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Book Chapter•DOI•

Damage Models for Concrete

Gilles Pijaudier-Cabot1, Jacky Mazars2•
École centrale de Nantes1, University of Paris2
01 Jan 2001-Vol. 2, pp 500-512
TL;DR: 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.

Summary (3 min read)

Jump to: [Introduction] – [1.1 VALIDITY] – [1.2 BACKGROUND] – [1.3 EVOLUTION OF DAMAGE] – [1.4 IDENTIFICATION OF PARAMETERS] – [2 NONLOCAL DAMAGE] – [2.2 PRINCIPLE] – [2.4 IDENTIFICATION OF THE INTERNAL LENGTH] – [2.5 HOW TO USE THE MODEL] – [3.1 VALIDITY] – [3.2 PRINCIPLE] – [3.3 DESCRIPTION OF THE MODEL] – [3.3.1 Evolution of Damage] – [6.13.3.3.3 Crack Closure Effects] and [3.5 HOW TO USE THE MODEL]

Introduction

  • Submitted on 6 Aug 2017 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not.
  • The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

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.
  • An enhancement of the relation which takes into account the effect of crack closure is possible.
  • 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.
  • Eij ¼ 1þ v0 E0ð1ÿ dÞ sij ÿ v0 E0ð1ÿ dÞ ½skkdij ð1Þ E0 and v0 are the Young’s modulus and the Poisson’s ratio of the undamaged material; eij and sij are the strain and stress components, and dij is the Kronecker symbol, also known as The stress-strain relation reads.
  • The elastic (i.e., free) energy per unit mass of material is rc ¼ 1 2 ð1ÿ dÞeijC0ijklekl ð2Þ where C0ijkl is the stiffness of the undamaged material.

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 *e ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX3 i¼1 ð eih iþÞ 2 r ð3Þ where h.i+ is the Macauley bracket and ei are the principal strains.
  • In the course of loading k assumes the maximum value of the equivalent strain ever reached during the loading history.
  • 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 atdt þ acdc ð7Þ where dt and dc are the damage variables in tension and compression, respectively.
  • Hence, dt and dc can be obtained separately from uniaxial tests.

1.4 IDENTIFICATION OF PARAMETERS

  • The Young’s modulus and Poisson’s ratio are measured from a uniaxial compression test.
  • A direct tensile test or threepoint bend test can provide the parameters which are related to damage in tension ðk0; At; BtÞ.
  • 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.
  • Table 1 presents the standard intervals for the model parameters in the case of concrete with a moderate strength.

2 NONLOCAL DAMAGE

  • The purpose of this section is to describe the nonlocal enhancement of the previously mentioned damage 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].

2.2 PRINCIPLE

  • Whenever strain softening is encountered, it may yield localization of strains and damage.
  • This localization corresponds to the occurrence of bifurcation, and a surface (in three dimension) of discontinuity of the strain rate appears and develops.
  • When such a solution is possible, strains and damage concentrate into a zone of zero volume, and the energy dissipation, which is finite for a finite volume of material, tends to zero.
  • In the nonlocal damage model, this length is incorporated in a modification of the variable which controls damage growth (i.e., the source of strain softening): a spatial average of the local equivalent strain.

2.4 IDENTIFICATION OF THE INTERNAL LENGTH

  • In fact, whenever the strains in specimen are homogeneous, the local damage model and the nonlocal damage model are, by definition, strictly equivalent ð%e ¼ *eÞ.
  • This can be viewed also as a simplification, since all the model parameters (the internal length excepted) are not affected by the nonlocal enhancement of the model if they are obtained from experiments in which strains are homogeneous over the specimen.
  • Since their failure involves the ratio of the size of the zone in which damage can localize versus the size of the structure, a size effect is expected because the former is constant while the later changes in size effect tests.
  • It should be stressed that such an identification procedure requires many computations, and, as of today, no automatic optimization technique has been devised for it.
  • An approximation of the internal length was obtained by Bazant and Pijaudier-Cabot [2].

2.5 HOW TO USE THE MODEL

  • The local and nonlocal damage models are easily implemented in finite element codes which uses the initial stiffness or secant stiffness algorithm.
  • The reason is that the constitutive relations are provided in a total strain format.
  • Compared to the local damage model, the nonlocal model requires some additional programming to compute spatial averages.
  • This table will be used for any subsequent computation, provided the mesh is not changed.
  • 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.

3.1 VALIDITY

  • In tension, microcracks are perpendicular to the tensile stress direction; in compression microcracks open parallel to the compressive stress direction.
  • The influence of crack closure is needed in the case of alternated loads: microcracks may close and the effect of damage on the material stiffness disappears.
  • Finally, plastic strains are observed when the material unloads in compression.
  • This anisotropic damage model has been compared to experimental data in tension, compression, compression–shear, and nonradial tension– shear.
  • It provides a reasonable agreement with such experiments [3].

3.2 PRINCIPLE

  • The basis of the model is the numerical interpolation of dðnÞ (called damage surface) which is approximated by its definition over a finite set of directions.
  • Depending on the interpolation of the damage variable dðnÞ, several forms of damage-induced anisotropy can be obtained.

3.3 DESCRIPTION OF THE MODEL

  • The variable dðnÞ is now defined by three scalars in three mutually orthogonal directions.
  • It is the simplest approximation which yields anisotropy of the damaged stiffness of the material.
  • In compression or tension–shear problems, plastic strains are also of importance and will be added in the model.
  • When the loading history is not monotonic, damage deactivation occurs because of microcrack closure.

3.3.1 Evolution of Damage

  • Note that the vectors n* are the three principal directions of the incremental strains whenever damage grows.
  • After an incremental 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.

6.13.3.3.3 Crack Closure Effects

  • Crack closure effects are of importance when the material is subjected to alternated loads.
  • During load cycles, microcracks close progressively and the tangent stiffness of the material should increase while damage is kept constant.
  • Since this new variable refers to the same physical state of degradation as in tension, dcðnÞ is directly deduced from dðnÞ.

3.5 HOW TO USE THE MODEL

  • The implementation of this constitutive relation in a finite element code follows the classical techniques used for plasticity.
  • An initial stiffness algorithm should be preferred because it is quite difficult to derive a consistent material tangent stiffness from this model.
  • Again, the evolution of damage is provided in a total strain format.
  • It is computed after incremental plastic strains have been obtained.
  • The difficulty is the numerical integration involved in Eq. 15, which is carried out according to Simpson’s rule or to some more sophisticated scheme.

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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
Citations
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Journal Article•DOI•
Multiscale analysis of solid, waffle, ribbed and hollowcore reinforced concrete slabs

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Arturo Moyeda1, Jacob Fish1•
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TL;DR: In this article, a multiscale plate element for the analysis of solid, waffle and hollowcore reinforced concrete slabs based on the higher order computational continua (HC 2 ) formulation is presented.

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Non‐linear seismic analysis of RC structures with energy‐dissipating devices

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P. Mata1, Alex H. Barbat1, Sergio Oller1, Rubén Boroschek2•
Polytechnic University of Catalonia1, University of Chile2
28 May 2009-International Journal for Numerical Methods in Engineering
TL;DR: In this article, the non-linear dynamic response of reinforced concrete (RC) buildings with energy dissipators is studied using advanced computational techniques, and a fully geometric and constitutive nonlinear model for the description of the dynamic behavior of framed structures is developed.
Abstract: The poor performance of some reinforced concrete (RC) structures during strong earthquakes has alerted about the need of improving their seismic behavior, especially when they are designed according to obsolete codes and show low structural damping, important second-order effects and low ductility, among other defects. These characteristics allow proposing the use of energy-dissipating devices for improving their seismic behavior. In this work, the non-linear dynamic response of RC buildings with energy dissipators is studied using advanced computational techniques. A fully geometric and constitutive non-linear model for the description of the dynamic behavior of framed structures is developed. The model is based on the geometrically exact formulation for beams in finite deformation. Points on the cross section are composed of several simple materials. The mixing theory is used to treat the resulting composite. A specific type of element is proposed for modeling the dissipators including the corresponding constitutive relations. Special attention is paid to the development of local and global damage indices for describing the performance of the buildings. Finally, numerical tests are presented for validating the ability of the model for reproducing the non-linear seismic response of buildings with dissipators.

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Cites background from "Damage Models for Concrete"

  • ...The list of works is extensive; however, representative examples are those of Chaboche [36, 37]; see also Krajcinovic and Fonseka [38] or, more recently, [39]; Pijaudier-Cabot and Mazars [40] have introduced a two damage variable model that couples the compressive and tensile components of the damage by means of empirical relations (see also references therein)....

    [...]

Journal Article•DOI•
Towards practical multiscale approach for analysis of reinforced concrete structures

[...]

Arturo Moyeda1, Jacob Fish1•
Columbia University1
01 Oct 2018-Computational Mechanics
TL;DR: A novel multiscale approach for analysis of reinforced concrete structural elements that overcomes two major hurdles in utilization of multiscales technologies in practice by means of reduced order homogenization customized for structural elements.
Abstract: We present a novel multiscale approach for analysis of reinforced concrete structural elements that overcomes two major hurdles in utilization of multiscale technologies in practice: (1) coupling between material and structural scales due to consideration of large representative volume elements (RVE), and (2) computational complexity of solving complex nonlinear multiscale problems. The former is accomplished using a variant of computational continua framework that accounts for sizeable reinforced concrete RVEs by adjusting the location of quadrature points. The latter is accomplished by means of reduced order homogenization customized for structural elements. The proposed multiscale approach has been verified against direct numerical simulations and validated against experimental results.

8 citations

Dissertation•DOI•
High velocity impact and fragmentation of concrete : numerical simulation

[...]

Barış İrhan
01 Jan 2014
TL;DR: In this paper, a three dimensional explicit Lagrangian finite element program has been developed for the simulation of contact, impact and fragmentation events based on mixed programming approach, in which a graphical user interface (GUI) was developed using C++ programming language to carry out pre- and post-processing tasks.
Abstract: In this study load-rate dependent behavior of plain concrete has been investigated by means of numerical methods. To accomplish this, a three dimensional explicit Lagrangian finite element program has been developed for the simulation of contact, impact and fragmentation events based on mixed programming approach. In this respect a graphical user interface (GUI) has been developed using C++ programming language to carry out pre- and post-processing tasks. On the other hand, another program has been developed using FORTRAN programming language to carry out finite element computations. Communication in between GUI and FORTRAN program has been established using standard function import/export mechanism. Microplane material model for concrete has been extended to account for large deformations, rate of loading and thermal effects. Stress locking observed under dominant tensile loading has been addressed by proper relaxation of the kinematical constraint. On the other hand mesh dependency, due to softening behavior present, has been tackled by crack-band regularization. Kinematical contact constraints in normal and tangential directions have been formulated in both total and rate forms. Predictor-corrector type algorithm has been employed as method of constraint enforcement. Requirements for the exact satisfaction of constraints have been discussed. Mohr-Coulomb type frictional constitutive behavior is adopted in tangential direction. Classical radial return mapping algorithm, frequently used for elastic-plastic materials, has been used to perform constitutive update in tangential direction. During high velocity contact-impact events, like projectile penetration, motion function loses its regularity around the impact region due to presence of very large deformations. In order to be able to continue simulations staying within Lagrangian framework, such material is simply removed with a technique based on adaptive element deletion. Maximum principal strain has been used as a deletion criterion. Topological data structures have been implemented to keep track of the evolving contact interface during simulations. Furthermore, to accelerate contact search a procedure based on so-called moving contact sphere has been developed. Predictive capability of the numerical techniques proposed has been assessed by comparisons with some relevant experimental results from literature. Main conclusions have been drawn out and future research directions have been recommended. In Rahmen dieser Arbeit wurde das lastrateabhangige Verhalten von Beton numerisch untersucht. In diesem Zusammenhang wurde ein dreidimensionales lagrangesches Finite-Elemente-Programm fur die Simulation von Kontakt- und Impaktverhalten sowie Fragmentierung auf Basis von gemischter Programmierung entwickelt. Unter Anwendung der C++ Programmiersprache wurde eine graphische Oberflache (GUI) fur die Durchfuhrung der Pre- und Post-processing-Aufgaben konzipiert. Ein weiteres Programm wurde fur die Finite-Elemente-Berechnungen mit FORTRAN Programmiersprache entwickelt. Die Kommunikation zwischen GUI und FORTRAN-Programm fand durch Standardfunktion Import/Export Mechanismus statt. Das Microplane-Modell fur Beton wurde erweitert, um die grosen Verformungen, die Belastungsrate und die thermischen Effekte zu berucksichtigen. Das bei der uberwiegenden Zugbelastung beobachtete „Stress-locking“ wurde durch eine geeignete Relaxation der kinematischen Bindungen vermieden. Die Netzempfindlichkeit, die Aufgrund der „Softening“ (Erweichung) des Betons entsteht, wurde durch Anwendung der Rissbandmethode berucksichtigt. Kinematische Bindungen fur Kontakt in normaler und tangentialer Richtung wurden in absoluter und anteilmasiger Form formuliert. Ein Pradiktor-Korrektor-Algorithmentyp wurde fur die Durchsetzung der Bindungen angewendet. Die Bedingungen fur die exakte Erfullung der Bindungen wurden angesprochen. Das konstitutive Reibungsverhalten des Mohr-Coulomb-Types wurde fur die tangentiale Richtung angenommen. Der klassische radiale Ruckprojektionsalgorithmus, der oft fur elastoplastische Materialen angewendet wird, wurde fur die Durchfuhrung der konstitutiven Aktualisierung in tangentialer Richtung eingesetzt. Wahrend der Kontakt–Impakt Hochgeschwindigkeitsereignisse, wie z.B. Geschosspenetration, verliert die Bewegungsfunktion ihre Regelmasigkeit in der Nahe des Impaktbereiches infolge der grosen Verformungen. Um die Simulationen in Rahmen des lagrangeschen Verfahrens fortsetzten zu konnen, wurde dieses Material geloscht, wobei eine Technik basierend auf der adaptiven Elementloschung eingesetzt wurde. Die maximalen Zugdehnungen wurden als Loschkriterium verwendet. Topologische Datenstrukturen wurden implementiert, um den Uberblick uber die entwickelnde Kontaktschnittstelle in Simulationen zu halten. Daruber hinaus wurde ein Verfahren basierend auf der sogenannten beweglichen Kontaktkugel entwickelt, um die Kontaktsuche zu beschleunigen. Die pradiktive Fahigkeit des vorgeschlagenen numerischen Verfahrens wurde durch Vergleich mit einigen relevanten experimentellen Ergebnissen aus der Literatur untersucht. Die wichtigsten Schlussfolgerungen wurden gezogen und kunftige Forschungsrichtungen wurden empfohlen.

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Journal Article•DOI•
Benchmarks for the validation of a non local damage model

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Ludovic Jason1, Shahrokh Ghavamian, Gilles Pijaudier-Cabot1, Antonio Huerta2•
École centrale de Nantes1, Polytechnic University of Catalonia2
28 Mar 2004
TL;DR: In this paper, the robustness of the FE implementation of a constitutive relation, its pertinence with respect to experiments, and quantitative and qualitative comparisons of structural elements are evaluated.
Abstract: The aim of this contribution is to present a series of organised benchmarks that helps at validating the robustness of the FE implementation of a constitutive relation, its pertinence with respect to experiments, and quantitative and qualitative comparisons of structural elements. It is applied to an isotropic damage model used for concrete, coupled with a non local gradient formulation to avoid a spurious description of strain localisation. After the elementary (uniaxial monotonic, cyclic or triaxial loading) and structural (three point bending tests) simulations, an industrial application is presented in the form of a representative structural volume of a containment building for French nuclear power plants.

8 citations

References
More filters
Journal Article•DOI•
Nonlocal damage theory

[...]

Gilles Pijaudier-Cabot1, Zdeněk P. Bažant1•
Northwestern University1
01 Oct 1987-Journal of Engineering Mechanics-asce
TL;DR: In this paper, the authors proposed a nonlocal damage theory, which is based on the nonlocal treatment of damage from the local treatment of elastic behavior, and 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 a material.
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 ...

1,672 citations

Book•
Theory of flow and fracture of solids

[...]

Arpád Nádai
01 Jan 1950

1,625 citations

"Damage Models for Concrete" refers methods in this paper

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Application de la mecanique de l'endommagement au comportement non lineaire et a la rupture du beton de structure

[...]

Jacky Mazars
09 May 1984
TL;DR: In this paper, the authors propose the construction of a MICROSTRUCTURE and a COMPORTATION MECANIQUE DU BETON, who is a member of the International Organization for Standardization (IOS).
Abstract: CETTE ETUDE PROPOSE LA CONSTRUCTION D'UN MODELE DE COMPORTEMENT S'APPUYANT SUR LES PROPRIETES MICROSTRUCTURELLES DU MATERIAU ET PERMETTANT LA SIMULATION NUMERIQUE DE L'ENDOMMAGEMENT. TROIS PARTIES COMPOSENT CE TRAVAIL : 1 - MICROSTRUCTURE ET COMPORTEMENT MECANIQUE DU BETON : IL S'AGIT DE L'ETUDE DE LA FORMATION ET DE L'EVOLUTION DE LA MICROSTRUCTURE DANS LE TEMPS ET SOUS SOLLICITATIONS MECANIQUES. DES MODELES MICROSTRUCTURAUX AIDENT A APPREHENDER LES PRINCIPAUX PHENOMENES DE COMPORTEMENT ET APPORTENT DE NOMBREUX ENSEIGNEMENTS A LA FORMULATION D'HYPOTHESES UTILISEES DANS LA SUITE. - 2 - MODELE DE COMPORTEMENT MECANIQUE DU BETON : ETABLI DANS LE CADRE DE LA THERMODYNAMIQUE DES PROCESSUS IRREVERSIBLES, ON PASSE EN REVUE LES ACQUIS ET ON JUSTIFIE LES APPORTS SPECIFIQUES A CETTE ETUDE. APRES AVOIR TRAITE DES PROBLEMES D'IDENTIFICATION, UNE ETUDE NUMERIQUE A L'ECHELLE DE L'ELEMENT DE VOLUME JUSTIFIE LES CHOIX EFFECTUES ET PERMET DE PROPOSER UNE INTERPRETATION ORIGINALE DE CERTAINS RESULTATS EXPERIMENTAUX. - 3 - ETUDE DE L'ENDOMMAGEMENT EN TRACTION DANS LES STRUCTURES : LA FORMATION DES FISSURES : L'ETUDE PRECEDENTE EST ADAPTEE PUIS INTEGREE A UN PROGRAMME DE CALCUL PAR ELEMENTS FINIS. UNE EXPERIMENTATION VISANT A CARACTERISER PHYSIQUEMENT ET MECANIQUEMENT LA NOTION D'ENDOMMAGEMENT ET DE RUPTURE PERMET DE VERIFIER LE BIEN FONDE DU MODELE PROPOSE ET D'ANALYSER SES PERFORMANCES DANS LE DOMAINE DU CALCUL PREVISIONNEL.

609 citations

Journal Article•DOI•
Measurement of Characteristic Length of Nonlocal Continuum

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Zdeněk P. Bažant, Gilles Pijaudier-Cabot1•
Rensselaer Polytechnic Institute1
01 Apr 1989-Journal of Engineering Mechanics-asce
TL;DR: The characteristic length of a heterogeneous brittle material such as concrete represents a material property that governs the minimum possible width of a zone of strain softening damage in nonlocal areas as mentioned in this paper.
Abstract: The characteristic length of a heterogeneous brittle material such as concrete represents a material property that governs the minimum possible width of a zone of strainsoftening damage in nonlocal...

412 citations

"Damage Models for Concrete" refers methods in this paper

  • ...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....

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

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Journal Article•DOI•
A nonlocal damage theory

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Shaobo Xia1, Guangxia Li1, Hao Lee1•
Huazhong University of Science and Technology1
01 Aug 1987-International Journal of Fracture
TL;DR: In this paper, a nonlocal damage field theory is proposed and a response model of nonlocal elastic damage is given, and a simulating experiment using aluminum alloy plates with arrays of holes shows that the load bearing capacity of a plate varies with angle γ between array and loading direction.
Abstract: A nonlocal damage field theory is proposed and a response model of nonlocal elastic damage is given. A simulating experiment using aluminum alloy plates with arrays of holes shows that the load bearing capacity of a plate varies with angle γ between array and loading direction. The characteristics and attenuation lengths appreciably affect the constitutive equations.

356 citations

"Damage Models for Concrete" refers methods in this paper

  • ...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]....

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Related Papers (5)
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Measurement of Characteristic Length of Nonlocal Continuum

[...]

01 Apr 1989-Journal of Engineering Mechanics-asce
Zdeněk P. Bažant, Gilles Pijaudier-Cabot
Nonlocal damage theory

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A description of micro- and macroscale damage of concrete structures

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A consistent characteristic length for smeared cracking models

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01 Feb 1989-International Journal for Numerical Methods in Engineering
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Frequently Asked Questions (12)
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. 

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. 

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

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. 

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

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

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. 

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]. 

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. 

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. 

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. 

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).