(a) (b)(c) (d)Figure 7: Damaged area expansion process at crack tips.121314151617181920(^2122232425)
(^2627)
28
(^2930)
31
EL.1049m^32
0102030
(m)
Figure 8: Distribution of cracks on typical elevation.
Darabi et al. [ 31 ]presentedathermoviscodamagemodel
to describe the damage evolution law, which could be stated
as
휙=Γ̇ 휑
0 (
푌(1 − 휙)
2푌 0)
푞
exp(푘휀To teff)퐺(푇), (14)where휀To teff=√휀푖푗휀푖푗is the effective total strain in the effective
configuration.휀푖푗includes both viscoelastic and viscoplastic
components.푞is the stress dependency parameter and푘is
a material parameter.Γ
휑
0 and푌 0 are the reference damage
viscosity parameter and the reference damage force, respec-
tively.퐺(푇)is a damage temperature function.푌 is the
damage driving force in the effective configuration, which can
beassumedtohaveamodifiedDrucker-Prager-typeformas푌=훼퐼 1 +휏,
휏=
√퐽 2
2
[
[
[
1+
1
푑
+(1−
1
푑
)
퐽 3
√퐽^32
]
]
]
,
(15)
where퐼 1 and퐽 2 are the invariants of the effective stress tensor.
푑gives the distinction of material behavior in compression
and extension loading conditions, where푑=1implies that휏=√퐽 2. Thus the damage driving force is expressed as푌=훼퐼 1 +√퐽 2. (16)
Damage evolution requires that푌>0;thatis,theequi-
librium stress field exceeds the yield function. As stated
above, the equivalent nodal force of stress exceeding the yield
function is unbalanced force, which is the driving force of
damageevolution.Unbalancedforceismoreexplicitthan
equivalent plastic strain in the sense of physical significance.