Page 216 - Mechanics of Asphalt Microstructure and Micromechanics

208 Ch a p t e r S i x

FIGURE 6.18 Illustration of Γ
the J integral concept.
Y
X

ds
Where m is a dimensionless constant and is approximately equal to 1.0 for plane
stress and 2.0 for plane strain. Please note that CTOD is related to the toughness of a
material and therefore a measure of the material’s resistance against fracturing.
6.6.9 The J Contour Integral
For non-linear materials, the J contour integral is a better measurement. Rice (1968)
showed that the following integral is path independent and it is equal to the energy
release rate:
⎛ u ∂ ⎞ dΠ
J = ⎜ ∫ wdy T i x ∂ i ds =− (6-214)
−
⎟
Γ ⎝ ⎠ da
Where w = ∫ ij ε σε ij
d is strain energy density; T i (T i = s ij n j ) and u i are the components
ij
0
of traction vector and displacement vector, respectively. This integral is independent of
the path of integral (see Figure 6.18).
6.6.10 Relations Between J and CTOD
Considering the relationship between energy release rate and the CTOD, one can have
the following relation:
J = mσδ
YS (6-215)
6.6.11 The C* Integral
For materials subjected to steady state creep, Landes and Begley (1976); Ohji et al. (1976)
and Nikbin et al. (1976) independently proposed the C* integral method.
⎛ u ∂ ⎞
C* = ⎜ ∫ ⎝ wdy T i x ∂ i ds ⎟ ⎠ (6-216)
−
ε Γ
where w = ∫ 0 kl σε ij
d
ij
It can be proved that C* is path independent. The dotted parameters in Equation
6-215 represent the time rate of the parameters.

6.6.12 Cracking in Viscoelasticity Materials
Schapery (1984) proposed a generalized J integral that is applicable to a wide range of
viscoelastic materials. To better understand his method, the fundamental viscoelastic
constitutive relationships are briefly reviewed. The hereditary integrals for viscoelastic
materials include the following both 1-D and 3D cases:
t σ t
ε() = ∫ ( C t − τ) d τ d σ() t = ∫ ( E t − τ) ε d τ d (6-217)
t
0 τ d 0 τ d
t d σ t ε d
τ
ε t() = ∫ C ( − τ) kl τ d σ t() = ∫ E ( − τ) kl τ d (6-218)
t
t
ij ijkl τ d ij ijkl τ d
0 0