ELASTICITY SOLUTIONS OF SPHERICALLY ISOTROPIC CONES UNDER CONCENTRATED LOADS AT APEX

ＥＬＡＳＴＩＣＩＴＹＳＯＬＵＴＩＯＮＳＯＦＳＰＨＥＲＩＣＡＬＬＹＩＳＯＴＲＯＰＩＣＣＯＮＥＳＵＮＤＥＲＣＯＮＣＥＮＴＲＡＴＥＤＬＯＡＤＳＡＴＡＰＥＸＤｉｎｇＨａｏｊｉａｎｇ（丁皓江）；ＺｏｕＤａｏｑｉｎ（邹道勤）；ＲｅｎＹｏｎｇｊｉａｎ；（任永坚）（...     

Bending analysis of functionally graded CNT reinforced doubly curved singly ruled truncated rhombic cone

The bending analysis of functionally graded carbon nanotube (CNT) reinforced doubly curved singly ruled truncated rhombic cone is investigated. In this study, a simple C⁰ isoparametric finite element formulation based on third order shear deformation theory is presented. To characterize the membrane-flexure behavior observed in a CNT reinforced truncated rhombic cone, a displacement field involving higher-order terms in in-plane fields is considered. The proposed kinematics field incorporates for transverse shear deformation and nonlinear variation of the in-plane displacement field through the thickness to predict the overall response of the CNT reinforced truncated rhombic cone in an accurate sense. The material properties of the CNT reinforced truncated rhombic cone are estimated according to the rule of mixture. The present model eliminates the need of shear correction factor and imposed zero-transverse shear strain at upper and lower surface of the truncated rhombic cone. The new feature in present model is simultaneous inclusion of twist curvature in strain field as well as curvature in displacement field that makes it suitable for moderately thick and deep truncated rhombic cone. The proposed new mathematical model is implemented in finite element code written in FORTRAN. The proposed model has been validated with analytical, experimental, and finite element results from the literature. This is first attempt to study bending response of CNT reinforced doubly curved singly ruled truncated rhombic cone. The effect of CNT distribution, boundary condition, loading pattern, and other geometric parameters are also examined.     

磁电弹性圆锥顶端作用集中荷载的解析解

当磁电弹性材料特征根互异时，用5个势函数表示的通解出发，对圆锥顶端作用集中扭矩Mx的扭转、集中力Px和点电荷Q的压缩、集中力Px和集中力矩My的弯曲变形问题，用一些调和函数的线性组合分别构造了势函数，并根据边界条件求出了势函数中的待定系数从而确定势函数，再将势函数代入通解得到磁电弹性圆锥顶端作用集中载荷时的位移、电势、磁势、应力、电位移和磁感应强度的三维解析解。此解形式简单便于应用。当圆锥角2α=π时，可退化得到半空间问题的解。     

Inhomogeneous deformations of a cone of a neo-Hookean solid

We study the inhomogeneous deformations of a cone due to squeezing or expanding the cone. Reminiscent of solutions to Jefferey-Hamel flows between intersecting planes and the deformation of elastic wedges, we find that depending on the cone angle we can have solutions that correspond to radial filaments being purely stretched or contracted, or solutions in which the displacement does not have the same sense, certain subparts being stretched, while others are contracted.     

Recovery of Stresses in a Beam from those in a Cone

We consider a trapezoidal thin plate, as shown in Figure 1, which has one end loaded by a concentrated transverse unit force and the other clamped at zero displacement. The stress is determined according to a method proposed by Galerkin [1] which involves the superposition of two plane elastic stress distributions. To access the accuracy of the method, the oblique sides of the trapezium are allowed to become parallel so that, in the limit, the trapezium becomes a rectangle of the same area. The limiting values of the maximum stress components are exactly those obtained by the Saint-Venant method for a bar. However, over the clamped end, the displacement vanishes only at three arbitrarily selected points. A similar procedure is applied to the transversely loaded truncated cone, but now the limiting values of the stress do not entirely match those obtained by the Saint-Venant theory for a circular cylinder. We also briefly discuss, for completeness, the same bodies under axial loadings, and arrive at similar conclusions which therefore hold for general loads. This revised version was published online in August 2006 with corrections to the Cover Date.     

Isolated crack in three-dimensional piezoelectric solid. Part II: Stress intensity factors for circular crack

This is part II of the work concerned with finding the stress intensity factors for a circular crack in a solid with piezoelectric behavior. The method of solution involves reducing the problem to a system of hypersingular integral equations by application of the unit concentrated displacement discontinuity and the unit concentrated electric potential discontinuity derived in part I [1]. The near crack border elastic displacement, electric potential, stress and electric displacement are obtained. Stress and electric displacement intensity factors can be expressed in terms of the displacement and the potential discontinuity on the crack surface. Analogy is established between the boundary integral equations for arbitrary shaped cracks in a piezoelectric and elastic medium such that once the stress intensity factors in the piezoelectric medium can be determined directly from that of the elastic medium. Results for the penny-shaped crack are obtained as an example.     

板锥网壳结构的连续化分析

板锥网壳结构是一种受力性能合理，技术经济效益良好的新型空间结构形式。本文将板锥网壳结构连续化为能共同作用的特殊形式的三层薄壳，按薄壳理论建立其位移法和混合法的基本微分方程。通过对微分方程的求解，计算其整体位移及结构内力。该法具有一定的精度，可宏观地了解结构的力学性能，并可用于板锥网壳结构的初步设计。     

The general solution of spherically isotropic magnetoelectroelastic media and its applications

A general solution of the three-dimensional equilibrium problem of spherically isotropic magnetoelectroelastic media is presented. Base on the obtained general solution, exact and compact form solutions are obtained for (1) a spherically isotropic magnetoelectroelastic cone subjected to concentrated force, concentrated couple, a point charge and a point electric current at its apex; (2) a spherically isotropic magnetoelectroelastic space with a concentrated force at the origin; (3) a spherical shell under spherically symmetric deformation; and (4) stress concentration around a spherical cavity subjected to remote uniform tensile force, electric charge and electric current.     

集中力作用下铝合金材料内部柱状缺陷检测的研究

工程中,构件的内部会因为各种原因不可避免地存在各种损伤或缺陷。本文通过弹性力学理论给出含柱状缺陷的铝合金材料在受到集中力作用下的离面位移场。同时运用电子散斑干涉方法成功地测量出试件缺陷覆盖层在受到集中力情况下的离面位移。理论和实验结果对比表明,利用电子散斑干涉术技术能够精确地测量含缺陷材料表面的离面位移。此实验方法可以作为一种新型的探测物体内部缺陷所处的位置和大小的评估办法。     

集中力拉伸作用下的不可压缩橡胶类锥体

分析了受集中力拉伸作用下不可压缩橡胶类锥体尖端附近的应力分布及形变行为。给出了锥体尖端应力场的渐近解，当锥角为１８０°时，即为非线性的Ｂｏｕｓｓｉｎｅｓｑ问题的解。     

Analysis of a rubber cone tensioned by a concentrated force

Using the constitutive equation of a rubber-like materials given by Gao (1997), this paper investigates the problem of a cone under tension of a concentrated force at its apex. Under consideration is the axial-symmetry case and the large strain is taken into account. The stress strain fields near the apex are obtained by both asymptotic analysis and finite element calculation. The two results are consistent well. When the cone angle is 180°, the solution becomes that of non-linear Boussinesq's problem for tension case.     

Elasticity solutions for a piezoelectric cone under concentrated loads at its apex

IntroductionTheproblemofaconesubjectedtoconcentratedloadsatitsapexisaclassicalprobleminthetheoryofelasticity.AnumberofscholarshavestUdiedtheproblem.LovereportedthesolutionstotheproblemofanisotropicconeunderconcentfatCdforcesatitsapex['].Lur'estudiedthisclassofproblemssystematicallybymeansofPapkovich-Neubergeneralsolution[2].LekniskiiandHu,byusingtheirrespectivegeneralsolutions,studiedcompressionandbendingproblemsofatransverselyisotropicconesubjectedtoaxialconcentfatedforcesandtfansverseconc…     

A note on line forces in gradient elasticity

The theory of gradient elasticity is applied to line forces. Line forces acting on a point within the body and a concentrated normal force (Flamant problem) which acts on a half plane are studied. Closed analytical solutions which have a simple form are obtained for displacement fields of these forces. The gradient elasticity solutions are free from undesirable displacement singularities predicted by classical elasticity.     

Displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space

This paper describes a displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space or full space. The formulation is based on hypersingular integral equations that relate displacement jumps and tractions along the crack. The integral kernels, which represent stress influence functions for ring dislocation dipoles, are derived from available axisymmetric dislocation solutions. The crack is discretized into constant-strength displacement discontinuity elements, where each element represents a slice of a cone. The influence integrals are evaluated using a combination of numerical integration and a recursive procedure that allows for explicit integration of hyper- and Cauchy singularities. The accuracy of the solution at the crack tip is ensured by adding corrective stresses across the tip element. The method is validated by a comparison with analytical and numerical reference solutions.     

Isolated crack in three-dimensional piezoelectric solid: Part I - Solution by Hankel transform

The fundamental solutions are obtained for a unit concentrated electric potential discontinuity and unit concentrated displacement discontinuity in a three-dimensional piezoelectric medium. Displacements and stresses are derived by application of the boundary integral equation method. These expressions are used to obtain the stress intensity factors for a circular crack in Part II of the study.     

压电激励圆形曲梁的静态响应及位移控制

研究压电激励圆形曲梁的静态位移响应及位移控制的参数特性。将压电夹层圆形曲梁等效为单层结构,基于一维小曲率曲梁理论,建立其控制方程。在集中弯矩和径向集中力以及电载荷作用下,分析了带压电激励器的圆形悬臂曲梁的静态响应。与有限元结果比较表明:本文的理论模型能够模拟压电激励的小曲率圆形曲梁的静态响应。压电夹层圆形曲梁在任意位置的径向集中力载荷作用下,控制其自由端径向位移响应为零,求得控制电压的解析表达,数值分析表明:随着集中力载荷的位置变化和梁长的变化,最优控制电压将出现峰值和反号。     

Elastic Displacement in a Half-Space Under the Action of a Tensor Force. General Solution for the Half-Space with Point Forces

The elastic displacement in an isotropic elastic half-space with free surface is calculated for a point tensor force which may arise from the seismic moment of seismic sources concentrated at an inner point of the half-space. The starting point of the calculation is the decomposition of the displacement by means of the Helmholtz potentials and a simplified version of the Grodskii-Neuber-Papkovitch procedure. The calculations are carried out by using generalized Poisson equations and in-plane Fourier transforms, which are convenient for treating boundary conditions. As a general result we compute the displacement in the isotropic elastic half-space with free surface caused by point forces with arbitrary structure and orientation, localized either beneath the surface (generalized Mindlin problem) or on the surface (generalized Boussinesq-Cerruti problems). The inverse Fourier transforms are carried out by means of Sommerfeld-type integrals. For forces buried in the half-space explicit results are given for the surface displacement, which may exhibit finite values at the origin, or at distances on the surface of the order of the depth of the source. The problem presented here may be viewed as an addition to the well-known static problems of elastic equilibrium of a half-space under the action of concentrated loads. The application of the method to similar problems and another approach to the starting point of the general solution are discussed.     

Green function on two-phase saturated medium by concentrated force in two-dimensional displacement field

The Green function on two-phase saturated medium by concentrated force has a broad and important use in seismology, seismic engineering, soil mechanics, geophysics, dynamic foundation theory and so on. According to the Green function on two-phase saturated medium by concentrated force in three-dimentional displacement field obtained by Ding Bo-yang et al. , it gives out the Green function in two-dimensional displacement field by infinite integral method along x3 -direction derived by De Hoop and Manolis. The method adopted in the thesis is simpler. The result will be simplified to the boundary element method of dynamic problem.     

Buckling of cantilever cones under external pressure

An experimental and numerical study was performed on the buckling behavior of clamped-free truncated cones under external pressure. A lap-jointed polyester cone was vertically settled in a water vessel and was clamped at the small radius end. Raising water level, the relation of normal displacement with water level was recorded and the buckling water level was determined. The experimental results were in good agreement with the finite-element solutions. The same finite-element program was then used for a parametric study on the buckling of clamped-free cones subjected to uniform external pressure. The results show that the ratio of the critical external pressure of a clamped-free cone with that of a simply supported equivalent cylinder is a function of only the taper ratio ψ of the cone.     

全长粘结锚杆在岩质边坡中的受力形式分析

基于空间一点受力的位移解形式--Mindlin 位移解, 结合已有文献对节理面位置处全长粘结锚杆受力的假想形式, 将边坡下滑力简化为作用在锚杆上的集中力, 推导出锚杆沿杆长方向的轴力、剪应力的表达形式. 将推导结果在岩质边坡中进行应用, 得出不同岩体类型岩质边坡锚杆沿杆长方向受力的轴力、剪应力图示.