Code_Aster ®
Version
7.2
Titrate:
SSLP101 - Taux of restitution of energy in plane constraints
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V3.02.101-D Page:
1/10

Organization (S): EDF-R & D/AMA

Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
Document: V3.02.101

SSLP101 - Taux of restitution of energy
in plane constraints

Summary:

It is about a test of breaking process in statics for a two-dimensional problem. One is considered
plate fissured in plane constraints, the functionalities tested are:

·
the rate of refund of energy G,
·
the rate of refund of energy calculated starting from the calculation of the coefficients of constraints K1 and K2.

The interest of the test is to compare the traditional value of G and the value of G (IRWIN) obtained starting from K1 and
K2. It also makes it possible to test the invariance of calculation compared to the crowns of integration.

This test contains 3 different modelings: the modeling A which treated calculation of Intégrale of Rice is not
more supported since Version 3.

Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLP101 - Taux of restitution of energy in plane constraints
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V3.02.101-D Page:
2/10

1
Problem of reference

1.1 Geometry

Rectangular plate with emerging fissure OC.
For reasons of symmetry, the model is tiny room to the half-structure y 0.

Y
I
v
H
U
With
O
C
X
has


Height plates: H = 250 mm

Width plates: I = 100 mm

Depth fissures: have = 37.5 mm (OC)

1.2
Material properties

E = 200000 MPa = 0.3
Assumption of the plane constraints.

1.3
Boundary conditions and loadings

·
Constraint imposed in Y = H:
= 1 MPa
·
Displacement for the edge CA defined by: has X L and y = 0
v = 0.
·
Not fixes a:
U = v = 0.

For modeling C one replaces the constraint imposed by a pressure on the lips of
fissure.
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLP101 - Taux of restitution of energy in plane constraints
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V3.02.101-D Page:
3/10

2
Reference solution

2.1
Method of calculation used for the reference solution

Reference solution of BROWN & STRAWLEY [bib1]:

J = F 2 has 2/E with F = 1 98
.
has
mm

in

2

and E

in

NR/mm

2.2
Results of reference for G

Results of reference G =
2 × ×
×
- 5 =
-
198
37 5 0 510
2 309310 3
.
.
.
.

I
The formula G (IRWIN) =
(2 2
K1 + K2) led, like K
0, with K
,
21 491
E
2 =
1 =

2.3
Results of reference for the derivative of G

While varying the Young modulus and the Fy loading, one notes that:


2
G
G = F
with
3

310
.
2
-
=
that is to say
=
F

2

Y
Y
F
Y


G

G
G =
with =
.
460 is
= -
E
E

E

2.4 Reference
bibliographical

[1]
BROWN-STAWLEY ASTM Special Technical Publication n° 410 (1966)

Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLP101 - Taux of restitution of energy in plane constraints
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V3.02.101-D Page:
4/10

3 Modeling
B

3.1
Characteristics of modeling

Radius
Center C


Ymax
Ymin
Xmin
Xmax


One calculates the field then the rate of refund of energy G, the coefficients of K1 constraints and K2,
the rate of refund of energy obtained by the formula of IRWIN, the direction of propagation of the fissure.

3.2
Characteristics of the grid

A number of nodes: 673

A number of meshs and types: 112 meshs QUAD8 and 142 meshs TRIA6

3.3 Functionalities
tested

Commands



MECHANICAL AFFE_MODELE
C_PLAN
TOUT
AFFE_CHAR_MECA



MECA_STATIQUE



CALC_THETA THETA_2D



CALC_G_THETA




Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLP101 - Taux of restitution of energy in plane constraints
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V3.02.101-D Page:
5/10

4
Results of modeling B

4.1 Values
tested

The values tested are the rate of refund of energy calculated by the method théta and the rate of
restitution of energy calculated by the formula of IRWIN starting from the coefficients of intensity of constraints
K1 and K2.

Identification Reference
Aster %
difference
Crown 1 G
2.3093 10­3 2.2994
10­3 4.
10­3
Crown 2 G
2.3093 10­3 2.2993
10­3 4.
10­3
Crown 3 G
2.3093 10­3 2.2993
10­3 4.
10­3
Crown 4 G
2.3093 10­3 2.2991
10­3 4.
10­3
Crown 5 G
2.3093 10­3 2.2981
10­3 5.
10­3
Crown 6 G
2.3093 10­3 2.2906
10­3 8.
10­3
Crown 1 G (IRWIN)
2.3093 10­3 2.2990
10­3 4.
10­3
Crown 2 G (IRWIN)
2.3093 10­3 2.2989
10­3 4.
10­3
Crown 3 G (IRWIN)
2.3093 10­3 2.2988
10­3 5.
10­3
Crown 4 G (IRWIN)
2.3093 10­3 2.2986
10­3 5.
10­3
Crown 5 G (IRWIN)
2.3093 10­3 2.2976
10­3 5.
10­3
Crown 6 G (IRWIN)
2.3093 10­3 2.2891
10­3 8.
10­3

4.2 Notice

1
The calculation of G, K1, K2, G (IRWIN) =
(2 2
1
K + K2) was carried out starting from 6 different fields,
E
correspondents each one with a circular ring centered out of C.

Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLP101 - Taux of restitution of energy in plane constraints
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V3.02.101-D Page:
6/10

5 Modeling
C

5.1
Characteristics of modeling

Radius
Center C


Ymax
Ymin
Xmin
Xmax


The loading differs:

·
one relieves the stress imposed in Y = H,
·
one imposes a pressure p = ­ 1 on the lips of the fissure.

5.2
Characteristics of the grid

A number of nodes: 673

A number of meshs and types: 112 meshs QUAD8 and 142 meshs TRIA6

5.3 Functionalities
tested

Commands



MECHANICAL AFFE_MODELE
C_PLAN ALL
AFFE_CHAR_MECA



MECA_STATIQUE



CALC_THETA THETA_2D



CALC_G_THETA



Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLP101 - Taux of restitution of energy in plane constraints
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V3.02.101-D Page:
7/10

6
Results of modeling C

6.1 Values
tested

Values of G

Identification Reference
Aster %
difference
Crown 1 G
2.3093 10­3 2.2994
10­3 4.
10­3
Crown 2 G
2.3093 10­3 2.2993
10­3 4.
10­3
Crown 3 G
2.3093 10­3 2.2993
10­3 4.
10­3
Crown 4 G
2.3093 10­3 2.2991
10­3 4.
10­3
Crown 5 G
2.3093 10­3 2.2981
10­3 5.
10­3
Crown 6 G
2.3093 10­3 2.2906
10­3 8.
10­3
Crown 1 G (IRWIN)
2.3093 10­3 2.2990
10­3 4.
10­3
Crown 2 G (IRWIN)
2.3093 10­3 2.2990
10­3 4.
10­3
Crown 3 G (IRWIN)
2.3093 10­3 2.2988
10­3 5.
10­3
Crown 4 G (IRWIN)
2.3093 10­3 2.2986
10­3 5.
10­3
Crown 5 G (IRWIN)
2.3093 10­3 2.2976
10­3 5.
10­3
Crown 6 G (IRWIN)
2.3093 10­3 2.2891
10­3 8.
10­3

6.2 Notice

1
2
2
The calculation of G, K1, K2 and G (IRWIN) =
(K +
1
K2)


was carried out starting from the same fields
E


that for preceding modeling. The results are identical.

Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLP101 - Taux of restitution of energy in plane constraints
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V3.02.101-D Page:
8/10

7 Modeling
F

7.1
Characteristics of modeling

Radius
Center C


Ymax
Ymin
Xmin
Xmax

One applies successively:

·
a surface force Fy=1 in Y = H,
·
a pressure p = ­ 1 on the lips of the fissure.

and one calculates dG/dFy and dG/dp on 2 different crowns.

In the 2 cases one calculates dG/on the same crowns.

7.2
Characteristics of the grid

A number of nodes: 673

A number of meshs and types: 112 meshs QUAD8 and 142 meshs TRIA6

7.3 Functionalities
tested

Commands



MECHANICAL AFFE_MODELE
C_PLAN ALL
AFFE_CHAR_MECA



MECA_STATIQUE



CALC_THETA THETA_2D



CALC_G_THETA_T SENSITIVITY



Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLP101 - Taux of restitution of energy in plane constraints
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V3.02.101-D Page:
9/10

8
Results of modeling F

8.1 Values
tested

Identification Reference
Aster %
difference
dg/of, crown n°1
­ 1.15E-8
­ 1.149E-8
0.03
dg/of, crown n°2
­ 1.15E-8
­ 1.149E-8
0.03
dg/dFy, crown n°1
4.6E-3
4.599E-3
0.03
dg/dFy, crown n°2
4.6E-3
4.599E-3
0.03
dg/dp, crown n°1
4.6E-3
4.599E-3
0.03
dg/dp, crown n°2
4.6E-3
4.599E-3
0.03

Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLP101 - Taux of restitution of energy in plane constraints
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V3.02.101-D Page:
10/10

9
Summary of the results

The calculation of G and its derivative is not sensitive to the choice of the field of integration.
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A

Outline document