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International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D. ch. 1.1, p. 13
Section 1.1.4.5.2.2. Tensors of higher rank
a
Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
A tensor of rank higher than 2 may be symmetric with respect to the indices of one or more couples of indices. For instance, by its very nature, the demonstration given in Section 1.1.1.4![[link]](/graphics/greenarr.gif)
shows that the tensors representing principal physical properties are of even rank. If n is the rank of the associated square matrix, the number of independent components is equal to ![[(n^{2} + n)/2]](/teximages/dach1o1/dach1o1fi225.svg)
. In the case of a tensor of rank 4, such as the tensor of elastic constants relating the strain and stress tensors (Section 1.3.3.2.1
), the number of components of the tensor is ![[3^{4} = 81]](/teximages/dach1o1/dach1o1fi226.svg)
. The associated matrix is a ![[ 9 \times 9]](/teximages/dach1o1/dach1o1fi227.svg)
one, and the number of independent components is equal to 45.
