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29
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
J. J. C. PITUBA | M. M. S. LACERDA
damage induces anisotropy in the concrete. Therefore, it is con-
venient to separate the damage criteria into two: the first one is
only used to indicate damage beginning, or that the material is
no longer isotropic and the second one is used for loading and
unloading when the material is already considered as transverse
isotropic. This second criterion identifies if there is or not evolution
of the damage variables. That division is justified by the difference
between the complementary elastic strain energies of isotropic and
transverse isotropic material. For identifying the damage beginning
it is suggested a criterion that compares the complementary elastic
strain energy
∗
e
W
, which is computed locally considering the me-
dium as initially virgin, isotropic and purely elastic, with a certain
reference value Y
0T
, or Y
0C
, obtained from experimental tests of
uniaxial tension, or compression, respectively. Accordingly, the cri-
terion for initial activation of damage processes in tension or com-
pression is given by:
(16)
f
T,C
(
s
) =
*
e
W - Y
0T,0C
< 0
then
D
T
= 0 (i. e., D
1
= D
4
= 0) for dominant tension states or
D
C
=
0 (i. e., D
2
= D
3
= D
5
= 0) for dominant compression states, where
the material is linear elastic and isotropic. The reference values
Y
0T
and Y
0C
are model parameters defined by
0
2
0
2
E
T
σ
and
0
2
0
2
E
C
σ
,
respectively, where σ
0T
and σ
0C
are the limit elastic stresses deter-
mined in the uniaxial tension and compression regimes.
It is important to note that the damaged medium presents a trans-
verse isotropy plane in correspondence to the current damage
level. Then, the complementary elastic energy of the damaged
medium is expressed in different forms, depending on whether
tension or compression strain states prevail. In the case of domi-
nant tension states (g(e,
D
T
,
D
C
) > 0) assuming that direction 1 in
the strain space be perpendicular to the transverse isotropy local
plane, it can be written:
(17)
0
33 22 0
1
0
33 11
22 11 0
0
2
33
2
22
2
1
0
2
11
e
E
)D1(E
)
(
E2
)
(
)D1(E2
W
ssn
-
-
ss+ssn
-
s+s
+
-
s
=
+
*
2
23
0
0
2
13
2
12
2
5
2
4
0
0
E
)
1( )
(
)D1()D1(E
)
1(
s
n+
+ s+s
-
-
n+
+
For the damaged medium in dominant compression states, the re-
lationships are similar to the tension case, where the complemen-
tary elastic energy is expressed in the following form:
(18)
2
3
0
33 22 0
3
2
0
33 11
22 11 0
2
3
0
2
33
2
22
2
2
0
2
11
e
)D1(E )D1)(D1(E
)
(
)D1(E2
)
(
)D1(E2
W
-
ssn
-
- -
ss+ssn
-
-
s+s
+
-
s
=
-
*
2
23
0
0
2
13
2
12
2
5
2
4
0
0
E
)
1( )
(
)D1()D1(E
)
1(
s
n+
+ s+s
-
-
n+
+
in that way the anisotropy and bimodularity induced by damage.
Those parameters are given by:
(12)
) ,
(
) (
)
)(
( ) ,
,
(
5 4 2
1 12
2
1
1
0
0
5 4 1 22
m2
l2
2 m2 l
l
DD
D
DD
DDD
-
- -
+ =
+
+
1 0
1 12
l
l
D D
=
+
) (
;
])
()
( [
) , (
2
5
2
4
0
5 4 2
1
1 1m2
m
D D
DD
-
- -
=
) ,
(
)
)(
( ) ,
,
,
(
3 2
12
2
2
2
0
0
5 4 3 2
22
l2
2 m2 l
l
DD
D D
DDDD
-
-
- -
+ =
) ,
(
) (
)
(
5 4 2
3
11
0
0
m2
l
n
1 n
DD
D
-
-
+
-
;
)
(
) (
2
3
3
0
3 11
2l
l
D D
D
-
=
-
)]
)(
( )
[(
) ,
(
3
2
2
3
0
3 2
12
1
1
1 l
l
D D
D
DD
-
- - - =
-
The different dyadic products in Eqs. (8), (10) and (11) have the
function of allocating the material constants in certain positions of
the stiffness constitutive tensors. For more details see Curnier [10]
and Pituba [11].
Observe that the bimodular character is taken into account by the
conditions g (ε,
D
T
,
D
C
) > 0 or g (ε,
D
T
,
D
C
) < 0, where g (ε,
D
T
,
D
C
)
is a hypersurface that contains the origin and divides the strain
space into a compression and tension sub-domains. A particular
form is adopted for the hypersurface in the strain space: a hyper-
plane g(ε,
D
) defined by the unit normal
N
and characterized by its
dependence of both the strain and damage states. To simplify the
presentation, the hyperplane will be here expressed as the one
obtained by enforcing the direction 1 in the strain space to be per-
pendicular to the transverse isotropy local plane. Thus, the hyper-
plane is given by:
(13)
g(
e
,
D
T
,
D
C
) =
N
(
D
T
,
D
C
) .
e
e
=
g
1
(D
1
,D
2
)
e
V
e
+
g
2
(D
1
,D
2
)
e
11
e
where g
1
(D
1
,D
2
) = {1+H(D
2
)[H(D
1
)-1]}h(D
1
)+{1+H(D
1
)[H(D
2
)-1]}
h(D
2
) and g
2
(D
1
,D
2
) = D
1
+D
2
. The Heaveside functions employed
above are given by:
(14)
H(D
i
) = 1 for D
i
> 0;
H(D
i
) = 0 for D
i
= 0 (i = 1, 2)
The h(D
1
) and h(D
2
) functions are defined, respectively, for the
tension and compression cases, assuming for the first one that
there was no previous damage of compression affecting the pres-
ent damage variable D
1
and analogously, for the second one that
has not had previous damage of tension affecting variable D
2
. The
functions can be written as:
(15)
h
(D
1
) =
3
2 3
2
1
1
D
D
- + -
;
h
(D
2
) =
3
2 3
2
2
2
D
D
- + -
As it has already been pointed out, in the model formulation the