Well-posedness of a model of strain gradient plasticity for plastically irrotational materials

The initial boundary value problem corresponding to a model of strain gradient plasticity due to [Gurtin, M., Anand, L., 2005. A theory of strain gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations. J. Mech. Phys. Solids 53, 1624–1649] is formulated as a variational inequality, and analysed. The formulation is a primal one, in that the unknown variables are the displacement, plastic strain, and the hardening parameter. The focus of the analysis is on those properties of the problem that would ensure existence of a unique solution. It is shown that this is the case when hardening takes place. A similar property does not hold for the case of softening. The model is therefore extended by adding to it terms involving the divergence of plastic strain. For this extended model the desired property of coercivity holds, albeit only on the boundary of the set of admissible functions.     

土的应力-应变关系的一种描述模式

通常将由土的剪切试验测得的应力-应变关系曲线(q～ε1曲线)分为应变硬化型和应变软化型,文中提出了一种能同时描述应变硬化型q～ε1曲线和应变软化型q～ε1曲线的新模式,并导出了统一的切线模量表达式,进一步探讨了该应力-应变关系曲线描述模式在拟合应变硬化型曲线时的简化形式,及简化形式中参数取值方法和参数与围压的关系等方面的问题。通过与邓肯-张模型和应变软化模型的对比,结果表明该应力-应变关系曲线描述模式能更好地与试验数据吻合,且用其简化形式拟合应变硬化型曲线时,可以通过调整其中一个参数值来表达出不同的曲线形式,从而体现出土体应力-应变关系的多样性。该应力-应变关系描述模式为发展更一般的非线性弹性模型提供了一定的理论基础。     

Recovery experimental techniques of tensile impact

Based on loading-unloading test, tensile impact recovery experimental techniques have been developed to obtain the isothermal stress-strain curves of materials under high strain rates. The thermal softening effect can be decoupled by comparing the isothermal stress-strain curves with the adiabatic stress-strain curves at the same strain rate. In the present paper, recovery experiments of brass have been carried out on a self-designed rotating disk tensile impact apparatus. According to the parabolic strain hardening power-law thermo-viscoplastic constitutive model, strain hardening parameter, strain rates strengthening parameter and thermal softening synthetical parameter have been decoupled from experimental results. Furthermore, from these parameters, one can determine the theoretical isothermal curves and adiabatic curves at high strain rates well-coinciding the experimental results respectively. It indicates that the recovery experimental techniques of tensile impact are effective and reliable and are important means for the study of thermo-mechanical coupling. The experimental results also reveals that brass is a typical thermo-viscoplastic material. The project supported by the National Natural Science Foundation of China     

Benchmark experiments and characteristic cyclic plasticity deformation

Key issues in cyclic plasticity modeling are discussed based upon representative experimental observations on several commonly used engineering materials. Cyclic plasticity is characterized by the Bauschinger effect, cyclic hardening/softening, strain range effect, nonproporitonal hardening, and strain ratcheting. Additional hardening is identified to associate with ratcheting rate decay. Proper modeling requires a clear distinction among different types of cyclic plasticity behavior. Cyclic hardening/softening sustains dependent on the loading amplitude and loading history. Strain range effect is common for most engineering metallic materials. Often, nonproportional hardening is accompanied by cyclic hardening, as being observed on stainless steels and pure copper. A clarification of the two types of material behavior can be made through benchmark experiments and modeling technique. Ratcheting rate decay is a common observation on a number of materials and it often follows a power law relationship with the number of loading cycles under the constant amplitude stress controlled condition. Benchmark experiments can be used to explore the different cyclic plasticity properties of the materials. Discussions about proper modeling are based on the typical cyclic plasticity phenomena obtained from testing several engineering materials under various uniaxial and multiaxial cyclic loading conditions. Sufficient experimental evidence points to the unambiguous conclusion that none of the hardening phenomena (cyclic hardening/softening, strain range effect, nonproportional hardening, and strain hardening associated with ratcheting rate decay) is isotropic in nature. None of the hardening behavior can be properly modeled with a change in the yield stress.     

On neck propagation in amorphous glassy polymers under plane strain tension

Unlike metals, necking in polymers under tension does not lead to further localization of deformation, but to propagation of the neck along the specimen. Finite element analysis is used to numerically study necking and neck propagation in amorphous glassy polymers under plane strain tension during large strain plastic flow. The constitutive model used in the analyses features strain-rate, pressure, and temperature dependent yield, softening immediately after yield and subsequent orientational hardening with further plastic deformation. The latter is associated with distortion of the underlying molecular network structure of the material, and is modelled here by adopting a recently proposed network theory developed for rubber elasticity. Previous studies of necking instabilities have almost invariably employed idealized prismatic specimens; here, we explicitly account for the unavoidable grip sections of test specimens. The effects of initial imperfections, strain softening, orientation hardening, strain-rate as well as of specimen geometry and boundary conditions are discussed. The physical mechanisms for necking and neck propagation, in terms of our constitutive model, are discussed on the basis of a detailed parameter study.     

Uniaxial ratcheting and failure behaviors of two steels

The strain cyclic characteristics, ratcheting and failure behaviors of 25CDV4.11 steel and SS304 stainless steel were experimentally studied under uniaxial cyclic tests and at room temperature. The cyclic hardening/softening features of the materials were first observed under uniaxial strain cycling; and then the ratcheting and failure behaviors of the materials were researched in detail under cyclic stressing. The effects of stress amplitude and mean stress on the ratcheting and failure were discussed under uniaxial asymmetrical stress cycling. It is concluded that the ratcheting and failure behaviors of the materials depend greatly on the cyclic softening/hardening features of the materials and the stress values of cyclic loading. Some conclusions useful to understand the fatigue failure of the materials presented under asymmetrical cyclic stressing are obtained.     

INVESTIGATIONS ON THE MECHANICAL CHARACTERISTICS OF NANOBEAMS BASED ON THE NONLOCAL STRAIN GRADIENT THEORY1)

基于非局部应变梯度理论,建立了一种具有尺度效应的高阶剪切变形纳米梁的力学模型. 其中,考虑了应变场和一阶应变梯度场下的非局部效应. 采用哈密顿原理推导了纳米梁的控制方程和边界条件,并给出了简支边界条件下静弯曲、自由振动和线性屈曲问题的纳维级数解. 数值结果表明,非局部效应对梁的刚度产生软化作用,应变梯度效应对纳米梁的刚度产生硬化作用,梁的刚度整体呈现软化还是硬化效应依赖于非局部参数与材料特征尺度的比值. 梁的厚度与材料特征尺度越接近,非局部应变梯度理论与经典弹性理论所预测结果之间的差异越显著.     

Prediction of the initial thickness of shear band localization based on a reduced strain gradient theory

Plastic flow localization in ductile materials subjected to pure shear loading and uniaxial tension is investigated respectively in this paper using a reduced strain gradient theory, which consists of the couple-stress (CS) strain gradient theory proposed by Fleck and Hutchinson (1993) and the strain gradient hardening (softening) law (C–W) proposed by Chen and Wang (2000). Unlike the classical plasticity framework, the initial thickness of the shear band and the strain rate distribution in both cases are predicted analytically using a bifurcation analysis. It shows that the strain rate is obviously non-uniform inside the shear band and reaches a maximum at the center of the shear band. The initial thickness of the shear band depends on not only the material intrinsic length l_cs but also the material constants, such as the yield strength, ultimate tension strength, the linear hardening and softening shear moduli. Specially, in the uniaxial tension case, the most possible tilt angle of shear band localization is consistent qualitatively with the existing experimental observations. The results in this paper should be useful for engineers to predict the details of material failures due to plastic flow localization.     

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.     

脆性固体损伤和局部软化的有限元分析

本文研究混凝土、岩石一类工程中常用的应变软化材料的有限元分析方法。在作者以往有关粘塑性损伤本构模型的工作基础上，给出了一组便于有限元计算的本构方程表达式。包括损伤弹性矩阵和局部损伤软化矩阵，分别运用于计算硬化和软化阶段的有限元刚度矩阵；对所提出的本构方程的实验验证计算和一些算例的有限元数值分析，表明文中给出的本构方程是可行的，相应的有限元算法能较好地对损伤固体的局部软化效应进行数值分析，并可成功地追踪应力应变响应的软化曲线     

A crystal plasticity model for strain-path changes in metals

A model is proposed that deals with the transient mechanical anisotropy during strain-path changes in metals. The basic mechanism is assumed to be latent hardening or softening of the slip systems, dependent on if they are active or passive during deformation, reflecting microstructural mechanisms that depend on the deformation mode rather than on the crystallography. The new model captures the experimentally observed behaviour of cross hardening in agreement with experiments for an AA3103 aluminium alloy. Generic results for strain reversals qualitatively agree with two types of behaviour reported in the literature – with or without a plateau on the stress–strain curve. The influence of the model parameters is studied through detailed calculations of the response of three selected parameter combinations, including the evolution of yield surface sections subsequent to 10% pre-strain. The mathematical complexity is kept to a minimum by avoiding explicit predictions related directly to underpinning microstructural changes. The starting point of the model is a combination of conventional texture and work hardening approaches, where an adapted full-constraints Taylor theory and a simple single-crystal work-hardening model for monotonic strain are used. However, the framework of the model is not restricted to these particular models.     

DAMAGE MODEL FOR HARDENING AND SOFTENING MATERIAL IN GENERAL STATE OF STRESS

The present paper is aimed to simulate progression of damage,hardeningand softening response in brittle materials such as concrete or rock in general state ofstress.Similar shape of surfaces for yield,failure and damage progressing areavailable,and softening strain is treated as plasticity.Then,the proposed model isapplied to solving several boundary value problems.     

岩石单轴压缩作用下变形局部化的梯度塑性解

采用梯度塑性理论研究单轴压缩作用下岩石变形局部化,得到了单轴压缩作用下岩石变形局部化带宽度的一维、二维解析解,为实验测定内部材料长度参数提供了理论依据.     

冲击载荷下盐岩的力学性能和本构关系研究

利用直径75mm的大尺寸SHPB实验装置开展盐岩的动态压缩性能实验研究,得到了其动态应力应变曲线。分析表明,在冲击压缩载荷作用下,盐岩材料有明显的损伤软化现象和应变率效应。针对实验曲线,通过引入描述材料强度随应变率强化的应变率增强因子和随不可逆变形发展而弱化的损伤因子,提出了含损伤率的相关动态本构模型,拟合得到的本构方程形式比较简单,能够较好地反映盐岩动态实验加载过程的主要特征,具有一定的工程应用价值。     

The uniaxial tension of particulate composite materials with nonlinear interface debonding

Debonding of particle/matrix interfaces can significantly affect the macroscopic behavior of composite material. We have used a nonlinear cohesive law for particle/matrix interfaces to study interface debonding and its effect on particulate composite materials subject to uniaxial tension. The dilute solution shows that, at a fixed particle volume fraction, small particles lead to hardening behavior of the composite while large particles yield softening behavior. Interface debonding of large particles is unstable since the interface opening (and sliding) displacement(s) may have a sudden jump as the applied strain increases, which is called the catastrophic debonding. A simple estimate is given for the critical particle radius that separates the hardening and softening behavior of the composite.     

Viscoplasticity theory based on overstress. The prediction of monotonic and cyclic proportional and nonproportional loading paths of an aluminum alloy

An isotropic formulation of the viscoplasticity theory for small strain and based on overstress with a differential growth law for the equilibrium stress is introduced. The four material constants and the two material functions of the theory are determined from uniaxial tensile tests involving strain-rate changes at room temperature and performed on a 6061 T6 Aluminum Alloy. Subsequently the theory is used to predict the biaxial behavior under axialtorsion loading. All tests are under strain control and involve proportional loading and axial followed by torsional straining (and vice versa). Cyclic histories include in-phase and out-of-phase cycling. The predictions of the theory are very reasonable for this cyclically neutral alloy. For cyclic hardening or softening materials a modification of the theory is necessary and is under development.     

Effect of crack-tip shape on the near-tip field in glassy polymer

The physical nature of a crack tip is not absolutely sharp but blunt with finite curvature. In this paper, the effects of crack-tip shape on the stress and deformation fields ahead of blunted cracks in glassy polymers are numerically investigated under Mode I loading and small scale yielding conditions. An elastic–viscoplastic constitutive model accounting for the strain softening upon yield and then the subsequently strain hardening is adopted and two typical glassy polymers, one with strain hardening and the other with strain softening–rehardening are considered in analysis. It is shown that the profile of crack tip has obvious effect on the near-tip plastic field. The size of near-tip plastic zone reduces with the increase of curvature radius of crack tip, while the plastic strain rate and the stresses near crack tip enhance obviously for two typical polymers. Also, the plastic energy dissipation behavior near cracks with different curvatures is discussed for both materials.     

Distortional hardening plasticity model for paperboard

A distortional hardening elasto-plastic model at finite strains suitable for modeling of orthotropic materials is presented. As a prototype material, paperboard is considered. An in-plane model is established. The model developed is motivated from non-proportional loading tests on paperboard where the paperboard is pre-strained in one direction and then loaded in the perpendicular direction. A softening effect is revealed in the pre-strained samples. The observed experimental findings cannot be accurately predicted by current models for paperboard. To be able to model the softening effects, a yield surface based on multiple hardening variables is introduced. It is shown that the model parameters can be obtained from simple uniaxial experiments. The model is implemented in a finite element framework which is used to illustrate the behavior of the model at some specific loading situations and is compared with strain fields obtained from Digital Image Correlation experiments.     

ON NONPROPORTIONAL CYCLIC PLASTIC BEHAVIOR OF STEEL 40

An experimental investigation was carried out on the flow characteristicand hardening of steel 40 subjected to complex combined axial-torsional cyclicstraining. For a specific cyclic strain path, the steel has mainly cyclic softeningbehavior when the strain amplitude is small. While with an increase of the effectivestrain amplitude, the softening becomes small, but there is the cyclic softening eventhough the steel is subjected to the cyclic loading by a square strain path. However, thesteel has cyclic additional hardening by a nonproportional path, compared with theproportional cycling. Generally, the additional hardening is small and its historicaleffect is not obvious at small strain amplitude. The additional hardening is remarkableby a cross-triangular strain path of large strain amplitude. The memory of the historyof nonproportional cyclic loading, the direction of plastic flow and the plastic modulusof the steel were also studied.     

Instabilities in power law gradient hardening materials

Tension and compression instabilities are investigated for specimens with dimensions in the micron range. A finite strain generalization of a higher order strain gradient plasticity theory is implemented in a finite element scheme capable of modeling power law hardening materials. Effects of gradient hardening are found to delay the onset of localization under plane strain tension, and significantly reduce strain gradients in the localized zone. For plane strain compression gradient hardening is found to increase the load-carrying capacity significantly.