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1. Introduction
In the Continuum Damage Mechanics (CDM), the damage effects
are evidenced in the stiffness constitutive tensor. The damage
leads to the reduction of several stiffness components, where the
damaged material can either keep its isotropic properties or to be-
come anisotropic.
For isotropic models the damage affects neither the direction nor
the initial number of symmetry planes presented by the material.
Thus, it does not matter if the medium is initially isotropic or aniso-
tropic with some degrees. In this case, those initial characteristics
are preserved during the damage process. Some constitutive dam-
age models have been proposed assuming the concrete as an iso-
tropic medium (Mazars [1], Mazars [2], Comi [3] and Berthaud [4]).
However, in the last decades anisotropic models which can modify
both the direction and the number of the material symmetry planes
have been proposed (Brünig [5], Pituba [6], Pietruszczak [7], Ibra-
himbegovic [8] and Dragon [9]).
Besides, another important characteristic presented by many fiber-
reinforced composite materials is the intrinsic bimodularity, i.e.,
distinct responses in tension and compression prevailing states.
On the other hand, brittle materials, such as concrete, are a kind of
composites that can be initially considered isotropic and unimodu-
lar. However when they have been damaged, those materials
would start to present some degree of anisotropy and bimodular-
ity. Assuming small deformations, a formulation of constitutive laws
for either initially isotropic or anisotropic elastic bimodular materi-
als was proposed by Curnier [10]. In order to incorporate damage
effects, the formulation of Curnier has been extended by Pituba
[11]. In particular, a constitutive model for concrete has been de-
rived. Accordingly, the material is initially considered as an isotro-
pic continuous medium with anisotropy and bimodularity induced
by the damage. On one side the class of anisotropy induced and
considered in the model (transversal isotropy) elapses from the
assumption that locally the loaded concrete always presents a dif-
fuse oriented damage distribution as appointed by experimental
observations (Van Mier [12]). On the other hand, the bimodularity
induced by damage is captured by the definition of two damage
tensors: one for dominant tension states and another one for domi-
nant compression states.
In order to use the Damage Mechanics in practical situations of
Structural Engineering, constitutive models presenting a reduced
number of parameters with easy identification and one-dimension-
al version are desirable. On the other hand, such models must
present reliable numerical results in order to estimate the mechani-
cal behavior of the structure as accurate as possible. In this work,
two examples of this kind of the damage models are used and
here, they are called simplified damage models.
This work intends to discuss the related problems to the numeri-
cal applications of the isotropic and anisotropic damage models in
the context of the one and two-dimensional analyses of reinforced
concrete structures. Besides, one intends to show the potentialities
of an anisotropic damage model recently proposed by Pituba [6].
Then, numerical responses supplied by the models are presented
and compared in order to evidence the difficulties and advantages
when one deals with this kind of modeling. Finally, some conclu-
sions about the employment of the simplified versions of these
kinds of damage models are discussed.
2. Isotropic Damage Model
This model has been proposed by Mazars [1] and the damage
is represented by the scalar variable D (with 0
≤
D
≤
1) whose
evolution occurs when the equivalent extension deformation
ε
~
is
bigger than a reference value. The plastic deformations evidenced
experimentally are not considered. The equivalent extension de-
formation is given by:
(1)
2
3
2
2
2
1
~
+
+
+
+ +
=
e
e
e
e
where
i
ε
is a principal deformation component, being
+
i
ε
its
positive part, i. e.:
[
]
i
i
i
ε
ε
ε
+ =
+
2
1
.
The damage activation occurs when
=
ε
~
e
d0
, being e
d0
the defor-
mation referred to the maximum stress of an uniaxial tension test.
Thus the criterion is given by:
(2)
0 DS
D, f
£ -= e
e
with S(0) =
e
d0
Considering the thermodynamics principles, the damage evolution
can be expressed by:
(3a)
0D
=
.
.
if f < 0 or f = 0 and f < 0
(3b)
.
.
.
( )
+
= e e
~~FD
if f = 0 and
0 f
=
where
D
.
= dD/dt
, i. e., D time derivative;
( )
ε
~
F
is written in
terms of
ε
~
and defined continuous and positive.
As the concrete behaves differently in tension and compression, the
damage variable D is obtained by combining properly the variables D
T
and D
C
, related to tension and compression, respectively, as follows:
(4)
C C T T
D D D
a a +
=
where
1
C T
= +a a
where D
T
and D
C
are given by:
(5a)
(
)
(
)
[
]
0d
T
T
T
0d
T
~B exp
A
~
A1
1 D
e e
e
e
-
-
-
-=
J. J. C. PITUBA | M. M. S. LACERDA
27
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1