Microscopic symmetric bifurcation condition of cellular solids based on a homogenization theory of finite deformation

In this paper, we establish a homogenization framework to analyze the microscopic symmetric bifurcation buckling of cellular solids subjected to macroscopically uniform compression. To this end, describing the principle of virtual work for infinite periodic materials in the updated Lagrangian form, we build a homogenization theory of finite deformation, which satisfies the principle of material objectivity. Then, we state a postulate that at the onset of microscopic symmetric bifurcation, microscopic velocity becomes spontaneous, yet changing the sign of such spontaneous velocity has no influence on the variation in macroscopic states. By applying this postulate to the homogenization theory, we derive the conditions to be satisfied at the onset of microscopic symmetric bifurcation. The resulting conditions are verified by analyzing numerically the in-plane biaxial buckling of an elastic hexagonal honeycomb. It is thus shown that three kinds of experimentally observed buckling modes of honeycombs i.e., uniaxial, biaxial and flower-like modes, are attained and classified as microscopic symmetric bifurcation. It is also shown that the multiplicity of bifurcation gives rise to the complex cell-patterns in the biaxial and flower-like modes.     

Simulation and interpretation of diffuse mode bifurcation of elastoplastic solids

The diffuse mode bifurcation of elastoplastic solids at finite strain is investigated. The multiplicative decomposition of deformation gradient and the hyperelasto-plastic constitutive relationship are adapted to the numerical bifurcation analysis of the elastoplastic solids. First, bifurcation analyses of rectangular plane strain specimens subjected to uniaxial compression are conducted. The onset of the diffuse mode bifurcations from a homogeneous state is detected; moreover, the post-bifurcation states for these modes are traced to arrive at localization to narrow band zones, which look like shear bands. The occurrence of diffuse mode bifurcation, followed by localization, is advanced as a possible mechanism to create complex deformation and localization patterns, such as shear bands. These computational diffuse modes and localization zones are shown to be in good agreement with the associated experimental ones observed for sand specimens to ensure the validity of this mechanism. Next, the degradation of horizontal sway stiffness of a rectangular specimen due to plane strain uniaxial compression is pointed out as a cause of the bifurcation of the first antisymmetric diffuse mode, which triggers the tilting of the specimen. Last, circular and punching failures of a footing on a foundation are simulated.     

Long-wave in-plane buckling of elastoplastic square honeycombs

In this study, two formulae derived for very long-wave in-plane buckling of elastic square honeycombs are extended and examined in the elastoplastic case. To this end, microscopic buckling and post-buckling behavior of elastoplastic square honeycombs subject to in-plane compression are numerically analyzed using a homogenization theory of the updated Lagrangian type. It is thus demonstrated that very long-wave buckling occurs just after the onset of macroscopic instability if periodic length is sufficiently long, and that plasticity can cause the localization of microscopic buckling accompanied by a significant decrease in macroscopic compressive stress. Then, the very long-wave buckling stresses computed are predicted by incorporating the effect of plasticity in the two formulae of elastic square honeycombs. It is shown that the resulting formulae are successful in predicting the very long-wave buckling stresses in the plastic range as well as in the elastic range.     

Post-buckling analysis of elastic honeycombs subject to in-plane biaxial compression

In this paper, employing the homogenization theory and the microscopic bifurcation condition established by the authors, we discuss which microscopic buckling mode grows in elastic honeycombs subject to in-plane biaxial compression. First, we focus on equi-biaxial compression, under which uniaxial, biaxial and flower-like modes may develop as a result of triple bifurcation. By forcing each of the three modes to develop, and by comparing the internal energies, we show that the flower-like mode grows steadily if macroscopic strain is controlled, while either the uniaxial or biaxial mode develops if macroscopic stress is controlled. Second, by analyzing several cases other than equi-biaxial compression, it is shown that a second bifurcation from either the uniaxial or biaxial mode to the flower-like mode, which is distorted, occurs under biaxial compression in a certain range of biaxial ratio under macroscopic strain control. Finally, the possibility of macroscopic instability under biaxial compression is discussed.     

Supercritical and subcritical buckling bifurcations for a compressible hyperelastic slab subjected to compression

We study the buckling bifurcation of a compressible hyperelastic slab under compression with sliding–sliding end conditions. The combined series-asymptotic expansions method is used to derive the simplified model equations. Linear bifurcation analysis yields the critical stress value of buckling, which gives a non-linear correction to the classical Euler buckling formula. The correction is due to the geometrical non-linearities coupled with the material non-linearities. Then through non-linear bifurcation analysis, the approximate analytical solutions for the post-buckling deformations are obtained. The amplitude of buckling is expressed explicitly in terms of the aspect ratio, the incremental dimensionless engineering stress, the mode of buckling and the material constants. Most importantly, we find that both supercritical and subcritical buckling could occur for compressible materials. The bifurcation type depends on the material constants, the geometry of the slab and the mode numbers.     

On the plastic bifurcation and post-bifurcation of axially compressed beams

The paper presents two new results in the domain of the elastoplastic buckling and post-buckling of beams under axial compression. (i) First, the tangent modulus critical load, the buckling mode and the initial slope of the bifurcated branch are given for a Timoshenko beam (with the transverse shear effects). The result is derived from the 3D J₂ flow plastic bifurcation theory with the von Mises yield criterion and a linear isotropic hardening. (ii) Second, use is made of a specific method in order to provide the asymptotic expansion of the post-critical branch for a Euler-Bernoulli beam, exhibiting one new non-linear fractional term. All the analytical results are validated by finite element computations.     

Post-bifurcation behavior in the plastic range

In the first part of the paper a simple model is used to introduce some of the analytical and physical features of post-bifurcation phenomena in continuous elastic-plastic systems. An analysis is presented for the initial post-bifurcation behavior of a class of elastic-plastic solids subject to loads characterized by a single load parameter. Bifurcations which occur at the lowest possible load are singled out for attention. The theory makes connection with Hill's general theory of bifurcation and uniqueness in elastic-plastic solids and Koiter's general approach to the initial post-buckling behavior of conservative elastic systems. Buckling of an axially compressed column in the plastic range is used to illustrate the theory.     

Bifurcation and post-bifurcation analysis in plasticity and brittle fracture

A mathematical model for the bifurcation and post-bifurcation behaviour of rate independent systems governed by normality laws is presented. The considered framework encompasses a large class of phenomena, including simple models of brittle fracture, brittle damage and metal plasticity. An associated method of asymptotic development of the bifurcated solution by integer series is discussed and illustrated by simple examples in plastic stability and fracture, i.e. the buckling of some simple elastic-plastic structures and the bifurcation of systems of interacting cracks. The proposed method furnishes a rigorous framework for the study of the bifurcation problem in the context of rate-independent dissipative systems obeying normality laws, using a different approach than the one adopted so far.     

On the first order change of bifurcation buckling loads due to structural geometry perturbations

It is shown that a simple method may be applied in order to determine the first order change of eigenmodes and critical loads at bifurcation buckling of slender structures when cavities or reinforcements are introduced. The perturbation procedure developed is applicable whenever a characteristic diameter of the structural geometry perturbation is one order less than the wave length of the unperturbed eigenmode and necessary computations involve at most the solution of two linear boundary value problems. Situations when the procedure may be further simplified are discussed and statements are made regarding requisite conditions for the buckling load definitely to increase or decrease due to geometry perturbations. Application of the method is illustrated by means of four cases of engineering interest, which involve perforated and reinforced beams and plates with holes and cracks.     

Elastoplastic bifurcation and collapse of axially loaded cylindrical shells

In this paper, a shell finite element is designed within the total Lagrangian formulation framework to deal with the plastic buckling and post-buckling of thin structures, such as cylindrical shells. First, the numerical formulation is validated using available analytical results. Then it is shown to be able to provide the bifurcation modes—possibly the secondary ones—and describe the complex advanced post-critical state of a cylinder under axial compression, where the theory is no longer operative.     

Buckling of a beam subjected to electromagnetic force and its stabilization by controlling the perturbation of the bifurcation

For a beam subjected to electromagnetic force, magnetoelastic buckling due to the increase of such force is theoretically investigated by taking account of the nonlinearity of the electromagnetic force and the elastic force of the beam. Using Liapunov-Schmidt method and center manifold theory, the equilibrium space, the bifurcation set and the bifurcation diagram are theoretically derived. Also, the effect of the higher modes other than the buckling mode on the mode shape of the postbuckling state is discussed. Furthermore, a control method to stabilize the magnetoelastic buckling is proposed, and the unstable equilibrium state of the beam in the postbuckling state, i.e., the straight position of the beam, is stabilized by controlling the perturbation of the bifurcation.     

Buckling modes of a compressed plate on an elastic substrate

The buckling modes of a homogeneously compressed elastic plate on a soft elastic substrate are studied. The critical compression is uniquely determined by the bifurcation equation, but this compression is associated with a wide set of buckling modes. It was proved that any solution of the Helmholtz equation satisfies the bifurcation equation. At the same time, in microelectronics, it is required to know which buckling mode is realized. Experimental and theoretical investigations show that the chessboard-like buckling mode should be expected. In what follows, this problem is discussed theoretically. The expected buckling mode can be found by analyzing the energy of the initial postcritical deformation, and the desired mode is determined from the condition of its minimum. The analytic expression of this energy is obtained. Its minimization results in the chessboard-like buckling mode.     

Onset of failure in finitely strained layered composites subjected to combined normal and shear loading

A limiting factor in the design of fiber-reinforced composites is their failure under axial compression along the fiber direction. These critical axial stresses are significantly reduced in the presence of shear stresses. This investigation is motivated by the desire to study the onset of failure in fiber-reinforced composites under arbitrary multi-axial loading and in the absence of the experimentally inevitable imperfections and finite boundaries.By using a finite strain continuum mechanics formulation for the bifurcation (buckling) problem of a rate-independent, perfectly periodic (layered) solid of infinite extent, we are able to study the influence of load orientation, material properties and fiber volume fraction on the onset of instability in fiber-reinforced composites. Two applications of the general theory are presented in detail, one for a finitely strained elastic rubber composite and another for a graphite-epoxy composite, whose constitutive properties have been determined experimentally. For the latter case, extensive comparisons are made between the predictions of our general theory and the available experimental results as well as to the existing approximate structural theories. It is found that the load orientation, material properties and fiber volume fraction have substantial effects on the onset of failure stresses as well as on the type of the corresponding mode (local or global).     

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.     

Elastoplastic buckling and post-buckling analysis of sandwich columns

Sandwich structures are widely used in many industrial applications thanks to their interesting compromise between lightweight and high mechanical properties. This compromise is realized thanks to the presence of different parts in the composite material, namely the skins which are particularly thin and stiff relative to the homogeneous core material and possibly core reinforcements. Owing to these geometric and material features, sandwich structures are subject to global but also local buckling phenomena which are mainly responsible for their collapse. The buckling analysis of sandwich materials is therefore an important issue for their mechanical design. In this respect, this paper is devoted to the theoretical study of the local/global buckling and post-buckling behavior of sandwich columns under axial compression. Only symmetric sandwich materials are considered with homogeneous and isotropic core/skin layers. First, the buckling problem is analytically addressed, by solving the so-called bifurcation equation in a 3D framework. The bifurcation analysis is performed using an hybrid model (the two faces are represented by Euler–Bernoulli beams, whereas the core material is considered as a 2D continuous solid), considering both an elastic and elastoplastic core material. Closed-form expressions are derived for the critical loadings and the associated bifurcation modes. Then, the post-buckling response is numerically investigated using a 2D finite element bespoke program, including finite plasticity, arc-length methods and branch-switching procedures. The numerical computations enable us to validate the previous analytical solutions and describe several kinds of post-critical responses up to advanced states, depending on geometric and material parameters. In most cases, secondary bifurcations occur during the post-critical stage. These secondary modes are mainly due to the modal interaction phenomenon and give rise to unstable post-buckled solutions which lead to final collapse.     

Diffuse mode bifurcation of soil causing convection-like shear investigated by group-theoretic image analysis

Diffuse mode bifurcation of soil under plane-strain compression test is shown, by means of an image analysis based on group-theoretic bifurcation theory, to trigger convection-like shear and to precede shear band formation. First digital photos of Toyoura sand specimens are processed by PIV (particle image velocimetry) to gather digitized images of deformation. Next bifurcation from a uniform state is detected by expanding these images into the double Fourier series and finding a predominant harmonic diffuse bifurcation mode based on that theory. This harmonic bifurcation mode, which is the mixture of a few harmonic functions, expresses complex convection-like shear. Last bifurcation from a non-uniform state is detected by decomposing each image into a few images with different symmetries to extract non-harmonic diffuse bifurcation modes. Diffuse modes of bifurcation, which hitherto were hidden behind predominant uniform compressive deformation, have thus been made transparent by virtue of the group-theoretic image analysis proposed. A possible course of deformation suggested herein is the evolution of diffuse mode bifurcation with a convection-like bifurcation mode breaking uniformity and symmetry, followed by the formation of shear bands through localization.     

Effects of third-order elastic constants on the buckling of thin plates

In the analysis of the bifurcation of thin orthotropic plates, the nonlinear terms associated with the third-order elastic constants are included in the stress-strain relation and large strain theory is used for the prebifurcation state. It is illustrated in an example that the second-order theory may affect considerably the buckling load (and mode).     

Postbuckling and Imperfection Sensitivity Analysis of Axially Stiffened Cylindrical Shells with Mode Interaction

Abstract

Interaction of nearly simultaneous buckling modes in the presence of imperfections is studied. The investigation is concerned with axially stiffened cylindrical shells under axial compression. In these structures, two modes are of particular interest, namely an overall long-wave and a local shortwave buckling mode. Numerical results show that in some cases bending of the stringers in the local mode postbuckling solution plays an important role. Exclusion of this effect, as was done in a previous study by Byskov and Hutchinson, may lead to an overestimation of the carrying capacity of the shell. Furthermore, it is found that apparently reasonable approximations to the postbuckling fields associated with both the local and the overall mode, as well as with the overall mode alone, may lead to inexact values of the buckling load.     

弹塑性梁在轴向应力波下的动态屈曲

本文考虑轴向应力波效应，利用分叉理论研究各种支承半无限长弹塑性梁的动态屈曲问题。在轴向阶梯载荷和脉冲载荷冲击下得到了梁的临界屈曲载荷及初始屈曲模态。其结果与实验现象相一致。同时也为研究结构动态屈曲问题提供了有效途径。     

Buckling of elastic-plastic cylindrical panel under axial compression

The buckling behaviour is investigated for an axially compressed elastic-plastic cylindrical panel of the type occurring in stiffened shells. The bifurcation stress is determined analytically and an asymptotically exact expansion is obtained for the initial post-bifurcation behaviour in the plastic range. For panels with small initial imperfections the behaviour is analysed asymptotically on the basis of the hypoelastic theory that results from neglecting the effect of elastic unloading. The imperfection-sensitivity of an elastic-plastic panel is also computed numerically by a linear incremental method, and the results are compared with the results of the asymptotic analysis. For a low hardening material the panel is found to be imperfection-sensitive in the whole range of curvatures considered, whereas for a high hardening material the panel is only imperfection-sensitive if the curvature exceeds a certain value.