Drucker’s sixth-order yield function for isotropic materials (J Appl Mech 16:349–357, 1949) was extended by Cazacu and Barlat (Math Mech Solids 6:613–630, 2001) for modeling orthotropic sheet metals via a generalization of the two stress invariants according to the theory of representation. The constant c in the original Drucker’s isotropic yield function was found by Dodd and Naruse (Int J Mech Sci 31:511–519, 1989) to be bound between –27/8 and 9/4 per the convexity requirement. In many subsequent modeling applications of orthotropic sheets, the same bound is also tacitly assumed for the constant c used in this class of the generalized Drucker’s yield function CB2001. No actual proof has, however, been presented in the literature that such a bound is indeed absolutely necessary if not sufficient to guarantee the convexity of the orthotropic CB2001 yield function. In this study, the validity of assuming such a convexity bound on the adjustable constant c is examined using a recently proposed numerical convexity certification algorithm. Representative orthotropic CB2001 yield functions whose material parameters have been calibrated and reported in the literature for some 12 FCC, BCC, and HCP sheet metals are evaluated. It is found that a single convexity bound between \(-27/8\) and 9/4 on the constant c does not hold at all for any of those yield functions. This is in contrast to another class of generalized Drucker’s yield function based on linearly transformed stresses where the original convexity bound on the constant c does still hold.

Drucker 的各向同性材料的六阶屈服函数 (J Appl Mech 16:349–357, 1949) 由 Cazacu 和 Barlat (Math Mech Solids 6:613–630, 2001) 扩展，用于通过对正交各向异性金属板的概括对两者进行建模根据表示理论应力不变量。Dodd 和 Naruse (Int J Mech Sci 31:511–519, 1989) 发现原始 Drucker 的各向同性屈服函数中的常数c根据凸性要求被限制在 –27/8 和 9/4 之间。在正交各向异性片的许多后续建模应用中，对于常数c也默认假定相同的界限用于此类的广义德鲁克屈服函数 CB2001。然而，在文献中没有提出实际证据表明这样的界限确实是绝对必要的，如果不足以保证正交各向异性 CB2001 屈服函数的凸性。在这项研究中，使用最近提出的数值凸度证明算法检查了在可调常数 c 上假设这种凸度界限的有效性。具有代表性的正交各向异性 CB2001 屈服函数，其材料参数已被校准并在文献中报告，用于评估一些 12 种 FCC、BCC 和 HCP 金属板。发现常数c上\(-27/8\)和 9/4之间的单个凸度界限对于这些屈服函数中的任何一个都不成立。这与另一类基于线性变换应力的广义德鲁克屈服函数形成对比，其中常数c上的原始凸性边界仍然成立。