Page 102 - MODELING OF ASPHALT CONCRETE

80 Cha pte r T h ree

method (Schapery 1962, 1965). Because there are different methods of estimating the
bulk and shear moduli of elastic composite, it is usually necessary to verify that the
conversion produces reliable results by comparison with actual measurements made
on the composite. A similar transformation can be used to convert the complex moduli
of the components into the effective complex moduli of the composite. In this way, the
formula for the elastic bulk modulus of the composite is converted into the formula for
the complex bulk modulus of the composite as in Eq. (3-13) (Christensen 1991).
K − K * 4 G + 3 K *
*
*
m = c m m (3-13)
K − K * i 4 [ G + 3 K + 3( K − Kc)]
*
*
*
*
*
i m m i m m i i
The K and G terms with the asterisks in this equation are the complex bulk and shear
modulus of the matrix and the inclusion, all of which have a real and an imaginary
component as in Eqs. (3-14a), (3-14b), and (3-14c) (Christensen 1991):
′′
K ()ω = ′ i K ()ω (3-14a)
*
K ()ω +
m m m
K ()ω = ′ i K ()ω (3-14b)
′′
*
K ()ω +
i i i
G ()ω = ′ m i G ()ω (3-14c)
′′
*
G ()ω +
m
m
If the material is nonlinear viscoelastic, the equations given above must be treated as
approximations, but they provide correct forms of equations that take into account the
strain-energy storage of each of the components of the composite material.
Effects of Microcracks on Stiffness
Because of the ability to use the correspondence principle to convert elastic solutions
into viscoelastic equations for asphalt concrete stiffness, it is possible to derive relations
using elastic theory with the confidence that they can be converted into the appropriate
viscoelastic form, either the creep compliance, the relaxation modulus, the complex
compliance, or the complex modulus. When a repeated load test is made on an asphalt
concrete, its stiffness appears to decrease with increasing numbers of load applications.
However, what is really happening is that small microcracks are forming in the material,
producing an apparently smaller modulus. This is illustrated in Fig. 3-8 with two straps
being subjected to the same tensile stress.
Microcrack size
Actual Apparent
E E′

2 c

Same stretch
E′ < E

FIGURE 3-8 Effect of microcrack size on apparent modulus.