Civil Engineering Reference
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For 0<H<H
c0
,
[7.5]
§
·
VH
H
2.0
¨
¸
VH
H
©
¹
c
0
c
0
c
0
For H
c0
<H,
[7.6]
V
1
Z
HH
c
0
V
c
0
On the plateau,
[7.7]
VV
pt
where V
c0
is the peak stress, H
c0
is the peak strain, Z is the slope after the peak and
V
pt
represents the residual stress.
Under cyclic loading, the compression non-linearity generally comes with
stiffness degradation and the appearance of plastic strains.
In traction, concrete has strength, yet after cracking, its behavior is brittle. The
law is linear for each traction part (Figure 7.6) and:
- up to the peak, the behavior is linear;
- softening is linear up to the ultimate strain.
V
c0
Z
V
pt
H
c0
b. Traction
a. Compression
Figure 7.6.
Example of compression and traction constitutive laws for concrete
7.3.2.
Damage model
Based on damage mechanics [LEM 90], [MAZ 86], the model [LAB 91] fits the
description of the damageable behavior generated by the creation of micro-cracks
(decrease of stiffness). During the cycles, the operation is linked to micro-crack re-
closing. Two scalar damage variables are used, one in traction
D
1
and the other in
compression
D
2
, acting on the stiffness of the material and generating permanent
strains. The 3D constitutive laws are given below:
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