A generally anisotropic hardening model for big offset-strain yield stresses

Summary In this paper, generally anisotropic quadric models for big offset-strain plastic yield and hardening of rolled sheet-steels are proposed. They are formulated in intrinsic forms such that the effects of isotropic and anisotropic yield coefficients to the hardening factors are naturally decomposed. Such decompositions are supported by the experimental observations and allow further simplifications. The simplified models are capable of describing changeable general anisotropy of hardening, and give predictions in excellent agreement with the experimental data. Two restricted models as special cases of the general model are also examined.     

基于Yoshida-Uemori材料模型的汽车结构件冲压回弹分析

Yoshida-Uemori随动硬化材料模型能够准确描述应变路径发生变化时材料性能的改变,从而较好地反映复杂加载情况下材料的各向异性.本文基于JSTAMP件分别采用Yoshida-Uemori随动硬化材料模型和各向同性硬化材料模型对汽车高强钢结构件的冲压成形进行了仿真分析与回弹预测,研究了不同材料硬化模型对回弹预测精度...     

Parameter identification of the homogeneous anisotropic hardening model using the virtual fields method

In this paper, the virtual fields method is applied to identify the constitutive parameters of the homogeneous yield function-based anisotropic hardening model. The procedure, previously developed for isotropic hardening law, is extended to the anisotropic case in this study. For validation purpose, a finite element model is developed to check the feasibility of the proposed methodology in retrieving the target constitutive parameters. The identification results are critically discussed. The suitability of this finite element model is also assessed with a view to future experiments.     

An analytical–numerical framework for the study of ageing in fibre reinforced polymer composites

In the last few years there has been available a growing amount of information on composites ageing that considers effects such as temperature, oxidation, UV radiation, permanent loading, etc. In this paper, an analytical–numerical framework adequate for the compilation, interpretation and application of experimental data to actual engineering analysis and design is discussed. The formulation proposed includes elastic anisotropic relations; ageing viscoelastic anisotropic constitutive equations in terms of state variables; age-adjusted failure and degradation criteria; all in a setting of large displacements with small strains. The essential differences between hardening and softening ageing processes are described together with the constitutive relations adequate for each case. These equations are written in a form suitable for numerical analysis in a finite elements context, using an incremental-iterative solution procedure that accounts for post-critical effects. Several examples, including elastic, viscoelastic and failure behaviour in a large displacement context are included to check the algorithms.     

A finite element approach to anisotropic damage of ductile materials in large deformations Part I: an anisotropic ductile elastic-plastic-damage model

During past decades, many material models using the continuum damage mechanics (CDM) approach have been proposed successfully in the small deformation regime to describe inelastic behaviors and fracturing phenomena of a material. For ductile materials, large deformation takes place at the level of damage appearance. Damage is anisotropic in nature. In this paper, the ductile damage at finite deformations is modeled as an anisotropic tensor quantity. Then, a fourth-order symmetric stress correction tensor is proposed for computationally efficient and easy implementation in the finite element formulations. Consequently, an explicit form of the fourth-order constitutive equations of the proposed elastic-plastic-damage model is derived. Both isotropic and kinematic hardening effects are included in the formulation. The new constitutive model can predict not only the elastic-plastic behaviors, but also the sequential variations of ductile materials. An evaluation of the constitutive and damage evolution equations is presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

Effect of anisotropy,kinematic hardening,and strain-rate sensitivity on the predicted axial crush response of hydroformed aluminium alloy tubes

There exists considerable motivation to reduce vehicle weight through the adoption of lightweight materials while maintaining energy absorption and component integrity under crash conditions. Aluminium and magnesium alloys, advanced high-strength steels, and composites are all proposed candidates for replacing mild steel in automotive structures. It was of particular interest to study the crash behaviour of lightweight tubular hydroformed structures. Thus, the current research has studied the dynamic crush response of hydroformed Al–Mg–Mn aluminium alloy tubes using both experimental and numerical methods. The research focused on axial crush structures that are designed to absorb crash energy by progressive axial folding. The main experimental parameter that was varied during the hydroforming process was the corner-fill radius of the tube. Numerical studies were carried out using explicit dynamic finite element models incorporating advanced constitutive material models to capture the measured forming and crash history. A constitutive model was implemented in the finite element models combining the Johnson–Cook strain-rate sensitivity model, a non-linear isotropic-kinematic hardening model, and the Yld2000-2d anisotropic model. Each effect was isolated, and it was shown that strain-rate sensitivity slightly increased the energy absorption capabilities while kinematic hardening and anisotropy effects decreased the energy absorption capabilities during axial crush. When including all three effects, the predicted energy absorption was less than the response predicted from simulations performed using the von Mises yield criterion and in reasonable agreement with measured data. It is recommended that a combined constitutive model be utilized for the study of materials that show sensitivity to the Bauschinger effect, strain-rate effects, and anisotropy.     

Assessment of anisotropic hardening models for conventional deep drawing processes

Assessing the predictive capabilities of recent advanced constitutive modelling approaches for processes with industrial complexity is a challenging task. Real process conditions such as blankholder pressure distribution, friction and tool elasticity sensitively affect experimental observations, making the isolation of constitutive effects difficult. A systematic approach is proposed in this work to assess the performance of anisotropic hardening models with the least possible disturbance from process conditions. Two deep drawing examples were used for these purpose (“cross die” and “lackfrosch”) in conjunction with a mild steel (DC05). Optically measured strain distributions have been compared to corresponding simulations, which have been calibrated to accurately match the measured blank draw-in. The effect of initial yield locus shape as well as anisotropic hardening effects have been discussed.     

Simulations of stress distributions in crankshaft sections under fillet rolling and bending fatigue tests

The residual stresses due to fillet rolling and the bending stresses near the fillets of crankshaft sections under bending fatigue tests are important driving forces to determine the bending fatigue limits of crankshafts. In this paper, the residual stresses and the bending stresses near the fillet of a crankshaft section under fillet rolling and subsequent bending fatigue tests are investigated by a two-dimensional plane strain finite element analysis based on the anisotropic hardening rule of Choi and Pan [Choi KS, Pan J. A generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials (in preparation)]. The evolution equation for the active yield surface during the unloading/reloading process is first presented based on the anisotropic hardening rule of Choi and Pan (in preparation) and the Mises yield function. The tangent modulus procedure of Peirce et al. [Peirce D, Shih CF, Needleman A. A tangent modulus method for rate dependent solids. Comput Struct 1984;18:875–87] for rate-sensitive materials is adopted to derive the constitutive relation. A user material subroutine based on the anisotropic hardening rule and the constitutive relation was written and implemented into ABAQUS. Computations were first conducted for a simple plane strain finite element model under uniaxial monotonic and cyclic loading conditions based on the anisotropic hardening rule, the isotropic and nonlinear kinematic hardening rules of ABAQUS. The results indicate that the plastic response of the material follows the intended input stress–strain data for the anisotropic hardening rule whereas the plastic response depends upon the input strain ranges of the stress–strain data for the nonlinear kinematic hardening rule. Then, a two-dimensional plane-strain finite element analysis of a crankshaft section under fillet rolling and subsequent bending was conducted based on the anisotropic hardening rule of Choi and Pan (in preparation) and the nonlinear kinematic hardening rule of ABAQUS. In general, the trends of the stress distributions based on the two hardening rules are quite similar after the release of roller and under bending. However, the compressive hoop stress based on the anisotropic hardening rule is larger than that based on the nonlinear kinematic hardening rule within the depth of 2 mm from the fillet surface under bending with consideration of the residual stresses of fillet rolling. The critical locations for fatigue crack initiation according to the stress distributions based on the anisotropic hardening rule appear to agree with the experimental observations in bending fatigue tests of crankshaft sections.     

Phenomenological evolution equations for the backstress and spin tensors

Summary A formulation of a constitutive model involving a new kinematic hardening rule is presented in the Eulerian reference system. The corotational stress and backstress rates involving spin tensors are discussed and incorporated in the evolution equations. The backstress evolution model is compared with other models and is found to be decomposable into a model that consists of two backstresses whose evolution is independent of each other.The elasto-plastic stiffness tensor is derived for all the models considered. It is shown in the proposed backstress evolution model we obtain an implicit system of differential equation in the stress and backstress rates. The theory is applied to two yield functions: one of the von-Mises type and the other of the anisotropic type. It is shown that for both these yield criteria the stress evolution is independent of the stress rate. As an example, the torsion of a cylindrical bar with fixed ends is investigated.     

AN EFFICIENT STRESS ALGORITHM WITH APPLICATIONS IN VISCOPLASTICITY AND PLASTICITY

This paper deals with two main topics. The first one concerns the equivalence of stress algorithms, based on a Backward-Euler-step applied on viscoplastic models of Chaboche-type, and their elastoplastic counterpart. Generally, the stress algorithm yields a system of non-linear algebraic equations and the corresponding consistent tangent operator, occurring in the principle of virtual displacements, leads to a system of linear equations. This procedure can be obtained utilizing only numerical methods. The second topic concerns a special constitutive relation based on a kinematic hardening model using a sum of Armstrong/Frederick terms, which is equivalent to a multi-surface plasticity model. Applying this model a so-called problem-adapted stress algorithm is derived, where only one non-linear equation must be solved. This result is independent of the number of terms in the hardening model. Furthermore, only the viscoplastic algorithm must be implemented, since it includes the elastoplastic constitutive model as a special case. © 1997 by John Wiley & Sons, Ltd.     

Modelling direction-dependent hardening in magnesium sheet forming simulations

In this work, a model capturing anisotropic hardening during plastic deformation under monotonic loading is proposed. For this purpose, the anisotropic plastic potential coefficients are assumed to be functions of a measure of the accumulated plastic strain. This model is applied to describe the plastic behavior of a magnesium alloy (ZM21) sheet at room temperature. The selected plastic potential accounts for the main features of Mg alloy plasticity, i.e., anisotropy and strength-differential (SD) effects. All the accumulated plastic strain dependent coefficients of the phenomenological model are determined from input data generated with a crystal plasticity approach. They are optimized to best capture the accumulated strain dependent potentials computed with crystal plasticity. The R-value (Lankford coefficient) anisotropy is used as an independent measure for the assessment of the approximation quality. This model is implemented into a finite element (FE) code and successfully validated through the numerical simulations of the cup drawing test. The calculated earing profile obtained with the proposed hardening model is compared to results assuming isotropic hardening for various plausible shapes of the plastic potential. Although the ear and valley numbers and positions are similar in all cases, the height differences between peaks and valleys are strongly dependent on the type of constitutive approach used in the simulation.     

Numerical implementation and analysis of a class of metal plasticity models coupled with ductile damage

This paper deals with a class of rate-independent metal plasticity models which exhibit non-linear isotropic hardening, non-linear kinematic hardening (Chaboche-Marquis model) and ductile damage (Lemaitre-Chaboche model). The backward Euler scheme is used to integrate the rate constitutive relations. The non-linear equations obtained are solved by the Newton method. The consistent tangent operator is obtained by exact linearization of the algorithm. Despite the complexity of the constitutive equations, closed-form expressions are derived, without any approximations. Analytical, numerical and experimental results are presented and discussed.     

Effective constitutive equations for fiber-reinforced viscoplastic composites exhibiting anisotropic hardening

Effective elastic-viscoplastic stress-strain relations are derived for fiber-reinforced composites whose constituents are elastic-viscoplastic materials displaying anisotropic hardening. The derivation is based on a recently developed high-order continuum theory with microstructure for the modeling of viscoplastic composites, and is generalized here to incorporate anisotropic hardening effects. A specific reduction of the theory gives the effective rate-dependent elastic-plastic behavior of the composite which exhibits plastic anisotropy. In the special case of perfectly elastic constituents, the approximate overall moduli of the fiber-reinforced composite are obtained. Rate-dependent average stress-strain curves are given for numerous modes of cyclic loading of the composite. The effective behavior of periodically bilaminated viscoplastic composites is determined as a special case.     

混凝土应力三轴比依赖的塑性损伤模型

提出了一种新的损伤塑性本构模型,定义了一个用应力不变量表示的应力三轴比,强调了应力三轴比对塑性屈服的影响。采用本研究设计的专门用于本构校验的计算机软件,对上述模型的应力应变加载曲线进行了数值模拟,数值结果与实验结果吻合得比较好。     

An anisotropic model of the Mullins effect

The Mullins effect in rubber-like materials is inherently anisotropic. However, most constitutive models developed in the past are isotropic. These models cannot describe the anisotropic stress-softening effect, often called the Mullins effect. In this paper a phenomenological three-dimensional anisotropic model for the Mullins effect in incompressible rubber-like materials is developed. The terms, damage function and damage point, are introduced to facilitate the analysis of anisotropic stress softening in rubber-like materials. A material parametric energy function which depends on the right stretch tensor and written explicitly in terms of principal stretches and directions is postulated. The material parameters in the energy function are symmetric second-order damage and shear-history tensors. A class of energy functions and a specific form for the constitutive equation are proposed which appear to simplify both the analysis of the three-dimensional model and the calculation of material constants from experimental data. The behaviour of tensional and compressive ground-state Young’s moduli in uniaxial deformations is discussed. To further justify our model we show that the proposed model produces a transversely anisotropic non-virgin material in a stress-free state after a simple tension deformation. The proposed anisotropic theory is applied to several types of homogenous deformations and the theoretical results obtained are consistent with expected behaviour and compare well with several experimental data.     

On the numerical integration of a class of pressure-dependent plasticity models including kinematic hardening

The algorithm proposed by Aravas to integrate a special type of elastic-plastic constitutive equations has been extended to incorporate kinematic hardening. Like in the case of isotropic hardening, the number of primary unknowns for the Newton iteration can be reduced to two scalar strain variables. Furthermore, the consistent tangent can be obtained explicitly. The modified algorithm has been applied to a Gurson-type model which takes into account kinematic hardening and the predictions of the Gurson-like model are compared with results obtained by unit cell calculations.The authors acknowledge the financial support from DFG-Project Br 521/11-3. Furthermore, the authors are grateful to Jean-Baptiste Leblond from the Université Pierre et Marie Curie, Paris for several helpful suggestions and hints.     

Fully implicit integration and consistent tangent modulus in elasto-plasticity

This paper deals with the numerical integration of a class of rate-independent elasto-plastic models. The backward Euler scheme is used to integrate the rate constitutive relations. The non-linear equations obtained are solved by the Newton method. The consistent tangent modulus is obtained by exact linearization of the algorithm. In the case of J₂ elasto-plasticity with non-linear isotropic hardening and non-linear kinematic hardening (Chaboche-Marquis model), explicit formulas are derived, without any approximations.