The stress and strain at any point in a material is a dyadic (an array of ordered vector pairs) or second-order tensor whose value is wholly independent of the coordinate system that is used to describe its components. The second-order stress tensor σ may, therefore, be written as (where summation over values 1, 2 and 3 is implied by repeated lower case suffices)
where σkl and σ′kl are the stress components referred to the two coordinate systems being considered. On defining m = cos ϕ and n = sin ϕ, it follows from (2.172) that
Thus, from (2.176) and (2.177), because σ′12=σ′21, σ′13=σ′31 and σ′23=σ′32
The inverse relationships are obtained by replacing ϕ by −ϕ (i.e. n is replaced by –n) so that
The relationships (2.178) and (2.179) are the standard transformations, arising from tensor theory, for the rotation of stress components about one axis of a right-handed rectangular set of Cartesian coordinates. Identical transformations apply when considering the strain tensor so that
with inverse relations
2.17 Transformations of Elastic Constants
On substituting the stress-strain relations (2.170) into (2.181)