Page 162 - Mechanics of Asphalt Microstructure and Micromechanics

154 Ch a p t e r Fiv e

then obtained the effective modulus of mixtures in the following equation, where air
void distribution has been considered:
)( −
v
2 E 1 ( − n 1 2 )
Ea () = 1 0 (5-169)
( −
0 n 1 ( + v ) + 2 1 2 v )
1 1
Lytton in Song et al., 2005, following the Energy Equivalency theorem, developed
equations to relate the modulus in a damaged material to the modulus of an undam-
aged material. The formations are related to whether the state is stress-controlled or
strain-controlled. For the strain controlled state, the following equations are obtained.
Cohesive:
⎡ ⎛ GG ⎞ ⎤ ⎡ ⎛ Δ c ⎞ ⎤
Δ
⎞
c
G' ⎛ m r 3 f f r GG f
−
⎥
−
f
= ⎢12π 2 ⎜ ⎟ ⎜ 1 − ⎟ = ⎢12πξ ⎜ 1 − ⎟ ⎥ (5-170)
2
G ⎢ ⎣ ⎝ A⎠ t ⎝ πτ 2 r ⎠ ⎥ ⎦ ⎦ ⎢ ⎣ t ⎝ πτ r ⎠ ⎥ ⎦

Adhesive:
⎡ 3 ⎛ ⎡ G ⎞ GG ⎤⎤ ⎡ ⎛ ⎡ ⎞ 4GG ⎤⎤
Δ
Δ
a
⎞
a
G' ⎛ m r f 4 f r G f f
⎢
⎢
−
−
= 1 π 2 ⎜ ⎟ ⎢ ⎜ 1 + ⎟ − ⎥⎥ = 1 πξ ⎢ ⎜ 1 + ⎟ − ⎥⎥ (5-171)
τ
2
G ⎢ ⎣ ⎝ A⎠ t ⎢ ⎣ ⎣ ⎝ G s ⎠ πτ 2 r ⎥⎥ ⎢ ⎣ t ⎣ ⎝ ⎢ G ⎠ πτ r ⎥⎥
⎦⎦
⎦⎦
s
Other Studies
Abbas, et al. (2000) compared micromechanics predictions using the Hashin and Sh-
trikman lower bound, the GSCS, and the inverse rule of mixture with their DEM sim-
ulations. Li, et al. (2005) compared micromechanics model predictions using the
Hirsch model, the Hashin Composite Sphere model, and the Christensen and Lo mod-
el with experimental observations. Chen and Peng (1998) applied the Hirsch model to
study the mastic modulus. Druta, et al. (2007a, b) also applied the model to predict the
modulus of mastic and mixture.
5.5 Doublet Mechanics
There have been significant developments in the area of micromechanics of granular
materials in the last two decades. Two different approaches have been generally fol-
lowed in micromechanics. One follows the routine of describing particle contact and
configuration (Christoffersen et al., 1981; Chang and Ma, 1991); the other, doublet me-
chanics (Granik and Ferrari, 1993; Ferrari et al., 1997) assumes a Bravis Lattice as the
microstructure and considers the doublet deformation and particle interaction by tak-
ing different orders of the deformation field expressed as a series of the coordinates.
When the particle size approaches zero, it becomes a continuum. Both methods could
explain the phenomenon of tensile stress induced under compressive forces. The fol-
lowing discussion presents a doublet mechanics explanation.
Granik et al. (1993) and Ferrari et al. (1997) assumed a face-centered cubic packing
of spheres as the microstructure model of granular materials (with bond among the
contacts) to solve the classical Flamant problem: a semi-infinite plate subjected to a
point force normal to the boundary (Todhunter and Pearson, 1960). The analytical solu-
tion by Granik and Ferrari (based on the microstructure model illustrated in Figure 5.6)