On Using Homogeneous Polynomials To Design Anisotropic Yield Functions With Tension/Compression Symmetry/Assymetry
Abstract
With few exceptions, nonquadratic 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), 1721 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, nonquadratic 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}
}

In the present paper, an improved strainenergydensity criterion presented recently for the commonly used fracture criterion, the minimum strainenergydensity criterion, is extended to the case of cyclic loading to predict mixedmode fatigue crack growth in materials with different yield strengths in tension and compression. The analysis of the mixedmode fatigue crack growth process is very complex. For the purpose of more precisely predicting the mixed mode fatigue crack growth process, the authors developed a numerical scheme in which the improved fatigue crack growth criterion is combined with the displacement discontinuity method, a boundary element method. In the fatigue crack growthmore »

Anisotropic strain relaxation of GaInP epitaxial layers in compression and tension
We have investigated the strain relaxation of intentionally lattice mismatched ({plus_minus}0.5{percent}) GaInP layers grown on GaAs substrates by organometallic vapor phase epitaxy. Double axis xray diffraction was used to measure the relaxation in these epitaxial layers in perpendicular {l_angle}110{r_angle} directions as a function of thickness. For samples in tension, the difference in relaxation between [1{bar 1}0] and [110] increases from 10{percent} to 48{percent} as the layer thickness increases from 7 to 28 times the critical thickness, {ital h}{sub {ital c}}. For samples in compression this difference is 28{percent} at 24{ital h}{sub {ital c}} while no relaxation is measured for amore » 
Approximating smooth functions using algebraictrigonometric polynomials
The problem under consideration is that of approximating classes of smooth functions by algebraictrigonometric polynomials of the form p{sub n}(t)+{tau}{sub m}(t), where p{sub n}(t) is an algebraic polynomial of degree n and {tau}{sub m}(t)=a{sub 0}+{Sigma}{sub k=1}{sup m}a{sub k} cos k{pi}t + b{sub k} sin k{pi}t is a trigonometric polynomial of order m. The precise order of approximation by such polynomials in the classes W{sup r}{sub {infinity}(}M) and an upper bound for similar approximations in the class W{sup r}{sub p}(M) with 4/3<p<4 are found. The proof of these estimates uses mixed series in Legendre polynomials which the author has introduced andmore » 
Accuracy Analysis of Anisotropic Yield Functions based on the RootMean Square Error
This paper evaluates the accuracy of popular anisotropic yield functions based on the rootmean square error (RMSE) of the yield stresses and the Rvalues. The yield functions include Hill48, Yld89, Yld91, Yld96, Yld20002d, BBC2000 and Yld200018p yield criteria. Two kind steels and five kind aluminum alloys are selected for the accuracy evaluation. The anisotropic coefficients in yield functions are computed from the experimental data. The downhill simplex method is utilized for the parameter evaluation for the yield function except Hill48 and Yld89 yield functions after the error functions are constructed. The yield stresses and the Rvalues at every 15 deg.more »