Two-dimensional buckling problem for a composite reinforced with a periodic row of collinear short fibers

A solution is found to the two-dimensional buckling problem for a composite material reinforced with a periodic row of collinear short fibers and compressed along the fibers. The problem formulation is based on the piecewise-homogeneous model and the three-dimensional theory of stability of deformable bodies. The dependence of the critical strain and buckling mode on the fiber spacing is studied for various material and geometrical characteristics of the composite components __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 6, pp. 90–100, June 2006.     

The stability of the interface between two bodies compressed along interface cracks. 2. Exact solutions for the case of equal roots

The problem of buckling of the interface between two bodies is considered for the case where several plane cracks are located in the interface and the bodies are compressed along the cracks (along the interface of two different materials). The studies were carried out for a plane problem using the three-dimensional linearized theory of stability of deformable bodies. The complex variables and potentials of the above-mentioned linearized theory are used. This problem is reduced to the problem of linear conjugation of two analytical functions of a complex variable. The exact solution of the above-mentioned buckling problem is obtained for the case where the roots of the basic equation are equal. Some mechanical effects are analyzed under general conditions (elastic, elastoplastic, compressible, incompresible, isotropic, and orthotropic bodies). S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 5, pp. 66–73, May, 2000.     

Exact solution and stability of postbuckling configurations of beams

We present an exact solution for the postbuckling configurations of beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions. We take into account the geometric nonlinearity arising from midplane stretching, and as a result, the governing equation exhibits a cubic nonlinearity. We solve the nonlinear buckling problem and obtain a closed-form solution for the postbuckling configurations in terms of the applied axial load. The critical buckling loads and their associated mode shapes, which are the only outcome of solving the linear buckling problem, are obtained as a byproduct. We investigate the dynamic stability of the obtained postbuckling configurations and find out that the first buckled shape is a stable equilibrium position for all boundary conditions. However, we find out that buckled configurations beyond the first buckling mode are unstable equilibrium positions. We present the natural frequencies of the lowest vibration modes around each of the first three buckled configurations. The results show that many internal resonances might be activated among the vibration modes around the same as well as different buckled configurations. We present preliminary results of the dynamic response of a fixed–fixed beam in the case of a one-to-one internal resonance between the first vibration mode around the first buckled configuration and the first vibration mode around the second buckled configuration.     

Setting up a theory of stability of fibrous and laminated composites

The results obtained in setting up a theory of stability of fibrous and laminated composites in the case where the plane Π is in an arbitrary position are analyzed. The plane Π is formed by the points of a buckling mode that have equal phases relative to the line of compression. This theory follows from the linearized three-dimensional theory of stability of deformable bodies and is used to determine the critical compressive load and the associated position of the plane Π. Numerical examples are presented. A brief historical sketch is given     

Three-dimensional theory of stability of a carbon nanotube in a matrix

The three-dimensional theory of stability of a carbon nanotube (CNT) in a polymer matrix is presented. The results are obtained on the basis of the three-dimensional linearized theory of stability of deformable bodies. Flexural and helical (torsional) buckling modes are considered. It is proved that the helical (torsional) buckling modes occur in a single CNT (the interaction of neighboring CNTs is neglected) and do not occur in nanocomposites (the interaction of neighboring CNTs is taken into account) __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 23–37, January 2006.     

A New Derivation of Jeffery’s Equation

In this article, we present a modern derivation of Jeffery’s equation for the motion of a small rigid body immersed in a Navier–Stokes flow, using methods of asymptotic analysis. While Jeffery’s result represents the leading order equations of a singularly perturbed flow problem involving ellipsoidal bodies, our formulation is for bodies of general shape and we also derive the equations of the next relevant order.      

The stability of the interface between two bodies compressed along interface cracks. 1. Exact solutions for the case of unequal roots

The buckling of the interface between two bodies is considered in the case where the interface contains several plane cracks and the bodies are compressed along them (along the interface of two different materials). The investigation is carried out for a plane problem using the three-dimensional linearized theory of stability of deformable bodies. Complex variables and potentials of the mentioned linearized theory are applied. This problem is reduced to the problem of linear conjugation of two analytical functions of complex variables. The exact solution is derived for the case of unequal roots of the basic equation. Some mechanical effects are analyzed for the general formulation of problems (elastic, elastoplastic, compressible, incompressible, isotropic, and orthotropic bodies). S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 69–79, April, 2000.     

Elasto-plastic buckling analysis of laminated plates including interfacial damage

Elasto-plastic buckling of orthotropic laminated plates, which include interfacial damage, is analyzed in detail. Firstly, a novel mixed hardening yield criterion, as an improvement of Hill’s counterpart, is proposed for the orthotropic materials on the basis of the plastic theory. And differing from Hill’s theory, the present yield criterion is related to the spherical tensor of stress. Then, the incremental elasto-plastic constitutive relations of the mixed hardening orthotropic materials are presented. Secondly, the incremental static equilibrium equations for laminated plates including interfacial damage are established based on Von-Karman type theory and the principle of minimum potential energy. Finally, the elasto-plastic buckling of laminated plates are solved by adopting the Galerkin method and iteration scheme. The numerical results show that buckling of the plate occurs easier due to the existence of interfacial damage, and the critical load trends to constant when the interfacial damage approaches a certain degree. Also, the effect of anisotropy on buckling is obvious and the analysis of elasto-plastic buckling is necessary.     

On out-of-plane buckling of anisotropic elastic plate subjected to a simple shear

Out-of-plane buckling of anisotropic elastic plate subjected to a simple shear is investigated. From exact 3-D equilibrium conditions of anisotropic elastic body with a plane of elastic symmetry at critical configuration, the eqution for buckling direction (buckling wave direction) parameter is derived and the shape functions of possible buckling modes are obtained. The traction free boundary conditions which must hold on the upper and lower surfaces of plate lead to a linear eigenvalue problem whose nontrivial solutions are just the possible buckling modes for the plate. The buckling conditions for both flexural and barreling modes are presented. As a particular example of buckling of anisotropic elastic plate, the buckling of an orthotropic elastic plate, which is subjected to simple shear along a direction making an arbitrary angle of θ with respect to an elastic principal axis of materials, is analyzed. The buckling direction varies with θ and the critical amount of shear. The numerical results show that only the flexural mode can indeed exist. Project supported by the National Natural Science Foundation of China (No. 19772032).     

Analysis and numerical simulation of magnetic forces between rigid polygonal bodies. Part II: Numerical simulation

The analysis of magnetoelastic phenomena is a field of active research. Formulae for the magnetic force in macroscopic systems have been under discussion for some time. In Popović et al. (Continum. Mech. Thermodyn. 2007), we rigorously justify several of the available formulae in the context of rigid bodies in two and three space dimensions. In the present, second part of our study, we investigate these formulae in a series of numerical experiments in which the magnetic force is computed in dependence on the geometries of the bodies as well as on the distance between them. In case the two bodies are in contact, i.e., in the limit as their distance tends to zero, we focus especially on a formula obtained in a discrete-to-continuum approximation. The aim of our study is to help clarify the question which force formula is the correct one in the sense that it describes nature most accurately and to suggest adequate real-life experiments for a comparison with the provided numerical data.      

薄壁圆筒壳初始几何缺陷不确定性量化的极大熵方法

针对薄壁圆筒壳结构轴压屈曲载荷的缺陷敏感性以及真实几何缺陷的不确定性,提出一种基于实测缺陷数据和极大熵原理的初始缺陷建模与屈曲载荷预测方法。首先,将初始几何缺陷视为二维随机场,并利用实测缺陷数据和Karhunen-Loève展开法将初始缺陷的随机场建模转化为随机向量的建模;其次,利用极大熵方法确定随机向量的概率分布;最后,基于所构建的初始缺陷随机模型,利用MCMC抽样方法和确定性屈曲分析方法,进行随机屈曲分析并给出基于可靠度的屈曲载荷折减因子。数值算例表明,与直接假设随机场相关结构的方法相比,本文方法的结果是对薄壁圆筒壳屈曲载荷的一个更无偏估计。     

Buckling analysis of woven fabric under uniaxial tension in arbitrary direction

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｏｒｄｅｒｔｏｄｅｖｅｌｏｐｔｈｅａｐｐａｒｅｌＣＡＤｓｙｓｔｅｍｔｈｅｂｕｃｋｌｉｎｇｏｆｗｏｖｅｎｆａｂｒｉｃｈａｓｒｅｃｅｉｖｅｄｍｕｃｈａｔｔｅｎｔｉｏｎｉｎｒｅｃｅｎｔｙｅａｒｓ[1,2 ].Ｔｈｅｆａｂｒｉｃｓｈｅｅｔｈａｓａｌｗａｙｓｂｅｅｎｒｅｇａｒｄｅｄａｓｅｌａｓｔｉｃｔｈｉｎｐｌａｔｅｉｎｔｈｅｆａｂｒｉｃｍｅｃｈａｎｉｃｓ,ｈｏｗｅｖｅｒ,ｔｈｅｒｅａｒｅｓｏｍｅｐｅｃｕｌｉａｒｂｕｃｋｌｉｎｇｐｈｅｎｏｍｅｎａｗｅｃａｎｓｅｅｉｎｔｈｅｆａｂｒｉｃｂ…     

Nonlinear shell-type to beam-type fea simplifications for composite frp poles

This paper presents a semi-analytical finite element analysis of pole-type structures with circular hollow cross-section. Based on the principle of stationary potential energy and Novozhilov’s derivation of nonlinear strains, the formulations for the geometric nonlinear analysis of general shells are derived. The nonlinear shell-type analysis is then manipulated and simplified gradually into a beam-type analysis with special emphasis given on the relationships of shell-type to beam-type and nonlinear to linear analyses. Based on the theory of general shells and the finite element method, the approach presented herein is employed to analyze the ovalization of the cross-section, large displacements, the P-Δ effect as well as the overall buckling of pole-type structures. Illustrative examples are presented to demonstrate the applicability and the efficiency of the present technique to the large deformation of fiber-reinforced polymer composite poles accompanied with comparisons employing commercial finite element codes.     

Micro-buckling in the nanocomposite structure of biological materials

Nanocomposite structure, consisting of hard mineral and soft protein, is the elementary building block of biological materials, where the mineral crystals are arranged in a staggered manner in protein matrix. This special alignment of mineral is supposed to be crucial to the structural stability of the biological materials under compressive load, but the underlying mechanism is not yet clear. In this study, we performed analytical analysis on the buckling strength of the nanocomposite structure by explicitly considering the staggered alignment of the mineral crystals, as well as the coordination among the minerals during the buckling deformation. Two local buckling modes of the nanostructure were identified, i.e., the symmetric mode and anti-symmetric mode. We showed that the symmetric mode often happens at large aspect ratio and large volume fraction of mineral, while the anti-symmetric happens at small aspect ratio and small volume fraction. In addition, we showed that because of the coordination of minerals with the help of their staggered alignment, the buckling strength of these two modes approached to that of the ideally continuous fiber reinforced composites at large aspect ratio given by Rosen's model, insensitive to the existing “gap”-like flaws between mineral tips. Furthermore, we identified a mechanism of buckling mode transition from local to global buckling with increase of aspect ratio, which was attributed to the biphasic dependence of the buckling strength on the aspect ratio. That is, for small aspect ratio, the local buckling strength is smaller than that of global buckling so that it dominates the buckling behavior of the nanocomposite; for comparatively larger aspect ratio, the local buckling strength is higher than that of global buckling so that the global buckling dominates the buckling behavior. We also found that the hierarchical structure can effectively enhance the buckling strength, particularly, this structural design enables biological nanocomposites to avoid local buckling so as to achieve global buckling at macroscopic scales through hierarchical design. These features are remarkably important for the mechanical functions of biological materials, such as bone, teeth and nacre, which often sustain large compressive load.     

A Motion Amplifier Using an Axially Driven Buckling Beam: I. Design and Experiments

For active materials such as piezoelectric stacks, which produce large force and small displacement, motion amplification mechanisms are often necessary – not simply to trade force for displacement, but to increase the output work transferred through a compliant structure. Here, a new concept for obtaining large rotations from small linear displacements produced by a piezoelectric stack is proposed and analyzed. The concept uses elastic (buckling) and dynamic instabilities of an axially driven buckling beam. The optimal design of the buckling beam end conditions was determined from a static analysis of the system using Euler's elastica theory. This analysis was verified experimentally. A stack-driven, buckling beam prototype actuator consisting of a pre-compressed PZT stack (140 mm long, 10 mm diameter) and a thin steel beam (60 mm× 12 mm× 0.508 mm) was constructed. The buckling beam served as the motion amplifier, while the PZT stack provided the actuation. The experimental setup, measuring instrumentation and method, the beam pre-loading condition, and the excitation are fully described in the paper. Frequency responses of the system for three pre-loading levels and three stack driving amplitudes were obtained. A maximum 16^∘ peak-to-peak rotation was measured when the stack was driven at an amplitude of 325 V and frequency of 39 Hz. The effects of beam pre-load were also studied.     

Finding minimum energy configurations for constrained beam buckling problems using the Viterbi algorithm

In this work, we present a novel technique to find approximate minimum energy configurations for thin elastic bodies using an instance of dynamic programming called the Viterbi algorithm. This method can be used to find approximate solutions for large deformation constrained buckling problems as well as problems where the strain energy function is non-convex. The approach does not require any gradient computations and could be considered a direct search method. The key idea is to consider a discretized version of the set of all possible configurations and use a computationally efficient search technique to find the minimum energy configuration. We illustrate the application of this method to a laterally constrained beam buckling problem where the presence of unilateral constraints together with the non-convexity of the energy function poses challenges for conventional schemes. The method can also be used as a means for generating “very good” starting points for other conventional gradient search algorithms. These uses, along with comparisons with a direct application of a gradient search and simulated annealing, are demonstrated in this work.     

Compound Bifurcations in the Buckling of a Delaminated Composite Strut

An analysis of the buckling and post-buckling of a delaminated composite strut is presented using a simple 4 degree of freedom nonlinear Rayleigh–Ritz formulation. Bifurcation analysis indicates that instability occurs in general at an asymmetric point of bifurcation. Depending on the depth of delamination both thin-film and overall buckling can occur in the post-buckling range, the transition being seen at a point of secondary bifurcation. For certain combinations of parameters this becomes a stellar bifurcation, associated with a double eigenvalue, where there are three possible subsequent routes for the post-buckling. The method used is fast and reliable and can be readily extended to modelling a composite with several layers.     

Navier��s Slip Problem for Motion of Inhomogeneous Incompressible Fluid-like Bodies

In this paper we are concerned with a flow of inhomogeneous incompressible fluid-like bodies (IIFB). The concept of IIFB is arised from the analysis of a certain type of granular flows. It is esssentially important to assign the so-called ‘slip’ boundary condition due to its behaviour at the surface, thus we take into account the Navier’s slip condition. Here, the theorem on the unique solvability, local in time, is proved.     

局部径向载荷作用下外压薄壁圆筒稳定性分析

为简化真空塔器外挂件支撑区局部失稳分析,提炼出局部径向载荷作用下外压薄壁圆筒稳定性计算模型. 以易拉罐为薄壁圆筒试件,对不同外压下试件的局部径向临界载荷进行了测试. 利用有限元法对各实验模型进行了特征值屈曲分析,结果与实验数据能较好地吻合. 采用正交设计及参数化计算,得到了各结构参数及外压下的局部径向临界载荷经验公式. 实例计算表明,所得经验公式稍有保守,可用于真空塔器外挂件支撑件区的局部稳定性分析.     

Buckling of nano-fibre reinforced composites: a re-examination of elastic buckling

Elastic buckling of layered/fibre reinforced composites is investigated. Assuming the existence of both shear and transverse modes of failure, the fibre is analysed as a layer embedded in a matrix. Interacting stresses, acting at the interfaces are determined from an exact derived stress field in the matrix. It is shown that buckling can occur only in the shear buckling mode and that the transverse buckling mode is spurious. As opposed to the well known Rosen shear buckling mode solution (predicated on an infinite buckling wavelength), shear buckling is shown to exist under two régimes: buckling of dilute composites with finite wavelengths and buckling of non-dilute composites with infinite wavelengths. Based on the analysis, a model is constructed which defines the fibre concentration at which the transition between the two régimes occurs. The buckling strains are shown to be (approximately) constant for dilute composites and, in the case of very stiff fibres, to have realistic values compatible with elastic behaviour. For the case of non-dilute composites, the strains are found to be in agreement with those given by the Rosen shear buckling solution. Numerical results for the buckling strains and stresses are presented and compared with the Rosen solution. These reveal that the Rosen solution is valid only for the case of non-dilute composites. The investigation demonstrates that elastic buckling may be a dominant failure mechanism of composites consisting of very stiff fibres fabricated in the framework of nano-technology.