Code_Aster
®
Version
4.0
Titrate:
Indicators of discharge and loss of proportionality of the loading
Date:
06/04/98
Author (S):
G. DEBRUYNE
Key:
R4.20.01-A
Page:
1/6
Manual of Reference
R4.20 booklet: Methods of analysis
HI-74/98/008/A
Organization (S):
EDF/IMA/MN
Manual of Reference
R4.20 booklet: Methods of analysis
Document: R4.20.01
Indicators of discharge and loss of
proportionality of the loading in elastoplasticity
Summary
One presents a whole of scalar parameters called indicators, allowing to appreciate a loss of
proportionality of a loading during its history. Two types of indicators are proposed:
indicators being appeared as scalar fields allowing to detect the zones of the structure
undergoing nonradial discharges or loadings, and total indicators integrated on an area of
the structure chosen by the user. The latter are more especially intended for the evaluation of the validity of
rate of refund of energy in nonlinear breaking process.
Code_Aster
®
Version
4.0
Titrate:
Indicators of discharge and loss of proportionality of the loading
Date:
06/04/98
Author (S):
G. DEBRUYNE
Key:
R4.20.01-A
Page:
2/6
Manual of Reference
R4.20 booklet: Methods of analysis
HI-74/98/008/A
1 Introduction
1.1
Definition of a loading proportional
Considering a structure subjected to a thermomechanical loading in the interval of time [0, T]
it will be said that this loading is proportional (or radial) with the material point P if the field of
stress represented in this point by the tensor
is proportional to a tensor independent of
the moment considered, the proportionality factor being a monotonous function of time.
Formally, that will be expressed by:
[]
=
>
0
0
0
, (,)
()
(), ()
T
P
P
monotonous function in [0, T].
This definition implies, in particular, that the main directions of the stresses remain constant,
at the point considered, throughout the way of loading (these directions can be of course variable
of a point with another).
1.2
Importance of the loading proportional and utility of indicators
For plastic materials, the mechanical fields depend on all the history run out during
the way of loading. The laws of flow thus have an incremental character and their integration
depends on each case of loading. A notable exception precisely relates to the loading
proportional for which the law of flow can be integrated once and for all. For example, the law
of plasticity of Prandtl-Reuss based on the criterion of Von Mises can be replaced by a law
nonlinear rubber band (called law of Hencky-Settings). Cases of loading strictly proportional
are rather rare. Indeed, it is necessary to fill a great number of conditions to carry out such a case [bib1]
and these last are seldom checked for the industrial structures. One can even say that
for structures presenting of the geometrical defects such as fissures, these conditions are not
never strictly checked.
When the loadings are multiaxial, cyclic, or thermo transitory mechanics, some
sections of the way of loading can be strongly nonproportional. It is then useful to identify
these sections and to evaluate the importance of the loss of proportionality, so for example adjusting
discretization in time of the elastoplastic problem for the section considered, or to measure the validity
certain postprocessings (in breaking process for example).
1.3
Various types of indicators of loss of proportionality
It seems difficult to define a single and simple size which could detect at the same time areas
space of loss of proportionality and the sections of path of loading (temporal areas)
in a material point. This is why we propose four scalar sizes having each one them
specificity: two, defined by fields measuring in each point the discharge and the deviation of
stresses between two pitches of time (indicating buildings), two others of more total nature,
characterizing in a given zone of the structure a history of loading nonproportional.
Code_Aster
®
Version
4.0
Titrate:
Indicators of discharge and loss of proportionality of the loading
Date:
06/04/98
Author (S):
G. DEBRUYNE
Key:
R4.20.01-A
Page:
3/6
Manual of Reference
R4.20 booklet: Methods of analysis
HI-74/98/008/A
2 Indicators
buildings
The goal of these indicators is to determine the zones of the structure where, at one particular moment, occurs
that is to say a discharge or a loss of radiality of the stress field. They are produced in
postprocessing of a static or dynamic calculation, 2D or 3D, using a law of behavior
rubber band or not. They are appeared as fields of scalars whose examination can
to be carried out by tracing their isovaleurs by a graphic post-processor.
2.1
Indicator of discharge
This indicator measures at the point M and between the moment
T
and
T
T
+
, relative variation of the standard of
stresses within the meaning of Von Mises. He is written formally:
I
M T
T
M T
M T
T
1
=
+
-
+
(,
)
(,)
(,
)
. This
quantity is negative in the event of local discharge at the point Mr. the standard
(
,)
M T
can be written
four ways different according to the choice from the modelisator:
1)
(,)
.
M T
D
D
=
3
2
, where
D
is the deviatoric part of the tensor of the stresses (this
standard is useful in plasticity with isotropic work hardening).
2)
(,)
.
M T
=
3
2
, where one considers the totality of the tensor of the stresses in order to detect by
example pressure decreases hydrostatic.
3)
(,)
(
).(
)
M T
D
D
=
-
-
3
2
X
X
, with
X
the tensor of the stress of recall in
case of an elastoplastic law with a kinematic work hardening.
4)
(,)
(
).(
)
M T
=
-
-
3
2
X
X
2.2
Indicator of loss of radiality
This indicator measures at the point M and between the moment
T
and
T
T
+
, variation of the direction of
stresses. It is written:
I
M T
M T
2
1
= -
(,).
(,)
, where the scalar product “. ” is associated the one of the four standards
the preceding ones. This quantity is null when the radiality is preserved during the increment of time.
Note:
These indicators are closely related to the discretization in time of the problem. In particular, if
this discretization is too coarse, one can not detect the discharge or the loss very well of
radiality occurring during the increment of time.
Code_Aster
®
Version
4.0
Titrate:
Indicators of discharge and loss of proportionality of the loading
Date:
06/04/98
Author (S):
G. DEBRUYNE
Key:
R4.20.01-A
Page:
4/6
Manual of Reference
R4.20 booklet: Methods of analysis
HI-74/98/008/A
3 Indicators
total
These indicators are intended to detect if, during the history of the structure and until the moment
current
T
, and for a zone of the structure chosen by the modelisator, there was loss of proportionality
loading (these indicators thus leave a trace of the history contrary to the indicators
buildings which are instantaneous). They are only usable within the framework of a behavior
elastoplastic with isotropic work hardening (in 2D or 3D).
3.1
Indicator on the parameters of plasticity
This quantity allows, in the case of the plasticity of Von Mises with isotropic work hardening, on the one hand
of knowing (on average about an area
S
field
) if stresses and deformations
plastics have the same directions and if the plastic threshold is reached at the current moment, and in addition if
during the history the plastic deformation changed direction. This quantity is written:
I
R p p D
S
p
Y
S
3
1
1
=
-
+
(
.
(
()) )
,
where
Y
is the initial plastic threshold,
R
extension of the surface of load related to work hardening and
p
cumulated plastic deformation. The scalar product is associated the standard within the meaning of Von Mises. This
indicator is standardized and has a value ranging between zero and one. It is null if the loading has
preserved its character proportional in each point of
S
throughout the past history.
Notice 1:
The indicator is not affected if during the history, there were discharges then refills
rubber bands without change of direction of the stresses when one reconsiders the threshold
(cf [Figure 3.1-a]).
() ()
p T
p T
1
2
=
I
3
0
=
I
3
0
()
Y
R p
+
Y
()
p T
1
()
p T
2
Appear 3.1-Error! Argument of unknown switch. : Path of loading between T
1
and T
2
in the plan deviatoric of the stresses
Notice 2:
In the formulation of this indicator three products related to plasticity intervene:
·
the variation enters the direction of the stresses and the current plastic deformations
(
.
p
p
),
·
the position of the stresses compared to the current threshold (
+
(
())
Y
R p
),
·
the variation enters the standard of the current plastic deformation and the cumulated plastic deformation
(
p
p
).
Code_Aster
®
Version
4.0
Titrate:
Indicators of discharge and loss of proportionality of the loading
Date:
06/04/98
Author (S):
G. DEBRUYNE
Key:
R4.20.01-A
Page:
5/6
Manual of Reference
R4.20 booklet: Methods of analysis
HI-74/98/008/A
A loss of proportionality could occur during the history without the indicator not detecting it
via the first two products (i.e. one can have at the end of the loading
coincidence of the directions of the stresses and the plastic deformations and being on the plastic threshold).
On the other hand, one will have
p
p
, and the indicator will be obligatorily higher than zero, consequently
the user will be informed loss of proportionality.
Notice 3:
If the indicator detects obligatorily a loss of proportionality in an area, in
practical it is necessary that the latter not contains sufficient material points with loading
proportional. Indeed, if one chooses a very vast area with few points concerned,
standardization of the indicator carried out with division by the volume of the area implies some
“crushing” towards zero of the value of the quantity. Typically, for a structure containing
a defect source of nonproportionality, one may find it beneficial to choose an area of integration
S
surrounding the defect with a weak vicinity in order to obtain a significant value of
the indicator.
3.2 Indicator
energetics
This indicator with the same function that the precedent, but is founded on the density of energy. It is written:
I
D
S
S
4
1
1
=
-
(
)
,
where
is the density of deformation energy defined by:
()
.
!
T
D
T
=
0
, and
is the density
of elastic energy associated the traction diagram if nonlinear elastic material were considered.
More precisely, this quantity is written:
()
(
)
()
()
()
(
)
µ
µ
(())
()
,
,
(())
()
,
,
T
Ktr
if
R p
T
Ktr
P
dq if
R p
Y
P
Y
=
+
<
+
=
+
+
=
+
1
2
2
3
1
2
6
2
2
2
2
0
R
R Q
with
K
the module of compressibility,
µ
the coefficient of shearing of Lamé,
R
the threshold of
traction diagram associated with the standard with the plastic deformation
P
p
=
(the aforementioned can be different
true plastic threshold, because
P
p
if the loading is nonproportional). This indicator is
also standardized between 0 and 1. It is null for a loading having always kept its character
proportional
(
)
=
.
Code_Aster
®
Version
4.0
Titrate:
Indicators of discharge and loss of proportionality of the loading
Date:
06/04/98
Author (S):
G. DEBRUYNE
Key:
R4.20.01-A
Page:
6/6
Manual of Reference
R4.20 booklet: Methods of analysis
HI-74/98/008/A
4
Environment of use of the various indicators
The indicators presented here are usable in postprocessing of a mechanical calculation and are
available for the finite elements of the continuous mediums in 2D (mode of plane deformations,
stresses plane or axisymmetric, triangular or quadrangular meshs) or 3D (meshs
hexahedral, tetrahedral, pentaedric or pyramids). Telegraphic elements, beams, plates and
hulls are excluded from this application.
4.1 Indicators
buildings
These indicators are accessible after a static or dynamic calculation whatever the law from
behavior of material. The operator
CALC_ELEM
present the options `
DCHA_ELGA_SIGM
'and
`
DCHA_ELNO_SIGM
'for the indicator of discharge
I
1
evaluated with the nodes or the points of Gauss of
the element, and the options `
RADI_ELGA_SIGM
'and `
RADI_ELNO_SIGM
'for the indicator of loss of
radiality
I
2
evaluated with the nodes or the points of Gauss of the element [bib2]. The key word
NORMALIZES
allows
to use one of the four standards described in paragraph 2.1, two last being used only if
one carried out as a preliminary an elastoplastic calculation with kinematic work hardening.
4.2 Indicators
total
These indicators are accessible only after one elastoplastic calculation with isotropic work hardening.
The operator
POST_ELEM
present the options `
INDIC_SEUIL
'and `
INDIC_ENER
'corresponding
respectively with the total indicators
I
3
and
I
4
. Those are evaluated on a group of mesh
previously defined by the user (for example by the control
DEFI_GROUP
).
5 Bibliography
[1]
J. LEMAITRE, J. - L.CHABOCHE, Mechanics of solid materials, Dunod 1985.
[2]
J. PELLET, Instruction manual of Code_Aster. Document [U4.61.02], 24/07/96.