Investigation on springback behaviours of hexagonal close-packed sheet metals

Highlights

• Novel analytical method for predicting bending and springback behaviours of HCP sheet.

• Bending moment-curvature relationships and springback of HCP metals are obtained analytically.

• Considering the nonlinearity in unloading, variable elastic modulus is implemented to improve springback prediction.

• The springback increases when increasing back force only under uniaxial loading condition.

• FE model is developed successfully by implementing this analytical method with abaqus.

Abstract

In this study, a novel analytical method for predicting bending and springback behaviours of hexagonal close-packed (HCP) sheet metals is presented. The proposed analytical approach is developed by using the Cazacu-Barlat 2004 asymmetric yield function and isotropic plastic hardening rule. This model can be used to determine bending moment-curvature relationships and springback of HCP metals under uniaxial and plane strain loading conditions. Furthermore, to capture the nonlinearity in unloading and to improve springback prediction, the variable elastic modulus approach is implemented in the proposed model. The proposed new model reveals that reverse effects of the back force on springback behaviours cannot be found under the plane strain condition, which could not be found by using any existing models. Moreover, the analytical model is implemented into Abaqus via UMAT subroutine for its application in complex cases, and a numerical model is then developed as a showcase. The proposed methods are validated by using those experimental results available in literature. The results show considerable improvements by considering the plane strain condition and nonlinear unloading.

Introduction

Hexagonal closed pack (HCP) metals are important structural materials due to their light weight, high strength-to-weight ratio, and unique material behaviours. Magnesium alloy is one of the most popular HCP metals utilised in industry. It is one of the lightest existing structure components with the HCP crystal structure and has become increasingly important for automotive manufacturing industries [9,21,40]. For the automotive industry, one of the most effective ways to improve fuel efficiency and reduce carbon dioxide (CO₂) emissions is to adopt lightweight metals for structural components of vehicles. However, in sheet metal forming, due to their crystalline structure, the number of slip system is limited and consequently, they exhibit poor formability [1,16]. In addition, due to their low elastic property, they exhibit large amounts of springback (Cubberly, [7,27]. Consequently, due to the high manufacturing cost, their applications are limited [46,16,11]. Hence, an accurate characterisation of the material behaviour of magnesium alloys is crucial to enable the possibility of enhancing mechanical properties and process optimisation to extend their applications.

Mechanical behaviour of magnesium alloys is unique in comparison to conventional cubic crystalline metals such as aluminium alloys and steels [29,27,24]. Mechanical properties of magnesium alloys can be different, depending on the loading path and initial texture. For example, in a rolling process, the strong basal texture is induced to wrought magnesium alloys [42,19]. Hence, they show unique material behaviours such as tension–compression asymmetry (TCA) or eccentricity in the initial yield points. Usually, the compressive strength is much lower than the tensile strength [3,13]. Moreover, the asymmetry can be observed in the subsequent plastic hardening, which is referred to as a “hardening differential” or “flow asymmetry” [27,53,24]. The strong basal texture of cold-rolled HCP metals is the main reason of the poor ductility and the strong tension–compression asymmetry [12,39,44]. In other words, the c-axes of the majority of grains are perpendicular to the plane of the sheet [14,52]. At room temperature, depends on the c/a ratio, the basal crystallographic texture restricts main deformation modes to either prismatic {101̅0} 〈112̅0〉 or basal {0001} 〈112̅0〉 slips along the basal direction 〈a〉 in plane (RD/TD) tension. The 〈a〉 type slips cannot cause any deformation along c-axis. A great anisotropy or large r-value was observed in magnesium alloy sheets due to these two main slip modes [34,16]. The pyramidal 〈c + a〉 slip system or compression twining {101̅1}<101̅2>, which are required for contractions of c-axis are very hard to activate at room temperature and consequently causing delay in yielding [20,48,10]. Easily activated tensile twining {101̅2} 〈<101̅1〉, which causes an extension along c-axis, is activated during tension along c-axis or under compression perpendicular to the c-axis [47,41]. This deformation mechanism is pre-dominant during the in-plane compression of magnesium alloy sheet and exhausted with continuous compression. Abrupt grain reorientation, creation and disappearance of twining boundaries lead to initiation of slip mechanism again, and consequently, the flow stress rises. Hence, due to this directional twining and exhausted twining, the stress–strain curve during in-plane compression shows an unusual concave-up or S-shaped curves and consequently the strong tension–compression asymmetry rapidly [26]. It is worth to indicate that for most HCP metals the thinning is negligible since the activation of pyramidal slip and compression twining are much harder than the main deformation mechanisms of basal slip and tension twining [6].

In order to capture the unique asymmetric behaviour of magnesium alloys, two approaches were used by researchers. One method is to consider an initial translation of yield surface in a hardening law, for example, applying non-zero back stress in a combined isotropic-kinematic hardening [26,25,24,34]. Lee et al. [26] applied a two-yield surface approach for AZ31B magnesium alloy sheets. Lee et al. [24] used the concepts of distortional hardening approach to introduce a model capable of extending the tension–compression asymmetry under different loading conditions. More recently, Park et al. [37] developed a method for anisotropic hardening based on non-associated flow rule to capture the strength differential effect. The second method applies an asymmetric yield function. For instance, Hosford [15] added linear stress terms into Hill's 48 yield criterion. Cazacu and Barlat [5] modified isotropic Drucker yield function [8] by applying the second and third invariants of stresses into the yield function. Yoon et al. [49] added the first stress invariant to an asymmetric yield function and presented a pressure-sensitive yield function.

Several experimental and numerical studies were conducted to capture the TCA in bending and springback of HCP metals. Lee et al. [28] used the constitutive model for the unique asymmetric hardening behaviour of magnesium alloy sheet presented by Lee et al. [26], to predict the springback of magnesium alloy. Tari et al. [45] performed experimental tests on magnesium AZ31B sheet at room temperature. They presented an evolving anisotropic/asymmetric continuum-based material model based on Cazacu et al. [6]. Härtel et al. [13] studied the effects of the TCA of magnesium alloy in the bending process numerically. They considered several modelling scenarios for modelling the material behaviour; including the plastic flow curve achieved from tensile, compressive or both curves. Badr et al. [2] used a strain path dependent constitutive model to describe the inelastic behaviour of titanium at room temperature. They used a homogenous yield function combined with the HAH anisotropic hardening model.

Along all the studies mentioned above, few analytical studies are available in literature regarding the bending and springback of asymmetric materials. Lee et al. [27] introduced a uniaxial analytical method to study the elastic-plastic pure bending and bending-under-tension of AZ31B magnesium alloy sheets. They used discrete linear hardening in each deformation region to approximate the stress–strain curve of the magnesium alloy sheet. It was found that the springback increases unusually with increasing the applied force. Kim et al. [18] presented a uniaxial semi-analytical model to study the sidewall curve of magnesium alloy in bending/unbending under tension. They found that the tensile force has a reverse effect on springback and sidewall curl of magnesium alloy in comparison to conventional sheet metals. Kuwabara et al. [22] and Maeda et al. [30] reported TCA in plastic behaviours of phosphor bronze and DP980. They used strain-stress curves, which were obtained from uniaxial tension–compression tests to calculate the bending moment-curvature diagrams. To find the stress in the sheet thickness, they divided the sheet thickness of the specimen into 100 layers.

All the above-mentioned analytical studies are uniaxial semi-analytical methods. They are based on piecewise fitting of the strain–stress curves of materials without using key factors of continuum plasticity such as yield surface or flow rule. Therefore, their methods are limited to uniaxial condition, relate to many parameters, and relay on an extensive amount of calculations. Mehrabi and Yang [31] presented a simple analytical solution for pure bending of HCP metals without considering the effects of neutral surface shift. Moreover, Mehrabi et al. [32] proposed new analytical and numerical methods to model the tension–compression asymmetry in the plastic behaviour of steels under bending. In this study, based on our previous works [31,32], a new analytical method for bending of HCP metals is proposed. The new bending elastic–plastic method is developed under uniaxial and plane strain conditions, based on the Cazacu-Barlat 2004 asymmetric yield function and an isotropic plastic hardening. It is suitable for bending of plates with a high width to thickness ratios. This new method can greatly simplify predicting springback without applying much complicated approaches. It can be used for both analytic study and numerical analysis by implementing it into the commonly-used commercial finite element analysis package – Abaqus/Standard via a user-subroutine UMAT. As a case study, the analytical model is applied to predict the moment-curvature and springback behaviours of AZ31B magnesium alloy sheets under uniaxial and plane strain loading conditions. Moment-curvature diagrams are obtained for both pure bending and bending under tension processes. Furthermore, springback values under various back forces are determined. To further improve the accuracy of predicted springback, the variable elastic modulus is applied for unloading. In order to capture the asymmetric behaviour of the material in the subsequent hardening, two different hardening laws based on Voce hardening rule are defined. The two hardening laws for tension and compression are different only in term of hardening parameters, which can be determined by tension and compression tests, respectively. In order to verify the results, the calculated results for the moment-curvature and springback are compared with the available results from the study conducted by [27]. Moreover, based on the developed UMAT code, finite element modelling and simulations with ideal loading conditions and the experimental conditions are performed, respectively, and the obtained results are compared for validation. Compared to exsiting models, one of the significant improvements of the current model is that the plane strain condition can be applied in deriving the bending and springback prediction equations. This condition is only applicable because of the novelty of the proposed analytical method on fusion of the asymmetric yield function and the isotropic plastic hardening. The results obtained by using this novel model show that the reverse effects of the applied tensile back forces on bending moment and springback, which was reported by [27], only happens in uniaxial condition. Under the plane strain condition, the bending moment decreases when increasing the tensile back force. Moreover, for the sheet with 2-mm thickness and under the uniaxial condition, the reverse effects only happen for a specific range of curvatures. In addition, the application of the plane strain condition and variable elastic modulus in the proposed model significantly improves the springback results. Finally, the numerical results based on the actual experimental conditions explain the differences between the predicted and measured values of springback due to the differences on the boundary condition settings.