Page 210 - Mechanics of Asphalt Microstructure and Micromechanics

202 Ch a p t e r S i x

FIGURE 6.13 Stress σ
analysis in an elliptic voids
in a thin plate.

2b A

2a

When 2a 2b
a
σ ≈ 2 σ (6-187)
A ρ

It was shown that even though the entire void is not elliptical, as long as its shape
around the tip (Point A) is elliptical, the above formulation gives a very good approxi-
⎛ σ ⎞
mation for the stress concentration ⎜ ⎝ σ ⎠ ⎟ . It should be noted that when the stress is
A
large enough the material close to the tip will yield and is dominated by plastic defor-
mation and will blunt the tip. While mathematically the radius of curvature could reach
zero, realistically this radius may not be smaller than the radius of an atom x 0 . There-

fore, one may conclude that the maximum stress is σ = 2 σ a . For actual materials,
A
x o

the stress concentration is much smaller than the above value due to yielding and plas-
tic deformation.
6.6.3 Energy Required to Create a Free Surface, an Atomic View of Fracture
Fracture ultimately goes to the dimension of atoms. Two atoms must be pulled apart so
that the attraction between the two atoms will no longer be able to push them back
without external forces.
Figure 6.14 illustrates the potential energy between two atoms of the same element.
The energy required to pull apart the two atoms x 0 (it is defined as the radius of the at-
oms) at equilibrium is called bond energy and can be calculated as:
b ∫
E = ∞ Pdx (6-188)
x o ⎛ π x⎞
If one approximates the force distribution between the two atoms as: P = P sin ⎜ ⎟
c ⎝ λ ⎠
The meanings of the terms are presented in Figure 6.14, the maximum force P c is
called the cohesive force. For small displacements, the force displacement relationship