Page 44 - Mechanics of Asphalt Microstructure and Micromechanics

Mechanical Proper ties of Constituents 37

elastic contact behavior, an approximate solution to the compression of an incompress-
ible elastic material between two rigid plates under full friction conditions for plane
strain and axisymmetric deformations was obtained by Nadai (1963). Nadai suggested
the use of a biharmonic stream function to obtain an approximate solution. If the width,
thickness, and length are 2a, 2h, and 2l, the elastic solution for the equivalent elastic
modulus of compression by Nadai is presented as Equation 2-8.
E
E = m A 2 (2-8)
ai
n 8
Where A = a and E m is the Young’s modulus of the material sandwiched. Obvi-
h
ously the equivalent modulus will increase when the film thickness is reduced. This is
consistent with the observation by Wang (2007).
For viscous binder material at ambient temperatures, the steady-state deformation
behavior of pure bitumen can be expressed using a generalized power law relationship
such as:

(2-9)

·
·
Where s ij is the deviatoric stress tensor, e ij is the deviatoric strain rate tensor, e e is
·
the effective deviatoric strain rate, and s 0 and e 0 are the reference deviatoric stress and
deviatoric strain rate, n is material constant. If n = 1, the material is linear viscous (New-
tonian). For many bitumen at modest stress levels, n > 2, indicating power law creeping
behavior. Equation 2-8 can be used to represent the viscous behavior of the film mate-
rial under multi-axial stress states (Cheung and Cebon, 1997a, 7b).
Consider the deformation of a plane strain contact in compression, for which the
·
two rigid plates approach each other at a constant velocity of −2h. The governing equa-
tion can be expressed as:
ε
σ n 1 A n+1 · 1
=
)
( n ) [( )(n + 2 ) ( ) n ]( · n n (2-10)
n
σ 2 n + 1 3 ε 0

FIGURE 2.2 Thin binder fi lm model (Cheung and Cebon, 1997b).