Experiments and modeling of anisotropic aluminum extrusions under multi-axial loading – Part I: Plasticity

Abstract

An extensive experimental program has been carried out to characterize the plastic behavior of 2 mm thick extruded aluminum AA6260-T6 sheets under large deformations. Using a newly-developed dual actuator system, combinations of normal and tangential loads are applied to a flat specimen to investigate the material response under more than 30 different multi-axial stress states. The Yld2000-2d yield criterion with an associated flow rule and an isotropic hardening model has been successfully used to describe the initial yield surface and its evolution. The comparison between the experimental results and finite element simulations shows that this constitutive model provides very accurate predictions for the material response under multi-axial loading. A special extension of the Yld2000-2d yield function for general three-dimensional stress states is also presented. The yield function for three-dimensional stress states is chosen such that it reduces to the Yld2000-2d yield function under plane stress conditions and makes use of the same anisotropy coefficients.

Introduction

The plasticity of aluminum alloys has been the focus of numerous studies in the past. It is well-known that conventional quadratic metal plasticity models such as the isotropic von Mises (1913) or the anisotropic Hill (1948) model provide only a poor approximation of the mechanical response of aluminum alloys. Woodthorpe and Pearce (1970) performed uniaxial tension and hydraulic bulge tests on cold-rolled purity aluminum sheets. Their results demonstrate that Hill’s theory underestimates the ratio of biaxial to uniaxial yield stress by more than 20%. Stout et al. (1983) subjected aluminum 2024T6 and T8 tubes to combinations of tension and internal pressure as well as tension and torsion. Their experiments confirmed that Hill’s quadratic model underestimates the equi-biaxial yield, while Hill’s (1979) non-quadratic criterion as well as Bassani’s (1977) criterion provided more accurate predictions of the anisotropic yield behavior of aluminum. Green et al. (2004) used a custom-made biaxial testing apparatus (Ferron and Makinde, 1988) to test aluminum 1145 cruciform specimens under different in-plane biaxial stretching conditions. Their numerical analysis suggests that the non-quadratic yield functions by Barlat and Lian (1989) and Hill (1990) can accurately describe the behavior of this alloy. Iadicola et al. (2008) made use of in situ X-ray diffraction to measure the stresses during modified Marciniak flat bottom ram tests (Marciniak and Kuczynski, 1967, Raghavan, 1995) on AA5754-O sheets. Their results for different biaxial loading paths revealed deficiencies of both quadratic yield functions (von Mises, 1913; Hill, 1948) and non-quadratic yield functions (Hosford, 1972, Barlat et al., 2003). Bai and Wierzbicki (2008) proposed a pressure and Lode angle dependent yield function to describe the plasticity of aluminum 2024-T351.

A comprehensive review of yield functions for metals is presented in the textbook of Banabic et al. (2000). One promising avenue towards the formulation of anisotropic yield functions is the use of linearly transformed stress tensors in isotropic yield functions (e.g. Cazacu and Barlat, 2001). Barlat et al. (1991) adapted the isotropic Hershey (1954) and Hosford (1972) yield function for anisotropic materials through the linear transformation of the stress tensor. In close analogy, Karafillis and Boyce (1993) obtained a more general anisotropic yield function by using the linear transformation of the stress tensor in conjunction with a linear combination of two isotropic yield functions. An extension of this yield function has been recently presented by Bron and Besson (2004) which makes use of two distinct linear transformation functions. Bron and Besson (2004) identified the material model parameters for aluminum 2024 based on U-notched tensile test results using simplex and SQP minimization algorithms. Based on the results from uniaxial tensile tests, Lademo (1999) concluded that none of the anisotropic models by Hill, 1990, Barlat and Lian, 1989 or Karafillis and Boyce (1993) is able to describe the anisotropy of both the yield stress and r-ratio of extruded AA6063-T1 or AA7108-T1 accurately. Lademo et al. (2002) reported satisfactory predictions of the anisotropic plastic response of different AA7108 tempers when using the Yld96 criterion of Barlat et al. (1997). The latter model has been enhanced further by Barlat et al. (2003). The result, the convex anisotropic Yld2000-2d function, is obtained from the isotropic Hosford criterion using two distinct linear transformations of the stress deviator. As an alternative to linear transformations of the stress tensor, flexible anisotropic plasticity models can be built with non-associated flow rules, in which the yield surface and the plastic flow potential are defined by different functions (Stoughton, 2002). It has been reported that quadratic anisotropic yield functions along with a non-associated quadratic flow rule can accurately predict the thickness evolution and the earing in cup drawing operations of aluminum alloys (Cvitanic et al., 2008, Taherizadeh et al., 2010), as well as the large deformation behavior of TRIP and DP steels under multi-axial loadings (Mohr et al., 2010). More recently, other approaches have been investigated to develop anisotropic constitutive models. Desmorat and Marull (2011) made use of the stress tensor spectral decomposition along Kelvin modes to develop a new class of anisotropic yield criteria. Paquet et al. (2011) have proposed a homogenization-based anisotropic continuum plasticity model for SDAS cast aluminum alloys which take microstructural aspects into account. In addition to describing the initial anisotropy with great accuracy, significant efforts are made to characterize and model the evolving anisotropy of aluminum alloys during straining (Khan et al., 2009, Khan et al., 2010, Stoughton and Yoon, 2009, Cardoso and Yoon, 2009, Rousselier et al., 2009, Barlat et al., 2011) as well as the effect of strain rate (e.g. Khan and Liu, 2012a, Khan and Liu, 2012b).

The Yld2000-2d model has been successfully employed by various authors. Naka et al. (2003) used this model to represent the yield surface of AA5083 that has been determined from cruciform tests. Kuwabara et al. (2005) characterized the deformation behavior of extruded AA5154-H112 tubes under combined tension/internal pressure. Their comparison of the experimental results with the Yld2000-2d model predictions indicates good agreement for the plastic work contours as well as for the direction of plastic flow. The uniaxial tests and hydrobulge tests by Jansson et al. (2005) confirmed the applicability of the Yld2000 material model to AA6063-T4 tubes. Lee et al. (2005) simulated the forming of AA5754-O and AA6111-T4 sheets. They found good agreement with the experiments using the Yld2000-2d model together with a Chaboche (1986) combined isotropic–kinematic hardening rule. Stoughton and Yoon (2005) used the Yld2000-2d model to simulate the deep drawing of AA6016-T4 sheets. Korkolis and Kyriakides, 2008, Korkolis and Kyriakides, 2009 performed a series of inflation and burst experiments on Al-6260-T4 tubes along proportional and non-proportional loading paths. They reported improved strain path predictions by the Yld2000-2d model in comparison with the Karafillis and Boyce (1993) model.

In view of applications with three-dimensional stress states, yield functions describing the anisotropic behavior in the full stress space are required. With the exception of the anisotropic yield functions proposed by Bron and Besson (2004) and Barlat et al. (2005), most yield functions presented in the open literature are either developed for plane stress only or are non-convex. Barlat et al. (2005) also proposed an 18-parameter yield function for general stress states (Yld2004-18p) which has advantages in cup-earing predictions (e.g. Yoon et al, 2006) and wall-thinning predictions during hydroforming (Korkolis and Kyriakides, 2011). However, the complete calibration of the Yld2004-18p model cannot be performed based on experimental results only and requires crystal plasticity simulations (Grytten et al., 2008). Furthermore, the computations associated with the solution of the constitutive equations (in particular the calculations of the derivatives) are far more complicated than with the Yld2000-2d model. Yoon and Hong (2006) pointed out that computations with the 8-parameter Yld2000-2d model are about four times faster than the Yld2004-18p model.

The material parameters of the Yld2000-2d model are typically determined from the uniaxial tensile test measurements of the yield stresses Y₀, Y₄₅, Y₉₀ and r-ratios r₀, r₄₅, r₉₀ as well as the equi-biaxial yield stress Y_b and equi-biaxial r-ratio r_b. In the present work, we make use of a newly-developed combined tension/compression and shear testing technique (Mohr and Oswald, 2008) to investigate the plasticity of 2 mm thick extruded aluminum AA6260-T6 sheets. In close analogy with the tension–torsion testing of thin-walled tubes (e.g. Taylor and Quinney, 1931), flat sheet specimens are subject to various combinations of tangential and normal loads. The experimental results are used to calibrate and validate the Yld2000-2d model. A series of uniaxial tensile tests is performed to confirm the prediction accuracy of the yield stress and r-ratio variations. In addition, an extension of the Yld2000-2d plane stress model to general three-dimensional stress states is presented.