Summary: It is shown that the fourth-order homogeneous polynomial yield function first considered by M. Gotoh [Int. J. Mech. Sci. 19, 505–512 (1977; Zbl 0373.73038)] for sheet metals can be recast into a particular general form in intrinsic variables. It leads to a method of generating many specific representations of a fourth-order R. Hill’s 1979 yield function [Math. Proc. Camb. Philos. Soc. 85, 179–191 (1979; Zbl 0388.73029)] generalized for off-axis loading. One may use the generalized fourth-order Hill’s yield function for modeling sheet metals of various degrees of planar anisotropy with ease. Furthermore, a sufficient condition on the positivity and convexity of a calibrated Hill’s yield function can also be verified straightforwardly. For six sheet metals whose nine material constants in a Gotoh’s yield function have been reported in the literature, their generalized Hill’s 1979 yield functions are obtained subsequently. The proposed sufficient condition on the positivity and convexity of Hill’s yield function is found to be satisfied by five of these six sheet metals. An approximate Hill’s 1979 yield function for the sixth sheet metal has also been identified, so the proposed sufficient condition on its positivity and convexity is also met. Generalizing fourth-order Hill’s 1979 yield functions may thus be used as an alternative approach for formulating a class of positive and convex yield functions with up to nine material constants for various orthotropic sheet metals.