A study on formulation of objective functions for determining material models

Optimisation techniques have been developed and used to determine material constants arising in unified creep/viscoplastic constitutive equations based on experimental data. Objective functions (OF) have been formulated as pointers to the quality of fit between the equations and experimental data and a set of criteria is presented to assess the suitability of the objective functions. Convergence features of two existing objective functions are analysed. The problems of using these objective functions are studied. To overcome difficulties arising in the existence of different scales of individual sub-objectives, a novel error definition is introduced, which has a natural unitless form and can provide a measure for “true” error. A novel weighting technique is introduced, which can also be chosen automatically to compensate the loss of credits in individual data points and curves. Using these techniques, a novel objective function is formulated, which meets the set criteria. The objective function, together with an evolutionary programming (EP) solver, are employed to determine material constants in three sets of unified constitutive equations which are formulated to match experimental data. The convergent features of the objective functions are compared and analysed.     

Modeling the powder compaction process using the finite element method and inverse optimization

This paper focuses on studying and adapting modeling techniques using the finite element method to simulate the rigid die compaction of metal powders. First, it presents the implementation of the cap constitutive model into ABAQUS FE software using the closest point projection algorithm. Then, an inverse modeling procedure was proposed to alleviate the problems raised by the interpretation of the experimental tests and to more accurately determine the material parameters. The objective function is formed, based on the discrepancy in density data between the numerical model prediction and the experiment. Minimization of the objective function with respect to the material parameters was performed using an in-house optimization software shell built on a modified Levenberg?CMarquardt method. Thus, an integrated simulation module consisting of an inverse optimization method and a finite element method was developed for modeling the powder compaction process as a whole. The simulation and identification module developed was applied to simulate the compaction of some industrial parts. The results reveal that the maximum absolute error between densities is 2.3%. It corresponds to the precision of the experimental method.     

基于连续介质损伤力学的7075铝合金高温成形极限理论预测与数值仿真

开展7075铝合金高温单向拉伸试验和成形极限试验,获得了不同温度和应变率的应力-应变曲线和成形极限曲线。结果表明,在较高的温度和应变率时7075铝合金的强度减小、成形性提高。为描述7075铝合金高温损伤演化过程,提出一种改进的连续介质损伤模型,并建立了耦合损伤的多轴统一黏塑性本构模型。基于试验结果,运用NSGAII遗传算法标定了模型中的参数,标定后的本构模型可以很好地预测7075铝合金的高温热力行为和极限应变。通过有限元软件Abaqus的用户材料子程序VUMAT,该本构模型被编入到Abaqus软件中进行数值仿真计算。结果表明,仿真获得的成形极限曲线和应变场分布与试验和理论结果吻合度好,进一步证明了所建立的耦合损伤的多轴本构模型的正确性及其在高温成形极限有限元仿真中的适用性。     

Modelling the plane strain compression test to obtain constitutive equations of aluminium alloys

Hot plane strain compression tests on 1050, 1198, 3003 and 3004 aluminium alloys have been conducted. Based on these experiments and on a set of internal type constitutive equations for hot working, the values of the parameters in the constitutive functions are determined. The constitutive equations proposed here, with the constitutive functions and material parameters associated, accurately reproduce the basic tests. The procedure used to fit the material parameters is improved, in comparison with classical slip line analysis, by using a finite element modelling of the plane strain compression test. It is demonstrated that accurate plane strain or three-dimensional large strain finite element analysis can be used to correct the friction and lateral spread effects. Furthermore, it is demonstrated from comparison with the experimental observations that microstructural parameters can be accurately determined from numerical modelling. The constitutive equations and finite element procedure proposed here can be useful for obtaining an improved analysis of hot rolling of aluminium alloys.     

A novel evolutionary algorithm for determining unified creep damage constitutive equations

The determination of material constants within unified creep damage constitutive equations from experimental data can be formulated as a problem of finding the global minimum of a well defined objective function. However, such an objective function is usually complex, non-convex and non-differentiable. It is difficult to be optimised by classical gradient-based methods. In this paper, the difficulties in the optimisation are firstly identified. Two different objective functions are proposed, analysed and compared. Then three evolutionary programming algorithms are introduced to solve the global optimisation problem. The evolutionary algorithms are particularly good at dealing with problems which are complex, multi-modal and non-differentiable. The results of the study shows that the evolutionary algorithms are ideally suited to the problem. Computational results of using the algorithms to determine the material constants in a set of physically based creep damage constitutive equations from experimental data for an aluminium alloy are presented in the paper to show the effectiveness and efficiency of the three evolutionary algorithms.     

CRITERIA FOR THE QUALITY ASSESSMENT OF CONSTITUTIVE EQUATIONS DEDICATED TO CUTTING MODELS

Determination of the flow stress under metal cutting conditions is difficult because high values of strains, strain rates and temperature exist in practical machining operations. Conventional tests for determining flow stress data cannot duplicate these deformation conditions and therefore, extrapolation will be required. In this study we have investigated critical issues regarding the constitutive equation determination by direct methods such as split Hopkinson's pressure bar bench (SHPB) tests. Quantitative analysis about the suitability of an experimentally determined constitutive equation for different purpose modeling is provided, leading to guidelines for the quality improvement of constitutive law dedicated to cutting, from the points of view of formulation and identification sequence.     

CRITERIA FOR THE QUALITY ASSESSMENT OF CONSTITUTIVE EQUATIONS DEDICATED TO CUTTING MODELS

Determination of the flow stress under metal cutting conditions is difficult because high values of strains, strain rates and temperature exist in practical machining operations. Conventional tests for determining flow stress data cannot duplicate these deformation conditions and therefore, extrapolation will be required. In this study we have investigated critical issues regarding the constitutive equation determination by direct methods such as split Hopkinson's pressure bar bench (SHPB) tests. Quantitative analysis about the suitability of an experimentally determined constitutive equation for different purpose modeling is provided, leading to guidelines for the quality improvement of constitutive law dedicated to cutting, from the points of view of formulation and identification sequence.     

Biomechanical behaviour of heel pad tissue: experimental testing, constitutive formulation, and numerical modelling

This paper deals with the constitutive formulation of heel pad tissue and presents a procedure for identifying constitutive parameters using experimental data, with the aim of developing a computational approach for investigating the actual biomechanical response. The preliminary definition of constitutive parameters was developed using a visco-hyperelastic formulation, considering experimental data from in vitro compression tests on specimens of fat pad tissue and data from in vivo tests to identify the actual trend of tissue stiffness. The discrepancy between model results and experimental data was evaluated on the basis of a specific cost function, adopting a stochastic/deterministic procedure. The parameter evaluation was upgraded by considering experimental tests performed on the fat pad tissues of a cadaveric foot using in situ indentation tests at 0.01 and 350 mm/s strain rates. The constitutive formulation was implemented in a numerical model. The comparison of data from in situ tests and numerical results led to an optimal domain of parameters based on an admissible discrepancy criterion. Numerical results evaluated for different sets of parameters inside the domain are reported and compared with experimental data for a reliability evaluation of the proposed procedure.     

Principes variationnels et exploitation de mesures de champs en élasticité

Sometimes parameters pertaining to constitutive materials or to the overall definition of a mechanical structure need to be identified in an indirect manner from experimental measurements. Field measurements provide rich data and, as such, are particularly well suited to material or structural parameter identification problems. In this article, identification techniques specifically devised for the exploitation of field measurements are presented. The identification of distributed elastic moduli is used as a model problem, but other identification problems can be treated along similar lines. This article emphasizes the fact that one is by no means limited to using output least-squares cost functions for solving such identification problems. Indeed, the availability of field measurements allow to define other criteria, well-suited to this type of data and directly linked to fundamental variational principles. Most of this article is devoted to two criteria of this family, namely the error in constitutive equation and the reciprocity gap functional. Various illustrative examples, generally of a synthetic (i.e., purely numerical) nature are provided.     

TC4-DT钛合金材料动态力学性能及其本构模型

利用同步组装的高温分离式Hopkinson压杆试验装置,对TC4-DT钛合金材料分别进行了常温下不同应变率（930~9700s-1）和应变率为5000s-1时不同温度下（20~800℃）的动态力学性能测试,获得了各种冲击载荷下的应力-应变曲线。试验数据表明,TC4-DT材料具有应变率增塑效应且存在着临界应变率值,当应变率高于此值时应变率敏感性增强明显,此外随着材料加热温度的升高,软化效应减弱。利用试验所得的数据拟合了基于Power-Law和Johnson-Cook两种热-黏塑性本构方程且获得这两种动态本构模型参数,并将所得的两种拟合曲线与试验所得数据进行对比分析,结果表明两曲线吻合度都较好,此外还对这两种曲线的拟合精度进行对比,对比结果表明两种模型的拟合误差相差不大,但是Power-Law模型拟合精度要略优于Johnson-Cook模型的拟合精度。