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Titrate:
Law of behavior
ENDO_ISOT_BETON
Date:
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EDF-R & D/AMA















Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
Document: R7.01.14



Law of behavior
ENDO_ISOT_BETON




Summary:

This documentation presents the theoretical writing and the numerical integration of the law of behavior
ENDO_ISOT_BETON
who describes an asymmetrical local damage mechanism of the concretes, with effect of
restoration of rigidity. In addition to the local model, the nonlocal formulation with regularized deformation is also
supported to control the phenomena of localization.
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Law of behavior
ENDO_ISOT_BETON
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Count
matters
1
Introduction ­ Applicability .................................................................................................... 3
2
Local law of behavior ................................................................................................................... 3
2.1
Theoretical writing ............................................................................................................................ 3
2.2
Taking into account of the withdrawal and the temperature .............................................................................. 6
2.3
Identification of the parameters .......................................................................................................... 6
2.3.1
Elastic parameters ............................................................................................................ 6
2.3.2
Parameters of damage ............................................................................................. 6
2.3.2.1
Use without dependence with containment ............................................................ 6
2.3.2.2
Use with dependence with containment ............................................................ 6
2.3.2.3
Passage of the values “user” to the values “models” ..................................... 7
2.4
Numerical integration ...................................................................................................................... 7
2.4.1
Evaluation of the damage ........................................................................................... 7
2.4.2
Calculation of the tangent matrix ................................................................................................. 8
2.4.2.1
Stamp tangent with constant damage ......................................................... 8
2.4.2.2
Term of the tangent matrix due to the evolution of the damage ....................... 9
2.4.3
Case of material completely damaged ...................................................................... 10
2.5
Description of the variables intern ................................................................................................. 10
3
Formulation with regularized deformation .......................................................................................... 11
3.1
Formulation .................................................................................................................................... 11
3.2
Integration of the law of behavior ........................................................................................... 11
3.3
Variables intern .......................................................................................................................... 11
4
Piloting by elastic prediction .......................................................................................................... 12
5
Bibliography ........................................................................................................................................ 12
Appendix 1
Demonstration of the clean reference mark of stress ........................................................ 13
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Law of behavior
ENDO_ISOT_BETON
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1
Introduction ­ Applicability
The law of behavior
ENDO_ISOT_BETON
aim at modelizing the most simply possible one
behavior of elastic concrete fragile. It can be seen like an extension of the law
ENDO_FRAGILE
[R5.03.18] (with which it keeps a proximity of unquestionable formulation) for
applications of Civil Engineering.
As for the law
ENDO_FRAGILE
, the material is isotropic. Rigidity can decrease, the loss of
rigidity measured by an evolving/moving scalar of 0 (healthy material) to 1 (completely damaged material).
On the other hand, contrary to
ENDO_FRAGILE
, the loss of rigidity distinguishes traction from
compression, to privilege the damage in traction. Moreover this loss of rigidity can
to disappear by return in compression, it is a question of the phenomenon of restoration of rigidity with
refermeture. It as should be noted as this law of damage aims at describing the rupture of the concrete in
traction; it is not thus adapted at all to the description of the nonlinear behavior of the concrete in
compression. It thus supposes that the concrete remains in a moderate compactness.
The law
ENDO_ISOT_BETON
present of softening, which generally involves a loss
of ellipticity of the equations of the problem and consequently a localization of the deformations, from where one
pathological dependence with the mesh. To mitigate this deficiency of the model, a formulation not
local must be adopted: for the law
ENDO_ISOT_BETON
, modeling
GRAD_EPSI
[R5.04.02],
based on the regularization of the deformation is usable. In this formulation, it should be noted that
only the relations of behavior are faded compared to a conventional local modeling;
consequently, the stresses preserve their usual direction.
Lastly, that one activates or not the nonlocal formulation, softening character of the behavior
also involve the appearance of instabilities, physics or parasites, which result in
snap-backs on the total answer and returns the piloting of the essential loading in statics.
piloting of the type
PRED_ELAS
[R5.03.80] then the mode of control of the loading it seems
more adapted.

2
Local law of behavior
2.1 Writing
theoretical
If one seeks to take account of the effect of refermeture, it is necessary to pay a great attention to
continuity of the stresses according to the deformations (what is an essential condition for
a law of behavior in a computation software by finite elements), cf [bib1]. Indeed, if one
modelize this effect in a too simplistic way, the law of behavior is likely great to present
a discontinuous answer.
To take account of the refermeture (i.e the transition between traction and compression), it is necessary to start
by finely describing what one calls traction and compression, knowing that in traction (resp.
compression) the fissure will be considered “open” (resp. “closed”). A natural solution is of
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):
()
()
+
=
I
I
2
2
tr
2
µ
éq 2.1-1
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
.
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ENDO_ISOT_BETON
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According to the rather reasonable principle according to - in a case of traction (“open fissure”), one corrects
the elastic energy of a factor of damage; in a case of compression (“closed fissure”),
one keeps the expression of elastic energy -, the free energy endommageable is written:
()
()
(
)
()
()
()




+
-
+
-
+




+
-
+
-
=
I
I
I
I
H
D
D
H
H
D
D
H
D
µ
1
1
tr
1
1
tr
tr
2
,
2
2
éq 2.1-2
It is noticed that the free energy is continuous with each change of mode. It is even
continuously derivable compared to the deformations, since it is nap of derivable functions (
function
(
)
X H X
2
is derivable) and the continuity of the partial derivative at the points
tr
=
0
and
I
=
0
is
immediate. One then clarifies the stresses (by knowing that they will be functions everywhere
continuous of the deformations). As in elasticity, the clean reference mark of the stresses coincides with
clean reference mark of the deformations, result shown in appendix.
One writes the stresses in the clean reference mark:
() (
)
()
(
)
()




+
-
+
-
+




+
-
+
-
=
II
II
II
II
H
D
D
H
H
D
D
H
µ
1
1
2
tr
1
1
tr
tr
éq
2.1-3
In this form, the continuity of the stresses with respect to the deformations is clear. The figure opposite
show the stress
1
in the plan
(
)
1
2
,
with constant damage (case 2D, deformation
plane). The effect of refermeture as well as the continuity of the stresses are quite visible.
Appear 2-a: illustration of continuity
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ENDO_ISOT_BETON
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The thermodynamic force
F
D
associated the internal variable of damage is written:
(
)
() ()
()


+
+
+
=
-
=
I
I
D
H
H
D
D
F
I
µ
2
2
2
tr
tr
2
1
1
éq
2.1-4
It remains to define the evolution of the damage. The diagram selected is that of the standard models
generalized. A criterion should be defined, that one takes in the form:
()
(
)
F F
F
D
K
D
D
=
-
,
éq
2.1-5
where K defines the threshold of damage. In order to take into account, on the level of the evolution of
the damage, the effect of containment, the threshold
K
depends on the state of deformation, in the form:
() (
)
tr
tr
1
0
-
+
=
H
K
K
K
éq
2.1-6
One compels oneself to remain in the field:
()
F F
D
0
éq 2.1-7
The evolution of the variable of damage is then determined by the conditions of Kuhn-Tucker:


=
<
=
0
for
0
0
for
0
F
D
F
D
&
&
éq 2.1-8
Note:
From a formal point of view, the generalized standard materials are characterized by one
potential of dissipation function positively homogeneous of degree 1, transformed
Legendre-Fenchel of the indicating function of the field of reversibility, which is thus worth here:
()
()
()
&
sup
&
&
&
/
D
F D
kd
I
D
F
F F
D
IR
D
D
=
=
+
+
0
éq
2.1-9
One will note the presence of an indicating function relating to
&d
, which ensures that the damage is
growing.
It still remains to take into account the fact that the damage is raised by 1. From a point of view
intuitive, that seems easy. To keep a writing completely compatible with the formalism
generalized standard, it is enough to introduce an indicating function of the acceptable field into
the expression of the free energy:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
µ
,
]
; ]
D
tr
H
tr
D
D H tr
H
D
D H
I
D
I
I
I
I
=
-
+ -
+




+
-
+ -
+




+
-
2
1
1
1
1
2
2
1
éq
2.1-10
The introduction of this indicating function prevents the damage from exceeding 1, indeed, for
D
=
1
,
F
D
D
= -
= -

, and the damage does not evolve/move any more.
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Law of behavior
ENDO_ISOT_BETON
Date:
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HT-66/05/002/A
2.2
Taking into account of the withdrawal and the temperature
The law of behavior takes into account a possible withdrawal of dessication, a possible withdrawal
endogenous and a possible thermal deformation. Deformation
in question in it
document being then “elastic strain”
Re
rd
HT
-
-
-
=
~
.
On the other hand, the parameters materials in question in the next paragraph are considered
like constants (in particular, they cannot depend on the temperature, in the state of
current development)
2.3
Identification of the parameters
The parameters of the law of behavior are 4 or 5 (see following paragraphs). They
are classically provided in the operator
DEFI_MATERIAU
.
2.3.1 Parameters
rubber bands
They are simplest: it is about the two conventional parameters, Young modulus and coefficient of
Poisson, provided under the key word
ELAS
or
ELAS_FO
of
DEFI_MATERIAU
.
2.3.2 Parameters
of damage
According to whether the user wants to use the dependence of the threshold with containment or not, 2 should be provided
or 3 parameters to control the law of damage.

2.3.2.1 Use without dependence with containment
In this case, one considers that the parameter
1
K
is null. It should be noted that the compactness of
concrete must remain moderate so that the law remains valid (compressive stress about
some times the stress with the peak of traction, in absolute value).
The user must inform, under the key word
BETON_ECRO_LINE
of
DEFI_MATERIAU
, values of:
·
SYT: limit of simple tensile stress,
·
D_SIGM_EPSI
: slope of the curve post-peak in traction.

2.3.2.2 Use with dependence with containment
In this case, the dependence with containment makes it possible the concrete to keep a realistic behavior in
compression until the order of magnitude of appearance of nonthe linearity in compression, given by
SYC, cf below (classically, a compressive stress of about ten times the stress
with the peak of traction, in absolute value)
The user must inform, under the key word
BETON_ECRO_LINE
of
DEFI_MATERIAU
, values of:
·
SYT: limit of simple tensile stress,
·
SYC: limit of the simple compressive stress,
·
D_SIGM_EPSI: slope of the curve post-peak in traction.
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Law of behavior
ENDO_ISOT_BETON
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2.3.2.3 Passage of the values “user” to the values “models”
For information, one obtains the values of
,
0
K
and possibly
1
K
(if the user informed
SYC) by the following formulas:
EPSI
SIGM
D
E
_
_
-
=




+
-
+


+
=
1
2
1
2
1
)
(
2
2
0
E
SYT
K
(
)
(
) (
)
(
) (
)
SYC
E
K
SYC
K
2
1
2
1
1
1
0
2
1
-
-
-
+
+
=

2.4 Integration
numerical
Two points are to be regulated before establishing the model: the first relates to the evaluation of
the damage; the second consists in calculating the tangent matrix, calculation made a little more delicate
that usually by the passage in a clean reference mark of deformation.
One places oneself here within the framework of the implicit integration of the laws of behavior. Dependence of
criterion according to containment [éq 2.1-6] is taken into account in explicit form, i.e the threshold
K
is entirely determined by the state of deformation of the preceding pitch, this to simplify integration
model.
2.4.1 Evaluation of the damage
As one will see it, a simple scalar equation makes it possible to obtain the damage, which allows
to avoid a recourse to the iterative methods.
One notes
D
-
the damage with the preceding pitch and
D
+
the evaluation of the damage to the pitch
running to the current iteration which will be the damage with the current pitch when convergence is
attack. Simplest to evaluate the damage of the current iteration is to suppose that one
reached the criterion at the current moment, which results in:
()
(
)
(
)
(
)
(
)
F F
D
tr
H tr
H
K
D
I
I
I
=
+
+
+




=
0
1
1
2
2
2
2
µ
éq
2.4.1-1
what gives:
(
)
(
)
(
)
D
K
tr
H tr
H
test
I
I
I
=
+
+


-


1
1
2
1
2
2
µ
éq
2.4.1-2
3 cases arise:
·
D
D
test
-
: that wants to say that at the moment running, the criterion is not reached, one concludes from it that
D
D
+
-
=
,
·
D
D
test
-
1
: the criterion is thus reached, the condition of coherence implies
D
D
test
+
=
,
·
D
test
1
: the material is then ruined in this point, from where
D
+
=
1
.
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Law of behavior
ENDO_ISOT_BETON
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P. BADEL
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HT-66/05/002/A
2.4.2 Calculation of the tangent matrix
The tangent matrix is the sum of two terms, the first expressing the relation
stress/deformation with constant damage, the second being resulting from the condition
F
=
0
. In
effect, one can write:
ij
kl
ij
kl
D C
ij
kl
F
you
D
D
=




+




=
=
0
éq
2.4.2-1
If the user asks for calculation with tangent matrix (cf documentation of
STAT_NON_LINE
,
[U4.51.03]), the law of behavior provides the expression given by [éq 2.4.2-1]. On the other hand, if
the user asks for calculation with the matrix of discharge, the law of behavior provides the matrix
secant, i.e. the first term of the member of straight line of [éq 2.4.2-1].


2.4.2.1 Stamp tangent with constant damage
As we underlined previously, the calculation of the tangent matrix is a little delicate
fact of the writing of the model in the clean reference mark of deformation. Thus, one knows easily
stamp tangent with constant damage in the clean reference mark of deformation, but what one
seek is this same tangent matrix in the total reference mark.
If the damage does not evolve/move, in the clean reference mark of deformation, the matrix
sought a simple relation of degraded elasticity expresses:
(
)
()
()
()




+
-
+
-
+




+
-
+
-
=






=
J
J
ij
C
D
J
I
H
D
D
H
tr
H
D
D
tr
H
you
µ
1
1
2
1
1
~
~
éq
2.4.2.1-1
It is now necessary to express the passage of the total reference mark to the clean reference mark of deformations, at least
if the eigenvalues of deformation are different. The tangent matrix not being
necessary that the algorithms of numerical resolution (diagram of Newton), one will allow oneself, at the time of
calculation of the tangent matrix (and only in this case) to disturb the possible ones numerically
identical eigenvalues (in order to make them distinct). It will be noticed in particular that that
allows, null damage, to find the matrix of elastic rigidity.
One notes with a tilde the tensors in the clean reference mark of deformation (which, one points out it, is too
the clean reference mark of stresses). By definition, while noting
I
U
the clean vector associated the i-ème
eigenvalue, the matrix basic change
(
)
3
2
1
U
U
U
Q
=
, one a:
m
jm
im
m
jm
im
m
jm
im
ij
T
Q
Q
Q
Q
Q
Q
Q
Q
~
~
~
~
+
+
=
=
If the eigenvalues of deformation are distinct, evolution of the clean vectors and
eigenvalues is given by (cf previously [§2]):
K
J
jk
K
J
U
U
~
~
~
-
=
&
&
for
J K

éq
2.4.2.1-2
II
I
&
&
~
~
=
for
J K
éq
2.4.2.1-3
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ENDO_ISOT_BETON
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One deduces some easily
Q
:
()
-
=
-
=
J
K
J
K
ik
K
J
jk
I
K
K
J
jk
ij
Q
U
Q
~
~
~
~
~
~
éq
2.4.2.1-4
While using then (the last expression being used only to obtain a clearly symmetrical matrix)
:
(
)
kl
kj
Li
lj
ki
kl
lj
ki
ij
Q
Q
Q
Q
Q
Q
+
=
=
2
1
~
One thus obtains:
kl
m
m
N L
K
N
m
kN
lm
jn
im
kl
m
m
N L
K
N
m
jm
kN
lm
in
kl
m
m
N L
K
N
m
km
jn
im
kl
m
m
N L
K
N
m
jm
km
in
L
K
N
m
kl
N
m
kN
jm
im
N
m
m
m
jm
im
m
jm
im
N
N
m
jm
im
ij
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
~
~
~
2
1
~
~
~
2
1
~
~
~
2
1
~
~
~
2
1
~
~
~
~
~
~
~
,
,
,
ln
,
ln
,
,
,
ln
,
-
+
-
+
-
+
-
+




=
+
+




=
éq 2.4.2.1-5
The tangent matrix with constant damage is thus written:
(
)
(
)
=




-
+
+
+




=
=
m
N N
m
m
m
N
im
jn
jm
in
kN
lm
km
N
m
C
D
N
m
kN
jm
im
kl
ij
ijkl
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
With
you
,
ln
,
ln
~
~
~
2
1
~
~
éq 2.4.2.1-6

2.4.2.2 Term of the tangent matrix due to the evolution of the damage
The expression to be evaluated is written:
ij
kl
F
D
D




=
0
éq
2.4.2.2-1
One writes the equation [éq 2.4.1-1] in the form:
(
)
(
)
[]
1
1
2
+
+
=
D
W
K
éq
2.4.2.2-2
with:
(
)
(
)
(
)
(
)
W
tr
H tr
H
I
I
I
µ
=
+
2
2
2
.
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Titrate:
Law of behavior
ENDO_ISOT_BETON
Date:
26/05/05
Author (S):
P. BADEL
Key
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:
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HT-66/05/002/A
While differentiating this expression, it comes:
(
)
(
)
(
)
(
)
-
+
+
+
+
+
=
2 1
1
1
1
0
3
2
D
W
D
D
el
éq
2.4.2.2-3
with:
el
W
=
One uses then the following equality:
(
)
D
D
el
=
+
+
1
1
2
éq
2.4.2.2-4
One concludes:
(
)
(
)
ij
kl
F
ijel
kl
el
D W




=
+
+
=
0
1
2 1
éq
2.4.2.2-5
2.4.3 Case of completely damaged material
In the case of the completely damaged material,
1
=
D
, the rigidity of the material point can
to cancel itself. That poses problem for the stress by no means; on the other hand, that can involve
null pivots in the matrix of rigidity. To mitigate this difficulty, one allows oneself to define a rigidity
minimal, for the tangent matrix or the matrix of discharge. This minimal rigidity does not affect
value of the damage (which can reach 1) or the stress (which can reach 0).
To preserve a reasonable conditioning of the matrix of rigidity, minimal rigidity is taken with
5
10
-
rigidity initale. An indicator
specify the behavior during the pitch of current time:
·
0
=
: no evolution of the damage during the pitch,
·
1
=
: evolution of the damage during the pitch,
·
2
=
: saturated damage
1
=
D
.

2.5
Description of the internal variables
The model has two internal variables:
·
VI (1): damage
D
,
·
VI (2): indicator
.
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3
Formulation with regularized deformation
3.1 Formulation
The approach with regularized deformation [R5.04.02] also makes it possible it to control the phenomena of
localization and for this reason seems an alternative to the formulation with gradient
of damage. But at the difference in the latter, this formulation has the advantage of
to resort to the standard algorithms for the nonlinear problems. Indeed, the only difference by
report/ratio with the local law of behavior lies in the data of two deformations instead of one,
local deformation
who intervenes in the forced relation ­ deformation and the deformation
regularized
who controls the evolution of the damage. The aforementioned results from the local deformation
by resolution of the system of partial derivative equations according to:



=
=
-
N
N
0
0
normal
of
edge
on
structure
in
2
B
L
éq
3.1-1
where the characteristic length
B
L
is again well informed under the key word
LONG_CARA
of
DEFI_MATERIAU
. Finally, the relation of behavior is written in the following way, the equation
(2-3) remains identical, while the equation [éq 2.1-5] takes into account the regularized deformation:
() (
)
()
(
)
()




+
-
+
-
+




+
-
+
-
=
II
II
II
II
H
D
D
H
H
D
D
H
µ
1
1
2
tr
1
1
tr
tr
éq
3.1-2
()
()
K
D
F
F
F
D
D
-
=
,
éq 3.1-3
3.2
Integration of the law of behavior
One of the advanced advantages for the nonlocal formulation with regularized deformation is the little of
amendments which it involves in the construction of the law of behavior. Indeed, the integration of
internal variables is completely controlled by the regularized deformation
.
The method of integration is exactly that described in the paragraph [§2.4.1], in condition of
to replace the deformation by the deformation regularized in the equations.
For the calculation of the tangent matrix, the expressions are the same ones as those given to
paragraph [§2.4.2], certain expressions of the deformation are to be replaced by the deformation
regularized (when the deformation concerns the criterion), while others do not change (when
deformation raises of the forced relation ­ deformation).

3.3 Variables
interns
They are the same internal variables as for the local law:
·
VI (1) damage
D
,
·
VI (2) indicating
.
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4
Piloting by elastic prediction
The piloting of the type
PRED_ELAS
control the intensity of the loading to satisfy some
equation related to the value of the function threshold
el
F
during the elastic test. Consequently, only them
points where the damage is not saturated will be taken into account. The algorithm which deals with
this mode of piloting, cf [R5.03.80], requires the resolution of each one of these points of Gauss of
the following scalar equation in which
is a data and
the unknown factor:
()
=
el
f~
éq 4-1
The function
el
f~
provides the value of the function threshold during an elastic test when the field of
displacement breaks up in the following way according to the scalar parameter
:
1
0
U
U
U
+
=
éq 4-2
where
0
U
and
1
U
are given. Thanks to the linearity in small deformations of the operators deformation
(calculation of the deformations starting from displacements) and regularized deformation, one also obtains
following decompositions:
1
0
1
0
+
=
+
=
and
éq
4-3

The function
el
F
presenting the good property to be convex, the equation [éq 4-1] presents zero, one or
two solutions, which are required as follows:
·
Determination of the number of solutions per study at the boundaries
±
and possibly (if
value at the two boundaries is each time positive) determination if
el
F
present a minimum
negative;
·
Determination of a framing of each solution starting from the preceding study
·
Determination of the solution (for a convex function knowing the framing, this
search is simple and fast)



5 Bibliography
[1]
P.B. BADEL: Contributions to the digital simulation of structures out of reinforced concrete. Thesis of
the University Paris VI, 2001.
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Appendix 1 Demonstration of the clean reference mark of stress

The term traces some in energy does not pose a problem: it is invariant by any change of reference mark.
Remain the term in
(
)
(
)
I
I
I
I
H
D
D H
2
1
1
-
+ -
+




.
Notation: one writes with an index (for example
I
) the i-ème eigenvalue of a tensor which is written (while clarifying
its two indices)
kl
.
·
If the eigenvalues of the deformation are all distinct, it is shown whereas
&
&
I
II
=
, with
&
kl
components of
&
in the fixed reference mark coinciding with the clean reference mark of deformation with
the moment considered (in this reference mark one thus has
kl
K
kl
=
).
Indeed, let us write the deformations in the form:
=
I
I
I
I
U
U
While differentiating this expression, it comes:
&
&
&
&
=
+
+
I
I
I
I
I
I
I
I
I
I
U
U
U
U
U
U
By using the fact that the clean vectors are orthonormés:
U U
U U
U U
I
J
ij
I
J
I
J
=
+
=
&
&
0
one obtains the variations of the eigenvalues and the clean vectors:
&
&
I
II
=
and
&
&
U U
J
K
jk
J
K
=
-
for
J K
This is obviously valid only if the eigenvalues are distinct (as one can see it
clearly on the expression of the variations of the clean vectors). That comes owing to the fact that them
clean vectors are not continuous functions of the elements of the matrix.
·
If two eigenvalues of deformations are equal (and apart from the case very
private individual where they are also null), they are either positive, or negative. Let us take the case
where they are positive (the other case lends itself to a demonstration in any similar point). Energy
concerning these two eigenvalues is written then:
=
3
2
2
I
I
(the two equal eigenvalues are
considered to have indices 2 and 3). By differentiating this expression, one obtains:
=
=
=
3
2
3
2
2
2
I
I
I
I
I
D
D
while noting
the common eigenvalue.
By invariance of the trace of a matrix, here the restriction of the deformation on the clean plan
considered, one obtains:
=
=
=
3
2
3
2
I
II
I
I
D
D
, whatever the evolution which underwent the clean reference mark at this time.
For the remaining eigenvalue (distinct from both others and index 1 with the selected notations),
one a:
11
1
D
D
=
.
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By gathering these expressions, one obtains:
()
()
(
)
()
()
=
=
=
=


+
=


3
1
3
2
2
1
2
1
3
1
2
2
I
II
II
II
I
I
I
I
I
I
D
H
H
D
H
D
H
D
In conclusion, that the eigenvalues are distinct or not, one obtains:
(
)
(
)
D
H
H
D
I
I
I
II
II
I
II
2
2




=
with the adopted notations.
This reasoning spreads easily with the case of three equal eigenvalues.
The differential of energy with constant damage is written then:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
D
D
tr D tr
H
tr
D
D H tr
D
H
D
D H
D C
II
II
I
II
II
you
µ
,
=
=
-
+ -
+




+
-
+ -
+




1
1
2
1
1
On this expression, one observes well that the clean reference mark of deformation is also reference mark
clean of stress.