Page 256 - Mechanics of Asphalt Microstructure and Micromechanics

248 Ch a p t e r E i gh t

M
Π = ∑ Π = 1 ∫ Duu d Ω − τ j u d Γ − e ∫ b ud Ω (8-30)
,
j
m
k i
,
j l
j
ijkl
j
Γ
m =1 2 Ω e ∫ e Ω Ω
The minimization of the strain energy requires the following conditions to be satis-
fied: M
δΠ = ∑ δΠ = 0 (8-31)
m
m =1
Which means that the displacement function should be properly selected so that the
overall strain energy is minimized. A consistent requirement at the element level will
need d Π m = 0.
The following will deal with the element-level minimization through the discretiza-
tion method:
1
Π = e ∫ u () B CBu d Ω − e ∫ N u d Γ − e e ∫ N bu dΩ (8-32)
τ
q T
T
q
q
T
q
T
e 2 Ω e e Γ e Ω e
∂Π
The minimization conditions will require that e e = 0 .
This will result in: ∂u i
e ∫ BCBu dΩ − e ∫ N dΓ −τ e ∫ N bdΩ (8-33)
q
T
T
T
Ω e Γ Ω
or
Ku = F e
q
e
e (8-34)
Where F = F + F : equivalent node forces.
e
e
e
B S
K = e ∫ B CBdΩ : element stiffness matrix.
T
e
Ω
S ∫
F = N d τ Γ : equivalent forces due to the surface tractions.
T
e
Γ e
B ∫
F = N bdΩ : equivalent forces due to distributed body forces.
T
e
Ω e
The superscript e denotes element-level variables.
Step 4: Formulate the System Equation
The domain decomposition equation also presents a way to assemble the element
properties into the overall property matrix, which relates the node displacements to the
forces applied on the nodes. Since elements share nodes, a specific order is necessary.
The general philosophy is to sum all the displacement contributed by the elements shar-
ing that node. Different orders will result in different required computational time.
Ku = F (8-35)
q

q
Where K is the overall stiffness matrix; u is the node displacement vector; and F is
the equivalent node force element. This is the system-level equilibrium equation.
There are many methods to assemble the system matrix from the element matrices.
They can be found in some of the reference books listed in Section 8.1.
Step 5: Application of the Boundary Conditions
There are basically two types of boundary conditions— displacement conditions
and force conditions (or surface tractions). Mathematically, it is equivalent to the modi-
q
fication of the node displacement vector u and the node force vector F.