Code_Aster
®
Version
7.2
Titrate:
SSLP101 - Rate of refund of energy in plane stresses
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V3.02.101-D
Page:
1/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A
Organization (S):
EDF-R & D/AMA
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
Document: V3.02.101
SSLP101 - Rate of refund of energy
in plane stresses
Summary:
It is about a test of breaking process in statics for a two-dimensional problem. One is considered
plate fissured in plane stresses, 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 stresses K1 and K2.
The interest of the test is to compare the conventional 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 the Integral of Rice is not
more supported since Version 3.
Code_Aster
®
Version
7.2
Titrate:
SSLP101 - Rate of refund of energy in plane stresses
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V3.02.101-D
Page:
2/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A
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.
H
I
With
has
C
Y
O
X
U
v
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 stresses.
1.3
Boundary conditions and loadings
·
Stress 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 stress imposed by a pressure on the lips of
fissure.
Code_Aster
®
Version
7.2
Titrate:
SSLP101 - Rate of refund of energy in plane stresses
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V3.02.101-D
Page:
3/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A
2
Reference solution
2.1
Method of calculation used for the reference solution
Reference solution of BROWN & STRAWLEY [bib1]:
2
NR/mm
in
and
mm
in
with
E
has
F
E
has
F
J
98
.
1
/
2
2
=
=
2.2
Results of reference for G
Results of reference
G
=
× ×
×
=
-
-
198
37 5 0 510
2 309310
2
5
3
.
.
.
.
The formula
G
(IRWIN) =
(
)
2
2
2
1
K
K
E
I
+
conduit, like
0
2
=
K
, with
491
,
21
1
=
K
2.3
Results of reference for the derivative of G
While varying the Young modulus and the Fy loading, one notes that:
2
Y
F
G
=
with
3
310
.
2
-
=
that is to say
Y
Y
F
F
G
2
=
E
G
=
with
.
460
=
that is to say
E
G
E
G
-
=
2.4 Reference
bibliographical
[1]
Special BROWN-STAWLEY ASTM Technical Publication n° 410 (1966)
Code_Aster
®
Version
7.2
Titrate:
SSLP101 - Rate of refund of energy in plane stresses
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V3.02.101-D
Page:
4/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A
3 Modeling
B
3.1
Characteristics of modeling
Radius
Center C
Y
max
X
min
Y
min
X
max
The field is calculated
then the rate of refund of energy G, coefficients of K1 stresses 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 mesh
A number of nodes: 673
A number of meshs and types: 112 meshs
QUAD8
and 142 meshs
TRIA6
3.3 Functionalities
tested
Controls
MECHANICAL AFFE_MODELE
C_PLAN
ALL
AFFE_CHAR_MECA
MECA_STATIQUE
CALC_THETA THETA_2D
CALC_G_THETA
Code_Aster
®
Version
7.2
Titrate:
SSLP101 - Rate of refund of energy in plane stresses
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V3.02.101-D
Page:
5/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A
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 stresses
K1 and K2.
Identification Reference
Aster %
difference
Crown 1 G
2.3093 10
3
2.2994
10
3
4.
10
3
Crown 2 G
2.3093 10
3
2.2993
10
3
4.
10
3
Crown 3 G
2.3093 10
3
2.2993
10
3
4.
10
3
Crown 4 G
2.3093 10
3
2.2991
10
3
4.
10
3
Crown 5 G
2.3093 10
3
2.2981
10
3
5.
10
3
Crown 6 G
2.3093 10
3
2.2906
10
3
8.
10
3
Crown 1 G (IRWIN)
2.3093 10
3
2.2990
10
3
4.
10
3
Crown 2 G (IRWIN)
2.3093 10
3
2.2989
10
3
4.
10
3
Crown 3 G (IRWIN)
2.3093 10
3
2.2988
10
3
5.
10
3
Crown 4 G (IRWIN)
2.3093 10
3
2.2986
10
3
5.
10
3
Crown 5 G (IRWIN)
2.3093 10
3
2.2976
10
3
5.
10
3
Crown 6 G (IRWIN)
2.3093 10
3
2.2891
10
3
8.
10
3
4.2 Notice
The calculation of G, K1, K2, G (IRWIN) =
(
)
1
12
22
E K
K
+
was carried out starting from 6 fields
different,
agents each one with a circular ring centered out of C.
Code_Aster
®
Version
7.2
Titrate:
SSLP101 - Rate of refund of energy in plane stresses
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V3.02.101-D
Page:
6/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A
5 Modeling
C
5.1
Characteristics of modeling
Radius
Center C
Y
max
X
min
Y
min
X
max
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 mesh
A number of nodes: 673
A number of meshs and types: 112 meshs
QUAD8
and 142 meshs
TRIA6
5.3 Functionalities
tested
Controls
MECHANICAL AFFE_MODELE
C_PLAN ALL
AFFE_CHAR_MECA
MECA_STATIQUE
CALC_THETA THETA_2D
CALC_G_THETA
Code_Aster
®
Version
7.2
Titrate:
SSLP101 - Rate of refund of energy in plane stresses
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V3.02.101-D
Page:
7/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A
6
Results of modeling C
6.1 Values
tested
Values of G
Identification Reference
Aster %
difference
Crown 1 G
2.3093 10
3
2.2994
10
3
4.
10
3
Crown 2 G
2.3093 10
3
2.2993
10
3
4.
10
3
Crown 3 G
2.3093 10
3
2.2993
10
3
4.
10
3
Crown 4 G
2.3093 10
3
2.2991
10
3
4.
10
3
Crown 5 G
2.3093 10
3
2.2981
10
3
5.
10
3
Crown 6 G
2.3093 10
3
2.2906
10
3
8.
10
3
Crown 1 G (IRWIN)
2.3093 10
3
2.2990
10
3
4.
10
3
Crown 2 G (IRWIN)
2.3093 10
3
2.2990
10
3
4.
10
3
Crown 3 G (IRWIN)
2.3093 10
3
2.2988
10
3
5.
10
3
Crown 4 G (IRWIN)
2.3093 10
3
2.2986
10
3
5.
10
3
Crown 5 G (IRWIN)
2.3093 10
3
2.2976
10
3
5.
10
3
Crown 6 G (IRWIN)
2.3093 10
3
2.2891
10
3
8.
10
3
6.2 Notice
The calculation of G, K1, K2 and G (IRWIN)
(
)
=
+
1
12
22
E K
K
was carried out starting from the same fields
that for preceding modeling. The results are identical.
Code_Aster
®
Version
7.2
Titrate:
SSLP101 - Rate of refund of energy in plane stresses
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V3.02.101-D
Page:
8/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A
7 Modeling
F
7.1
Characteristics of modeling
Radius
Center C
Y
max
X
min
Y
min
X
max
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 mesh
A number of nodes: 673
A number of meshs and types: 112 meshs
QUAD8
and 142 meshs
TRIA6
7.3 Functionalities
tested
Controls
MECHANICAL AFFE_MODELE
C_PLAN ALL
AFFE_CHAR_MECA
MECA_STATIQUE
CALC_THETA THETA_2D
CALC_G_THETA_T SENSITIVITY
Code_Aster
®
Version
7.2
Titrate:
SSLP101 - Rate of refund of energy in plane stresses
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V3.02.101-D
Page:
9/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A
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
Code_Aster
®
Version
7.2
Titrate:
SSLP101 - Rate of refund of energy in plane stresses
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V3.02.101-D
Page:
10/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/04/005/A
9
Summary of the results
The calculation of G and its derivative is not sensitive to the choice of the field of integration.