ON THE TENSOR FUNCTION REPRESENTATIONS OF 2ND-ORDER AND 4TH-ORDER TENSORS .1.

In this paper, we establish the complete representations for isotropic scalar-, 2nd-order tensor- and 4th order tensor-valued functions of any finite number of 2nd- and 4th-order tensors in both 2- and 3-dimensional spaces, and prove the irreducibility of these representations. The fourth-order tensors under consideration are those regarded as skew-symmetric linear transformations of both symmetric and skew-symmetric 2nd-order tensors in 3-dimensional space, symmetric linear transformations of skew-symmetric 2nd-order tensors in 3-dimensional space, and both symmetric and skew-symmetric linear transformations of both symmetric and skew-symmetric 2nd-order tensors in 2-dimensional space. Complete and irreducible isotropic tensor function representations in 3-dimensional space involving 4th-order tensors as symmetric linear transformations of symmetric 2nd-order tensors are derived in a continued paper (part II). These representations allow general forms of constitutive laws involving 4th-order tensor agencies (damage tensors, internal variables) of isotropic materials to be developed.