Code_Aster
®
Version
7.4
Titrate:
Relations of behavior of the discrete elements
Date:
14/04/05
Author (S):
J.M. PROIX, B. QUINNEZ, G.DEVESA
Key
:
R5.03.17-C Pag
E:
1/20
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Organization (S):
EDF-R & D/AMA, SINETICS
Manual of Reference
R5.03 booklet: Nonlinear mechanics
Document: R5.03.17
Relations of behavior of the discrete elements
Summary:
This document describes the nonlinear behaviors of the discrete elements which are called by the operators
of resolution of nonlinear problems
STAT_NON_LINE
[R5.03.01] .ou
DYNA_NON_LINE
[R5.05.05].
More precisely, the behaviors described in this document are:
·
the behavior of the Von Mises type to isotropic work hardening used for the modeling of
threaded assemblies, implemented in
MACR_GOUJ 2e_CALC
, accessible by the key words
DIS_GOUJ 2e_PLAS
and
DIS_GOUJ 2e_ELAS
key word COMP_INCR [U4.51.11],
·
the behavior of unilateral the contact type with friction of Coulomb (in translation), and it
behavior of the Von Mises type to isotropic or kinematic work hardening linear (in rotation), used
to modelize the behavior within the competences of connection - pencil of the fuel assemblies roasts,
accessible by the key word
DIS_CONTACT
key word COMP_INCR [U4.51.11],
·
the behavior of the shock type with friction of Coulomb, accessible by the key word
DIS_CHOC
of
key word COMP_INCR [U4.51.11].
The integration of the models of behavior mentioned above is implicit.
Other behaviors relating to the discrete elements are available, but nonhere detailed:
·
Armament of the lines (Relation
ARM
) [R5.03.31],
·
Nonlinear assembly of angles of pylons (Relation
ASSE_CORN
) [R5.03.32],
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior of the discrete elements
Date:
14/04/05
Author (S):
J.M. PROIX, B. QUINNEZ, G.DEVESA
Key
:
R5.03.17-C Pag
E:
2/20
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R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Count
matters
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior of the discrete elements
Date:
14/04/05
Author (S):
J.M. PROIX, B. QUINNEZ, G.DEVESA
Key
:
R5.03.17-C Pag
E:
3/20
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
1
General principles of the relations of behavior of
discrete elements
1.1 Nonlinear relations of behavior (of the discrete elements)
currently available
The relations available in Code_Aster for the discrete elements are relations of
behavior incremental data under the key word factor
COMP_INCR
by the key word
RELATION
in the nonlinear operators
STAT_NON_LINE
and
DYNA_NON_LINE
. One distinguishes:
·
the behavior of the Von Mises type to isotropic work hardening used for the modeling of
threaded assemblies, implemented in
MACR_GOUJ 2e_CALC
and accessible by the words
keys
DIS_GOUJ 2e_PLAS
and
DIS_GOUJ 2e_ELAS
,
·
the behavior of unilateral the contact type with friction of Coulomb, used for
to modelize the behavior in translation within the competences of connection roasts - pencil of
fuel assemblies, accessible by the key word
DIS_CONTACT
,
·
the behavior of the Von Mises type to isotropic or kinematic work hardening linear used
to modelize behavior in rotation within the competences of connection - pencil roasts of
fuel assemblies, also accessible by the key word
DIS_CONTACT
, of
STAT_NON_LINE
.
And the following behaviors, which are not here detailed:
·
Armament of the lines (Relation
ARM
) [R5.03.31],
·
Nonlinear assembly of angles of pylons (Relation
ASSE_CORN
) [R5.03.32],
·
Contact with shocks (Relation
DIS_CHOC
).
The parameters necessary to these relations are provided in the operator
DEFI_MATERIAU
by
key words:
Behavior in
STAT_NON_LINE
DYNA_NON_LINE
Type of element (modeling)
in
AFFE_MODELE
Key words in
DEFI_MATERIAU
AFFE_CARA_ELEM
key words under
DISCRETE
DIS_GOUJ 2e_ELAS
DIS_GOUJ 2e_PLAS
2d_DIS_T:
discrete element 2D
with two nodes in translation
TRACTION
CARA: “K_T_D_L'
DIS_CONTACT
friction of Coulomb
DIS_T
or
DIS_TR:
element
discrete 3D with two nodes in
translation or translation/
rotation
DIS_CONTACT: (
COULOMB:
RIGI_N_FO:
EFFO_N_INIT:
)
CARA: “K_T_D_L'
or
CARA: “K_TR_D_L'
if rotation
DIS_CONTACT
rotation
DIS_TR
discrete element 3D with
two nodes in translation/
rotation
DIS_CONTACT:
RELA_MZ: curve
CARA: “K_TR_D_L'
DIS_CHOC
contact and
shock with friction
of Coulomb
DIS_T
:
discrete element 3D with
two nodes in translation
DIS_CONTACT: (
COULOMB:
RIGI_NOR:
RIGI_TAN:
AMOR_NOR:
AMOR_TAN:
DIST_1
DIST_2
PLAY
)
CARA: “K_T_D_L'
To be able to use
stamp rigidity
rubber band, option
RIGI_MECA
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior of the discrete elements
Date:
14/04/05
Author (S):
J.M. PROIX, B. QUINNEZ, G.DEVESA
Key
:
R5.03.17-C Pag
E:
4/20
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
These behaviors were developed and are used for the following applications:
DIS_GOUJ 2e_PLAS:
Relation effort - displacement of the Von Mises type with work hardening
isotropic for the modeling of the threaded assemblies
DIS_GOUJ 2e_ELAS:
Relation effort - linear displacement deduced from the curve effort -
displacement characterizing L `assembly
DIS_CONTACT
Elastoplastic behavior in rotation,
unilateral contact with friction of Coulomb in translation
DIS_CHOC
To take into account the shocks as well as friction enters one
structure and its supports or between the structures.
Contrary to the models of behavior 1D [bib3], these relations bind the efforts directly and
displacements, instead of being formulated between stresses and deformations. They are not valid
that in small deformations.
One describes for each relation of behavior the calculation of the field of efforts starting from an increment of
displacement given (cf algorithm of Newton [R5.03.01]), the calculation of the nodal forces
R
and of
stamp tangent.
1.2
Calculation of the deformations (small deformations)
For each finite element of Code_Aster, in
STAT_NON_LINE
, the total algorithm (Newton)
provides to the elementary routine, which integrates the behavior, an increase in field in
displacement [R5.03.01]
For the discrete elements, one deduces from it the increase in strain (in translation) or rotation,
between nodes 1 and 2 of the element, which is equivalent to the calculation of the increase in deformation
in the case of continuous mediums or of the behaviors 1D.
=
-
U
U
2
1
,
1.3
Calculation of the efforts and the nodal forces
For integration of the behavior, it is necessary to provide to the total algorithm (Newton) a vector containing
generalized efforts, on the one hand, and on the other hand a vector containing the nodal forces
R
[R5.03.01] in total reference mark (X, Y, Z).
For the discrete elements, the resolution of the nonlinear local problem provides the efforts directly
in the element (uniforms in the element), in local reference mark (X, y, Z), which is form:
F
F
F
=
1
2
1
2
(
)
(
)
node
node
with
in 2D:
F
F
F
F
X
y
1
2
=
=
in 3D:
F
F
F
F
F
X
y
Z
1
2
=
=
in translation alone,
F
F
F
F
F
M
M
M
X
y
Z
X
y
Z
1
2
=
=
in
translation and rotation.
X
y
N1
N2
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior of the discrete elements
Date:
14/04/05
Author (S):
J.M. PROIX, B. QUINNEZ, G.DEVESA
Key
:
R5.03.17-C Pag
E:
5/20
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
The vector
R
equivalent nodal forces (which is expressed in the total reference mark) is deduced from
F
by change of reference mark:
R
P R P
R
F
F
T
loc
loc
=
= -
with
node
node
1
2
1
2
(
)
(
)
where
P
is the matrix of change of reference mark, allowing the passage of the total reference mark towards the reference mark
room, as for the elements of beam [R3.08.01].
2
Relation of behavior of the threaded assemblies
2.1 Notations
general
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
Q
Q
Q
Q
=
+
=
+
=
+
-
-
T
T
T
T
2.2
Equations of the model
DIS_GOUJ 2e_PLAS
They are deduced from the behavior 3D
VMIS_ISOT_TRAC
[R5.03.02]: a relation there is represented
of behavior of the elastoplastic type to isotropic work hardening, binding the efforts in the element
discrete with the difference in displacement of the two nodes in the local direction y.
In local direction X, the behavior is elastic, linear, and the coefficient connecting the Fx effort to
Dx displacement is the Kx stiffness provided via
AFFE_CARA_ELEM
.
The nonlinear behavior relates to only the local direction y.
While noting
=
-
U
U
y
y
1
2
and
=
=
F
F
y
y
1
2
The relations are written in the same form as the relations of Von Mises 1D [R5.03.09]:
(
)
()
()
()
()
&
&
&
&
p
p
eq
eq
eq
p
E
R p
R p
R p
p
R p
p
=
=
-
-
=
-
-
< =
-
=
0
0
0
0
0
In these expressions, p represents a “cumulated plastic displacement”, and the function of work hardening
isotropic
()
R p
is closely connected per pieces, data in the form of a curve effort - displacement definite
point by point, provided under the key word factor
TRACTION
of the operator
DEFI_MATERIAU
[U4.43.01].
The first point corresponds at the end of the linear field, and is thus used to define at the same time the limit of
linearity (similar to the elastic limit), and
E
who is the slope of this linear part (
E
is
independent of the temperature). The function
()
R p
is deduced from a curve characteristic of
the assembly (modeling of some nets) expressing the effort on the pin according to
difference in average displacement between the pin and the flange [bib1]:
F
F U v
=
-
(
)
.
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior of the discrete elements
Date:
14/04/05
Author (S):
J.M. PROIX, B. QUINNEZ, G.DEVESA
Key
:
R5.03.17-C Pag
E:
6/20
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
2.3
Integration of the relation
DIS_GOUJ 2e_PLAS
By direct implicit discretization of the relations of behavior, a way similar to integration 1D
[R5.03.09] one obtains:
(
)
(
)
(
)
E
E p
R p
p
R p
p
p
R p
p
p
-
=
+
+
+
-
+
+
-
+
<
=
+
-
+
=
-
-
-
-
-
-
-
-
0
0
0
0
0
Two cases arise:
·
(
)
-
-
+
<
+
R p
p
in this case
p = 0
that is to say
= E
thus
()
-
+
<
-
R p,
·
(
)
-
-
+
=
+
R p
p
in this case
p 0
thus
()
-
-
+
R p.
One deduces the algorithm from it from resolution:
let us pose
E
=
+
-
if
()
E
R p
p
=
-
then
0
and
=
if
()
E
R p
>
-
then it is necessary to solve:
E
E p
=
+
+
+
+
-
-
-
(
)
E
E p
=
+
+
+
-
-
1
thus by taking the absolute value:
E
E p
=
+
+
+
-
-
1
maybe, while using
(
)
-
-
+
=
+
R p
p.
(
)
E
R p
p
E p
=
+
+
-
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior of the discrete elements
Date:
14/04/05
Author (S):
J.M. PROIX, B. QUINNEZ, G.DEVESA
Key
:
R5.03.17-C Pag
E:
7/20
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
By taking account of the linearity per pieces of
()
R p
, one can solve explicitly this
equation to find
p
. Once calculated
p
one deduces some
in the following way:
()
E
E
R p
=
then:
(
)
()
()
=
+
=
=
+
-
E
E
E
R p
E p
R p
1
Moreover, the option
FULL_MECA
allows to calculate the tangent matrix
K
in
with each iteration.
The tangent operator who is used for building it is calculated directly on the preceding discretized system.
One obtains directly:
()
() ()
if
if not
>
-
E
T
R p
E
E R p
E R p
E
=
=
+
=
.
2.4 Variables
interns
The relation of behavior
DIS_GOUJ 2e_PLAS
product two internal variables: “displacement
plastic cumulated “p, and an indicator being worth 1 if the increase in plastic deformation is nonnull
and 0 in the contrary case.
3
Relation of behavior within the competences of connection roasts
fuel pin
The behavior
DIS_CONTACT
, used to modelize the behavior in translation of the springs
of connection roasts - pencil of the fuel assemblies covers in fact two behaviors
distinct, relating to different degrees of freedom:
·
the behavior of unilateral the contact type with friction of Coulomb, relates to them
directions X and y local,
·
the behavior of the Von Mises type to isotropic or kinematic work hardening linear used
to modelize behavior in rotation relates to rotation around Z local and that
around X local.
In the other directions (translations according to Z local, rotation around there local), the behavior is
rubber band, defined by the stiffnesses provided in
CARA
:
“K_T_D_L'
or
CARA
: `
K_TR_D_L'
key word
factor
DISCRETE
control
AFFE_CARA_ELEM
.
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior of the discrete elements
Date:
14/04/05
Author (S):
J.M. PROIX, B. QUINNEZ, G.DEVESA
Key
:
R5.03.17-C Pag
E:
8/20
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
3.1
Model of contact with friction of Coulomb
3.1.1 Presentation of the model of contact - friction
The connection roasts - pencil is ensured by a flexible blade and two bosses. This system allows one
slip with friction of the pencil in the axial direction. In addition, the neutron irradiation has for
effect to soften this connection: the gripping force decreases according to time. To modelize this
connection, one introduced a behavior which applies for discrete elements to two nodes
MECA_DIS_TR_L
and
MECA_DIS_T_L
That is to say an element with two nodes. That is to say NR direction carried by the element (X local) and T a direction
perpendicular (in this case, it corresponds to the axial direction of the pencil): it is the direction
y local. Are the efforts in the directions NR and T (cf [Figure 3.1.1-a]).
The relation of behavior of elastic type perfectly plastic and is characterized by
[Figure 3.1.1-b]:
·
K
Te
an “elastic” slope,
·
K
NR
elastic rigidity in the direction NR,
·
a threshold of friction defined by
R
R
T
NR
=
µ
where
µ
is the coefficient of friction of
Coulomb,
·
a module “of fictitious work hardening”
K
Tl
to avoid a slip not control,
·
R
NR
0
an initial voltage in the direction
NR
,
·
a function of time
()
F T
or of the fluence
()
G
, standardized (“of decrease”) of
rigidity of the connection in the direction
NR
.
NR
T
Center pencil
Roast
U
T
Node 2
Node 1
U
N1
U
N2
Appear 3.1.1-a
K
Te
K
Tl
R
T
&
U
T
R
S
Appear 3.1.1-b
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior of the discrete elements
Date:
14/04/05
Author (S):
J.M. PROIX, B. QUINNEZ, G.DEVESA
Key
:
R5.03.17-C Pag
E:
9/20
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
These data are provided in
DEFI_MATERIAU
:
DIS_CONTACT: (
COULOMB:
µ
(coefficient of friction)
EFFO_N_INIT
:
RN0
(initial voltage of the spring)
/
RIGI_N_FO
:
F (T)
(standardized function of time)
/
RIGI_N_IRRA
:
G (
)
(standardized function of the fluence)
.
KT_ULTM
:
Ktt
(slope of work hardening)
)
Elastic characteristics of the springs
K
K
NR
Te
and
are provided under the key word factor
DISCRETE
control
AFFE_CARA_ELEM
. Modeling supposes that the direction T of
slip is the there local one and that X local is the normal direction NR with the contact (to direct the element
discrete, the key word factor is used
ORIENTATION
of
AFFE_CARA_ELEM
). The contact with friction
is written in directions X and Y. For third direction Z and the degrees of freedom of rotation,
the law of behavior is purely elastic and the characteristics of rigidity do not vary with
time.
The slope of work hardening is a slope of regularization which simulates a nonperfect slip, but which
allows to obtain a solution if the pencil is subjected to no imposed displacement and enters one
mode of pure slip.
3.1.2 Equations of the model
The model of contact - friction is similar to a behavior of Von Mises in perfect plasticity:
()
(
)
(
)
(
)
S
T
S
T
T
T
T
p
T
NR
S
Tl
S
T
T
T
T
p
T
T
Te
T
NR
NR
NR
NR
NR
NR
R
R
R
R
R
R
R
U
R
R
K
R
R
U
U
U
U
U
K
R
U
U
K
R
T
F
R
R
<
=
=
>
=
=
-
=
+
-
=
-
=
-
+
=
if
0
if
0
with
)
sgn (
.
.
with
0
1
2
1
2
0
µ
&
Tl
K
is defined by:
Tt
Te
Tt
Te
Tl
K
K
K
K
K
-
=
.
3.1.3 Integration of the relation
It is made on the basis of purely implicit integration.
One supposes known the solution at the previous moment
T
-
:
R
T
-
,
R
NR
-
and displacements and increases in displacements of iteration N of the algorithm of Newton of
STAT_NON_LINE
, noted:
U
U
U
T
T
T
=
-
2
1
and
()
()
U
U
T
U
T
NR
NR
NR
=
-
2
1
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior of the discrete elements
Date:
14/04/05
Author (S):
J.M. PROIX, B. QUINNEZ, G.DEVESA
Key
:
R5.03.17-C Pag
E:
10/20
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
The problem is written then:
()
(
)
R
F T R
K U
R
K
U
R
R
R
R
K
R
R
R
R
NR
NR
NR
NR
T
Te
T
T
T
T
S
Tl
T
S
T
S
=
+
=
-
+
>
=
=
<
0
0
0
0
with
if
if
.
Phase of prediction:
One poses:
R
R
K
U
Te
T
Te
T
=
+
-
.
One can find directly
()
(
)
{
}
R
F T R
K U
NR
NR
NR
NR
=
+
min,
0
0
Resolution:
·
If there is contact, then
R
NR
0
and:
- If
R
R
Te
S
<
then there is no slip:
= 0
and
R
R
T
Te
=
-
If not, there is slip:
To solve, one writes like usually:
(
)
R
R
R
R
K
U
K
R
R
R
K
R
R
R
R
K
T
T
T
T
Te
T
Te
T
T
Te
Te
T
T
T
S
Tl
=
+
=
+
-
=
-
=
+
+
-
-
-
.
.
.
thus by gathering the terms:
(
)
R
K
R
R
R
R
K
T
Te
T
Te
T
S
Tl
1
1
+
=
=
+
+
-
.
.
that is to say still
(
)
R
R
R
K
R
T
T
T
Te
Te
+
=
.
(
)
R
R
K
T
S
Tl
=
+
+
-
.
(
)
thus R
R
K
Te
T
Te
=
+
.
, which makes it possible to find
()
:
(
)
()
() (
)
()
R
R
K
K
R
K
K
K
Te
S
Tl
Te
S
Tl
Te
Tl
=
+
+
+
=
+
+
+
-
-
.
.
.
.
then to find
R
T
by:
(
)
(
)
R
R
K
R
R
T
S
Tl
Te
Te
=
+
+
-
.
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior of the discrete elements
Date:
14/04/05
Author (S):
J.M. PROIX, B. QUINNEZ, G.DEVESA
Key
:
R5.03.17-C Pag
E:
11/20
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
·
If there is no more contact (
R
NR
= 0
) then:
R
T
= 0
For the calculation of the tangent matrix, the option
FULL_MECA
allows to calculate the tangent matrix
K
in
with each iteration. The tangent operator who is used for building it is calculated directly on the system
discretized preceding. One obtains directly:
- if
R
NR
0
and
> 0
then
()
=
=
T
Tl
NR
NR
K
F T K
and
.
- if not
()
=
=
T
NR
NR
F T K
0
and
.
Note:
If there is separation (
R
NR
= 0
), it is necessary to take guard with the fact that the part which is
normally “held” by the discrete element is not it any more. For example, in the case of one
fuel pin, if none the springs is more in contact, the aforementioned can “fall”. In
practical, to avoid these situations, it is necessary that the spring modelized by the discrete element is
always in compression. That can be specified by the user using the initial effort, of
coefficient of rigidity and the function F (T).
3.2
Elastoplastic model of ball joint
3.2.1 Equations of the model
They are deduced from the behavior 3D
VMIS_ECMI_TRAC
[R5.03.16]: indeed, it is here about
to represent a relation of behavior of the elastoplastic type to unspecified isotropic work hardening,
superimposed on a linear kinematic work hardening, binding the Mz moment (or MX) in the discrete element
with the difference in rotation of the two nodes around Z local (or X local). One is not thus interested here
that with rotation around Z (or X) local.
While noting
=
-
Z
Z
2
1
and
=
=
M
M
Z
Z
1
2
, (resp.
=
-
X
X
2
1
and
=
=
M
M
X
X
1
2
)
The relations are written here still in the same form as the relations of Von Mises 1D [R5.03.09].
They can be deduced from the behavior
VMIS_ECMI_TRAC
[R5.03.16], by noticing that in
the uniaxial case, the constant of Prager
C
must be multiplied by 3/2. In any rigor it would be necessary to write:
X
C
p
= 32
, but here, one writes directly:
X
C
p
=
. It is provided via the key word
PRAGER
key word factor
DIS_CONTACT
of the operator
DEFI_MATERIAU
.
(
)
()
(
)
()
(
)
()
=
=
-
-
<
-
-
=
-
-
-
=
-
-
=
p
eq
eq
p
p
C
X
p
R
X
p
p
R
X
p
p
R
X
E
X
X
p
0
0
0
0
0
if
if
&
&
&
&
Code_Aster
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Titrate:
Relations of behavior of the discrete elements
Date:
14/04/05
Author (S):
J.M. PROIX, B. QUINNEZ, G.DEVESA
Key
:
R5.03.17-C Pag
E:
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HT-66/05/002/A
p represents here a “plastic rotation cumulated” around each local directions Z and X.
isotropic function of work hardening
()
R p
is closely connected per pieces, data in the form of a function
defined point by point under the key word factor
RELA_MZ
key word factor
DIS_CONTACT
of
the operator
DEFI_MATERIAU
. It is supposed implicitly that the relation used for rotation around
of Z is identical to that used for rotation around X local.
The function
()
R p
is deduced from a curve characteristic of the spring during a deflection test
of a pencil in a grid, curves which expresses the moment applied with the pencil according to
rotation of the pencil
F
F
= ()
who is translated with our notations into:
= F ()
.
()
R p
is deduced from
F
,
as in [bib2] by taking account of the linearity per pieces of
F
. The first point corresponds
at the end of linearity, and thus at the same time the limit of linearity similar to the elastic limit defines and
E
who is the slope of this linear part (independent of the temperature for this development).
3.2.2 Integration of the relation
By direct implicit discretization of the relations of behavior, a way similar to integration 1D
[R5.03.09] one obtains:
()
(
)
(
)
(
)
-
-
=
+
-
-
-
+
-
=
+
-
-
+
-
-
+
-
-
=
+
=
+
-
-
<
+
-
-
-
-
-
-
-
-
-
-
-
-
X
R p
X
X
R p
p
E
E p
X
X
X
X
p
X
X
R p
p
p
X
X
R p
p
-
0
0
0
if
if
Two cases arise:
·
(
)
-
<
+
-
X
R p
p
in this case
p
=
=
0
that is to say
thus
()
-
+
<
-
R p,
·
(
)
-
=
+
-
X
R p
p
in this case
p 0
thus
()
-
-
+
R p.
One deduces the algorithm from it from resolution:
let us pose
E
X
=
+
-
-
-
if
()
E
R p
-
then
p = 0
and
=
if
()
E
R p
>
-
then it is necessary to solve:
(
)
E
X
X
X
E p
X
X
X
E C
p
X
X
=
+
-
-
+
+
-
-
= -
+
+
-
-
-
-
because
X
C p
X
X
=
-
-
Code_Aster
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7.4
Titrate:
Relations of behavior of the discrete elements
Date:
14/04/05
Author (S):
J.M. PROIX, B. QUINNEZ, G.DEVESA
Key
:
R5.03.17-C Pag
E:
13/20
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R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
one thus obtains:
(
)
(
)
E
E C
p
X
X
=
+
+
-
-
1
and by taking the absolute value:
(
)
E
E C
p
X
X
=
+
+
-
-
1
maybe, while using
(
)
-
=
+
-
X
R p
p
.
(
)
(
)
E
R p
p
E C
p
=
+
+
+
-
By taking account of the linearity per pieces of
()
R p
, one can solve explicitly this
equation to find
p
. Once calculated
p
one deduces some
in the following way:
one with the relation of proportionality:
()
E
E
X
R p
=
-
where
X
is calculated using:
()
(
)
X
C p
X
X
C p
C p R p E C p
E
E
E
=
-
-
=
=
+
+
from where
()
p
R
X
E
E
+
=
Moreover, the option
FULL_MECA
allows to calculate the tangent matrix
K
in
with each iteration.
The tangent operator who is used for building it is calculated directly on the preceding discretized system.
One obtains directly:
()
()
(
)
()
if
if not
>
-
E
T
R p
E
E R p
C
E R p
C
E
=
=
+
+
+
=
.
3.3 Variables
interns
The relation of behavior
DIS_CONTACT
product 3 internal variables:
·
The first relates to the contact - friction: it is worth:
-
1 if there is slip,
-
0 so not slip,
-
1 if separation.
·
The two following ones relate to elastoplastic behavior in rotation:
-
the second internal variable is equal to p around local direction Z,
-
the third internal variable is equal to p around the direction X local.
Code_Aster
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Titrate:
Relations of behavior of the discrete elements
Date:
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Key
:
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E:
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HT-66/05/002/A
4
Modeling of the shocks and friction:
DIS_CHOC
The behavior
DIS_CHOC
translated the contact with shock and friction between two structures, via two
types of relations:
·
the relation of unilateral contact which expresses the non-interpenetrability between the solid bodies,
·
the relation of friction which governs the variation of the tangential stresses in the contact. One
will retain for these developments a simple relation: the law of friction of
Coulomb.
4.1
Relation of unilateral contact
Are two structures
1
and
2
. One notes
D
NR 1 2
/
the normal distance enters the structures,
F
NR 1 2
/
force normal reaction of
1 out of 2.
The law of the action and the reaction imposes:
F
F
NR
NR
2 1
1 2
/
/
= -
éq 4.1-1
The conditions of unilateral contact, still called conditions of Signorini [bib5], are expressed
following way:
D
F
D
F
and F
F
NR
NR
NR
NR
NR
NR
1 2
1 2
1 2
1 2
2 1
1 2
0
0
0
/
/
/
/
/
/
,
,
=
= -
éq
4.1-2
Code_Aster
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Titrate:
Relations of behavior of the discrete elements
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Key
:
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D
NR
1/2
F
NR
1/2
Appear 4.1-a: Graph of the relation of unilateral contact
This graph translates a relation force-displacement which is not differentiable. It is thus not
usable in a simple way in a dynamic calculation algorithm.
4.2
Law of friction of Coulomb
The law of Coulomb expresses a tangential mechanical force limiting
2
/
1
T
F
of tangential reaction of
1
on
2
. That is to say
2
/
1
T
u&
the relative speed of
1
compared to
2
in a point of contact and is
µ
coefficient of friction of Coulomb, one has [bib5]:
0
.
,
0
,
,
0
2
/
1
2
/
1
2
/
1
2
/
1
=
=
-
S
F
S
NR
µ
T
T
T
F
U
F
&
=
éq
4.2-1
and the law of the action and the reaction:
2
/
1
1
/
2
T
T
F
F
-
=
éq 4.2-2
Ý
U
T
1/2
F
T
1/2
Appear 4.2-a: Graph of the law of friction of Coulomb
The graph of the law of Coulomb is him also nondifferentiable and is thus not simple to use in
a dynamic algorithm.
2
/
1
T
u&
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Titrate:
Relations of behavior of the discrete elements
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4.3
Approximate modeling of the relations of contact by penalization
4.3.1 Model of normal force of contact
The principle of the penalization applied to the graph of the figure [Figure 4.3.1-a] consists in introducing one
univocal relation
()
2
/
1
2
/
1
F
NR
NR
D
F
=
by means of a parameter
. The graph of
F
must tend towards
the graph of Signorini when
tends towards zero [bib6].
One of the possibilities consists in proposing a linear relation enters
2
/
1
NR
D
and
2
/
1
NR
F
:
if not
0
;
0
1
2
/
1
2
/
1
2
/
1
2
/
1
=
-
=
NR
NR
NR
NR
F
D
if
D
F
éq
4.3.1-1
If one notes
1
=
NR
K
called commonly “stiffness of shock”, one finds the conventional relation,
modelizing an elastic shock:
2
/
1
2
/
1
NR
NR
NR
D
K
F
-
=
éq
4.3.1-2
The approximate graph of the law of contact with penalization is as follows:
D
NR
1/2
F
NR
1/2
Appear 4.3.1-a: Graph of the relation of unilateral contact approached by penalization
To take account of a possible loss of energy in the shock, one introduces a “damping of
shock "
C
NR
The expression of the normal force of contact is expressed then by:
2
/
1
2
/
1
2
/
1
NR
NR
NR
NR
NR
U
C
D
K
F
&
-
-
=
éq
4.3.1-3
where
2
/
1
NR
u&
is the relative normal speed of
1
compared to
2
. To respect the relation of
Signorini (not of blocking), one must on the other hand check a posteriori that
F
NR 1 2
/
is positive
or null. Only the positive part will thus be taken
+
expression [éq 4.3.1-3]:
0
0
0
<
=
=
+
+
X
X
X
X
X
if
if
Code_Aster
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Titrate:
Relations of behavior of the discrete elements
Date:
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Key
:
R5.03.17-C Pag
E:
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HT-66/05/002/A
The complete relation giving the normal force of contact which is retained for the algorithm is
following:
2
/
1
1
/
2
2
/
1
2
/
1
2
/
1
2
/
1
,
0
NR
NR
NR
NR
NR
NR
NR
NR
F
F
U
C
D
K
F
D
-
=
-
-
=
+
&
if
if not
F
F
NR
NR
2 1
1 2
0
/
/
.
=
=
éq
4.3.1-4
4.3.2 Model of tangential force of contact
The graph describing the tangential force with law of Coulomb is not-differentiable for the phase
of adherence
(
)
0
2
/
1
=
T
u&
. One thus introduces a univocal relation binding relative tangential displacement
2
/
1
T
D
and the tangential force
()
2
/
1
2
/
1
F
T
T
D
F
=
by means of a parameter
. The graph of
F
must
to tend towards the graph of Coulomb when
tends towards zero [bib6].
One of the possibilities consists in writing a linear relation enters
D
T1 2
/
and
F
T1 2
/
:
while noting
0
has
the value of a quantity has at the beginning of the pitch of time:
(
)
0
2
/
1
2
/
1
0
2
/
1
2
/
1
1
T
T
T
T
D
D
F
F
-
-
=
-
éq
4.3.2-1
If one introduces a “tangential stiffness”
K
T
= 1
, the relation is obtained:
(
)
0
2
/
1
2
/
1
0
2
/
1
2
/
1
T
T
T
T
D
D
F
F
-
-
=
T
K
éq
4.3.2-2
For numerical reasons, related to the dissipation of parasitic vibrations [bib7] in phase
of adherence, one is brought to add a “tangential damping”
C
T
in the expression of the force
tangential. Its final expression is:
(
)
2
/
1
1
/
2
2
/
1
0
2
/
1
2
/
1
0
2
/
1
2
/
1
,
T
T
T
T
T
T
T
F
F
U
D
D
F
F
-
=
-
-
-
=
&
T
T
C
K
éq
4.3.2-3
It is necessary moreover than this force checks the criterion of Coulomb, that is to say:
2
/
1
1
/
2
2
/
1
2
/
1
2
/
1
2
/
1
2
/
1
2
/
1
,
apply
one
if not
T
T
T
T
T
T
F
F
U
U
F
F
-
=
-
=
&
&
NR
NR
F
F
µ
µ
éq
4.3.2-4
Code_Aster
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Titrate:
Relations of behavior of the discrete elements
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The approximate graph of the law of friction of Coulomb modelized by penalization is as follows:
Ý
U
T1/2
F
T1/2
K
T
Appear 4.3.2-a: Graph of the law of friction approached by penalization
4.4
Definition of the parameters of contact
One specifies the key words here allowing to define the parameters of contact, damping and
friction [U4.43.01]
The operand
RIGI_NOR
is obligatory, it allows to give the value of normal stiffness of shock
K
NR
.
The other operands are optional.
The operand
AMOR_NOR
allows to give the value of normal damping of shock
C
NR
.
The operand
RIGI_TAN
allows to give the value of tangential stiffness
K
T
.
The operand
AMOR_TAN
allows to give the tangential value of damping of shock
C
T
.
The operand
COULOMB
allows to give the value of the coefficient of Coulomb.
The operand
DIST_1
allows to define the distance characteristic of matter surrounding the first
node of shock
The operand
DIST_2
allows to define the distance characteristic of matter surrounding the second
node of shock (shock between two mobile structures)
The operand
PLAY
defines the distance between the node of shock and an obstacle not modelized (case of a shock
between a mobile structure and an indeformable and motionless obstacle).
Code_Aster
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5 Bibliography
[1]
J. ANGLES: “Modeling of the threaded assemblies…” Note HI74-99-020A
[2]
J.M. PROIX, B. QUINNEZ, P. MASSIN, P. LACLERGUE: “Fuel assemblies under
irradiation. Feasibility study ". Note HI-75/97/017/0
[3]
G. JACQUART: “Methods of Ritz in non-linear dynamics - Application to systems
with shock and friction localized " - Report/ratio EDF DER HP61/91.105
[4]
Mr. JEAN, J.J. MOREAU: “Unilaterality and dry friction in the dynamics off rigid bodies
collection " Proceedings off the International Mechanics Contact Symposium - ED. A. CURNIER
- Polytechnic Presses and French Academics - Lausanne, 1992, p 31-48
[5]
J.T. ODEN, J.A.C. MARTINS: “Models and computational methods for dynamic friction
phenomena " - Computational Methods Appl. Mech. Engng. 52, 1992, p 527-634
[6]
B. BEAUFILS: “Contribution to the study of the vibrations and the wear of the beams of tubes in
cross-flow " - Thesis of doctorate PARIS VI
[7]
Fe WAECKEL, G. DEVESA: “File of specifications of a model of shock in
order DYNA_NON_LINE of Code_Aster “. Note HP-52/97/026/B
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