Page 260 - Mechanics of Asphalt Microstructure and Micromechanics

252 Ch a p t e r E i gh t

Step 4: By summing up all the element equations, one obtains a set of equations
with all the node displacements as independent variables. In this example, there are
48 unknown node displacements. It should be noted that the only differences for the
element stiffness are differences in node coordinates for different elements. Therefore,
one can develop a subroute to calculate the stiffness and simply replace i, j, and k coor-
dinates through coordinate transfer in this case.
1
1
2
u uuu ,
u = ( , 2 , 2 2 ,.... u , u )
1 2 1 2 24 24
×
⎣
K = ⎡48 48 ⎤ ⎦
2
2
2
1
1
2
,
,
,
,
F = ( F FFF ,.... F F )
1 2 1 2 24 24
Step 5: In addition, the known displacements (for example, zero displacements) for
all the nodes should be placed in the system equation. In this example, (u 1 , v 1 ), (u 2 , v 2 ),
(u 3 , v 3 ), and (u 4 , v 4 ) are all equal to zero.
S ∫
By surface integral, one can obtain: F = Γ e N d τ Γ
T
e
−
F = ( ,02ρ gAh/ ,.. , − P,..., , −ρ gAh/ )
3
0
3
0
Step 6: With the above preparations, everything is ready to solve Equation (8-35).
This is actually a linear algebra equation. One can use MathCad or Matlab to solve it. As
a matter of fact, there are quite a few books focusing on using Matlab to solve FEM
problems (Kwon and Bang, 2000; Kattan, 2007).
Ku = F
q
Solving for the above system equation, one can obtain the displacements for all the
nodes. Then one can calculate the strains and stresses.
Step 7: Check the Convergence Criteria
The author suggests that readers manually solve this problem so an in-depth un-
derstanding of the FEM procedure can be obtained. There are many books on FEM, so
detailed descriptions on FEM will not be presented. Nevertheless, it is anticipated that
concise presentations will allow readers to understand the fundamentals in FEM.
There are also matured methods for accounting non-linear problems, including both
geometric and material non-linearity. Interested readers may refer to books by Bonet
and Wood (2008) and Crisfield (1997). The following sections will focus on several top-
ics of interest to researchers in the AC area, including: 1) implementation of interface
models; 2) use of rigid element; 3) use of infinite element; and 4) implementing constitu-
tive models (it involves a local integration process for material models only). More re-
cent developments in areas such as XFEM (Moës et al., 1999; Mohammadi, 2008), Mesh-
Free FEM (Belytschko and Chen, 2007), micromechanics FEM (Zohdi and Wriggers,
2005), and Multiscale FEM (Efendiev and Hou, 2008) will not be covered.
8.3 Interface Element
Interfaces often demonstrate different properties (including geometric characteristics)
from those of the two bulk materials forming the interfaces. The deformation format