numerically optimized stacking sequences. Characteristic lamination patterns have been obtained. FEM analyses have confirmed the corresponding significant increases of buckling pressures with respect to initial design solutions. Experiments on thin glass/epoxy and carbon/epoxy cylinders have been performed. The measured buckling pressures appear to be in good agreement with numerical results and demonstrate the gains due to the optimized laminations.     

Optimality criteria-based topology optimization of a bi-material model for acoustic–structural coupled systems

This article investigates topology optimization of a bi-material model for acoustic–structural coupled systems. The design variables are volume fractions of inclusion material in a bi-material model constructed by the microstructure-based design domain method (MDDM). The design objective is the minimization of sound pressure level (SPL) in an interior acoustic medium. Sensitivities of SPL with respect to topological design variables are derived concretely by the adjoint method. A relaxed form of optimality criteria (OC) is developed for solving the acoustic–structural coupled optimization problem to find the optimum bi-material distribution. Based on OC and the adjoint method, a topology optimization method to deal with large calculations in acoustic–structural coupled problems is proposed. Numerical examples are given to illustrate the applications of topology optimization for a bi-material plate under a low single-frequency excitation and an aerospace structure under a low frequency-band excitation, and to prove the efficiency of the adjoint method and the relaxed form of OC.     

考虑气动弹性约束的复合材料支撑机翼优化设计

基于遗传-敏度混合算法对复合材料支撑机翼开展考虑气动弹性约束的优化设计,并与常规机翼构型进行比较。在严重载荷状态下,以结构质量最小化为目标,以翼尖变形、屈曲稳定性和颤振速度为约束,设计复合材料机翼铺层和支撑结构参数,并研究不同支撑点位置对于优化设计结果的影响。结果表明,复合材料支撑机翼构型能大幅减少弯曲方向上的铺层材料,有明显的减重优势。支撑点位置对于结构质量、屈曲稳定性和扭转刚度分布有较大影响,支撑结构的屈曲破坏在复合材料支撑机翼的结构设计中要引起重视。     

Application of third order shear deformation theory in buckling analysis of 2D-functionally graded cylindrical shell reinforced by axial stiffeners

This paper presents buckling analysis of a two-dimensional functionally graded cylindrical shell reinforced by axial stiffeners (stringer) under combined compressive axial and transverse uniform distributive load. The shell material properties are graded in the direction of thickness and length according to a simple power law distribution in terms of the volume fractions of the constituents. Primarily, the third order shear deformation theory (TSDT) is used to derive the equilibrium and stability equations. Since there is no closed form solution, the numerical differential quadrature method, (DQM), is applied for solving the stability equations. Initially, the obtained results for an isotropic shell using DQM were verified against those given in the literature for simply supported boundary conditions. The effects of load, geometrical and stringer parameters along with FG power index in the various boundary conditions on the critical buckling load have been studied. The study of results confirms that, stringers have significant effects on critical buckling load.     

复合材料支撑机翼撑杆位置与结构综合优化设计

针对复合材料支撑机翼,发展了一种撑杆位置和结构综合优化设计的方法。在两种严重设计载荷状态下,考虑气动弹性效应和复合材料铺层结构的不确定性,以结构质量最小化为目标,以翼尖垂直变形、翼尖扭角、撑杆屈曲稳定性、颤振速度和强度要求为约束,在一个优化过程中实现了撑杆位置和结构参数的同步优化设计和鲁棒优化设计。结果表明,翼尖垂直变形和颤振速度要求对于撑杆位置影响明显,最优的撑杆展向位置都靠近翼根一侧,同时撑杆的总体稳定性成为同步优化设计的关键约束。鲁棒优化设计得到的撑杆位置和结构参数的最优组合对铺层结构的不确定性摄动具有良好的抗干扰性,鲁棒优化得到的最优撑杆位置会随着设计变量摄动范围而变化,翼尖垂直变形成为鲁棒优化设计的关键约束。     

一种基于复合材料加筋板结构效率的稳定性优化方法

提出一种以承载效率最高作为目标的新设计方法, 对复合材料加筋板的承载能力进行优化。讨论了不同压缩与弯曲刚度的匹配模式与加筋板临界失稳载荷的关系。将全局失稳载荷、局部失稳载荷与静载荷的接近程度作为结构承载效率的量化标准, 通过静载荷的控制, 使结构的稳定性向着效率最高的方向优化。以宏观的加筋板压缩与弯曲刚度参数作为设计变量, 构建了一种可用于结构效率优化的代理模型, 避免了局部最优点的出现, 更便于数值寻优。通过有限元分析验证, 优化后壁板的临界失稳载荷与所施加的静载荷基本一致, 反映出较高的效率, 从而验证了该方法的可靠性。      

Uncertainty modelling of critical column buckling for reinforced concrete buildings

Buckling is a critical issue for structural stability in structural design. In most of the buckling analyses, applied loads, structural and material properties are considered certain. However, in reality, these parameters are uncertain. Therefore, a prognostic solution is necessary and uncertainties have to be considered. Fuzzy logic algorithms can be a solution to generate more dependable results. This study investigates the material uncertainties on column design and proposes an uncertainty model for critical column buckling reinforced concrete buildings. Fuzzy logic algorithm was employed in the study. Lower and upper bounds of elastic modulus representing material properties were defined to take uncertainties into account. The results show that uncertainties play an important role in stability analyses and should be considered in the design. The proposed approach is applicable to both future numerical and experimental researches. According to the study results, it is seen that, calculated buckling load values are stayed in lower and upper bounds while the load values are different for same concrete strength values by using different code formula.     

Analysis of elastic wave propagation in a functionally graded thick hollow cylinder using a hybrid mesh-free method

In this paper, a hybrid mesh-free method based on generalized finite difference (GFD) and Newmark finite difference (NFD) methods is presented to calculate the velocity of elastic wave propagation in functionally graded materials (FGMs). The physical domain to be considered is a thick hollow cylinder made of functionally graded material in which mechanical properties are graded in the radial direction only. A power-law variation of the volume fractions of the two constituents is assumed for mechanical property variation. The cylinder is excited by shock loading to obtain the time history of the radial displacement. The velocity of elastic wave propagation in functionally graded cylinder is calculated from periodic behavior of the radial displacement in time domain. The effects of various grading patterns and various constitutive mechanical properties on the velocity of elastic wave propagation in functionally graded cylinders are studied in detail. Numerical results demonstrate the efficiency of the proposed method in simulating the wave propagation in FGMs.     

Material tailoring for orthotropic elastic rotating disks

The tailoring of elastic moduli in the radial direction is studied to design a fiber-reinforced orthotropic linear elastic rotating disk with constant radial or hoop stress or constant in-plane shear stress. For fibers arranged in concentric circles the axes of material symmetry coincide with the radial and the circumferential directions. However, when fibers are aligned along helices, the orientation of material principal axes varies with the radial coordinate of a point. For a solid disk made of an orthotropic material with Young’s moduli proportional to each other, we give explicit expressions for the required variations of the elastic moduli with the radius to attain a given state of stress. For a rotating annular disk composed of a fiber-reinforced composite with fibers placed along concentric circles, the required radial variation of the volume fraction of fibers is calculated numerically and exhibited graphically. For fibers of known volume fraction laid along helices, the radial variation of the fiber orientation angle is determined. We have also analyzed the material tailoring problem for a disk of variable thickness. Results presented herein should help structural engineers and material scientists optimally design rotating disks composed of radially inhomogeneous materials.     

Optimum design of hybrid composite multi-ring flywheel rotor based on displacement method

A structure optimum design based on the displacement method has been performed to maximize the energy storage capacity of a hybrid composite multi-ring flywheel rotor. In the process of optimal design, the preload stress generated by interference assembly, the fiber material failure and the delamination between two adjacent rings under high speed rotation are all considered. Four types of the optimal schemes of energy storage capacity, energy per unit mass (EPM), energy per unit volume (EPV), energy per unit cost (EPC) and energy per unit mass and cost (EPMC) are proposed to satisfy the needs of different applications and optimal designs are carried out by using a sequential quadratic programming (SQP). The optimal results show that all composed rings of the hybrid flywheel rotor can nearly reach the limits of strength in both radial and circumferential directions, and simultaneously the rotor is at the critical state of delamination. The radius parameters and the maximum allowed rotational speed of the hybrid composite flywheel are closely related to the optimal schemes. Considering the effects of angular acceleration and gravity on the delamination will result in the decreasing of energy storage capacities for four typical applications.     

Minimum weight design of non‐linear elastic structures with multimodal buckling constraints

It well known that multimodal instability is an event particularly relevant in structural optimization. Here, in the context of non‐linear stability theory, an exact method is developed for minimum weight design of elastic structures with multimodal buckling constraints. Given an initial design, the method generates a sequence of improved designs by determining a sequence of critical equilibrium points related to decreasing values of the structural weight. Multimodal buckling constraints are imposed without repeatedly solving an eigenvalue problem, and the difficulties related to the non‐differentiability in the common sense of state variables in multimodal critical states, are overcome by means of the Lagrange multiplier method. Further constraints impose that only the first critical equilibrium states (local maxima or bifurcation points) on the initial equilibrium path of the actual designs are taken into account. Copyright © 2002 John Wiley & Sons, Ltd.     

Hydrodynamic stability of viscous flow between curved porous channel with radial flow

A linear stability analysis has been presented for the flow between long concentric stationary porous cylinders driven by constant azimuthal pressure gradient, when a radial flow through the permeable walls of the cylinders is present. The radial Reynolds number, based on the radial velocity at the inner cylinder and the inner radius is varied from −100 to 30. The linearized stability equations form an eigenvalue problem which are solved using a numerical technique based on classical Runge-Kutta scheme combined with a shooting method, termed as unit disturbance method. It is observed that radially outward flow and strong inward flow have a stabilizing effect, while weak inward flow has a destabilizing effect on the stability. Profiles of the relative amplitude of the perturbed radial velocities show that radially outward flow shifts the vortices toward the outer cylinder, while radially inward flow shifts the vortices toward the inner cylinder.     

Buckling of functionally graded conical panels under mechanical loads

This paper presents an analytical approach to investigate the linear buckling of truncated conical panels made of functionally graded materials and subjected to axial compression, external pressure and the combination of these loads. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and linear stability equations in terms of displacement components for conical panels are derived by using the classical thin shell theory. Approximate analytical solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to obtain closed-form relations of bifurcation type buckling loads. An analysis is carried out to show the effects of material and geometrical properties and combination of loads on the linear stability of conical panels.     

Post microbuckling mechanics of fibre-reinforced shape-memory polymers undergoing flexure deformation

The buckling mechanics of fibre-reinforced shape-memory polymer composites (SMPCs) under finite flexure deformation is investigated. The analytical expressions of the key parameters during the buckling deformation of the materials were determined, and the local post-buckling mechanics of the unidirectional fibre-reinforced SMPC were further discussed. The cross section of SMPC under flexural deformation can be divided into three areas: the non-buckling stretching area, non-buckling compression area and buckling compression area. These areas were described by three variables: the critical buckling position, the neutral plane position and the fibre buckling half-wavelength. A strain energy expression of the SMPC thermodynamic system is developed. According to the principle of minimum energy, the analytical expressions of key parameters in the flexural deformation process is determined, including the critical buckling curvature, critical buckling position, position of the neutral plane, wavelength of the buckling fibre, amplitude of the buckling fibre and macroscopic structural strain of the composite material. The results showed that fibre buckling occurred in the material when the curvature increasing from infinitesimal to the critical value. If the curvature is greater than the critical curvature, the neutral plane of the material will move towards the outboard tensile area, and the critical buckling position will move towards the neutral plane. Consequently, the half-wavelength of the buckling fibre was relatively stabilised, with the amplitude increasing dramatically. Along with the increasing of the shear modulus, the critical curvature and buckling amplitude increase, while the critical half-wavelength of the fibre buckling decrease and the critical strain of the composite material increase. Finally, we conducted experiments to verify the correction of the key parameters describing SMPC materials under flexural deformation. The values determined by the experiments proved that the theoretical prediction is correct. Additionally, the buckling deformation of the carbon fibre generated a large macroscopic structural strain of the composite material and obtained a resulting large flexural curvature of the structure with minimal material strain of the carbon fibre.     

Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes

In this paper, the mechanical buckling of a functionally graded nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes (SWCNTs) subjected to uniaxial and biaxial in-plane loadings is investigated. The material properties of the nanocomposite plate are assumed to be graded in the thickness direction and vary continuously and smoothly according to two types of the symmetric carbon nanotubes volume fraction profiles. The material properties of SWCNT are determined according to molecular dynamics (MDs), and then the effective material properties at a point are estimated by either the Eshelby–Mori–Tanaka approach or the extended rule of mixture. The equilibrium and stability equations are derived using the Mindlin plate theory considering the first-order shear deformation (FSDT) effect and variational approach. The results for nanocomposite plate with uniformly distributed CNTs, which is a special case in the present study, are compared with those of the symmetric profiles of the CNTs volume fraction. A numerical study is performed to investigate the influences of the different types of compressive in-plane loadings, CNTs volume fractions, various types of CNTs volume fraction profiles, geometrical parameters and different types of estimation of effective material properties on the critical mechanical buckling load of functionally graded nanocomposite plates.     

复合材料层合板屈曲稳定性优化设计及其灵敏度计算方法

笔者在有限元分析基础上研究了以屈曲稳定性作为约束条件或优化目标的复合材料层合板结构优化设计及其灵敏度分析方法,重点讨论了屈曲临界荷载灵敏度对内力场和载荷的依赖关系及其在铺层优化、尺寸优化和形状优化问题中的不同计算方法,并在JIFEX软件中实现了复杂结构复合材料层合板优化设计方法。数值算例验证了本文算法和程序的有效性。     

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.     

Differential quadrature analysis of functionally graded circular and annular sector plates on elastic foundation

The main purpose of this study is to investigate buckling and free vibration behaviors of radially functionally graded circular and annular sector thin plates subjected to uniform in-plane compressive loads and resting on the Pasternak elastic foundation. In-plane compressive loads may be applied to either radial, circumferential, or all edges of circular/annular sector plates. Based on the classical plate theory (CPT), critical buckling loads and fundamental frequencies of the circular/annular sector plates under simply-supported and clamped boundary conditions are obtained by using differential quadrature method (DQM). The inhomogeneity of the plate is characterized by taking exponential variation of Young’s modulus and mass density of the material along the radial direction whereas Poisson’s ratio is considered to be constant. Convergence study is carried out to demonstrate the stability of the present method. To confirm the excellent accuracy of the present approach, a few comparisons are made for limited cases between the present results and those available in literature. Critical buckling load and fundamental frequency parameters of the circular/annular sector thin plates are computed for different boundary conditions, various values of the material inhomogeneity constants, sector angles, and inner to outer radius ratios.