Numerical Modeling of the Crashworthiness of Thin-Walled Cast Magnesium Components

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1 Numerical Modeling of the Crashworthiness of Thin-Walled Cast Magnesium Components Cato Dørum 1, Odd S. Hopperstad 2, Magnus Langseth 2, Hariaokto Hooputra 3, Stefan Weyer 3 1 SINTEF Materials and Chemistry, No-7465 Trondheim, Norway 2 Department of Structural Engineering, No-7491 Trondheim, Norway 3 BMW Group, Forschungs- und Innovationszentrum, D Munich, Germany Abstract: A through-process methodology for numerical simulations of the crashworthiness of thin-walled cast magnesium components is presented. The methodology consists of casting process simulations using MAGMAsoft, mapping of data from the process simulation onto a FEmesh (shell elements) and numerical simulations using the explicit FE-code ABAQUS. In this work, generic High Pressure Die Cast (HPDC) AM60 components have been studied using 3- point bending and axial crushing tests. The experimental data are applied to obtain a validated methodology for finite element modeling of thin-walled cast components subjected to quasi-static loading. The cast magnesium alloy is modeled using a user-defined material model consisting of an elastic-plastic model based on a J2-flow theory and the Cockcroft-Latham fracture criterion. The fracture criterion is coupled with an element erosion algorithm available in ABAQUS. The constitutive model and fracture criterion are calibrated both with data from material tests and data from the process simulation using MAGMAsoft. Keywords: Castings, Constitutive Model, Crashworthiness, Failure. 1. Introduction As the lightest structural engineering metal available, magnesium is very attractive for structural automotive applications where weight saving is a matter of serious concern. With High Pressure Die Casting of magnesium and aluminum alloys, components with very complex, thin-walled geometry, like instrument panels, A and B pillars and front end structures, can be cast with a high production rate. The challenge with this production method is to optimize the process parameters with respect to the part design and the solidification characteristics of the alloy in order to obtain a sound casting without casting defects. Unbalanced filling and lack of thermal control can cause porosity and surface defects due to turbulence and solidification shrinkage. These defects can give low ductility compared to extruded materials, for instance. The HPDC method also leads to a 2006 ABAQUS Users Conference 203

2 skin-effect, where the microstructure of the castings near the free surfaces differs significantly from the interior as the skin region has much finer microstructure [1]. Design and production of thin-walled cast structural components for the automotive industry are challenging tasks that involve the development of alloys and manufacturing processes, structural design and crashworthiness analysis. In order to reduce the lead time to develop a new product it is necessary to use finite element analysis to ensure a structural design that exploits the material. Accurate description of the material behavior is essential to obtain reliable results from such analyses. In order to minimize the weight of the structural component while maintaining the safety in a crash situation, the ductility of the material has to be utilized without risking un-controlled failure. Hence, a reliable failure criterion is also required, giving limits for the plastic deformations under various loading combinations. Very precise and validated constitutive models and failure criteria are available for materials such as extruded aluminum and rolled steel [2]. However, much work is still to be done in this area for thin-walled cast materials. The long-term objective of this work is to develop design and modeling tools that allow the structural behavior of thin-walled cast components to be predicted when subjected to static and dynamic loads. In the current study, the structural behavior of generic structural HPDC components, shown in Figure 1, has been investigated using of axial crushing and 3-point bending experiments. The components were cast of magnesium alloy AM60 at Hydro's Research Centre in Porsgrunn, Norway with a Bühler SC42D 420-ton cold chamber die casting machine. Figure 1: Generic profile geometry ABAQUS Users Conference

3 2. Material Characterization The first step in the development of the through process strategy for numerical modeling of thinwalled castings is material characterization. In this work, the overall material hardening properties have been characterized using uniaxial tensile specimens cut from the 80 mm wide flange of the generic AM60 components illustrated in Figure 1. The specimens were aligned with the longitudinal direction of the components. Contrary to what is observed with extruded aluminum alloys, the test specimens were found to fail before the point of diffuse necking. The AM60 material has also been characterized using uniaxial compression tests, since previous work [3, 4] revealed different tensile and compressive behavior for magnesium alloys. For details, it is referred to Dørum et al. [5]. The experimental true stress - true plastic strain curves for both uniaxial tension and uniaxial compression are shown in Figure 2. No significant differences were found between the specimens regarding the observed stress level. Thus, the stress-strain curves given in Figure 2, can be viewed as representative for the overall hardening properties. Since the tensile tests [6] did reveal very small differences in stress level, it is assumed that the hardening properties are uniform throughout the component σ [MPa] Tension, AM60 Compression, AM ε p Figure 2: Typical experimental true stress true plastic strain curves for AM ABAQUS Users Conference 205

4 4.2% 2.3% 5.9% 22% 13% 12% 14% Figure 3: Experimental tensile elongation distribution. The scatter in fracture elongation is quite large. The poorest area is the outer stiffener on the outlet side, where values as low as 2-3% (effective plastic strain) have been measured. On the other hand, the best areas were in the 80 mm flange in front of the gates, where fracture elongation values as high as 22% effective plastic strain have been measured [6]. An overview of the tensile elongation distribution is provided in Figure NUMERICAL SIMULATIONS Numerical simulations of the generic structural High Pressure Die Cast components are performed using ABAQUS. The components are modeled using shell elements. Both 3-point bending and axial crushing tests are simulated, and the numerical results are then compared with experimental force-displacement characteristics. 3.1 CONSTITUTIVE MODEL Due to the different types of behavior in compression and tension, it has been decided to model the structural behavior of the AM60 components using a modified J2-flow theory as proposed by Dørum et al. [7]. However, it should be noted that since the strength differential (SD) effect in the investigated AM60 alloy is rather small, approximately 10 % comparing the compressive and tensile yield stress, the classical J2-flow theory would give quite similar results. The nonassociated extended J2-flow theory, which is adopted in this work, is designed to model the behavior of metals with strength differential effects. Therefore, the material model will now be referred to as the SD model. In the SD model, the von Mises yield surface is modified along the axis of normalized pressure such that different hardening properties can be given for two different user-defined values of triaxiality (e.g for uniaxial tension and uniaxial compression). The yield surface for the SD model can be expressed as: ( σ, ε ) σ κ( σ σ ε ) Φ = tr, = 0 e e e e ABAQUS Users Conference

5 where σ e is the effective von Mises stress, and ε e is the corresponding effective plastic strain. The term trσ σ represents a normalized pressure or a triaxiality parameter. κ represents the current e yield stress, or the hardening properties, and is given as a function of both the effective plastic strain and the normalized pressure. The material model is calibrated based on stress-strain curves obtained from uniaxial tensile and compression tests with specimens machined from the midsection of the 80 mm wide flange of the AM60 components. Consequently, the porosity and void growth in the cast material is directly taken into account in the plastic domain. For simplicity, the elastic properties are assumed to be unaffected by the porosity Fracture modeling The constitutive model used for the numerical simulations, briefly discussed above, includes the Cockcroft-Latham fracture criterion [9]. The criterion can be expressed as: ( σ ) W = d W max 1,0 εe c where σ 1 is the maximum principal stress and ε e is the true effective plastic strain. This criterion implies that fracture is a function of the tensile stresses and equivalent plastic strain and W c has the dimensions of work per unit volume. If the maximum principal stress is compressive, fracture cannot occur. Furthermore, neither stresses nor strains alone are sufficient to cause fracture. The critical value of plastic work, W c (or the corresponding value for the fracture elongation, ε f ), can then easily be calibrated against uniaxial tension tests. However, to take into account the variations in local tensile ductility throughout the cast AM60 components, a reliable distribution of W c is required. Here, this distribution is based on a porosity distribution predicted by casting process simulations using MAGMAsoft [10]. An overview of the predicted porosity distribution is provided in Figure ABAQUS Users Conference 207

6 The predicted porosity distribution shows that the porosity mainly is below 1% throughout the whole component. Consequently, relating the numerical porosity predictions to the experimental results found from the uniaxial tensile tests, it is assumed that bulk material with 0% porosity has a fracture elongation of 22% while bulk material with 1% porosity has a fracture elongation of 1.5%. For porosity levels above 1%, no further decrease in tensile fracture elongation is assumed. For material with porosity between 0% and 1%, the fracture elongation is assumed to be linearly dependent on the porosity. This range of ductility can be regarded as a conservative assumption. It Figure 4: Predicted porosity distribution (component shown from outlet side). should be noted that the predicted porosity distribution is simply viewed as a defect map for the local ductility of the bulk material. The tensile ductility of the bulk material is probably not dependent only upon the porosity but on other variables such as grain size and oxide layers. However, it is assumed that these other defects are of second order importance. 4. GENERIC COMPONENT TESTS The structural behavior of the cast generic AM60 components was studied using 3-point bending and axial crushing tests. For both the 3-point bending tests, two different load cases were defined. Depending on how the load was applied, on the 80 mm wide flange or on the small outer stiffeners, the load cases are referred to as the n-mode or the u-mode. The two different load cases are illustrated in Figure 5 and Figure 6, respectively. Numerical simulations were carried out using the explicit FE-code ABAQUS. The FE-model of the component consists of 5000 shell elements with one point integration in the plane and five through the thickness. This gives an average element size of approximately 4.0mm x 4.0mm. The relatively fine mesh was chosen to give a good description of buckling as well as being able to include a reasonable description of the material inhomogeneities ABAQUS Users Conference

7 In the simulations of the bending tests, the deformation was applied through rigid body motion of cylinders and plates, respectively, in which the material was assumed rigid. The deformation was applied smoothly but much faster than in the experiments. However, the analysis can be regarded as quasi-static, and therefore comparable with the experimental results, provided the kinetic energy and its variation are both negligible throughout the simulation. This was checked for all simulations. The comparisons of force-displacement characteristics obtained from the numerical analyses and the experiments are shown in Figure 7 and Figure 8 for 3-point bending tests and in and Figure 9 for axial crushing. The numerical data has been smoothened using a 9-point running average (total number of data points for each of the simulations is 1000) to give a better visualization of the force-displacement characteristics after fracture has occurred. It is seen from the experimental results that the components subjected to bending in the n-mode show a relatively large scatter in structural ductility. This is because the structural capacity of the components here is limited by tensile failure of the small outer stiffeners. When loaded in the n-mode, the small outer stiffeners experience approximately uniaxial tension. The inhomogeneous micro-mechanical structure present in the HPDC AM60 components causes large differences in local tensile ductility. Consequently, as a large scatter was found in tensile elongation from uniaxial tensile tests [6], the scatter in structural response must be expected to be of the same order. Most significant is the scatter for the case of 3-point bending in the n-mode, where fracture always occurs at the small outer stiffeners at or close to the midpoint of component. Figure 5: Test set-up for 3-point bending, load case (n) ABAQUS Users Conference 209

8 Figure 6: Test set-up for 3-point bending, load case (u). 12 ABAQUS Experiment 8 F [kn] w [mm] Figure 7: Force-displacement behavior for 3-point bending (n-mode) ABAQUS Users Conference

9 12 ABAQUS Experiment 8 F [kn] w [mm] Figure 8: Force-displacement behavior for 3-point bending (u-mode). For the components loaded in the u-mode, a very small scatter in force-displacement characteristics is found. Here, the first part of the force-displacement characteristics, i.e. before any fracture takes place, is limited by local buckling of the 40 mm side webs and the compression stiffeners. The buckling phenomenon is controlled by the material hardening, slenderness and initial imperfections of the profile. After the buckling load has been reached, the plastic deformation in the buckles increases rapidly as the buckles develop further and fracture occurs. Thus, relative to the global force-displacement behavior, the variations in local tensile ductility are of little importance, and the small scatter should be expected. For the case of axial crushing, the observed behavior is similar to the behavior found in the bending tests in u-mode. The first part of the force-displacement behavior is limited by buckling. After the buckling load has been reached, fracture occurs almost instantly and the load carrying capacity of the components is lost after small amounts of additional deformation. The figures further show that the structural response of the components subjected to bending is captured very well in the numerical simulations. Furthermore, it is seen that the failure criterion adopted in the SD model predicts failure quite accurately for both the cases of axial crushing and 3-point bending experiments in u-mode. For the 3-point bending loaded in n-mode, it is seen that the simulations using the SD model predict a structural response that is in the brittle range of the experimental results. This is in agreement with the choice of conservative assumptions for the fracture elongation distribution. It is seen that the fracture occurs at the small outer stiffeners that experience approximately uniaxial tension ABAQUS Users Conference 211

10 60 ABAQUS Experiment F [kn] w [mm] Figure 9: Force-displacement behavior for axial crushing. Experimental work performed by Sannes et al. [6] show that the tensile ductility varies significantly throughout the cast component. The poorest area is the small stiffeners where elongations as low as 2-3% have been measured. The 80 mm wide flange, from where the specimens for calibration of the failure criterion were taken, showed much better properties with elongation in the range of 12-16%. Thus, in the bending experiments in n-mode, the largest strains in the components are found where the mechanical properties are poorest. Examination of the tested components in this case has shown that the fracture always occurred close to the gating channels, probably due to a notch effect. However, a proper modeling of the notch is not possible when using shell elements. The fracture criterion proposed by Cockcroft and Latham seems to describe the structural ductility very well. More details concerning the experimental work is given by Dørum et al. [5] and Sannes et al. [6]. 5. CONCLUSION In the present study, High Pressure Die Cast AM60 components were subjected to 3-point bending and axial crushing. Laboratory experiments and explicit finite element simulations with shell elements were performed. Only quasi-static loading conditions were considered. An elastic-plastic constitutive model based on a non-associated extended J2-flow theory, referred to as the SD model, was implemented in the FE-code ABAQUS. The SD model is designed to model the ABAQUS Users Conference

11 behavior of metals with strength differential (SD) effects, and is calibrated against uniaxial tension and uniaxial compression tests. The fracture criterion proposed by Cockcroft and Latham [9] was adopted for the SD model, and was calibrated using both experimental results (uniaxial tensile testing and 3- and 4-point plate bending) and numerical predictions of the material quality resulting from casting process simulations. The experimental data from the uniaxial tensile tests was used together with porosity predictions to give a conservative map for the bulk material ductility. In general, experimental results presented both in this work and by Dørum et al. [5, 7] show that there exists a relatively large scatter in the force-displacement behavior when the HPDC components are subjected to deformation modes where the failure depends on the local tensile ductility of the material. For deformation modes where the force-displacement behavior is controlled by local buckling much less scatter is found. The fracture criterion proposed by Cockcroft and Latham [9] linked to results from casting process simulation gave very promising results. Especially the structural ductility of components subject to bending in the u-mode was quite accurately predicted using a constant value of W c. The reason for the good agreement in these cases is related to the evolution of local plastic strains with increasing global deformation after the buckling load has been reached. Here, a small increase in global deformation results in a relatively large increase in local plastic strains. Thus, the accuracy of the prediction of failure in terms of global force-displacement behavior is not very dependent on the accuracy of the W c parameter. For the case of 3-point bending in n-mode, the simulations gave a prediction in the brittle range of the experiments. This is in good agreement with the conservative assumptions made for the material ductility. In order to further improve the simulations with respect to the description of structural ductility, the quality of the material ductility map should be further examined. ACKNOWLEDGEMENT The authors would like to acknowledge Hydro Aluminium and the Research Council of Norway for their support through the NorLight project. REFERENCES 1. Shan, Z., Gokhale, A. M., Utility of micro-indentation technique for characterization of the constitutive behavior of skin and interior microstructures of die-cast magnesium alloys, Materials Science and Engineering A361, Lademo, O.-G., Engineering Models of Elastoplasticity and Fracture for Aluminium Alloys, Dr. Ing. Thesis. Department of Structural Engineering, Norwegian University of Science and Technology. 3. Hydro Magnesium, Data Sheet, Die Cast Magnesium Alloys. Norsk Hydro ABAQUS Users Conference 213

12 4. Kelley, E.W., Hosford, H.F., The Deformation Characteristics of Textured Magnesium, Transactions of the Metallurgical Society of AIME 242, Dørum, C., Hopperstad, O.S., Lademo, O.-G., Langseth, M., Aluminium and magnesium castings - experimental work and numerical analyses, International Journal of Crashworthiness 8, Sannes, S., Svalestuen, J.M., Dørum, C., Lademo, O.-G., Die Casting of Magnesium Alloys - Crash Requirements and Testing of Part Performance. Magnesium Automotive and End User Seminar, Aalen. 7. Dørum, C., Hopperstad, O.S., Lademo, O.-G., Langseth, M., Numerical modelling of the structural behaviour of thin-walled cast magnesium components, accepted for publication in International Journal of Solids and Structures. 8. Simo, J.C., Taylor, R.L., Consistent Tangent Operators for Rate-Independent Elastoplasticity, Computer Methods in Applied Mechanics and Engineering 48, Cockcroft, M.G., Latham, D.J., Ductility and the Workability of Metals, Journal of the Institute of Metals 96, Sannes S., Dørum C., Westengen, H., Gjestland H., Lohne O., The use of quality mapping to predict performance of thin-walled magnesium castings, 2005 SAE World Congress, Detroit ABAQUS Users Conference