Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
1/18
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Organization (S):
EDF-R & D/AMA, IRSN
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
Document: R7.01.12
Modeling of the thermo hydration, drying
and of the shrinking of the concrete
Summary
One describes two types of phenomena here occurring at periods distinct from the life of a concrete:
·
on the one hand a reaction of thermo hydration generating a withdrawal known as endogenous, appearing with
youth of the concrete (the first 100 days),
·
in addition an evaporation of part of the water not used in the process of hydration,
phenomenon called drying and involving a withdrawal of desiccation. This phenomenon can last,
according to dimensions of the structure of concrete implemented, a few months to several years.
These phenomena are modelized in Code_Aster in the form of equations of dissemination whose solution is
represented by new variables allowing to calculate the deformations of the endogenous withdrawal directly
(of with the hydration) and of the withdrawal of desiccation (of with drying).
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Count
matters
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
3/18
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
1 Introduction
The behavior of the concrete, fragile in extension, heterogeneous and porous material is governed by
many and complex physicochemical phenomena. Losses of prestressings induced by
behavior differed from the concrete (withdrawal and creep) reduce in the course of time the field of
loading which the structure can support. These differed deformations which appear in
concrete during the life of the aforementioned, are composed by the withdrawal at the youth (endogenous withdrawal
specific to the hydration and thermal withdrawal), by the withdrawal of desiccation with the modeling of
drying, and as soon as it undergoes stresses, by clean creep and the creep of desiccation.
In the rules of dimensioning, the deformations differed from the concrete are generally based
on empirical rules fixed on a great number of results resulting from the literature, fascinating in
count the main parameters, like the temperature, moisture, the content of aggregate,
proportion water/cement. The kinetics of the phenomena uses times equivalent calculated to aid
of a law of Arrhenius to take into account ageing and the temperature.
A fine analysis of the physicochemical phenomena which are at the origin of the various deformations
differed from the concrete allows to propose a modeling on the basis of model of the medium type
continuous equivalent [bib2], which was introduced into Code_Aster (clean creep and the creep of
desiccation are not treated here).
1.1 Phenomenologic aspects of the behavior of the concrete to the young person
age: the thermo hydration
One defines the youth as the first 100 days of the life of the concrete. Endogenous withdrawal or withdrawal
of hydration, and the thermal withdrawal intervene as of the first moments of the catch (at the youth),
for one duration going from a few hours to a few days, for the thermal withdrawal, and of some
month at one year, for the withdrawal of hydration, in general finished at the time of the setting in prestressing.
phenomena of prevented withdrawals or differential withdrawals, under formwork, can be at the origin of
stresses or of fissures which should be evaluated. In liquid phase, the concrete is a viscous fluid in
which the solid matter constituents are in suspension in the hydraulic binder containing of the solid particles
(cements…). Following the formation of the first hydrates, the catch of the concrete intervenes, ten
hours after its manufacture, which corresponds to the establishment of related bridges hydrates between
cement grains in the totality of material. With the whole beginning, the grains are relatively dispersed
in mixing water. In the course of time, the hydration of the cement grains is accompanied by one
consumption of this mixing water. In experiments, it is noted that the voluminal assessment of
reaction is negative; it is the contraction of Chatelier. Known as simply, the total volume of the hydrates
is lower of almost 10% than the total volume of these components. Mechanically, on a scale
cement grains, the phenomenon stops when the bridges of hydrates formed between the grains are
sufficient rigid to prevent a possible relative bringing together of the grains. Consequences
macroscopic on the works are practically non-existent since in all the duration of this
phase, the concrete is still deformable, and that any contraction is compensated by a readjustment
granular of material against the walls of the formwork. Although of relatively weak width, and effect
mechanics insufficient to generate a real cracking of the concrete, stresses generated with
the interface of two consecutive liftings can start of 50% the margin of resistance in traction of
material.
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
4/18
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
The catch of the concrete accompanied by the hydration of cement involves an exothermic reaction. In
massive structures the temperature can then rise with more 50°C. The hydration is one
activated thermo reaction, i.e. the speed of hydration increases with the temperature.
When the speed of hydration decrease, the temperature decreases, involving a thermal withdrawal. Moreover
the mechanical properties of the concrete vary according to its degree of hydration, and finally
water consumption occurring during the hydration involves a capillary withdrawal. The different ones
withdrawals can cause stresses much higher than resistance (weak) in traction of the concrete
and to bring to a cracking of material.
The calculation of the fields of temperature and degree of hydration is available with the control
THER_NON_LINE
(cf [U4.30.02]). The calculation of the mechanical fields taking of account withdrawal
endogenous is carried out with the control
STAT_NON_LINE
.
1.2
Drying and withdrawal of desiccation
To modelize drying is important owing to the fact that the physicochemical and mechanical properties of
material are strongly dependant on moisture inside this last. The objective is to propose
a macroscopic modeling of the drying of the concrete starting from a restricted number of parameters,
easily measurable in experiments, starting from a law of transitory dissemination nonlinear of
the moisture, chained at the temperature, while freeing itself from the mechanical complexes couplings,
physics and chemical, on a material scale.
To the dismantling, the concrete is plunged in an external environment which presents a degree in general
of moisture of about 60 to 80% HR (relative humidity = report/ratio of the steam pressure on
steam pressure saturating for a given temperature). It undergoes a true hydrous shock then
(by analogy with a thermal shock). The concrete is then in thermodynamic unbalance with
atmosphere. Drying will enable him to find a hydrous balance with the external medium.
Physically, drying brings into play complex phenomena closely coupled the ones with
others, depend on the heterogeneous and granular structure of the concrete. With the macroscopic scale, it is
possible [bib2] to modelize drying like a nonlinear phenomenon of dissemination, with dissemination
in liquid phase of Darcy type, as long as there is continuity of the liquid phase, and with dissemination in phase
gas of Fick type, for the water vapor.
The withdrawal of desiccation is the macroscopic consequence first of the drying of the concrete. It is
direct prolongation of the phenomena of capillary voltage which are at the origin of the endogenous withdrawal. By
its intensity, deformations being about 400.10
6
to 800.10
6
for 50% of hygroscopy and for
current concretes, it is of one to three times the more important than the elastic strain for one
loading close to 10 MPa.
One initially presents the modeling of the thermo hydration in the operator of
nonlinear thermics of Code_Aster, then the modeling of drying, and finally, the introduction of
endogenous withdrawal and of the withdrawal of desiccation in the nonlinear operator of mechanics.
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
5/18
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
2
Formalization of the thermo hydration
2.1
Equation of the thermo hydration
The modeling of the phenomenon of thermo hydration consists in enriching the equation by the heat of
following way:
C dT
dt
div
Q D T
dt
S
grad T
p
+
=
+
= -
Q
Q
()
éq
2.1-1
where
Q
is the heat of hydration per degree of hydration
(presumedly constant in J/m
3
),
S
is one
source interns heat (J/m
3
S),
C
p
voluminal heat with pressure constant (J/m
3
°K) and
conductivity thermal (W/m
2
°K), these two last quantities being independent of the temperature
T (°K).
The law of evolution of the degree of hydration is given by:
D
dt
With
E
RT
has
=
-
() exp (
)
éq
2.1-2
with
E
R
with the constant of Arrhenius (variable rather empirical parameter between 4000 and 7000°K, and being
regarded as being equal to 4000°K in the absence of additional information [bib2]),
temperature T being expressed in Kelvin degree.
With (
) is a function depending on the degree of hydration and the composition of the concrete, given with
the aid of a calorimetric test. Determination of function A (
) is done starting from the data of
the calorimetric test. The test being adiabatic, the only data of the change of the temperature is enough
to determine function A (
). Heat of hydration
Q
is determined by the difference of
temperature of the sample at the end and the beginning of the test:
Q
C T
T
p
=
-
(
)
0
The originality of the processing compared to a conventional thermal calculation is the joint presence of one
term of source depending on the temperature (equation [éq 2.1-1]) and on a law of evolution of one
parameter (in fact the degree of hydration
) (equation [éq 2.1-2]).
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
6/18
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
2.2 Exploitation of the calorimetric test for the determination
function A (
)
Function A (
) must be provided in data by the user. The goal of this paragraph is to show
how one can exploit a test quasi-adiabat to determine A (
). A concrete sample
expenses is plunged in a calorimeter and one measures the change of the temperature in the course of time
until hardening of the concrete. The test being adiabatic,
div Q
= 0
, and one has then:
()
(
()
)
T
C T
T
T
Q
p
AD
=
-
0
, i.e.
()
()
T
T
T
T
T
T
AD
AD
=
-
-
0
0
, where
T
AD
is the final temperature of
the adiabatic test,
T
the 0 initial temperature (one makes the assumption that (0) =0). One determines then
required function:
With
T
T
dT
T
dt
E
RT
T
AD
AD
has
AD
()
() exp (
())
=
-
1
0
.
When
AD
T
T
=
,
1
=
.
In fact, one can generally define the degree of hydration in
each moment T as being the report/ratio of the quantity of heat released until the moment T on
quantity of total heat released at the end of the process of hydration:
)
(
))
(
(
)
(
=
T
Q
T
T
Q
T
.
3
Discretization of the problem of thermo hydration
3.1
Choice of the method of resolution
The selected method consists in solving overall the nonlinear equation [éq 2.1-1] while putting at
profit the nonlinear algorithm of thermics of Code_Aster and to locally solve the equation [éq 2.1-
2] which represents the law of evolution of a kind of variable interns representing the degree of hydration,
this law expressing itself by a function of the thermal state of the system. Indeed, there is no operator
differential spaces some for the variable
in the equations and thus not need for finite element.
relation [éq 2.1-2] represents a local law as in plasticity. The same number then is preserved
degrees of freedom that for conventional thermics. Such an uncoupled process involves nevertheless
the calculation of the same quantities several times. Indeed, let us suppose that
that is to say discretized with the nodes
elements. Let us consider the example schematized by [Figure 3.1-a].
1
3
2
4
1
Appear 3.1-a
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
On node 1, the equation of evolution [éq 2.1-2] will be integrated four times. A possible solution would have
summer that local calculations can be done on fields with the nodes (Aster concept
CHAM_NO
) and
not on fields of nodes by element (concept Aster
CHAM_ELEM
, option
ELNO
), which is
currently impossible.
The solution which was finally adopted, consists in calculating
at the points of gauss of the element, it
who is all the more natural as for mechanical calculation the Young modulus depends explicitly
of
. This generates nevertheless much local calculations except strongly under-integrating the element
finished. For example, if one considers a mesh comprising NR hexahedral elements with 20 nodes, it
exist about 4N nodes and 27N points of Gauss.
3.2
Algorithm of resolution
By taking again the notations of documentation Aster [bib5], the weak formulation of the equation
[éq 2.1-1] is written in the following way:
)
'
1
(
,
.
.
)
(
.
.
)
(
).
(
*
*
*
*
*
*
T
D
T
D
T
E
QA
D
T
S
D
T
T
T
D
T
T
RT
E
has
+
+
=
+
-
&
The development of the thermo hydration within the general algorithm of thermics not
linear in Code_Aster thus consists in discretizing in an explicit way in the second member
the term
-
D
T
E
QA
RT
E
has
*
.
)
(
. While noting respectively
-
-
+
+
,
,
,
T
T
, variables of hydration
and of temperature at the beginning and the end of the pitch of time, one calculates in each point of Gauss
quantity
-
-
-
RT
E
has
E
QA
)
(
who is integrated directly in the second member. After each resolution
pitch running, the variables is reactualized
(
,
)
+
-
+
-
=
=
T
T
. The test of convergence is not
credit that on the temperature, the variable
not entering the iterative process of Newton used
in nonlinear thermics. The taking into account of the hydration is in fact only the taking into account
of a heat source known at the beginning of the pitch of time. This purely explicit discretization
thus require to use pitches of sufficiently small times.
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
4
Formalization of drying
This part refers to the document of specification of the development of drying in
Code_Aster [bib3], like with the thesis of L. Granger [bib2].
4.1
Modeling and equations of drying
Modelings of thermics or the thermo hydration and drying are uncoupled at the time of
the resolution. Drying is then presented like an operation chained at thermics. Like
equations making it possible to solve drying and thermal nonlinear are similar for it to
coefficients close, this decoupling makes it possible to integrate the resolution of the calculation of drying in
Code_Aster, by directly using the module of resolution of nonlinear thermics, without adding
new phenomena, new types of elements nor new options of calculation, and in
thus minimizing the volume of added and duplicated code.
The concentration or water content, variable of calculation in the modeling of drying, is comparable,
in term of the type of variable, at a temperature (standard
TEMP
). The transitory field of temperature,
intervening in the equation of drying, is only one auxiliary parameter on which depends
possibly the coefficient of dissemination.
Phenomena of thermics and drying, within the framework of a modeling uncoupled between
thermics and drying, is governed by the following equations:
·
equation of “conventional” thermics:
()
()
C dT
dt
div
Q D T
dt
S T
T grad T
p
+
=
+
= -
Q
Q
()
éq
4.1-1
(
C
p
voluminal heat with constant pressure,
, thermal conductivity,
Q
heat
of hydration per degree of hydration
and S the second member).
·
equation characterizing drying:
(
)
[
]
C
T
Div D C T
C
-
=
,
0
éq
4.1-2
where
C
(m
3
/m
3
or L/m
3
) is the variable of calculation (concentration or water content
voluminal),
T
is the variable of input of calculation (the temperature), variable auxiliary of the resolution
drying,
D
(m
2
/S) is a coefficient of dissemination, characterizing nonthe linearity of the equation, and
depending at the same time on the variable on calculation, C, and auxiliary variable,
T
. This law
of dissemination is given in various forms, according to the model selected, (law of Bazant,
law of Mensi, cf [§4.3] and [bib2]).
The equations [éq 4.1-1] and [éq 4.1-2] correspond to a thermal chained calculation/drying. One can
thus to calculate
T
without knowing the water concentration, then to calculate the latter, for which
T
is then a parameter, (by making the assumption that thermal conductivity
does not depend on
water concentration
C
). Also let us note that the phenomenon of drying is uncoupled from the evolutions
mechanics of the concrete.
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
4.2
Coefficient of dissemination
The material is described by the coefficient of dissemination
D
, characteristic of material, dependant with
time of the temperature
T
and of the water concentration
C
. The equation of the migration of moisture
in the concrete is resulting from those of the mechanics of the porous environments. One will refer to [bib2] for
more precision. Classically, a law of dissemination expresses a flow like the product of one
size characteristic of material by the gradient of an intensive size. The different ones
sizes considered are defined by an average on representative elementary volume, for
as far as one can define this average for material considered, so that operators
of derivation a direction has. One thus makes in general the assumption which consists in supposing that them
phases liquid and gas are related:
·
for the dissemination of the vapor, one leaves the positivity of dissipation associated with transport
gas phase, by differentiating two phenomena, a phenomenon of permeation type
(Darcy), related to gradients of pressure, and a phenomenon of type dissemination (Fick), related to
gradients of concentration,
·
for the dissemination of liquid water, the positivity of dissipation associated with transport with water
fluid, and the law of Darcy, makes it possible to express the flow of fluid according to the pressure of
fluid. The law of Kelvin describing the coexistence of the two phases liquid and gas by
the writing of the equality of the mass free enthali leads to the expression of flow in function
gradient of the percentage of moisture.
From the two preceding results, one obtains the expression of total flow according to the gradient of the degree of
water concentration. Conventional experimental methods in the problems of drying
generally access to the water concentration gives, and very seldom with the relative humidity. It is
thus preferable to express flow according to the water content, while using classically
the isotherm of desorption of the concrete, which connects the water content,
C
, and relative humidity,
H
. Moisture
relative is the relationship between steam pressure and saturating steam pressure.
The postulate of the local state stipulates that the current state of a homogeneous system in unspecified evolution
can be characterized by the same variables as with balance, and than it is independent speeds
of evolution. In other words, water content
C
, and relative humidity
H
, are well connected by
even relation that with balance. What leads to the conventional equation of the dissemination:
(
)
[
]
C
T
Div D C T
C
-
=
,
0
éq
4.2-1
This equation highlights the nonlinear character of the dissemination of moisture in the concrete.
In the industrial cases, the temperature is in general not uniform in the structure. It is thus
necessary to take into account a coefficient of dissemination of the moisture which depends on
temperature. In practice, in the literature, the most known authors (Bazant cf [bib2]) propose
an expression of the coefficient of dissemination of the type:
(
)
()
D C T
D C T
T
T E
Qs
R
T T
,
,
=
-
-
0
0
1
1
0
éq
4.2-2
with Q
S
/R = 4700 K
- 1
and T in °K
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
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Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
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HT-66/05/002/A
Note:
Way in which the things are presented, it would seem that one did not use the fact that
drying is a phenomenon coupled with the mechanics, (i.e. it is the cause of one
withdrawal of desiccation). Actually, we made the assumption of a decoupling of
phenomena, when we used the curve of sorption/desorption. In fact, at the time of
measure loss in weight with balance according to H, the body of test carries out a withdrawal.
At the microscopic level, all occurs like if the withdrawal, modifying porosity, went
to interact on the relative hygroscopy inside the sample, since pressure of
vapor and H increase. This withdrawal of desiccation being very weak, it is usual of
to neglect in calculations of the water content. There is thus only one chaining between the calculation of
water content and the mechanical calculation of withdrawal of desiccation.
4.3
Usual laws of dissemination
The law of dissemination, function of the two parameters,
C
and
T
, can be freely defined by the user
in the form of a tablecloth. However, usual expressions of the law of dissemination, which one finds
in the literature are as follows:
Law suggested by Granger:
(
)
(
)
D C T
With E
T
T E
B C
Q
R
T T
S
,
.
.
=
-
-
0
1 1
0
éq
4.3-1
With (m
2
/S), B, T
0
, Q
S
, and R (Qs/R in °K) are coefficients chosen by the user.
D
is a function of
temperature and of the water concentration.
Law of Mensi:
()
(
)
D C
With E
B C
=.
.
éq
4.3-2
With
and
B
are coefficients chosen by the user.
D
is a function only of the concentration
out of water.
Law of Bazant:
The law of Bazant is expressed starting from the percentage of moisture
H
, which is connected to the water concentration by
curve of sorption/desorption. The form of this law is as follows:
()
-
-
+
-
+
=
N
C
H
D
H
D
75
.
0
1
)
(
1
1
1
1
éq
4.3-3
Usually,
D
1
= 3.10
- 10
m
2
/S
lies between 0.025 and 0.1,
N
is about 6.
()
H C
is the percentage of moisture, which is expressed according to the water concentration with
the aid of the curve of sorption/desorption.
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
11/18
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
The curve of sorption/desorption can be introduced in the form of a tabulée standard function,
knowing that actually, this curve has a hysteresis, but can be regarded as being
invertible, if one takes account only of one direction of course.
4.4
Modeling of the boundary conditions
The boundary conditions are expressed in general by a nonlinear relation between the flow of
water concentration (L/m
3
X ms
1
)
W
fl
and water concentration. These conditions are thus
analogues in the conditions known as of exchange in thermics. One will be able for example to use the formula
proposed by L. Granger [bib2] page 181:
Its expression is as follows:
(
)
(
)
[
]
(
)
W
C
C
C
C
C
C C
fl
eq
eq
eq
=
-
-
-
-
0 5
2
0
2
0
.
.
éq
4.4-1
where
C
eq
is the water concentration for a moisture of 50%HR,
C
0
is the water concentration for a moisture of 100%HR,
(L/m
3
X m/s) is a coefficient, which can be defined in experiments and can evolve/move according to
the cracking of the heat-transferring surface ([bib2]),
and
C
is the current concentration (unknown) on the heat-transferring surfaces.
5
Integration of drying in Code_Aster
These developments relate to the axisymmetric elements 2D and elements, as well as
elements 3D isoparametric, of a number of nodes unspecified, linear and quadratic.
5.1 Introduction of the concept of behavior into the operator of
nonlinear thermics
The operator
THER_NON_LINE
was reserved exclusively for the nonlinear thermics, which will remain
the option of calculation per defect. But one uses the same module of resolution to solve them
problems of drying and hydration, because of analogy of the equations.
The concept of behavior was added in the nonlinear operator of thermics, with one
nomenclature and a syntax analogues with those of the operator of nonlinear mechanics. It
imply for drying a concept of entity topological, to which this behavior is applied.
This can be useful, when there are several types of possible laws of dissemination, or when one wants to make
a purely thermal calculation on part of the mesh, whereas on another part one does one
calculation of thermo hydration (on the other hand, the simultaneous use on the same mesh of
behaviors of the drying type, and behaviors of the thermal type or hydration would not have
direction).
A behavior “drying” is associated each law of dissemination, such as one can find them
in the literature, just as a specific material is associated each law of dissemination,
to define the characteristic coefficients of them. The resolution of drying is identical, with
coefficients close, with that of nonlinear thermics, and no amendment was made to
the algorithm of resolution.
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
12/18
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
For drying, four distinct behaviors are defined under the key words
“SECH_GRANGER”
,
“SECH_BAZANT”
,
“SECH_MENSI”
, or
“SECH_NAPPE”
, to characterize each law of
possible dissemination. They can be allotted to parts complementary to the mesh, during one
even calculation. The simultaneous definition several behaviors “drying” associated with entities
topological different requires several occurrences of the key word
“BEHAVIOR”
. Then, entity
topological will have to be identified by informing one of the key words
GROUP_MA
or
NET
.
In parallel of the four behaviors “drying”, in the operator
DEFI_MATERIAU,
four
materials initially make it possible to define the values of the coefficients of the laws of
dissemination, nonlinear functions of the water content and the temperature. The user can choose
(or them) law (S) of its choice, and defines the value which it wishes for each one of these coefficients.
The key word
SECH_GRANGER
allows to define the law of dissemination of liquid and gas water under its
form most conventional among the expressions of the literature. Four coefficients like one
temperature of reference T
0
characterize this law.
Key words
SECH_MENSI
and
SECH_BAZANT
allow to define the laws of Mensi and Bazant, with
the aid of the coefficients which are appropriate. The law of Bazant, expressing itself starting from the percentage of moisture,
require to define a curve of desorption allowing to convert the water content into degree
of moisture within the framework of this modeling.
Lastly, the key word
SECH_NAPPE
allows to use a law of dissemination, starting from a function tabulée of
two variables, which will be interpolated in calculations starting from the values of the water concentration and
temperature. This last possibility presents the disadvantage of not raising ambiguity enters
these two variables associated with an identical type,
“TEMP”
.
It is necessary, for drying, to introduce in input of calculation a concept of the type [
evol_ther
],
representing the evolution of the field of temperature of the concrete structure, within the framework of a calculation
chained thermal/drying. Indeed, the calculation of drying requires the preliminary calculation of
temperature and possibly of the hydration, because the coefficient of dissemination
(
)
D C T
,
depends on
temperature.
5.2
Implementation of the boundary conditions for drying
5.2.1 Expression of the boundary conditions
The boundary conditions are expressed in the form of flow of moisture on surfaces in contact with
external medium following the expression [éq 4.4-1].
5.2.2 Delimitation of the calculation of drying using the boundary conditions
The calculation of drying is defined on the totality of the mesh where finite elements are affected. For
to make effective the calculation of drying that on a portion of the mesh (this with an aim of preserving it
even model for calculations of drying and mechanical calculations and to facilitate them
“continuations” of calculation Aster [bib4]), one will use the boundary conditions. Indeed, drying does not take place
that if there is exchange with outside. It is thus the attribution of the boundary conditions which allows
“to locate” calculation. The absence of drying on a portion of the structure will be expressed by the absence of
boundary conditions on the heat-transferring surfaces concerned.
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
13/18
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
5.2.3 Implementation in Aster
The boundary conditions can be defined, as in thermics, in the form of normal flow not
linear formulated starting from a function tabulée of the variable of calculation, and interpolated during calculations.
That makes it possible to avoid creating new options of calculation, similar to the options of thermics not
linear
char_ther_flunl
and
resi_ther_fluxnl
who calculate the first and the second member, and
who can be used directly for drying. It is then enough to choose a tabulée function
corresponding to the expression of flow, given by the equation [éq 4.4-1].
Using a preset function (
FORMULATE
), the expression of flow, given in polynomial form and
function of the variable of calculation, is transformed into tabulée curve, via the operators
Aster (
CALC_FONC_INTERP)
. One thus does not create a new option of calculation for the processing of
boundary conditions.
The calculation of new options would have the advantage of being optimal in term of result (because of absence
interpolations and because of “exact” calculations of derivative), but would require to develop two
new options of calculation, similar to the options
char_ther_flunl
and
resi_ther_fluxnl
.
5.2.4 Example of working of the boundary conditions
The sequence of controls, described in the example which follows and whose numerical values are
fictitious, will be adopted, for the creation of a boundary condition
charsech
on a group of meshs
limit
.
Note:
“FORMULA”
Aster is the numerical expression of flow of the normal water concentration which
the equation [éq 4.4-1] begins again.
beta
=
DEFI_VALEUR
(R8:
0.25)
c_0
=
DEFI_VALEUR
(R8:
0.30)
c_eq = DEFI_VALEUR
(R8:
0.70)
!FORMULATE (REAL: (
fon_humi
(REALITY: temp) =
(0.5 * beta
/
((c_0
-
c_eq) ** 2)
* (temp - (2 * c_0 - c_eq))* (temp - c_eq)) ));
pas0
=
DEFI_VALEUR
(R8: EVAL (1. /20. ));
LIST0 = DEFI_LIST_REEL (BEGINNING: 0.0
INTERVAL: (JUSQU_A: 1.0
NOT: pas0)) ;
flu_humi = CALC_FONC_INTERP (
FUNCTION:
fon_humi
LIST_PARA
:
list0
PROL_GAUCHE:
“EXCLUDED”
PROL_DROIT
:
“EXCLUDED”
Interpol
:
“LINE”
TITRATE
:
'flow
of humidité');
charsech = AFFE_CHAR_THER_F (MODEL:
modether
FLUX_NL
:
(
GROUP_MA
: limit
FLUN
:
flu_humi)) ;
Note:
It is important that the interpreted function and the tabulée function do not bear the same name,
so that the interpolations on the right and on the left are suitably defined, because them
exclusions on the right and on the left “do not overload” not the prolongations of a function
interpreted, transformed using the operator
CALC_FONC_INTERP
.
In the example above, the tabulée function fon_humi is defined outside the interval [0.0,
1.0] but it is not defined apart from the interval in the interpreted function flu_humi.
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
14/18
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
5.3
Numerical integration of drying
The equation of heat
(
)
()
C dT
dt
Div grad T
S T
p
-
=
or
()
[
]
()
&
-
=
Div
T
T
S T
conduit,
in the case of a boundary condition in normal flow on the border
with the variational formulation:
(
)
(
)
(
)
(
)
T
T
D
T
T
D
S T
D
T
T
N
D
.
.
.
.
.
+
=
+
éq
5.3-1
In a similar way, the equation governing drying
(
)
[
]
C
T
Div D C T
C
-
=
,
0
conduit, in the case
of a boundary condition in normal flow on the border
with the variational formulation:
(
)
(
)
C
T
D
D C T
C
D
D C T
C
N
D
.
,
.
.
,
.
+
= +
0
éq
5.3-2
The resolution of drying is integrated into the operator
THER_NON_LINE
, while replacing
p
C
by
constant function equalizes with the identity, and conductivity by the dissemination
(
)
D C T
,
, the temperature
intervening like a constant in calculations (auxiliary variable). According to the law of dissemination
chosen, it is necessary to calculate the value of the coefficient of dissemination like its derivative, according to the temperature
and water concentration at the moment running, the current point.
One will refer to the documentation of the nonlinear operator of thermics [R5.02.02] for moreover
full details on the numerical integration of nonlinear thermics.
Within the framework of drying, the boundary conditions are given in term of normal flow, and
lead, as in thermics, with a term in the first member, associated the option of calculation
rigi_ther_fluxnl
, and in the term in the second member, associated the option
char_ther_fluxnl
.
6
Formalization of the endogenous withdrawal and desiccation
6.1
Withdrawal in Code_Aster
In the framework of a formalization of the withdrawal in term of deformation, the total increment of deformation
can break up thermal component all in all, of a component representing it
endogenous withdrawal, and of a component representing the withdrawal of desiccation, added to
mechanical component (elasticity, creep,…).
One can modelize the withdrawal of desiccation in the form:
(
)
D
one
dessiccati
I
C
C
.
0
-
-
=
éq
6.1-1
where
C
is the water concentration,
0
C
initial water concentration.
and
a coefficient characterizing the withdrawal, depending mainly on the water concentration.
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
15/18
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
One can modelize the endogenous withdrawal in the form:
D
endogenous
I.
.
-
=
éq
6.1-2
where
is the hydration,
and
a coefficient characteristic of the material whose dependences are badly known.
The withdrawals of desiccation and endogenous can thus intervene in any law of
behavior by replacing the usual terms there
- T I
D
.
by
-
-
-
T I
D
desiccation
endogenous
.
. In the current state of Code_Aster, this is done only for
linear elasticity and for the behaviors elastoplastic of Von Mises type with work hardening
linear or kinematic. One has then for example in elasticity 1D:
()
(
)
=
+
-
-
1
E
T
C
.
.
éq
6.1-3
Mechanical parameters
E
(Young modulus) and
(thermal dilation) depend
mainly of the variable of hydration
.
This formulation of the withdrawal of desiccation and the endogenous withdrawal has the advantage of using directly
water content
C
, that one can connect to the loss of weight by simple integration on volume. If one
used the relative humidity
H
, it would have to be retranslated in term of water content by the means of
the isotherm of desorption of each various concrete.
For Code_Aster, these parameters can be defined within a relatively general framework, like
functions of the various variables of calculation and variables auxiliary (temperature, hydration,
water concentration, or constants) to leave the choice to the user to define them freely
dependences of the parameters. It remains with the load of the user to use the functions of Code_Aster
to reproduce the expression of the Young modulus given in the equation [éq 6.1-3].
For more detail on these formulations, and the means of calculating the coefficients
and
, one
will defer to the thesis of L. Granger, [bib2], on pages 99 and following, and pages 210 and following.
For mechanical calculation variables
(the hydration) and
C
(water concentration) are
data, like the east the temperature during a thermomechanical calculation.
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
16/18
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
6.2
Integration of the withdrawal in the law of behavior mechanics
Thermics and drying are uncoupled from the mechanical resolution, just like drying is
an operation chained with thermics and the hydration. This decoupling makes it possible to integrate the withdrawal
in the operator of resolution of nonlinear mechanics, without adding new phenomena,
behaviors, types of elements and options of calculation. Moreover, it makes it possible to introduce the withdrawal in way
simple in all the nonlinear laws of behavior. The syntax of the operators of mechanics
STAT_NON_LINE
and
MECA_STATIQUE
is not modified.
In the current version of the nonlinear operator of mechanics, the withdrawal was integrated into
elastic behavior (
ELAS
), with the elastoplastic behavior of linear work hardening type and
kinematics (
VMIS_ISOT_TRAC
and
VMIS_ISOT_CINE
) and with the models specific to the concretes:
MAZARS, ENDO_ISOT_BETON, BETON_DOUBLE_DP, GRANGER, BETON_UMLV_FP, BAZANT_FD. It
consist in removing the terms of withdrawal to the total deflection, before the resolution of the equations
of balance at the points of Gauss, in the same way which is taken into account thermal dilation.
Coefficients
and characterizing the withdrawals endogenous and of desiccation are defined under the word
key
“ELAS_FO”
, like constants. Other mechanical characteristics, coefficient of
Poisson, modulus Young, thermal expansion factor can also be defined like
functions of the new variables
HYDR
and
SECH
, which was added to the catalogs of both
operators
DEFI_FONCTION
and
DEFI_NAPPE
.
Two new options of loading were added to the operator
AFFE_CHAR_MECA
, in order to
to define under the key words
SECH_CALCULEE
and
HYDR_CALCULEE
, the concepts results, of type
[
evol_ther
], resulting from a calculation from thermics nonlinear, or thermo hydration, and of a calculation of
drying. They correspond respectively to the thermo fields hydrous of type
“TEMP/HYDR”
, and with
field drying of the type
“TEMP”
, calculated previously with the mechanical resolution. These options
(as their syntax) are similar to the option
TEMP_CALCULEE
, which defines the thermal field.
They allow:
·
to calculate the withdrawals endogenous and of desiccation, if characteristics
material associated will have been before defined in
DEFI_MATERIAU
,
·
to interpolate the Young modulus, the Poisson's ratio, and the expansion factor
thermics, when those are functions of the variables hydration or drying.
Note:
In the presence of a field of drying, it is necessary to inform key word SECH_REF in
control AFFE_MATERIAU. This value defines the value of SECH for which withdrawal of
desiccation is null.
It is thus necessary to take care to be coherent with values SECH_CALCULE used (in particular at the moment
initial!).
6.3 Stamp
tangent
The calculation of the tangent matrices of the various laws of nonlinear behavior is not affected
by the addition of the endogenous withdrawal and withdrawal of desiccation, because one neglects the derivative compared to
variables of hydration and drying, of the terms of the equilibrium equations, just as are
usually neglected the derivative compared to the temperature of these same terms. These derivative
intervene with the second command.
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
17/18
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
7 Bibliography
[1]
J. PELLET, Instruction manual of Code_Aster. Document [U4.61.02], 24/07/96.
[2]
L. GRANGER: Behavior differed from the concrete in the chambers of nuclear thermal power station:
analyze and modeling. Thesis of Doctorate of the ENPC (February 1995).
[3]
B. CIREE: Specifications of the development of the drying of the concrete in Code_Aster.
Report/ratio CS IF DSFN/128EE1/RAP/98.044 Version 1.1
[4]
B. CIREE: Specifications of the development of the endogenous withdrawal and the withdrawal of desiccation
in Code_Aster. Report/ratio CS IF DSFN/128HJ1/RAP/98.088. Version 1.0
[5]
C. DURAND: Nonlinear thermics. Manual of Reference of Code_Aster. Document
[R5.02.02]
Code_Aster
®
Version
7.4
Titrate:
Modeling of the thermo hydration, the drying and the shrinking of the concrete
Date:
26/05/05
Author (S):
G. DEBRUYNE, B. CIREE
Key
:
R7.01.12-B
Page
:
18/18
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Intentionally white left page.