href="#ulink_38d37496-8230-5c94-970d-31585c3ea00d">(5.14) indicates that the double products of j 's can be seen as matrix products of the corresponding unit vectors, according to
(5.16)
(5.17)
(5.18)
(5.19)
(5.20)
(5.21)
(5.22)
and
In view of Eqs. (5.15)–(5.23), one may rewrite Eq. (5.14) as
(5.24)
– or, in a more condensed fashion,
(5.25)
provided that i and j denote x (i = 1 or j = 1), y (i = 2 or j = 2), or z (i = 3 or j = 3).
One may now briefly refer to the multiplication of scalar α by tensor τ – represented by
(5.26)
which may be rewritten as
(5.27)
in view of Eq. (5.2); the distributive and commutative properties of multiplication of scalar by matrix as per Eq. (4.34) produces
(5.28)
or, in condensed form,
(5.29)
– which is equivalent to writing
(5.30)
in parallel to Eq. (4.20).
The scalar (or dot) product of a vector u by a tensor τ is a vector defined by
(5.31)
inspired by Eq. (3.52) and taking advantage of Eq. (4.47) – or, in condensed form,
if the order of multiplication is reversed, one obtains
(5.33)
that may be rephrased as
– so inspection of Eq. (5.34) vis‐à‐vis with Eq. (5.32) indicates that
(5.35)
except if τ is symmetric (as τji = τij under such circumstances).
The scalar product of two tensors, σ and τ (also known as double dot product, :), is a scalar defined as
(5.36)
where straightforward algebraic rearrangement was used to advantage – or, in condensed form,
(5.37)
Here tensor σ abides to
(5.38)
while (1 × 3) row vectors σ_x, σ_y, and σ_z are defined as
(5.39)
(5.40)
and
(5.41)
respectively, whereas (3 × 1) column vectors τx_, τy_, and τz_ abide to
(5.42)
