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Organization (S): EDF-R & D/AMA
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Document: R7.01.09
Law of behavior ENDO_ORTH_BETON
Summary:
This documentation presents the theoretical writing and the numerical integration of the law of behavior
ENDO_ORTH_BETON developed by [bib1], which describes the anisotropy induced by the damage in the concrete,
as well as the unilateral effects (different behavior in traction and compression). Validation of the model
compared to experimental results is also proposed in this document.
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Count
matters
1 Introduction ............................................................................................................................................ 4
1.1 Characteristics of the damage of the concrete ............................................................................ 4
1.2 Objectives of law ENDO_ORTH_BETON ............................................................................................ 4
2 Expression of the free energy .................................................................................................................. 6
2.1 Taking into account of refermeture of the fissures .............................................................................. 6
2.2 Introduction of the variables of damage ................................................................................ 6
2.3 Terminals of the damage .......................................................................................................... 7
2.4 Introduction of a blocked energy ................................................................................................. 8
2.5 Final expression of the free energy, the constraints and the associated thermodynamic forces
with the variables of damage .................................................................................................... 9
3 Law of evolution of the variables of damage ................................................................................ 11
3.1 Law of evolution ................................................................................................................................ 11
3.2 Function threshold depend on the deformation .................................................................................. 13
4 Study of the parameters .......................................................................................................................... 15
4.1 Influence parameter ............................................................................................................... 15
4.1.1 Uniaxial tests ................................................................................................................... 16
4.1.2 Warning ....................................................................................................................... 17
4.1.3 Identification of the parameter ................................................................................................ 18
4.2 Influence parameters B and D .................................................................................................. 19
4.2.1 Simple tensile test ........................................................................................................ 19
4.2.2 Unconfined compression test ............................................................................................... 20
4.2.2.1 Warning ........................................................................................................... 20
4.2.2.2 Combination of blocked energy and the threshold depend on the deformations .......... 21
4.2.3 Identification of the parameters ............................................................................................... 22
4.3 Influence parameters of the function threshold ............................................................................... 23
4.3.1 Simple traction: influence parameter k0 ......................................................................... 23
4.3.2 Compression: influence parameters k1 and k2 ................................................................ 24
4.3.2.1 Role of the parameter k1 ............................................................................................... 24
4.3.2.2 Role of the parameter k2 ............................................................................................... 24
4.3.3 Identification of the parameters ............................................................................................... 26
4.4 Assessment on the study of the parameters ................................................................................................... 27
5 numerical Establishment ....................................................................................................................... 27
5.1 Evaluation of the damage .................................................................................................. 27
5.2 Calculation of the tangent matrix ........................................................................................................ 28
5.2.1 Term with constant damage .................................................................................... 29
5.2.2 Term related to the evolution of the damage ....................................................................... 30
6 Validation ............................................................................................................................................. 32
6.1 Identification of the parameters ......................................................................................................... 32
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6.2 Traction simple .............................................................................................................................. 32
6.3 Simple compression ...................................................................................................................... 33
6.3.1 Hognestad and Al [1955] ........................................................................................................ 33
6.3.2 Ramtani [1990] ..................................................................................................................... 34
6.4 Simple traction followed by a simple compression ......................................................................... 35
6.5 Biaxial tests .............................................................................................................................. 36
6.5.1 Answer stress-strain .......................................................................................... 36
6.5.2 Wrap field of rupture ....................................................................................... 37
Appendix 1 ................................................................................................................................................. 39
Appendix 2 ................................................................................................................................................. 42
7 Bibliography ........................................................................................................................................ 45
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1 Introduction
1.1
Characteristics of the damage of the concrete
The concrete is a complex material, formed of aggregates and a paste ensuring cohesion between these
aggregates, and in which preexist of the microscopic cracks formed at the time of the various stages of
manufacture. The concrete is generally regarded as an initially isotropic material, them
microscopic cracks not having privileged orientation. This isotropy is preserved if the loading
applied remains in the elastic range. Starting from a certain level of loading, microscopic cracks
will develop in particular directions, which induces the appearance of the anisotropy in
non-linear phase. The fissures develop preferentially orthogonally with the directions of
greater traction or of smaller compression. The process of damage is thus translated
by a loss of rigidity generated by the decoherence of the matter. Rigidity can be found
when the fissures are closed (unilateral effect). It is added to that another strong dissymmetry of
behavior enters traction and compression: the constraints supported in compression are
10 times (even more) higher than the constraints supported in traction. Let us announce finally others
phenomena like the formation of unrecoverable deformations (caused for example by blocking
lips of fissures by friction or the presence of matter degraded between these lips) or of
phenomena of dissipation of energy by friction of the lips of fissures.
1.2
Objectives of law ENDO_ORTH_BETON
In spite of the existence of various models of anisotropic damage for the concrete, the models
isotropic are always exclusively used in the engineering and design departments to account for
behavior of the concrete structures. This is due, for the anisotropic models, with the complexity of
their implementation numerical, with the difficulty in identifying their sometimes many parameters, with not
agreement of the objectives of the model and the industrial study, with the difficulty of coupling the model with
other physical phenomena (creep, plasticity), and in the majority of the cases at the calculating times
important needs by the anisotropic models. The use of the anisotropic models is not
moreover not necessary whenever the isotropic models describe the same behavior of
the structure. There is however cases where the anisotropic models can prove to be interesting, as of
at the time they predict a behavior different from the isotropic models.
The objective is to have a simple anisotropic model (low number of parameters),
numerically robust, and complying with the rules of thermodynamics (positive dissipation). It is
obvious that a certain number of phenomena observed in experiments could not be taken
in account. A schedule of conditions was thus defined before the development of the model so
to define the framework of it.
Two categories of requirements can be distinguished, the objective being to obtain a reasonable result
whatever the loading. One relates to the physical coherence of the prediction of the model (1) with
4) ), and the other relates to the numerical robustness (5) and 6)). The framework of our model is made up
following points:
1) Taking into account of the anisotropy grace the introduction of a symmetrical tensor of command 2
representing the effects of the damage. It is thus more exact to speak about model
orthotropic insofar as the use of such a tensor makes it possible to define only three
clean directions of the damage. A tensor of a higher nature (4 even 8) is
necessary to account for the complete anisotropy.
2) Cancellation of the constraint to the ruin. That leads us to define a free energy, function
deformations, rather than a free enthalpy, function of the constraints, because it seems more
easy to obtain a null constraint starting from deformations finished rather than the reverse.
3) Increasing and limited eigenvalues of the damage. This point accounts for
irreversible character of the process of damage (growth of the eigenvalues) of which
the ruin constitutes the limit (limited eigenvalues). It makes it possible of more than reach the ruin
in several directions.
4) Taking into account of the unilateral behavior of the concrete: refermeture of the fissures in
compression, dissymmetry of the thresholds of damage between traction and compression.
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5) Continuity of the forced answer/deformation, particularly with the open-closed passage
fissures. In addition to a discontinuity would be physically doubtful, it would involve
problems of convergence of the numerical algorithm.
6) Respect of the framework of generalized standard materials. That makes it possible to ensure coherence
thermodynamics of the model (positive dissipation) and that provides properties
pleasant mathematics for the resolution numerical (existence and unicity of the solution of
problem of the calculation of the constraints and the final damage with increment of deformation
fixed, said “local problem of projection” by analogy with plasticity).
Note:
The framework of generalized standard materials (CSMG) such as it here is heard is not strictly
that defined by Halphen and Nguyen [bib11]. Indeed, the strict CSMG ensures the existence and unicity
total solution of the problem if energy is convex compared to all the variables
at the same time. This cannot in the case of be checked the lenitive laws of behavior.
The “degraded” CSMG that we define ensures only the existence and the unicity of
solution of the local problem of projection (calculation of the evolution of the damage with deformation
fixed). To respect the CSMG, one must check the convexity of the free energy, on the one hand by
report/ratio with the deformation, and in addition compared to the internal variables:
· Convexity compared to the deformation is necessary to ensure the stability of
elastic problem.
· Convexity compared to all the variables intern simultaneously is
necessary to have the good mathematical properties for the local problem of
projection. If several internal variables are used, separate convexities
compared to each one of these variables are not sufficient.
· Total convexity compared to the deformation and with the variables intern simultaneously
is not necessary since the increment of deformation is fixed for local projection
allowing to calculate the evolution of the internal variables. It seems impossible besides
to obtain this total convexity in the case of lenitive laws of behavior.
The framework which one defined omits a certain number of physical phenomena associated
the damage of the concrete:
· appearance of unrecoverable deformations
· voluminal dilation of material in compression
· behavior hysteretic for cycles load-discharge, generated by friction
between the lips of fissures.
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2
Expression of the free energy
2.1
Taking into account of refermeture of the fissures
If one seeks to take account of the effect of refermeture, it is necessary to pay a great attention to
continuity of the constraints according to the deformations (what is an essential condition for
a law of behavior in a computation software by finite elements), cf [bib2]. Indeed, if one
model this effect in a too simplistic way, the law of behavior is likely great to present
a discontinuous answer. A solution is to finely describe what one calls traction and
compression, knowing that in traction (resp. compression) the fissure will be considered “open”
(resp. “closed”). A natural solution is to place itself in a clean reference mark of deformation. In
such a reference mark, the elastic free energy is written (and µ indicating the coefficients of Lamé):
()
= (tr) 2 + µ 2
I éq
2.1-1
2
I
One can then define:
· a traction or voluminal compression, according to the sign of tr,
· a traction or compression in each clean direction, according to the sign of I.
The elastic free energy can then be written:
(
) = ([tr) 2+ + (tr) 2 -] + µ [tr (2
+) + tr (2
)]
éq
2.1-2
2
-
with the following definitions for the parts positive and negative:
(tr) = H (tr) tr
2
2
2
2
+
; (tr) = H (- tr) tr
-
; tr () =
+
H (I) I; tr () =
-
H (- I) I
I
I
where H is related to Heaviside.
Note:
A more detailed study of the properties of the parts positive and negative of a tensor is made in
appendix 1.
2.2
Introduction of the variables of damage
Taking into account the complexity of the damage mechanisms, and having noted that it was
difficult to describe the behavior of the concrete by using only one variable of damage,
we chose to introduce two variables of damage:
· A tensor D of a nature 2 relating to the damage created in traction
· A scalar D relating to the damage created in compression
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Note:
The choice of a tensor of command 2 to model the damage of traction is relatively
traditional and intuitive. It makes it possible orthogonally to describe the privileged orientation of the fissures with
direction of greater traction. The question of the damage created in compression
(orthotropic or isotropic?) is much less clear.
Let us suppose to simplify that the fissures can be generated only in the plans
orthogonal with er, er or er. In the case of a simple traction in the direction er, the fissures
1
2
3
1
are created in the orthogonal plan with er, which involves a loss of rigidity in the direction
1
er. If one exerts then a simple traction in the direction er, one “does not see” not the fissure bus
1
2
the loading is parallel to the plan of the fissure and rigidity is not affected.
The damage is thus clearly anisotropic in traction. In the case of a compression
simple in the direction er, one creates fissures in the orthogonal plans with er and er. If one
1
2
3
charge then in compression in the direction er, one “sees” the orthogonal fissures with er.
2
3
rigidity in the direction er is thus lower than not degraded normal rigidity, but it is
2
stronger than rigidity in the direction er because compression in the direction er is sensitive to
1
1
all fissures, i.e. orthogonal with er and er. Consequently, the damage in
2
3
compression seems anisotropic than in traction, without being however completely isotropic.
In the absence of physical argument clearly on the isotropic or anisotropic character of
the damage in compression, we chose to take it isotropic for reasons of
simplicity.
One poses B = I - D representing the integrity of material in traction. One introduces the damage of
traction in the “positive” terms of the free energy [éq 2.1-2] and the damage of compression
in the negative terms. The free energy is now defined as follows:
(
, D) = ([trB) 2 +
2
2
1
2
2
2
+
(1 - D) (tr) -]
+ µ
tr ((+ B)+) + (1 - D) tr (-)
éq 2.2-1
2
4
Note:
The convexity of the free energy opposite, of the deformation on the one hand, and the variables
of damage simultaneously in addition, is respected strictly. One will refer to [bib2] and
[bib1] for a demonstration.
2.3
Terminals of the damage
Ruin, or creation of a fissure crossing the matter element completely considered, imposes
a limit upper than the damage. This terminal is imposed on each eigenvalue of
the damage of traction (D
where D indicate the eigenvalues of D), which
I
[] 1
,
0
I
allows to reach the ruin in 3 orthogonal directions. A convex indicating function by report/ratio
with the damage, I]
I
-] for D or
for B, is thus used to control each one of
1
,
[0, [
eigenvalues of the damage (cf [bib2]). In the same way, one uses an indicatrix on the value of
the scalar damage of compression. One obtains the expression of the following free energy:
(
1
, B, D) =
([trB) 2+ + (1 - D) 2 (tr) 2] + µ tr (
(B+B)2+) + (1-d) 2tr (2)
2
4
éq 2.3-1
+ I [0, [[min (Bi)]+ I] -] 1, [D]
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Note:
It results from this expression which once the damage is worth 1 in a clean direction,
this clean direction from now on is blocked and the damage cannot evolve/move any more but in
the plan perpendicular to this direction. The demonstration of this point is in appendix 2.
2.4
Introduction of a blocked energy
We propose to introduce an energy blocked in order to better control the evolution of
the damage according to the loading, in the form used by [bib4]. The idea consists with
to introduce an additional term into energy depending only on the damage, and not of
the state of deformation. It then results from it an additional term in derivation from the forces
thermodynamic which controls the evolution of the damage (cf section [§3]). This term
additional on the other hand no modification of the expression of the constraint implies.
Energy is written in the following way:
(
2
2
2
1
, B, D) =
([trB) +
+
(1 - D) (tr) -] + µ tr (
(B+B)2+) + (1-d) 2tr (2)
2
4
éq 2.4-1
+ I [0, [[min (B)] + I] -] 1, [D]
blocked
I
+
(B, D)
where
blocked
(B, D) is a convex function of the damage. One chooses to take this
null additional energy when the material is healthy. It must moreover be expressed with the means
invariants of the tensor of damage. One wishes finally that the additional term in
the expression of the thermodynamic forces depends on the damage, which makes it possible to eliminate it
choice of a linear term in damage for blocked energy
(B, D).
We chose the following expression:
blocked
(
B, D)
B
=
tr ((I - B)2)
D
2
+
D
2
2
éq
2.4-2
B
=
tr (2
D)
D
2
+
D
2
2
where B and D are parameters of the model.
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Note:
One can put the question of the physical significance of this additional energy.
The evolution of the damage indeed gives place in energy to the appearance of an energy
“blocked” by the system, and this energy is not null when the material is ruined. By
consequent, if one makes the energy balance of the creation of a fissure, a part only of
the power consumption corresponds to the dissipation of energy in the form of heat, whereas one
another part remains blocked.
One can find an origin physical of this energy blocked in the case of compression.
Indeed, if the macroscopic constraint is slackened, fissures opened by compression
are not closed again completely, as the existence of a residual deformation proves it
macroscopic. There are thus probably deformations and residual stresses
local which stores energy (cf [Figure 2.4-a]), by analogy with plasticity. (One recalls
that law ENDO_ORTH_BETON does not describe the unrecoverable deformations.).
Appear 2.4-a
Apart from the physical considerations, interest of this blocked energy, equivalent to one
energy of work hardening, is that the model remains well within the framework of standard materials
generalized. The advantage, compared to an energy of work hardening, is that this energy will have
an effect different on the two variables from damage (we will explain this point in
following section).
2.5 Final expression of the free energy, the constraints and the forces
thermodynamic associated the variables of damage
The final expression of the free energy is written:
(
2
2
2
1
, B, D) =
([trB) +
+
(1 - D) (tr) -] + µ tr (
(B+B)2+) + (1-d) 2tr (2)
2
4
éq 2.5-1
+ I [0, [[min (B)] + I] -] 1, [D]
blocked
I
+
(B, D)
The expression of the constraints results from energy by derivation compared to the deformations:
(, B, D)
=
= ([trB) B +
2
+
(1 - D) (tr) - I]
éq 2.5-2
+
1
µ
(B + B) B +
2
+
B (B + B)+) + 2 (1 - D)
-
2
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The continuity of the constraint results from the continuity of the eigenvalues of a matrix with respect to
this matrix (cf theorem of Ostrowski in [bib4]).
Note:
A defect appears however in the “open-closed” passing. Fields of activation and of
desactivation of the terms of opening and closing of the fissures do not coincide. This is due
with the fact that the opening and closing are not associated positive separation in parts and
negative of the same sizes (opening: damage combined with the deformation,
closing: deformation alone). This does not affect however the property of continuity. It appears
moreover than this defect is limited to a zone close to the origin of the deformations, where it does not generate
no the physical aberration, and which it has no incidence apart from this interval (cf [bib2]).
formalism suggested in [bib3], allowing to ensure the continuity of the constraint while taking
in account the effect of refermeture of the fissures, presents the same defect.
One in addition deduces from the free energy the expression of the thermodynamic forces associated
damages:
B
µ
F (, B) = -
= - (trB) -
+
([B + B) +
+
(B + B)+] + B (I - B) éq 2.5-3
B
2
F D (, D) = -
= (1 - D) (tr) 2 + 2µ
2
-
(1 - D) tr (-) - D éq
2.5-4
D
D
Each thermodynamic force is consisted of the two parts:
· A part depending on the deformation and the damage, which is derived from the part
rubber band of energy.
· A part depending only on the damage, which is derived from blocked energy. It
term will play the part of a work hardening, and makes it possible to control the answer
stress-strain. It is seen that the terms deriving from the energy blocked in each one
thermodynamic forces are independent one of the other, which makes it possible to control
more easily evolution of each variable of damage.
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3
Law of evolution of the variables of damage
3.1 Law
of evolution
So that model ENDO_ORTH_BETON enters within the framework of generalized standard materials, one
must obtain a potential of dissipation. For reasons of simplicity, one defines the field rather
of reversibility, as for law ENDO_ISOT_BETON (cf Doc. [R7.01.04]).
Note:
From a formal point of view, the generalized standard materials are characterized by a potential
of dissipation function positively homogeneous of degree 1, transformed of Legendre-Fenchel of
the indicating function of the field of reversibility. One can thus choose to define, that is to say a potential
of dissipation, that is to say a field of reversibility.
The first idea is to define two criteria of evolution corresponding to each variable
interns. This solution is completely possible insofar as the thermodynamic forces
are dissociated one of the other. It proves however that this solution has two disadvantages:
· First is of a “physical” nature. Let us consider a sample subjected to a compression
uniaxial. One can imagine that the criterion of compression alone is reached and that the criterion of
traction is not activated. Only the damage of compression evolves/moves then. If one subjects
this sample, after discharge, with a traction in an orthogonal direction with the axis of
compression precedent, the material behaves according to the model like a healthy material,
in spite of creation, actually, microscopic cracks parallel to the axis of compression, therefore
perpendicular to the axis of traction.
· The second disadvantage is of command practical. It is indeed easier to treat numerically
only one criterion, utilizing only one multiplier of Lagrange (cf [Figure 3.1-a]), rather
that two separated criteria, utilizing two multipliers of Lagrange and being able to create
zones where the direction of flow is not a priori defined (cf [Figure 3.1-b]).
Appear 3.1-a
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Appear 3.1-b
One thus decides to introduce a single criterion coupling the evolution of the two variables
of damage:
G (
2
2
B
F, F D) =
B
F
+
D
-
(1) (+
F) - K () 0 éq
3.1-1
where K () is a threshold depend on the state of deformation (this point will be commented on in the section
[§3.2]).
The evolution of the variables of damage is then determined by the conditions of Kuhn-Tucker:
D & = 0
for G < 0
&B = 0 &D = 0
éq
3.1-2
D & 0
for G = 0
B & 0 D & 0
I
I
Note:
The criterion utilizes only the positive part
D
F+ of D
F and negative B
-
F of B
F so
to impose the growth of the damage. This condition is ensured in the potential of
dissipation by the introduction of the indicating functions I
I - B&
+ (D &) and
((equivalent to
I)
IR
IR
I + (D &).
I)
IR
Note:
The elliptic criterion is convex in the space of the thermodynamic forces, which ensures
convexity of the potential of dissipation.
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Within the framework of generalized standard materials, the evolution of the internal variables follows the law
of flow associated with the criterion via the principle with normality:
B
G
-
B & = &
=
F
&
éq
3.1-3
B
2
F
B
-
F
+ (1 -) (D
F+) 2
G
(1) D
F
D
+
& = &
= &
éq
3.1-4
D
2
F
B
-
F
+ (1 -) (D
F+) 2
One can integrate the common denominator in the multiplier of Lagrange to have a relation
simpler:
G
B
B & = &
= & F éq
3.1-5
B
-
F
G
D & = &
= & (1 -) D
F éq
3.1-6
D
+
F
This system utilizes a single plastic multiplier &.
The equations of evolution ensure the positivity of the potential of dissipation:
2
B
D
2
B
D
F: B & + F D & = & F
+
-
(1)
(+F) 0
éq
3.1-7
3.2
Function threshold depend on the deformation
In order to better control the dissymmetry of the behavior enters traction and compression (report/ratio 10
rupture limits), we introduced a function threshold depend on the state of deformation
in the criterion [éq 3.1-1]. The role of this function threshold is to push back the elastic limit in
compression. One wishes moreover than the breaking stress in simple traction cannot be
exceeded during a biaxial test (cf [bib1] for a detailed study of the function threshold).
The function that we propose is as follows:
K ()
tr
= K -
0
1
K (tr)
()
-
- arctan -
éq
3.2-1
k2
This function threshold introduces 3 parameters for the model.
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Note:
This function was not optimized for the case of the loadings of triaxial compression. One
recall for this reason that law ENDO_ORTH_BETON was conceived to describe in a finer way
the damage of traction, the description of the remaining damage of compression
isotropic. Another law of behavior must thus be used for applications making
to intervene of the loadings of strong triaxial compression.
When the trace of the deformations is positive, the threshold remains constant: K () = K. The threshold increases
0
when one passes in compression, which makes it possible to push back the elastic limit, and consequently
rupture limit. It is noted that the function “arctan” was introduced for represented the envelope better
of rupture in the case of biaxial tests (a detailed study is in [bib1]).
[Figure 3.2-a] the comparison between the elastic limit shows us provides by a constant threshold and
that obtained with a threshold depend on the trace of the deformations in the case of biaxial tests in
plane constraint.
Appear 3.2-a: Enveloppe of the field of elasticity for biaxial loadings
in plane constraint.
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4
Study of the parameters
In addition to elastic parameters traditional E (Young modulus) and (Poisson's ratio), it
model utilizes 6 additional parameters:
Code_Aster
Function Dimension *
Identification **
ALPHA
Parameter of coupling
Without
1
k0
K0
Constant part of the threshold
MPa
2
k1
K1
Parameter of the threshold
MPa
3
k2
K2
Parameter of the threshold
Without
3
B
ECROB
Blocked energy relating to traction
MJ/m3=MPa 2
D
ECROD
Blocked energy relative to
MJ/m3=MPa 3
compression
* One multiplies the parameters in MégaPascals (MPa) by 106 if one works in Pascals (Pa).
** The parameters must be gauged in the following order:
· 1 one fixes the parameter
· 2 identification of k0 and B on a simple tensile test
· 3 identification of k1, k2 and D on an unconfined compression test and a biaxial test
(= with =-0,2 to check that the breaking stress in traction is not
1
2
exceeded)
Let us study now a little more in detail the influence of the various parameters on the answer of
model.
4.1
Influence parameter
The role of the parameter is to control the report/ratio of influence of the two thermodynamic forces
associated in the criterion of evolution. A parameter close to 1 privileges the evolution of
the damage of traction and a parameter close to 0 privileges the evolution of
the damage of compression. We decided to take a constant parameter for
reasons of simplicity.
If one damages in simple compression in direction 1, one creates fissures in the plans
orthogonal with er and er. If one makes then a traction in direction 2 or 3, one “sees” these
2
3
fissures. To obtain this effect in the model, it is necessary that a compression generates not only one
damage of compression, but also of traction. If one starts on the other hand with one
traction in direction 1, the fissures in the plan perpendicular to er will be not very open
1
(weak deformation due to the rupture), therefore one can think that it them “will not be seen” not if one does then one
compression in direction 2 or 3. One concludes from it that it is necessary to take 0 and 1 to have one
coupling, and close to 1 to support the evolution of the damage of traction at the time of
compressions. There will be also a small evolution of the damage of compression at the time of
tractions, deprived of physical direction, but that will not be awkward if this damage remains
weak.
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4.1.1 Tests
uniaxial
One proposes here to observe the influence of the parameter in the case of the simple traction and of
simple compression. The other parameters of the model are taken constant for our series of tests:
E = 28800 MPa, = 0.2, k0 = 3.10-4 MPa, k1 = 10 MPa, k2 = 2.10-4, B = 0 MJ/m3, D = 0..06 MJ/m3.
The parameter has an influence
relatively weak on the peak of constraint
in traction for the range of values
regarded as one can observe it on
[Figure 4.1.1-a]. One remains indeed in
the case where the thermodynamic force
associated the damage of traction
is dominating in the criterion (it
would not be the case if one took near
from 0)
Appear 4.1.1-a: Influence of the parameter has
in simple traction
For compression, one observes an important difference of peak of constraint (cf [Figure 4.1.1-b]).
The closer is to 1, the more the constraint threshold is high. This phenomenon is accentuated when one
takes a threshold depend on the deformations as it is the case on [Figure 4.1.1-b].
Note:
The fact that the constraint threshold is more sensitive in compression than in traction comes owing to the fact that
one takes a value of close relation of 1, privileging the damage of traction. The effect would be
reversed if one took near to 0.
The parameter also influences the relative speed of evolution of damages via the law of
normality of the flow. The closer is to 1, the more the side damage of Dyy traction
increase quickly compared to the damage of compression D as one sees it on
[Figure 4.1.1-c].
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Be reproduced 4.1.1-b: Influence of the parameter on a simple compression
Appear 4.1.1-c: Evolution of the damage of traction per report/ratio
with the damage of compression for a simple compression
4.1.2 Warning
In an unconfined compression test, fissures are created orthogonally with the directions of
positive deformation, and influences later behavior in traction. We introduced it
coupling in order to represent this phenomenon. One however sees on the unconfined compression test that
the side damage of traction does not reach the ruin (
lim
D
< 1) when the damage of
yy
compression D tends towards 1. This represents a limitation of the model.
It is however possible to reach the ruin for a value moreover nearer to 1.
The damage of traction then will evolve/move more quickly than the damage of compression
(cf [Figure 4.1.2-a]). Unfortunately the answer stress-strain then reveals one
snap-back (cf [Figure 4.1.2-b]), deprived of physical direction, so that it is necessary to exclude these values from.
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Be reproduced 4.1.2-a: Effet of a parameter very close to 1 on the answer
stress-strain in simple compression
Be reproduced 4.1.2-b: Effet of a parameter very close to 1 on the relative evolution
damages of traction and compression in simple compression
4.1.3 Identification of the parameter
There is a breaking value of parameter beyond which one falls on the disadvantages
statements in the section [§4.1.2]. This breaking value depends on the other parameters but us
let us not have an empirical formula allowing to find it. One will thus prefer to use a value of
around 0.9 which makes evolve/move the damage of compression more quickly than
the damage of traction in the unconfined compression test, so that one then cannot
to observe the ruin in traction in the directions perpendicular to that of compression.
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4.2
Influence parameters B and D
The introduction of a blocked energy depend on the variables of damage makes it possible to control
the speed of evolution of the damage, and this fact makes it possible to control the shape of the curve
stress-strain.
4.2.1 Simple tensile test
In the simple tensile test, only the part of blocked energy relating to the damage of
traction has a true influence. The stress-strain curve is represented on
[Figure 4.2.1-a] for various values of B. One sees that the larger B is, the more the constraint of
rupture is large, and more the peak is broad. This is due to the fact that blocked energy slows down the evolution
damage, as shows it it [Figure 4.2.1-b].
Appear 4.2.1-a: Influence of the energy blocked on a simple tensile test
Appear 4.2.1-b: Influence of the energy blocked on the speed of evolution of the damage
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4.2.2 Unconfined compression test
In the unconfined compression test, only the part of blocked energy relating to the damage
of compression a true influence has. The parameter D is however less easy to gauge fact
that the threshold of the criterion depends on the state of deformation. We immediately will explain this point
in the following warning.
Note:
Theoretically, the warning that we will expose is also valid in traction. In
practical, we let us not be confronted there because the damage of traction is always privileged
by taking a parameter of coupling close to 1, and one does not take a value of D too much
large (the behavior of the concrete in traction is quasi-fragile).
4.2.2.1 Warning
The introduction of blocked energy must make it possible to slow down the evolution of the damage of
compression, and of this fact must make it possible to round the shape of the peak in a test of compression
simple. A warning must however be formulated concerning the use of this energy
blocked. The thermodynamic force which controls the evolution of the damage of compression is
summon of two terms: one corresponding to the derivation of elastic energy, depend on
deformation and of the damage, positive, and the other corresponding to the derivation of energy
blocked, depending only on the damage, negative. However, when the damage increases, it
may be that the term corresponding to blocked energy is too large in absolute value compared to
that corresponding to elastic energy, which can prevent the evolution of the damage.
This problem appears when one considers a constant threshold in the criterion. [Figure 4.2.2.1-a] us
show in this case the influence of the introduction of energy blocked on the forced curve
deformation in an unconfined compression test. It is observed that the introduction of this energy, in
slowing down the evolution of the damage of compression, allows well initially
to increase the height of the peak of constraint as well as the width of the peak. One notices however
what appears a snap-back on the two curves where a blocked energy was added. This is due to
fact that one slows down the evolution of the damage of compression but which one does not act on
the damage of traction. This is illustrated on [Figure 4.2.2.1-b], which shows a stabilization of
the damage of compression combined to a fast increase in the damage of
traction.
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Appear 4.2.2.1-a: Influence of the energy blocked in an unconfined compression test
Appear 4.2.2.1-b: Evolution of the side damage of traction
according to the damage of compression
4.2.2.2 Combination of blocked energy and the threshold depend on the deformations
When one uses a threshold depend on the state of deformation, the force associated with elastic energy
remain more important than that associated with blocked energy because, for the same state
of damage, the criterion is reached for more important levels of deformation. One sees in
this case the influence of the introduction of an energy blocked on the response of the model in compression.
That makes it possible well to control the shape of the peak (cf [Figure 4.2.2.2-a]), i.e. the constraint of
rupture as well as the strain at failure. One in addition sees on [Figure 4.2.2.2-b] that more it
parameter associated with blocked energy is large, plus the side damage of traction evolves/moves by
report/ratio with the damage of compression. It is however obvious that if a parameter is taken
D too large, one finds the problems stated before.
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Appear 4.2.2.2-a: Influence of the energy blocked in an unconfined compression test
Appear 4.2.2.2-b: Evolution of the side damage of traction
according to the damage of compression
4.2.3 Identification of the parameters
The parameter B is identified on the simple tensile test. It makes it possible to regulate the height and the width of
peak for this test. It must be regulated at the same time as the parameter k0 threshold.
The parameter D is identified on the unconfined compression test. It makes it possible to regulate the height and
width of the peak for this test. It must be regulated at the same time as the parameters k1 and k2 of the threshold.
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4.3
Influence parameters of the function threshold
The function threshold which one uses utilizes three parameters:
K ()
tr
= K -
0
1
K (tr)
()
-
- arctan -
éq
4.3-1
k2
In the case of a simple traction, only the parameter k0 intervenes. Since the trace of
deformations becomes negative, as it is the case in simple compression, the three parameters
intervene. The parameter k0 must thus be gauged beforehand on a simple tensile test.
parameters k1 and k2 will be then gauged on an unconfined compression test and a biaxial test,
with k0 fixed.
4.3.1 Simple traction: influence parameter k0
If one takes a blocked energy not depending on the damage of traction (B=0), the value
breaking stress is completely determined by the value of k0, and the parameters
rubber bands:
K E
2
0
=
éq
4.3.1-1
rupture
4
+ 2 (1 -) (1+) 2
Appear 4.3.1-a: Dépendance of the breaking stress in traction
with respect to the parameter k0 (E=32000 MPa, =0.2, =0.87)
In an obvious way, more k0 is large, more the breaking stress in traction is large like
show it [Figure 4.3.1-a].
There does not exist unfortunately of analytical expression of the breaking stress when one B0.
dependence of the response with respect to the parameter B is studied in the paragraph [§4.2.1].
parameters k0 and B must be gauged simultaneously.
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4.3.2 Compression: influence parameters k1 and k2
The role of the function threshold, which depend on the negative trace of the deformation, is to increase
rupture limit in compression, in order to better control the dissymmetry of the behavior of the concrete
between traction and compression.
Moreover, we made the choice take a function threshold allowing to describe the envelope of
rupture of the concrete in the case of biaxial loadings. This choice was made for two reasons:
We have for these tests experimental results (cf [bib5]). These tests concern certainly
of course only the concretes used by [bib5], but present common characteristics that one
seems it can to generalize.
That makes it possible to widen the range of use of the model. The uniaxial tests are indeed insufficient
to ensure the relevance of the model in the case of calculations in 3D.
Note:
The model should not in the case of be used strong triaxial compressions, the free energy and
the function threshold not having been established to treat this case. It is pointed out that the principal objective of
model is to describe the damage of traction in the concrete.
4.3.2.1 Role of the parameter k1
The parameter k1 is the parameter which makes it possible to increase the rupture limit in compression.
[Figure 4.3.2.1-a].
Appear 4.3.2.1-a: Dépendance of the response in simple compression with respect to k1
(E=32000MPa, =0.2, k0=3.10-4MPa, k2= 6.10-4, d=6.10-2MJ/m3)
4.3.2.2 Role of the parameter k2
To include/understand the role of the parameter k2, let us reconsider the advance which led us to choose
the function threshold [éq 3.2-1].
The first function threshold having been tested is the linear function:
K () = K - K
éq
4.3.2.2-1
0
1 (tr) -
This function makes it possible well to increase the breaking stress in simple compression, but it poses
problem when one is interested in biaxial tests. [Figure 4.3.2.2-a] the envelope shows us of
elastic range for biaxial tests.
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Note:
The envelope of the field of elasticity is different from the envelope of rupture (for which us
let us have experimental results). We study the envelope of the field of elasticity because it
can be calculated analytically, contrary to the envelope of rupture, and bus it is from
this one which we chose our function threshold. The user will have to gauge however well its
parameters on the envelope of rupture.
Appear 4.3.2.2-a: Enveloppe of the field of elasticity for biaxial tests
with a linear function threshold
The linear variation of the threshold with the deformation does not seem not adapted since one sees appearing one
swelling in the zone (>,
0 < 0 which gives ultimate stresses in traction too much
1
2
)
important.
We thus modified the function threshold:
K ()
tr
= K -
0
1
K (tr)
()
-
- arctan -
éq
4.3.2.2-2
k2
The fact that the function arctan presents a bearing makes it possible to find a linear threshold when the trace
deformations increases in absolute value. The fact that it is null in the beginning makes it possible to slow down
increase in the threshold close to this origin.
One can see on [Figure 4.3.2.2-b] the effect of the introduction of the new function threshold compared to
linear function on the field of elasticity for the same value of the parameter k1. This news
function threshold makes it possible to avoid the phenomenon of swelling observed with the linear function when one
increase the value of the parameter k2 (the case k2=0 corresponds to the linear function). The other effect of
parameter k2 is to decrease the constraint of initiation of the damage, which implies one
reduction in the breaking stress (opposite effect of k1) as shows it it [Figure 4.3.2.2-c].
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Appear 4.3.2.2-b: Domaine of elasticity with the function threshold in two parameters
Appear 4.3.2.2-c: Dependence of the response in simple compression with respect to k2
(E=32000MPa, =0.2, k0=3.10-4MPa, k1= 10.5 MPa, d=6.10-2MJ/m3)
4.3.3 Identification of the parameters
The parameters D, k1 and k2 are identified simultaneously. The parameters k1 and D make it possible to regulate
the breaking stress in simple compression and the parameter k2 make it possible to avoid the “swelling” of
the envelope of the elastic range (and thus of the envelope of rupture) for the biaxial tests. More one
takes k1 large, more it is necessary to take k2 large.
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4.4
Assessment on the study of the parameters
In spite of the relative simplicity of the model, and the low number of parameters to be identified (6), the number of
experimental data available to the engineer is often, even always, lower than the number of
parameters to be identified. This implies that the arbitrary character of the choice of certain parameters.
Here, in short the command in which the parameters must be selected:
· For the parameter, a value between 0,85 and 0,9 are recommended.
· The parameters k0 and B must be identified simultaneously on a simple tensile test
(not of analytical formula in the case B0).
· The parameters D, k1 and k2 must be identified simultaneously. Simplest is to fix it
parameter k2 to 6.10-4 (value for which it is probable that one does not observe swelling
envelope of rupture (cf [§4.3.2.2]) for biaxial tests), and to gauge D and k1 on one
unconfined compression test. One checks then if the envelope of rupture is correct, one
modify k2 if need be and one starts again for D and k1.
Examples of sets of parameters are in section 6 (validation on tests
experimental). One will find moreover in the document [V6.04.176], the case-test allowing to identify
parameters.
5 Establishment
numerical
For the integration of the law of behavior in Code_Aster, we placed ourselves in
tally of the implicit integration of the laws of behavior.
5.1
Evaluation of the damage
One notes -
deformation and -
B and -
D variables of damage at the end of the step of time
precedent (after convergence). One wishes to determine the evolution of the damage when one
apply an increment of deformation
. Final deformation = - +
is thus fixed.
The criterion of damage is evaluated:
2
F (, B
2
, D -) = BFR +
-
(1) (F d+) - K ()
éq
5.1-1
If F 0, the variables of damage do not evolve/move and one can pass to the following iteration for
mechanical balance.
If F > 0, the variables of damage evolve/move, by respecting at the same time the criterion and the flow
normal. One thus seeks (B, D,
) solution of the system:
- B +
B
-
F (-
B + B) 0
R (B, D,) = - D + (1 -) d+
F (-
D + D)
0
éq
5.1-2
F (
-
, B +
-
B, D + D)
=
0
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This system is nonlinear and requires the use of an iterative method. We chose
to use a method of Newton-Raphson. Index N represents the iterations of Newton. That is to say
N
R (
N
N
N
B, D
,
) the residue of the system to iteration N, the linearization of the system are written:
N +1
N
B
- B
n+1
R (
n+1
n+1
n+1
N
N
N
N
N
N
N
N
B, D,
) = R (B, D,) + (R (B, D,)
:
N +
D
1 - N
D
N +1
N
-
éq 5.1-3
- N
B + N
B N
F
-
where
N
R = - N
D + N
(1 -) D N
F
F (-
, B + N
-
B, D + N
D)
B
B
N F
F
-
B
- I +
:
0
F
B
-
F
B
N
F D
N
F D
D
and R
=
0
- 1+ (1 -)
+
(1)
F
.
D
+
F D
B
F
B
F
1 F D
F D
-
(-)
+
:
0
2
B
F
+
2
2
2
1 F D
B
B
F
1
D
D
F
-
(-)
+
+
-
(-)
+
One solves then the system linearized Rn+1 (B
n+1, n+1
D
, n+1
) = 0. One obtains
(N 1+ N 1+ N 1
B, D,
+
). This procedure is reiterated until the residue is lower than one
parameter of convergence.
5.2
Calculation of the tangent matrix
The tangent matrix is the tensor M of a nature 4 defined by:
ij
= M
M =
éq
5.2-1
ij
ijkl
kl
ijkl
kl
This matrix is not calculated on the continuous problem but on the incremental problem. One seeks
thus effects of a variation of the increment of deformation between two steps of successive times on
variation of final constraint, taking into account the fact that the internal variables can also evolve/move. One
has with the first command:
ij (, B, D)
ij
ij
=
+
B
ij
+
D
kl
éq
5.2-2
B
kl
D
kl
kl
B, D
, D
, B
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that is to say:
B
D
, B, =
+
+
ij (
D)
ij
ij
mn
ij
kl éq 5.2-3
kl
B
mn
kl
D
kl
B, D
, D
, B
With
The notation
employee here means that one derives A compared to B for C and D constant.
B
C, D
One thus breaks up the tangent matrix into two parts, one with constant damage, and the other
representing the evolution of the internal variables:
cst
evol
M = M + M
éq
5.2-4
B
D
with
cst
ij
M
=
and
evol
ij
mn
ij
M
=
+
ijkl
ijkl
B
D
kl B, D
mn
kl
, D
, B
kl
5.2.1 Term with constant damage
The matrix
cst
M
is the derivative of the constraint compared to the deformation with damage
constant:
cst
M
= H trB B B + F (D) H - tr
ijkl
(
) ij kl
(
) ij kl
µ A
+ip
+
(B + B + B + B B
mk nl
mk
nl
ml nk
ml
nk) pj
4 A
mn
µ
With
éq
5.2.1-1
+ pj
+ B
B + B + B + B
IP
(mk nl mk nl ml nk ml nk)
4
With
mn
- ij
+ 2µf (D) kl
With
where A = B + B
, H is related to the Heaviside and the derivative
- and
+ are defined in
With
appendix 1.
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5.2.2 Term related to the evolution of the damage
The matrix
evol
M
is calculated by differentiating the criterion and the law of normality. This derivation is done on
the incremental problem and not on the continuous problem.
The criterion first of all is differentiated:
B
B
D
D
B
B
F F
F (
2
D
D
+ - F F
F, F) = F
+
F
K
K
-
(1 -) (+) 2 - ()
:
-
(1)
0
+
-
=
diff
2
B
F
+
F D
-
(-) (+)
0
1
2
éq 5.2.2-1
One differentiates then the law of flow of the incremental problem, discretized in an implicit way:
B = B
F
(1 -) F dB = B
F
D
+
-
D
= (1 -) F D
+
éq 5.2.2-2
(1 -) F dB + 1
B B
F
B
F
+
(-) Fd =
D
+
D
+
-
-
diff
One seeks the relation between B, D and. One can express the variations of the forces
thermodynamic according to the variations of the deformations and the variables of damage:
B
BFR BFR
BFR
F =
-:
:
: B
-
B
+
F
B
éq
5.2.2-3
D
F D F D
F D
F =
+
:
D
+
D
F
D
The system of equations defined by [éq 5.2.2-1], [éq 5.2.2-2] and [éq 5.2.2-3] leads to the expressions
following:
B
= - 1: :
éq
5.2.2-4
- 1
D = -
+: : :
with
D
B
B
B
B
B
F
B
F
F
F
F
F
1
+
-
B
=
-
F
-
:
:
+ D
-:
- 1 - F +
ijkl
-
(
) D
D
D
D
+ (
B
B
ik
jl
it
jk)
F
F
2
+
(1)
D
F
F
B
F
B
kl
ijkl
F
D
ij
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B
D
B
B
F
F
B
2
2
-
+
B
F
F
F
K
=
-
F
-
:
:
+ 1 - F
- F
+ 1 - F
ijkl
-
(
)
D
D
B
+
-
(
)
D
D
D
B
(d+)
(
1 -)
D
F
F F
F
+
kl
F
D
ij
+ (1 -)
D
D
B
B
F
F
F
F
+
B
- D
-:
D
ij
B
F
F
kl
ijkl
B
B
2
2
B
F
F
F
K
= F
-
:
:
+ 1 - F
- F
+ 1 - F
ij
-
(
)
D
D
B
+
-
(
)
B
(d+)
F
ij
B
B
B
F
F
= F
-
:
:
ij
-
B
F
B
ij
= (1 -)
D
F D
F+
D
- 1
1
1
such as -
= +
ijkl
klmn
(im jn in jm)
2
B
D
One thus obtains the expressions of
and of
.
Moreover, according to the definition of the constraint and the thermodynamic forces, which derive from one
even energy, one a:
B
F
D
=
F
and
=
éq
5.2.2-5
B
D
, D
B
, B
D
what enables us to calculate the part of the tangent matrix relating to the evolution of
the damage:
B
evol
ij
D
mn
ij
M
=
+
éq
5.2.2-6
ijkl
B
D
mn
kl
, D
, B
kl
Note:
It should be noted that the tangent matrix is not symmetrical. This is due to the fact that the threshold
on damage depends on the deformation. In the case of a constant threshold, the matrix is well
symmetrical, since the model quite standard is then generalized and a diagram is used
of implicit integration.
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6 Validation
The first stage of the validation of the model is the comparison between the prediction and different
experimental results for simple tests. Taking into account the simplifying assumptions which have
summer made for our modeling, we will not be able of course to reproduce all them
experiments, in particular experiments of triaxial compression. One concentrates here on the tests
of simple traction, simple compression, like on biaxial tests.
6.1
Identification of the parameters
The identification of the parameters must be done in three successive phases:
· One must choose a value for the constant of coupling. It must be taken of such kind
that the scalar damage of compression remains negligible in a tensile test.
One decides to take it equal to 0,87 for all the tests of this section in order to avoid it
phenomenon of snap-back in compression (cf [§4.1.2]).
· The second phase is the identification of the parameters k0 and B. Ces parameters can be
identified directly on the tensile test simple bus other parameters of the model
do not intervene in this test.
· The third phase is the identification of the parameters D, k1 and k2 on the tests of
simple compression and biaxial tests.
In any rigor, the identification of the parameters of our model for a material requires
to have the experimental curves in simple traction, simple compression and under loading
biaxial. Unfortunately, all these results are generally not available simultaneously
for a material. We will have to thus make a certain number of assumptions to gauge our
parameters. For example, we will choose the parameter k2 for all the tests in such a way that
tensile stresses in the biaxial tests do not exceed the breaking stress in
traction. Moreover, we do not have for the tests of compression the breaking stress in
traction for studied materials. We will thus take in an arbitrary way of the parameters k0 and B
equal to those which we calculated for the simple tensile test.
6.2 Traction
simple
The experiments aiming at observing the behavior of the concrete under loading of traction are
extremely difficult to realize, which explains the relatively low number of results of tests in
simple traction. The difficulty lies in the fact that the damage concentrates in tapes
of localization corresponding to fissures, which causes to make inhomogenous the specimen
studied. Since strong heterogeneities appear, it becomes impossible to deduce one
stress-strain curve starting from the curve force-displacement, and thus to establish a law of
behavior for material. The measuring apparatus PIED ([bib6], [bib7]) allows to measure one
relatively homogeneous deformation and stress field by limiting the localization. It is
why we use the results obtained by [bib6] testing our model.
In the case of simple traction, only 3 of the 6 parameters of the model will play a part:
· The parameter of coupling.
· The constant of threshold k0.
· The constant of blocked energy associated the damage of traction B.
· The parameters k1 and k2 do not have any influence on this test and the influence of the parameter D is
negligible.
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The parameter k0 governs the height of the peak of constraint and the parameter B controls its width as well as
slope post-peak. The larger B is, the less the damage evolves/moves quickly, which reveals
non-linearity before the peak and increases the deformation with the peak and its width. [Figure 6.2-a] shows
response of the model, compared with the experimental data of [bib6]. Calculation is carried out on one
only element not to meet a phenomenon of localization. The parameters are as follows:
k0 (Mpa)
B (kJ/m3)
0.87 3.10-4 7
Appear 6.2-a: simple Essai of tensile, comparison with the experimental data
of Bazant and Pijaudier-Cabot [1989]
6.3 Compression
simple
The fact that the experimental data are more numerous than for the tensile tests is due to
fact that they are easier to realize. The phenomenon of localization is much less important there
that for a loading of traction, at least when the damage remains relatively weak. One
use the results of Hognestad and Al [bib8] and the test of Ramtani [bib9] to validate our model on
unconfined compression tests. In spite of the fact that the results of [bib8] are relatively old,
we use them because the experiment was undertaken for several different concretes. We use too
results of [bib9] to show that the model remains valid for more recent experiments.
As we said to the paragraph [§6.1], we do not have the breaking stress in
traction for these various tests, this is why we use the same parameters as those
obtained in the paragraph [§6.2]: = 0.87, k0 = 3.10-4 Mpa, B = 7 kJ/m3.
6.3.1 Hognestad and Al [1955]
[Figure 6.3.1-a] shows us the comparison between the experimental results of [bib8] and
prediction of our model in the case of three materials of which the maximum constraint in value
absolute FC is noted:
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We used the following parameters:
Ultimate constraint FC
20.7 Mpa
32.1 Mpa
42.8 Mpa
E
17000 27000 36000
0.2.0.2.0.2
D (kJ/m3)
60 60 60
k1 (Mpa)
4.8 10. 18
k2
7.10-4 7.10-4 7.10-4
Appear 6.3.1-a: simple Essais of compression of Hognestad and Al [1955]
6.3.2 Ramtani
[1990]
The test of [bib9] is a cyclic test of compression. It highlights the creation of deformations
irreversible and the phenomenon of hysteresis. We do not describe these phenomena. Us
thus let us satisfy for our part to calculate the response under monotonous loading. Parameters
used are as follows:
E
D (kJ/m3)
k1 (Mpa)
K2
33700 0.2
60
20.5 7.10-4
Appear 6.3.2-a: simple Essais of compression of Ramtani [1990]
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6.4
Simple traction followed by a simple compression
One proposes here to compare our model with the experimental results of [bib9] on a test of
traction simple follow-up of a simple compression, in order to highlight the restoration of rigidity
generated by refermeture of the fissures.
The parameters used are as follows:
E
k0 (Mpa) B (kJ/m3) D (kJ/m3)
k1 (Mpa)
k2
16400 0.2 0.87
7.
10-5 0.3
40
5.5
6.10-4
The experimental results show the appearance of unrecoverable deformations in the phase of
simple traction (cf [Figure 6.4-a]). These unrecoverable deformations are not described by our
model, this is why the loss of rigidity generated by the damage seems over-estimated. It
problem does not seem to have of incidence when the fissures are closed. One observes indeed on
[Figure 6.4-b] a good correspondence of the model with the experimental results in the phase
of compression. It seems as well as the restoration of the rigidity obtained thanks to the model is very
near to that obtained in experiments.
Appear 6.4-a: simple Phase of traction in the test of Ramtani [1990]
Appear 6.4-b: Essais of traction followed by a simple compression (Ramtani [1990])
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6.5 Tests
biaxial
This paragraph is devoted to the study of the biaxial tests of [bib5]. One seeks on the one hand to observe
the answer stress-strain in the case of a uniaxial loading and of two loadings
biaxial in plane constraint, and in addition to describe the envelope of the field of rupture in space
constraints in the case of biaxial loadings in plane constraints.
Note:
The experiments of [bib5] are relatively old. These biaxial tests require
to carry out a great number of tests, this is why one finds few more recent results on
this kind of tests. We use them because they always represent a reference for
modélisateurs.
The maximum constraint in absolute value of the uniaxial pressing, noted p in [bib5] is worth 4650
psi (32.1 Mpa). We standardize our response by this breaking stress. Parameters that
we use are as follows:
E
k0 (Mpa)
B (kJ/m3)
D (kJ/m3)
K1 (Mpa)
K2
32000 0.2 0.87 3.10-4 1
60 10.5 6
6.5.1 Answer
stress-strain
One traces the answer stress-strain initially in the case of a loading
uniaxial and of two biaxial loadings in plane constraint:
= with = (0, 0.52, 1) and
2
1
= 0
3
.
The model enables us to obtain the answers represented [Figure 6.5.1-b] which one compares with
results of [bib5] represented [Figure 6.5.1-a]. As one saw in the preceding paragraph, it is
possible to correctly describe behavior in uniaxial pressing in the direction of
compression. One however sees for this test that the lateral distortion predicted by the model
decrease when the damage occurs whereas it actually increases. This is due to the fact that
one did not take into account the existence of unrecoverable deformations dependant on
the damage, which seems to be at the origin of voluminal dilation in compression. It
phenomenon is even more important in the case of the tests of bicompression. One observes indeed
with the model that the threshold of rupture in bicompression is higher than that in simple compression
(less than the experimental results), but that the strains at failure are less
important in the case of bicompression that for compression, which does not correspond to
experimental results.
Appear 6.5.1-a: Essai biaxial of Kupfer and Al [1969]
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Appear 6.5.1-b: Réponse of the model for the biaxial tests
6.5.2 Wrap field of rupture
One is interested now in the envelope of the field of rupture for biaxial tests in constraint
plane. The experimental results obtained by [bib5] for various concretes are represented on
[Figure 6.5.2-a]. One observes a relative similarity of the shape of the envelope of standardized rupture
for various materials.
We took again the parameters used in the paragraph [§6.5.1] and we compared the prediction
of our model for the envelope of rupture of the biaxial tests with the experimental results
(cf [Figure 6.5.2-b]).
Appear 6.5.2-a: Enveloppe of rupture for biaxial tests
in plane constraints Kupfer and Al [1969]
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Appear 6.5.2-b: Prédiction of the model for the envelope of rupture
biaxial tests in plane constraints
One observes a relatively satisfactory correspondence of the prediction compared to the results
experimental. The most important variation is in the zone of bicompression. This problem is not
not astonishing insofar as we took the party to use only two parameters for
threshold, which controls the pace of the envelope of rupture as well as the response of the model in compression.
A thorough study of the function threshold would probably make it possible to better approach them
experimental results, at the price of the introduction of new parameters.
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Appendix 1
A1.1 Définition of the spectral decomposition and the positive parts and
negative of a tensor
That is to say X a tensor of command 2 symmetrical. Not to weigh down the notations, X will be also noted,
~
wrongly, the matrix of this tensor in the base fixes observer. Maybe moreover X the matrix (diagonal)
this tensor in its own base:
~
X
0
0
1
~
X =
~
0
X
0
éq A1.1-1
2
~
0
0
X
3
While noting U the clean vector associated the ième eigenvalue, and Q = (U, U, U the matrix of passage enters
1
2
3)
I
the base fixes and the clean base of X, one with the relation:
~ T
X = Q X
. Q
.
éq A1.1-2
The parts positive and negative of tensor X are defined by:
H (~ ~
X X
0
0
1)
~
1
T
~
~ ~
X = P: X = Q X
.
Q
.
X =
0
H X X
0
+
(
éq
A1.1-3
2)
+
+
+
with
2
0
0
H (~ ~
X X
2)
3
H (- ~ ~
X X
0
0
1)
~
1
T
~
~ ~
X = P: X = Q X
.
Q
.
X =
0
H
X X
0
-
(-
éq A1.1-4
2)
-
-
-
with
2
0
0
H (- ~ ~
X X
2)
3
where H is related to Heaviside.
A1.2 Calcul of the derivative
For the calculation of the tangent matrix, like for the calculation of the evolution of the damage, we have
need to evaluate the derivative of the parts positive and negative of a tensor compared to this last. It is enough for
that to imagine that tensor X depends on time and to calculate the tensors M
M
+ and
- defined by:
X & = M: X
X & = M:
-
-
&
+
+
& and
X
éq A1.2-1
The differentiation of the equation [éq A1.1-2] gives us:
~
T
~
T
~
T
X = Q X&
&
.
Q
.
+ Q & X
.
Q
.
+ Q X
.
Q
. &
+
+
+
+
éq A1.2-2
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Assumption:
It is necessary to establish the expression of Q
&. The demonstration which will follow is valid only if the eigenvalues of X
are distinct. Insofar as the calculation of M
M
+ and
- will be used only in algorithms of
numerical resolution, we will allow ourselves to disturb possible eigenvalues numerically
identical in order to make them distinct and to be able to use the results below.
To calculate Q
&, one needs to express the derivative of the clean vectors U &. For that, according to the step of
I
[bib11], the expression of X is differentiated:
X = ~
~
~
X U U X =
X&
&
~
U
U
X U&
U
X U
U & éq
A1.2-3
I
I
I
+
+
I
I
I
I
I
I
I
I
I
I
I
Moreover, one has the following relations between the clean vectors and their derivative:
U U
.
=, U & .U +U .U & = 0
éq A1.2-4
I
J
ij
I
J
I
J
By contracting the equation [éq A1.2-3] on the left and on the right by the clean vectors, and by using the relations
[éq A1.2-4], the following relations are obtained:
~
U. &
~
XU
.
X = X&
&
éq A1.2-5
I
I
() II I
~
~
~
~
~
U. &XU
.
&
=
. & +
&.
=
-
&.
for
éq A1.2-6
J
K
(X) X U U X U U
jk
K
J
K
J
J
K
(X X
J
K) U U
J K
J
K
(~X&)
In these expressions, there is no summation on the indices, them
the components of X indicate
&
jk
(~X&) =U .X&U. , and X&~
jk
J
K)
in the fixed base coinciding with the clean base of X at the moment considered
I
indicate the derivative of the eigenvalues of X (not the eigenvalues of derived X
&.
One deduces from the relation [éq A1.2-6] the expression of U &:
J
(~X&)
U & =
U & U
.
U
.
~
~ U
éq A1.2-7
J
(J K) =
K
jk
K
K J
K J (X - X
J
K)
This enables us to express Q
&:
(~X&)
&Q = (U &, U &, U&
&
éq A1.2-8
1
2
3) Q
=
jk
~
~ Q
ij
ik
K J (X - X
J
K)
(~X&)
, components of X
& in the fixed base coinciding with the clean base of X at the moment considered,
jk
~
can express itself according to the components of X
& in the fixed base. Thus, X & indicating the matrix of
(~X&), one a:
jk
~
X & QT
=
X
. & Q
.
éq A1.2-9
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One deduces from the expression [éq A1.2-8] the expression of Q
&:
Q Q X&
Q & =
~
~
Q
éq A1.2-10
ij
mj nk mn ik
K J m, N (X - X
J
K)
There are finally the obvious relations between the eigenvalues of X+ and X:
~
~ ~
~
~
X = H X X X&
~
= H X X&
+
éq
A1.2-11
I
(I) I
+i
(I) I
One deduces from the relations [éq A1.2-5], [éq A1.2-9] and [éq A1.2-11], the following relation:
~
~ ~
~ ~
X&
~
= H X X & = H X X & = H X Q Q X&
+
éq
A1.2-12
I
(I) I (I) () II (I) ji ki jk
The relations [éq A1.2-10], [éq A1.2-11] and [éq A1.2-12] allow us to express the relation [éq A1.2-2]:
~
Q Q
~ ~
km
ln
X&
Q Q Q Q H X
X&
~
~ H X X Q Q
Q Q
X&
+
éq A1.2-13
ij
=
im jm km lm (m)
+
kl
(m) m (
+
in
jm
im
jn)
K, L m
kl
K, L
m, N X
-
X
m
N
mn
As X is a symmetrical tensor, one a:
1
1
X = (
T
X + X) X & = (
T
X & + X &)
éq
A1.2-14
2
2
This allows us récrire the equation [éq A1.2-13]:
~
X&
Q Q Q Q H X
X&
+ij
=
im jm km lm (m)
kl
K, L m
~ ~
éq
A1.2-15
+ 1
H X X
m
m
&
2
(Q Q + Q Q
~
~
Q Q
Q Q
X
km
ln
kN
lm)
() (
+
in
jm
im
jn)
kl
K, L
m, N
X -
X
m
N
mn
One deduces the expression from it from the components of M+:
~
1
~ ~
H X X
M
=
Q Q Q Q H X +
Q Q +
m
m
Q Q
~
~
Q Q + Q Q
+
ijkl
im jm km lm (m) (km ln kN lm) () (in jm im jn)
m
2 m, N
X - X
m
N
mn
éq
A1.2-16
By analogy, one easily deduces the expression from it from the components of m:
~
1
~ ~
H - X X
M
=
Q Q Q Q H - X +
Q Q +
m
m
Q Q
~
~
Q Q + Q Q
-
ijkl
im jm km lm (m) (km ln kN lm) (
) (in jm im jn)
m
2 m, N
X - X
m
N
mn
éq A1.2-17
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Appendix 2
One wishes to show that when an eigenvalue of the damage reaches 1 in a direction, then this
direction is blocked and the damage cannot evolve/move any more but in the orthogonal plan with this direction.
One places oneself in the clean base of the damage:
~
D = D E E éq A2-1
I I
I
I
~
where E indicates the clean vector associated with eigenvalue D.
I
I
Adopted energy is the sum of an elastic energy and an indicating function of the eigenvalues of
the damage:
(, D)
el
= (, D)
~
+ I - 1, [max (D
éq A2-2
I)]
]
]
The criterion of evolution of the damage is written:
F (D
F) = tr ((D
F) 2
K
+) -
0
éq A2-3
The evolution of the internal variable follows the following law of flow obeying to the principle of normality:
D
F
D & = &
=
F+
&
.
0
éq A2-4
D
D
F
F: D
F
+
+
The derivative D
& is thus colinéaire with D
F+.
The thermodynamic force derives from the free energy [éq A2-2]:
~
el
I] -] (max
1
(Di)
FD = -
= -
-
éq A2-5
D
D
D
Assumption:
~
It is supposed that the damage is worth 1 in direction 1: D = 1 and which it is different from 1 in the others
1
directions. The demonstration would be similar if the damage is worth 1 in two directions
orthogonal.
The derivative of the indicating function compared to the damage is written:
I (
(~
max Di)
I (~
~
~
D
I D
D
D
1)
(1)
=
=
1
11
~
éq A2-6
D
D
D
D
D
1
11
~
I (~
D
~
D
D
~
1)
However one a:
~
= + bus D = 1;
11
=;
1 =1 bus (D)
D&
&
=
(see Annexe 1).
D
1
I
1
1 J
D
D
11
1
1
ij
11
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This enables us to write the derivative of the indicatrix compared to the damage:
T 0 0
I (~
Di)
-
= 0 0 0 with T = -
éq A2-7
D
0 0 0
The derivative of the elastic part of energy is with finished values. One writes it in the shape of a matrix 3x3
symmetrical:
has
has
has
11
12
13
el
-
=a
has
has
éq A2-8
12
22
23
D
has
has
has
13
23
33
This enables us to write the expression of the thermodynamic force:
T + has
has
has
11
12
13
D
F = has
has
has
éq A2-9
12
22
23
has
has
has
13
23
33
One now seeks to calculate the positive part of the thermodynamic force. For that, one must calculate them
eigenvalues of D
F and associated clean vectors. To carry out this calculation, us conveniently
let us regard the term T as very large negative (tending towards -) but not strictly infinite.
There are then T >> has for all indices I, J. One can thus write the matrix D
F in the form:
ij
1
O (1/T) O (1/T)
D
F T O (1/T) O (1/T) O (1/T)
éq
A2-10
O (1/T) O (1/T) O (1/T)
That is to say an eigenvalue of D
F and U the associated clean vector, then the components of U are solutions
following system:
U + O 1/T U + O 1/T U = U
(I)
1
(
) 2
(
) 3
1
T
D
F .U = U
(1/T) U + O 1/T U + O 1/T U = U
(II) éq A2-11
1
(
) 2
(
) 3
2
T
O (1/T) U + O 1/T U + O 1/T U = U
(III)
1
(
) 2
(
) 3
3
T
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Law of behavior ENDO_ORTH_BETON
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Among the clean vectors, there are some at least for which the U1 component is nonnull, because them
clean vectors form a base of R3. Let us consider this clean vector. Then equations of the system
[éq A2-11] impose:
(I)
= T
(I) and (II)
U = 0
2
éq A2-12
(I) and (III)
U = 0
3
One deduces from it that T is an eigenvalue of D
F and that the vector e1 is the associated clean vector.
Moreover, the clean base of a symmetrical matrix being orthogonal, two other clean vectors of D
F
are in the plan defined by e2 and e3.
Thus the U1 component of these clean vectors is null. Under these conditions, the equation (I) does not provide, in
limit T -, that the identity 0=0, and the equations (II) are reduced, after the multiplication per T, with:
U + has U = U has
22 2
23
3
2
éq A2-13
U + has U = U has
23
2
33
3
3
Thus clean vectors of D
F distinct from e1 and the associated eigenvalues are the clean vectors and
has
has
has
11
12
13
has
has
22
23
the eigenvalues of projection has
=
matrix has = has
has
has
in the plan (E
2D
has
has
12
22
23
2, e3).
23
33
has
has
has
13
23
33
It results from what precedes that the positive part of D
F is worth:
0 0
0
D
F = 0
+
éq A2-14
0
(2D has)
+
Like D
& is colinéaire with D
F+, this implies that only the components D22, D23, D33 still can
to evolve/move, the D11 components, D12, D13 from now on being fixed.
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7 Bibliography
[1]
V. GODARD: Modeling of the anisotropic damage of the concrete with taking into account
unilateral effect: Application to the digital simulation of the containments.
Thesis of Université Paris VI, 2005.
[2]
P.B. BADEL: Contributions to the digital simulation of structures out of reinforced concrete. Thesis of
Université Paris VI, 2001.
[3]
R. DESMORAT: Dissymmetry of the anisotropic elastic behavior coupled or not with
the damage. C.R. Acad. Sci., t.328, Iib Series, 445-450, 2000.
[4]
J.J. MARIGO: Damage: Localization and Stabilité. School of numerical summer of analysis,
EDF/ECA/INRIA, July 2002.
[5]
H. KUPFER, H.K. HILSDORF, H. RUSCH: Behavior off concrete under biaxial stress, ACI
Newspaper, n°66-52, 656-666, August 1969.
[6]
Z.P. BAZANT, G. PIJAUDIER-CABOT:Measurement off characteristic length off not room
continuum, ASCE J. off Engng. Mech., 115, 755-767, 1989.
[7]
J. MAZARS, Y. BERTHAUD: An experimental technique applied to the concrete to create one
diffuse damage and to put in écidence its unilateral character, C.R. Acad. Sci.,
t. 308, Series II, 579-584, 1989.
[8]
E. HOGNESTAD, N.W. HANSON, D. McHENRY: Concrete stress distribution in ultimate
strength design, J. Am. Concr. Inst. 66, 656-666, 1955.
[9]
S. RAMTANI
:Contribution to the modeling of the multiaxial behavior of the concrete
damaged with description of the unilateral character, Thèse of doctorate of Université Pierre
and Marie Curie (Paris VI), 1990.
[10]
J.B. LEBLOND: With constitutive inequality for hyperelastic materials in finite strain, Eur.
J. Mech. With/Solids, vol. 11 (4), 447-466, 1992.
[11]
B. HALPHEN, Q.S. NGUYEN: On generalized standard materials, Journal of
Mechanics, vol. 14 (1), 39-63, 1975.
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Law of behavior ENDO_ORTH_BETON
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:
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