Damage Models for Concrete

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

Introduction

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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.

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).

Full text

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Submitted on 6 Aug 2017
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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).