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Code_Aster
®
Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
U2.05.01-A
Page
:
1/52
Instruction manual
U2.05 booklet: Damage and breaking process
HT-66/03/002/A
Organization (S):
EDF-R & D/AMA, EDF-DPN/UTO















Instruction manual
U2.05 booklet: Damage and breaking process
U2.05.01 document



Applicability of the operators of
breaking process of Code_Aster and consultings
of use




Summary:

The characterization of the state of the fissured parts is based on the determination of the rate of refund of energy and
stress intensity factors, bases of many criteria in fragile breaking process
(priming in bottom of fissure, propagation of defects, methods simplified). This document presents these
functionalities, available in Code_Aster, indicates their field of validity and gives consultings
of use.
One also presents new formulations resulting from recent research tasks but not yet
validated, like GTP and Gp.
The reading of this document can be done on two levels:
·
for a new user in breaking process, wanting to know the methods used and them
controls of Code_Aster necessary to the realization of its study,
·
for a user more informed, in the search of consultings of use to solve certain points
delicate and eager to take note of recent research tasks.

He is constantly referred to the Reference and Instruction manuals, whose reading remains
essential. The bibliography must also make it possible to the reader to look further into the subject which interests it.
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Code_Aster
®
Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
U2.05.01-A
Page
:
2/52
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U2.05 booklet: Damage and breaking process
HT-66/03/002/A
Count
matters
1
Tally of use of the functionalities available in breaking process in Code_Aster..3
1.1
Theoretical framework: principle of the method théta .............................................................................. 3
1.2
Formulation of the functionalities of breaking process in Code_Aster ......................... 4
1.3
Field of validity of the functionalities of breaking process in Code_Aster ............ 7
1.4
Approach energy the elastoplastic rupture and formulation of the parameter Gp ................. 11
2
Methodology and recommendations of use ................................................................................... 13
2.1
Mesh of the fissured structure .................................................................................................... 13
2.2
Introduction of the field théta .......................................................................................................... 14
2.3
Standardization of the total rate of refund G in Code_Aster ............................................... 17
2.4
Method of interpolation in 3D ....................................................................................................... 18
2.5
Calculation of G for a non-linear problem ................................................................................... 20
3
Implementation of a calculation in breaking process in Code_Aster ..................................... 26
3.1
Methodology ................................................................................................................................. 26
3.2
Example 1: Calculation of G, K1 and K2 for a linear elastic problem in 2D ............................ 31
3.3
Example 2: Calculation of G and G (S) local for an elastic thermo problem in 3D ....................... 33
3.4
Example 3: calculation of Gp for an elastoplastic problem in 2D ............................................. 37
4
Documentation of Code_Aster relating to the fragile breaking process ..................................... 49
5
Bibliography ........................................................................................................................................ 50
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Code_Aster
®
Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
U2.05.01-A
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:
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U2.05 booklet: Damage and breaking process
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1 Framework of use of the functionalities available in
breaking process in Code_Aster
1.1
Theoretical framework: principle of the method théta
One considers a fissured elastic solid occupying the field
. Are:
U
the field of displacement,
T
the field of temperature,
F
the field of voluminal forces applied to
,
G
the field of surface forces applied to a part
S
of
,
U
the field of displacements imposed on a part
S
D
of
.
the tensor of the stresses,
the tensor of the deformations,
HT
the tensor of the deformations of thermal origin,
()
,
T
density of free energy.
S
S
D
F
G

Let us consider the energy approach of the rupture of Griffith. The results are rigorous only in
thermo linear elasticity but of the extensions are possible with the nonlinear problems.
For a fissured elastic solid, the criterion of propagation of Griffith results in:
G
>
2
where
is
binding energy per unit of area.
G
, called rate of refund of energy, is defined by
opposite of derived from the potential energy to balance
()
W
U
compared to the field
:
G
W
= -
()
U
with:
()
()
(
)
W
T D
D
D
S
U
U
F U
G U
=
-
-
,
The difficulty of the calculation of the rate of refund of energy comes from derivation compared to the field
of an integral depending on this same field. A rigorous method is the method théta, which
is a Lagrangian method of derivation of the potential energy. It consists in introducing one
field
and to consider transformations
()
F
: M
M
M
+
area of reference
in a field
who correspond to propagations of the fissure. These transformations
do not have to modify the edges of the field except the bottom of fissure.
This method is detailed in [bib38] and the use of the field théta in Code_Aster is described
with [§2.2].
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Code_Aster
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Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
U2.05.01-A
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In plane linear elasticity (assumption of the plane strains or plane stresses), the field
of displacement U can break up into a singular part and a regular part. The part
singular, also called singularity, contains the coefficients of intensity of stresses
K
I
and
K
II
:
U
U
U
U
=
+
+
R
I
S
I
II
S
II
K
K
In plane linear elasticity, the coefficients of intensity of stresses are connected to the rate of refund
of energy by the formula of IRWIN:
(
)
(
)
plane
S
stress
in
plane
NS
déformatio
in
1
1
2
2
2
2
2
II
I
II
I
K
K
E
G
K
K
E
G
+
=
+
-
=

1.2
Formulation of the functionalities of breaking process in
Code_Aster
1.2.1 Rate of refund of energy G
With the method théta, the rate of refund of energy
G
is solution of the variational equation:
() () ()
()
G S
S
S ds
O
=
m
G
,
where
m
is the unit normal at the bottom of fissure
O
located in the tangent plan at
and re-entering in
, and where
()
G
is defined by the opposite of derived from the potential energy
()
()
W U
with balance by
report/ratio with the initial evolution of the bottom of fissure
:
()
()
()
G
= -
=
D W
D
U
0
One notes
conditions to fill by the field
(see [§2.2.1]).
In dimension 2, bottom of fissure
O
brings back itself to a point, and one can choose a field
of such
left that the variational equation is reduced to
()
G
=
G
.
In dimension 3 dependence of
G
()
with respect to the field
on the bottom of fissure is more
complex. In Code_Aster, one can calculate:
·
the total rate of refund
G
agent with a uniform progression of the fissure
(control
CALC_G_THETA
[U4.82.03]). The user must choose a unit field théta with
vicinity of the checking bottom of fissure
() ()
S
S
S
=
m
1,
and then:
()
()
G L
G S ds
O
=
=
G
where L is the length of the bottom of fissure,
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Code_Aster
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Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
U2.05.01-A
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O

·
the rate of refund of energy room
()
G S
solution of the preceding variational equation
(control
CALC_G_LOCAL
[U4.82.04]). In this case, the user does not give a field
théta, fields
I
necessary to the resolution of the variational equation and the calculation of
G S
()
are calculated automatically.
For a linear elastic problem the thermo or not-linear expression of
()
G
is:
()
G
=
-
-
+
+
+
+
-
-
(
)
(
)
(
(
))
,
,
,
,
,
,
,
,
,
ij
I p
p J
K K
K
K
I
I
K K
I K
K
I
I K
S
K
I
I
I
K K
K
K
ij
S
J
I K
K
U
T T
D
F U
F
U D
G
U
G U
N N
D
N U
D
D
If one places oneself on the assumption of the great transformations, the term should be replaced
ij
I p
p J
U
D
,
,
by
F S U
D
ik
kj
I p
p J
,
,
with
S the tensor of the stresses of Piola-Lagrange called still second tensor of Piola-Kirchoff,
F the gradient of the transformation which makes pass from the configuration of reference to the configuration
current.
If one takes account of the initial deformations
ij
0
and of the initial stresses
0
ij
, it is necessary to add it
term:
°
°
°
°




-
-
-


-
D
K
K
ij
ij
HT
ij
ij
K
ij
ij
ij
,
,
2
1
2
1
For a problem the thermo elastoplastic expression of
()
G
retained in Code_Aster is:
()
(
)
()
G
F,
conventional
terms
+


-
+

+
+
-
+
-
=
D
p
T
T
U
K
p
K
ij
ij
K
ij
ij
K
y
K
K
K
J
K
K
I
ij
,
,
,
,
,
,
,
~
R
~
~
G
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Code_Aster
®
Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
U2.05.01-A
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with:
~
total mechanical energy,
p
the tensor of the plastic deformations,
p
the variable interns scalar isotropic work hardening (cumulated plastic deformation),
one or more tensorial or scalar variables of kinematic work hardening,
y
initial linear elastic limit,
R
the radius of the surface of load for isotropic work hardening.
For a radial and monotonous loading:
(
)
ij
ij K
p
y
K
ij
ij K
p
,
,
,
R
~
=
+
+
and one finds
the expression of
()
G
in thermo nonlinear elasticity [R7.02.03].
1.2.2 Coefficients of intensity of stresses K1 and K2 deduced from the calculation of G.
In thermo linear elasticity, the rate of refund of energy
G
is a symmetrical bilinear form of
field of displacement
U
:
()
G
G
=
U U
,
. By using the method théta, the bilinear form
()
G
,
associated
G
is defined by:
()
(
)
(
)
()
G
B
B
B
D
U v
U
v
v
U
U v
,
,
div
=
+ -




1
2
while limiting themselves at the end conventional and while noting:
()
(
)
U
density of energy elastic:
()
(
)
()
()
()
U
U
U
U U
=
=
1
2
1
2
: :
,
B
the tensor of elasticity,
B
the symmetrical bilinear form defined by:
()
()
()
B U v
U
v
,
: :
=
In the method established in Code_Aster (control
CALC_G_THETA
[U4.63.03]), for
to uncouple the modes from rupture
I
and
II
and to calculate the coefficients
K
I
and
K
II
, this form is used
bilinear symmetrical
G
,
()
and decomposition of the field of displacement
U
in parts regular
R
U
and singular
S
U
:
II
S
II
I
S
I
R
K
K
U
U
U
U
+
+
=
(
U
S
I
and
U
S
II
are known explicitly):
(
)
(
)
K
E G
K
E G
I
S
I
II
S
II
=
=




U U
U U
,
,
= -
1
2
in plane deformations and
=
1
in plane stresses.
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Code_Aster
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Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
U2.05.01-A
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:
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1.2.3 Coefficients of intensity of stresses K1, K2 and K3 obtained by extrapolation
field of displacements.
For a plane fissure in an elastic, homogeneous and isotropic material, one can also reach
with the values of K1, K2 and K3 by extrapolation of the jumps of displacements on the lips of this
fissure (control
POST_K1_K2_K3
).
Contrary to the preceding approach (calculation of Ki by the bilinear form of the rate of refund
of energy), one can thus calculate these coefficients in axisymmetric geometry and 3D and reach
K3 coefficient. For each coefficient of intensity of stress, method, less precise than
method G_THETA [R7.02.01], provides two values framing the solution. One can however be done
an idea of the precision of the results by recomputing G by the formula of Irwin, starting from the values of
K1, K2 and K3, and by comparing this value with that obtained with G_THETA. Precision of
results is clearly improved if elements touching the bottom of fissure (quadratic elements)
nodes mediums located at the quarter of the edges have.
1.2.4 Propagation
Lagrangian
It is possible with Code_Aster to calculate the rate of refund of energy for different
lengths of fissure (in 2D and 3D) by using only one mesh representing a length of fissure
fix reference. These developments are available in linear elasticity, for the elements of
continuous medium 2D and 3D, in the situations where the variations of geometry do not affect the edges
charged.
Any calculation using this method requires, to ensure the passage of the real field studied
area of reference, the preliminary creation of a field théta, using the control
CALC_THETA
[U4.82.02]. The formulation developed in Code_Aster does not take account of the terms
thermics, of the loadings on the lips of the fissure nor of the forces of volume in general, except
initial deformations which are taken into account in 2D only.
For more precise details on this option see document [R7.02.04].

1.3 Field of validity of the functionalities of breaking process
in Code_Aster
1.3.1 Model
The calculation of the rate of refund of energy G is valid for modelings of the continuous mediums 2D
plane strains or plane stresses (
D_PLAN
,
C_PLAN)
, axisymmetric 2D (
AXIS)
and 3D (
3D
).
These modelings correspond for a two-dimensional medium to triangles to 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 nodes or 27 nodes, of the pentahedrons with 6 or 15 nodes, of the tetrahedrons with 4 or 10
nodes, of the pyramids with 5 or 13 nodes, of the faces with 4, 8 or 9 nodes.
The calculation of the rate of refund of energy room G (S) has direction only for the modeling of the mediums
continuous 3D.
The calculation of the stress intensity factors K1, K2 deduced from the bilinear form
()
G
,
is
valid only for modelings of the continuous mediums 2D plane deformations or
plane stresses (
D_PLAN
,
C_PLAN)
. The calculation of the mode antiplan K3 is not available.
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Code_Aster
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Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
U2.05.01-A
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:
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On the other hand, the method of extrapolation of displacements makes it possible to calculate Ki (of which K3) in
axisymmetric and 3D when the fissure is plane.
D_PLAN C_PLAN AXIS
3D
G
·
·
·
·
G (S) local
-
-
-
·
K1, K2
calculation of
G
·
·
- -
K1, K2, K3
extrapolation of
U
·
·
·
·
Modelings available
1.3.2 Characteristics of material
For the calculation of the rate of refund of energy, the characteristics of the material (Young modulus,
, thermal and possibly limit expansion factor Poisson's ratio elastic, modulates
on work hardening) can depend on the temperature. Calculation is valid for a homogeneous material
isotropic or for an isotropic bimatériau (fissure with the interface of two materials to the characteristics
different).
For the calculation of the coefficients of intensity of stresses at a given moment, the characteristics of
material must be independent of the temperature. Calculation is valid only for one
isotropic homogeneous material (possibly for a bimatériau if the point of fissure is not
located at the interface of two materials).

Modulate
of Young E (T)
Coefficient of
Poisson
(T)
Coefficient of
dilation
thermics
()
Elastic limit
y (T)
Modulate
of work hardening
D_SIGM_EPSI
G
·
·
·
·
·
G (S) local
·
·
·
·
·
K1, K2
-
-
-
-
-
Dependence of the characteristics at the temperature
Characteristics
y (T) and
D_SIGM_EPSI
(T) are treated only for one elastic problem not
linear with linear isotropic work hardening of Von Mises and the option of calculation of the rate of
restitution of energy.
Material
homogeneous
Bimatériau (fissure with the interface)
G
·
·
G (S) local
·
·
K1, K2
·
-
Homogeneity of material

Material
isotropic
Orthotropic material
G
·
-
G (S) local
·
-
K1, K2
·
-
Isotropy of material
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Code_Aster
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Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
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1
er
case: There is a bimatériau but the point of fissure is in only one material.
material 1
E
1
,
1
,
1
material 2
E
2
,
2
,
2
R
R
If one is assured that the crown, definite enters the radii inferior
R
inf
and superior
R
sup
(in
order
CALC_THETA
[U4.82.02]), has like support of the elements of same material, calculation is
valid some is the selected option. If not only the calculation of the rate of refund of energy is valid.
2
ème
case: There is a bimatériau where the point of fissure is with the interface.
material 1
E
1
,
1
,
1
material 2
E
2
,
2
,
2
R
R
To date, only the option of calculation of the rate of refund of energy is valid. The calculation of coefficients
of intensity of stresses K
1
and K
2
is false in this case.
1.3.3 Relation of behavior used in postprocessing of mechanics of
rupture
For the calculation of the rate of refund of energy, the possible relations of behavior are:
·
thermo linear elasticity,
·
thermo nonlinear elasticity (hyperelasticity),
·
thermo elastoplasticity (criterion of Von Mises with isotropic or kinematic work hardening).
The calculation of the coefficients of intensity of stresses is possible only in thermo elasticity
linear on the assumption of the small deformations.
The relation of behavior is selected in the controls
CALC_G_THETA
[U4.82.03] and
CALC_G_LOCAL
[U4.82.04] via the key words factors
COMP_ELAS
(thermo elasticity
linear or not linear) or
COMP_INCR
(
thermo elastoplasticity).
Relations treated under the key word factor
COMP_ELAS
are:
ELAS
: thermo linear elasticity,
ELAS_VMIS_LINE
: Von Mises with linear isotropic work hardening,
ELAS_VMIS_TRAC
: Von Mises with isotropic work hardening given by a traction diagram.
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Code_Aster
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Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
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I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
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Relations treated under the key word factor
COMP_INCR
are:
VMIS_ISOT_LINE
: Von Mises with linear isotropic work hardening,
VMIS_ISOT_TRAC
: Von Mises with isotropic work hardening given by a traction diagram,
VMIS_CINE_LINE
: Von Mises with linear kinematic work hardening.

RELATION
G or G (S) local
K1, K2
COMP_ELAS “ELAS”
·
·
“ELAS_VMIS_LINE”
·
-
“ELAS_VMIS_TRAC”
·
-
COMP_INCR “ELAS”
·
·
“VMIS_ISOT_TRAC”
·
-
“VMIS_ISOT_LINE”
·
-
“VMIS_CINE_LINE”
- -
Relation of behavior used in breaking process

The nonlinear relation of thermo behavior elastic can be used with the large ones
displacements and of great rotations (with the proviso of having only dead loads). This
functionality is started by the key word
DEFORMATION = “GREEN”
. The deformations are them
deformations of Green-Lagrange [R7.02.03 §2.1]:
()
(
)
ij
I J
J I
K I
K J
U
U
U
U
U
=
+
+
1
2
,
,
,
,

1.3.4 Loading
Loadings currently supported by various modelings and for the calculation of
functionalities of breaking process are as follows (see
AFFE_CHAR_MECA (_F)
[U4.44.01]
for more details):
C_PLAN, D_PLAN, AXIS
3D
K1, K2
G
G and G (S) local
TEMP_CALCULEE
·
·
·
FORCE_INTERN
·
·
·
PRES_REP
·
·
·
FORCE_CONTOUR
·
·
///
FORCE_FACE
//////
·
FORCE_NODALE
- -
-
FORCE_ARETE
//////
-
GRAVITY
·
·
·
ROTATION
·
·
·
EPSI_INIT
- -
-
DDL_IMPO
(on fissure)
- -
-
FACE_IMPO
(on fissure)
- -
-
·
mean possible and available
///option without object means
-
mean option possible but nonavailable
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Code_Aster
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Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
U2.05.01-A
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:
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These loadings can depend on the geometry, the moment of calculation and possibly
to apply to the lips of the fissure.
The loadings not supported by an option are ignored.
It is important to note that the only loadings to be taken into account in a calculation of mechanics
rupture with the method
are those applied to the elements inside the crown
(between R
inf
and R
sup
for an elastic thermo behavior linear or not linear [R7.02.01 §3.3],
between the bottom of fissure and R
sup
for an elastoplastic thermo relation [R7.02.07]).
If one makes a calculation in great transformations (key word
DEFORMATION
=
“GREEN”
under the key word
factor
COMP_ELAS
) the supported loadings must be died loads, typically one
force imposed and not a pressure [R7.02.03 §2.4].

1.3.5 State
initial
It is possible to take account of an initial state (either of the initial stresses, or of the deformations
initial) for the calculation of the rate of refund of energy. Two possibilities are offered to the user:
·
to define initial deformations with the key word
EPSI_INIT
in the control
AFFE_CHAR_MECA (_F)
[U4.44.01] and to recover them under the key word
CHARGE
in
controls
CALC_G_THETA
[U4.82.03] or
CALC_G_LOCAL
[U4.82.04],
·
to recover a stress field or initial deformations resulting from a mechanical calculation
(
evol_noli
resulting from the control
STAT_NON_LINE
[U4.51.03]) with the key word
ETAT_INIT
.

1.3.6 Contact
The calculation of the sizes of breaking process in Code_Aster is not valid if there is
contact with friction between the faces of the fissure. Indeed the calculation of the rate of refund of energy
does not take into account the dissipative phenomena.
On the other hand if the elements of contact are beyond the crown defined between R
inf
and R
sup
calculations of G, G (S), K1 and K2 are valid.
On the other hand, it is possible for the calculation of G and G (S) to only take into account
conditions of contact without friction to avoid the interpenetration of the lips of the fissure.
1.4 Approach energy the elastoplastic rupture and formulation
Gp parameter
The conventional global solution presents important limits:
·
the loading must be monotonous,
·
more generally, the loading must be proportional and radial (see chapter [§2.5.2]),
·
one cannot simulate great propagations,
·
one cannot take into account a residual stress field (see chapter
[§1.3.5]).
The application of the global solution apart from its field of validity led to problems of
“transferability” of test-tubes with structures.
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Another approach was then considered with EDF-R & D: energy approach.
This new approach was developed, on the one hand, within the framework of the ductile tear [bib57], and,
in addition, within the framework of brittle fracture by cleavage.
In the case of brittle fracture by cleavage, one leaves the theory of Frankfurt Marigo in elasticity
[bib56]. This theory is a generalization of the criterion of Griffith for fragile elastic materials.
One applies the principle of minimization of energy, to predict the initiation or the propagation of one
fissure of surface S of a surface created dS. One defines, starting from elastic energy, a parameter
Gel/freezing, rate of refund of energy in elasticity [bib58] by the following formula:
Gel/freezing = - [We (dS) - We (0)] /Surface (dS).
One extends then this approach to plasticity, by making the assumption that plastic dissipation and
dissipation related to the rupture are independent.
One can then define a plastic parameter G [bib58], noted Gp, like a rate of refund of
energy in plasticity incremental [bib58] by the following formula:
Gp = - [W (dS) - W (0)] /Surface (dS)
where W is total energy (free energy + energy of work hardening + plastically dissipated energy).
But one finds oneself then confronted with 2 paradoxes of the theory of Griffith [bib62]:
·
the paradox of Rice,
·
scale effects of the theory of Frankfurt-Marigo induced by the assumption of Griffith.
One makes the choice then modelize the defect in the form of notch and not of fissure.
One defines a rate of refund of Gp energy applicable to a fissure represented in notch, in
being based on the formulation of Frankfurt-Marigo and mechanics damage continues,
with the help of some additional assumptions.
Note:
Another alternative consists in being directed towards a theory of Frankfurt-Marigo based on one
other models that that of Griffith, like that of Barenblatt.
It is supposed that this notch with the shape of a cigar, bottom of notch (
) being represented
by a half-circle of radius R. the area corresponding to the propagation of the notch is noted
Ze (L) (damaged Area) and depends on L, outdistances propagated, in accordance with the figure
below:






The Gp parameter is defined by the following formula:
()
]
[(
L
L
L
Gp
=
/
)
.
(
max
)
.
dS
W
Notch
E
where We is elastic energy.
This parameter makes it possible to predict:
·
progressive propagation of the notch (when the maximum is obtained for L = 0)
·
brutal propagation of the notch (when the maximum is obtained for L 0).
One evacuates in this case the 2 paradoxes of the theory of Griffith.
L
Ze (L)
notch
(
): melts of notch
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One can use Gp to analyze situations of brittle fracture per cleavage, when the Gtheta approach
is not valid. They can be problems with:
·
discharges [bib59],
·
loadings nonproportional,
·
residual stresses,
·
effect small defect [bib60], [bib61].
One introduces the implementation of the calculation of Gp in Code_Aster in the chapter [§2.5.2.3] and one
illustrate this approach on an example in the chapter [§3.4].

2
Methodology and recommendations of use
2.1
Mesh of the fissured structure
2.1.1 Tool for mesh of fissured block
Maillor GIBI comprises a parameterized automatic procedure which makes it possible to conceive
mesh of blocks fissures in 3D. This procedure was developed by EDF-R & D and was validated
to ensure the good quality of the mesh. One obtains a mesh with the format GIBI which can recognize
Code_Aster (control
PRE_GIBI
). The user informs a certain number of parameters
geometrical (dimensions of fissure, cuts block,…) or topological (modeling of the basic core
of fissure in crowns, sectors and sections, déraffinement, a number of elements,…) and software
generate a block fissures, which can then be integrated in another structure.
The user has indicators of quality of mesh to adjust the parameters as well as possible.
2.1.2 Methodology
Quality of the mesh depends numerical quality on the results resulting from mechanical calculation
(displacements and stresses) and by consequence of the quality of the sizes in mechanics of
rupture. In the presence of a fissure it is thus necessary to refine in the vicinity of the bottom of fissure to collect with
better singularities. But it is not necessary to refine exaggeratedly: interest of the method
théta is to utilize the singular terms on elements between R
inf
and R
sup
and not on those with
vicinity of the bottom of fissure (except for a calculation in thermo elastoplasticity, for this case
private individual to refer to [§2.5.2]).
Calculations of the sizes of breaking process are valid for linear elements or
quadratic, but it is strongly advised to use quadratic elements, in particular in 3D.
The calculation of these sizes indeed requires to determine with a good approximation them
deformation and stress fields which strongly vary in the vicinity of the bottom of fissure. However, with
an identical number of nodes, the quadratic elements give better results that them
linear elements, undoubtedly because they are ready to represent this type of variation. Let us add
that in 3D, it is necessary to carry out a compromise between a sufficient refinement in bottom of fissure
on the one hand, and a reasonable size of problem on the other hand. The quadratic choice of elements
contribute to carry out such a compromise.
A radiant mesh in bottom of fissure is not obligatory: the radii R
inf
and R
sup
are not dependant
with the mesh and the crown can be “with horse” on several elements. Nevertheless practice
show that a radiant mesh in bottom of fissure gives good numerical results.
The radiant mesh has in particular the advantage of making it possible to impose a constant cutting according to
the polar angle, around the bottom of fissure, and in the immediate vicinity of the aforementioned, cutting well
adapted to the asymptotic representation of the fields in bottom of fissure. Indeed, this variation according to
the polar angle does not depend on the distance from the point considered at the bottom from fissure.
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In the case of the calculation of the coefficients of intensity of stress by the method of extrapolation of
jumps of displacements on the lips of this fissure (control
POST_K1_K2_K3
), it is strongly
advised to position the nodes mediums of the quadratic elements concerning the bottom of fissure with
quarter of the edges (mesh of the type “BARSOUM”). Thus dependence in
R
field of
displacement is represented better and the quality of the results is improved. Values of
Ki coefficients obtained by this method tend towards those deduced from the calculation of G
(
CALC_G_THETA_T
option
CALC_K_G
) with the refinement of the mesh around the bottom of fissure.
2.1.3 Estimators
errors
To assess the quality of the mesh it is advised to carry out an elastic design and to use them
estimators of errors of discretization: estimators of errors of ZHU-ZIENKIEWICZ in elasticity
2D [R4.10.01] or the estimator of error by residue [R4.10.02].
These estimators are established in Code_Aster in the control
CALC_ELEM
[U4.81.01]. They
are activated starting from the following options:
ERRE_ELEM_NOZ1
for ZZ1,
ERRE_ELEM_NOZ2
for ZZ2
and
ERRE_ELGA_NORE
for the estimator in residue by element.

2.2
Introduction of the field théta
2.2.1 Definition of the field théta and conditions to respect
The field théta is a field of vectors, definite on the fissured solid, which represents the transformation
field during a propagation of fissure. It is reminded the meeting that the rate of refund of energy
G
is
solution of the variational equation:
() () ()
()
G S
S
S ds
O
=
m
G
,
where
m
is the unit normal at the bottom of fissure
O
located in the tangent plan at
(i.e.
tangent in the plan of cracking in 3D or the lips of the fissure in 2D) and re-entering in
. One notes
conditions to fill by the field
Ci below:
m
O
The transformation should modify only the position of the bottom of fissure and not the edge of the field
.
field
must thus be tangent with
(in particular lips of the fissure), i.e while noting
N
normal with
:
on
0
.
=
N
.
The field
must be locally in the tangent plan with the lips of the fissure and in normal 3D with
the edge to which it belongs. This corresponds to the direction of propagation of the fissure.
The field
must also be continuous on
.
Quantity
m
.
represent the normal speed of the bottom of fissure.
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2.2.2 Choice of the field théta in Code_Aster
Because of the singularity of the field of displacement, one uses fields
m
.
constant with
vicinity of the bottom of fissure, thus cancelling in this vicinity the singular terms
µ
ij
I p
p K
K K
,
,
,
-
in
()
G
.
The field théta is defined in the following way: in each node of the bottom of fissure, one is given
2 radii
()
R
S
inf
and
()
R
S
sup
. In on this side
()
R
S
inf
the module of the field théta is constant, with beyond it is
no one and it are linear between the two.
R
sup
0
R
inf
0
R
inf
0
N
R
sup
The construction of the field théta is described precisely in [R7.02.01]. It is established in
order
CALC_THETA
in 2D and 3D for the calculation of the total rate of refund G, and in
order
CALC_G_LOCAL
for the calculation of the local rate of refund G (S).
In 2D and axisymmetric the bottom of fissure
O
limits itself to a point. The user defines:
·
radii
inf
R
and
R
sup
,
·
the module
O
in bottom of fissure,
·
direction of propagation of the fissure Mr.
In 3D the user defines:
·
radii
()
R
S
inf
and
()
R
S
sup
,
·
directions of propagation of the fissure only at the ends of the bottom of fissure
(key words
DTAN_ORIG
and
DTAN_EXTR
in the control
DEFI_FOND_FISS
[U4.82.01]),
·
the topology of the bottom of fissure: opened or closed according to if the fissure is emerging or not,
·
the module
O
in bottom of fissure (only for the calculation of G total if not P
fields
I
necessary to the resolution of the variational equation and the calculation of
G S
()
are
calculated automatically according to the family of functions of selected interpolation: Lagrange or
Legendre, to see [§2.4]).
The directions of the field théta except ends are calculated automatically starting from the lips of
fissure, but the user can possibly define them itself by using the key word
DIRE_THETA
,
to see [§2.2.3].
The field
is then built so that:
()
()
()

R S
if R S
R
S
R S
if R S
R
S
R S
R
R
R
R
if R
S
R S
R
S
O
O
()
()
()
()
()
()
()
()
()
()
sup
inf
sup
sup
inf
inf
sup
=
=
=
-
-
0
m
m
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2.2.3 Problem of the discretization in 3D
Problem of the emerging fissure in a nonperpendicular way: at the end emerging of
melts of fissure, the field
cannot simultaneously be normal with the edge to which it belongs
(in the tangent plan of the lips of the fissure) and to check the condition
=
N 0
on
.
Advised solution: To define the direction of the field
O
on all the nodes of the bottom of fissure with
key word
DIRE_THETA
in the controls
CALC_THETA
[U4.82.02] or
CALC_G_LOCAL
[U4.82.04].
In the vicinity of the emerging end to choose like direction for the field
O
the average enters
direction checking
=
1
N
0
on
and normal direction with the edge
2
.
N
1
2

Problem of the choice of
inf
R
and
sup
R
: The calculation of the sizes of breaking process is
independent of the choice of the crown of integration, i.e. choice of
R
inf
and
R
sup
. Nevertheless it
is preferable to comply with some rules:
·
never not to take
R
inf
=
0
or too small compared to dimensions of the problem because them
singular displacements are badly calculated in the vicinity of the bottom of fissure (valid also
in 2D),
·
in 3D it is necessary to find a compromise enters
R
inf
not too small and
R
sup
not too large. Indeed if
one analyzes the algorithm of construction of the field théta (see [R7.02.01]), one notes that
to know the direction of the field théta in an unspecified point of the solid, it is necessary to project
this point on the bottom of fissure (i.e. to determine the X-coordinate of the basic point fissures it
nearer) and to associate the same direction to him. If one considers a point too far away from the bottom
of fissure, it may be that the algorithm of search of the basic point of fissure nearest
give an “erroneous” point: the direction of the field théta is badly calculated, with the direction where it
does not correspond to the propagation of awaited fissure.
Solutions:
·
to check by visualizing the mesh that, for
R
sup
chosen, one is not likely to have points
“badly” projected,
·
to take several crowns to check the invariance of G, preferably which are followed
[R1, R2], [R2, R3], [R3, R4],…
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2.3
Standardization of the total rate of refund G in Code_Aster
2.3.1 2D forced plane and plane deformations
In dimension 2 (plane stresses and plane deformations), the bottom of fissure is tiny room to a point and
the value
()
G
exit of the control
CALC_G_THETA
is independent of the choice of the field
:
()
G
=
G
,
2.3.2 Axisymetry
Into axisymmetric it is necessary to standardize the value
()
G
obtained with Aster:
()
G
R
=
1
G
where
R
is the distance from the bottom of fissure to the axis of symmetry [R7.02.01, §2.3.3].
2.3.3 3D
In dimension 3, the value of
()
G
for a field
given by the user is such as:
()
() () ()
G
=
G S
S
S ds
O
m
In the control
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
field
unit in the vicinity of the bottom of fissure, one a:
() ()
S
S
=
m
1
and:
()
()
G
=
G S D
O
That is to say
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
:
()
G
=
G L
in 3D
2.3.4 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
modelize that half of the solid compared to the fissure (or to specify the key word
SYME_CHAR =
“SYME”
or
“ANTI”
in the controls concerned).
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2.4
Method of interpolation in 3D
2.4.1 Tally
General
The rate of refund of energy room
()
G S
is solution of the variational equation
() () ()
()
G S
S
S ds
O
=
m
G
,
.
To solve this equation, the scalar field
()
G S
is discretized on a basis which we note
()
()
p S
J
J NR
1
.
That is to say
G
J
components of
()
G S
in this base:
()
()
G S
G p S
J
J
NR
J
=
=
1
O
0
S

It is also necessary to define
P
fields
I
independent discretized on a noted basis
()
(
)
Q S
K
K M
1
:
()
()
I
K
I
K
M
K
S
Q S
=
=
1
G
J
are given by solving the linear system with
P
equations and
NR
unknown factors:
() () ()
()
G has
B
I
P
has
p S Q S
S ds
B
ij
J
J
NR
I
ij
K
I
K
M
J
K
I
I
O
=
=
=
=
=
=






1
1
1
,
,
with
m
G
This system has a solution if one chooses
P
fields
I
independent such as:
P NR
and if
M
NR
.
It can comprise more equations than unknown factors, in which case it is solved within the meaning of least
squares.
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2.4.2 Methods of smoothing of G and Théta: polynomials of Legendre, functions of
form nodes
In Code_Aster, one chose two families of bases (cf [§2.2]):
·
polynomials of LEGENDRE
()
J
S
of degree J (0 J 7),
·
functions of form of the node K of
O
:
()
K
S
(1
K
NO = a number of nodes of
O
)
(of degree 1 for the linear elements and of degree 2 for the quadratic elements).
G S
()
is broken up:
·
maybe according to the polynomials of
LEGENDRE
:
LISSAGE_G = “LEGENDRE”
·
maybe according to the functions of forms of the nodes of the bottom of fissure:
LISSAGE_G = “LAGRANGE”
·
maybe according to the functions of forms of the nodes of the bottom of fissure with simplification of
stamp to reverse:
LISSAGE_G = “LAGRANGE_NO_NO”
()
I
S
are broken up:
·
maybe according to the polynomials of
LEGENDRE
:
LISSAGE_THETA = “LEGENDRE”
·
maybe according to the functions of forms of the nodes of the bottom of fissure:
LISSAGE_THETA = “LAGRANGE”
Attention, all the combinations between the families of functions of smoothing for G and THETA are not
not authorized:
G S
()
LEGENDRE
G S
()
LAGRANGE
G S
()
LAGRANGE_NO_NO
I
S
()
LEGENDRE
·
not not
I
S
()
LAGRANGE
·
·
·
Théta method: Legendre/G: Legendre: the resolution of the linear system gives:
()
()
()
G S
S
J
J
NDEG
J
=
=
G
0
Théta method: Lagrange/G: Legendre: one is reduced to the resolution of the linear system to
NO
equations and with
NDEG
+1 unknown factors:
() ()
()
J
I
J
J
NDEG
I
S
S dS
G
I
NO
O




=
=
=
0
1
G
,
,
In this case, one must have
NO
NDEG
, that is to say
NDEG
min (7,
NO
) where
NO
is the number of
nodes of the bottom of fissure.
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Théta method: Lagrange/G: Lagrange: one is reduced to the resolution of the square linear system
() ()
()
J
I
J
J
NO
I
S
S dS
G
I
NO
O




=
=
=
0
1
G
,
,
Simplified method known as
LAGRANGE_NO_NO
for the smoothing of G with diagonaliser the matrix consists
thus obtained by summation of the horizontal terms.
Remarks and consultings of use:
·
The user does not give a field théta, the fields
I
necessary to the calculation of
()
G S
are
calculated automatically according to the method specified in the control
CALC_G_LOCAL
[U4.82.04].
·
Choice of the maximum degree of the polynomials of Legendre: this choice depends on the number of
nodes in bottom of fissure. If one has a low number of nodes (ten) it is useless of
to take a degree higher than 3 (one conceives easily that the results are poor if one
try to find a polynomial of degree 7 passer by by 10 points). Beyond a score of
nodes in bottom of fissure one can use degrees going up to 7. The experiment shows
that the choice of a degree equal to 5 gives good results in the majority of the cases.
·
Choice of the method: it is difficult to give a preference to one or the other method. In
principle both give equivalent numerical results. Nevertheless method
Théta: Lagrange is a little more expensive in time CPU than the Théta method: Legendre.
For the first calculation, the use of the two methods and the comparison of the results,
allows to consolidate the validity of the model. If the bottom of fissure is a closed curve,
problems of continuity of the solution at the arbitrarily selected point like X-coordinate
curvilinear origin prohibit the use of the polynomials of Legendre. If the bottom of fissure were
declared “closed” in
DEFI_FOND_FISS
[U4.82.01], one must use the functions of form
(Lagrange) to describe the functions G and Théta.
·
Problem of nonthe respect of symmetry: if one modelizes only half of the solid by
report/ratio with the fissure, one must in theory have a curve G (S) whose slope of the tangent is
null with the interface of symmetry. This is not respected by the two methods. Values
G (S) obtained at the ends of the bottom of fissure must always be interpreted with
prudence, especially if the fissure is emerging in a nonperpendicular way (see [§2.2.3]).
·
Problem of the oscillations of the solution with smoothing of G by the polynomials of Legendre,
in particular if G (S) = 0 or constant. If one tries to interpolate a constant function by one
polynomial of raised degree, one expects this problem.

2.5
Calculation of G for a non-linear problem
The essential problem in the nonlinear situations comes from the difficulty in separating them
various energy contributions. It is necessary to consider two very distinct classes of problems:
·
that where, in spite of nonthe geometrical linearities or of behavior, one can exhiber one
potential for the interior and external actions (nonlinear elasticity or hyperelasticity),
·
that where such a potential does not exist (thermo elastoplasticity).
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For the first class, one can extend the criterion of Griffith by using the potential energy to
balance, and to calculate the rate of refund of energy as in thermo linear elasticity.
For the second class of problem, the essential difficulty comes owing to the fact that dissipation is not
only due to the propagation of the fissure itself. One cannot distinguish which share any more
restored energy is used for the propagation and which share is directly used by another
dissipative phenomenon (plasticity in fact).
2.5.1 Thermo nonlinear elasticity [R7.02.03]

2.5.1.1 Not linearity of behavior
The relation of nonlinear elastic behavior is described in [R5.03.20]. It should be noted that the law
elastoplastic of Hencky-Von Mises (isotropic work hardening) in the case of a radial loading and
monotonous is equivalent to the non-linear elastic law. Material hyperelastic A a behavior
reversible mechanics, i.e. any cycle of loading does not generate any dissipation. The EC
fact the relation of behavior of material derives from the free potential energy and one can give one
feel at the rate of refund of energy within the framework of the energy approach of Griffith.

2.5.1.2 Not geometrical linearity
One extends the relation of behavior to great deformations, insofar as it derives from one
potential (hyperelastic law). This functionality is started by the key word
DEFORMATION =
“GREEN”
in the controls
CALC_G_THETA
[U4.82.03] and
CALC_G_LOCAL
[U4.82.04].
The behavior of the solid is supposed to be hyperelastic, namely that the tensor of deformations of
Green-Lagrange
E
is connected to the field of displacement
U
measured compared to the configuration of
reference
O
by:
()
(
)
E
U
U
U U
ij
I J
J I
I K
K J
U
=
+
+
1
2
,
,
,
,
and that the relation of behavior derives from the free potential energy
()
E
:
S
E
ij
ij
=
S
being the tensor of the stresses of Piola-Lagrange called still second tensor of
Piola-Kirchoff
Such a relation of behavior makes it possible in any rigor to take into account the large ones
deformations. However, one confines oneself with great displacements and great rotations, but one
remain in small deformations. That to ensure the existence of a solution and to be identical to one
elastoplastic behavior under a monotonous radial loading [R5.03.20 §2.1].
2.5.2 Thermo elastoplasticity [R7.02.07]
The field of validity of the calculation of the rate of refund of energy is limited to the elastic thermo framework
linear or not-linear. To deal with the elastoplastic problem, two solutions are possible:
·
to bring back itself to a thermo problem elastic non-linear with restrictive assumptions,
·
to use another formulation, like that of the energy approach.
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2.5.2.1 Equivalence enters a thermo problem elastic nonlinear and a thermo problem
elastoplastic
The relation of nonlinear elastic behavior makes it possible to deal with the problems of
breaking process by approaching the elastoplastic thermo behavior. In the case of one
monotonous radial loading, it makes it possible to obtain strains and stresses of
structure similar to those which one would obtain if the material presented an isotropic work hardening.
The use of the indicators of discharge and loss of radiality makes it possible to be ensured of equivalence
laws of behavior.
But conditions of loadings proportional and monotonous, essential to ensure
coherence of the model with actual material, lead to important restrictions of the field of
capable problems being dealt with by this method (thermal in particular can lead it to
local discharges).

2.5.2.2 Formulation of parameter GTP
Caution:
This formulation results from recent research tasks and parameter GTP does not have yet
experimental validity.
Within the elastoplastic thermo framework, dissipated energy is distributed on the one hand in rupture and of other
leaves in plasticity without it being possible to separately quantify a priori these two types of
dissipation. The choice suggested in Code_Aster consists in deriving total mechanical energy for
to obtain a rate of refund of energy, which we will call parameter of rupture GTP. It
parameter makes it possible to analyze the nonmonotonous situations of loadings of the defect, for
irreversible material behaviors. The relations of elastoplastic thermo behavior are
described in detail in the document [R5.03.02].
How to make a calculation of GTP in thermo plasticity?
·
The presence of the key word factor
COMP_INCR
, and of the key word factor
RELATION
=
“VMIS_ISOT_LINE”
(or
“VMIS_ISOT_TRAC”
) in the controls
CALC_G_THETA
and
CALC_G_LOCAL
indicate that it is necessary to recover the field of
displacements
U
, stresses
, and characteristics of elastoplastic material. It is
also necessary to recover the fields of the tensors of plastic deformation by
the operator
CALC_ELEM
[U4.81.01].
·
Modeling by a notch: The defect must be modelized by a notch and not by
a fissure.
Indeed the formulation of G for an elastoplastic thermo relation is valid only
for a notched solid and not for a fissured solid: the main difficulty in
the establishment of this formulation is impossibility of showing the existence of
derived from total mechanical energy for a field comprising a fissure, and this
mainly by the absence of knowledge of the singularities of the fields in plasticity.
It is important to note that the terms taken into account in a calculation
thermo elastoplastic with the method théta are those supported by the elements
between the point of fissure and Rsup (in opposition to calculation in thermo elasticity not
linear where only the terms between Rinf and Rsup are nonnull).
Form of possible notch:
OK
OK
NOT
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Within the framework of the method théta one considers that the notch is propagated while keeping
even form (even if that does not have physical significance for a notch of the type
pelletizes).
The type of the notch and the radius in bottom of notch do not have an influence on the values of
GTP provided that the thickness of the notch is low compared to dimensions of
structure. If one modelizes by a pointed notch (conventional fissure) the results must
to be regarded as forgery (the terms of gradient of the plastic deformations are
badly calculated numerically).
It is necessary to use a fine mesh with quadratic elements in the vicinity
bottom of the notch to have reliable results in the cases of discharge.

·
Difficulties:
The smoothness of the mesh can lead to important calculating times.
The modeling of a fissure by a notch is delicate in 3D.
Which interpretation to make results obtained with this parameter of rupture GTP? With
run of the discharge the values of GTP are initially decreasing then then
increasing: this is in conformity with the definition of GTP which integrates all accumulation
plastic in bottom of defect. If one places oneself on the assumption of Griffith, one could
thus to have propagation of the fissure in discharge, which is problematic. Like one
the problem sees remains open and still requires the validation of a criterion of rupture
by experimental tests.

2.5.2.3 Approach energy the elastoplastic rupture and fomulation of the Gp parameter
This formulation results from recent research tasks [bib62].
One defines a rate of refund of energy in plasticity called Gp applicable to a fissure represented
in notch, while being based on the formulation of Frankfurt-Marigo for the fragile mediums and on
mechanics continues damage (see the chapter [§1.4]).
The Gp parameter is defined by the following fomule:
()
]
[(
L
L
L
Gp
=
/
)
.
(
max
)
.
dS
W
Notch
E
where
E
W
is elastic energy.
How to make a calculation of Gp in thermo plasticity?
Mesh:
The user must carry out a mesh of the structure with a defect modelized in the form of a notch
and not in the form of a fissure. The notch with the shape of a cigar or even the shape of a fissure
prolonged of a circle in its end.
OK
OK
Two types of authorized notches
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Bottom of notch (
) is represented by a half-circle of radius R. the area corresponding to
propagation of the notch is noted Ze (
L) (damaged Area) and depends on L, outdistances propagated,
in accordance with the figure below:




Notch with chips

The potential damaged area is modelized by a stacking of a hundred chips which
will allow to make the calculation of energy.
The refinement of the mesh close to the bottom of the notch must be extremely fine. Indeed, one advises
to choose the following geometrical data:
·
the radius of the circle in bottom of notch must be about R = 50 microns, according to material
considered,
·
each chip must have a thickness equal to 1/5 R is deltal = 10 microns.
One varies the propagated distance L while varying the number of chips considered: L = K deltal.
Only one mesh is enough.

Note:
The Gp parameter does not depend pathologically on the mesh.
Difficulties:
The mesh must be parameterized of kind to being able to carry out postprocessings
automatically with loops on the kth chip considered.
Because of smoothness of the mesh, calculations can be rather long and require
place memory.
The mesh of a notch in 3D is rather delicate to realize.
Calculation:
One makes a calculation with
STAT_NON_LINE
(). The calculation of energy in Code_Aster is done
simply thanks to the control
POST_ELEM
() with the option
ENER_ELAS
.
It is necessary then for each moment of nonlinear calculation, to calculate Gp (K), for each value
L corresponding to K chips by the formula:
Gp (K) = Eelas (K)/(K deltal)

One determines then for each moment of calculation, the ligament where the maximum of Gp is
obtained and in particular at the moment corresponding to the rupture.
L
Ze (
L)
notch
(
): melts of notch
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2.5.3 Indicators of discharge and loss of radiality
These indicators make it possible to locate the local discharges and the loss of radiality. Attention with
the interpretation of the indicators of discharge and loss of radiality: the value given to time Ti
corresponds to the diagnosis from what occurs between Ti and ti+1. Thus, the computed value with the last pitch of
time does not have a direction. The indicator of discharge is negative to indicate a local discharge, and
the indicator of radiality is worth 0 for a radial way.
2.5.4 The Councils of use of the law of behavior
Calculation in thermo linear elasticity:
Before carrying out a calculation into non-linear it is advised to carry out the first thermo calculation
linear rubber band and post-to treat the results to have a first idea of about size
results.
Calculation into non-linear:
Insofar as it is possible it is preferable to make an elastoplastic thermo calculation and of
to compare the results obtained with those of a thermo calculation elastic nonlinear. That allows
to make sure that the loading is radial and monotonous, with possibly a certain approximation
(use of the indicators of discharge and loss of radiality). If such is not the case, the problem
remain open, and one can then be directed worms of postprocessings of the type “approaches local”.
Even if nothing prohibits in Code_Aster to carry out a calculation with a law of behavior and of
post-to treat with another law, the results are generally to question and the user must
thus to be very attentive on this point.
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3
Implementation of a calculation in breaking process in
Code_Aster
3.1 Methodology
3.1.1 Controls
Code_Aster
One presents the main controls here to be implemented in Code_Aster Version 6. for
to carry out a postprocessing in fragile breaking process. For more precise details on
controls concerned, one will be able to refer to the documentation of use.
Acquisition of the data of the file of mesh:
LIRE_MAILLAGE
[U4.21.01]
For a mesh 3D it is necessary to think as of the generation of the mesh of naming the nodes of the bottom of
fissure and the meshs of the lips of the fissure. The nodes of the bottom of fissure must be ordered for
to define the direction of course of the curvilinear X-coordinate of the bottom of fissure. One can order the nodes of
melts of fissure with the control
DEFI_FOND_FISS
[U4.82.01].
The Council:
Obligatorily to use quadratic elements (for an elastoplastic problem and strongly
advised for a 3D problem).
For more details to consult [§2.1].
Definition of the model:
AFFE_MODELE
[U4.41.01]
The modelized physical phenomenon is mechanical (
PHENOMENE=' MECANIQUE'
). Modeling is
chosen among modelings of the continuous mediums 2D plane deformations or plane stresses,
axisymmetric 2D and 3D (
D_PLAN, C_PLAN, AXIS, 3D
).
Characteristics of material:
DEFI_MATERIAU
[U4.43.01] and
AFFE_MATERIAU
[U4.43.03]
The behavior is is elastic linear (key word factor
ELAS
or
ELAS_FO
) is nonlinear (word
key factor
ECRO_LINE
or
ECRO_LINE_FO
or
TRACTION
). Characteristics of materials
to define are the modulus Young, the Poisson's ratio, possibly the expansion factor
thermics and in the nonlinear case elastic limit and the module of work hardening or the curve of
traction.
These characteristics can depend on the temperature for the calculation of the rate of refund
of energy.
For the calculation of the stress intensity factors the characteristics must be defined on all
materials, including on the elements of edge, because of method of calculation [R7.02.05]. For
to ensure itself so it is advised to do one
AFFE=_F (TOUT=' OUI')
in the control
AFFE_MATERIAU
[U4.43.03], even if it means to use the rule of overload then.
Assignment of the mechanical loadings:
AFFE_CHAR_MECA (_F)
[U4.44.01]
The mechanical loadings are those of the continuous mediums. One will take care that the loadings
used either supported well by the operators of breaking process (voir§1.3.4) if not they are
been unaware of.
For a problem where thermics intervenes, one recovers the thermal loading of origin by the word
key
TEMP_CALCULEE
in the control
AFFE_CHAR_MECA
[U4.44.01]. For the possible resolution
thermal problem, it is necessary to define the thermal model with
AFFE_MODELE
[U4.41.01] (
selected modeling is the same one as that of the mechanical model). The thermal loadings are
those of the continuous mediums and are defined with
AFFE_CHAR_THER (_F)
[U4.44.02]. The resolution is
made with
THER_LINEAIRE
[U4.54.01] or
THER_NON_LINE
[U4.54.02].
Resolution of the mechanical problem:
MECA_STATIQUE
[U4.51.01] or
STAT_NON_LINE
[U4.51.03]
If the problem is elastic linear, the total operator is used
MECA_STATIQUE
who calculates them
displacements starting from the model, of the material field, the boundary conditions and the loading.
The concept produced by this operator is of type
evol_elas
.
If the problem is non-linear, the total operator is used
STAT_NON_LINE
who produces a concept of
type
evol_noli
.
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It is possible to use the control
CREA_CHAMP (OPERATION=' EXTR')
[U4.72.04] to recover
the field of displacements to the nodes (necessary for the postprocessing of mechanics of
rupture). But one can also use the concept directly
evol_elas
and
evol_noli
in
controls of breaking process, by specifying the desired sequence numbers.
The Council: For an expensive non-linear calculation in memory and time CPU, it is advised of
to constitute a base and to continue the study for postprocessings (in particular in mechanics of
rupture). For more details to consult the documents
BEGINNING
[U4.11.01],
CONTINUATION
[U4.11.03] and
END
[U4.11.02]. It is then necessary to be vigilant on the compatibility of the versions of Code_Aster between two
connected executions.
Postprocessing in breaking process
Definition of the characteristics of the bottom of fissure:
DEFI_FOND_FISS
[U4.82.01]
This control makes it possible to define:
·
in 2D the node of the bottom of fissure and the normal with the fissure,
·
in 3D nodes of the bottom of fissure and meshs of the lips of the fissure.
In 2D this control is obligatory only for the calculation of the coefficients of intensity of
stresses. In the case of a symmetrical structure where half of the fissure is represented, the single one
lip must be defined by
LEVRE_SUP
. If the fissure does not emerge, it is not then of course
necessary to define the directions of théta at the ends by
DTAN_ORIG
and
DTAN_EXTR
.
Assignment of the field théta:
CALC_THETA
[U4.82.02]
This control makes it possible to affect the field théta necessary to the calculation of the rate of refund of energy
or of the stress intensity factors. The field théta is a field with the nodes defined on all it
mesh.
The user must define the characteristics of the field théta:
·
the module (equal to 1. a priori),
·
direction of propagation: equalize with that of the bottom of fissure in 2D, calculated
automatically in 3D starting from the directions of propagation of the nodes in bottom of fissure
(these directions are recovered by the concept of the type
fond_fiss
product by the operator
DEFI_FOND_FISS
or by the key word
DIRE_THETA
),
·
the Rinf radii and Rsup of the crowns surrounding the bottom of fissure and used in
method théta: in 2D the bottom of fissure is tiny room to a node and the crowns are
circulars. In 3D the radii can be variable with the curvilinear X-coordinate of the bottom of
fissure and R
inf
, R
sup
two deformed and variable cylinders define then surrounding the bottom
of fissure.
This control is not necessary if one carries out a calculation of the rate of refund of energy room:
field théta is calculated automatically starting from the bottom of fissure resulting from
DEFI_FOND_FISS
, of
radii R
inf
and R
sup
and of the method of interpolation defined in the control
CALC_G_LOCAL
.
Choice of the radii R
inf
and R
sup
:
·
The choice of the radii R
inf
and R
sup
is independent of the topology of the mesh (even if it is
preferable, one is not obliged to have a radiant mesh at a peak of fissure).
·
Never not to use a definite field théta with a radius lower R
inf
no one. Indeed fields
displacements are singular in bottom of fissure and introduce results vague in
postprocessing of breaking process.
·
In thermo elastoplasticity, one uses a fissure as notch. One will make sure that the radius
inferior R
inf
is quite higher than the radius of the notch.
·
In 2D the radius higher R
sup
can be as large as one wants in condition of course that
crown thus defined either contained in the solid.
·
In 3D the problem is more delicate: it is necessary to find a compromise between R
inf
not too small
(results vague because of the fields of singular displacements evil calculated in bottom of
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fissure) and R
sup
not too large (direction of the field théta can be badly calculated). To see it
[§2.2.3] for more details.
·
To forget only the loadings applied beyond R
sup
have a null contribution
in postprocessings of breaking process. This can be useful if one is applied
loading not supported like
FORCE_NODALE
,
DDL_IMPO
(in 2D) or
FACE_IMPO
(in
3D) to see [§1.3.4].
·
To take several consecutive crowns to check [R1, R2], [R2, R3], [R3, R4],…
Calculation of the rate of refund of energy in 2D or 3D:
CALC_G_THETA (_T)
[U4.82.03]
The control
CALC_G_THETA
allows to calculate
the rate of refund of energy G in 2D or 3D
by the method théta in the case of an elastic thermo problem linear or not linear.
For this calculation the user must specify obligatorily:
·
the model,
·
the material field,
·
the field of displacements (starting from a field with the nodes or of a result),
·
the field théta,
and possibly:
·
loading (if the voluminal, surface loading on the lips of the fissure or origin
thermics),
·
the relation of behavior (by defect thermo linear elasticity),
·
plastic deformations (if the behavior is thermo elastoplastic).
The control
CALC_G_THETA
also the calculation of the rate of refund of energy allows with
Lagrangian propagation (i.e. for an extension of the fissure by using the same mesh)
in 2D or 3D in the case of a thermo problem elastic linear (option
CALC_G_LAGR
). For more
precise details see document [R7.02.04].
Calculation of the coefficients of intensity of stresses in 2D:
CALC_G_THETA (_T)
[U4.82.03]
The control
CALC_G_THETA
allows to calculate
coefficients of intensity of stresses in 2D
(plane stresses or plane deformations) by the method théta in the case of a problem
thermo linear rubber band. The option should be specified
CALC_K_G
under the key word
OPTION
.
For this calculation the user must specify obligatorily:
·
the model,
·
the material field,
·
the field of displacements (starting from a field with the nodes or of a result),
·
the field théta,
·
bottom of fissure,
and possibly loading (if the voluminal, surface loading on the lips of the fissure or
of thermal origin).
Calculation of the coefficients of intensity of stresses by extrapolation of the field of
displacements:
POST_K1_K2_K3.
The control
POST_K1_K2_K3
allows to calculate the coefficients of intensity of stresses (y
included/understood K3) in 2D (forced plane or plane deformations), 3D and axisymmetric the case of one
fissure planes in a homogeneous and isotropic elastic material.
For this calculation the user must obligatorily specify the fields of displacement on each lip,
provided in the form of tables extracted the concept result
evol_elas
by the control
POST_RELEVE_T
.
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Calculation of the rate of refund of energy room in 3D:
CALC_G_LOCAL (_T)
[U4.82.04]
The control
CALC_G_LOCAL
allows to calculate
the rate of refund of energy G in 2D or 3D
by the method théta in the case of an elastic thermo problem linear or not linear.
For this calculation the user must specify obligatorily:
·
the model,
·
the material field,
·
the field of displacements (starting from a field with the nodes or of a result),
·
bottom of fissure,
·
the radii R
inf
and R
sup
defining the crowns surrounding the bottom of fissure,
and possibly:
·
loading (if the voluminal, surface loading on the lips of the fissure or origin
thermics),
·
the relation of behavior (by defect thermo linear elasticity),
·
method of discretization of the field théta in bottom of fissure (per defect method of
Legendre, degree 5),
·
plastic deformations (if the behavior is thermo elastoplastic).
It will be noted that the field théta is calculated starting from the bottom of fissure and the radii R
inf
and R
sup
(useless
to use the control
CALC_THETA
except for the particular case of the Lagrangian propagation).
The control
CALC_G_THETA
also the calculation of the rate of refund of energy room allows with
Lagrangian propagation in 3D (option
CALC_G_LGLO
) in the case of an elastic thermo problem
linear [R7.02.04].
Calculation of energy for the calculation of the rate of refund of energy in Gp plasticity:
The control is used
POST_ELEM
(), with the option
ENER_ELAS
. One obtains the calculation of Gp by one
post_traitement manual (see chapter [§1.4] and chapter [§2.5.2.3]).
3.1.2 Traps to be avoided
Mesh:
In 3D the nodes of the bottom of fissure must be ordered.
For an elastoplastic thermo problem (parameter GTP and Gp) it is necessary to modelize the fissure by one
notch and to use quadratic elements.
Loading:
During a thermal calculation, one should not forget to introduce into the operand
CHARGE
of
CALC_G_THETA
or
CALC_G_LOCAL
the thermal load of origin.
The not supported loadings are ignored. No message of alarm is transmitted, one will refer
thus with [§1.3.4] to make sure that the loadings used have a direction in breaking process
and are well treated.
If the list of the loads comprises more than one load, a loading of comparable nature cannot appear
that in only one load. In the contrary case, only the last load is taken into account.
If the field of displacement were calculated by a load with a multiplying coefficient different from
1., one will have, to obtain G corresponding to the good loading, to introduce into the operand
CHARGE
of
CALC_G_THETA
or
CALC_G_LOCAL
the load in question multiplied by this coefficient (see
COMB_CHAM_NO
[U4.72.02] for this problem).
If one makes a calculation in great transformations (key word
DEFORMATION
=
“GREEN”
under the key word
factor
COMP_ELAS
) the supported loadings must be died loads, typically one
force imposed and not a pressure [R7.02.03 §2.4].
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Law of behavior:
Nothing prohibits in practice to solve the mechanical problem with a law of behavior (in
MECA_STATIQUE
or
STAT_NON_LINE
) and to carry out postprocessing with another law of
behavior: to avoid.
Symmetry of the loading and standardization:
In the controls
CALC_G_THETA
and
CALC_G_LOCAL
the key word
SYME_CHAR
allows to indicate if
the loading is symmetrical or antisymmetric in the case or one only half of the solid modelizes
compared to the fissure.
This key word is essential if the option is used
“CALC_K_G”
to calculate the factors of intensity of
stresses: it makes it possible to assign K2 to 0 if the loading is symmetrical compared to the fissure or K1 with
0 if it is antisymmetric.
It also makes it possible to multiply by 2, the values of the rate of refund of energy G if one does not modelize
that half of the solid compared to the fissure.
“WITHOUT” “ANTI” “SYME”
G G
ASTER
2.*
G
ASTER
2.*
G
ASTER
K1 K1
ASTER
K1
ASTER
0.
K2 K2
ASTER
0. K2
ASTER
Caution:
Not to forget that in certain configurations, a manual postprocessing is necessary
to obtain the standardization of the value of the rate of refund of energy. In particular in
axisymmetric, G should be divided
ASTER
by the distance from the bottom of fissure to the axis of symmetry and in
3D by the length of the bottom of fissure [§2.3].
Definition of the bottom of fissure and the Rinf radii and Rsup in 3D:
When the fissure is emerging, to define the directions of the field théta well at the ends of the bottom of
fissure using the key words
DTAN_ORIG
and
DTAN_EXTR
in the control
DEFI_FOND_FISS
[U4.82.01]. See 2.2.2.
Attention with the choice of the Rinf radii and Rsup of the crown. See [§2.2.3].
Calculation of energy for the calculation of the rate of refund of energy in Gp plasticity:
It should be taken care that energy is calculated with sufficient precision because one carries out for
calculation of Gp a difference between very small quantities.
3.1.3 Checks concerning postprocessings of breaking process
It is important to have an idea of about size of the results before beginning any calculation
numerical (simplified model, test of reference, bibliography,…).
It is advised to use the controls successively
CALC_G_THETA
or
CALC_G_LOCAL
with with
less 3 fields théta of different crowns to ensure itself of the stability of the results. In the event of
important variation (higher than 5-10%) it is necessary to wonder about the good taking into account of all
modeling. This stability is a condition necessary (but not sufficient) for the validity of
results.
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3.2 Example 1: Calculation of G, K1 and K2 for an elastic problem
linear in 2D
It is about a test of breaking process for a linear elastic problem in plane stresses
SSLP101 [V3.02.101]. One calculates the rate of refund of energy and the coefficients of intensity of
stresses for a linear elastic problem in plane stresses.
3.2.1 Geometry
Rectangular plate with emerging fissure OC.
For reasons of symmetry, the model is tiny room to the half-structure Y
0.
H
I
With
has
C
Y
O
X
U
v
Height plates: H = 250 mm
C = N668
Width plates: I = 100 mm
Depth fissures: have = 37.5 mm (OC)

3.2.2 Material properties
E = 200000 NAKED MPa = 0.3
Assumption of the plane stresses.

3.2.3 Boundary conditions and loadings
Stress imposed in Y = H:
= 1 MPa
Displacement for the edge (has
X
I, Y = 0):
v = 0.
Not fixes a:
U = v = 0.
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3.2.4 Command file
BEGINNING ()
MA=LIRE_MAILLAGE ()
MO=AFFE_MODELE (MAILLAGE=MA,
AFFE=_F (ALL = “YES”,
PHENOMENON = “MECHANICAL”,
MODELING = “C_PLAN”))
MAT=DEFI_MATERIAU (ELAS=_F (E = 200000., NAKED = 0.3, RHO = 1.))
CHMAT=AFFE_MATERIAU (MAILLAGE=MA,
AFFE=_F (ALL = “YES”, MATER = CHECHMATE))
CH=AFFE_CHAR_MECA (MODELE=MO, DDL_IMPO= (
_F (GROUP_NO = “GRNM5”, DY = 0.),
_F (NODE = “N451”, DX = 0.)),
FORCE_CONTOUR=_F (GROUP_MA = “GRMA1”, FY = 1.) )
FCONT = FORMULA (REEL= """ (REAL:X, REALITY:Y) =1.""")
CHFONC=AFFE_CHAR_MECA_F (MODELE=MO,
FORCE_CONTOUR=_F (GROUP_MA = “GRMA1”,
FY = FCONT))
CHAMDEPL=MECA_STATIQUE (MODELE=MO, CHAM_MATER=CHMAT,
EXCIT=_F (LOAD = CH))
DEP=CREA_CHAMP (OPERATION=' EXTR', TYPE_CHAM=' NOEU_DEPL_R',
NOM_CHAM=' DEPL', RESULTAT=CHAMDEPL,
NUME_ORDRE=1)
THETA1=CALC_THETA (MODELE=MO,
THETA_2D=_F (NODE = “N668”, MODULE = 1.,
R_INF = 22.04078,
R_SUP = 30.),
DIRECTION= (1., 0., 0.,))
FOND=DEFI_FOND_FISS (MAILLAGE=MA,
FOND=_F (NODE = “N668”),
NORMALE= (0., 1., 0.,))
G1=CALC_G_THETA_T (MODELE=MO,
DEPL=DEP,
THETA=THETA1,
CHARGE=CHFONC,
SYME_CHAR=' SYME',
COMP_ELAS=_F (RELATION = “ELAS”,
DEFORMATION = “SMALL”),
CHAM_MATER=CHMAT)
GK1=CALC_G_THETA_T (MODELE=MO,
DEPL=DEP,
THETA=THETA1,
FOND_FISS=FOND,
SYME_CHAR=' SYME',
CHARGE=CHFONC,
CHAM_MATER=CHMAT,
OPTION=' CALC_K_G')
PRECISION=1.E-4)
END ()
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3.3 Example 2: Calculation of G and G (S) local for a thermo problem
rubber band in 3D
It is about a test of breaking process into thermomechanical for a three-dimensional problem
HPLV103 [V7.03.103]. One considers a circular fissure plunged in an elastic thermo medium.
One imposes a uniform temperature on the lips of the fissure. This test makes it possible to calculate the rate of
restitution of energy total G and the rate of refund G local in various points of the bottom of fissure.
3.3.1 Geometry
One considers a circular fissure plunged in an elastic thermo medium:
X
y
Z
H
D
E
With
F
I
C
O
G
C
B
The radius of the fissure is: OA = OB = 1.0
The medium is modelized by a parallelepiped of dimensions: OE = OD = OC = 30.0
3.3.2 Material properties
Thermal conductivity:
= 1.
Thermal expansion factor:
= 10
- 6
/°C
Young modulus:
E = 2.10
5
MPa
Poisson's ratio:
= 0.3
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3.3.3 Boundary conditions and loadings
O
Y
Z
X
has
T
0
= constant = - 1
3.3.4 Command file
BEGINNING ()
M=LIRE_MAILLAGE ()
M=DEFI_GROUP (reuse=M, MAILLAGE=M, CREA_GROUP_NO= (
_F (GROUP_MA = “LEVREINF”),
_F (GROUP_MA = “SSUP_S'),
_F (GROUP_MA = “SAV_S'),
_F (GROUP_MA = “SLAT_S'),
_F (GROUP_MA = “SINF”),
_F (GROUP_MA = “SAR”),
_F (GROUP_MA = “SLAT”),
_F (NAME = “INFINITE”,
UNION = (“SINF”, “SAR”, “SLAT”,)))
)
#--------------------------------------------------------------------
# BEGINNING OF THERMICS #
#--------------------------------------------------------------------
MOTH=AFFE_MODELE (MAILLAGE=M,
AFFE=_F (ALL = “YES”,
PHENOMENON = “THERMAL”,
MODELING = “3D”)
)
MATH=DEFI_MATERIAU (THER=_F (RHO_CP = 0., LAMBDA = 1.) )
CMTH=AFFE_MATERIAU (MAILLAGE=M,
AFFE=_F (ALL = “YES”,
MATER = MATHS)
)
CHTH=AFFE_CHAR_THER (MODELE=MOTH, TEMP_IMPO= (
_F (GROUP_NO = “INFINITE”,
TEMP = 0.0),
_F (GROUP_NO = “LEVREINF”,
TEMP = 1.))
)
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THLI=THER_LINEAIRE (MODELE=MOTH,
CHAM_MATER=CMTH,
EXCIT=_F (LOAD = CHTH)
)


TEMP=CREA_CHAMP (OPERATION=' EXTR', TYPE_CHAM=' NOEU_TEMP_R',
NOM_CHAM=' TEMP', RESULTAT=THLI,
INST=0.0
)
#--------------------------------------------------------------------
# END OF THERMICS #
#--------------------------------------------------------------------

MO=AFFE_MODELE (MAILLAGE=M,
AFFE=_F (ALL = “YES”,
PHENOMENON = “MECHANICAL”,
MODELING = “3D”)
)

MA=DEFI_MATERIAU (ELAS=_F (E = 200000.,
NAKED = 0.3,
ALPHA = 0.000001)
)

#

CM=AFFE_MATERIAU (MAILLAGE=M,
AFFE=_F (ALL = “YES”,
MATER = MA,
TEMP_REF = 0.)
)

#

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

#

MEST=MECA_STATIQUE (MODELE=MO,
CHAM_MATER=CM,
EXCIT=_F (LOAD = CH)
)

#
DEPLA=CREA_CHAMP (OPERATION=' EXTR', TYPE_CHAM=' NOEU_DEPL_R',
NOM_CHAM=' DEPL', RESULTAT=MEST,
INST=0.0
)
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#--------------------------------------------------------------------
# CALCULATION OF G
#--------------------------------------------------------------------

FF=DEFI_FOND_FISS (MAILLAGE=M,
FOND=_F (GROUP_NO = “LFF”),
NORMALE= (0., 0., 1.,),
DTAN_ORIG= (1., 0., 0.,),
DTAN_EXTR= (0., 1., 0.,)
)

#
#
THETA1=CALC_THETA (MODELE=MO,
FOND_FISS=FF,
THETA_3D=_F (ALL = “YES”,
MODULATE = 1.0,
R_INF = 0.07,
R_SUP = 0.2)
)

#

G1=CALC_G_THETA_T (MODELE=MO,
DEPL=DEPLA,
CHAM_MATER=CM,
THETA=THETA1,
CHARGE=CH,
COMP_ELAS=_F (RELATION = “ELAS”,
DEFORMATION = “SMALL”)
)


#--------------------------------------------------------------------
# CALCULATION OF GLOCAL #
#--------------------------------------------------------------------
GLOC1=CALC_G_LOCAL_T (MODELE=MO,
DEPL=DEPLA,
CHAM_MATER=CM,
FOND_FISS=FF,
CHARGE=CH,
DEGRE=6,
R_INF=0.07,
R_SUP=0.2,
LISSAGE_THETA=' LAGRANGE',
LISSAGE_G=' LEGENDRE',
COMP_ELAS=_F (RELATION = “ELAS”,
DEFORMATION = “SMALL”)
)
END ()
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3.4
Example 3: calculation of Gp for an elastoplastic problem in 2D
One carries out a calculation in breaking process for an elastoplastic problem in 2D deformations
plane.
One implements the calculation of the Gp parameter resulting from recent research tasks (see chapter
[§2.5.2.3]) to highlight “the effect small defect”.
Context and objective:
Tenacity is a parameter determined in experiments on a test-tube CT fissured in traction,
who is supposed to represent the breaking strength of material. But on the test-tubes, the fissures
are big sizes compared to the real cases. The effects of triaxiality are important and plasticity
weak.
On the contrary on real cases, the fissures are smaller sizes, the effects of triaxiality are more
weak, and plasticity is stronger. Measured tenacity would be then larger, from where a gain of
margins potential. The size of the fissure thus has an effect on the measured value of tenacity. It is this
effect which is called “effect small defect”.
One applies here the energy approach based on the calculation of Gp parameter to the interpretation of the effect
small defect.
One considers on the one hand a test-tube SENB with a great defect (SENB1) and on the other hand one
test-tube SENB with a small defect (SENB2).
3.4.1 Geometry
Rectangular plate with small or great defect. One represents only half of the structure.
Height plates H = 50 mm
Width plates L = 420 mm
Spacing between two supports S = 370 mm
Cut defect af = 25 mm (SENB1) or 3.8 mm (SENB2).














3.4.2 Properties of materials
Young modulus: E = 208510
Poisson's ratio: Naked = 0.3
Traction diagram with nonlinear work hardening (behavior
VMIS_ISOT_TRAC
)
3.4.3 Boundary conditions and loadings
One applies the condition of support to Lappui Dy = 0.
One applies the condition of symmetry dx = 0 to the ligament of defect LIGR.
One charges in displacement out of Dy on edge LPILOT.
H
S/2
L/2
af
Lappui
Lpilot
Ligr
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3.4.4 2 command files for each of the 2 cases

TEST-TUBE SENB1 GREAT DEFECT af/H = 0.5 IS af = 25 MM
BEGINNING ()
PRE_GIBI ()
MA=LIRE_MAILLAGE ()
MA=DEFI_GROUP (reuse =MA,
MAILLAGE=MA,
CREA_GROUP_NO=_F (TOUT_GROUP_MA=' OUI',),)
#
# MODELING OF THE MESH
#
MOD=AFFE_MODELE (MAILLAGE=MA,
AFFE=_F (TOUT=' OUI',
PHENOMENE=' MECANIQUE',
MODELISATION=' D_PLAN',),)
#
# DEFINITION OF MATERIAL
#
SIGM_F = DEFI_FONCTION (NOM_PARA = “EPSI”,
VALE= (
2.74E-03, 571.32,
1.29E-02, 609.42,
2.31E-02, 647.52,
3.33E-02, 685.62,
4.34E-02, 715,
5.36E-02, 746,
6.37E-02, 775,
7.38E-02, 797,
8.39E-02, 814,
9.40E-02, 831.66,
0.10405, 844.47,
0.11411, 856.22,
0.12416, 867.1,
0.14425, 886.7,
0.16434, 904.04,
0.18441, 919.62,
0.20448, 933.78,),
PROL_DROITE = “CONSTANT”,
PROL_GAUCHE = “CONSTANT”,
)

ACIER=DEFI_MATERIAU (ELAS=_F (E=208510.,
NU=0.3,
ALPHA=0.0,),
TRACTION=_F (SIGM = SIGM_F),
)

CH_MAT=AFFE_MATERIAU (MAILLAGE=MA,
AFFE=_F (GROUP_MA=' SENB',
MATER=ACIER,
TEMP_REF=0.0,),)
#
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# DEFINITION Of a LIST Of MOMENTS AND a SLOPE
#
LIST=DEFI_LIST_REEL (DEBUT=0.0,
INTERVALLE= (_F (JUSQU_A=22.0,
NOMBRE=22,),
_F (JUSQU_A=27.0,
NOMBRE=5,),
_F (JUSQU_A=32.0,
NOMBRE=5,),
_F (JUSQU_A=37.0,
NOMBRE=5,),
_F (JUSQU_A=42.0,
NOMBRE=5,),
_F (JUSQU_A=47.0,
NOMBRE=5,),
_F (JUSQU_A=52.0,
NOMBRE=5,),
_F (JUSQU_A=57.0,
NOMBRE=5,),
_F (JUSQU_A=61.0,
NOMBRE=4,),
_F (JUSQU_A=65.0,
NOMBRE=4,),
_F (JUSQU_A=70.0,
NOMBRE=5,),
_F (JUSQU_A=76.0,
NOMBRE=6,),
_F (JUSQU_A=82.0,
NOMBRE=6,),
_F (JUSQU_A=88.0,
NOMBRE=6,),
_F (JUSQU_A=94.0,
NOMBRE=6,),
_F (JUSQU_A=100.0,
NOMBRE=6,),),)
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RAMPE=DEFI_FONCTION (NOM_PARA=' INST',
VALE= (0.0, 0.0, 100.0, 100.0),
PROL_DROITE=' LINEAIRE',
PROL_GAUCHE=' LINEAIRE',)
#
# LOADING AND CONDITIONS LIMITING
#
CHAR=AFFE_CHAR_MECA (MODELE=MOD,
DDL_IMPO= (_F (GROUP_NO=' LIGR',
DX=0.0,),
_F (GROUP_NO=' LAPPUI',
DY=0.0,),
_F (GROUP_NO=' LPILOT',
DY=-0.04,),),)
#
# APPLICATION OF THE LOAD & CALCULATION OF THE STRESSES
#
RESU=STAT_NON_LINE (MODELE=MOD,
CHAM_MATER=CH_MAT,
EXCIT=_F (CHARGE=CHAR,
FONC_MULT=RAMPE,),
COMP_INCR=_F (RELATION=' VMIS_ISOT_TRAC',
DEFORMATION=' PETIT',
GROUP_MA=' SENB',),
INCREMENT=_F (LIST_INST=LIST,
NUME_INST_FIN=30,),
NEWTON=_F (PREDICTION=' TANGENTE',
MATRICE=' TANGENTE',
REAC_ITER=4,),
RECH_LINEAIRE=_F (RESI_LINE_RELA=1.E-3,
ITER_LINE_MAXI=3,),
CONVERGENCE=_F (RESI_GLOB_MAXI=1.E-08,
RESI_GLOB_RELA=1.E-08,
ITER_GLOB_MAXI=20,),
SOLVEUR=_F (METHODE=' MULT_FRONT',
RENUM=' METIS',),)
#
# CALCULATION OF G
#
THETA1=CALC_THETA (MODELE=MOD,
DIRECTION= (0.0, 1.0, 0.0),
THETA_2D=_F (GROUP_NO=' O',
MODULE=1.0,
R_INF=0.25,
R_SUP=0.5,),)
G1=CALC_G_THETA_T (MODELE=MOD,
CHAM_MATER=CH_MAT,
THETA=THETA1,
RESULTAT=RESU,
TOUT_ORDRE=' OUI',
SYME_CHAR=' SYME',
COMP_ELAS=_F (RELATION=' ELAS_VMIS_TRAC',
DEFORMATION=' PETIT',),)
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IMPR_TABLE (TABLE=G1, FORMAT = “AGRAF”)


THETA2=CALC_THETA (MODELE=MOD,
DIRECTION= (0.0, 1.0, 0.0),
THETA_2D=_F (GROUP_NO=' O',
MODULE=1.0,
R_INF=0.50,
R_SUP=1.0,),)
G2=CALC_G_THETA_T (MODELE=MOD,
CHAM_MATER=CH_MAT,
THETA=THETA2,
RESULTAT=RESU,
TOUT_ORDRE=' OUI',
SYME_CHAR=' SYME',
COMP_ELAS=_F (RELATION=' ELAS_VMIS_TRAC',
DEFORMATION=' PETIT',),)

IMPR_TABLE (TABLE=G2, FORMAT = “AGRAF”)

THETA3=CALC_THETA (MODELE=MOD,
DIRECTION= (0.0, 1.0, 0.0),
THETA_2D=_F (GROUP_NO=' O',
MODULE=1.0,
R_INF=1.0,
R_SUP=2.0,),)

G3=CALC_G_THETA_T (MODELE=MOD,
CHAM_MATER=CH_MAT,
THETA=THETA3,
RESULTAT=RESU,
TOUT_ORDRE=' OUI',
SYME_CHAR=' SYME',
COMP_ELAS=_F (RELATION=' ELAS_VMIS_TRAC',
DEFORMATION=' PETIT',),)

IMPR_TABLE (TABLE=G3, FORMAT = “AGRAF”)

END ()
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Code_Aster
®
Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
U2.05.01-A
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:
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Instruction manual
U2.05 booklet: Damage and breaking process
HT-66/03/002/A
CONTINUATION (PAR_LOT=' NON')

ENEE= [Nun] * 200
ENET= [Nun] * 200
#
deltal = 0.01
#
importation bone
f2=open (“fort.44”, “W”)
f3=open (“fort.45”, “W”)
f2.write (“brutal Propagation - Test-tube SENB1 - Mesh M1 \ N”)
f2.write (“Work hardening diagram traction ECA \ N”)
f2.write (“Propagation - elastic Energy - G plastic (dW/DLL) \ N”)

for K in arranges (1,101):

LIG = “COPS_ % I” % (K)
print “ligament number: ”, K
print “cumulated propagation: ”, K * deltal, “millimetres”

ENEE [K] = POST_ELEM (MODELE=MOD,
RESULTAT=RESU,
CHAM_MATER=CH_MAT,
TOUT_ORDRE = ' OUI',
ENER_ELAS=_F (GROUP_MA=LIG),
TITER=' Energy élastique',
)
IMPR_TABLE (TABLE=ENEE [K],
FORMAT_R='1PE18.11')


# End of the iterations

for J in arranges (1,31):
f2.write (“Urgent: % F \ N” % (J))
gpmax = 0.
for K in arranges (1,101):
ETOT=ENEE [K] [“TOTAL”, J]
GP = 2.0 * (ETOT)/(K * deltal)
yew GP > gpmax:
gpmax = GP
kmax = K * deltal
f2.write (“% F % 0.11f % 3f \ N” % ((K * deltal), ETOT, GP))
f3.write (“% F % 3f % 3f \ N” % (J, kmax, gpmax))

f2.close ()
f3.close ()

END ()
TEST-TUBE SENB2 SMALL DEFECT af/H = 0.076 IS af = 3.8 MM
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Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
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Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
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U2.05 booklet: Damage and breaking process
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BEGINNING ()

PRE_GIBI ()

MA=LIRE_MAILLAGE ()

MA=DEFI_GROUP (reuse =MA,
MAILLAGE=MA,
CREA_GROUP_NO=_F (TOUT_GROUP_MA=' OUI',),)
#
# MODELING OF THE MESH
#

MOD=AFFE_MODELE (MAILLAGE=MA,
AFFE=_F (TOUT=' OUI',
PHENOMENE=' MECANIQUE',
MODELISATION=' D_PLAN',),)
#
# DEFINITION OF MATERIAL
#

SIGM_F = DEFI_FONCTION (NOM_PARA = “EPSI”,
VALE= (
2.74E-03, 571.32,
1.29E-02, 609.42,
2.31E-02, 647.52,
3.33E-02, 685.62,
4.34E-02, 715,
5.36E-02, 746,
6.37E-02, 775,
7.38E-02, 797,
8.39E-02, 814,
9.40E-02, 831.66,
0.10405, 844.47,
0.11411, 856.22,
0.12416, 867.1,
0.14425, 886.7,
0.16434, 904.04,
0.18441, 919.62,
0.20448, 933.78,
),
PROL_DROITE = “CONSTANT”,
PROL_GAUCHE = “CONSTANT”,
)
ACIER=DEFI_MATERIAU (ELAS=_F (E=208510.,
NU=0.3,
ALPHA=0.0,),
TRACTION=_F (SIGM = SIGM_F),
)
CH_MAT=AFFE_MATERIAU (MAILLAGE=MA,
AFFE=_F (GROUP_MA=' SENB',
MATER=ACIER,
TEMP_REF=0.0,),)
#
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Code_Aster
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Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
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Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
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# DEFINITION Of a LIST Of MOMENTS AND a SLOPE
#
LIST=DEFI_LIST_REEL (DEBUT=0.0,
INTERVALLE= (_F (JUSQU_A=22.0,
NOMBRE=22,),
_F (JUSQU_A=27.0,
NOMBRE=5,),
_F (JUSQU_A=32.0,
NOMBRE=5,),
_F (JUSQU_A=37.0,
NOMBRE=5,),
_F (JUSQU_A=42.0,
NOMBRE=5,),
_F (JUSQU_A=47.0,
NOMBRE=5,),
_F (JUSQU_A=52.0,
NOMBRE=5,),
_F (JUSQU_A=57.0,
NOMBRE=5,),
_F (JUSQU_A=61.0,
NOMBRE=4,),
_F (JUSQU_A=65.0,
NOMBRE=4,),
_F (JUSQU_A=70.0,
NOMBRE=5,),
_F (JUSQU_A=76.0,
NOMBRE=6,),
_F (JUSQU_A=82.0,
NOMBRE=6,),
_F (JUSQU_A=88.0,
NOMBRE=6,),
_F (JUSQU_A=94.0,
NOMBRE=6,),
_F (JUSQU_A=100.0,
NOMBRE=6,),),)

RAMPE=DEFI_FONCTION (NOM_PARA=' INST',
VALE= (0.0, 0.0, 100.0, 100.0),
PROL_DROITE=' LINEAIRE',
PROL_GAUCHE=' LINEAIRE',)
#
# LOADING AND CONDITIONS LIMITING
# ---------------------------------------------------------------------
#


CHAR=AFFE_CHAR_MECA (MODELE=MOD,
DDL_IMPO= (_F (GROUP_NO=' LIGR',
DX=0.0,),
_F (GROUP_NO=' LAPPUI',
DY=0.0,),
_F (GROUP_NO=' LPILOT',
DY=-0.04,),),)
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Code_Aster
®
Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
U2.05.01-A
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:
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Instruction manual
U2.05 booklet: Damage and breaking process
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#
# APPLICATION OF THE LOAD & CALCULATION OF THE STRESSES
#
RESU=STAT_NON_LINE (MODELE=MOD,
CHAM_MATER=CH_MAT,
EXCIT=_F (CHARGE=CHAR,
FONC_MULT=RAMPE,),
COMP_INCR=_F (RELATION=' VMIS_ISOT_TRAC',
DEFORMATION=' PETIT',
GROUP_MA=' SENB',),
INCREMENT=_F (LIST_INST=LIST,
NUME_INST_FIN=95,),
NEWTON=_F (PREDICTION=' TANGENTE',
MATRICE=' TANGENTE',
REAC_ITER=4,),
RECH_LINEAIRE=_F (RESI_LINE_RELA=1.E-3,
ITER_LINE_MAXI=3,),
CONVERGENCE=_F (RESI_GLOB_MAXI=1.E-08,
RESI_GLOB_RELA=1.E-08,
ITER_GLOB_MAXI=20,),
SOLVEUR=_F (METHODE=' MULT_FRONT',
RENUM=' METIS',),)

#
# CALCULATION OF G
#

THETA1=CALC_THETA (MODELE=MOD,
DIRECTION= (0.0, 1.0, 0.0),
THETA_2D=_F (GROUP_NO=' O',
MODULE=1.0,
R_INF=0.25,
R_SUP=0.5,),)

G1=CALC_G_THETA_T (MODELE=MOD,
CHAM_MATER=CH_MAT,
THETA=THETA1,
RESULTAT=RESU,
TOUT_ORDRE=' OUI',
SYME_CHAR=' SYME',
COMP_ELAS=_F (RELATION=' ELAS_VMIS_TRAC',
DEFORMATION=' PETIT',),)

IMPR_TABLE (TABLE=G1, FORMAT = “AGRAF”)


THETA2=CALC_THETA (MODELE=MOD,
DIRECTION= (0.0, 1.0, 0.0),
THETA_2D=_F (GROUP_NO=' O',
MODULE=1.0,
R_INF=0.50,
R_SUP=1.0,),)
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Code_Aster
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Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
U2.05.01-A
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:
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Instruction manual
U2.05 booklet: Damage and breaking process
HT-66/03/002/A

G2=CALC_G_THETA_T (MODELE=MOD,
CHAM_MATER=CH_MAT,
THETA=THETA2,
RESULTAT=RESU,
TOUT_ORDRE=' OUI',
SYME_CHAR=' SYME',
COMP_ELAS=_F (RELATION=' ELAS_VMIS_TRAC',
DEFORMATION=' PETIT',),)

IMPR_TABLE (TABLE=G2, FORMAT = “AGRAF”)

THETA3=CALC_THETA (MODELE=MOD,
DIRECTION= (0.0, 1.0, 0.0),
THETA_2D=_F (GROUP_NO=' O',
MODULE=1.0,
R_INF=1.0,
R_SUP=2.0,),)

G3=CALC_G_THETA_T (MODELE=MOD,
CHAM_MATER=CH_MAT,
THETA=THETA3,
RESULTAT=RESU,
TOUT_ORDRE=' OUI',
SYME_CHAR=' SYME',
COMP_ELAS=_F (RELATION=' ELAS_VMIS_TRAC',
DEFORMATION=' PETIT',),)

IMPR_TABLE (TABLE=G3, FORMAT = “AGRAF”)

#

END ()

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Code_Aster
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Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
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Author (S):
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Key
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CONTINUATION (PAR_LOT=' NON')

ENEE= [Nun] * 200;
ENET= [Nun] * 200;
#
deltal = 0.01
#
#
importation bone
f2=open (“fort.44”, “W”)
f3=open (“fort.45”, “W”)
f2.write (“brutal Propagation - Test-tube SENB2 - Mesh M1 \ N”)
f2.write (“Work hardening diagram traction ECA \ N”)
f2.write (“Propagation - elastic Energy - G plastic (dW/DLL) \ N”)

for K in arranges (1,101):

LIG = “COPS_ % I” % (K)
print “ligament number: ”, K
print “cumulated propagation: ”, K * deltal, “millimetres”

ENEE [K] = POST_ELEM (MODELE=MOD,
RESULTAT=RESU,
CHAM_MATER=CH_MAT,
TOUT_ORDRE = ' OUI',
ENER_ELAS=_F (GROUP_MA=LIG),
TITER=' Energy élastique',
)
IMPR_TABLE (TABLE=ENEE [K],
FORMAT_R='1PE18.11')


# End of the iterations

for J in arranges (1,96):
f2.write (“Urgent: % F \ N” % (J))
gpmax = 0.
for K in arranges (1,101):
# f2.write (“Deltal: % F \ N” % (K * deltal))
# f2.write (“Nb chips: % I \ N” % (K))
ETOT=ENEE [K] ['', J]
GP = 2.0 * (ETOT)/(K * deltal)
yew GP > gpmax:
gpmax = GP
kmax = K * deltal
f2.write (“% F % 0.11f % 3f \ N” % ((K * deltal), ETOT, GP))
f3.write (“% F % 3f % 3f \ N” % (J, kmax, gpmax))

f2.close ()
f3.close ()

END ()
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Code_Aster
®
Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
U2.05.01-A
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:
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U2.05 booklet: Damage and breaking process
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3.4.5 Interpretation
For large fissure (SENB1)
This case corresponds to calculation on test-tube CT.
One identifies the moment T1 = 30 S corresponding to the arrow with the experimental rupture of 1.21 Misters One
determine at this moment G theta which is stable for various crowns:
Gtheta
SENB1
= 47.86
One then determines for every moment the ligament where Gp is maximum
and in particular at moment T1: Gp = 0.606 on ligament 26 for DLL = 0.26 Misters.
For small fissure (SENB2)
One determines in this case the moment when Gpmax is worth also 0.606.
It is about t2 = 80 S on ligament 16 for DLL = 0.16 Misters.
One calculates at the moment t2 the value of Gtheta
SENB2
= 153.79.
One thus deduces an effect from it small defect which is expressed in the form:
(epd)
2
= Gtheta
SENB2
/Gtheta
SENB1
= 3.21 is epd = K
SENB2
/K
SENB1
= 1.79
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Code_Aster
®
Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
U2.05.01-A
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:
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U2.05 booklet: Damage and breaking process
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4 Documentation
Code_Aster relating to the mechanics of
brittle fracture
Key
Titrate document
Reference documents:
[R7.02.01]
Rate of refund of energy in thermo linear elasticity
[R7.02.03]
Rate of refund of energy in thermo nonlinear elasticity
[R7.02.04] Representation
Lagrangian of variation of field
[R7.02.05]
Calculation of the coefficients of intensity of stresses in plane linear thermoelasticity
[R7.02.07]
Rate of refund of energy in thermo elastoplasticity
[R7.02.08]
Calculation of the coefficients of intensity of stresses by extrapolation of the field of
displacements
Documents of Use:
[U4.82.01] Operator
DEFI_FOND_FISS
[U4.82.02] Operator
CALC_THETA
[U4.82.03] Operator
CALC_G_THETA_T
[U4.82.04] Operator
CALC_G_LOCAL_T
[U4.82.05] Operator
POST_K1_K2_K3
[U4.81.22] Operator
POST_ELEM
Documents of Validation:
SSLP101
Rate of refund of energy in plane stresses
SSLP102
Rate of refund of energy with initial deformations (Lagrangian propagation)
SSLP103
Calculations of the stress intensity factors KI and KII for a fissured circular plate
in linear elasticity
SSLP310
Biblio_18 Fissures pressurized in an unlimited plane field
SSLP311
Biblio_65 fissures central oblique in a finished rectangular plate, with two materials,
subjected to uniform traction
SSLP313
Fissure inclined in an unlimited plate, subjected to a uniform traction ad infinitum
SSLV110
Rate of refund of energy for a semi-elliptic fissure in an infinite medium
SSLV112
Calculation of G by the Lagrangian method for a circular fissure
SSLV134
Fissure circular in infinite medium
SSNP102
Rate of refund of energy for a plate notched in elastoplasticity
SSNP311
Biblio_131 Cracking in mode II of an elastoplastic test-tube
SSNP312
DMT94.132 Fissures parallel with the interface in a bimetallic test-tube CT
HPLA310
Biblio_49 Fissures radial external in a circular bar subjected to a thermal shock
HPLA311
Murakami 11.39 Fissures circular in the center of a sphere subjected to a temperature
uniform on the lips
HPLP100
Calculation of the rate of refund of the energy of a plate fissured in thermo elasticity
HPLP101
Plate fissured in thermoelasticity (forced plane)
HPLP310
Biblio_35 Fissures radial intern in a thick cylinder under pressure and loading
thermics
HPLP311
Murakami 11.17: Fissure in the center of a rectangular thin section making obstacle with one
uniform heat transfer rate in isotropic medium
HPLV102
Rate of refund of energy in thermo elasticity for a circular fissure in infinite medium
HPLV103
Calculation of G thermo elastic 3D for a circular fissure
* These tests result from the validation independent of version 3 in breaking process and are
disseminated in electronic documentation.
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Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
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Author (S):
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Key
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:
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5 Bibliography
[1]
BABUSKA I., Accurate and applicable determination off edge and vertex stress intensity factors,
ICAS-90-4.9.2, 1990
[2]
BABUSKA I., PAPADAKIS J., PANAGIOTIS J., Determination off some quantities related to
K1-K2, Comput. Methods Appl. Mech. Eng., 122, 1995, 69-92
[3]
BONNAMY Mr., WADIER Y., BLOCFISS Mesh 2D and 3D of blocks fissures, HI-74/96/007/0,
octobre96
[4]
BUI H.D., fragile Breaking process, Masson, 1977
[5]
BUI H.D., AMESTOY Mr., DANG VAN K., infinitesimal Deviation of a fissure in one
arbitrary direction, C.R. Acad. SC Paris, T. 289, series B, No 8, p99, 1979
[6]
BUI H.D., mixed Decoupling of the modes of rupture by the use of 2 new integrals of
contour, C.R. Acad. SC Paris, T. 295, series B, p521, 1982
[7]
BUI H.D., THIN H., Stress intensity factor drawn from the mechanical magnitudes
total, C.R. Acad. SC Paris, T. 306, series B, p1213-1216, 1988
[8]
COOR. MECHANICS, ANDRIEUX S., MIALON P., Co, Breaking process Session,
Seminar of Code_Aster formation, IMA/MN, HI-74/94/019/0
[9]
DAUGE Mr., Coefficients of the singularities for the problem of Dirichlet on a polygon, C.R.
Acad. SC Paris, T. 304, series A, No 16, 1987
[10]
DAUGE Mr., Problems in extreme cases elliptic on fields with corners (1), C.R. Acad. SC
Paris, T. 304, series A, Nos 17,18 and 19, 1987
[11]
DAUGE Mr., COSTABEL Mr., Computation off corner singularities in linear elasticity, IRMAR,
Prépublication 93-23, October 1993
[12]
DEBRUYNE G., Problem of the finished propagation of a fissure, Lagrangian representation,
IMA/MN, HI-73/8193, 1992
[13]
DEBRUYNE G., Propagation of fissures and residual stresses, IMA/MN, HI-
74/94/083/0, 1995
[14]
DEBRUYNE G., Proposal for an energy parameter of ductile rupture in
thermoplasticity, IMA/MN, HI-74/95/027/0, 1995
[15]
DESTUYNDER pH, DJAOUA Mr., Equivalence of the integral of Rice and G in mechanics of
rupture, C.R. Acad. SC Paris, T. 290, series A, p347, 1980
[16]
DESTUYNDER pH., DJAOUA Mr., On an interpretation of the integral of Rice in theory of
brittle fracture, Mathematics Methods in the Applied Sciences, vol. 3, pp. 70-87, 1981.
[17]
DESTUYNDER pH., DJAOUA Mr., LESCURE S., New method of the calculation of the singularity
for an elliptic problem, C.R. Acad. SC Paris, T. 293, series A, p343, 1981
[18]
DESTUYNDER pH., OUSSET Y., STACKLER C., On the singularities of stresses in
adhesive bonded joints, Newspaper of Mechanics Theoretical and Applied, vol. 7, No 6, 899-926, 1988
[19]
ERDOGAN F., Stress intensity factors, Newspaper off Applied Mechanics, Flight 50, 993, 1983
[20]
EYMARD I., elastic Singularities of stresses to the junction of two materials,
IMA/MN, HI-74/93/099, 1994
[21]
GRIFFITH A., The phenomena off Rupture and flow in solids, Mr. Eng. (off the Royal Aircraft
Establishment), communicated by Taylor, F.R.S., 1920
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Code_Aster
®
Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
U2.05.01-A
Page
:
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Instruction manual
U2.05 booklet: Damage and breaking process
HT-66/03/002/A
[22]
GRISVARD P., Elliptic problems in nonsmooth domains, Monograph and Studies in
Mathematics 24, Pitman, 1985
[23]
GRISVARD P., Problems in extreme cases in the polygons, Instructions - EDF - Bulletin of
the Management of the Studies and Search, Series C, 1, 1986 pp. 21-59.
[24] HUTCHINSON
J.W.,
Fundamentals off the phenomenological theory off non-linear fracture
mechanics, Newspaper off Applied Mechanics, Flight 50, Dec. 1983
[25]
LEGUILLON D., anisotropic brittle Fracture. Collateral microscopic cracks with a fissure
main, C.R. Acad. SC Paris, T. 313, series B, 139-144, 1991
[26]
LEGUILLON D., Asymptotic and numerical analysis off has ace branching in non-isotropic
materials, Newspaper off Mechanics, A/Solids, 1991
[27]
LEGUILLON D., Analysis off the brittle fracture in composites using singularities,
Micromechanics off materials, 1993
[28]
LEGUILLON D., Asymptotic and numerical analysis off has ace branching in non-isotropic
materials, Eur. J. Mech., A/Solids, 12, No 1, 33-51, 1993
[29]
LENCZNER Mr., Lagrangian Formulation of the equations of microscopic cracks and calculation of
stress intensity factors, Thesis of doctorate of the Pierre university and Marie Curia,
Paris VI, July 4, 1991
[30]
LENCZNER Mr., Characterization of the dual singular functions for the Laplacian 3D, C.R.
Acad. SC Paris, T. 314, series A, 265-270, 1992
[31]
LESCURE S., Studies of the criteria of propagation right or deviated of a fissure of one
bimatériau, IMA/MN, HI/4567-07, 1983
[32]
MAUGIN A., TRIMARCO C., Bui' S path independent integral in finite elasticity, Meccanica 30,
139-145, 1995
[33]
MAUGIN G.A., Eshelby stress in elastoplasticity and ductile fracture, International Newspaper off
Plasticity, vol. 10, No 4, 393-408, 1994
[34]
MEISTER E., WADIER Y., Study of the applicability of the global solution on a structure
bimatériau, IMA/MN, HI-73/7949, 1992
[35]
MEISTER E., Elements bibliographical on the breaking process of the interfaces
bimatériau, IMA/MN, HI-74/93/116, 1994
[36]
MEISTER E., Approach with two parameters in breaking process, IMA/MN,
HI-74/94/020/0, 1994
[37]
MIALON P., Study of the rate of refund of energy in a direction marking an angle
with a fissure, notes intern EDF, HI/4740-07, 1984.
[38]
MIALON P., Calculation of derived from a size compared to a bottom of fissure by
method théta, EDF, Bulletin of the direction of the studies and search, Series C, 1988
[39]
MIALON P., SCREWS E., Rate of refund of energy in great displacements, Deuxième
national conference in calculation of the structures, Giens, May 16-19, 1995
[40]
MOUSSAOUI Mr., AMARA Mr., Approximation off solutions and singularities coefficients for year
elliptic problem, Laboratory Maths., E.N.S.L., Lyon
[41]
NEEDLEMAN A., SHIH c.f., LI F.Z., A comparison off methods for calculating energy release
spleens, Engineering Fractures Mechanics, Vol 21, p 405-421, 1985
[42]
NGUYEN Q.S., On the use of the criteria of energy in Rupture and, C.R. Acad.
SC Paris, T. 286, series A, p483, March 13, 1978
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Code_Aster
®
Version
6.4
Titrate:
Applicability of the operators of breaking process
Date:
03/04/03
Author (S):
I. DEBOST, G. DEBRUYNE, Y. WADIER, E. SCREWS
Key
:
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:
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HT-66/03/002/A
[43]
NGUYEN Q.S., energy Methods in breaking process, Newspaper of mechanics,
Vol. 19, No 2, 1980
[44]
NGUYEN Q.S., A thermodynamic description off the running ace problem, IUTAM, Dourdan,
France, 2-5 June 1980
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