Page 286 - Mechanics of Asphalt Microstructure and Micromechanics

278 Ch a p t e r E i gh t

Therefore, the displacements in the domain can be calculated using Equation (8-144)
and those on the boundary can be calculated using Equation (8-145). Once the displace-
ment is obtained, the strains and stresses can be calculated. The major efforts for BEM
*
are therefore the evaluation of the surface integrals ∫ Γ (τ u ik * − τ u k ) Γd , ∫ t ij * ( , ) ( ) Γξx u x d
k
j
ik
Γ
ij ∫
and ux (, ) ( )ξ t x dΓ . There are many numerical methods to develop such evaluations.
*
j
Γ
Gaul et al. (2003), Becker (1992), Paris and Canas (1997), and Brebbia et al. (1984) have
excellent descriptions on these techniques. By far the fundamental mechanisms of BEM
have been presented.
8.8.3 BEM Applications to Modeling AC
BEM application to AC is very limited. The major reason may be due to the lack of com-
mercially available powerful software to allow convenient applications. Another barrier
might be the difficulties in finding the fundamental solutions. Nevertheless, it may
have significant advantages in modeling AC behavior with multiple aggregate inclu-
sions.
The displacement discontinuity boundary element method (DD-BEM) has the po-
tential to simulate cracking in granular materials such as AC. In comparison to the
continuum-based fracture-mechanics-based approaches, this method explicitly models
the crack initiation and propagation in AC.
The DD-BEM method simulates cracking by creating a numerical model with two
types of elements: exterior boundary elements and potential crack elements (Sangpetn-
gam, 2003). The exterior boundary elements along the boundary of a problem simulate
the edge of specimen and random potential crack elements are placed inside a speci-
men to simulate predefined crack paths that are assumed to occur along a grain bound-
ary or pass through a grain.
When applied, DD-BEM is usually coupled with various tessellation techniques
that have been widely used to represent granular structure in simulation of fracture
process. Two basic tessellation schemes, Delaunay and Voronoi, have been used to sim-
ulate granular structure of brittle rocks by Peirce and Napier (1995) and Steen et al.
(2001), and produced realistic failure patterns. Peirce and Napier (1995) developed a
new boundary element solution technique (the multi-pole method) using the two tes-
sellation schemes (Delaunay and Voronoi) for three levels of grain densities. The Vor-
onoi assemblies outperform the Delauney triangulations in that the former are less
prone to shed load.
Birgisson et al. (2002a) applied the 2D DD-BEM to simulate crack growth in Super-
pave IDT specimens, using exterior boundary elements with randomly positioned po-
tential crack elements (Voronoi tessellation) inside the specimen. The numerical model
can realistically capture stress-strain responses and crack pattern with an appropriate
set of material parameters for local failure at potential crack elements. Birgisson et al.
(2002b) further found that Voronoi tessellation with internal fracture path provided re-
alistic simulation of crack growth in asphalt mixture as aggregate particles are allowed
to break down at high load level. The two crack growth rules evaluated, i.e., the sequen-
tial and parallel, were found to give similar results. Using the same numerical schemes
as Birgisson et al. (2002b), it was found that the method was capable of predicting re-
sults that matched vertical compressive stress-strain curve, horizontal tensile stress-
strain curve, crack pattern, tensile strength, and fracture energy of the mixes during an
IDT test.