Bending analysis of a functionally graded rotating disk based on the first order shear deformation theory

The theoretical formulation for bending analysis of functionally graded (FG) rotating disks based on first order shear deformation theory (FSDT) is presented. The material properties of the disk are assumed to be graded in the radial direction by a power law distribution of volume fractions of the constituents. New set of equilibrium equations with small deflections are developed. A semi-analytical solution for displacement field is given under three types of boundary conditions applied for solid and annular disks. Results are verified with known results reported in the literature. Also, mechanical responses are compared between homogeneous and FG disks. It is found that the stress couple resultants in a FG solid disk are less than the stress resultants in full-ceramic and full-metal disk. It is observed that the vertical displacements for FG mounted disk with free condition at the outer surface do not occur between the vertical displacements of the full-metal and full-ceramic disk. More specifically, the vertical displacement in a FG mounted disk with free condition at the outer surface can even be greater than vertical displacement in a full-metal disk. It can be concluded from this work that the gradation of the constitutive components is a significant parameter that can influence the mechanical responses of FG disks.     

Bending of thick functionally graded plates resting on Winkler–Pasternak elastic foundations

The static response of simply supported functionally graded plates (FGP) subjected to a transverse uniform load (UL) or a sinusoidally distributed load (SL) and resting on an elastic foundation is examined by using a new hyperbolic displacement model. The present theory exactly satisfies the stress boundary conditions on the top and bottom surfaces of the plate. No transverse shear correction factors are needed, because a correct representation of the transverse shear strain is given. The material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of material constituents. The foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second parameter is zero. The equilibrium equations of a functionally graded plate are given based on the hyperbolic shear deformation theory of plates presented. The effects of stiffness and gradient index of the foundation on the mechanical responses of the plates are discussed. It is established that the elastic foundations significantly affect the mechanical behavior of thick functionally graded plates. The numerical results presented in the paper can serve as benchmarks for future analyses of thick functionally graded plates on elastic foundations.     

Dispersion analysis of generalized thermo elastic plate of polygonal cross-sections

In this paper, wave propagation in generalized thermo elastic plate of polygonal cross-sectional plate is studied using the Fourier expansion collocation method. The equations of motion based on two-dimensional theory of elasticity is applied under the plane strain assumption of generalized thermo elastic plate of polygonal cross-sections composed of homogeneous isotropic material. The frequency equations are obtained by satisfying the boundary conditions along the surface of the polygonal plate using Fourier expansion collocation method. The numerical calculations are carried out for triangular, square, pentagonal and hexagonal cross sectional plates. Dispersion curves are drawn using the computed non-dimensional frequencies for longitudinal and flexural (symmetric and antisymmetric) modes of vibration.     

Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory

The main objective of this research work is to present analytical solutions for free vibration analysis of moderately thick rectangular plates, which are composed of functionally graded materials (FGMs) and supported by either Winkler or Pasternak elastic foundations. The proposed rectangular plates have two opposite edges simply-supported, while all possible combinations of free, simply-supported and clamped boundary conditions are applied to the other two edges. In order to capture fundamental frequencies of the functionally graded (FG) rectangular plates resting on elastic foundation, the analysis procedure is based on the first-order shear deformation plate theory (FSDT) to derive and solve exactly the equations of motion. The mechanical properties of the FG plates are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents, whereas Poisson’s ratio is set to be constant. First, a new formula for the shear correction factors, used in the Mindlin plate theory, is obtained for FG plates. Then the excellent accuracy of the present analytical solutions is confirmed by making some comparisons of the results with those available in literature. The effect of foundation stiffness parameters on the free vibration of the FG plates, constrained by different combinations of classical boundary conditions, is also presented for various values of aspect ratios, gradient indices, and thickness to length ratios.     

Dynamic stability of a porous rectangular plate

The study is devoted to a axial compressed porous-cellular rectangular plate. Mechanical properties of the plate vary across is its thickness which is defined by the non-linear function with dimensionless variable and coefficient of porosity. The material model used in the current paper is as described by Magnucki, Stasiewicz papers. The middle plane of the plate is the symmetry plane. First of all, a displacement field of any cross section of the plane was defined. The geometric and physical (according to Hook's law) relationships are linear. Afterwards, the components of strain and stress states in the plate were found. The Hamilton's principle to the problem of dynamic stability is used. This principle was allowed to formulate a system of five differential equations of dynamic stability of the plate satisfying boundary conditions. This basic system of differential equations was approximately solved with the use of Galerkin's method. The forms of unknown functions were assumed and the system of equations was reduced to a single ordinary differential equation of motion. The critical load determined used numerically processed was solved. Results of solution shown in the Figures for a family of isotropic porous-cellular plates. The effect of porosity on the critical loads is presented. In the particular case of a rectangular plate made of an isotropic homogeneous material, the elasticity coefficients do not depend on the coordinate (thickness direction), giving a classical plate. The results obtained for porous plates are compared to a homogeneous isotropic rectangular plate. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

微极热弹性无限板的轴对称自由振动

研究了在应力自由和刚性固定边界条件下,无能量耗散的均匀、各向同性微极热弹性无限板的轴对称自由振动波的传播,导出了相应的对称和斜对称模态波传播的闭合式特征方程和不同区域的特征方程.对短波的情况,应力自由热绝缘和等温板中对称和斜对称模态波传播的特征方程退化为Rayleigh表面波频率方程.根据导出的特征方程得到了热弹性、微极弹性和弹性板的结果.在对称和斜对称运动中计算了板的位移分量幅值、微转动幅值和温度分布,给出了对称和斜对称模式的频散曲线,并示出了位移分量和微转动幅值和温度分布的曲线.能够发现理论分析和数值结论是非常一致的.     

Analysis of stresses in an infinite elastic body with a locally curved covered nanofiber

Within the framework of a piecewise homogeneous body model, with the use of the three-dimensional geometrically nonlinear exact equations of the theory of elasticity, the method developed for determining the stress distribution in nanocomposites with unidirectional locally curved covered nanofibers is used to investigate the normal stresses acting along nanofibers. The investigation is carried out for an infinite elastic body containing a single locally curved covered nanofiber in the case where there exists a bond covering cylinder of constant thickness between the nanofiber and the matrix material. It is assumed that the body is loaded at infinity by uniformly distributed normal forces in the fiber direction. Upon formulation and mathematical solution of the boundary value problem, the boundary form perturbation method is used. Numerical results for the stress distribution in the body and the influence of geometrical nonlinearity on this distribution are presented and interpreted.     

平面弹性裂纹分析的一种有效边界元方法

提出了一种简单而有效的平面弹性裂纹应力强度因子的边界元计算方法．该方法由Crouch与Starfield建立的常位移不连续单元和闫相桥最近提出的裂尖位移不连续单元构成A·D2在该边界元方法的实施过程中,左、右裂尖位移不连续单元分别置于裂纹的左、右裂尖处,而常位移不连续单元则分布于除了裂尖位移不连续单元占据的位置之外的整个裂纹面及其它边界．算例(如单向拉伸无限大板中心裂纹、单向拉伸无限大板中圆孔与裂纹的作用)说明平面弹性裂纹应力强度因子的边界元计算方法是非常有效的．此外,还对双轴载荷作用下有限大板中方孔分支裂纹进行了分析．这一数值结果说明平面弹性裂纹应力强度因子的边界元计算方法对有限体中复杂裂纹的有效性,可以揭示双轴载荷及裂纹体几何对应力强度因子的影响．     

Homogenization of microstructures in dynamics using periodic boundary conditions

A homogenization method is used to calculate effective material properties of periodic microstructures subjected to small, time-harmonic strain fields. The method is based on the equivalence of averaged micro-field quantities in dispersive heterogeneous media and those in effective homogeneous media with frequency dependent elastic constants. Solutions to the elastodynamic boundary value problem (BVP) are obtained from a boundary element method where periodic boundary conditions on the unit-cell are incorporated using Lagrangian multipliers. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A BEM for transient thermoelastic analysis of a functionally graded layer on a homogeneous substrate under thermal shock

Transient thermoelastic analysis of isotropic and linear thermoelastic bimaterials, which are constituted by a functionally graded (FG) layer attached to a homogeneous substrate, subjected to thermal shock is presented in this paper. For this purpose, a boundary element method for transient linear coupled thermoelasticity is developed. The material properties of the FG layer are assumed to be continuous functions of the spatial coordinates. The boundary-domain integral equations are derived by using the fundamental solutions of linear coupled thermoelasticity for the corresponding isotropic, homogeneous and linear thermoelastic solids in the Laplace-transformed domain. For the numerical solution, a collocation method with piecewise quadratic approximation is implemented. Numerical results for the dynamic stress intensity factors are presented and discussed. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A boundary element analysis for stress intensity factors of multiple circular arc cracks in a plane elasticity plate

In this paper, a numerical approach for analyzing interacting multiple cracks in infinite linear elastic media is presented. By extending Bueckner’s principle suited for a crack to a general system containing multiple interacting cracks, the original problem is divided into a homogeneous problem (the one without cracks) subjected to remote loads and a multiple crack problem in an unloaded body with applied tractions on the crack surfaces. Thus, the results in terms of the stress intensity factors (SIFs) can be obtained by considering the latter problem, which is analyzed easily by means of the displacement discontinuity method with crack-tip elements proposed recently by the author. Test examples are given to illustrate that the numerical approach is very accurate for analyzing interacting multiple cracks in an infinite linear elastic media under remote uniform stresses. In addition, the displacement discontinuity method with crack-tip elements is used to analyze a multiple crack problem in a finite plate. It is found that the boundary element method is also very accurate for investigating interacting multiple cracks in a finite plate. Specially, a generalization of Bueckner’s principle and the displacement discontinuity method with crack-tip elements are used to analyze multiple circular arc crack problems in infinite plate in tension (including: Two Collinear Circular Arc Cracks, Three Collinear Circular Arc Cracks, Two Parallel Circular Arc Cracks, Three Parallel Circular Arc Cracks and Two Circular Arc Cracks) in a plane elasticity plate. Many results are given.     

Elasticity Solutions for Cylindrical Bending of Functionally Graded Piezoelectric Material Plates北大核心CSCD

功能梯度压电材料（FGPM）同时兼具功能梯度材料和压电材料特性,可为多功能或智能化轻质结构设计提供支撑,在诸多领域有着广泛的应用前景.将Mian和Spencer功能梯度板理论由功能梯度弹性材料推广到功能梯度压电材料,解析研究了FGPM板的柱面弯曲问题,其中,材料弹性常数、压电和介电参数沿板厚方向可以任意连续变化.最终,给出了FGPM板受横向均布荷载作用下柱面弯曲问题的弹性力学解.通过算例分析,重点讨论了压电效应对FGPM板静力响应的影响.     

多变量样条元法分析弹性地基板的弯曲,振动与稳定问题

本文应用双三次乘积型二元B样条函数来构造弹性地基板的位移、弯矩和扭矩等多种场函数，由混合变分原理导出多变量样条无法方程．文中，对弹性地基板的弯曲、振动与稳定问题作了分析与计算．由于，本文方法设定了各自独立的场函数，因此，所算得的场未知量如位移、弯矩和扭矩值的精度均比较高。     

Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory

Buckling analysis of functionally graded micro beams based on modified couple stress theory is presented. Three different beam theories, i.e. classical, first and third order shear deformation beam theories, are considered to study the effect of shear deformations. To present a profound insight on the effect of boundary conditions, beams with hinged-hinged, clamped–clamped and clamped–hinged ends are studied. Governing equations and boundary conditions are derived using principle of minimum potential energy. Afterwards, generalized differential quadrature (GDQ) method is applied to solve the obtained differential equations. Some numerical results are presented to study the effects of material length scale parameter, beam thickness, Poisson ratio and power index of material distribution on size dependent buckling load. It is observed that buckling loads predicted by modified couple stress theory deviates significantly from classical ones, especially for thin beams. It is shown that size dependency of FG micro beams differs from isotropic homogeneous micro beams as it is a function of power index of material distribution. In addition, the general trend of buckling load with respect to Poisson ratio predicted by the present model differs from classical one.     

Exact solutions for coupled analysis of thin-walled functionally graded beams with non-symmetric single- and double-cells

In the present work, the exact solutions for coupled analysis for bending and torsional case thin-walled functionally graded (FG) beams with non-symmetric single- and double-cells are presented for the first time. For this purpose, an accurate and efficient method is proposed to obtain the FG member stiffness matrix based on the series expansions of displacement components. Three types of material distributions are considered and the beam mechanical properties are graded along the wall thickness according to a power law of the volume fraction. The present beam model is on the basis of the Euler-Bernoulli beam theory and the Vlasov one for bending and torsional problems, respectively. The explicit expressions for displacement parameters are derived using the power series approach from the four coupled equilibrium equations. Finally, the FG member stiffness matrix is determined from the seven force-displacement relations. In order to show the accuracy and super convergence of the thin-walled FG beam element developed by this study, the numerical solutions are presented and compared with results obtained from the finite beam element based on the approximate interpolation polynomials and other available results. Especially, the effects of various structural parameters such as material distribution type, volume fraction index, boundary condition, and material ratio on the spatially coupled responses of FG box beams with non-symmetric single- and double-cells are parametrically investigated.     

面内功能梯度三角形板等几何面内振动分析

基于平面应变理论,利用等几何有限元方法分析了弹性边界条件下面内功能梯度三角形板的面内振动特性.板的材料属性沿厚度方向呈均匀分布,而在面内方向呈任意指数梯度变化.采用非均匀有理B样条(NURBS)基函数对三角形结构进行等几何建模和位移描述,实现了三角形板几何设计和振动分析的无缝衔接.在三角形板边界上引入虚拟弹簧约束并通过调节虚拟弹簧刚度,实现任意边界条件的施加.通过不同的单元细化方案和对比算例,验证了等几何方法的灵活性、准确性和快速收敛性.系统研究了边界条件、材料属性和几何参数对三角形板振动特性的影响.同时给出了弹性边界条件下面内功能梯度三角形板的振动特性解,具有重要参考价值.     

Free vibration of piles embedded in soil having different modulus of subgrade reaction

The soil that the pile is embedded in is idealized by a Winkler model and is assumed to be two layered. The part of the pile extending above the ground is called the first region, and the parts embedded in the soil are called the second and the third regions, respectively. The dynamic displacement function of the pile subjected to an axial force is obtained as a fourth order partial differential equation by taking account of the effects of bending moment and shear force. It is assumed that the behavior of the material is linearly elastic and axial force along the pile length to be constant. Shear effects are included in the differential equations by the second derivative of the elastic curve function with respect to shear deformation. Normalized natural circular frequencies of the pile are calculated using a carry-over matrix and the secant method for non-trivial solution of the linear homogeneous system of equations obtained for a specific value of the axial force, and for two combinations of boundary conditions:     

Micromechanical analysis of the stress transfer in single-fiber composite: The influence of the uniform and graded interphase with finite-thickness

A theoretical model is developed to analyze the stress transfer between fiber and matrix through the interphase with finite thickness. The Young's modulus of interphase is assumed to be homogeneous uniform or power-graded along radial direction while other material parameters are constants. The bonds between fiber and interphase as well as between interphase and matrix are perfect. The geometrical equations are strictly satisfied except that the radial displacement gradient with respect to the axial direction is neglected, as its magnitude is much smaller than that of the axial displacement gradient with respect to the radial direction. The equilibrium equations along radial direction are strictly satisfied, while the equilibrium equations along axial direction are satisfied in the integral forms. In addition, both the interfacial displacement and stress continuity conditions as well as stress boundary conditions are enforced exactly. Two coupled 2nd-order ordinary differential equations can be obtained in terms of average axial stresses in fiber and matrix. Finite element analysis (FEA) with refined mesh for single-fiber composite containing uniform interphase with finite thickness is developed to validate the present model. Series of parameter studies are performed to investigate the influence of interphase properties and thickness as well as the fiber volume content and model length on the stress distribution in composites.     

Nonlinear finite element analysis of functionally graded plates integrated with patches of piezoelectric fiber reinforced composite

In this paper, a nonlinear static finite element analysis of simply supported smart functionally graded (FG) plates in the presence/absence of the thermal environment has been presented. The substrate FG plate is integrated with the patches of piezoelectric fiber reinforced composite (PFRC) material which act as the distributed actuators of the plate. The material properties of the FG substrate plate are assumed to be temperature dependent and graded along the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The derivation of this nonlinear thermo-electro-mechanical coupled finite element model is based on the first order shear deformation theory and the Von Karman type geometric nonlinearity. The numerical solutions of the nonlinear equations of the finite element model are obtained by employing the direct iteration method. The numerical illustrations suggest the potential use of the distributed actuator made of the PFRC material for active control of nonlinear deformations of smart FG structures. The effects of volume fraction index of the FG material of the substrate plates and the locations of the PFRC patches on the control authority of the patches are investigated. Emphasis has also been placed on investigating the effect of variation of piezoelectric fiber orientation angle in the PFRC patches on their actuation capability for counteracting the large deflections of FG plates.     

Weak solutions for time‐dependent boundary integral equations associated with the bending of elastic plates under combined boundary data

The existence, uniqueness, stability, and integral representation of distributional solutions are investigated for the equations of motion of a thin elastic plate with a combination of displacement and moment‐stress components prescribed on the boundary. Copyright © 2004 John Wiley & Sons, Ltd.