ON TENSOR FUNCTIONS OF 2ND-ORDER TENSORS AND VECTORS UNDER N-GONAL CLASSES D-NH

In three-dimensional space there are seven series of classes, i.e., D- nh, D-n, D-nd, L-nv, L-nh, L-n, and L-ni, in general n-gonal systems f or n = integer greater than or equal to 1. The classes D-nh are of mos t practical interest, they not only cover the usual orthotropy D-2h, t etratropy D-4h, and hexatropy D-6h, but they contain the other sia: se ries of classes, too. Furthermore, one may refer to the overall materi al symmetry of a fibre-reinforced composite with fibre reinforcements equivalently in the 0, pi/s, 2 pi/s,...,(s - 1) pi/s directions for s greater than or equal to 3 as the symmetry D-2sh, and this kind of com posite is known as a quasi-isotropic material since its planar linear elasticity is isotropic. In this paper we establish the complete and i rreducible representation for scalar-valued; vector-valued, and second -order tensor-valued functions of any finite number of second-order te nsors and vectors with respect to the infinitely many classes D-nh, in supplement to the known representation with respect to the orthotropy D-2h. These results allow Sor a general invariant formulation of cons titutive equations for crystals in the tetragonal, trigonal, and hexag onal systems, as well as quasi-isotropic fibre-reinforced composites.