Page 285 - Mechanics of Asphalt Microstructure and Micromechanics

F inite Element Method and Boundar y Element Method 277

displacements in i direction (i = 1, 2, 3) due to the forces in the j direction (j = 1, 2, 3). The
*
fundamental solution u* j (x,x ) should also have nine components as ux (, )ξ , which is
ij
the solution to the equation Lu x (, )ξ =− δ x (, )ξ δ .
*
ik ij ij
u ()ξ = ∫ (τ u − τ * u d ) Γ + ∫ b u dΩ
*
*
i Γ k ik ik k Ω j ij (8-144)
For simplification, this discussion is stopped here. Interested readers can read more
details in many excellent books referred in Section 8.1.
Equation (8-144) is the basis for BEM approach. However, it still has a domain inte-
gral (it can be resolved conveniently as b j and u* ij are known (the fundamental solution
is the Kevin solution).
There are the multiple reciprocity method, the dual reciprocity method, and the
direct transformation method (using higher order solutions) to convert the domain in-
tegral to the boundary integral. For simplification, one may neglect the body force term
if it is not so critical (for example, the traffic loading is much larger than the weight of
the materials for the surface course). For simplification, the body force in Equation
(8-144) will be dropped out.
Equation (8-144) calculates the displacement for a point within the domain. It can be
extended to the calculation of the displacement of points on the boundary. It can be es-
timated in such a way as to extend a small boundary of a circle of a sphere that is cen-
tered on that point (Figure 8.6). If the additional boundary integral is assessed and the
circle or sphere radius approaches zero, the solution for the displacement of the point
on the boundary is obtained.
For the fundamental solution (the Kelvin solution, see Chapter 1), it can be shown
1 1
*
*
that u ~ (weakly singular), t ~ (strongly singular) for r → 0. By evaluating the
ij
r ij r 2
integrals around the point on the surface, one can obtain the following boundary inte-
gral equation (BIE):
ij ∫
ij ∫
c () ()ξ u ξ + t x ( , ) ( )ξ u x d =Γ u x ( , )ξ ttx d() Γ
*
*
ij j j j (8-145)
Γ Γ
ij ∫
tx (, ) ( )ξ u x dΓ is a Cauchy principal value integral.
*
j
Γ
ij ∫
c ()ξ = δ + lim t x ( , )ξ dΓ
*
ij ij r→0 (8-146)
Γ
FIGURE 8.6 Illustration of x 2
the scheme to obtain the
displacements on the
boundary. Γ e
e
Γ e*
Γ '
x 1
Γ