On the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach

In finite element methods that are based on position and slope coordinates, a representation of axial and bending deformation by means of an elastic line approach has become popular. Such beam and plate formulations based on the so-called absolute nodal coordinate formulation have not yet been verified sufficiently enough with respect to analytical results or classical nonlinear rod theories. Examining the existing planar absolute nodal coordinate element, which uses a curvature proportional bending strain expression, it turns out that the deformation does not fully agree with the solution of the geometrically exact theory and, even more serious, the normal force is incorrect. A correction based on the classical ideas of the extensible elastica and geometrically exact theories is applied and a consistent strain energy and bending moment relations are derived. The strain energy of the solid finite element formulation of the absolute nodal coordinate beam is based on the St. Venant-Kirchhoff material: therefore, the strain energy is derived for the latter case and compared to classical nonlinear rod theories. The error in the original absolute nodal coordinate formulation is documented by numerical examples. The numerical example of a large deformation cantilever beam shows that the normal force is incorrect when using the previous approach, while a perfect agreement between the absolute nodal coordinate formulation and the extensible elastica can be gained when applying the proposed modifications. The numerical examples show a very good agreement of reference analytical and numerical solutions with the solutions of the proposed beam formulation for the case of large deformation pre-curved static and dynamic problems, including buckling and eigenvalue analysis. The resulting beam formulation does not employ rotational degrees of freedom and therefore has advantages compared to classical beam elements regarding energy-momentum conservation.     

Analytical solution and semi-analytical solution for anisotropic functionally graded beam subject to arbitrary loading

Analytical and semi-analytical solutions are presented for anisotropic functionally graded beams subject to an arbitrary load, which can be expanded in terms of sinusoidal series. For plane stress problems, the stress function is assumed to consist of two parts, one being a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (y), and the other a linear polynomial of x with unknown coefficients depending on y. The governing equations satisfied by these y-dependent functions are derived. The expressions for stresses, resultant forces and displacements are then deduced, with integral constants determinable from the boundary conditions. While the analytical solution is derived for the beam with material coefficients varying exponentially or in a power law along the thickness, the semi-analytical solution is sought by making use of the sub-layer approximation for the beam with an arbitrary variation of material parameters along the thickness. The present analysis is applicable to beams with various boundary conditions at the two ends. Three numerical examples are presented for validation of the theory and illustration of the effects of certain parameters. Supported by the National Natural Science Foundation of China (Grant Nos. 10472102, 10432030, and 10725210)     

Higher order representation of the beam cross section deformation in large displacement finite element analysis

Most existing beam formulations assume that the cross section of the beam remains rigid regardless of the amplitude of the displacement. The absolute nodal coordinate formulation (ANCF); however, allows for the deformation of the cross section and leads to a more general beam models that capture the coupling between different modes of displacement. This paper examines the effect of the order of interpolation on the modes of deformation of the beam cross section using ANCF finite elements. To this end, a new two-dimensional shear deformable ANCF beam element is developed. The new finite element employs a higher order of interpolation, and allows for new cross section deformation modes that cannot be captured using previously developed shear deformable ANCF beam elements. The element developed in this study relaxes the assumption of planar cross section; thereby allowing for including the effect of warping as well as for different stretch values at different points on the element cross section. The displacement field of the new element is assumed to be cubic in the axial direction and quadratic in the transverse direction. Using this displacement field, more expressions for the element extension, shear and the cross section stretch can be systematically defined. The change in the cross section area is measured using Nanson’s formula. Measures of the shear angle, extension, and cross section stretch can also be systematically defined using coordinate systems defined at the element material points. Using these local coordinate systems, expressions for a nominal shear angle are obtained. The differences between the cross section deformation modes obtained using the new higher order element and those obtained using the previously developed lower order elements are highlighted. Numerical examples are presented in order to compare the results obtained using the new finite element and the results obtained using previously developed ANCF finite elements.     

Complex eigensolutions of coupled flexural and longitudinal modes in a beam with inclined elastic supports with non-proportional damping

Structure borne vibration and noise in an automobile are often explained by representing the full vehicle as a system of elastically coupled beam structures representing the body, engine cradle and body subframe where the engine is often connected to the chassis via inclined viscoelastic supports. To understand more clearly the interactions between a beam structure and isolators, this article examines the flexural and longitudinal motions in an elastic beam with intentionally inclined mounts (viscoelastic end supports). A new analytical solution is derived for the boundary coupled Euler beam and wave equations resulting in complex eigensolutions. This system is demonstrated to be self-adjoint when the support stiffness matrices are symmetric; thus, the modal analysis is used to decouple the equations of motion and solve for the steady state, damped harmonic response. Experimental validation and computational verifications confirm the validity of the proposed formulation. New and interesting phenomena are presented including coupled rigid motions, modal properties for ideal angled roller boundaries, and relationships between coupling and system modal loss factors. The ideal roller boundary conditions when inclined are seen as a limiting case of coupled longitudinal and flexural motions. In particular, the coupled rigid body motions illustrate the influence of support stiffness coupling on the eigenvalues and eigenfunctions. The relative modal strain energy concept is used to distinguish the contribution of longitudinal and flexural deformation modes. Since the beam is assumed to be undamped, the system damping is derived from the viscoelastic supports. The support damping (for a given loss factor) is shown to be redistributed between the system modes due to the inclined coupling mechanisms. Finally, this article provides valuable insight by highlighting some technical issues a real-life designer faces when balancing modeling assumptions such as rigid or elastic formulations, proportional or non-proportional damping, and coupling terms in multidimensional joint properties.     

Evolution of substructures under microlevel and mesolevel plastic deformation

The existence of a hierarchy of structural levels of plastic deformation can be considered to be an experimentally and theoretically proven fact [1–3]. Mescheryakov [1] showed that a noncrytallographic level of deformation arises in elastoplastic waves, manifesting itself as macrofluxes of particles of the medium; the velocity of the particles relative to each other at velocity has dispersion and the particles move in the direction of the wave propagation. Displacement of macrofragments of the crystal, which is also a manifestation of noncrystallographic structural levels of deformation, has been detected in highly excited systems [2]. The relaxation approach used increasingly to describe plastic deformation assumes that defects are created, move, and are restructured during deformation in a way so that the level of stresses inside the material drops. The nonuniformity of the stress field gives rise to nonuniform plastic deformation and local shears and rotations at points of stress concentration. These concepts make it possible to use the principles of synergetics to build specific theoretical models and to consider loaded material as a nonequilibrum dissipative structure [3]. To date, however, the construction of the theory describing multilevel plastic deformation processes has not been completed. In particular, it is not yet known what levels are added, depending on the rate and duration of the loading and on how the levels are linked.St. Petersburg Branch of the A. A. Blagonravov Institute of Mechanical Engineering. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 7–12, October, 1992.     

焊接变形光学测量技术研究

针对薄板焊接变形测量难的问题,提出了一种非接触式三维光学实时测量薄板焊接变形的新方法。首先通过2个高分辨率高速数字相机从不同观察点同时拍摄物体运动的视频图像;然后运用三维重建技术,包括共线方程、基于共面方程的像片定向、直接线性变换解法、外极线几何约束和光束平差解法等重建待测标志点的三维坐标;最后依据同名点将不同时刻薄板的变形状态关联起来,计算出待观测点的焊接变形量,并实时绘制显示出变形曲线。与传统方法相比,该方法具有不受焊接温度及强烈干扰影响和实时、高效、高精度等优点。     

Vibration of continuous systems with compliant boundaries

A theoretical study of two types of continuous systems with a general form of compliant boundary conditions is presented. The systems considered are elastic beams and circular plates with elastic damped edge constraints. Beam studies are restricted to those with identical boundary conditions at each end. The method of solution consists of formulating the edge condition of the system in terms of the impedance of the compliant boundary material and of using classical solution techniques to solve the equations of motion. The result of matching the boundary conditions of the system with constraining conditions is the system frequency equation in terms of the constraint impedances.A discussion is presented giving the influence of the compliant material on the vibration of the structure. The models give numerically the effect of elasticity and damping of the supports on the resonant frequencies of the systems. Parameters are obtained which indicate when one may assume simply supported or clamped boundaries for the actual case of elastic damped constraints without introducing large errors in the natural frequencies.     

Dynamic analysis of an axially moving finite-length beam with intermediate spring supports

This paper presents the analysis for the transverse vibration of an axially moving finite-length beam inside which two points are supported by rotating rollers. In this study, the rollers are modeled as uniaxial springs in the transverse direction. Hamilton?s principle is applied to derive the equations of motion and boundary conditions of the system. The equations of motion include translational and rotational motions as well as flexible motion. These equations are discretized using Galerkin?s method, and then the dynamic characteristics of a flexible beam with spring supports are studied by solving an eigenvalue problem. The veering phenomenon of natural frequency loci and mode exchanges are investigated for different positions of the springs and various values of the spring stiffness. In addition, the mode localization is also analyzed using the peak amplitude ratio. It is found in this study that the first mode is localized in one of the beam spans if an appropriate value of the spring constant is selected. Furthermore, it is shown that mode localization can be used to reduce the vibration transferred from one span to the other span while a beam moves axially.     

Surface microrelief smoothing mechanisms in a target irradiated by an intense charged particle beam

Physical mechanisms and a mathematical model are proposed to describe the smoothing of the microrelief of an irradiated surface. Before melting, smoothing is caused by elastoplastic deformation of the material; after the melting of the surface layer of the irradiated target, smoothing is caused by the dynamics of the target boundary under the action of surface tension forces, viscous forces, and inertial forces. The results of numerical simulation agree qualitatively and quantitatively with experimental data. The proposed mathematical model can be used to predict changes in the microrelief of the irradiated surface and to choose irradiation conditions when solving the problems of radiation technologies.     

Grain Boundary Dynamics: A Novel Tool for Microstructure Control

The reaction of grain boundaries to a wide spectrum of forces is reviewed. Curvature, volume energy and mechanical forces are considered. The boundary mobility is strongly dependent on misorientation, which is attributed to both grain boundary structure and segregation. In magnetically anisotropic materials grain boundaries can be moved by magnetic forces. For the first time a directionality of boundary mobility is reported. Flat boundaries can also be moved by mechanical forces, which sheds new light on microstructure evolution during elevated temperature deformation. Curvature driven and mechanically moved boundaries can behave differently. A sharp transition between the small and large angle boundary regime is observed. It is shown that grain boundary triple junctions have a finite mobility and thus, may have a serious impact on grain growth in fine grained materials. The various dependencies can be utilized to influence grain boundary motion and thus, microstructure evolution during recrystallization and grain growth.     

像散光束的光束参数与光强二阶矩的关系

采用光束的二阶强度矩的方法讨论三维像散光束的光束特性。通过计算像散光束在自由空间、球面光学系统中的传输特性以及坐标系旋转时的变换特性 ,给出了光束的束腰位置、束腰半径、瑞利长度、有效波面曲率半径、近场和远场的方位角、光束的光束参数积等参数与光强二阶矩之间的关系。     

散斑位移量与照明激光束特性关系的实验研究

激光照射到粗糙物体的表面会形成激光散斑，散斑随物体的运动而运动，特别是物体在自身平面内运动时散斑的同步运动比较显著，因此可以用这一现象测物体的微小位移，散斑运动规律与照明光束的特性有关，本文通过实验研究了不同光束照明下散斑位移的规律，验证了散斑位移理论，并指出散斑位移测量中应注意的问题。     

Free vibrations of plate structures with intermediate frames

A method is proposed for vibration analysis of a composite structure comprising a plate and frames. This method takes account of the effect of the rigidity of the beams or columns in the frames, which is incorporated in terms of restraining forces acting on the plate or beams in such a way that the continuity of deformation between the plate and frames at the intermediate supporting beams or columns is preserved. The resulting compatibility equations and boundary conditions of the plate and beams give the desired natural frequency equations. A unidirectionally continuous rectangular plate on a flexible simple beam, a unidirectionally continuous rectangular plate on a rigid support, and a rectangular plate on an intermediate frame are treated to illustrate application of the method.     

Constraint on linear, spatiotemporally nonlocal, spatiotemporally nonhomogeneous constitutive relations

The structure of modern electromagnetic theory imposes a fundamental constraint on the constitutive relations which describe linear material media. Here, in extension to previous work, we provide a proof of this algebraic constraint for the most general type of linear medium: one that is bianisotropic and spatiotemporally nonhomogeneous as well as spatiotemporally nonlocal.     

An improved penalty immersed boundary method for fluid–flexible body interaction

An improved penalty immersed boundary (pIB) method has been proposed for simulation of fluid–flexible body interaction problems. In the proposed method, the fluid motion is defined on the Eulerian domain, while the solid motion is described by the Lagrangian variables. To account for the interaction, the flexible body is assumed to be composed of two parts: massive material points and massless material points, which are assumed to be linked closely by a stiff spring with damping. The massive material points are subjected to the elastic force of solid deformation but do not interact with the fluid directly, while the massless material points interact with the fluid by moving with the local fluid velocity. The flow solver and the solid solver are coupled in this framework and are developed separately by different methods. The fractional step method is adopted to solve the incompressible fluid motion on a staggered Cartesian grid, while the finite element method is developed to simulate the solid motion using an unstructured triangular mesh. The interaction force is just the restoring force of the stiff spring with damping, and is spread from the Lagrangian coordinates to the Eulerian grids by a smoothed approximation of the Dirac delta function. In the numerical simulations, we first validate the solid solver by using a vibrating circular ring in vacuum, and a second-order spatial accuracy is observed. Then both two- and three-dimensional simulations of fluid–flexible body interaction are carried out, including a circular disk in a linear shear flow, an elastic circular disk moving through a constricted channel, a spherical capsule in a linear shear flow, and a windsock in a uniform flow. The spatial accuracy is shown to be between first-order and second-order for both the fluid velocities and the solid positions. Comparisons between the numerical results and the theoretical solutions are also presented.     

标准镜的高精度柔性支撑镜框的优化设计

为了实现干涉仪标准镜中光学元件的高精度定位,设计了一种柔性支撑镜框,研究了该结构的力学模型、结构参数、定位精度和透镜变形。首先,根据材料力学原理将柔性镜框等效为一个弹簧系统;根据力学方程和几何关系,建立了透镜中心位置与柔性结构的挠度之间的二元方程。然后,分析了安装位置、温度、结构参数对透镜位置以及作用力的影响。最后,应...     

Characteristic of energy input for laser forming sheet metal

Laser forming is a process in which laser-induced thermal deformation is used to form sheet metal without a hard forming tool or external forces. The energy input of laser beam is the key factor for the temperature and stress distribution of sheet metal. The purpose of this work is to investigate the influence of energy input condition on heat input and deformation angle for two-dimension laser forming. Variations in heat input resulting from material deformation was calculated and discussed in this paper at first. Furthermore, in laser forming under the condition of constant laser energy input, the effects of energy input mode on deformation angle and temperature field were investigated.     

Nonlinear response of shear deformable beams on tensionless nonlinear viscoelastic foundation under moving loads

In this paper, a boundary element method is developed for the geometrically nonlinear response of shear deformable beams of simply or multiply connected constant cross-section, traversed by moving loads, resting on tensionless nonlinear three-parameter viscoelastic foundation, undergoing moderate large deflections under general boundary conditions. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse moving loading as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Three boundary value problems are formulated with respect to the transverse displacement, to the axial displacement and to a stress functions and solved using the Analog Equation Method, a Boundary Element based method. Application of the boundary element technique yields a system of nonlinear Differential-Algebraic Equations, which is solved using an efficient time discretization scheme, from which the transverse and axial displacements are computed. The evaluation of the shear deformation coefficient is accomplished from the aforementioned stress function using only boundary integration. Analyses are performed to illustrate, wherever possible, the accuracy of the developed method, to investigate the effects of various parameters, such as the load velocity, load frequency, shear deformation, foundation nonlinearity, damping, on the beam displacements and stress resultants and to examine how the consideration of shear and axial compression affects the response of the system.     

An immersed boundary energy-based method for incompressible viscoelasticity

Incompressible viscoelastic materials are prevalent in biological applications. In this paper we present a method for incompressible viscoelasticity in which the elasticity of the material is described in Lagrangian form (i.e. in material coordinates), and Eulerian (spatial) coordinates are used for the equations of motion and to enforce the incompressibility condition. The elastic forces are computed directly from an energy functional without the use of stress tensors, and the immersed boundary method is used to communicate between Lagrangian and Eulerian variables. The method is first applied to a warm-up problem, in which a viscoelastic incompressible material fills a two-dimensional periodic domain. For this problem, we study convergence of the velocity field, the deformation map, and the Eulerian force density. The numerical results indicate that the velocity field and deformation map converge strongly at second order and the Eulerian force density converges weakly at second order. Incompressibility is well maintained, as indicated by area conservation in this 2D problem. Finally, the method is applied to a three-dimensional fluid–structure interaction problem with two different materials: an isotropic neo-Hookean model and an anisotropic fiber-reinforced model.