International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 1, Pages 87-96

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

Josef Betten

Rheinisch-Westfälische Technische Hochschule Aachen, Templergraben 55, Aachen D-5100, Germany

Received 8 July 1981

Copyright © 1982 Josef Betten. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.