Orthonormal decomposition of symmetric second rank tensors

Abstract

In this paper, a new orthonormal decomposition method for symmetric second rank tensors namely as, orthonormal tensor basis is presented. Complex variable representation method is developed by using the existing theories in literature. For comparison purposes, a brief review of the spectral method is given. It is shown that stress tensor, as an example to symmetric second rank tensors, is decomposed into six orthonormal parts by orthonormal tensor basis and complex variable representation methods. The matrix forms of these decomposed parts are given. This is the first time in literature that physical meanings of each six decomposed parts which are obtained from the orthonormal decomposition of stress tensor by orthonormal tensor basis and complex variable representation methods, different from the traditionally form, are emphasized. Illustrative applications on orthonormal tensor basis and complex variable representation decomposition methods are given. Finally, it is proved that the spectral method is a non-linear decomposition method which yields three non-linear orthonormal decomposed parts. This case is a significant innovation in decomposition procedures for symmetric second rank tensors in literature.