An evolving non-associated Hill-Barlat yield criterion for anisotropic hardening with closed-form analytical calibration and guaranteed convexity

International Journal of Plasticity, vol. 204, pp. 104762

Abstract

Accurate description of anisotropic hardening is critical for the predictive simulation of sheet metal forming. Within a non-associated flow rule framework, the evolving Hill48 family of models provides an attractive analytical route, in which the anisotropy parameters are obtained directly from the directional flow curves without optimization-based recalibration. Their quadratic structure, however, limits the curvature of the yield surface between the uniaxial and equibiaxial tension states. Higher curvature flexibility has been introduced through the coupled quadratic and non-quadratic (CQN) family, but the resulting yield functions require strain- and material-dependent verification of convexity, which can fail for realistic materials. This work proposes an evolving non-associated Hill48-Yld91 yield criterion that resolves both limitations. The Yld91 yield function controls the curvature through its non-quadratic exponent, while the quadratic Hill48 plastic potential captures the r -value anisotropy. Convexity is inherited intrinsically from the Yld91 structure, with no subsequent verification. The mathematical core of the work is an explicit closed-form analytical solution for the four Yld91 anisotropy parameters at exponent n =2, derived directly in terms of the Hill48 calibration coefficients, which reduces the evolving calibration to a single algebraic step without numerical iteration. By construction, the anisotropy parameters and the curvature-controlling exponent are obtained independently, so the deployment exponent can be chosen separately from the calibration. The proposed model is validated against the directional anisotropic hardening of two steels (718AT and MP980) and two aluminum alloys (AA5182-O and AA6022-T43), and is further tested for convexity against cases (QP1180 and an artificial material) for which CQN-type formulations have been reported to lose convexity. The framework provides a simple, analytically calibrated, and convexity-guaranteed route to evolving anisotropic hardening.

Authors 3

1. Guijia Li

   Aalto University

   Affiliation as printed

   Materials to Products, Department of Energy and Mechanical Engineering, Aalto University, 02150 Espoo, Finland

2. Junhe Lian corresponding Aachen Institute of Metal Forming

   RWTH Aachen University · Aalto University

   Affiliation as printed

   Institute of Metal Forming, RWTH Aachen University, Intzestraße 1, 52072, Aachen, Germany

   Materials to Products, Department of Energy and Mechanical Engineering, Aalto University, 02150 Espoo, Finland

3. Jeong Whan Yoon corresponding

   Korea Advanced Institute of Science and Technology · Deakin University

   Affiliation as printed

   Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea

   School of Engineering, Deakin University, Waurn Ponds, VIC 3220, Australia