Page 283 - Mechanics of Asphalt Microstructure and Micromechanics

F inite Element Method and Boundar y Element Method 275

8.8 Boundary Element Method
8.8.1 Theoretical Basis
The basic FEM Equation (8-18) is copied as follows:

∫ Du w d Ω = ∫ Duw n d Γ − ∫ Du wdΩ
Ω ijkl k il , j Γ ijkl k i , j l Ω ijkl k ki , j l , (8-18)
Considering the identity u w = ( uw ), − uw and applying the divergence the-
k i , j l , k j l , i k j li ,
orem, the following equation can be derived:
∫ Duw dΩ = ∫ Du w n dΓ − ∫ Du wwdΩ (8-133)
,
Ω ijkl k i , j l , Γ ijkl k j l i Ω ijkl k jli ,
Replacing the second term on the right side of Equation (8-18) with the right side of
8-133, the following equation can be derived:
Ω ∫ D ( u w − u w ) d Ω = ∫ D ( u w n − u wn dΓ (8-134)
)
,
,
ijkl k il j k j li , Γ ijkl k i , j l k k j l i
If one selects the fundamental solution (corresponding to the displacement field
caused by a concentrated force) as the weighting function w = u* and denotes the cor-
responding field variables as s *, t *, e *, and so on, and considering:
Du = σ Dw = σ * and the stress-surface traction relationship, one has:
ijkl k il , ki l , ijkl j li ji l ,
,
Ω ∫ D ( u w − u w ) d Ω = Ω ∫ (σ ki l , u −σ udΩ
*
*
)
k
l
k il
,
ji,l
j
j li
k
,
j
ijkl
And
Ω ∫ (σ u * −σ u )d Ω Γ ∫ (τ u * − u τ * )d Γ
*
ki ,l j , ji l k j j k k (8-135)
It can be conveniently proved that this is actually Betti’s reciprocal work principle,
which is concerned with the two states of stress and strain within a domain (Ω, Γ) con-
tained in a large domain (Ω*, Γ*). Betti’s principle states that the virtual work (virtual
strain energy) done by s ki over e * ki is equal to the virtual work done by s * ki over e ki , or:
σε = ε σ * (8-136)
*
ki ki ki ki
This can be conveniently proved as:
σ = D ε and σ = D ε *
*
ki ijkl jl ki ijkl jl
And thus one has:
Ω ∫ σε d Ω Ω ∫ σε d Ω
*
*
ki ki
ki ki
Ω ∫ σ ud Ω Ω ∫ σ ud Ω (8-137)
*
*
or ki k i , ki k i ,
By identities σ u = ( σ u ) − σ u and σ u = σ ( * k ki u ) −σ * ki i , u k
*
*
*
*
k i
,
k i
ki i
,
ki
k i
ki
k i
ki
,
k