对弹性静力学中外力功表达式及相关问题的探讨

外力功、势能等概念是弹性力学能量原理的重要内容. 但笔者发现, 经典弹性理论著作中给 出的外力功、势能的表达式并非真实的功、真实势能值. 本文根据弹性静力学的基本假定和 应变能定理对此进行了论证, 并以一维弹簧受压问题进行了说明. 对总势能、外力势能、外 力功及保守力的定义及其相互关系进行了详细分析, 建议在介绍系统总势能定义的同时, 应 一方面说明它不是系统的真实总势能, 另一方面要补充介绍这样定义总势能的原因是保证最 小势能原理与平衡方程等价. 通过厘清上述概念之间的关系, 以期给弹性力学的初学者以明 晰的概念.     

基于余能原理的有限变形问题有限元列式

利用基面力概念,推导了一种基于余能原理的有限变形问题显式有限元列式,可应用于结构的大位移、大转动问题。以基面力为状态变量来表达单元的余能,将有限变形情况下的单元余能分解为变形余能部分和转动余能部分,利用Lagrange乘子法推导出余能原理有限元的控制方程,编制了相应的非线性有限元程序。通过算例分析,说明该列式和程序的可靠性和精确性。     

小特征尺寸材料结构的演化

现代的电子和光子器件都是小特征尺寸的固体结构.在制造和使 用中，扩散过程能够重置物质，因此结构随时间演化.薄膜可能破裂成 小滴(droplets)，导线上可能长出孔穴.人们早就知道应力和电流是驱 动这些变化的力.这些力尽管重要，但越来越多的事实表明，它们不足 以说明各种不同的试验现象，这说明其它物理来源的力也起作用.在结 构中，原子、电子和光子的集体作用都对结构的自由能作出贡献.结构 一旦改变其构形，其自由能也发生变化.自由能的改变定义了一种热力 学的力，它反过来驱动结构的构形变化.本文通过一些特别的现象来阐 明这些概念.重点放在不同来源的力的物理描述上，这些来源包括弹性、 静电、毛细作用、电流、成分梯度、光子散射和电子约束.其中一些力 的作用在小特征尺寸例如几nm到几百nm的结构中尤为重要.随着仪器的 小型化，对这些力的认识变得日益重要.这一领域的研究极有希望带来 固体力学的革新.     

Efficient Evaluation of the Elastic Forces and the Jacobian in the Absolute Nodal Coordinate Formulation

This paper develops a new procedure for evaluating the elastic forces, the elastic energy and the jacobian of the elastic forces in the absolute nodal coordinate formulation. For this procedure, it is fundamental to use some invariant sparse matrices that are integrated in advance and have the property of transforming the evaluation of the elastic forces in a matrix multiplication process. The use of the invariant matrices avoids the integration over the volume of the element for every evaluation of the elastic forces. Great advantages can be achieved from these invariant matrices when evaluating the elastic energy and calculating the jacobian of the elastic forces as well. The exact expression of the jacobian of the differential system of equations of motion is obtained, and some advantages of using the absolute nodal coordinate formulation are pointed out. Numerical results show that there is important time saving as a result of the use of the invariant matrices.     

高速冲击下薄壁组合结构吸能特性研究

研究了高速冲击下组合薄壁结构的吸能特性。采用显式动力有限元方法模拟薄壁结构在轴向冲击下的动态屈曲并分析其吸能特性,通过在结构上增加诱导缺陷和提出新型组合截面来提高其抗冲击吸能性能。同时对多种模型对比分析,研究了不同诱导缺陷、截面形式对吸能特性的影响,以得到最优化的结构。     

Regulation of Cell Adhesion Free Energy by External Sliding Forces

Cells experience a variety of forces and external velocities in vivo. While a number of continuum based models have addressed cell substrate interactions in these dynamic environments, a molecular picture of adhesion under external sliding forces and velocities is lacking. Using a molecular thermodynamic model, that incorporates entropic and steric repulsions, molecular conformations and constraints, penetrable substrates and explicit binding interactions, we study the effect of external sliding velocities on cell adhesion. We map the free energy landscapes under a broad range of external forces, binding energies and receptor surface coverages. Our calculations predict the regimes where free energy landscapes become resistant to external forces. Our model shows good agreement with experimental studies and lays out a scalable framework for analyzing and quantifying a broad spectrum of in vivo and in vitro adhesion studies.     

Configurational forces in discrete elastic systems

The concept of configurational forces is applied to a simple, one-dimensional problem that is solved by finite elements. Both the exact solution and its finite-element approximation are provided in closed form. The total energy according to the approximate solution depends on the choice of the nodes. Any virtual shift of a node results in a virtual change of energy, which can be interpreted as the virtual work done by a configurational force acting on that node. It is shown that, in the case of equidistant nodes, the configurational forces acting on the interior nodes vanish. Also, the relation between the nodal configurational forces and the Eshelby stress resultant along the rod is investigated.     

Dislocation equilibrium conditions revisited

If there is an equilibrium arrangement of a given collection of dislocations, each having a fixed size and shape, in an externally loaded or unloaded elastic body, the corresponding potential energy will be stationary with respect to infinitesimal perturbations of the dislocation positions. This leads to the dislocation equilibrium conditions: the Peach–Koehler forces along the dislocation line of each dislocation due to externally applied stress and the interaction of the dislocation with other dislocations and its own image field is a set of self-equilibrated forces. The earlier proof of this result presented in the literature was based on an incomplete expression for the elastic strain energy. This is modified here by using the elastic strain energy expression that accounts for all dislocation core energy.     

不同宽厚比H形截面钢梁抗冲击性能

基于经实验校核的非线性有限元模型,对受横向冲击作用的H形钢梁进行了有限元分析。设计不同宽厚比组配的H形钢梁,分析H形钢梁跨中受横向冲击的动态响应和应力发展过程,并研究宽厚比对H形钢梁抗冲击性能的影响,重点讨论了腹板厚度、翼缘厚度对冲击力平台值和峰值以及耗能的影响。分析结果表明,两端铰接H形钢梁在跨中受冲击载荷作用下的变形模式主要为弯曲变形。相同冲击能量下,冲击力平台值主要受翼缘厚度的影响,冲击力峰值主要受腹板厚度的影响。翼缘厚度对钢梁抗冲击性能的影响要大于腹板厚度。本研究可为不同宽厚比H形钢梁的抗冲击设计提供依据和参考。     

Accurate modeling,comparative analysis,and performance enhancement of broadband piezoelectric energy harvesters with single and dual magnetic forces

The accurate electromechanical modeling and effectiveness of a piezoelectric energy harvester having dual magnetic forces at its tip is evaluated for low-frequency excitation purposes. To accurately represent the magnetic force, the exact finite element magnetic force is accurately fitted with a fifth-order polynomial representation in order to use it for low spacing distances between the two magnets. A comparative study is then carried out between the performances of piezoelectric energy harvesters having single and dual magnetic forces. It is indicated that hardening nonlinear behaviors take place in the dual attractive magnets compared to softening behaviors in the single magnet design. Static and eigenvalue problem analyses are also performed in order to determine the impacts of the spacing distance between the magnets on the static deflection and fundamental natural frequency of the energy harvester. The results show that the inclusion of a second magnetic force with an attractive interaction results in a delay in the static pull-in and a decrease in the fundamental natural frequency for same spacing distances. After that, a nonlinear distributed-parameter model is derived using the Galerkin discretization and used to study the performance and effectiveness of a piezoelectric energy harvester with dual attractive magnetic forces. The results show that the attractive dual magnets with same spacing distances lead to the presence of broadband resonance regions when the spacing distance between the magnets decreases to lower values. In addition, unlike the dual magnetic forces configuration which has only cubic nonlinearity, it is demonstrated that the single magnetic force configuration results in the presence of the quadratic nonlinearity which allows the energy harvester to have a softening behavior.     

Elastostatics with non absolutely continuous data

The main problem of classical elastostatics is considered when given body forces and surface forces act on the body. The problem is studied in the case when the given forces are represented by general vector measures, i.e. when concentrated loads may be given on subsets either of the body or of its boundary. An existence and uniqueness (up to rigid displacements) theorem is proved. Conditions under which the energy integral is finite are given.     

A direct approach to fiber and membrane reinforced bodies. Part I. Stress concentrated on curves for modelling fiber reinforced materials

An approach is outlined to the equilibrium in fiber-reinforced materials in which the fibers are modeled as curves or lines with concentrated material properties. The system of forces representing the interaction of the fibers with the bulk matter is analyzed, and equilibrium of forces is derived from global laws. The displacements of the bulk matter are assumed to have continuous extension to the fibers. This forces the set of admissible deformations superquadratically integrable. This in turn forces the energy of the bulk of superquadratic growth. The material of the bulk matrix therefore cannot be linearly elastic. The energy of fibers can have a slower growth and can be quadratic. A formal set of assumptions is given under which an equilibrium state of minimum energy exists in the given external conditions. A weak form of equilibrium equations is derived for this equilibrium state. An explicitly calculable axisymmetric example is presented with an isotropic and quadratic energy of the matrix (linear elasticity) and linearly stretchable fiber. Since the superquadratic growth assumption is not satisfied, some peculiar features of the solution arise, such as the infinite limit of the radial displacement near the fiber. Nevertheless, from the obtained solution, we can compute the normal force in the fiber and the shear stress at the interface.     

A preliminary study of 3D difference scheme with energy dynamic equilibrium

In this paper, a difference scheme with energy dynamic equilibrium (DS-EDE) is presented, which can be used for the simulation of long-term atmosphere and sea motion. Based on three dimensional nonlinear evolution equations for atmosphere and sea motion, a three dimensional compact upwind scheme (CUWS) is constructed, as the basis of the DS-EDE. The DS-EDE satisfies the following condition of energy dynamic equilibrium (EDE): the total work of external forces on the region boundary is equal to the sum of the total effective variation of the kinetic energy and the energy dissipation in the average flow motion and the effective variation of the potential energy per unit time within the region of interest. It really reflects the basic mechanism of the action of external forces and dissipation in atmosphere and sea movement. Therefore, the DS-EDE developed in this paper is a suitable model for simulating long-term atmosphere and sea movement with forcing and dissipation.     

A solar system model with dissipation

A problem of motion for an arbitrary number of planets is discussed with consideration of the forces of gravitational interaction according to the law of universal gravitation. The planets are assumed to be homogeneous viscoelastic spheres. In the process of motion, the planets are deformed and the dissipation of energy takes place due to internal viscous forces. On the basis of the motion separation method, an approximate system of equations is obtained to describe the motion of planet centers of mass and the variation of planet angular momenta with respect to the centers of mass. The equations of motion contain small conservative corrections to the law of universal gravitation and small dissipative forces whose influence causes a decrease of the total mechanical energy. The motion under consideration admits the following first integral: the law of angular momentum conservation for the system with respect to the centers of mass. When the system executes the steady motion corresponding to its rotation with a constant angular velocity as a rigid body, the dissipative forces do not perform work, since the deformed planets have no time-dependent deformations.     

Material electromagnetic fields and material forces

Electromagnetic fields address configurational forces in a natural way through an energy–stress tensor, which reduces to the Maxwell tensor in the simplest case. This tensor is related to physical forces and to the Cauchy traction in a continuum. Material forces, as opposed to physical forces, are of a different nature as they act upon a site of a continuum where the possible material inhomogeneity is located. A material energy–stress tensor, which is reminiscent of the Maxwell stress, is associated with these forces. Through appropriate balance laws, a material momentum is also associated with material forces. The material momentum is of particular interest in electromagnetic materials as it is intimately related to the pseudomomentum of light [Peierls in Highlights of Condensed Matter Physics, pp. 237–255 (1985) and in Surprises in Theoretical Physics, pp. 91–99 (1979); Thellung in Ann. Phys. 127, 289–301 (1980)]. The balance law for the material momentum can be derived either from the classical physical laws or independently of them. This derivation, which is based on the material electromagnetic potentials and the related gauge transformations, is discussed and commented on for an electromagnetic body.     

泡沫铝填充金属薄壁圆管侧向压缩的实验和理论

利用实验研究与理论分析相结合方法研究了泡沫铝填充金属薄壁圆管在准静态侧向压缩下的力学响应。基于能量法,建立了泡沫铝填充圆管和金属薄壁圆管在侧向均匀压缩时的瞬时侧向力、平均侧向力和总吸能的理论公式。对泡沫铝填充管与金属薄壁圆管进行了准静态侧向压缩实验,并且将实验结果与理论公式进行了对比,结果表明理论预测值与实验结果吻合较好。基于建立的理论分析模型,研究了管的几何尺寸以及泡沫铝材料的密度对结构的瞬时侧向力、平均侧向力、总吸能和比吸能的影响。结果表明,在准静态侧向压缩下,泡沫铝填充管的总吸能大于对应的金属薄壁圆管;泡沫铝填充管的侧向压缩力和总吸能随管长度、壁厚和直径的增加而增大;当填充材料泡沫铝密度增大时,填充管的总吸能与侧向压缩力均增加。     

多边形截面薄壁管撕裂卷曲吸能研究

对薄壁管在准静态载荷下的卷曲撕裂进行了理论分析,并采用显示有限元方法进行了数值模拟。通过分析薄壁管的撕裂能、塑性变形能和摩擦能,讨论了冲头的锥形角、壁厚以及边数等因素对结构吸能的影响。     

Symmetric Bifurcation of Plane Frames through a Modified Potential Energy Approach

Abstract

The modified potential energy approach to the nonlinear analysis of slender elastic beams is employed to investigate the buckling and postbuckling behavior of two symmetric plane frames, namely a hinged portal frame and a hinged two-bar frame, The beams are assumed to be inextensible and shear undeformable. The “hybrid” functional on which the present analysis is based according to a Koiter-type energy approach is obtained from the total potential energy functional by adding extra terms which account for the nonlinear energy associated with the internal forces applied at the beam ends. Thus the nonlinear constraints on the potential energy (expressed in terms of the cross-section rotation φ only), which are associated with the continuity condition relative to the axial and lateral displacement across the joints, are accounted for by introducing the corresponding joint forces as Lagrangian multipliers. The inherent simplicity of the proposed beam model facilitates the postbuckling analysis of the two structures up to and including fourth order terms. A number of diagrams illustrate the results obtained.     

基面力概念在几何非线性余能有限元中的应用

以基面力为基本未知量描述一个弹性系统的应力状态并表征单元的余能,将大变形的余能分解为变形余能部分和转动余能部分,采用Lagrange乘子法放松单元的平衡方程,利用已有的弹性大变形余能原理建立了一种几何非线性显式有限元模型,编制了相应的几何非线性余能原理有限元程序. 数值算例表明:该方法具有较好的收敛性和计算精度,可进行大载荷步的大位移、大转动计算.      

考虑内力分项贡献度的框架构件重要性评价

提出了一种水平和竖向荷载共同作用下的框架结构构件重要性评价方法。首先,基于线弹性材料假设,利用应变能方程和节点-外荷载关系计算梁柱构件的轴力、剪力和弯矩;然后,定义构件内力分项计算值与抗力分项值之比为构件分项重要性系数,用于体现构件抗力利用水平;再组合各内力分项系数,针对不同外荷载形式定义构件综合重要性系数,用于体现构件对框架体系整体承载力的贡献;最后,通过两榀框架算例验证了方法的可行性,可为框架的结构优化设计和连续性倒塌分析提供定量依据。