Page 211 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 203

Potential Potential
and Cohesive
Force Force
Force
k
Area=Bond
Energy
λ Bond Distance
Energy
x 0

Equilibrium Spacing

FIGURE 6.14 Potential, bond energy and cohesive force.

⎛ π x⎞
can be approximated as a linear relationship P = P c ⎜ ⎟ . The bond stiffness can then be
obtained as: ⎝ λ ⎠
π
k = P (6-189)
c λ
Multiplying both sides of the above equation by the number of bonds per unit area
and the gage length x 0 leads to the following equation:
π
kNx = Nx P (6-190)
0 0 c λ
One can obtain the following equation:
E λ
σ = (6-191)
c πx
o
Considering l and x 0 are at a similar magnitude, the above equation can be ap-
proximated as:
E
σ ≈
c π
And the surface energy can be calculated as:
1 λ ⎛ πx⎞ λ
γ = ∫ σ sin dx = σ (6-192)
S 2 0 c ⎜ ⎝ λ ⎠ ⎟ c π

And therefore:
γ E
σ = s (6-193)
c x
o