Code_Aster
®
Version
5.0
Titrate:
Integration of the elastoplastic relations
Date:
20/03/01
Author (S):
J.M. PROIX, E. LORENTZ, P. MIALON
Key:
R5.03.02-C
Page:
1/26
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Organization (S):
EDF/MTI/MN, RNE/MTC
Manual of Reference
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 the 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 the 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.
Code_Aster
®
Version
5.0
Titrate:
Integration of the elastoplastic relations
Date:
20/03/01
Author (S):
J.M. PROIX, E. LORENTZ, P. MIALON
Key:
R5.03.02-C
Page:
2/26
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
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 stresses 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 stresses 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 Work hardening in plane stresses ........................................................................... 25
Code_Aster
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Version
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Titrate:
Integration of the elastoplastic relations
Date:
20/03/01
Author (S):
J.M. PROIX, E. LORENTZ, P. MIALON
Key:
R5.03.02-C
Page:
3/26
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
1 Introduction
1.1
Relations of behaviors described in this document
In the 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
method of Newton [R5.03.01 § 2.2.2.2] one must calculate the nodal forces
R U
I
N
()
=
Q
T
I
N
(options
RAPH_MECA
and
FULL_MECA
) stresses
in
being calculated in
each point of integration of each element starting from displacements
U
I
N
via
relation of behavior. One must also build the tangent operator to calculate
K
I
N
(option
FULL_MECA
).
Before the first iteration, for the phase of prediction, one calculates
K
I
-
1
(option
RIGI_MECA_TANG
).
The calculation of
K
I
-
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
K
I
N
by the option
FULL_MECA
, with
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,
K
I
-
1
, then the calculation of
stress field starting from an increment of deformation, the calculation of the nodal forces
R
and of
stamp tangent
K
in
.
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Titrate:
Integration of the elastoplastic relations
Date:
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Author (S):
J.M. PROIX, E. LORENTZ, P. MIALON
Key:
R5.03.02-C
Page:
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
2
General notations and assumptions on the deformations
All the quantities evaluated at the previous moment are subscripted by
-
.
Quantities evaluated at the moment
T
+
T
are not subscripted.
The increments are indicated by
. 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 stresses.
operator déviatoire:
~
ij
=
ij
-
1
3
kk
ij
.
()
eq
equivalent value of Von Mises:
eq
ij
ij
=
3
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.
3
3
2
K
=
+
µ
modulate compressibility
thermal expansion factor 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
has
:
[
]
“has
=
T
xx
yy
zz
xy
xz
yz
has
has
has
has
has
has
2
2
2
The hydrostatic vector is introduced
“1
and stamps it deviatoric projection
P
:
[
]
“1
=
T
1 1 1 0 0 0
P
Id
1 1
=
-
1
3
““
Code_Aster
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Titrate:
Integration of the elastoplastic relations
Date:
20/03/01
Author (S):
J.M. PROIX, E. LORENTZ, P. MIALON
Key:
R5.03.02-C
Page:
5/26
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
2.1
Partition of the deformations (small deformations)
One writes for any moment:
T
()
=
E
T
()
+
HT
T
()
+
p
T
()
,
with
()
()
()
()
E
T
T T
T
=
-
With
1
with
()
()
()
()
(
)
HT
ref.
T
T T
T T
T
=
-
Id
or in a more general way:
()
()
(
) () (
)
()
(
)
()
HT
def
ref.
ref.
def
ref.
HT
ref.
T
T T T
T
T
T
T T T
T
=
-
-
-
=
-
=
#
and
0
With
depends on the moment
T
via the temperature. The thermal expansion factor
()
()
T T
is an average expansion factor which can depend on the temperature
T
.
temperature
T
ref.
is the temperature of reference, i.e. that for which thermal dilation
is supposed to be null if the average expansion factor is not known compared to
T
ref.
, one can
to use a temperature of definition of the average expansion factor
T
def
(defined by the key word
TEMP_DEF_ALPHA
of
DEFI_MATERIAU
) different from the temperature of reference [R4.08.01].
What leads to:
()
()
()
()
()
()
!
!
!
T
T T
T
T
T
HT
p
=
+
+
-
·
With
1
$
%
&&
'
&&
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
+
With
-
1
(
)
+
HT
with:
With
-
1
(
)
=
With
-
1
T
-
+
T
(
)
-
+
(
)
-
With
-
1
T
-
()
-
and
HT
=
T
-
+
T
(
)
T
-
T
ref.
(
)
-
T
-
()
T
-
-
T
ref.
(
)
(
)
Id
Code_Aster
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Version
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Titrate:
Integration of the elastoplastic relations
Date:
20/03/01
Author (S):
J.M. PROIX, E. LORENTZ, P. MIALON
Key:
R5.03.02-C
Page:
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
2.2 Reactualization
In
STAT_NON_LINE
, under the key word factor
COMP_INCR
, three modes of calculation of the deformations
are possible:
·
“SMALL”
·
“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
:
One writes
X
=
X
O
+
U
I
-
1
+
U
in
, the calculation of the gradients of
U
in
is thus made with the geometry
X
instead of the initial geometry
X
O
.
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
preceding calculation at the specified moment.
Code_Aster
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Titrate:
Integration of the elastoplastic relations
Date:
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Author (S):
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Key:
R5.03.02-C
Page:
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
3
Relation of Von Mises with isotropic work hardening
3.1
Expression of the relations of behavior
These relations are obtained by the key words
VMIS_ISOT_LINE
and
VMIS_ISOT_TRAC
.
For these two relations, the mode of calculation of the deformations is
DEFORMATION: “SMALL”
:
()
()
()
(
)
!
!
~
!
!
!
!
! :
:
:
.
p
eq
HT
eq
eq
eq
p
HT
HT
ref.
p
With
R p
p
R p
p
R p
p
T T
=
=
-
-
-
=
-
<
-
=
=
-
-
·
3
2
0
0
0
0
0
1
$ % '
if
if
speed of plastic deformation,
cumulated plastic deformation,
thermal deformation of origin
:
Id
The function of work hardening
R p
()
is deduced from a simple test tensile monotonous and isothermal
In this case:
()
=
=
=
=
-
-
L
eq
L
LP
L
L
L
p
E
R p
0 0
0
0 0
0
0 0
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 the operator
DEFI_MATERIAU
[U4.43.01].
/ECRO_LINE: (D_SIGM_EPSI:
E
T
SY:
y
)
/ECRO_LINE_FO: (D_SIGM_EPSI:
E
T
SY:
y
)
ECRO_LINE_FO
corresponds if
E
T
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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Integration of the elastoplastic relations
Date:
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Key:
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In this case the traction diagram is as follows:
E
E
T
y
L
L
I.e.:
L
=
E
L
if
L
<
y
E
L
=
y
+
E
T
L
-
y
E
if
L
y
E
.
Note:
y
is the elastic limit (the choice of
y
fall 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
-
R
L
-
L
E
=
0,
from where
R p
()
=
E
T
E
E
-
E
T
p
+
y
.
3.1.2 Relation
VMIS_ISOT_TRAC
The data of material are those provided under the key word factor
TRACTION: (SIGM: F)
, of
the 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.
Code_Aster
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Titrate:
Integration of the elastoplastic relations
Date:
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Key:
R5.03.02-C
Page:
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HI-75/01/001/A
The interpolation compared to the temperature is carried out in the following way:
That is to say
the temperature considered, if there exists
K
such as
K
,
K
+
1
[
]
where
K
indicate the index of
traction diagrams contained in the tablecloth, one point by point builds the traction diagram with
temperature
while interpolating compared to
X-coordinates and ordinates of the points of both
extreme traction diagrams.
K
+
1
K
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 numbers of points of discretization of the traction diagram with
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.
Code_Aster
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Version
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Titrate:
Integration of the elastoplastic relations
Date:
20/03/01
Author (S):
J.M. PROIX, E. LORENTZ, P. MIALON
Key:
R5.03.02-C
Page:
10/26
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
In all the cases, the traction diagram considered is a linear function per pieces:
=
I
+
I
+
1
-
I
I
+
1
-
I
-
I
(
)
for
I
,
I
+
1
[
]
for
I
+
1
N.
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).
y
=
1
E
2
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
E
I
I
I
=
-
.
The function of work hardening is then:
R p
()
=
I
+
I
+
1
-
I
p
I
+
1
-
p
I
p
-
p
I
(
)
for
p
p
I
, p
I
+
1
[
]
.
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 the key word
NEWTON:(MATRIX:“ELASTIC”)
is
present in
STAT_NON_LINE
) by the key words:
/ELAS: (NAKED:
E:
E
)
/ELAS_FO: (NAKED:
E:
E
)
Code_Aster
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Titrate:
Integration of the elastoplastic relations
Date:
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Key:
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
3.2
Tangent operator. Option
RIGI_MECA_TANG
The goal of this paragraph is to calculate the tangent operator
K
I
-
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 previous moment
T
I
-
1
.
For that, if the tensor of the stresses with
T
I
-
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 stresses with
T
I
-
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 previous moment
T
I
-
1
, which is them
only known at the moment of the phase of prediction. One thus obtains:
(
)
(
)
!
!
!
~!
!
~!
!
!
!
!
!,
F
F
F
p p
F
F
p p
F
F
p p
F
F
p p
P
P
=
+
=
+
=
-
+
=
-
+
µ
µ
µ
µ
2
2
2
2
because
F
is deviative.
With
()
-
-
-
-
=
=
T
i-1
,
()
-
-
-
-
=
=
T
i-1
,
()
-
-
-
-
p
p
p
I
T
=
=
1
and
()
p
p
p T
-
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].
Code_Aster
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Titrate:
Integration of the elastoplastic relations
Date:
20/03/01
Author (S):
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Key:
R5.03.02-C
Page:
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
One has then:
()
3
2
2
2
3
2
0
~
!
!
~
!
'
----
µ
µ
eq
eq
p
R p p
-
=
what leads to:
()
()
()
!
~. !~
'
p
R p
eq
=
+
3
3
µ
µ
thus
()
()
()
()
(
)
!
~. !~
~,
(,)
,
!
!
!~
!
'
p
eq
eq
eq
ij
kk
ij
ij
ij
p
R p
F
p
R p
R p
K
=
+
=
-
=
-
<
=
+
-
9
2 3
0
0
0
2
2
µ
µ
µ
if
if
Note:
Information
()
eq
R p
-
-
-
=
0
is preserved in the form of an internal variable
who is worth 1
in this case and 0 if
()
eq
R p
-
-
<
.
The tangent operator binds the vector of virtual deformations
*
with a vector of virtual stresses
*
.
The matrix of tangent rigidity is written for an elastic behavior:
(
)
=
+
K “
“
1 1
P
2
µ
and for a plastic behavior:
(
)
=
+
-
K
C
p
““
1 1
P
S
S
2
µ
with
S
the vector of the deviatoric stresses associated
-
defined by:
(
)
S
T
=
-
-
-
-
-
-
~, ~, ~,
~,
~,
~
11
22
33
12
23
31
2
2
2
and:
()
()
C
R
p
eq
=
+
µ
µ
3
1
3
2
2
'
()
=
-
=
-
-
1
0
0
if
if not
eq
R p
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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HI-75/01/001/A
3.3
Calculation of the stresses and the internal variables
The decomposition of the deformations makes it possible to write:
(
)
=
+
+
-
p
HT
With
1
Maybe, by taking the spherical and deviatoric parts:
~
~
=
+
p
2
µ
because
~
.
HT
=
0
tr
K
tr
HT
=
+
tr
3
because
tr
p
=
0.
By direct implicit discretization of the relations of behavior for isotropic work hardening, one
obtains then:
(
)
(
)
2
3
2 2
2
2
µ
µ
µ
µ
~
~
~
~
~
~
-
+
=
+
+
-
-
-
-
-
-
p
eq
tr
tr
tr
tr
=
+
-
-
-
3
3
3
3
K
K
K
K
HT
(
)
(
)
(
)
(
)
(
)
(
)
-
-
-
-
-
-
+
-
+
=
+
<
+
+
=
+
eq
eq
eq
R p
p
p
R p
p
p
R p
p
0
0
0
if
if
One defines, to simplify the notations, the tensor
E
such as:
~
~
~
E
=
+
-
-
2
2
2
µ
µ
µ
and
tr
tr
E
=
.
Two cases arise:
·
(
)
(
)
-
-
+
<
+
eq
R p
p
in this case
p
E
=
+
=
0
that is to say
~
~
~
~
====
-
-
-
-
thus
()
()
~
E
eq
R p
<
-
·
(
)
(
)
-
-
+
=
+
eq
R p
p
in this case
p
0
thus
()
()
~
E
eq
R p
-
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One deduces the algorithm from it from resolution:
·
if
()
~
eq
E
R p
-
then
p
E
=
+
=
0
that is to say
~
~
~
~
====
-
-
-
-
·
if
~
eq
E
>
R p
-
()
then it is necessary to solve:
~
E
=
~
-
+
~
+
3
2 2
µ
p ~
-
+
~
+
(
)
eq
thus while factorizing
~
~
-
+
and by taking the equivalent value of Von Mises
(
) (
)
µ
eq
E
p
=
+
+
+
-
-
1 32 2
eq
eq
that is to say:
(
)
µ
eq
E
R p
p
p
=
+
+
-
3
because:
(
)
(
)
eq
eq
R p
p
=
+
=
+
-
It is a scalar equation in
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:
p
=
eq
E
-
y
-
R
'
p
-
R
'
+
3
µ
with:
R
'
=
E E
T
E
-
E
T
.
·
If work hardening is given by a traction diagram, one benefits from the linearity
by pieces to determine exactly
p
to see [§An1].
Once
p
determined, one calculates
by:
~
~
. ~
-
+
=
-
µ
eq
E
eq
E
E
p
3
and
(
)
tr
tr
-
+
=
E
.
Options
RAPH_MECA
and
FULL_MECA
both carry out the preceding calculation, which clarifies it
calculation of
R U
I
N
()
. It is noticed that actually,
R U
I
N
()
=
Q
T
I
N
where
I
N
is calculated not in function
of
U
I
N
, but of
I
-
1
and
U
in
.
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Date:
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Key:
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HI-75/01/001/A
Note:
Particular case of the plane stresses.
The model of Von Mises with isotropic work hardening (
VMIS_ISOT_LINE
or
VMIS_ISOT_TRAC
) is
also available in plane stresses, 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
FULL_MECA
allows to calculate the tangent matrix
K
I
N
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 stresses is on the border of the field,
F
=
0
then one has, in
differentiating the expression of the law of normality in
~
~
~
=
+
-
:
2
2
3
2 2
3
2
3
µ
µ
µ
====
p
~
~
~
~
~: ~.~
-
=
+
-
p
p
p
eq
eq
eq
where
p
~ ~
represent infinitesimal increases around the solution in
incremental elastoplastic problem obtained previously.
Like:
()
3
2
~: ~
'
eq
R p p
=
by carrying out the tensorial product of the first equation by
~
one a:
2
2
µ
µ
~: ~ ~: ~
.
-
=
p
eq
,
while eliminating
p
of the two last equations:
()
~: ~
~: ~
'
µ
µ
=
+
2
1
3
R p
.
·
So on the other hand if the tensor of the stresses is inside the field,
F
<
0
, then
the tangent operator is the operator of elasticity.
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Key:
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HI-75/01/001/A
While expressing
p
and
~: ~
in the first equation, one obtains:
(
)
2
3
µ
µ
~
~
~
. ~: ~
~,
-
=
+
+
p
C
eq
p
with:
()
()
C
R p
p
R p
p
eq
eq
=
-
+
9
1
1
3
2
2
µ
µ
'
'
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
has
C
has
p
while posing:
(
)
has
p
R p
p
=
+
+
1
3
µ
The tangent operator binds the vector of virtual deformations
*
with a vector of virtual stresses
*
.
The matrix of tangent rigidity is written for an elastic behavior:
(
)
=
+
K “
“
1 1
P
2
µ
and for a plastic behavior:
=
+
-
K
has
C
has
p
““
1 1
P
S
S
2
µ
with
S
the vector of the deviatoric stresses associated
-
defined by:
(
)
S
T
=
-
-
-
-
-
-
~, ~, ~,
~,
~,
~
11
22
33
12
23
31
2
2
2
and:
=
1
0
0
if
conduit with a plasticization and
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
p
and
has
.
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3.5
Produced internal variables
Relations of behavior
VMIS_ISOT_LINE
and
VMIS_ISOT_TRAC
two variables produce
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 the key word
VMIS_CINE_LINE
key word factor
COMP_INCR
.
It is written:
(
)
(
)
(
)
(
)
(
)
(
)
!
!
~
~
!
~
!
!
!
!
P
eq
eq
HT
p
HT
ref.
eq
y
eq
y
eq
y
p
X
p
X
C
T
T
p
p
=
-
-
=
-
-
=
-
-
=
=
-
-
-
=
-
-
-
-
=
-
·
3
2
3
2
0
0
0
0
0
1
X
X
With
X
Id
X
X
X
$ % '
if
if
éq 4.1-1
y
is the elastic limit (the choice of
y
fall 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 stresses uniaxial, tensor of plastic deformations isochoric and
orthotropic):
=
=
-
-
L
L
L
L
X
X
X
0 0
0
0 0
0
0 0
0
0
0
2
0
0
0
2
X
(
)
-
=
-
=
=
-
X
eq
L
L
L
L
P
L
L
X
X
C
C
E
3
2
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E
T
y
L
E
L
The data materials are those provided under the key word factor
ECRO_LINE
or
ECRO_LINE_FO
of
the operator
DEFI_MATERIAU
:
/ECRO_LINE (D_SIGM_EPSI:
E
T
SY:
y
)
/ECRO_LINE_FO (D_SIGM_EPSI:
E
T
SY:
y
)
ECRO_LINE_FO
corresponds if
E
T
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
.
For
L
>
y
E
L
=
y
+
E
T
L
-
y
E
,
but one also has:
L
L
y
L
L
L
X
X
C
E
-
=
=
-
3
2
from where, while eliminating
X
L
and while identifying:
C
=
2
3
E E
T
E
-
E
T
.
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4.2
Tangent operator. Option
RIGI_MECA_TANG
The goal of this paragraph is to calculate the tangent operator
K
I
-
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 previous moment
T
I
-
1
.
For that, if the tensor of the stresses with
T
I
-
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
eq
y
=
=
-
-
-
,
-
-
-
-
-
-
-
-
-
-
-
-
X
X
So on the other hand the tensor of the stresses with
T
I
-
1
is inside the field,
F
<
0
, then
the tangent operator is the operator of elasticity.
One poses:
(
)
Dev.
eq
y
=
-
=
-
-
=
-
-
-
-
~
(
)
X
X
and
if
information given by the internal variable
if not
1
0
0
The problem of speed is written in this case:
(
)
(
)
(
)
(
)
(
)
(
)
!
~
. !~ ~
!
!
!~
!
p
y
y
y
ij
kk
ij
ij
ij
p
C
K
=
-
-
+
-
-
=
-
-
<
=
+
-
1
2
3
2
2
2
0
0
0
2
2
µ
µ
µ
µ
X
X
X
X
if
if
eq
The tangent operator binds the vector of virtual deformations
*
with a vector of virtual stresses
*
.
The matrix of tangent rigidity is written for an elastic behavior:
(
)
=
+
K “
“
1 1
P
2
µ
and for a plastic behavior:
(
)
=
+
-
K
C
p
““
1 1
P
S
S
2
µ
with
S
the vector of the deviatoric stresses associated
Dev.
defined by:
S
T
=
11
Dev.
,
22
Dev.
,
33
Dev.
, 2
12
Dev.
, 2
23
Dev.
, 2
31
Dev.
(
)
.
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HI-75/01/001/A
and:
C
p
=
3
2
2
µ
y
2
1
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 stresses and variables internal
The direct implicit discretization of the continuous relations results in solving:
(
)
(
)
(
)
2
2
2
2
3
2 2
0
0
3
3
3
3
µ
µ
µ
µ
µ
p
y
p
eq
y
eq
y
HT
p
C
C
C
p
p
K
K
K
K
=
+
-
=
-
=
+
-
=
-
<
+
=
+
-
-
-
-
-
-
-
-
~
~
~
~
tr
tr
tr
tr
X
X
X
X
X
if
if not
One still poses:
~
~
~
E
C
C
=
+
-
-
-
-
-
2
2
2
µ
µ
µ
X
.
The first equation is also written:
2
2
2
3
2 2
µ
µ
µ
µ
++++
~
~
~
~
+
=
-
-
-
p
y
X
while cutting off
X
X
=
+
-
-
C
C
C
p
has each term, one obtains:
2
2
2
3
2 2
µ
µ
µ
µ
+
+
+
+
~
~
~
~
+
-
= -
- +
-
-
-
-
C
C
p
C
y
X
X
X
P
or, by using the law of flow:
(
)
(
)
~
~
1
1
1
1 ++++
E
X
=
-
+
3
2 2
µ
C
p
y
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One still obtains a scalar equation in
p
by taking the equivalent values of Von Mises:
(
)
µ
eq
E
y
C
p
=
+
+
+
+
+
3
2 2
what gives directly:
(
)
p
C
eq
E
y
=
-
+
µ
3
2 2
And
is obtained by:
~
~
~
=
+
-
-
-
2
2
2
2
µ
µ
µ
µ
p
By noticing that:
----
p
X
=
=
3
2
3
2
p
p
y
E
eq
E
~
~
because
~
~
----
X
y
E
eq
E
=
one thus has:
(
)
~
~
~
. ~
=
+
-
+
-
-
-
+
2
2
2
2
2
µ
µ
µ
µ
µ
C
eq
E
y
eq
E
E
Internal variables
X
are calculated by:
X
X
X
E
=
+
=
+
-
-
-
-
C
C
C
C
C
C p
p
eq
E
3
2
~
Note:
Particular case of the plane stresses.
The direct taking into account of the assumption of the plane stresses 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
,
PIPE
,
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
The option
FULL_MECA
allows to calculate the tangent matrix
K
I
N
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.
One poses
Dev.
=
~
-
X
and
(
)
=
-
1
0
0
0
if
if not
>
and
p
~
. ~
X
The tangent operator binds the vector of virtual deformations
*
with a vector of virtual stresses
*
.
Then the matrix of tangent rigidity is written:
(
)
=
+
-
K
has
C
p
““
1 1
P
S S
2
2
µ
with
S
the vector of stresses associated with
Dev.
by:
S
T
=
11
Dev.
,
22
Dev.
,
33
Dev.
, 2
12
Dev.
, 2
23
Dev.
, 2
31
Dev.
(
)
.
and:
C
p
=
3
2
2
µ
y
2
1
2
µ +
Turnover
1
has
1
=
1
1
+
3
2
2
µ +
C
(
)
p
y
has
2
=
has
1
1
+
3
2 C
p
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, Elements of analysis and numerical resolution of the relations of elastoplasticity.
EDF - Bulletin of the Management of the Studies and Search - Series 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. Manual of reference of the course
”, Note
EDF/DER/HI-74/96/013
Code_Aster
®
Version
5.0
Titrate:
Integration of the elastoplastic relations
Date:
20/03/01
Author (S):
J.M. PROIX, E. LORENTZ, P. MIALON
Key:
R5.03.02-C
Page:
23/26
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Appendix 1 Relation
VMIS_ISOT_TRAC
: complements on
integration
Implicit discretization of the relation of behavior led to solve an equation in
p
[§5].
eq
E
-
3
µ
p
-
R p
-
+
p
(
)
=
0.
One solves the equation exactly while firing 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
N
is a possible solution.
For that:
·
if
(
)
µ
eq
E
N
N
p
p
+
-
-
-
3
0
then one is in the following situation:
p
-
+
p
R p
-
+
p
(
)
eq
E
+
3
µ
p
-
(
) -
3
µ
p
-
+
p
(
) = 0
p
N
eq
-
if the prolongation on the right is linear then:
that is to say
(
)
N
N
N
N
N
N
N
N
N
p
p
H
p
p
-
-
-
-
-
-
-
-
=
-
-
=
+
-
1
1
1
1
1
1
1
then:
p
H
eq
E
N
N
=
-
+
-
-
µ
1
1
3
-
if the prolongation is constant:
p
=
eq
E
-
N
3
µ
-
if not an error message is transmitted,
Code_Aster
®
Version
5.0
Titrate:
Integration of the elastoplastic relations
Date:
20/03/01
Author (S):
J.M. PROIX, E. LORENTZ, P. MIALON
Key:
R5.03.02-C
Page:
24/26
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
·
if not, the solution p is to be sought in the interval
[
]
p p
I
I
,
+
1
such as:
(
)
µ
I
eq
E
I
p
p
+
-
+
>
+
-
1
1
3
and
(
)
µ
I
eq
E
I
p
p
+
-
-
3
(
)
µ
eq
E
I
p
p
+
-
=
-
3
0
p
I
+
1
p
I
(
)
µ
eq
E
I
p
p
+
-
=
-
+
3
0
1
p
(
)
µ
eq
E
p
p
+
-
=
-
3
0
(
)
I
I
I
I
I
I
I
I
I
p
p
H
p
p
I
N
=
-
-
=
+
-
=
-
+
+
-
1
1
1
1
for
with
then,
p
is such as:
[
]
p
H
p
p
p p
eq
E
I
I
I
I
=
-
+
+
-
+
µ
3
1
and
,
Code_Aster
®
Version
5.0
Titrate:
Integration of the elastoplastic relations
Date:
20/03/01
Author (S):
J.M. PROIX, E. LORENTZ, P. MIALON
Key:
R5.03.02-C
Page:
25/26
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Isotropic appendix 2 Work hardening in plane stresses
In this case, the system to be solved comprises an equation moreover:
33
0
=
.On obtains it then
following system:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
2
3
2 2
3
0
0
0
0
33
µ
µ
~
~
~
~
tr
tr
-
=
+
+
=
+
-
+
=
+
<
+
+
=
+
=
-
-
-
-
-
-
-
-
p
K
R p
p
p
R p
p
p
R p
p
eq
eq
eq
eq
if
if
With this assumption,
is not entirely known:
33
cannot be calculated only from
U
I
N
.
Note:
In the case of modelings other than
C_PLAN
, therefore for example for modelings of hulls
(
DKT
,
COQUE_3D
), assumptions on the transverse terms of shearing
13
and
23
are defined
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 from
U
I
N
and of elasticity, therefore
(
)
33
11
22
1
0 0
0
0 0
0
0 0
Q
Q
Q
y
y
= - -
+
=
and
is unknown.
Compared to the preceding system, there is an additional unknown factor,
y
.
·
If
(
)
(
)
~
~
-
-
+
<
+
=
eq
R p
p
p
then
0
thus
2
µ
~
~,
=
i.e.
y
=
0.
·
If not, the technique of resolution consists in expressing
y
according to
p
. One then is obtained
nonlinear scalar equation in
p
.
One poses:'
~
~
~
E
Q
=
+
-
-
2
2
2
µ
µ
µ
. In the same way that for integration except plane stresses, one
obtains:
(
)
~
~
~
~
E
y
p
R p
p
+
=
+
+
+
-
2
1
3
µ
µ
.
Code_Aster
®
Version
5.0
Titrate:
Integration of the elastoplastic relations
Date:
20/03/01
Author (S):
J.M. PROIX, E. LORENTZ, P. MIALON
Key:
R5.03.02-C
Page:
26/26
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
But this expression utilizes an additional unknown factor
y
: In particular:
(
)
~
~
~
~
µ
µ
E
y
p
R p
p
33
2
1
3
33
33
33
+
=
+
+
+
-
however
~
33
y
=
2
3
y
and
tr
-
+
(
)
=
3 K tr
Q
+
3 K
+
y
+
3 K
+
3 K
-
tr
-
-
3 K
+
HT
.
Like:
~
33
-
+
~
33
=
33
-
+
33
-
tr
+
-
(
)
3
=
0
-
tr
+
-
(
)
3
.
One obtains an equation flexible
y
and
p
:
~
E
33
+
2
µ
2
3
y
=
1
+
3
µ
p
R p
+
p
(
)
-
tr
E
-
3 K
y
3
with
tr
E
=
3 K
3 K
-
tr
-
+
3 K tr
Q
-
3 K
HT
.
That is to say:
(
)
(
)
y
K
p
R p
p
p
R p
p
E
E
4
3
1
3
3
1
3
33
µ
µ
µ
+
+
+
= -
-
+
+
-
-
~
tr
by noticing that:
~
33
E
=
33
E
-
tr
E
3
=
0
-
tr
E
3
and while clarifying
µ
, K
, one obtains:
(
)
() (
)
y
p
E p
R p
p
E
=
-
+
-
+
3 1 2
2 1
33
~
to defer in the equation in
p
(identical to the preceding cases)
~
E
+
2
µ
µ
µ
µ
~
y
(
)
eq
-
3
µ
p
-
R p
-
+
p
(
)
=
0.
where
y
express yourself according to
p
since:
~
y
y
=
-
-
3
1
1
2
The scalar equation in
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
y
then
.