Shakedown reduced kinematic formulation,separated collapse modes,and numerical implementation

Our shakedown reduced kinematic formulation is developed to solve some typical plane stress problems, using finite element method. Whenever the comparisons are available, our results agree with the available ones in the literature. The advantage of our approach is its simplicity, computational effectiveness, and the separation of collapse modes for possible different treatments. Second-order cone programming developed for kinematic plastic limit analysis is effectively implemented to study the incremental plasticity collapse mode. The approach is ready to be used to solve general shakedown problems, including those for elastic–plastic kinematic hardening materials and under dynamic loading.     

Elasto-plastic vibrational analysis of tapered bars under uniform axial loading considering shear deformation and rotary inertia

The analysis of structures in the plastic regime is important in order to develop an adequate and competitive design in engineering. This paper presents a study of the small-amplitude free vibration of tapered bars under pre-stress in the post-elastic regime due to a uniform axial loading. The plastic behavior is taken into account using an energy approach. The method does not require an iterative procedure, unlike conventional methods used in plasticity. The Timoshenko beam theory and the dynamic version of the principle of virtual work are used to derive the eigenvalue problem. The solution is carried out using beam finite elements. The results are validated using 3D finite element software and results from the open literature. A variety of numerical results are given in order to analyze the influence of plastic behavior for various bar geometries and material parameters. The combined effect of the stiffening due to the axial loading and the plastic softening may produce an increase or decrease of the natural frequencies as the tensile load increases. The plastic softening effect is seen to be pronounced for short bars and for high taper ratios. In addition, axial normal modes are more affected than bending modes.     

A modified upper bound approach to limit analysis for plane strain deeply cracked specimens

The classical upper bound approach of limit analysis is based on assumption of rigid blocks of deformation that move between lines of tangential displacement discontinuity. This assumption leads to considerable simplification but often at cost of higher estimate of the actual load. Moreover, in many cases, it does not give a correct shape of the plastic field. In order to overcome these limitations a modified upper bound approach is proposed in this article. The proposed approach is basically an energetic approach but unlike the classical upper bound approach it is capable of including presence of statically governed stress field. As an application, of proposed approach, theoretical plane strain solutions are presented for deeply cracked fracture mechanics specimens (single edge cracked specimen in pure bending – SE (PB), single edge cracked specimen in three-point bending – SE (B), and compact tension – C (T) specimens). Plane strain plasticity problem in rigid elastic–plastic mono-material (homogeneous) was solved to evaluate useful parameters like limit load, plastic eta function (η_p) and plastic rotation factor (r_p) and in bi-material (mismatch welds) to evaluate mismatch limit load, for deeply cracked specimens. New kinematically admissible velocity fields are proposed for SE (B) and C (T) specimens. Proposed theoretical solutions were confirmed by classical slip-line field solutions, wherever available, and by detailed elastic–plastic finite element analysis with Von-Mises yield criterion. Good agreement was found between proposed solutions and results obtained from the classical slip-line field theory and finite element analysis.     

The near tip fields and temperature distribution around a crack in a body of hardening material containing small damage

In this paper, the following conclusions are reached: The influence of damage on the stress and strain feilds can be neglected in an asymptotic sense for the solutions of damage field in a plastic solid containing small damage. The formulation of the problem is simplified with an uncoupled approach. Based on experimental results of plastic damage, most of the damage in the material are considered as small damage with the critacal damage variable ω _c ≪1. Using this approach, closed form expressions of the near tip damage fields for mode III, mode I and the temperature distribution induced by plastic dissipation in a hardening material containing damage are deduced. We point out that the temperature distribution in the process zone is strongly dependent on the damage of materials even for the small damage case. The results of the predicted value of the temperature rise near the tip region ignoring the damage effect is appreciably higher than the observed data. The main reason of this discrepancy is the presence of damage dissipation and the fact that its influence on the calculation of plastic dissipation have not been appropriately taken account of. The calculation is improved by taking into account the damage effect on the temperature rise, then theT _max value is in better accord with the experimental value. The project supported by the National Natural Science Foundation of China.     

A comparison of cohesive zone modeling and classical fracture mechanics based on near tip stress field

A mode III crack with a cohesive zone in a power-law hardening material is studied under small scale yielding conditions. The cohesive law follows a softening path with the peak traction at the start of separation process. The stress and strain fields in the plastic zone, and the cohesive traction and separation displacement in the cohesive zone are obtained. The results show that for a modest hardening material (with a hardening exponent N = 0.3), the stress distribution in a large portion of the plastic zone is significantly altered with the introduction of the cohesive zone if the peak cohesive traction is less than two times yield stress, which implies the disparity in terms of the fracture prediction between the classical approach of elastic–plastic fracture mechanics and the cohesive zone approach. The stress distributions with and without the cohesive zone converge when the peak cohesive traction becomes infinitely large. A qualitative study on the equivalency between the cohesive zone approach and the classical linear elastic fracture mechanics indicates that smaller cracks require a higher peak cohesive traction than that for longer cracks if similar fracture initiations are to be predicted by the two approaches.     

Crack tip singular fields in ductile crystals with taylor power-law hardening. I: Anti-plane shear

Asymptotic singular solutions of the HRR type are presented for anti-plane shear cracks in ductile crystals. These are assumed to undergo Taylor hardening with a power-law relation between stress and strain at sufficiently large strain. Results are given for several crack orientations in fcc and bcc crystals. The neartip region divides into angular sectors which are the maps of successive flat segments and vertices on the yield locus. Analysis is simplified by use of new general integrals of crack tip singular fields of the HRR type. It is conjectured that the single crystal HRR fields are dominant only over part of the plastic region immediately adjacent to the crack tip, even at small scale yielding, and that their domain of validity vanishes as the perfectly plastic limit is approached. This follows from the fact that while in the perfectly plastic limit the HRR stress states approach the correct discontinuous distributions of the complete elasticideally plastic solutions for crystals (Rice and Nikolic, J. Mech. Phys. Solids33, 595 (1985)), the HRR displacement fields in that limit remain continuous. Instead, the complete elastic-ideally plastic solutions have discontinuous displacements along planar plastic regions emanating from the tip in otherwise elastically stressed material. The approach of the HRR stress fields to their discontinuous limiting distributions is illustrated in graphical plots of results. A case examined here of a fcc crystal with a crack along a slip plane is shown to lead to a discontinuous near-tip stress state even in the hardening regime.Through another limiting process, the asymptotic solution for the near-tip field for an isotropic material is also derived from the present single crystal framework.     

端部受斜撞击作用悬臂梁的弹塑性动力响应模式

基于大变形动力控制方程并利用有限差分离散分析,研究了斜撞击作用下弹塑性悬臂梁的动力响应．通过对屈服函数以及弯矩、轴力在动力响应过程中分布规律的分析,阐明了斜撞击下恳臂梁的弹塑性动力响应模式和斜撞击的轴向分量对变形机制的影响．研究表明,弹塑性响应过程可划分为四个阶段,对应的变形模式为：“压缩塑性区扩展”模式,“广义移行塑性铰”和“压缩塑性区收缩”混合模式,“驻定塑性铰”模式,“弹性自由振动”模式．与刚塑性分析所假定的两相变形模式比较,弹塑性应响分析证实了响应早期的瞬态轴向压缩模式和梁根部“驻定塑性铰”模式的存在性,肯定了刚塑性分析所假定变形模式的主要特征．斜撞击的轴向分量在撞击发生的瞬时主导了梁的变形,使梁呈现同承受横向冲击明显小同的变形规律．随着响应的深入,轴向分量迅速衰减,其对截面屈服的贡献非常微弱,由横向分量引起的弯曲挠动在大部分时间内主导和控制梁的变形．数值计算结果表明,斜撞击载荷的质量、撞击速度和角度是影响梁动力响应的重要因素．     

Hutchinson模型的塑性后分叉和缺陷敏感性

本文通过对Ｈｕｔｃｈｉｎｓｏｎ模型的初始塑性后分叉和缺陷敏感性的分析，介绍了一个分析塑性后屈曲的一般方法，这一方法非常类似Ｋｏｉｎｔｅｒ的弹性后屈曲理论。用这一方法分析缺陷敏感性和初始后分叉的过程是一致的，所得展开在渐近的意义上是精确的，和数值解符合得相当好。     

Multi-scale non-local approach for geomaterials

A thermodynamically consistent multi-scale, rate dependent, non local approach is developed in this work for geo-materials in conjunction with the anisotropic modified Cam Clay model. The gradient for the micro-structure is incorporated through the micro level gradient of the back-stress and volumetric strain while the gradient for macro-structure is incorporated through the macro level gradient of back-stress and volumetric plastic strain. Gradient results in the regularization of the local behavior. Visco-plasticity is also incorporated for an additional regularization of the local behavior. Therefore, the effects of two separate regularizations are naturally separated. The plastic spin is incorporated to separate the effect of micro-structural rotation from the gradient effect. The flow characteristics of the soil is also incorporated in order to separate the viscosity effect from the flow effect.Through this multi scale non local approach, a more realistic simulation of large strain problems such as shear band formation can be achieved.     

Elastic-plastic contact model for rough surfaces based on plastic asperity concept

A mathematical formulation for the contact of rough surfaces is presented. The derivation of the contact model is facilitated through the definition of plastic asperities that are assumed to be embedded at a critical depth within the actual surface asperities. The surface asperities are assumed to deform elastically whereas the plastic asperities experience only plastic deformation. The deformation of plastic asperities is made to obey the law of conservation of volume. It is believed that the proposed model is advantageous since (a) it provides a more accurate account of elastic-plastic behavior of surfaces in contact and (b) it is applicable to model formulations that involve asperity shoulder-to-shoulder contact. Comparison of numerical results for estimating true contact area and contact force using the proposed model and the earlier methods suggest that the proposed approach provides a more realistic prediction of elastic-plastic contact behavior.     

Approximate yield criteria for anisotropic metals with prolate or oblate voidsCritères macroscopiques pour des métaux plastiques anisotropes contenant des cavités non sphériques

Following the study of Gologanu et al. (1997) which has extended the well-known approach of Gurson (1975), we propose approximate yield criteria for anisotropic plastic voided metals containing non spherical cavities. The plastic anisotropy of the matrix is described by means of Hill's quadratic criterion. The procedure to establish the closed form expression of approximate macroscopic criteria, in which void shape and plastic anisotropic effects are included, is detailed. The new criteria allow us to recover existing results in the cases of spherical and cylindrical voids in an Hill type plastic matrix. Moreover, they agree with previous criteria for non spherical voids in an isotropic plastic matrix. Finally, for validation purposes, we provide, in the general case of non spherical cavities in the anisotropic matrix, a comparison with the numerical exact two field criteria. To cite this article: V. Monchiet et al., C. R. Mecanique 334 (2006).     

A FE formulation for elasto-plastic materials with planar anisotropic yield functions and plastic spin

In this article a stress integration algorithm for shell problems with planar anisotropic yield functions is derived. The evolution of the anisotropy directions is determined on the basis of the plastic and material spin. It is assumed that the strains inducing the anisotropy of the pre-existing preferred orientation are much larger than subsequent strains due to further deformations. The change of the locally preferred orientations to each other during further deformations is considered to be neglectable. Sheet forming processes are typical applications for such material assumptions. Thus the shape of the yield function remains unchanged. The size of the yield locus and its orientation is described with isotropic hardening and plastic and material spin.The numerical treatment is derived from the multiplicative decomposition of the deformation gradient and thermodynamic considerations in the intermediate configuration. A common formulation of the plastic spin completes the governing equations in the intermediate configuration. These equations are then pushed forward into the current configuration and the elastic deformation is restricted to small strains to obtain a simple set of constitutive equations. Based on these equations the algorithmic treatment is derived for planar anisotropic shell formulations incorporating large rotations and finite strains. The numerical approach is completed by generalizing the Return Mapping algorithm to problems with plastic spin applying Hill’s anisotropic yield function. Results of numerical simulations are presented to assess the proposed approach and the significance of the plastic spin in the deformation process.     

Parametric finite-volume micromechanics of periodic materials with elastoplastic phases

The recently incorporated parametric mapping capability into the finite-volume direct averaging micromechanics (FVDAM) theory has produced a paradigm shift in the theory’s development. The use of quadrilateral subvolumes made possible by the mapping facilitates efficient modeling of microstructures with arbitrarily shaped heterogeneities, and eliminates artificial stress concentrations produced by the rectangular subvolumes employed in the standard version. Herein, the parametric FVDAM theory is extended to the inelastic domain by implementing additional formulation required to accommodate plastic and thermal loading. Two different approaches of implementing plasticity have been investigated. The first approach is based on the treatment employed in previous versions of the theory wherein plastic strain fields are represented by a series expansion in Legendre polynomials. The second approach is based on direct surface-averaging of plastic strains calculated at a number of collocation points along the quadrilateral subvolumes’ surfaces, and offers substantial simplification in the parametric finite-volume theory’s elastic–plastic framework. Moreover, substantial reductions in execution times without loss of accuracy are realized due to the elimination of redundant plastic strain calculations in the subvolumes’ interiors employed in the evaluation of the Legendre polynomial coefficients. Numerical studies demonstrate the advantages of the parametric FVDAM theory relative to the standard version, together with new results that highlight its modeling capabilities vis-a-vis an emerging class of periodic lamellar materials with wavy microstructures and the thus-far undocumented architectural effects amplified by plasticity.     

板材多点成形过程的有限元分析

多点成形过程采用静力隐式格式进行数值模拟是比较合适的。本文建立了用于多点成形过程分析的静力隐式弹塑性大变形有限元方法 ,给出了对稳定迭代收敛过程效果较好的板壳有限单元模型、处理多点不连续接触边界的接触单元方法以及增量变形过程中应力及塑性应变计算的多步回映计算方法。基于这些方法编制了计算软件 ,应用该软件进行了矩形板的液压胀形过程及球形模具拉伸成形过程的有限元分析 ,数值计算结果与典型的实验结果及计算结果吻合很好。最后给出了球形、圆柱形目标形状的实际多点成形过程的数值模拟结果。     

脆塑性岩石破坏后区应力跌落效应数值模拟

采用三轴试验和数值模拟研究了岩石类脆塑性材料的应力跌落效应,并用塑性流动因子λ来描述应力跌落效应。为简化计算,给出了一种应力陡降过程中伴随的非零应变增量的工程近似处理方法,基于此针对性地编制了处理脆塑性材料应力跌落的有限元分析程序代码。数值模拟结果验证了塑性流动因子λ以及该近似处理方法的有效性。     

Constitutive relations for isotropic or kinematic hardening at finite elastic–plastic deformations

The rate-type constitutive relations of rate-independent metals with isotropic or kinematic hardening at finite elastic–plastic deformations were presented through a phenomenological approach. This approach includes the decomposition of finite deformation into elastic and plastic parts, which is different from both the elastic–plastic additive decomposition of deformation rate and Lee’s elastic–plastic multiplicative decomposition of deformation gradient. The objectivity of the constitutive relations was dealt with in integrating the constitutive equations. A new objective derivative of back stress was proposed for kinematic hardening. In addition, the loading criteria were discussed. Finally, the stress for simple shear elastic–plastic deformation was worked out.     

Interpretation of the behavior of metals under large plastic shear deformations: comparison of macroscopic predictions to physically based predictions

A large plastic shear problem is analyzed by application of a macroscopic anisotropic plasticity model (Kuroda, M., 1997. Interpretation of the behavior of metals under large plastic shear deformations: a macroscopic approach. Int. J. Plasticity 13, 359–383), and the results are compared to predictions based on crystal plasticity with the Taylor assumption. It is found that these two different-scale models provide very similar predictions. The interpretations for such similarities are pursued in detail. The present macroscopic model reproduces quite well the change in orientation of anisotropy, which is directly predicted in the crystal plasticity analyses as a macroscopic manifestation of texture development. Consequently, the predictions for the rotation of the yield surface by the different-scale models become very similar. It is clearly shown that, in a macroscopic sense, the rotation of the anisotropic yield surface is a main cause of the axial effects in large plastic shear deformation.     

Dislocation dynamics and acoustic emission during plastic deformation of crystals

A soliton approach to acoustic emission during plastic deformation of crystals is presented. The approach is based on a microscopic Frenkel-Kontorova model where the rigidity of the substrate is removed in order to establish the interaction mechanism between a dislocation and both longitudinal and transverse acoustic waves. It is shown that this interaction is described by a sine-Gordon-d' Alembert system. Within the framework of this system, two basic mechanisms of acoustic emission are investigated both analytically and numerically. One mechanism is related to nonstationary dislocation motion and the other one to the annihilation of dislocation kink-antikink pairs during Frank-Read source operation. In both cases, computer simulations are obtained which illustrate graphically the analytical considerations and model the acoustic radiation. The obtained results are in agreement with existing experimental data and may provide a better physical insight to the acoustic emission mechanisms during plastic deformation of crystals.