Page 100 - MODELING OF ASPHALT CONCRETE

78 Cha pte r T h ree

the asphalt concrete stiffness. It is fortunate that the disciplines of micromechanics and
viscoelastic fracture mechanics have provided principles that may be used to determine
how these factors will affect the mixture stiffness. An overview of these principles as
applied to an asphalt concrete mixture is presented in the next section.

Characterization of Asphalt Concrete Stiffness
Characterization of asphalt concrete stiffness refers to the mathematical relations between
stress, strain, temperature, loading rates, moisture, and composition that are found in
the disciplines of mechanics, including micromechanics and fracture mechanics. The
stiffness of asphalt in an undamaged state is different from that in a damaged state and
this difference is directly a result of the surface energy, adhesive and cohesive fracture
and healing, plasticity and viscoplasticity, and moisture damage characteristics of the
material. There is a large body of technical literature relating to the micromechanics of
representing the properties of a composite material when the properties of the component
materials are known. The discussion that follows will provide some of the high points
that are relevant to the characterization of both damaged and undamaged asphalt
concrete stiffness.

Undamaged Stiffness
There are several approaches in micromechanics, all of which require that the strain
energy that is put in to a composite material when it is being loaded is completely
accounted for in being stored in the component materials of the composite. The same is
true of energy that is released when the composite is unloaded. The objective is to arrive
at a single stiffness that treats the composite as a homogeneous material but one that
absorbs and releases strain energy in the same way that the composite does. Several of
the principal micromechanics approaches give an upper and lower bound to this
stiffness. One result of this mathematical process is the ratio of the shear modulus of the
*
composite G to the shear modulus of the matrix in which a solid inclusion is imbedded
G . The relation is in Eq. (3-7) (Christensen 1991; Aboudi 1991):
m
⎡ ⎛ G ⎞ ⎤
15 1 − ν m ⎢ 1 − ⎜ i ⎟ ⎥ c
(
)
m ⎦
G ⎣ ⎝ G ⎠ i
= 1 − (3-7)
G ⎛ G ⎞
m 7 − 5ν m + 224( − 5ν m ⎜ ⎝ G ⎠ ⎟
i
)
m
where G,G = shear moduli of the matrix and the inclusion, respectively
d
i
m
,
n = Poisson s ratio of the matrix
r
m
c = fraction of the total volume occupied by the inclusion
l
i
Two well-known results are when the inclusion is rigid and when it is a void,
corresponding roughly to aggregate particles in an asphalt matrix and air voids in an
asphalt medium. The rigid particle relation is in Eq. (3-8):
G 5
= 1 + c (3-8)
G m 2 i
This relation was first published by Einstein (Einstein 1956). It has been used in a
somewhat modified form in the Shell nomograph to estimate the stiffness of a mixture