Page 13 - Mechanics of Asphalt Microstructure and Micromechanics

6 Ch a p t e r O n e

Multiplication of a vector by a scalar is equivalent to extension or contraction of the
vector along its original orientation (may reverse its direction, or make it a zero vector,
which may not have a physical meaning).
λv = λν e (1-4)
ii
The dot (scalar) product of two vectors results in a scalar (the magnitude of the
projection of one vector on the other) (Figure 1.1c).

uv•= v u•= u v cosθ = u v (1-5)
ii
Where q is the angle between the two vectors.
The cross product of two vectors is a vector with its magnitude equal to uv sinθ
and its orientation normal to the plane formed by the two vectors and following the
right-hand convention (Figure 1.1d). The magnitude is equal to the area of the parallel-
isms as shown in Figure 1.1d.

×
uv×= − v u = uv sinθ e (1-6)
The indicial format of the cross-product will be presented in the next section as it
requires the use of some tensors.

1.5.3 Tensor
Many quantities in mechanics require more than three independent components to rep-
resent. The stresses and strains are good examples. They can be represented as second-
2
order tensor (rank of two), with a total of nine components (3 = 9). The nine compo-
nents may not necessarily be independent. In this chapter, capital characters such as T,
Q, and C are used to represent tensors. The indicial format of the tensors (second-order)
may be represented as T ij and Q ij .
Two important tensors, the Kronecker Delta and the Permutation Tensor, are first
defined as they are often used in other vector and tensor operations.

1.5.3.1 Kronecker Delta
Kronecker Delta is equivalent to the unit (often represented as unit isotropic tensor I) or
one in scalar. It also serves as an operator (some tensors such as the transformation ten-
sor also serve as operators).

1 for = , = 1-3; = 1-3i j i j
δ = (1-7)
ij 0 for i ≠ ,j i = 1-3; = 1-3j

It can be conveniently verified that:
e • e = δ (i, j = 1,2,3)
i j ij
∂x
i = δ
∂x ij
j
δ e = e
ij j i
The last property represents the replacement operation.