Page 270 - Mechanics of Asphalt Microstructure and Micromechanics

262 Ch a p t e r E i gh t

Where A = E tt / 1+ ( μ )(E / E 1− ( μ ) − μ ⎤ ⎦
⎡
2 ) . E tt and E nn are interpreted as the
⎣
tt
tt
tt
nt
nn
Young’s moduli for loading in the tangential plane (t-o’-n), E tt being along the axis t and
E nn along the normal axis n, respectively. The variable m tt is the Poisson’s ratio relating
loading along one axis in the tangential plane to the strain along the other axis in the
plane. The relationship between extensional strains in the tangential plane and normal
loading, or between normal extensional strains and tangential loading, is controlled by
the other Poisson’s ratio m tn or m nt . The variable G nt relates shear stresses out of the tan-
gential plane to shear strains out of the tangential plane, E tt and m tt can be obtained from
conventional laboratory tests. E nn and G nt are calculated approximately from the rela-
tions (Sharma and Desai, 1992):
E = k d,and G = k d (8-61)
nt n nt t
The contribution to the in-plane stress s tt from normal strain e nn is often neglected
(Sharma and Desai, 1992). Therefore, m nt is set to be zero. The special Mohr-Coulomb
yielding criterion with the associated flow rule is employed, which is an elasto-per-
fectly plastic model, and is literally described by Chan et al. (1997).
As described at the beginning, tangential shearing along an interface is the pre-
ferred direction of yielding due to the anisotropy of interface materials (Chan et al.,
1997). This implies that a general yielding criterion in which yielding may take place in
any direction is not applicable to the continuous interface modeling. Yielding along the
tangential direction of the interface should be prescribed in the yield criterion for the
continuous interface elements.

8.4 Infinite Element and Rigid Element

8.4.1 Infinite Element
In solving the boundary value problems of an infinite domain or problems whose far-
field solutions may not so significantly affect the region of interest (ROI), those far-field
areas may be represented by infinite elements. The ABAQUS manual and the manual
for 3D computer aided pavement analysis system (CAPA-3D) by Scarpas present excel-
lent references.

8.4.2 Rigid Element
The principal advantage to represent portions of a model with rigid bodies rather than
deformable finite elements is computational efficiency. Element-level calculations are
not performed for elements that are part of a rigid body. Although some computational
effort is required to update the motion of the nodes of the rigid body and to assemble
concentrated and distributed loads, the motion of the rigid body is determined com-
pletely by a maximum of six degrees of freedom at the reference node.

8.5 Constitutive Model Implementation
8.5.1 Mathematics Representation of the FEM Solution Process
In the FE solution for a stress-strain problem, equilibrium and boundary conditions are
properly combined into the weak-form boundary value problem through the principle