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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
1/60
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
Organization (S):
EDF-R & D/AMA, SINETICS, UTO/LOCATED














Manual of Reference
R7.02 booklet: Breaking process
R7.02.01 document



Rate of refund of energy in thermo elasticity
linear




Summary:

One presents the calculation of the rate of refund of energy by the method theta in 2D or 3D for a problem
thermo linear rubber band. It is explained how the field theta is introduced into Code_Aster and how it
rate of refund of energy is established.
Studies mechanic-reliability engineers of evaluation of probability of priming of the rupture requires, moreover, its derivative
compared to a variation of field controlled by another field. The establishment of this option is detailed
in the code.
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
2/60
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
Count
matters
1
Calculation of the rate of refund of energy by the method theta in thermo linear elasticity ................... 4
1.1
Relation of behavior .............................................................................................................. 4
1.2
Potential energy and relations of balance ....................................................................................... 6
1.3
Lagrangian expression of the rate of refund of energy ............................................................. 6
2
Discretization of the rate of refund of energy ..................................................................................... 12
2.1
Method theta in dimension 2 ...................................................................................................... 12
2.2
Method theta in dimension 3 ...................................................................................................... 12
2.3
Choice in Aster of the discretization of G in dimension 3 .......................................................... 14
2.4
Establishment of G in thermo linear elasticity in Aster ........................................................... 23
2.4.1
Types of elements and loadings .................................................................................. 23
2.4.2
Environment necessary ................................................................................................... 23
2.4.3
Calculations of the various terms of the rate of refund of energy ............................................ 24
2.4.3.1
Elementary conventional term ................................................................................... 24
2.4.3.2
Term forces voluminal ............................................................................................ 25
2.4.3.3
Term forces surface ............................................................................................ 25
2.4.3.4
Thermal term ...................................................................................................... 25
2.4.3.5
Deformations term and initial stresses ............................................................. 25
2.4.4
Standardization of the rate of refund of energy in Aster ................................................... 26
2.4.4.1
Axisymetry ............................................................................................................... 26
2.4.4.2
Other cases ................................................................................................................ 27
2.5
Parameter setting of the controls ...................................................................................................... 27
3
Introduction of the field theta into Aster .............................................................................................. 29
3.1
Conditions to fill ....................................................................................................................... 29
3.2
Choice of the field theta in dimension 3 .......................................................................................... 29
3.2.1
Method of construction ...................................................................................................... 29
3.2.2
Calculation algorithms ........................................................................................................... 30
3.3
Choice of the field theta in dimension 2 .......................................................................................... 34
3.4
Another method ............................................................................................................................... 34
4
Derived from the rate of refund of energy compared to a variation of field .............................. 35
4.1
Problems ................................................................................................................................ 35
4.2
Opening remarks ............................................................................................................... 37
4.2.1
Theorem of transport ......................................................................................................... 37
4.2.2
Loadings and materials ................................................................................................... 39
4.2.3
Form ............................................................................................................................ 40
4.3
Calculations of the various terms of derived from the rate of refund of energy ................................ 42
4.3.1
Derived from the elementary conventional term .............................................................................. 42
4.3.2
Derived from the thermal term ................................................................................................. 44
4.3.3
Derived from the terms forces voluminal and surface ............................................................ 44
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
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:
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R7.02 booklet: Breaking process
HT-66/05/002/A
4.3.4
Derived from the deformations term and initial stresses ........................................................ 45
4.4
Establishment in Code_Aster .................................................................................................. 46
4.4.1
Perimeter of use ........................................................................................................... 46
4.4.2
Environment necessary .................................................................................................. 48
4.4.3
Standardization ....................................................................................................................... 48
5
Bibliography ........................................................................................................................................ 49
Appendix 1
Calculation of the derived seconds of the quadratic elements 2D ............................... 50
Appendix 2
Calculation of the term forces surface and of its derivative in 2D ....................................... 55
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
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R7.02 booklet: Breaking process
HT-66/05/002/A
1
Calculation of the rate of refund of energy by the method
theta in thermo linear elasticity
1.1
Relation of behavior
One considers a fissured elastic solid occupying the field
space
R
2
or
R
3
. That is to say:
·
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
.
S
S
D
F
G
Appear fissured elastic Solid 1.1-a:
To simplify, one places oneself in linear elasticity and small deformations, but this approach
generalize without sorrow with plasticity [R7.02.07], the great deformations, dynamics
[R7.02.02]…
One indicates by:
·
the tensor of the deformations,
·
°
the tensor of the initial deformations,
·
HT
the tensor of the deformations of thermal origin,
·
the tensor of the stresses,
·
°
the tensor of the initial stresses,
·
(
)
, °, °, T
density of free energy,
·
the tensor of elasticity.
is connected to the field of displacement
U
by:
()
(
)
U
=
+
1
2 U
U
I J
J I
,
,
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
5/60
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R7.02 booklet: Breaking process
HT-66/05/002/A
Density of free energy
(
)
, °, °, T
is identified by a dilation and tensile test in
small deformations.
(
)
, °, °, T
is a convex and derivable function.
(
)
(
) (
) (
)
, °, °,
°
°
° °
° °
T
HT
HT
HT
=
-
-
-
-
+
-
-
+
1
2
1
2
The law of behavior of an elastic material is written in the form:
(
)
(
)
(
)
-
=
° °
=
- ° + °
=
-
,
, T
T T
HT
ijth
ref.
ij
with
The constant term
1
2
° °
has a null contribution on the calculation of the rate of refund of energy, but
from a numerical point of view it makes it possible to find exactly the same value for a calculation
rubber band of
G
by having any intermediate elastic initial state:
° =
°
.
One finds:
(
)
(
) (
)
(
)
,
,
° °
=
-
-
=
-
T
HT
HT
HT
1
2
If initial deformations
°
and the initial stresses are null, the density of energy
free is written:
()
()
(
)
(
)
µ
, T
K T T
K
T T
II
ij
ij
ref.
kk
ref.
=
+
-
-
+
-
1
2
3
9
2
2
2
2
The relation of behavior is written:
ij
=
kk
ij
+ 2
µ
ij
- 3K
T
- T
ref.
(
)
ij
and
µ
are the coefficients of BLADE.
is the thermal expansion factor.
T
ref.
is the temperature of reference.
K
, voluminal module of compressibility, is connected to the coefficients of BLADE by
:
3K
= 3
+ 2
µ
.
The relation of behavior starting from the YOUNG modulus
E
and of the Poisson's ratio
is:
ij
=
E
1
+
ij
+
1
- 2
tr
()
ij
-
E
1
- 2
T
- T
ref.
(
)
ij
with:
=
E
1
+
(
)
1
- 2
(
)
µ
=
E
2 1
+
(
)
3K
=
E
1
- 2
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
6/60
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
1.2
Potential energy and relations of balance
One defines spaces of the fields kinematically acceptable
V
and
V
O
.
V
=
v
acceptable
,
v
=
U
on
S
D
{
}
V
O
=
v
acceptable
,
v
=
0
on
S
D
{
}
With the assumptions of [§1.1] (and for
° = ° = 0
), relations of balance in weak formulation
are:
U
V
ij
v
I, J
D
=
F
I
v
I
D
+ G
I
S
v
I
D
, v V
O

They are obtained by minimizing the total potential energy of the system:
W v
()
=
v
()
,
T
(
)
D
- F
I
v
I
D
- G
I
S
v
I
D
Indeed, if this functional calculus is minimal for the field of displacement
U
, then:
W
=
ij
ij
D
- F
I
v
I
D
- G
I
S
v
I
D
=
ij
1
2
v
I, J
+
v
J, I
(
)
D
- F
I
v
I
D
- G
I
S
v
I
D
=
ij
v
I, J
D
- F
I
v
I
D
- G
I
S
v
I
D
= 0
We thus find the equilibrium equations and the relation of behavior while having posed:
ij
=
ij
.
1.3
Lagrangian expression of the rate of refund of energy
By definition [bib1] the rate of refund of local energy
G
is defined by the opposite of derived from
potential energy compared to the field
:
G
= -
W
This rate of refund is calculated in Code_Aster by the method theta, which is a method
Lagrangian of derivation of the potential energy [bib4] [bib2]. Transformations are considered
()
M
M
M
:
+
F
area of reference
in a field
modelizing propagations
fissure, which at a material point
P
make correspond a space point
M
. These transformations
must modify that the position of the bottom of fissure
O
. Fields
must thus be tangent with
, i.e. while noting
N
the normal with
:
=
µ
such as
µ N = 0
on
{
}
Notice
This family of functions of transformation must be sufficiently regular. In particular, it
must be at least twice derivable per pieces in
P
and in
(so that the derivative
partial seconds switch over) and to carry out a diffeomorphism for each value of the parameter
(that ensures the reversibility of the process).
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
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HT-66/05/002/A
That is to say
m
the unit normal with
O
located in the tangent plan at
(i.e. tangent in the plan of
fissure) and re-entering in
.
m
O
Plan of
fissure
Appear 1.3-a: Melts of fissure in 3D
According to proposal 7 of [bib4], the rate of refund of local energy
G
is solution of the equation
variational:
()
G
O
=
m
G
,
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
= -
= -
=
&
W
D W
D
U
0
Quantity
m
represent the normal speed of the bottom of fissure. In addition,
G
()
with same
value which it is of a right propagation [Figure 1.3-b] (A) or about a curved propagation
[Figure 1.3-b] (b) insofar as that Ci with the same tangent at the beginning (then one can anything of it
to say). On the other hand, one can nothing say case of the propagation in a direction marking one
angle [bib5] [Figure 1.3-b] (c).
has
B
C
Appear 1.3-b: Various geometries of propagations
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
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:
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Thereafter, when no confusion is possible, one will indicate par. the Lagrangian derivative
in a virtual propagation of fissure speed
. That is to say
(
)
, M
a space field (or eulérien)
unspecified definite on
R
+
×
, we will note his material representation (or Lagrangian)
()
()
(
)
,
,
P
P
=
F
and its derivative particulate (or Lagrangian) compared to this propagation
virtual
&
=
=0
.
Remarks [bib6]:
· The fact of adopting two different visions (eulérienne and Lagrangian) introduced structurally
concepts of cross derivabilities. Thus, this particulate derivative of a space field
called Lagrangian derivative consists in deriving
(
)
, M
by fixing the material point
()
()
P
F
M
=
-
1
. One transposes the field of Lagrangian representation, then it is derived
compared to
before reconverting it of representation eulérienne.
· It is reminded the meeting that this Lagrangian derivative is related to derived the eulérienne
by the relation
&
.
=
+
Notice [bib4]:
The derivative eulérienne
depends only on
restricted with, i.e. trace of on
melts of fissure.
With these notations, the rate of refund of energy in this propagation
is written (by using it
theorem of transport of Reynolds cf [§4.2.1]):
()
(
)
}
-
=
-


+
-
-
+
-




·
·
G
F U
F U
D
G U
G U
N N
D
I
I
I
I
K K
I
I
I
I
K K
K
K
S
6 7
4 8
4
,
,
However
T
T
T
ij
ij
ij
ij
ij
ij
&
&
&
&
&
+
°
°
+
°
°
+
=
°
°
)
,
,
,
(
T
,
,
,
F G
° °
being supposed independent of
, i.e. being the restriction on
(or
) of
fields defined on
R
3
, there are the following relations:
&
&
&
&
&
,
,
,
,
,
T
T
F
F
G
G
K
K
I
I K
K
I
I K
K
ij
ij K K
ij
ij K K
=
=
=
=
=
°
°
°
°
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
9/60
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R7.02 booklet: Breaking process
HT-66/05/002/A
Indeed, if one considers loadings and materials
who are the restriction on
geometry
(or part of its border) of fields defined on
R
3
entire:
=
/
Derivation compared to the parameter
switching over with this restriction, there is the result
= =
=
0
0
Note:
· This assumption is checked only for sufficiently regular fields (for example
belonging to spaces of Sobolev of
). Their definition should not be impacted by
variation of border.
· In the case of the derivation of the rate of refund of energy compared to a variation of
field (cf [§4]) the derivative eulérienne of the field of temperature could not be neglected any more.
In addition, one as supposed as the derivative eulériennes of the characteristics materials
are null, which is true only on the problem discretized with the current functionalities of
the operator
DEFI_MATERIAU
. Their gradient on each element is also null by construction (they
are discretized
P
0
i.e. constant by finite elements), it results from this that the derivative
Lagrangian is null:
{
{
}
&
.
,
=
+
=
=
0
0
123 for
E
T
ref.
Caution:
With characteristics variable materials within finite elements of the crown theta of
calculation, this simplification is not licit any more.
(
)
(
) (
) (
)
(
)
(
)
(
)
K
K
K
K
ij
ij
HT
ij
ij
K
K
ij
ij
ij
ij
ij
ij
HT
ij
ij
ij
ij
HT
ij
ij
ij
ij
ij
ij
kl
HT
kl
kl
ijkl
ij
ij
ij
kl
HT
kl
kl
ijkl
ij
HT
HT
HT
T
T
T
,
,
,
.
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
,
,
,
+


-
-
+


-
+
=
-
-
=
+
-
-
=
-
=
-
-
-
-
=
=
+
-
-
=
°
°
+
°
°
-
-
+
°
-
-
°
-
-
=
°
°
°
°
°
°
°
°
°
°
°
°
°
°
°
°
&
&
or
of
has
one
Like
In addition, according to proposal 2 of [bib4]:
}
I J
I J
I p
p J
,
,
,
,
&
·




=
-
(
) (
)
&
&
&
,
,
,
,
,
,
ij
I J
J I
I p
p J
J p
p I
U
U
U
U
=
+
-
+
1
2
1
2
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
10/60
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
And one can eliminate
&u
expression of
G
()
by noticing that
&u
is kinematically acceptable
and by using the equilibrium equation:
ij
I J
I
I
I
S
I
ij
S
J
I K
K
U
D
F U D
G U D
N U
D
D
=
+
+
&
&
&
,
,
from where:
()
(
)
(
)
(
)
-
=
-
-
+
-
-
+
-
-




=
-
-
-
-
+
+
+
-
G
&
&
&
&
&
&
&
&
&
&
,
,
,
,
,
,
,
F U
F U
F U
D
G U
G U D
G U
N N
D
U
F U D
G U D
F U D
U
U
D
T T
F U
I
I
I
I
I
I
K K
I
I
I
I
S
I
I
S
K K
K
K
ij
I J
I
I
I
S
I
I
I
ij
I p
p J
J p
p I
I
1
2
(
)
I
K K
I
S
I
I
I
K K
K
K
ij
ij
ij K K
ij
ijth
ij
ij K K
D
G U
G U
N N
D
D
,
,
,
,
&
-
+
-




+
-




+
-
-




°
°
°
°
1
2
1
2
and finally:
()
G
=
-
-
+
-




-
-
-




+
+
+
+
-




-
°
°
°
°
ij
I p
p J
K K
K
K
ij
ij
ij K
K
ij
ijth
ij
ij 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
U
T T
D
D
F U
F
U D
G
U
G U
N N
D
N U
D
,
,
,
,
,
,
,
,
,
,
,
1
2
1
2
K
K
D
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
11/60
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
Note:
· In deformations of Euler-Lagrange the first intégrande becomes
J
p
p
I
ij
p
I
U
H
,
,
,
with
H
U
I J
I J
I J
,
,
,
=
+
.
· In axisymetry, there is the formal analogy
()
()
X y
R Z
,
,
and all components of
gradients implying the component orthoradiale are null except
,
=
R
R
. Moreover
the element of surface is multiplied by
R
to take into account the calculation of the integral for one
unit of radian.
· The possibility of taking into account fields of imposed displacements was not
developed. Those are not constrained besides by the propagation of fissure since they
appear via the equilibrium condition.
· In the surface term there are normal derivations on the surface which do not have a direction
for the elements of skin used in Code_Aster. One thus has recourse to the geometry
differential and with derived the contravariantes for better apprehending this intégrande on
surface calculation (cf [Appendix 2]).
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Code_Aster
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Version
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Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
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:
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Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
2
Discretization of the rate of refund of energy
2.1
Method theta in dimension 2
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
and re-entering
in
,
·
{
}
=
µ
µ
=
on
that
such
0
N
.
In dimension 2, bottom of fissure
O
brings back itself to a point
M
0
, and one can choose a field
unit in the vicinity of this point, so that:
()
()
G
M
0
=
G
O
m
Appear 2.1-a: Melts of fissure in 2D
2.2
Method theta in dimension 3
Dependence of
G
()
with respect to the field
on the bottom of fissure is more complex. The field
scalar
G S
()
can be discretized on a basis which we will note
p
J
S
()
()
1
jN
.
O
0
S
Appear 2.2-a: Discretization of the bottom of fissure in 3D (curvilinear X-coordinate)
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
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Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
That is to say
G
J
components of
G S
()
in this base:
G S
()
=
G
J
J
=1
NR
p
J
S
()
In the same way, fields
I
(pertaining to
) can be discretized on a basis which we will note
Q
K
S
()
(
)
1
km
. Let us indicate by
I
the trace of the field
I
on the bottom of fissure
O
:
I
S
()
=
I
S
()
and
by
ki
components of
I
S
()
in this base:
I
S
()
=
K
I
K
=1
M
Q
K
S
()
G S
()
being solution of the variational equation
() ()
()
()
G S
S
S ds
O
=
m
G
,
, them
G
J
check:
()
()
(
)
()
()
[]
,
,
O
G p S
Q S
S ds
I
P
J
J
NR
J
ki
K
K
M
I
=
=
=
1
1
1
m
G
that is to say:
() () ()
()
[]
J
NR
ki
K
M
J
K
J
I
p S Q S
S ds G
I
P
O
=
=


=
1
1
1
m
G
,
,
G
J
can thus be 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
ki
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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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
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Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
2.3 Choice
in
Aster of the discretization of G in dimension 3
In dimension 2, there is no problem bus by choosing a field
unit in the vicinity of
melts of fissure, one obtains the relation
()
G
=
G
. The rate of refund of energy is independent of
field
.
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 value of
G
()
for a field theta given by the user (cf orders
CALC_G_THETA_T
[U4.82.03]). It is interesting to choose the unit field theta with
vicinity of the bottom of fissure and such as:
S
()
m S
()
= 1, S
curvilinear X-coordinate of
O
O
Appear 2.3-a: Discretization of the bottom of fissure in 3D (normal)
One obtains in this case a total rate of refund
G
agent with a uniform progression of
the fissure such as:
()
()
Gl
G S ds
O
=
=
G
where
L
is the length of the upper lip or lower of the fissure.
·
The rate of refund of energy room
G S
()
solution of the variational equation
() ()
()
()
G S
S
S ds
O
=
m
G
,
In this case, the user does not give a field theta, the fields
I
necessary to calculation
of
G S
()
(cf are calculated automatically orders
CALC_G_LOCAL_T
[U4.82.04]).
In Code_Aster, one chose two families of bases (cf [§2.2]):
·
Polynomials of LEGENDRE
()
J
S
of degree
J
(
)
0
J
Deg
max
.
·
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).
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
15/60
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
Let us recall that the polynomials of LEGENDRE constitute a not normalized orthogonal family. They
are obtained by the relation of recursion:
N
+1
(
)
P
N
+1
T
()
- 2n +1
(
)
T P
N
T
()
+ N P
N
- 1
T
()
= 0
In particular:
P
0
T
()
= 1
P1 T
()
= T
P
2
T
()
=
3 T
2
- 1
(
)
/2
P
3
T
()
=
5 T
3
- 3t
(
)
/2
In Code_Aster, one normalizes them in the form:
J
S
()
=
2 J
+1
L
P
J
2s
L - 1
where:
·
S
is the curvilinear X-coordinate of
O
,
·
L
the length of the bottom of fissure
O
.
2
S
()
1
S
()
O
S
()
S
= 0
S
= L
O
0
0
L
L
Appear 2.3-b: Polynomials of Legendre
In Code_Aster, one limits oneself to
Deg
max
= 7
like maximum degree.
Functions of forms
K
S
()
are associated the discretization of
O
.
3
S
()
2
S
()
1
S
()
K
= 1
K
= 2
K
= 3
Appear 2.3-c: Functions of form of the bottom of fissure (linear elements)
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
16/60
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
Let us recall that one is brought to discretize
G S
()
and fields
I
S
()
(trace of the field
I
on the bottom
of fissure
O
).
G S
()
=
G
J
J
=1
NR
p
J
S
()
I
S
()
=
K
I
K
=1
M
Q
K
S
()
There is thus several possible choices of discretizations, summarized in the table below:
Polynomials of LEGENDRE
Functions of form
G S
()
G
J
J
=0
NDEG
J
S
()
G
J
J
=1
NO
J
S
()
I
S
()
ki
K
=0
NDEG
K
S
()
()
ki
K
NO
K
S
=
1
Table 2.3-1: Choice of the discretization
NO
:
a number of nodes of the bottom of fissure
O
NDEG
: maximum degree of the polynomials of LEGENDRE chosen by the user
(
)
7
max
=
Deg
NDEG
In the control
CALC_G_LOCAL_T
(cf [U4.82.04]) key words
LISSAGE_THETA
and
LISSAGE_G
allow to choose the discretization of
I
G
and
.
The options available in Aster are summarized in the following table:
I
S
()
Polynomials of LEGENDRE
Functions of form

G S
()
Polynomials of
LEGENDRE
LISSAGE_THETA: “LEGENDRE”
LISSAGE_G: “LEGENDRE”
(1
Er
case)
LISSAGE_THETA: “LAGRANGE”
LISSAGE_G: “LEGENDRE”
(2
Nd
case)
Functions of
form
Nonavailable
LISSAGE_THETA: “LAGRANGE”
LISSAGE_G: “LAGRANGE”
or
“LAGRANGE_NO_NO”
(3
ème
case)
Table 2.3-2: Options of discretization of Code_Aster
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
17/60
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
First case:
G S
()
and fields
I
S
()
are broken up according to the polynomials of LEGENDRE.
G S
()
=
G
J
J
=0
NDEG
J
S
()
I
S
()
=
K
I
K
=0
NDEG
K
S
()
NDEG
components
G
J
are given by solving the linear system with
P
equations:
() ()
()
()
G has
B
I
P
has
S
S
S ds
B
ij
J
J
NDEG
I
ij
K
NDEG
J
K
ki
I
I
O
=
=
=
=
=
=










0
0
1
,
,
with
m
G
One makes the choice in Code_Aster take, like fields
I
, them
NDEG
fields
I
such as:
()
()
()
I
I
S
S
S
=
m
where
I
S
()
is the polynomial of LEGENDRE of degree
I
.

The linear system is simplified then in a system of
P
= NDEG
equations with
NDEG
unknown factors:
()
() ()
G has
I
NDEG
has
S
S ds
ij
J
J
NDEG
I
ij
J
I
ij
O
=
=
=
=
=


0
1
G
,
,
with
because the polynomials of Legendre form a base orthonormée on
0
.
Thus
()
G
J
J
=
G
and thus
()
()
()
G S
S
J
J
NDEG
J
=
=
G
0
.
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
18/60
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
Second case:
G S
()
is broken up according to the polynomials of LEGENDRE.
I
S
()
is defined by the functions of form of the nodes of the bottom of fissure.
()
()
()
()
G S
G
S
S
S
J
J
NDEG
J
I
ki
K
NO
K
=
=
=
=
0
1
One makes the choice in Code_Aster take, like fields
I
, them
NO
fields
I
such as:
()
()
()
I
I
S
S
S
=
m
where
()
I
S
is related to form of the node
I
bottom of fissure.
That is to say:
K
I
K
=1
NO
K
S
()
m S
()
=
I
S
()
and one has
NO
equations with
NDEG
unknown factors:
() ()
()
G has
B
I
NO
has
S
S dS
B
ij
J
J
NDEG
I
ij
J
I
I
I
O
=
=
=
=
=






0
1
,
,
with
G
In this case, one must have
NDEG
NO
, that is to say
NDEG
min 7, NO
(
)
where
NO
is the number of
nodes of the bottom of fissure.

Third case:
()
G S
and
()
I
S
are defined by the functions of form of the nodes of the bottom of fissure.




=
=
=
=
NO
K
K
I
K
I
NO
J
J
J
S
S
S
G
S
G
1
1
)
(
)
(
)
(
)
(
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
19/60
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
The system to be solved is as follows:




=
=
=
=
0
)
(
)
(
with
)
,
1
(
)
0
ds
S
S
has
NO
I
G
has
J
I
ij
NO
J
J
ij
I
G (
NO
: a number of nodes of the bottom of fissure
I
: function of form of the node
I
If there are linear elements:
()
(
)
()
(
)
1
2
1
2 1
1
2 1
X
X
X
X
=
-
=
+
1
2
1
1
0
- 1 0
Element of reference
,
0
)
(
)
(
=
=
+
-
J
I
I
J
I
I
has
has
if
2
J
() ()
() ()
(
)
() ()
(
)
(
)
(
)
(
)
()
(
)
()
(
)
(
)
(
)
(
)
(
) (
)
[
]
+
-
+
-
-
+
+
-
+
-
+
-
+
-
+
-
+
-
-
-
-
-
-
-
-
+
-
=
-
-
+
+
-
=
-
+
-
=
-
=
-
-
=
-
=
=
=
-
1
1
1
1
1
1
2
1
2
1
1
1
1
1
2
1
1
2
2
1
1
1
1
1
1
2
1
2
1
1
1
1
)
1
(
3
1
1
4
1
2
1
4
1
2
2
2
6
1
1
4
1
2
2
1
I
I
I
I
I
I
I
I
I
I
I
I
II
I
I
I
I
I
I
S
S
I
I
I
I
I
I
S
S
S
S
dx
X
S
S
dx
X
S
S
dx
X
S
S
dx
X
S
S
has
S
S
dx
X
S
S
dx
X
X
S
S
ds
S
S
ds
S
S
has
O
I
I
I
- 1 I
I
+1
Appear 2.3-d: Linear functions of form
The matrix
With
ij
is thus written:
(
) (
)
(
) (
) (
)
(
) (
) (
)
(
) (
)
1
6
2
0
0
2
0
0
2
0
0
2
2
1
2
1
2
1
3
1
3
2
3
2
4
2
4
3
4
3
5
3
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
-
-
-
-
-
-
-
-
-
-










L
L
L
L
M
M
M
M
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
20/60
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
If there are quadratic elements:
()
(
)
()
(
)
() (
) (
)
1
2
3
1
2
1
1
2
1
1
1
X
X X
X
X X
X
X
X
=
-
=
+
= -
+
Appear 2.3-e: Quadratic functions of form (Element of reference)
It is necessary to distinguish node node and node medium
·
I
= node node:
,
0
)
(
)
(
=
=
+
-
J
I
I
J
I
I
has
has
if
3
J
() ()
() ()
(
)
() ()
(
)
(
)
(
)
() ()
() ()
(
)
() ()
(
)
(
) (
)
(
)
()
(
)
()
(
)
+
-
-
+
+
-
+
-
+
-
-
-
+
-
-
-
-
-
-
+
-
+
-
-
-
-
-
-
-
=
-
+
-
=
-
+
=
-
+
-
=
-
=
=
=
-
-
=
-
-
=
-
=
=
=
-
-
1
1
2
2
2
1
1
1
2
2
2
2
1
1
2
2
2
1
1
3
2
2
1
1
)
1
(
2
1
1
1
1
2
2
2
2
1
2
2
2
)
2
(
15
2
2
2
15
1
1
2
1
2
2
30
1
4
1
2
2
2
2
I
I
I
I
I
I
II
I
I
I
I
I
I
S
S
I
I
I
I
I
I
I
I
I
I
I
I
S
S
I
I
I
I
I
I
S
S
dx
X
S
S
dx
X
S
S
has
S
S
dx
X
X
X
S
S
dx
X
X
S
S
ds
S
S
ds
S
S
has
S
S
dx
X
X
S
S
dx
X
X
S
S
ds
S
S
ds
S
S
has
O
I
I
O
I
I
Appear 2.3-f: Node node
i-2 i-1 I i+1 i+2
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
21/60
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
·
I = node medium:
2
0
)
(
)
(
=
=
+
-
J
has
has
J
I
I
J
II
if
() ()
(
) () () (
)
(
) (
) (
)
(
)
()
(
) () ()
(
)
+
-
-
+
+
-
+
+
-
+
-
+
-
+
-
-
-
=
+
-
-
=
=
-
=
+
-
-
-
=
-
=
=
+
-
+
-
1
1
1
1
2
2
1
2
1
1
1
1
1
1
1
1
3
1
1
1
)
1
(
15
8
1
1
2
15
1
1
1
2
1
2
2
1
1
1
1
I
I
I
I
S
S
I
II
I
I
I
I
I
I
S
S
I
I
I
I
S
S
dx
X
X
S
S
ds
S
has
S
S
dx
X
X
X
X
S
S
dx
X
X
S
S
ds
S
S
has
I
I
I
I
Appear 2.3-g: Node medium
The matrix
With
ij
is written:
(
) (
) (
)
(
)
(
) (
)
(
) (
) (
) (
) (
)
(
)
(
) (
)
(
) (
) (
)
1
30
4
2
0
0
2
16
2
0
0
2
4
2
0
0
0
2
16
2
0
0
0
2
4
0
0
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
5
1
5
3
5
3
5
3
5
3
5
3
5
3
5
3
7
3
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-














L
L
L
M
M
M
Particular case:
S
S
cste L
I
I
+
- =
=
2
= length of an element
+
+
L
L
30
4
2
1 0
0
2
16
2
0
0
1 2
8
2
1
0
0
2
16
2
0
0
1 2
8
-
-
-
-














L
L
L
L
L
M
M
M
M
node node of edge
node medium
node node
i-2 i-1
I i+1
i+2
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
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Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
Method “node by node”:
This method results from the Lagrange-Lagrange method but it is simplified: one replaces
resolution of the linear system by multiplying the values
()
G
I
by a weighting coefficient.
()
G =
I
I
I
G
1
2
3
4
5
NR
Appear 2.3-h: Method “node by node”
Moreover if
G (
) = cte = B
,
I
and that one considers one G constant per element (this method does not have
no vectorial significance), one a:
·
node node of edge:
(
)
B
L
G
B
G
L
6
1
2
4
30
that is to say
=
=
-
+
·
node node:
(
)
B
L
G
B
G
L
3
1
8
2
2
1
30
that is to say
=
=
-
+
+
+
-
·
node medium:
(
)
B
L
G
B
G
L
2
3
2
16
2
30
that is to say
=
=
+
+
L
6b
L
3b
L
3
2
B
L
What gives if the elements do not have constant lengths:
·
node node of edge:
(
)
(
)
1
3
1
2
6
6
=
-
=
-
-
S
S
S
S
NR
NR
NR
or
·
node node: for example
(
) (
) (
)
3
3
1
5
3
5
1
6
6
=
-
+
-
=
-
S
S
S
S
S
S
that is to say:
(
)
I
I
I
S
S
=
-
+
-
6
2
2
or:
(
)
I
I
I
I
S
S
'
=
-
+
3
1
·
node medium:
(
)
I
I
I
S
S
=
-
+
3
2
1
To activate this method it is necessary to specify in
CALC_G_LOCAL_T
:
LISSAGE_G: “LAGRANGE_NO_NO”
LISSAGE_THETA: “LAGRANGE”
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Code_Aster
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Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
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R7.02 booklet: Breaking process
HT-66/05/002/A
2.4
Establishment of G in thermo linear elasticity in Aster
2.4.1 Types of elements and loadings
In Code_Aster, it is possible to calculate in thermo linear elasticity:
·
the rate of refund of energy
G
()
in 2D and 3D, associated a field of propagation
virtual of the fissure
(given by the user using the control
CALC_THETA
[U4.82.02]): order
CALC_G_THETA_T
[U4.82.03],
·
the rate of refund of local energy
G S
()
in 3D, where
S
is the curvilinear X-coordinate of the bottom of
fissure: order
CALC_G_LOCAL_T
[U4.82.04].
These calculations are valid for following modelings:
·
D_PLAN
·
C_PLAN
·
AXIS
·
3D
and for the thermo loadings mechanical following applying to a two-dimensional medium
(affected to triangles with 3 or 6 nodes, quadrangles with 4, 8 or 9 nodes and segments with 2 or
3 nodes) or on a three-dimensional medium (affected with hexahedrons with 8, 20 nodes or 27 nodes,
pentahedrons with 6 or 15 nodes, of the tetrahedrons with 4 or 10 nodes, of the faces with 3 or 6 nodes and of the faces
to 4, 8 or 9 nodes):
·
F
, field of voluminal forces applied to
(mechanical loads of the type
GRAVITY
,
ROTATION
,
FORCE_INTERN
),
·
G
, field of surface forces applied to a part
S
of
(including on the lips
fissure:
PRES_REP
,
FORCE_FACE
),
·
U
, field of displacements imposed on part
S
D
of
(Not developed to date),
·
T
, field of temperature (
TEMP_CALCULEE),
·
, initial field of defomation (
EPSI_INIT
).
These loadings can depend on time and space.
Characteristics of material (
E
,
and
) can depend on the temperature
T
and of
space while remaining constant by elements.

2.4.2 Environment
necessary
For the calculation of the rate of refund of energy
G
()
by the method
in the case of a problem
thermo rubber band, the field
must obligatorily be created before (either by the control
CALC_THETA
[U4.82.02], that is to say by the control
AFFE_CHAM_NO
[U4.44.11]).
For the calculation of the rate of refund of local energy
G S
()
, fields
I
necessary to calculation are
generated automatically.
In both cases, it is about a postprocessing only starting from the field of solution displacement
calculation on the model considered. In particular, the density of free energy and the stresses are
calculated starting from the field of displacement and the characteristics of material.
For calculation in 3D, it is necessary to define, starting from an ordered list of nodes, a bottom of fissure of one
mesh 3D, and starting from two lists of meshs, the upper lip and the lower lip of this
fissure control
DEFI_FOND_FISS
[U4.82.01]. This operator creates a concept usable by
operators
CALC_THETA
and
CALC_G_LOCAL_T
. In 2D, the bottom of fissure is tiny room to a point and this
operator is not necessary for the calculation of
G
()
.
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Code_Aster
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Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
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R7.02 booklet: Breaking process
HT-66/05/002/A
2.4.3 Calculations of the various terms of the rate of refund of energy
The expression supplements
G
()
is given to [§1.3]. We will detail each term. The field
is null apart from a disc of radius
R
sup
S
()
defined in chapter 3 [Figure 3.3-a].
Let us notice that as all the terms utilize
or its gradient, elementary terms
are null apart from this disc of radius
R
sup
S
()
. In the controls
CALC_G_THETA_T
and
CALC_G_LOCAL_T
, it is thus not necessary to specify the loadings which do not apply
in this area.

2.4.3.1 Elementary conventional term
()
(
)
TCLA
U
U T
ij
I p
p J
K K
=
-
,
,
,
,
Density of energy elastic
()
(
)
U T
,
is written in thermo linear elasticity:
·
in 3D and in
AXIS
:
()
(
)
()
µ
U, T
II
ij
ij
HT
=
+
-
1
2
2
·
in DP:
()
(
)
(
)
(
) (
)
(
)
(
) (
)
(
)
U, T
E
E
E
xx
yy
xx
yy
xy
HT
=
-
+
-
+
+ +
-
+ +
-
1
2 1
1 2
1
1 2
1
2
2
2
·
in CP:
()
(
)
(
)
(
)
(
)
(
)
U, T
E
E
E
xx
yy
xx
yy
xy
HT
=
-
+
+ -
+ +
-
2 1
1
1
2
2
2
2
2
with
(
)
(
)
HT
ref.
II
ref.
K T T
K
T T
=
-
-
-
3
9
2
2
2
where:
3K
=
E
1
- 2
;
=
E
1
+
(
)
1
- 2
(
)
; 2
µ
=
E
1
+
E
: YOUNG modulus
: Poisson's ratio
,
µ
: coefficients of BLADE
: thermal dilation
Density of energy elastic
()
(
)
U T
,
can be written in a general way in the form:
()
(
)
(
)
(
)
µ
U T
K
T T
kk
ref.
eq
eq
ijD ijD
ijD
ij
kk ij
,
=
-
-
+
=
=
-
1
2
3
2
3
3
2
1
3
2
2
2
with
and
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Code_Aster
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Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
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E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
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HT-66/05/002/A
(
)
()
(
)
(
)
(
)
(
)
(
)
2
2
2
2
2
2
2
2
2
2
9
3
2
3
2
9
3
2
1
,
3
2
1
ref.
kk
ref.
ij
ij
kk
kk
ij
ij
ref.
kk
ref.
kk
kk
ij
ij
eq
T
T
K
T
T
K
T
T
K
T
T
K
K
T
U
-
+
-
-
+
=
-
+
-
+
-
-
=
-
=
µ
µ
µ
and
that is to say

2.4.3.2 Term forces voluminal
TFOR
= F
I
U
I
K, K
+ F
I, K
K
U
I

2.4.3.3 Term forces surface
TSUR
= G
I, K
K
U
I
+ G
I
U
I
K, K
-
N
K
N
K


Note:
In this surface term there are normal derivations on the surface which do not have a direction for
elements of skin used in Code_Aster. One thus has recourse to the differential geometry
and with derived the contravariantes for better apprehending this intégrande on the surface of calculation
(cf [Appendix 2]).

2.4.3.4 Term
thermics
THER
= -
T T
, K
K
with:
()
(
)
()
(
)
(
)
()
(
)
(
)
(
)
T
U T
dK T
dT
T T
K
D T
dT
T T
T T
kk
ref.
ref.
kk
ref.
,
=
-
-
-
+
-








-
-
1
2
3
3
3

2.4.3.5 Deformations term and initial stresses
TINI
ij
ij
ij K
ij
ijth
ij
ij K
K
=
-




-
-
-








°
°
°
°
1
2
1
2
,
,
One can notice that if
° =
°
then:
(
)
(
)
=
-
- ° + ° =
-
=
HT
HT
TINI
and
0
Note:
Taking into account the various digital processings carried out at the time of the establishment in the source of
the operator, it is not licit to cumulate stress fields and initial deformations,
because this term is then not cancelled. The user will have to return either of the initial stresses, or of
initial deformations but not both.
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Code_Aster
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Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
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E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
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:
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Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
2.4.4 Standardization of the rate of refund of energy in Aster
2.4.4.1 Axisymetry
G
()
such as it is established here, the restitution of energy in the kinematics defined calculates by
. It
can be necessary to standardize it (with the hand! it is not done automatically in the code)
to be able to compare with an intrinsic value with material, in particular into axisymmetric.
Let us consider the case of an inclined fissure, whose bottom of fissure is at a distance
R
axis of
symmetry:
Y
X
R
L
Appear 2.4.4.1-a: Melts of fissure in axisymetry
In Aster, the axis
OY
is the axis of symmetry in modeling
“AXIS”
and the rate of refund of
energy calculated is:
G
()
= - dW
D L
where
W
is the potential energy per unit of radian.
However the intrinsic value of the rate of refund of energy is:
G
= - dW
total
dA
where:
·
W
total
is total potential energy,
·
dA
is the variation of surface of the fissure.
with:
L
D
R
dA
W
W
total
2
2
=
=
from where:
dW
dA
dW
D L
D L
dA
R
dW
D L
total
=
=
2
1
and thus
()
G
R
= 1
G
in axisymetry.
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Code_Aster
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Titrate:
Rate of refund of energy in thermo linear elasticity
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Key
:
R7.02.01-D
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2.4.4.2 Others
case
In dimension 3, the value of
G
()
for a field
given by the user is such as:
G
()
=
G
O
S
()
S
()
m S
()
ds
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
()
m S
()
= 1, S
curvilinear X-coordinate of
O
and:
G
()
=
G
O
S
()
D
That is to say
G
the total rate of refund of energy. To have its value per unit of length, it is necessary
to divide the value obtained by the length of the fissure
L
:
()
G
L
=
G
in 3D
In dimension 2 (
C_PLAN
and
D_PLAN
), the bottom of fissure is tiny room to a point and the value of
G
()
is
independent of the choice of the field
(with
and
unit in the vicinity of the bottom of fissure).
()
G
=
G
,
2.5
Parameter setting of the controls
The table below proposes a summary of the parameter setting of the controls
CALC_G_LOCAL_T
CALC_G_THETA_T
. For more precision one will refer to [U4.82.03] and [U4.82.04].
Controls Key word
Value
by
defect
Ref.
CALC_G_LOCAL_T
MODEL
[§2.4]
“D_PLAN”
“C_PLAN”
“AXIS”
“3D”
CHAM_MATER
[§2.4.1] Def.
materials
MELTS
[§2.4.1]
Def. bottom of
fissure
DEPL
Recup.
of one
field of depl.
RESULT
EXCIT
[§2.4.1] Standard
charg.
SYME_CHAR “WITHOUT”
“WITHOUT”
“SYME”
“ANTI”
LISSAGE_THETA “LEGENDRE”
[§2.2]
“LEGENDRE”
“LAGRANGE”
LISSAGE_G “LEGENDRE”
[§2.2]
“LEGENDRE”
“LAGRANGE”
“LAGRANGE_NO_NO”
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Code_Aster
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Titrate:
Rate of refund of energy in thermo linear elasticity
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Key
:
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DEGREE 5
[§2.2]
INFORMATION 1
TITRATE
OPTION “CALC_G”
“CALC_G”
[§2.4]
“CALC_G_LGLO”
R_INF
[§3.2]
R_SUP
[§3.2]
R_INF_FO
[§3.2]
R_SUP_FO
[§3.2]
COMP_ELAS
COMP_INCR
ETAT_INIT
[§2.4.3]
CALC_G_THETA_T
MODEL
[§2.4]
“D_PLAN”
“C_PLAN”
“AXIS”
“3D”
CHAM_MATER
[§2.4.1] Def.
materials
THETA
[§2.4.2] Def.
theta
MELTS
[§2.4.1]
Def. bottom of
fissure
DEPL
Recup.
of one
field of depl.
RESULT
EXCIT
[§2.4.1] Standard
charg.
SYME_CHAR “WITHOUT”
“WITHOUT”
“SYME”
“ANTI”
INFORMATION 1
TITRATE
OPTION “CALC_G”
“CALC_G”
[§2.4]
“CALC_G_LAGR”
“CALC_K_G”
“G_BILINEAIRE”
“CALC_G_MAX”
“CALC_DG”
[§4] Behavior
COMP_ELAS
COMP_INCR
ETAT_INIT
[§2.4.3]
Table 2.5-1: Parameter setting of the controls
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Titrate:
Rate of refund of energy in thermo linear elasticity
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3
Introduction of the field theta into Aster
3.1
Conditions to fill
The field theta is a field of vectors, definite on the fissured solid, which represents the transformation
field during a propagation of fissure within the meaning of [§1]. The transformation should only modify
the position of the bottom of fissure and not the edge of the field
, i.e.:
N = 0
on
(
N
normal with
). Moreover, the field theta must be regular on
[bib4].
Because of the singularity of the field of displacement, it is interesting from the numerical point of view
to use fields
constant in a vicinity of
O
, thus cancelling in this vicinity them
singular terms
K K
ij
I p
p K
U
,
,
,
-
in
G
()
.

3.2
Choice of the field theta in dimension 3
3.2.1 Method of construction
One must build a field
checking:
N
= N = 0
on the edge of the field
(
N
is the normal with
)
=
O
given on the bottom of fissure
O
where
represent the trace of
on
O
.
Two volumes are given
T
and
S
(deformed cylinders) surrounding the bottom of fissure
O
.
Appear 3.2.1-a: Construction of the field theta in 3D (overall picture)
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One notes
()
R
S
inf
the variable radius of
T
and
()
R
S
sup
that of
S
.
T
S
O
R
R
sup
R
inf
P
Appear 3.2.1-b: Construction of the field theta in 3D (plane of cut)
In any point of
O
, identified by its curvilinear X-coordinate
S
, one can define a normal plan
P
in
which the field
is introduced in the following way:
·
()
()
()
N
O
R S
S
=
for
()
()
0
R S
R
S
inf
·
()
()
N
R S
= 0
for
()
()
R S
R
S
sup
·
N
vary linearly compared to the radius
R S
()
in the crown
()
(
)
()
(
)
S R
S
T R
S
sup
inf
\
·
N
is continuous in
()
(
)
S R
S
sup
.
This manner of introducing
is geometrical. It amounts giving itself two radii
R
inf
S
()
and
R
sup
S
()
, and to carry out calculations of distance from a point running at the bottom of fissure to determine
value of
in this point.
3.2.2 Calculation algorithms
The method requires the data of the field
O
on the bottom of fissure
O
and of the two radii
()
R
S
inf
and
()
R
S
sup
who can depend on the position of the point on
O
. The user introduces these data
node by node on
O
in the following way:
Nodes of
O
O
R
inf
R
sup
NR
1

M
NR
I
O I
R
inf
I
R
sup
I

M
Table 3.2.2-1: Data for construction of the field theta in 3D
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Titrate:
Rate of refund of energy in thermo linear elasticity
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Key
:
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The program is given the responsability to calculate the field
in any point of
according to the following procedure:
·
Calculation of the field theta
O
in each point of
O
: The module
O
being given (by
the user or by the method theta, to see [§2.3]), the problem is to determine the direction of
O
.
must be locally in the tangent with the lips of the fissure and normal plan to the edge with
which it belongs.
being calculated with the nodes, in the general case (bottom of fissure not
plan) direction of
will be realized on the 2 edges of
O
having the joint node.
N
1
N
2
N
2
N
O
T
1
F
1
M
N
1
T
2
F
2
Appear 3.2.2-a: Construction of the field theta in 3D (normals)
Are
F
1
and
F
2
two faces belonging to the lips of the fissure and including/understanding the successive edges
T
1
and
T
2
of
O
. One calculates initially the normal
N
1
with the edge
T
1
in the plan of the face
F
1
then
normal
N
2
with the edge
T
2
in the plan of the face
F
2
.
N
1
and
N
2
being unit normals, one deduces some
N
N
N
=
+
1
2
2
then
()
()
M
M
=
N
for
M
O
.
It is considered that the faces
F
I
are right:
·
If
F
I
is a triangle, the plan of the face
F
I
is defined.
·
If
F
I
is a quadrangle, one cuts out
F
I
in 2 triangles
F
i1
and
F
i2
. One must
then to calculate the equations of the two plans containing the faces
F
i1
and
F
i2
and to make two
calculations of normal per edge
T
I
.
This calculation requires to know the faces belonging to the lips of the fissure and including/understanding one
edge of
O
. In Code_Aster, the user re-enters all the surface elements belonging to
lips of the fissure. These faces appear in one or more groups of meshs and are described in
connectivities of the elements of surfaces. The algorithm sorts these faces to preserve only those
having 2 nodes on
O
. The stages of the algorithm are as follows:
1) For each node of
O
, one extracts the meshs belonging to the lips from the fissure,
2) Of these meshs, one tri those having two nodes on
O
,
3) One recovers the type of the face (
SORTED
or
QUAD
) and one calculates the equation of the plan (S)
tangent (S),
4) For each edge of
O
nodes
I
NR
,
1
+
I
NR
calculation of the normals
N
I, 1
,
N
I
+1,1
,
2
,
I
N
and
2
,
1
+
I
N
.
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
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Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
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Lastly,
is calculated according to the following algorithm:
Loop on the nodes
NR
I
of
O
:
(
)
() ()
N
N
N
N
I
I
I
I
I
I
NR
NR
=
+
=
1
2
1
2
,
,
End of the loop on the nodes
NR
I
of
O
Algorithm 1: Calculation of
N
I, 2
N
i+1,2
N
i+1,1
NR
I
NR
i+1
N
I, 1
Appear 3.2.2-b: Notations of the normals in the bottom of fissure
·
Calculation of the field
in each point of
:
Loop on the nodes
M
Calculation of projection
M
of
M
on
O
(Gives in fact the nodes
M
I
and
M
I
+1
such as
[
]
M
MR. M
I
I
+
,
1
and
[]
S
0 1
,
such as
MR. M
MR. M
I
I
I
S
=
+1
- Calculation
of
D
= D M, M
(
)
- Calculation
of
()
M
by linear interpolation:
()
(
)
()
(
)
M
S
M
S
M
I
I
=
-
+
+
1
1
- Calculation
of
()
R
S
inf
and
()
R
S
sup
by linear interpolation:
()
(
)
R
S
S R
S R
I
I
inf
inf inf
=
-
+
+
1
1
()
(
)
R
S
S R
S R
I
I
sup
sup sup
=
-
+
+
1
1
-
()
()
/
,
sup
If
D
R
S
M
>
=
0
()
()
()
/
,
inf
If
D
R
S
M
M
<
=
()
()
/
,
inf
sup
inf ()
sup ()
inf ()
If
R
S
D
R
S
D
R
R
R
S
S
S
=
-
-
and
()
(
)
()
M
M
=
-
1
F
insi
Algorithm 2: Calculation of the field theta in 3D
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
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O
S
M
D
M
I
M
I
+1
M
Appear 3.2.2-c: Calculation of the field theta in 3D
We detail the calculation of projection below
M
of
M
on
O
:
For each node
M
:
-
Recovery of the co-ordinates of
M
-
Loop on the nodes
M
I
of
O
(
)
I
NO
=
-
1
1
,
Recovery of the co-ordinates of
M
I
and
M
I
+1
Calculation of
S
I
I
I
I
I
I
=
+
+
MR. M
MR. M
MR. M
1
1
.
/
S
I
< 0:
S
I
= 0
/
S
I
> 1:
S
I
= 1
Calculation of the co-ordinates of
M
O
S
I
I
I
I
I
I
:
M
OM
MR. M
=
+
+1
Calculation of
(
)
D
D MR. M
I
I
=
,
Fine loops
- Recovery
of
J
such as
()
D
D
J
I
I
= min
- Knowing
J
one recovers
M
M
S
J
J
J
,
,
+1
and projection
M
of
M
on
O
such as:
MR. M
MR. M
J
J
J
J
S
=
+1
Algorithm 3: Calculation of projections on the bottom of fissure
M
M
I
M
I
+1
M
I
S
I
D
i-1
D
I
S
i-1
Appear 3.2.2-d: Projection of the points on the bottom of fissure
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Titrate:
Rate of refund of energy in thermo linear elasticity
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Key
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3.3
Choice of the field theta in dimension 2
It is about a particular case of dimension 3.
O
limits itself to a point, the user chooses the radii
R
inf
and
R
sup
, the module
in bottom of fissure
O
and the field
is built so that:
0
)
(
=
R
if
sup
R
R
N
0
)
(
=
R
if
inf
R
R
N
0
inf
sup
sup
)
(
R
R
R
R
R
-
-
=
if
sup
inf
R
R
R
R
sup
0
R
inf
0
R
inf
0
N
R
sup
^
Appear 3.3-a: Calculation of the field theta in 2D


3.4 Other
method
The user can enter itself the field
, by using the control
AFFE_CHAM_NO
[U4.44.11] of
Code_Aster which makes it possible to affect
node by node or group of nodes.
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
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Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
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4
Derived from the rate of refund of energy compared to one
variation of field
Initially one points out the problems mechanic-reliability engineer justifying the introduction of this
option then one summarizes, through an example, its procedure in Code_Aster. Afterwards
some preliminaries on the theoretical implications of the derivation installation (supplementing those
[§1.3]) one details the calculation of each integral term of
()
G
F
S
S


=0
where
F
is the field
theta used in the preceding paragraphs. To conclude, one is interested in the establishment of this
functionality in the code and with its perimeter of use.
4.1 Problems
Studies mechanic-reliability engineers require the derivative of the rate of refund of energy compared to
a variation of field. By coupling Code_Aster with software PROBAN, one can thus
to know the probability of priming of the rupture for a distribution of variation of field
data. For example, within the framework of project PROMETE [bib7], one sought to determine
probability of rupture of a tank ITEM by regarding the thickness of its lining as
a random variable.
Until now this type of application required expensive parametric studies to determine,
with each pitch of calculation of PROBAN, sensitivity of the mechanical thermo fields and rate of
restitution of energy to a variation thickness of the coating. From now on, with this option of
Code_Aster, one determines in only one calculation the value of these derivative.
Beyond the aspect performance, that largely simplifies the process of obtaining of derived and
improve their reliability. One thus avoids having to re-mesh and requalify infinitesimal alternatives of
the initial structure. There are not any more states of core to have as for the relevance of the parameter
of variation
of thickness. Indeed, calculation by finished differences (paradoxically, to validate the step
analytical on real cases, one is well obliged y to have recourse!) can depend on the variable with
to differentiate, to be sensitive to the mesh and, in a general way, to the errors of any kind (elements
stop, discontinuity, conditioning, programming…).
The technique of derivation selected is completely analytical (taking into account its architecture
software, Code_Aster cannot be differentiated by automatic tools (such ODYSSEY) for
to solve this type of problem) and rests on the direct derivation of the equations expressed in form
variational. The variation of field is then modelized by a function theta sensitivity
noted
S
, not to confuse with the function theta fissures noted
F
. In practice, although one
be interested that with derivative eulériennes, one handles also derivative Lagrangian because they
intervene naturally in the results of derivation of integral (theorems of transport of
Reynolds). Moreover, one calculates the first using the seconds.
These studies of sensitivity are for the moment accessible only in 2D for plane modelings
or axisymmetric in thermo linear elasticity and with loadings (and materials)
independent of the temperature and the variation of field. But they can spread with
3D, with non-linear elasticity, plasticity…
Thus let us consider a plane structure subjected to a pressure distributed on its edge higher and than
displacements and of the imposed temperatures. For carrying out thermomechanical calculation, it is necessary
to define the field theta sensitivity. In our example, it decrease between the X-coordinates
X
1
and
X
2
of sound
vertical support and it are directed along the X-axis. It gathers all the material points of
configuration which will move virtually according to the transformation:
()
M
M
M
S
S
S
F
+
:
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
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R7.02 booklet: Breaking process
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The aforementioned answers the same properties of regularity as the transformation related to the field theta fissures
that we will note henceforth
()
M
M
M
F
F
F
F
+
:
(cf [§1.3]). The variation thus is materialized
of field on the left edge of the structure.
y
Field
S
sensitivity
Field
F
fissure
X
Fissure
X
2
X
1
F
Appear 4.1-a: Derived from
G
(
F
)
compared to a variation of field controlled by
S
Then, one provides this field thermal theta sensitivity to the operators and mechanics which go
to solve, in addition to their problem direct, of the “pseudo” assistant systems built by
derivation terms with terms of the first [R4.03.01]. The resolution of these systems makes it possible to exhume
the derivative Lagrangian of the temperature and displacement, noted respectively,
&
&
T
and
U
.
By assembling these derivative Lagrangian during the calculation of the rate of refund of energy, one deduces some
then the derivative compared to the variation of field. Well-sure only the parts intervene of
supports of the fields included in the crown of calculation.
On the figure above this crown is centered on the bottom of fissure
F
and it corresponds to one
linear decrease of the module of theta fissures, radially, from the center towards the circumference.
Note:
This technique of derivation is related deployed technique of representation
Lagrangian of variation of field [R7.02.04]. In both cases, one avoids the expensive ones
parametric studies by using a mesh fixes reference and by modelizing the variations
virtual of field by suitable functions theta.
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Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
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E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
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The procedure (cf [§4.4.1]) of a calculation of sensitivity can be thus schematized in the form
following:
T
T
&
,
T
T
&
,
U
U
&
,
Calculation of
(
)
G
F
and of
(
)
G
F
S
S

=0
CALC_G_THETA_T
Calculation
Thermics
THER_LINEAIRE
Calculation
Mechanics
MECA_STATIQUE
Calculation
of

S
CALC_THETA
Calculation of

F
CALC_THETA

S

S

F
Appear 4.1-b: Procedure of the derivation of
G

4.2 Remarks
preliminaries
4.2.1 Theorem of transport
The expression of
()
G
F
established in the code comprises five integral terms in accordance with
definition of [§2.4.3] of the type (or its during into surface):
I
v
S
S
D
=
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Code_Aster
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Titrate:
Rate of refund of energy in thermo linear elasticity
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Key
:
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To derive them compared to their support (in the vicinity of the support of reference), one uses one of
alternatives of the theorem of transport of Reynolds by supposing that all conditions of
regularity are checked: it is necessary that the transformation
F
S
modelizing the movement of the border of
volume moving
S
, and the intégrande (tensor of command 0, 1 or 2)
are all of class C
1
.
One has then, while noting
&
S
the Lagrangian derivative (in the vicinity of the origin) compared to the field
theta sensitivity:
I
div
v
S
S
S
S
S
S
S
D






=
+








=
=
0
0
&
F
This theorem is declined in several versions, according to whether one considers a material or space volume
and that one places of Lagrangian description or eulérienne. However like one derives with
vicinity of the origin
I
S
S
S






=0
all these alternatives are equivalent (“Philosophically” it
result is reassuring because it makes it possible not to privilege neither the matter strain (volume geometrical), nor
appearance of matter (material volume), in the interpretation of this variation of field).
The necessary theoretical regularities are far from being checked in practice, but these flat are
majority of the times “embedded” in the errors due to arithmetic finished, the method of the elements
stop and with numerical integrations. Thus the border moving, as in example Ci
above, often presents “corners” and the transformation
F
F
is not always C
1
(
F
S
is, but
not
F
F
, which has two surfaces of discontinuity on the borders
R
inf
and
R
sup
crown).
Indeed, the field theta fissure is defined in the form of a first order polynomial in
crown and of a constant polynomial outside (it is thus
C
0
) whereas the field theta sensitivity
is a combination of third order students'rag processions which return it
C
2
except in the middle of its support
(where it is right
C
1
). During the calculation of
G
one calls directly upon the derived first of the field
theta fissures, whereas for obtaining his derivative one uses the derivative indirectly second
theta sensitivity (for obtaining the Lagrangian derivative of the tensor of the deformations, cf p.44).
A compromise was thus found between the theoretical command required by derivations and the precision of
finite elements modelizing calculation (One did not have to penalize the calculation of
G
with elements
linear). One uses functions theta of a command of regularity just lower than the theoretical command.
Note:
· During numerical tests one substituted for the functions theta sensitivity and fissure a spline
cubic natural particular (with condition of connection of the derived type first null with
edges) due to R. Wodicka (R. Wodicka. Carryforward off the institutes für Geometry and Praktische
mathematik, RWTH Aachen, 1977), which filled all desired conditions of regularity.
But the aforementioned brings only marginal gains unless refining to the extreme the mesh and
to circumscribe the area of calculation around discontinuities of the fields theta used.
· In practice, for better apprehending the cubic variations of the function theta sensitivity and
to ensure a better convergence of the solution, the user is obliged to lead his calculation
of sensitivity with complete or incomplete quadratic finite elements (SEG3, TRIA6,
QUAD8 and QUAD9). Whatever their command, these elements of the Lagrange type us
guarantee that a regularity
C
0
at the borders. The use of elements of Hermit would have been more
adapted to bring this continuity to the level of the derivative first.
· The derivation of the integral reveals two terms: the first corresponds to derived from
the intégrande calculated as if its parameter setting were distinct from that defining the support of
the integral; the second evaluates the rate of
through the mobile border (term of convection
particulate derivative).
· The result is unchanged when the integral is surface (or linear in a PLANE problem
or AXIS). It is just necessary to replace the voluminal divergence by surface. From a point of view
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Titrate:
Rate of refund of energy in thermo linear elasticity
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Key
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numerical, it is besides to better calculate the latter via derivative contravariantes on
surface:
div
S
=
that to approximate it, by a voluminal divergence and a normal derivative brought back to
surface calculation:
(
)
div
div
S
=
- N N
.
Indeed, this term reveals normal derivations on the surface which do not have a direction for
elements of skin used in Code_Aster. They would have to be approximated while projecting
voluminal calculations on the surface element. To cure it one has recourse to the derivative
contravariantes which makes it possible to express this divergence with the aid only of sizes
surface. One finds these problems in all the surface calculations set up in
the rate of refund of energy and its derivative.
The terms of the rate of refund of energy, except function theta, can pose problem. It is
for this reason that the function theta fissures revêt the shape of a crown of constant value in
its center. That makes it possible to dam up discontinuities of the gradient of field of displacement on
melts of fissure likely to penalize the term conventional elementary.
Thereafter, as long as confusion will not be possible, we will note by a simple point
&
Lagrangian derivative related to the variation of field. One will not be interested any more but in this transformation.
On the other hand one will continue to distinguish the various fields theta. Before approaching calculations us
let us close these rather qualitative remarks by examining the loadings and materials.
4.2.2 Loadings and materials
Let us take again the same remarks as those formulated with [§1.3]. We thus remind the meeting that:
&
, &
,
&
,
&
,
&
,
&
,
&
,
&
F
F
G
G
I
I
S
I
I
S
ij
ij
S
ij
ij
S
S
S
S
ref.
ref.
S
E
E
T
T
=
=
=
=
=
=
=
=
°
°
°
°
Indeed, that is to say
the loading or the material considered, then there exists a field
of
R
3
such as:
=
/
Derivation compared to the parameter
switching over with this restriction, there is the result:
= =
=
0
0
Note:
This assumption is checked only for sufficiently regular fields (for example
belonging to spaces of Sobolev of
). Their definition should not be impacted by
variation of border.
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Titrate:
Rate of refund of energy in thermo linear elasticity
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Key
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On the other hand, for this derivation, the derivative eulérienne of the field of temperature is nonnull, because
the trace of
S
on the edge of the structure moving cannot be neglected any more [bib4].
The derivative Lagrangian of the characteristics materials are null for the problem
discretized taking into account the preceding remark and owing to the fact that one defines them constant by elements
finished. When one allows them to depend on the temperature, their Lagrangian derivative will not be any more
null. One will have, for example, for the Young modulus:
()
(
)
S
T
T
T
E
E
E
T
E
.
.
0
-
=
+
=
=
&
3
2
1
&
Note:
· One can use characteristics variable materials within finite elements or
dependant on the temperature, provided that that is apart from
()
()
supp
supp
F
S
I
.
· For the calculation of derived from
G
, the derivative Lagrangian of the loadings (and even
those their gradients) do not intervene apart from
()
()
supp
supp
F
S
I
.
· For the moment, one does not take into account loadings whose definition is impacted by
variation of field. It is for example the case for a function whose support is defined
according to geometrical characteristics of the moving border, or, for a field of
initial deformations builds starting from displacement resulting from a thermo calculation
mechanics. One could envisage specific options of calculations in the operators
concerned to exhume the derivative eulériennes missing and to instill them into the operator
CALC_G_THETA_T
via a second operand of the key word
SENSITIVITY
.
4.2.3 Form
Setting with share the relation between the Lagrangian derivative and eulérienne:
&
.
=
+
S
éq
4.2.3-1
one uses only the formula giving the Lagrangian derivative of the gradient of a field according to
gradients of the field and its Lagrangian derivative:
&
&
= -
S
éq
4.2.3-2
In Cartesian co-ordinates, one can apply these formulas component by component when
the field is represented by a vector or a matrix. The second term is then a simple product
matrix-vector (in theory, it is about the contracted product of two tensors). For example in the case
of a vector or a tensor one a:
}
}
I J
I J
I K K J
S
ij K
ij K
ij L L K
S
,
,
,
,
,
,
,
,
&
&
·
·
=
-
=
-
éq
4.2.3-3
from where the Lagrangian derivative of the divergence:
}
div
·
=
-
&
,
,
,
I I
I K K I
S
éq
4.2.3-4
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Code_Aster
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Titrate:
Rate of refund of energy in thermo linear elasticity
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E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
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In cylindrical co-ordinates, the things become complicated a little if the field considered is one
tensor of command 1 or 2. It is then necessary to take account of the component orthoradiale gradients. By
example, for a tensor of command 1 the first relation comprises the complementary term
R
R
who
finally does not intervene because it is multiplied by the component orthoradiale field theta sensitivity
who is null. On the other hand, in the second, a complementary term appears due to the derivation of
R
R
precedent:
}
=






=






-
+
+
+
·
·
R R
R Z
R
Z R
Z Z
R R
R Z
R
Z R
Z Z
R R R R
S
R Z Z R
S
R R R Z
S
R Z Z Z
S
R rs
Z R R R
S
Z Z Z R
S
Z
R
R
R
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
&
&
&
&
&
0
0
0
0
0
0
0
0
0
0
0
0
2
R R Z
S
Z Z Z Z
S
,
,
,
+








éq
4.2.3-5
Note:
· In the calculation of derived from
G
, this relation intervenes only for the particulate derivative of
gradient of theta fissures (thus that of its divergence) and for that of the gradient of
displacements (thus for those of the tensors of the strains and the stresses), because all
the other gradients are multiplied by the component orthoradiale theta fissures which is
null.
· In axisymetry, with the help of the complementary terms, the Cartesian formulas can
to apply directly with the formal analogy
()
()
X y
R Z
,
,
. Moreover the element of surface
is multiplied by
R
to take into account the calculation of the integral for a unit of radian.
When one is interested in the Lagrangian derivative of the gradient of a loading such as it is taken in
currently count, derivative second appear. Thus, in the case of a vector
it
comes:
}
{
I J
I J
I L
L J
S
I
I K
ks
J
I L
L J
S
I K J
ks
,
,
,
,
,
,
,
,
,
&
·
=
=
-
=
+






-
=
0
éq
4.2.3-6
The derivative second of the functions of form of the quadratic elements not being available
in the code, they thus should have been set up. Their introduction on the element of reference is
quasi-immediate, but their transcription on the element real 2D is harder (cf [Appendix 1]). On
the elements 1D (for the surface loadings), one has recourse to the differential geometry and to
derived contravariantes for better apprehending the intégrandes on the surface of calculation
(cf [Appendix 2] and [§4.2.1]).
background image
Code_Aster
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Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
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:
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Note:
· One could have freed oneself from this establishment by carrying out integrations by parts (via
theorem of Green), but those having to apply to factors made up of three
terms, that complicated much the formulation (without counting the taking into account of
integrals of border).
· In spite of the low regularity of the elements used, numerical tests showed the maid
quality of these derivative second (for polynomial fields).
· The analytical calculation of these derivative second was set up for the elements
quadratic in modelings plane or axisymmetric related to the mechanical phenomena and
thermics. This calculation concerns only the first family of points of Gauss and, for
reasons of data-processing stability, values of these derivative second (at the points of Gauss)
were stored at the end of object JEVEUX dedicated to the functions of forms.
4.3 Calculations of the various terms of derived from the rate of refund
of energy
One wishes to calculate the various terms of
()
G
F
S
S


=0
. One takes again the nomenclature of [§2.4.3]
by handling only the intégrandes and by detailing each term in co-ordinates first of all
Cartesian (indifferently 2D or 3D) and in small displacements. Thereafter, on a case-by-case basis, one
specify the possible amendments justified by the axisymetry and great displacements.
Let us notice that they utilize all
F
or its gradient: they are thus null apart from the disc of
radius
R
sup
. In the control
CALC_G_THETA_T
it is thus not necessary to specify them
loadings which do not apply in this area.
4.3.1 Derived from the elementary conventional term
According to the theorem of transport of [§4.2.1], the intégrande corresponding to derived from the term
conventional elementary is written:
()
()
(
)
}
}
TCLA
U
U T
TCLA
U
U
U
TCLA
S
F
ij
I p
p J
F
F
S
ij
I p
ij
I p
p J
F
ij
I p
p J
F
F
F
S
S




=
-
+
=
+




+
-
-
+
=
·
·
·
·
0
,
,
,
,
,
,
,
,
&
&
div
div
div
div
div
6
7
44444
4
8
444444
678
It is thus necessary to calculate
}}
&,
,
, &
,
,
ij
I p
p J
F
F
U
·
·
·
and div
678
.
First of all, taking into account the regularity of
F
S
, one can show [bib4], [bib9] that there is a field
Lagrangian (this remark simplifies much calculations and consists in changing theta formally
fissure for each
S
)
F
representing theta fissures such as:
()
()
(
)
F
F
S
F
P
P
P
=
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Rate of refund of energy in thermo linear elasticity
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thus the Lagrangian derivative of theta fissure is null:
()
(
)
()
&
F
F
S
S
F
S
F
S
S
=

=



=
=
=
P
P
0
0
0
[éq 4.2.3-6] led then to:
}
p J
F
p L
F
L J
S
F
K L
F
L K
S
,
,
,
,
,
·
·
= -
= -
and div
678
While applying [éq 4.2.3-3] to the field of displacement it occurs:
}
()
U
U
U
I p
I p
I K
K p
S
,
,
,
,
&
·
=
-
from where the calculation of the tensor of the deformations which will enable us to obtain the derivative Lagrangian
stress field and density of free energy:
()
()
(
)
()
(
)
()
ij PD
I J
J I
I K
K J
S
J K
K I
S
ij GD
ij PD
I K
I p
p K
S
J K
J K
J p
p K
S
I K
U
U
U
U
U
U
U
U
U
U
·
·
·
=
+




-
+
=
+
-
+
-








678
678 678
1
2
1
2
1
2
&
&
&
&
,
,
,
,
,
,
,
,
,
,
,
,
,
,
(small deformations)
(great deformations)
Since one limited oneself to linear elasticity and the preceding remarks on
loadings and of [éq 4.2.3-3]:
}
(
)
(
)
(
)
(
)
µ
ij
ijkl
kl
kl
kl m m
S
ij m m
S
kk
kk
ref.
eq eq
eq
ijD
ijD
ijD
ij
kk kl
T
K
T
T T
·
=
-
-
+
=
-
-
-
+
=
=
-
&
&
&
&
&
&
&
&
&
&
&
,
,
0
0
3
3
4
3
3 6
12
1
3
with
)
(tensor are equivalent
(tensor deviatoric)
In axisymetry, in accordance with the remarks of the paragraph [§4.2.3], one applies the formulas
Cartesian on the first two variables with the formal analogy
()
()
X y
R Z
,
,
, that one
supplements by the “orthoradiaux” terms following:
}
}
,
,
,
,
,
,
,
&
F
RF
S
rs
R
F
RF rs
R
R rs
R
R
U
U
R
R
U
R U
U
R
=
=
=
= -
=
-






·
·
2
1
and
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Titrate:
Rate of refund of energy in thermo linear elasticity
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from where the Lagrangian derivative of the component 3x3 of the tensor of the deformations:
}
}
PD
GD
PD
R
U
U
U
R
·
·
·
·
·
=
=
+
678
678 678
,
,
(small deformations)
(great deformations)

4.3.2 Derived from the thermal term
According to the theorem of transport of [§4.2.1], the intégrande corresponding to derived from the term
thermics is written:
()
}
()
(
)
THER
T T
THER
T T
T
T
T
THER
S
F
K KF
S
K
K
L L K
S
KF
S
S




= -
+
= -
+
-




+
=
·
·
0
,
,
,
,
,
&
6 7
4
8
4
div
div
It thus remains us to calculate the Lagrangian derivative of derived compared to the temperature from
density of free energy:
}
(
)
T
K
T
kk
·
= -
-
3
3
&
&

4.3.3 Derived from the forces terms voluminal and surface
According to the theorem of transport of [§4.2.1], the intégrande corresponding to derived from the term forces
voluminal is written:
()
(
)
(
)
}
TFOR
U F
F
TFOR
U F
F
U F
F
F
TFOR
S
F
I
I K KF
I
F
S
I
I K KF
I
F
I
I K KF
I L ls
F
I
F
S
S




=
+
+
=
+
+
+
+


+
=
·
·
·
0
,
,
,
,
&
div
div
div
div
div
div
6
7
444
4
8
4444
678
The only term which it remains to calculate is the Lagrangian derivative of the gradient of the voluminal forces:
}
F
F
I K
I jk
J
S
,
,
·
=
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Rate of refund of energy in thermo linear elasticity
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Key
:
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Note:
The surface theorem of transport led to replace the voluminal divergence by one
surface divergence. The aforementioned and the Lagrangian derivative of the intégrande reveals
normal derivations on the surface which do not have a direction for the elements of skin used in
Code_Aster. One thus has recourse to the differential geometry and derived the contravariantes for
to better apprehend the intégrandes on the surface of calculation (cf [Appendix 2]).

4.3.4 Derived from the deformations term and initial stresses
According to the theorem of transport of [§4.2.1], the intégrande corresponding to derived from the term
“initial strains and stresses” is written:
()
(
)
}
TINI
T T
TINI
S
F
ij
ij
ij K
ij
ref.
ij
ij
ij K
KF
S
KF
ij
ij
ij K
ij
ij
ij K
S




=
-




+
-
-
-








+
=
-




+
-




+
=
·
·
0
0
0
0
0
0
0
0
0
1
2
1
2
1
2
1
2
,
,
,
,
&
&
&
6
7
44444444444
8
44444444444
div
(
)
}
ij
ij
ij
ij K
KF
ij
ref.
ij
ij
ij K
S
T
T T
TINI
-
-






+
-
-
-




+
·
&
&
,
,
1
2
1
2
0
0
0
0
div
Only the derivative which was not exhumed yet are those of the gradients of the tensors of
deformations and of the initial stresses which, according to [éq 4.2.3-6], are written:
}
}
ij K
ij lk
ls
ij K
ij lk
ls
,
,
,
,
0
0
0
0
·
·
=
=
Note:
Taking into account the various digital processings carried out at the time of the establishment in the source of
the operator, it is not licit to cumulate stress fields and initial deformations,
because this term “strains and stresses initial” is then not cancelled. The user will have
to re-enter either of the initial stresses, or of the initial deformations but not both.
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Code_Aster
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Titrate:
Rate of refund of energy in thermo linear elasticity
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E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
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4.4
Establishment in Code_Aster
4.4.1 Perimeter
of use
The calculation of derived from the rate of refund of energy is obtained in supplement of the value of the rate
of energy with the option `
CALC_DG'
. This option enriched the total card by a fourth noted field
`
DG
'.
----------------------------------------------------------------------
ASTER 6.00.19 CONCEPT G CALCULATES THE 07/12/2000 OF TYPE TABL_CALC_G_TH
NUME_ORDRE INST G DG
1 0.00000E+00 3.622622E-01 1.889340E-03
----------------------------------------------------------------------
Example 1: Trace total card
Its perimeter of application limits to linear elastic calculations thermo 2D resting on
complete or incomplete quadratic finite elements (SEG3, TRIA6, QUAD8 and QUAD9). Options
sensitivity allowing preliminary calculations of the derivative Lagrangian of the temperature and of
field of displacement were installation in the operators
MECA_STATIQUE
and
THER_LINEAIRE
.
Caution:
·
The calculation of sensitivity in thermics is restricts with the linear 2D, stationary case or
transient, with voluminal sources and conditions of imposed temperature, flow
normal imposed and of convectif exchange. Conditions of exchange between wall and of
radiation are not taken yet into account [R4.03.01] [U4.54.01].
·
In mechanics, the calculation of sensitivity is restricted, for the moment, with linear case D_PLAN or
AXIS with conditions limit of imposed displacement type, connections uniform and pressure
external [R4.03.01] [U4.51.01].
This calculation of sensitivity is based on modelings 2D:
D_PLAN
and
AXIS
. They are taken in
count in the entirety of the process of derivation (
THER_LINEAIRE
,
MECA_STATIQUE
and
CALC_G_THETA_T
). On the other hand, configuration
C_PLAN
is taken into account only in postprocessing
calculation of mechanics. In fact, it should appear only after the calculation of sensitivity of
MECA_STATIQUE
who supports only modelings
D_PLAN
and
AXIS
(with this option).
data-processing developments corresponding to this taking into account in a calculation of sensitivity
were not still carried out. In such a configuration, the user is of course an only judge of
relevance of its results.
Taking into account the preceding remarks, it is clear that one is interested only in materials
isotropic rubber bands independent of the temperature. They can be heterogeneous provided that
their characteristics remain constant by finite elements.
One can use the same loadings as for the rate of energy provided that they are
independent of the variation of field in their intrinsic definitions as in those of
their supports. In other words, their derivative eulérienne must be null.
In addition, only loadings of the pressure type distributed (
PRES_REP
) and calculated temperature
(
TEMP_CALCULEE
) are usable in the totality of the process. This software restriction is not due
that with the limited development of the option
SENSITIVITY
in the operator
MECA_STATIQUE
. Like
for modeling
C_PLAN
, other types of loading (
FORCE_INTERN
,
FORCE_CONTOUR
,
EPSI_INIT
,
GRAVITY
and
ROTATION
) are taken into account only in postprocessing of the calculation of
mechanics. They cannot and they should intervene only for the assembly of the terms of
derived from G. They are thus modelized by
AFFE_CHAR_MECA
or
AFFE_CHAR_MECA_F
inserted
enter
MECA_STATIQUE
and
CALC_G_THETA_T
.
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Titrate:
Rate of refund of energy in thermo linear elasticity
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E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
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One can summarize these operations by the following table:

Modeling
D_PLAN
-
AXIS
Elements
SEG3
-
TRIA6
-
QUAD8
-
QUAD9
Materials Rubber band
isotropic
Loading
PRES_REP
TEMP_CALCULE
THERMAL MECHANICS
C.L.
EPD. Imposed - uniform Connections
Chgt. thermics
: TEMP_CALCULEE
C. exchanges between wall: refused
C. of radiation: refused
Calculation
MECA_STAT
THER_LINE




Configuration
usable
only in
postprocessing
(relevance
left with the free one
choice of
the user)
Modeling
C_PLAN
Type of loading
FORCE_INTERN
FORCE_CONTOUR
EPSI_INIT
GRAVITY
ROTATION








Via the key word
ETAT_INIT
one can also take into account a stress field or of
initial displacements in the calculation of the rate of energy. This possibility was extended to calculation of
its derivative with the same restrictions as for the loadings. For the same reasons these
initial fields are taken into account only in postprocessing of the calculation of mechanics.
For more information on the field of validity of the options of calculation and to take as a starting point examples
of use one will be able to refer to the user's manual [U4.82.03] and the case test HPLP100B
[V7.02.100].
CALC_G_THETA_T
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4.4.2 Environment
necessary
As for the calculation of the rate of refund of energy, the field theta fissures
F
must obligatorily
to be created before (either by the control
CALC_THETA
[U4.82.02], that is to say by the control
AFFE_CHAM_NO
[U4.44.11]). For obtaining its derivative, it is necessary in more to have constituted the field
theta sensitivity
S
before thermomechanical calculation (since it is provided in input of these operators
via the operand
SENSITIVITY
).
It is him also a field of vector 2D in each node of the mesh. It is directed along the axis of
X-coordinates. It can be affected directly with the control
AFFE_CHAM_NO
but, in practice,
it results generally from the specific control
CALC_THETA
with the option
THETA_BANDE
who
allows to seize the module (key word
MODULATE
) and X-coordinates
X
1
and
X
2
(key word
R_INF
and
R_SUP
) of
points delimiting its vertical support. It is reminded the meeting that this field decrease value
MODULATE
with
zero value enters the X-coordinates
X
1
and
X
2
, and that it is null everywhere else. These X-coordinates can be
negative but one must have
X
1
<
X
2
. [Figure 4.1-a] an example of this type of field theta illustrates.
Caution:
The field theta sensitivity is thus for the solidified moment, colinéaire with the unit vector of the axis of
X-coordinates (and in the same direction). This preliminary construction of
S
by the operator
CALC_THETA
corresponded to the specifications of project PROMETE [bib7]. But nothing prevents
taking into account of unspecified directions to be able to simulate derivations compared to
tilted variations of fields.

4.4.3 Standardization
In axisymetry, to carry out comparisons, it is necessary to standardize with the hand (it is not
automatically not made) the derivative provided by Code_Aster. As for the rate of refund
of energy (cf [§2.4.4.1]) it is necessary to divide the numerical value obtained by the radius
R
bottom of fissure
(equal to its distance to the axis of symmetry
y
):
()
()
G
G
intrinsic
F
F
S
S
S
S
R


=

=
=
0
0
1
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Titrate:
Rate of refund of energy in thermo linear elasticity
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5 Bibliography
[1]
BUI H.D., fragile Breaking process, Masson, 1977.
[2]
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.
[3]
GRISVARD P., “Problems in extreme cases in the polygons”, Instructions - EDF - Bulletin
management of the Studies and Search, Series C, 1, 1986 pp. 21-59.
[4]
MIALON P., “Calculation of derived from a size compared to a bottom of fissure by
method theta ", EDF - Bulletin of the Management of the Studies and Search, Series C, n°3 1988
pp1-28.
[5]
MIALON P., Study of the rate of refund of energy in a direction marking an angle
with a fissure, intern EDF, HI/4740-07-1984 notes.
[6]
GURTIN Mr. E. Year introduction to continuum mechanics. Mathematics in science and
engineering. Academic Near, 1981.
[7]
VENTURINI V. and Probabilistic Al Study of the tank by a coupling Mechanic-reliability engineer. Assessment
P1-97-04 project, HP-26/99/012/A, Nov. 1999.
[8]
DHATT.G and TOUZOT.G. A presentation of the finite element methods. ED. Maloine,
1984.
[9]
MURAT.F and SIMON.J. On control by a geometrical field. University of Paris VI,
1976.
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Rate of refund of energy in thermo linear elasticity
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Appendix
1
Calculation of the derived seconds of the elements
quadratic 2D
Initially one expresses the derivative second functions of form on the element of
reference, then are used they to determine those of the real element which are the only ones to intervene
indeed in the calculation of the elementary terms. One preserves here the notations of the code for
isoparametric elements [R3.01.01]. We will carry out the exercise only in 2D but it spreads
without sorrow with the 3D.
To calculate the derivative second on linear elements one has recourse to the geometry
differential and with derived the contravariantes (cf [Appendix 2]). They make it possible to better apprehend
intégrandes on the surface of calculation so that they do not reveal normal derivations
who do not have a direction for the elements of skin used in Code_Aster.
A1.1 Derived seconds on the element from reference
A1.1.1 Segment (element of edge)
The derivative second of the three functions of forms are written:
N1
- 1
N3
0
N2
1
()
()
()
2 1
2
2
2
2
2
3
2
1
1
2
NR
NR
NR
=
=
= -
Appear A1.1.1-a: Segment of reference
A1.1.2 Triangle (element of face)
The derivative second of the six functions of forms are written:
N2
N5
N3
N4
N6
N1
Appear A1.1.2-a: Triangle of reference
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Rate of refund of energy in thermo linear elasticity
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I
NR
NR
NR
I
I
I
2
2
2
2
2
1
0
1
0
2
1
1
1
3
1
0
0
4
0
2
1
5
2
0
1
6
0
0
1
-
-
-
-
Appear A1.1.2-b: Derived seconds from the triangle of reference

A1.2 complete or incomplete Quadrangle (element of face)
N8
N4
N5
N9
N1
N7
N2
N6
N3
Appear A1.2-a: Quadrangle of reference
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Rate of refund of energy in thermo linear elasticity
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:
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The derivative second of the eight (resp. nine) functions of forms of the incomplete quadrangle
(resp. complete) are written:
I
NR
NR
NR
I
I
I
2
2
2
2
2
1
1
2
1
2
2
1
4
2
1
2
1
2
2
1
4
3
1
2
1
2
2
1
4
4
1
2
1
2
2
1
4
5
0
1
6
1
0
7
0
1
8
1
0
+
-
- -
-
-
- - +
-
+
- + -
+
+
+ +
-
-
- -
-
- -
-
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
I
NR
NR
NR
I
I
I
2
2
2
2
2
2
2
2
2
1
1
2
1
2
2
1 2
1
4
2
1
2
1
2
2
1 2
1
4
3
1
2
1
2
2
1 2
1
4
4
1
2
1
2
2
1 2
1
4
5
1
1
1 2
6
1
1
1 2
7
1
1
2
1
8
1
1
+
-
+
-
-
-
-
-
-
+
-
+
+
+
+
+
-
-
-
-
-
-
-
-
+
-
+
-
+
-
-
(
)
(
)
(
)
2
1
9
2
1
2
1
4
2
2
+
-
-
Appear A1.2-1: Derived seconds from the complete and incomplete quadrangle of reference
A1.3 Derived seconds on the real element
Problematic A1.3.1
The elementary terms to discretize are written in the real field, even if they are transcribed on
the element of reference via the change of variable using the jacobien. Their intégrandes uses
thus derivative in
X
. However the nodal approximation on the real element being often too intricate (
geometrical function of interpolation
:
X
admits a reciprocal bijection but its construction
is hard as of the QUAD4. One will note
[J]
its matrix jacobienne and
det (J)
its jacobien. Of other
code, such N3S, chose however, for reasons of performance, to work exclusively on
the real element), one prefers his expression to him on the element of reference:
()
()
()
()
()
{}




NR
NR
NR
N
N
N
E
E
1
1
noted
with:
1
,
2
,….,
values of
with
N
E
points of interpolation and
NR
1
(
)
,
NR
2
(
),…, NR
(
)
their functions of form associated on the element with reference.
It is thus necessary to transcribe these derivative compared to
X
in derived compared to
via description
direct of the geometrical interpolation
.
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Rate of refund of energy in thermo linear elasticity
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Key
:
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A1.3.2 two-dimensional Case
By using derivation in channel, one writes the derivative first of all in
=
(
,
)
from those in
X
= (
X, y
):
{}
[]
{}









=














=
X
y
X
y
X
y
J
X
noted
éq A1.3.2-1
By reversing this system (like
) one is bijective can thus deduce the derivative from it in
X
from
those in
:
{}
[]
{}
X
J
=
- 1
éq A1.3.2-2
and by deriving them formally one obtains:
[]
[]
{}
[]
{}
[]
{}

2
2
2
2
2
1
2
2
2
2
2
2
2
1
2
2
X
y
X y
T
T
T
T
X








=








+












=
+
noted
éq A1.3.2-3
In addition while deriving [éq A1.3.2-1] compared to
, while taking account of [éq A1.3.2-2], it comes:
{}
[]
{}
[]
{}
[]
[]
{}
[]
{}
2
1
2
2
1
1
2
2
=
+
=
+
-
C
C
C
J
C
X
X
X
by deferring the expression obtained in [éq A1.3.2-3]:
{}
[] [] []
[]
(
)
{}
[] []
{}
[] []
[] [] []
[]
X
X
T
T C J
T C
T
C
T
T C J
2
1
2
1
1
2
2
2
2
2
1
1
2
1
1
=
+
+
=
= -



-
-
-
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Rate of refund of energy in thermo linear elasticity
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:
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Matrices
[C
1
]
and
[C
2
]
being obtained easily, via
, there are thus a constructive process allowing
to deduce the matrices
[T
1
]
and
[T
2
]
sought:
[]
[]
C
X
y
X y
X
y
X y
X X
y y
y X
X y
C
X
y
X
y
X
y
2
2
2
2
2
1
2
2
2
2
2
2
2
2
2
2
2
2
=
















+


















=









[]
()
[]
()
[] []







=








-








-
-
-
+
+


















= -
-
-
T
J
y
y
y y
X
X
X X
y X
y X
y X
X y
T
J T C
y
y
X
X
2
2
2
2
2
2
1
2
1
1
2
2
1
det
det






Thus, for example, the first derived second in
X
expressed on the element of reference is written:
()
()
()
()
()
()
()
()
()
()
()
2
2
1
1
2
2
2
2
2
2
2
2
11
1 2
11
1 2
1 3
X
T
T
T
T
T
=
+
+
+
+
,
,
,
,
,
For further information, one will be able to refer to the excellent work of G.Dhatt and G.Touzot
[bib8] pp51-57.
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Appendix 2 Calculation of the term forces surface and of its derivative in
2D
A2.1 Introduction
According to the paragraphs [§2.4.3.3] and [§4.3.3] the term forces surface and its derivative compared to
variation of field are written (before projection of the operators on the Cartesian basis):
(
)
()
(
)
(
) (
)
TSUR
TSUR
F
F
S
S
F
S
F
S
F
F
S
F
F
S
S
=
+




=
+
+


+
+
+
=
·
F
F
U
F
F
F
U
F
F
U U
div
D
div
div
div
div
D
S
S
S
S
S
.
.
. &
0
678
They thus reveal clearly derivative normals on the surface of calculation. However in
Code_Aster, one chose to calculate these elementary terms (had with the surface efforts) on
“elements of skin” for which this normal variation does not have a direction. To cure it one has
resort to the differential geometry which makes it possible to express these intégrandes only using
surface sizes.
We will carry out the exercise only in 2d-PLAN but it spreads without sorrow with the 3D. In our
case, the surface of calculation is reduced to a curve (in the plan
(X, y)
of calculation) and the forces are not
more than linear. In addition, according to whether modeling is plane or axisymmetric, it is necessary
to take into account complementary terms, because in the first case it is a question of a calculation per unit
of length, whereas in the second, it is per unit of radian.
We now will introduce a curvilinear parameter setting of the vicinity of the curve of work
S
and
of its associated fundamental reference marks. That is to say
an acceptable parameter setting of
S
. To describe it
volume made up of a vicinity of this curve by using an orthogonal reference mark, one associates two to him
other variables
and
.
y
X
Z
, G
3
G
2
G
1
S
O
M (
)
Appear curvilinear A2.1-a: Parameter setting of the vicinity of the curve of work
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The figure above illustrates the natural base covariante (
G
1
,
G
2,
G
3
) associated the parameters
,
and
.
The vectors of this curvilinear base are written in the total reference mark
(O, X, y, Z)
in the form (in one
not
M (X (
), y (
))
unspecified of
S
)
G
G
G
1
2
3
0
0
0
0
1
=
= -
=
X
y
y
X
From where the metric tensor
G
and its reciprocal tensor
G
- 1
, while noting
J
the jacobien of the transformation:
[] [
]
[] []
2
2
2
2
1
1
2
2
y
X
=
J
1
0
0
0
0
0
0
.
1
0
0
0
0
0
0
.




+








=
=
=




=
=
=
-
-
-
-
with
J
J
G
J
J
G
ij
J
I
ij
ij
ij
J
I
ij
ij
G
G
G
G
G
G
Reciprocal metric tensor one deduces bases it contravariante (
G
1
,
G
2,
G
3
) which proves very useful for
to calculate the derivative covariantes:
G
G
G
G
G
G
G
G
I
ij
J
G
J
J
=
=
=
=
- 1
1
2
1
2
2
2
3
3
1
1
,
,
Note:
That modeling is plane or axisymmetric, these tensors remain diagonal since them
selected bases are orthogonal. On the other hand the value of the elementary element of integration
differ
D
J D
R J D
=
in 2D - PLAN
in 2D - AXI
to take account of integration per unit of radian in axisymetry. Taking into account the analogy
formal
()
()
X y
R Z
,
,
, the jacobien of the transformation is written:
J =
R




+
2
2
Z

A2.2 Terme forces surface
Let us break up this term of into two intégrandes:
(
)
(
)
(
)
TSUR
TSUR
TSUR D
TSUR
TSUR
S
F
F
=
+
=
=
1
2
1
2
with
and
F
U
F
U
.
.
div
S
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Rate of refund of energy in thermo linear elasticity
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A2.2.1 Calculation of
TSUR
1
By expressing the gradient by derivative covariantes and by breaking up the vector theta fissures and it
vector displacement on the basis covariante, one obtains (after some tensorial operations of
base)
(
)
(
)
{}
I
fj
J
I
it
jk
L
fk
J
I
L
L
K
fk
J
I
J
I
F
U
F
J
G
U
F
U
F
TSUR
j=
I, K, L
2
1
1
2
,
1
.
.
=
=
=
=
with
G
G
G
G
U
F
It remains to determine
I
fj
F
J
I
U
F
and
,
via the base contravariante to obtain:
(
)
()
()
TSUR
J
J
X
y
U
F
U
F
F
xf
yf
X
X
y
y
1
2
1
1
1
2
2
2
=




+






=
+




+




-


.
.
.
.
.
G
F G U G
F G U G
Note:
In axisymetry, taking into account the nullity of the component orthoradiale of the field theta fissures, it
does not have there complementary term.
A2.2.2 Calculation of TSUR
2
By expressing the surface divergence as the trace of the surface gradient
(
)
(
)
with
1
,
tr
tr
div
=
=
=
J
I
J
I
J
fi
F
S
F
S
G
G
and by breaking up the vector theta fissures and the vector displacement on the basis covariante, one
obtains (after some basic tensorial operations (to take the trace of a tensor of the second command
amounts carrying out its contraction)) :
(
)
(
)
(
)
(
)
{}
K
J
fj
K
kl
ij
L
J
fi
K
L
L
J
I
J
fi
K
K
F
S
U
F
J
G
U
F
U
F
TSUR
j=
I
K, L
2
2
1
,
2
,
1
.
tr
.
tr
=
=
=
=
with
G
G
G
G
U
F
It remains to determine
J
fj
K
F
,
and
K
U
with the base contravariante to obtain:
() () () ()
{
}
(
)
TSUR
J
J
X
y
F U
F U
F
xf
yf
X X
y y
2
2
1
1
1
2
2
2
=




+
=
+


+
-
.
.
.
.
.
G
F G
U G
F G
U G
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Note:
In axisymetry it is necessary to take account of the nonnull component orthoradiale gradient at the time of
calculation of the surface divergence:
F
R
F
Z
F
R
F
R
Fr
F
S
R
Z
R
J
div
1
2
+




+
=
+
=
-
A2.3 Dérivée from the term forces surface
Let us break up this term of into five intégrandes:
()
(
)
(
)
(
)
(
)
(
)
(
)
(
)
TSUR
DTSUR
DTSUR
DTSUR
DTSUR
DTSUR
S
F
S
S
F
S
F
F
S
F
S
S
F
F
F
F
S
S
DTSUR
DTSUR
DTSUR
DTSUR
DTSUR




=
+
+
+
+
=
+
+
=
·
=
=
=
=
=
0
1
2
3
4
5
1
2
3
4
5
D
div
div
tr
div
div
div
S
S
S
S
S
with
F
U
F
U
F U
F U
F
F
U
F
F
U
.
.
.
.
. &
.
678
A2.3.1 Calculation of DTSUR
1
By expressing the double gradient by derivative covariantes and by breaking up the vector theta
fissure, the vector theta sensitivity and the vector displacement on the basis covariante, one obtains (afterwards
some basic tensorial operations):
(
)
(
)
(
)
(
)
{}
I
fj
sk
jk
I
in
jm
kl
N
Fm
SSL
jk
I
N
N
m
m
F
L
L
S
K
J
I
jk
I
F
S
U
F
J
G
U
F
U
F
DTSUR
k=
J
N
m
I, L
2
1
1
,
2
,
1
,
,
.
.
=
=
=
=
with
G
G
G
G
G
G
U
F
It remains to determine
I
sk
S
fj
jk
I
U
F
and
,
,
with the base contravariante to deduce some:
(
) (
)
()
()
TSUR
J
J
X
y
X
y
U
F
U
F
F
S
xf
yf
xs
ys
X
X
y
y
1
2
1
1
2
2
1
1
2
2
2
2
4
2
2
2
2
=






+









=
+




+




+


-


.
.
.
.
.
.
G
G
F G U G
F G U G
Note:
In axisymetry, taking into account the nullity of the component orthoradiale of the field theta
sensitivity, it does not have there a complementary term.
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A2.3.2 Calculation of DTSUR
2
It is obtained immediately by taking again the result of the calculation of TSUR
1
(after having replaced the field
theta fissures by the field theta sensitivity) and by multiplying it by the expression of the divergence
surface of the field theta fissures TSUR
2
.




+




+




+
=
-
y
X
F
U
F
U
y
X
J
DTSUR
F
y
F
X
y
y
X
X
S
y
S
X
4
2
Note:
In axisymetry it is necessary to take account of the nonnull component orthoradiale gradient at the time of
calculation of the surface divergence
div
J
R
Z
R
S F
RF
zf
RF
=
+






+
- 2
A2.3.3 Calculation of DTSUR
3
By expressing the surface divergence as the trace of the surface gradient and by breaking up it
vector forces linear and the vector displacement on the basis covariante, one obtains (after some
basic tensorial operations (to take the trace of a tensor of the second command amounts carrying out its
contraction):
(
)
(
)
(
) (
)
(
)
(
)
{}
K
I
sj
J
fi
K
kl
in
jm
L
N
Sm
J
fi
K
L
L
N
m
J
I
N
Sm
J
fi
K
K
S
S
F
S
U
F
J
G
U
F
U
F
TSUR
n=
m
J
I
K, L
2
3
1
,
,
,
2
,
1
.
tr
.
tr
=
=
=
=
with
G
G
G
G
G
G
U
F
It remains to determine
K
I
sj
J
fi
K
U
F
and
,
,
with the base contravariante
() () () ()
{
}
(
)
y
y
X
X
S
y
S
X
F
y
F
X
S
F
U
F
U
F
y
X
y
X
J
J
TSUR
+




+




+
=
+








=
-
4
2
2
1
1
1
1
2
3
.
.
.
.
.
.
G
U
G
F
G
U
G
F
G
G
Note:
In axisymetry it is necessary to take account of the nonnull component orthoradiale derivative
Lagrangian of the surface gradients (cf [§4.2.2])
div
J
R
Z
R
Z
R
S
F
R
F
Z
F
R
S
Z
S
R
F
S
F
·
-
=
+




+




-
678
4
2
background image
Code_Aster
®
Version
7.4
Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
E. CRYSTAL, O. BOITEAU, E. SCREWS
Key
:
R7.02.01-D
Page
:
60/60
Manual of Reference
R7.02 booklet: Breaking process
HT-66/05/002/A
A2.3.4 Calculation of DTSUR
4
It is enough to summon the expressions of TSUR1 and TSUR2 by replacing the components of the field of
displacement
U uu
X
y
by those of its Lagrangian derivative
& &&
U uu
X
y
.

A2.3.5 Calculation of DTSUR
5
It is enough to multiply sum TSUR1+TSUR2 by the expression of the surface divergence of the field
theta sensitivity
div
J
X
y
S
S
X
S
y
S
=
+






- 2
Note:
In axisymetry it is necessary to take account of the component orthoradiale not no one of the gradient at the time of
calculation of the surface divergence
div
J
R
Z
R
S
S
R
S
Z
S
R
S
=
+




+
- 2