Page 264 - Mechanics of Asphalt Microstructure and Micromechanics

256 Ch a p t e r E i gh t

interface elements denote those with a finite thickness and the special strain descrip-
tions. It is noteworthy that those continuous interface elements are renewable based on
the current configurations of interfaces.
8.3.2 Formulation of Strains in Continuous Interface Elements
As noted previously, with the emergence of large shear deformation along an interface,
distortions of original continuous interface elements are very likely to occur if they re-
main unchanged. Therefore, interface elements need to be renewed based on the up-
dated coordinates. The renewal is different from the technique for the simulation of
shear band inception and propagation, in which new nodal points lead to the change in
global degrees of freedom (Wang et al., 2003). In this method, for the sake of simplicity,
triangular and quadrilateral interface elements are automatically generated to circum-
vent the element distortions. The contact band element technique, which is used to
simulate shear band inception and propagation by Wang et al. (1995, 2002) is employed
to update interface elements. Contact band elements are generated through regular
solid elements by breaking those solid elements, and are not prescribed. They also dif-
fer from continuous interface elements in constitutive descriptions. A brief description
of the formation of new continuous interface elements is included.
A portion of an interface is shown in Figure 8.4. With an increase in the applied
shearing load, configuration of the original interface elements gets greatly distorted as
shown in Figure 8.4(a). As a result, numerical solutions will be drastically degraded due
to the large distortion. However, if coordinates of the nodal points are updated, and
they are reorganized to produce new interface elements based on the updated coordi-
nates, the renewed interface elements have much better configurations, as illustrated in
Figure 8.4(b). Consequently, they give a more accurate description of displacements
and strains in the interface elements with three or four nodes. The following example is
given to illustrate the formation of new interface elements.
As shown in Figure 8.4 (a) or (b), the interface is originally discretized with several
initial continuous interface elements. The discretization gives m nodes on the upper
boundary, S IJ , and n nodes on its lower boundary, S JI , which are denoted by:
ND :={ N N ,...., N ; N N ,...., }
1
2
1
2
m
n
N
,
,
u u u l l l (8-43)
j
Where N u denotes point i on the upper boundary, and N l represents point j on the
i
lower boundary. Considering a nodal point K on S IJ at time t, the distance from node K
to any node L on S JI is defined by:
t = t S IJ t S JI
d KL X K − X L (8-44)
2
Node K forms two angles a and b with line segment MN on S JI:
⎡ t S IJ t S JI ) ( t S JI t S JI ⎤
T
( ⎢ X K − X M X N − X M ) ⎥
α = cos -1 ⎢ S S S ⎥ (8-45)
t
t
⎢ t X − X M JI t X N JI − X M JI ⎥
J
S IJ
K
⎣ 2 2 ⎦
⎡ t S IJ t S JI ) ( t S JI t S JI ⎤
T
( ⎢ X K − X N X M − X N ) ⎥
β = cos -1 ⎢ S S S ⎥ (8-46)
t
t
⎢ t X − X N JI t X M JI − X N JI ⎥
S IJ
J
K
⎣ 2 2 ⎦