Code_Aster
®
Version
8.1
Titrate:
SSNA115 Arrachement of a rigid reinforcement
Date
:
25/11/05
Author (S):
J. LAVERNE
Key
:
V6.01.115-A
Page:
1/6
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A
Organization (S):
EDF-R & D/AMA
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
Document: V6.01.115
SSNA115 Arrachement of a rigid reinforcement with
elements with discontinuity
Summary:
This case test has as an aim the numerical study of the wrenching of a rigid reinforcement embedded in a cylinder
hollow. Decoherence is modelized starting from elements with internal discontinuity with a cohesive law CZM_EXP
(see documentation [R7.02.12]) by using modeling AXIS_ELDI. To validate the results us
will support on the analytical solution developed in [bib3]. The interested reader will also be able y
to defer for a thorough study of this case test.
Code_Aster
®
Version
8.1
Titrate:
SSNA115 Arrachement of a rigid reinforcement
Date
:
25/11/05
Author (S):
J. LAVERNE
Key
:
V6.01.115-A
Page:
2/6
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A
1
Problem of reference
1.1 Geometry
and
loading
That is to say a hollow roll length
L
, of interior radius
F
R
and of external radius
R
. That is to say one
brace rigid circular section of radius
F
R
embedded in its center. One notes
I
and
E
surfaces interior and external of the hollow roll (see [Figure 1.1-a]). The loading consists with
to apply, at the node of the rigid reinforcement, a displacement
I
U
Z
E
(
0
I
U
>
) as well as a displacement
no one on the external edge
E
.
Appear 1.1-a: Diagram of the field and loading
The assumption of an axisymmetric solution is made what enables us to restrict our study has one
rectangular field 2D
. Dimensions of the field are as follows
:
0.5mm,
5.5mm,
10 mm
F
R
R
L
=
=
=
. The loading on the rigid reinforcement will be taken into account
by applying imposed displacement
I Z
U.E.
on all the side
I
field 2D like one
null displacement on the side
E
to take into account the embedding of the cylinder. Finally one imposes
a radial displacement no one on the faces lower and higher of the field in order to avoid a singularity
dependant on a change of boundary condition at the points
With
and
With
(see [Figure 1.1-a]). These conditions
in extreme cases will lead to a anti-plane solution (independent of
Z
) what makes it possible to obtain more
simply an analytical solution.
1.2 Parameters
Material
Values of the Young modulus, the Poisson's ratio, the critical stress and the tenacity of
material are taken in the following way:
- 1
1.5 MPa,
0,
1.1 Mpa,
0.9 N.mm
C
C
E
G
=
=
=
=
(They are of course values “tests” which does not correspond to any material in particular.)
With
I
E
R
E
Z
E
I
U
=
Z
U
E
0
R
U
=
0
R
U
=
With
R
L
F
R
Code_Aster
®
Version
8.1
Titrate:
SSNA115 Arrachement of a rigid reinforcement
Date
:
25/11/05
Author (S):
J. LAVERNE
Key
:
V6.01.115-A
Page:
3/6
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A
2
Reference solution
The reference solution is an analytical solution draw from [bib3], it even inspired from a study
unidimensional proposed in [bib1] and in a more general way being based on the approach
energetics of the rupture suggested per G.A. Francfort and J.J. Marigo [bib2]. We will not return
in the details of the calculation of this solution, we will present just the analytical value of the answer
total of the structure: imposed displacement
U
according to the corresponding force
F
:
()
()
1
2
2
F
F
F
F L
U F
sign F
R L
R L
µ
-
=
+
éq
2-1
where
µ
indicate the coefficient of Lamé (
2
E
µ
=
here),
density of energy of cracking (see
documentation [R7.02.12]) and where
(
)
ln
F
F
L
R
R R
=
is a length structural feature
decisive for the brutal or progressive evolution of decoherence.
3 Modeling
With
3.1
Characteristics of modeling
Simulation is carried out into axisymmetric. The elements with internal discontinuity allow
to represent the fissure along
I
. The latter have as a modeling AXIS_ELDI and one
cohesive behavior CZM_EXP. The other elements of the mesh are QUAD4 with one
elastic behavior ELAS in modeling AXIS.
3.2
Characteristics of the mesh
One carries out a mesh structured in quadrangles of the field
with 76 meshs in the height and
28 meshs in the radial direction. One lays out a layer of elements with discontinuity interns length
of
I
using control CREA_MAILLAGE and key word CREA_FISS (see documentation
[U4.23.02]). The orientation of the elements with discontinuity is carried out so that the direction
normal is directed according to
R
- E
(the tangential direction is thus according to
Z
- E
). The remainder of the field
is divided into linear meshs QUAD4 (see [Figure 3.2-a]).
Appear 3.2-a: Mesh of the field
I
E
Elements with discontinuity
QUAD4
R
E
Z
E
Code_Aster
®
Version
8.1
Titrate:
SSNA115 Arrachement of a rigid reinforcement
Date
:
25/11/05
Author (S):
J. LAVERNE
Key
:
V6.01.115-A
Page:
4/6
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A
3.3 Functionalities
tested
Controls
STAT_NON_LINE COMP_INCR
RELATION
CZM_EXP
AFFE_MODELE MODELING AXIS_ELDI
DEFI_MATERIAU RUPT_FRAG
SIGM_C
SAUT_C
CREA_MAILLAGE CREA_FISS
3.4
Sizes tested and results
The tangential stress
T
along the fissure (i.e in the elements with discontinuity) corresponds to
opposite of the force
F
divided by the surface of decoherence:
2
F
R L
. Moreover while resting on
form density of energy of surface
defined in [R7.02.12] and according to [éq 2-1] one deduces
following relation:
()
()
ln
T
T
C
T
T
C
C
L
G
U
sign
µ
= -
+
éq
3.4-1
The latter will enable us to carry out tests summarized in the table below.
Size tested
Theory
Code_Aster
Difference (%)
Tangential stress: VI7
PG1 of mesh MJ38
Moment: 6.00070E+00
7.69747E-01
7.6974726277784E-01
3.41E-05
Tangential stress: VI7
PG1 of mesh MJ38
Moment: 1.20004E+01
4.34935E-01
4.3493490987033E-01
- 2.07E-05
Tangential stress: VI7
PG1 of mesh MJ38
Moment: 1.93334E+01
1.28483E-01
1.2848319446210E-01
1.51E-04
Displacement DY
N5 node
Moment: 1.20004E+01
1.57674E+00
1.5767415306566E+00
9.71E-05
4
Summary of the results
It is noted that the element with discontinuity allows a good prediction of decoherence, indeed this
last develops in an identical way on all the height of the cylinder moreover the results
numerical are very close to the analytical solution. In addition modeling suggested allows
to correctly reproduce the brutal or progressive evolution of cracking according to
characteristic lengths
L
structure and
C
C
G
behavior. This last point is not
highlighted here but the interested reader will be able to refer to [bib3] for more details.
Code_Aster
®
Version
8.1
Titrate:
SSNA115 Arrachement of a rigid reinforcement
Date
:
25/11/05
Author (S):
J. LAVERNE
Key
:
V6.01.115-A
Page:
5/6
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A
5 Bibliography
[1]
CHARLOTTE Mr., FRANKFURT G.A., MARIGO J.J. and TRUSKINOVSKY L.: Revisiting
brittle fracture have year energy minimization problem: comparison off Griffith and Barenblatt
surface energy models. Proceedings off the Symposium one “Continuous Ramming and
Fracture " The dated science library, Elsevier, edited by A. BENALLAL, Paris, pp. 7-18, (2000).
[2]
FRANKFURT G.A. and MARIGO J.J.: Revisiting brittle fracture have year energy minimization
problem. J. Mech. Phys. Solids, 46 (8), pp. 1319-1342 (1998).
[3]
LAVERNE J.: Energy formulation of the rupture by models of cohesive forces:
numerical considerations theoretical and establishments, Thesis of Doctorate of the University
Paris 13, November 2004.
Code_Aster
®
Version
8.1
Titrate:
SSNA115 Arrachement of a rigid reinforcement
Date
:
25/11/05
Author (S):
J. LAVERNE
Key
:
V6.01.115-A
Page:
6/6
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A
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