Code_Aster ®
Version
3
Titrate:
SSLP103 Calcul of K I and KII for a fissured circular plate
Date:
24/08/99
Author (S):
E. SCREWS
Key:
V3.02.103-A Page:
1/6
Organization (S): EDF/IMA/MN
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
Document: V3.02.103
SSLP103 - Calcul of the coefficients of intensity of
constraints KI and KII for a circular plate
fissured in linear elasticity
Summary
It is about a test of breaking process in static linear elasticity for a two-dimensional problem. One
consider a circular plate fissured (with a tilted fissure of 30 degrees compared to the axis of
X-coordinates) for which one calculates:
· coefficients of intensity of constraints KI and KII,
· the rate of refund of energy G starting from the formula of IRWIN.
The interest of the test is to know the analytical solution which gives the coefficients of intensity of constraints and
to have a tilted fissure.
This test includes/understands a modeling which treats successively the plane strains and the plane stresses
(elements of continuous mediums).
The numerical results do not deviate more than 1 to 2% from the values of reference.
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HI-75/96/016 - Ind A

Code_Aster ®
Version
3
Titrate:
SSLP103 Calcul of K I and KII for a fissured circular plate
Date:
24/08/99
Author (S):
E. SCREWS
Key:
V3.02.103-A Page:
2/6
1
Problem of reference
1.1 Geometry
It is about a circular plate of radius 0A = 100 mm, with a tilted fissure of 30 degrees by
report/ratio with the X-axis.
Y
30°
0
X
With
1.2
Material properties
The characteristics of material are as follows:
E = 200.000 MPa
= 0.3
1.3
Boundary conditions and loadings
Displacements are imposed on the contour of the plate. They result from the analytical solution
singular in mixed mode (with KI = 2. and KII = 1.).
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HI-75/96/016 - Ind A

Code_Aster ®
Version
3
Titrate:
SSLP103 Calcul of K I and KII for a fissured circular plate
Date:
24/08/99
Author (S):
E. SCREWS
Key:
V3.02.103-A Page:
3/6
2
Reference solution
2.1
Method of calculation used for the reference solution
Y
x2
M
R

x1

O
X
In plane strains or plane stresses, the distribution of displacements is given in it
locate (0, x1, x2) by:

1 +
R



U
=
K cos (K - cos) +

K
sin
(K cos 2)
1

E
I
II
2
2

2
-
+





1 +
N



U =
K sin
(K - cos) -

K
cos
(K cos 2)
2

E
I
II
2
2

2
+
-





with K = 3 - 4 in plane deformations
3 -
K = 1+ in plane constraints
U

= cos u1 - sin U
X
2
or in the reference mark (O, X, Y) by: U
= sin u1 + cos U
Y
2
On the contour of the plate, one a: R = 0A = 100 Misters.
One chooses to take KI = 2. and KII = 1. and to impose displacements on the contour of the plate
circular.
2.2
Results of reference
KI = 2.
KII = 1.
G = 2.275 10­5
in plane deformations
G = 2.5 10­5
in plane constraints
2.3 References
bibliographical
[1]
H.D. BUI Mécanique of Brittle fracture - ED. Masson 1978
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HI-75/96/016 - Ind A

Code_Aster ®
Version
3
Titrate:
SSLP103 Calcul of K I and KII for a fissured circular plate
Date:
24/08/99
Author (S):
E. SCREWS
Key:
V3.02.103-A Page:
4/6
3 Modeling
With
3.1
Characteristics of modeling
Calculation is carried out in plane constraints (C_PLAN) then in plane deformations (D_PLAN).
0
With
3.2
Characteristics of the grid
A number of nodes: 737
A number of meshs and types: 204 meshs QUAD8, 30 meshs TRIA6
3.3 Functionalities
tested
Commands
Keys
CALC_G_THETA
CALC_K_G
U4.63.03
CALC_G_THETA
CALC_G
U4.63.03
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HI-75/96/016 - Ind A

Code_Aster ®
Version
3
Titrate:
SSLP103 Calcul of K I and KII for a fissured circular plate
Date:
24/08/99
Author (S):
E. SCREWS
Key:
V3.02.103-A Page:
5/6
4
Results of modeling A
4.1 Values
tested
The values tested are the coefficients of intensity of constraints KI and KII and the rate of refund
of energy G calculated by the formula of IRWIN:
Identification
Reference
Aster
% difference
Plane constraints
K
2.0
2.0067
0.33
I
K
1.0
0.9877
1.23
II
G
2.5 10-5
2.5213 10-5
0.85
Plane deformations
K
2.0
2.0030
0.15
I
K
1.0
0.9960
0.39
II
G
2.275 10-5
2.2968 10-5
0.96
4.2 Remarks
(1 - 2) 2 2
The formula of IRWIN gives:
G =
(K +K
I
II)
in plane deformations
E
1
2
2
and
G =
(K +K
I
II)
in plane constraints
E
Calculations are carried out with a crown of lower integration of radius 10.0 and radius
superior 20.0.
4.3 Parameters
of execution
Version: 3.06
Machine: CRAY C98
System: UNICOS
8.0
Obstruction memory:
8 MW
Time CPU To use:
22 seconds
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HI-75/96/016 - Ind A

Code_Aster ®
Version
3
Titrate:
SSLP103 Calcul of K I and KII for a fissured circular plate
Date:
24/08/99
Author (S):
E. SCREWS
Key:
V3.02.103-A Page:
6/6
5
Summaries of the results
Numerical values of the coefficients of intensity of constraints and the rate of refund of energy
do not deviate more than 1 to 2% from the values of reference, which is satisfactory.
The grid could be improved, in particular in the vicinity of the bottom of fissure.
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HI-75/96/016 - Ind A