Page 164 - Mechanics of Asphalt Microstructure and Micromechanics

156 Ch a p t e r Fiv e

For a distributed load ranging from –a to a, the micro-stresses at location (x, y) can
be obtained by integration as in Equations 5-174a, b, and c. Figure 5.7 graphically pres-
ents the stress distributions.
⎡ 3 −− 3 −+ + ⎞ − ⎞ ⎤
qy
1 ∫
F = − a p du = 2 3π ⎢ ⎢ y x a − yx a − 1 arctg ⎛ xa ⎟ + 1 arctg ⎛ xa ⎟ ⎥ (5-174a)
⎜
⎜
)
2
1
2
+
−
a
2
⎣ ⎣ ( xa) + y ( xa) + y 2 y ⎝ y ⎠ y ⎝ y ⎠ ⎥ ⎦
⎡ 3 ++ 3 +− + ⎞ − ⎞ ⎤
2 ∫
F = a p du = 2 qy ⎢ y x a − yx a + 1 arctg ⎛ xa ⎟ − 1 arctg ⎛ xa ⎟ ⎥ (5-174b)
⎜
⎜
)
3π ⎢
2
−
2
2
+
−
2
a
⎣ ⎣ ( xa) + y ( xa) + y 2 y ⎝ y ⎠ y ⎝ y ⎠ ⎥ ⎦
⎡ x 2− 2 − + ⎞ − ⎞ ⎤
3 ∫
F = a p du = 2 qy ⎢ 2 2 a − x 2 a − 1 arctg ⎛ xa ⎟ + 1 arctg ⎛ xa ⎟ ⎥ (5-174c)
⎜
⎜
3π ⎢
2
−
+
+
−
a
3
2
⎣ ⎢ ( xa) + y ( xa) + y 2 y ⎝ y ⎠ y ⎝ y ⎠ ⎥ ⎦
The micro-stresses are related to the configurations of the particles. This indicates that
the microstructure of the material must be considered in the micromechanics analysis.
FIGURE 5.7 Stress y
2a
distribution under a strip
of loading.
x
F1 distribution in the x-y plane
10-20
0-10
-10-0
-20--10
-30--20
-40--30
-50--40
-60--50
-70--60
-80--70
F2 distribution in the x-y plane
70-80
60-70
50-60
40-50
30-40
20-30
10-20
0-10
-10-0
-20--10
F3 distribution in the x-y plane
10-20
0-10
-10-0
-20--10
-30--20
-40--30
-50--40