Page 115 - Mechanics of Asphalt Microstructure and Micromechanics

Experimental Methods to Characterize the Heterogeneous Strain F ield 107

macro-strains like those defined in the finite element method can be calculated. The
calculation is illustrated as follows:
For each particle i, assume its two nearest neighbor particles are particles j and k.
The displacements of the three particles (calculated from the differences between the
coordinates of the centroids in un-deformed and deformed configurations) are u i , u j , u k ,
v i , v j , v k (i, j, and k are arranged counter-clockwise).
For a triangle, the valid displacement format can be assumed as:
u = a + a x a y (4-2)
+
0 1 2
+
v = b + b x b y (4-3)
0 1 2
Where x and y are coordinates in the image frame of the un-deformed configura-
tion.
u = relative displacement in x direction.
v = relative displacement in y direction.
a 0 , a 1 , a 2 and b 0 , b 1 , b 2 are constants for a triangle but usually vary from triangle to
triangle.
They can be represented conveniently by Equations 4-4 and 4-5.
1,,xy ⎧ ⎫ ⎧ u ⎫
a
0
i i ⎪ ⎪ ⎪ i ⎪
⎪
⎪
1,,xy ⎨ ⎬ = ⎨ u ⎬ (4-4)
a
j j 1 j
⎪ ⎪ ⎪ ⎪
,
1,xy k ⎩ a 2 ⎭ ⎩ ⎪ u ⎪
k k ⎭
k
1,,xy ⎧ ⎫ ⎧ v ⎫
b
i i ⎪ ⎪ ⎪ i ⎪
0
⎪
⎪
1,,xy ⎨ ⎬ = ⎨ v ⎬ (4-5)
b
j j ⎪ ⎪ ⎪ j ⎪
1
1,xy ⎩ ⎭ ⎩ ⎪ v ⎪
b
,
k k ⎭
k k 2
For small deformations, the strains can be computed as follows:
ε = ∂u = a (4-6)
x ∂x 1
∂v
ε = = b (4-7)
y 2
∂y
∂v ∂u
ε = ( + )/2 = ( b + a )/2 (4-8)
xy ∂x ∂y 1 2

4.2.4 Experimental Results and Analysis
The permanent macro strains for the eight images immediately beneath and adjacent to
the rutting wheel were determined. As noted, the macro-strains reflect an averaged or
homogenized strain because no distinction is made between the deformable mastic and
the non-deformable aggregate particles. The x and y direct strains, the shear strains, and
the volumetric strains for Image 9 are illustrated in Figure 4.7. The values of the strain