正交各向异性圆柱体在轴压作用下的应力场

基于材料体积不可压假设,对轴向压缩作用下圆柱试件在加载面内的环向和径向应力分布进行理论分析,计算结果表明:当试件材料本构为正交各向异性时,环向和径向应力分布为半径的幂函数形式;试件材料为横观各向同性时,环向和径向应力为半径的二次函数.在圆柱试件轴线上环向和径向应力相等,且均具有最大值;试件圆周边界上径向应力为0,环向应力具有极小值.通过最大拉伸应变破坏理论对试件环向应变进行分析,获得了产生环向拉伸破坏时的临界轴向载荷;并采用Hill-蔡强度理论对试件圆周边界上计算得到的应力参量进行描述,得到了轴压作用下圆柱试件的Hill-蔡强度理论表达式,其不仅取决于轴向应力和试件材料的基本力学性能,还与试件轴向变形的应变率及应变率随时间的变化率相关.     

柔性圆环壳在子午面内整体弯曲的复变量方程及细环壳的一般解

在З.Л.Аксельрад(E.L.Axelrad)非轴对称载荷下柔性旋转壳线性方程的基础上,导出了圆环壳在子午面内整体弯曲的复变量方程和相应的细环壳方程.该方程可与钱伟长给出了一般解的В.В.Новожилов(V.V.Novozhilov)轴对称环壳方程相类比.通过类比,给出了细环壳在子午面内整体弯曲的一般解.所给出的解可以用来计算波纹管整弯曲的应力和端面位移.     

矩形巷道围岩应力分布特征

为得出矩形巷道围岩应力分布特征,根据复变函数理论推导得出了矩形巷道围岩应力解析式并采用巷道围岩沿线分布和matlab软件对巷道围岩应力进行仿真分析两种方法对巷道围岩应力分布特征进行分析.以沿水平线φ=0的围岩应力分布规律为例进行巷道围岩沿线应力分布分析,得出了侧压系数对巷道径向应力和环向应力大小影响规律;侧压系数对径向应力和环向应力峰值出现位置影响规律;围岩至巷道右帮距离与径向应力和环向应力变化规律.以侧压系数对围岩应力空间分布影响为例采用matlab软件对巷道围岩应力仿真分析,得得到了巷道围岩应力空间分布图,可比较方便直观的得到巷道围岩任意位置应力分布情况,还得出了不同巷道尺寸及不同侧压下应力集中系数峰值位置均出现在矩形巷道的四角上.研究结果为巷道支护设计提供了理论依据.     

无限圆柱体旋转运动时的热应力

研究旋转对确定边界条件下无限圆柱体的影响.当热荷载沿径向作用时,给出了旋转圆柱体中热应力、位移和温度的分析过程.当无限弹性圆柱体部分弯曲界面有常温作用,而其余界面维持零温度时,讨论其热动应力的分布.圆柱体表面绝缘材料熔化时出现这种情况.得到了应力分量、位移分量和温度的解和数值结果.提出的半解析法所得到的结果,与早期采用方法所得到的结果比较,发现两者显示出很好的一致性.     

旋转壳在子午面内整体弯曲的几何非线性摄动有限元法及在波纹管计算中的应用(Ⅰ)

为切实有效地计算波纹管,建立了旋转壳在子午面内整体弯曲几何非线性问题的摄动有限元法。以结构环向应变的均方根为摄动小参数,将有限元节点位移列式和节点力列式直接展开。通过摄动小参数将非线性有限元的载荷分级和迭代过程有机地统一起来,即载荷的分级是有约束的,每一级载荷增量和所对应的位移增量之间的关系是已知的,每一级的计算一步到位。为叙述方便并具实用性,将旋转壳用截锥壳单元进行离散。位移分量和载荷分量沿环向按Fourier级数展开,沿子午线用多项式插值,端面弯矩和横向力化成载荷分量离散到节点上。采用Sanders中小转角非线性几何方程和各向同性广义Hooke定律。对多层材料叠合而成的旋转壳按各层薄膜应变、弯曲应变、扭转应变相等的原则进行处理,该方法能方便有效地计算单层和多层波纹管整体纯弯曲、横向弯曲的几何非线性问题。并为有限元处理非线性问题提供了一条新途径。     

求解双材料裂纹结构全域应力场的扩展边界元法

在线弹性理论中,复合材料裂纹尖端具有多重应力奇异性,常规数值方法不易求解.该文建立的扩展边界元法(XBEM)对围绕尖端区域位移函数采用自尖端径向距离r的渐近级数展开式表达,其幅值系数作为基本未知量,而尖端外部区域采用常规边界元法离散方程.两方程联立求解可获得裂纹结构完整的位移和应力场.对两相材料裂纹结构尖端的两个材料域分别采用合理的应力特征对,然后对其进行计算,通过计算结果的对比分析,表明了扩展边界元法求解两相材料裂纹结构全域应力场的准确性和有效性.     

板弯曲断裂计算的边界配置解法

采用复变函数理论和边界配置方法,分析计算了Kirchhoff板的弯曲断裂问题.假设了位移及内力的复变函数式,它们能满足一系列的基本方程和支配条件,例如域内的平衡方程、裂纹表面的边界条件、裂纹尖端的应力奇异性质.这样,仅板边界的边界条件需要考虑.它们可用边界配置法和最小二乘法近似满足.对不同边界条件和载荷情形进行了分析计算.数值算例表明,本文方法精度较高,计算量小,是一种有效的半解析、半数值计算方法.     

探讨一种新的复合型断裂判据——塑性区最短距离r_(min)判据

裂纹尖端塑性区的存在是抗裂的重要因素,在同一个塑性区上,哪个方向上塑性区距离最短(指裂纹尖端到塑性区边缘的距离),裂纹就最容易从哪个方向上扩展.将复合型裂纹尖端应力分量代入R.von Mises屈服条件,得到裂纹尖端塑性区的边界方程:r表示从裂纹尖端到塑性区边缘的距离,裂纹沿rmin方向扩展,即由条件确定裂纹扩展方向.材料破坏的形式之一是在剪应力作用下产生滑动,在复杂受力下,八面体剪应力就是促使这种破坏的作用力.裂纹尖端附近某一点的八面体剪应力为:可令上式中B是一个和裂纹大小、形状以及外加应力有关的量.它的大小反映了裂纹尖端附近应力场的强弱.显然它可作为裂纹临界扩展的判据.当B达到临界值BC时,裂纹开始扩展,即B=BC,用本判据理论研究Ⅰ型裂纹问题,得到本判据理论是根据弹性力学八面体剪应力和R.von Mises屈服条件建立起来的,它把断裂力学和传统的力学理论联系了起来,概念清楚、计算简便.从电算结果来看,它比S判据、(σθ)_max判据,G判据更合理、更准确些.     

复合材料加筋薄壁圆锥壳体有限变形的混合型理论

本文利用变分原理和平均筋条刚度法,建立了在任意载荷作用下纵向和环向密加筋复合材料圆锥壳体有限变形的Donnell型理论.考虑了面板最一般的弯曲拉伸耦合关系和加筋筋条的偏心效应的影响.导出了平衡条件、边界条件和变形协调方程.给出了以应力函数和挠度函数表示的耦合形式的非性性变系数偏微分方程组.对于一些特殊情况,给出了相应的简化方程.     

周向加肋非圆柱壳谐振分析的一个新矩阵方法

基于一类柱壳谐振控制方程呈一阶常微分矩阵方程形式以及傅立叶级数展开,提出了一种新矩阵方法,求解两端简支具有环肋加强非圆柱壳在谐外压作用下的稳态响应．该方法和以往同类方法相比,有两个突出的优点:1) 矩阵微分方程的解采用齐次扩容精细积分法替代龙格-库塔法,提高了精度;其中传递矩阵能实现计算机精确计算．2) 环肋作用力借助Dirac-δ函数和三角级数逼近可以解析求出;除法向作用力外,还考虑了切向作用力．通过数值计算,还研究了外激励频率对壳体位移和应力的影响规律．对比有限元分析与其它方法的计算结果,表明了该方法的正确性和有效性．     

Static analysis of rectangular nano-plate using three-dimensional theory of elasticity

Analytical solution for bending of a simply supported rectangular graphene sheets based on three dimensional theory of elasticity, is studied employing non-local continuum mechanics. By applying the Fourier series solution to the both displacement and stress field along the in-plane rectangular coordinates direction, and to the governing equation and constitutive relations, the three-dimensional governing equations in term of displacement components can be obtained. Closed form solution for the bending behavior of nano-plate is obtained by exerting the surface tractions on the state equations. To validate the accuracy and convergence of the present approach, numerical results are presented and compared with the results available in the open literature. Effect of non-local parameter, aspect ratio, thickness to length ratio and half wave numbers in the bending behavior of plate are examined. Furthermore, these results may also serve as benchmark to further results into the two-dimensional plate theories.     

A linear model for the static and dynamic analysis of non-homogeneous curved beams

A simple one-dimensional mechanical model is presented to analyse the static and dynamic feature of non-homogeneous curved beams and closed rings. Each cross-section is assumed to be symmetrical and the “resultant loads” are acted in the plane of symmetry. The internal forces in a cross-section are replaced by an equivalent force–couple system at the origin of the cylindrical coordinate system used. The equations of motion and the boundary conditions are expressed in terms of two kinematical variables. The first kinematical variable is the radial displacement of cross-sections and the second one is the rotation of the cross-sections. Each of them depends on the time and the polar angle. Assumed form of the displacement field assures the fulfillment of the classical Bernoulli–Euler beam theory. Rotary inertia is included in the equations of motion. Natural frequencies for simply supported laminated composite curved beams and non-homogeneous circular rings are obtained by exact solutions. The application of the model presented is illustrated by examples.     

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.     

夹芯梁的精确解法

夹芯梁与普通梁的本质区别在于剪切引起芯层横截面严重的而又不均匀的翘曲变形,其应力分布已远非初等理论所能描述,而正在广泛应用的经典夹层理论却都建立在平面假设基础上,尤其不能正确反映弱芯的轻质夹层结构的行为,本文放弃了不合理的假设,将夹芯梁视为一般层状弹性体,严格按弹性理论导出了既满足控制方程又同时满足全部边界条件、层间的应力及位移的连续条件的封闭解.它可确切地反映夹芯梁的位移形态和应力分布,并从不同角度,包括多种实验和FEM数值解,验证了它的正确性.     

3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory

This paper addresses a 3D elasticity analytical solution for static deformation of a simply-supported rectangular micro/nanoplate made of both homogeneous and functionally graded (FG) material within the framework of modified couple stress theory. The plate is assumed to be resting on a Winkler–Pasternak elastic foundation, and its modulus of elasticity is assumed to vary exponentially along thickness. By expanding displacement components in double Fourier series along in-plane coordinates and imposing relevant boundary conditions, the boundary value problem (BVP) of plate system, including its governing partial differential equations (PDEs) of equilibrium are reduced to BVP consisting only ordinary ones (ODEs). Parametric studies are conducted among displacement and stress components developed in the plate and FG material gradient index, length scale parameter, and foundation stiffnesses. From the numerical results, it is concluded that the out-of-plane shear stresses are not necessarily zero at the top and bottom surfaces of plate. The results of this investigation may serve as a benchmark to verify further bending analyses of either homogeneous or FG micro/nanoplates on elastic foundation.     

LS model on thermal shock problem of generalized magneto-thermoelasticity for an infinitely long annular cylinder with variable thermal conductivity

In the present paper, an estimation is made to investigate the transient phenomena in magneto-thermoelastic model in the context of the theory of generalized thermoelasticity LS model with variable thermal conductivity. FEM is proposed to analyze the problem and obtain the numerical solutions for the displacement, temperature, and radial and hoop stresses. The boundary conditions for the mechanical and Maxwell’s stresses at the internal and outer surfaces is considered. An application of an infinitely long annular cylinder is investigated for the inner surface is traction free and subjected to thermal shock, while the outer surface is traction free and thermally isolated. Finally, the displacement, incremental temperature, the stress components are obtained and then presented graphically.     

圆形三向网架非线性动力稳定性分析

用拟板法将网架简化为平板,给出表层应变与中面位移的非线性关系．根据薄板的非线性动力学理论,建立了在直角坐标系中三向网架的非线性动力学方程,又将此方程转化为极坐标系轴对称非线性动力学方程．在周边固定条件下,引入异于等厚度板的无量纲量,对基本方程无量纲化．利用Galerkin法得到一个三次非线性振动方程,在无外激励情况下,讨论了稳定性与分岔问题．在外激励情况下,用Melnikov方法研究了圆形三向网架可能发生的混沌运动．通过数字仿真绘出了发生混沌的相平面图．     

Students’ challenges with polar functions: covariational reasoning and plotting in the polar coordinate system

Covariational reasoning has been the focus of many studies but only a few looked into this reasoning in the polar coordinate system. In fact, research on student's familiarity with polar coordinates and graphing in the polar coordinate system is scarce. This paper examines the challenges that students face when plotting polar curves using the corresponding plot in the Cartesian plane. In particular, it examines how students coordinate the covariation in the polar coordinate system with the covariation in the Cartesian one. The research, which was conducted in a sophomore level Calculus class at an American university operating in Lebanon, investigates in addition the challenges when students synchronize the reasoning between the two coordinate systems. For this, the mental actions that students engage in when performing covariational tasks are examined. Results show that coordinating the value of one polar variable with changes in the other was well achieved. Coordinating the direction of change of one variable with changes in the other variable was more challenging for students especially when the radial distance r is negative.     

A dynamical model of storm surges along island coasts

A time-independent dynamical model of storm surge along island coasts using orthogonal curvilinear coordinates is presented. The curved annulus between an island coast and an arbitrary deep-water boundary is mapped conformally onto a rectangular image. Two configurations of island coasts are investigated; circular and elliptic coasts. The corresponding coordinates are circular polar and elliptic respectively. The linearized vertically-integrated equations of motion are used to model storm surges with two assumptions: (i) bottom stress is proportional to horizontal transport, and (ii) storm forces are shear stresses on water surface. Analytical solutions are presented for three dynamical cases: (i) a constant-depth basin acted upon by a uniform storm stress, (ii) variable-depth basin acted upon by a uniform-direction variable-magnitude stress, and. (iii) a basin with closed depth contours acted upon by vortex-shaped storm stress. The obtained solutions clarify the relative importance of the various parameters and variables that affect surge height distribution along island coasts. These solutions may be used to test a time-dependent, numerical dynamical storm model.     

Analytical study on dynamic responses of a curved beam subjected to three-directional moving loads

Considering the warping resistance, inertia force and moving three-directional loads, a more comprehensive set of governing equations for vertical, torsional, radial and axial motions of the curved beam are derived. The analytical solutions for vertical, torsional, radial and axial responses of the curved beam subjected to three-directional moving loads are obtained, using the Galerkin method to discretize the partial differential equations and the modal superposition method to decouple the ordinary differential equations. The analytical results are compared with the numerical integration and a published work to verify the validity of the proposed solutions. Effects of Galerkin truncation terms and damping ratio on solution convergence are also discussed. Considering first-mode and higher-mode truncation respectively, the conditions of resonance and cancellation are analyzed for vertical, torsional, radial and axial motions of the curved beam. Taking a curved bridge under passage of a vehicle as an example, the influences of system parameters, such as vehicle speed, braking acceleration, bridge curve radius, bridge span and bridge deck elastic modulus, on bridge midpoint vibration are explored. The proposed approach and results may be beneficial to enhance understanding the three-directional vehicle-induced dynamic responses of curved bridges. It is shown that when the axial motion, or the multiple moving loads are involved, the first-order truncation are not accurate enough and one should use higher-mode truncation to study the responses of curved beams. In addition, it is necessary to consider damping in the vibration study of curved beams.