On the Certification of Positive and Convex Gotoh’s Fourth-Order Yield Function

Abstract

Gotoh proposed in 1977 a fourth-order homogeneous polynomial of three plane stress components as a yield function to model anisotropic yielding and plastic flow of sheet metals. The yield function admits up to eight experimental inputs from uniaxial tension tests and one measurement from an equal biaxial tension test for calibrating its nine material constants and can model the formation of up to eight ears in deep cup drawing. However, the superior Gotoh’s yield function has not been widely adopted in sheet metal forming analyses in both academy and industry, especially outside Japan. One major concern is uncertain about the positivity and convexity of a calibrated Gotoh’s yield function. Here the problem of certifying a calibrated Gotoh’s yield function to be strictly positive and convex is first described and its resolutions are summarized based on some very recent research results. Algebraic necessary and sufficient conditions for a positive and convex Gotoh’s yield function with only two non-zero stress components are first presented. Sufficient conditions in terms of semi-algebraic and algebraic inequalities for establishing the positivity and convexity of tri-component plane stress Gotoh’s yield function are then summarized. Finally, the complete necessary and sufficient conditions for a positive and convex Gotoh’s yield function in plane stress are realized as a fully numerical minimization problem. Examples of successfully applying those conditions in certifying positive and convex Gotoh’s yield functions of various sheet metals are also given.