Page 12 - Mechanics of Asphalt Microstructure and Micromechanics

Introduction and Fundamentals for Mathematics and Continuum Mechanics 5

compresses one or two of the three dimensions to zero, one can have two-dimensional
(2D) vectors and a 1D vector (a scalar). A vector is also called a first-order tensor, having
1
3 = 3 independent bases. Mathematically, a set of mutually independent N variables
can be considered as an N dimensional vector. Lower-case characters such as v and u
are used to represent vectors in this chapter. Vectors can be also represented in indicial
format as v i and u i (i = 1~3). Vectors in rigorous sense (2D and 3D) follow the triangle
rule for “add” and “subtract.” Abstract extension to an N dimensional vector may not
follow such rules.
A vector can be represented in its indicial format as follows:
3
ν = ν e + ν e + ν e = ∑ ν e (1-1)
11 2 2 3 3 ii
i =1
Where e i are the three bases (with unit length). By following the Einstein convention
(the dummy index or the repeaded index will sum running from 1 to 3), it can be simply
represented as Equation 1-2:
n = n i e i (1-2)
v = v + v + v is the magnitude of a vector or the “normal/mod/length” of a
2
2
2
1 2 3
vector (see Figure 1.1a). The vector can be generally represented as scalar (magnitude)
in an orientation where e is the orientation. It is this orientation that needs a reference
system to measure it. The above representation (Equation 1-1) is for a Cartesian refer-
ence system.
Addition or subtraction of vectors will follow the following rules (Figure 1.1b):
u v w e =,
w =± ( u ± ) (1-3)
v e
ii i i i
X 2

b
a
a 3 a
c
a 1
a 2
X 1 a + b = c
X 3 (a) (b)

b c
b

a
c a
a · b = c a b = c
(c) (d)

FIGURE 1.1 Illustration of typical vector operations.