Page 212 - Mechanics of Asphalt Microstructure and Micromechanics

204 Ch a p t e r S i x

And the remote stress at failure will be:
/
⎛ γ E ⎞ 12 ⎛ γ E ⎞ 12
/
σ = s , σ = α s (6-194)
f ⎜ ⎝ 4 a ⎠ ⎟ f ⎜ ⎝ a ⎠ ⎟

There are quite a few assumptions in the above calculations. There is no possibility
that only two atoms are interacting with each other; the yielding process may weaken
the bonds significantly when the displacements reach a certain level; other factors such
as dislocations and defects are more dominating.

6.6.4 The Griffith Energy Criteria for Crack Opening
Griffith (1920) applied the First Law of Thermodynamics, which states that a system
going from a non-equilibrium state to equilibrium will have a net decrease in energy.
Therefore, a critical condition is that the total external energy input is equal to the en-
ergy consumed in creating new fractured surfaces (cracking opening or propagation) so
the total energy in the system is equal to zero.
dE = d∏ + dW s = 0 (6-195)
dA dA dA
Where E is the total energy, Π is the potential energy provided by internal strain en-
ergy and external forces, and W s is the work required to create new surfaces. Therefore:
d ∏ dW
− = s (6-196)
dA dA
Griffith used the stress analysis by Inglis (1913) and showed that for a crack of
width a (b approaches zero) embedded in a thin plate of thickness B under external
stress s , the driving potential energy is:
d ∏ πσ 2 a
− = (6-197)
dA E
To create crack surfaces (2aB on each surface, or the 2A ), the energy required:
W = 4 aBγ (6-198)
s s
Therefore,
dW
s = 2γ (6-199)
dA s
The stress that is required to drive the crack (fracture stress) is:
/
⎛ 2 γ E ⎞ 12
σ = s
f ⎜ ⎝ πa ⎠ ⎟ (6-200)

The fracture stress for a penny-shaped flaw can be shown as:
1
⎛ πγ E ⎞ 2
σ = s (6-201)
f ⎜ ⎝ 21 ( − va⎠ ⎟
2
)

Where n is Poisson’s ratio.