Page 282 - Mechanics of Asphalt Microstructure and Micromechanics

274 Ch a p t e r E i gh t

p
Where Δe is plastic strain increment, from time t to time t + Δt. t+Δt s , the stress
tr
predictor, takes the form:
t+Δ t σ tr t + e Δε (8-126)
= σ C:
If the full Newton scheme is employed to solve the nonlinear equilibrium equa-
tions, one needs to calculate the so-called tangential linearization moduli right after the
stress calculation, which is consistent with the stress integration algorithm by applying
total stresses at time t+Δt. It is simply written as:
t+Δ t ⎛ σ
t+Δ t = ∂ ⎞
C ⎜ ⎝ ε ⎟ (8-127)
∂ ⎠

One can decompose the stress tensor s into hydrostatic and deviatoric components:

σ= σ IS + (8-128)
+ = −pIS
m
Where I is the identity tensor and s m denotes the mean stress, i.e., the average of the
three normal stresses. Pressure p, holding an opposite sign with the mean stress, and
deviatoric stress components S are expressed as (Wang, et al., 2004):

t+Δ t p = t+Δ t p + 1 Δε p
tr
a m (8-129)
M
t+Δ t S = t+Δ t S − 1 e Δ p
tr
ij ij a ij (i, j = 1, 2, 3) (8-130)
E
tr
Stress predictors t+Δt tr t+Δt S ij are computed as:
p and
ε
t+Δ t p = 1 t+Δ t ''
tr
a m
M (8-131)
t+Δ t S = 1 t+Δ t e ''
tr
ij a ij
E (8-132)
t+Δ t ε " = t+Δ t ε t p t+Δ t '' t+Δ t e − t p p
e =
Where m m − ε , ij ij e are known values by which the elastic
m
ij
trial stresses are defined. t+Δt e m and t+Δt e ij are total mean strain and deviatoric strains at
time t + Δt, respectively. Both a M and a E are elastic constants, which are written as:
t p
a M = (1-2ν)/E, a E = (1 + ν)/E, respectively. e m and e ij are plastic mean strain and
t p
plastic deviatoric strain components at time t, respectively. Their incremental forms are
p
p
denoted by Δe m and Δe ij , respectively. The use of the non-associated flow rule enables
p
one to express the plastic strain increment Δe as:
⎛ g ∂ ⎞
Δεε = Δλ ⎜ ⎝ σ ⎟
p
∂ ⎠
For more details on the implicit implementation, one can see Wang et al. (2004) on
the example of the J3 dependent constitutive model.