A Description of Three-Dimensional Yield Surfaces by Cubic Polynomials

A description of yield surfaces in the axial-torsional-hoop stress space, which is able to capture the yield surface expansion/contraction, translation, rotation, affine deformation, and distortion, is constructed under a three-dimensional point of view. Starting from the yield surface theory of cubic polynomials in two-dimensional spaces, the authors first clarify the relations between it and the Kurtyka and Życzkowski theory which exhibits excellent performance of fitting the experimental data of yield points in two-dimensional stress space. Because an arbitrary cubic polynomial does not guarantee that its zero-level locus is closed, the authors apply projective transformation of the Weierstrass normal form of cubic curves in the two-dimensional space in order to obtain closed cubic yield loci. Following the derivation of closed cubic yield loci, the authors investigate the convexity of the yield loci. Then the two-dimensional theory is extended into a standard formulation of convex closed cubic polynomial in the three-dimensional stress space with consideration of preserving the closure and the convexity of yield surface, and the orthotropic and isotropic symmetry of the convex closed cubic polynomial is explored. Furthermore, a geometrically meaningful identification with three stages is proposed to estimate the values of coefficients of the polynomials from the experimental data of yield points in the axial-torsional-hoop stress space. Validation is performed by checking the estimated results of yield surface and experimental yield points and shows that almost all probed yield points are located directly on or near the estimated evolving yield surfaces. Therefore the convex closed cubic polynomial and three-stage identification provides reliable visualization of the yield surfaces which unite individual yield points to present the global information.