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
Organization (S): EDF-R & D/AMA, SINETICS, UTO/LOCATED
Handbook of Référence
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 starting 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
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 Lagrangienne 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 traditional 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 constraints ............................................................. 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 commands ...................................................................................................... 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 traditional 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 ®
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Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
26/05/05
Author (S):
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:
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4.3.4 Derived from the deformations term and initial constraints ........................................................ 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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Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
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:
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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 of space R2 or R3. 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 Sd part of.
F
S
G
Sd
Appear elastic 1.1-a: fissured Solide
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 constraints,
·
° the tensor of the initial constraints,
·
(, °, °, T) density of free energy,
·
the tensor of elasticity.
is connected to the field of displacement U by:
1
(U) =
(U +u
I, J
J I,)
2
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Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
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Author (S):
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:
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The density of free energy (, °, °, T) are identified by a dilation and tensile test in
small deformations. (, °, °, T) is a convex and derivable function.
(
1
1
, °, °, T)
(HT °) (HT °) (HT
=
-
-
-
-
+
-
- °) °+ °°
2
2
The law of behavior of an elastic material is written in the form:
=
(, °, °, T) = (- HT - °) + °
with HT =
ij
(T - Tréf) ij
1
The constant term ° ° has a null contribution on the calculation of the rate of refund of energy, but
2
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: ° = °.
1
(, °, °, T) = (- HT) (- HT)
One finds:
2
= (- HT)
If the initial strains ° and the initial stresses are null, density of energy
free is written:
(
1
2
9
2
, T) =
() + µ - 3
K (T - T) +
K 2 (T - T
II
ij
ij
ref.
kk
ref.)
2
2
The relation of behavior is written:
(
)
ij
= kk ij + 2µ ij - 3K T - Tréf ij
and µ is the coefficients of LAME.
is the thermal dilation coefficient.
Tréf is the temperature of reference.
K, module of compressibility voluminal, is connected to the coefficients of LAME by
:
3K = 3 + 2µ.
The relation of behavior starting from YOUNG E and the Poisson's ratio modulus is:
()
E (
)
ij
=
E
tr
T - T
1 +
ij +
1 - 2
ij - 1 - 2
ref.
ij
with:
=
E
1
(+) 1
(- 2)
µ
E
=
2 1
(+)
E
3K = 1 - 2
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Titrate:
Rate of refund of energy in thermo linear elasticity
Date:
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:
R7.02.01-D Page
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1.2
Potential energy and relations of balance
One defines spaces of the fields kinematically acceptable V and Vo.
V =
v
{acceptable, v = U on S}
D
V
{
}
O
=
v acceptable, v = 0 on Sd
With the assumptions of [§1.1] (and for ° = ° = 0), relations of balance in weak formulation
are:
U
V
ij VI, J D
=
F
I VI D + G
I VI D, vVo
S
They are obtained by minimizing the total potential energy of the system:
W v
() =
((v), T) D - F
I VI D - G
I VI D
S
Indeed, if this functional calculus is minimal for the field of displacement U, then:
W =
D - F v
v
ij
I I D G
I I D
ij
S
=
1
(
)
ij
v
D - F
v
v
2
I, J + vj, I
I
I D -
gi I D
S
=
ij VI, J D F
I VI D - G
I VI D = 0
S
We thus find the equilibrium equations and the relation of behavior while having posed:
ij =.
ij
1.3
Lagrangienne 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:
W
G = -
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
F M
:
M + (M) of the area of reference in a modelling field of the propagations
fissure, which at a material point P make correspond a space point Mr. Ces transformations
must modify that the position of the bottom of fissure O. The fields must thus be tangent with
, i.e. by 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 out of P and in (so that the derivative
partial seconds commutate) and to carry out a diffeomorphism for each value of the parameter
(that ensures the reversibility of the process).
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That is to say m the unit normal with O located in the tangent plan at (i.e. tangent at the plan of
fissure) and re-entering in.
O
m
Plan of
fissure
Appear 1.3-a: Fond 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 m
G (),
=
O
where
(())
G
() is defined by the opposite of derived from the potential energy W U in balance by report/ratio
with the initial evolution of the bottom of fissure:
D W (U ()
G () = - &
W = -
D
=0
The quantity m represents 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: Différentes geometries of propagations
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Rate of refund of energy in thermo linear elasticity
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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) = (
,
, F (P)) 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
- 1
P = (F) (M). One transposes the field of Lagrangian representation, then it is derived
compared to before reconverting it of representation eulérienne.
· It is pointed out that this Lagrangian derivative is related to derived the eulérienne by the relation
& =
+
.
Notice [bib4]:
The derivative eulérienne does not depend that 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]):
·
·
6 7
4 8
4
G ()
F U
(F U)
}
-
=
-
D -
,
G U + G U
-
N D
I
I
I
I
K K
I
I
I
I
K, K
K
+
-
nk
S
However
&
(, °, °, T) =
& +
& ° +
&
° +
T &
ij
ij
ij
°
°
T
ij
ij
ij
T, F, G, °, ° being supposed independent of, i.e. being the restriction on (or) of
fields defined on R3, there are the following relations:
&T =
,
T K K
&f = F
I
I, K K
&g
= G
I
I, K K
&°
°
ij = ij, K K
&°
°
ij = ij, K K
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Indeed, if one considers loadings and materials which are the restriction on
geometry (or part of its border) of fields defined on entire R3:
/=
Derivation compared to the parameter commutating with this restriction, one has 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
operator DEFI_MATERIAU. Their gradient on each element is also null by construction (they
are discretized P0 i.e. constant by finite elements), it results from this that the derivative
Lagrangian is null:
& =
+. for {E, Tref}
{123
= 0
= 0
Caution:
With characteristics variable materials within finite elements of the crown theta of
calculation, this simplification is not licit any more.
Like (,
°
1
1
,
° T) = (
HT
- - °) (
HT
- - °) + (
HT
- - °) ° +
° °
2
2
has
one
=
- - ° + ° =
ijkl (
HT
kl
kl
kl)
ij
ij
ij
= - - - ° 1
- °
1
= ° -
°
ijkl (
HT
kl
kl
kl)
ij
ij
ij
2
2
ij
=
°
(
HT
- - °
1
+ ° = -
1
- °
ij
ij
ij)
HT
ij
ij
ij
ij
2
2
ij
.
1 °
°
1 ° °
HT
of or
& = & + -
+ - -
+
T
ij ij
ij
ij
ij, K
K
ij
ij
ij
ij, K
K
, K
K
2
2
T
}
·
In addition, according to proposal 2 of [bib4]:
= & -
I, J
I, J
I, p
p, J
1
1
&
=
(&u, + &u,) - (U, +u,
ij
I J
J I
I p
p J
J p
p, I)
2
2
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And one can eliminate &u from the expression of G
() by noticing that &u is kinematically acceptable
and by using the equilibrium equation:
U & D =
F U
,
& D + G U & D + N U
ij
I J
I
I
I
I
ij
J
I, K
K D
S
Sd
from where:
- G () =
&
- fi &u - &f U +
I
I
I
(- F U
I
I)
D -
K, K
(gi &u +
I
&g U
I
I) D
S
- G U
-
N
D
I
I
K, K N
K
K
S
1
=
ij &u -
I,
F
J
I &
U D - G
I
I &
U D -
&f U D -
I
I I
ij (U +
I, p
p,
U
J
J,
p
p, I) D
2
S
+
&T +
(-
I)
- &
+
-
F U
D
G U
G U
N
D
T
I
K, K
I
I
I
I
K, K
K
N
S
K
1
HT
1
+
°
°
°
°
ij - ij ij, K K
ij ij
ij ij, K K D
2
+
-
-
2
and finally:
G () =
U
-
-
T
ij
I, p
p, J
K, K
, K
K D
T
1
1
+ -
° °
HT
°
°
ij
ij
2
-
-
-
ij, K
K
ij
ij
ij
2
ij, K K D
+ F U
+
F
I
I
K, K
I, K
K ui D
+ G U + G U
-
N
I, K K I I I
D
K, K
K
nk
S
- ij nj Ui, K K D
Sd
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Titrate:
Rate of refund of energy in thermo linear elasticity
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Note:
· In deformations of Euler-Lagrange the first intégrande becomes H U
with
I, p
ij
I, p
p, J
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. Moreover
R
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 [Annexe 2]).
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Rate of refund of energy in thermo linear elasticity
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2
Discretization of the rate of refund of energy
2.1
Method theta in dimension 2
It is pointed out that the rate of refund of energy G is solution of the variational equation:
G (S) (S)
(
m S) ds
G (),
=
O
where:
·
m is the unit normal at the bottom of fissure O located in the tangent plan at and re-entering
in,
·
{
= µ such as
µ N =
0 on
}
.
In dimension 2, the bottom of fissure O is brought back to a M0 point, and one can choose a field
unit in the vicinity of this point, so that: G (M0) = G ()
m
O
Appear 2.1-a: Fond 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 J NR
0
S
O
Appear 2.2-a: Discrétisation of the bottom of fissure in 3D (curvilinear X-coordinate)
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That is to say Gj components of G S
() in this base:
NR
G S
() = G ()
J pj S
j=1
In the same way, fields I (pertaining to) can be discretized on a basis which we will note
Q
(
)
()
I
K S
()
. Let us indicate by I the trace of field I on the bottom of fissure
=
S
() and
1k M
O: I S
by I
()
K components of I S in this base:
M
I (S) = ik qk S ()
k=1
G S
() being solution of the variational equation G (S) (S) m (S) ds G (),
=
, Gj
O
check:
NR
M
G p (S)
I
I
J
(Q (S)
K
K
) m (S)
ds = G (), I [, 1 P
J
]
j=1
K =1
O
that is to say:
NR
M
I
p (S) Q (S) m (S)
I
K
J
K
ds G J
G (),
I [1, P]
=
j=1 k=1
O
Gj can thus be given by solving the linear system with P equations and NR
unknown factors:
NR
G has
= B
, I = 1, P
ij
J
I
j=1
M
with A
I
=
p (S) Q (S) m (S) ds
ij
K J
K
K =1
O
B
I
I
= G ()
This system has a solution if one chooses P independent fields I such as: P NR and if MR. 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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Rate of refund of energy in thermo linear elasticity
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:
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2.3 Choice
in
Aster of the discretization of G in dimension 3
In dimension 2, there is no problem bus by choosing a unit field 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 X-coordinate curvilinear of O
O
Appear 2.3-a: Discrétisation of the bottom of fissure in 3D (normal)
One obtains in this case a total rate of refund G corresponding to a uniform progression of
the fissure such as:
Gl = G (S) ds = G ()
O
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) m (S) ds G (),
=
O
In this case, the user does not give a field theta, fields I necessary to calculation
G S
() are calculated automatically (cf 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 Degmax).
·
Functions of form of the node K of O: K S
() (1 K NNO = a number of nodes of
O) (of degree 1 for the linear elements and of degree 2 for the quadratic elements).
Handbook of Référence
R7.02 booklet: Breaking 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
: 15/60
Let us recall that the polynomials of LEGENDRE constitute a not normalized orthogonal family. They
are obtained by the relation of recurrence:
N
(+1) P () (
)
n+1 T - 2n + 1 T Pn T
() + N Pn-1 T () = 0
In particular:
P0 T () = 1
P1 (T) = T
P
(
)
2 T
() = 3 t2 - 1/2
P
(
)
3 T
() = 5 T3 - 3t/2
In Code_Aster, one normalizes them in the form:
()
2
S
J S
=
2 J + 1 P
- 1
L
J L
where:
·
S is the curvilinear X-coordinate of O,
·
L the length of the bottom of fissure O.
()
2 (S)
1 S
O (S)
0
S = 0
S = L O
L
0
L
Appear 2.3-b: Polynômes de Legendre
In Code_Aster, one limits oneself to Degmax = 7 like maximum degree.
Functions of forms K S
() are associated the discretization of O.
3 (S)
()
1 S
2 S
()
K = 1
K = 2
K = 3
Appear 2.3-c: Functions of form of the bottom of fissure (linear elements)
Handbook of Référence
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Code_Aster ®
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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 Page
: 16/60
Let us recall that one is brought to discretize G S
() and fields I (S) (trace of field I on the bottom
of fissure O).
NR
G (S) = Gj pj S ()
J =1
M
I (S) = I ()
K qk S
k=1
There is thus several possible choices of discretizations, summarized in the table below:
Polynomials of LEGENDRE
Functions of form
G S
()
NDEG
NNO
G ()
()
J J S
Gj J S
j= 0
j=1
I (S)
NDEG
NNO
I
()
I
K K S
()
K
K S
k= 0
K =1
Table 2.3-1: Choice of the discretization
NNO:
a number of nodes of the bottom of fissure O
NDEG: maximum degree of the polynomials of LEGENDRE chosen by the user
(NDEG deg = 7
max
)
In command CALC_G_LOCAL_T (cf [U4.82.04]) key words LISSAGE_THETA and LISSAGE_G
allow to choose the discretization of I and G.
The options available in Aster are summarized in the following table:
I (S)
Polynomials of LEGENDRE
Functions of form
Polynomials of LISSAGE_THETA: “LEGENDRE”
LISSAGE_THETA: “LAGRANGE”
LEGENDRE
LISSAGE_G: “LEGENDRE”
LISSAGE_G: “LEGENDRE”
G S
()
(1st case)
(2nd case)
Functions of
Nonavailable
LISSAGE_THETA: “LAGRANGE”
form
LISSAGE_G: “LAGRANGE”
or “LAGRANGE_NO_NO”
(3rd case)
Table 2.3-2: Options of discretization of Code_Aster
Handbook of Référence
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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
First case:
G S
() and fields I (S) are broken up according to the polynomials of LEGENDRE.
NDEG
G (S) =
Gj J S ()
J =0
NDEG
I (S) =
I ()
K K S
K = 0
The component NDEG Gj are given by solving the linear system with P equations:
NDEG
G has
= B
, I = 1, P
ij
J
I
j=0
NDEG
has
=
(S) (S) I m (S) ds
ij
J
K
K
with
K =0
O
B
I
I = G ()
One makes the choice in Code_Aster take, like fields I, the NDEG fields I such as:
I (S) m (S) = (S)
I
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 unknown NDEG:
NDEG
G has
I
ij
J
=
(
G), I = 1, NDEG
j=
0
with A
=
(S) (S) ds
ij
J
I
=
I
J
O
because the polynomials of Legendre form a base orthonormée on 0.
NDEG
Thus G
J
J
J = G () and thus G (S) =
G () (S)
J
.
j=0
Handbook of Référence
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Code_Aster ®
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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 Page
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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.
NDEG
G (S) =
G (S)
J
J
j=0
NNO
I
(S)
I
=
(S)
K
K
K =1
One makes the choice in Code_Aster take, like fields I, the NNO fields I such as:
I (S) m (S) = (S)
I
where ()
I S is related to form of node I of the bottom of fissure.
That is to say:
NNO
I
()
K K S
() m S = I S ()
k=1
and there are NNO equations with unknown NDEG:
NDEG
G has
= B
, I = 1, NNO
ij
J
I
j=0
has
=
(S) (S) dS
ij
J
I
with
O
B
I
I
=
G ()
In this case, one must have NDEG NNO, that is to say NDEG min (7, NNO) where NNO 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.
NNO
G (S) = G (S)
J
J
j=1
NNO
I
I
(S) = (S)
K
K
K =1
Handbook of Référence
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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
The system to be solved is as follows:
NNO
G (
G has =
ij
J
)
(I =,
1 NNO)
I
j=0
with A =
(S) (S) ds
ij
I
J
0
NNO: a number of nodes of the bottom of fissure
I: function of form of node I
If there are linear elements:
(X) 1
= (1 - X)
1
1
1
2
2
Element of reference
(X) 1
= (1+ X)
0
2
2
- 1 0 1
has
= has
=,
0 if J 2
I (I J)
I (i+ J)
has
=
S S ds = if S S ds
I (I)
1
I () i-1 ()
I ()
- 1 ()
(
S
I
O
I - 1
S - S
1
S - S
I
I
I
I
1
1
1) +
=
X X dx
1 X dx
S
S
1 () 2 ()
(
- 1)
+
=
1
- 2
=
-
- 1
(
)
(I I)
-
-
1
1
(2
2
4
6
S - S
1
S
S
I
I 1
2
-
-) +
has
=
I
I
X dx
X dx
II
2 ()
(+1
) +
+
1
2
1 ()
- 1
- 1
(2
2
S - S
1
S
S
I
I
1
I
I
1
1
1
2
-
-) +
=
(1+ X)
(+1
) +
dx +
1
(1 - X) 2dx = ([S - S + S - S
i+1
I)
(I I)]
- 1
-
-
1
1
2
4
2
4
3
I - 1 I
I + 1
Appear 2.3-d: Linear functions of form
The Aij matrix is thus written:
(
2
2
S - 1
S) (2
S - 1
S)
0
0
L
(2
S - 1
S)
(2 3s - 1s) (3s - 2s)
0
L
1
0
(
3
S - 2
S)
(2 4s - 2s) (4s - 3s)
6
L
0
0
(
4
S - 3
S)
(2 5s - 3s) L
M
M
M
M
Handbook of Référence
R7.02 booklet: Breaking 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
: 20/60
If there are quadratic elements:
(X) 1
= X (X -)
1
1
2
(X) 1
= X (X +)
2
1
2
(X) = (1 - X) (1+ X)
3
Appear 2.3-e: Quadratic functions of form (Element of reference)
It is necessary to distinguish node node and node medium
·
I = node node:
has
= has
=,
0 if J 3
I (I J)
I (i+ J)
has
=
S
S ds = if S
S ds
I (i-2)
I () i-2 ()
I ()
- 2 ()
O
(
S
I
I - 2
S - S
1
S - S
I
I
1
2) +
=
I
I
X X dx
X X
1 dx
1 () 2 ()
(
- 2)
+
=
1
2
2 -
- 1
(
)
- 1
(2
2
4
S - S
I
i-2)
= -
30
S
I
S -
has
=
S S ds =
S S ds =
S
1
I
I 2
X X dx
I (I)
1
I () i-1 ()
I ()
- 1 ()
(
-) +
2 ()
()
S
I
O
I -
- 1
3
(
2
2
S - S
1
S
S
I
I
1
2
2
-
-) +
=
X (X +)
1 (1 - X)
(I i-2)
dx = +
- 1
2
2
15
S - S
+1
S
S
I
I
I
I
2
2
2
-
-
has
=
X dx
X dx
S
S
II
2 ()
(+2
) +
+
1
2
1 ()
=
(-
i+2
I
)
- 1
-
-
1
2
2
2
15
i-2 i-1 I i+1 i+2
Appear 2.3-f: Node node
Handbook of Référence
R7.02 booklet: Breaking process
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Code_Aster ®
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 Page
: 21/60
·
I = node medium:
has
= has
= 0 if J 2
(II J)
I (i+ J)
has
=
si+1 S S ds
I (I)
1
I ()
- 1 ()
S
I
I - 1
+1 (S
- S
S
- S
i+
I
I
I
1
1
- 1)
=
X X dx
X X 1 1 X 1 X dx
3 () 1 ()
(+1
) +
=
1
(-) (-) (+)
- 1
- 1
(
2
2
2
S
- S
i+1
i-1)
=
15
has
=
si+
S
S
1
2
S ds
X
X dx
S
S
II
()
(-
i+
I
8
1
) +
=
1 (1 -) 2 (1+) 2 = (-
i+1
I
)
S I
I - 1
-
-
1
1
2
15
i-2 i-1
I i+1
i+2
Appear 2.3-g: Node medium
The Aij matrix is written:
(
4
3
S - 1
S)
(2 3s - 1s) - (3s - 1s)
0
0
L
(
2 3
S - 1
S) 1 (
6 3
S - 1
S)
(2 3s - 1s)
0
0
L
1 - (
3
S - 1
S)
(2 3s - 1s) (4 5s - 1s) (2 5s - 3s) - (5s - 3s) 0
30
0
0
(2
5
S - 3
S) 1 (
6 5
S - 3
S)
(2 5s - 3s) 0
0
0
-
(5s - 3s) (2 5s - 3s) (4 7s - 3s)
L
0
0
M
M
M
Particular case: S
S = cste = L
i+ -
2
I
= length of an element
+
+
L
4
2
- 1 0
0
L
node node of edge
2
16
2
0
0
L
node medium
L - 1 2
8
2
- 1
node node
L
30 0
0
2
16
2
L
M
0 0 - 1 2 8
L
M
M
M
Handbook of Référence
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Code_Aster ®
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 Page
: 22/60
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 (
)
3
4
5
…
2
1
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:
L
6
·
node node of edge:
(4+ 2 -) 1G = bsoitG = B
30
L
L
3
·
node node:
(- 1+ 2 + 2 + 8 -) 1G = bsoitG = B
30
L
L
3
·
node medium:
(2+16+ 2) G = bsoitG = B
30
L
2
L
6b
3b
3b
L
L
2l
What gives if the elements do not have constant lengths:
6
6
·
node node of edge: = (
or
=
S - S)
1
NR
3
1
(S - S
NR
NR - 2)
6
6
·
node node: for example 3 = (
=
3
S - 1
S) + (5
S - 3
S) (5
S - 1
S)
6
that is to say: I = (
I
S
-
+2
I
S - 2)
3
or: '= (
S
- S
+1
)
I
I
I
I
3
·
node medium: I = (
2 is -
+1
I
S)
To activate this method it is necessary to specify in CALC_G_LOCAL_T:
LISSAGE_G: “LAGRANGE_NO_NO”
LISSAGE_THETA: “LAGRANGE”
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A
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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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 command 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 Sd part of (Non developed to date),
·
T, field of temperature (TEMP_CALCULEE),
·
, initial field of defomation (EPSI_INIT).
These loadings can depend on time and space.
The characteristics of the material (E, and) can depend on the temperature T and on
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 command
CALC_THETA [U4.82.02], is by command 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 constraints 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
grid 3D, and starting from two lists of meshs, the upper lip and the lower lip of this
fissure command 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
().
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A
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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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 Rsup S
() defined in chapter 3 [Figure 3.3-a].
Let us notice that as all the terms utilize or its gradient, the elementary terms
are null apart from this disc of radius Rsup S
(). In commands CALC_G_THETA_T and
CALC_G_LOCAL_T, it is thus not necessary to specify the loadings which do not apply
in this zone.
2.4.3.1 Elementary traditional term
TCLA = U
- ((U), T
ij
I p
p J
)
,
,
K, K
The density of energy elastic ((U), T) is written in thermo linear elasticity:
·
in 3D and AXIS:
(
1
(U), T) =
(2
II) + µ -
ij
ij
HT
2
·
in DP:
1 - E
(
E
E
(U), T)
(
)
=
(2 +2
2
xx
yy)
(
+
+
-
2 1 +) (1 -
2)
(1+) (1 -
2) xx yy
(1+) xy
HT
·
in CP:
E
(
E
E
(U), T) =
(
2 + 2 +
+
2 -
2 1 - 2) (xx
yy)
(1-2) xx yy (1+) xy HT
9
with
= 3
K (T - T) -
2
2
HT
ref.
II
K
(T - rTéf)
2
where:
E
E
E
3K =
;
=
1 - 2
1
(+) 1
(- 2); 2µ = 1+
E: YOUNG modulus
: Poisson's ratio
, µ: coefficients of LAME
: thermal dilation
The density of energy elastic ((U), T) can be written in a general way in the form:
1
2
2µ
((U), T) = K (-
3 (T - T
2
kk
ref.)
+
eq
2
3
3
1
2
with
= D D and D = -
eq
ij ij
ij
ij
kk ij
2
3
Handbook of Référence
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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
: 25/60
2
1
that is to say eq = (
2
3ijij - kk)
2
and ((
µ
U), T) 1
2
= K
- 3
kk
K (T - ref.
T
)
9
2
kk + K (T - ref.
T
) 2
2
+ µ -
2
2
ij ij
3 kk
2
=
+ µ - 3
kk
ij ij
K (T - ref.
T
)
9
2
kk + K (T - ref.
T
) 2
2
2
2.4.3.2 Term forces voluminal
TFOR =
fi ui K, K + fi, K K ui
2.4.3.3 Term forces surface
TSUR = G
I, K K ui + gi ui
K, K -
N
N K
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 [Annexe 2]).
2.4.3.4 Term
thermics
THER = -
T
T, K K
with:
(
1 dK T
D T
(U), T)
()
=
(- 3 (T - T)
()
- 3K +
(T - T)
- 3
-
2
(
(T T
kk
ref.
ref.
kk
ref.)
T
dT
dT
2.4.3.5 Deformations term and initial constraints
1 ° °
1 °
TINI
HT
=
°
ij - ij ij
, K
I
J
I
J
ij
ij, K K
2
-
-
-
2
One can notice that if ° = ° then:
= (- HT -) ° + °= (- HT) and TINI = 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 constraints, or of
initial deformations but not both.
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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, calculates the restitution of energy in definite kinematics par. 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 of the axis of
symmetry:
Y
R
L
X
Appear 2.4.4.1-a: Fond of fissure in axisymetry
In Aster, 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:
dW
G = -
total
dA
where:
·
Wtotale is total potential energy,
·
dA is the variation of surface of the fissure.
W
=
2 W
with: total
dA=
2 Rdl
dWtotale
dW D L
1 dW
from where:
= 2
=
dA
D L dA
R D L
1
and thus G
=
G () in axisymetry.
R
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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
(S) (S) m S () ds
O
In command CALC_THETA [U4.82.02], the user defines the direction of the field in bottom of
fissure. By defect, it is the normal at the bottom of fissure in the plan of the lips. By choosing one
unit field in the vicinity of the bottom of fissure, one a:
(S) m S
() = 1, S X-coordinate curvilinear of O
and:
G
() =
G
(S) D
O
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 ()
G =
in 3D
L
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 commands
The table below proposes a summary of the parameter setting of commands CALC_G_LOCAL_T
CALC_G_THETA_T. For more precision one will refer to [U4.82.03] and [U4.82.04].
Commands Key word
Value
by
Ref.
defect
CALC_G_LOCAL_T
MODELE
[§2.4]
“D_PLAN”
“C_PLAN”
“AXIS”
“3D”
CHAM_MATER
[§2.4.1] Def.
materials
FOND
[§2.4.1]
Def. bottom of
fissure
DEPL
Recup.
of one
field of depl.
RESULTAT
EXCIT
[§2.4.1] Standard
charg.
SYME_CHAR “WITHOUT”
“SANS”
“SYME”
“ANTI”
LISSAGE_THETA “LEGENDRE”
[§2.2]
“LEGENDRE”
“LAGRANGE”
LISSAGE_G “LEGENDRE”
[§2.2]
“LEGENDRE”
“LAGRANGE”
“LAGRANGE_NO_NO”
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DEGRE 5
[§2.2]
INFO 1
TITER
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
MODELE
[§2.4]
“D_PLAN”
“C_PLAN”
“AXIS”
“3D”
CHAM_MATER
[§2.4.1] Def.
materials
THETA
[§2.4.2] Def.
theta
FOND
[§2.4.1]
Def. bottom of
fissure
DEPL
Recup.
of one
field of depl.
RESULTAT
EXCIT
[§2.4.1] Standard
charg.
SYME_CHAR “WITHOUT”
“SANS”
“SYME”
“ANTI”
INFO 1
TITER
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 commands
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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 constant fields in a vicinity of O, thus cancelling in this vicinity them
singular terms
- U
K, K
ij
I, p
p, K in G
().
3.2
Choice of the field theta in dimension 3
3.2.1 Method of construction
One must build a checking field:
N = N = 0 on the edge of the field (N is the normal with)
= O given on the bottom of fissure O
where represents the trace of on O.
One gives oneself two volumes 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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R is noted
(S)
inf
the variable radius of T and R
(S)
sup
that of S.
P
S
T
R
O
Rinf
Rsup
Appear 3.2.1-b: Construction of the field theta in 3D (plane of cut)
In any point of O, located by his curvilinear X-coordinate S, one can define a normal plan P in
which the field is introduced in the following way:
·
N (R (S)) = (S)
O
for 0 R (S) R
(S)
inf
·
N (R (S)) = 0 for R (S) R (S)
sup
·
()
S (R
S
sup
) \ T (R (S)
inf
)
N varies linearly compared to the radius R S in the crown
()
·
S (R
S
sup
)
N is continuous in
().
This manner of introducing is geometrical. It amounts giving itself two radii R ()
inf S and
Rsup 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
Rinf
Rsup
N1
M
NR
R
R
I
O I
inf I
sup I
M
Table 3.2.2-1: Data for construction of the field theta in 3D
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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) the direction of will be realized on the 2 edges of O having the joint node.
n1
N2
N
O
T1
n1 N
F
2
1
T2
Mr. F2
Appear 3.2.2-a: Construction of the field theta in 3D (normals)
Are F1 and F2 two faces belonging to the lips of the fissure and including/understanding the successive edges
T1 and T2 of O. One calculates initially the normal n1 with edge T1 in the plan of the F1 face then
normal N2 with the T2 edge in the plan of the F2 face.
N + N
N
1
2
1 and N2 being unit normals, one deduces some N =
then (M) = (M) N for
2
M O.
It is considered that the Fi faces are right:
·
If Fi is a triangle, the plan of the Fi face is defined.
·
If Fi is a quadrangle, one cuts out Fi in 2 triangles Fi1 and Fi2. One must
then to calculate the equations of the two plans containing the faces Fi1 and Fi2 and to make two
calculations of normal per edge Ti.
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 (TRIA or QUAD) and one calculates the equation of the plan (S)
tangent (S),
4) For each edge of O of nodes NR, NR calculation of normals N
N and
I
I 1
+
I, 1, ni+1,1,
I, 2
N
.
I,
1
+ 2
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Lastly, is calculated according to the following algorithm:
Loop on the nodes Nor of O:
1
nor = (N + N
I,
I,)
2
1
2
(Ni) = (Neither) nor
End of the loop on the nodes Nor of O
Algorithm 1: Calculation of
ni+1,1
nor, 2
N
N
i+1,2
I, 1
NR
NR
I
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 [
M, M
I
i+1] and
S [0]
1
, such as MR. M = S MR. M
I
I
i+1
D = D M
(, M)
- Calculation
of
- Calculation
of
(M) by linear interpolation:
(M) = (1 - S) (M) + S (M
I
i+1)
- Calculation
of
R (S)
inf
and R
(S)
sup
by linear interpolation:
R (S) = (1 - S) R
+ S R
inf
inf I
inf I
+1
R
(S) = (1 - S) R
+ S R
sup
S
I
I
up sup
+1
-
/If D > R
(S), (M)
sup
= 0
/If D < R (S), (M)
inf
= (M)
D - R
/
S
If
R (S) D R
(S),
inf ()
inf
sup
=
and (M) = (1 -) (M)
S
R up (S) - I
R nf (S)
Finsi
Algorithm 2: Calculation of the field theta in 3D
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M
O
M I
D
M
S
M i+1
Appear 3.2.2-c: Calculation of the field theta in 3D
We detail below the calculation of projection M of M on O:
For each node M:
-
Recovery of the co-ordinates of M
-
Loop on the Mi nodes of O (I = 1, NNO -)
1
Recovery of the co-ordinates the Semi one and Mi+1
MR. M
. MR. M
Calculation of S
I
I 1
I
I
=
+
MiMi+1
/if < 0: if = 0
/if > 1: if = 1
Calculation of the co-ordinates of M I: OMi = OM + S
I
I Semi Mi+1
Calculation of D
= D (M, M
I
I)
Fine loops
- Recovery
of
J such as D = min (D
J
I)
I
- Knowing
J one recovers M, M
, S
J
j+1
J and projection M of M on O such as:
MR. M = S MR. M
J
J
J
j+1
Algorithm 3: Calculation of projections on the bottom of fissure
M
di-1
di
if-1
if
Mi+1
M
M I
I
Appear 3.2.2-d: Projection of the points on the bottom of fissure
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3.3
Choice of the field theta in dimension 2
It is about a particular case of dimension 3. O is limited to a point, the user chooses the radii
Rinf and Rsup, the module in bottom of fissure O and the field are built so that:
(R) = 0 if R R
sup
(R) = N if R R
0
inf
R
- R
sup
(R) =
N if R R R
0
R
- R
inf
sup
sup
inf
Rsup
0
Rinf N
0
0
R
R
inf
sup
^
Appear 3.3-a: Calcul of the field theta in 2D
3.4 Other
method
The user can enter itself the field, by using command AFFE_CHAM_NO [U4.44.11]
Code_Aster which makes it possible to affect node by node or group of nodes.
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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
G
(F)
[§1.3]) one details the calculation of each integral term of
where F
is the field
S
s=0
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 starting 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 REP 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 step 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 grid 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 modelled 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 x1 and x2 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:
F: MR. M
S
+
S
S
(M)
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This one answers the same properties of regularity as the transformation related to the field theta fissures
that we will note F henceforth: MR. M
F
+
(cf [§1.3]). The variation thus is materialized
F
F
(M)
of field on the left edge of the structure.
y
Field S
sensitivity
Field F
fissure
X
F
Fissure
x1
x2
Appear 4.1-a: Dérivée of 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 grid fixes reference and by modelling the variations
virtual of field by suitable functions theta.
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The procedure (cf [§4.4.1]) of a calculation of sensitivity can be thus schematized in the form
following:
Calculation
of S
CALC_THETA
S
S
Calculation
T, T&
Calculation
Thermics
Mechanics
THER_LINEAIRE
MECA_STATIQUE
Calculation of F
T, T&
CALC_THETA
U, U&
F
G
(
F)
Calculation of G (
F) and of
S
s=0
CALC_G_THETA_T
Appear operational 4.1-b: Mode 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):
Is =
v
D
S
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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 Fs transformation modelling the movement of the border of
volume moving
S, and the intégrande (tensor of command 0, 1 or 2) are all of C1 class.
One has then, while noting &
S the Lagrangian derivative (in the vicinity of the origin) compared to the field
theta sensitivity:
I S
S
F
S
&
div
=
+
v
D
S
S
=0
=0
S
S
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
I S
vicinity of the origin
all these alternatives are equivalent (“Philosophiquement” it
S s=0
result is reassuring because it makes it possible not to privilege neither the matter elongation (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 FF is not always C1 (Fs is, but
not FF, which has two surfaces of discontinuity on the Rinf borders and Rsup of the 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 C0) whereas the field theta sensitivity
is a combination of third order students'rag processions which return it safe C2 in the middle of its support
(where it is right C1). 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 modelling 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 Geometrie and Praktische
mathematik, RWTH Aachen, 1977), which filled all desired conditions of regularity.
But this one brings only marginal gains unless refining to the extreme the grid and
to circumscribe the zone 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 C0 regularity at the borders. The use of elements of Hermite 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 problem PLAN
or AXIS). It is just necessary to replace the voluminal divergence by surface. From a point of view
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:
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numerical, it is besides to better calculate the latter via derivative contravariantes on
surface:
divs =
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 assistance 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 traditional 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 point out that:
&f = F S
,
S
=
I
I
&g
G
,
I
I
&°
°
S
=,
°
°
S
=
ij
ij
&
,
ij
ij
&
S
E
= E
, &
S
=,
&
S
=
,
&
S
T
= T
ref.
ref.
Indeed, that is to say the loading or the material considered, then there exists a field of R3 such as:
/=
Derivation compared to the parameter commutating with this restriction, one has 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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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:
E& (T)
E
E
=
+ E
. =
(
S
T & - T
. )
3
2
1
T
=0
Note:
· One can use characteristics variable materials within finite elements or
dependant on the temperature, provided that that is apart from
(
supp F)
(
supp 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 F)
(
supp 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 key word SENSIBILITE.
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:
}
·
= & - S
I, J
I, J
I, K K, J éq
4.2.3-3
}
·
= &
- S
ij, K
ij, K
ij L, L, K
from where the Lagrangian derivative of the divergence:
}
·
div = & - S
I, I
I, K K, I éq
4.2.3-4
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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 which
R
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 preceding:
R
·
S
S
S
S
0
0
R R
,
R Z
,
&
0
R R
,
&r Z, +
+
R R
, R R
,
R Z
, Z R
,
R R
, R, Z
R Z
,
Z Z
,
}
·
&
S
=
R
0
0 =
R
0
0 -
R R
0
0
éq
4.2.3-5
R
R
r2
0
Z R
,
Z Z
, &
0
Z R
,
&z, Z S + S
0
S
S
Z R, R R,
Z Z
, Z R
,
Z, rr Z +z Z
,
, Z Z
,
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:
}
·
= & - S
I, J
I, J
I L,
L, J
I
=
+ S - S
éq
4.2.3-6
I, K
K
I L,
L, J
{
= 0
, J
=
S
I, K J
K
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 [Annexe 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 [Annexe 2] and [§4.2.1]).
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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
G
(F)
One wishes to calculate the various terms of
. One takes again the nomenclature of [§2.4.3]
S
s=0
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 modifications 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
Rsup radius. In command CALC_G_THETA_T it is thus not necessary to specify them
loadings which do not apply in this zone.
4.3.1 Derived from the elementary traditional term
According to the theorem of transport of [§4.2.1], the intégrande corresponding to derived from the term
traditional elementary is written:
·
TCLA
6
7
44444
4 444448
4
(F)
= U F -
F
S
ij
I, p
p, J
((U), T)
div
+ TCLA D
iv
S
s=0
·
}
·
}
= & U + U F + U F
ij I, p
ij
I, p p, J
ij
I, p
p, J
·
678
- & div F - div F + TCLA
S
div
·
·
}
·
}
678
F
It is thus necessary to calculate &
,
,
,
F
U,
,
&
and di
ij
I p
p J
v
.
First of all, taking into account the regularity of Fs, 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 (P)
F
= ((P)
S
F
) P
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thus the Lagrangian derivative of theta fissure is null:
F
F
F P
P
F
(()
S
)
()
& =
=
0
=
S
S
=0
S =0
S
[éq 4.2.3-6] led then to:
}
·
·
678
F = F
-
S and di F
v
= - F S
p, J
p L, L, J
K L, L, K
While applying [éq 4.2.3-3] to the field of displacement it occurs:
}
·
U, = (U &) - U
S
I p
I
, p
I, K
K, p
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:
·
678
PD
1
1
S
S
ij
=
(u&i) + u&
U
U
(small deformations)
, J
(J)
(
,
,
J, K
K I,)
2
I
I K
K J
, -
+
2
·
·
678 678
GD
1
= PD
S
S
ij
ij
+
(&
,
,
,
&
(great deformations)
2 (ui) - U
U
+ U
- U
U
, K
I p
p K) J K (J)
, K
J, p
p, K
I, K
Since one limited oneself to linear elasticity and the preceding remarks on
loadings and of [éq 4.2.3-3]:
}
·
0
S
0
S
ij = ijkl (& - &
T
-
kl
kl
kl, m m) +
ij, m m
4µ
& = K (& -
kk
3 T&) (-
kk
3 (T - R
T EFF) +
eq &eq
3
3 6 D
&
ij
&eq =
(tensor are equivalent)
12
D
with
ij
D
1
& = & - &
ij
ij
kk kl
(tensor deviatoric)
3
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
S
F
R
U
=
, S
R
R
,
, =
,
U, =
R
R
R
}
·
·
F S
}
S
1
F
U
R
R
R R
and
, = -
2
U, = &ur -
R
R
R
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from where the Lagrangian derivative of the component 3x3 of the tensor of the deformations:
·
678
}
·
PD
=
U, (small deformations)
·
·
678.678}
·
GD
PD
ur
=
+
U,
(great deformations)
R
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:
·
6 7
4
8
4
THER
(
F)
= -
T F + THER
S
div
T, K K
S
s=0
}
·
= -
T
S
F
S
, K +
(T & - T
,
,
+ THER
div
T
T (), K
L L K) K
It thus remains us to calculate the Lagrangian derivative of derived compared to the temperature from
density of free energy:
}
·
= -
3 K (& -
3 T
kk
&)
T
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:
·
64447
4 4448
4
TFOR
(F)
= U
F
F
S
I (F
+ F
I, K K
I div
) + TFORd
iv
S
s=0
·
·
678
=
U
F
F
F
S
F
F
&i (F + F
I, K K
I div
)
}
+ U F + F D
iv
+ F
I I, K K
I L, L
I div
+ TFOR
S
div
The only term which it remains to calculate is the Lagrangian derivative of the gradient of the voluminal forces:
}
·
F
= F
S
I, K
I, jk J
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Note:
The surface theorem of transport led to replace the voluminal divergence by one
surface divergence. This one 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 [Annexe 2]).
4.3.4 Derived from the deformations term and initial constraints
According to the theorem of transport of [§4.2.1], the intégrande corresponding to derived from the term
“initial strains and stresses” is written:
·
64444444444 7
4
8
44444444444
TINI
1 0 0
1 0
(F)
=
-
0
F
S
ij
ij
ij, K
ij
(T Tref)
+ TINI
div
ij
ij
ij, K K
2
+
-
-
-
2
S
S =0
}
·
1
1
1
= F
0
& - 0
& 0
0 0
- &
T
- & 0
K
ij
ij
ij, K
ij
ij
ij, K
&
+
2
+
-
2
+ ij
ij
ij
ij, K
2
·
F
1
-
0
0
S
K
ij
(T - Re
T F)
}
-
TINI
ij
ij
ij, K
div
2
+
Only the derivative which was not exhumed yet are those of the gradients of the tensors of
deformations and of the initial constraints which, according to [éq 4.2.3-6], are written:
}
·
0 = 0 S
ij, K
ij L, K
L
}
·
0 = 0 S
ij, K
ij L, K
L
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 constraints, or of the initial deformations but not both.
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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 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 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 which 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 option SENSIBILITE in operator MECA_STATIQUE. Like
for modeling C_PLAN, the other types of loading (FORCE_INTERN, FORCE_CONTOUR,
EPSI_INIT, PESANTEUR 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. Ils are thus modelled by inserted AFFE_CHAR_MECA or AFFE_CHAR_MECA_F
between MECA_STATIQUE and CALC_G_THETA_T.
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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. Imposés - Liaisons uniforms
C. exchanges between wall: refused
Chgt. thermics: TEMP_CALCULEE
C. of radiation: refused
Calculation
MECA_STAT
THER_LINE
Configuration
Modeling
C_PLAN
usable
only in
postprocessing
(relevance
left with the free one
choice of
the user)
Type of loading FORCE_INTERN
FORCE_CONTOUR
EPSI_INIT
PESANTEUR
ROTATION
CALC_G_THETA_T
Via 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].
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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 command CALC_THETA [U4.82.02], or by the command
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 operand SENSIBILITE).
It is him also a field of vector 2D in each node of the grid. It is directed along the axis of
X-coordinates. It can be affected directly with command AFFE_CHAM_NO but, in practice,
it results generally from specific command CALC_THETA with the option THETA_BANDE which
allows to seize the module (key word MODULE) and the X-coordinates x1 and x2 (key word R_INF and R_SUP) of
points delimiting its vertical support. It is pointed out that this field decrease value MODULE with
zero value between the X-coordinates x1 and x2, and that it is null everywhere else. These X-coordinates can be
negative but one must have x1 < x2. [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 of the bottom of fissure
(equal to its distance to the axis of symmetry y):
Gintrinseque
(F)
1 G
(F)
=
R
S
S
=0
=0
S
S
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5 Bibliography
[1]
BUI H.D., Mécanique of brittle fracture, Masson, 1977.
[2]
DESTUYNDER pH, DJAOUA Mr., Sur 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., “Problèmes in extreme cases in the polygons”, Mode of employment - EDF - Bulletin
of Direction from Etudes and Recherches, Série C, 1, 1986 pp. 21-59.
[4]
MIALON P., “Calcul of derived from a size compared to a bottom of fissure by
method theta ", EDF - Bulletin of Direction of Etudes and Recherches, Série C, n°3 1988
pp1-28.
[5]
MIALON P., Etude 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 Al Etude PRObabiliste 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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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 [Annexe 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 Dérivées seconds on the element of reference
A1.1.1 Segment (element of edge)
The derivative second of the three functions of forms are written:
2
2
1
NR (
NR
)
2
= 1
() = 1
2
2
N1
N3
N2
2 NR
- 1
0
1
3 () = 2
-
2
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:
N1
N4
N6
N2
N5
N3
Appear A1.1.2-a: Triangle of reference
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2 NR
2 NR
2 NR
I
I
I
I
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: Dérivées seconds of the triangle of reference
A1.2 Quadrangle complete or incomplete (element of face)
N1
N8
N4
N7
N5
N9
N2
N6
N3
Appear A1.2-a: Quadrangle of reference
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The derivative second of the eight (resp. nine) functions of forms of the incomplete quadrangle
(resp. complete) are written:
2 NR
2 NR
2 NR
I
I
I
I
2 NR
2 NR
2 NR
2
2
I
I
I
I
2
2
1 +
- 1
(
2 +)
1 (
2 -)
1
1
1 +
1 -
- 1
2
2
4
1
-
2
2
2
4
- 1
- 1
(
2 -)
1 (
2 -)
1
1 -
1 -
- - 1
2
2
+
2
2
4
2
2
2
4
- 1
1 +
(
2 -)
1 (
2 +)
1
1 -
1 +
- + 1
3
3
-
2
2
4
2
2
2
4
1 +
1 +
(
2 +)
1 (
2 +)
1
1 +
1 +
+ 1
4
4
+
2
2
2
4
2
2
4
5
1 - 2
(1)
(1 -
2)
5
0
- 1
6
(1 -)
1 - 2
(
1 -
2)
6
- 1
0
7
1 - 2
- (
+)
1
- (
2 +)
1
7
0
- - 1
-
8
- (+)
1
1 - 2
- (
2 +)
1
8
- - 1
0
-
9
(22 -) 1
(22 -) 1
4
Appear A1.2-1: Derived seconds from the complete and incomplete quadrangle of reference
A1.3 Dérivées seconds on the real element
A1.3.1 Problématique
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 complicated (
geometrical function of interpolation: X admits a reciprocal bijection but its construction
is hard as of the QUAD4. One will note [J] his matrix jacobienne and det (J) his 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:
1
() N1 ()… NR E ()…
N
noted () NR () {N}
with:
1, 2,…., them values of at the points of interpolation and
N1 (), N2 (),…, Nne () 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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A1.3.2 Cas two-dimensional
By using derivation in chain, one first of all writes the derivative in = (,) from those in
x= (X, y):
X
y
X
= X
y
éq A1.3.2-1
y
noted {} = [J] {X}
By reversing this system (as is bijective) one can thus deduce the derivative in X from it from
those in:
{
- 1
X} = [J]
{} éq A1.3.2-2
and by deriving them formally one obtains:
2
2
2
X
2
2
2
= T
+ T
2
[1] [2]
y
2
éq A1.3.2-3
2
2
X y
noted {2} = [
2
1
T] {} + [2
T
X
] {}
In addition while deriving [éq A1.3.2-1] compared to, while taking account of [éq A1.3.2-2], it comes:
{2
2
} = [1
C] {X} + [C2] {X}
= [
1
-
2
1
C] [J] {} + [C2] {X}
by deferring the expression obtained in [éq A1.3.2-3]:
{2
1
-
2
X} = [
(T1] + [T2] [C1] [J]) {} + [T2] [C2] {X}
[
-
T
1
2] = [C2]
[T
1
-
1] = [
- T2] [C1] [J]
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The matrices [C1] and [C2] being obtained easily, via, there are thus a constructive process allowing
to deduce matrices [T1] and [T2] required:
X 2
y 2
X y
2
X
2
y
2
2
2
2
2
[
X
y
X y
2
X
2
y
C
2] =
2
[C1] =
2
2
X X there there there X
X y
2
X
2
y
+
y 2
y 2
y y
- 2
2
2
[
1
X
X
X X
T2]
=
2
2
det (J)
-
y
X
y X
y X X y
-
-
+
+
y
y
[
1
T
1]
-
-
=
[T2] [C1]
det (J)
X
X
-
Thus, for example, the first derived second in X expressed on the element of reference is written:
2
2
2
2
() = T (1,)
1
() + 1T (12,) () + 2
T (1,)
1
() + T (12,) () + T (1,)
3
()
2
1
X
2
2
2
2
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 Calcul 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 = (
F
F
F + F div
S
) .u D
S
·
TSUR
678
(F) =
(
S
F) F
S
F
F
+ F div + F div .u
S
S
+
S
=0 S
S
(F
F
F + F div
S
).(
S
U & + udiv
S
) D
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. Pour 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.
M ()
G
G
1
2
S
y
, g3
X
O
Z
Appear curvilinear A2.1-a: Paramètrage of the vicinity of the curve of work
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The figure above illustrates the natural base covariante (g1, g2, g3) 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)
X
y
0
y
X
G =
G = -
G
1
2
3 = 0
0
0
1
From where the metric tensor G and its reciprocal tensor g-1, by noting J the jacobien transformation:
2
2
J
0
0
-
J
0
0
G = [G
G G
G
G G
ij]
= [
-
-
I
J
-
.i J]
2
1
= 0
J
0
=
ij
ij
[1
gij] = [. ]
2
= 0
J
0
ij
ij
0
0
1
0
0
1
2
2
X
y
with J =
+
Reciprocal metric tensor one deduces bases it contravariante (g1, g2, g3) which proves very useful for
to calculate the derivative covariantes:
1
1
gi = g-1 G
g1 =
G, g2 =
G, g3 = G
ij
J
J 2 1
J 2 2
3
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
J D in 2D - PLAN
D =
R J D
in 2D - AXI
to take account of integration per unit of radian in axisymetry. Taking into account the analogy
R 2 2
Z
formal (X, y) (R, Z), the jacobien of the transformation is written: J =
+
A2.2 Terme forces surface
Let us break up this term of into two intégrandes:
TSUR = (TSUR + TSUR
1
2) D
S
with
TSUR =
1
(F
F) .u and TSUR =
2
(
F
F div
S
) .u
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A2.2.1 Calcul of TSUR1
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, K, L,
1 2
TSUR = F
.u =
G G
G. G
with
1
(F
)
(I
J
fk
F
U
J
I
K) L
{}
L
j=1
I
fk
L
= F
U G
J
jk
it
2
I
fj
I
= J F
U
J
It remains to determine I
fj
I
F
,
and
U via the base contravariante to obtain:
J
F
F
F
TSUR = J 2
1
1
1
2
2
1
(F .g) .g (.ug) + .g (.ug)
X
y
F
F
= J - 2
F
F
+
U
X + U
y
X
y
X
y
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 Calcul of TSUR2
By expressing the surface divergence as the trace of the surface gradient
div
F
= tr
G
G
S
(F
S
) = tr (fi J
J
I
) with I, J = 1
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, L
,
1 2
TSUR =
F tr
.u =
G tr
G G. G
with
2
((F
F
U
S
) (K K (fi
J
J
I
) L
{}
L
I, j=1
K
fi
L
= F
U G
J
ij
kl
2
K
fj
K
= J F
U
J
It remains to determine K
fj
F
,
and K
U with the base contravariante to obtain:
J
F
TSUR = J 2
1
1
1
2
2
.g
2
{(f.g) (.ug) + (f.g) (.ug)}
F
X
F
-
y y
= J 2
X
+
(F U + F U
X X
y y)
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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
Fr
F
div =
+
S
R
F
F
- 2
R Z
1
R
Z
F
= J
+
+ R
R
A2.3 Dérivée of the term forces surface
Let us break up this term of into five intégrandes:
TSUR
(F) = (DTSUR + DTSUR + DTSUR + DTSUR + DTSUR D
1
2
3
4
5)
S
=0
S
S
with
DTSUR = (F S) F) .u
1
DTSUR = (F S) .u div F
2
S
·
678
DTSUR = F .u div F = F .u tr
S
(F S
S
S
)
3
DTSUR = (F F + F div F .u
S
) &
4
DTSUR = (F F + F div F .u
S
) div S
5
S
A2.3.1 Calcul of DTSUR1
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, L
, m, N,
1 2
DTSUR = F
.u =
G G G
G
G. G with
1
(
S
) F
)
(I
J
K
L
F
U
jk
I
S
L) m
F
m) N
{}
N
J, k=1
I
SSL
Fm
N
= F
U G
jk
kl
jm
in
2
I
sk
fj
I
= J F
U
jk
It remains to determine I
fj
sk
I
F
,
,
and
U with the base contravariante to deduce some:
jk
S
2
2
F
F
TSUR = J 2
1
1
1
1
2
2
.g .g
.g.
U G
.g
1
(F) (S)
+
.
U G
2
(
) 2 ()
X
y
X
y
2
F
2
F
= J - 4
F
F
S
S
+
+
U
X + U
y
X
y
X
y
X
2
y
2
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 Calcul of DTSUR2
It is obtained immediately by taking again the result of the calculation of TSUR1 (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 TSUR2.
4
S X
S y
F
F
X
y F
X
X fy y
DTSUR = -
J
+
U
+ U
+
2
X
y
X
y
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
div
F
= J - 2
R
Z
R
S
+
+ R
A2.3.3 Calcul of DTSUR3
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, L
,
1 2
TSUR =
F tr
.u =
G tr
G G G G
. G with
3
((F
S
F
U
S
S
) (K K (fi Sm
J
N (
J
I
) (
N
m
)) L
{}
L
I, J, m, n=1
K
fi
Sm
L
= F
U G
J
N
jm
in
kl
2
K
fi
sj
K
= J F
U
J
I
It remains to determine K
fi
sj
K
F
,
,
and
U with the base contravariante
J
I
F
S
2
1
TSUR = J
1
.g
.g
f.g
.
U G
f.g
.
U G
3
({1) (1) + (2) (2)}
F
F
S
S
4
-
X
y
X
y
X
y
X
y
= J
+
+
(F U + F U
X
X
y
y)
Note:
In axisymetry it is necessary to take account of the nonnull component orthoradiale derivative
Lagrangian of the surface gradients (cf [§4.2.2])
·
F
678
-
R
F
Z
S
R
S
Z
F
F
div F
= J 4
R
Z
R
Z
R
S
+
+
-
S
R 2
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A2.3.4 Calcul of DTSUR4
It is enough to summon the expressions of TSUR1 and TSUR2 by replacing the components of the field of
ux
&ux
displacement U
U
U by those of its Lagrangian derivative &
.
y
&uy
A2.3.5 Calcul of DTSUR5
It is enough to multiply sum TSUR1+TSUR2 by the expression of the surface divergence of the field
theta sensitivity
S
X
S
y y
div S
= J - 2
X
+
S
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
S
R
S
Z
S
div S
= J - 2
R
Z
R
+
+
S
R
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A
Outline document