On the Tensor Function Representations of 2nd‐Order and 4th‐Order Tensors. Part I

Abstract

AbstractIn 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.