Necking behavior of AA 6022-T4 based on the crystal plasticity and damage models

Abstract

In order to analyze necking behavior with a continuum level approach, initial imperfection or bifurcation algorithm should be employed. The initial imperfection is arbitrary and has a great effect on the necking behavior of sheet metals. Most polycrystalline materials have natural imperfection such as orientation mismatches across the grain boundaries. In polycrystal plasticity, orientation mismatch is considered as material imperfection instead of geometric imperfection including thickness unevenness. The main idea of this study is to investigate the possibility that the stress concentration on a grain boundary (caused by orientation mismatch) works as imperfection and finally causes necking. For the consideration of stress concentration along grain boundaries, a crystal plasticity model was introduced in three-dimensional finite element analysis of a tensile test. A small region of a tensile specimen was divided into sufficient number of octahedral grains and each grain was discretized by fine tetrahedron elements. The same orientation angle was allocated for all the elements in each grain, but different orientations for grain by grain. Using this crystal plasticity-based analysis, stress concentration in the grain boundaries can be predicted. Also, four damage models based on the critical strain and stress were proposed to consider material softening due to void initiation and growth under the framework of crystal plasticity theory. The damage parameters for each model were determined based on the stress–strain relations obtained from experiment. Void nucleation, growth and coalescence behaviors during necking were reasonably predicted. Finally, the predicted necking behaviors of AA 6022-T4 were compared with the experimental results in terms of necking strain, deformed shape, and necking direction.

Introduction

Finite element analysis based on crystal plasticity model is widely used to understand the plastic behaviors of materials. Most researches have been focused on developing the constitutive relations of single crystal (Kim and Oh, 2003, Borg, 2007, Li et al., 2008) and polycrystalline materials (Dao and Asaro, 1996, Yoon et al., 2005, Zamiri et al., 2007, Li et al., 2004, Li et al., 2008). Yoon et al. (2005) investigated the anisotropic hardening behavior of cube textured aluminum alloy sheets using a crystal plasticity model. Recently, various studies have been carried out predicting the twinning behavior of hexagonal close packed (HCP) materials (Choi et al., 2010, Zhang et al., 2007, Bridier et al., 2009, Beausir et al., 2007, Knezevic et al., 2010, Wang et al., 2010, Wang et al., 2013). Among them, Choi et al. (2010) analyzed the stress concentration at the grain boundary and the twinning behaviors of magnesium alloys. They allocated the grain orientation obtained from electron backscatter diffraction (EBSD) to a regular element. Knezevic et al. (2010) investigated effect of anisotropic strain hardening on twinning behavior of AZ31 using three-dimensional crystal plasticity finite element method.

For computational effectiveness, crystal plasticity finite element analyses were sometimes carried out in two-dimensional plane stress or plane strain framework. However, 2D analysis is not capable of predicting three dimensional deformations including out of plane (Rossiter et al., 2010). Moreover, the localized deformation after necking is three dimensional, and this localized deformation cannot be properly modeled in 2D analysis (Simha et al., 2008). In this study, the necking behavior of AA 6022-T4 is analyzed using crystal plasticity finite element method (CPFEM) in three-dimensional framework. Using a conventional finite element method based on continuum mechanics, a localized necking cannot be predicted without initial imperfections (Marciniak and Kuczynski, 1967, Wang et al., 2011, Bettaieb and Abed-Meraim, 2015) or bifurcation algorithm (Hill and Hutchinson, 1975, Okazawa, 2010, Yoshida and Kuroda, 2012). However, in crystal plasticity-based finite element analysis, the stress concentration from orientation mismatch near the grain boundaries can cause initial voids and initiate the localized deformation. In polycrystalline metals, each grain has its own orientation. Therefore, the orientation of one grain does not coincide with that of its neighboring grains and stress concentration may take place during deformation. To analyze the stress concentration near grain boundaries during necking, crystal plasticity finite element method is introduced by conducting tensile test simulation.

There are small and large scale CPFEM regarding the scale of volume a material point represents. A recent comprehensive review of CPFEM is done by Roters et al. (2010). In small scale CPFEM simulations, the size of the finite element mesh is equal to or smaller than the grain size. With such sub-grain resolutions, heterogeneity of deformation with a single crystal is taken into account (Zhang et al., 2009, Saito et al., 2012, Mayeur et al., 2013, Rossiter et al., 2013, Sabnis et al., 2013, Eisenlohr et al., 2013, Choi et al., 2013, Choi et al., 2014). Large scale refers to simulations with more than one crystal assigned to one integration point and the number of grains is usually large enough to study the average behavior of the material such as the texture evolution. Homogenization schemes are needed to connect a material point to each constituent grain. Commonly used homogenization schemes were described in detail in the literatures by Lebensohn and Tomé, 1993, Segurado et al., 2012 and Qiao et al. (2014). Especially, a crystal plasticity-based forming limit prediction in large scale CPFEM approach was presented by Inal et al., 2005, Neil and Agnew, 2009 and Franz et al. (2013). Effect of grain orientation on sheet necking was investigated by Wu et al. (2007).

Asaro and Needleman (1985) showed that localized plastic deformation is strongly influenced by microstructure. In many studies of crystal plasticity finite element method it is assumed that many grains exist in an element and homogenization or averaging scheme is used to integrate a single crystal constitutive relation into a polycrystalline constitutive relation (Zamiri et al., 2007, Yoon et al., 2005, Dao and Li, 2001). Using a homogenization scheme in a large scale CPFEM, therefore, the size of an element can be far greater than the one of a grain, and a large scale problem such as sheet forming can be analyzed. However, the localized deformation and stress concentration between grains cannot be predicted using a homogenization method. To overcome this difficulty, Wang et al. (2011) introduced initial imperfections, proposed by Marciniak and Kuczynski (1967), to predict the localized necking and forming limit diagram (FLD) for a magnesium alloy sheet using crystal plasticity finite element method.

In this study, grains are discretized into fine elements in order to describe the stress concentration near grain boundaries and to predict the localized necking deformation. In the current approach, a large scale problem such as sheet metal forming cannot be solved because of computational power including computer memory and computation time. However, the initiation and progress of necking can be predicted in detail. Rossiter et al. (2010) also discretized a grain using many elements and simulated material deformation in three-dimensional microstructure. They used the microstructure information obtained from EBSD and investigated the effects of strain rate, strain path, and thermal softening on the formation of localized deformation. Kanjarla et al. (2010) also discretized grains using many elements and investigated the plastic deformation fields near the grain boundaries and influence of grain interaction on intra-grain deformations. Lin et al. (2011) also modeled the grains near crack tips using finite elements and predicted a crack propagation path under cyclic loads. In this study, each grain is discretized by sufficient number of elements and, then a large deformation up to necking is analyzed using crystal plasticity finite element method with damage models. Void nucleation, growth and coalescence behaviors are investigated.

Without the consideration of damage evolution, i.e. material softening due to damage, a sudden drop of load carrying capacity after necking during tensile tests cannot be described by finite element analysis (Nielsen and Tvergaard, 2009). There are two main approaches to damage mechanics. The first one is a micromechanics-based damage model that was proposed by Gurson (1975). In micromechanics-based approach (Gurson, 1975, Tvergaard and Needleman, 1984, Zhang et al., 2000), damage evolution was described by void nucleation, growth and coalescence. Void nucleation and growth are modeled and the related coefficients have to be determined using experimental data. Nielsen and Tvergaard (2009) successfully analyzed the necking and fracture behavior of friction stir welded sheets using the modified Gurson model (Tvergaard and Needleman, 1984). The other approach to damage analysis is the Continuum Damage Mechanics (CDM). In CDM frameworks (Cockcroft and Latham, 1968, Johnson and Cook, 1985, Lemaitre, 1985, Lemaitre and Chaboche, 1990), fracture strain is determined using stress, pressure, temperature, and stress triaxiality. After these studies, many improved models were proposed to include the effect of Lode angle (Teng, 2008, Xue, 2009, Malcher et al., 2012, Malcher et al., 2013) and anisotropic damage (Ekh et al., 2004, Mengonia and Ponthot, 2014). Malcher and Mamiya (2014) proposed an improved damage evolution law by developing a denominator of damage function.

In addition to these studies, there were a few recent researches about damage incorporating CPFEM. Bieler et al. (2009) showed the role of heterogeneous deformation on damage nucleation at the grain boundaries in single phase metals. Recently, Tasan et al. (2014) showed the strain localization and damage in dual phase steels by coupled in-situ experiment and crystal plasticity simulation. Nguyen et al. (2015) proposed a nonlocal coupled damage-plasticity model for the analysis of ductile failure.

In this study, several simple damage models are proposed with the function of critical shear strain, maximum shear strain, effective strain, principal strain etc within the framework of CPFEM. Therefore, the damage models used in this study are based on Continuum Damage Mechanics (CDM). However, the void initiation, growth and coalescence behavior can be predicted by incorporating CPFEM. Furthermore, the material weakening from damage is used to describe a sudden drop of load carrying capacity after necking. In the previous works by the authors (Kim et al., 2012, Kim and Yoon, 2013), the possibility of necking prediction by CPFEM was briefly explored using 2D shaped grains (Kim et al., 2012) and 3D shaped grains (Kim and Yoon, 2013). In the works, analyses were carried out by an explicit time integration scheme and it opened the possibility that the necking behavior is able to be predicted from CPFEM. However, the necking shape and load–displacement curve were not predicted. In this study, for an accurate prediction of necking and load–displacement curve, analyses were carried out with an implicit time integration scheme and more reasonable load–displacement curve could be obtained. Moreover, experiments were carried out and were compared with the analysis results. Finally, four different damage models with coefficient calibration were newly proposed, and a model which best describes the necking behavior is proposed.

The major study points in the present work can be summarized as follows:

• Crystal plasticity finite element method is implemented for the analysis of a tensile test via VUMAT in ABAQUS/Explicit.

• Stress concentration near grain boundaries from orientation mismatch is investigated.

• Several damage models are proposed with the functions of critical shear strain, maximum shear strain, effective strain, and principal strain.

• Void initiation, growth, and coalescence behaviors are predicted using CPFEM (Crystal Plasticity FEM) and damage models.

• Necking behavior of AA 6022-T4 is analyzed using three-dimensional grain shapes and, then the results are compared to experimental results.