background image
Code_Aster
®
Version
7.4
Titrate:
Calculation of the coefficients of intensity of stresses
Date:
02/05/05
Author (S):
E. CRYSTAL
Key
:
R7.02.05-B
Page
:
1/14
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
Organization (S):
EDF-R & D/AMA















Manual of Reference
R7.02 booklet: Breaking process
Document: R7.02.05



Calculation of the coefficients of intensity of stresses
in plane linear thermoelasticity




Summary:

One presents the method of calculation of the coefficients of intensity of stresses K
I
and K
II
and in thermoelasticity
linear plane. The formulation regards the rate of refund of energy as a symmetrical bilinear form
field of displacement
U
and uses the explicit expressions of the fields of singular displacements
known in plane linear elasticity.



Key words:

Breaking process, coefficient of intensity of stresses, thermoelasticity.
background image
Code_Aster
®
Version
7.4
Titrate:
Calculation of the coefficients of intensity of stresses
Date:
02/05/05
Author (S):
E. CRYSTAL
Key
:
R7.02.05-B
Page
:
2/14
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
Count
matters
1
Expressions of the stress intensity factors K
I
and K
II
in linear thermoelasticity 2D .............. 3
1.1
Presentation ..................................................................................................................................... 3
1.2
Formulate IRWIN and rate of refund of energy G ........................................................................ 5
1.3
Decoupling of the modes of rupture I and II ........................................................................................ 6
2
Establishment of K
I
, K
II
in linear thermoelasticity 2D in Aster ........................................................ 8
2.1
Types of elements and loadings .............................................................................................. 8
2.2
Environment necessary for the calculation of K
I
, K
II
......................................................................... 8
2.3
Bilinear form symmetrical G (. .).................................................................................................. 8
2.3.1
Elementary conventional term ................................................................................................. 8
2.3.2
Term forces voluminal ......................................................................................................... 11
2.3.3
Thermal term .................................................................................................................. 12
2.3.4
Term forces surface ........................................................................................................ 12
2.4
Fields of singular displacements and their derivative ................................................................. 12
2.5
Postprocessing of the results of K
I
and K
II
....................................................................................... 12
3
Bibliography ........................................................................................................................................ 14
background image
Code_Aster
®
Version
7.4
Titrate:
Calculation of the coefficients of intensity of stresses
Date:
02/05/05
Author (S):
E. CRYSTAL
Key
:
R7.02.05-B
Page
:
3/14
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
1
Expressions of the stress intensity factors K
I
and K
II
in linear thermoelasticity 2D
1.1 Presentation














Are the axes of Cartesian co-ordinates O
X
1
in the prolongation of the fissure and O
X
2
perpendicular with the fissure. The problem is plane. We will express the Cartesian components
displacements and stresses according to the polar co-ordinates
R
and
.
In linear elasticity, the system of the equilibrium equations, without voluminal force, and the conditions
in extreme cases homogeneous on the fissure, the null stresses ad infinitum, admit a noncommonplace solution
form
()
U
G
I
I
R
=
. The stresses are infinite at the bottom of the fissure like
R
- 12
[bib3].
For an unspecified problem in plane linear elasticity (plane strains or plane stresses),
the field of displacement
U
can break up into a singular part and a regular part.
singular part, also called singularity, is that clarified above, it contains them
coefficients of stresses. In linear elasticity, modes of rupture
I
and
II
are separate:
U
U
U
U
=
+
+
R
I
IF
II
Software house
K
K
with:
()
(
)
()
(
)
U
E
R
K
U
E
R
K
IF
IF
1
1 2
2
1 2
1
2
2
1
2
2
=
+
-
=
+
-


/
/
cos
cos
sin
cos
()
(
)
()
(
)
U
E
R
K
U
E
R
K
Software house
Software house
1
1 2
2
1 2
1
2
2
2
1
2
2
2
=
+
+
+
= - +
+
-


/
/
sin
cos
cos
cos
X
2
X
1
R
0
background image
Code_Aster
®
Version
7.4
Titrate:
Calculation of the coefficients of intensity of stresses
Date:
02/05/05
Author (S):
E. CRYSTAL
Key
:
R7.02.05-B
Page
:
4/14
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
with:
K
= -
3 4
in plane deformations
D_PLAN
(
) (
)
K
= -
+
3
1
/
in plane stresses
C_PLAN
and:
E
YOUNG modulus
Poisson's ratio

The distribution of the singular stresses in the vicinity of the fissure is given by the formulas:
11
11
11
12
12
12
22
22
22
S
I
I
II
II
S
I
I
II
II
S
I
I
II
II
K
K
K
K
K
K
=
+
=
+
=
+




with:
(
)
(
)
(
)
11
1 2
12
1 2
22
1 2
1
2
2
1
2
3
2
1
2
2
2
3
2
1
2
2
1
2
3
2
I
I
I
R
R
R
=


-








=








=


+














/
/
/
cos
sin
sin
cos
sin
cos
cos
sin
sin
(
)
(
)
(
)
11
1 2
12
1 2
22
1 2
1
2
2
2
2
3
2
1
2
2
1
2
3
2
1
2
2
2
3
2
II
II
II
R
R
R
= -


+










=


-








=














/
/
/
sin
cos
cos
cos
sin
sin
sin
cos
cos
background image
Code_Aster
®
Version
7.4
Titrate:
Calculation of the coefficients of intensity of stresses
Date:
02/05/05
Author (S):
E. CRYSTAL
Key
:
R7.02.05-B
Page
:
5/14
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
1.2
Formulate IRWIN and rate of refund of energy G
In plane linear elasticity, the stress intensity factors are connected to the rate of refund
of energy
G
by the formula of IRWIN:
(
)
(
)
(CP)
plane
S
stress
in
(DP)
plane
NS
déformatio
in
2
2
2
2
2
1
1
II
I
II
I
K
K
E
G
K
K
E
G
+
=
+
-
=
The demonstration of these formulas can be made starting from the expression of the rate of refund
of energy
G
established in Code_Aster and known under the name of the method theta [bib5].
Let us recall that
G
is defined by the opposite of derived from the potential energy compared to the evolution
bottom of fissure.

F
F
F
U

In the Lagrangian method of derivation of the potential energy, one considers transformations
M
M
M
()
+
area of reference
0
in a field
who correspond to
propagations of the fissure. With these families of configuration of reference thus defined
correspond of the families of deformed configurations whose fissure was propagated. Energy
potential definite on
is brought back on
0
.
The surface forces are considered
F
and voluminal
F
applied respectively to
F
and
0
. One
note
()
(
)
U
density of free energy,
U
the field of displacement,
T
the field of temperature and
the field of vectors describing the direction of transport in
= 0
, then the general expression of
rate of refund of energy
G
[bib5] is:
background image
Code_Aster
®
Version
7.4
Titrate:
Calculation of the coefficients of intensity of stresses
Date:
02/05/05
Author (S):
E. CRYSTAL
Key
:
R7.02.05-B
Page
:
6/14
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
() (
)
()
(
)
[
]
(
)
(
)
[
]
(
)
(
) (
)
U
F
D
D
D
D
T
T
D
G
U
F
-




-
+
+
+
+
-
-
=
on
imposed
ts
déplacemen
with
had
Term
on
S
surface
forces
with
had
Term
on
voluminal
forces
with
had
Term
thermics
with
had
Term
conventional
Term
U
N
N
N
U
F
U
F
U
F
U
F
U
U
U
F
F
div
div
div
:
In linear elasticity,
G
can be regarded as the symmetrical bilinear shape of the field of
displacement
U
. Density of energy elastic
()
(
)
U
is written:
()
(
)
()
()
()
U
U
U
U U
=
=
1
2
1
2
: :
,
B
while noting:
the tensor of elasticity
B
the symmetrical bilinear form defined by:
()
()
()
B U v
U
v
,
: :
=
and the bilinear form
()
G
,
associated
G
is defined by:
()
(
)
(
)
()
(
) (
) (
)
[
]
G
B
B
B
D
D
U
v
U
v
U v
U
v
v
U
U v
F
v
F
U
F v F U
,
,
div
div
=
+ -




+
+
+
+
1
2
1
2
while limiting itself under the terms conventional and due to the voluminal forces
F
.
One has
()
G
G
= U U
,
if
U
is solution of the elastic problem.
1.3
Decoupling of the modes of rupture I and II
In the method established in Code_Aster, to uncouple the modes from rupture
I
and
II
and
to calculate the coefficients
K
I
and
K
II
, the symmetrical bilinear form is used
()
G
,
and
decomposition of the field of displacement
U
in parts regular and singular [bib7].
(
)
(
) (
)
(
)
(
)
(
)
(
) (
)
(
)
(
)
G
G
K
K
G
K G
K G
G
G
K
K
G
K G
K G
S
I
R
I
IF
II
Software house
IF
R
IF
I
IF
S
I
II
S
II
S
I
S
II
R
I
IF
II
S
II
Software house
R
Software house
I
IF
Software house
II
Software house
S
II
U U
U
U
U
U
U
U
U U
U
U
U U
U
U
U
U
U
U
U U
U
U
,
,
,
,
,
,
,
,
,
,
=
+
+
=
+
+
=
+
+
=
+
+



background image
Code_Aster
®
Version
7.4
Titrate:
Calculation of the coefficients of intensity of stresses
Date:
02/05/05
Author (S):
E. CRYSTAL
Key
:
R7.02.05-B
Page
:
7/14
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
One shows in plane linear thermoelasticity that
U
IF
and
U
Software house
are orthogonal for the product
scalar defined by the bilinear form
()
G
,
, that terms utilizing the regular part
cancel themselves and finally:
(
)
(
)
(
)
(
)
G
K G
G
K G
IF
I
S
I
IF
Software house
II
S
II
Software house
U U
U U
U U
U
U
,
,
,
,
=
=
Moreover, by writing the rate of refund of energy in the form:
()
(
)
G
G
G
K
K
K
K
R
I
IF
II
Software house
R
I
IF
II
S
II
=
=
+
+
+
+
U U
U
U
U
U
U
U
,
,
and like:
(
)
(
)
(
)
(
)
G
G
G
G
IF
Software house
S
II
IF
R
IF
R
Software house
U U
U
U
U
U
U
U
,
,
,
,
=
=
=
=
0
0
the formula of IRWIN is found:
()
(
)
(
)
G
K G
K G
I
IF
IF
II
S
II
Software house
U U
U
U
U
U
,
,
,
=
+
2
2
with:
(
)
(
)
(
)
(
)
C_PLAN
D_PLAN
in
in
E
G
G
E
G
G
II
S
II
S
I
S
I
S
II
S
II
S
I
S
I
S
1
,
,
1
,
,
2
=
=
-
=
=
U
U
U
U
U
U
U
U
Finally:
()
()
()
()
C_PLAN
D_PLAN
in
in



=
=




-
=
-
=
II
S
II
I
S
I
II
S
II
I
S
I
G
E
K
G
E
K
G
E
K
G
E
K
U
U
U
U
U
U
U
U
,
,
,
1
,
1
2
2

Establishment of the calculation of the coefficients of intensity of stresses in plane linear thermoelasticity
in Code_Aster is realized starting from the expression of the rate of refund of energy
G
in elasticity
linear 2D, written in symmetrical bilinear form, by introducing the known expressions of
singular displacements, and by using the method theta.
background image
Code_Aster
®
Version
7.4
Titrate:
Calculation of the coefficients of intensity of stresses
Date:
02/05/05
Author (S):
E. CRYSTAL
Key
:
R7.02.05-B
Page
:
8/14
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
2
Establishment of K
I
, K
II
in linear thermoelasticity 2D in
Aster
2.1
Types of elements and loadings
To calculate the coefficients of intensity of stresses
K
I
and
K
II
in linear elasticity 2D, it is necessary
to use the option
CALC_K_G
control
CALC_G_THETA
. This option is available for all
thermomechanical loading applying to a model of affected two-dimensional continuous medium to
triangles with 3 or 6 nodes, quadrangles with 4, 8 or 9 nodes, and segments with 2 or 3 nodes.
It is valid for a modeling '
C_PLAN
“or”
D_PLAN
'.
Note:
One does not take account of the term due to the displacements imposed on
U
, one thus should not
to impose conditions of DIRICHLET on the lips of the fissure.
2.2
Environment necessary for the calculation of KI, KII
The control
CALC_G_THETA
allows to recover the model of the problem, the characteristics of
material, the field of displacements and the field theta. For the calculation of the coefficients of intensity of
stresses, it is necessary to add the key word
FOND_FISS
, which makes it possible to recover a concept of
type
fond_fiss
where the basic node of fissure and the normal with the fissure are defined.
When that the fissure is laid out along an axis of symmetry, one can also specify
symmetry of the loading by the key word
SYME_CHAR
. By defect one supposes that there is no symmetry.
If one affects the value '
SYME
'with the key word
SYME_CHAR
, that means that only mode I of rupture acts
(opening of the lips of the fissure) and one automatically affects the zero value to
K
II
. If one affects
the value '
ANTI
', then only mode II is active (slip of a lip compared to the other) and
K
I
is
no one.
Let us insist on the need for assigning to all the elements (including those of edges) the values of
YOUNG moduli
E
and of the Poisson's ratio
, because they are used in the calculation of
singular displacements. These values must be homogeneous on all the support of the field theta.
2.3
Bilinear form symmetrical G (. .)
Note:
The routine
GBILIN
calculate the rate of refund of energy
G
in the bilinear form
symmetrical
()
G U v
,
in thermo linear elasticity planes (plane strains or stresses)
for the isoparametric elements 2D.
2.3.1 Elementary conventional term
()
(
)
()
(
)
TCLA
=
-
U
U
U
:
div
Density of energy elastic
()
(
)
U
is written in thermo linear elasticity:
background image
Code_Aster
®
Version
7.4
Titrate:
Calculation of the coefficients of intensity of stresses
Date:
02/05/05
Author (S):
E. CRYSTAL
Key
:
R7.02.05-B
Page
:
9/14
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
in
D_PLAN
:
()
(
)
(
)
(
) (
)
(
)
(
) (
)
U
=
-
+
-
+
+ + -
+ +
-
1
2 1
1 2
1
1 2
1
2
2
2
E
E
E
xx
yy
xx yy
xy
HT

in
C_PLAN
:
()
(
)
(
)
(
)
(
)
(
)
U
=
-
+
+ -
+ +
-
E
E
E
xx
yy
xx yy
xy
HT
2 1
1
1
2
2
2
2
2
with
HT
= Density of energy due to thermics:
(
)
HT
ref.
K
T T
=
-
3
tr
where:
reference
of
E
températur
NS
déformatio
of
tensor
thermics
dilation
=
=
=
-
=
ref.
T
E
K
2
1
3
and in a general way, one can write:
()
(
)
(
)
2
2
4
2
1
2
2
2
3 2
U
=
+
+
+
-
C
C
C
xx
yy
xx yy
xy
HT
with:
(
)
(
) (
)
(
) (
)
(
)
(
)
(
)
.
in
in
C_PLAN
D_PLAN
1
2
1
1
;
1
2
2
1
1
2
2
1
1
1
3
2
2
2
1
3
2
1




+
=
-
=
-
=




=
+
=
=
-
+
=
+
=
-
+
-
=
µ
µ
E
C
E
C
E
C
E
C
E
C
E
C
background image
Code_Aster
®
Version
7.4
Titrate:
Calculation of the coefficients of intensity of stresses
Date:
02/05/05
Author (S):
E. CRYSTAL
Key
:
R7.02.05-B
Page
:
10/14
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
Therefore, while noting
()
(
)
()
,
U
U U
=
, one has
()
2
1
1
U v
,
=
-
S
S TH
with:
(
)
()
(
)
()
(
)
U
v
v
U
tr
tr
3
1
1
3
2
1
ref.
ref.
y
X
y
X
X
y
y
X
y
y
X
X
T
T
T
T
K
TH
S
X
v
y
v
X
U
y
U
C
X
v
y
U
y
v
X
U
C
y
v
y
U
X
v
X
U
C
S
-
+
-
=






+




+
+








+






+










+




=
where
T
U
is the temperature associated with the field with displacement
U
by the relation:
()
(
)
=
-
U
HT
where
(
)
ijth
ref.
ij
T
T
=
-
and
the equilibrium equations check.
In the same way, the term
()
(
)
U
U
:
can be written:
()
(
)
U
U
:
=
-
S
S TH
2
2
with:






+
+
+




+
+






+
+




+
+






+
+




+
=
X
y
U
X
X
U
y
y
U
y
X
U
X
U
y
U
C
X
y
U
X
X
U
y
U
y
y
U
y
X
U
X
U
C
y
y
U
y
X
U
y
U
X
y
U
X
X
U
X
U
C
S
y
y
X
y
y
X
X
X
y
X
y
X
X
X
y
y
y
X
y
X
y
y
X
y
y
y
X
X
X
X
3
2
1
2

Terms
U
X
U
X
X
X
become in the symmetrical bilinear form
U
X
v
X
X
X
and terms of
type
U
y
U
X
X
y
become
1
2
U
y
v
X
v
y
U
X
X
y
X
y
+




.
background image
Code_Aster
®
Version
7.4
Titrate:
Calculation of the coefficients of intensity of stresses
Date:
02/05/05
Author (S):
E. CRYSTAL
Key
:
R7.02.05-B
Page
:
11/14
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
(
)
(
)


-
-
=
=


+
+
+
-
+


+
+
+
-
=
C_PLAN
D_PLAN
in
in
where
1
2
1
1
1
1
3
2
1
3
2
1
2
TH
TH
y
X
U
X
y
U
y
y
U
X
X
U
T
T
K
TH
y
X
v
X
y
v
y
y
v
X
X
v
T
T
K
TH
TH
S
X
y
y
X
y
y
X
X
ref.
X
y
y
X
y
y
X
X
ref.
v
U

and finally the conventional term is written:
(
)
(
)
TCLA
S
S TH
S
S TH
=
-
-
-
2
2
1
2
1
1
div
2.3.2 Term forces voluminal
(
)
TFOR
=
+
F
U F U
div
In any rigor, the symmetrical bilinear expression of
TFOR
is written in
()
U v
,
:
()
(
) (
) (
)
[
]
TFOR
U
v
U
v
U v
F
v
F
U
F v F U
,
div
=
+
+
+
1
2
where
F
U
are the voluminal forces associated the field of displacement
U
for the elastic problem.
but as the expressions which we are brought to calculate are of the type
()
TFOR U U
,
and
(
)
TFOR
S
U U
,
, where
U
and
U
S
are respectively the field of displacement and the singular field, and
that:
()
(
)
F
U
U
S
S
=
=
div
0
on
One limits oneself to write:
()
(
)
[
]



=
=
=
=
+
=
U
v
U
v
v
F
v
F
v
U
1
5
.
0
div
.
.
,
CS
CS
CS
TFOR
S
U
U
with
background image
Code_Aster
®
Version
7.4
Titrate:
Calculation of the coefficients of intensity of stresses
Date:
02/05/05
Author (S):
E. CRYSTAL
Key
:
R7.02.05-B
Page
:
12/14
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
Finally:
()
TFOR
CS v
F
X
F
y
F
v
F
X
F
y
F
X
X X
X y
X
y
y
X
y
y
y
U, v
=
+
+






+
+
+




div
div
The same remark is valid for the thermal conventional term, the additional term due to
thermics and terms due to the surface forces.
2.3.3 Term
thermics
One makes the assumption that the characteristics of material
(
)
,
,
E
do not depend on
temperature.
(
)
TTHE
T
T
K
T
X
T
y
X
y
= -
=
+




1
2 3
tr
2.3.4 Term forces surface
In 2D, for the isoparametric elements of edge, one introduced the loadings of the pressure type
shearing and force distributed of real type.
The term forces surface is written in the same way that the voluminal term from:
(
)
TSUR
=
+




F
U F U
N
N
-
div

.
2.4
Fields of singular displacements and their derivative
Singular fields
U
IF
and
U
Software house
, respectively associated with the modes
I
and
II
, are known
explicitly like their derivative. They are written according to the polar co-ordinates in
reference mark related to the fissure. The knowledge of the co-ordinates of the basic node of fissure and of its
normal makes it possible to calculate them in the total reference mark
0xy
.
The successive introduction of these fields
U
IF
and
U
Software house
allows, as indicated in [§1], calculation
elementary of the coefficients of intensity of stresses
K
I
and
K
II
.
2.5
Postprocessing of the results of K
I
and K
II
Knowing the values of the coefficients of intensity of stresses
K
I
and
K
II
for a fissure
data, formulas of AMESTOY - BUI and DANG-VAN, allow the calculation of the angle of
propagation of the fissure according to 3 criteria (
K
I
maximum,
K
II
and
G
maximum) [bib6].
That is to say
m
a field identical to
except that the fissure is prolonged in the direction of angle
m
of a segment of straight line length
.
background image
Code_Aster
®
Version
7.4
Titrate:
Calculation of the coefficients of intensity of stresses
Date:
02/05/05
Author (S):
E. CRYSTAL
Key
:
R7.02.05-B
Page
:
13/14
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
U
m
F
F
m
F
m
O
Are
()
K
m
I
,
,
()
K
m
II
,
,
()
G
m
,
stress intensity factors and the rate of refund
of energy of
m
subjected to the same loading as
.
One poses:
()
()
()
()
()
()
K m
K
m
K m
K
m
G m
G
m
I
I
II
II
*
*
*
lim
,
lim
,
lim
,
=
=
=
0
0
0
Criteria quoted by AMESTOY - BUI and DANG-VAN [bib6] are:
·
to choose
m
O
such as
()
K m
I
O
*
that is to say maximum,
·
to choose
m
O
such as
()
K m
II
O
*
that is to say no one,
·
to choose
m
O
such as
()
G m
O
*
that is to say maximum.
These criteria give very nearby results [bib8].
The results are given in the form of a table of 4 coefficients
K
11
,
K
21
,
K
12
,
K
22
allowing
to calculate
K
I
*
and
K
II
*
in all the cases of loading:
K
K
K
K
K
K
K
K
I
II
I
II
*
*




=


11
12
21
22
background image
Code_Aster
®
Version
7.4
Titrate:
Calculation of the coefficients of intensity of stresses
Date:
02/05/05
Author (S):
E. CRYSTAL
Key
:
R7.02.05-B
Page
:
14/14
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
Angle
m
(°)
K
11
K
21
K
12
K
22
0
10
20
30
40
50
60
70
80
90
1
0,9886
0,9552
0,9018
0,8314
0,7479
0,6559
0,5598
0,4640
0,3722
0
0,0864
0,1680
0,2403
0,2995
0,3431
0,3696
0,3788
0,3718
0,3507
0
-- 0,2597
-- 0,5068
-- 0,7298
-- 0,9189
-- 1,0665
-- 1,1681
-- 1,2220
-- 1,2293
-- 1,1936
1
0,9764
0,9071
0,7972
0,6540
0,4872
0,3077
0,1266
-- 0,0453
-- 0,1988
(
)
()
(
)
()
(
)
()
(
)
()
K
m
K m K
m
K
m K
m
K
m K
m
K
m
11
11
21
21
12
12
22
22
-
=
-
= -
-
= -
-
=
,
,
,
The search of the angle
m
O
in
CALC_G_THETA
is made of 10 degrees in 10 degrees. The angle of
propagation is not calculated and is printed out (in the file MESSAGE) only if INFORMATION is worth 2.



3 Bibliography
[1]
H.D. BUI, J.M. PROIX
: “Law of conservation in thermo linear elasticity”
- C.R.
Acad.Sc.Paris, t.298, Series II, n° 8, 1984.
[2]
H.D. BUI: “Associated path independent J-Integrals for separating mixed modes” - J. Mech.
Phys. Solids, vol. 31, N° 6, pp. 439-448, 1983.
[3]
H.D. BUI: “Breaking process fragile” - Masson, 1977.
[4]
P;. DESTUYNDER, Mr. DJAOUA: “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.
[5]
P. MIALON: “Calculation of derived from a size compared to a bottom of fissure by
method théta " - E.D.F. Bulletin of Management of the Studies and Search, Series C, n° 3,
1988, pp. 1-28.
[6]
Mr. AMESTOY, H.D. BUI, Ky DANG-VAN: “Deviation infinistésimale of a fissure in one
arbitrary direction " - C.R. Acad. Sc Paris, t.289 (September 24, 1979), Série B-99.
[7]
E. SCREWS: “Calculation of the stress intensity factors in plane linear elasticity” - Note
intern EDF-DER-MMN, HI-75505D of the 05/07/94.
[8]
P. MIALON: “Study of the rate of refund of energy in a direction marking an angle
with a fissure ", intern E.D.F. HI/4740-07 - 1984 notes.