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28
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
Simplified damage models applied in the numerical analysis of reinforced concrete structures
and compression when damaged. This model has been proposed
by Pituba [6] and it follows the from the formalism presented in
Pituba [11]. Moreover, the model respects the principle of energy
equivalence between damaged real medium and equivalent con-
tinuous medium established in the CDM (Lemaitre [13]).
Now, the damage model is shortly described. Initially, for dominant
tension states, a damage tensor is given by:
(9)
D
T
= f
1
(D
1
, D
4
, D
5
)
)
(
AA
Ä
+ 2 f
2
(D
4
, D
5
)
)]
( )
[(
AA AI IA
Ä -Ä+Ä
where f
1
(D
1
, D
4
, D
5
) = D
1
– 2 f
2
(D
4
, D
5
) and f
2
(D
4
, D
5
) = 1 – (1-D
4
)
(1-D
5
).
The variable D
1
represents the damage in direction orthogonal to
the transverse isotropy local plane of the material, while D
4
is rep-
resentative of the damage due to the sliding movement between
the crack faces. The third damage variable, D
5
, is only activated
if a previous compression state accompanied by damage has
occurred. In the Eq. (9), the tensor
I
is the second-order identity
tensor and the tensor
A
, by definition, Curnier [10], is formed by
dyadic product of the unit vector perpendicular to the transverse
isotropy plane for himself.
On the other hand, for dominant compression states, it is proposed
the other damage tensor:
(10)
D
C
=f
1
(D
2
,D
4
,D
5
)
)
(
AA
Ä
+f
2
(D
3
)
)]
( )
[(
AA I I
Ä -Ä
+2f
3
(D
4
,D
5
)
)]
( )
[(
AA AI IA
Ä -Ä+Ä
where f
1
(D
2
, D
4
, D
5
) = D
2
– 2 f
3
(D
4
, D
5
) ,f
2
(D
3
) = D
3
and f
3
(D
4
, D
5
)=
1 – (1-D
4
) (1-D
5
).
Note that the compression damage tensor introduces two addi-
tional scalar variables in its composition: D
2
and D
3
. The variable
D
2
(damage perpendicular to the transverse isotropy local plane
of the material) reduces the Young’s modulus in that direction and
in conjunction to D
3
(that represents the damage in the transverse
isotropy plane) degrades the Poisson’s ratio throughout the per-
pendicular planes to the one of transverse isotropy.
Finally, the constitutive tensors are written as follows:
(11a)
=
T
E
]
[
]
[
I I
I I
Ä +Ä
1
11
m2
l
) ,
, (
5 4 1 22
l
DDD
+
-
]
[
AA
Ä
) (
1 12
l
D
+
-
[
A
Ä
I
+
I
Ä
A
]
) ,
(
5 4 2
m
DD
-
]
[
AI IA
Ä+Ä
(11b)
=
C
E
]
[
]
[
I I
I I
Ä +Ä
1
11
m2
l
) ,
,
,
(
5 4 3 2
22
l
DDDD
-
-
]
[
AA
Ä
) ,
(
3 2
12
l
DD
-
-
[
A
Ä
I
+
I
Ä
A
]
) (
3 11
l
D
-
-
[
I
Ä
I
] -
) (
)
(
3
11
0
0
l
n
n21
D
-
-
[
I
Ä
I
]
) , (
5 4 2
m
DD
-
]
[
AI IA
Ä+Ä
where
0
11
l l =
and
0
1
µ
µ
=
are Lamè constants. The re-
maining parameters will only exist for no-null damage, evidencing
(5b)
(
)
(
)
[
]
0d
C
C
C
0d
C
~B exp
A
~
A1
1 D
e e
e
e
-
-
-
-=
In Eqs (5) A
T
and B
T
are parameters related to uniaxial tension tests
while A
C
and B
C
are obtained from uniaxial compression tests. To
compute the a
T
and a
C
values defined in Eq.(4), we have to obtain,
initially, the deformations e
T
and e
C
associated, respectively, to ten-
sion and compression states as follows:
(6a)
I
E
E
1
i
*
i
*
T
+
+
å
-
+
=
s
u
s
u
e
(6b)
I
E
E
1
i
*
i
*
C
-
-
å
-
+
=
s
u
s
u
e
where
I
is the identity tensor, E the elastic modulus of a non-dam-
aged material,
+
*
σ
and
−
*
σ
are, respectively, positive and
negative parts of the stress tensor
*
σ
obtained from the relation
ε
σ
0
*
D
=
, where
0
D
is the elastic fourth order tensor of the
non-damaged material.
Thus the coefficients
T
α
and
C
α
are obtained by the following
expression:
(7a)
+
+
å
=
V
i Ti
T
e
e
a
(7b)
+
+
å
=
V
i
Ci
C
e
e
a
where
+
Ti
e
and
-
Ci
e
are, respectively, positive and nega-
tive parts of the deformations e
T
and e
C
defined in Eq. (6);
+
V
ε
is
given by:
å
+
+
+
+
=
i
Ci
Ti
V
e
e
e
.
Finally, the constitutive relation can be expressed in terms of the
actual deformation tensor as follows:
(8)
( )
e
s
0
DD1
- =
3. Anisotropic Damage Model
In this model, it is assumed that the concrete belongs to a category
of materials that can be considered initially isotropic and unimodu-
lar, however they start to present different behaviours in tension