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Titrate:
Relation of behavior of Bazant for the creep of desiccation
Date:
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Author (S):
J. EL GHARIB Key
:
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Organization (S): EDF-R & D/AMA
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
R7.01.05 document
Relation of behavior of Bazant
for the intrinsic creep of desiccation of the concrete
Summary:
Contrary to the clean creep which is the share of the creep measured on a test-tube protected from
external desiccation, the creep of desiccation is calculated on a mechanically charged test-tube and
subjected to drying simultaneously.
This document presents the model of intrinsic creep of desiccation of Bazant (1985). One details there
also the writing and digital processing of the model.
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Relation of behavior of Bazant for the creep of desiccation
Date:
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Author (S):
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:
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Count
matters
1 Introduction ............................................................................................................................................ 3
2 Partition of the deformation ....................................................................................................................... 4
3 constitutive Law ....................................................................................................................................... 5
4 Discretization ......................................................................................................................................... 5
5 Integration of the law of behavior ................................................................................................... 6
5.1.1 Deviatoric part .................................................................................................................. 6
5.1.2 Hydrostatic part ................................................................................................................ 7
6 tangent Matrix .................................................................................................................................... 8
6.1 Phase of prediction ......................................................................................................................... 8
6.2 Reactualization of the tangent matrix ........................................................................................... 8
6.3 Variables of state .............................................................................................................................. 10
7 Implementation of a calculation of creep of desiccation ......................................................................... 10
8 Bibliography ........................................................................................................................................ 11
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Relation of behavior of Bazant for the creep of desiccation
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:
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1 Introduction
One points out the deformations differed from a concrete structure to locate the share of the deformation
calculated in this document:
· at the youth:
- endogenous withdrawal (1j - 1 year),
caused by a reaction of thermo hydration
- thermal withdrawal (1h 1j).
· in the medium term without load: withdrawal of desiccation (qq m qq year) according to dimensions' of
structure caused by the drying which results in an evaporation of part of water not
used in the process of hydration.
· long-term under load:
- clean creep (without exchange of moisture with outside thus without drying),
- the creep of desiccation (with drying which assigns the behavior of the concrete to the scale
microscopic, which is translated the macroscopic scale by creep of desiccation).
The differed deformations constitute a significant part of the deformations which appear in
concrete during its life. Among its differed deformations, withdrawals endogenous and thermal with short
term, withdrawal of desiccation caused by medium-term drying. One quotes also them
deformations differed under long-term load like clean creep and creep from desiccation.
The model presented here relates to the modeling of the deformation differed associated creep from
intrinsic desiccation. The creep of desiccation in complement to clean creep is the share of
total creep directly related to the water departure affecting the concrete which undergoes a mechanical loading
on the one hand and drying on the other hand. In other words, the deformation which one measures in one
test-tube which dries is directly related to the drying under constraints which carries not of creep of
desiccation.
The model suggested here is that of Bazant (1985) and adopted by L. Granger in its thesis (1995).
It is a law of the viscoelastic type linear which holds in account of the effect of the variation of
the hygroscopy. One presents the details of the numerical integration of this law in Code_Aster.
In Code_Aster, this model is used under the name of BAZANT_FD.
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Relation of behavior of Bazant for the creep of desiccation
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2
Partition of the deformation
Into small deformations, the increment of the total deflection is broken up into several terms
relating to the mechanisms considered. If one holds account in the partition of the increments of
deformations, thermics, associated the thermal, endogenous withdrawal and with the withdrawal of desiccation, then:
E
HT
Re
Re
fl
=
+
+
end +
as of +
éq
2-1
The increment of the deformation of creep
fl
breaks up into two components, corresponding
with clean creep and the creep of desiccation:
fl
fl
fl
=
Pr +
as of éq 2-2
The creep of desiccation
fl
as of as for him, breaks up into two intrinsic and structural part:
fl
fl
fl
=
+
as of
as of _ int
as of _ struc
It is agreed that the structural deformation is not a component of deformation in oneself,
thus in this document the only component of the creep of desiccation relates to the part
intrinsic:
fl
fl
=
as of
as of _ int
éq 2-3
with:
E =
H
HT
= (T - Tref) I
Re
= - I
end
: hydration
Re
= - Ci
as of
C: water concentration
H: stamp elastic, thermal dilation, coefficients related on the withdrawals endogenous and the withdrawal
of desiccation are data material.
Here, one wants to model
fl
Des.
Note:
This partition of the deformations is purely numerical. For the calculation of each one of these
components, the experimenters consider a combination different from
components of deformation (Voir [bib1] and [bib2]).
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3 Law
constitutive
Bazant et al. (1985) suggest that the drying and the application of a loading in compression
at the same time are responsible for the microcomputer-diffusion of the molecules between the macro-pores and them
micropores. The microcomputer-diffusion of the water molecules would support the rupture of the connections between
particles of freezing inducing the deformation of creep of desiccation. It is one of the phenomena
physicochemical most complicated to model resulting from a coupling between the constraint, it
clean creep and drying. They propose the following equation to take into account the creep of
intrinsic desiccation at the elementary level:
fl
&
= &
éq 3-1
as of
H
with:
fl
,
as of the deformation of the intrinsic creep which evolves/moves in time,
, a parameter material [1
Pa],
,
H the relative humidity which evolves/moves in time, fact of the case of evolution.
This expression is similar to the rheological model of the damping device:
fl
éq 3-2
as of
&
=
Note:
By preoccupation with a lightening of notations, one uses fl
to replace fl
as of in the continuation of
document.
4 Discretization
The evolution of the relative humidity is approached by a function closely connected per pieces (Benboudjema and
Al, 2001d). This discretization according to (Bazant, 1982) makes it possible to increase the precision of calculations
numerical in a considerable way compared to an approximation by bearing (Heaviside function)
especially if the size of the step of time is important:
,
H (T)
(T - T
T
T T
N)
[N n+1]
= N
H +
N
H
with
éq
4-1
tn
N
H = N
H +1 - N
H
according to the equation [éq 3-1], one can write:
fl = fl
N
N +
+
hn+ - h.
N tn +
T with
1
1
(
)
[] 1
,
0
éq
4-2
For = 1/2, semi-implicit diagram which makes it possible to have a better quadratic convergence of
solution, one obtains:
fl
fl
fl
fl
(N + N 1+)
=
or
N
N +
N
H
- h.
N N +
=
N
N +
N
H
- h.
1
+
1
+
éq
4-3
2
1
+
1
+
N
2
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5
Integration of the law of behavior
As in this document one is interested in integration of the intrinsic creep of desiccation, one goes
to consider for creep the only component
fl
dess_int but to simplify the writing one goes
to call
fl
. One poses in the same way:
With
HT
Re
Re
=
+
end +
as of éq
5-1
By employing the following notations: Has, A, A
for quantity A evaluated at the known moment tn, with
the moment tn 1
+ and its increment T
, respectively.
It is a question of expressing the constraint at time + according to the constraint at time and of the increment
of deformation at time -. One seeks initially the expression of the deviatoric component and then
the expression of the hydrostatic component of the constraint.
5.1.1 Part
deviatoric
One seeks a relation between the deviatoric constraint ~ and the variation of the deformation
deviatoric ~
at time +:
The constraint at time + is written:
~
~e
2µ ~-
~e
= 2µ =
+ 2µ
éq
5.1.1-1
2µ -
The elastic prediction of the deviatoric constraint is written:
E
2
~
µ
=
~ - + µ ~
2
éq
5.1.1-2
2µ -
Like the component
With
do not have a deviatoric part, one can write:
2
~
µ ~-
~
~ fl
=
+ 2µ - 2µ éq
5.1.1-3
2µ -
While using [éq 4-3], one obtains:
2
~
µ
=
~ - + µ ~
H
H
2 - 2µ
~ - - 2µ
~ éq 5.1.1-4
2µ -
2
2
1 4
4 2 4
4 3
~e
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from where,
2µ
~
H
-
~
~-
E
+ 2
µ - 2µ
~
H ~
-
-
-
2µ
~
= 2µ
2
=
2
éq
5.1.1-5
H
H
1 + 2µ
1 + 2µ
2
2
5.1.2 Part
hydrostatic
One seeks a relation between tr () and tr (
) at time +:
The constraint at time + is written:
3
tr () = 3
(E) K
Ktr
=
tr (- 3
éq
5.1.2-1
-
) + Ktr (E
)
3K
The elastic prediction of the hydrostatic constraint is:
() = 3K
tr E
tr (- 3Ktr
éq
5.1.2-2
-
) +
(
)
3K
from where,
3
tr () = 3
(E) K
Ktr
=
tr (- 3
3
3
éq
5.1.2-3
-
) + Ktr (
) - Ktr (fl
) - Ktr (A
)
3K
According to [éq 4-3], one can express the hydrostatic part of
fl
:
3
tr (
K
H
H
) =
tr (-
éq 5.1.2-4
-
) +3Ktr () - 3Ktr (A
) - 3K
tr (-) - 3K
tr ()
3K
2
2
from where,
3K tr (
H
H
-
3Ktr
3Ktr
With
3K
tr
tr E
3Ktr
With
3K
tr
-
) +
() -
()
-
(-) () - ()
-
(-) éq 5.1.2-5
tr () = 3K
2
=
2
H
H
1 + 3K
1 + 3K
2
2
One thus deduces the total constraint from it by combining the two components deviatoric and
hydrostatic at time +:
tr
~
()
= +
éq
5.1.2-6
ij
ij
3
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6 Matrix
tangent
6.1
Phase of prediction
The option used is RIGI_MECA_TANG, the tangent operator calculated in each point of Gauss is known as
in speed:
&ij = D
ijkl &kl,
in this case, Dijkl is a viscoelastic operator calculated starting from the not discretized equations.
6.2
Reactualization of the tangent matrix
The option used is FULL_MECA, when one reactualizes the tangent matrix with each iteration in
updating the internal constraints and variables:
D ij = ijkl
With dkl,
in this case, Aijkl is a viscoelastic operator calculated starting from the discretized equations
implicitly.
~ 1 tr
() D
=
+
I
éq
6.2-1
3
~
~ 1 tr
() tr
() D
=
+
éq
6.2-2
~
3 tr
()
I
~
1 tr
ij
ij
(ij)
1
=
-
= -
ik
jl
ij
kl
3
3
kl
kl
kl
tr
(ij)
=
ij
kl
kl
According to [éq 5.1.1-5]:
~
H
1 + 2
µ
= 2µ éq
6.2-3
~
2
According to [éq 5.1.2-5]:
tr
()
H
1 + 3
= 3
éq
6.2-4
tr
()
K
K
2
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Writing of speed:
~
H
~
1 + 2µ
= 2µ.
éq
6.2-5
T
2
T
tr
()
H
tr
()
1 + 3K
= 3K.
éq
6.2-6
T
2
T
Thus while returning to [éq 6.2-2], one can deduce the writing of speed:
2µ
D 1 D
D
1
K
=
3
I
éq
6.2-7
4 -
I I +
(D D
I I)
H
3
3
H
1 + 2µ
1 + 3K
2
2
Linearization:
H
~
1 + 2
µ
= 2µ ~
.
éq
6.2-8
2
tr () 1+ 3 H
K
= 3K.tr () éq
6.2-9
2
Like
H
is independent of the constraint, it is the same writing that one finds afterwards
linearization from where the form of the tangent matrix:
K
4µ
K
2µ
K
2µ
+
-
-
0
0
0
H
H
H
H
H
H
1+ 3
K
3 1+ 2µ
1+ 3
K
3 1 + 2µ
1+ 3
K
3 1+ 2µ
2
2
2
2
2
2
K
2µ
K
4µ
K
2µ
-
+
-
0
0
0
H
H
H
H
H
H
1+ 3
K
3 1+ 2µ
1 + 3
K
3 1 + 2µ
1+
3
K
3 1+ 2µ
2
2
2
2
2
2
K
µ
K
µ
K
µ
11
2
2
4
11
-
-
+
0
0
0
H
H
H
H
H
H
22
1 + 3
K
3 1+ 2µ
1+ 3
K
3 1 + 2µ
1+ 3
K
22
3 1+ 2µ
2
2
2
2
2
2
33
=
33
2
2µ
2
12
0
0
0
0
0
12
2
H
2
23
1 + 2µ
23
2
2
2
31
31
2µ
0
0
0
0
0
H
1+ 2µ
2
2µ
0
0
0
0
0
H
1+ 2µ
2
1
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
2
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
In this case, the tangent operator is the same one for RIGI_MECA_TANG and FULL_MECA:
With = D. It has a writing similar to the elastic matrix with dependant coefficients
ijkl
ijkl
of H
and of.
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Relation of behavior of Bazant for the creep of desiccation
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6.3 Variables
of state
The variables of state are:
·
: tensor of the constraints,
·
: tensor of the deformations,
·
C: water concentration.
The internal variables of this law of behavior is the value of the hygroscopy at the current moment.
7
Implementation of a calculation of creep of desiccation
In a way similar to the clean model of creep of Granger, GRANGER_FP, established already in
Code_Aster, this law constitutive depends on H, the relative humidity, which evolves/moves in time.
1)
To make a mechanical calculation of creep of desiccation with this law, it is necessary to have
relative humidity. The user can confront himself with two situations:
Has
The user knows moisture H or the water content C of the structure at various moments,
initial and final in the majority of the cases.
In this case, it can with CREA_CHAM and key word AFFE to affect the field of
temperature “TEMP” with the structure. It must repeat command CREA_CHAM with each
desired moment. Then, with command CREA_RESU, creates a structure of
data result starting from the fields already defined in the corresponding moments.
B
The user does not know the distribution of the field of moisture of the structure.
In this case, it must carry out a calculation of drying. The field of drying is given
thanks to the command THER_NON_LINE, but which is comparable in term of variable with
a temperature (standard TEMP) of field NOEU_TEMP_R.
Once defined (A) or calculated (B) a field of temperature “compared to a field of drying”, it
is necessary to begin mechanical calculation:
2)
Initially by creating the loading corresponding under AFFE_CHAR_MECA and the key word
SECH_CALCULEE. On the level of STAT_NON_LINE, which one put in SECH_CALCULEE is
regarded from now on as a field of drying with variable “SECH”.
However the law is written according to the hygroscopy H and not according to the water content C,
They is the same the case of the clean law of creep of Granger. One proceeds in the same way, it
is necessary:
3)
To define the curve sorption-desorption which allows the passage of the water content C
the hygroscopy curved h. Cette must be indicated by the user with DEFI_FONCTION and
NOM_PARA = SECH.
4) To define
under
DEFI_MATERIAU, the key word BAZANT_FD in which it is necessary to give like words
obligatory keys: LAM_VISC which is a parameter material and FONC_DESORP which is one
function defined before and which connects H the hygroscopy to C the water content.
5)
Mechanical calculation is carried out thanks to command STAT_NON_LINE with like relation
in the key word COMP_INCR = _F (RELATION = “BAZANT_FD”).
One of the evolutions to be envisaged is the use in RELATION_KIT of the two laws of creep of
Granger:
GRANGER_FP and BAZANT_FD.
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8 Bibliography
[1]
L. GRANGER: Behavior differed from the concrete in the enclosures of nuclear thermal power stations:
analyze and modeling. Thesis of Doctorat of the ENPC (1995).
[2]
F. BENBOUDJEMA, F. MEFTAH, J.M. TORRENTI, Y. LE-PAPE: Taking into account of the effects
drying on the deformations of the concrete noncharged and charged. Note HS-DG/AA/NNN/A
(2002).
[3]
A. RAZAKANAIVO: Modeling of the behavior of Granger for the clean creep of
concrete. Doc. [R7.01.01], Code_Aster (2001).
[4]
G. DEBRUYNE, B. CIREE: Modeling of the thermo hydration, drying and the withdrawal
concrete. Doc. [R7.01.12], Code_Aster (2001).
[5]
J. EL GHARIB: Comparison of the processing of the deformations differed between the model from
Granger and model LGCU. CR-AMA-02.125.
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