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Book ChapterDOI

Damage Models for Concrete

01 Jan 2001-Vol. 2, pp 500-512

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.
Topics: Crack closure (54%), Constitutive equation (51%)

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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
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
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

Citations
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Journal ArticleDOI
Abstract: A modified non-local damage model with evolving internal length, inspired from micromechanics, is developed. It is shown in particular that the non-local influence between two points in the damaged material depends on the value of damage at each of these points. The resulting weight function is non-symmetric and truncated. Finite element results and strain localization analysis on a one-dimensional problem are presented and compared to those of the original non-local damage model. It is shown that in the course of damage localization, the incremental strain profiles expand according to the modified non-local model, instead of shrinking according to the original constitutive relation. Comparisons with experimental data on model materials with controlled porosity are also discussed. Acoustic emission analyses provide results with which the theoretical model is consistent qualitatively. This model also opens the path for durability mechanics analyses, where it has been demonstrated that the internal length in the non-local model should evolve with environmentally induced damage.

93 citations


Cites background from "Damage Models for Concrete"

  • ...It is the perturbation stress field due to a circular inclusion loaded by an internal pressure obtained from the well-known Eshelby solution (see e.g. Reference [8])....

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  • ...The experimental programme which is briefly recalled here (for details, see Reference [5] or Reference [6]), aimed at determining the mechanical characteristics (compressive strength, elastic modulus and fracture energy) of mortar specimens with a controlled microstructure....

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  • ...These evolutions of the damage localization profiles can be further explained with the help of a one-dimensional bifurcation analysis on an infinite bar in an initial state of homogeneous tensile strain and damage ðe0; d0Þ: The analysis follows the steps described, for instance, in Reference [15]....

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  • ...The rest of the constitutive model, and in particular the equations governing damage growth, can be found for instance in Reference [9]....

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  • ...In particular, the terms l2ik are functions of the square of the void radius a2i which can be related to damage, according to many homogenization schemes (see e.g. Reference [10])....

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Abstract: The original model proposed present many advantages for applications which require cracks opening and reclosure management. It is easy to implement in most of the finite element codes based on a displacement formulation. It uses only measurable input data, like elasticity coefficients, tensile and compressive strengths, fracture energies and strains at the peak of the uni-axial stress-strain experimental curves. It supplies the crack opening perpendicular to the localized cracks directly, which can be plotted on the mesh. The orthotropic damage formulation on which it is based allows crack energy dissipation to be realistically verified and consequently ensures the objectivity of the FEM analysis toward the mesh size. Unilateral behavior and hysteretic dissipation due to crack re-closing phenomena are also considered in this framework. The latter aspect not only realistically predicts behavior during cyclic loading but, overall, provides a thermodynamic free energy form that can be generalized to treat quasi-unilateral aspects in various brittle materials concerned by crack re-closure problems. (C) 2012 Elsevier Ltd. All rights reserved.

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Book ChapterDOI
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Abstract: Anisotropic damage evolution and crack propagation in elastic-brittle materials is analyzed by the concepts of Continuum Damage Mechanics (CDM) and Finite Element Method FEM (ABAQUS). The original total formulation of the Murakami-Kamiya (MK) model of elastic-damage material is extended and used for damage anisotropy and fracture prediction in concrete. The incremental formulation of the stress-strain equations is developed by the use of the tangent elastic-damage stiffness. The Helmholtz free energy representation is discussed. The unilateral crack opening/closure effect is incorporated in such a way that the continuity requirement during unloading holds. The general failure criterion is proposed by checking the positive definiteness of the Hessian matrix of the free energy function. The Local Approach to Fracture (LAF) by FEM is applied to both the pre-critical damage evolution that precedes the crack initiation, and the post-critical damage/fracture interaction. Crack is modeled as the assembly of failed finite elements in the mesh, the stiffness of which is reduced to zero when the critical points at stress-strain curves are reached. Another way to model crack consists in releasing of the kinematic constrains in the nodes. The developed constitutive model is capable of capturing anisotropic damage evolution and crack growth in 2D structure subjected to the quasistatic or cyclic mechanical or thermal loadings. Different damage evolution in tension or compression, as well as the corresponding fracture modes may be analysed.

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References
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Book
01 Jan 1950

1,608 citations


"Damage Models for Concrete" refers methods in this paper

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Abstract: In the usual local finite element analysis, strain softening causes spurious mesh sensitivity and incorrect convergence when the element is refined to vanishing size. In a previous continuum formulation, these incorrect features were overcome by the imbricate nonlocal continuum, which, however, introduced some unnecessary computational complications due to the fact that all response was treated as nonlocal. The key idea of the present nonlocal damage theory is to subject to nonlocal treatment only those variables that control strain softening, and to treat the elastic part of the strain as local. The continuum damage mechanics formulation, convenient for separating the nonlocal treatment of damage from the local treatment of elastic behavior, is adopted in the present work. The only required modification is to replace the usual local damage energy release rate with its spatial average over the representative volume of the material whose size is a characteristic of the material. Avoidance of spurious mesh ...

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  • ...An approximation of the internal length was obtained by Bazant and Pijaudier-Cabot [2]....

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