Page 266 - Mechanics of Asphalt Microstructure and Micromechanics

258 Ch a p t e r E i gh t

larger deformation, as shown in Figure 8.4b. The number of nodal points for an inter-
face element varies and is dependent on the locations of the nodal points. The continu-
ous interface elements are re-formed after converged results are obtained at each load
step with node coordinates updated in the wake of each global equilibrium iteration.
Extra equilibrium iterations are performed before a new load increment is applied for
the new interface elements. Due to the continuous updates of the interface elements,
stresses, strains, and other state variables at each nodal point should be tracked down.
Initial stresses, strains, etc. at Gaussian points within a new interface element will be
obtained by interpolating from those nodal values at the beginning of each load step
(Wang et al., 2002).
After the formation of the continuous interface elements, the strains need to be for-
mulated for triangular and rectangular interface elements, respectively. For large defor-
mation analyses of interfaces or joints, triangular and quadrilateral interface elements
are necessary, but not well documented. No considerable attention has been paid to
characterize their strain distributions. In the next subsection, strains in the triangular
and quadrilateral interface elements are defined and derived. The discussions in the
previous section are fundamentals to the new strain descriptions. Shear and normal
strains across interfaces are related to relative displacements between the upper and
lower surfaces. The strains stated above are assumed to be uniform in the directions
normal to an interface.

8.3.3 Triangular Interface Element
As shown in Figure 8.3a, at certain load steps, points K, A, and B form a new triangular
element. The relative displacement rates of point K to segment AB at solution time t+ Δt
are defined as:
+
t+Δt •
⎧ ⎫
⎪ U A ⎪
t+Δ t t+Δ t ⎪ • ⎪
⎧ • ⎫ ⎧ • ⎫
⎪Δ W t ⎪ ⎪ U K ⎪ ⎡ + t Δ t N C 0 + t Δ t N C 0 ⎤ ⎪V A ⎪
−
⎨ • ⎬ = ⎨ • ⎬ − ⎢ A + t Δ t C B + t Δ t C ⎥ ⎨ • ⎬
⎪ ⎣ ⎢
⎪ Δ W n ⎭ ⎪ V K ⎭ 0 N A 0 N B ⎦ ⎥ ⎪ ⎪ U B ⎪
⎪
⎩
⎩
⎪
⎪ • ⎪
⎩ V B ⎭
+ t Δ t
⎡ • ⎤ ⎤
⎢ U A ⎥
⎢ • ⎥
⎢V ⎥ (8-48)
A
⎢ • ⎥
− ⎡ + t Δ t N C 0 − + t Δ t N C 0 1 0⎤ ⎢ U ⎥
= ⎢ ⎢ A + t Δ B + t Δ ⎥ ⎢ B ⎥
⎣ ⎢ 0 − t N C A 0 − t N C B 0 1 ⎥ ⎦ ⎢ • ⎥
⎢ V B ⎥
⎢ • ⎥
⎢ U ⎥
K
⎢ • ⎥
K ⎥
⎢V ⎦
⎣

+ t Δ t ⎧ ⎫
•
= + t Δ t ⎡N int ⎤ ⎦ ⎨ ⎬
U
⎣
⎩ ⎭