Page 142 - Mechanics of Asphalt Microstructure and Micromechanics

134 Ch a p t e r Fiv e

By these considerations, it is assumed that the effective aggregate is a linearly elastic
material observing the following constitutive equation:

τ = λε + 2 με (5-25)
sij skk sij
Where l and m are the Lame’s constants; t sij and e sij are the stress and the strain ten-
sor in the real material, and the subscript “s” represents solids.
It is assumed that the air voids behave as an ideal compressible gas observing the
following constitutive equation:

τ =− δ p (5-26)
aij ij
p = γ RT (5-27)
a
Where R = the universal gas constant, T = the absolute temperature, g a = the density
of air, and t aij = the stress in the air; the subscript a represents air.
The stress calculated from the above two equations is the actual stress in the mate-
rial. It is related to the partial stress by the following equation:

σ = φτ (5-28)
sij sij
)
σ = (1 − φ τ (5-29)
aij aij
Momentum interaction
With the consideration of static loading, Equation 5-22 can be simplified as:
p =−ξγ grad +φ ξ γ gradφ (5-30)
s 11 s 12 a

p =−ξγ grad +φ ξ γ gradφ (5-31)
a 21 s 22 a
Boundary conditions
There are many methods proposed to handle the boundary condition problem (Rajago-
pal, 1996). Krishnan and Rao (2000) also followed Rajagopal’s method. The basic idea is
to use the relation that the area fraction is equal to the volume fraction and distribute
the mixture surface traction to the partial surface traction of constituents according to
their area fraction.
σ n = t (5-32)
α s α
t = φ t (5-33)
α α
Where s a is the partial stress; n s is the normal to the surface, and t a is the partial trac-
tion.
By the distribution proportions expressed in Equation 5-33, it is easy to show that
N
N
∑ t α ∑ φ = t . Formally, it satisfies the requirements. However, it should be noted
=
t
α
α=1 α=1
that Equation 5-33 might not hold universally. For example, the shear components of
the surface traction cannot be allocated to the air constituent at all. In general, bound-
ary conditions might be case dependent. In the two-constituent situation (solids and