Page 402 - Mechanics of Asphalt Microstructure and Micromechanics

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394 Ch a p t e r w e l v e

simplicity. It also has some theoretical basis in that the initial stresses at
t = 0 has significant influence on the subsequent viscous and plastic deformation and
the stress analysis can be performed using elasticity theory through the elastic-viscoelas-
tic correspondence principle. In this chapter, an anistropic elasticity analysis of a pave-
ment under wheel load will be presented. Although AC may demonstrate general an-
isotropy, only cross anisotropy or orthotropy is considered for the purpose of simplifica-
tion, and due to the experimental evidence. For the orthotropic case, AC is considered to
have significant differences only in vertical and horizontal directions due to the aniso-
tropic compaction, aggregate orientation, restraint conditions, and gravity direction. In
orthotropic elasticity, there are only five material constants. However, the general aniso-
tropic elasticity has 21 material constants and is not realistic for modeling and character-
ization. The five material constants of the orthotropic elasticity are E v , E h , n vh , n hh , and G vh .
Where E v and E h are elastic modulus in vertical and horizontal directions respectively; n vh
and n hh are Poisson’s ratios for vertical-horizontal and horizontal-horizontal responses,
respectively; and G vh is the shear modulus along the vertical plane. Hooke’s Law for the
orthotropic case can be expressed as follows:

Δε = 1 Δσ − ν 1 Δσ − ν 1 Δσ (12-1a)
x E x hh E y vh E z
h h v
1 1 1
Δε = Δσ − ν Δσ − ν Δσ (12-1b)
y E y hh E x vh E z
h h v
1 1 1
Δε = Δσ − ν Δσ − ν Δσ (12-1c)
z E y vh E x vh E y
v h v
1
Δγ = Δτ (12-1d)
yz G yz
vh
Δγ = 1 Δτ (12-1e)
zx G zx
vh
(1 + ν )
Δγ = hh Δτ (12-1f)
xy E xy
h
Where Δe x , Δe y , Δe z are normal strain increments and Δs x , Δs y , Δs z are normal stress
increments; Δg yz , Δg zx and Δg xy are shear strain increments; and Δt yz , Δt zx and Δt xy are the
corresponding shear stress increments. It should be noted that in the above formula-
tions, the properties in tension and compression are considered the same.

12.3 Boussinesq’s Solution for Orthotropic Materials
For a full-depth AC pavement of isotropic materials, if the distributed tire load could be
approximated as a point load, the stress field could be approximated by the Boussin-
esq’s solution, a half-space subjected to a concentrated load P as illustrated in Figure
12.1. The corresponding Boussinesq’s solution for orthotropic material was obtained by
Wolf (1935). The analytical expressions for two of the four stress components, s θ and t yz ,
are presented in Equations 12-2a and 12-2b. The solution is based on the cylindrical