Page 281 - Mechanics of Asphalt Microstructure and Micromechanics

F inite Element Method and Boundar y Element Method 273

acceptability of a proper approximation of the Jacobian in case of difficulties in obtain-
ing an accurate expression. In this case (Wang and Li, 2008) the stress Jacobian is addi-
tive for the viscous component and the elastoplasticity component. For the viscous
component, the average of the elasticity modulus at the start and at the end of an inter-
val was used, while for the elastoplasticity component the average elastoplasticity
modulus at the start and at the end of an interval was used. Wang and Li (2008) demon-
strated the effectiveness of the semi-implicit method.

8.7 Full Implicit Implementation of the Druker-Plager Model
In this section, the implementation mechanisms for fully implicit methods are demon-
strated based on Wang et al. (2004). The additive decomposition of strain rate is ad-
opted for the infinitesimal strain assumption. For the closest-point projection approxi-
mation of a rate-independent problem, the behavior of an elastoplastic material can be
generally characterized by the following equations:
dε = dε + d ε p (8-118)
e

e
σ = C : ε e (8-119)
φσ ,S,H ) = (
(
φσ,H ) = ( φ I , , ,H ) = 0 (8-120)
J
J
α m α 1 2 3 α
g ∂
dε = Δ λ (8-121)
p
∂ σ
dH = ( p t H ) = h d ( εε , I J J , t H )
h dεε ,,σ
p
,
,
α β 1 2 3 β (8-122)
p
e
Where e and e are elastic and plastic strains, respectively, while f and g denote
yield and plastic potential functions, respectively. H a is a set of scalar state variables
t
governing the hardening yield surface, and H b denotes the state variables at time t.
They are both assumed twice differentiable functions in order to get second derivatives,
which are necessary in the stress integration process. Δl , a positive multiplier, is called
a plastic consistency parameter satisfying the Kuhn-Tucker complementary conditions
(Simo and Hughes, 1998):
Δλ ≥ 0, φ( ,óH α ) ≤ 0, Δλφ( ,óH α ) = 0 (8-123)

e
C represents the fourth order elasticity tensor. It is given by:
C e = λδ δ + 2 Gδ δ (8-124)
ijkl ij kl ik jl
2
Where λ = K − G , K and G are elastic bulk and shear modulus, respectively, and
3
d ij is the Kronecker delta.
In the closest point return mapping algorithm or the Euler backward algorithm, for
any given strain increments Δe, the corresponding stress increments must be computed
iteratively. The stresses at time t+Δt can be written in the following form:
t+Δ t σ = σ + Δσ = t+Δ t σ tr e Δε p (8-125)
t
− C: