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Titrate:
SSLP313 - Fissure inclined in an unlimited plate


Date:
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J.M. PROIX, I. CORMEAU, E. LECLERE Key
:
V3.02.313-A Page:
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Organization (S): EDF/MTI/MN, CS IF
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
V3.02.313 document

SSLP313 - Fissure inclined in a plate
unlimited, subjected to a uniform traction ad infinitum

Summary:

This test results from the validation independent of version 3 in breaking process.

One calculates KI, KII and the rate of refund of energy for a right fissure, tilted of an angle, in one
large-sized plate subjected to a uniform traction. The model is two-dimensional in constraints
plane. The material is elastic linear isotropic. This test of reference in 2D makes it possible to check separability
KI and KII in a mixed mode.
The reference solution, given for a theoretically unlimited field, is analytical.

In addition to the energy method (CALC_G_THETA), one tests the method of calculation of the factors of intensity
constraints by extrapolation of displacements (POST_K1_K2_K3). Modeling B makes it possible to test
this last method with a type of grid recommended (nodes mediums with the quarter) to obtain a solution
precise.
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HI-75/01/010/A

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Titrate:
SSLP313 - Fissure inclined in an unlimited plate


Date:
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:
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1
Problem of reference

1.1 Geometry

y
has
X
has



One allots an unspecified value to the slope, = 37 degrees.
One chooses has = 1.E-3 Mr.

1.2
Properties of material

The material is elastic linear isotropic, of Young E and Poisson's ratio modulus.
E = 2.E11 Pa, = 0.3.
Assumption of the plane constraints

1.3
Boundary conditions and loadings

·
Arbitrary limits of the field with a grid:
- xmax X xmax with xmax = 10a
- ymax y ymax with ymax = 20a

·
Boundary conditions:
In order to block the 3 plane rigid modes exclusively.
UX = UY = 0 with the left lower corner of the complete model.
UY = 0 with the corner lower right of the complete model.
On the lower edge, we impose UY = 0

·
Loading: uniform tension yy = 0 on the higher edge:
The value of 0 is worth 100MPa, in plane constraints.
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HI-75/01/010/A

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SSLP313 - Fissure inclined in an unlimited plate


Date:
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:
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2
Reference solution

2.1
Method of calculation used for the reference solution

Function of constraint of Airy.

2.2
Results of reference

K =
has
2
I
O cos
K =
has
II
O sin cos

1
G
=
(K 2 + K 2
ref.
) in plane constraints
E
I
II

2.3
Uncertainty on the solution

Exact analytical solution (Irwin) in unlimited medium.

2.4 References
bibliographical

[1]
Y. MURAKAMI Stress intensity factors handbook, box 4.2, page 188. The Society off
Materials Science, Japan, Pergamon Near, 1987.

Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HI-75/01/010/A

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SSLP313 - Fissure inclined in an unlimited plate


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3 Modeling
With

3.1
Characteristics of modeling


Complete model
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SSLP313 - Fissure inclined in an unlimited plate


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After symmetrization and orientation
Handbook of Validation
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SSLP313 - Fissure inclined in an unlimited plate


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Initial block 2D

radius
center


The radius is worth 7,5E-5 m

3.2
Characteristics of the grid

The grid consists of 14888 nodes and 6674 elements, including 1392 elements QUA8 and 5282
elements TRI6.

3.3
Functionalities tested

Commands



MECHANICAL AFFE_MODELE
C_PLAN
TOUT
AFFE_CHAR_MECA FORCE_CONTOUR


MECA_STATIQUE


CALC_THETA THETA_2D

CALC_G_THETA_T OPTION
CALC_G

CALC_G_THETA_T OPTION
CALC_K_G

POST_K1_K2_K3


Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HI-75/01/010/A

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Titrate:
SSLP313 - Fissure inclined in an unlimited plate


Date:
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4
Results of modeling A

4.1 Values
tested

4.1.1 Results obtained with CALC_G_THETA_T
Identification Reference
Aster
% difference
G 1.0019E+02
1.0126E+02
1.07
KI 3,5750E+6
3,6038E+6
0,81
KII 2,6939E+6
2,7003E+6
0,24

4.1.2 Results obtained with POST_K1_K2_K3
Identification Method Reference
Aster %
difference
K1_MAX 1
3.57E+06
3.54E+06
­ 1.04
K1_MIN 1
3.57E+06
3.19E+06
­ 10.72
K2_MAX 1
2.69E+06
2.62E+06
­ 2.82
K2_MIN 1
2.69E+06
1.92E+06
­ 28.62
G_MAX 1
1.00E+02
9.69E+01
­ 3.33
G_MIN 1
1.00E+02
6.94E+01
­ 30.70
K1_MAX 2
3.57E+06
3.51E+06
­ 1.76
K1_MIN 2
3.57E+06
3.33E+06
­ 6.79
K2_MAX 2
2.69E+06
2.61E+06
­ 3.12
K2_MIN 2
2.69E+06
2.25E+06
­ 16.49
G_MAX 2
1.00E+02
9.57E+01
­ 4.50
G_MIN 2
1.00E+02
8.08E+01
­ 19.32

4.2
Remarks on the 2 methods of POST_K1_K2_K3:

Two methods are programmed in POST_K1_K2_K3:

·
Method 1: one calculates the jump of the field of displacements squared and one divides it by R.
Various values of K1 (resp. K2, K3) are obtained (except for a factor) by extrapolation
in R =0 of the segments of straight lines thus obtained. If the solution were perfect (field
asymptotic analytical everywhere), one should obtain a line. Actually, one is obtained
curve, therefore values different of K1, K2, K3. In order to give an indication of
quality of the result, one lists the values maximum and minimum obtained on the unit of
discussed items (that one names here K1_MAX, K1_MIN, etc…)
·
Method 2: one traces the jump of the field of displacements squared according to R. Les
approximations of K1 are (always except for a factor) equal to the slope of the segments connecting
the origin at the various points of the curve. There still, one obtains various values of K1,
K2, K3.et one lists the values maximum and minimum obtained on the unit of the points
treaties (named K1_MAX, K1_MIN, etc…)
·
To provide to compare the solution obtained with that provided by CALC_G_THETA_T, one
calculate G starting from K1 and K2 by the formula of Irwin, which gives G_MAX and G_MIN.

4.3 Parameters
of execution

Version: 5.3


Machine: SGI/ORIGIN 2000


Obstruction memory:
128 Mo
Time CPU To use: 22 seconds
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HI-75/01/010/A

Code_Aster ®
Version
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Titrate:
SSLP313 - Fissure inclined in an unlimited plate


Date:
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Author (S):
J.M. PROIX, I. CORMEAU, E. LECLERE Key
:
V3.02.313-A Page:
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5 Modeling
B

5.1
Characteristics of modeling

Even form of grid that previously, but modification of the co-ordinates of the nodes mediums
edges touching the bottom of fissure, to move them with the quarter of these edges (method of
Barsoum).

This modification of the co-ordinates of the nodes is carried out by an accessible procedure GIBI in
to card-index data of grid (SSLP313B.datg).

5.2
Characteristics of the grid

The grid consists of 14888 nodes and 6674 elements, including 1392 elements QUA8 and 5282
elements TRI6.

5.3
Functionalities tested

Commands



MECHANICAL AFFE_MODELE
C_PLAN
TOUT
AFFE_CHAR_MECA FORCE_CONTOUR


MECA_STATIQUE


CALC_THETA THETA_2D


CALC_G_THETA OPTION
CALC_G

CALC_G_THETA OPTION
CALC_K_G

POST_K1_K2_K3


Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HI-75/01/010/A

Code_Aster ®
Version
5.3
Titrate:
SSLP313 - Fissure inclined in an unlimited plate


Date:
15/10/01
Author (S):
J.M. PROIX, I. CORMEAU, E. LECLERE Key
:
V3.02.313-A Page:
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6
Results of modeling B

6.1 Values
tested

6.1.1 Results obtained with CALC_G_THETA_T
Identification Reference
Aster
% difference
G 1.0019E+02
1.0135E+02
1.16
KI 3,5750E+6
3.6033E+06
0.79
KII 2,6939E+6
2.6996E+06
0.21

6.1.2 Results obtained with POST_K1_K2_K3
Identification Method Reference
Aster %
difference
K1_MAX 1
3.57E+06
3.6089E+06
0.95
K1_MIN 1
3.57E+06
3.5995E+06
0.69
K2_MAX 1
2.69E+06
2.7035E+06
0.36
K2_MIN 1
2.69E+06
2.6944E+06
0.02
G_MAX 1
1.00E+02
1.0142E+02
1.23
G_MIN 1
1.00E+02
1.0120E+02
1.01
K1_MAX 2
3.57E+06
3.6027E+06
0.78
K1_MIN 2
3.57E+06
3.5344E+06
­ 1.14
K2_MAX 2
2.69E+06
2.6927E+06
­ 0.05
K2_MIN 2
2.69E+06
2.6478E+06
­ 1.71
G_MAX 2
1.00E+02
1.0115E+02
0.96
G_MIN 2
1.00E+02
9.7512E+01
­ 2.67

6.2
Remarks on the 2 methods of POST_K1_K2_K3:

Two methods are programmed in POST_K1_K2_K3:

·
Method 1: one calculates the jump of the field of displacements squared and one divides it by R.
Various values of K1 (resp. K2, K3) are obtained (except for a factor) by extrapolation
in R =0 of the segments of straight lines thus obtained. If the solution were perfect (field
asymptotic analytical everywhere), one should obtain a line. Actually, one is obtained
curve, therefore values different of K1, K2, K3. In order to give an indication of
quality of the result, one lists the values maximum and minimum obtained on the unit of
discussed items (that one names here K1_MAX, K1_MIN, etc…)
·
Method 2: one traces the jump of the field of displacements squared according to R. Les
approximations of K1 are (always except for a factor) equal to the slope of the segments connecting
the origin at the various points of the curve. There still, one obtains various values of K1,
K2, K3.et one lists the values maximum and minimum obtained on the unit of the points
treaties (named K1_MAX, K1_MIN, etc…)
·
To provide to compare the solution obtained with that provided by CALC_G_THETA_T, one
calculate G starting from K1 and K2 by the formula of Irwin, which gives G_MAX and G_MIN.

6.3 Parameters
of execution

Version: 5.3


Machine: SGI/ORIGIN 2000


Obstruction memory:
128 Mo
Time CPU To use:19 seconds
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HI-75/01/010/A

Code_Aster ®
Version
5.3
Titrate:
SSLP313 - Fissure inclined in an unlimited plate


Date:
15/10/01
Author (S):
J.M. PROIX, I. CORMEAU, E. LECLERE Key
:
V3.02.313-A Page:
10/10

7
Summary of the results

With this choice of the limits of the field of calculation, we obtain variations of about 1% on
coefficients KI and KII, and on the rate of refund of energy G.

With regard to method POST_K1_K2_K3, the results are further away from the reference with
a standard grid (of ­ 1% with ­ 30% of variation), on the other hand, with a grid of the type Barsoum (nodes
mediums with the quarter on the sides), recommended for this type of method, the differences are included/understood between ­ 3% and
+1.2%, which is relatively precise.
Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HI-75/01/010/A

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