Formability and fracture in deep drawing sheet metals: Extended studies for pre-strained anisotropic thin sheets

Highlights

• LDR and fracture cup height was experimentally evaluated for different sheet materials.

• Necking and fracture limits were theoretically estimated using MK and BW model.

• LDR, thinning profile estimated using MK model incorporating anisotropy yield theory.

• Anisotropic BW curve was used into GISSMO platform to predict fracture cup height.

• PEPS based failure limits used to identify necking and fracture of pre-strained sheet.

Abstract

In this study, experimental and numerical investigations were conducted to predict the formability and fracture behavior of as-received and pre-strained sheet materials during deep drawing process. In this context, various laboratory scale experimental setups were developed to impart different types and amounts of pre-strain such as 5% and 10% equi-biaxial pre-strains (5% EBP and 10% EBP), 10% plane strain pre-strain (10% PSP), and 10% uni-axial pre-strain (10% UP) on the extra deep drawing (EDD) steel and aluminum alloy (AA5052) sheets of 1.2 mm thickness. Further, all the pre-strained sheet samples were deformed using a cylindrical deep drawing setup. The forming limit diagrams (FLDs) of as-received sheets were predicted by the Marciniak–Kuczyński (MK) model incorporating different anisotropic yield functions such as Hill48 models identified based on r-values (Hill48-r), and yield stresses (Hill48-σ), and the non-quadratic plane stress Yld2000-2d model. Also, the Bao–Wierzbicki (BW) fracture curve was calibrated using different anisotropic yield functions. Subsequently, the formability in terms of limiting drawing ratio (LDR) was predicted using the MK-FLD. The BW fracture curve was incorporated into the generalized incremental stress state dependent damage model (GISSMO) platform in LS-Dyna software and the fracture behavior was predicted in terms of failure location and cup height at the onset of fracture. It was also found that the incorporation of Barlat Yld2000-2d yield function into the FE simulation efficiently predicted the necking and fracture behavior of as-received sheets. Furthermore, the concept of path independent polar effective plastic strain (PEPS) based failure model was used to predict LDR, thinning profile and fracture cup height of all the different pre-strained sheets. Finally, the strain paths and experimental fracture strains were plotted in 3D fracture locus to get insight into the deformation behavior during the deep drawing experiments.

Introduction

The finite element (FE) simulation of the sheet forming processes is very efficient to achieve precise prediction of part performances and post-forming characteristics such as the surface strain distributions and thickness distributions. Moreover, the FE approach can help to save energy and resources, which increases the productivity of the assembly line. However, the precise failure prediction is a challenging task to the engineers during forming of the sheet metal components. Different material attributes such as constitutive models and failure models are required to perform a reliable FE simulation. The modelling of plastic deformation requires proper identification of yield functions, flow rules and hardening laws. Also, different theoretical models are used to predict the necking and fracture strains of sheet materials during plastic deformation. Several theoretical models, such as maximum force based models, bifurcation theory based models and geometrical imperfection based models, have been proposed to predict the necking limit strains or forming limit diagram (FLD) of sheet materials [1], [2], [3]. Among the flow localization models, Marciniak and Kuczyński model (MK model) [3] has been widely used to predict the analytical FLD of the sheet materials. As another approach, the fracture during the sheet metal forming operation can be estimated by using ductile fracture models coupled with different plasticity theories [4], [5], [6]. Among the different fracture models, Bao and Wierzbicki [7] developed a complete shape of the fracture curve from a number of experimental trials. Recently, researchers are interested in implementing the different anisotropic yield functions into the failure models for precise prediction of the failure strains. Butuc et al. [8] showed the influences of different yield functions such as von-Mises, Hill48, Hill79, and Barlat Yld96 on the MK-FLD prediction of AA6016-T4 sheet material. Panich et al. [9] demonstrated the effect of the hardening model on the MK-FLD, and concluded that the implementation of a non-quadratic anisotropic yield function, i.e. Yld2000-2d model, along with the Swift hardening model best predicted the strain at localized necking. On the other hand, Park et al. [10], [11] predicted the fracture strain of anisotropic dual phase steel by incorporating Hill48 and Barlat Yld91 models into the Lou-Huh damage model. More recently, Basak and Panda [12] implemented the Barlat Yld2000-2d function into MK-FLD and Bao-Wierzbicki fracture model to predict the necking and fracture strain, respectively, of the anisotropic sheet material. Anderson et al. [13] implemented the Lou-Huh fracture surface into the phenomenological GISSMO platform in LS-Dyna software to predict the failure for DP780 sheet metal. Also, there are some recent analytical models, based on continuum damage mechanics (CDM) approach, have been developed for more accurate prediction of the forming processes. These advanced anisotropic models are associated with nonlinear kinematic and isotropic/anisotropic hardening with an associated/non-associated flow rule [14], [15], [16], [17]. However, from the literature studies, it is evident that limited research works have been performed on the implementation of the anisotropic fracture curve into the GISSMO platform for the prediction of fracture strain during the sheet forming process.

Deep drawing is one of the popular sheet forming processes in the manufacturing industries to produce metal parts by pushing a sheet metal into a forming die cavity with a desired punch shape. In addition to the purpose of successfully manufacture various industrial components, a deep drawing experiment also serves as a basic laboratory scale test for evaluating the formability of sheet materials. The formability or maximum drawability of a sheet material during the deep drawing experiment is quantified with a limiting drawing ratio (LDR), which is defined as the ratio of the maximum diameter of the circular blank which can be drawn safely to the punch diameter. The LDR depends upon several parameters such as strain hardening exponent, normal anisotropy of the sheet metal, frictional condition, tooling geometry etc. [18], [19]. Leu [20] developed an analytical model based on a force equilibrium condition to predict the LDR of a sheet metal. During deep drawing, it is found that different stress and strain states are acting at different portion of the drawn cup. The cup flange, cup wall and cup bottom are subjected to tension-compression, plane strain and tension-tension deformation modes, respectively. Hence, numerical studies were carried out by different investigators [21], [22] to predict the drawing force, LDR, thinning profile, cup height etc. The LDR is often restricted by the localized necking and/or tearing/fracture during deep drawing experiment. Therefore, a precise prediction of the failure is very important to reduce the scrap during the production of the deep drawn components in the press shop floor.

For manufacturing a complex shaped automotive part, a multi-stage forming process is often practiced. Due to the adaption of different die-punch configurations and process variables at each stage, a prominent change in the strain path usually takes place which affects the failure strains of the sheet metals. Therefore, it is necessary to have prior knowledge on the failure limits when the sheet metals are subjected to different combinations of loading paths. The dynamic nature of the necking and fracture limit diagram was established through different experimental and numerical studies in previous literature [23], [24], [25]. To overcome the dependency of the conventional strain based forming limit diagram, several path independent failure models such as stress based limit [26], [27] and polar effective plastic strain (PEPS) based limit [28], [29] were implemented. However, the effect of pre-strain on the LDR during deep drawing experiments has not been well addressed in the available open literature. Also, the path independent failure limit for evaluating LDR and fracture is an interesting area of research which has not been explored in the previous studies. Therefore, in the present study, a cylindrical deep drawing experimental setup was designed to predict the deep drawing behavior of different pre-strained extra deep drawing (EDD) steel sheets and aluminum alloy AA5052 sheets. In this context, experiments were conducted to impart different amount and direction of pre-strain on the EDD steel and AA5052 sheets. Subsequently, the LDR, thickness profiles and cup height at the onset of fracture were experimentally evaluated for as-received and pre-strained sheet materials. Moreover, the LDR and thickness profiles of as-received sheets were predicted from FE simulations with different anisotropic yield functions and MK-FLD as a failure model. The fracture cup height was predicted by implementing a newly developed anisotropic BW fracture surface into the GISSMO platform in LS-Dyna software. In addition, the concept of path independent polar effective plastic strain (PEPS) based failure model was applied for the prediction of the necking and fracture behavior of pre-strained deep drawn cups. Lastly, the strain path and the fracture data points obtained from the deep drawing experiments of as-received and pre-strained specimens were analyzed in the 3-dimensional stress triaxiality locus to get insight into the predictive efficiency of the newly developed anisotropic fracture surface.