Damage Models for Concrete

TLDR: In this paper, a constitutive relation for standard concrete with a compression strength of 30-40 MPa is proposed to capture the response of the materials subjected to loading paths in which extension of the material exists (uniaxial tension, uniaxial compression, bending of structural members).

Abstract: This constitutive relation is valid for standard concrete with a compression strength of 30–40 MPa. Its aim is to capture the response of the material subjected to loading paths in which extension of the material exists (uniaxial tension, uniaxial compression, bending of structural members) [4]. It should not be employed (i) when the material is confined ( triaxial compression) because the damage loading function relies on extension of the material only, (ii) when the loading path is severely nonradial (not yet tested), and (iii) when the material is subjected to alternated loading. In this last case, an enhancement of the relation which takes into account the effect of crack closure is possible. It will be considered in the anisotropic damage model presented in Section 3. Finally, the model provides a mathematically consistent prediction of the response of structures up to the inception of failure due to strain localization. After this point is reached, the nonlocal enhancement of the model presented in Section 2 is required.

Figures

FIGURE 1 Uniaxial response of the model.

TABLE 1 STANDARD Model Parameters

FIGURE 2 Uniaxial tension–compression response of the anisotropic model (longitudinal [1], transverse [2], and volumetric [v] strains as functions of the compressive stress).

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Distributed under a Creative Commons Attribution| 4.0 International License
Damage Models for Concrete
Gilles Pijaudier-Cabot, Jacky Mazars
To cite this version:
Gilles Pijaudier-Cabot, Jacky Mazars. Damage Models for Concrete. Jean Lemaitre Handbook of
Materials Behavior Models, 2, Elsevier, pp.500-512, 2001, Failures of materials, 978-0-12-443341-0.
�10.1016/B978-012443341-0/50056-9�. �hal-01572309�

Damage Models for Concrete
GILLES PIJAUDIER-CABOT
1
and JACKY MAZARS
2
1
Laboratoire de G
!
eenie Civil de Nantes Saint-Nazaire, Ecole Centrale de Nantes, BP 92101,
44321 Nantes Cedex 03, France
2
LMT-Cachan, ENS de Cachan, Universite
´
Paris 6, 61 avenue du Pre
´
sident Wilson, 94235,
Cachan Cedex, France
1

1 ISOTROPIC DAMAGE MODEL
1.1 VALIDITY
This constitutive relation is valid for standard concrete with a compression
strength of 30–40 MPa. Its aim is to capture the response of the material
subjected to loading paths in which extension of the material exists (uniaxial
tension, uniaxial compression, bending of structural members) [4]. It should
not be employed (i) when the material is confined ( triaxial compression)
because the damage loading function relies on extension of the material only,
(ii) when the loading path is severely nonradial (not yet tested), and (iii)
when the material is subjected to alternated loading. In this last case, an
enhancement of the relation which takes into account the effect of crack
closure is possible. It will be considered in the anisotropic damage model
presented in Section 3. Finally, the model provides a mathematically
consistent prediction of the response of structures up to the inception of
failure due to strain localization. After this point is reached, the nonlocal
enhancement of the model presented in Section 2 is required.
1.2 BACKGROUND
The influence of microcracking due to external loads is introduced via a single
scalar damage variable d ranging from 0 for the undamaged material to 1 for
completely damaged material. The stress-strain relation reads:
e
ij
¼
1 þ v
0
E
0
ð1 ÿ dÞ
s
ij
ÿ
v
0
E
0
ð1 ÿ dÞ
½s
kk
d
ij

ð1Þ
E
0
and v
0
are the Young’s modulus and the Poisson’s ratio of the undamaged
material; e
ij
and s
ij
are the strain and stress components, and d
ij
is the
Kronecker symbol. The elastic (i.e., free) energy per unit mass of material is
rc ¼
1
2
ð1 ÿ dÞe
ij
C
0
ijkl
e
kl
ð2Þ
where C
0
ijkl
is the stiffness of the undamaged material. This energy is assumed
to be the state potential. The damage energy release rate is
Y ¼ÿr
@c
@d
¼
1
2
e
ij
C
0
ijkl
e
kl
with the rate of dissipated energy:
’
ff ¼ÿ
@rc
@d
’
dd
2

Since the dissipation of energy ought to be positive or zero, the damage rate is
constrained to the same inequality because the damage energy release rate is
always positive.
1.3 EVOLUTION OF DAMAGE
The evolution of damage is based on the amount of extension that the
material is experiencing during the mechanical loading. An equivalent strain
is defined as
*
ee ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
3
i¼1
ð e
i
hi
þ
Þ
2
r
ð3Þ
where h.i
+
is the Macauley bracket and e
i
are the principal strains. The loading
function of damage is
fð
*
ee; kÞ¼
*
ee ÿ k ð4Þ
where k is the threshold of damage growth. Initially, its value is k
0
, which can
be related to the peak stress f
t
of the material in uniaxial tension:
k
0
¼
f
t
E
0
ð5Þ
In the course of loading k assumes the maximum value of the equivalent
strain ever reached during the loading history.
If fð
*
ee; kÞ¼0 and
_
ffð
*
ee; kÞ¼0; then
d ¼ hðkÞ
k ¼
*
ee
(
with
’
dd  0; else
’
dd ¼ 0
’
kk ¼ 0
(
ð6Þ
The function hðkÞ is detailed as follows: in order to capture the differences of
mechanical responses of the material in tension and in compression, the
damage variable is split into two parts:
d ¼ a
t
d
t
þ a
c
d
c
ð7Þ
where d
t
and d
c
are the damage variables in tension and compression,
respectively. They are combined with the weighting coefficients a
t
and a
c
,
defined as functions of the principal values of the strains e
t
ij
and e
c
ij
due to
positive and negative stresses:
e
t
ij
¼ð1 ÿ dÞC
ÿ1
ijkl
s
t
kl
; e
c
ij
¼ð1 ÿ dÞC
ÿ1
ijkl
s
c
kl
ð8Þ
3

a
t
¼
X
3
i¼1
e
t
i

e
i
hi
*
ee
2

b
; a
c
¼
X
3
i¼1
e
c
i

e
i
hi
þ
*
ee
2

b
ð9Þ
Note that in these expressions, strains labeled with a single indicia are
principal strains. In uniaxial tension a
t
¼ 1 and a
c
¼ 0. In uniaxial
compression a
c
¼ 1 and a
t
¼ 0. Hence, d
t
and d
c
can be obtained separately
from uniaxial tests.
The evolution of damage is provided in an integrated form, as a function of
the variable k:
d
t
¼ 1 ÿ
k
0
ð1 ÿ A
t
Þ
k
ÿ
A
t
exp½B
t
ðk ÿ k
0
Þ
d
c
¼ 1 ÿ
k
0
ð1 ÿ A
c
Þ
k
ÿ
A
c
exp½B
c
ðk ÿ k
0
Þ
ð10Þ
1.4 IDENTIFICATION OF PARAMETERS
There are eight model parameters. The Young’s modulus and Poisson’s ratio
are measured from a uniaxial compression test. A direct tensile test or three-
point bend test can provide the parameters which are related to damage in
tension ðk
0
; A
t
; B
t
Þ. Note that Eq. 5 provides a first a pproximation o f the
initial threshold of damage, and the tensile strength of the material can be
deduced from the compressive strength according to standard code formulas.
The parameters ðA
c
; B
c
Þ are fitted f rom t he r esponse o f t he m aterial to
uniaxial compression. Finally, b should be fitted f rom t he r esponse o f the
material to shear. This type of test is difficult to implement. The usual value is
b ¼ 1, which underestimates the shear strength of the material [7].
Table 1 presents the standard intervals for the model parameters in the case
of concrete with a moderate strength.
TABLE 1 STANDARD Model Parameters
E
0
30,000–40,000 MPa
v
0
0.2
k
0
1 10
ÿ4
0.74A
t
41.2
10
4
4B
t
45 10
4
14A
c
41.5
10
3
4B
c
42 10
3
1.04b41.05
4

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Damage models for concrete" ?

HAL this paper is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. 

Q2. What is the importance of the averaging procedure?

Attention should also be paid to axes of symmetry: as opposed to structural boundaries where the averaging region lying outside the structure is chopped, a special averaging procedure is needed to account for material points that are not represented in the finite element model. 

Q3. What is the robust way of calibrating the internal length?

The most robust way of calibrating the internal length is by a semi-inverse technique which is based on computations of size effect tests. 

Q4. How is the influence of microcracking introduced?

The influence of microcracking due to external loads is introduced via a single scalar damage variable d ranging from 0 for the undamaged material to 1 for completely damaged material. 

Q5. What is the effect of crack closure on the material?

During load cycles, microcracks close progressively and the tangent stiffness of the material should increase while damage is kept constant. 

Q6. What is the internal length of a microcrack?

In tension, microcracks are perpendicular to the tensile stress direction; in compression microcracks open parallel to the compressive stress direction. 

Q7. What is the evolution of the damage surface?

After anincremental growth of damage, the new damage surface is the sum of two ellipsoidal surfaces: the one corresponding to the initial damage surface, and the ellipsoid corresponding to the incremental growth of damage. 

Q8. What is the purpose of the modification of the model?

This modification of the model is necessary in order to achieve consistent computations in the presence of strain localization due to the softening response of the material [8]. 

Q9. What is the purpose of the nonlocal model?

This model, however, enables a proper description of failure that includes damage initiation, damage growth, and its concentration into a completely damaged zone, which is equivalent to a macrocrack. 

Q10. What is the damage energy release rate?

The damage energy release rate isY ¼ ÿr @c @d ¼ 1 2 eijC0ijkleklwith the rate of dissipated energy:’f ¼ ÿ @rc @d ’dSince the dissipation of energy ought to be positive or zero, the damage rate is constrained to the same inequality because the damage energy release rate is always positive. 

Q11. How can The authorspeed up the computation of a gauss point?

To speed the computation, a table in which, for each gauss point, its neighbors and their weight are stored can be constructed at the time of mesh generation. 

Q12. What is the relationship between the stress and the effective stress?

For a linear displacement interpolation, a is the solution of the following equality where the states of strain and stresses correspond to uniaxial tension:hf ¼ Gf ; with f ¼ Z 10 Z O ½ ’dð~nÞnkstklnlni njdOdeij ð24Þwhere f is the energy dissipation per unit volume, Gf is the fracture energy, and h is related to the element size (square root of the element surface in a two-dimensional analysis with a linear interpolation of the displacements).