Prediction of failure in bending of an aluminium sheet alloy
Introduction
The prediction of rupture for ductile materials in sheet metal forming has been an active field of research for the last decades. From an industrial point of view, numerical process validation is classically based on forming limit diagrams (FLD), which are a representation of the maximum principal strains in the sheet plane before the onset of necking. Such tools were shown to be dependent on the thickness of the sheet and on the strain path [1], [6], [2] and will not be further considered because they are not designed to predict failure in bending and hemming. With an extension beyond necking, fracture can be predicted using fracture forming limit diagrams (FFLD) or more generally by macroscopic rupture criteria based on stress and strain invariants, e.g. [3]. Such an approach neglects the microscopic mechanisms of ductile damage but considers the condition of the stress and strain states for a first macroscopic crack occurrence. This study is limited to fracture criteria based on a full description of the stress state and which have been developed making use of the Mohr-Coulomb (M-C) fracture criterion [8], [22], [5], [4] given by Eq. (1):where , the friction coefficient and , the shear resistance, are two material parameters. and are respectively the normal stress and the corresponding shear stress of the cutting plane. The macroscopic fracture criteria use the following stress tensor invariants: the stress triaxiality η, the Lode angle and the equivalent stress or equivalent plastic strain. The M-C criterion was used for its explicit dependence on the Lode angle parameter, which is missing in most of the other models of ductile fracture [6], [33] and was successfully applied to ductile fracture of several metallic alloys [6], [8], [33], [21], [5], [9], [7]. Tests on sheet specimens combining tension and shear were performed and rupture was predicted using the Modified M-C criterion (MMC) except [39], [10] for the Hosford-Coulomb model. Bai and Wierzbicki [6] added mechanical tests on smooth and notched round bars in tension and compression and on a square bar in tension. Beese et al. [8], Dunand and Mohr [9] and Li et al. [7] added punch tests for validation of the criterion. Marcadet and Mohr [10] accounted for variable pre-strains and, as a result, non-proportional loading paths. They investigated both tension and compression of DP780 steel samples, cut from thin sheets, with the help of an anti-buckling device. The non-linear strain path came from the fact that the samples were subjected to several values of compression and then were deformed in tension up to the rupture of the sample.
Thin aluminium alloys sheets are now commonly used in the automotive industry, in particular for outer panels; during the forming and assembling steps, the sheets are subjected to bending over small radii and there is a strong interest in the numerical prediction of rupture under such mechanical conditions. Hudgins et al. [11], Kim et al. [12], Li et al. [13] and Luo and Wierzbicki [14] used experimental bending processes including simultaneous tension and bending, qualified as stretch bending, where the metallic sheet is clamped at its two ends between the lower die and a binder while the upper die moves down. This stretch bending process is used to represent at best the first stage of a typical stamping process where fracture can occur along the bent areas of small radii and are referred to as “shear fracture” [11]. This kind of fracture is extremely dependent on the radius of the die over the sheet thickness (R/t) ratio; when the ratio is low, fracture occurs in the bent section. Concerning the stress triaxiality, a complex history at some critical points, where rupture later occurred, was observed for DP780 steel thin sheets [14], which calls for the use of a stress state dependent fracture model. Le Maoût et al. [15] studied hemming of thin sheets of an aluminium alloy.
Hemming is an assembly process that consists in folding the edge of a thin sheet over an inner reinforcing one. Hence, the apex of the folded area is subjected to high strains. As opposed to the stretch bending and in order to better represent pure bending, another type of bending called free bending or air bending was studied, for example [26], [29], [37], [13]. The blank is supported at the outer edges without any clamping, as opposed to stretch bending. The bending radius is here determined by the displacement of the die and not by its own radius, reducing the force required for forming. This bending process allows for the plane strain state assumption if the width to thickness ratio is higher than 8 [18]. Also, through experiments and simulations, the authors showed that fracture occurred at the apex of the convex free surface of the sheet and described its propagation and patterns.
The fracture in bending has been modeled taking into account void growth and coalescence with the use of a damage model, for example the Gurson-Tvergaard-Needleman damage model, and applied to DP1000 steel and aluminium alloys of series 5000 and 6000 [15], [37], [7]. Other studies have been accomplished with the use of macroscopic criteria, for example the MMC criterion [33], [12], [28] applied to DP780 steel and aluminium AA2024-T351. Also, Mishra and Thuillier [16] applied several macroscopic fracture criteria, e.g. Cockroft and Latham [21], Brozzo [22], Ayada [23] and Rice and Tracey [24] to the rupture in bending of DP980 steel sheets. The authors showed that the two criteria that are explicitly dependent on the stress triaxiality presented the best results, strengthening the choice of a M-C type fracture criterion.
However, applying pure bending for AA6016 thin sheets does not lead to fracture, in other words, the strain applied by the air bending process alone is not sufficient to achieve rupture of thin sheets samples [25]. Hence, aluminium sheets were pre-strained before being bent in order to obtain fracture in bending. Such a two steps process is of interest, as representative of hemming of parts that were deep drawn; indeed, an equivalent plastic strain of 0.15 can be reached for an automotive hood, prior to hemming [26]. It should be emphasized that though necking is prohibited in automotive parts [27], high pre-strain values were used, in order to extend the range of the validation, as bending occurred over a small area centered on the necking zone. Moreover, though having or not unloading between the pre-strain and the subsequent path is clearly an issue when considering forming limit curves [30], [17], unloading between the pre-strain and the subsequent strain path was considered, as representative of the unloading of the industrial part in-between the deep drawing and the hemming stage. The succession of two processes implies non-linear or non-proportional loading path, which is to be taken into account in the fracture criterion. As mentioned earlier, Marcadet and Mohr [10] investigated fracture in tension of pre-strained thin sheets using the damage evolution law given by Eq. (2).where is the equivalent plastic strain and denotes the equivalent strain to fracture under proportional loading. The parameter m emphasizes the effect of the stress state at the early stage of loading or the effect of the stress state right before fracture initiation, while the case of a linear damage accumulation rule can be retrieved for m=1. Papasidero et al. [30] also worked on pre-strained samples and presented a detailed description of the influence of parameter m using various pre-strained specimens coupling compression and torsion.
The mechanical behavior of AA 6016 thin sheet was already characterized experimentally in uniaxial tension, simple shear and biaxial tension as well as in bending without any pre-straining [7], [35] and with tensile pre-straining [25]. The purpose of the present work is to obtain an accurate and reliable description of this mechanical behavior, in order to predict the rupture in bending. An anisotropic elastic-plastic model is associated to a fracture model for aluminium alloy AA 6016. This model includes (i) an anisotropic yield function, (ii) a mixed (isotropic and kinematic) hardening law and (iii) a fracture criterion. Such a representation, though demanding in terms of model implementation and material parameter identification, should be an asset within a virtual design strategy. The first section recalls briefly the constitutive equations and in a second step, the inverse identification methodology developed to determine material parameters is detailed. Finally, finite element simulations of the tensile pre-strain and of air bending tests are presented with the goal of validating the model against experimental results.
Section snippets
Constitutive modeling and rupture
AA6016 alloy presents anisotropic mechanical characteristics, as for a large number of aluminium alloy sheets obtained by rolling. To take into account the anisotropy, the Yld2004-18p yield function was chosen [32]. This function gives a high flexibility when considering a wide range of experiments, due to its important number of parameters. The constitutive equations are written in a large strain framework. The yield function is defined by Eq. (3).
Material parameter identification
There are a total of 22 material parameters, to be identified over an experimental database including 8 mechanical tests. However, 18 outputs were chosen to represent the material behavior within different strain ranges, the highest equivalent plastic strain being for the hydraulic bulge test. The identification of the parameters was then performed within an inverse approach, which consisted of minimizing iteratively the gap between numerical predictions and experimental results quantified by
Validation in bending with and without pre-strain
The parameter identification presented in the previous section is now used to predict the rupture in air bending of thin sheets. Neither rupture nor the initiation of rupture are reached when bending non pre-strained samples [25]. In order to overcome this limitation, sheet samples are subjected to various values of tensile pre-strain before being used for the air bending tests. Above a given pre-strain level, fracture is successfully reached in bending. The aim of this section is to briefly
Conclusion
Large strain tension – bending fracture experiments were performed on thin sheets of aluminium alloy (AA 6016). Pre-strain tensions were achieved from moderate up to high strains, with local strain of the order of 0.45. Then, rectangular samples were cut out of the tensile specimen and air bending tests were carried out. A mechanical model based on a mixed hardening law, to account for the Bauschinger effect at the tension – bending transition, and an anisotropic yield criterion [32] is used
Acknowledgments
The authors thank R. Valente and T. Grilo, from University of Portugal, Aveiro, for providing the first version of the UMAT program.
References (40)
- et al.
Generalised forming limit diagrams showing increased forming limits with non-planar stress states
Int J Plast
(2009) - et al.
Path independent forming limits in strain and stress spaces
Int J Solids Struct
(2012) - et al.
Calibration and evaluation of seven fracture models
Int J Mech Sci
(2005) - et al.
Prediction of shear-induced fracture in sheet metal forming
J Mater Process Technol
(2010) - et al.
Partially coupled anisotropic fracture model for aluminum sheets
Eng Fract Mech
(2010) - et al.
On the predictive capabilities of the shear modified Gurson and the modified Mohr-Coulomb fracture models over a wide range of stress triaxialities and Lode angles
J Mech Phys Solids
(2011) - et al.
Effect of compression-tension loading reversal on the strain to fracture of dual phase steel sheets
Int J Plast
(2015) - et al.
Predicting instability at die radii in advanced high strength steels
J Mater Process Technol
(2010) - et al.
The shear fracture of dual-phase steel
Int J Plast
(2011) - et al.
Theoretical failure investigation for sheet metals under hybrid stretch-bending loadings
Int J Mech Sci
(2015)