Page 27 - Mechanics of Asphalt Microstructure and Micromechanics

20 Ch a p t e r O n e

1.6.3 Stresses
1.6.3.1 Cauchy Stress
Considering an area element ΔS of current configuration, having applied forces Δf i and
momentum ΔM i on it, the Cauchy stress principle asserts that the following limits (as
the area ΔS approaches zero) exist:

Δf df
lim i = i = t () n (1-98)
ΔS→0 ΔS dS i
It is named the stress vector or traction vector.
ΔM
lim i = 0 (1-99)
ΔS→0 ΔS
This means that the distributed momentum is equal to zero.

1.6.3.2 Stress Tensor
Consider an infinitesimal tetrahedron in Figure 1.5 at point P in the current configura-
tion, the three stress vectors can be represented as:
t e () = t e () e + t e () e + t e () 1 e
1
1
1
1 1 2 2 3 3
t e ( 2 ) = t e ( 2 ) e + t e ( 2 ) e + t e ( 2 ) e
1 1 2 2 3 3
t e () = t e () 3 e + t e () e + t e () 3 e
3
3
1 1 2 2 3 3 (1-100)
Using summation convention, these stress vectors can be represented as:
t () = t () e , ( i=1,2,3)
e i
e i
j j
In Figure 1.5, the minus sign denotes the opposite direction of the stress vectors.
The Newton second law for the tetrahedron can be represented as:
=
−
n ()
t dS t e () 1 dS − t e ( 2 ) dS − t e () 3 dS + ρ b dV = ρadV (1-101)
i i 1 i 2 i 3 i i
FIGURE 1.5 Free-body e 3
diagram for illustrating
stress tensor.
C
n
t
t 2
3 B
e 2
2
t
2
P t 1 2

A
e 1