自然弯扭梁的非线性分析

研究了两端边界均为完全约束的自然弯扭梁在小应变，大位移和大转动情况下的非线性性质，并考虑了横向剪切变形和扭转翘曲变形的影响，分析中还包括了拉伸，弯曲和扭转的各种弹性耦合。由最小势能原理可以导出所给问题的平衡方程。这里欧拉角可以用来表示任意大的转动。该方法还可方便地推广到其他各种不完全约束边界的情况。此外，利用上述结果还可以得到该梁在小位移理论中的基本方程和有关公式。     

Generalized variational principle of dynamic analysis on naturally curved and twisted box beams for anisotropic materials

An incomplete generalized variational functional for naturally curved and twisted composite box beams with complete constrained boundaries at two ends is established by means of Lagrange multiplier method. The equations of motion governing the dynamic behavior of the beams and corresponding boundary conditions are derived from the stationary condition of the functional. The non-classical influences relevant to the beams are those due to transverse shear deformations, torsion-related warping and several elastic couplings that can arise in composite beams. In order to demonstrate the correctness of the theory developed the natural frequencies and normal mode shapes of the beams under in-plane free vibration are evaluated and compared with the results using PATRAN’s beam elements.     

A theory of shells with small strain accompanied by moderate rotation

This paper is concerned with a constrained theory of shells in the presence of small strain accompanied by moderate rotation. The constrained theory accounts for the effect of transverse normal strain and includes, of course, the special case (corresponding to the Kirchhoff-Love theory of shells) in which the effect of transverse normal strain is absent. After precise estimates for (local) moderate rotation and relative displacement gradients in terms of infinitesimal strain have been effected, a complete theory is formulated with the use of linear constitutive equations. The nature of the complete theory is further examined when initially the shell-like body is a plate; and it is shown that our kinematical formulae (strain-displacement relations), as well as the relevant differential equations of the theory in the absence of the effect of transverse normal strain, systematically reduce to those used in the von Kármán plate equations. Also, in the light of the present results, an assessment of kinematical aspects of previously developed theories of shells undergoing small strain and moderate rotation is indicated.     

C~0型高阶锯齿理论及复合材料厚梁热应力分析

基于已有锯齿理论构造单元时,需使用满足单元间C1连续的插值函数,难于构造多节点高阶单元,而且精度较低。针对已有锯齿理论存在的问题,本文首先发展了C0型锯齿理论。通过虚位移原理推导出在热载荷作用下复合材料梁的平衡方程,并给出了简支复合材料层合梁解析解。基于发展的锯齿理论分析了复合材料夹层梁和层合梁热膨胀问题,并与其他理论结果对比。数值结果表明,发展的C0型锯齿理论能克服已有锯齿理论的难题。     

Dynamic stiffness method for free vibration of an axially moving beam with generalized boundary conditions

Axially moving beams are often discussed with several classic boundary conditions, such as simply-supported ends, fixed ends, and free ends. Here, axially moving beams with generalized boundary conditions are discussed for the first time. The beam is supported by torsional springs and vertical springs at both ends. By modifying the stiffness of the springs, generalized boundaries can replace those classical boundaries.Dynamic stiffness matrices are, respectively, established for axially moving Timoshenko beams and Euler-Bernoulli(EB) beams with generalized boundaries. In order to verify the applicability of the EB model, the natural frequencies of the axially moving Timoshenko beam and EB beam are compared. Furthermore, the effects of constrained spring stiffness on the vibration frequencies of the axially moving beam are studied. Interestingly, it can be found that the critical speed of the axially moving beam does not change with the vertical spring stiffness. In addition, both the moving speed and elastic boundaries make the Timoshenko beam theory more needed. The validity of the dynamic stiffness method is demonstrated by using numerical simulation.     

Non-linear theories of beams and plates accounting for moderate rotations and material length scales

The primary objective of this paper is to formulate the governing equations of shear deformable beams and plates that account for moderate rotations and microstructural material length scales. This is done using two different approaches: (1) a modified von Kármán non-linear theory with modified couple stress model and (2) a gradient elasticity theory of fully constrained finitely deforming hyperelastic cosserat continuum where the directors are constrained to rotate with the body rotation. Such theories would be useful in determining the response of elastic continua, for example, consisting of embedded stiff short fibers or inclusions and that accounts for certain longer range interactions. Unlike a conventional approach based on postulating additional balance laws or ad hoc addition of terms to the strain energy functional, the approaches presented here extend existing ideas to thermodynamically consistent models. Two major ideas introduced are: (1) inclusion of the same order terms in the strain–displacement relations as those in the conventional von Kármán non-linear strains and (2) the use of the polar decomposition theorem as a constraint and a representation for finite rotations in terms of displacement gradients for large deformation beam and plate theories. Classical couple stress theory is recovered for small strains from the ideas expressed in (1) and (2). As a part of this development, an overview of Eringen׳s non-local, Mindlin׳s modified couple stress theory, and the gradient elasticity theory of Srinivasa–Reddy is presented.     

An Adaptive Mesh Rezoning for FSI Problems

In the problems of fluid-structure interaction (FSI) the mesh updating scheme plays a key role. We have developed an adaptive mesh rezoning technique which is applicable to the three-dimensional FSI problems. In order to prevent the inversion of elements in the mesh and to maintain a well-conditioned shape for successive time-step calculations, we introduce constrained conditions of dilatational strain in the least-square form as well as the gradient of displacement vectors, in relatively small elements. By the present mesh rezoning technique, even under the large deformation of boundaries concerned, we can reduce the use of the process of mesh generation and switching of nodal values at the interboundary of time slabs. These steps require rather significant CPU time and induced projection errors of nodal values from the previous mesh to the current one. The case of collapsing tube problems shows the remarkable potential of our method. The present method is entirely general in that it can be applied to structured and unstructured meshes, effectively.     

Thermal post-buckling and nonlinear vibration analysis of FGM beams based on physical neutral surface and high order shear deformation theory

Thermal post-buckling and nonlinear vibration behaviors of FGM beams are analyzed by using concept of physical neutral surface, von Kármán strain-displacement relationships and high order shear deformation theory. Material properties are assumed to be temperature dependent and vary along the thickness. The prominent character of physical neutral surface higher-order shear deformation beam theory is that stretching-bending couplings are eliminated in constitutive equations, and governing equations have the similar forms as homogeneous isotropic beams. Approximate solutions are given out by Ritz method, and influences played by different supported boundaries, thermal environmental conditions and volume fraction index are discussed in detail.     

Elasticity solution of clamped-simply supported beams with variable thickness

This paper studies the stress and displacement distributions of continuously varying thickness beams with one end clamped and the other end simply supported under static loads. By introducing the unit pulse functions and Dirac functions, the clamped edge can be made equivalent to the simply supported one by adding the unknown horizontal reactions. According to the governing equations of the plane stress problem, the general expressions of displacements, which satisfy the governing differefitial equations and the boundary conditions attwo ends of the beam, can be deduced. The unknown coefficients in the general expressions are then determined by using Fourier sinusoidal series expansion along the upper and lower boundaries of the beams and using the condition of zero displacements at the clamped edge. The solution obtained has excellent convergence properties. Comparing the numerical results to those obtained from the commercial software ANSYS, excellent accuracy of the present method is demonstrated.     

多孔功能梯度材料Timoshenko梁的自由振动分析

基于Timoshenko梁理论研究多孔功能梯度材料梁(FGMs)的自由振动问题.首先,考虑多孔功能梯度材料梁的孔隙率模型,建立了两种类型的孔隙分布.其次,基于Timoshenko梁变形理论,给出位移场方程、几何方程和本构方程,利用Hamilton原理推导多孔功能梯度材料梁的自由振动控制微分方程,并进行无量纲化,然后应用微分变换法(DTM)对无量纲控制微分方程及其边界条件进行变换,得到含有固有频率的等价代数特征方程.最后,计算了固定-固定(C-C)、固定-简支(C-S)和简支-简支(S-S)三种不同边界下多孔功能梯度材料梁自由振动的无量纲固有频率.将其退化为均匀材料与已有文献数据结果对照,验证了正确性.讨论了孔隙率、细长比和梯度指数对多孔功能梯度材料梁无量纲固有频率的影响.     

Analysis of micropolar elastic beams

In this paper, a linear theory for the analysis of beams based on the micropolar continuum mechanics is developed. Power series expansions for the axial displacement and micro-rotation fields are assumed. The governing equations are derived by integrating the momentum and moment of momentum equations in the micropolar continuum theory. Body couples and couple stresses can be supported in this theory. After some simplifications, this theory can be reduced to the well-known Timoshenko and Euler–Bernoulli beam theories. The nature of flexural and longitudinal waves in the infinite length micropolar beam has been investigated. This theory predicts the existence of micro-rotational waves which are not present in any of the known beam theories based on the classical continuum mechanics. Also, the deformation of a cantilever beam with transverse concentrated tip loading has been studied. The pattern of deflection of the beam is similar to the classical beam theories, but couple stress and micro-rotation show an oscillatory behavior along the beam for various loadings.     

考虑横法向热应变的C~0型整体-局部高阶层合梁理论

通过考虑横法向热变形,本文建立了预先满足层间应力连续的C0型整体-局部高阶层合梁理论,并用于分析复合材料层合梁热膨胀和热弯曲问题。虽然考虑了横法向应变,不增加额外的位移变量。此理论位移场不含有横向位移一阶导数,便于构造多节点高阶单元。基于虚功原理推导了复合材料层合梁平衡方程,并分析了简支多层复合材料梁热膨胀和热弯曲问题。数值结果表明,建立的模型能准确分析复合材料层合梁热膨胀和热弯曲问题,忽略横法向应变的理论分析热膨胀问题误差较大。     

On beams made of a phase-transforming material

This paper presents a theory for the mechanics of beams made of single crystals of shape memory alloys. The behavior of such beams can be quite unexpected and complicated due to the presence and propagation of phase boundaries. It is shown that the usual laws of mechanics do not fully determine the propagation of phase boundaries and that there is a need for additional constitutive information in the form of a kinetic relation. A simple experiment to measure this kinetic relation is proposed. Finally, a strategy to use such beams for propulsion at small scales is presented.     

Higher-order free vibrations of sandwich beams with a locally damaged core

The dynamical equations describing the free vibration of sandwich beams with a locally damaged core are derived using the higher-order theory approach. The nonlinear acceleration fields in the core are accounted for in the derivations, which is essential for the vibration analysis of the locally damaged sandwich beams. A local damage in the core, arbitrarily located along the length of the sandwich beam, is assumed to preclude the transition of stresses through the core between the undamaged parts of the beam. The damage is assumed to exist before the vibration starts and not to grow during oscillations. The numerical analysis based on the derived equations has been verified with the aid of the commercial finite element software ABAQUS. The numerical simulations reveal that a small local damage causes significant changes in the natural frequencies and corresponding vibration modes of the sandwich beams. An important practical consequence of the present work is that the vibration measurements can be successfully used as a nondestructive damage tool to assess local damages in sandwich beams.     

求解接触问题的一种新的实验误差法

提出了一种带松弛因子的UZAW算法求解实验误差法中给定状态下的位移和接触力满足的等式方程，并证明了该算法是R超线性收敛的。整个区域被划分为多个子区域，不同子区域位移场的求解是独立的。还提出了一种带参数的以不完全因子分解为基础的预条件子共轭梯度法求解不同子区域位移场，该算法在块体规模较大时更加有效。     

一维六方准晶层合简支梁自由振动与屈曲的精确解

伪Stroh型公式能够将多场耦合材料的控制方程转化为线性特征系统来求解,从而获得多层结构简支边界条件的精确解.本文利用伪Stroh型公式,研究一维六方准晶层合简支梁的自由振动和屈曲问题,通过传递矩阵法,获得准晶层合梁自由振动固有频率与临界屈曲载荷的精确解.通过与已有梁的剪切变形理论结果比较,验证了本文伪Stroh型公式的正确性和有效性.通过数值算例,分析由两种不同准晶材料组成的三明治层合梁的叠层方式、高跨比、层厚比及层数对梁的固有频率、临界屈曲载荷及其模态的影响规律.结果表明,叠层顺序和梁的高跨比、层厚比对准晶层合梁的自由振动固有频率和临界屈曲载荷有很大影响,可通过调整梁的几何尺寸和叠层顺序得到准晶层合梁的最佳固有频率和临界屈曲载荷.本文给出的精确解可为工程上研究准晶梁的各种数值解法和实验方法提供理论参考.     

A new simple third-order shear deformation theory of plates

An improved simple third-order shear deformation theory for the analysis of shear flexible plates is presented in this paper. This new plate theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the author; a system of 10th-order differential equilibrium equations in terms of the three generalized displacements of bending plates; five boundary conditions at each edge of plate boundaries. Although the resulting displacement field is the same as that proposed by Murthy, the variational consistent governing equations and the associated proper boundary conditions are derived and identified in this work for the first time in the literature. The applications and accuracy of the present shear deformation theory of plates are demonstrated by analytically solving the differential governing equations of a twisting plate, a bending beam and two bending plates to which the 3-D elasticity solutions are available, and excellent agreements are achieved even for the torsion of a plate with square cross-section as well the local effects of stresses at plate boundaries can be characterized accurately. These analytical solutions clearly show that the simple third-order shear deformation theory developed in this work indeed gives better results than the first-order shear deformation theories and other simple higher-order shear deformation theories, since the present third-order shear flexible theory is based on a more rigorous kinematics of displacements and consists of not only a system of variational consistent differential equations, but also a group of consistent boundary conditions associated with the differential equations. The present simple third-order shear deformation theory can easily be applied to the static and dynamic finite element analysis of laminated plates just like the applications of other popular shear flexible plate theories, and improved results could be obtained from the present simple third-order shear deformable theories of plates.     

An efficient coupled layerwise theory for static analysis of piezoelectric sandwich beams

Summary An efficient one-dimensional model is developed for the statics of piezoelectric sandwich beams. Third-order zigzag approximation is used for axial displacement, and the potential is approximated as piecewise linear. The displacement field is expressed in terms of three primary displacement variables and the electric potential variables by satisfying the conditions of zero transverse shear stress at the top and bottom and its continuity at layer interfaces. The deflection field accounts for the piezoelectric transverse normal strain. The governing equations are derived using a variational principle. The present results agree very well with the exact solution for thin and thick highly inhomogeneous simply supported hybrid sandwich beams. The developed theory can accurately model open and closed circuit boundary conditions. The first author is grateful to DST, Government of India, for financial support for this work.     

Improved torsional analysis of laminated box beams

The improved torsional analysis of the laminated box beams with single- and double-celled sections subjected to a torsional moment is performed by introducing 14 displacement parameters. For this, a thin-walled laminated box beam theory considering the effects of shear and elastic couplings is presented. The governing equations and the force-displacement relations are derived from the variation of the strain energy. The system of linear algebraic equations with non-symmetric matrix is constructed by introducing the displacement parameters and by transforming the higher order simultaneous differential equations into first order ones. This numerical technique determines eigenmodes corresponding to 12 zero and 2 non-zero eigenvalues and derives displacement functions for displacement parameters based on the undetermined parameter method. Finally, the element stiffness matrix is determined using the member force-displacement relations. The theory developed by this study is validated by comparing several torsional responses from the present approach with those from the finite element beam model using the Lagrangian interpolation polynomials and three-dimensional analysis results using the shell elements of ABAQUS for coupled laminated beams with single- and double-celled sections.