Non-normality and bifurcation in plane strain tension and compression

The bifurcations of a rectangular block subject to plane strain tension or compression are investigated. The block material is taken to be incompressible and is characterized by an incrementally linear constitutive law for which “normality” does not necessarily hold. The consequences of non-normality regarding bifurcation are given primary emphasis here. The characteristic regimes of the governing equations (elliptic, parabolic and hyperbolic) are detennined. In each of these regimes both symmetric and antisymmetric diffuse bifurcation modes are available. Additionally, in the hyperbolic and parabolic regimes, bifurcation into a localized shear band mode is also possible. Particular attention is given to the limiting cases of long wavelength and soon wavelength diffuse bifurcation modes. The range of parameter values is identified for which bifurcation into some localized mode may precede bifurcation into a long wavelength diffuse mode. Some difficulties associated with employing a linear incremental solid in a bifurcation analysis, when primary interest is in the bifurcation of an underlying elastic-plastic solid, are also discussed.     

On interfacial properties in gradient damaging continua

The bifurcation problem near an interface is considered for a heterogeneous body made of two different materials that damage following gradient constitutive relations. The roles of internal length scales on bifurcation are studied especially in the shortwavelength regime. It is shown that the interfacial complementing condition is always satisfied meaning that a minimum wavelength exists for the bifurcation mode. The regularization properties of gradient damage models are underlined. A simple plane strain problem is used to illustrate the results. The interface bifurcated modes are explicitly computed: their wavelengths turn out to be fixed by the gradient coefficient; the influence of the interface behaviour is also highlighted. To cite this article: A. Benallal, C. Comi, C. R. Mecanique 333 (2005).     

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.     

Necking of biaxially stretched elastic-plastic circular plates

The necking of an elastic-plastic circular plate under uniform radial tensile loading is investigated both within the framework of the three-dimensional theory and within the context of the plane-stress approximation. Attention is restricted to axisymmetric deformations of the plate. The material behavior is described by two different constitutive laws. One is a finite-strain version of the simplest flow-theory of plasticity and the other is a finite-strain generalization of the simplest deformationtheory, which is employed as a simple model of a solid with a vertex on its yield surface. For an initially uniform plate made of an incompressible material, bifurcation from the uniformly stretched state is studied analytically. The regimes of stress and moduli where the governing axisymmetric three-dimensional equations are elliptic, parabolic or hyperbolic are identified. The plane-stress local-necking mode emerges as the appropriate limiting mode from the bifurcation modes available in the elliptic regime. In the elliptic regime, the main qualitative features of the bifurcation behavior are revealed by the plane-stress analysis, although three-dimensional effects delay the onset of necking somewhat. For the deformation theory employed here, the first bifurcation modes are encountered in the parabolic regime if the hardening-rate is sufficiently high. These bifurcations are not revealed by a plane-stress analysis. For a plate with an initial inhomogeneity, the growth of an imperfection is studied by a perturbation method, by a plane-stress analysis of localized necking, and by numerical computations within the framework of the three-dimensional theory. When bifurcation of the corresponding perfect plate takes place in the elliptic regime, the finite element results show that the plane-stress analysis gives reasonably good agreement with the numerical results. When bifurcation of the corresponding perfect plate first occurs in the parabolic regime, then a bifurcation of the imperfect plate is encountered, that is, the finite element stiffness matrix ceases to be positive definite.     

Bifurcation in compressible elastic/plastic cylinders under uniaxial tension

The bifurcation problem of a circular cylinder of elastic/plastic material under uniaxial tension is investigated, with particular reference to the usual engineering criterion that necking is initiated when the load on the specimen reaches a maximum. The material considered is compressible, with a smooth yield surface and associated flow rule. A lower bound analysis shows that for the particular constitutive equation chosen bifurcation cannot occur under a range of loading conditions while the stress is less than a certain value which is itself slightly less than the stress at the maximum load point. Diffuse axisymmetric necking modes under the commonly assumed loading conditions of prescribed axial components of velocity and shear-free traction-rates on the ends are, however, found to be initiated always after maximum load, the delay depending on the same factors shown for an incompressible material in reference [1]. The effect of the elastic compressibility assumption is to reduce the delay for a wide range of geometries, but to increase it for very slender specimens, as compared with the incompressible case. Surface modes are also found, but at stresses of an unrealistically high order of magnitude.     

Exact analytical solutions for the local and global buckling of sandwich beam-columns under various loadings

Sandwich structures are widely used in many industrial applications, due to the attractive combination of a lightweight and strong mechanical properties. This compromise is realized thanks to the presence of different parts in the composite material, namely the skins and possibly core reinforcements or thin-walled core structure which are both thin/slender and stiff relative to the other parts, namely the homogeneous core material, if any. The buckling phenomenon thus becomes mainly responsible for the final collapse of such sandwiches. In this paper, classical sandwich beam-columns (with homogeneous core materials) are considered and elastic buckling analyses are performed in order to derive the critical values and the associated bifurcation modes under various loadings (compression and pure bending). The two faces are represented by Euler–Bernoulli beams, whereas the core material is considered as a 2D continuous solid. A set of partial differential equations is first obtained from a general bifurcation analysis, using the above assumptions. Original closed-form analytical solutions of the critical loading and mode of a sandwich beam-column are then derived for various loading conditions. Finally, the proposed analytical formulae are validated using 2D linearized buckling finite element computations, and parametric analyses are performed.     

Prediction of the initial thickness of shear band localization based on a reduced strain gradient theory

Plastic flow localization in ductile materials subjected to pure shear loading and uniaxial tension is investigated respectively in this paper using a reduced strain gradient theory, which consists of the couple-stress (CS) strain gradient theory proposed by Fleck and Hutchinson (1993) and the strain gradient hardening (softening) law (C–W) proposed by Chen and Wang (2000). Unlike the classical plasticity framework, the initial thickness of the shear band and the strain rate distribution in both cases are predicted analytically using a bifurcation analysis. It shows that the strain rate is obviously non-uniform inside the shear band and reaches a maximum at the center of the shear band. The initial thickness of the shear band depends on not only the material intrinsic length l_cs but also the material constants, such as the yield strength, ultimate tension strength, the linear hardening and softening shear moduli. Specially, in the uniaxial tension case, the most possible tilt angle of shear band localization is consistent qualitatively with the existing experimental observations. The results in this paper should be useful for engineers to predict the details of material failures due to plastic flow localization.     

Discontinuous bifurcation analysis in fracture energy-based gradient plasticity for concrete

Conditions for discontinuous bifurcation in limit states of selective non-local thermodynamically consistent gradient theory for quasi-brittle materials like concrete are evaluated by means of both geometrical and analytical procedures. This constitutive formulation includes two internal lengths, one related to the strain gradient field that considers the degradation of the continuum in the vicinity of the considered material point. The other characteristic length takes into account the material degradation in the form of energy release in the cracks during failure process evolution.The variation from ductile to brittle failure in quasi-brittle materials is accomplished by means of the pressure dependent formulation of both characteristic lengths as described by Vrech and Etse (2009).In this paper the formulation of the localization ellipse for constitutive theories based on gradient plasticity and fracture energy plasticity is proposed as well as the explicit solutions for brittle failure conditions in the form of discontinuous bifurcation. The geometrical, analytical and numerical analysis of discontinuous bifurcation condition in this paper are comparatively evaluated in different stress states and loading conditions.The included results illustrate the capabilities of the thermodynamically consistent selective non-local gradient constitutive theory to reproduce the transition from ductile to brittle and localized failure modes in the low confinement regime of concrete and quasi-brittle materials.     

Finite elasto-viscoplastic modeling of polymers including asymmetric effects

Asymmetric effects between compression and tension are a pronounced behavior for glassy polymers such as polycarbonate. For its simulation an elasto-viscoplastic framework is formulated within a geometrically nonlinear theory. Here a new approach within the concept of stress mode dependent weighting functions is used, where each material parameter is additively decomposed into a sum of weighted stress mode-related quantities. The characterization of the stress modes is obtained in the octahedral plane of the deviatoric stress space in terms of the mode angle, such that stress mode dependent scalar weighting functions can be constructed. The constitutive equations are formulated for large strains in terms of logarithmic Hencky strains and its work conjugated Hill stresses. The resulting evolution equations are updated using a semi-implicit Euler scheme, and the algorithmic tangent operator is derived for the finite element equilibrium iteration. The numerical implementation is also used to identify the material parameters thus resulting into a good agreement with experimental data. Furthermore, the model is used to simulate the cold drawing processes for a dumbbell-shaped specimen in tension and a perforated strip in compression and tension.     

THE VISUAL EXPERIMENTAL TEACHING METHOD FOR MATERIAL FAILURE1)

基于数字图像相关技术,提出材料破坏过程可视化的实验教学方法,并以混凝土材料为例,介绍该方法在劈拉与单轴压缩实验中的应用及其效果。通过监测混凝土劈拉与压缩破坏过程,分析了该材料破坏模式,揭示了混凝土材料劈拉与压缩破坏机理。应用数字图像相关方法获得试件表面应变场分布,验证了材料破坏机理,并提出适用的破坏强度理论。     

基于参变量变分原理的直升机系留载荷高性能计算方法

为了保证直升机在舰船上的安全性,必须使用系留设备将直升机系留在舰船上。直升机的系留问题可简化为由机身刚体、索具和起落架组成的杆件系统,索具只承受拉力而不承受压力,起落架只承受压力而不承受拉力。因此,直升机系留问题为典型的强非线性问题,需要发展有效的求解算法。在考虑大变形的情况下,基于参变量变分原理建立了求解直升机系留载荷的高性能计算方法。 该方法利用参变量变分原理能够准确判断索具和起落架的拉压状态,并将材料非线性静力问题转换为线性静力互补问题求解,极大地提高了结果的收敛性。数值算例中,通过与有限元通用软件NASTRAN和ABAQUS计算结果比较,证实了该方法的精确性、收敛性及高效性。     

A non-local model with tension and compression damage mechanisms

A non-local isotropic damage model is proposed which can be used to predict the behaviour of rock-like materials up to failure. Two isotropic damage variables account for the progressive degradation of mechanical properties under stress states of prevailing tension and compression and two internal lengths, one for tension and the other for compression, are introduced as localisation limiters. A linear bifurcation analysis highlights the regularisation properties of the non-local model. An iterative scheme for the numerical solution of the finite-step problem consisting of a linear global predictor, an averaging phase and a non-linear local corrector is presented. Some illustrative examples of tension and splitting tests show the effectiveness of the proposed model.     

Numerical investigation of different modes of internal circulation in spherical drops: Fluid dynamics and mass/heat transfer

The results of detailed, three-dimensional numerical simulations of fixed spherical drops in a uniform flow are presented. The fluid dynamics outside and inside of the drops as well as the internal problem of mass (or heat) transfer are studied. Liquid drops in both a liquid and a gaseous ambient phase are considered. Special emphasis is put on the investigation of different modes of internal circulation.At low Reynolds numbers of the inner fluid, the flow field inside the drop resembles the well known Hill’s vortex solution. However, at higher internal Reynolds numbers, stable steady or quasi-steady alternative modes of internal circulation are found. As these modes are not cylindrical symmetric around the streamwise axis, the often applied assumption of a two-dimensional, axisymmetric flow field is not justified in these cases. Thus, major discrepancies to previous numerical studies are obtained. However, it is shown that experimental results support our findings.For liquid drops surrounded by a liquid, a major influence of the state of internal circulation on the drag is discovered, whereas the drag is nearly unaltered in the case of a liquid drop in gas.Concerning the internal problem of mass/heat transfer, the various internal flow modes show different characteristics. At low internal Peclet numbers, higher Sherwood numbers are reached for the Hill’s vortex-like cases, whereas at higher Peclet numbers, the transfer is faster for the alternative modes. For cases with a Hill’s vortex-like solution, asymptotic Sherwood numbers for very high Peclet numbers of around 20 are found, whereas no upper limit for cases with alternative modes can be determined. In the present study a maximum internal Sherwood number of 130 is reached, more than six times the maximum value for a case with a Hill’s vortex-like internal solution.     

The initiation of necking in rectangular elastic/plastic specimens under uniaxial and biaxial tension

Necking in a rectangular parallelepiped of incompressible elastic/plastic material under uniaxia tension is studied as a bifurcation problem. Approximate upper-bound bifurcation stresses are found which show that bifurcation can occur immediately after the load on the specimen reaches a maximum if the length of the specimen is sufficiently great compared to the width and thickness. A simple formula applicable to sufficiently thin specimens is obtained for the approximate bifurcation stress. Sufficient conditions for uniqueness are found for elastic/plastic solids subjected to a general homogeneous stress-field. The particular case of the rectangular specimen under equal biaxial tension is investigated further, and the magnitude of the bifurcation stress is found to be very sensitive to the particular boundary conditions imposed.     

On the bifurcations of the Lamé solutions in plane-strain elasticity

We consider the in-plane bifurcations experienced by the Lamé solutions corresponding to an elastic annulus subjected to radial tension on the curved boundaries. Numerical investigations of the relevant incremental problem reveal two main bifurcation modes: a long-wave local deformation around the central hole of the domain, or a material wrinkling-type instability along the same boundary. Strictly speaking, the latter scenario is related to the violation of the Shapiro–Lopatinskij condition in an appropriate traction boundary-value problem. It is further shown that the main features of this material instability mode can be found by using a singular-perturbation strategy.     

Influence of void nucleation on ductile shear fracture at a free surface

An approximate continuum model of a ductile, porous material is used to study the influence of the nucleation and growth of micro-voids on the formation of shear bands and the occurrence of surface shear fracture in a solid subject to plane strain tension. Bifurcation into diffuse modes is analysed for a plane strain tensile specimen described by these constitutive relations, which account for a considerable plastic dilatancy due to void growth and for the possibility of non-normality of the plastic flow law. In particular, bifurcation into surface wave modes and the possible influence of such modes triggering shear bands is investigated. For solids with initial imperfactions such as a surface undulation, a local material inhomogeneity on an inclusion colony, the inception and growth of plastic flow localization is analysed numerically. Both the formation of void-sheets and the final growth of cracks in the shear bands is described numerically. Some special features of shear band development in the solid obeying non-normality are studied by a simple model problem.     

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.     

Bifurcation of inflated circular cylinders of elastic material under axial loading—I. Membrane theory for thin-walled tubes

Bifurcations of circular cylindrical elastic tubes subjected to inflation combined with axial loading are analysed. Membrane tubes are considered in detail as a background to the more difficult analysis of thickwalled tubes described in the companion paper (Part II). Our results for membranes reinforce and extend those given by R.T. Shield and his co-workers.Two modes of bifurcation are investigated: firstly, a bulging (axisyrmmetric) mode; secondly, a prismatic mode in which the cross-section of the tube becomes non-circular. Necessary and sufficient conditions for the existence of modes of either type are given in respect of an arbitrary (incompressible isotropic) form of elastic strain-energy function. For a closed tube with a fixed axial loading many features of the results have close parallels with recent findings by D.M. Haughton and R.W. Ogden for spherical membranes. On the other hand, some results for tubes with fixed ends have no such parallel. In particular, bifurcation may, under certain conditions, occur before the inflating pressure reaches a maximum. A combination of the two modes is interpreted in terms of bending for a tube under axial compression, and the relative importance of the bending and bulging modes is discussed in relation to the length to radius ratio of the tube. The analytical results are illustrated for specific forms of strain-energy function. Corresponding analysis is given for thick-walled tubes in Part II.     

复合材料层合板面内渐进损伤分析的CDM模型简

基于连续介质损伤力学,提出了一个预测复合材料层合板面内渐进损伤分析的模型,它包括损伤表征、损伤判定和损伤演化3 部分. 模型能够区分纤维拉伸断裂、纤维压缩断裂、纤维间拉伸损伤和纤维间压缩损伤4 种损伤模式,定义了与4 个损伤模式对应的损伤状态变量,导出了材料主轴系下损伤前后材料本构之间的关系. 损伤起始采用Puck 准则判定,损伤演化由特征长度内应变能释放密度控制. 假定材料服从线性应变软化行为,建立了损伤状态变量关于断裂面上等效应变的渐进损伤演化法则. 模型涵盖了复合材料面内损伤起始、演化直至最终失效的全过程. 完成了含孔[45/0/-45/90]_2S 层合板在拉伸和压缩载荷下失效分析,结果表明该模型能合理进行层合板的强度预测和损伤失效分析.     

Magnetoelastic buckling of a rectangular block in plane strain

Of interest here is the stability of a rectangular block subjected to a uniform magnetic field perpendicular to its longitudinal axis. The two ends of the block are frictionless and kept parallel to each other. This boundary value problem is motivated by the classical problem of magnetoelastic buckling in which a cantilever beam subjected to a transverse magnetic field buckles when the applied field reaches a critical value.This work presents a finite strain continuum mechanics formulation of the stability problem of a homogeneous, compressible, magnetoelastic rectangular block in plane strain subjected to a uniform transverse magnetic field. The applied variational approach employs an unconstrained energy minimization recently proposed by the authors.The analytical solution for the critical buckling fields for both the antisymmetric and symmetric modes are obtained for three different constitutive laws. The corresponding result for thin beams is extracted asymptotically for a special material and the solution is compared to previously published results. The critical magnetic field is shown to increase monotonically with the block's aspect ratio for each material and mode type. Antisymmetric modes are always the critical buckling modes for stress saturated and neo-Hookean materials, except for a narrow range of moderate aspect ratios (about 0.25) where symmetric modes become critical. For strain-saturated solids no buckling is possible above a maximum aspect ratio.