Page 268 - Mechanics of Asphalt Microstructure and Micromechanics

260 Ch a p t e r E i gh t

are only the functions of local coordinate r, and are irrelevant to the other local coordi-
nate, s. They can be written as:
t+Δ t
⎡ • ⎤
⎢ U 3 ⎥
• ⎢ ⎥
⎢ V 3 ⎥
• ⎢ ⎥
⎢ ⎥
t+Δ t ⎢ U 4 ⎥
⎧ • ⎫ ⎢ • ⎥
⎪Δ W nt ⎪ ⎡ N 0 N 0 − N 0 − N 0 ⎤ ⎢ V 4 ⎥
⎨ ⎬ = ⎢ 3 4 1 2 ⎥ (8-53)
⎪ • ⎪ ⎣ 0 N 3 0 N N 4 0 − N 1 0 − N 2 ⎦ ⎢ • ⎥
⎩ Δ W nn ⎭ ⎢ U 1 ⎥
• ⎢ ⎥
⎢ V 1 ⎥
⎢ • ⎥
⎢ ⎥
⎢ U 2 ⎥
⎢ • • ⎥
2
⎣ V ⎦
t+Δ t ⎧ ⎫ t+Δ t ⎡ • • • • • • • • ⎤
•
Let ⎨ U ⎬ denote matrix ⎢ U 3 V 3 U 4 V 4 U 1 V 1 U 2 V 2 , the corre-
⎥
⎩ ⎭ ⎣ ⎦
sponding strain rates would read:
t+Δ t
⎧ • ⎫ t+Δ t t+Δ t T
⎪ε nt ⎪ 1 ⎡ N 0 N 0 − N 0 − N 0 ⎤ ⎧ ⎫
•
⎨ ⎬ = ⎢ 3 4 1 2 ⎥ ⎨ U ⎬ (8-54)
•
⎪
⎪ ε nn ⎭ d ⎣ 0 0 N 3 0 N 4 0 − N 1 0 − N 2 ⎦ ⎩ ⎭
⎩
On the upper and lower boundaries of the continuous interface element, local coor-
dinate s is equal to 1 and –1, respectively. Hence,
1− r 1+ r
N = = N ; N = = N (8-55)
1 2 4 2 2 3
The normal strain in the direction of the interface length is still taken as in a regular
solid element. Therefore, total strain rates in continuous quadrilateral interface ele-
ments are described as:
t+Δ t t+Δ t
⎧ • ⎫ ⎡ ⎤
⎪ ε nt ⎪ ⎢ N 0 0 N 0 − N 0 − N 0 ⎥
⎪
•
⎪ • ⎪ ⎪ 1 ⎢ 3 4 1 2 ⎥ t+Δ t ⎧ ⎫
⎨ ε nn ⎬ = ⎢ 0 N 3 0 N 4 0 − N 1 0 − N ⎥ ⎨ U ⎬
⎬
2
⎪ • ⎪ d ⎢ ∂ s ∂ s ∂ s ∂ s ⎥ ⎩ ⎭
⎪ε tt ⎪ ⎢ d N 3 0 d N 4 0 d d N 1 0 d N 2 0 ⎥
⎩ ⎪ ⎭ ⎪ ⎣ ⎢ t ∂ t ∂ t ∂ t ∂ ⎦ ⎥
+ t Δ t ⎧ ⎫
•
+ t Δ
= 1 t ⎡N ' ⎤ ⎦ ⎨ ⎬
U
⎣
d ⎩ ⎭
(8-56)
s
s
s
s
Where N , N , N and N are regular shape functions for a quadrilateral solid ele-
1 2 3 4
ment.