Page 272 - Mechanics of Asphalt Microstructure and Micromechanics

264 Ch a p t e r E i gh t

So that all the state variables satisfy constitutive equations. Both this update process
and the global update process are called time integration as the solution is concerned
with the integration of differential equations driven by fictitious or real time.
8.5.2 Time Integration Algorithms
A general method concerning the solution of a partial differential Equation (8-64) is the
generalized Euler method represented by Equation (8-65) (Lubliner, 1990).
.
ξ = φ ξ(, ) (8-64)
t

Δξ = Δt[(1 − θ φ ξ t) + θφ ξ + Δξ +t, Δt)] (8-65)
(
)
,
(

where x – could be a scalar, a vector or a tensor, or combined quantities
t – could be time or fictitious time
q – weight parameter
q = 0, Explicit method; q ≠ 0 Implicit method. In implicit methods, several well
known methods include: Backward Euler method (q = 1), Trapzoidal method (q = 0.5),
and Galerkin method (q = 2/3).
An iteration method producing faster convergence is the Newton-Raphson method,
by which the equation will be solved with the following iteration:
.
Defi ne ψ(Δ ξ ξ, ,Δt ≡− ( t (8-66)
,
t
)
ξ φ ξ, )

∂ψ ∂φ
J = ( ∂ξ ) Δ ξ 0 = I −θΔ t( ∂ξ ) ξ + ξ 0 (8-67)
Δ
Δ

Δξ ( ) 1 = Δξ () + J 1 − ψ( Δξ ) (8-68)
0
()
0

Where I is a unit matrix. The Newton-Raphson method is the most commonly used
method in implicit FE algorithm.
Generally, the numerical implementation of the solution to a set of constitutive
equations requires the consideration of accuracy, stability, and efficiency. A compromise
should be reached between these factors for a large scale or a long sequence FE prob-
lem. Ortiz et al. (1985) summarized several important findings about the generalized
Euler method:
a. (q > 0.5), unconditionally stable; q < 0.5, conditionally stable
b. q = 0.5, better accuracy
c. The equivalency of small-scale stability in energy normal is equivalent to large-
scale stability in the associated geodesic distance. Stability analysis may be
confined to the assessment of small-scale stability.

Although viscoplasticity constitutive equations are usually highly nonlinear and
numerically very stiff, a small variation of the independent variables will produce a
great change of the dependent variables (Krieg, 1977; Shin et al., 1977). Both explicit and
implicit methods could be used for the time integration. Successful examples of apply-
ing explicit method were presented by Zienkiewics et al. (1974), however, the stability