# On Using Homogeneous Polynomials To Design Anisotropic Yield Functions With Tension/Compression Symmetry/Assymetry

## Abstract

With few exceptions, non-quadratic homogeneous polynomials have received little attention as possible candidates for yield functions. One reason might be that not every such polynomial is a convex function. In this paper we show that homogeneous polynomials can be used to develop powerful anisotropic yield criteria, and that imposing simple constraints on the identification process leads, aposteriori, to the desired convexity property. It is shown that combinations of such polynomials allow for modeling yielding properties of metallic materials with any crystal structure, i.e. both cubic and hexagonal which display strength differential effects. Extensions of the proposed criteria to 3D stress states are also presented. We apply these criteria to the description of the aluminum alloy AA2090T3. We prove that a sixth order orthotropic homogeneous polynomial is capable of a satisfactory description of this alloy. Next, applications to the deep drawing of a cylindrical cup are presented. The newly proposed criteria were implemented as UMAT subroutines into the commercial FE code ABAQUS. We were able to predict six ears on the AA2090T3 cup's profile. Finally, we show that a tension/compression asymmetry in yielding can have an important effect on the earing profile.

- Authors:

- Department of Mechanical and Aerospace Engineering, University of Florida's Graduate Engineering and Research Center, Shalimar, FL 32579 (United States)
- Materials Science Division, Alcoa Inc., Alcoa Technical Center, 100 Technical Drive, Alcoa Center, PA 15069 (United States)

- Publication Date:

- OSTI Identifier:
- 21061733

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: AIP Conference Proceedings; Journal Volume: 908; Journal Issue: 1; Conference: NUMIFORM 2007: 9. international conference on numerical methods in industrial forming processes, Porto (Portugal), 17-21 Jun 2007; Other Information: DOI: 10.1063/1.2740877; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 36 MATERIALS SCIENCE; A CODES; ALUMINIUM ALLOYS; ANISOTROPY; ASYMMETRY; COMPRESSION; COMPUTERIZED SIMULATION; CRYSTAL STRUCTURE; CYLINDRICAL CONFIGURATION; DRAWING; FINITE ELEMENT METHOD; POLYNOMIALS; STRESSES; SYMMETRY

### Citation Formats

```
Soare, S., Cazacu, O., and Yoon, J. W.
```*On Using Homogeneous Polynomials To Design Anisotropic Yield Functions With Tension/Compression Symmetry/Assymetry*. United States: N. p., 2007.
Web. doi:10.1063/1.2740877.

```
Soare, S., Cazacu, O., & Yoon, J. W.
```*On Using Homogeneous Polynomials To Design Anisotropic Yield Functions With Tension/Compression Symmetry/Assymetry*. United States. doi:10.1063/1.2740877.

```
Soare, S., Cazacu, O., and Yoon, J. W. Thu .
"On Using Homogeneous Polynomials To Design Anisotropic Yield Functions With Tension/Compression Symmetry/Assymetry". United States.
doi:10.1063/1.2740877.
```

```
@article{osti_21061733,
```

title = {On Using Homogeneous Polynomials To Design Anisotropic Yield Functions With Tension/Compression Symmetry/Assymetry},

author = {Soare, S. and Cazacu, O. and Yoon, J. W.},

abstractNote = {With few exceptions, non-quadratic homogeneous polynomials have received little attention as possible candidates for yield functions. One reason might be that not every such polynomial is a convex function. In this paper we show that homogeneous polynomials can be used to develop powerful anisotropic yield criteria, and that imposing simple constraints on the identification process leads, aposteriori, to the desired convexity property. It is shown that combinations of such polynomials allow for modeling yielding properties of metallic materials with any crystal structure, i.e. both cubic and hexagonal which display strength differential effects. Extensions of the proposed criteria to 3D stress states are also presented. We apply these criteria to the description of the aluminum alloy AA2090T3. We prove that a sixth order orthotropic homogeneous polynomial is capable of a satisfactory description of this alloy. Next, applications to the deep drawing of a cylindrical cup are presented. The newly proposed criteria were implemented as UMAT subroutines into the commercial FE code ABAQUS. We were able to predict six ears on the AA2090T3 cup's profile. Finally, we show that a tension/compression asymmetry in yielding can have an important effect on the earing profile.},

doi = {10.1063/1.2740877},

journal = {AIP Conference Proceedings},

number = 1,

volume = 908,

place = {United States},

year = {Thu May 17 00:00:00 EDT 2007},

month = {Thu May 17 00:00:00 EDT 2007}

}