轴对称任意梯度圆筒的弹性动力学解

基于轴对称平面应变问题的运动方程及弹性梯度材料的应力和位移关系,通过将圆筒分层使材料性质离散为分段常数函数,同时在时域内应用有限差分格式,求得了材料性质沿径向梯度变化的圆筒弹性动力学解。本文解不仅适合任意梯度的弹性圆筒,而且容易满足多种形式的初始条件和边界条件。通过对材料性质沿径向为连续函数分布和分段函数分布的梯度圆筒数值分析,并与已有文献结果比较,得出本文解与已有文献的解吻合较好,验证了本文解的正确性和有效性。对材料性质为分段函数的三层组合圆筒分析发现,中间功能梯度层的指数分布因子对圆筒的径向位移和应力随时间变化都会产生显著影响。     

功能梯度梁与均匀梁静动态解间的相似转换

基于Euler-Bernoulli 梁理论, 研究了功能梯度材料梁的弯曲、屈曲和自由振动. 通过分析和比较功能梯度材料梁 和均匀梁的控制方程, 得到了功能梯度材料梁与均匀梁的解之间的相似转换关系. 在给定功 能梯度材料梁的材料性质在横向按幂函数分布的情况下, 导出了解之间的相似转换系数的解 析表达式. 该系数集中反映了功能梯度梁的材料非均匀性. 因此, 可将功能梯度材料梁的静 动态问题的求解转换为同样载荷和边界条件下均匀梁的静动态问题求解以及相似转换系数的 计算.     

考虑尺度效应的微梁静态弯曲数值分析

基于修正偶应力理论及表面弹性理论,本文提出了一种新的双曲线剪切变形梁模型,用于均匀微尺度梁的静态弯曲分析。该理论可以直接利用本构关系获得横向剪切应力,满足梁顶部和底部的无应力边界条件,避免了引入剪切修正因子。根据广义Young-Laplace方程建立了梁的内部与表面层的应力连续性条件,单一的变量场可以描述梁的位移模式。通过在位移场中考虑表面层厚度以及表面层的应力连续条件,可以使新模型能够更准确地预测微尺寸和表面能相关的尺度效应。通过Hamilton原理推导出了梁的控制方程和边界条件。应变能除了考虑经典弹性理论,还要考虑微结构效应和表面能。Navier-type的解析解适用于简支边界条件,而基于拉格朗日插值的微分求积法(DQEM)可以研究在不同边界条件下的力学响应。把该数值解与Navier方法得出的解析解作了对比,得出:微尺度梁在考虑表面能或微尺寸效应、不同载荷和梁高变化下的响应一致;当不考虑微结构相关性和表面能效应时,该模型退化为经典的欧拉梁模型。     

含固支端梁的解析解

采用弹性力学的应力函数法求含固支端梁的应力和位移时,无法严格满足固支端的实际边界条件,需要采用简化的固支边界条件。本文对已有的简化固支边界条件进行了改进,基于新的简化固支边界条件,推导出了四种含固支端梁的应力和位移的解析解,并进行了相应的数值计算,对几种固支边界条件进行了讨论。由本文方法得到的上表面受均布载荷作用悬臂梁的位移u和v的解析解与有限元解的最大误差分别为3.0%和1.0%,两端固支梁的应力σx的解析解与有限元解的最大误差为5.3%。通过理论与数值结果的比较表明,本文改进的固支边界条件是对固支端一种很好的简化。     

一种特殊梯度分布的功能梯度圆柱壳的二维分析

功能梯度材料是一种新型材料,其结构分析已成为当今力学研究的热点。本文对一种特殊梯度分布的功能梯度材料圆柱壳进行了二维精确分析。从弹性力学平面应变问题的基本方程出发,引入应力函数,导出功能梯度材料圆柱壳受静载作用下的控制微分方程。假设材料的杨氏模量沿半径方向呈幂函数分布,泊松比为常数,利用分离变量法,导出了简支边界情况下功能梯度圆柱壳的精确解。通过算例分析了不同梯度变化时,功能梯度圆柱壳内的应力和位移变化规律。计算结果表明不同梯度分布的圆柱壳结构中的应力、位移沿厚度方向的变化规律是不同的,有时甚至差别很大。因此对于材料性质梯度变化的功能梯度材料圆柱壳,必须针对其自身特点,建立相应的理论分析模型。     

具有梯度界面层的双材料悬臂梁解析解

采用应力函数法,求得了具有弹性模量沿高度线性变化的梯度界面层的双材料悬臂梁在均布载荷作用下的应力和位移解析解。该解可退化为双材料梁、弹性模量沿整个梁高线性变化的梯度梁以及均质材料梁的情况,退化为均质材料梁时与已有结果一致。通过一具体算例将得到的解析解与有限元解进行了比较,两者吻合较好。并讨论了梯度界面层的高度变化对梁中的应力和梁端挠度的影响。结果表明,在梁的总高度不变的情况下,增加梯度界面层的高度可减小弯曲应力和梁端挠度,而对挤压应力和切应力的影响很小。     

压电、压磁材料球对称问题的通解

研究压电、压磁材料在球坐标系下,不计体力、体电荷和体电流的情况下,由平衡方程、梯度方程、压电和压磁的本构方程导出应力、应变、位移、电位移、电场强度、电位势、磁感强度和磁位势各未知量的通解．考虑不同的边界条件,将其通解应用到应力、电学短路以及位移、电学开路和磁场分布的边界条件中,得到不同边界条件下问题的解。     

任意光滑梯度变化的功能梯度材料纯弯曲梁的弹塑性分析

假设功能梯度材料为理想弹塑性材料,其屈服强度和弹性模量均沿梁的高度方向按任意光滑函数连续变化,在小变形及平截面假定下,导出了功能梯度材料纯弯曲梁弹性极限弯矩及塑性极限弯矩的解析表达式,建立了弹塑性应力状态下截面弯矩和截面的弹、塑性应力分布之间的解析关系．研究表明,功能梯度材料梁存在多种可能的屈服模式,其最先屈服的点不一定位于截面应力最大处,而可能位于截面的其他任意位置;屈服强度及弹性模量的梯度变化对梁的弹塑性力学性能有很大影响．研究结果可为功能梯度材料纯弯曲梁的弹塑性问题研究提供一定的参考．     

SiC/Al梯度功能材料紧凑拉伸试件裂纹扩展分析

SiC/Al梯度功能材料各梯度层由不同体积浓度的陶瓷和金属组成,由于材料组分梯度变化,克服了双材料界面的应力突变问题,获得了优异的使用性能.本文首先采用激光云纹干涉法,对具有四个梯度层的SiC/Al梯度功能材料紧凑拉伸试件在机械荷载作用下的拉伸实验位移场进行记录,进而获得紧凑拉伸试件裂纹口位移P-V曲线以及材料断裂韧度实验值;然后根据层合梯度功能材料理论分析模型,建立FGM紧凑拉伸试件的渐近分网有限元平面应变模型,采用通用有限元软件的单元删除模块对试件模型进行裂纹扩展单元的应变模拟分析,从计算所得的裂纹尖端应变图像分析得出梯度材料试件的裂纹扩展规律.     

HRR理论的近似解析解

本文根据文献[3]给出的HRR 解的角函数数值表,利用数值拟合方法,给出了HRR理论的近似解析解.本文结果不仅形式简单,而且和精确解符合得很好.对幂硬化指数n在0和1之间的任何弹塑性材料均适用.给进一步研究裂纹尖端附近的弹塑性应力应变场以及工程安全评定提供了一个方便的解析工具.为了计算方便,我们把应力、应变、位移无量纲化,它们与实际应力、应变、位移的关系为     

Two-dimensional elasticity solution for bending of functionally graded beams with variable thickness

This paper studies the stress and displacement distributions of functionally graded beam with continuously varying thickness, which is simply supported at two ends. The Young’s modulus is graded through the thickness following the exponential-law and the Poisson’s ratio keeps constant. On the basis of two-dimensional elasticity theory, the general expressions for the displacements and stresses of the beam under static loads, which exactly satisfy the governing differential equations and the simply supported boundary conditions at two ends, are analytically derived out. The unknown coefficients in the solutions are approximately determined by using the Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the beams. The effect of Young’s modulus varying rules on the displacements and stresses of functionally graded beams is investigated in detail. The two-dimensional elasticity solution obtained can be used to assess the validity of various approximate solutions and numerical methods for the aforementioned functionally graded beams.     

Static analysis of anisotropic functionally graded magneto-electro-elastic beams subjected to arbitrary loading

This paper considers the analytical and semi-analytical solutions for anisotropic functionally graded magneto-electro-elastic beams subjected to an arbitrary load, which can be expanded in terms of sinusoidal series. For the generalized plane stress problem, the stress function, electric displacement function and magnetic induction function are assumed to consist of two parts, respectively. One is a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (z), and the other a linear polynomial of x with unknown coefficients depending on z. The governing equations satisfied by these z-dependent functions are derived. The analytical expressions of stresses, electric displacements, magnetic induction, axial force, bending moment, shear force, average electric displacement, average magnetic induction, displacements, electric potential and magnetic potential are then deduced, with integral constants determinable from the boundary conditions. The analytical solution is derived for beam with material coefficients varying exponentially along the thickness, while the semi-analytical solution is sought by making use of the sub-layer approximation for beam with an arbitrary variation of material parameters along the thickness. The present analysis is applicable to beams with various boundary conditions at the two ends. Two numerical examples are presented for validation of the theory and illustration of the effects of certain parameters.     

BENDING OF ORTHOTROPIC SANDWICH PLATES WITH A FUNCTIONALLY GRADED CORE SUBJECTED TO DISTRIBUTED LOADINGS

Based on the Reissner assumptions, this paper is concerned with the bending analysis of simply supported sandwich plates with functionally graded core and orthotropic face sheets subjected to transverse distributed loadings. First, the expressions of the displacements, stresses and internal forces of the sandwich plate are presented according to the constitutive relations and stress states of the core and face sheets. Then, the solutions of bending equilibrium equations are derived by expanding the deflection w, transverse shearing forces Q_x and Q_y with double trigonometric series that satisfy the simply supported boundary conditions. Finally, the proposed solution is validated by comparing the results with available elasticity solutions for a square sandwich plate with an isotropic core and finite element simulations for one with functionally graded core. The Young's modulus of the functionally graded core is assumed to be graded by a power law distribution of volume fractions of the constituents, and the Poisson's ratio is held constant. And the effects of the core's top-bottom Young's modulus ratio ? and volume fraction exponent n₀ on the variation of the displacements of the functionally graded sandwich plate are also examined.     

An inverse problem for a functionally graded elliptical plate with large deflection and slightly disturbed boundary

This paper deals with the inverse problem of a functionally graded material (FGM) elliptical plate with large deflection and disturbed boundary under uniform load. The properties of functionally graded material are assumed to vary continuously through the thickness of the plate, and obey a simple power law expression based on the volume fraction of the constituents. Based on the classical nonlinear von Karman plate theory, the governing equations of a thin plate with large deflection were derived. In order to solve this non-classical problem, a perturbation technique was employed on displacement terms in conjunction with Taylor series expansion of the disturbed boundary conditions. The displacements of in-plane and transverse are obtained in a non-dimensional series expansion form with respect to center deflection of the plate. The approximate solutions of displacements are solved for the first three terms, and the corresponding internal stresses can also be obtained.     

Analysis of functionally graded rotating disks with variable thickness

Elastic solutions for axisymmetric rotating disks made of functionally graded material with variable thickness are presented. The material properties and disk thickness profile are assumed to be represented by two power-law distributions. In the case of hollow disk, based on the form of the power-law distribution for the mechanical properties of the constituent components and the thickness profile function, both analytical and semi-analytical solutions are given under free–free and fixed-free boundary conditions. For the solid disk, only semi-analytical solution is presented. The effects of the material grading index and the geometry of the disk on the stresses and displacements are investigated. It is found that a functionally graded rotating disk with parabolic or hyperbolic convergent thickness profile has smaller stresses and displacements compared with that of uniform thickness. It is seen that the maximum radial stress for the solid functionally graded disk with parabolic thickness profile is not at the centre like uniform thickness disk. Results of this paper suggest that a rotating functionally graded disk with parabolic concave or hyperbolic convergent thickness profile can be more efficient than the one with uniform thickness.     

Analytical solutions for single- and multi-span functionally graded plates in cylindrical bending

A recently developed plate theory using the concept of shape function of the transverse coordinate parameter is extended to determine the stress distribution in an orthotropic functionally graded plate subjected to cylindrical bending. The transfer matrix method is presented to derive the shape function. The equations governing the plate deformation are then solved analytically using the transfer matrix method for arbitrary boundary conditions. For a simply supported functionally graded plate, a comparison of the present solution with the exact elasticity solution, the first- and third-order shear deformation plate theories is presented and discussed. It is demonstrated that the present method yields more accurate stresses than the first- and third-order shear deformation theories. The effect of boundary conditions and inhomogeneity of material on the displacements and stresses in functionally graded plates are investigated. A multi-span functionally graded plate with arbitrary boundary conditions is further considered to demonstrate the efficiency of the present method.     

Thermo-magnetic analysis of thick-walled spherical pressure vessels made of functionally graded materials

This study presents an analytical solution of thermal and mechanical displacements, strains, and stresses for a thick-walled rotating spherical pressure vessel made of functionally graded materials(FGMs). The pressure vessel is subject to axisymmetric mechanical and thermal loadings within a uniform magnetic field. The material properties of the FGM are considered as the power-law distribution along the thickness. Navier's equation, which is a second-order ordinary differential equation, is derived from the mechanical equilibrium equation with the consideration of the thermal stresses and the Lorentz force resulting from the magnetic field. The distributions of the displacement,strains, and stresses are determined by the exact solution to Navier's equation. Numerical results clarify the influence of the thermal loading, magnetic field, non-homogeneity constant, internal pressure, and angular velocity on the magneto-thermo-elastic response of the functionally graded spherical vessel. It is observed that these parameters have remarkable effects on the distributions of radial displacement, radial and circumferential strains, and radial and circumferential stresses.     

Three-dimensional analysis of transient thermal stresses in functionally graded plates

An analytical solution is presented for three-dimensional thermomechanical deformations of a simply supported functionally graded (FG) rectangular plate subjected to time-dependent thermal loads on its top and/or bottom surfaces. Material properties are taken to be analytical functions of the thickness coordinate. The uncoupled quasi-static linear thermoelasticity theory is adopted in which the change in temperature, if any, due to deformations is neglected. A temperature function that identically satisfies thermal boundary conditions at the edges and the Laplace transformation technique are used to reduce equations governing the transient heat conduction to an ordinary differential equation (ODE) in the thickness coordinate which is solved by the power series method. Next, the elasticity problem for the simply supported plate for each instantaneous temperature distribution is analyzed by using displacement functions that identically satisfy boundary conditions at the edges. The resulting coupled ODEs with variable coefficients are also solved by the power series method. The analytical solution is applicable to a plate of arbitrary thickness. Results are given for two-constituent metal-ceramic FG rectangular plates with a power-law through-the-thickness variation of the volume fraction of the constituents. The effective elastic moduli at a point are determined by either the Mori–Tanaka or the self-consistent scheme. The transient temperature, displacements, and thermal stresses at several critical locations are presented for plates subjected to either time-dependent temperature or heat flux prescribed on the top surface. Results are also given for various volume fractions of the two constituents, volume fraction profiles and the two homogenization schemes.     

Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam

In this paper, an exact analytical solution is presented for a transversely isotropic functionally graded magneto-electro-elastic (FGMEE) cantilever beam, which is subjected to a uniform load on its upper surface, as well as the concentrated force and moment at the free end. This solution can be applied for any form of gradient distribution. For the basic equations of plane problem, all the partial differential equations governing the stress field, electric, and magnetic potentials are derived. Then, the expressions of Airy stress, electric, and magnetic potential functions are assumed as quadratic polynomials of the longitudinal coordinate. Based on all the boundary conditions, the exact expressions of the three functions can be determined. As numerical examples, the material parameters are set as exponential and linear distributions in the thickness direction. The effects of the material parameters on the mechanical, electric, and magnetic fields of the cantilever beam are analyzed in detail.