Technical Brief

Algebraic Convexity Conditions for Gotoh's Nonquadratic Yield Function

[+] Author and Article Information
Wei Tong

Department of Mechanical Engineering,
Lyle School of Engineering,
Southern Methodist University,
Dallas, TX 75275-0337
e-mail: wtong@smu.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 12, 2017; final manuscript received March 31, 2018; published online May 8, 2018. Assoc. Editor: A. Amine Benzerga.

J. Appl. Mech 85(7), 074501 (May 08, 2018) (7 pages) Paper No: JAM-17-1636; doi: 10.1115/1.4039880 History: Received November 12, 2017; Revised March 31, 2018

A necessary and sufficient condition in terms of explicit algebraic inequalities on its five on-axis material constants and a similarly formulated sufficient condition on its entire set of nine material constants are given for the first time to guarantee a calibrated Gotoh's fourth-order yield function to be convex. When considering the Gotoh's yield function to model a sheet metal with planar isotropy, a single algebraic inequality has also been obtained on the admissible upper and lower bound values of the ratio of uniaxial tensile yield stress over equal-biaxial tensile yield stress at a given plastic thinning ratio. The convexity domain of yield stress ratio and plastic thinning ratio defined by these two bounds may be used to quickly assess the applicability of Gotoh's yield function for a particular sheet metal. The algebraic convexity conditions presented in this study for Gotoh's nonquadratic yield function complement the convexity certification based on a fully numerical minimization algorithm and should facilitate its wider acceptance in modeling sheet metal anisotropic plasticity.

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Grahic Jump Location
Fig. 1

A flow chart for the efficient convexity certification andadjustment of a calibrated yield fucntion based on a cascade of necessary, sufficient, and necessary and sufficient conditions

Grahic Jump Location
Fig. 2

Domains of admissible yield stress and plastic strain ratios for planarly isotropic Gotoh's yield function




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