Determination of strain hardening parameters of tailor hardened boron steel up to high strains using inverse FEM optimization and strain field matching

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Abstract

In this article, an inverse FEM optimization strategy is proposed for identification of the strain hardening parameters of boron alloyed steel 22MnB5 in five different hardness grades. In the proposed elasto-plastic constitutive model, the strain hardening is represented by a nonlinear combination of the Swift hardening law and a modified version of the Voce law. Initial fits of these two classical strain hardening equations are constructed based on experimental data from uni-axial tensile tests up to the point of diffuse necking. The strain hardening response beyond the point of diffuse necking is determined from a 3D strain field analysis of notched tensile and equibiaxial tension tests. Both the measured force–displacement curves and the strain fields are used as inputs for the optimization algorithm that identifies suitable material model parameters by minimizing the differences between experimental and simulated results. In order to show the contribution of the different parts of the elasto-plastic model for representing the real material response, three simplified versions of the proposed model and the parameter identification procedure are applied on two selected hardness grades, confirming the importance of a flexible strain hardening law, suitable yield criteria and accurate experimental data up to high plastic strains. The calibrated model was shown to accurately capture the elasto-plastic response of 22MnB5 in different hardness grades, with an excellent representation of the strain fields up to the point of fracture.

Introduction

The automotive industry is continuously working on the weight reduction of their vehicles in order to lower both fuel consumption and CO2 emissions, while maintaining or even improving the crashworthiness in accordance with increasing safety demands. In recent years, the hot stamping process has gained popularity as a manufacturing method for crash-relevant structural components. In the conventional hot stamping process, boron steel blanks are fully austenitized in a furnace, after which they are simultaneously formed and quenched in a cooled stamping tool. Due to the high cooling rates during the forming process, the austenitic microstructure transforms into martensite, causing the tensile strength of the material to increase from an initial 600 MPa to 1500 MPa in the final state. Karbasian and Tekkaya (2010) provide a general overview on current hot stamping technologies, a more in-depth view on the underlying microstructural transformations and on modeling approaches for simulation of the hot stamping process can be found in Naderi (2007) and Åkerström (2006), respectively.

Although hot stamped components benefit from an exceptionally high strength, car manufacturers are exploring new production methods that allow the introduction of regions of reduced strength and higher ductility for local, improved energy absorption. A well-known application is the B-pillar: for optimal performance in a side crash, the bottom part should show a high energy absorption capacity, while the upper part should ensure a high intrusion resistance (Maikranz-Valentin et al., 2008). Common methods to produce parts with such ‘tailored’ mechanical properties are: local reduction of the in-die cooling rate through the use of tool materials with varying thermal conductivities or using increased die temperatures (heated/tailored tooling), or by post tempering fully hardened parts with differential heating in the furnace (George et al., 2012).

To be able to fully exploit the possibilities of such tailor hardened components, it is of paramount importance to attain accurate predictive models of their crash response. In particular the modeling of transition zones between harder and softer regions is a challenging task. In previous work of the authors, it was found that the strain hardening behavior has a crucial influence on the onset and location of fracture initiation in hardness transition zones: a small decrease in the gradient of the strain hardening curve leads to a higher probability of localized necking, which will ultimately lead to fracture (Eller et al., 2014). Before considering the fracture characteristics of tailor hardened 22MnB5, it is thus required to obtain an appropriate elasto-plastic constitutive model with accurately determined parameters.

Several approaches concerning the modeling of the elasto-plastic response of boron steels and identification of the corresponding parameters have been presented in literature. Mohr and Ebnoether (2009) used a planar isotropic version of the Hill (1948) yield surface in combination with a piecewise linear hardening equation k(ɛ¯p) to model the plastic behavior of fully hardened 22MnB5. The strain hardening function was obtained from a pure shear experiment, which yielded experimental data up to an engineering shear strain of 0.28. Although the model provided excellent results for specimens with the same geometry under different loading angles, the stress level of the uni-axial tensile test was overestimated by 10%. Bardelcik et al. (2012) performed miniature tensile tests at four different strain rates from 0.003 s−1 to 1075 s−1 with 22MnB5 in five different hardness grades ranging from 268 HV to 466 HV. For the quasi-static tests, experimental true stress–strain data reached up to strains of 0.035 for the hardest grade and 0.07 for the softest grade. A modified Voce (1955) law was fitted to the data to model the strain hardening behavior in dependence of strain rate and Vickers hardness. Ten Kortenaar et al. (2013) quenched 22MnB5 samples to fully martensitic and fully bainitic microstructures in order to determine the fracture strains under different stress triaxialities using tensile specimens with different notch geometries. Their piecewise linear hardening model was calibrated using flow stress curves from uni-axial tensile tests up to diffuse necking, after which extrapolation was applied based on the apparent hardening rate at this point. When using the calibrated model to simulate the notched tensile tests, it was found that the stresses were overestimated and that the extrapolation of post-uniform hardening behavior is critical to the accuracy of the model. Östlund et al. (2014) calibrated the localization and failure behavior of three grades of 22MnB5 with yield strengths of 400, 550 and 800 MPa. They used a digital image correlation system to perform full field measurements of tensile tests and calibrated piecewise linear flow and localization curves with these data. The full field measurements allowed for evaluation of mechanical properties at different analysis lengths, providing parameters for a model which accounts for shell element size. The objective of their work was not to model the phenomenon of localized necking, but rather its effect on load response and subsequent rupture, which is why no direct comparisons of measured and simulated strain fields are shown.

In the present study, an inverse FEM optimization routine for determination of the strain hardening parameters is applied to five different hardness grades of quench-hardenable boron steel 22MnB5. In the proposed model, the strain hardening is represented by a nonlinear combination of the Swift hardening law and a modified version of the Voce law. The non-quadratic Yld2000-2D yield function (Barlat et al., 2003) with an extension for general three-dimensional stress states by Dunand et al. (2012) with αk = 1 is used and compared to results obtained with the Von Mises yield criterion. For fitting of the initial stress–strain response, force–displacement curves of uni-axial tensile tests are used up to the point of necking initiation. Because the model is planned to be used for calibration of a strain based fracture criterion, it should be able to accurately represent the strain fields under different loading conditions up to fracture initiation, which for the uni-axial case is far beyond the point of uniform elongation. For this purpose, full field strain measurements are performed using a 3D digital image correlation (DIC) system. These measurements are used as input for the optimization routine that optimizes the strain hardening parameters in the post-necking regime. The uni-axial tensile test is not suited for this purpose, because localization may occur anywhere in the relatively long gauge section and often initiates outside the DIC measuring window. Therefore, strain fields of notched tensile tests are used, which have a very well defined strain distribution. For calibration up to even higher strains than obtained from the notched tensile tests, an equibiaxial tension test is used.

Section snippets

Material description

The material used for this study is 22MnB5. Known by the commercial name Usibor® 1500 P, ArcelorMittal has provided the 22MnB5 steel grade with an aluminum-silicon coating that protects the metal against oxidation and decarburization during the press hardening process (ArcelorMittal, 2012). In the as-delivered state 22MnB5 has a ferritic/pearlitic microstructure, an ultimate tensile strength of 600 MPa, and a uniform elongation of 0.22. After quenching in cooled stamping tools, a fully

Material preparation

In order to create material samples with different hardnesses, the as-delivered sheets are first fully austenitized in a furnace at 950 °C for approximately 6 min, after which they are subjected to carefully controlled cooling processes in cooled and heated stamping tools. After opening the furnace and taking the sheets out, transfer from furnace to tool takes on average 3.8 s, after which another 5.8 s pass until full closure of the tool. During the total transfer time of 9.6 s, the sheets cool

Model assumptions

In the early design phase of a car, the rolling direction is generally not known. For full vehicle crash simulations, it is thus preferred to use isotropic constitutive models. As the experimental results presented in Section 3.4.1 revealed that all five considered hardness grades show only minor anisotropy of the stress–strain response, and that all r-values are in the range 0.75 < r < 1, an isotropic model will be used. The optimization routine for determination of the strain hardening parameters

Application of the method to other material hardness grades

To check the performance of the proposed parameter identification procedure, it is applied to all five available hardness grades presented in Section 3. With hardnesses between 183 and 497 HV0.1, these grades cover the full range of possible hardness values of 22MnB5, from a soft ferritic/pearlitic microstructure to fully hardened martensite. The results presented in this section have been obtained with the standard settings presented in Sections 4.2 Lower and upper bounds, 4.3 Parameter

Conclusions

An elasto-plastic constitutive model for five different hardness grades of boron steel 22MnB5 has been proposed and calibrated that provides an accurate prediction of the strain fields up to the point of fracture initiation. In the proposed model, a nonlinear combination of the Swift hardening law and a modified Voce law are used to represent the strain hardening behavior in combination with an isotropic version of the Yld2000-2D yield function. Initial fits of the two classical strain

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    The uniaxial tensile tests according to ISO6892-1:2016 standard are carried out for five kinds of specimens. Since the sheets are heated to be fully austenitized during industrial application, it is believed that the mechanical properties of all the specimens extracted at 0°, 45° and 90° along the rolling direction are the same, which has been verified by Eller et al. [13]. The force displacement curves measured by standard uni-axial tensile tests are shown in Fig. 5.

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