Page 284 - Mechanics of Asphalt Microstructure and Micromechanics

276 Ch a p t e r E i gh t

And using the Gauss theorem, one can convert Equation (8-137) into the following
equation:
Ω ∫ σ ud Ω ∫ Γ σ u n d Γ − ∫ Ω σ u d Ω
*
*
*
k
i
,
ki i
ki
ki
k i
k
,
∫ σ ud Ω = ∫ σ u n d Γ − ∫ σ u d Ω (8-138)
*
*
*
Ω ki k i , Γ ki k i Ω ki i , k
Therefore, the following equations can be obtained:
∫ σ un d Γ − ∫ σ ud Ω ∫ σ un d Γ − ∫ σ udΩ (8-139)
*
*
*
*
Γ ki k i Ω ki i , k Γ ki k i Ω kiii , k
Or
Ω ∫ (σ u * −σ u )d Ω ∫ (τ u * − u τ * )d Γ
*
ki ,l j , ji l k Γ j j k k (8-140)
It is the same as Equation (8-135). This is actually the basis equation for BEM.
8.8.2 Fundamental Solution
The fundamental solution is the solution corresponding to an infinite domain subjected
to a unit force at point x or δ(x − ξ)n (the unit force in j direction). Since there are three
j
displacement components and three unit force directions, the displacement responses
*
can be represented as a tensor u (displacement in the ith direction due to the forces
ij
applied in the jth direction) or:
b = δ x (, )ξ n (Ω * → ∞ Γ * → ) ∞
*
,
j j
Considering the two equilibrium equations (static situations for less complexity in
understanding the fundamental mechanisms):
Du =− , * =− b * (8-141)
b Du
,
ijkl k li , j ijkl k li j
Then
∫ σ ud Ω ∫ b ud Ω, ∫ σ ud Ω − ∫ b uudΩ (8-142)
*
*
*
*
Ω ki i , k Ω k k Ω ki i , k Ω k k
Ω ∫ * * Ω Γ ∫ * τ *
Equation (8-140) (σ u j −σ u k )d (τ u j − u k k )d Γ becomes:
, ji l
ki
,l
j
∫ Γ (τ u * k − τ u k ) Γd ∫ Ω (b u j − b u * j ) Ωd
*
*
j
k
j
k
ξ
Since b = δ x (, )ξ n , and considering the identity ∫ bu dΩ = u () , one has:
*
*
n
j j Ω j j j j
j ∫
u ()ξ n = (τ u − τ * u d ) Γ + ∫ b u dΩ
*
*
j Γ k k k k Ω j j (8-143)
Equation (8-143) is the well-known Somigliana’s identity. It is basically the jth dis-
placement component at point x caused by a unit force in the jth (1, 2, 3) direction.
Considering that there are three components of the displacements and forces, Equation
(8-143) actually has nine equations and displacements can be represented as u ij (x ),