On the bifurcation and postbifurcation analysis of elastic-plastic solids under general prebifurcation conditions

In this work we have studied the bifurcation and postbifurcation of elastic-plastic solids whose behavior near the critical point could not be idealized as hypoelastic and thus the “hypoelastic comparison solid” concept of R. Hill's theory is no longer applicable. First a simple continuous model is considered in order to illustrate the different possibilities in the stability behavior of the structures considered here. Next, a general three-dimensional stability analysis for a broad class of rate independent elastic-plastic solids is presented. It is found that for all the constitutive theories considered and for all possible prebifurcation solutions, the bifurcation functional is a simple generalization of Hill's. A completely different postbifurcation analysis is needed, however, in the case where the “hypoelastic comparison solid” concept cannot be used.     

Bounds to bifurcation stresses in solids with non-associated plastic flow law at finite strain

In the present paper, Hill's theory of bifurcation and stability in solids obeying normality is generalized to include a non-associated flow law. A one-parameter family of linear comparison solids has been found that admits a potential and has the property that if uniqueness is certain for the comparison solid then bifurcation and instability are precluded for the underlying elastic-plastic solid. The uniqueness criterion derived may be used as a device to determine lower bounds to the magnitudes of primary bifurcation and instability stresses which are ordinarily unknown. A second linear solid is introduced whose constitutive relations have the same form as the elastic-plastic solid “in loading”. The first eigenstate of this solid gives an upper bound to the primary bifurcation state of the underlying elastic-plastic solid. The search for the genuine primary bifurcation state is therefore replaced by a search for upper and lower bounds in the situation when normality fails to hold. The theory is applied to problems of homogeneous stress states.     

Elastoplastic microscopic bifurcation and post-bifurcation behavior of periodic cellular solids

In this study, a general framework is developed to analyze microscopic bifurcation and post-bifurcation behavior of elastoplastic, periodic cellular solids. The framework is built on the basis of a two-scale theory, called a homogenization theory, of the updated Lagrangian type. We thus derive the eigenmode problem of microscopic bifurcation and the orthogonality to be satisfied by the eigenmodes. The orthogonality allows the macroscopic increments to be independent of the eigenmodes, resulting in a simple procedure of the elastoplastic post-bifurcation analysis based on the notion of comparison solids. By use of this framework, then, bifurcation and post-bifurcation analysis are performed for cell aggregates of an elastoplastic honeycomb subject to in-plane compression. Thus, demonstrating a basic, long-wave eigenmode of microscopic bifurcation under uniaxial compression, it is shown that the eigenmode has the longitudinal component dominant to the transverse component and consequently causes microscopic buckling to localize in a cell row perpendicular to the loading axis. It is further shown that under equi-biaxial compression, the flower-like buckling mode having occurred in a macroscopically stable state changes into an asymmetric, long-wave mode due to the sextuple bifurcation in a macroscopically unstable state, leading to the localization of microscopic buckling in deltaic areas.     

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.     

An analysis of the imperfection sensitivity of square elastic-plastic plates under axial compression

For a simply supported elastic-plastic square plate under axial compression the post-bifurcation behaviour and the sensitivity to initial imperfections are investigated. An exact asymptotic expansion is given for the initial post-bifurcation behaviour of a perfect plate compressed into the plastic range. The imperfection sensitivity is studied through an asymptotic analysis of the behaviour of the hypoelastic plate that results from neglecting the effect of elastic unloading. The results of the asymptotic analyses are compared with results of a numerical incremental solution by means of a combined finite element—Rayleigh Ritz method. The paper considers the effect of different in-plane boundary conditions and the effect of various degrees of strain hardening.     

Imperfection-sensitivity in the plastic range

The effect of small imperfections on the buckling of continuous structures loaded into the plastic range is studied. A simple model study is presented and several additional examples are discussed. The rôle of the load at which elastic unloading first occurs is emphasized, and a general asymptotic analysis is given for the behavior prior to the onset of elastic unloading for a class of elastic-plastic solids subject to loads characterized by a single load parameter. Asymptotic imperfection-sensitivity formulae are obtained whose features are similar to analogous formulae for elastic structures.     

Elastic-plastic behavior of fibrous composites

The elastic-plastic behavior of fibrous composites is explored with self-consistent models. The overall and local yield surfaces, instantaneous moduli, thermal coefficients, plastic strains and thermal microstresses are calculated for selected material systems, using R. Hill's (1965) model, and a modified scheme. Axisymmetric mechanical loads and uniform thermal changes are considered and the extension to shear loads is discussed. A limited comparison of the calculated microstresses is made with available experimental results.     

Analysis of necking based on a one-dimensional model

Dimensional reduction is applied to derive a one-dimensional energy functional governing tensile necking localization in a family of initially uniform prismatic solids, including as particular cases rectilinear blocks in plane strain and cylindrical bars undergoing axisymmetric deformations. The energy functional depends on both the axial stretch and its gradient. The coefficient of the gradient term is derived in an exact and general form. The one-dimensional model is used to analyze necking localization for nonlinear elastic materials that experience a maximum load under tensile loading, and for a class of nonlinear materials that mimic elastic-plastic materials by displaying a linear incremental response when stretch switches from increasing to decreasing. Bifurcation predictions for the onset of necking from the simplified theory compared with exact results suggest the approach is highly accurate at least when the departures from uniformity are not too large. Post-bifurcation behavior is analyzed to the point where the neck is fully developed and localized to a region on the order of the thickness of the block or bar. Applications to the nonlinear elastic and elastic-plastic materials reveal the highly unstable nature of necking for the former and the stable behavior for the latter, except for geometries where the length of the block or bar is very large compared to its thickness. A formula for the effective stress reduction at the center of a neck is established based on the one-dimensional model, which is similar to that suggested by Bridgman (1952).     

On buckling of axisymmetric thin elastic-plastic shells

A buckling criterion for shells with an axisymmetric middle surface and subjected to edge loads and hydrostatic surface pressure loading is formulated starting from Hill's three-dimensional continuum theory for uniqueness of deformation of inelastic solids. It turns out that a physically consistent two-dimensional set of equations may be derived for a quite general class of strain-hardening elastic-plastic solids, the only essential restriction being that of a smooth yield function. The intrinsic errors inherent in the derived rate equations, being an integral part of an eigenvalue problem, are discussed in relation to a circular cylinder under axial compression. Analytical results are given which illustrate the influence of the constitutive properties and the boundary contraints on the magnitude of the critical load.     

Numerical modelling of shocks in solids with elastic-plastic conditions

Results are presented for a range of one- and two-dimensional shock-wave problems in elastic-plastic and hydrodynamic metals. These problems were solved numerically using the Flux-Corrected Transport (FCT) technique which achieves high resolution without non-physical oscillations, especially at shock fronts, and has not been used before in elastic-plastic solids. The two-dimensional problems were solved using both operator- and non-operator-split techniques to highlight the significant differences between these techniques when solving shock-wave problems in elastic-plastic solids. Comparisons of the elastic-plastic solutions with the hydrodynamic solutions are made and illustrate the importance of including elastic-plastic conditions when modelling the behaviour of solids. Also, the errors in these solutions that are due to the initial conditions are discussed in detail.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

受压U形截面柱的塑性后分叉

对压缩到塑性阶段的简支Ｕ形截面柱，本文详细讨论了对应于最低分叉截荷的非对称初始后分叉问题。在分析中考虑了弹性卸载对两条不同后分叉路径的影响，求出了两种不同初始后分叉的载荷与分叉模态值之间的渐近关系式，并求出了渐近展开式中的高阶项。     

Arrest characteristics of an impacted crack dissipating plastic energy

When a structural member is accidentally struck, an initial defect may grow and then arrest. An estimate of its size increase is made by considering the geometry of a centrally cracked panel subjected to a step function load in time. Two load amplitudes differing by a factor of five are considered. Under impact, the crack accelerates and then decelerates prior to arrest. The dynamic characteristics depend on the interaction of the elastic-plastic stress waves intervening with the physical boundaries. This effect is assessed quantitatively by computing for the energy stored in a unit volume of material and by incorporating sliding nodes in the finite element method. The energy dissipated by plastic deformation must be accounted for as it is no longer available for creating new macrocrack surface. Obtained are the near tip normal stresses that are found to change from compression to tension. Their magnitude is considerably larger than the corresponding static values. Increase in crack length changes from 0.87% to 20.6% when the magnitude of the impact load is raised five times. The rate of change of the strain energy density factor ΔS with crack growth Δa is found to be governed by the condition ΔS/Δa = const. during loading while dynamic relaxation corresponded to a nonlinear behavior. The physical implication of this remains to be clarified in view of the fact that plasticity theory may not adequately explain the near tip crack behavior.     

Elastodynamic behavior of laminated orthotropic plates under initial stress

A finite element method of analysis of the vibrational and wave propagational characteristics is presented for a laminated orthotropic plate under initial stress. The plate may have an arbitrary number of bonded elastic orthotropic layers, each with distinct thickness, density and mechanical properties, and the analysis is capable of treating a completely arbitrary three-dimensional state of initial stress. Biot's theory for incremental elastic deformations of a stressed solid forms the basis for this study. A homogeneous, isotropic plate under two different states of initial stress was analyzed and their numerical results showed excellent correlation with those from an exact solution. Further examples of a three layer composite plate and a sandwich plate are offered to add some general insight to the physical behavior of such plates.     

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.     

Buckling of eccentrically stiffened elastic-plastic panels on two simple supports or multiply supported

Buckling due to axial compression is investigated for elastic-plastic, stiffened wide panels either continuous in the longitudinal direction over several transverse supports or finite and supported along the two edges. An analytical treatment is given of the bifurcation behaviour and of the initial post-bifurcation behaviour of perfect panels compressed into the plastic range. The behaviour of initially imperfect panels is computed numerically using an incremental method. In each increment a linear problem is solved by a combined Rayleigh Ritz-finite element method. Computed examples show a considerable imperfection-sensitivity, both for panels that bifurcate in the plastic range, and for panels with a yield stress a little above the elastic bifurcation stress.     

三维轮轨法向接触力学行为弹塑性理论分析

基于Hertz接触理论和双线性强化模型,建立了轮轨法向接触弹塑性理论分析模型,分析了轮轨法向接触力学响应特征,讨论了轴重对接触压力和接触变形的影响规律。同时,基于三维轮轨接触有限元模型模拟了轮轨接触力学行为,并引入理论误差系数分析了弹性模型和双线性强化模型对轮轨接触力学响应预测结果的差异性。结果表明,轮轨最大接触压力和接触变形量均随轴重的增大而增大;双线性强化模型的理论误差系数较小,采用双线性强化分析模型能较准确地预测轮轨接触弹塑性力学行为。研究结果可为轮轨系统安全服役和损伤评估提供理论和技术支持。     

Creep buckling and postbuckling of circular cylindrical shells under axial compression

An infinitely long, axially compressed, circular cylindrical shell with an imperfection in the shape of the axisymmetric classical buckling mode, undergoing steady or non-steady creep, is analyzed. The axisymmetric problem is solved incrementally using nonlinear shell equations The ratio of the applied stress to the classical buckling stress determines if the shell will collapse axisymmetrically or if it will bifurcate into a nonaxisymmetric mode, and whether or not bifurcation will result in instantaneous collapse. The bifurcation problem is formulated exactly and the initial postbuckling behavior is investigated via an asymptotic elastic analysis, based on Koiter's general theory Numerical results are compared with available experimental data.     

Finite-element formulations for problems of large elastic-plastic deformation

An Eulerian finite element formulation is presented for problems of large elastic-plastic flow. The method is based on Hill's variational principle for incremental deformations, and is ideally suited to isotropically hardening Prandtl-Reuss materials. Further, the formulation is given in a manner which allows any conventional finite element program, for “small strain” elastic-plastic analysis, to be simply and rigorously adapted to problems involving arbitrary amounts of deformation and arbitrary levels of stress in comparison to plastic deformation moduli. The method is applied to a necking bifurcation analysis of a bar in plane-strain tension.The paper closes with a unified general formulation of finite element equations, both Lagrangian and Eulerian, for large deformations, with arbitrary choice of the conjugate stress and strain measures. Further, a discussion is given of other proposed formulations for elastic-plastic finite element analysis at large strain, and the inadequacies of some of these are commented upon.     

Elastic-plastic behavior of elastically homogeneous materials with a random field of inclusions

A multiphase material is considered, which consists of a homogeneous elastic-plastic matrix containing a homogeneous statistically uniform random set of ellipsoidal elastic-plastic inclusions. The elastic properties of the matrix and the inclusions are the same, but the so-called “stress-free strains”, i.e. the strain contributions due to temperature loading, phase transformations, and the plastic strains, fluctuate. A general theory of the yielding for arbitrary loading (by the stress and by the temperature) is employed. The realization of an incremental plasticity scheme is based on averaging over each component of the nonlinear yield criterion. Usually, averaged stresses are used inside each component for this purpose. In distinction from this usual practice physically consistent assumptions about the dependence of these functions on the component's values of the second stress moments are applied. The application of the proposed theory to the prediction of the thermomechanical deformation behavior of a model material is shown.     

Constitutive modeling of plastic-bonded explosives

Typically, elastic and elastic-plastic theory are used in structural-analysis computer programs to model the mechanical behavior of high explosives; these models, however, do not fit the observed behavior of plastic-bonded explosives. This paper discusses the development of an equation-of-state creep model and a linear viscoelastic model for the analysis of these material systems and shows comparisons between experimental results and analytical-model predictions.