Page 267 - Mechanics of Asphalt Microstructure and Micromechanics

F inite Element Method and Boundar y Element Method 259

Where Δ denotes the difference between the upper and lower surface quantities,
• •
• denotes the rate of quantities, U and V are the tangential and normal displacement
• •
rates under coordinate system t-o′-n, ΔW t and ΔW n are the relative tangential and nor-
C
C
mal displacement rates, respectively. t+Δt N A and t+Δt N B are given by:
t+Δ t N =− t+Δ t r t+Δ t N = t+Δ t r (8-49)
C
C
1
A c B c
In which t+Δt r c is simply obtained as the ratio of the length of segment AC to the
length of segment AB. Here, point C is the image of point K on boundary AB. The loca-
tion of point C is determined by assuming that KC and AB in Figure 8.3(a) are inter-
sected at a right angle. As a result, the corresponding strain rates in the element as de-
scribed in Section 2 take the form:
t+Δ t
⎧ • ⎫ t+Δt
Δ
⎪ε nt ⎪ 1 t+Δ t ⎧ ⎫
•
⎨ • ⎬ = t+Δ t ⎡N int ⎤ ⎦ ⎨ U ⎬
⎣
⎪ ε nn ⎭ ⎪ d ⎩ ⎭
⎩ (8-50)
Where thickness t+Δt d of the interface element, written as d hereafter, is calculated
from current coordinates of points K and C, and will be updated continuously. As men-
tioned before, the normal strain parallel to the interface is taken as:
t+Δ t • ⎡ ∂ N ∂ N ∂ N ⎤ t+Δ t ⎧ ⎫
•
ε tt = ⎢ t ∂ A 0 t ∂ B 0 t ∂ K 0 ⎥ ⎨ U ⎬
⎬
⎣ ⎦ ⎩ ⎭ (8-51)
N A , N B , and N K , different from t+Δt N A or t+Δt N B ¸ are regular shape functions for a tri-
C
C
angular element. Therefore, for a triangular interface element, the three strain rates are
given by:
+ t Δ t
⎧ • ⎫ ⎫
⎪ U A ⎪
t+Δ t ⎡ ⎤ ⎪ • ⎪
⎧ • ⎫ ⎪V ⎪
A
⎪ ε nt ⎪ ⎢ − t+Δ t N C 0 − t+Δ t N C 0 1 0 ⎥ ⎪ • ⎪
B ⎪
⎪
⎪ • ⎪ ⎪ 1 ⎢ A A t+Δ B t+Δ ⎥ ⎥ ⎪ ⎪U ⎪
⎨ ε nn ⎬ = ⎢ 0 − t N C 0 − t N C 0 1 ⎥ ⎨ ⎬
⎪ • ⎪ d ⎢ ⎢ ⎛ ∂ N ⎞ A ⎛ ⎞ B ⎛ ⎞ ⎥ ⎥ ⎪ • ⎪
⎪ε tt ⎪ ⎢ d ⎜ ⎜ A ⎟ 0 d ⎜ ∂N B ⎟ 0 d ⎜ ∂N K ⎟ 0 ⎥ ⎪ V B ⎪
⎩ ⎪ ⎭ ⎪ ⎣ ⎝ t ∂ ⎠ ⎝ ∂ ⎠ ⎝ ∂ ⎠ ⎦ ⎪ • ⎪
t
t
⎪ U K ⎪
⎪ • ⎪
K
⎩ ⎪V ⎭ ⎪
+ t Δ t
•
1 + t Δ t ⎧ ⎫
= ⎡N ' ⎤ ⎦ ⎨ ⎬
U
⎣
d ⎩ ⎭
(8-52)
8.3.4 Quadrilateral Interface Element
Sometimes it would be more suitable to use quadrilateral elements to accommodate
deformed configurations of interfaces. As shown in Figure 8.3b, relative displacement
rates at time (t+Δt) between the upper and lower surfaces at any point in the element