Page 14 - Mechanics of Asphalt Microstructure and Micromechanics

Introduction and Fundamentals for Mathematics and Continuum Mechanics 7

Using the Kronecker Delta, the scalar product can be represented as:
•
u v = • = uv e • e = uv δ = uv = u v = u v + u v + u v
v
v u
i j i j i j ij i i j j 11 2 2 3 3
1.5.3.2 Permutation Tensor
3
The permutation tensor, x ijk , is a rank three tensor. It has 3 = 27 components as defined
in the following:
1 if ijk appearsasin the sequence 1123123
ξ =−1 if ijk appears as in the sequence 3211321
ijk (1-8)
0 if ijk appears as in any other sequencees

One can conveniently verify that: x miq x jkq = d mj d ik – d mk d ij and x ijk = – x jik
It can also be conveniently verified that the cross product of the bases can be repre-
sented as:
e × e = ξ e (i, j, k = 1, 2, 3)
i j ijk k
Therefore, the cross product of two vectors can be expressed as:
×= (
u v u e ×) ( v e =) uv e ×( e =) uv ξ e
i i j j i j i j i j ijk k
×
and uv×= − v u , making use of ξ =− ξ .
jik
ijk
1.5.3.3 Dyadic Product of Two Vectors
The dyadic product of two vectors, a b, is defined as a linear operation that makes two
vectors into a tensor. It has the following features (a, b, and c are vectors; a is a scalar).
) b =
(αa a (αb ) α= (a b ) (1-9)

(1-10)
(1-11)

(1-12)
In more rigorous mathematics, a tensor is defined as a linear combination of the
basis dyads (Equation 1-13).
(1-13)

1.5.3.4 Product of Dyad with Vector
The inner product of a dyad with a vector is defined as . Its indicial
format is as follows:

(1-14)
a b c + b c + b c e ) + a b c + b c + b c e )
(
(
2 1 1 2 2 3 3 2 3 1 1 2 2 3 3 3