Imperfection sensitivity of the post-buckling behavior of higher-order shear deformable functionally graded plates

This paper investigates the sensitivity of the post-buckling behavior of shear deformable functionally graded plates to initial geometrical imperfections in general modes. A generic imperfection function that takes the form of the product of trigonometric and hyperbolic functions is used to model various possible initial geometrical imperfections such as sine type, local type, and global type imperfections. The formulations are based on Reddy’s higher-order shear deformation plate theory and von Karman-type geometric nonlinearity. A semi-analytical method that makes use of the one-dimensional differential quadrature method, the Galerkin technique, and an iteration process is used to obtain the post-buckling equilibrium paths of plates with various boundary conditions that are subjected to edge compressive loading together with a uniform temperature change. Special attention is given to the effects of imperfection parameters, which include half-wave number, amplitude, and location, on the post-buckling response of plates. Numerical results presented in graphical form for zirconia/aluminum (ZrO₂/Al) graded plates reveal that the post-buckling behavior is very sensitive to the L2-mode local type imperfection. The influences of the volume fraction index, edge compression, temperature change, boundary condition, side-to-thickness ratio and plate aspect ratio are also discussed.     

Thermal post-buckling analysis of imperfect laminated plates using a higher-order shear deformation theory

Thermal post-buckling analysis is presented for a simply supported, composite laminated plate subjected to uniform or non-uniform tent-like temperature loading. The initial geometrical imperfection of the plate is taken into account. The formulations are based on the Reddy's higher-order shear deformation plate theory, and include thermal effects. The analysis uses a mixed Galerkin-perturbation technique to determine thermal buckling loads and post-buckling equilibrium paths. Numerical examples cover the performances of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated plates. The effects played by transverse shear deformation, thermal load ratio, plate aspect ratio, total number of plies, fiber orientation and initial geometrical imperfections are studied. Typical results are presented in dimensionless graphical form.     

Finite element simulation on the mechanical properties of MHS materials

Finite element simulations are carried out to examine the mechanical behavior of the metallic hollow sphere (MHS) material during their large plastic deformation and to estimate the energy absorbing capacity of these materials under uniaxial compression. A simplified model is proposed from experimental observations to describe the connection between the neighboring spheres, which greatly improves the computation efficiency. The effects of the governing physical and geometrical parameters are evaluated; whilst a special attention is paid to the plateau stress, which is directly related to the energy absorbing capacity. Finally, the empirical functions of the relative material density are proposed for the elastic modulus, yield strength and plateau stress for FCC packing arrangement of hollow spheres, showing a good agreement with the experimental results obtained in our previous study.     

Thermomechanical Postbuckling of Imperfect Moderately Thick Plates on Nonlinear Elastic Foundations*

ABSTRACT

A postbuckling analysis is presented for a moderately thick rectangular plate subjected to combined axial compression and uniform temperature loading, and resting on a softening nonlinear elastic foundation. The cases of (1) thermal postbuckling of initially compressed plates and (2) compressive post-buckling of initially heated plates are considered. The initial geometrical imperfections of the plates are taken into account. Formulations are based on Reissner-Mindlin plate theory, considering first-order shear deformation effects, and including plate-foundation interaction and thermal effects. The analysis uses a deflection-type perturbation technique to determine buckling loads and postbuckling equilibrium paths. Numerical examples include the performance of perfect and imperfect, moderately thick plates resting on softening nonlinear elastic foundations. Typical results are presented in dimensionless graphical form.     

Thermomechanical instability associated with heat release during plastic flow

The effect of heat generation accompanying plastic deformation on instability of uniform plastic flow is considered. Linear stability analysis is performed taking into account both classical necking (geometrical effect) and thermomechanical coupling. Two types of conditions imposed in tensile testing are considered: those of constant (time and coordinate independent) imposed load and constant imposed cross-head velocity. A central issue in the paper is instability of plastic flow at extremely low temperatures (4.2 K to 10 K). Instability conditions are formulated, taking into account the temperature dependence of the thermal parameters. An example of tensile deformation of a thin Al-Mg rod is considered in some detail. Available experimental data are explained on the basis of stability analysis.     

Numerical investigation of plastic localization bands in aluminium rings subjected to static axial compression

Different modes of plastic localization bands initiating in aluminium rings under static compression are obtained numerically for different geometrical ratios. The deformation modes, compared with published experimental data, show a good agreement.     

Finite element simulations on the mechanical properties of MHS materials

Finite element simulations are carried out to examine the mechanical behavior of the metallic hollow sphere (MHS) material during their large plastic deformation and to estimate the energy absorbing capacity of these materials under uniaxial compression. A simplified model is proposed from experimental observations to describe the connection between the neighboring spheres, which greatly improves the computation efficiency. The effects of the governing physical and geometrical parameters are evaluated; whilst a special attention is paid to the plateau stress, which is directly related to the energy absorbing capacity. Finally, the empirical functions of the relative material density are proposed for the elastic modulus, yield strength and plateau stress for FCC packing arrangement of hollow spheres, showing a good agreement with the experimental results obtained in our previous study. The project supported by the Hong Kong Research Grant Council (RGC) (HKUST 6079/00E) and the National Natural Science Foundation of China (10532020).     

IMPERFECTION SENSITIVITY ANALYSIS OF A RECTANGULAR COLUMN COMPRESSED INTO THE PLASTIC RANGE

I.IntroductionSinceelasticunloadingoccursinthedeformationprocess,theimperfectionsensitivityanalysisofstructuresloadedintotheplasticrangeismuchcomplicated.TheproblemisfirststudiedintheorybyHutchinson(1973,1974)l"ZI.HeusedKoiter'stheorytoanalysethebehaviorpriortotheonsetofelasticunloadingandtoobtaintheloadandthedisplacementatwhichelasticunloadingbegins.NeedlemanandTvergaard(1982)l'jpresentedananalysisofimperfeCtionsensitivityintheplasticrangethatignoreselasticunloading,thatis,substituteshypo-…     

Prediction of ductile fracture in tension by bifurcation,localization, and imperfection analyses

In this investigation, it is shown that the onset of ductile fracture in tension can be interpreted as the result of a supercritical bifurcation of homogeneous deformation and that this fact can be applied to predict ductile fracture initiation of materials with general imperfections or flaws. We focus on one dimensional quasi-static simple tension for rate-independent isotropic plastic materials. For deformation beyond the bifurcation point, multiple equilibrium paths appear. The homogeneous deformation, as one of the equilibrium paths, loses stability while the inhomogeneous paths are stable, thus indicating the occurrence of strain localization. This investigation also provides a physical example for the application of the Lambert W function in material localization analyses. Material instability is treated as the instability of a static system with dynamic perturbation. We also address the presence of microvoids in a power law plastic material as an unfolding of the supercritical pitchfork bifurcation. The imperfect system, idealized as spherical voids within the plastic matrix, is analyzed using the familiar Gurson model which is based on the presumption of a randomly voided material and characterized by the volume fraction of voids. If, in addition, the sizes of the microvoids are known, this then provides a length scale for the imperfection zone. In this manner, relevance to the sample size effects of strain-to-failure for ductile fracture initiation is addressed by considering separate zones with variations in void volume fractions. Fracture initiation predictions are presented and compare very well to existing experimental results.     

Effect of initial imperfections in geometry on the elastic-plastic stability of a thin annular plate

In this paper,Neale’S generalized variational principle aboutincremental boundarg-value problems is utilized to study theeffect of initial imperfections in geometry on tbe criticalloads of elasticoplastic buckling of thin annular plates.Thecalculations show that.if the effect of initial imperfectionsin geometrg is taken into account in the solutions by J_2 in-cremental theorg.the results are very close to the bifurca-tional buckling loads of the perfect annular Plates accordingto the plastic deformation theorg.     

Buckling of imperfect functionally graded cylindrical shells under axial compression

Buckling behaviors of axially compressed functionally graded cylindrical shells with geometrical imperfections are investigated in this paper using Donnell shell theory and the nonlinear strain-displacement relations of large deformation. The analysis is based on the nonlinear prebuckling consistent theory. Both the prebuckling effects and the temperature-dependent material properties are taken into account. The buckling condition for imperfect functionally graded cylindrical shells is obtained by using the Galerkin method. Numerical results show various effects of imperfection, structural type, power law exponent, temperature and dimensional parameters on buckling. The present theoretical results are verified by those in literature.     

Finite-element analysis of indentation experiments in surface-coated materials

There are only a few methods suitable for a quantitative characterization of the mechanical properties of surface-coated materials; indentation testing is one of those methods. Within the last decade, a great deal of effort has been made to improve the indentation test and to gather information on the complete deformation process from experimental and numerical investigations. Following this line, this contribution concentrates on the numerical calculation of the elastic-plastic field of deformation in the specimen during the indentation process and the corresponding load-depth of indentation curves in dependence on the dominating parameters. The basic idea is to determine the influence of geometrical imperfections of the indenter on the—experimentally obtained—mechanical properties such as hardness and to provide methods which enable one to distinguish between properties of the system used for testing and the material investigated. Results obtained for uncoated and coated materials are compared. Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 8–11.     

考虑周长约束的圆柱薄壳初始几何缺陷随机场建模方法

利用随机场对圆柱薄壳结构的初始几何缺陷进行建模,并据此建立了一种用于含初始几何缺陷轴压圆柱薄壳屈曲分析的随机分析方法。首先,指出已有将圆柱薄壳初始几何缺陷表征为二维高斯随机场的方法会导致与实际不相符的初始几何缺陷,如圆柱周长显著增大或缩小的几何缺陷。其次,提出一种考虑周长不变约束的随机场建模方法,以剔除与实际不相符的随机几何缺陷。最后,基于所建立的初始几何缺陷随机场模型,利用非干涉多项式混沌展开法进行圆柱薄壳的随机屈曲分析,给出临界屈曲载荷的概率分布。数值试验结果表明,基于随机场理论的初始几何缺陷建模方法可有效刻画几何缺陷对结构承载能力的影响,而提出的约束随机场建模方法又能有效减小结果的分散性。     

Ermittlung der Beultraglasten ausgesteifter Schalen mit Hilfe von Übertragungsmatrizen

Übersicht Bei Beultraglastsberechnungen von ausgesteiften Kreiszylinderschalen unter Axialbelastung im plastischen Bereich werden das Deformationsverfahren und das Eigenwertverfahren mit Hilfe von übertragungsmatrizen verwendet, wobei die Übertragungsmatrix im Deformationsverfahren aus der nach der Finite-Element-Methode gewonnenen Steifigkeitsmatrix hergeleitet wird. Bei gleichzeitiger Berücksichtigung der Vorverformungen und der Restspannungen bzw. bei alleiniger Berücksichtigung der Restspannungen werden Rechenergebnisse von längsausgesteiften Schalen nach beiden Verfahren vorgelegt und verglichen.

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Study of the buckling load of stiffened shells by the transfer matrix method

Summary The deformation and eigenvalue method using the transfer matrix are applied to ultimate strength analyses of axially compressed, stiffened cylindrical shells in the plastic range. The transfer matrix in the deformation method is obtained by suitable transformation of the stiffness matrix used in the finite element method, and the transfer matrix in the eigenvalue method is described by the linearized differential equation of shells. In the deformation method geometrical imperfections and residual stresses are taken into consideration, while in the eigenvalue method the former are not considered. The numerical results of stiffened shells obtained by both methods are illustrated and compared with each other.

     

Experimental investigation of the buckling of shallow spherical shells

In the present paper, the buckling behavior of clamped thin shallow spherical shells under external pressure is studied. Seventy-nine plastic shells formed by thermovacuum process were tested. The distributions of initial geometrical imperfections and vertical displacements were minutely measured with a differential transformer. It was possible to control the symmetrical initial geometrical imperfection of each specimen.Results indicate that the buckling phenomena of shallow spherical shells vary greatly with the symmetrical initial imperfection parameter η. In the case of the geometrical parameter λ larger than 5.5, the amplitude of the asymmetrical displacement component with the bifurcation buckling wave calculated by Huang becomes large immediately before buckling. The validity of Huang's theory for an initially perfect shell is experimentally demonstrated.     

几何缺陷浅拱的动力稳定性分析

研究了几何缺陷对粘弹性铰支浅拱动力稳定性能的影响。从达朗贝尔原理和欧拉-贝努利假定出发推导了粘弹性铰支浅拱在正弦分布突加荷载作用下的动力学控制方程,并采用Galerkin截断法得到了可用龙格-库塔法求解的无量纲化非线性微分方程组。同时引入能有效追踪结构动力后屈曲路径的广义位移控制法,对含几何缺陷浅拱的响应曲线进行几何、材料双重非线性有限元分析。用这两种方法分析了前三阶谐波缺陷对浅拱动力稳定性能的影响,其中动力临界荷载由B-R准则判定。主要结论有:材料粘弹性使浅拱动力临界荷载增大且结构响应曲线与弹性情况差别很大;二阶谐波缺陷影响显著,它使动力临界荷载明显下降且使得浅拱粘弹性动力临界荷载可能低于弹性动力临界荷载。     

On the Stress Field of a Nonlinear Elastic Solid Torus with a Toroidal Inclusion

Standard measures of local deformation such as deformation gradient, strain, elastic deformation, and plastic deformation are dimensionless. However, many macroscopic and submacroscopic geometrical changes observed in continuous bodies result in the formation of zones across whose boundaries significant changes in geometry can occur. In order to predict the sizes of such zones and their influence on material response, theories of elasticity and plasticity have been employed in which second gradients of deformation, gradients of strain, as well as gradients of elastic or of plastic deformation are taken into account. The theory of structured deformations provides additive decompositions of first deformation gradient and of second deformation gradient, valid for large deformations of any material, in which each term has a multiscale geometrical interpretation corresponding to the presence or absence of submacroscopic disarrangements (non-smooth geometrical changes such as slips and void formation). This article provides a field theory that broadens the earlier field theory, elasticity with disarrangements, by including energetic contributions from submacroscopic “gradient-disarrangements” (limits of averages of jumps in gradients of approximating deformations) and by treating particular kinematical conditions as internal constraints. An explicit formula is obtained showing the manner in which submacroscopic gradient-disarrangements determine a defectiveness density analogous to the dislocation density in theories of plasticity. A version of the new field theory incorporates this defectiveness density to obtain a counterpart of strain-gradient plasticity, while another instance of elasticity with gradient-disarrangements recovers an instance of strain-gradient elasticity with symmetric Cauchy stress. All versions of the new theory included here are compatible with the Second Law of Thermodynamics.     

Effects of inertia on dynamic neck formation in tensile bars

Neck localization during high rate extension of round bars is analyzed numerically. An axisymmetric problem formulation is given and the material is described as rate independent and elastic–plastic. The time history of neck development is investigated and effects of geometry and initial thickness imperfections are visualized. Studies of both weakly and strongly developed necks are performed, revealing the occurrence of multiple necks in some loading cases. The influence on neck formation from background inertia (lateral inertia) corresponding to the inertia originating from homogeneous deformation of the bar is examined somewhat qualitatively by introducing an artificial volume load. Calculations show that in the present analysis, background inertia does not have any noticeable influence on the necking pattern unless the effect is artificially magnified by three orders of magnitude so that it becomes comparable to the yield stress.     

Experimental buckling of GRP cylindrical shells

During the setup of an experiment, errors may occur. Sources of such errors may be due to several factors which sometimes accumulate and then cause erroneous results. An experimental investigation on buckling of GRP (glass-reinforced-plastic) cylindrical shells, subject to axial compression and/or external pressure loading, has been carried out. At the beginning of the experiment, the initial geometrical imperfections were measured. Because of the small size of these quantities and the great effect these imperfections had on buckling loads, any small errors in the measurement procedure may lead to unreasonable results. Attempts have been made to detect these errors, and to identify and minimize their effect on experimental results. Tables are provided to show a comparison between the final experimental results and the corresponding theoretical ones.     

Kármán-type equations for a higher-order shear deformation plate theory and its use in the thermal postbuckling analysis

Kármán-type nonlinear large deflection equations are derived occnrding to the Reddy’s higher-order shear deformation plate theory and used in the thermal postbuckling analysis The effects of initial geometric imperfections of the plate areincluded in the present study which also includes th thermal effects.Simply supported,symmetric cross-ply laminated plates subjected to uniform or nomuniform parabolictemperature distribution are considered. The analysis uses a mixed GalerkinGolerkinperlurbation technique to determine thermal buckling louds and postbucklingequilibrium paths.The effects played by transverse shear deformation plate aspeclraio, total number of plies thermal load ratio and initial geometric imperfections arealso studied.