Prediction of failure in bending of an aluminium sheet alloy

https://doi.org/10.1016/j.ijmecsci.2016.09.033Get rights and content

Highlights

  • Prediction of the mechanical behavior up to rupture of an aluminium alloy thin sheet.

  • Inverse parameter identification over a large database.

  • Anisotropy, mixed-hardening, macroscopic rupture criterion.

  • Tensile pre-strain followed by unconstrained bending up to rupture.

  • Prediction and comparison with experiments of the load variation and displacement at rupture in bending.

Abstract

This work is dedicated to numerical prediction of the bending of thin aluminium alloy sheets, with a focus on the material parameter identification and the prediction of rupture with or without pre-strains in tension prior to bending. The experimental database consists of i) mechanical tests at room temperature, such as tension and simple shear, performed at several orientations to the rolling direction and biaxial tension ii) air bending tests of rectangular samples after (or not) pre-straining in tension. The mechanical model is composed of the Yld2004-18p anisotropic yield criterion (Barlat et al. [3]) associated with a mixed hardening rule. The material parameters (altogether 21) are optimized with an inverse approach, in order to minimize the gap between experimental data and model predictions. Then, the Hosford-Coulomb rupture criterion is used in an uncoupled way, and the parameters are determined from tensile tests, both uniaxial and biaxial, with data up to rupture. In a second step, numerical simulations of the bending tests are performed, either on material in its original state or after pre-straining in tension, with pre-strain magnitudes increasing from 0.19 up to 0.3. The comparisons are performed on different outputs: load evolution, strain field and prediction of the rupture. A very good correlation is obtained over all the tests, in the identification step as well as in the validation one. Moreover, the fracture criterion proves to be successful whatever the amount of pre-strain may be. A convincing representation of the mechanical behavior at room temperature for an aluminium alloy is thus obtained.

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):(τ+c1σn)f=c2where c1, the friction coefficient and c2, the shear resistance, are two material parameters. σn and τ 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 θ 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).dD=m(ϵ̅pϵ̅fpr[η,θ̅])m1dϵ̅pϵ̅fpr[η,θ̅]where ϵ̅p is the equivalent plastic strain and ϵ̅fpr 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 Φ is defined by Eq. (3).Φ(S˜(i))=[14j,k1,3|S˜j(1)S˜k(2)|a]1a,i=1,2

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.

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