Code_Aster ®
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Titrate:
Model of damage of Mazars


Date:
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:
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Organization (S): EDF-R & D/AMA
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R7.01.08 document

Model of damage of Mazars

Summary:

This documentation presents the model of behavior of Mazars which makes it possible to describe the behavior
rubber band-endommageable of the concrete. This model is 3D, isotropic and is based on a criterion of damage
writing in deformation and describing dissymmetry traction and compression. It does not take into account the possible ones
plastic deformations or viscous effects which can be observed during deformations of a concrete.

Two versions of the models are established: the local version (with risk of dependence to the grid) and
not-local version where the damage is controlled by a not-local deformation. It is also possible
to take into account the dependence of the parameters of the law with the temperature, the hydration and drying.
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Model of damage of Mazars


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Count

matters

1 Introduction ............................................................................................................................................ 3
1.1 A law of behavior élasto-endommageable .......................................................................... 3
1.2 Limits of the local approach and methods of regularization ........................................................... 3
1.3 Coupling with thermics ............................................................................................................ 4
1.4 Law of Mazars in the presence of a field of drying or hydration ........................................... 4
2 Description of the model ........................................................................................................................... 5
2.1 The model of Mazars ...................................................................................................................... 5
2.2 Example in 1D ................................................................................................................................ 6
3 Identification ........................................................................................................................................... 8
4 numerical Resolution .......................................................................................................................... 10
4.1 Evaluation of the damage .................................................................................................. 10
4.2 Calculation of the constraint ................................................................................................................... 10
4.3 Calculation of the tangent matrix ........................................................................................................ 10
4.4 Stored internal variables ........................................................................................................... 11
4.5 Modelings compatible with the law of MAZARS ...................................................................... 11
5 Bibliography ........................................................................................................................................ 12

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Model of damage of Mazars


Date:
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1 Introduction

1.1
A law of behavior élasto-endommageable

The model of behavior of MAZARS ([bib1]) is a model simple, considered robust, based on
mechanics of the damage [bib2], which makes it possible to describe the reduction in the rigidity of material
under the effect of the creation of microscopic cracks in the concrete. It is based on only one internal variable
scalar D, describing the isotropic damage of way, but distinguishing despite everything
the damage of traction and the damage of compression.
Contrary to model ENDO_ISOT_BETON, this model does not allow to translate the phenomenon of
refermeture of the fissures (restoration of rigidity). In addition, the model of Mazars does not take in
count the possible plastic deformations or viscous effects which can be observed with the course
deformations of a concrete.

1.2
Limits of the local approach and methods of regularization

Like all the lenitive laws, the model of Mazars raises difficulties related to the phenomenon
of localization of the deformations.
Physically ([bib3]), the heterogeneity of the microstructure of the induced concrete of the remote interactions
between the formed fissures. Thus, the deformations locate in a metal strip, called tape
of localization, to form macrofissures. The state of the constraints in a material point cannot any more
to be only described by the characteristics at the point but must also take into account sound
environment. In the case of the local model, no indication is included concerning the scale of
cracking. Consequently, no information is given over the bandwidth of
localization which becomes null then. This leads to a mechanical behavior with rupture without
dissipation of energy, physically unacceptable.
Mathematically ([bib4]), the localization returns the problem to be solved badly posed because softening
cause a loss of ellipticity of the differential equations which describe the process of deformations.
The numerical solutions do not converge towards physically acceptable solutions in spite of
refinements of grid.
Numerically, one observes a dependence of the solution to the network extremely prejudicial
(cf [R5.04.02]).

A method of regularization thus becomes necessary. Several are possible. The choice which was
fact here is to take again one of the developments already made for models ENDO_FRAGILE and
ENDO_ISOT_BETON, by using a tensor of nonlocal deformation which checks the equation
characteristic [R5.04.02]:
=
2
-
C


where the scalar a.c. the dimension a length to the square one.

Note:

Let us announce that this model not-room does not correspond to the version initially proposed
by J. Mazars and G. Pijaudier-Cabot [bib5] and which is in particular established in Castem
2000. The delocalization is obtained while using as equivalent deformation, the average
local equivalent deformation on a volume V:
(X) = 1 -

Vr (X)
(X S) eq (S) ds

where is the volume of the structure
Vr (X) is representative volume as in point X: Vr = (X - S) ds

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Model of damage of Mazars


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2
4 X - S
(X - S)


is a weight function: (X - S) =

exp -
2


LLC

LLC is the internal length of the not-continuous medium (traditionally estimated at three times
cut larger aggregate).

Numerical tests made it possible to connect the 2 parameters of delocalization LLC and C
in the case of the model of Mazars. The following relation was obtained:
4 C LLC

The model of Mazars is thus available in Code_Aster under 2 versions:

· the local version of the model for which the dependence of the solution to the network is
observable as for all the lenitive models.
· a nonlocal version which uses a tensor of nonlocal deformation.

1.3
Coupling with thermics

For certain studies, it can be interesting to be able to take into account the modification of
parameters materials under the effect of the temperature. This is possible in Aster (MAZARS_FO
compound or not with ELAS_FO). The assumptions made for the coupling with thermics are them
following:

· thermal dilation is supposed to be linear is HT
= (T - T I with = constant or
ref.) D
function of the temperature,
· one does not take into account mechanical thermo interactions, i.e. one does not model
not the effect of the mechanical state of stress on the thermal deformation of the concrete,
· concerning the evolution of the parameters materials with the temperature, one considers that
those depend not on the current temperature but on the maximum temperature seen
by material during its history,
· only elastic strain (mechanical) induced of the damage.

Note:

Because of data-processing constraints, the initial value of Tmax is initialized to 0. In
consequence, one cannot use the parameters materials defined for temperatures
negative (if necessary, one can however circumvent this problem while re-entering all them
temperatures in Kelvin instead of °C)

One initially presents the writing of the model then some data on the identification of
parameters. To finish, one exposes the principles of numerical integration in Code_Aster.

1.4
Law of Mazars in the presence of a field of drying or hydration

The use of ELAS_FO and/or MAZARS_FO under operator DEFI_MATERIAU makes it possible to make depend
the parameters materials of drying or the hydration.
In addition, the deformations related on the withdrawal of dessication and the endogenous withdrawal are taken in
count in the model, in the following form:
Re
= - I and = - C
(
- C) I
D
rd
ref.
D
where is the hydration, C, the water concentration (field of drying in the terminology
Code_Aster), Cref initial water concentration (or drying of reference). Finally is the coefficient
of endogenous withdrawal and the coefficient of withdrawal of desiccation to be informed in DEFI_MATERIAU,
key word factor ELAS_FO, operands B_ENDO and K_DESSIC. As one said to the paragraph
precedent, the choice which was made in the establishment of the model of Mazars, it is that only
elastic strain induced of the damage. Consequently, if a test-tube is modelled
out of concrete which dries freely or which is hydrated freely, one a field of deformation will obtain well
not no one and a stress field perfectly no one.
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Model of damage of Mazars


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2
Description of the model

2.1
The model of Mazars

The model of Mazars was elaborate within the framework of the mechanics of the damage.
The constraint is given by the following relation:

E
= 1
(- D) a:







éq 2.1-1
with: With the matrix of Hooke
D the variable of damage
E
elastic strain E
HT
rd
Re
= - - -
HT
= (T - T I thermal dilation
ref.) D
Re
= - I endogenous withdrawal (related to the hydration)
D
= - C
(
- C) I withdrawal of desiccation (related to drying)
rd
ref.
D

The variable of damage D results from a combination of a damage of traction T
D and
of a damage of compression C
D:
D


= T T
D + 1
(- T) C
D éq
2.1-2
· The coefficient is a parameter material which makes it possible, when it is higher than 1, to improve
the response in shearing.
· The damages of traction and compression are defined by the equations
following since eq d0:
d0 (1 - C
With)
With
D = 1
C
-
-
(limited between 0 and 1)
eq
[C
exp B (
-)
C
eq
D 0]
d0 (1 - T
With)
With
D = 1
T
-
-
(limited between 0 and 1)
eq
[T
exp B (
-)
T
eq
D 0]
where A, A, B, B,
C
T
C
T D 0 are parameters materials to be identified.

· The damage is controlled by the equivalent deformation eq which makes it possible to translate one
triaxial state by an equivalence in a uniaxial state. As the extensions are of primary importance
in the phenomenon of cracking of the concrete, the introduced equivalent deformation is defined in
to leave the positive eigenvalues of the tensor of the deformations, is:

eq = +: + or in the principal reference mark of the tensor of deformations
2
2
2
=
1
eq
+ 2 + 3
+
+
+

knowing that the positive part < >

+ is defined so that if
I is the deformation
principal in direction I


I = I if I 0

+


I
= 0 if I < 0
+
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Note:

In the case of a thermomechanical loading, only elastic strain
E
HT
= - contributes to the evolution of the damage from where: eq =
E
E
:
.
+
+

· The coefficient T which carries out the coupling between traction and damage is such as in
the total absence of traction,
and in the total absence of compression
. It is
T = 1
T = 0
defined by the following relation:
3
[
I
Ti
+
]
I 1

=
=
(limited between 0 and 1)
T
2
eq

where Ti is the deformation created by the positive principal constraints is:

1+

T =
+ - tr (+)
E
E

2.2
Example in 1D

One shows here the answer obtained with the model of Mazars (or not-local local version) when one
an element of volume subjects to a uniaxial loading.
The parameters materials used are as follows:

E = 32.000 MPa, = 0.2, d0 = 9.375 10­5, C
With = 1.15, T
With = 0.8, C
B = 1391.3, T
B = 10.000, = 1

The loading is applied in several stages:

1) compression until ­ 0.3%
2) discharge
3) compression until ­ 0.4%
4) discharge then traction up to 0.035%
5) discharge
6) traction up to 0.07%
7) discharge

On the following figure, one represented axial stress according to the axial deformation thus
that according to the lateral distortion. Initial rigidity was also represented in order to better
to visualize the effect of the damage.
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6
7
4
2
3
1

Appear 2.2-a: Réponse forced deformation of the model of Mazars for a stress 1D

The preceding figure makes it possible to visualize a certain number of characteristics of the model of Mazars,
with knowknowing:

· the damage affects the rigidity of the concrete,
· there are no unrecoverable deformations,
· the answers in traction and compression are quite dissymmetrical,
· although one distinguishes the damage due to compression and that due to traction,
effective variable of damage is well a linear combination of both. In
consequence, when one préendommage the material in compression (resp. in traction), and
that one charges then in traction (resp. in compression), it is well rigidity with
damage which is observed and not initial rigidity.

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3 Identification

In addition to the parameters thermo elastic E, the model of Mazars utilize 6 parameters
material: WITH, B, A, B,
C
C
T
T D 0 and.

·
d0 is the threshold of damage. It acts obviously on the constraint with the peak but
also on the shape of the curve post-peak. Indeed, the fall of constraint is of as much
less brutal than d0 is small. In general d0 is included/understood into 0.5 and 1.5 10­4.

The coefficients A and B make it possible to modulate the shape of the curve post-peak:

·
A introduces a horizontal asymptote which is the axis of for A = 1 and the horizontal one
passing by the peak for A = 0 (cf [Figure 3-a]). In general, C
A lies between 1 and 1.15 and
T
With between 0.7 and 1.
·
B according to its value can correspond to a brutal fall of constraint (B > 10.000) or one
preliminary phase of increase in constraint followed, after passage by a maximum,
of a more or less fast decrease as one can see it on [Figure 3-b]. In
General C
B lies between 1000 and 2000 and T
B between 10.000 and 100.000.


Appear 3-a: Influence of parameters A [bib1]
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Appear 3-b: Influence of the parameters B [bib1]

· is a corrective factor allowing to improve the response in shearing compared to
initial version of the model (which corresponds to =1). The advised value is 1.06.

A means simple to obtain a set of parameters is to have the uniaxial test results in
compression and of deflection tests (for several sizes of beams to determine the parameter C in
not-room)
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4 Resolution
numerical

4.1
Evaluation of the damage

The procedure of obtaining the damage is simple. The stages are as follows:

· calculation of the elastic strain and thermal
· calculation of the equivalent deformation
· calculation of T, C
D and T
D for this state of deformation
·

calculation of
test
D

= T
T
D + 1
(- T)
C
D
- if
-
Dtest D, then the criterion is not reached and +
-
D = D
- if
-
Dtest > D, then the criterion is reached and +
test
D = D

- if test
D
= 1, then the material is completely damaged and +
D = 1 (makes some (1 -) for
to avoid the numerical problems).

4.2
Calculation of the constraint

After evaluation of D, one calculates simply:

E
= 1
(- D) A

4.3
Calculation of the tangent matrix

One seeks the tensor M such as & = M & knowing that = 1
(- D)
A. The matrix is thus
summon of two terms, one with constant damage, the other due to the evolution of
the damage:

& = (1 - D) A & - A D
&

The first term is easy, it acts simply of the operator of Hooke, multiplied by (1 - D).
The second requires the evaluation of the increment of damage D &.
Knowing that the damage is defined by the relation D

= D

+ 1
(-) D, it comes
T
T
T
C
·
·
D & =
D & + 1
(-

) D & +
D + 1
(-

) D
T
T
T
C
T
T
T
C
If it is supposed that the loading is radial &
([bib1]), it is then enough to express D &.
T = 0
C T,
D
~
C T,
&Dc T, =
Tr
~
&

~
with
+
=



~
D 1
(
0
- A)
WITH B

C, T
C, T C, T
from where &
D

C, T =
+

+
Tr
~

2
~
~ &


[
exp B (
C, T
- D)
0]





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Note:

1. Being given made simplifications, in the general case the tangent matrix is not
consistent. Also, it can happen that the reactualization of the tangent matrix at the course
iterations of Newton does not contribute to convergence. In this case, it is enough to use
only the secant matrix by imposing STAT_NON_LINE (NEWTON =
_F (REAC_ITER =0)).
2. In the general case, the tangent matrix is not-symmetrical. In version 6, the matrix
tangent was symmetrized. It is not any more the case starting from version 7 (even if it is
always possible to do it thanks to key word SOLVEUR=_F (SYME = “OUI”) of
STAT_NON_LINE).
3. Concerning the nonlocal approach, the processing of the boundary conditions is such as one
could be brought, in the case of symmetrical structures, to treat the calculation of
the whole of the structure and not of the “representative” part (cf [R5.04.02]).
4. The analytical expression of the tangent matrix is valid only for loadings
radial (dt = cd. = 0). In the other cases, the quadratic convergence of the method
is not guaranteed any more.
Knowing that in any case, the use of the law of damage in form
directly integrated ([éq 2.1-2]) is theoretically valid only under this same
assumption, in the case of nonradial loadings, one will prefer other models with
model of Mazars, in particular of modelings taking into account
anisotropy of the behavior or the effect of refermeture of the fissures (stresses
alternated), adapted to this type of loading.

4.4
Stored internal variables

We indicate in the table according to the internal variables stored in each point of Gauss
for the model of Mazars:

Internal variable
Feel physical
V1
D: variable of damage
Indicating V2
of damage
(0 so elastic, 1 if damaged i.e. as soon as D is not null any more)
V3
: maximum temperature attack at the point of gauss

4.5
Modelings compatible with the law of MAZARS

The law of Mazars is usable in Aster with various modelings:

· traditional version: 3D, D_PLAN, AXIS, C_PLAN,
· not-local version: 3D_GRAD_EPSI, D_PLAN_GRAD_EPSI, C_PLAN_GRAD_EPSI,
· coupled with the models of THHM (cf [R7.01.11]).

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5 Bibliography

[1]
Mazars J. (1984). Application of the mechanics of the damage to the behavior not
linear and with the rupture of the structural concrete. Thesis of doctorate of state of Université Paris VI.
[2]
Lemaitre J. and Chaboche J.L. (1988). Mechanics of solid materials. ED. Dunod.
[3]
H. Askes, space Advanced discretization strategies for localized failure, mesh adaptivity and
meshless methods, PhD thesis, Delft University off Technology, Faculty off Civil Engineering
and Geosciences, 2000.
[4]
R.H.J. Peerlings, R. of Borst, W.A. Mr. Brekelmans, J.H.P. of Vree, I. Spee, Some
observations one localization in nonroom and gradient ramming models, Eur. J. Mech. With/Solids,
15, N°6, 937-953, 1996.
[5]
Pijaudier-Cabot G., Mazars J. and Pulikowski J. (1991). Steel-concrete jump analysis with
not room continuous ramming, J. Structural Engrg ASCE 117, 862-882.

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