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Relation of behavior UMLV for the clean creep of the Date concrete
:
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Author (S):
Y. The Key POPE
:
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: 1/16
Organization (S): EDF-R & D/MMC
Handbook of Référence
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R7.01.06 document
Relation of behavior UMLV for creep
clean of the concrete
Summary:
This document presents the clean model of creep UMLV, which is a way of modelling the clean creep of
concrete.
One also details there the writing and the digital processing of the model.
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:
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:
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Count
matters
1 Introduction ............................................................................................................................................ 3
2 Assumptions ............................................................................................................................................ 3
3 Description of the model [bib1] ................................................................................................................. 4
3.1 Description of the spherical part ................................................................................................... 4
3.2 Description of the deviatoric part ............................................................................................... 5
4 Discretization of the equations constitutive of the model ........................................................................... 6
4.1 Discretization of the equations constitutive of spherical creep ...................................................... 6
4.2 Discretization of the equations constitutive of creep deviatoric ................................................ 10
5 tangent Matrix .................................................................................................................................. 12
6 Description of the variables intern ....................................................................................................... 14
7 Notations .............................................................................................................................................. 14
8 Bibliography ........................................................................................................................................ 15
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Code_Aster ®
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Relation of behavior UMLV for the clean creep of the Date concrete
:
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Author (S):
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:
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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.
Models of creep of the existing concretes (e.g.: model of Granger to see [bib4] and [R7.01.01]) have
summer developed in optics to predict the longitudinal deflections of creep under
uniaxial constraints. The generalization of these models, in order to take into account a state of
multiaxial constraints, is done then via a Poisson's ratio of creep arbitrary,
constant and equal, or close, of the elastic Poisson's ratio. However, determination a posteriori of
Poisson's ratio of effective creep shows his dependence with respect to the path of loading.
In addition, concrete of certain works of Parc EDF, the such containments of
nuclear engine, is subjected in a state of biaxial stresses. This report led to the setting to
not law of deformations of clean creep UMLV (marl-the-Vallée Université, partner in
the development of this model) for which the Poisson's ratio of creep is one
direct consequence of the calculation of the principal deformations.
In Code_Aster, the model is used under the name of BETON_UMLV_FP.
2 Assumptions
Assumption 1 (H.P.P.)
The law is written within the framework of the small disturbances.
Assumption 2 (partition of the deformations)
Into small deformations, the tensor of the total deflections is broken up into several terms
relating to the processes considered. Being the description of the various mechanisms of
deformations differed from the concrete, it is admitted that the total deflection is written:
E
FP
fd
Re
rd
HT
= {
+ {
+ {
+ {
+ {
+ {
éq
2-1
N
déformatio
creep
of
creep
withdrawal
of
withdrawal
N
déformatio
rubber band
clean
one
dessiccati
endogenous
one
dessiccati
thermics
Within the framework of this documentation, one will limit oneself to the description of clean creep. With ends of
simplification of writing, the exhibitor F will indicate the clean deformation of creep so that [éq 2-1]
reduces to:
E
F
= +
éq 2-2
N.B.:
In the continuation the term “creep” will indicate clean creep exclusively.
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Assumption 3 (decomposition of the components of creep)
In a general way, clean creep can be modelled by combining the elastic behavior of
solid and the viscous behavior of the fluid. For the law presented, creep is described like
combination of the elastic behavior of the hydrates and the aggregates and the viscous behavior
water.
In the case of law UMLV, one carries out the assumption that creep can be broken up into one
process uncoupling a spherical part and a deviatoric part. The tensor of the deformations
total of creep is written then:
F
fs
fd
=
1
1 +
with fs
F
= tr
éq
2-3
{
{
3
part
part
spherical
ue
déviatoriq
The tensor of the constraints can be developed according to a similar form:
S
D
= 1 +
éq 2-4
{
{
part
part
spherical
ue
déviatoriq
The law of creep UMLV supposes a total decoupling between the spherical components and
deviatoric: the deformations induced by the spherical constraints are purely spherical and
the deformations induced by the deviatoric constraints are purely deviatoric. To hold
count effect of internal moisture, the constraints are multiplied by internal relative moisture:
S
= H F (S
)
D
= H F (D
)
and
éq 2-5
Or H indicates internal relative moisture.
The condition [éq 2-5] makes it possible to check a posteriori that the deformations of clean creep are
proportional to the relative humidity.
3
Description of the model [bib1]
3.1
Description of the spherical part
The spherical constraints are at the origin of the migration of the water adsorbed with the interfaces between
hydrates on the level of the macroporosity and absorptive within microporosity in porosity
thin cable. Diffusion of water interlamellaire of the pores of hydrates towards capillary porosity
be carried out in an irreversible way. The total spherical deformation of creep is thus written like
summon of a reversible part and an irreversible part:
fs
fs
fs
=
+
{
R
éq
3.1-1
{
I
part
part
reversible
irréversib
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The process of deformation spherical of creep is controlled by the system of coupled equations
according to (equations [éq 3.1-2] and [éq 3.1-3]):
fs
& = 1 [
S
S
fs
H - Kr R]
fs
- I
éq
3.1-2
S
&
R
where S
Kr indicates rigidity connect associated with the skeleton formed by blocks with hydrates on the scale
mesoscopic;
and S
R viscosity connect associated with the mechanism with diffusion within capillary porosity.
fs
= 1
&
I
[sk fs -
R
(sk + S
R
I
K) fs
I] - [
S
H
- S
fs
Kr R] +
éq
3.1-3
if
where
S
ki indicates rigidity connect intrinsically associated with the hydrates on the scale
microscopic;
and S
I viscosity connects associated with the interfoliaceous mechanism of diffusion.
+
+
1
In [éq 3.1-3], hooks
appoint the operator of Mac Cauley: X
= (X + X)
2
Appear 3.1-a: phenomenologic Modèle associated with the spherical part of clean creep
3.2
Description of the deviatoric part
The deviatoric constraints are at the origin of a mechanism of slip (or mechanism of quasi
dislocation) of the layers of HSC in nano-porosity. Under deviatoric constraint, creep
be carried out with constant volume. In addition, the law of creep UMLV supposes the isotropy of creep
deviatoric. Phénoménologiquement, the mechanism of slip comprises a contribution
reversible viscoelastic of the water strongly adsorbed with the layers of HSC and a contribution
irreversible viscous of interstitial water:
fd
fd
fd
=
+
éq
3.2-1
{
{
R
{
I
contribution
N
déformatio
one
contributi
water
ue
déviatoriq
water
absobée
total
free
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The principal component jème of the total deviatoric deformation is governed by the equations
[éq 3.2-2] and [éq 3.2-3]:
D
D, J
D
D, J
D, J
& + K = H éq
3.2-2
R
R
R
R
where D
Kr indicates rigidity associated with the capacity with water adsorbed to transmit loads (load
bearing toilets);
and D
R viscosity associated with the water adsorbed by the layers with hydrates.
D D, J
D J
I
& =
I
H
,
éq 3.2-3
where D
I indicates the viscosity of interstitial water.
Appear 3.2-a: phenomenologic Modèle associated with the deviatoric part of clean creep
4
Discretization of the equations constitutive of the model
4.1
Discretization of the equations constitutive of spherical creep
One carries out a linearization with the first command of the product of the constraints and moisture:
T - tn
T () H T () N N
H +
(N nh +n nh)
éq
4.1-1
T
N
After discretization of the constraints and relative humidity by functions closely connected, the deformation
spherical of clean creep is discretized by the following equation:
fs
= S
+ S has
B S
+ S
C S
F
S
S
S
N
N
N
N
N N 1
+
(tr) =3a +b tr +c
N
N
N tr
éq 4.1-2
N
N 1
+
where S
and S
N
N 1
+ are the spherical constraints at the beginning and the end of the step of current time.
It is necessary to distinguish two cases from figures according to whether the unrecoverable deformation must be taken into account or
not.
1st case: the deformation of spherical creep irreversible is not taken into account, the equation
[éq 4.1-2] can put itself in the form (simple chain of Kelvin):
S
fs
& (T) + K S fs
(T) = H (T) S
(T)
R R
R R
R
éq
4.1-3
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After discretization, the preceding equation can be put in the form:
fs
S
S
S
S
S
R, N = rear, N + R
B, N N + Cr, N N 1
+
éq
4.1-4
With:
T
S
has
= exp
1
R, N
-
N
- fs
D
R, N
R
1
2
T
2
T
S
S
S
S
S -
R
R
N
R
R
N
B =
1 H
H
exp
1 H
H
éq
4.1-5
R, N
D -
+
+
N
n+1
-
+
-
-
N
n+1
S
K
T
T
T
T
R
N
N
R
N
N
1 S
T
S
- T
S
C
=
exp
H
H
R, N
R
-
N
- R
N
N
N
D
S
K
T
T
R
N
R
N
The unrecoverable deformation, as for it, does not vary:
S
has
= 0
fs
I, N
= 0
S
B
éq 4.1-6
I, N
= 0
I, N
S
C = 0
I, N
2nd case: the deformation of spherical creep irreversible must be taken into account.
Using the linearization [éq 4.1-1],], the system of coupled equations is written:
1
T - T
fs
& T () + fs
2 & T () =
H
H
H
K
T
()
R
I
S
+
N
N
N
(+
N
N
N
N) -
S
fs
R
R
T
R
N
+
éq 4.1-7
1
T - T
fs
& T () = -
-
S
fs
2k
T
() + S fs
K
T
(
)
H +
N
H
H
I
R
R
I
I
N
N
(+
N
N
N
N)
S
T
I
N
This system can be put in the form:
&
&
fs
fs T ()
fs
S
fs
S
S
S
S
T
() has
T
() has
T
() B
C T T
& T () = R
A:
T
() B
T T C
fs
=
fs
+ +
R
(- N)
=
+
+
+
R
rr
R
laughed I
R
R (- N)
& T ()
& T () has T () has T () B C T T
I
fs
= S fs
+ S S
+ S + S
I
ir
R
II I
I
I (- N)
éq 4.1-8
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With, B and C are defined as follows:
S
S
S
K
K
K
4
2
S
S
- R -
R
I
S
S
S
has
has
With = rr
ir
S
S =
R
I
I
has
has
S
S
K
K
ir
II
I
2
- I
S
S
I
I
1
2
S
+
S
S
B
B = R
H
éq
4.1-9
S =
R
I
N
N
B
1
I
-
S
I
1
2
S
+
S
S
C H + H
C = rs =
N
N
N
N R
I
C
T
1
I
N
-
S
I
The preceding system of equations can be uncoupled and solved in space of the clean vectors.
The system of equations is written indeed:
*
*
*
*
*
*
& (T) = (T) + B + C
éq
4.1-10
K
K
K
K
K (T - T N)
&
with
& = 1
P
* =
- 1
&
&2
Thus, in the space of the clean vectors, the model of creep becomes equivalent to a double chain
of Kelvin. It is necessary to know the solution of the homogeneous equation (without second member),
as well as a particular solution in order to solve the preceding differential equation. The solution
homogeneous of each of the two equations is as follows:
*
T
K
T () = µ E
éq
4.1-11
K
K
where µ is a parameter depend on the initial condition. A particular solution is obtained by
K
method of variation of the constant (µ = µ (T)). The following solutions then are obtained:
K
K
*
1
*
*
1
T () =
T
µ E K -
B
C T T
éq
4.1-12
K
K
+
K
K -
-
N
K
K
The spherical deformations of reversible and irreversible creep are then equal to:
H +
H
fs
(T) = N
N
n+1
N +
1tn 1
2tn 1
X µ E
µ E
R
n+1
(
+
+
+
1
1
2
)
S
K
R
éq
4.1-13
H +
H
fs
(T) = N
N
n+1
N +
1tn 1
2tn 1
µ E
X µ E
I
n+1
(
+
+
+
S
1
2
2
)
ki
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After simplification, one then obtains the following expressions for the values of µ:
K
1
H +
H
H +
H
µ =
(
)
(
)
1
fs
X
T
- N
N
n+1
N
-
fs T
- N
N
n+1
N
2
R
n+1
I
n+1
(X X - 1 1
1
2
) T
E N
S
S
K
K
R
I
1
H +
H
H +
H
fs
N
N
n+1
N
fs
N
N
n+1
N
µ =
(
)
(
)
2
(
T
X
T
X X - 1 2
1
2
)
-
-
+
-
R
n+1
1
I
n+1
tn
S
S
E
K
K
R
I
éq 4.1-14
The equation [éq 4.1-2] can thus be put in the form, after discretization:
fs
= S
has
+ S
B
S
+ S
C
S
R, N
R, N
R, N
N
I, N
n+1
éq
4.1-15
fs
S
S
S
S
S
= + B has + C
I, N
I, N
I, N
N
I, N n+1
With:
S
X X
1
E
tn -
2
E
tn
1 tn
2 tn
1
2
fs
E
-
E
has
=
fs
R, N
-
1
- X
R, N
1
max (I, K)
X X - 1
1
2
X X - 1
1
2
kN
H
S
- X X
1
E
tn +
2
E
tn
H
1 tn
2 tn
N
1
2
N
E
-
E
B
=
R, N
+
1 +
X
1
éq
4.1-16
K Sr
X X - 1
S
1
2
ki
X X - 1
1
2
H
S
- X X
1
E
tn +
2
E
tn
H
1 tn
2 tn
N
1
2
N
E
-
E
C =
R, N
+
1 +
X
1
S
S
Kr
X X - 1
1
2
ki
X X - 1
1
2
1
S
E
tn -
2
E
tn
fs
X X
2
E
tn -
1
E
tn
has
= X
fs
I, N
2
-
R, N
1 2
-
1 max (I, K)
X X - 1
1
2
X X - 1
1
2
kN
H
2
S
N
E
tn -
1
E
tn H - X X
2
E
tn +
1
E
tn
B =
X
N
I, N
2
+
1 2
+
1
éq
4.1-17
K Sr
X X - 1
S
1
2
ki
X X - 1
1
2
H
2
S
N
E
tn -
1
E
tn H - X X
2
E
tn +
1
E
tn
C =
X
N
I, N
2
+
1 2
+
1
S
S
Kr
X X - 1
1
2
ki
X X - 1
1
2
In the equations [éq 4.1-16] and [éq 4.1-17] the parameters, X and X are a function of
1
2
1
2
intrinsic parameters of material. With each step of calculation, it is necessary to back up two
variables intern fs
, last reversible spherical deformation obtained and max (fs
, i.e.
I, K)
R, N
kN
fs
, greatest reversible spherical deformation obtained in the history of the element. The choice of
I, N
to retain the expressions [éq 4.1-5] and [éq 4.1-6] (not of deformation unrecoverable), or the expressions
[éq 4.1-16] and [éq 4.1-17] (existence of unrecoverable deformations) to determine the increment of
total spherical deformation is carried out a posteriori according to the sign of
fs
in [éq 4.1-15].
I, N
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:
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1
3
1
2
4
1
1
2
4
3
3
4
2
Illustration of the numerical answers obtained by using the discretized expressions [éq 4.1-3] to
[éq 4.1-17] for four stories of loading: 1 level of unit constraint to moisture
constant (100%), 2 level of unit constraint to linearly decreasing moisture of 100% with
50%, 3 level of unit constraint during half of the duration of the calculation followed by a recouvrance
with half of the initial constraint on the second part of calculation; moisture is supposed to be constant
(100%), 4 the mechanical loading is identical to 3; moisture decrease linearly of 100% to 50%.
Appear 4.1-a
To carry out simulations of [Figure 4.1-a] the following parameters were retained:
K S =,
2 0e + 5 [MPa]; S
=,
4 0e +10 [MPa.s]; K S =,
1 0e + 4 [MPa]; S
=,
1 0e +11 [MPa.s].
R
R
I
I
calculation comprises 200 intervals of 5000 [S].
4.2
Discretization of the equations constitutive of creep deviatoric
After discretization of the constraints and relative humidity by functions closely connected, the tensor
diverter of the deformations of clean creep is discretized by the following equation:
fd
D
D
D
D
D
= + N has
B + N
C
éq
4.2-1
N
N
N
N 1
+
where D
and D
are the tensors of the deviatoric constraints at the beginning and the end of the step of time
N
N 1
+
running.
The stages carried out are:
·
One calculates the parameters compared to the deformation of clean creep deviatoric
reversible, whose model is:
D
fd
& (T) + kd fd
(T) = H (T) D
(T)
R
R
éq
4.2-2
R
R
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After discretization, the preceding equation can be put in the form:
fd
D
D
D
D
D
= has
+ R
B, N + R
C, N
éq
4.2-3
R, N
R, N
N
N 1
+
With:
D
T
has
=
exp -
N -
1 F, D
R, N
D
,
R
R N
D
1
D
2
D
T D
2
D
-
R
R
N
R
R
tn
B
=
R, N
-
+
1 H +
N
N
H +1
exp -
+
-
1 H -
N
N
H +1 éq 4.2-4
D
D
K
T
T
T
T
R
N
N
R
N
N
D
D
1
D
T
-
C
=
R
exp -
N H - R
tn
R, N
N
N
H
D
D
K
T
T
R
N
R
N
Note:
The equation [éq 4.2-4] (left reversible creep deviatoric) is similar to the equation
[éq 4.1-5] (left reversible creep in the absence of unrecoverable deformations). They
correspond to the discretization of a single chain of Kelvin.
One calculates the parameters compared to the deformation of clean creep deviatoric, of which the model
is:
D
F, D
& (T) = H (T) D (T)
I
éq
4.2-5
I
After discretization, the preceding equation can be put in the form:
F, D
D
D
D
D
D
= has + ib, N + ic, N
éq
4.2-6
I, N 1
+
I, N
N
N 1
+
With:
D
has
=
0
I, N
D
T H
B
=
N
n+1
R, N
éq
4.2-7
D
2 I
D
T H
C
=
N
N
R, N
D
2 I
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:
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1
1
3
2
2
4
1
1
3
2
3
4
2
4
Illustration of the numerical answers obtained by using the discretized expressions [éq 4.2-1] to
[éq 4.2-7] for four stories of loading: 1 level of unit constraint to constant moisture
(100%), 2 level of unit constraint to linearly decreasing moisture of 100% to 50%, 3
unit level of constraint during half of the duration of the calculation followed by a recouvrance to
half of the initial constraint on the second part of calculation; moisture is supposed to be constant
(100%), 4 the mechanical loading is identical to 3; moisture varies linearly decreasing
100% to 50%.
Appear 4.2-a
To carry out simulations of [Figure 4.2-a], the following parameters were retained:
K D =,
5 0e + 4 [MPa]; D
=,
1 0e +10 [MPa.s]; D
=,
1 0e +11 [MPa.s]. Calculation comprises 1000
R
R
I
intervals of 1000 [S].
5 Matrix
tangent
By introducing the elastic modulus of rigidity µ, the diverter of the constraints at the moment N + 1
is written according to the diverter elastic strain:
D
ED
D
D
F, D
= 2µ
= + 2µ
- 2µ
éq
5-1
N 1
+
N 1
+
N
N
N
In substituent the deviatoric part of the clean deformation of creep by the expression [éq 4.2-1], it
rise the following relation:
D
(1+2d
µc)
D
= (1 - 2D
B
µ) + 2
D
µ
- 2
D
µa 1 éq
5-2
N 1
+
N
N
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Expression which induces by derivation compared to D
:
N 1
+
DNN 1+ (1+2µ cd.) = 2 1µ éq 5-3
DNN 1+
By taking a similar step for the spherical part and by introducing the module of rigidity with
dilation K, it follows the three following relations:
E
tr
= 3K tr
= tr + 3K tr (
) - 3K tr (F
) éq
5-4
n+1
n+1
N
N
N
tr
(
S
1+ 3Kc) = tr (
S
1 - 3Kb) + 3K tr ()
S
- Ka éq
5-5
n+1
N
N
(tr
)
+
éq 5-6
(
N 1
S
1+ 3
= 3
tr
) (Kc) K
N 1
+
The tangent matrix is written finally:
D
D
D
1 (tr)
1 (tr) (tr)
=
+
1 =
+
éq
5-7
3
D
(tr)
1
3
I.e.:
2µ
1
=
K
1 - 11 +
11
éq
5-8
1+ 2µ D
C
3
1+ 3 S
Kc
4
1 4
2 3
4
1 4
2 3
After linearization, the tangent matrix develops as follows:
2
1
1
+ - - 0 0 0
11
3
3
3
11
1
2
1
22 - + - 0 0 0 22
3
3
3
33
=
1
1
2
33
éq
5-9
2
0 0 0
12 -
-
+
2
3
3
3
12
2
0
0
0
0
0
13
213
2
0
0
0
0
0
23
2 23
0
0
0
0
0
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6
Description of the internal variables
The following table gives the correspondence between the number of the internal variables accessible by
Code_Aster and their description:
Number of the variable
Description
1
Reversible spherical deformation
2
Irreversible spherical deformation
3
Reversible deviatoric deformation, component 11
4
Irreversible deviatoric deformation, component 11
5
Reversible deviatoric deformation, component 22
6
Irreversible deviatoric deformation, component 22
7
Reversible deviatoric deformation, component 33
8
Irreversible deviatoric deformation, component 33
9 -
10 -
11 -
12
Reversible deviatoric deformation, component 12
13
Irreversible deviatoric deformation, component 12
14
Reversible deviatoric deformation, component 13
15
Irreversible deviatoric deformation, component 13
16
Reversible deviatoric deformation, component 23
17
Irreversible deviatoric deformation, component 23
18 -
19 -
20 -
7 Notations
tensor of the total deflections
F
tensor of the deformations of clean creep
E
tensor of the elastic strain
fs
1 spherical part of the tensor of the deformations of clean creep
fs
1 reversible spherical part of the tensor of the deformations of clean creep
R
fs
1 irreversible spherical part of the tensor of the deformations of clean creep
I
fd
deviatoric part of the tensor of the deformations of clean creep
fd
reversible deviatoric part of the tensor of the deformations of clean creep (contribution of water
R
absorptive)
fd
irreversible deviatoric part of the tensor of the deformations of clean creep (contribution of
I
interstitial water)
tensor of the total constraints
S
1 spherical part of the tensor of the constraints
D
deviatoric part of the tensor of the constraints
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H relative humidity internal
K modulates elastic rigidity with dilation
S
Kr rigidity connects associated with the skeleton formed by blocks with hydrates on a mesoscopic scale
S
ki rigidity connects intrinsically associated with the hydrates on a microscopic scale
D
Kr rigidity associated with the capacity with water adsorbed to transmit loads (load bearing toilets)
µ elastic modulus of rigidity
S
I viscosity connects associated with the mechanism with diffusion interlamellaire
S
R viscosity connect associated with the mechanism with diffusion within capillary porosity
D
I viscosity of interstitial water.
D
R viscosity associated with the water adsorbed by the layers with hydrates
X, X, X respectively indicate a scalar, a vector and a tensor of command 2.
N
X, N
X
+,
1
N
X indicate respectively the value of quantity X at time tn, at time tn 1
+ and
variation of X during the interval [T;
]
N tn 1
+.
8 Bibliography
[1]
BENBOUDJEMA F.: Modeling of the deformations differed from the concrete under stresses
biaxial. Application to the buildings engines of nuclear thermal power stations, Mémoire of D.E.A.
Materials Avancés Ingénierie of Structures and Enveloppes, 38 p. (+ appendices) (1999).
[2]
BENBOUDJEMA F., MEFTAH F., HEINFLING G., POPE Y.: Numerical study and
analytical of the spherical part of the clean model of creep UMLV for the concrete, notes
technique HT-25/02/040/A, 56 p (2002).
[3]
BENBOUDJEMA F., MEFTAH F., TORRENTI J.M., POPE Y.: Algorithm of the model of
clean creep and of desiccation UMLV coupled to an elastic model, notes technical
HT-25/02/050/A, 68 p (2002).
[4]
GRANGER L.: Behavior differed from the concrete in the enclosures of nuclear thermal power station:
analyze and modeling, Thèse de Doctorat of the ENPC (1995).
[5]
RAZAKANAIVO A.: Relation of behavior of Granger for the clean creep of the concrete,
Documentation Code_Aster [R7.01.01], 16 p (2001).
Handbook of Référence
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:
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:
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: 16/16
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