Code_Aster
®
Version
5.0
Titrate:
SSLP310 - Biblio_18. Fissure pressurized in an unlimited plane field
Date:
05/11/02
Author (S)
:
S. GRANET, I. CORMEAU, B. KURTH
Key:
V3.02.310-A
Page:
1/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/02/001/A
Organization (S)
: EDF-R & D/AMA, CS IF
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
V3.02.310 document
SSLP310 - Biblio_18. Fissure pressurized in one
unlimited plane field
Summary:
This test results from the validation independent of version 3 in breaking process.
It is about a two-dimensional test in statics (plane strains or stresses) which aims at
checking of G and KI under loading by pressure distributed not uniform on the lips, in unlimited medium. One
also check the nullity of KII with the option
CALC_K_G
.
The behavior of the structure is elastic linear isotropic.
The case test includes/understands only one plane modeling 2D in which one studies the influence of the parameter C
intervening in the loading.
Code_Aster
®
Version
5.0
Titrate:
SSLP310 - Biblio_18. Fissure pressurized in an unlimited plane field
Date:
05/11/02
Author (S)
:
S. GRANET, I. CORMEAU, B. KURTH
Key:
V3.02.310-A
Page:
2/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/02/001/A
1
Problem of reference
1.1 Geometry
X
y
- 1
1
p (X, c)
One considers the rectilinear fissure - 1
X
1 in the unlimited plane field.
1.2
Properties of material
The material is elastic linear homogeneous of Young E and Poisson's ratio modulus
.
E = 1000 Mpa,
= 0,3
1.3
Boundary conditions and loadings
The loading car-being balanced, the model is limited to the half space y
0.
Boundary conditions
Linear relation UX (- 1,0) + UX (1,0) = 0
Condition of symmetry UY = 0 for X
- 1, X
1 and y = 0.
Loading n° 1
p (X) = 1
Loading n° 2
p (X, c) = exp (cx) where C is a parameter
Loading n° 3
p (X, c) = HS (cx) where C is a parameter
Loading n° 4
p (X, c) = CH (cx) where C is a parameter
Loading n° 5
p (X, c) = cos (cx) where C is a parameter
Code_Aster
®
Version
5.0
Titrate:
SSLP310 - Biblio_18. Fissure pressurized in an unlimited plane field
Date:
05/11/02
Author (S)
:
S. GRANET, I. CORMEAU, B. KURTH
Key:
V3.02.310-A
Page:
3/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/02/001/A
2
Reference solution
2.1
Method of calculation used for the reference solution
Exact calculation symbolic system using software MAPLE V [bib1].
2.2
Results of reference
Loading n° 1
K X
I
(
)
= =
1
Loading n° 2
(
)
()
()
(
)
K X
C
I C
I C
I
=
=
+
1
0
1
,
where I
0
and I
1
are related to Bessel modified of first
species of indices 0 and 1 [bib2].
Loading n° 3
(
)
()
K X
C
I C
I
=
=
1
1
,
Loading n° 4
(
)
()
K X
C
I C
I
=
=
1
0
,
Loading n° 5
(
)
()
K X
C
J C
I
=
=
1
0
,
where J
0
is related to Bessel of first species of index 0 [bib2].
In all the cases of loading
G
K
E
I
=
2
in plane stresses
(
)
G
K
E
I
= -
1
2
2
in plane deformations
2.3 References
bibliographical
[1]
There the evaluation off stress intensity factors for is simple ace under parametric loading.
Technical notes. N.I. IOKADIMIS and G.T. ANASTASSELOS. Computers and Structures, 51,
n°6, 791-794, 1994.
[2]
Handbook off mathematical functions, Chapter 9. Mr. ABRAMOWITZ and I.A. STEGUN
(Editors). United States Dept. off Trade, National Office off Standard.
Code_Aster
®
Version
5.0
Titrate:
SSLP310 - Biblio_18. Fissure pressurized in an unlimited plane field
Date:
05/11/02
Author (S)
:
S. GRANET, I. CORMEAU, B. KURTH
Key:
V3.02.310-A
Page:
4/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/02/001/A
3 Modeling
With
3.1
Characteristics of modeling
The model is limited to the half space y
0 and with the finished area - X
max
X
X
max
,
0
y
ymax with X
max
= y
max
= 15.
It consists of 578 quadrangles with 8 nodes and 1699 triangles with 6 nodes.
It comprises 5230 nodes.
One uses the hyphotèse plane stresses.
3.2
Characteristics of the mesh
Use of procedure FISS 2d_V1.
The topological parameters concerning refinement around the bottom of fissure are:
·
nc = 4 (a number of crowns)
·
NS = 8 (a number of sectors)
·
NT = 1 (a number of crowns of déraffinement)
The radiant fine mesh is limited at the right end of the fissure.
The partly current density of the mesh of the fissure is selected in order to be able to discretize
suitably the loading p (X, c).
Code_Aster
®
Version
5.0
Titrate:
SSLP310 - Biblio_18. Fissure pressurized in an unlimited plane field
Date:
05/11/02
Author (S)
:
S. GRANET, I. CORMEAU, B. KURTH
Key:
V3.02.310-A
Page:
5/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/02/001/A
3.3 Functionalities
tested
Calculation of the rate of refund of energy G by the method THETA for various crowns.
Controls
DEFI_FOND_FISS
MELTS
GROUP_NO
NORMAL
CALC_THETA
THETA_2D
GROUP_NO
CALC_G_THETA_T
SYME_CHAR
CALC_K_G
Code_Aster
®
Version
5.0
Titrate:
SSLP310 - Biblio_18. Fissure pressurized in an unlimited plane field
Date:
05/11/02
Author (S)
:
S. GRANET, I. CORMEAU, B. KURTH
Key:
V3.02.310-A
Page:
6/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/02/001/A
4
Results of modeling A
4.1 Values
tested
Crown 0 (triangles)
Rinf = 0 mm, Rsup = 0,02 mm
Identification Reference
Aster %
difference
G, loading n°1
3,14158E-3
3,04077E-3
-
3,209
KII, loading n°1
0
0
0
KI, loading n°1
1,77245
1,6559
-
6,574
G, loading n°2, c=1
1,05349E-2
1,01411E-2
-
3,738
KII, loading n°2, c=1
0
0
0
KI, loading n°2, c=1
3,24576
3,03108
-
6,614
G, loading n°2, c=5
8,356742
7,9065
-
5,387
KII, loading n°2, c=5
0
0
0
KI, loading n°2, c=5
91,41522
85,4189
-
6,559
G, loading n°3, c=1
1,00344E-3
9,6006E-4
-
4,323
KII, loading n°3, c=1
0
0
0
KI, loading n°3, c=1
1,00172
0,93505
-
6,655
G, loading n°3, c=5
1,86052
1,760148
-
5,395
KII, loading n°3, c=5
0
0
0
KI, loading n°3, c=5
43,13380
40,33829
-
6,481
G, loading n°4, c=1
5,03571E-3
4,86064E-3
-
3,477
KII, loading n°4, c=1
0
0
0
KI, loading n°4, c=1
2,24404
2,09602
-
6,596
G, loading n°4, c=5
2,331095
2,20566
-
5,381
KII, loading n°4, c=5
0
0
0
KI, loading n°4, c=5
48,28142
45,08068
-
6,629
G, loading n°5, c=1
1,839487E-3
1,78707E-3
-
2,849
KII, loading n°5, c=1
0
0
0
KI, loading n°5, c=1
1,356277
1,267569
-
6,541
G, loading n°5, c=2,4048255577
0
4,1738E-8
-
KII, loading n°5, c=2,4048255577
0
0
0
KI, loading n°5, c=2,4048255577
0
2,0383E-3
-
Code_Aster
®
Version
5.0
Titrate:
SSLP310 - Biblio_18. Fissure pressurized in an unlimited plane field
Date:
05/11/02
Author (S)
:
S. GRANET, I. CORMEAU, B. KURTH
Key:
V3.02.310-A
Page:
7/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/02/001/A
Crown 1 (quadrangles)
Rinf = 0,02 mm, Rsup = 0,04 mm
Identification Reference
Aster
% difference
G, loading n°1
3,14158E-3
3,1669E-3
0,807
KII, loading n°1
0
0
0
KI, loading n°1
1,77245
1,78079
0,471
G, loading n°2, c=1 1,05349E-2
1,056655E-2
0,30
KII, loading n°2, c=1
0
0
0
KI, loading n°2, c=1
3,24576
3,256597
0,334
G, loading n°2, c=5
8,356742
8,25545
-
1,212
KII, loading n°2, c=5
0
0
0
KI, loading n°2, c=5
91,41522
91,528
0,123
G, loading n°3, c=1
1,00344E-3
1,000804E-3
-
0,263
KII, loading n°3, c=1
0
0
0
KI, loading n°3, c=1
1,00172
1,003475
0,175
G, loading n°3, c=5
1,86052
1,83815
-
1,202
KII, loading n°3, c=5
0
0
0
KI, loading n°3, c=5
43,13380
43,2091
0,175
G, loading n°4, c=1 5,03571E-3
5,06348E-3
0,552
KII, loading n°4, c=1
0
0
0
KI, loading n°4, c=1
2,24404
2,25312
0,405
G, loading n°4, c=5
2,331095
2,302636
-
1,221
KII, loading n°4, c=5
0
0
0
KI, loading n°4, c=5
48,28142
48,3188
0,078
G, loading n°5, c=1 1,839487E-3
1,86066E-3
1,152
KII, loading n°5, c=1
0
0
0
KI, loading n°5, c=1
1,356277
1,363914
0,563
G, loading n°5, c=2,4048255577
0
3,98377E-8
-
KII, loading n°5, c=2,4048255577
0
0
0
KI, loading n°5, c=2,4048255577
0
4,721938E-3
-
Code_Aster
®
Version
5.0
Titrate:
SSLP310 - Biblio_18. Fissure pressurized in an unlimited plane field
Date:
05/11/02
Author (S)
:
S. GRANET, I. CORMEAU, B. KURTH
Key:
V3.02.310-A
Page:
8/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/02/001/A
Crown 2 (quadrangles)
Rinf = 0,04 mm, Rsup = 0,06 mm
Identification Reference
Aster
% difference
G, loading n°1
3,14158E-3
3,1678E-3
0,835
KII, loading n°1
0
0
0
KI, loading n°1
1,77245
1,78075
0,468
G, loading n°2, c=1 1,05349E-2
1,056949E-2
0,328
KII, loading n°2, c=1
0
0
0
KI, loading n°2, c=1
3,24576
3,256529
0,332
G, loading n°2, c=5
8,356742
8,257967
-
1,182
KII, loading n°2, c=5
0
0
0
KI, loading n°2, c=5
91,41522
9,1527E1
0,123
G, loading n°3, c=1
1,00344E-3
1,001087E-3
-
0,234
KII, loading n°3, c=1
0
0
0
KI, loading n°3, c=1
1,00172
1,0034589
0,174
G, loading n°3, c=5
1,86052
1,838717
-
1,172
KII, loading n°3, c=5
0
0
0
KI, loading n°3, c=5
43,13380
43,2088
0,174
G, loading n°4, c=1 5,03571E-3
5,064887E-3
0,579
KII, loading n°4, c=1
0
0
0
KI, loading n°4, c=1
2,24404
2,25307
0,402
G, loading n°4, c=5
2,331095
2,30333
-
1,191
KII, loading n°4, c=5
0
0
0
KI, loading n°4, c=5
48,28142
48,31838
0,077
G, loading n°5, c=1 1,839487E-3
1,86117E-3
1,179
KII, loading n°5, c=1
0
0
0
KI, loading n°5, c=1
1,356277
1,363877
0,560
G, loading n°5, c=2,4048255577
0
4,0008E-8
-
KII, loading n°5, c=2,4048255577
0
0
0
KI, loading n°5, c=2,4048255577
0
4,711869E-3
-
Code_Aster
®
Version
5.0
Titrate:
SSLP310 - Biblio_18. Fissure pressurized in an unlimited plane field
Date:
05/11/02
Author (S)
:
S. GRANET, I. CORMEAU, B. KURTH
Key:
V3.02.310-A
Page:
9/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/02/001/A
Crown 3 (quadrangles)
Rinf = 0,06 mm, Rsup = 0,08 mm
Identification Reference
Aster
% difference
G, loading n°1
3,14158E-3
3,16794E-3
0,839
KII, loading n°1
0
0
0
KI, loading n°1
1,77245
1,78078
0,471
G, loading n°2, c=1 1,05349E-2
1,05699E-2
0,333
KII, loading n°2, c=1
0
0
0
KI, loading n°2, c=1
3,24576
3,2566
0,334
G, loading n°2, c=5
8,356742
8,25837
1,177
KII, loading n°2, c=5
0
0
0
KI, loading n°2, c=5
91,41522
91,5293
0,125
G, loading n°3, c=1
1,00344E-3
1,001132E-3
-
0,230
KII, loading n°3, c=1
0
0
0
KI, loading n°3, c=1
1,00172
1,003481
0,176
G, loading n°3, c=5
1,86052
1,838809
-
1,167
KII, loading n°3, c=5
0
0
0
KI, loading n°3, c=5
43,13380
43,20984
0,176
G, loading n°4, c=1 5,03571E-3
5,065103E-3
0,584
KII, loading n°4, c=1
0
0
0
KI, loading n°4, c=1
2,24404
2,25312
0,405
G, loading n°4, c=5
2,331095
2,303447
-
1,186
KII, loading n°4, c=5
0
0
0
KI, loading n°4, c=5
48,28142
48,31948
0,079
G, loading n°5, c=1 1,839487E-3
1,86124E-3
1,183
KII, loading n°5, c=1
0
0
0
KI, loading n°5, c=1
1,356277
1,363907
0,563
G, loading n°5, c=2,4048255577
0
4,00631E-8
-
KII, loading n°5, c=2,4048255577
0
0
0
KI, loading n°5, c=2,4048255577
0
4,71155E-3
-
Code_Aster
®
Version
5.0
Titrate:
SSLP310 - Biblio_18. Fissure pressurized in an unlimited plane field
Date:
05/11/02
Author (S)
:
S. GRANET, I. CORMEAU, B. KURTH
Key:
V3.02.310-A
Page:
10/10
Manual of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/02/001/A
4.2 Remarks
Constant 2,4048… is first zero of the function of Bessel J
0
.
Crown 0 (surrounding the bottom of fissure consists of triangles) gives poor results
compared to the other crowns.
The relative variations maxima between crowns 1,2 and 3 for G and KI are given below for
various loadings.
Loading 1 Loading 2 Loading 3 Loading 4 Loading 5
Variation on G
0,03%
0,03% 0,03% 0,03% 0,03%
Variation on KI 0,002% 0,002% 0,002% 0,002% 0,002%
The variations on G and KI are negligible.
Comment on the results of the loading n°5 with C = 2,4048….
The loading n°5 can be compared in order of magnitude with the loading n°1. Indeed, the amplitude
loading n°5 is worth 1 and its resultant is worth 1,14 to compare with the unit constant value
loading n°1.
The absolute deviation for the loading n°1 and crowns it n°1 is of:
·
2,53E-5 for G
·
8,34E-3 for KI
For the loading n°5, the Aster values which are compared with a null value of reference are:
·
3,98E-8 for G
·
4,72E-3 for KI
The order of magnitude on the absolute deviations is similar for the two loadings.
One can thus regard as correct the Aster results.
5
Summary of the results
Except for the results obtained on crown 0, calculations of K and G are very close to
exact theoretical solution. Indeed, the variations are always lower than 1,2% for the calculation of G and
lower than 0,6% for the calculation of KI.