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Titrate:
Calculation of the cyclic limiting states with Method ZAC
Date:
11/05/01
Author (S):
S. TAHERI, J. ANGLES
Key:
R7.06.01-A
Page:
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R7.06 booklet:
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EDF/MTI/MN
Manual of Reference
R7.06 booklet:
Document: R7.06.01
Calculation of the cyclic limiting states with Method
ZAC
Summary
This method is based on linear kinematic work hardening. It gives an approximation for the adapted state
and two approximations for the adapted state of the characteristics in stress and deformation of the cycles
limits, for a structure under thermomechanical periodic loading in the field of plasticity. That
is made using three elastic thermo calculations at weak costs calculation.
Code_Aster
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Titrate:
Calculation of the cyclic limiting states with Method ZAC
Date:
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S. TAHERI, J. ANGLES
Key:
R7.06.01-A
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HI-75/01/001/A
1 Introduction
Under a periodic loading, evolution of the deformation, or displacement, according to
a many cycles of the various points of a structure are concretized in three forms: the adaptation, where
all the points of the structure reach an elastic stable absolute limit; accommodation, where at least
a point of the structure reaches an elastoplastic stable absolute limit; and the ratchet where for at least one
not there is a constant increment of the deformation to each cycle. For a material describes by a law
from linear kinematic work hardening one can obtain only accommodation or adaptation.
To study the problems of fatigue or progressive deformation, for a structure, one has
generally need to know the stress and strain state to the absolute limit, i.e. them
values of the amplitudes of strain and stress as well as the mean stress for fatigue,
and the value of the maximum deformation for the progressive deformation. These values can be
obtained using a cyclic law of behavior. Nevertheless if the absolute limit is obtained after one
a significant number of cycles calculations can be very long.
The post-processor
POST_ZAC
provides an approximation of the characteristics in stress and in
deformation quoted above. More precisely it proposes two estimates for each
characteristic in stress or deformation. The duration of calculations corresponds to 3 or 4 calculations
rubber bands. It should be noted that in the model suggested the constants are independent of
temperature. If the temperature varies it is necessary to choose the constants of material to one
optimal temperature for calculations.
2 Materials with linear kinematic work hardening under
periodic loading
One in the case of places an elastoplastic structure at linear kinematic work hardening strictly
positive and the following assumptions are made:
·
Quasi-static evolution;
·
Infinitesimal deformations;
·
The material is with linear elasticity independent of the temperature;
·
Periodic loadings (thermics
T
, internal forces
F
, external forces
F
on
F
,
imposed displacements
U
on
U
) on the structure
and of border
;
·
In addition, it is supposed that the field of elasticity is defined by the criterion of Von Mises:
(
) (
)
3
2
~
~
~
-
-
-
y
y
,
with the diverter of the stresses
~
()
=
-
ij
ij
tr
1 3
, elastic limit
y
, the internal variable
=
C
p
where
C
H
=
2 3
(
H
E E
E E
T
T
=
-
/(
)
where
E
T
is the slope of the traction diagram
during work hardening and where
p
is the plastic deformation.
Code_Aster
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Titrate:
Calculation of the cyclic limiting states with Method ZAC
Date:
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Key:
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F
U
F
Appear 2-a: Types of loadings of the structure
Within this framework and under the assumptions above there is the following theorem [bib1]:
Theorem: (Generalized Melan). Any solution of the problem of evolution tends towards a solution
periodical in stress and deformation. If the local amplitude of plastic deformation is null one
known as that there is local adaptation, if not it is said that there is local accommodation. So at least a point of
structure is adapted one says that the structure is adapted.
3
Presentation of the simplified method, ZAC
Method ZAC [bib2], [bib3] is a method which is based on the kinematic model of work hardening
linear and which gives in a simplified and inexpensive way of the approximations of the characteristics of
cycle limit in stress and plastic deformation.
For the adapted case, one has for each component the values of: the amplitude of stress, of
mean stress and of the limiting plastic deformation.
For the case adapted for each component, one has two values for the amplitude of the stress,
two values for the amplitude of the plastic deformation, a value for the mean stress and
a value for the average plastic deformation.
One can summarize the method in the following way:
One breaks up the stress as the sum of a first term representing the calculated stress
with an elastic behavior
el
and of a second called residual stress
. In term of
diverters, that is written:
~
~
~
=
+
el
.
The use of the modified internal parameter will make it possible to build in each point the absolute limit of
structure, independently of what occurs for the other points.
The criterion of plasticity being:
~
-
y
the idea-key of the method is to make the calculations uncoupled in each point from the structure. For it
to make, one introduces the modified parameter:
!
~
= -
.
is written then:
~
!
el
y
-
.
Code_Aster
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Titrate:
Calculation of the cyclic limiting states with Method ZAC
Date:
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S. TAHERI, J. ANGLES
Key:
R7.06.01-A
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HI-75/01/001/A
The simplification introduced by method ZAC lies in the way of calculating
!
with the absolute limit, in
each point independently from/to each other and to bring back calculation to 3 or 4 elastic designs.
The approximation of
!
with the absolute limit,
!
lim
, is differently given according to whether there is adaptation or
accommodation. This approximation is obtained starting from the initial value of the tensor
!
, which is
noted
!
0
.
Thus, to summarize, if the elastoplastic thermo problem of origin is written:
(
)
(
)
,
,
,
(
),
P
div
F
N
F
U
U
U
K
F
T
U
p
HT
1
1 2
0
=
=
=
+
=
=
-
-
on
on
+ standard law of elastoplastic behavior in linear kinematics.
The thermo problem elastic associated is written:
(
)
(
)
,
,
,
(
),
P
div
F
N
F
U
U
U
K
el
el
F
el
el
T
el
el
U
el
el
HT
2
1 2
0
=
=
=
+
=
=
-
on
on
And by making the difference between these two assemblies of equations by using the relation
!
~
=
-
C
p
,
as well as the following definitions:
= -
el
,
innate
el
= -
and
U
U U
innate
el
= -
the following problem is found:
(
)
(
)
(
)
()
,
,
,
~/
!
P
div
N
U
U
U
K
C
C
K
F
innate
innate
T
innate
innate
U
innate
el
el
p
=
=
=
+
=
=
+
+
= -
=
-
-
-
0
0
1 2
0
1
on
on
It is an elastic problem with an initial deformation equalizes with
!
C
where
,
innate
and
U
innate
are them
unknown factors.
Code_Aster
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Titrate:
Calculation of the cyclic limiting states with Method ZAC
Date:
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Author (S):
S. TAHERI, J. ANGLES
Key:
R7.06.01-A
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HI-75/01/001/A
4 Passage in the space of the modified parameters and
local construction of the solution
In the continuation one will distinguish the cases where there is an initial state no one or not within the meaning of method ZAC.
In the case of an initial state no one the tensor
!
0
is taken equal to zero. One will distinguish in the same way the cases where
the way loading in each point of the solid closely connected (is often named radial in the case of this
method) or not closely connected. Consequently, the definition of these two concepts is given.
Definition 1: Initial State not no one within the meaning of method ZAC.
It is said that one has an initial state not no one when one makes an elastoplastic incremental calculation until one
level of loading given and which one calculates
!
~
0
=
-
C
p
. The level of the loading considered
is generally the level of the maximum loading (It is desirable to take into account a state
initial not no one when, for example, the first cycle of loading involves a deformation
important of the structure.)
Definition 2: Way of loading refines within the meaning of method ZAC.
It is said that a way loading is closely connected if in each point of the structure, for a behavior
rubber band, the way of the stresses is closely connected.
4.1
Adaptation and accommodation in the case of a loading closely connected
In a point
X
structure
, one defines
F
X
el
()
, such as:
F
X
X T
X T
X
X
el
T T
el
el
()
max ~ (,)
~ (,)
~
()
~
()
,
=
-
-
0 1
0
1
max
el
min
el
,
where
~ (,)
el
X T
is the tensor deviatoric of the elastic stresses at the point
X
and at the moment
T
and where them
moments
T
0
and
T
1
correspond to the extrema of the cycle of loading. The moment
T
0
can be equal to
zero. One defines
F
el
such as:
F
F
X
X
X
el
X
el
X
el
el
=
=
-
max
()
max ~
()
~
()
max
min
.
The comparison of
F
el
with the value of the elastic limit of material allows to know if the state
limit of the structure is of adapted or adapted type:
F
el
y
2
adaptation
F
el
y
>
2
accommodation.
4.1.1 Case of the adaptation
If the structure is adapted, there is a limiting, fixed field in time, modified parameters
!
, noted
!
lim
, such as
!
lim
belongs to the intersection
C
L
of the two convex ones of center
~
min
el
and
~
max
el
and of
radius
y
, to see the figure [Figure 4.1-a], [bib3], [bib4].
Code_Aster
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Titrate:
Calculation of the cyclic limiting states with Method ZAC
Date:
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Key:
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C
L
y
y
~
min
el
~
max
el
Appear 4.1-a: the intersection of the two convex ones
C
L
, in the case of the adaptation.
In the case of the adaptation the value limits
!
, noted
!
lim
is determined by projection
orthogonal of
initial, noted
!
0
on the convex one
C
L
according to the rules presented on the figures
[Figure 4.1-b], [Figure 4.1-c], [Figure 4.1-d].
el
min
el
max
!
!
lim
=
0
C
L
~
~
Appear 4.1.b: Cas where
!
0
is strictly included in the convex one
C
L
!
0
el
min
C
L
el
max
!
lim
~
~
Appear 4.1-c: Case where
!
0
does not belong to the convex one
C
L
,
but belongs to the cone of node
~
max
el
, (or
~
min
el
)
!
lim
is on the edge of the convex one
C
L
Code_Aster
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Titrate:
Calculation of the cyclic limiting states with Method ZAC
Date:
11/05/01
Author (S):
S. TAHERI, J. ANGLES
Key:
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Page:
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!
0
max
el
min
el
!
lim
C
L
~
~
Appear 4.1-d: Case where
!
0
does not belong nor to the convex one
C
L
,
nor with the cone of node
~
max
el
,
!
lim
is on the edge of the convex one
C
L
In the case of the adaptation, one can take an initial state no one
!
0
0
=
, or a calculation is made
elastoplastic incremental which makes it possible to have an initial state not no one
!
~
0
=
-
C
p
to find
often a better result.
4.1.2 Case of accommodation
When the structure is adapted, i.e. in at least one of the points of the structure them
two convex of center
~
min
el
and
~
max
el
have an empty intersection, one is resulted in defining the three
sizes which follow, at the points where there is accommodation:
1) the parameter of average modified internal work hardening, noted
!
moy
;
2) the amplitude of the parameter of modified internal work hardening lower, noted
!
inf
;
3) the amplitude of the parameter of modified internal work hardening higher, noted
!
sup
.
These three sizes are represented on the figure [Figure 4.1-e].
!
max
!
min
!
sup
el
min
el
max
!
moy
!
inf
~
~
Appear 4.1-e: Representation of
!
moy
,
!
inf
and
!
sup
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The first two sizes are calculated
!
moy
and
!
inf
as follows:
(
)
!
~
~
!
~
~
~
~
inf
max
min
max
min
moy
el
el
el
el
y
el
el
=
+
=
-
-
-
min
max
2
1
2
and
Calculation of
!
sup
The calculation of
!
sup
represented on the figure [Figure 4.1-e] is given below:
In each point of the structure, if there is plasticization, one a:
“
~
~
p
=
-
-
with
>
0
One can then write, in term of modified parameter, that:
“
! ~
! ~
p
el
el
= -
-
-
, with
>
0
.
Like
~
! ~
-
-
=
el
y
, in each point where there is plasticization,
!
is on the edge of one
swell of center
~
el
and of radius
y
[Figure 4.1-e]. Size
“
p
is the interior normal with this
swell at the point
!
.
It is supposed that if one goes very far in work hardening, one a:
!
“~”
=
el
, [bib3].
For a loading refines, in other words, when one passes from
min
el
with
max
el
one has
!
~
~
~
=
-
=
max
min
el
el
el
, i.e. direction of displacement of
!
is equal to
~
el
.
Resolution of the homogeneous elastic problem
()
P
with the modified constants and the deformation
initial
~
el
C
, cf [§3], one deduces
p
and thus a direction for
“
p
. By holding the same one
reasoning when one passes from
~
max
el
with
~
min
el
, one obtains
!
~
= -
el
, which gives to
“
p
direction opposed to the preceding case, (cf [Figure 4.1-e]). One takes finally
!
!
!
sup
min
max
=
-
.
Code_Aster
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Titrate:
Calculation of the cyclic limiting states with Method ZAC
Date:
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S. TAHERI, J. ANGLES
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4.2 Adaptation and accommodation in the case of a loading not
closely connected
When the loading is nonclosely connected method ZAC is operational only in the case of
the adaptation.
Case of the adaptation
At the point
X
one considers the intersection of the balls of center
~ (,)
el
X T
over one period. In the case of
the adaptation this intersection is nonempty and convex. One thus defines, as in the case refines,
!
lim
by the projection of
!
0
on this convex, [bib3], [bib4]. Method of projection used in
Code_Aster is successive projection on the intersection two to two of the spheres.
Case of accommodation
In this case the rules suggested are not rigorous [bib5] and there is no option associated in
Code_Aster. One will use approximations closely connected of the loading in this case.
5
Return in the space of the parameters of origin
Once one calculated
!
lim
,
!
moy
,
!
inf
and
!
sup
there is an initial deformation for
problem
()
P
[§3] which will be able to take the following values in turn:
!
lim
C
,
!
moy
C
,
!
inf
C
and
!
sup
C
.
The problem then is solved
()
P
with the initial deformations to obtain above:
1) in the case of the adaptation:
U
lim
,
lim
,
lim
and
lim
p
;
2) in the case of accommodation:
U
moy
,
moy
,
moy
,
moy
p
,
U
inf
,
inf
,
inf
,
inf
p
,
U
sup
,
sup
,
sup
and
sup
p
.
Code_Aster
®
Version
5.0
Titrate:
Calculation of the cyclic limiting states with Method ZAC
Date:
11/05/01
Author (S):
S. TAHERI, J. ANGLES
Key:
R7.06.01-A
Page:
10/10
Manual of Reference
R7.06 booklet:
HI-75/01/001/A
6 Operator
POST_ZAC
in Code_Aster
The operator post-processor
POST_ZAC
need for the following data has: the model, the material, and them
two moments of the loading. If the initial state is null (
!
0
0
=
) the concept is used
EVOL_ELAS
. If the initial state is not null,
!
~
0
=
-
C
p
being obtained starting from a calculation
elastoplastic, one uses in more the concept
EVOL_NOLI
.
At the exit
POST_ZAC
give:
U
lim
,
lim
,
lim
and
lim
p
if it there has adaptation and gives:
U
moy
,
moy
,
moy
,
moy
p
,
U
inf
,
inf
,
inf
,
inf
p
,
U
sup
,
sup
,
sup
and
sup
p
in
case where there is accommodation. The document U associated is [U4.74.05].
The case test associated with the method is: SSNA100 [V6.01.100].
7 Bibliography
[1]
“Quasistatic Problems in viscoplasticity” of B. Halphen, Thèse of doctorate of state be
mathematical sciences presented at the university Pierre and Marie Curia Paris VI, 1978.
[2]
J. Rack, Thesis of doctor-engineer, Paris 1977.
[3]
“Method ZAC” of S. TAHERI, Note EDF-DER HI-71/6139, 1989.
[4]
“Studies on method ZARKA for an analysis with fatigue” of Mr. F. ROBBE and
B. AUTRUSSON, report ECA/DRN/DMT 93/361, 1993.
[5]
“Simplified Analysis of the structures élasto-visco-plastics under cyclic loadings” of
G. INGLEBERT, Thesis of state Paris VI, 1983.
[6]
“RCC-MR Rules of design and construction of the hardware of the nuclear boilers
RNR AFCEN “, edition 1985.