Page 147 - MODELING OF ASPHALT CONCRETE

Complex Modulus fr om the Indir ect Tension Test 125

where
⎡ l l ⎤
ν
B = ⎢ (ν − ) ∫ n y dy −( ) (1 + ) ∫ m y dy ⎥ (5-16)
( )
1
⎣ ⎢ l − l − ⎦ ⎥
with
2
(1 − y / )sin 2α
2
R
my() = (5-17)
−
4
2
2
12 y / R cos 2α + y / R 4
2
⎧ 1 + y / R 2 ⎫
and ny () = tan −1 ⎨ − 2 2 tanα ⎬ (5-18)
⎩ 1 y / R ⎭
By equating Eqs. (5-11) and (5-15), one can obtain
(5-19)
⋅
AV t = B U t()
⋅
()
Then, one may derive the expression for Poisson’s ratio as follows:
β Ut() − γ V t()
ν = 1 1 (5-20)
− β Ut() + γ V t()
2
2
where
l l
∫
∫
()
()
β =− n y dy − m y dy
1
−l −l
l l
2 ∫ ∫
()
β = ny dy − my()ddy
−l l −
l l
γ = ∫ f x dx −() ∫ g x dx
()
1
l − l −
l −l
∫
2 ∫
γ = f x dx + g x dx (5-21)
(
)
(
)
−l −l
Combining Eqs. (5-11) and (5-15) yields a single form of the dynamic modulus, as
shown below:
P sin( wt − )φ AV t + P sin( wt − )φ BU t ( )
( )
∗
E = 0 0 (5-22)
⋅
π ad V t ( )⋅Ut
( )
)
After substituting Eqs. (5-12) and (5-16) into Eq. (5-22), one can obtain
P sin( wt − )φ βγ − β γ
∗
E = 2 0 1 2 2 1 (5-23)
π ad γ Vt − β U t ()
()
2 2