**Article**

# Plane strain transversely anisotropic analysis in sheet metal forming simulation using 6-component Barlat yield function

## Authors

- First online:
- Received:
- Accepted:

DOI: 10.1007/s10999-012-9198-2

## Abstract

In most FEM codes, the isotropic-elastic and transversely anisotropic-elastoplastic model using Hill’s yield function has been widely adopted in 3D shell elements (modified to meet the plane stress condition) and 3D solid elements. However, when the 4-node quadrilateral plane strain or axisymmetric element is used for 2D sheet metal forming simulation, the above transversely anisotropic Hill model is not available in some FEM code like Ls-Dyna. A novel approach for explicit analysis of transversely anisotropic 2D sheet metal forming using 6-component Barlat yield function is elaborated in detail in this paper, the related formula between the material anisotropic coefficients in Barlat yield function and the Lankford parameters are derived directly. Numerical 2D results obtained from the novel approach fit well with the 3D solution.

### Keywords

4-Node quadrilateral element Transversely anisotropic Sheet metal forming## 1 Introduction

In order to accurately simulate sheet metal forming processes, it is essential to describe correctly the material constitutive behaviors. Since most sheet metals exhibit anisotropic material behaviors, the use of appropriate anisotropic yield criterion is important to predict material behaviors accurately. Moreover, anisotropy has an important effect on the strain distribution in sheet metal forming process, and it is closely related to thinning and formability of sheet metal, so the anisotropy of the material should be properly considered to capture the realistic material behaviors.

The influence of plastic anisotropy on sheet metal forming has been studied with the help of FEM codes combined with appropriate anisotropic yield functions. Many such functions have been proposed. The quadratic yield function by Hill (1948) has long been one of the popular choices to represent planar anisotropy and has been widely used in FEM forming simulation. Several non-quadratic criteria were developed by Hill (1979, 1990), Hershey (1954), Hosford (1972), Bassani (1977), Gotoh (1977), Logan and Hosford (1980), Barlat and Lian (1989), Karafillis and Boyce (1993), Bron and Besson (2004), Banabic et al. (2005) and Barlat et al. (1991, 1997, 2003, 2005). In general, Hill (1948) has been useful for explaining phenomena associated to anisotropic plasticity particularly for steels, the others can be used to improve the yielding description of aluminum alloys. But in many circumstances Yld89 (Barlat and Lian 1989), Yld91 (Barlat et al. 1991) can be used for steels or aluminum alloys. In particular, the yield criteria Yld89 for planar anisotropy, and Hill (1990) have three stress components and are applicable to plane stress condition. The analytical forms of these criteria are relative simple. The criteria Yld91 account for six stress components and can be applied to general 3D elasto-plastic continuum codes. Barlat et al. (2007) have made detailed discussion about the relation between these yield criteria. In fact, Yld91 is a particular case of Yld2004-18p (Barlat et al. 2005), Yld89 is a particular case of Yld2000-2d (Barlat et al. 2003), the yield function proposed by Banabic et al. (2005) is identical to Yld2000-2d. In some special conditions, Yld91 can reduce to Hill (1948), Mises or Tresca yield function.

Section analysis provides a faster and more efficient alternative procedure for analyzing complex part shapes in some cases. In this procedure, a cross section is selected from the tooling along a direction of interest. The problem is then analyzed in 2D, assuming plane strain or axisymmetric conditions. Although most sections in sheet forming do not completely satisfy such assumptions, there are still many local sections in a complicated sheet forming process which can be successfully simulated by 2D section analysis. In general, this method can provide a quicker analysis and designers can modify local geometry of tools with experience at preliminary design stages.

The present study attempts to conduct an explicit section analysis of 2D sheet metal forming using the 4-node quadrilateral plane strain element. In most FEM codes, the isotropic-elastic, transversely anisotropic-elastoplastic model proposed by Hill (1948) has been widely used in 3D shell elements (modified to meet the plane-stress condition) and 3D solid elements. However, the above transversely anisotropic model is not available for simulating 2D sheet metal forming process using 4-node quadrilateral plane strain or axisymmetric element in some FEM code like Ls-Dyna (Hallquist 2007). In this paper, a novel approach for the explicit analysis of transversely anisotropic 2D sheet metal forming using 6-component Barlat yield function (Barlat et al. 1991) is proposed, the related formula and parameters are derived directly in detail, the numerical 2D results obtained from the novel approach fit well with the 3D solution.

## 2 Fundamental theory

### 2.1 Basic finite element equation

**F**^{ ext }is the vector of external force,

**F**^{ c }the vector of contact force,

**F**^{ int }the vector of internal force, and

*the mass matrix. In the explicit algorithm, Eq. (1) can be solved as follows:*

**M***t*

_{ n }can be solved by Eq. (2) furthermore, the displacement vector

**U**_{ n+1}at

*t*

_{ n+1}can be solved by Eqs. (3) and (4).

### 2.2 Transversely anisotropic model

#### 2.2.1 The general Hill orthotropic anisotropic yield criteria

*F*,

*Q*,

*H*,

*L*,

*M*and

*N*are anisotropic constants relating with the material yield behaviors. When

*F*=

*Q*=

*H*=

*L*/3 =

*M*/3 =

*N*/3, (5) reduces to Mises criteria. For the general anisotropic material behaviors, they meets the follow condition:

#### 2.2.2 Transversely anisotropic criteria

### 2.3 6-Component Barlat model

*S*

_{ k }(

*k*= 1–3) represents the three eigenvalues of the following matrix:

**I**is the identity matrix,

*I*

_{1},

*I*

_{2}and

*I*

_{3}are the following tensor invariants:

### 2.4 The relation between material anisotropic coefficients

In general, the Lankford parameters R in three directions (i.e., 0°, 45° and 90°) should be available when sheet metals are provided. On the other hand, the six material anisotropic coefficients a, b, c, f, g and h are generally not available. In order to use the above derived model, it is necessary to obtain their quantitative values from the Lankford parameters.

*R*

_{ϕ}can be described as:

*R*

_{ϕ}and the yield function can be derived:

*R*

_{ϕ}and the six material anisotropic coefficients a, b, c, f, g and h, for simplicity and not losing generality, it can be obtained by assuming m = 2:

_{ yz }, σ

_{ zx }, σ

_{ xy }). For isotropy materials, the six coefficients a, b, c, f, g and h are all equal to 1. When the anisotropy of the sheet metal is described, it can approximately be made that f = g = h = 1. It is now possible to solve a, b and c in the assembled Eqs. (29)–(31).

## 3 Numerical example

## 4 Conclusions

The explicit analysis of transversely anisotropic plane strain sheet metal forming is implemented in Ls-Dyna code using 6-component Barlat yield function. The related formula and parameters has been derived explicitly. The transversely anisotropic property of sheet metal can be described by the three parameters a, b, and c. Numerical results demonstrated the validity of this approach.

## Acknowledgments

This work was financially supported by Shanghai Leading Academic Discipline Project (J51402) and Shanghai University of Engineering Science Scientific Development Fund (2009xy07).

### Open Access

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