To obtain the general form of constitutive equations for fiber-reinforced composites with different kind of symmetry (viz., transversely isotropic, orthotropic, tetratropic, hexatropic or octotropic), one needs complete and irreducible representations for scalar-, symmetric tensor-, skew- symmetric tensor- and vector-valued functions of any finite number of symmetric tensors, skew-symmetric tensors and vectors possessing the prescribed kind of material symmetry. Such complete and irreducible representations are obtained and presented in tables 1-5 of this paper, for the case of composites under plane deformation.
The algebraic analysis leading to these useful results is presented in the following order: (1) the tensors \(P\) characterizing the symmetry groups of tetratropy, hexatropy, and octotropy are given; (2) complete isotropic tensor function representations of argument vectors and (symmetric and skew-symmetric) tensors in conjunction with \(P\) are obtained and shown to be irreducible; (3) the principle of isotropy of space then implies that these representations are also complete and irreducible orthotropic, tetratropic, hexatropic and octotropic tensor function representations of the argument vectors and tensors (excluding \(P\)), when \(P\) is the tensor characterizing orthotropy, tetratropy, hexatropy and octotropy, respectively.