Page 214 - Mechanics of Asphalt Microstructure and Micromechanics

206 Ch a p t e r S i x

Model I Model II Model III

FIGURE 6.16 The stress modes of cracks.

In general, the stress fields ahead of a crack tip in an isotropic linear elastic material
can be expressed as follows:
K
θ
limσ () I = I f () I ( )
r→0 ij 2π r ij
K
limσ () = II f () ( )
θ
II
II
r→0 ij 2π r ij
K
θ
limσ (III ) = III f (III ) ( )
r→0 ij 2π ij
r (6-205)
σ ( total) = σ I ( ) + σ ( II) + σ ( III)
ij ij ij ij (6-206)
Where K I , K II , K III are the corresponding stress intensity factors; f ()θ , f ()θ and
I
II
ij
ij
III
f ()θ are functions representing stress distributions. There are methods to calculate
ij
the stress intensity of mixed modes.
6.6.7 Crack Tip Plasticity
The elasticity analysis of the stresses around a tip results in infinite stresses when r ap-
proaches zero. Real material will yield at a much lower stress. Considering the stress on
the plane of crack (q = 0), one has:
K K
σ = I σ = I (6-207)
xx yy
2 πr 2 πr
By setting s yy = s YS one can have an approximate solution for the plastic zone.
1 ⎛ K ⎞ 2
r = ⎜ I ⎟ (6-208)
⎝
y 2πσ YS ⎠