Plane stress yield function for aluminum alloy sheets—part 1: theory

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Abstract

A new plane stress yield function that well describes the anisotropic behavior of sheet metals, in particular, aluminum alloy sheets, was proposed. The anisotropy of the function was introduced in the formulation using two linear transformations on the Cauchy stress tensor. It was shown that the accuracy of this new function was similar to that of other recently proposed non-quadratic yield functions. Moreover, it was proved that the function is convex in stress space. A new experiment was proposed to obtain one of the anisotropy coefficients. This new formulation is expected to be particularly suitable for finite element (FE) modeling simulations of sheet forming processes for aluminum alloy sheets.

Introduction

Over the years, yield functions were developed to describe the plastic anisotropy of sheet metals, for instance, Hill, 1948, Hill, 1979, Hill, 1990, Hill, 1993. Other references can be found in the work by Barlat et al. (1997b) and Banabic (2000). Some of these yield functions were particularly intended for aluminum alloy sheets (Barlat and Lian, 1989, Barlat et al., 1991a, Barlat et al., 1991b, Karafillis and Boyce, 1993, Barlat et al., 1997a, Barlat et al., 1997b). The strain rate potential is another possible concept that can describe plastic anisotropy (Hill, 1987, Barlat et al., 1993, Barlat et al., 1998, Barlat and Chung, 1993). However, the strain rate potential is not discussed in this paper.

The yield functions referenced above have been implemented into some finite element (FE) codes to simulate sheet forming processes (Chung and Shah, 1992, Yoon et al., 1999, Tucu and Neale, 1999, Inal et al., 2000, Worswick, 2000). Although theoretical problems still remain, particularly, in relation to the rotation and distortion of the initial anisotropic reference frame (Tucu et al., 1999), reasonable assumptions have led to very successful forming process simulations (Yoon et al., 1999). It is the perception of the authors that, at the present time, one of the most accurate anisotropic yield functions for aluminum and its alloys is the function denoted Yld96 (for instance, Lademo, 1999, Lademo et al., 1999). This yield function (Barlat et al., 1997a, Barlat et al., 1997b) takes seven parameters into account in the plane stress condition. These parameters can be computed from σ0, σ45, σ90, r0, r45 and r90, the uniaxial yield stresses and r values (width-to-thickness strain ratio in uniaxial tension) measured at 0, 45 and 90° from the rolling direction, and σb the balanced biaxial yield stress measured with the bulge test. There are three problems associated with Yld96 with respect to FE simulations:

  • There is no proof of convexity, which is an important requirement in numerical simulations to ensure the uniqueness of a solution.

  • The derivatives are difficult to obtain analytically which is inconvenient again for FE simulations.

  • The plane stress implementation in FE codes does not provide any particular problems and leads to good simulation results. However, for full stress states, some numerical problems, which might be difficult to solve because of the relative complexity of the Yld96 formulation, have been encountered (Becker, 1998, Szabo, 2001).

The objective of this paper is to explore better incompressible anisotropic plasticity formulations that can guarantee convexity, make FE implementation and application simpler, and take σ0, σ45, σ90, r0, r45, r90 and σb into account for plane stress.

Section snippets

General considerations

If an isotropic function is convex with respect to the principal stresses, then this function is also convex with respect to the stress components expressed in any set of material axes (see proof in the Appendix). For instance, Φ1 and Φ2 are two isotropic functions, convex with respect to the three principal stresses and, consequently, with respect to the six components of the stress tensor expressed in any material frameΦ1=s1−s2a+s2−s3a+s3−s1a=2σ̄aΦ2=s1a+s2a+s3a=2a+23aσ̄a

In , , σ̄ is the

Yield function Yld2000–2d

An isotropic yield function that reduces to the function proposed by Hershey (1954) and Hosford (1972) in Eq. (1) can be simplyφ=φ′+φ″=2σ̄awhereφ′=s1−s2a φ″=2s2+s1a+2s1+s2a

Because, as mentioned above, a plane stress state can be described by two principal values only, φ′ and φ″ are two isotropic functions since it is possible to permute the (in-plane) indices 1 and 2 in each function. Therefore, , describe the yield function φ as a particular case of Eq. (9) with κ=2. For the anisotropic case,

Validation on binary Al–Mg alloy sheet sample

In this paper, x, y and z represent the rolling, transverse and normal directions of the sheet, respectively. In order to show the flexibility of the formulation, the model was applied to the case of a binary aluminum alloy containing 2.5 wt.% of magnesium (Mg). Because Mg is in solid solution in the aluminum matrix at room temperature, the alloy is a single phase material, i.e., an ideal polycrystal. The yield surface of this sheet sample was measured (Barlat et al., 1997b) using stacked sheet

Conclusions

The plane stress yield function Yld2000-2d was proposed and validated with experimental and polycrystal data obtained on a binary Al–2.5 wt.%Mg alloy sheet. This yield function provides a simpler formulation than Yld96 with at least the same accuracy. Its convexity is proven, and its implementation into FE codes appears to be straightforward. An efficient Newton–Raphson procedure was developed to numerically calculate the yield function coefficients using experimental data collected from a few

Acknowledgements

The authors would like to thank Dr. R.C. Becker (Lawrence Livermore National Laboratory) who provided the motivation to conduct this work and Dr. M.E. Karabin, Alcoa Technical Center, for a thorough review of the manuscript.

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