Page 254 - Mechanics of Asphalt Microstructure and Micromechanics

246 Ch a p t e r E i gh t

Considering Du = σ , the above equation becomes an energy conservation
ijkl k i , jl
equation: work by the stress over the strain is equal to work by surface forces and body
forces over surface and body displacements. In general, it is actually the virtual work
mechanism.
The strain energy of the entire system will be:
1
Π = ∫ Duu d Ω − ∫ τ u d Γ − ∫ b u d Ω (8-23)
2 Ω ijkl k i , j l , Γ j j Ω j j
It is understood that the minimization of the strain energy will result in equilibrium
conditions.
If one subdivides Ω into a set of finite domains (finite elements) Ω m (m = 1,N), each
subdomain has r m nodes, one can have the following domain sum:
∫ ... Ω +…d ∫ ... Ω +…d ∫ ... Ωd ∫ ... Ωd (8-24)
Ω 1 Ω m Ω N Ω
For the surface integral, the inter-element surface integral will cancel out, so only
the external surface integral will play a role in the above equations. The above equation
is the foundation for the FEM approach. The major techniques and procedures are pre-
sented as follows.
Step 1: Discretization of the Continuum
Considering the geometry of the object, the properties of the materials, the defects
(cracking) and interface conditions, and the loading conditions, the object can be subdi-
vided into subdomains. It is also directly related to Step 3: selection of element type.
Usually, a fine element mesh is used for locations of concentrated loading, cracking tips,
and interfaces.
Step 2: Selection of Interpolation Functions
Depending on the shapes of the discretized domains (elements), accuracy require-
ments and computational efficiency, there are a variety of interpolation functions (shape
functions) to select. Typically, polynomials are selected for the convenience of integra-
tion and differentiation. Shape functions must satisfy requirements such as continuity
on element or domain boundaries and permission for rigid movement of the element.
This step is actually to obtain the displacement field in any of these finite domains
through interpolating by using the displacements of nodes:

2 ∑
1 ∑
3 ∑
u = r N xx x u ) q i 1 , u = r N xx ,xxu) q i 2 , u = r N x x xu q i 3 (8-25)
(,
(,
,
,
(
,
)
3
3
1
3
2
i
2
i
i
1
2
1
=
=
=
i 1 i 1 i 1
Where r is the number of nodes; u 1 , u 2 , and u 3 are the three displacements in the do-
main including edges and surfaces. Continuing, q represents the corresponding dis-
q
placements of the nodes and N i are shape functions. Denoting u and u as displacement
vectors and N as a matrix, Equation (8-26) can be obtained.
⎧ u ⎫ ⎧ u ⎫
⎪ 1 ⎪ ⎪ ⎪
u = ⎨ u ⎬ = ⎨ ⎬
v
⎪ 2 ⎪ ⎪ ⎪
w
u
⎩ 3 ⎭ ⎩ ⎭
q
q
q
q
q
q
q
q T
q
q
q
q q
u v w ,...,
,
u = ( , q , q u v w ) or u = ( u , u , u ,..., u , u , u ) T
,
1 1 1 r r r 11 21 31 r 1 r 2 r 3