Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures

Localization phenomena in one-dimensional imperfect continuous structures are analyzed, both in dynamics and buckling. By using simple models, fundamental concepts about localization are introduced and similarities between dynamics and buckling localization are highlighted. In particular, it is shown that strong localization of the normal modes is due to turning points in which purely imaginary characteristic exponents assume a non zero real part; in contrast, if turning points do not occur, only weak localization can exist. The possibility of a disturbance propagating along the structure is also discussed. A perturbation method is then illustrated, which generalizes the classical WKB method; this allows the differential problem to be transformed into a sequence of algebraic problems in which the spatial variable appears as a parameter. Applications of the method are worked out for beams and strings on elastic soil. All these structures are found to have nearly-defective system matrices, so their characteristic exponents are highly sensitive to imperfections.     

非完全相关荷载下钢筋混凝土框架结构体系可靠度分析

目前框架结构体系可靠度分析大多是在荷载完全相关的假定下完成的,对于非完全相关荷载作用下的体系可靠度研究则鲜有涉及,对于非理想弹塑性或弹脆性的结构尤其如此.本文在随机系统分析的概率密度演化理论的基础上,结合结构非线性全过程分析的位移控制算法,推导了基于归一化位移参数的结构非线性发展概率密度演化方程和承载力裕度的概率密度演化方程.采用纤维梁柱单元进行非线性分析,研究了钢筋混凝土框架结构在非完全相关荷载下的体系可靠度,并与Monte Carlo法进行了对比分析,证明了文中建议方法的可行性.基于文中方法,分析了荷载相关性对可靠度的影响,计算结果表明,荷载相关性对体系可靠度有比较明显的影响.     

Magnetoelectric effects in multiferroic laminated plates with imperfect interfaces

Two-dimensional(2D)equations for multiferroic(MF)laminated plates with imperfect interfaces are established in this paper.The interface between two adjacent sublayers,which are not perfectly bonded together,is modeled as a general spring-type layer.The mechanical displacements,and the electric and magnetic potentials of the two adjacent layers are assumed to be discontinuous at the interface.As an example,the influences of imperfect interfaces on the magnetoelectric(ME)coupling effects in an MF sandwich plate are investigated with the established 2D governing equations.Numerical results show that the imperfect interfaces have a significant impact on the ME coupling effects in MF laminated structures.     

Influences of imperfect interfaces on effective properties of multiferroic composites with coated inclusion

In order to predict the effective properties of multiferroic composite materials, the effective material constants of multiferroic composites with the coated inclusion and imperfect interface are investigated. Based on the generalized self-consistent theory, the closed-form solutions of the effective material constants are derived. For the composites with piezomagnetic inclusion, piezoelectric coating and polymer matrix, numerical calculations are performed to present the influences of the imperfect interface cooperating with the coating on the effective material constants. From the results, it can be observed that the effective constants can be enhanced by the coating but reduced by the imperfect interface. Moreover, the coating has the shielding effects on the imperfect interface for the composite structures with its higher filling ratio.     

Optimum design of shell structures with random geometric,material and thickness imperfections

The optimum design of isotropic shell structures with random initial geometric, material and thickness imperfections is investigated in this paper and a robust and efficient methodology is presented for treating such problems. For this purpose, the concept of an initial “imperfect” structure is introduced involving not only geometric deviations of the shell structure from its perfect geometry but also a spatial variability of the modulus of elasticity as well as of the thickness of the shell. An efficient reliability-based design optimization (RBDO) formulation is proposed. The objective function is considered to be the weight of the structure while both deterministic and probabilistic constraints are taken into account. The overall probability of failure is taken as the global probabilistic constraint for the optimization procedure. Numerical results are presented for a cylindrical panel, demonstrating the efficiency as well as the applicability of the proposed methodology in obtaining rational optimum designs of imperfect shell-type structures.     

Actuation of the Kagome Double-Layer Grid. Part 2: Effect of imperfections on the measured and predicted actuation stiffness

The actuation stiffness of a set of steel Kagome Double-Layer Grid (KDLG) structures with brazed joints is measured experimentally and compared with predictions by the finite element method. The predicted actuation stiffnesses for the perfect KDLGs much exceed the measured values, and it is argued that the low values of observed actuation stiffness are due to the presence of geometric imperfections introduced during manufacture. In order to assess the significance of geometric defects upon actuation stiffness, finite element calculations are performed on structures with a stochastic dispersion in nodal position from the perfectly periodic arrangement, and on structures with wavy bars. It is found that bar waviness has the dominant effect upon the actuation stiffness. The predicted actuation stiffness for the imperfect structures are in satisfactory agreement with the measured values assuming the same level of imperfection between theory and experiment.     

Buckling Models for Higher Catastrophes∗

Abstract

An illustrative study of higher order catastrophes (elliptic umbilic and two kinds of double cusp) is presented on a simple elastic structure using polar co-ordinates. The bifurcation paths are analyzed for both perfect and imperfect structures.     

Imperfect symmetry: A new approach to structural optima via group representation theory

While imperfect symmetry (i.e., small deviation from symmetry) is abundant in Nature, it is not common among engineering structures. Here we investigate whether, and under which conditions the response of engineering structures with given symmetry group Γ may be improved by adding small perturbations x_i to the geometry. We will prove that a naturally emerging representation of Γ in the space of the variables {x_i} plays a key role and, based on this representation, we formulate exact conditions of improvability, utilizing classical representation theory of finite groups. We also present various examples among which optimal structures with imperfect symmetry emerge, somewhat counter to the engineer’s intuition.     

Non-linear buckling of simple models with tilted cusp catastrophe

Non-linear buckling universal solutions of simple multiple-parameter discrete models are discussed via a comprehensive and readily employed procedure using Catastrophe Theory. Attention is focused on perfect models whose total potential energy (TPE) function, upon small disturbance breaking symmetry, reduces to the universal unfolding of the tilted cusp catastrophe. A local analysis based on simple approximations allows us to classify the TPE universal unfolding of any model to one of the seven elementary Thom's catastrophes by defining the corresponding control parameters. Subsequently, using global analyses one can obtain exact results for establishing the non-linear equilibrium paths: (a) of the “perfect” perturbed model (due to the small effect of an extra parameter) with the corresponding “imperfect” bifurcation and limit points(s), and (b) of the imperfect models (resulting after inclusion of the effect of normal imperfection parameters) together with the corresponding to each parameter limit points. Moreover, conditions for the direct evaluation of (non-degenerate) hysteresis points associated with a tilted cusp point in the control parameter plane, are established. Numerical results illustrate the methodology proposed herein.     

Imperfection sensitivity of hilltop branching points of systems with dihedral group symmetry

Imperfection sensitivity of a hilltop branching point occurring as a coincidence of a limit point and a double bifurcation point of a finite-dimensional, elastic, conservative system equivariant to the dihedral group is investigated. In the neighborhood of this point, the potential is expanded into a power series of independent state variables, loading parameter and imperfection magnitude. The form of the expansion is determined through exploitation of dihedral-group symmetry. For the perfect system, the hilltop branching point and bifurcated paths are shown to be all unstable. For an imperfect system, equilibrium paths in general break into a series of paths: including fundamental, complementary and aloof paths. The imperfection sensitivity laws for maximum (critical) points of loading on these paths are obtained as a novel finding of this paper. Critical points on the fundamental and complementary paths enjoy a piecewise linear law, which is less severe than a one-half or two-thirds power law for the double bifurcation point. By contrast, maximum points on aloof paths suffer more severe sensitivity. The hilltop branching point thus displays complex system of imperfection sensitivities. As numerical examples, imperfection sensitivity of simple structural models with the hilltop point is investigated to ensure the validity of the present formulation.     

Dynamic Buckling of Autonomous Systems Having Potential Energy Universal Unfoldings of Cuspoid Catastrophe

This work deals with dynamic buckling universal solutions of discrete nondissipative systems under step loading of infinite duration. Attention is focused on total potential energy functions associated with universal unfoldings of cuspoid type catastrophes with one active coordinate. The fold, dual cusp and tilted cusp catastrophes under statically applied loading occurring via limit points, asymmetric/symmetric bifurcations and nondegenerate hysteresis points are extended to the case of dynamic loading. Catastrophe manifolds of these types showing imperfection sensitivity under both types of loading are fully assessed. Important findings regarding dynamic buckling of imperfect systems generated by perfect systems associated with imperfect bifurcations are explored. The analysis is supplemented by a numerical application of a system exhibiting imperfect bifurcation when it is perfect as well as a hysteresis point associated with a tilted cusp catastrophe, when it becomes imperfect.     

Modified Monte Carlo method for buckling analysis of nonlinear imperfect structures

In this paper, we propose a modified Monte Carlo method for analysis of buckling of an imperfect beams on softening nonlinear elastic foundation. Such structures exhibit considerable imperfection sensitivity, i.e. reduction in the maximum load that the structure is able to support in contrast to classical buckling load of the perfect structure. The initial imperfections are treated as random functions of axial coordinate. In order to reduce the needed number of simulations, the Monte Carlo method is coupled with maximum likelihood methodology and the Kolmogorov–Smirnov test.     

非完善板屈曲路径的有限元增量摄动法

本文以Thompson一般稳定性理论为基础,提出非完善结构(有初始几何缺陷)屈曲平衡路径分析的有限元增量摄动法,它克服了增量迭代法及摄动法各自的缺点,利用增量摄动法,还建立了非完善板屈曲平衡路径的数学分析模型,对非完善矩形板在各种边界条件下的屈曲路径实现了数值程序分析,得到了符合实验的结果。     

Three-dimensional solution of smart laminated anisotropic circular cylindrical shells with imperfect bonding

This paper presents an exact solution for a simply-supported and laminated anisotropic cylindrical shell strip with imperfect bonding at the off-axis elastic layer interfaces and with attached anisotropic piezoelectric actuator and sensor subjected to transverse loading. In this research, the imperfect interface conditions are described in terms of linear relations between the interface tractions in the normal and tangential directions, and the respective discontinuities in displacements. The solution for an elastic (or piezoelectric) layer of the smart laminated cylindrical shell strip is obtained in terms of the six-dimensional (or eight-dimensional) pseudo-Stroh formalism, solution for multilayered system is then derived based on the transfer matrix method. Finally, a numerical example is presented to demonstrate the effect of imperfect interface on the static response of the smart laminated cylindrical shell. The derived solutions can serve as benchmark results to assess various approximate shell theories and numerical methods.     

Electroelastic interaction between piezoelectric screw dislocation and circularly layered inclusion with imperfect interfaces

The interaction between a piezoelectric screw dislocation and an interphase layer in piezoelectric solids is theoretically investigated.Here,the dislocation located at arbitrary points inside either the matrix or the inclusion and the interfaces of the interphase layer are imperfect.By the complex variable method,the explicit solutions to the complex potentials are given,and the electroelastic fields can be derived from them.The image force acting on the dislocation can be obtained by the generalized PeachKoehler formula.The motion of the piezoelectric screw dislocation and its equilibrium positions are discussed for variable parameters.The important results show that,if the inner interface of the interphase layer is imperfect and the magnitude of degree of the interface imperfection reaches the certain value,two equilibrium positions of the piezoelectric screw dislocation in the matrix near the interface are found for the certain material combination which has never been observed in the previous studies(without considering the interface imperfection).     

Micromechanical analysis of fibrous piezoelectric composites with imperfectly bonded adherence

In this work, two-phase parallel fiber-reinforced periodic piezoelectric composites are considered wherein the constituents exhibit transverse isotropy and the cells have different configurations. Mechanical imperfect contact at the interface of the piezoelectric composites is studied via linear spring model. The statement of the problem for two-phase piezoelectric composites with mechanical imperfect contact is given. The local problems are formulated by means of the asymptotic homogenization method, and their solutions are found using complex variable theory. Analytical formulae are obtained for the effective properties of the composites with spring imperfect type of contact and different rhombic cells. Using the concept of a representative volume element (RVE), a finite element model is created, which combines the angular distribution of fibers and imperfect contact conditions (spring type) between the phases. Periodic boundary conditions are applied to the RVE, so that effective material properties can be derived. The fibers are distributed in such a way that the microstructure is characterized by a rhombic cell. The presented numerical homogenization technique is validated by comparing results with theoretical approach reported in the literature. Some studies of particular cases, numerical examples, and comparisons between the two aforementioned methods with other theoretical results illustrate that the model is efficient for the analysis of composites with presence of rhombic cells and the aforementioned imperfect contact.     

Snap-through buckling of two simple structures

The snap-through buckling of two simple structures subjected to quasistatic loading is analyzed by use of the elastica theory of prismatic bars. In the first problem, the deformation of perfect and imperfect three-hinged deep trusses is considered and the results of the previous experimental observations are explained analytically. In the second problem, the snap-through behavior of a column restrained by an elastic wire is studied and a comparison of critical loads are made with the approximate solutions obtained recently by Nachbar [1].     

A theory of plasticity for carbon nanotube reinforced composites

Carbon nanotubes (CNTs) possess exceptional mechanical properties, and when introduced into a metal matrix, it could significantly improve the elastic stiffness and plastic strength of the nanocomposite. But current processing techniques often lead to an agglomerated state for the CNTs, and the pristine CNT surface may not be able to fully transfer the load at the interface. These two conditions could have a significant impact on its strengthening capability. In this article we develop a two-scale micromechanical model to analyze the effect of CNT agglomeration and interface condition on the plastic strength of CNT/metal composites. The large scale involves the CNT-free matrix and the clustered CNT/matrix inclusions, and the small scale addresses the property of these clustered inclusions, each containing the randomly oriented, transversely isotropic CNTs and the matrix. In this development the concept of secant moduli and a field fluctuation technique have been adopted. The outcome is an explicit set of formulae that allows one to calculate the overall stress-strain relations of the CNT nanocomposite. It is shown that CNTs are indeed a very effective strengthening agent, but CNT agglomeration and imperfect interface condition can seriously reduce the effective stiffness and elastoplastic strength. The developed theory has also been applied to examine the size (diameter) effect of CNTs on the elastic and elastoplastic response of the composites, and it was found that, with a perfect interface contact, decreasing the CNT radius would enhance the overall stiffness and plastic strength, but with an imperfect interface the size effect is reversed. A comparison of the theory with some experiments on the CNT/Cu nanocomposite serves to verify the applicability of the theory, and it also points to the urgent need of eliminating all CNT agglomeration and improving the interface condition if the full potential of CNT reinforcement is to be realized.     

光纤陀螺敏感线圈的温度漂移特性与绕圈技术研究

敏感线圈的热非互易性是引起光纤陀螺温度漂移的重要因素之一，因而人们多采用四极对称技术（ＱＡＤ）绕制光纤线圈。本文结合光纤陀螺研制中的实际状态，对Ｍｏｈｒ温度模型进行了修正和推广，研究了非理想ＱＡＤ环圈结构的温度漂移特性。这对于光纤陀螺及其环圈的优化设计具有指导意义。