ON TRANSVERSELY ISOTROPIC, ORTHOTROPIC AND RELATIVE ISOTROPIC FUNCTIONS OF SYMMETRICAL TENSORS, SKEW-SYMMETRICAL TENSORS AND VECTORS .1. 2-DIMENSIONAL ORTHOTROPIC AND RELATIVE ISOTROPIC FUNCTIONS AND 3-DIMENSIONAL RELATIVE ISOTROPIC FUNCTIONS

In this part, we derive the complete and irreducible representations for two dimensional orthotropic functions and two and three dimensional relative isotropic (i.e. the symmetry group is the proper orthogonal tensor group) functions of symmetric tensors, skew-symmetric tensors and vectors. The functions may be any general (not only polynomial) scalar-valued, vector-valued, symmetric tensor-valued and skew-symmetric tensor-valued ones. The complete representations for three dimensional transversely isotropic and orthotropic functions are derived in Parts II and IV respectively and their irreducibility is proved in Parts III and V respectively.