Page 101 - MODELING OF ASPHALT CONCRETE

Overview of the Stif fness Characterization of Asphalt Concr ete 79

from the stiffness of the bitumen (Heukelom and Klomp 1964, Van der Poel 1954). The
air void relation is in Eq. (3-9):
G 5
= 1 − c i (3-9)
G m 2
It is possible to use these equations to get a rough estimate of how the shear stiffness of
the asphalt concrete is related to the shear stiffness of the asphalt [Eq. (3-8)] or how air
bubbles in the asphalt will alter the stiffness of the asphalt [Eq. (3-9)]. A corresponding
relation for the bulk modulus of the composite is in Eq. (3-10) (Christensen 1991):
K − K 4 ( G + 3 K )
m = c m m (3-10)
K − K m i 4 [ G + 3 K + ( K − K )c ]
3
c
i
i
i
m
m
i
where K = bulk modulus of the composite
K , K = bulk modulus of the matrix and inclusion, respectively
m i
G = shear modulus of the matrix material
m
c = volume fraction occupied by the inclusion
i
There are other, different relations predicting the bulk modulus of the composite that
have been developed by using slightly different mathematical approaches, all of which
have their limitations. Using the bulk and shear moduli of the composite as estimated from
one of these formulas, it is possible to make further estimates of the Young’s modulus and
Poisson’s ratio. Recognizing these limitations, the Method of Cells was developed to
provide a numerical method for estimating these and other material properties, taking into
account the one-, two-, or three-dimensional geometry of the inclusion (Aboudi 1991). The
numerical results are able to match measured results very closely. Other properties that are
estimated by use of these micromechanics methods include the coefficient of thermal
expansion, thermal conductivity, creep compliance and relaxation modulus, the time-
temperature shift function, electrical conductivity and dielectric constant, and anisotropic
yield strength of the composite among others (Christensen 1991).
Because asphalt concrete stiffness is a viscoelastic, rather than an elastic property, it is
frequently necessary to convert the viscoelastic properties of the components of the
composite into the effective viscoelastic property of the composite. This can be done using
the same micromechanics formulas and the correspondence principle which is illustrated
in Eqs. (3-11), (3-12), and (3-13). The effective elastic bulk modulus formula in Eq. (3-10) is
rewritten with bars over the modulus terms as in Eq. (3-11) (Christensen 1991):
K − 4 Gm + 3
K m = c K m (3-11)
K i − K m i 4 [ Gm+3 K i+3( K m − K c ]
)
i
i
The meaning of the bar over either K or G is the Laplace transform of the relaxation modulus
multiplied by s (Carson transform), the Laplace transform parameter in Eq. (3-12).
Ks() = s ∫ o ∞ K t e dt (3-12)
−
st
)
(
Equation (3-11) is solved for the Laplace transform of the composite Ks(), and the entire
expression is inverted to produce the effective bulk relaxation function of the composite
K(t). A similar exercise will produce the effective shear relaxation function of the
composite G(t). The inversion of this expression is usually done numerically although
closed forms are possible using Schapery’s approximate inverse Laplace transform