Microsoft Word - V603128a. Doc.

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8.1

Titrate:

SSNP128 Validation of the element with discontinuity on a plane plate

Date

25/11/05

Author (S):

J. LAVERNE

Key

V6.03.128-A

Page:

1/8

Manual of Validation

V6.03 booklet: Nonlinear statics of the plane systems

HT-66/05/005/A

Organization (S):

EDF-R & D/AMA

Manual 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.

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Version

8.1

Titrate:

SSNP128 Validation of the element with discontinuity on a plane plate

Date

25/11/05

Author (S):

J. LAVERNE

Key

V6.03.128-A

Page:

2/8

Manual of Validation

V6.03 booklet: Nonlinear statics of the plane systems

HT-66/05/005/A

Problem of reference

1.1 Geometry

In the Cartesian frame of reference

()

X y

, let us consider a rectangular plane plate

rubber band noted

] []

[

(see [Figure 1.1-a]). Let us note

{}

]

[

0, H

the left face of

field and

the part complementary to the edge.

Appear 1.1-a: Diagram of the plate

Dimensions of the field

1 mm, H=2 mm

1.2

Material properties

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

- 1

10 MPa,

1.1 MPa,

0.9 N.mm

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

. The loading corresponds to a displacement imposed on the edges of the plate: (see

[Figure 1.3-a]).

(

)

X y

()

U U

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Version

8.1

Titrate:

SSNP128 Validation of the element with discontinuity on a plane plate

Date

25/11/05

Author (S):

J. LAVERNE

Key

V6.03.128-A

Page:

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Manual of Validation

V6.03 booklet: Nonlinear statics of the plane systems

HT-66/05/005/A

Appear 1.3-a: Diagram of the loading

Values

U U

and

are defined during the construction of the reference solution in the part

following.

U U

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Titrate:

SSNP128 Validation of the element with discontinuity on a plane plate

Date

25/11/05

Author (S):

J. LAVERNE

Key

V6.03.128-A

Page:

4/8

Manual of Validation

V6.03 booklet: Nonlinear statics of the plane systems

HT-66/05/005/A

Reference solution

In this part one exhibe an analytical solution with a nonconstant jump along

, then one

give a condition of unicity of the solution.

2.1 Solution

analytical

The function of Airy

()

X y

controlled by the equation

, if efforts

outsides are null, leads to stresses satisfying the compatibility and equilibrium equations
in elasticity (see

ung [bib1]). Components of the stress

and

derive from

()

X y

in the following way:

and

X y

= -

éq

2.1-1

Let us choose a function Bi-harmonic

()

X y

defined by:

(

)

(

)

X y

with

and

arbitrary real constants. One deduces some according to [éq 2.1-1] the field from

stress:

= -

éq

2.1-2

By integrating the elastic law, if one notes

the Young modulus and

the Poisson's ratio (that one

takes no one), one deduces the field from it from displacement in

checking balance:

()

(

)

éq

2.1-3

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Titrate:

SSNP128 Validation of the element with discontinuity on a plane plate

Date

25/11/05

Author (S):

J. LAVERNE

Key

V6.03.128-A

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5/8

Manual of Validation

V6.03 booklet: Nonlinear statics of the plane systems

HT-66/05/005/A

Let us note respectively

and

displacements on

and

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 stress

that one notes

()

, it is obtained

jump of displacement

()

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]):

()

= -

for all

[

]

0, H

. Thus, the new displacement imposed on

generating such a jump is equal

with

. One thus built an analytical solution of the plane plate checking the equations

of balance and compatibility with a discontinuity in

along which the jump of displacement

is not constant. Let us point out the boundary conditions of the problem:

(

)

X y

()

U U

éq

2.1-4

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:

éq 2.2-1

Dimensions of the plate and the parameters material previously given check this
condition.

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Titrate:

SSNP128 Validation of the element with discontinuity on a plane plate

Date

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Author (S):

J. LAVERNE

Key

V6.03.128-A

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Manual of Validation

V6.03 booklet: Nonlinear statics of the plane systems

HT-66/05/005/A

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

. The latter have as a modeling PLAN_ELDI and one

behavior CZM_EXP. The other elements of the mesh 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

0 MPa.mm,

1 4

MPa.mm,

1 2 MPa and

0 MPa

3.2

Characteristics of the mesh

One carries out a mesh 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 one

with the normal

directed according to

- uur

. This is carried out using key word CREA_FISS of CREA_MAILLAGE (see

documentation [U4.23.02]).

3.3 Functionalities

tested

Controls

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
On element MJ15

4.9315E-01 4.9363194272125E-01

0.098

Variable threshold: VI1
On element MJ45

1.075 1.0757405707848

0.069

Normal stress: VI6
On element MJ30

4.489E-01 4.4850081901631E-01

- 0.089

Code_Aster

Version

8.1

Titrate:

SSNP128 Validation of the element with discontinuity on a plane plate

Date

25/11/05

Author (S):

J. LAVERNE

Key

V6.03.128-A

Page:

7/8

Manual of Validation

V6.03 booklet: Nonlinear statics of the plane systems

HT-66/05/005/A

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 mesh was
realized in [bib2]. It is noted that the error made on the jump of displacement decrease when one refines
mesh. 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 mesh.

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, Thesis of Doctorate of the University
Paris 13, November 2004.