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e p
[7.8]
İİİ
: partition of the plastic and permanent strain tensors
V
V
Q
e
H
V
V
Tr
1
[7.9]
EDEDE
1
1
0
1
0
2
0
wV
f
E
D
E
D
p
11
2 2
ED
[7.10]
H
1
ED
1
w V
1
0
1
0
2
where
E
0
is the Young's modulus for the healthy material and Q is the Poisson's
coefficient.
, with
V
and
V
are respectively the “traction stress” and the
“compression stress”;
D
1
and
D
2
are respectively the traction and compression
damage variables. Their evolution beyond a specific threshold, between (healthy
material) and 1 (broken material), is linked to the evolution of the local elastic
energy.E
1
and E
2
are constants whose values calibrate the evolution of the
permanent strains, and
VVV
f
V
is the function of the crack opening-closing up
adjustment.
Figure 7.7.
Uniaxial response of the model with unilateral damage (from [LAB 91])
Figure 7.7 gives the uniaxial response of the model with a compression-traction
alternate loading. Beyond the peak in traction, we can see that damage (D
1
) evolves,
unloading reveals a Young's modulus affected by crack opening, E
0
(1-D
1
), and a
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