Industrial challenges for finite element and multiscale methods for material modeling

The automotive industry promotes lightweight design to reduce the CO2-emission and enhances the passenger’s safety using high strength steel grades. One limiting factor to the accuracy of modern stamping simulation are the empirical constitutive models. In particular for high strength multiphase steels the modelling techniques like multi-scale methods are becoming more interesting. However they should meet the industrial needs. Not only the accuracy but also features like time, costs and complexity are rapidly increasing. The challenge is the development of finite element technologies and multi-scale methods in an appropriate framework for industrial projects. The crystal plasticity finite element method bridges the gap between the micro level and macroscopic mechanical properties that opens the way for more profound consideration of metal anisotropy in stamping process simulation. Nevertheless new empirical constitutive models are favourable for spring back prediction in forming simulations, even if the number of material parameters and the amount of tests for their identification increases. In this paper the application of crystal plasticity FEM within the concept of virtual material testing with a representative volume element (RVE) is demonstrated.     

Concepts for Integrating Plastic Anisotropy into Metal Forming Simulations

Modern metal forming and crash simulations are usually based on the finite element method. Aims of such simulations are typically the prediction of the material shape, failure, and mechanical properties during deformation. Further goals lie in the computer assisted lay‐out of manufacturing tools used for intricate processing steps. Any such simulation requires that the material under investigation is specified in terms of its respective constitutive behavior. Modern finite element simulations typically use three sets of material input data, covering hardening, forming limits, and anisotropy. The current article is about the latter aspect. It reviews different empirical and physically based concepts for the integration of the elastic‐plastic anisotropy into metal forming finite element simulations. Particular pronunciation is placed on the discussion of the crystallographic anisotropy of polycrystalline material rather than on aspects associated with topological or morphological microstructure anisotropy. The reviewed anisotropy concepts are empirical yield surface approximations, yield surface formulations based on crystallographic homogenization theory, combinations of finite element and homogenization approaches, the crystal plasticity finite element method, and the recently introduced texture component crystal plasticity finite element method. The paper presents the basic physical approaches behind the different methods and discusses engineering aspects such as scalability, flexibility, and texture update in the course of a forming simulation.     

Anisotropic viscoplasticity constitutive model for directionally solidified superalloy

Abstract

The macroscopic deformation behaviour of a Ni-based directionally solidified (DS) superalloy was experimentally investigated, and an anisotropic constitutive model of the material was developed. Monotonic and creep tests were performed on uniaxial test specimens machined from DS plates so that the angle between the loading direction and the solidified grain direction varied between 0 and 90°. Tension-torsion creep tests were also conducted to examine the anisotropic behaviour under multiaxial stress conditions. The material exhibited marked anisotropy under elastic and viscous deformation conditions, whereas it showed isotropy under plastic deformation conditions of high strain rates. Then crystal plasticity analyses were carried out to identify slip systems under creep loading conditions, assuming the anisotropic creep behaviour of the DS material. A viscoplastic constitutive model for expressing both the anisotropic elasticity-viscosity and the isotropic plasticity was proposed. The elastic constants were determined using a self-consistent approach, and viscous parameters were modelled by crystal plasticity analyses. The calculation results obtained using the constitutive model were compared with the experimental data to evaluate the validity of the model. It was demonstrated that the constitutive model could satisfactorily describe the anisotropic behaviour under uniaxial and multiaxial stress conditions with a given set of material parameters.     

高强钢TRB高温下热流变特性

为了准确仿真高强钢板热冲压成形过程,获得高强钢高温下的材料本构关系模型,利用Gleeble3500热模拟试验机在不同温度和应变速率下对不同厚度的高强钢B1500HS钢板进行了单向拉伸试验,获得各种工艺条件下的应力-应变曲线,并基于变形抗力数学模型,引入板材厚度参数,通过最小二乘法进行数据拟合获得高强钢TRB高温下的材料本构关系.利用试验结果对本构关系模型进行的拟合验证表明,拟合程度较好,说明建立的材料本构关系能很好地描述高强钢TRB在高温下的应力-应变关系.     

The equivalent plastic strain-dependent Yld2000-2d yield function and the experimental verification

An equivalent plastic strain-dependent anisotropic material model was developed for 5754O aluminum alloy sheet. In the developed model, the anisotropy coefficients for Barlat’s Yld2000-2d anisotropic yield function were established as a function of the equivalent plastic strain. The developed anisotropic material model was implemented into the commercial FEM code ABAQUS as a user material subroutine (UMAT) for simulations. In order to evaluate the accuracy of the developed material model, biaxial tensile tests were carried out using cruciform specimens and a biaxial loading testing machine. The results show that the developed material model predicts the experimental results better than the other three material models (Yld2000-2d, Mises and Hill48 yield functions). It is also found that the developed material model describes the uniaxial tensile test curves better than Yld2000-2d yield function. The deep drawing test for 5754O aluminum alloy sheet was carried out and was simulated with different material models. The comparison between the experimental and simulation results indicates that the developed material model predicts the earing profile better than other material models. It is concluded that the equivalent plastic strain-dependence of the material coefficients should be considered for the accurate prediction of the anisotropic deformation behavior of materials.     

Phenomenological and crystal plasticity approaches to describe the mechanical behaviour of Ti6Al4V titanium alloy

This paper presents a multiscale study of the quasi-static behaviour of a Ti6Al4V titanium alloy sheet. Tensile and compressive tests were carried out on specimens along several orientations from the rolling direction in order to characterise the material anisotropy. In parallel, X-Ray diffraction texture measurements were performed before and after deformation in tension. A phenomenological model (CPB06exn) and a multiscale crystal plasticity model (Multisite) were investigated to describe the mechanical behaviour of the tested material. The identification of the material parameters provides good predictions of the plastic anisotropy using both tensile and compressive data. The crystal plasticity model is in good agreement with the experiments in tension but it was observed that some improvements should be done to take into account the tension-compression asymmetry displayed by the material. Moreover both models lead to a good prediction of the Lankford’s coefficients and yield strength.     

解析Poly6-I屈服准则对3104-H19铝合金拉深制耳预测的研究

目的 研究解析Poly6-I屈服准则预测具有高各向异性的3104-H19铝合金本构关系的能力,并将其应用于有限元仿真分析中,以实现对3104-H19铝合金拉深制耳的精确预测。方法 分析解析Poly6-I屈服准则的表达形式,减少计算参数所需的试验个数,并与经典的Yld2004-18p屈服准则进行对比,验证它对高各向异性力学性能预测的能力,将其嵌入到有限元软件中进行杯型件拉深制耳模拟,验证模型的精确性和有效性。结果 对于高各向异性材料,解析Poly6-I屈服准则所使用的试验个数可以减少到11,它预测的3104-H19铝合金屈服轨迹的各向异性系数曲线和单向拉伸曲线与Yld2004-18p屈服准则预测的结果基本相同,杯型件拉深有限元模拟结果与试验结果基本一致。结论 与Yld2004-18p屈服准则相比,考虑高各向异性特性的解析Poly6-I屈服准则所使用的试验数据更少,且无须使用优化软件求取参数,更为方便。解析Poly6-I屈服准则能精确地预测3104-H19铝合金材料在杯型件拉深试验中的制耳个数及杯型件杯壁的成形高度。     

Constitutive modeling of ferritic stainless steel

In this work, constitutive models, including phenomenological and crystal plasticity, were used to simulate the anisotropy behavior and texture evolution of two ferritic stainless steel sheets, AISI409L and AISI430. Uniaxial tension, hydraulic bulge and disk compression tests were performed to characterize the mechanical properties of the two materials, and to determine the yield surfaces at different amounts of plastic work. Meanwhile, X-ray diffraction and electron backscatter diffraction techniques were used to analyze the texture of undeformed and deformed specimens. Crystal plasticity simulations were performed to determine the plastic behavior in selected deformation paths. The analysis of the mechanical test results showed that the yield surface shapes were changing during deformation. Crystal plasticity results indicated that texture evolution was mainly responsible for the yield surface shape change, i.e., anisotropic work-hardening, in AISI409L. For AISI430, the results were not completely consistent. More work is needed to understand the plastic behavior of this material.     

纯钼薄板各向异性拉压不对称性本构建模研究

目的 对纯钼薄板的各向异性、拉压不对称性以及本构关系进行研究.方法 对纯钼薄板进行不同方向的单拉实验、中心带孔试样单拉实验,和纯钼薄板V形弯曲实验,同时结合有限元模拟反推材料力学性能以及对CPB06屈服准则及Swift强化模型进行参数标定,并进行模型可靠性验证.结果 纯钼薄板具有一定的面内各向异性、拉压不对称性以及显著...     

Nonlinear fracture mechanics and plasticity of the split cylinder test

The split cylinder test is subjected to an analysis combining nonlinear fracture mechanics and plasticity. The fictitious crack model is applied for the analysis of splitting tensile fracture, and the Mohr-Coulomb yield criterion is adopted for modelling the compressive crushing/sliding failure. Two models are presented, a simple semi-analytical model based on analytical solutions for the crack propagation in a rectangular prismatic body, and a finite element model including plasticity in bulk material as well as crack propagation in interface elements. A numerical study applying these models demonstrates the influence of varying geometry or constitutive properties. For a split cylinder test in load control it is shown how the ultimate load is either plasticity dominated or fracture mechanics dominated. The transition between the two modes is related to changes in geometry or constitutive properties. This implies that the linear elastic interpretation of the ultimate splitting force in term of the uniaxial tensile strength of the material is only valid for special situations, e.g. for very large cylinders. Furthermore, the numerical analysis suggests that the split cylinder test is not well suited for determining the tensile strength of early age or fibre reinforced concrete.     

新型铝锂合金板材塑性流动与各向异性性能试验研究

通过单向拉伸试验,研究了新型Al-Li-Cu-Mg合金板材的基本成形性能.针对材料显著的各向异性性能,选取屈服强度、抗拉强度、延伸率以及厚向异性指数等材料性能参数进行对比分析,绘制了7个不同取样方向的单向拉伸曲线,研究了材料各向异性的规律.基于对本构方程、各向异性屈服准则的研究及对比,建立了新型Al-Li-Cu-Mg合金的本构模型,并根据实验曲线计算得到Hill48、Barlat89屈服准则中的各向异性参数,结合各屈服准则绘制了新型Al-Li-Cu-Mg合金屈服轨迹.对比分析各试件的断口方向,并结合第一、第三强度理论,分析了材料的各向异性.利用SEM观察试样的断口形貌,分析对比试件断口的韧窝特征及带状特征.研究发现:试件的延伸率越大,其韧窝特征越明显;反之,其带状特征越明显.从微观角度印证了Al-Li-Cu-Mg合金板材存在的各向异性.     

FEM-simulation of the hardening behavior of FCC single crystals

Summary Although the deformation behaviour of single crystals has been investigated experimentally and theoretically for several decades, a generally accepted theory for the underlying hardening processes has not been found yet. In the last years, computer simulations offered additional investigation possibilities, whereby especially the use of constitutive equations in combination with the finite element method has delivered important results. The publication reports on an FEM crystal plasticity model-which was previously used for multicrystals-and its application to single crystals. The model is designed for the low temperature behaviour of pure fcc metal crystals. The rate dependent equations include kinematic and isotropic hardening, with formulations founded on the responsible slip system processes. The obtained simulation results coincide very well with typical single crystal experiments like tensile or latent hardening tests, which confirms the chosen mathematical approaches. It is shown that both kinematic and isotropic processes determine hardening, whereby the underlying slip system interactions play an important role. Moreover, experimentally not visible processes can be studied in detail and are discussed concerning the metal physics theories, which finally contributes to a better understanding of metal deformation behaviour.     

Multiscale prediction of mechanical behavior of ferrite–pearlite steel with numerical material testing

Multiscale mechanical behaviors of ferrite–pearlite steel were predicted using numerical material testing (NMT) based on the finite element method. The microstructure of ferrite–pearlite steel is regarded as a two‐component aggregate of ferrite crystal grains and pearlite colonies. In NMT, the macroscopic stress–strain curve and the deformation state of the microstructure were examined by means of a two‐scale finite element analysis method based on the framework of the mathematical homogenization theory. The microstructure of ferrite–pearlite steel was modeled with finite elements, and constitutive models for ferrite crystal grains and pearlite colonies were prepared to describe their anisotropic mechanical behavior at the microscale level. While the anisotropic linear elasticity and the single crystal plasticity based on representative characteristic length have been employed for the ferrite crystal grains, the constitutive model of a pearlite colony was newly developed in this study. For that reason, the constitutive behavior of the pearlite colony was investigated using NMT on a smaller scale than the scale of the ferrite–pearlite microstructure, with the microstructure of the pearlite colony modeled as a lamellar structure of ferrite and cementite phases with finite elements. On the basis of the numerical results, the anisotropic constitutive model of the pearlite colony was formulated based on the normal vector of the lamella. The components of the anisotropic elasticity were estimated with NMT based on the finite element method, where the elasticity of the cementite phase was numerically evaluated with a first‐principles calculation. Also, an anisotropic plastic constitutive model for the pearlite colony was formulated with two‐surface plasticity consisting of yield functions for the interlamellar shear mode and yielding of the overall lamellar structure. After addressing the microscopic modeling of ferrite–pearlite steel, NMT was performed with the finite element models of the ferrite–pearlite microstructure and with the microscopic constitutive models for each of the components. Finally, the results were compared with the corresponding experimental results on both the macroscopic response and the microscopic deformation state to ascertain the validity of the numerical modeling. Copyright © 2011 John Wiley & Sons, Ltd.     

Modeling short- and long-term time-dependent nonlinear behavior of polyethylene

A new methodology for developing macromechanical constitutive formulations for time-dependent materials is presented in this article. In particular, two phenomenological constitutive models for polymer materials are illustrated, describing time-dependent and nonlinear mechanical behavior. In this new approach, short-term creep test data are used for modeling both short-term and long-term responses. The differential form of a model is used to simulate typical nonlinear viscoelastic polymeric behavior using a combination of springs and dashpots. Unified plasticity theory is then used to develop the second model, which is a nonlinear viscoplastic one. Least squares fitting is applied for the determination of material parameters for both models, based on experimental results. Due to practical constraints, experimental data are usually available for short-term time-frames. In the presented proposed formulation, the material parameters determined from short-term testing are used to obtain material parameter relationships for predicting the long-term material response. This is done by extending short-term information for longer time frames. Finally, theoretical and experimental results of tensile tests on polyethylene subjected to various load levels and test times are compared and discussed. Very good agreement of the modeling results with experimental data shows that the developed formulation provides a flexible and reliable framework for predicting load responses of polymers.     

Determining the stress-strain behaviour at large strains from high strain rate tensile and shear experiments

To characterise the high strain rate mechanical behaviour of metals, split Hopkinson bar experiments are frequently used. These experiments basically yield the force and elongation history of the specimen, reflecting not only the specimen material behaviour but also the specimen structural behaviour. Calculation of the real material behaviour from this global response is not straightforward, certainly for materials such as Ti6Al4V where due to low strain hardening, the specimen deformation is very inhomogeneous. However, for fundamental material research and constitutive material modelling, knowledge of the true effective stress versus plastic strain, strain rate and temperature is essential.In this contribution, a combined experimental-numerical approach for extraction of the strain rate and temperature dependent mechanical behaviour from high strain rate experiments is presented. The method involves the identification of the material model parameters used for the finite element simulations. The technique is applied to determine the stress-strain behaviour of Ti6Al4V using both high strain rate in-plane shear and tensile test results. For the tensile tests, even stress-strain data beyond diffuse necking are retrieved. A comparison is made between the material behaviour extracted from the tensile and the shear experiments. The material behaviour is modelled with the Johnson-Cook constitutive relation. It is found that the simultaneous use of tensile and shear tests to identify the model parameters gives a more generally applicable model. Validation of the material model and the finite element simulations is done by local strain measurements in the shear and tensile test by means of digital image correlation.     

Constitutive modeling of woven composites considering asymmetric/anisotropic, rate dependent, and nonlinear behavior

The mechanical performance of woven composites was analyzed focusing on their nonlinear and rate dependent asymmetric/anisotropic deformation behavior. Three key characteristics were identified which are indispensable for realistically simulating the mechanical performance of woven composites: the asymmetric material behavior between tension and compression, its anisotropic and nonlinear evolution and rate dependency. To include all three characteristics into the nonlinear finite element analysis for woven composites, a phenomenological constitutive equation was developed based on an elasto-viscoplastic theory using the modified Drucker–Prager yield criterion and, in particular, developing the anisotropic nonlinear hardening law. A characterization method using both uniaxial tensile and compressive tests at different strain rates was proposed to determine the material properties for the constitutive equation. Then, the developed constitutive equation was incorporated into a finite element code and was validated by comparing the finite element simulation of the three points bending test with experiments.     

电子陶瓷材料的数字模拟与建模

电子陶瓷材料正由经验研究和实物展示向虚拟设计和测试转变，从材料到器件性能的计算机数字模拟和建模也应运而生。本文介绍了电子陶瓷材料领域技术数字模拟与建模的理论背景，着重讨论了有限元方法(FEM)在压电及其相关陶瓷及换能器建模中的应用，以及分子动力学模拟在电子陶瓷材料电子结构、点缺陷和晶界特性等电子陶瓷材料中的研究进展和发展趋势。     

An inverse approach to determine the constitutive model parameters from axial crushing of thin-walled square tubes

Knowledge of the behaviour of structural components is essential for their design under crash consideration. Constitutive models describe their material behaviour in finite element (FE) codes. These constitutive models are in relation to the material parameters which have to be determined. The strain rates commonly observed in crash events are in the range of 0–500 s^-1. Classic experimental devices such as Hopkinson’s bars do not easily cover this range of strain rates. An inverse numerical approach based on the experimental quasi-static and dynamic axial crushing of thin-walled square tubes has therefore been developed to determine the constitutive model’s parameters. The inverse method is applied in this paper in two stages to determine the power type elastic–plastic constitutive model’s parameters and the Cowper–Symonds constitutive model’s parameters. The identified power law is compared with the results obtained by quasi-static tensile tests and shows that the identified parameters are intrinsic to the material behaviour. The Cowper– Symond’s parameters identified by this method are then used in FE simulation to predict the dynamic response of the same square tube subjected to bending loading. The results obtained show a good correlation between the experimental and numerical results.     

Hyperelastic modelling for mesoscopic analyses of composite reinforcements

A hyperelastic constitutive law is proposed to describe the mechanical behaviour of fibre bundles of woven composite reinforcements. The objective of this model is to compute the 3D geometry of the deformed woven unit cell. This geometry is important for permeability calculations and for the mechanical behaviour of the composite into service. The finite element models of a woven unit cell can also be used as virtual mechanical tests. The highlight of four deformation modes of the fibre bundle leads to definition of a strain energy potential from four specific invariants. The parameters of the hyperelastic constitutive law are identified in the case of a glass plain weave reinforcement thanks to uniaxial and equibiaxial tensile tests on the fibre bundle and on the whole reinforcement. This constitutive law is then validated in comparison to biaxial tension and in-plane shear tests.