A hardening model for anisotropic materials

Results from proportionate-loading experiments under plane-stress states illustrate the existence of a field of uniform-hardening potentials for the yield and creep deformation behavior of isotropic, slightly anisotropic, aelotropic and orthotropic polycrystalline materials in the initially strain-free condition. For two different plane-stress states, it is shown that a linear functional relationship holds between the plastic-strain increment ratio and the stress ratio in these materials and that, consequently, the field is adequately modeled by a uniform-hardening anisotropic function that is quadratic in the components of deviatoric stress. Anisotropic plane-stress yield functions are formulated for any stage in the deformation process by combining the uniform-hardening function with the kinematic-hardening rule. The resulting functions, which correspond to rigid translations of initial yield loci according to Ziegler's rule, provide good agreement with experimental observations on a marked Bauschinger effect and an absence of cross hardening.     

A generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials

In this paper, a generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials is derived. The evolution equation for the active yield surface with reference to the memory yield surface is obtained by considering the continuous expansion of the active yield surface during the unloading/reloading process. The incremental constitutive relation based on the associated flow rule is then derived for a general yield function for pressure insensitive and sensitive materials. Detailed incremental constitutive relations for materials based on the Mises yield function, the Hill quadratic anisotropic yield function and the Drucker–Prager yield function are derived as the special cases. The closed-form solutions for one-dimensional stress–plastic strain curves are also derived and plotted for materials under cyclic loading conditions based on the three yield functions. In addition, the closed-form solutions for one-dimensional stress–plastic strain curves for materials based on the isotropic Cazacu–Barlat yield function under cyclic loading conditions are summarized and presented. For materials based on the Mises and the Hill anisotropic yield functions, the stress–plastic strain curves show closed hysteresis loops under uniaxial cyclic loading conditions and the Masing hypothesis is applicable. For materials based on the Drucker–Prager and Cazacu–Barlat yield functions, the stress–plastic strain curves do not close and show the ratcheting effect under uniaxial cyclic loading conditions. The ratcheting effect is due to different strain ranges for a given stress range for the unloading and reloading processes. With these closed-form solutions, the important effects of the yield surface geometry on the cyclic plastic behavior due to the pressure-sensitive yielding or the unsymmetric behavior in tension and compression can be shown unambiguously. The closed form solutions for the Drucker–Prager and Cazacu–Barlat yield functions with the associated flow rule also suggest that a more general anisotropic hardening theory needs to be developed to address the ratcheting effects for a given stress range.     

Modified kinematic hardening rule for multiaxial ratcheting prediction

A modified kinematic hardening rule is proposed in which one biaxial loading dependent parameter δ′ connecting the radial evanescence term [(α:n)ndp] in the Burlet–Cailletaud model with the dynamic recovery term of Ohno–Wang kinematic hardening rule is introduced into the framework of the Ohno–Wang model. Compared with multiaxial ratcheting experimental data obtained on 1Cr18Ni9Ti stainless steel in the paper and CS1026 steel conducted by Hassan et al. [Int. J. Plasticity 8 (1992) 117], simulation results by modified model are quite well in all loading paths. The simulations of initial nonlinear part in ratcheting curves can be improved greatly while the evolutional parameter δ′ related to plastic strain accumulation is added into the modified model.     

Nonlocal gradient-dependent modeling of plasticity with anisotropic hardening

This work addresses the formulation of the thermodynamics of nonlocal plasticity using the gradient theory. The formulation is based on the nonlocality energy residual introduced by Eringen and Edelen (1972). Gradients are introduced for those variables associated with isotropic and kinematic hardening. The formulation applies to small strain gradient plasticity and makes use of the evanescent memory model for kinematic hardening. This is accomplished using the kinematic flux evolution as developed by Zbib and Aifantis (1988). Therefore, the present theory is a four nonlocal parameter-based theory that accounts for the influence of large variations in the plastic strain, accumulated plastic strain, accumulated plastic strain gradients, and the micromechanical evolution of the kinematic flux. Using the principle of virtual power and the laws of thermodynamics, thermodynamically-consistent equations are derived for the nonlocal plasticity yield criterion and associated flow rule. The presence of higher-order gradients in the plastic strain is shown to enhance a corresponding history variable which arises from the accumulation of the plastic strain gradients. Furthermore, anisotropy is introduced by plastic strain gradients in the form of kinematic hardening. Plastic strain gradients can be attributed to the net Burgers vector, while gradients in the accumulation of plastic strain are responsible for the introduction of isotropic hardening. The equilibrium between internal Cauchy stress and the microstresses conjugate to the higher-order gradients frames the yield criterion, which is obtained from the principle of virtual power. Microscopic boundary conditions, associated with plastic flow, are introduced to supplement the macroscopic boundary conditions of classical plasticity. The nonlocal formulation developed here preserves the classical assumption of local plasticity, wherein plastic flow direction is governed by the deviatoric Cauchy stress. The theory is applied to the problems of thin films on both soft and hard substrates. Numerical solutions are presented for bi-axial tension and simple shear loading of thin films on substrates.     

Plastic analysis of thin plates with anisotropic hardening

In this paper we discuss the adoption of the anisotropic hardening model due to the existence of Bauschinger effect when thin plate is applied by repeated loading. The loading condition of thin plates for linear kinematic hardening has been deduced in terms of generalized forces and generalized plastic deformation. And it can be extended to nonlinear kinematic hardening and mixed hardening. Finally as an example the numerical results are obtained for a circular plate.     

On the calibration of the Chaboche hardening model and a modified hardening rule for uniaxial ratcheting prediction

A systematic mathematical approach is developed in the context of uniaxial cyclic ratcheting for the parameter determination of the decomposed Chaboche hardening rule. This is achieved by deriving the relation between the evolution of the backstress and the plastic strain accumulation. Unlike current calibration techniques where a trial–error approach is employed to fit the simulation results to experimental data, the proposed method determines the parameters directly from uniaxial ratcheting experiments. Numerical results indicate that Chaboche’s hardening model is much more efficient than what has been demonstrated before. Finally, as an improvement to the decomposed model, a modification is made to one of the backstress components. This improved component enables the model to predict uniaxial ratcheting with more accuracy.     

Law of nonlinear flow in saturated clays and radial consolidation

It was derived that micro-scale amount level of average pore radius of clay changed from 0.01 to 0.1 micron by an equivalent concept of flow in porous media.There is good agreement between the derived results and test ones.Results of experiments show that flow in micro-scale pore of saturated clays follows law of nonlinear flow.Theoretical analyses demonstrate that an interaction of solid-liquid interfaces varies inversely with permeability or porous radius.The interaction is an important reason why nonlinear flow in saturated clays occurs.An exact mathematical model was presented for nonlinear flow in micro-scaie pore of saturated clays.Dimension and physical meanings of parameters of it are definite.A new law of nonlinear flow in saturated clays was established.It can describe characteristics of flow curve of the whole process of the nonlinear flow from low hydraulic gradient to high one.Darcy law is a special case of the new law.A math- ematical model was presented for consolidation of nonlinear flow in radius direction in saturated clays with constant rate based on the new law of nonlinear flow.Equations of average mass conservation and moving boundary,and formula of excess pore pressure distribution and average degree of consolidation for nonlinear flow in saturated clay were derived by using an idea of viscous boundary layer,a method of steady state in stead of transient state and a method of integral of an equation.Laws of excess pore pressure distribution and changes of average degree of consolidation with time were obtained.Re- suits show that velocity of moving boundary decreases because of the nonlinear flow in saturated clay.The results can provide geology engineering and geotechnical engineering of saturated clay with new scientific bases.Calculations of average degree of consolidation of the Darcy flow are a special case of that of the nonlinear flow.     

Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheets

Magnesium alloy sheets have been extending their field of applications to automotive and electronic industries taking advantage of their excellent light weight property. In addition to well-known lower formability, magnesium alloys have unique mechanical properties which have not been thoroughly studied: high in-plane anisotropy/asymmetry of yield stress and hardening response. The reason of the unusual mechanical behavior of magnesium alloys has been understood by the limited symmetry crystal structure of HCP metals and thus by deformation twinning. In this paper, the phenomenological continuum plasticity models considering the unusual plastic behavior of magnesium alloy sheet were developed for a finite element analysis. A hardening law based on two-surface model was further extended to consider the general stress–strain response of metal sheets including Bauschinger effect, transient behavior and the unusual asymmetry. Three deformation modes observed during the continuous in-plane tension/compression tests were mathematically formulated with simplified relations between the state of deformation and their histories. In terms of the anisotropy and asymmetry of the initial yield stress, the Drucker–Prager’s pressure dependent yield surface was modified to include the anisotropy of magnesium alloy. The numerical formulations and characterization procedures were also presented and finally the correlation of simulation with measurements was performed to validate the proposed theory.     

A framework for multiplicative elastoplasticity with kinematic hardening coupled to anisotropic damage

The objective of this contribution is the formulation and algorithmic treatment of a phenomenological framework to capture anisotropic geometrically nonlinear inelasticity. We consider in particular the coupling of viscoplasticity with anisotropic continuum damage whereby both, proportional and kinematic hardening are taken into account. As a main advantage of the proposed formulation standard continuum damage models with respect to a fictitious isotropic configuration can be adopted and conveniently extended to anisotropic continuum damage. The key assumption is based on the introduction of a damage tangent map that acts as an affine pre-deformation. Conceptually speaking, we deal with an Euclidian space with respect to a non-constant metric. The evolution of this field is directly related to the degradation of the material and allows the modeling of specific classes of elastic anisotropy. In analogy to the damage mapping we introduce an internal variable that determines a back-stress tensor via a hyperelastic format and therefore enables the incorporation of plastic anisotropy. Several numerical examples underline the applicability of the proposed finite strain framework.     

Material instabilities of anisotropic saturated multiphase porous media

This paper analyses the material instability of fully saturated multiphase porous media. On account of the fact that anisotropic mechanical behaviours are widely observed in saturated and partially saturated geomaterials, the anisotropic constitutive model developed by Rudnicki for geomaterials is used to model the anisotropic mechanical behaviour of the solid skeleton of saturated porous geomaterials in axisymmetric compression test. The inertial coupling effect between solid skeleton and pore fluid is also taken into account in dynamic cases. Conditions for static instability (strain localisation) and dynamic instability (stationary discontinuity and flutter instability) of fully saturated porous media are derived. The critical modulus, shear band angle for strain localisation, and the bound within which flutter instability may occur are given in explicit forms. The effects of material parameters on material instability are investigated in detail by numerical computations.     

叠加型A-F类随动强化模型塑性应变的数值计算法

以Chaboche随动强化模型为例,在M isses屈服准则及正交流动准则的前提下,推导了叠加型A rm-strong-F rederick(A-F)类随动强化模型塑性应变的数值计算法,联合利用四阶龙格-库塔法与径向返回法实现数值计算中的内部平衡迭代。同时推导了统一切向矩阵以便确定每一平衡迭代后的试算应变。利用AN SY S提供的U PF s将算法嵌入到AN SY S有限元程序,实现了叠加型A-F类随动强化模型塑性应变的数值计算,并利用四边形单元模拟了单轴循环加载时的棘轮应变,计算结果能够很好地与实验值吻合。     

A large strain anisotropic elastoplastic continuum theory for nonlinear kinematic hardening and texture evolution

In this paper we present a continuum theory for large strain anisotropic elastoplasticity based on a decomposition of the modified plastic velocity gradient into energetic and dissipative parts. The theory includes the Armstrong and Frederick hardening rule as well as multilayer models as special cases even for large strain anisotropic elastoplasticity. Texture evolution may also be modelled by the formulation, which allows for a meaningful interpretation of the terms of the dissipation equation.     

THE ELASTO-PLASTIC STRAIN ANALYSIS OF NOTCHED PLATE UNDER CYCLIC LOADING—AN APPLICATION OF PLASTIC ANISOTROPIC HARDENING THEORY

In order to predict the life of engineering structures, it is necessary to investigate the strain distribution in notched members. In general, the Bauschinger Effect of materials under cyclic loading is not negligible, and so the anisotropic hardening model has been suggested. From the comparison between the calculated and experimental results in this paper, we can see that even the linear kinematic hardening model is quite suitable for strain analysis under cyclic loading.     

316L不锈钢随动强化模型改进研究

本文对316L不锈钢进行了单轴与多轴非比例路径下的应力控制棘轮试验,考察了应力幅值、平均应力和加载历程对棘轮特性的影响。同时进行了应变控制循环试验以研究材料的应力松弛特性。试验结果表明轴向棘轮效应在对称剪切荷载下效果明显,同时棘轮应变随应力幅值和平均应力的增加而增加。研究了Chen-Jiao随动强化模型与Jiang-Sehitoglu随动强化模型采用的单轴与多轴参数对背应力分量增量方向的影响,将Chen-Jiao模型中的多轴系数替换为界面饱和率,并在此基础上引入新的参数对塑性模量系数进行修正,计算结果表明修正后的模型能提升应力控制下多轴棘轮的预测精度,并能很好的预测应力松弛现象,表明了新模型的正确性与有效性。     

Analysis of stress-strain relations by use of an anisotropic hardening plastic potential

A method of analyzing plastic behavior by use of an anisotropic hardening plastic potential is proposed. The plastic potential surface in deviatoric stress space is assumed to be the same as the equi-plastic-strain surface. Stress-strain relations in combined loading and in multi-axial cyclic loading are calculated by use of the anisotropic hardening plastic potential and the normality rule of the plastic strain increment vector to the plastic potential surface, which are experimentally determined or confirmed by subjecting thinwalled tubular test specimens of $\text{60}\text{40}$ brass to combined axial load, internal pressure and torsion. The calculated results agree fairly well with the experimental observations.     

A novel approach for anisotropic hardening modeling. Part II: Anisotropic hardening in proportional and non-proportional loadings,application to initially isotropic material

The modeling of anisotropic hardening, in particular for non-proportional loading paths, is a challenging task for advanced macroscopic models. The complex distortion of the yield locus is related to the activation and cross-hardening of different slip systems, depending on crystallographic orientations. These physical mechanisms can be taken into account in polycrystalline models but the computation times are enormous. The novel approach detailed in Part I (Rousselier et al., 2009) consists in: (i) drastically reducing the number of crystallographic orientations to save the computation cost, (ii) applying a parameter calibration procedure to obtain a good agreement with the experimental database. This methodology is first applied here to the anisotropic hardening in the proportional loadings of the strongly anisotropic aluminum alloy of Part I. Very good modeling is achieved with only eight crystallographic orientations. Different levels of additional hardening in biaxial proportional loading as compared to uniaxial loading can be modeled with the same polycrystalline model. For this, only the parameter calibration has to be performed with different databases. The same methodology has also been applied for the modeling of isotropic behavior. The best compromise between model accuracy and numerical cost is obtained with fourteen orientations. The deviations from isotropy are acceptable in all loading directions. Different levels of hardening in orthogonal loading: simple shear followed by simple tension, are achieved without any modification of the model equations. Only the parameter calibration has to be performed with different hardening levels in the database. FE calculations of a deep drawing test have been performed. The CPU time of the polycrystalline model is only five times larger than that with the simple von Mises model. The CPU time with texture evolution is further increased by a factor of two. The effects of texture evolution in rolling of the initially isotropic fcc material have been investigated. The resulting texture and hardening are qualitatively good.