Integrity basis for a second-order and a fourth-order tensor

Abstract

In this paper a scalar-valued isotropic tensor function is considered, the variables of which are constitutive tensors of orders two and four, for instance, characterizing the anisotropic properties of a material. Therefore, the system of irreducible invariants of a fourth-order tensor is constructed. Furthermore, the joint or simultaneous invariants of a second-order and a fourth-order tensor are found. In a similar way one can construct an integrity basis for a tensor of order greater than four, as shown in the paper, for instance, for a tensor of order six.

Keywords

• theory of algebraic invariants
• integrity basis under a subgroup
• isotropic tensor functions
• representation-theory
• irreducible basic and principal invariants of a fourth-order tensor
• constitutive tensors
• characteristic polynomial
• alternation process
• integrity basis for a tensor of order greater than four
• bilinear operator
• construction of simultaneous or joint invariants
• Hamilton-Cayley's theorem
• isotropic constitutive tensors.