Code_Aster ®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 1/16

Organization (S): EDF-R & D/AMA
Handbook of Référence
R7.01 booklet: Modeling for Génie Civil and the géomatériaux ones
R7.01.01 document

Relation of behavior of Granger
for the clean creep of the concrete

Summary:

This document presents the clean model of creep of “Granger”, which is a way of modelling creep
clean of the concrete.
One also details there the writing and the digital processing of the model.

Handbook of Référence
R7.01 booklet: Modeling for Génie Civil and the géomatériaux ones
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 2/16

Count

matters

1 Introduction ............................................................................................................................................ 3
2 Recall on behavior in creep of a linear viscoelastic material [bib3] ............................ 3
2.1 Principle of superposition of Boltzmann ......................................................................................... 3
2.2 Model of Kelvin in series ............................................................................................................... 4
3 Presentation of the clean model of creep of Granger [bib1] ................................................................ 4
3.1 Experimental properties of the clean creep of the concrete in uniaxial loading .............................. 4
3.2 Modeling by a series connection of models of Kelvin ................................................... 4
3.3 Effect of the temperature .................................................................................................................... 5
3.4 Effect of the hygroscopy ....................................................................................................................... 6
3.5 Effect of ageing ...................................................................................................................... 6
3.6 Modeling 3D ............................................................................................................................... 7
3.7 Superposition on the constraint, the temperature and the hygroscopy (1D) ............................................. 7
4 Relations of behavior Code_Aster ............................................................................................... 8
5 numerical Integration of the model .......................................................................................................... 9
5.1 Discretization (1D) ........................................................................................................................... 9
5.2 Integration of the relation of behavior ................................................................................... 12
5.3 Variables of state .............................................................................................................................. 13
5.4 Stamp tangent ............................................................................................................................ 14
6 Bibliography ........................................................................................................................................ 16

Handbook of Référence
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HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 3/16

1 Introduction

Within the framework of the studies of the long-term behavior of structures out of concrete, a share
dominating of the deformations measured on structure relates to the differed deformations which
appear in the concrete during its life. They comprise the withdrawals at the youth, the withdrawal of
desiccation, clean creep and the creep of desiccation.

The model presented here is dedicated to the modeling of the differed deformation associated creep
clean. Clean creep is, in complement of the creep of desiccation, the share of creep of the concrete
that one would observe during a test without exchange of water with outside. In experiments concrete in
clean creep presents a growing old viscous behavior. The deformation of creep observed is
proportional to the constraint of loading, depends on the temperature and the hygroscopy.
longitudinal deflection is accompanied as in elasticity by a transverse deformation by sign
opposed.
The selected model is that proposed by L. Granger [bib1]. It is model of a viscoelastic type which
takes into account the effect of ageing as well as the history of constraint, temperature and of
the hygroscopy. It thus allows this fact of modelling the experimental facts quoted above.

One initially carries out a short recall on the linear viscoelastic models and one
present then the model itself like its numerical integration in Code_Aster.

In Code_Aster, 3 versions are available: GRANGER_FP_V the complete model, GRANGER_FP
who does not take into account the effect of ageing and GRANGER_FP_INDT, which in more does not depend
temperature.

2
Recall on behavior in creep of a material
viscoelastic linear [bib3]

The traditional curve of creep represents the evolution according to the time of the deformation of one
material subjected to a constant unidimensional constraint. The deformation of creep fl is,
in opposition to the instantaneous strain, the share of deformation which evolves/moves with time.
If a material has a linear viscoelastic behavior, then whatever the constant load
applied as from the time of loading Tc, the deformation of creep (1D) can be written:
fl (T) = F (T - T)







éq 2-1
C
where J (T, T) = F (T -)
C
Tc is related to creep, increasing function of (T - Tc) and null for (T - Tc)
negative.

2.1
Principle of superposition of Boltzmann

The relation [éq 2-1] is valid only for one constant loading. For a history of loading
nonconstant the principle of superposition of Boltzmann is applied; history of loading (T)
is broken up into increments of load:
N
(T) = H (T -)
I
Ti
i=0

where
H
Heavyside.

of

function



is
N
One can then write: fl
T () = F T (- T)
I
I
i=0
Handbook of Référence
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Code_Aster ®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 4/16

what uninterrupted gives:
T
fl


(T) =
F (T -)

D = F
= (F)

éq
2.1-1

T
=0
where the product of convolution represents.

2.2
Model of Kelvin in series

One can show that any linear viscoelastic body can be modelled by a series connection
models of Kelvin and that the function of creep can then be put in the form

R
T
F (T) = J 1
.(- exp (-
))
S


S 1
=
S

S and Js are plus coefficients identified on the experimental curves of creep.

3
Presentation of the clean model of creep of Granger [bib1]

3.1
Experimental properties of the clean creep of the concrete in loading
uniaxial

The clean creep tests on test-tube reveal the following properties:

·
in a range of constraint lower than 50% of the breaking strength, clean creep
is proportional to the constraint,
·
the clean creep of a test-tube with hygroscopy ext.
H is almost proportional to ext.
h.
clean creep of a no-slump concrete is almost null and it is maximum for a concrete saturated with water,
·
when the temperature T increases one has an acceleration of creep,
·
clean creep is a strongly growing old phenomenon,
·
a longitudinal deflection of creep is accompanied by a transverse deformation by
sign opposite (effect Poisson).

One chooses to model the clean creep of the concrete with a linear viscoelastic model which will have in
to more take into account the dependence of creep with respect to the temperature and the hygroscopy.

3.2
Modeling by a series connection of models of Kelvin

One uses a series connection of models of Kelvin whose coefficients are identified from
experimental curves of creep. It is shown in practice that one reproduces in a satisfactory way them
curves of concrete creep with R = 8 models in series.
Handbook of Référence
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Code_Aster ®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 5/16

The following function of creep is thus used:
8
T T
J (T, T) =
C


-
J
. 1 - exp -
C
S

éq
3.2-1

s=1


S
In practice it is very difficult to determine at the same time the J S and S as soon as the number of series of
Kelvin exceeds 2. One thus makes generally a choice a priori on the S,
1
=
S
S
1 10 -

and one
then determine by linear regression the J S.
The expression [éq 3.2-1] is related to basic creep of the model. One shows below how
taken into account of the effect of the temperature, the hygroscopy and ageing is integrated in
final model.

3.3
Effect of the temperature
To take account of the effect of the temperature on the kinetics of creep, one defines a “time
equivalent “T (T)
eq
who will replace time T in the model.

T
U

1
1
T
T

C
() =
exp -

-
ds

eq



éq
3.3-1
R T (S) T

S T

ref.
=


C

Note:

U
·
The temperature and the term of activation of the law of Arrhénius
C are expressed in
R
degrees K.

·
To model thus the effect of the temperature T exploits only the kinetics of creep. For
to really utilize T on the amplitude of the phenomenon of creep, in particular on
the level of the value ad infinitum of the function of creep, T is also introduced into
the expression of J like a multiplicative function of the coefficients of creep such as:

T - T
(
-
R
)
45
T
T
J T
(, T, T) =
ref.

C


-
eq
C
J
. 1 - exp-

S


éq
3.3-2


45
s=1


S

·
T is the temperature of reference. It is chosen by the user. It is
ref.
generally taken equalizes with 20°C. In the continuation of the document T will be taken equalizes with
ref.
20°C.
·
For the version independent of the temperature, there are simply T
T
() = T and
eq
R
T
T
J T
(, T, T) =
C


-
eq
C
J
. 1 - exp-

S

.


s=1


S


Handbook of Référence
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Code_Aster ®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 6/16

3.4
Effect of the hygroscopy

Into the model, H is also introduced like a multiplicative parameter of the coefficients of creep
so that:
T -
R
248
T
T
J T
(, T, T, H) = H

J. 1 exp

éq
3.4-1
C


-
eq
C
-
-

S


45

s=1


S

Note:

It is the noted variable drying C that one has at the end of Code_Aster calculation of drying and it is
isothermal curve of sorption-desorption which makes it possible to pass from the variable C to the hygroscopy of
ambient conditions h. Soit C the isothermal curve of desorption: C = C (H) and H = C-1 (C). The curve
H = C-1 (C) must be indicated by the user.

3.5
Effect of ageing

For a growing old viscoelastic material, the function of creep varies for two times of
loading different. Ageing is associated the hydration at the youth and others
phenomena like polymerization for the old concrete. The effect of ageing is modelled in
multiplying the coefficients of creep by a function of ageing K (T depend on time on
c)
loading. Modeling chosen to take into account ageing associated with the hydration
is that of the CEB [bib2]:

28 2.
0
+ 1
.
0
K T
() =
T

in

expressed

is
day.
C
2
.
0
C
T +1
C

To reveal a sensitivity of the phenomenon of ageing compared to the temperature one
also a time of loading are equivalent Tc (T) defines which replaces T in the function of
eq
C
C
ageing.

Tc
U 1
1
Tc T

v
() =
exp -

-
ds

eq
C

R T (S) T
s=t0

ref.


T: corresponds to the age of the concrete at the youth, it
0
is generally taken equal to 28 days
T: the time or age of loading expressed in
C
days
Note:

U
·
T and v are in degrees K,
R
·
for the old concrete it would be necessary to use another equivalent time and another function of
ageing,
·
if one does not take into account ageing, one has simply K (T) = 1.
C

The function of creep, which will be related final to creep of the model, is written then:

T -
N
248
T
T
J T
(, T, T, H) = H
K Tc
(
)
J. 1 exp

éq
3.5-1
C
eq


-
eq
C
-
-

S


45

s=1


S

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Code_Aster ®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 7/16

3.6 Modeling
3D

The traditional assumption consists in supposing the existence of a Poisson's ratio of constant creep
and equal to the elastic Poisson's ratio, that is to say =
2
.
0. From where for, T, H constant:
F

fl
(T) = J (T, T, T, H) 1
[(+ v) - v tr () I]
C
F
F

and thus:

~
~

fl T () = J T (, T, T, H) 1
(+
)

C
F


fl
tr
(
T
()) = J T
(, T, T, H) 1
(-
2)
tr
C
F

3.7
Superposition on the constraint, the temperature and the hygroscopy (1D)

To simplify the demonstration, one takes in this part like function of creep one of
components of the series of Kelvin, without taking into account of the effect of ageing, nor of time
equivalent parameterizing the temperature, is:

T - 248

T - T
J T
(, T, T, H) = H
J .1 - exp-
C

C
S


45

S

T 248
T T
fl
-

-
deformation of then being written creep:
= H
J .1 - exp-
C
.
S


45

S

It is pointed out that this writing of the deformation of creep is valid for, T and H constant
(in this case the model is equivalent in fact to take a Young modulus decreasing according to
time).
For a history of loading, temperature and hygroscopy nonconstant one applies it
principle of superposition of Boltzmann.

Let us suppose that for an element of volume given, one knows at time tn the sizes
(fl

fl
N, T, H)
N
N
N. At time tn 1
+ the sizes will be (N 1
+,
, T
, H
)
N 1
+
N 1
+
N 1
+
.

For T < T <
N
tn 1
+ one proposes to calculate the deformation of creep in the following way:

fl
fl
N 1
+ (T) = N (T) - J (T, T, T, H) +
J (T, T, T
, H
)
N
N
N
N 1
+
N 1
+
N 1
+

N
N

i.e.:


T
- 248
T - T
fl
fl
n+1

N
n+1 T () = N T () +

H
J 1 - exp-
n+1
n+1
S
45







S

T - 248
T - T
N

N
-
H J 1
N
N
S
45

exp





S
Handbook of Référence
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Code_Aster ®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 8/16

The superposition is thus considered not only on the constraint but also on the temperature and
the hygroscopy which is treated mathematically in the same way. From where:

N

T - T
T - 248
fl

I
T () = J .1 - exp-



H
S


i=0



45

S
I

One has then in integral writing, the deformation of creep of a component S of the series of Kelvin:

T



fl
T -
T - 248

T () =
J
. 1 - exp-

éq
3.7-1
S
S

D
H



45
=t0

S


4
Relations of Code_Aster behavior

One introduces into Code_Aster three relations of behavior associated with clean creep:

·
GRANGER_FP_V
·
GRANGER_FP
·
GRANGER_FP_INDT

The first takes account of the whole of the effects (forced, temperature, hygroscopy and
ageing), the second does not take account of the phenomenon of ageing and the last does not hold
count neither ageing nor of the effect of the temperature. They are available in modeling 2D,
3D and plane constraints.

The various parameters of the model are indicated in DEFI_MATERIAU. Are well informed under
the key word GRANGER_FP, of which the use is common to the relations of behavior GRANGER_FP
and GRANGER_FP_V, the characteristics materials following:


GRANGER_FP:

·
(2x8) constant characteristics of
J1: J
1
function of creep J,
I
I
TAUX_1:
1

.
.
.
J8: J
8
TAUX_8:
8

·
the curve of sorption-desorption giving H
FONC_DESORP: C-1 (C)
according to the variable drying C
·
the constant of energy of activation for
CPU
time-temperature equivalence.
QSR_K:

R

If one uses the growing old relation of behavior then one informs in more the key word
V_GRANGER_FP under which the characteristics associated with ageing are indicated, with
to know energy of activation for the calculation of the time of equivalent loading and the function of
ageing K (Tc).
eq

V_GRANGER_FP:
U
QSR_VEIL:
v
R
FONC_V: K (Tc)
eq
Handbook of Référence
R7.01 booklet: Modeling for Génie Civil and the géomatériaux ones
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Code_Aster ®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 9/16

For law GRANGER_FP_INDT, the parameters to be informed under key word GRANGER_FP_INDT of
DEFI_MATERIAU are:


GRANGER_FP_INDT:

·
(2x8) constant characteristics of
J1: J
1
function of creep J,
I
I
TAUX_1:
1

.
.
.
J8: J
8
TAUX_8:
8

·
the curve of sorption-desorption giving H
FONC_DESORP: C-1 (C)
according to the variable drying C

5
Numerical integration of the model

5.1 Discretization
(1D)

T - 248
Let us pose S = T H
T =

45

The expression [éq 3.7-1] is thus written:

T
-
fl
T


S
T
() =
J 1
.(- exp-
.)
.



D
S
S


=t
S
0

The discretization in time is such as for T [T
T] one considers a linear evolution of S
N,
1 N
(decomposition of S (T) in linear functions per piece). One has then:

N

-
fl
S
Ti
T

T
I
() =
.
J
N
.1 - exp-



D
S
N


1
i=
T
S

I =t1

S
I -

N
S
S
T
T
T
T
fl

N

-
-

T () =
J
T
J
1
exp
exp

S
N
I -
S
I

I


- N
I
-
- N
I




S S




1
1


i=
T
T
I
i= I


S


S


N
N


T
T
T
fl
-


T () = J
S
S J
exp
1 exp

éq
5.1-1
S
N
S
- I S
N
I
I
I
S

-

-
-



1
1


i=
i=
Ti

S


S

Note:

Notation X = X - X
.
I
I
I 1
-

8
Now let us consider the 8 models of Kelvin in series one a: fl
T () =
T ()
T ()
N
fl
=
S
N
fls N
s=1
S
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Code_Aster ®
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7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 10/16


T
One can then break up the deformation of creep [éq 5.1-1] on the basis of;
1


exp -

and

S
to carry out a recurrence on the coefficients of this base. According to [éq 5.1-1] one has with T:
N

N
N


-
fl


S
tn Ti
Ti
0
S
() =
-

exp-

S
tn
J S
If
If
J S
1 - exp-
= J






S N
With - N
With
T



i=1
i=1
I


S


S
3
2
1
1
4
4
4
4
4
4
4
4
4
2
4
4
4
4
4
4
4
4
4
3
A0
Have
N
N

With T
one can also write:
N 1
+

N
N
1

T
T
T
fl
-


T () = J
S
S
J
1
exp
1 exp
S
N 1
S
-
I

n+
I
I
+
I
S
S

-

-
-



1
1


i=
i=
Ti


S


S


T
+ S J - S
S
J
n+1
1 exp
n+1
S
n+1
S

-
-


T

n+1

S


that is to say:

n+1
N
1

fl
T

T - T

T
T () = J
1
S
N 1
S S -
S
J exp- n+
N
I
I
+
I
I
S
S

exp-

1 - exp-



T



i=1
i=1
I

S


S


S


T

- S

S
J
n+1
n+1
S 1 - exp-


tn+1

S

One can thus write:

fl
T () = J A0 + S J
S
n+1
S
N
n+1
S
T



T

- S
With exp -
n+1
- S

S
J
n+1
1 exp
N




n+1
S

-
-





T

S

n+1

S


8
Let us pose J = J, one has then:
S
s=1

8
8
fl
(T) = J 0
With -
With and
T
J WITH
With

N
N
S
fl (
) = 0 -
S
N
n+1
n+1
n+1
s=1
s=1

with

A0 = A0 + S
n+1
N
n+1

T

T

S
S




With = A exp (- n+1) + S
S
J
n+1
1 exp
n+1
N
n+1
S -
-




T

S
n+1

S

Handbook of Référence
R7.01 booklet: Modeling for Génie Civil and the géomatériaux ones
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 11/16

More precisely, if:

·
one takes into account equivalent time for the temperature and ageing,
·
during a step of time the parameter T is evaluated in the middle of the step of time for
calculation of equivalent times T and Tc, its linear evolution being supposed during
eq
eq
this step of time,

then one a:

8
fl
(T) = J 0
With
-
S
With

n+1
n+1
n+1
s=1

with
U 1
1
T (T) - T (T) = dt (T) = exp - C
-
T
eq
N 1
+
eq
N
eq
N 1
+
N 1
+



R T
T

N 1+/2

ref.

U 1
1
Tc (T) - Tc (T) = dtc (T) = exp - v
-
T
eq
N 1
+
eq
N
eq
N 1
+




N 1
R T
293
+


N 1+/2


A0 = A0 + K Tc
(
T
(
)) S
n+1
N
eq
n+1/2
n+1

T

T

eq

eq
S
With
= S
With exp (-
n+1) + S

S
J K Tc
(
T
(
))
n+1
1 exp
n+1
N
n+1
S
eq
n+1/2

-
-




T

S
n+1

S


Note:

N 1
+
·
If one does not take then account of ageing K 0
With
=
S S
,
N
=
1
+
I
N 1
+
I 1
=
X
+ X
·
one noted
N 1
N
X
=
+
N 1
+
,
/2
2
·
T was noted
= T T
.
eq
eq (
1
+)
N 1
+
N

To have at time T
, one should not store that A
With step of previous time, are 9
fl
N 1
+
0 and them
S
variables. In 3D A and
A are tensors. One will associate the two relations then
0

S
clean behavior of creep (9x6) variable interns corresponding to the components of the tensors
A. Elles characterize the advance of creep.

The writing in increment of deformation, nearer to the programming gives as for it:


teq

fl
(T) = fl
(T) - fl
(T) = S
n+
To 1 exp (
1)
S
n+1
S
n+1
S
N
N -
-



S



teq
+ S
K (Tc (T
)) J
S
N +
1
1
(
exp
1)
n+1
eq
n+1/2
S

-
-
-



T

n+1
S

Handbook of Référence
R7.01 booklet: Modeling for Génie Civil and the géomatériaux ones
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 12/16

5.2
Integration of the relation of behavior

1
That is to say the increment of deformation =
((U
)
T
+ (U
)).
2
If one holds account, in the partition of deformation, of the thermal deformation, the deformations
associated the endogenous withdrawal and the withdrawal of desiccation, then:

E
fl
HT
retightens
retdés

=
+

+

+

+



where:


E
= H
: elastic strain
HT

= T
(- ref.
T
I
) D
: thermal deformation

retightens
= - I
: endogenous deformation of withdrawal
D

retdes
= - C
(
-
ref.
C I
) D: deformation of withdrawal of desiccation

with:

: hydration,
C: water concentration.
ref.
T
and Cref: temperature and drying of reference
H,
: characteristics materials

Note:

In the continuation of the document, one will note
With
HT
ret end
ret of


=

+

+

.

~
~
µ
E
2 ~-
~
~ fl
= 2µ =
+ 2µ - 2µ
2 -
µ
and
~ fl

8
~ 0
~ S
(T) = 1
(+
) J A
-
With

n+1
F
n+1



n+1
s=1


it results that:






T T ()
1

~
S
eq
n+

1+ (2µ) (1+ H T K Tc
J
~
1
1 exp

µ ~
2

F) (
(eq
)



+1/2

-
-
-
=
- +


S
N












S
T


-
n+1


S





~
T T ()
S
eq
n+1
- (2µ 1
) (+
~
)
To 1 exp
(H T) K Tc

F


-
-
- - - -
N
(eq)







n+


1/2
S


S





T T ()
S
eq
n+1
J 1
1 - exp -

S










S
T

n+1


S



Handbook of Référence
R7.01 booklet: Modeling for Génie Civil and the géomatériaux ones
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 13/16

In the same way:

HT
3K
tr () = 3K tr () =
tr (-) + 3K tr (
E

) - 3K tr (
fl

) - 3K tr (A

)
3K -
and
fl

8
0
S
tr () = 1
(- 2
) J A -
With

F
N


N
s=1

from where:





T (T)
+

K
S
eq
N 1
3
tr () 1
+ 3K 1
(- 2) (H T K (Tc
)
+
J 1
1 - exp -
=
tr (-)
F
eqn 1/2
S










-
S
T +


K
N 1
S
3



T

N +1

T

+ 3K tr (E
) - 3K (1 - 2

With - -
- - H T - K Tc


F)
tr (Sn)
(
)
1 exp
eq
(tr
) (eq
)
N +1/2





S
S








T
(T)
+

J 1
S
-
1 - exp
eq
N 1
-

S








S
tn 1+

S



- 3K tr (
With
)
1
One deduces some then since = ~ + tr
ij
ij
ij
3

5.3 Variables
of state

The variables of state of the two relations of behavior are thus:

·
: tensor of the constraints,
·
: tensor of the deformations,
·
T: temperature,
·
C: water concentration,
·
: hydration,
·
A: tensors characteristic of the advance of creep, are 6x9 variable,
S
·
Tc: time of equivalent loading, characteristic of the age of the concrete.
eq

A and Tc are internal variables of the laws of behavior, which thus comprise
S
eq
55 internal variables.
Handbook of Référence
R7.01 booklet: Modeling for Génie Civil and the géomatériaux ones
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 14/16

5.4 Stamp
tangent

~ 1 (tr)
=
+
I
D

3
~ ~ ~
~ij
=
=
1
-
~

ik
jl
ij kl


3
kl
(tr) (tr) (tr)
()
=
tr
=
ij

(tr)
ij

Iteration of Newton:

~


T (T)
S
eq
n+

1
+ 2µ 1
(+) .h T K (Tc
)
J

F
eqn+
1 (
1
) (- exp (
1
-
))
S

= 2µI
~
1/2

S
T



N 1
+
S


with I
=
ijkl
ik
jl

(tr)


T

T
(
)
+

1
+ 3K 1
(- 2
H
). T K Tc
(
)
+
J
S
eq
N
1 - (
1
) (- exp (
1
-
))



= 3KI
F
eq
S
N
(tr)
1/2




S
T


N 1
+
S


Phase of prediction for the step of time [tn, tn+1]

Note:

fl

-

S
With
S
In 1D:
S

=
- J K Tc
(
)

.
T
S
eq


T


S
tn

Writing of speed at the moment T:
N

~

~

-
-

1+2µ 1
(+) (J K Tc
(
) T
H)

2µ 1
F


=
-

S
eqn

(+ F)
T

S

T

~-
With
T
D
dh

S - J K Tc
(
) ~ - -
H
- J K Tc
(
) ~-
T
S
eq
S
eq


N
N




S
dt
dt
S



Handbook of Référence
R7.01 booklet: Modeling for Génie Civil and the géomatériaux ones
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 15/16

linearization for the phase of prediction of the step of time [tn, tn +]
1:


~
-
-
1+2µ 1
(+) (J K Tc
(
) T
H)
F


=

S
eqn


S


~-
With

µ~
2
- 2µ 1
(+)
T J K Tc
(
~
)
H T J K Tc
(
~
)
T
H
F S -

- - -
- -
S
eq
S
eq
N
N


S
S


Writing of speed at the moment tn:

(tr)
-
-
(tr)
1+3K 1
(- 2) (J K (Tc) T H) =3K
T
F
S
eq





S
T
(tr -
With)


S
-
-
T
D
-
- dh
- 3K 1
(- 2)
- J K (Tc) (tr) H
- J K (Tc) (tr) T

F
S
eq
S
eq

N
N

S
dt
dt
S

dT
D
cd.
- 3K 3
(
) +3K 3
(
) +3K 3
(
)
dt
dt
dt

linearization for the phase of prediction of the step of time [tn, tn +]
1:


-
-
(tr
) 1+3K 1
(- 2) (J K (Tc) T H) =3K (tr

)


F
S
eq

N

S

(tr -
With)

- 3K 1
(- 2)
S

T
- J K (Tc) (tr -
) H T
- J K (Tc) (tr -
) T H

F
S
eq
S
eq

N
N


S
S

- 3K 3
(T
) +3K 3
() +3K 3
()
C

Creep thus introduces a specific term of second member at the time of the phase of prediction which in
fact is neglected, without consequence on the results.

Handbook of Référence
R7.01 booklet: Modeling for Génie Civil and the géomatériaux ones
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
Key S. MICHEL-PONNELLE
:
R7.01.01-C Page
: 16/16

6 Bibliography

[1]
L. GRANGER: Behavior differed from the concrete in the enclosures of nuclear thermal power station:
analyze and modeling. Thesis of Doctorat of the ENPC (February 1995).
[2]
CEB FIP Model (1990) General task group n°9, Evaluation off the time behavior off concrete.
[3]
J. LEMAITRE, J-L CHABOCHE: Mechanics of solid materials. Dunod.
Handbook of Référence
R7.01 booklet: Modeling for Génie Civil and the géomatériaux ones
HT-66/05/002/A

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