Page 103 - MODELING OF ASPHALT CONCRETE

Overview of the Stif fness Characterization of Asphalt Concr ete 81

On the left is the actual strap with a crack in it of length 2c. On the right is an intact
strap with an apparent elastic modulus E′. If it is required to find the modulus E′ that
will store as much strain energy as the real strap on the left where the actual modulus E
remains unchanged, the ratio between the apparent and the real modulus of the material
is given by Eq. (3-15):
′ E 1
= (3-15)
E c 2 ⎡ 8Γ E ⎤
12π ⎢ 1 − ⎥
+
bh ⎣ σ 2 c ⎦
where c = half crack length
b, h = width and length of the strap, reespectively
Γ =fracture suurface energy of the material
E, E′ = undamaged and apparently damaged modulus of the material,
d
respectively
σ =tensile stress applied to the strap
l
If the same strap is loaded repeatedly and the crack grows with each load repetition,
the relation between the real and the apparent modulus is in Eq. (3-16):

′ E 1
=
E 2 ⎡ 1 ⎤ (3-16)
4
+
12π c ⎢ 1 − 2Γ ⎛ At ⎞ ⎟ 1 +n ⎥
bh ⎢ ⎜ dW ⎥
⎢ ⎢ ⎜ ⎟ ⎠ ⎥
⎣ ⎝ dN ⎦
where t = thickness of the strap
=
A, n = Paris-Erdogan fracture law parameters
dW
= rate of change of dissipated strain eneergy per load cycle
N
dN
e
N =number of load repetitions
If, instead of there being a single large crack of length 2c, there is a distribution of
microcracks of various sizes with a crack density of (m/bh), the ratio between the
apparent and the real modulus is given in Eq. (3-17):
′ E 1
=
E ⎡ 1 ⎤ (3-17)
⎢
4
12π c 2 ⎛ m ⎞ ⎟ 1 2Γ ⎛ At ⎞ ⎟ 1 +n ⎥ ⎥
−
+
⎜
⎠
⎝ bh ⎢
⎜
⎢ ⎜ dW ⎟ ⎟ ⎠ ⎥
⎣ ⎝ dN ⎦
where m = number of microcracks in theestrap
(/ ) = microcrack density
mb
h
c = mean crrack size
Equations (3-16) and (3-17) both show that the apparent loss of stiffness depends
upon the fracture properties of the material, A and n, the microcrack density (m/bh), the
instantaneous rate of change of the dissipated strain energy per load cycle (dW/dN),
and the average microcrack size c.