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SSLP310 - Biblio_18. Fissure pressurized in an unlimited plane field
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Key: V3.02.310-A Page: 1/10

Organization (S): EDF-R & D/AMA, CS IF
Handbook 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 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.


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1
Problem of reference

1.1 Geometry
y
p (X, c)
X
- 1
1


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
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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 =)
1 =
I

Loading n° 2

K (X = 1, c) = (I
where I and I are related to Bessel modified of first
0 (c) + I1 (C
I
))
0
1
species of indices 0 and 1 [bib2].

Loading n° 3

K (X = 1, c) = I

1 (C
I
)

Loading n° 4

K (X = 1, c) = I

0 (C
I
)

Loading n° 5

K (X = 1, c) = J
where J is related to Bessel of first species of index 0 [bib2].
0 (C
I
)
0

In all the cases of loading

K 2
G
I
=
in plane constraints
E

(1 - 2) K2

G
I
=
in plane deformations
E

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, Chapitre 9. Mr. ABRAMOWITZ and I.A. STEGUN
(Editors). United States Dept. off Commerce, National Bureau off Standards.
Handbook of Validation
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3 Modeling
With

3.1
Characteristics of modeling

The model is limited to the half space y 0 and the finished area - X
X X
,
max
max
0 y ymax with X = y = 15.
max
max

It consists of 578 quadrangles with 8 nodes and 1699 triangles with 6 nodes.
It comprises 5230 nodes.

One uses the hyphotèse plane constraints.

3.2
Characteristics of the grid

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 grid is limited at the right end of the fissure.
The partly current density of the grid of the fissure is selected in order to be able to discretize
suitably the loading p (X, c).



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3.3 Functionalities
tested

Calculation of the rate of refund of energy G by method THETA for various crowns.

Commands


DEFI_FOND_FISS
FOND
GROUP_NO


NORMALE


CALC_THETA
THETA_2D
GROUP_NO


CALC_G_THETA_T
SYME_CHAR


CALC_K_G


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Key: V3.02.310-A Page: 6/10


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
-
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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
-
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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
-

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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
-

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SSLP310 - Biblio_18. Fissure pressurized in an unlimited plane field
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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.

Handbook of Validation
V3.02 booklet: Linear statics of the plane systems
HT-66/02/001/A

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