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 Page:
1/20
Organization (S): EDF-R & D/AMA, SINETICS
Handbook of Référence
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 of 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 model the behavior within the competences of connection - pencil of the fuel assemblies roasts,
accessible by key word DIS_CONTACT from key word COMP_INCR [U4.51.11],
·
the behavior of the shock type with friction of Coulomb, accessible by key word DIS_CHOC from
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 ARME) [R5.03.31],
·
Nonlinear assembly of angles of pylons (Relation ASSE_CORN) [R5.03.32],
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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 Page:
2/20
Count
matters
1 general Principles of the relations of behavior of the discrete elements ......................................... 3
1.1 Nonlinear relations of behavior (of the discrete elements) currently available ........ 3
1.2 Calculation of the deformations (small deformations) .............................................................................. 4
1.3 Calculation of the efforts and the nodal forces ......................................................................................... 4
2 Relation of behavior of the threaded assemblies ............................................................................. 5
2.1 General notations ......................................................................................................................... 5
2.2 Equations of model DIS_GOUJ 2e_PLAS ..................................................................................... 5
2.3 Integration of relation DIS_GOUJ 2e_PLAS ................................................................................. 6
2.4 Variables intern ............................................................................................................................ 7
3 Relation of behavior within the competences of connection roasts fuel pin ...................................... 7
3.1 Model of contact with friction of Coulomb ............................................................................. 8
3.1.1 Presentation of the model of contact - friction .................................................................... 8
3.1.2 Equations of the model .............................................................................................................. 9
3.1.3 Integration of the relation ......................................................................................................... 9
3.2 Elastoplastic model of knee joint ................................................................................................... 11
3.2.1 Equations of the model ............................................................................................................ 11
3.2.2 Integration of the relation ....................................................................................................... 12
3.3 Variables intern .......................................................................................................................... 13
4 Modeling of the shocks and friction: DIS_CHOC ......................................................................... 14
4.1 Relation of unilateral contact ........................................................................................................ 14
4.2 Law of friction of Coulomb ....................................................................................................... 15
4.3 Approximate modeling of the relations of contact by penalization .............................................. 16
4.3.1 Model of normal force of contact .................................................................................... 16
4.3.2 Model of tangential force of contact .............................................................................. 17
4.4 Definition of the parameters of contact ............................................................................................ 18
5 Bibliography ........................................................................................................................................ 19
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R5.03 booklet: Nonlinear mechanics
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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 Page:
3/20
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 key word RELATION
in 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
assemblies threaded, 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 model the behavior in translation within the competences of connection roasts - pencil of
fuel assemblies, accessible by key word DIS_CONTACT,
·
the behavior of the Von Mises type to isotropic or kinematic work hardening linear used
to model behavior in rotation within the competences of connection - pencil roasts of
fuel assemblies, also accessible by key word DIS_CONTACT, of
STAT_NON_LINE.
And the following behaviors, which are not here detailed:
·
Armament of the lines (Relation ARME) [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 operator DEFI_MATERIAU by
key words:
Behavior in Type of element (modeling) key Mots in
AFFE_CARA_ELEM
STAT_NON_LINE
in AFFE_MODELE
DEFI_MATERIAU
key words under DISCRET
DYNA_NON_LINE
DIS_GOUJ 2e_ELAS
2d_DIS_T: discrete element 2D TRACTION
CARA: “K_T_D_L'
DIS_GOUJ 2e_PLAS
with two nodes in translation
DIS_CONTACT
DIS_T or DIS_TR: element DIS_CONTACT: (
CARA: “K_T_D_L'
friction of discrete Coulomb 3D with two nodes in COULOMB:
or
translation or translation/
RIGI_N_FO:
CARA: “K_TR_D_L'
rotation
EFFO_N_INIT: if rotation
)
DIS_CONTACT
DIS_TR discrete element 3D with DIS_CONTACT:
CARA: “K_TR_D_L'
rotation
two nodes in translation/RELA_MZ: curve
rotation
DIS_CHOC contact and DIS_T: discrete element 3D with DIS_CONTACT: (
CARA: “K_T_D_L'
shock with friction two nodes in translation
COULOMB:
To be able to use
of Coulomb
RIGI_NOR:
stamp rigidity
RIGI_TAN:
rubber band, option
AMOR_NOR:
RIGI_MECA
AMOR_TAN:
DIST_1
DIST_2
JEU
)
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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 Page:
4/20
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 constraints 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 elongation (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 (
1
node)
F = 1
F
with
2 (node 2)
Fx
in 2D: F = F = F
1
2
y
X
y
Fx
N2
F
F
y
X
Fz
in 3D: F = F = F
=
=
1
2
there of translation alone, F
F
1
2
in
M
F
X
N1
Z
My
M
Z
translation and rotation.
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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 Page:
5/20
The vector R of the equivalent nodal forces (which is expressed in the total reference mark) is deduced from
F by change of reference mark:
- F (node 1)
R = PT R P
R
with
=
1
loc
loc
F
(node
2
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 par. One has as follows:
Q = (
Q +) = Q () + (
Q) = Q
T
T
T
T
+ Q
2.2
Equations of 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 = u1 - u2 and = F 1 = F 2
y
y
y
y
The relations are written in the same form as the relations of Von Mises 1D [R5.03.09]:
p
& = &p
= E (- p)
- R p =
- R p 0
eq
()
()
- R p < 0 p = 0
eq
()
&
- R p = 0 p 0
eq
()
&
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 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 which is the slope of this linear part (E are
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 support [bib1]: F = F (U - v).
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Titrate:
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Date:
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Author (S):
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:
R5.03.17-C Page:
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2.3
Integration of 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) 0
- + - (
R p + p) < 0 p = 0
- + - (
R p + p) = 0 p 0
Two cases arise:
·
-
R (-
+
<
p + p) in this case p = 0 is =
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) then
p
= 0 and =
if E > R (p) then it is necessary to solve:
- +
E = - +
+ E p
-
+
E p
E
= +
(-
1
)
+
+
-
thus by taking the absolute value:
E p
E = +
-
1
+
+
-
maybe, while using
-
R (-
+
=
p + p).
E = R (p + p
) + E p
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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 Page:
7/20
By taking account of the linearity per pieces of R (p), one can solve explicitly this
equation to find p. Une fois calculated p one deduces some in the following way:
E
=
E
R (p)
then:
E
E
= (- +
) =
R (p) =
E
E p
1 + R (p)
Moreover, the option
N
FULL_MECA makes it possible to calculate the tangent matrix K I with each iteration.
The tangent operator who is used for building it is calculated directly on the preceding discretized system.
One obtains directly:
E.R p
if E > R (-
p)
()
= E =
T
E + R (p)
if not
=
E
2.4 Variables
interns
The relation of behavior DIS_GOUJ 2e_PLAS produces 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
Behavior DIS_CONTACT, used to model 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 model 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' of the key word
factor DISCRET of command AFFE_CARA_ELEM.
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Relations of behavior of the discrete elements
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Author (S):
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:
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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 model 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]:
·
KTe an “elastic” slope,
·
Elastic kN rigidity in the direction NR,
·
a threshold of friction defined by R
= µ R
T
NR where µ is the coefficient of friction of
Coulomb,
·
a module “of work hardening” fictitious KTl to avoid a slip not control,
·
RN0 an initial tension in the direction NR,
·
a function of time F (T) or fluence G (), standardized (“of decrease”) of
rigidity of the connection in the direction NR.
T
Center pencil
UT
Node 1
Node 2
Roast
NR
UN1
UN2
Appear 3.1.1-a
RT
R
K
S
Tl
KTe
&
UT
Appear 3.1.1-b
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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 Page:
9/20
These data are provided in DEFI_MATERIAU:
DIS_CONTACT: (
COULOMB: µ
(coefficient of friction)
EFFO_N_INIT
:
RN0
(initial tension 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
and K
NR
Te are provided under the key word factor
DISCRET of command 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, one uses the key word factor ORIENTATION of AFFE_CARA_ELEM). The contact with friction
is written in directions X and Y. Pour third direction Z and for 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:
R 0
NR
R = F T R
+ K U
- U
NR
() (N0
NR (
N2
N1)
R = K U - U
U
with
= U - U
T
Te (
p
T
T)
T
T 2
T1
R R + K.
T
S
Tl
R = - µ R
S
NR
R
p
T
U & = .SGN (R) =
T
T
RT
with >
if
0
R = R
T
S
=
if
0
R < R
T
S
K K
K is defined by:
Te
Tt
K =
.
Tl
Tl
K - K
Te
Tt
3.1.3 Integration of the relation
It is made on the basis of purely implicit integration.
One supposes known the solution at the moment previous T -: R -, R -
T
NR
and displacements and increases in displacements of iteration N of the algorithm of Newton of
STAT_NON_LINE, noted:
U = U - U and U = U (T) - U (T)
T
T 2
T1
NR
N2
NR 1
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Relations of behavior of the discrete elements
Date:
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Author (S):
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:
R5.03.17-C Page:
10/20
The problem is written then:
R = F (T)
+
0
NR
(R
K U
NR 0
NR
NR)
R
R
= K U
T
-
T
Te
T
R
T
R R +. K
T
S
Tl
with > 0 if R = R
T
S
= 0 if R < R
T
S
Phase of prediction:
One poses: R
= R + K. U
Te
T
Te
T
One can find R directly =
{
min,
0 F (T) (R + K U
NR
NR 0
NR
NR)}
Resolution:
·
If there are contact, then R 0 and:
NR
- If
R < R
Te
S then it does not have there slip:
= 0 and R = R
T
Te
-
If not, there is slip:
To solve, one writes like usually:
R
R
R = R + R
= R + K
U
- K
T
.
= R - K
T
.
T
T
T
T
Te
T
Te
R
Te
Te
R
T
T
R = R + K
-
. +
T
S
Tl (
)
thus by gathering the terms:
1
R 1 + K
.
R
T
Te
R = Te
T
R = R + K
-
. +
T
S
Tl (
)
that is to say still
RT (R + K. =
T
Te
) R
R
Te
T
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 by:
T
R
R
-
Te
=
+
+
.
T
(R K
S
Tl (
)) RTe
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Relations of behavior of the discrete elements
Date:
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Author (S):
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:
R5.03.17-C Page:
11/20
·
If there is no more contact (R =
=
NR
0) then: RT 0
For the calculation of the tangent matrix, the option
N
FULL_MECA makes it possible to calculate the tangent matrix K I
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 0 and > 0 then
K and
F (T). K
NR
=
Tl
=
NR
T
NR
- if not
0 and
F (T)
=
=
. K
NR
T
NR
Note:
If there is separation (R
=
NR
0), it are 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, this one can “fall”. In
practical, to avoid these situations, it is necessary that the spring modelled 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 knee 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 = - and = M = M, (resp. = - and = M
= M)
z2
z1
z1
z2
x2
x1
x1
x2
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 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:
3
X
C p
=
, but here, one writes directly: X
C p
=. It is provided via the key word
2
PRAGER of the key word factor DIS_CONTACT of operator DEFI_MATERIAU.
p
- X
& = p & -
X
= E (- p
)
- X - R (p) 0
p & = 0 if (- X) - R (p) <
eq
0
p & 0 if (- X) - R (p)
=
eq
0
X =
p
C
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Relations of behavior of the discrete elements
Date:
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Author (S):
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:
R5.03.17-C Page:
12/20
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 of the key word factor DIS_CONTACT of
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 () which 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 defines at the same time the limit of linearity similar to the elastic limit 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) 0
- +
- -
X -
X
E
-
=
E p
- +
- -
X - X
p
0
if - +
- -
X - X = R (-
p + p)
p
= 0 if - +
- -
X - X <
-
R (p + p)
Two cases arise:
·
-
< (-
X
R p + p
) in this case p = 0 are =
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:
- X
- X
E = - + - X -
- X + X + E p
= - X + (E + C)
p
-
X
- X
- X
because X =
C p -
X
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one thus obtains:
E + C
p
E
(
)
= 1 +
(X)
-
X
-
and by taking the absolute value:
(E + C) p
E = 1 +
-
X
-
X
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. Une fois calculated p one deduces some in the following way:
one with the relation of proportionality:
E
- X
=
E
R (p)
where X is calculated using:
- X
E
E
X =
C p
=
C p
=
C p
-
X
E
R (p) + (E + C) p
from where
E
=X +
R (p)
E
Moreover, the option
N
FULL_MECA makes it possible to calculate the tangent matrix K I with each iteration.
The tangent operator who is used for building it is calculated directly on the preceding discretized system.
One obtains directly:
E.R p + C
if E > R (-
p)
(())
=
=
AND
E + R (p) + C
if not
=
E
3.3 Variables
interns
The relation of behavior DIS_CONTACT produces 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.
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4
Modeling of the shocks and friction: DIS_CHOC
Behavior DIS_CHOC translates 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
1/2
1/2
1 and 2. D NR is noted
the normal distance enters the structures, FN
force normal reaction of 1 out of 2.
The law of the action and the reaction imposes:
F 2 1
/= - F 1/2
NR
NR
éq 4.1-1
The conditions of unilateral contact, still called conditions of Signorini [bib5], are expressed
following way:
D 1/2 0, F 1/2 0, D 1/2 F 1/2 = 0 and F 2 1
/= - F 1/2
NR
NR
NR
NR
NR
NR
éq
4.1-2
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F 1/2
NR
D 1/2
NR
Appear 4.1-a: Graphe 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 limitation of effort
1/2
F
of tangential reaction of
T
1 on
. That is to say 1/2
U
compared to in a point of contact and is µ it
2
&
the relative speed of
T
1
2
coefficient of friction of Coulomb, one has [bib5]:
1/2
1/2
S = F
- µ F
,
0
1/2
1/2
u&
= F
,
0 .s = 0
T
NR
T
T
éq
4.2-1
and the law of the action and the reaction:
2/1
1/2
F
= - F
éq 4.2-2
T
T
F1/2
T
Ý
U 1/2
1/2
u&T
Appear 4.2-a: Graphe 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.
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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
1/2
F
= F
NR
(
1/2
D NR
)
univocal relation
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
1/2
D
and
1/2
F
:
NR
NR
1/2
1
1/2
1/2
F
= - D
if D
;
0
1/2
F
= 0 if not
éq
4.3.1-1
NR
NR
NR
NR
1
If K is noted
= called commonly “stiffness of shock”, one finds the traditional relation,
NR
modelling an elastic shock:
1/2
1/2
F
= - K D
éq
4.3.1-2
NR
NR
NR
The approximate graph of the law of contact with penalization is as follows:
F 1/2
NR
D 1/2
NR
Appear 4.3.1-a: Graphe 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 " CN the expression of the normal force of contact is expressed then by:
1/2
1/2
1/2
F
= - K D
- C U
éq
4.3.1-3
NR
NR
NR
NR
&N
where
1/2
u&
is normal speed relative from report/ratio to. To respect the relation of
NR
1
2
Signorini (not of blocking), one must on the other hand check a posteriori that F 1/2
NR
is positive
+
or null. Only the positive part will thus be taken
expression [éq 4.3.1-3]:
+
X
= X if X 0
+
X
= 0 if X < 0
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The complete relation giving the normal force of contact which is retained for the algorithm is
following:
1/2
1/2
1/2
1/2 +
2/1
1/2
if D
0F
= - K D
- C u&
,
NR
NR
NR
NR
NR
NR
NR
F
= - NR
F
if not
F 2 1
/= F 1/2
NR
NR
= 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 (1/2
u&
= 0. One thus introduces a univocal relation binding relative tangential displacement
T
)
1/2
D
1/2
F
= F
T
(
1/2
dT)
and the tangential force
by means of a parameter. The graph of F must
T
to tend towards the graph of Coulomb when tends towards zero [bib6].
One of the possibilities consists in writing a linear relation between D 1/2
1/2
T
and FT:
by noting 0
with the value of a quantity has at the beginning of the step of time:
0
1/2
1/2
1
F
- F
= - D
- D
éq
4.3.2-1
T
T
(
0
1/2
1/2
T
T
)
1
If one introduces a “tangential stiffness” KT =, one obtains the relation:
0
1/2
1/2
F
= F
- K D
- D
éq
4.3.2-2
T
T
T
(
0
1/2
1/2
T
T
)
For numerical reasons, related to the dissipation of parasitic vibrations [bib7] in phase
of adherence, one is brought to add a “tangential damping” CT in the expression of the force
tangential. Its final expression is:
0
1/2
1/2
F
= F
- K D
- D
- C U
éq
4.3.2-3
T
T
&
F
= - F
T
T
(
0
1/2
1/2
T
T
)
1/2
2/1
1/2
,
T
T
T
It is necessary moreover than this force checks the criterion of Coulomb, that is to say:
1/2
1/2
1/2
1/2
1/2
u&
2/1
1/2
F
µ F
apply
one
if not
T
F
= - µ F
, F
= - F
éq
4.3.2-4
T
NR
T
NR
1/2
T
T
u&T
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The approximate graph of the law of friction of Coulomb modelled by penalization is as follows:
F 1/2
T
KT
Ý
U 1/2
T
Appear 4.3.2-a: Graphe 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]
Operand RIGI_NOR is obligatory, it makes it possible to give the value of normal stiffness of shock kN.
The other operands are optional.
Operand AMOR_NOR makes it possible to give the value of normal damping of shock CN.
Operand RIGI_TAN makes it possible to give the value of tangential stiffness KT.
Operand AMOR_TAN makes it possible to give the tangential value of damping of shock CT.
The COULOMB operand makes it possible to give the value of the coefficient of Coulomb.
Operand DIST_1 makes it possible to define the distance characteristic of matter surrounding the first
node of shock
Operand DIST_2 makes it possible to define the distance characteristic of matter surrounding the second
node of shock (shock between two mobile structures)
Operand JEU defines the distance between the node of shock and an obstacle not modelled (case of a shock
between a mobile structure and an indeformable and motionless obstacle).
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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 with systems
with shock and friction localized " - Rapport 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 Contact Mechanics International Symposium - ED. A. CURNIER
- Presses Polytechniques and Universitaires Romandes - 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
transverse flow " - Thèse 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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