Numerical plane stress elastic–perfectly plastic crack analysis under Tresca yield condition with comparison to Dugdale plastic strip model

A number of plane stress numerical analyses of the mode I elastoplastic fracture mechanics problem have been performed in the past using the Huber–Mises yield criterion. This study employs instead the Tresca yield condition using an incremental theory of plasticity for a stationary crack. A commercial finite element program is used to solve the opening mode of fracture problem (mode I) for a square plate containing a central crack under generalized plane stress loading conditions. A biaxial uniform tensile traction is applied to the edges of a thin plate composed of a linear elastic non-work hardening material under small strain assumptions. The finite element results are compared with the analytical predictions of the Dugdale plastic strip model for a crack in an infinite plate subject to a biaxial uniform load at infinity.     

Two-parameter failure criterion for elastoplastic bodies with mode I cracks

The generalized Dugdale crack model is used to formulate two-parameter failure criteria for the cases of quasibrittle state and developed plastic zones at a mode I crack tip. The failure criteria relate the fracture strength characteristics and the stress mode at the crack tip through the plastic constraint factor. The critical state of bodies with cracks under uni-and biaxial loading is analyzed in the cases of plane stress and plane strain using the Tresca and von Mises yield criteria. A small-scale yield criterion, which is an analytic relation between the critical stress intensity factor and T-stresses, is established __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 47–57, July 2007.     

An experimental study on subsequent yield surface after finite shear prestraining

An equimodulus surface is introduced and the subsequent yield surface after large finite shear prestraining is experimentally investogated. Fully annealed, thin-walled copper tubular specimens were subjected to large torsional loading and partial unloading; strain gages were carefully mounted on the specimen after the application of pure shear loading. Specimens were then subjected to various combined tension-torsion loadings. Influences of he von Mises and Tresca equivalent offset strains on the subsequent yield surfaces are studied. On examining the experimental results reported in this article, it was found that the smaller the offset strains, the more distorted are the subsequent yield surfaces. At the torsional preloading point, a rounded corner was developed, whereas in the region opposite to the preloading point, the subsequent yield surface was flattened. When large von Mises offset strains were used, the corresponding subsequent yield surfaces passed through the von Mises loading surface. But this was not the case when Tresca offset strains were used. The subsequent yield surface determined by the back extrapolation method was almost completely outside the von Mises loading surface. On the other hand, the subsequent yield surface determined by the back extrapolation method was close to the Tresca loading surface. It is also found that the equimodulus surface is distorted and cannot simply be described by the combined kinematic and isotropic hardening rule.     

Stress Analysis of an Orthotropic Work-Hardening Cylinder with Body Force*

ABSTRACT

An elastoplastic analysis of an axisymmetric cylinder subjected to linear body forces is presented. The effect of reinforcement and anisotropy are also included. Classical plasticity and familiar assumptions of plane stress and strain are used to arrive at closed-form solutions for the case of linear body forces. The problem is solved for the general case in which orthotropy is considered in the elastic range. For the case of plasticity, first the isotropic yield functions (von Mises and Tresca) are used and then the problem is extended to the case of Hill's yield criterion. Closed-form solutions are found for both the von Mises (plane strain) and Tresca (plane stress and strain) cases.     

Mesoscale plastic flow generation and development for polycrystals

The traditional yield criteria of plasticity such as Mises, Tresca, etc. make use of averaged macroparameters while mesomechanics consideration is based on the physical notion of plastic deformation mechanisms. They may involve the development of plastic shears on the surfaces and interfaces of internal structure elements involving stress concentration and relaxation. A criterion of plastic flow is proposed; it is based on the stress–strain state in a cell of computational grid as well as in the neighboring cells. An algorithm of plastic shear generation is developed for the progressive propagation of the plastic shears over the crystal. Test calculations of the crystal behavior under tension are made and the results are presented.     

A plasticity model for pressure-dependent anisotropic cellular solids

The initial and subsequent yield surfaces for an anisotropic and pressure-dependent 2D stochastic cellular material, which represents solid foams, are investigated under biaxial loading using finite element analysis. Scalar measures of stress and strain, namely characteristic stress and characteristic strain, are used to describe the constitutive response of cellular material along various stress paths. The coupling between loading path and strain hardening is then investigated in characteristic stress–strain domain. The nature of the flow rule that best describes the plastic flow of cellular solid is also investigated. An incremental plasticity framework is proposed to describe the pressure-dependent plastic flow of 2D stochastic cellular solids. The proposed plasticity framework adopts the anisotropic and pressure-dependent yield function recently introduced by Alkhader and Vural [Alkhader M., Vural M., 2009a. An energy-based anisotropic yield criterion for cellular solids and validation by biaxial FE simulations. J. Mech. Phys. Solids 57(5), 871–890]. It has been shown that the proposed yield function can be simply calibrated using elastic constants and flow stresses under uniaixal loading. Comparison of stress fields predicted by continuum plasticity model to the ones obtained from FE analysis shows good agreement for the range of loading paths and strains investigated.     

Localization in elasto-plastic materials: Influence of an evolving yield surface in biaxial loading conditions

The influence of the plasticity yield surface – and of its evolution with plastic deformation – on the development of instabilities in metals is analyzed. Conditions for the activation of slip bands are taken as an instability criterion. They are exhibited in stress states identical to the ones encountered in a flat plate in biaxial tension. The classical bifurcation criterion is replaced by a criterion on the growth of a perturbation at a time scale comparable to the one of the homogeneous solution. This second criterion reveals less severe than the bifurcation one which is reached for the limit case of an infinite growth rate in the perturbation approach. The growth rate is a decreasing function of the biaxiality of the loading which is in agreement with previous studies. The possible destabilizing effect of texture evolution is also exhibited by using an evolving yield surface the curvature of which increases in the neighborhood of the homogeneous solution.     

简支圆板在复杂荷载作用下的塑性极限荷载统一解析解

本文采用双剪统一屈服准则对受线性荷载和边缘弯矩联合作用下的简支圆板进行了塑性极限分析，考虑了联合作用的两种形式，分别给出了统一的解析解。得到了极限荷载随不同屈服准则的变化曲线。对于不同的材料，本文均能给出相应的极限荷载。已有的Tresca准则、Von Mises准则、双剪应力准则的解答是文中解答的特例或逼近。本文得到的一系列有规则变化的解析解，可以适用于各种拉压强度相同材料的简支圆板的塑性极限荷载求解。文中统一解大于Tresca单剪理论解，它可以更好地发挥材料的强度潜力，工程应用可以取得明显的经济效益。     

Multiaxial ratcheting with advanced kinematic and directional distortional hardening rules

Ratcheting is defined as the accumulation of plastic strains during cyclic plastic loading. Modeling this behavior is extremely difficult because any small error in plastic strain during a single cycle will add to become a large error after many cycles. As is typical with metals, most constitutive models use the associative flow rule which states that the plastic strain increment is in the direction normal to the yield surface. When the associative flow rule is used, it is important to have the shape of the yield surface modeled accurately because small deviations in shape may result in large deviations in the normal to the yield surface and thus the plastic strain increment in multi-axial loading. During cyclic plastic loading these deviations will accumulate and may result in large errors to predicted strains.This paper compares the bi-axial ratcheting simulations of two classes of plasticity models. The first class of models consists of the classical von Mises model with various kinematic hardening (KH) rules. The second class of models introduce directional distortional hardening (DDH) in addition to these various kinematic hardening rules. Directional distortion describes the formation of a region of high curvature on the yield surface approximately in the direction of loading and a region of flattened curvature approximately in the opposite direction. Results indicate that the addition of directional distortional hardening improves ratcheting predictions, particularly under biaxial stress controlled loading, over kinematic hardening alone.     

The yield-point loads of symmetrically-notched metal strips

Theoretical plastic yield-point loads are calculated for symmetrically-notched metal strips in plane stress, assuming that the material obeys the von Mises yield criterion. Deep-notch solutions due to R. Hill are extended to cover all notch depths. The numerical results are presented in simple empirical formulae. The purpose of the work is to provide a way of discriminating between metals that obey the Tresca or the von Mises yield criteria.     

Variational formulations for the plane strain elastic-plastic problem for materials governed by the von Mises criterion

Path-dependent materials, complying with Drucker's postulate requirements and governed by an internal variable rate plasticity model, are considered. A variational principle for the small strain, rate plasticity problem is established in this context and extended to cover finite loading steps. Results are subsequently specialized to plane strain solids made of elastically isotropic materials with a plastic behavior governed by the von Mises criterion, accounting for combined isotropic and kinematic hardening. By exploiting previous results, the formulation is fully reduced to the plane. Further generalizations of the statements are also provided, which can be regarded as extensions to the elastic-plastic, plane strain problem of the Hu-Washizu principle in elasticity.     

冲击荷载作用下简支圆板的塑性动力响应统一解

采用统一强度理论求解了简支圆板在中等脉冲荷载作用下的动力响应问题,得出了统一的动力塑性极限荷载、内力场和速度场,并给出了上限解和下限解。讨论了静力许可条件和运动许可条件。利用本文的解还得出了简支圆板在静力荷载作用下的极限荷载、内力场和速度场。根据选择不同的拉压比参数,本文所给出的解可以适用于各种拉压异性和拉压同性材料。Tresca解、Mohr Coulomb解和双剪统一屈服准则解是本文的特例,Mises解是本文当=1和b=0.5时的线性逼近。研究结果表明,拉压比和强度理论参数b对动力解的影响要大于对静力解的影响,所以,根据材料的不同选择合适的强度理论,对于更好的发挥材料的强度潜力,减轻结构的重量具有重要的意义。     

A cyclic plasticity model for single crystals

The Armstrong–Frederick type kinematic hardening rule was invoked to capture the Bauschinger effect of the cyclic plastic deformation of a single crystal. The yield criterion and flow rule were built on individual slip systems. Material memory was introduced to describe strain range dependent cyclic hardening. The experimental results of copper single crystals were used to evaluate the cyclic plasticity model. It was found that the model was able to accurately describe the cyclic plastic deformation and properly reflect the dislocation substructure evolution. The well-known three distinctive regimes in the cyclic stress–strain curve of the copper single crystals oriented for single slip can be reproduced by using the model. The model can predict the enhanced hardening for crystals oriented for multislip, showing the model's ability to describe anisotropic cyclic plasticity. For a given loading history, the model was able to capture not only the saturated stress–strain response but also the detailed transient stress–strain evolution. The model was used to predict the cyclic plasticity under a high–low loading sequence. Both the stress–strain responses and the microstructural evolution can be appropriately described through the slip system activation.     

Strain gradient crystal plasticity effects on flow localization

In metal grains one of the most important failure mechanisms involves shear band localization. As the band width is small, the deformations are affected by material length scales. To study localization in single grains a rate-dependent crystal plasticity formulation for finite strains is presented for metals described by the reformulated Fleck–Hutchinson strain gradient plasticity theory. The theory is implemented numerically within a finite element framework using slip rate increments and displacement increments as state variables. The formulation reduces to the classical crystal plasticity theory in the absence of strain gradients. The model is used to study the effect of an internal material length scale on the localization of plastic flow in shear bands in a single crystal under plane strain tension. It is shown that the mesh sensitivity is removed when using the nonlocal material model considered. Furthermore, it is illustrated how different hardening functions affect the formation of shear bands.     

Texture development and plastic anisotropy of B.C.C. strain hardening sheet metals

A Taylor-like polycrystal model is adopted here to investigate the plastic behavior of body centered cubic (b.c.c.) sheet metals under plane-strain compression and the subsequent in-plane biaxial stretching conditions. The <111> pencil glide system is chosen for the slip mechanism for b.c.c. sheet metals. The {110} <111> and {112} <111> slip systems are also considered. Plane-strain compression is used to simulate the cold rolling processes of a low-carbon steel sheet. Based on the polycrystal model, pole figures for the sheet metal after plane-strain compression are obtained and compared with the corresponding experimental results. Also, the simulated plane-strain stress—strain relations are compared with the corresponding experimental results. For the sheet metal subjected to the subsequent in-plane biaxial stretching and shear, plastic potential surfaces are determined at a given small amount of plastic work. With the assumption of the equivalence of the plastic potential and the yield function with normality flow, the yield surfaces based on the simulations for the sheet metal are compared with those based on several phenomenological planar anisotropic yield criteria. The effects of the slip system and the magnitude of plastic work on the shape and size of the yield surfaces are shown. The plastic anisotropy of the sheet metal is investigated in terms of the uniaxial yield stresses in different planar orientations and the corresponding values of the anisotropy parameter R, defined as the ratio of the width plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial tensile loading. The uniaxial yield stresses and the values of R at different planar orientations from the polycrystal model can be fitted well by a yield function recently proposed by Barlat et al. (1997b).     

The effect of load biaxiality on crack growth in non-linear materials

The influence of load biaxiality on the stress field and fracture behavior of a cracked plate is investigated. Considered is a square plate containing a central through the thickness crack and subjected to a biaxial loading perpendicular and parallel to the crack plane. The stress field of the plate is analyzed by a finite element code based on incremental plasticity and the von Mises yield condition. A method based on the strain energy density theory is used to determine the critical stress for crack initiation. It was found that the equi-biaxial loading mode induces the smallest plastic zones, while the critical applied stress for crack initiation becomes maximum. Quite the contrary happens for the shear loading system which causes the largest plastic zones and the minimum applied stress values fro crack growth. Results showing the dependence of the above quantities on the biaxiality of the applied stress are presented in graphical form.     

屈服准则及切线模量修正的弹塑性计算模型

为提高不同应力状态下金属弹塑性行为的模拟精度,采用含应力三轴度修正的von Mises屈服准则,材料弹塑性本构关系在等效应力$\!$-$\!$-$\!$等效应变曲线基础上,提出一个切线模量为主应力函数的理论修正项. 将这两个修正项结合,以子程序形式编程嵌入ABAQUS主程序,以此模拟几种不同形状试样的弹塑性行为,并将其他屈服准则在单一曲线假设下编程与之对比. 模拟结果与真实试验结果对比发现,对于屈服而言,含应力三轴度修正的vonMises屈服准则比其他屈服理论准确;对于弹塑性阶段计算而言,提出的切线模量为主应力函数这一假设比单一曲线假设更加接近真实试验.      

Perfectly plastic caustics for the opening mode of fracture

Exact expressions for the caustics generated by the reflection of light surrounding crack tips in perfectly plastic materials under plane stress loading conditions and tensile tractions at infinity (mode I) are derived. Two individual cases are examined involving two different yield criteria. The first case uses an approximation of the Mises yield condition, where in the principal stress plane two intersecting parabolas replace the standard ellipse. The second case uses the Tresca yield condition where the mode I caustic is obtained as a limit of an elliptical hole in a perfectly plastic material. In both cases, kinematically admissible velocity fields are employed to obtain strain fields from which the theoretical caustics are predicted.     

Limit loads of edge-restrained shallow shells

Limit analysis of edge-restrained rigid, perfectly plastic shallow spherical shells under external pressure is considered. A numerical solution, based on the von Mises yield criterion, is shown to be in good agreement with existing approximate solutions based on the Tresca yield criterion.