On representations of yield functions for crystals, quasicrystals and transversely isotropic solids

The material symmetry of a solid places invariance restrictions on the form of the yield function of this solid, the latter being a scalar-valued function of the stress tensor according to the usual stress formulation. The main objective of this article is from such material symmetry restrictions to derive general reduced forms of yield functions for the infinitely many classes of crystals and quasicrystals with the material symmetry groups D2m+2h, C2m+2h, D2m+1d and S4m+2 for all integers m greater than or equal to 1, as well as the transversely isotropic solids and the cubic crystals. For the infinitely many classes of crystals and quasicrystals just mentioned, the results are provided in unified forms valid for all integers m greater than or equal to 1. It is shown that eight polynomial invariants are enough to determine the general reduced form of the yield function for each aforementioned crystal class and quasicrystal class except the cubic crystal class T-h and the transverse isotropy. For pressure-independent yielding and plane stress yielding, the numbers of invariants required are reduced to seven and three respectively. Further, each presented result is shown to be irreducible in the sense that it contains no redundant invariant as an argument entering the general reduced form of yield functions. It seems that the results for quasicrystals have been presented here for the first time. For crystal classes, the presented results are either more compact than (for S-6, C-4h, D-4h, C-6h and T-h) or as compact as (for D-3d, D-6h and O-h) the corresponding established results. (C) Elsevier, Paris.