Premium
On VOIGT and KELVIN Matrix Notations of Second‐Order Tensors
Author(s) -
Schlebusch Rainer
Publication year - 2019
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201900389
Subject(s) - matrix (chemical analysis) , orthonormal basis , scalar (mathematics) , matrix multiplication , notation , invariant (physics) , algebra over a field , homogeneous space , mathematics , pure mathematics , physics , geometry , quantum mechanics , mathematical physics , arithmetic , materials science , composite material , quantum
The implementation of constitutive models into a computer software necessitates the conversion of abstract tensors into real numbers that can be processed by a computer. This is accomplished (1) by introducing a basis, and (2) by arranging the scalar tensor components into matrices. In literature, such conversion processes are often used ad hoc without any rigorous mathematical derivation or justification. Probably most prominent are the VOIGT and KELVIN matrix notations. These matrix notations use existing symmetries of the tensors and allow to remove any redundant information from the matrix representations. Thus, in this special but very common case, the effective size of the matrices are significantly reduced and with this the computational effort. In this contribution, the mathematical background for the two mentioned conversion processes is explained. It is shown that VOIGT and KELVIN matrix representations are not restricted to orthonormal bases, and with this not to Cartesian tensors. Moreover, it is demonstrated that the value of a duality pairing is invariant with respect to the conversion process.