Code_Aster ®
Version
8.2
Titrate:
Operator CALC_G_THETA_T


Date:
31/01/06
Author (S):
E. CRYSTAL, O. BOITEAU, G. Key NICOLAS
:
U4.82.03-H1 Page:
1/24

Organization (S): EDF-R & D/AMA, SINETICS

Handbook of Utilization
U4.8- booklet: Postprocessing and dedicated analyzes
Document: U4.82.03

Operator CALC_G_THETA_T

1 Goal

To calculate the rate of refund of energy in 2D or 3D and the stress intensity factors in 2D.

This operator calculates the following sizes of breaking process:

·
the rate of refund of energy G in 2D or 3D by the method in the case of a problem
thermo or not linear linear rubber band [R7.02.01] and [R7.02.03], in statics or in
dynamics [R7.02.02]
·
the bilinear form G, function of a series of displacements, such as G (U, U) =G (U),
·
stress intensity factors K1 and K2 in 2D (plane or forced deformations
plane) by the method of singular displacements in the case of a problem
thermo linear rubber band [R7.02.05].

For studies mechanic-reliability engineers of evaluation of probability of starting of the rupture, one calculates in
more of the rate of refund of energy G, its derivative compared to a variation of field controlled by
a suitable function theta [R7.02.01] [R4.03.01]. This option is limited to the thermo problems
linear rubber bands 2D being pressed on quadratic finite elements.

Before a first use, it is advised to refer to the consulting and reference documents
of correspondents use, in particular the document [U2.05.01].

Functionalities concerning the rate of refund of energy with Lagrangienne propagation
(i.e. for an extension of the fissure by using the same grid) in 2D or 3D in
cases of a thermo problem elastic linear are described in the document [R7.02.04].

This operator generates a concept of the type counts.
Handbook of Utilization
U4.8- booklet: Postprocessing and dedicated analyzes
HT-62/06/004/A

Code_Aster ®
Version
8.2
Titrate:
Operator CALC_G_THETA_T


Date:
31/01/06
Author (S):
E. CRYSTAL, O. BOITEAU, G. Key NICOLAS
:
U4.82.03-H1 Page:
2/24

2 Syntax

[tabl_ *] = CALC_G_THETA_T
(
MODELE
=
Mo,
[model]


CHAM_MATER = to subdue,
[cham_mater]



THETA
=
theta,
[cham_no_DEPL_R]

# recovery of the field of displacements




/
DEPL =
depl,
[cham_no_DEPL_R]
/
VITE =
quickly,
[cham_no_DEPL_R]
ACCE = acce,
[cham_no_DEPL_R]

/
RESULTAT
=
resu,
/
[evol_elas]
/
[evol_noli]
/
[dyna_trans]
/
[mode_meca]

# if RESULTAT of the evol_elas type, evol_noli or dyna_trans




/TOUT_ORDRE = “YES”,
[DEFAUT]
/
NUME_ORDRE =
l_ordre,
[l_I]
/
LIST_ORDRE =
read
,
[listis]
/
INST =
l_inst, [l_R8]
/
LIST_INST
=
l_reel, [listr8]

# if RESULTAT of the mode_meca type




/TOUT_MODE



=
“OUI”,
[DEFAUT]
/
NUME_MODE




=
l_ordre,
[l_I]
/
LIST_MODE




= read
,
[listis]
/
FREQ =
l_inst, [l_R8]
/
LIST_FREQ
=
l_reel, [listr8]




| PRECISION
=
/
prec,
[R]
/
1.0D-6, [DEFAUT]





| CRITERION =/“RELATIVE”, [DEFECT]
/
“ABSOLU”
,

# loading



EXCIT = (_F (LOAD
=
charge
,
[char_meca]

















[char_cine_meca]







FONC_MULT = fmult,
[function]
[formula]
),)


SYME_CHAR
=
/
“SANS”
,
[DEFAUT]
/
“SYME”
,
/
“ANTI”
,

# behavior



/COMP_ELAS
=_F
(




RELATION
=
/
“ELAS”, [DEFECT]
/
“ELAS_VMIS_LINE”,
/
“ELAS_VMIS_TRAC”,




DEFORMATION
=
/
“PETIT”,
[DEFAUT]
/
“GREEN”,




/ALL = “YES”,
[DEFAUT]
/
|
GROUP_MA
=
lgrma,
[l_gr_maille]
|
MAILLE
=
lma
,
[l_maille]








),



Handbook of Utilization
U4.8- booklet: Postprocessing and dedicated analyzes
HT-62/06/004/A

Code_Aster ®
Version
8.2
Titrate:
Operator CALC_G_THETA_T


Date:
31/01/06
Author (S):
E. CRYSTAL, O. BOITEAU, G. Key NICOLAS
:
U4.82.03-H1 Page:
3/24


/
COMP_INCR
=_F
(




RELATION
=
/
“ELAS”, [DEFECT]
/
“VMIS_ISOT_TRAC”,
/
“VMIS_ISOT_LINE”,




DEFORMATION =/“SMALL”, [DEFECT]
/
“PETIT_REAC”
,




/ALL = “YES”
,
[DEFAUT]
/
|
GROUP_MA
=
lgrma
,
[l_gr_maille]
|
MAILLE
=
lma
,
[l_maille]








),


ETAT_INIT
=_F
(



/ | DEPL = EPD
,
[cham_no_DEPL_R]





| SIGM =
sig
,
/
[carte_SIEF_R]
/
[cham_elem_SIEF_R]





),

# option requested: - calculation of the rate of refund of energy G
#
-
calculation

coefficients
of intensity
of
constraints
K1
and
K2
#
-
calculation
of
G
with
propagation
Lagrangian
#
-
calculation
of

form
bilinear
G
#





- calculation of derived from G compared to one
variation of field.



OPTION
=
/
“CALC_G”
,
[DEFAUT]
#






/“CALC_K_G”
,







FOND_FISS
= fiss, [fond_fiss]
#





/“K_G_MODA”
,







FOND_FISS
= fiss, [fond_fiss]

#






/
“CALC_G_LAGR”,







PROPAGATION = alpha
, [l_Kn]
#
/
“G_BILINEAIRE”
,
#






/
“CALC_G_MAX”,







BORNES
=_F
(
NUME_ORDRE = num, [I]








VALE_MIN
= qmin
, [R]








VALE_MAX
= qmax
, [R]
),









SENSIBILITE
=
(… to see [U4.50.02])


TITER
=
titrate
, [l_Kn]

# impression
information


INFORMATION =/1, [DEFECT]






/2,

)
Handbook of Utilization
U4.8- booklet: Postprocessing and dedicated analyzes
HT-62/06/004/A

Code_Aster ®
Version
8.2
Titrate:
Operator CALC_G_THETA_T


Date:
31/01/06
Author (S):
E. CRYSTAL, O. BOITEAU, G. Key NICOLAS
:
U4.82.03-H1 Page:
4/24

3 Operands

3.1 Operand
MODELE


MODELE =
Mo

Mo is the name of the model on which G, K1 and K2 are calculated. It is generated by the command
AFFE_MODELE [U4.41.01].

The name of the model is:
· Optional if the field of displacement is given with key word RESULTAT and if
structure of data resu is of type EVOL_ELAS, EVOL_NOLI:
- If the name of the model misses, the operator takes that which is present in
structure of data resu;
- If the name of the model is provided by the user, the operator checks if it is identical to
that present in the structure of data resu, the contrary case an error
fatal is emitted.
· Obligatory in all the other cases.

The calculation of the rate of refund of energy G (option “CALC_G”) is valid for modelings
following:
·
D_PLAN,
·
C_PLAN,
·
AXIS,
·
3D.

These modelings correspond:

·
for a two-dimensional medium with triangles with 3 or 6 nodes, quadrangles with 4, 8 or
9 nodes and of the segments with 2 or 3 nodes,

·
for a three-dimensional medium with hexahedrons with 8, 20 or 27 nodes, pentahedrons with
6 or 15 nodes, of the tetrahedrons with 4 or 10 nodes, of the pyramids with 5 or 13 nodes, of
faces with 3, 4, 8 or 9 nodes.

The calculation of the stress intensity factors K1, K2 (option “CALC_K_G”) is possible for
following modelings:
·
D_PLAN,
·
C_PLAN.

The calculation of the stress intensity factors in 3D starting from the rate of refund of energy is
possible for linear elements with option “CALC_K_G” of the operator
CALC_G_LOCAL_T. To date, the option is not developed for axisymmetric modelings.

For a plane fissure in an elastic, homogeneous and isotropic material, one also can
to reach the values of K1, K2 and K3 by extrapolation of the jumps of displacements on the lips of
this fissure: order POST_K1_K2_K3 [U4.82.05].

The calculation of derived from the rate of refund of energy compared to a variation of field
is licit only for modelings 2D (D_PLAN, AXIS and C_PLAN) in thermo linear elasticity.

Caution:

With this option, the configuration forced plane is taken besides into account only in
postprocessing of the calculation of mechanics, i.e. for the determination of the tensors of
deformations and of the constraints starting from displacements. They should not appear at the time
calculation of sensitivity of MECA_STATIQUE which supports only modelings D_PLAN
and AXIS. In such a configuration the user is of course an only judge of the relevance of
its results.
Handbook of Utilization
U4.8- booklet: Postprocessing and dedicated analyzes
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Code_Aster ®
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Titrate:
Operator CALC_G_THETA_T


Date:
31/01/06
Author (S):
E. CRYSTAL, O. BOITEAU, G. Key NICOLAS
:
U4.82.03-H1 Page:
5/24

3.2 Operand
CHAM_MATER


CHAM_MATER =
to subdue

to subdue is the field of material generated by command AFFE_MATERIAU [U4.43.03].

The name of the material field is:
· Optional if the field of displacement is given with key word RESULTAT and if
structure of data resu is of type EVOL_ELAS, EVOL_NOLI:
- If the name of the material field misses, the operator takes that which is present
in the structure of data resu;
-
If the name of the material field is provided by the user, the operator checks if it is
identical to that present in the structure of data resu. In the contrary case,
an alarm is emitted and calculation continues with the material field provided by
the user.
· Obligatory in all the other cases.

The material field makes it possible to recover the characteristics of material:

·
YOUNG modulus E,
·
NAKED Poisson's ratio,
·
thermal dilation coefficient ALPHA (for a thermomechanical problem),
·
elastic limit SY (for a nonlinear elastic problem),
·
slope of the traction diagram D_SIGM_EPSI (for a nonlinear elastic problem
with linear isotropic work hardening).

These characteristics can depend on the temperature only for option “CALC_G”.
They must be independent of the temperature for the calculation of the factors of intensity of
constraints (option “CALC_K_G”).

The calculation of sensitivity was developed only for elastic materials independent of
temperature. They can on the other hand be heterogeneous.

Characteristics SY and D_SIGM_EPSI are treated only for one elastic problem not
linear with work hardening of von Mises and the option of calculation of the rate of refund of energy
“CALC_G”. The calculation of the coefficients of intensity of constraints is treated only in elasticity
linear.

Note:

For the calculation of the stress intensity factors (option “CALC_K_G”), them
characteristics must be defined on all materials, including on the elements of
edge, because of method of calculation [R7.02.05]. To be ensured so it is advised of
to make AFFE = _F (TOUT = “OUI”) in command AFFE_MATERIAU [U4.43.03],
even if it means to use the rule of overload then.
Handbook of Utilization
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Code_Aster ®
Version
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Titrate:
Operator CALC_G_THETA_T


Date:
31/01/06
Author (S):
E. CRYSTAL, O. BOITEAU, G. Key NICOLAS
:
U4.82.03-H1 Page:
6/24

Problem of Bi-material:

1st case: There is a Bi-material but the point of fissure is in only one material, cf [Figure 3.2-a]. If
one is assured that the crown, definite enters the radii inferior R and higher R (the command
CALC_THETA [U4.82.02]), has like support of the elements of same material, calculation is possible
whatever the selected option. If not only option “CALC_G” is possible.
material 1
R
E
R
1, 1, 1
material 2
E2, 2, 2

Appear 3.2-a: Bi-material: 1st case
2nd case: There is a Bi-material where the point of fissure is with the interface, cf [Figure 3.2-b]. To date,
only the option of calculation of the rate of refund of energy (option “CALC_G”) is available. The calculation of
coefficients of intensity of constraints K1 and K2 is not possible in this case.

material 1
E1, 1, 1
R
R
material 2
E2, 2, 2

Appear 3.2-b: Bi-material: 2nd case

3.3 Operand
THETA

THETA = theta

The field is a field of vector in each node of the grid. It is a concept of the type
cham_no_DEPL_R. It can be affected directly with command AFFE_CHAM_NO [U4.44.11].
In practice, it results generally from the specific command CALC_THETA [U4.82.02] which
allows to affect the module, the direction of the field theta and the radii of the crown surrounding it
melts of fissure.

For more precise details to refer to [R7.02.01 §3].

The Councils:

·
To avoid using a field theta defined with a radius lower no one. Fields of
displacements are singular in bottom of fissure and introduce results vague in
postprocessing of breaking process.

·
It is advised to use successively command CALC_G_THETA_T with at least
3 fields theta of different crowns to ensure itself of the stability of the results. In
case of important variation (higher than 5-10%) it is necessary to wonder about the good catch in
count of all modeling.

·
In 2D, this field theta making it possible to determine the zone of calculation around the fissure is
completely independent of the field theta related to the calculation of sensitivity. The option takes in
count their possible recoveries of supports, even the displacement of the one by
report/ratio with the other.
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Titrate:
Operator CALC_G_THETA_T


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:
U4.82.03-H1 Page:
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3.4 Operands
DEPL/QUICKLY/ACCE/RESULT

These operands make it possible to recover the field of displacement (and speed and acceleration for
a calculation in dynamics) starting from a field with the nodes or extract of a result.

3.4.1 Operand
DEPL

/DEPL = depl

depl is a field with the nodes solution of calculation on Mo.

3.4.2 Operand
QUICKLY/ACCE

/
VITE = quickly


ACCE = acce

quickly and acce is respectively a field speed and a field of acceleration. It is
fields with the nodes solution of a dynamic calculation on Mo.
These two operands must be simultaneously present to calculate the rate of refund
energy into elastodynamic [R7.02.02].

3.4.3 Operand
RESULTAT

/RESULTAT = resu

Name of a concept result of the evol_elas type, evol_noli, dyna_trans or mode_meca.

3.4.4 Operands TOUT_ORDRE/NUME_ORDRE/LIST_ORDRE/INST/
LIST_INST/TOUT_MODE/NUME_MODE/LIST_MODE/FREQ/
LIST_FREQ/PRECISION/CRITERION

These operands are used with operand RESULTAT. See [U4.71.00].

3.5 Word
key
EXCIT and operands CHARGE/FONC_MULT

EXCIT = _F (LOAD
= load






FONC_MULT = fmult)

Key word EXCIT makes it possible to recover a list of loadings charges, resulting from the commands
Multiplying AFFE_CHAR_MECA or AFFE_CHAR_MECA_F [U4.44.01], and coefficients fmult.

Key word EXCIT is optional.

If displacements are provided by key word RESULTAT and that the structure of
data resu is of type EVOL_ELAS, EVOL_NOLI, the loading taken into account is is that
provided by the user, that is to say that extracted from resu if it misses command. If the loading
provided in resu (coherence of the name and the number of loads is different from that present, of
couples load-function), an alarm is emitted and calculation continues with the loading
indicated by the user.

In all the cases, it should be taken care that the loads indicated here were indeed taken into account
in the preceding mechanical calculation which produced the field of displacements.
Handbook of Utilization
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Code_Aster ®
Version
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Titrate:
Operator CALC_G_THETA_T


Date:
31/01/06
Author (S):
E. CRYSTAL, O. BOITEAU, G. Key NICOLAS
:
U4.82.03-H1 Page:
8/24

The loadings currently supported by various modelings are as follows:

Option Modeling
Loading
key word of AFFE_CHAR_MECA
or AFFE_CHAR_MECA_F
CALC G C_PLAN, D_PLAN, AXIS TEMP_CALCULEE
FORCE_CONTOUR
FORCE_INTERN
PESANTEUR
PRES_REP
ROTATION
EPSI_INIT

CALC_G,
3D TEMP_CALCULEE
FORCE_FACE
CALC_G_MAX,
FORCE_INTERN
PESANTEUR
G_BILINEAIRE PRES_REP
ROTATION
EPSI_INIT

CALC_K_G,
C_PLAN, D_PLAN
TEMP_CALCULEE
PESANTEUR
K_G_MODA
FORCE_INTERN
ROTATION
PRES_REP


FORCE_CONTOUR


EPSI_INIT

CALC_K_G,
AXIS, 3D
Modelings nonavailable
K_G_MODA
(in 3D: to see operator CALC_G_LOCAL_T)
Table 3.5-a: Périmètre, by modeling, of the licit loadings.

Note:

The loadings not supported by an option are ignored. To date, loadings
following being able to have a direction in breaking process is not treated:
·
FORCE_NODALE
·
FORCE_ARETE
·
DDL_IMPO on the lips of the fissure
·
FACE_IMPO

It is important to note that the only loadings taken into account in a calculation of mechanics
rupture with the method are those supported by the elements inside the crown,
where the field of vectors theta is nonnull (between Rinf and Rsup [R7.02.01 §3.3]). Only types of
load likely to influence the calculation of G are thus the voluminal loadings
(gravity, rotation), a nonuniform field of temperature or efforts applied to
lips of the fissure.

Caution:

·
A loading of comparable nature (for example voluminal force) can appear only in
only one load. In the contrary case, calculation finishes in error.
·
One observes also a rule of exclusion at the time of the simultaneous presence of a field of
deformations (via “EPSI_INIT”) and of initial displacements (via “ETAT_INIT/DEPL”
cf [§3.9]). Only one of both must remain.
·
It is not possible to date to associate a load defined as a function
(AFFE_CHAR_MECA_F) and a multiplying coefficient (FONC_MULT). In this case, it
calculation finishes in error.
·
For option CALC_K_G, if a loading is imposed on the lips of the fissure
(PRES_REP or FORCE_CONTOUR), then it is obligatorily necessary to direct them correctly
meshs of those (by using ORIE_PEAU_2D) before the calculation of K.
·
If one makes a calculation in great transformations (key word DEFORMATION: “GREEN” under
the key word factor COMP_ELAS or DEFORMATION = “PETIT_REAC” under the key word
factor COMP_INCR) the supported loadings must be died loads,
typically an imposed force and not a pressure [R7.02.03 §2.4]; these loads
must be declared like nonfollowing in STAT_NON_LINE.
Handbook of Utilization
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Code_Aster ®
Version
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Titrate:
Operator CALC_G_THETA_T


Date:
31/01/06
Author (S):
E. CRYSTAL, O. BOITEAU, G. Key NICOLAS
:
U4.82.03-H1 Page:
9/24

During calculation of derived from G compared to a variation from field (calculation of sensitivity),
only loadings PRES_REP and TEMP_CALCULEE are usable in the totality of
process. This software restriction is due only to the limited development of the option
SENSIBILITE in operator MECA_STATIQUE. As for modeling C_PLAN, them
other types of loading are taken into account only in postprocessing of the calculation of
mechanics. They cannot and they should intervene only for the assembly of the terms of
derived. They are thus modelled by inserted AFFE_CHAR_MECA or AFFE_CHAR_MECA_F
between MECA_STATIQUE and CALC_G_THETA_T (cf [§3.10], [§5.3]).
In addition one can, for the moment, only to handle loadings independent of
variation of field, in their intrinsic definitions as in those their supports.
In other words, their derivative eulériennes must be null.

3.6 Operand
SYME_CHAR

SYME_CHAR
=/“SANS”
, [DEFAUT]
/
“SYME”
,
/
“ANTI”
,

Key word SYME_CHAR makes it possible to indicate if the loading is symmetrical or antisymmetric in
case where one models only half of the solid compared to the fissure. This key word can be
essential if one uses option “CALC_K_G” to calculate the stress intensity factors: it
allows to assign K2 to 0. if the loading is symmetrical compared to the fissure or K1 with 0. if it is
antisymmetric.

This key word also makes it possible to multiply by 2, the values of the rate of refund of energy G and its
possible derivative, if one models only half of the solid compared to the fissure.


“WITHOUT” “SYME”
“ANTI”
G
G () 2.D0 * G () 2.D0 * G ()
K1
K1
K1 0.D0
K2
K2 0.D0
K2
Table 3.6-a: Prise in account of symmetry

3.7 Word
key
COMP_ELAS

COMP_ELAS
=

This key word factor makes it possible to define the relation of behavior of material used for it
postprocessing of breaking process.

By defect the relation of behavior is elastic linear in small deformations.

The calculation of derived from G compared to a variation of field is restricts with elasticity
linear (into pre and postprocessing), on the other hand it was also extended to the deformations of
Green-Lagrange.

Note:
· The calculation of the rate of refund of energy G or not has direction only in linear elasticity
linaire (COMP_ELAS). It is however possible to calculate in elastoplasticity
(COMP_INCR) a parameter G then defined as the total flow of energy (plasticity and
rupture) through the defect. In the case of elastoplasticity, the defect must be modelled
by a notch.
· Nothing prohibits to affect a behavior different during calculation from displacements (by
elastoplastic example) then to carry out this postprocessing with another relation (by
non-linear elastic example). The user is responsible for interpretation for
results obtained [R7.02.03].
· If the loading is perfectly radial monotonous, calculations in nonlinear elasticity and
in elastoplasticity lead to the same results. For this type of loading (and
only in this case), it is also possible to make an elastoplastic calculation on
a fissure.

For more precise details, to refer to [U2.05.01].
Handbook of Utilization
U4.8- booklet: Postprocessing and dedicated analyzes
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Code_Aster ®
Version
8.2
Titrate:
Operator CALC_G_THETA_T


Date:
31/01/06
Author (S):
E. CRYSTAL, O. BOITEAU, G. Key NICOLAS
:
U4.82.03-H1 Page:
10/24

3.7.1 Operand
RELATION

RELATION =

/
“ELAS”

Relation of elastic behavior linear i.e. the relation between the deformations
and the constraints considered is linear [R7.02.01 §1.1].

/
“ELAS_VMIS_LINE”

Relation of nonlinear elastic behavior, von Mises with isotropic work hardening
linear. The data materials necessary of the field material are provided in
operator DEFI_MATERIAU (cf the operator STAT_NON_LINE [U4.51.03] and the key word
VMIS_ISOT_LINE) [R7.02.03 §1.1] and [R5.03.20].

/
“ELAS_VMIS_TRAC”

Relation of nonlinear elastic behavior, von Mises with isotropic work hardening not
linear. The data materials necessary of the field material are provided in
operator DEFI_MATERIAU (cf the operator STAT_NON_LINE [U4.51.03] and the key word
VMIS_ISOT_TRAC) [R7.02.03 §1.1] and [R5.03.20].

3.7.2 Operand
DEFORMATION


DEFORMATION =
/
“PETIT”

The deformations used in the relation of behavior are the linearized relations:

(
1
U)
ij
=
(iu J + uj I)
2
,
,


/
“GREEN”

The deformations used in the relation of behavior are the deformations of
Green-Lagrange [R7.02.03 §2.1]:

(
1
U)
ij
=
(iu J + uj I + ku I ku J)
2
,
,
,
,


Caution:

·
The supported loadings are those supported in linear rubber band provided that it
are dead loads: charge imposed or nonfollowing pressure.
·
Displacements and rotations can be large but it is preferable to be limited
with small deformations if one wishes a coherence with actual material. For more
precise details to refer to [R7.02.03 §2.5].

3.7.3 Operands
ALL/GROUP_MA/MESH



/
TOUT =
“OUI”
,


/ | GROUP_MA = lgrma,
|
MAILLE
=
lma
,

Specify the meshs or the nodes on which the relation of behavior is used.
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3.7.4 Relation of behavior available for each option


“CALC_G”
“CALC_K_G”,
“K_G_MODA”
COMP_ELAS “ELAS”
“PETIT”


“PETIT”


“GREEN”
“ELAS_VMIS_LINE”
“PETIT” not
disp.


“GREEN”
“ELAS_VMIS_TRAC”
“PETIT” not
disp.


“GREEN”
Table 3.7.4-a: Disponibilité, by option, of the relations of behavior.

It is possible for these relations of behavior to calculate the rate of refund of energy G in
great transformations [R7.02.03 §2] with the proviso of having only dead loads.

3.8 Word
key
COMP_INCR

COMP_INCR
=

The relation of behavior elastoplastic is associated a criterion of von Mises with
isotropic or kinematic work hardening.


RELATION
=

/
“ELAS”

Relation of elastic behavior incremental [U4.51.03].

/
“VMIS_ISOT_LINE”

von Mises with linear isotropic work hardening ([U4.51.03] and [R5.03.20]).

/
“VMIS_ISOT_TRAC”

von Mises with isotropic work hardening given by a traction diagram [U4.51.03].

DEFORMATION
=

/
“PETIT”

Linearized deformations: = (U
) = 1/2 U

+ U


ij
ij
(I, J
J, I)

/
“PETIT_REAC”


U
U


I
J


ij = 1/2
+
(
[U4.51.03].
X + U)
(X + U)


J
I


ALL/GROUP_MA/MESH

The meshs specify on which the incremental relation of behavior is used.

3.9 Word
key
ETAT_INIT

ETAT_INIT
=

Initial State of reference selected. By defect, all the fields are identically null. The data
of an initial state does not have direction (and is not thus taken into account) only for the treated part of the field
in incremental behavior (COMP_INCR): if calculation is elastic (COMP_ELAS) that does not have
no incidence.
Taking into account the formula established in the source of CALC_G_THETA_T, it is not licit of
to cumulate an initial deformation (either directly in the load with EPSI_INIT, or like
below, in the form of displacement), with an initial stress field.
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Caution:

· If one wants to take into account an initial state in elasticity, it is key word ELAS located under
COMP_INCR which should be used.
· One observes a rule of exclusion at the time of the simultaneous presence of a field of
displacements (via CHARGE/EPSI_INIT cf [§3.5]) and of deformations initial (via
ETAT_INIT/DEPL). Only one of both must remain.

During calculation of derived as for the loadings, these initial states are not taken in
count that in postprocessing calculation of mechanics. They cannot and they do not have
to intervene that for the assembly of the terms of this derivative.
In addition one can, for the moment, only to handle states independent of the variation of
field, in their intrinsic definitions as in those their supports. In others
terms, their derivative eulériennes must be null (cf [§3.10] and [§5.3]).

/SIGM
=
sig
,
/DEPL
=
depl,

Respectively, displacement and stress fields taken in an initial state. They can by
being example resulting from command RECU_CHAMP, or to be read in a file with the format
I-DEAS by command LIRE_RESU. Either one gives an initial displacement, or a constraint
initial. Attention, if the load transmitted in operand CHARGE contains an initial deformation
(key word EPSI_INIT of AFFE_CHAR_MECA_F), this one will be taken into account in the same way
that depl displacement provided here; it is then illicit to give an initial state with the key word
DEPL.

3.10 Operand
OPTION

OPTION
=/“CALC_G”,

calculation of the rate of refund of energy G
/
“CALC_K_G”,

calculation of the coefficients of intensity of constraints K1, K2
/
“K_G_MODA”,

calculation of the modal coefficients of intensity of constraints
/
“CALC_G_LAGR”,
calculation of G with Lagrangienne propagation
/
“G_BILINEAIRE”,
calculation of the bilinear matrix G
/
“CALC_G_MAX”,
maximization of G under constraints terminals.

3.10.1 OPTION = “CALC_G” [R7.02.01] and [R7.02.03]

It is the default option. It allows the calculation of the rate of refund of energy G by the method theta
in 2D or 3D for an elastic thermo problem linear or not linear. In 3D, it is necessary to divide
gross amount of G given by Aster by the length of the fissure. In the same way into axisymmetric, it is necessary
to divide by the radius into bottom of fissure.

3.10.2 OPTION = “CALC_K_G” [R7.02.05]

This option allows the calculation of the coefficients of intensity of K1 constraints and K2 in thermo elasticity
linear planes (modeling C_PLAN or D_PLAN) by the method of the singular fields (use of
the bilinear form of G). It also calculates the rate of traditional refund of energy G is too
calculated.

If option “CALC_K_G” is used it is necessary to provide information on the bottom of fissure by the operand
FOND_FISS (cf [§11]).

3.10.3 OPTION = “K_G_MODA” [R7.02.05]

This option allows the calculation of the modal coefficients of intensity of constraints K1 and K2, i.e them
factors of intensity of the constraints associated with the clean modes with vibration with the structure.

Calculations are carried out in thermo linear elasticity planes (modeling C_PLAN or D_PLAN) by
method of the singular fields (use of the bilinear form of G), starting from a structure of
data RESULTAT of the mode_meca type only. The rate of refund of energy G is too
calculated.

If option “K_G_MODA” is used it is necessary to provide information on the bottom of fissure by the operand
FOND_FISS (cf [§11]).
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3.10.4 OPTION = “CALC_G_LAGR” [R7.02.04]

This option relates to only the propagation Lagrangienne [R7.02.04]. It is necessary to provide the value of
propagation behind key word PROPAGATION.

3.10.5 OPTION = “G_BILINEAIRE” [R7.02.01]

For a series of displacements (U,…, U), this option allows the calculation of the bilinear form
1
N
G (U, U for I J; if I = J then G (U, U) = G (U). The results are stored in a table
I
J)
comprising two indices I and J in reference to displacements U and U ordered in the list
I
J
contained in the structure of data result under key word RESULTAT.

Caution:

This option of calculation is valid only for linear elastic designs where
superposition of loading by linear combination is possible.

3.10.6 OPTION = “CALC_G_MAX” [R7.02.05]

This option relates to only the maximization of G in 3D under constraints terminals
[R7.02.05]. It is necessary to provide the value of the constraints terminals behind the key word BORNES, cf paragraph
3.14.

3.11 Operand
SENSIBILITE

SENSIBILITE = theta

Name of the significant parameter by report/ratio to which one derives (see [U4.50.02]).

With this operand one has access, in addition to the value of the rate of refund of energy such as it is
provided with “CALC_G”, with its derivative compared to a variation of field described by the field
theta sensitivity (cf [§3.10]).
Its perimeter of application limits to linear elastic calculations thermo 2D, resting on
complete or incomplete quadratic finite elements (SEG3, TRIA6, QUAD8 and QUAD9). They
support various modelings (cf [§3.1]), loadings (cf [§3.5]) and states initial (cf [§3.9]) in pre-or
postprocessings of mechanical calculation. The materials can be heterogeneous but they must be
independent of the temperature.

Note:

The field of investigation of this option is related among that of optionCALC_G_LAGR.
In both cases one avoids expensive parametric studies by using a fixed grid
of reference and by modelling the virtual variations of field by functions theta
adapted.
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The field S = theta is a field of vector 2D in each node of the grid. It is directed according to
the X-axis. It is a concept of the type CHAM_NO_DEPL_R. It can be affected directly with
command AFFE_CHAM_NO [U4.44.11].
In practice, it results generally from specific command CALC_THETA [U4.82.02] with
the option THETA_BANDE which makes it possible to seize the module (key word MODULE) and the X-coordinates x1 and x2
(key word R_INF and R_SUP) of the points delimiting its vertical support. It is pointed out that this field decrease
value MODULE with the zero value between the X-coordinates x1 and x2, and that it is null everywhere else. These
X-coordinates can be negative but one must have x1 < x2.
y
Field S
sensitivity
Field F
fissure
X
x1
X X
2
2

Appear 3.11-a: Dérivée of G (F) compared to a variation of field controlled by S

Note:

· Contrary to the field theta fissures which is just continuous and defined in the form of one
first order polynomial, this field theta is a combination of students'rag processions of
third command and it are of safe C2 class in the middle of its support (where it is right C1).
· Indeed, during the calculation of G one calls only upon the derived first of the field theta
fissure, whereas for obtaining his derivative one uses the derivative second theta
sensitivity. A compromise was thus found between the theoretical command required by
derivations and precision of the finite elements modelling calculation. Thus it is necessary to have recourse
with quadratic elements to estimate this derivative.

3.12 Operand
FOND_FISS

FOND_FISS = fiss,

This key word is obligatory if one uses option “CALC_K_G”. If not it is not used.

fiss is a concept of the fond_fiss type resulting from command DEFI_FOND_FISS. It allows
to recover the basic node of fissure and the normal with the fissure [U4.82.01].

3.13 Operand
PROPAGATION

PROPAGATION = alpha

This key word is obligatory if one uses option “CALC_G_LAGR”. If not it is not used.

alpha is the value of the propagation [R7.02.04].

3.14 Key word
BORNES

BORNES
=

This key word factor is obligatory if one uses option “CALC_G_MAX”. If not it is not used. It
allows to define couples of constraints terminals (Q, q+
I
I) for each sequence number of
structure of data result. One then seeks to define the combination of loading more
penalizing in term of rate of refund D `energy:
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NR


max G Q Q = max
I I
ij
G qiq J where IQ are the NR associated unit loadings
Q Q q+


I
I
I
I
I, j=1
with the various Ui displacements contained in the structure of data result, and
G = G (U, U
ij
I
J) bilinear form of G.


NUME_ORDRE = num

Sequence number in the structure of data result associated with the values of constraints
terminals.


VALE_MIN = qmin

Value minimal of the coefficient applied to the loading associated with the result stored in
sequence number num of the structure of data resu.


VALE_MAX = qmax

Value maximum of the coefficient applied to the loading associated with the result stored in
sequence number num of the structure of data resu.

Note:

·
The user must as many give couples of terminals of sequence numbers contained
in the structure of data result under penalty of fatal error.
·
This option of calculation is valid only for elastic designs linear where
superposition of loading by linear combination is possible.
·
An example of use of this option to maximize G in the presence of constraints
signed and not signed is given in the 5.4.

3.15 Operand
TITER

TITER = title

[U4.03.01].

3.16 Operand
INFO

INFO
=
/1,
[DEFAUT]



/2,

Level of messages in file “MESSAGE”.

Note:

For option “CALC_K_G”, if INFO is worth 2, one generates calculation and the impression (in the file
MESSAGE) of the angle of propagation of the fissure. This angle, calculated according to 3 criteria (K1 or G
maximum, minimal K2) according to the formulas of AMESTOY, BUI and DANG-VAN [R7.02.05 §2.5],
is given with a margin of 10 degrees.

3.17 Count
produced

Command CALC_G_THETA_T generates a concept of the type counts. This one contains the rate of
restitution of energy then possibly, according to the options, its derivative or factors of intensity of
constraints.

Command IMPR_TABLE [U4.91.03] makes it possible to print the results with the desired format.
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4
Standardization of the total rate of refund G

4.1
2D forced plane and plane deformations

In dimension 2 (plane constraints and plane deformations), the bottom of fissure is tiny room to a point and
the value G () resulting from command CALC_G_THETA_T is independent of the choice of the field:

G = G (),

4.2 Axisymetry

Into axisymmetric it is necessary to standardize the value G () obtained with Aster:

1
G =
G ()
R
where R is the distance from the bottom of fissure to the axis of symmetry [R7.02.01 §2.4.4].

4.3 3D

In dimension 3, the value of G () for a given field is such as:

G () =
() () m ()
G S S
S ds
O

In command CALC_THETA [U4.82.02], the user defines the direction of the field in bottom of
fissure. By defect, it is the normal at the bottom of fissure in the plan of the lips. By choosing one
unit field in the vicinity of the bottom of fissure, one a:

(S) m (S) = 1


and:
G () =
()
G S D
O

Either G the total rate of refund of energy, to have the value of G per unit of length, it is necessary
to divide the value obtained by the length of the fissure L:

1
G= G () in
3D
L

4.4
Symmetry of the model

Not to forget to multiply by 2, values of the rate of refund of energy G or G (S) if one
model that half of the solid compared to the fissure (or to specify the key word SYME_CHAR = “SYME”
or “ANTI” in the commands concerned).

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5 Examples

5.1 Example of calculation of the rate of refund of energy (option
“CALC_G”) in 2D

5.1.1 Calculation of G in linear elasticity

G1
=
CALC_G_THETA_T
(
MODELE
=
Mo,
CHAM_MATER =
chma,
THETA
=
theta,
DEPL =
depl,
EXCIT

=
_F (CHARGE
=
CH),)

One calculates the rate of refund of energy G on the model Mo, with the field of depl displacement
solution of the elastic problem with:

·
the material chma field produces by AFFE_MATERIAU,
·
the field theta produced by CALC_THETA or AFFE_CHAM_NO,
·
the CH load produced by command AFFE_CHAR_MECA or AFFE_CHAR_MECA_F.

This example results from test SSLP101 [V3.02.101]. It is about a plate fissured in traction.


Appear 5.1.1-a: Plaque fissured in traction


Appear 5.1.1-b: Half-plate fissured in traction

If one models only half of the plate as on [Figure 5.1.1-b] it should be added the key word:

SYME_CHAR
=
“SYME”

to specify the symmetry of the loading compared to the fissure. The values of G are thus
multiplied automatically by 2.

To note that in this case key word EXCIT is not essential: indeed the loading of traction
CH does not apply between the crowns Rinf and Rsup. On the other hand if there is a pressure on the lips of
the fissure or a voluminal loading, it should be taken into account in the calculation of G.
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5.1.2 Calculation of G in nonlinear elasticity in 3D

G2
= CALC_G_THETA_T (RESULTAT = resu,
TOUT_ORDRE=
“OUI”,
THETA
=
theta,
COMP_ELAS
=
_F (RELATION=' ELAS_VMIS_TRAC'),)

One calculates the rate of refund of energy G. The model, the field material and the loads are
recovered in the structure of data resu, resulting from STAT_NON_LINE. The rate of refund of
energy is calculated at every moment of calculation of the nonlinear elastic problem.

The relation of behavior is elastic nonlinear of von Mises with isotropic work hardening.

For other examples in 3D one will be able to refer to the tests:

SSLV110 [V3.04.110] semi-elliptic Fissure in infinite medium
SSLV112 [V3.04.112] circular Fissure in infinite medium
HPLV103 [V7.03.103] Thermoélasticité with circular fissure in infinite medium

5.1.3 Calculation of G in great transformations

G3
= CALC_G_THETA_T (MODEL
= Mo,
RESULTAT
=
resu,
INST =
(1.,
2.,
3.),
CHAM_MATER=
chma,
THETA
=
theta,
EXCIT

=

_F (
CHARGE
=
CH,
FONC_MULT
= fmult,)
COMP_ELAS
=
_F (
RELATION = “ELAS”
),
DEFORMATION
=
“GREEN”),)

One calculates the rate of refund of energy G for 3 moments of a nonlinear calculation into large
transformations starting from a concept resu resulting from STAT_NON_LINE. The coherence of the model Mo, of
field of material chma and the CH excitation/fmult with the information contained in
structure of data resu is checked.
The relation of behavior is elastic “linear” (the relation between the deformations and them
constraints is linear) but the behavior of the solid is hyperelastic
(DEFORMATION = “GREEN”). For more precise details to refer to [R7.02.03 §2.1].

5.2
Example of use of option “CALC_K_G”

y
N
N01
X
Appear 5.2-a: Calcul of the stress intensity factors.

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my
=
LIRE_MAILLAGE
()
Mo
=
AFFE_MODELE
(
MAILLAGE = my,
AFFE
=_F (
TOUT
=
“OUI”,
PHENOMENE
= ' MECANIQUE',
MODELISATION
=
“D_PLAN”))

theta = CALC_THETA (MODELE
=
Mo,
THETA_2D
=_F (
NOEUD
= “NO1”,
MODULE
=
1.,
R_INF
=
2.0,
R_SUP
=
3.0),
DIRECTION
=
(1.
0.),)

FF
=
DEFI_FOND_FISS
(
NOEUD
=
“N01”,
NORMALE =
(0.
1.),)

G4
=
CALC_G_THETA_T
(
MODELE
=
Mo,
DEPL =
depl,
CHAM_MATER =
chma,
THETA
=
theta,
EXCIT

=
_F (CHARGE
=
CH,),
SYME_CHAR
=
“SYME”,
FOND_FISS
=
FF,
OPTION
=
“CALC_K_G”
INFO = 2,)

IMPR_TABLE (TABLE

= G4)

One calculates the stress intensity factors K1 and K2 on the model Mo, with displacement
depl solution of the elastic problem with:

·
the material chma field produces by AFFE_MATERIAU,
·
the CH load produced by command AFFE_CHAR_MECA.

One recovers the basic node of N01 fissure and the normal with the fissure by the concept fond_fiss. One
specify that the total loading is symmetrical compared to the fissure thanks to the key word
SYME_CHAR.

Command IMPR_TABLE makes it possible to print in file RESULTAT the various sizes
calculated with option CALC_K_G, namely the rate of refund of energy G, factors of intensity
constraints K1 and K2, and G_IRWIN, rate of refund of energy obtained starting from the formula
of Irwin:
1
G _ IRWIN =
(2
2
K + K
1
2)
plane

S
constraint
in
E
2
1

G _ IRWIN =
(2
2
K + K
1
2)
plane

NS
déformatio
in
E
with E modulus Young and Poisson's ratio. The comparison between G and G_IRWIN allows
to ensure itself of the coherence of the results.

As INFO is worth 2, the angle of propagation of the fissure is also calculated, and the result is
printed in file MESSAGE with the following format:

Basic node of fissure: N01

Co-ordinates of the basic node of fissure: 0. 0.

Co-ordinates of the normal to the fissure: 0. 1.

K1
K2 G
(IRWIN)
2.14364E+01 0.0000E+00 1.14880E03

Rate of refund of energy G: 1.14907E03

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Direction of the deviation of the fissure (in degrees):

According to the maximum K1 criterion
: 0 with K1 max
: 2.14364E+01
According to null the K2 criterion
: 0 with null K2
: 0.0000E+00
According to the maximum criterion G
: 0 with Gmax
: 1.1488E03

From the stress intensity factors K1 and K2, one can indeed calculate the K1 coefficients * and
K2 * correspondent with a propagation of fissure given (according to work of AMESTOY, BUI and
DANG-VAN [R7.02.05 §2.5]).

The direction of the deviation of the fissure is calculated according to these results and according to 3 K1 criteria *
maximum, K2 * no one and G * maximum. The angle of propagation, given in degree, is calculated compared to
prolongation of the fissure.

y

X
Appear 5.2-b: Angle of propagation

Note:

·
For a thermal loading, coefficients characteristic of the material (E,…)
must be independent of the temperature.
·
Attention with the orientation of the normal to the fissure.

1
+ 2
N
2

Appear 5.2-c: Orientation of the normal to the fissure

For other examples one will be able to refer to the tests:

HPLP100 [V7.02.100] Plaque fissured in thermoelasticity
SSLP103 [V3.02.103] circular Plaque fissured
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5.3 Example of calculation of derived from the rate of refund of energy
for a variation of field

Let us preserve the same configuration that at the preceding paragraph and calculate this time the derivative of
G (F) compared to the variation of field controlled by S.


y Champ S
sensitivity
Field F
fissure
X
N1
x1
X
X
2
2

Appear 5.3-a: Dérivée of G (F) compared to a variation of field controlled by S

After having built the models Mo and moth in modeling “D_PLAN” and the field theta sensitivity
S (thetas), via key word THETA_BANDE of CALC_THETA, one affects loadings
thermics of the temperatures type imposed on the flat rims and left of the structure. Then, one
carry out the thermal calculation itself which uses thetas (provided via key word SENSIBILITE)
to calculate the field of temperature and its Lagrangian derivative.

Note:

·
The calculation of sensitivity in thermics is restricts with the linear 2D, stationary case or
transient, with voluminal sources and conditions of imposed temperature, flow
normal imposed and of convectif exchange. Conditions of exchange between wall and of
radiation are not taken yet into account [R4.03.01] [U4.54.01].
·
The finite elements supporting the grid must be quadratic.

Mo
=
AFFE_MODELE
(
MAILLAGE = my,
AFFE
=_F (
TOUT
=
“OUI”,
PHENOMENE
= ' MECANIQUE',
MODELISATION
=
“D_PLAN”),)

moth =
AFFE_MODELE
(
MAILLAGE = my,
AFFE
=_F (
TOUT
=
“OUI”,
PHENOMENE
= ' THERMIQUE',
MODELISATION
=
“PLAN”),)

thetas
= CALC_THETA (MODEL
=
Mo,
OPTION
=
“BANDE”,
THETA_BANDE
=
_F (MODULE =
1.,
R_INF
=
X1,
R_SUP
=
X2),)

chther= AFFE_CHAR_THER (
MODELE
=
moth,
TEMP_IMPO
=
_F
(GROUP_NO
=
“bordd”,
TEMP
=-100)
TEMP_IMPO
=
_F
(GROUP_NO
=
“bordg”,
TEMP
=
100))

resth=
THER_LINEAIRE (
MODELE
=
moth,
SENSIBILITE
=
thetas,
CHAM_MATER
=
cmth,
EXCIT
=
(CHARGE
=
chther),)
Handbook of Utilization
U4.8- booklet: Postprocessing and dedicated analyzes
HT-62/06/004/A

Code_Aster ®
Version
8.2
Titrate:
Operator CALC_G_THETA_T


Date:
31/01/06
Author (S):
E. CRYSTAL, O. BOITEAU, G. Key NICOLAS
:
U4.82.03-H1 Page:
22/24

Before carrying out thermomechanical calculation, one affects a pressure distributed on the higher edge.
This loading, just like the calculated temperature, is used in the totality of the process of
sensitivity, contrary to the loadings of gravity and the later initial deformations. These
the last will intervene only in postprocessing of the calculation of mechanics.

Caution:

·
With such a taking into account of the loadings, the user is only responsible for
the interpretation of the results.
·
To be fully usable, the calculation of the sensitivity of G will have to be extended to
principal loadings for all the process.
·
The calculation of sensitivity in mechanics is restricted, for the moment, with linear case D_PLAN or
AXIS with conditions limit uniform imposed displacement types, connections and
external pressure [R4.03.01] [U4.51.01].

chmeca= AFFE_CHAR_MECA (
MODELE
=
Mo,
TEMP_CALCULEE
=
resth,
DDL_IMPO
=
_F
(GROUP_MA=' bordi', DY
=
0),
PRES_REP
=
_F
(GROUP_MA=' bords', PRES
=100),)

resme=
MECA_STATIQUE (
MODELE
=
Mo,
SENSIBILITE
=
thetas,
CHAM_MATER
=
cm,
EXCIT
=
_F
(CHARGE
=
chmeca))

a1
=
1.E-8
a2
=
2.E-6
a3
=
5.E-2

FONC1 = FORMULA (NOM_PARA = (“X”, “Y”), VALE = “A1 * (X ** 2) +A2 * Y+a3”)

FONC2 = FORMULA (NOM_PARA = (“X”, “Y”), VALE = “A1 * (Y ** 2) +A2 * X+a3”)

FONC3 = FORMULA (NOM_PARA = (“X”, “Y”), VALE = “A1 * (X * Y) +A3”)

f=
AFFE_CHAR_MECA_F (MODEL
=
Mo,
PESANTEUR
=
(10.
0.
- 1.
0.),
EPSI_INIT
=
_F
(
ALL = “YES”,
EPXX
=
fonc1,
EPYY
=
fonc2,
EPXY
=
fonc3))

As in the preceding examples one builds the field theta fissures F (thetaf) which will play it
role of a function test during the calculation of the integral of the rate of refund of energy G (F) while delimiting
the zone of calculation centered around the N1 point.

thetaf
= CALC_THETA (MODEL
=
Mo,
THETA_2D
=_F (
NOEUD
= “N1”,
MODULE
=
1.,
R_INF
=
r1,
R_SUP
=
r2,)
DIRECTION
=
(1.
0.
0.),)

G5
=
CALC_G_THETA_T
(
MODELE
=
Mo,
RESULTAT
=
resme,
SENSIBILITE
=
thetas,
CHAM_MATER =
chma,
THETA
=
thetaf,
EXCIT

=
(_F (CHARGE =
chmeca),
_F (CHARGE
= F)),
SYME_CHAR
=
“SYME”,
COMP_ELAS
=
_F (RELATION
=
“ELAS”,

DEFORMATION
=
“PETIT”),









)

For other examples in D_PLAN one will be able to refer to case-test HPLP100B [V7.02.100]. One y
will find in particular examples of sequences of CREA_CHAMP allowing to build
analytical stress fields and to relocate a grid (in order to simulate a difference finished in
variation of field).
Handbook of Utilization
U4.8- booklet: Postprocessing and dedicated analyzes
HT-62/06/004/A

Code_Aster ®
Version
8.2
Titrate:
Operator CALC_G_THETA_T


Date:
31/01/06
Author (S):
E. CRYSTAL, O. BOITEAU, G. Key NICOLAS
:
U4.82.03-H1 Page:
23/24

5.4 Maximization of G in the presence of constraints not signed with
option “CALC_G_MAX”

The purpose of this example is to specify how to maximize the rate of refund of energy for one
linear problem with at the same time of the constraints signed (actual weight, internal pressure) and of
constraints which one does not know the sign a priori (seism). The studied problem is in modeling
3D, with a linear elastic behavior. The contact on the lips of the fissure is not made in
count.

Let us suppose for example, that in addition to the boundary conditions of blocking CHCL, there is a loading
of pressure signed not signed CHPRESS, and two loadings applying to groups of meshs
distinct from the model, CH_NS1 and CH_NS2:

CHCL=AFFE_CHAR_MECA (MODELE=MO,
DDL_IMPO= (_F (GROUP_NO = “SSUP_S', DZ = 0.),
_F (GROUP_NO = “SLAT_S', DX = 0.),
_F (GROUP_NO = “SAV_S', DY = 0.),),)

CHPRES=AFFE_CHAR_MECA (MODELE=MO,
PRES_REP=_F (GROUP_MA = “SINF”, CLOSE = - 1.E6),)

CH_NS1=AFFE_CHAR_MECA (MODELE=MO,
FORCE_NODALE=_F (GROUP_NO = “SLAT”, FZ = 1540),)

CH_NS2=AFFE_CHAR_MECA (MODELE=MO,
FORCE_NODALE=_F (GROUP_NO = “SINF”, FX = 2100),)

One calculates the solution of the problem associated with each one of these loadings while defining
multiplying functions:

F0=DEFI_FONCTION (NOM_PARA=' INST',
PROL_GAUCHE=' LINEAIRE',
PROL_DROITE=' CONSTANT',
VALE= (1., 1., 2., 0.,
3., 0.,),)

F1=DEFI_FONCTION (NOM_PARA=' INST',
PROL_GAUCHE=' LINEAIRE',
PROL_DROITE=' CONSTANT',
VALE= (1., 0., 2., 1.,
3., 0.,),)

F2=DEFI_FONCTION (NOM_PARA=' INST',
PROL_GAUCHE=' LINEAIRE',
PROL_DROITE=' CONSTANT',
VALE= (1., 0., 2., 0.,
3., 1.,),)

LIST=DEFI_LIST_REEL (DEBUT=0.E+0,
INTERVALLE=_F (JUSQU_A = 3., NOMBRE = 3),)

RESU=MECA_STATIQUE (MODELE=MO,
CHAM_MATER=CHMAT,
EXCIT= (_F (LOAD = CHCL),
_F (LOAD = CHPRES, FONC_MULT = F0),
_F (LOAD = CH_NS1, FONC_MULT = F1),
_F (LOAD = CH_NS2, FONC_MULT = F2),),



LIST_INST
=
LIST,)

One defines the bottom of fissure and the crown théta for the calculation of G:

FOND=DEFI_FOND_FISS (MAILLAGE=MA,

FOND_FISS=_F (GROUP_MA = “LFF”),

NORMALE= (0., 0., 1.,),

DTAN_ORIG= (1., 0., 0.,),
Handbook of Utilization
U4.8- booklet: Postprocessing and dedicated analyzes
HT-62/06/004/A

Code_Aster ®
Version
8.2
Titrate:
Operator CALC_G_THETA_T


Date:
31/01/06
Author (S):
E. CRYSTAL, O. BOITEAU, G. Key NICOLAS
:
U4.82.03-H1 Page:
24/24


DTAN_EXTR= (0., 1., 0.,),)

THETA=CALC_THETA (MODELE=MO,
FOND_FISS=FOND,
THETA_3D=_F (ALL = “YES”,

MODULE = 1.,


R_INF = 0.2,


R_SUP = 0.5),)

The maximization of G is done by option CALC_G_MAX of CALC_G_THETA_T (G total) or of
CALC_G_LOCAL_T (G local). The coefficient of the signed loading is worth 1, the coefficients of
not signed loadings vary between ­ 1 and 1:


G_MAX=CALC_G_THETA_T (THETA=THETA,
RESULTAT=RESU,
BORNES= (_F (NUME_ORDRE = 1,
VALE_MIN = 1., VALE_MAX = 1.),
_F (NUME_ORDRE = 2,
VALE_MIN = - 1., VALE_MAX = 1.),
_F (NUME_ORDRE = 3,
VALE_MIN = - 1., VALE_MAX = 1.),),
OPTION=' CALC_G_MAX',)

IMPR_TABLE (TABLE = G_MAX)

GL_MAX=CALC_G_LOCAL_T (RESULTAT=RESU,
FOND_FISS=FOND,
R_INF=0.2,
R_SUP=0.5,
BORNES= (_F (NUME_ORDRE = 1,
VALE_MIN = 1., VALE_MAX = 1.),
_F (NUME_ORDRE = 2,
VALE_MIN = - 1., VALE_MAX = 1.),
_F (NUME_ORDRE = 3,
VALE_MIN = - 1., VALE_MAX = 1.),),
OPTION=' CALC_G_MAX',)

IMPR_TABLE (TABLE = GL_MAX)

The table produced by CALC_G_THETA_T is as follows:

# ASTER 8.02.01 CONCEPT GMAX CALCULATES THE 21/12/2005 A 15:49:17 OF TYPE
# TABL_CALC_G_TH
Q_1 Q_2 Q_3 G G_MAX
1.00000E+00 1.00000E+00 - 1.00000E+00 3.91703E+03 3.91703E+03
1.00000E+00 1.00000E+00 - 1.00000E+00 3.91703E+03 3.91703E+03
1.00000E+00 - 1.00000E+00 - 1.00000E+00 3.63507E+03 -
1.00000E+00 - 1.00000E+00 - 1.00000E+00 3.63507E+03 -
1.00000E+00 - 1.00000E+00 1.00000E+00 2.92029E+03 -
1.00000E+00 - 1.00000E+00 1.00000E+00 2.92029E+03 -
1.00000E+00 1.00000E+00 1.00000E+00 2.68007E+03 -
1.00000E+00 1.00000E+00 1.00000E+00 2.68007E+03 -

Thus, the maximum rate of refund is obtained for the combination of the loading of pressure with
CH_NS1 with a sign “+” and CH_NS2 with a sign `- `.
For other examples, one can refer to the case test SSLV134E/F [V3.04.134].

Handbook of Utilization
U4.8- booklet: Postprocessing and dedicated analyzes
HT-62/06/004/A