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Titrate:
Integration of the elastoplastic relations
Date:
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Author (S):
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Key:
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Organization (S): EDF/MTI/MN, RNE/MTC
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
R5.03.02 document
Integration of the relations of behavior
elastoplastic of Von Mises
Summary:
This document describes the quantities calculated by operator STAT_NON_LINE necessary to the implementation of
the quasi static nonlinear algorithm describes in [R5.03.01] in the case of the elastoplastic behaviors.
These quantities are calculated by the same subroutines in operator DYNA_NON_LINE in the case
of a dynamic stress [R5.05.05].
This description is presented according to the various key words which make it possible the user to choose the relation
of behavior wished. The relations of behavior treated here are:
· the behavior of Von Mises with isotropic work hardening (linear or not linear)
· the behavior of Von Mises with linear kinematic work hardening (model of Prager)
The method of integration used is based on a direct implicit formulation. From the initial state, or to leave
moment of preceding calculation, one calculates the stress field resulting from an increment of deformation. One
also calculate the tangent operator.
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Titrate:
Integration of the elastoplastic relations
Date:
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Author (S):
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Key:
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Contents
1 Introduction ............................................................................................................................................ 3
1.1 Relations of behaviors described in this document .............................................................. 3
1.2 Integration ........................................................................................................................... 3 drank
2 general Notations and assumptions on the deformations ...................................................................... 4
2.1 Partition of the deformations (small deformations) ........................................................................... 5
2.2 Reactualization ................................................................................................................................ 6
2.3 Initial conditions ........................................................................................................................... 6
3 Relation of Von Mises with isotropic work hardening ...................................................................................... 7
3.1 Expression of the relations of behavior .................................................................................... 7
3.1.1 Relation VMIS_ISOT_LINE ................................................................................................... 7
3.1.2 Relation VMIS_ISOT_TRAC ................................................................................................... 8
3.2 Tangent operator. Option RIGI_MECA_TANG ............................................................................. 11
3.3 Calculation of the constraints and the variables intern ........................................................................... 13
3.4 Tangent operator. Option FULL_MECA ........................................................................................ 15
3.5 Produced internal variables .......................................................................................................... 17
4 Relation of Von Mises with linear kinematic work hardening ................................................................. 17
4.1 Expression of the relation of behavior .................................................................................. 17
4.2 Tangent operator. Option RIGI_MECA_TANG ............................................................................. 19
4.3 Calculation of the constraints and variables intern .................................................................................. 20
4.4 Tangent operator. Option FULL_MECA ........................................................................................ 22
4.5 Produced internal variables .......................................................................................................... 22
5 Bibliography ........................................................................................................................................ 22
Appendix 1 Relation VMIS_ISOT_TRAC: complements on integration ................................................. 23
Isotropic appendix 2 Ecrouissage in plane constraints ........................................................................... 25
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Integration of the elastoplastic relations
Date:
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Author (S):
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1 Introduction
1.1
Relations of behaviors described in this document
In operator STAT_NON_LINE [U4.51.03] (or DYNA_NON_LINE [U4.53.01]), two types of
behaviors can be treated:
· the incremental behavior: key word factor COMP_INCR,
· the behavior in nonlinear elasticity: key word factor COMP_ELAS.
For each behavior one can choose:
· the relation of behavior: key word RELATION,
· mode of calculation of the deformations: key word DEFORMATION.
For more details, to consult the document [U4.51.03] user's manual, the behaviors described
here raising only of the key word factor COMP_INCR.
The relations treated in this document are:
VMIS_ISOT_LINE:
Von Mises with linear isotropic work hardening,
VMIS_ISOT_TRAC:
Von Mises with isotropic work hardening given by a traction diagram,
VMIS_CINE_LINE:
Von Mises with linear kinematic work hardening.
1.2
Integration drank
To solve the nonlinear total problem posed on the structure, the document [R5.03.01] described
the algorithm used in Aster for nonlinear statics (operator STAT_NON_LINE) and it
document [R5.05.05] described the method used for nonlinear dynamics (operator
DYNA_NON_LINE).
These two algorithms are based on the calculation of local quantities (in each point of integration of
each finite element) which results from the integration of the relations of behavior.
With each iteration N of the method of Newton [R5.03.01 § 2.2.2.2] one must calculate the nodal forces
R one
()
N
N
I
= QT I (options RAPH_MECA and FULL_MECA) constraints I being calculated in
each point of integration of each element starting from displacements linked via
relation of behavior. One must also build the tangent operator to calculate Kni (option
FULL_MECA).
Before the first iteration, for the phase of prediction, one calculates Ki - 1 (option RIGI_MECA_TANG).
The calculation of Ki - 1, which is necessary to the phase of initialization [R5.03.01 §2.2.2.2] corresponds to
calculation of the tangent operator deduced from the problem of speed.
This operator is not identical to that which is used to calculate Kni by option FULL_MECA, to
run of the iterations of Newton. Indeed, this last operator is tangent with the problem discretized of
implicit way.
One describes here for the relations of behavior VMIS_ISOT_LINE, VMIS_ISOT_TRAC and
VMIS_CINE_LINE, the calculation of the tangent matrix of the phase of prediction, Ki - 1, then the calculation of
stress field starting from an increment of deformation, the calculation of the nodal forces R and
stamp tangent Kni.
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Key:
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2
General notations and assumptions on the deformations
-
All the quantities evaluated at the previous moment are subscripted par.
The quantities evaluated at the moment T + T are not subscripted.
The increments are indicated par. One has as follows:
Q = Q T
(+ T) = Q T () + Q = Q + Q.
For the calculation of the derivative, one will note: !
Q derived from Q compared to time
tensor of the constraints.
operator déviatoire: ~
ij = ij - 1
3 kk ij.
() eq
3 ~ ~
equivalent value of Von Mises:
=
eq
ij ij
2
increment of deformation.
With
tensor of elasticity.
, µ, E, v, K
moduli of the isotropic elasticity, respectively: coefficients of Lamé,
Young modulus, Poisson's ratio and module of compressibility.
3K = 3 + 2µ
modulate compressibility
thermal dilation coefficient average.
T
time.
T
temperature.
()
positive part.
+
To calculate the tangent operators, one will adopt the convention of writing of the symmetrical tensors
of command 2 in the form of vectors with 6 components. Thus, for a tensor a:
“T
a= [axx ayy azz
2axy
2axz
2ayz]
“
One introduces the hydrostatic vector 1 and stamps it deviatoric projection P:
“1=t [1 1 1 0 0] 0
1 ““
P = Id - 1 1
3
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2.1
Partition of the deformations (small deformations)
One writes for any moment:
T () = E T () + HT T () + p T (),
with
E (T) =
-
To 1 (T (T)) (T)
with
HT (T) = (T (T)) (T (T) - T Id
ref.)
or in a more general way:
HT (T)
= (T) (T - T - T
T - T
def) (ref.) (ref.
def)
= # (T) (T - Tref)
and
HT (T
= 0
ref.)
A depends on the moment T via the temperature. The thermal dilation coefficient
(T (T)) is an average dilation coefficient which can depend on the temperature T.
temperature T is the temperature of reference, i.e. that for which thermal dilation
ref.
is supposed to be null if the average dilation coefficient is not known compared to T, one can
ref.
to use a temperature of definition of the dilation coefficient average T
(defined by the key word
def
TEMP_DEF_ALPHA of DEFI_MATERIAU) different from the temperature of reference [R4.08.01].
·
$ & %
&
'
&&
What leads to: !(T)
-
To 1 (T (T)) (T)!HT (T)! p
=
+
+
(T)
This choice is made by preoccupation with a coherence with elasticity: it is necessary to be able to find the same solution in
elasticity (operator MECA_STATIQUE) and in elastoplasticity (operator STAT_NON_LINE) when them
characteristics of material remain elastic. This choice leads to the discretization:
= p + A-1
() + HT
with:
A-1
() = A-1 T + T
(
) -
(+
) - A 1 T
() -
and
HT = T
(+ T
(
) T (- T)
(
)
ref.
- T -
() T - Tref Id
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2.2 Reactualization
In STAT_NON_LINE, under the key word factor COMP_INCR, three modes of calculation of the deformations
are possible:
· “PETIT”
· “SIMO_MIEHE” [R5.03.21] (which carries out calculation in great deformations for one
isotropic work hardening)
· “PETIT_REAC” (which is a substitute with calculation in great deformations, valid for the small ones
increments of load, and for small rotations [bib2]).
This last possibility consists in reactualizing the geometry before calculating:
X is written
= X
N
N
O + ui - 1 + ui, the calculation of the gradients of ui is thus made with geometry X
instead of the initial geometry xo.
2.3 Conditions
initial
They are taken into account via -, p, U.
In the event of continuation or resumption of a preceding calculation, there is directly the initial state -, p, U in
on the basis of, p, U of preceding calculation at the specified moment.
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3
Relation of Von Mises with isotropic work hardening
3.1
Expression of the relations of behavior
These relations are obtained by key words VMIS_ISOT_LINE and VMIS_ISOT_TRAC.
For these two relations, the mode of calculation of the deformations is DEFORMATION: “PETIT”:
~
·
$ % '
p
3
!
=
- 1
!p
=! - A -!HT
2
eq
eq - R (p) 0
!p = 0 if eq - R (p) <
0
!p 0 if eq - R (p) = 0
! p: speed of plastic deformation,
p: cumulated plastic deformation,
HT: thermal deformation of origin: HT = (T - ref.
T).
Id
The function of work hardening R (p) is deduced from a simple test tensile monotonous and isothermal
In this case:
0
eq = L
0
L
= 0 0
0
p = P
L
L
= L -
.
E
0 0 0
L - R (p) 0
The user can choose a linear work hardening (relation VMIS_ISOT_LINE) or a traction diagram
data by points (relation VMIS_ISOT_TRAC).
3.1.1 Relation
VMIS_ISOT_LINE
The data of the material characteristics are those provided under the key word factor
ECRO_LINE or ECRO_LINE_FO of operator DEFI_MATERIAU [U4.43.01].
/ECRO_LINE: (D_SIGM_EPSI: AND
SY: y)
/ECRO_LINE_FO: (D_SIGM_EPSI: AND
SY: y)
ECRO_LINE_FO corresponds if AND and y depend on the temperature and are then calculated
for the temperature of the point of current Gauss.
The Young modulus E and the Poisson's ratio are those provided under the key words factors
ELAS or ELAS_FO.
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In this case the traction diagram is as follows:
L
AND
y
E
L
I.e.:
y
L = E L
if L < E
.
y
y
L = y + AND L -
E if L E
Note:
y is the elastic limit (the choice of falls on it on the user: it can correspond to the end
of linearity of the real traction diagram, either lawful elastic limit or
conventional. At all events, one uses here the single value defined under ECRO_LINE).
When the criterion is reached one a:
()
L - R p
= 0,
thus
L
L - R L -
E = 0,
from where
E
R (p) =
T E p +
E - E
y.
T
3.1.2 Relation
VMIS_ISOT_TRAC
The data of material are those provided under the key word factor TRACTION: (SIGM: F), of
operator DEFI_MATERIAU.
F is a function with one or two variables representing the simple traction diagrams. The first
variable is obligatorily the deformation, the second if it exists is the temperature (parameter
of a tablecloth). For each temperature, the traction diagram must be such as:
· the X-coordinates (deformations) are strictly increasing,
· the slope between 2 successive points is lower than the elastic slope between 0 and the first point
curve.
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The interpolation compared to the temperature is carried out in the following way:
That is to say the temperature considered, if there is K such as
[
]
K, K +1 where K indicates the index of
traction diagrams contained in the tablecloth, one point by point builds the traction diagram with
temperature while interpolating compared to the X-coordinates and the ordinates of the points of both
extreme traction diagrams.
K
K + 1
If is apart from the intervals of definition of the traction diagrams, one extrapolates conformément
with the prolongations specified by the user in DEFI_NAPPE [U4.31.03] and according to the principle
precedent.
Note:
It is disadvised and dangerous to extrapolate the traction diagrams for values of
temperature very far away from the extreme temperatures to which the curves are defined. It is
always preferable to provide traction diagrams for values of temperature framing
temperatures of calculation.
If the numbers of points of discretization of the traction diagram to K and K + 1 are different, one
interpolate between the last point of the poorest curve with all the remaining points of the curve
richer. Consequently, it is preferable enough to have a number of points of discretization
homogeneous for the various temperatures.
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In all the cases, the traction diagram considered is a linear function per pieces:
=
I +1 - I (
)
[
]
I +
- for
for I + 1 N.
I
I, I + 1
I +1 - I
N being the number of points of interpolation with a linear extrapolation, constant or excluded according to
the choice carried out in DEFI_FONCTION by the user (cf [U4.31.02] for more precise details on
extrapolation considered).
2
y = 1
E
1
2
The first point makes it possible to define:
y = 1
E = 1.
1
It is this Young modulus who is used in the integration of the relation of behavior.
One thus has for any I:
p
I
= -
.
I
I
E
The function of work hardening is then:
I +1 - I
R (p) =
(
)
[
]
I +
p - p for p p
.
p
I
I, pi + 1
I +1 - pi
The user must also give the Poisson's ratio, and a fictitious Young modulus yg (which is only useful
to calculate the elastic matrix of rigidity if key word NEWTON:(MATRICE:“ELASTIQUE”) is
present in STAT_NON_LINE) by the key words:
/ELAS: (NAKED: E: E)
/ELAS_FO: (NAKED: E: E)
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3.2
Tangent operator. Option RIGI_MECA_TANG
The goal of this paragraph is to calculate the tangent operator Ki - 1 (option of calculation RIGI_MECA_TANG
called with the first iteration of a new increment of load) starting from the results known with
the moment previous Ti - 1.
For that, if the tensor of the constraints with Ti - 1 is on the border of the field of elasticity, one writes
condition:
!F = 0
who must be checked (for the continuous problem in time) jointly in the condition:
F
= 0
with:
F
(, p) =
()
eq - R p.
So on the other hand the tensor of the constraints with Ti - 1 is inside the field, F < 0, then
the tangent operator is the operator of elasticity.
The quantities intervening in this expression are calculated at the moment previous Ti - 1, which are them
only known at the moment of the phase of prediction. One thus obtains:
!
F
F
F
F
F
F
F
=
! +
!
~
p =
! +
!p =
(~
2 µ! - 2 µ! P
) +
!p
p
p
p
F
=
(
F
2 µ! - 2 µ! P
) +
!p,
p
F
because is deviative.
With
-
=
= (T
p
p -
,
-
=
= (T, = = p (T and p
= p = (
p Ti-1)
i-1)
i-1)
i-1)
Note:
One does not hold account in this expression of the variation of the elastic coefficients with
temperature. It is an approximation, without important consequence, since this operator
is useful that to initialize the iterations of Newton. On the other hand, dependence of the tangent operator by
report/ratio with the thermal deformations is well taken into account on the level of the total algorithm
[R5.03.01].
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3 ~
3 ~
One has then:
2µ
! - 2µ!
'
p
R
- (p) p = 0
2
2
!
eq
eq
(3 µ) (~ ~
. !)
what leads to: !p =
thus
'
3 µ + R (p
eq
)
9µ
(~ ~
. !)
~
if
2
=
-
=
p
,
F (, p)
R p
0
!
'
= 2 3 µ + R (p)
eq
()
eq
0,
if
- R p < 0
eq
()
~
!
p
= K
+ 2 µ
-
ij
!kk ij
(!ij!ij)
Note:
Information -
-
eq - R (p) = 0 is preserved in the form of an internal variable which is worth 1
in this case and 0 if -
-
eq < R (p).
The tangent operator binds the vector of virtual deformations * to a vector of virtual constraints
*.
The matrix of tangent rigidity is written for an elastic behavior:
““
(K 1 1 2Μ P)
=
+
and for a plastic behavior:
““
(K 1 1 2µ PC S S
p
)
=
+
-
with S the vector of the deviatoric constraints associated - defined by:
St
= (~ - - -
-
-
-
, ~, ~,
~
2
,
~
2
,
~
2
11 22 33
12
23
31)
and:
(3 µ) 2
1
CP = (2 3 +
µ
'
eq)
R
if -
eq - R (-
1
p)
= 0
= 0 if not
In the case of the first increment of loading, therefore if the state at the previous moment corresponds to one
nonconstrained initial state, the tangent operator is identical to the operator of elasticity.
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3.3
Calculation of the constraints and the internal variables
The decomposition of the deformations makes it possible to write:
= p + (-) + HT
To 1
Maybe, by taking the spherical and deviatoric parts:
~
p
~ = +
HT
= 0
because
~
.
2µ
tr
tr =
tr
HT
because tr p
= 0.
K
3 +
By direct implicit discretization of the relations of behavior for isotropic work hardening, one
obtains then:
3
-
-
~
~
~
~
~
+
2 µ
2 µ
(
)
2 µ
-
-
p
-
~
-
+
= 2
(- +
)
2 µ
eq
3 K
tr =
tr -
HT
-
+ 3 K tr
- 3 K tr
3 K
(- +
) - R (-
p + p
) 0
eq
p
= 0 if (- +
) < R p
p
eq
(- +)
p
0
if (- +
)
= R p
p
eq
(- +)
One defines, to simplify the notations, the tensor E such as:
~
µ
E
2
~ -
~
=
+ 2 µ
E
-
and tr = tr.
2 µ
Two cases arise:
·
(-
)
R (-
+
<
p + p
eq
)
in this case
p
0 are ~
~
= - ~
~ E
=
+ =
thus
(~e) < R p
eq
()
·
(-
)
R (-
+
=
p + p
eq
)
in this case
p 0
thus
(~e) R p
eq
()
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One deduces the algorithm from it from resolution:
·
~
if
E R p then p
0
~
~ -
~
~
that is to say
=
E
=
+ =
eq
()
· if
~
eeq > R p
()
then it is necessary to solve:
~
~
3
- + ~
E = ~ - + ~ + 2 µ p
2
(+
) eq
thus by factorizing ~ -
~
+ and by taking the equivalent value of Von Mises
3
2
p
E
µ
= +
-
1
eq
2 (+
) (
) eq
eq
+
-
that is to say:
E = R p + p
+ 3
p
eq
(
) µ
because:
-
eq
= (+
)
= R (p + p
)
eq
It is a scalar equation out of p, linear or not according to R (p). p is obtained analytically, because
R is a linear function per pieces.
· If work hardening is linear (relation VMIS_ISOT_LINE), one obtains directly:
E -
p =
eq
y - R' p
R' + 3 µ
with:
E E
R'
=
T.
E - AND
· If work hardening is given by a traction diagram, one benefits from the linearity
by pieces to determine p exactly to see [§An1].
Once p determined, one calculates by:
E
~
eq - 3 µ p
-
~
+
=
. ~
E
eeq
and
tr (- +
) = tr E.
Options RAPH_MECA and FULL_MECA carry out both the preceding calculation, which clarifies it
calculation of R one
()
N
()
N
N
I. It is noticed that actually, R ui
= QT I where I is calculated not in function
N
of linked, but of I - 1 and ui.
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Note:
Particular case of the plane constraints.
The model of Von Mises with isotropic work hardening (VMIS_ISOT_LINE or VMIS_ISOT_TRAC) is
also available in plane constraints, i.e. for modelings C_PLAN, DKT,
COQUE_3D, COQUE_AXIS, COQUE_D_PLAN, COQUE_C_PLAN, PIPE, TUYAU_6M.
In this case, the system to be solved comprises an additional equation. This calculation is detailed in
appendix 2.
3.4
Tangent operator. Option FULL_MECA
The option
N
FULL_MECA makes it possible to calculate the tangent matrix Ki with each iteration. The operator
tangent which is used for building it is calculated directly on the preceding discretized system (one notes
to simplify: ~
= ~
- + ~
, p = p + p) and one writes the expressions only in
isothermal case.
· If the tensor of the constraints is on the border of the field, F
= 0 then one have, in
differentiating the expression of the law of normality in ~
~ -
~
= +:
3
~
~
~
: ~
p
~
~
3
2 µ = 2 µ
2 µp
p
p
.~
-
=
+
-
2
eq
2
3
eq
eq
where p ~
~ represents infinitesimal increases around the solution in
incremental elastoplastic problem obtained previously.
Like:
3 ~
: ~
'
= R (p) p
2 eq
by carrying out the tensorial product of the first equation by ~
one a:
2 µ ~: ~ ~
-: ~ = 2 µ.
p,
eq
by eliminating p from the two last equations:
~
~
2 µ: ~
: ~
=
3 µ.
1 + '
R (p)
· So on the other hand if the tensor of the constraints is inside the field, F
< 0, then
the tangent operator is the operator of elasticity.
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By expressing p and ~: ~
in the first equation, one obtains:
3 µ p
2 µ ~ -
~ =
~ + C .p (~: ~) ~,
+
eq
with:
9 2
µ
R' (p) p
1
C
=
1
2
-
p
R'
+ 3 µ
eq
eq
(p)
~
The positive part (: ~
) allows to gather in only one equation the two conditions:
+
· that is to say
F
< 0, which implies p = 0
· that is to say F
= 0
One obtains then:
2 µ
C
~
~
p (~: ~) ~
=
-
+
has
has
while posing:
3 µ p
= 1 + R (p has + p
)
The tangent operator binds the vector of virtual deformations * to a vector of virtual constraints
*.
The matrix of tangent rigidity is written for an elastic behavior:
““
(K 1 1 2Μ P)
=
+
and for a plastic behavior:
2µ
C
““
p
K 1 1
P
S
S
=
+
-
has
has
with S the vector of the deviatoric constraints associated - defined by:
St
= (~ - - -
-
-
-
, ~, ~,
~
2
,
~
2
,
~
2
11 22 33
12
23
31)
and:
1
~
if led to a plasticization and. ~
0
= 0 if not
It is noted that the tangent operator with the system resulting from the implicit discretization differs from the operator
tangent with the problem of speed (RIGI_MECA_TANG). One finds it while making: p = 0 in
expressions of C and A.
p
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3.5
Produced internal variables
The relations of behavior VMIS_ISOT_LINE and VMIS_ISOT_TRAC produce two variables
interns: p and (useful for the calculation of the tangent operator).
4
Relation of Von Mises with linear kinematic work hardening
4.1
Expression of the relation of behavior
This relation is obtained by key word VMIS_CINE_LINE of the key word factor COMP_INCR.
It is written:
~
~
~
- X
3
-
·
3
X
$ % '
P
!
=
!p
- 1
With
2
(
=
p
=
-
-
- X)
!
HT
2
- X
eq
(
)
!
!
eq
X = C p
HT = (T - T Id
ref.)
(-
éq 4.1-1
X) -
0
eq
y
!p = 0 if (- X) -
0
eq
y
!p 0 if (- X) -
= 0
eq
y
y is the elastic limit (the choice of falls on it on the user: it can correspond to the end of
linearity of the real traction diagram, either lawful elastic limit or
conventional… At all events, one uses here the single value defined under ECRO_LINE).
C is the coefficient of work hardening deduced from the data by a simple tensile test.
In this case (tensor of constraints uniaxial, tensor of plastic deformations isochoric and
orthotropic):
X
0
0
L
0
0
L
X
= 0 0
0
X = 0
- L
0
2
0 0 0
X
0
0
- L
2
(
3
- X)
=
X
L -
eq
L
2
P
L
X
C
C
L
=
L =
L -
E
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L
AND
y
E
L
The data materials are those provided under the key word factor ECRO_LINE or ECRO_LINE_FO of
operator DEFI_MATERIAU:
/ECRO_LINE (D_SIGM_EPSI: AND SY: y)
/ECRO_LINE_FO (D_SIGM_EPSI: AND SY: y)
ECRO_LINE_FO corresponds if AND and y depend on the temperature and are then calculated
for the temperature of the point of current Gauss.
The Young modulus E and the Poisson's ratio are those provided under the key words factors ELAS
or ELAS_FO.
y
y
For
L >
E
L
= y + AND L -
E,
but one also has:
3
L -
X L = y
2
L
X L = C L -
E
from where, by eliminating XL and while identifying:
E E
C = 2
T.
3rd - AND
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4.2
Tangent operator. Option RIGI_MECA_TANG
The goal of this paragraph is to calculate the tangent operator Ki - 1 (option of calculation RIGI_MECA_TANG
called with the first iteration of a new increment of load) starting from the results known with
the moment previous Ti - 1.
For that, if the tensor of the constraints with Ti - 1 is on the border of the field of elasticity, one writes
condition:
!F = 0
who must be checked (for the continuous problem in time) jointly in the condition:
F
= 0
with
F = F (- -
X) = (-
-
,
-
- X) -
eq
y
So on the other hand the tensor of the constraints with Ti - 1 is inside the field, F
< 0, then
the tangent operator is the operator of elasticity.
One poses:
-
-
1
0
Dev.
~ -
-
if (- X) -
= (information given by the variable interns)
X and
eq
y
=
-
= 0 if not
The problem of speed is written in this case:
2
1 3 2 µ (~ - X) ~
. !) (~ - X)
if
- X -
= 0
p
()
!
y
= 2 µ 2
C + 2 µ
y
0
if (- X) - < 0
y
eq
~
!
p
= K + 2 µ -
ij
!kk ij
(!ij!ij)
The tangent operator binds the vector of virtual deformations * to a vector of virtual constraints
*.
The matrix of tangent rigidity is written for an elastic behavior:
““
(K 1 1 2Μ P)
=
+
and for a plastic behavior:
““
(K 1 1 2µ PC S S
p
)
=
+
-
with S the vector of the deviatoric constraints associated Dev. defined by:
St =
Dev. Dev. Dev.
Dev.
Dev.
Dev.
(
)
11, 22, 33,
2 12, 2 23, 2 31.
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and:
2
2
µ
1
C
p
= 3
.
2 y 2 µ + C
In the case of the first increment of loading, therefore if the state at the previous moment corresponds to one
nonconstrained initial state, the tangent operator is identical to the operator of elasticity.
4.3
Calculation of the constraints and variables internal
The direct implicit discretization of the continuous relations results in solving:
~ -
~
~
~
- X
p
3
2
µ
= 2 µ +
-
-
=
2 µ p
2 µ
2 µ
2
y
C
X =
-
X
-
+ C p
(
C
-
X)
eq
y
p
= 0 if (- X) <
eq
y
p
0 if not
tr (
K
- +
)
3
=
tr -
tr
tr
-
+ 3 K
- 3 K HT
3 K
One still poses:
~
2 µ
C
E
~ -
~
=
-
2 µ
X
-
+
-
.
2 µ
C
The first equation is also written:
2
~
~
µ
3
~
X
-
~
-
2 µ +
p
-
= +
2 µ
2 µ
2
y
C
by cutting off X
=
X + C
p
-
has each term, one obtains:
C
2
~
~
µ
C
3
~
X
-
-
~
-
2 µ
+
X
X
p
C
P
- -
-
= - +
2 µ
+
2 µ
C
2
y
or, by using the law of flow:
~
3
p
E
= (~
- X) 1
+
(2 µ + C)
2
y
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One still obtains a scalar equation out of p by taking the equivalent values of Von Mises:
3
eeq = y +
(2 µ + C) p
2
what gives directly:
E
-
p
eq
y
= 3 (2µ+C)
2
2 µ
And is obtained by: ~
~ -
~
=
2 µ
2 µ
p
-
+
-
2 µ
By noticing that:
3
~
~
X
3
E
p
-
= p
= p
2
2
E
y
eq
because
~
~
- X
E
=
E
y
eq
one thus has:
E
2 µ
2 µ
(eq - y)
~
~ -
~
=
+ 2 µ
-
+.~
E
-
2 µ
2 µ + C
eeq
The variables intern X are calculated by:
~ E
C
C
3
X =
X
p
-
C
C p
-
+
=
X
-
+
C
C
E
2
eq
Note:
Particular case of the plane constraints.
The direct taking into account of the assumption of the plane constraints in the integration of the model of Von
Settings with linear kinematic work hardening was not made in Code_Aster.
On the other hand, to take into account this assumption, i.e. to use VMIS_CINE_LINE with
modelings C_PLAN, DKT, COQUE_3D, COQUE_AXIS, COQUE_D_PLAN, COQUE_C_PLAN, TUYAU,
TUYAU_6M, one can use the method of condensation static (due to R. of Borst [R5.03.03]) which
allows to obtain a plane state of stresses with convergence of the total iterations of the algorithm of
Newton.
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4.4
Tangent operator. Option FULL_MECA
N
Option FULL_MECA makes it possible to calculate the tangent matrix Ki with each iteration. The operator
tangent which is used for building it is calculated directly on the preceding discretized system (one notes
to simplify: ~
= ~
- + ~
, p = p + p) and one writes the expressions only in
isothermal case.
1
~
if p > 0 and (- X). ~
0
Dev. is posed
= ~
- X and = 0sinon
The tangent operator binds the vector of virtual deformations * to a vector of virtual constraints
*.
Then the matrix of tangent rigidity is written:
““
(K 1 1 2µa PC S S
2
p
)
=
+
-
with S the vector of constraints associated to Dev. by:
St =
Dev. Dev. Dev.
Dev.
Dev.
Dev.
(
)
11, 22, 33,
2 12, 2 23, 2 31.
and:
2
2
µ
1
C
p
= 3
.a
2
1
y 2 µ + C
a1 =
1
2
(µ + C) p
1 + 32
y
3 p
has
2
= a1 1 + C
2
y
4.5
Produced internal variables
The variables intern are 7:
·
tensor
X stored on 6 components,
· the scalar variable.
5 Bibliography
[1]
P. MIALON, Eléments of analysis and numerical resolution of the relations of elastoplasticity.
EDF - Bulletin of Direction of Etudes and Recherches - Série C - N° 3 1986, p. 57 - 89.
[2]
E.LORENTZ, J.M. PROIX, I.VAUTIER, F.VOLDOIRE, F.WAECKEL “
Initiation with
thermo plasticity in Code_Aster. Handbook of reference of the course
”, Note
EDF/DER/HI-74/96/013
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Appendix 1 Relation VMIS_ISOT_TRAC: complements on
integration
Implicit discretization of the relation of behavior led to solve an equation out of p [§5].
eeq - 3 µ p - R p + p
(
) = 0.
One solves the equation exactly while drawing left the linearity per pieces.
One examines initially if the solution could be apart from the terminals of the points of discretization of the curve
R (p), i.e., if p p is a possible solution.
N
For that:
· if
E +
p
3
- p -
0
eq
µ (
N)
N
then one is in the following situation:
eq
R (p + p)
eeq + 3µp -
(
) - 3µ (p - + p) = 0
p
p
N
+ p
-
if the prolongation on the right is linear then:
that is to say
-
N
N - 1
1
=
H 1 =
1 +
p
1
- p
N -
N -
N -
N - (
N - 1)
p - p
N
N - 1
then:
E
- H
p
eq
N
=
- 1
3
1
µ
-
+
N
-
if the prolongation is constant:
E -
p =
eq
N
3 µ
-
if not an error message is transmitted,
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· if not, the solution p is to be sought in the interval [p, p
such as:
I
I + 1]
E
3
1 >
+
p - p
i+
eq
µ (
i+1)
and
E
+
p
3
- p
I
eq
µ (
I)
E
-
eq + 3 µ (p - I
p) = 0
E
-
eq + µ
3 (p - p) = 0
E
-
eq + 3 µ (p - I
p +) = 0
1
p
p
p
I
i+1
+1 -
I
I
=
H
=
+
p - p
for I
= 1 with N - 1
I
I
I I (
I)
p +1 - p
I
I
then, p is such as:
E
- H
p
eq
I
=
and
p + p
,
3
[p p
I
I + 1]
+ µ
I
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Isotropic appendix 2 Ecrouissage in plane constraints
In this case, the system to be solved comprises an equation moreover:
= 0 .On obtains it then
33
following system:
~ -
~
3
+
2 µ
~ - ~ =
2 µ p
2
(- +
) eq
tr = 3 K tr
(- +
) - R (-
p + p) 0
eq
p = 0 if (- +
) < R
eq
(- p + p)
p 0
if (- +
)
= R
eq
(- p + p)
= 0
33
With this assumption, is not entirely known:
N
33 cannot be calculated only starting from ui.
Note:
In the case of modelings other than C_PLAN, therefore for example for modelings of hulls
(DKT, COQUE_3D), the assumptions on the transverse terms of shearing and are defined
13
23
by these modelings (in general, the behavior related to transverse shearing is linear, elastic and
uncoupled from the equations above). These terms thus do not take into consideration here.
One poses =
Q +
y with Q entirely known starting from linked and of elasticity, therefore
0 0 0
Q
Q
Q
y
33
= -
11 +
and
=
- (
22
0 0 0 are unknown.
1
)
0 0 y
Compared to the preceding system, there are an additional unknown factor, Y.
· If
(~ - ~
)
R (-
+
<
p + p
) then p
= 0 thus 2 µ ~ = ~,
eq
i.e. y = 0.
· If not, the technique of resolution consists in expressing y according to p. One obtains one then
nonlinear scalar equation in p.
2 µ
One poses:'~ E
~ -
~
2 µ
-
Q
=
+
. In the same way that for integration except plane constraints, one
2 µ
obtains:
y
~
~
~-
~
3 µ p
2 µ
1
E +
= +
+
.
R (p + p
)
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But this expression utilizes an additional unknown factor y: In particular:
y
~
~
~
-
~
3 µ p
+ 2
33
=
+
33
1
E
µ
33
33
+
R (p + p
)
however y
~
33 = 2 y
3
3 K+
and tr -
(+
) = 3 K tr Q + 3 K+ y + 3 K tr - - 3 K+ HT.
Like:
tr + -
(
)
tr + -
(
)
~
-
-
33 + ~
33 = 33 + 33 -
= 0 -
.
3
3
One obtains an equation binding y and p:
~
3 µ p
-
tr E - 3 K y
E
+ 2 µ 2 y =
1 +
33
3
R (p + p)
3
with
3 K
tr E = 3 K tr - + 3 K tr Q - 3 K HT.
That is to say:
4 µ
3 µ p
~
tr
3 µ
E
E
p
y
+ K1 +
1
3
-
-
R (p + p)
33
= -
-
+
3
(RP + p)
by noticing that:
~
tr
tr
E
E
E
E
33
= 33 -
= 0 -
3
3
and by clarifying µ, K, one obtains:
(31 - 2) p
y
E
=
~
E p + (
2 1 -) R (p + p) 33
to defer in the equation out of p (identical to the preceding cases)
~
E + 2 µ ~ y
(
) - 3µ p R p + p
(
) = 0.
eq
1
y
-
y
where expresses itself there according to p since:
~
=
- 1
3
2
The scalar equation out of p thus obtained is always nonlinear. This equation is solved by a method
of search for zeros of functions, based on an algorithm of secant (cf [R6.03.02]). Once the solution
p known one calculates there then.
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