User-friendly Gotoh’s 1977 fourth-order yield function

The homogeneous polynomial of fourth order with nine coefficients as originally developed by Gotoh in 1977 is a simple non-quadratic yield function that can model fairly well an orthotropic sheet metal under both uniaxial and biaxial stress states. Unlike Yld2000-2d yield function with eight anisotropic material parameters, however, a Gotoh’s 1977 yield function calibrated by the conventional direct method may not be convex and has thus not been as widely used in sheet metal forming analyses. Four user-friendly methods to develop a strictly convex fourth-order yield function are presented and evaluated in this study. The first three methods decompose Gotoh’s 1977 quartic yield function into a sum of two or more fourth-order polynomials whose convexity may be rather easily established. The last method uses a certain algebraic certificate of sum-of-squares (SOS) convexity to verify directly the convexity of a calibrated Gotoh’s yield function. More importantly, a least-square minimization (with algebraic convex constraints as needed) may also be readily implemented using these methods towards parameter identification of a Gotoh’s yield function with guaranteed convexity. These four methods were subsequently applied to verify the convexity of many calibrated Gotoh’s yield functions reported in the literature. Convexity-constrained parameter identification was also carried out successfully using a set of twelves experimental inputs for one aluminum sheet and one steel sheet. Except the method based on a sum of squares of Hill’s 1948 quadratic yield functions, results of our current study showed that other three methods achieved nearly equal effectiveness. As the SOS-convex method is easiest to use, it is thus recommended as the preferred user-friendly method in developing convex Gotoh’s yield functions for industrial sheet metal forming applications.