Page 93 - MODELING OF ASPHALT CONCRETE

Overview of the Stif fness Characterization of Asphalt Concr ete 71

B
C

A D
I
1

J′
2

FIGURE 3-2 Loading and unloading stress path for an isotropic stress sensitive material.

components of the stress tensor. Because of the shape of the particles and the way that
an asphalt concrete mixture is compacted, there is a vertical modulus, a horizontal
modulus, and a shear modulus. There are also two Poisson’s ratios, one in the vertical
plane and the other in the horizontal plane. Testing and analysis of triaxial test data
have been able to determine all five of the moduli of this cross-anisotropic case. The fact
of cross-anisotropy has several important implications for the way that pavements
resist both fracture and plastic deformation.
In a stress-sensitive material like asphalt concrete, an elastic material is one in which
loading and unloading follow a different stress path but one which closes upon itself as
is illustrated schematically in Fig. 3-2. The figure is a graph of the first stress invariant
plotted against the second deviatoric stress invariant and illustrates how the invariants
during loading and unloading follow the path A-B-C-D-A.
The elastic work potential for this kind of material is given by (Lade and Nelson 1989)

l
2
W = ∫ ABCDA ⎝ ⎛ ⎜ IdI 1 + dJ′ ⎞ ⎟ (3-1)
G ⎠
K
2
9
where W = elastic work potential
I = first invariant of the stress tensor
1
′ J 2 = second invariant of the ddeviatoric stress tensor
K,G =bulk and shear modulus of the material
k
If the elastic modulus of the material depends upon the same two stress invariants
according to the equation,
J′
E = K I () ( ) K 3 (3-2)
K 2
2
1
1
The requirement that the elastic work potential results in no net work produces a partial
differential equation that Poisson’s ratio must satisfy (Lytton et al. 1993).
2 ∂ν 1 ∂ν ⎛ 2 k k ⎞ ⎞ ⎛ 1 k k ⎞
− + = ν ⎜ 3 + 2 ⎟ + − 3 + 2 ⎟ (3-3)
⎜
2
3 ∂ ′ J 2 I 1 ∂I 1 ⎝ 3 ′ J 2 I 1 ⎠ ⎝ 3 ′ J 2 I 1 2 ⎠