Page 271 - Mechanics of Asphalt Microstructure and Micromechanics

F inite Element Method and Boundar y Element Method 263

of virtual work. The spatial discretization of the weak form equations uses finite ele-
ment and some spatial interpolation scheme for the displacement u, i.e.:
u = Nu q (8-62)

q
Where u is the displacement at the nodes, N is the interpolating function. Through
the strain-displacement relation and constitutive model, the equilibrium equation
could be represented as a set of equations with node displacements as independent
variables, or:
q
gu () = 0
(8-63)
For a quasi-static initial boundary value problem in a specific time interval, the
solution of Equation (8-63) is obtained incrementally and successively by dividing the
time interval into an appropriate number of time steps. For each step, the solution
procedure for a displacement driving FE system could be summarized as the follow-
ing three steps:

1. Global computation (equilibrium equations). Under the defined material
behavior, loading condition, kinematic conditions, Equation (8-63) could be
solved iteratively. For example, for the FE code ABAQUS, Newton’s method is
used. Using Δu, the strain increment Δe can be obtained.
2. Local computation (constitutive equations). At each material calculation point,
stress and the set of internal state variables are integrated and updated under
specified initial conditions and given strain increment Δe. For ABAQUS and
user’s materials, this is accomplished through UMAT. The following sections
deal with this computation.
3. Residual force computation and convergence check. Determine the residual
stress of the FE equilibrium equations and check convergence condition. If not
reached, repeat 1 and 2.

Two basic strategies exist for the step 1 and 2 computations: (1) simultaneous solu-
tion of the equilibrium and constitutive equations, or (2) separate solution of the two
sets of equations requiring iteration procedures at the global level. Among the tech-
niques of the first type, the method by Kanchi et al. (1978) and Marques et al. (1983) and
the method by Hughes et al. (1985) are two different possibilities. The technique by
Kanchi and Marques evaluates state variables at time t k+1 by Taylor series expansion at
time t k without further iteration, while the technique by Hughes uses Newton-Raphson
iterations. Among the techniques of the second type, there are two techniques also: (a)
use the same time step for both equilibrium and constitutive equations (Snyder et al.,
1981; Heaps et al., 1986) and (b) use different time steps for equilibrium and constitutive
equations with smaller time steps for constitutive equations (Ottosen and Gunneskov
1985). Most of these techniques could be applied to local computations.
In local computations, FE solution could be treated as the following mathematical
equivalency:

Given at Step k: state variables, s k and Δe, Δt, t k
Find at Step k+1: state variables and Δs, σ k+1 = σ + Δ σ t k+1 = t + Δ t
,
k
k