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Titrate:
SSNP128 ­ Validation of the element with discontinuity on a plate planes Date
:
25/11/05
Author (S):
J. LAVERNE Key
:
V6.03.128-A Page:
1/8

Organization (S): EDF-R & D/AMA
Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
Document: V6.03.128
SSNP128 ­ Validation of the element with discontinuity
on a plane plate

Summary:

The goal of this test is of exhiber an analytical solution in order to validate the quality of the element with discontinuity (see
documentation [R7.02.12] for details on this element). The objective of this test is to check that this model
conduit with a good prediction of the value of the jump of displacement along a fissure. With this intention, one
seek an analytical solution presenting a nonconstant jump along a discontinuity which one compares
with the solution obtained numerically. In addition when one seeks to validate a numerical method it is
preferable to ensure itself of the unicity of the required solution. We will see that it is the case for the solution
analytical presented if a condition relating to the maximum size of the field studied according to
parameters of the model is checked.
Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/05/005/A

Code_Aster ®
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Titrate:
SSNP128 ­ Validation of the element with discontinuity on a plate planes Date
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Author (S):
J. LAVERNE Key
:
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1
Problem of reference

1.1 Geometry

In the Cartesian frame of reference (X, y), let us consider a rectangular plane plate
rubber band noted =] 0, [
L ×] 0, H [(see [Figure 1.1-a]). Let us note = 0 × 0, H the left face of
0
{}]
[
field and the 0 part complementary to the edge.




H
0

Y

X


L

Appear 1.1-a: Schéma of the plate

Dimensions of the field:

L = 1 mm, H=2 mm

1.2
Material properties

The material is elastic with a critical stress and a tenacity arbitrarily chosen:

- 1
E = 10 MPa, = 0, = 1.1 MPa, G = 0.9 N.mm
C
C

1.3
Boundary conditions and loadings

The boundary conditions are determined by the analytical solution presented in the part
following so that they lead to a fissure having a nonconstant jump along
0. The loading corresponds to a displacement imposed on the edges of the plate: (see
[Figure 1.3-a]).

U = U (X, y) on 0


U = U0 (y) - (y) on 0

Handbook of Validation
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Code_Aster ®
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Titrate:
SSNP128 ­ Validation of the element with discontinuity on a plate planes Date
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Author (S):
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U = U

0

U = U -

U = U

0

Y

X

U = U
Appear 1.3-a: Schéma of the loading

The values U, U0 and are defined during the construction of the reference solution in the part
following.

Handbook of Validation
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Titrate:
SSNP128 ­ Validation of the element with discontinuity on a plate planes Date
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2
Reference solution

In this part one exhibe an analytical solution with a nonconstant jump along 0, then one
give a condition of unicity of the solution.

2.1 Solution
analytical

The function of Airy (X, y) controlled by the equation = 0 on, if efforts
outsides are null, leads to constraints satisfying the compatibility and equilibrium equations
in elasticity (see Fung [bib1]). Components of the constraint,
and derive from
xx
yy
xy
(X, y) in the following way:

2
2
2




=
, =
and = -

éq
2.1-1
xx
2
yy
2
xy
y

X

X
y


Let us choose a function Bi-harmonic (X, y) defined by:

3
2
(
y
y
X, y) =
+ (X +)
+ xy
6
2

with, and arbitrary real constants. One deduces some according to [éq 2.1-1] the field from
constraint:


= X + y +
xx


= 0

éq
2.1-2
yy


=
- y -
xy

By integrating the elastic law, if one notes E the modulus Young and the Poisson's ratio (that one
takes no one), one deduces the field from it from displacement in checking balance:


1
x2


- y2 + X (y +)
U (X, y)
E

2





U =^

éq
2.1-3
v (X, y) =



1 x2



-
+
2 X




E 2


Handbook of Validation
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Code_Aster ®
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Titrate:
SSNP128 ­ Validation of the element with discontinuity on a plate planes Date
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Respectively let us note U0 and U the displacements on 0 and 0 given by [éq 2.1-3]. These
the last correspond to the boundary conditions leading to the stress fields [éq 2.1-2]. With
to start from these data, it is easy to build a field of displacement with a discontinuity on
the edge. Indeed, knowing the normal constraint .n on which one notes F (y), one obtains it
0
0
jump of displacement (y) by reversing the exponential law of behavior of Barenblatt type:
CZM_EXP (see documentation on the elements with internal discontinuity and their behavior:
[R7.02.12]):

G F y
F y
C
()
(y)
()
= -


F y


C
() ln
C



for all y in [0, H]. Thus, the new displacement imposed on generating such a jump is equal
0
with U -. One thus built an analytical solution of the plane plate checking the equations
0
of balance and compatibility with a discontinuity in 0 along which the jump of displacement
is not constant. Let us point out the boundary conditions of the problem:

U = U (X, y) on 0


éq
2.1-4
U = U0 (y) - (y) on 0

2.2
Unicity of the solution

After having built an analytical solution it is important to make sure that the latter is single
to be able to compare it with the numerical solution. One shows, to see [bib2], that unicity is guaranteed
as soon as the following condition, on the geometry of the field like on the parameters material, is
checked:
2 G
µ C
L <
.









éq 2.2-1
2
C
Dimensions of the plate and the parameters material previously given check this
condition.

Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/05/005/A

Code_Aster ®
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Titrate:
SSNP128 ­ Validation of the element with discontinuity on a plate planes Date
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Author (S):
J. LAVERNE Key
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3 Modeling
With

3.1
Characteristics of modeling

The idea is to carry out a digital simulation corresponding to the problem presented in the part
the preceding one and to compare the results obtained. The elements with discontinuity allow
to represent the fissure along 0. The latter have as a modeling PLAN_ELDI and one
behavior CZM_EXP. The other elements of the grid are elastic QUAD4 in
modeling D_PLAN.

The values of the parameters of the function of Airy for the construction of the analytical solution are taken
arbitrarily:

- 1
- 1
= 0 MPa.mm, = 1 4 MPa.mm, = 1 2 MPa and = 0 MPa

3.2
Characteristics of the grid

One carries out a grid of the plate structured in quadrangles with 20 meshs in the width and
50 in the height. One has the elements with discontinuity along the with dimensions 0 with the normal
uur
directed according to - X. This is carried out using key word CREA_FISS of CREA_MAILLAGE (see
documentation [U4.23.02]).

3.3 Functionalities
tested

Commands



STAT_NON_LINE COMP_INCR
RELATION
CZM_EXP
AFFE_MODELE MODELING PLAN_ELDI
DEFI_MATERIAU RUPT_FRAG
SIGM_C

SAUT_C

CREA_MAILLAGE CREA_FISS



3.4
Sizes tested and results

Size tested
Theory
Code_Aster
Difference (%)
Variable threshold: VI1
4.9315E-01 4.9363194272125E-01
0.098
On element MJ15
Variable threshold: VI1
1.075 1.0757405707848
0.069
On element MJ45
Normal constraint: VI6
4.489E-01 4.4850081901631E-01
- 0.089
On element MJ30

Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/05/005/A

Code_Aster ®
Version
8.1
Titrate:
SSNP128 ­ Validation of the element with discontinuity on a plate planes Date
:
25/11/05
Author (S):
J. LAVERNE Key
:
V6.03.128-A Page:
7/8

4
Summary of the results

These results enable us to conclude that the element with internal discontinuity leads to good
approximation of the analytical solution. Moreover, one study on the dependence with the grid was
realized in [bib2]. It is noted that the error made on the jump of displacement decrease when one refines
grid. That makes it possible to conclude that, in spite of a constant jump by element, this model allows
to correctly reproduce a fissure with a nonconstant jump by refining the grid.

5 Bibliography

[1]
FUNG Y.C.: Foundation off Solid Mechanics, Prentice-Hall, (1979).
[2]
LAVERNE J.: Energy formulation of the rupture by models of cohesive forces:
numerical considerations theoretical and establishments, Thèse de Doctorat of Université
Paris 13, November 2004.
Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/05/005/A

Code_Aster ®
Version
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Titrate:
SSNP128 ­ Validation of the element with discontinuity on a plate planes Date
:
25/11/05
Author (S):
J. LAVERNE Key
:
V6.03.128-A Page:
8/8

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Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
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