Page 106 - MODELING OF ASPHALT CONCRETE

84 Cha pte r T h ree

Changes this slight with one loading and unloading cycle can accumulate with
repeated load cycles and the damage can grow accordingly. In order for the pseudostrain
energy to match actual measured energy expenditures, the reference modulus E must be
R
a real modulus of the material which is found by dividing the maximum measured stress
by the maximum measured strain. The rate of change of this dissipated pseudostrain area
with respect to the change of microcrack surface area is defined as the pseudo-J-Integral and
is used in the fundamental law of fracture mechanics from which all fracture predictions
are derived. This fundamental law was stated by Schapery (Schapery 1984) to be
Γ
2= ED t () J (3-18)
R α R
where Γ = cohesive fracture surface energy of the material
J = pseudo-J-integral
R
D(t ) = compliance of the material
a
t = time required for the crack to grow through the distance of the fracture
a
process zone which is of length, a
E = reference modulus of the material
R
Several examples of the application of this fundamental law of fracture are given in Chap. 12
and will not be repeated here. The law given in Eq. (3-18) is for cohesive fracture. There is a
slightly different formulation for adhesive fracture which takes into account the interaction of
the materials that are bonded at an interface, and even include the effect of a third material,
such as water, which may be present on the interface. The calculation of the adhesive and
cohesive bond strength from the individual surface energies of the component materials is
also illustrated in Chap. 12. Thus the expression of the ratio between the damaged and
undamaged modulus of a nonlinear viscoelastic material, taking into account the rate of
change of dissipated pseudostrain energy in repeated loading is given in Eq. (3-19):
⎧ ⎡ ⎡ 1 ⎤ ⎫
⎢
E ⎛ m ⎞ ⎪ 2 12Γ ⎛ 4 At ⎞ 1 + n ⎥ ⎪
−
= 12+ π ⎜ ⎟ ⎨ ∫ c ⎢ ⎜ dW ⎟ ⎥ pc dc ⎬ (3-19)
()
E′ ⎝ bh ⎠ ⎪ ⎢ ⎜ R ⎟ ⎥ ⎪
⎩ ⎣ ⎝ dN ⎠ ⎦ ⎭
If a material that has been damaged by cracking is allowed time between repeated loads
for the cracks to close and to reform the broken bonds, it will heal. Healing is a
complementary process to fracture except that the polar and nonpolar components of
wetting or healing surface energy play differing roles than they do in fracture. In
fracture, they both resist the fracture. In healing, the stronger the polar wetting surface
energies, the more they assist in forming healing bonds whereas the stronger the
nonpolar surface energies of wetting or healing, the more they resist the reforming of
the broken bonds. The nonpolar surface energies, which are primarily Lifshitz-van der
Waals forces, affect the rate of short term healing, in the range of seconds. The polar
surface energies, which are primarily hydrogen bonds, affect the long-term rate of
healing in the range of minutes and hours. Thus, it is possible for a pavement to have
microcracks and even large propagating shear cracks develop under repeated traffic
loading, as illustrated in Fig. 3-13. But if the pavement is built of a good asphalt concrete
which both resists fracture and heals well, it is possible to have the microcracks and
even some of the propagating cracks to heal substantially and to recover much of the
original strength and stiffness of the material during low traffic periods.