A three-dimensional model of anisotropic hardening in metals and its application to the analysis of sheet metal formability

The hardening model proposed by Z. Mróz based on the uniaxial fatigue behavior of many metals is adopted to derive an incremental constitutive equation for general three-dimensional problems. This constitutive law is then employed in the analysis of metal forming problems to assess the influence of loading cycles, of the types involved in standard forming processes, on the ultimate formability of sheet metals. The predicted forming limit curves differ quantitatively from results obtained via an isotropie hardening model and differ qualitatively from those obtained via a kinematic model. Also investigated are the effects of such loading cycles on material response to simple tensile loading, which is often used to characterize a material. Significant differences between the present model and the other two models considered are observed in such characterizers of simple tensile behavior as the stress-strain curve, the anisotropy parameter and the uniform elongation. These differences suggest a rather simple experiment to identify the proper material model to be used in analyses of problems which involve loading cycles. Comparisons with some experimental results reveal that the employment of an anisotropic hardening model, such as the generalized Mróz model derived herein, is indeed crucial in accurately predicting material response to complicated loading histories.     

On the consistency of nested surfaces models and their kinematic hardening rules

In this paper we briefly address the consistency of formulations for nested surfaces plasticity and their kinematic hardening translation rules. Some requirements for these formulations are then given. It is shown and discussed that multilayer plasticity based on nested yield surfaces present some inconveniences when modelling multiaxial cyclic loading. On the other hand, the use of hardening surfaces, instead of yield surfaces solves the problem partially. It is also shown that multilayer plasticity based on the Mróz kinematic rule yields clearly different multiaxial predictions when using the same uniaxial bilinear curve and different number of surfaces, which is a priori inconsistent since the same monotonic stress–strain curve should not result into a clearly different multiaxial behavior simply because of the discretization employed by the user. It is demonstrated that, in contrast with the Mróz translation rule, multilayer plasticity using the Prager translation rule results in the same prediction regardless of the number of surfaces in which the hardening part of the bilinear curve is discretized. This rule is also compliant with the principle of maximum dissipation. It also allows for a vanishing elastic region without falling into theoretical or numerical difficulties. Hence, it should not be discarded when developing advanced models.     

A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite

A 3-D constitutive model for polycrystalline shape memory alloys (SMAs), based on a modified phase transformation diagram, is presented. The model takes into account both direct conversion of austenite into detwinned martensite as well as the detwinning of self-accommodated martensite. This model is suitable for performing numerical simulations on SMA materials undergoing complex thermomechanical loading paths in stress–temperature space. The model is based on thermodynamic potentials and utilizes three internal variables to predict the phase transformation and detwinning of martensite in polycrystalline SMAs. Complementing the theoretical developments, experimental data are presented showing that the phase transformation temperatures for the self-accommodated martensite to austenite and detwinned martensite to austenite transformations are different. Determination of some of the SMA material parameters from such experimental data is also discussed. The paper concludes with several numerical examples of boundary value problems with complex thermomechanical loading paths which demonstrate the capabilities of the model.     

Numerical Modeling of Hydraulic Hysteresis in Unsaturated Soils

This article presents the implementation of the constitutive model of Wheeler (Geotechnique 53(1):41–54, 2003) for coupling of hydraulic hysteresis and mechanical behavior of unsaturated soils in a fully coupled transient hydro-mechanical finite element (FE) model (computer code UNSATEX) developed by the authors. The constitutive model considers the effects of irreversible changes of degree of saturation on stress–strain behavior and the influence of plastic volumetric strains on the water retention behavior. The mathematical framework and the numerical implementation of the constitutive model are presented and discussed. The FE model is verified and validated against analytical predictions [obtained using the model of Wheeler (Geotechnique 53(1):41–54, 2003] as well as experimental results from the literature involving unsaturated soils undergoing various combinations of drying, wetting, loading, unloading, and reloading paths. Comparison of the results shows that the developed FE model can be used to predict various aspects of the behavior of unsaturated soils under drying and wetting as well as loading and unloading paths. The merits and limitations of the FE model are highlighted.     

Behaviors of three BCC metal over a wide range of strain rates and temperatures: experiments and modeling

The elastic–plastic behaviors of three body-centered cubic metals, tantalum, tantalum alloy with 2.5% tungsten, and AerMet 100 steel, are presented over a wide range of strains (15%), strain rates (10⁻⁶–10⁴ s⁻¹) and temperatures (77–600°F). Johnson-Cook and Zerilli-Armstrong models were found inadequate to describe the observations. A new viscoplastic model is proposed based on these experimental results. The proposed constitutive model gives good correlations with these experimental results and strain-rate jump experiments. In the next paper (Liang, R., Khan A.S., 2000. Behaviors of three BCC metals during non-proportional multi-axial loadings and predictions using a recently proposed model. International Journal of Plasticity, in press), multi-axial loading results on these materials and comparison with the proposed model will be presented.     

Damage model for metal–matrix composite under high intensity loading

A damage model for a composite structure under high intensity dynamic loading is presented. The model is based on a thermodynamic micromechanic approach, which is formulated using the conservation laws and the energy balance equations (the first and second laws of thermodynamics). A homogenization or averaging technique is implemented in the development to simplify the representation of the non-homogeneous material. The metal–matrix composite's inelastic response is modeled using elastic–plastic constitutive relations considering finite plastic strain and damage effect. The damage model is validated with experimental data available in the literatures, and it shows fairly good agreement. A parametric study demonstrating the characteristics of the damage model is also presented.     

A computational micromechanics constitutive model for the unloading behavior of paper

In this study, a computational micromechanics material model for the unloading behavior of paper and other nonwoven materials is presented. The asymptotic fiber and bond (AFB) model for paper elastic–plastic behavior [Sinha, S.K., Perkins, R.W., 1995. Micromechanics constitutive model for use in finite element analysis, In: Proceedings of the 1995, Joint ASME Applied Mechanics and Materials Summer Meeting, Los Angeles, CA, USA, Jun 28–30, 1995] has been extended to model the unloading process through a computational algorithm and implemented using the UMAT subroutine in ABAQUS finite element code. For every unloading increment, the material model assumes elastic unloading with a slope equal to the initial elastic modulus. The Jacobian matrix of the constitutive model is updated at every unloading increment by applying the incremental form of AFB model for a planar element with an elastic fiber and bond condition. A uniaxial tensile and a biaxial Mullen burst loading–unloading experiments were carried out for a paperboard sample and simulated using the model. The stress–strain curve and residual strain for the uniaxial loading were in good agreement with experimental results. The finite element model of the burst test with the AFB unloading material model predicted the general shape of the pressure versus deflection curve. However, the model over predicted the residual deflection by more than 50%. The loading portion of the pressure–deflection curve had a significant offset from experimental curves, and the nonlinearity in the unloading curve towards the end was not predicted. The discrepancies with experimental results are attributed to the burst test itself, model parameter estimation inadequacies, boundary conditions used in the FEA, and neglecting time-dependant effects. Nevertheless, the model can be useful in parametric studies relating microstructure to unloading behavior in structural problems.     

Orthotropic elastic–plastic material model for paper materials

Paper and paperboard generally exhibit anisotropic and non-linear mechanical material behaviour. In this work, the development of an orthotropic elastic–plastic constitutive model, suitable for modelling of the material behaviour of paper is presented. The anisotropic material behaviour is introduced into the model by orthotropic elasticity and an isotropic plasticity equivalent transformation tensor. A parabolic stress–strain relation is adopted to describe the hardening of the material. The experimental and numerical procedures for evaluation of the required material parameters for the model are described. Uniaxial tensile testing in three different inplane material directions provides the calibration of the material parameters under plane stress conditions. The numerical implementation of the material model is presented and the model is shown to perform well in agreement with experimentally observed mechanical behaviour of paper.     

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.     

QUASI－FLOW CORNER THEORY ON LARGE PLASTIC DEFORMATION OF DUCTILE METALS AND ITS APPLICATIONS

ＱＵＡＳＩ－ＦＬＯＷＣＯＲＮＥＲＴＨＥＯＲＹＯＮＬＡＲＧＥＰＬＡＳＴＩＣＤＥＦＯＲＭＡＴＩＯＮＯＦＤＵＣＴＩＬＥＭＥＴＡＬＳＡＮＤＩＴＳＡＰＰＬＩＣＡＴＩＯＮＳＨｕＰｉｎｇ（胡平）ＬｉｕＹｕｑｉ（柳玉启）ＧｕｏＷｅｉ（郭威）ＴａｉＦｅｎｇ（台风）（Ｒ...     

转动惯量法在多轴疲劳本构关系预测中的应用

一些金属材料在承担多轴非比例加载过程时,会产生额外非比例附加强化或软化现象,这一现象往往会导致在评估疲劳寿命时因为材料本构关系的不确定而引起预测结果出现较大误差．因此基于单轴疲劳理论得出的寿命预测模型并不能准确地预测多轴非比例疲劳加载下的材料寿命．针对此问题,本文阐述了非比例附加强化效应产生的原因及结果,结合转动惯量法的理论和塑性增量法,建立了预测多轴低周疲劳加载下循环应力-应变曲线的数值计算模型．利用316L 不锈钢试样在5 种加载路径下的实验数据对预测结果进行了验证,结果表明该模型具有良好的预测有效性及精度．     

Elastic-plastic finite element polycrystal model

A finite element polycrystal model is formulated with the initial strain method where the stiffness matrix in FEM is based on the elastic modulus. For the determination of time-independent slips, a new numerical scheme, “successive integration method,” is proposed, which uses only Schmid's law. The numerical result for a simple tension of nonhardening FCC metal is compared with other theories. Crystal lattice rotations are presented for some examples of loading. A numerical experiment is done to show the evolution of anisotropy due to plastic deformation. The numerical code of the present model is quite simple and can be applied to arbitrary loading paths.     

Finite viscoplasticity of amorphous glassy polymers in the logarithmic strain space

The paper outlines a new constitutive model and experimental results of rate-dependent finite elastic–plastic behavior of amorphous glassy polymers. In contrast to existing kinematical approaches to finite viscoplasticity of glassy polymers, the formulation proposed is constructed in the logarithmic strain space and related to a six-dimensional plastic metric. Therefore, it a priori avoids difficulties concerning with the uniqueness of a plastic rotation. The constitutive framework consists of three major steps: (i) A geometric pre-processing defines a total and a plastic logarithmic strain measures determined from the current and plastic metrics, respectively. (ii) The constitutive model describes the stresses and the consistent moduli work-conjugate to the logarithmic strain measures in an analogous structure to the geometrically linear theory. (iii) A geometric post-processing maps the stresses and the algorithmic tangent moduli computed in the logarithmic strain space to their nominal, material or spatial counterparts in the finite deformation space. The analogy between the formulation of finite plasticity in the logarithmic strain space and the geometrically linear theory of plasticity makes this framework very attractive, in particular regarding the algorithmic implementation. The flow rule for viscoplastic strains in the logarithmic strain space is adopted from the celebrated double-kink theory. The post-yield kinematic hardening is modeled by different network models. Here, we compare the response of the eight chain model with the newly proposed non-affine micro-sphere model. Apart from the constitutive model, experimental results obtained from both the homogeneous compression and inhomogeneous tension tests on polycarbonate are presented. Besides the load–displacement data acquired from inhomogeneous experiments, quantitative three-dimensional optical measurements of the surface strain fields are carried out. With regard to these experimental data, the excellent predictive quality of the theory proposed is demonstrated by means of representative numerical simulations.     

Overall softening and anisotropy related with the formation and evolution of dislocation cell structures

In this work, a model, based on a representation of the dislocation cell microstructures by a non-local two-phase material with evolving microstructures, is proposed for the elastic–plastic behavior of metals under monotonic and sequential loading. The first phase represents the cell interior and the second one, the cell walls. The evolution of the microstructure is taken into account considering the cell-wall interfaces as free boundaries. Finally, the accumulation within walls of dislocations crossing the cells defines a non-local hardening process. Assuming a piecewise uniform plastic strain field and assuming ellipsoidal cells, the free energy of the system is calculated. The driving and critical forces associated with the plastic flow of the two-phases and the morphology of the cells are established. In a third part, numerical results are presented for monotonic and sequential loading. The results show an overall softening related to the destabilization of the dislocation microstructures which occurs in sequential as well as monotonic paths.     

Analytical and numerical analysis of frictional damage in quasi brittle materials

Frictional sliding and crack growth are two main dissipation processes in quasi brittle materials. The frictional sliding along closed cracks is the origin of macroscopic plastic deformation while the crack growth induces a material damage. The main difficulty of modeling is to consider the inherent coupling between these two processes. Various models and associated numerical algorithms have been proposed. But there are so far no analytical solutions even for simple loading paths for the validation of such algorithms. In this paper, we first present a micro-mechanical model taking into account the damage-friction coupling for a large class of quasi brittle materials. The model is formulated by combining a linear homogenization procedure with the Mori–Tanaka scheme and the irreversible thermodynamics framework. As an original contribution, a series of analytical solutions of stress–strain relations are developed for various loading paths. Based on the micro-mechanical model, two numerical integration algorithms are exploited. The first one involves a coupled friction/damage correction scheme, which is consistent with the coupling nature of the constitutive model. The second one contains a friction/damage decoupling scheme with two consecutive steps: the friction correction followed by the damage correction. With the analytical solutions as reference results, the two algorithms are assessed through a series of numerical tests. It is found that the decoupling correction scheme is efficient to guarantee a systematic numerical convergence.     

考虑路径相关性的非比例循环塑性本构模型

根据非比例加载下金属材料响应的延迟特性及加载路径相关性,选取沿应力迹法向的塑性应变的累积量作为非比例加载影响的度量,相应给出反映非比例附加强化的变量,并假设其模量和强化率与加载路径的几何参数相关．为反映由于非比例加载而引起的材料强化的异向效应,在Valanis的塑性内时响应方程中引入与加载路径几何性质有关的应力项,构成非比例循环塑性本构关系．对316和304不锈钢材料在一些典型非比例循环加载路径下的应力响应进行了理论预测,与Benallal等及McDowell的实验结果取得了良好的一致．     

非比例循环塑性内时本构关系研究

在Valanis的内时本构理论的框架中,引入内结构张量以反映由于非比例加载而引起金属材料的附加等向强化及异向强化效应,同时提出材料强化程度的度量采用沿路径法线方向的塑性应变分量来描述.这些考虑的有效性已经通过用所建模型对304不锈钢材料在一些典型非比例循环加载路径下的响应进行的理论预测得到了验证;将该模型应用于U71Mn材料室温单轴棘轮行为描述中,结果显示内结构张量的引入不仅能较好地反映应变控制下的非比例附加效应,而且也能较好地反映应力控制下塑性应变的累积及变化率.     

An elastic–plastic constitutive law for the description of uniaxial and multiaxial ratchetting

An elastic–plastic constitutive model is proposed to describe 1-D and 2-D ratchetting. The model is also able to give correct results for 2-D ratchetting when only uniaxial identification is used, while no special threshold or parameter is used for the case of non-proportional loading. The original feature of this model consist in the introduction of a ratchetting stress (material characteristic) along with the maximal stress supported in the history of loading and the plastic strain at the last unloading. In this paper uniaxial and 3-D formulations have been described based on a numerical implementation in the software Code_Aster. Uniaxial and also multiaxial identifications have been used. Simulations have been realized for proportional and non-proportional homogeneous cases, as well as for structures under anisothermal thermomechanical loading. The results of a benchmark on a structure, comparing experiment, simulations by this model and some other phenomenological models, and a polycrystalline model are presented. An analysis of error margin due to the choice of Mises criterion is exposed.