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.     

Dynamic buckling of a simple elastic-plastic model under pulse loading

The dynamic buckling of an elastic-plastic imperfection-sensitive model subjected to rectangular- and triangular-shaped loading pulses is examined to provide some insight into the dynamic buckling behaviour of structures. The loading pulse is expressed as a function of the horizontal displacement, which allows an analytical method to be used for determining the stability domains for both pulses. The estimates obtained are compared with some previously published results on the dynamic elastic-plastic buckling of the same model under a step loading. It transpires that for the pulse loading of models with large imperfections dynamic instability occurs either elastically or plastically depending on the pulse duration, while for a step loading only an elastic instability is possible for the parameters examined.     

应变损伤材料平面应力动态裂纹尖端场的渐近解

采用塑性动力学方程,对应变损伤材料的平面应力动态裂纹尖端场进行了渐近分析。假定损伤规律服从反比例关系,对平面应力问题,导出了本构方程,并给出了动态弹塑性场的渐近解,揭示了场的渐近特性。     

Exact integration of constitutive equations of kinematic hardening materials and its extended applications

In this paper, an exact formula for the integration of the constitutive equations of kinematic hardening material is presented. Its algorithms are simple and clear. For isotropic hardening or mixed hardening material, the formula is still an exact solution for the case of radial loading, and it is an approximate solution with reasonable accuracy for the case of non-radial loading. The computation results show that the procedure proposed in this paper improves both accuracy and efficiency of numerical integration schemes adopted widely in elastic-plastic finite element analysis.     

Research note on a reexamination of neutral loading experiments

In the examination of the published results from neutral loading experiments, the question as to whether plastic deformation occurs is found to depend on both the material and initial loading strain. Provided that initial loading is elastic, then a subsequent stress path that follows the boundary of the initial yield surface for a hardening material is truly neutral with a wholly elastic response. However, when initial loading is elastic-plastic, then further plastic deformation is produced from a subsequent stress path that follows an isotropic expansion of the initial yield surface. These results enable the appropriateness of the kinematic hardening rule and more recent developments in plasticity theory to be appraised. Neutral loading of a non-hardening material produces plastic flow. Whether the absence of hardening is inherent or induced by plastic prestrain, it is shown that the Prandtl-Reuss theory then represents the observed behavior. In general, the purely elastic and nonhardening solutions provide respectively lower and upper bounds on the deformation.     

Imperfection sensitivity of axially compressed stringer reinforced cylindrical sandwich panels

This paper presents some numerical results of the effects of several nondimensional parameters on the buckling and initial post buckling behaviors of shallow sandwich panels under axial compression. Results are presented that show these effects due to transverse shearing resistance of the core material, different face-sheet thicknesses, and different core thicknesses. Further effects on the buckling and initial postbuckling behaviors of sandwich panels are presented due to the torsional resistance of longitudinal edge stiffeners.The results show that the range of flatness parameter, δ/d, for which sandwich panels remain imperfection-insensitive increases with increases in transverse shearing resistance of the core material and with larger core thicknesses. These results also indicate that this range of δ/d is smallest when the face-sheet thicknesses are equal. Finally, as in the case of homogeneous panels, torsional resistance of the longitudinal edge stiffeners has the effect of making the sandwich panel less imperfection-sensitive.     

Effect of thickness inhomogeneities in internally pressurized elastic-plastic spherical shells

The behaviour of elastic-plastic spherical shells under internal pressure is investigated numerically for thickness-to-radius ratios ranging from cases of thin shells to very thick shells. The shells under consideration are made of strain-hardening elastic-plastic material with a smooth yield-surface. Attention is restricted to axisymmetric deformations, and results are presented for initial thickness inhomogeneities in various axisymmetric shapes. For smooth thickness-variations in the shape of the critical bifurcation mode, the reduction in maximum pressure is studied together with the distribution of deformations in the final collapse mode. Also, the possibility of flow localization due to more localized, initially thin regions on a spherical shell is investigated.     

On shakedown theory for elastic-plastic materials and extensions

The idea that an elastic-plastic structure under given loading history may shake down to some purely elastic state (and hence to a safe state) after a finite amount of initial plastic deformation, can apply to many sophisticated material models with possible allowable changes of additional material characteristics, as has been done in the literature. Despite some claims to the contrary, it is shown; however, that the shakedown theorems in a Melan-Koiter path-independent sense have been extended successfully only for certain elastic-plastic hardening materials of practical significance. Shakedown of kinematic hardening material is determined by the ultimate and initial yield stresses, not the generally plastic deformation history-dependent hardening curve between. The initial yield stress is no longer the convenient one (corresponding to the plastic deformation at the level of 0.2%) as in usual elastic-plastic analysis but to be related to the shakedown safety requirement of the structure and should be as small as the fatigue limit for arbitrary high-cycle loading. Though the ultimate yield strength is well defined in the standard monotonic loading experiment, it also should be reduced to the so-called “high-cycle ratchetting” stress for the path-independent shakedown analysis. A reduced simple form of the shakedown kinematic theorem without time integrals is conjectured for general practical uses. Application of the theorem is illustrated by examples for a hollow cylinder, sphere, and a clamped disk, under variable (including quasiperiodic dynamic) pressure.     

A 3/D finite element approach for metal matrix composites based on micromechanical models

Based on analytical considerations by Dvorak and Bahel-El-Din, a 3/D finite element material law has been developed for the elastic-plastic analysis of unidirectional fiber-reinforced metal matrix composites. The material law described in this paper has been implemented in the finite element code ABAQUS via the user subroutine UMAT. A constitutive law is described under the assumption that the fibers are linear-elastic and the matrix is of a von Mises-type with a Prager-Ziegler kinematic hardening rule. The uniaxial effective stress-strain relationship of the matrix in the plastic range is approximated by a Ramberg-Osgood law, a linear hardening rule or a nonhardening rule. Initial yield surface of the matrix material and for the fiber reinforced composite are compared to show the effect of reinforcement. Implementation of this material law in a finite element program is shown. Furthermore, the efficiency of substepping schemes and stress corrections for the numerical integration of the elastic-plastic stress-strain relations for anisotropic materials are investigated. The results of uniaxial monotonic tests of a boron/aluminum composite are compared to some finite element analyses based on micromechanical considerations. Furthermore a complete 3/D analysis of a tensile test specimen made of a silicon-carbide/aluminum MMC and the analysis of an MMC inlet inserted in a homogenous material are shown.     

An elastic-plastic analysis of an asymmetric cracked specimen subjected to longitudinal shear

Some recent elastic-plastic analyses of cracked specimens subjected to symmetric mode III loading are extended to include asymmetric loading and geometry. Solutions are given for arbitrary work hardening behaviour in any specimen that is amenable to a linear elastic analysis. It is shown that asymmetry has a major influence on the shape of the plastic zone, but does not affect the J-integral unil the loading is well into the large scale yielding range. In particular the “plastic zone corrected” estimate of J, obtained by elastically solving a problem for a crack longer than the actual one, is shown to remain a valid two-term asymptotic expansion in the presence of asymmetry. The general results are applied to a crack at an angle to a uniform stress field in a power law hardening material. The growth of the plastic zone is displayed graphically for various hardening exponents and crack orientations. No other asymmetric solution is available, but values of J are compared with those obtained from a fully plastic analysis in the symmetric case.     

The effect of strain softening in the matrix material during void growth

A Finite element analysis has been employed to investigate the growth of an initially spherical void embedded in a cylinder of elastic-plastic material. The boundary displacement of this cylindrical cell is regulated by the value of a parameter α which controls the radial shrinkage of the cell as it elongates. A large strain analysis was used and results for both strain hardening and strain softening (after an appropriate amount of hardening has taken place) have been obtained. The effects of different mean tensile stresses, equivalent strains and initial void volume fractions have also been included. The numerical work shows relationships between the mechanical and geometrical variables that may favour ductile fracture by void coalescence or by shear decohesion.     

On an elastic-plastic transition zone in cellular metals

Summary A theory of plasticity is proposed for cellular metals to describe their elastic-plastic transition zone at small strain. Under certain conditions, only a plane strain test is necessary to determine the yield surface. The method to derive the elastic–plastic behaviour [14, 15] was originally proposed for classical metals. A simple cubic model of a cellular metal is used to demonstrate the method by the finite element method. Recommendations for the numerical simulation are given. The influence of the relative density and the hardening behaviour of the cell wall material is investigated.     

Simulation of curing-induced viscoplastic deformation: a new approach considering chemo-thermomechanical coupling

A modelling and simulation approach for plastic deformation effects in curing resins is presented. For this purpose known rheological models of viscoelasticity and viscoplasticity are combined and a thermochemical element is added to account for chemical shrinkage and thermal expansion. The degree of cure of the resin has a major influence on the behaviour of the curing material, and therefore, the material model is formulated depending on the degree of cure. It affects the viscoelastic behaviour as well as the chemical shrinkage and the yield function of the viscoplastic part of the model. For the yield function a von Mises approach with isotropic hardening is chosen, where the initial yield stress as well as the yield surface depends on the degree of cure and the temperature.     

Finite element analysis of mode III steady crack growth in anisotropic hardening materials

In this paper, the steady crack growth of mode III under small scale yielding conditions is investigated for anisotropic hardening materials by the finite element method. The elastic-plastic stiffness matrix for anisotropic materials is given. The results show the significant influences of anisotropic hardening behaviour on the shape and size of plastic zone and deformation field near the crack tip. With a COD fracture criterion, the ratio of stress intensity factorsk _ss/k_c varies appreciably with the anisotropic hardening parameterM and the hardening exponentN.     

不同硬化指数的幂次硬化结合材料的界面端奇异性分析

从位移匹配的观点出发,本文认为对任意结合角度的幂次硬化材料界面端弹塑性问题,如果两种材料的硬化指数不相同,则应力场的奇异次数应由硬化指数较高一方材料的材料性质和几何形状决定。进一步分析表明,奇异次数只与该材料的硬化指数n及界面端角度有关,与比例常数α等其它材料常数无关。通过边界元数值计算对上述结论进行了验证,并且发现随着硬化指数的提高,应力奇异次数降低。     

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.     

Restrictions on dynamically propagating surfaces of strong discontinuity in elastic-plastic solids

For dynamic three-dimensional deformations of elastic-plastic materials, we elicit conditions necessary for the existence of propagating surfaces of strong discontinuity (across which components of stress, strain or material velocity jump). This is accomplished within a small-displacement-gradient formulation of standard weak continuum-mechanical assumptions of momentum conservation and geometrical compatibility, and skeletal constitutive assumptions which permit very general elastic and plastic anisotropy including yield surface vertices and anisotropic hardening. In addition to deriving very explicit restrictions on propagating strong discontinuities in general deformations, we prove that for anti-plane strain and incompressible plane strain deformations, such strong discontinuities can exist only at elastic wave speeds in generally anisotropic elastic-ideally plastic materials unless a material's yield locus in stress space contains a linear segment. The results derived seem essential for correct and complete construction of solutions to dynamic elastic-plastic boundary-value problems.     

Elastic-plastic analysis of a flat plate with a circular rigid inclusion

Summary An elastic-plastic solution is presented for a circular rigid inclusion in a unidirectionally stressed flat plate of linearly strain hardening material. The results are compared to the results obtained for the free hole. It is postulated that for intermediate constraints the plastic flow would be less pronounced than in these limiting cases.     

Shear waves in an elastic-plastic material due to a supersonic surface step load

A step shear load moves steadily on the surface of an elastic-plastic half space at a speed exceeding the elastic shear wave speed of the material. The orientation of the shear traction is such that the deformation is two-dimensional antiplane strain. Two different representations of the rate independent elastic-plastic material response are considered. The first material model is based on the associated flow rule and the Mises yield condition with isotropic hardening, whereas the second model is based on a particular flow theory of plasticity which represents incremental behavior at a corner of the instantaneous yield surface. Both models predict the same response under the same proportional loading. The stress history experienced by a typical material particle during passage of the load step is determined, and the variation of final strain with the magnitude of the load step is calculated. One conclusion resulting from comparison of results for the two material models for this problem is that the influence of yield surface vertex formation is not significant.     

A general analytic solution for plane strain bending under tension for strain-hardening material at large strains

A new analytic solution for plane strain bending under tension of a sheet is proposed for elastic-plastic, isotropic, incompressible, strain-hardening material at large strains. Numerical treatment is only necessary to calculate ordinary integrals and solve transcendental equations. No restriction is imposed on the hardening law. All governing equations and boundary conditions are exactly satisfied. The only exception is that the actual stress distribution over the ends of the sheet is replaced with a concentrated force and a concentrated bending moment. The through-thickness distribution of residual stresses and a measure of springback are also found. The range of validity of the solution is determined. An illustrative example is provided for Swift’s hardening law.