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Titrate:
SSLV110 - Elliptic Fissure in an infinite medium


Date:
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:
V3.04.110-C Page:
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Organization (S): EDF-R & D/AMA

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
Document: V3.04.110

SSLV110 - Elliptic Fissure in an infinite medium

Summary:

It is about a test in statics for a three-dimensional problem. This test makes it possible to calculate the rate of refund
of energy total and local on the bottom of fissure by the method.

The radii of the crowns of integration are variable along the fissure, and the rate of refund of energy
room is calculated according to 2 different methods (LEGENDRE and LAGRANGE).

The interest of the test is the validation of the method in 3D and the following points:

·
comparison between the results and an analytical solution,
·
stability of the results according to the crowns of integration,
·
comparison between 2 methods different for calculation from G local,
·
2 cases of equivalent loadings (pressure distributed and voluminal loading).

This test contains 4 different modelings.

The 3rd modeling tests the derivative of G compared to the parameters material and loading.

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SSLV110 - Elliptic Fissure in an infinite medium


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1
Problem of reference

1.1 Geometry

It is about an elliptic fissure plunged in a presumedly infinite medium. Only one eighth is modelled
of a parallelepiped:

y
P
Z
X
120 mm
6 mm
m
0
1250 m
25 mm
725 mm


0: melts of elliptic fissure

1.2 Properties
materials

E= 210.000.MPa

= 0.3

1.3
Boundary conditions and loadings

Symmetry compared to the 3 principal plans:

Ux = 0. in the plan X = 0.

UY = 0. in the plan Y = 0.

UZ = 0. in the plan Z = 0. out of the fissure

The conditions of loadings are is:

P = 1 MPa in the plan Z = 1250 mm (modelings A and B)

that is to say:

FZ = 8.10­4 NR/mm3 on all the elements of volume (loading are equivalent to the precedent)
(modelings C and D).
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SSLV110 - Elliptic Fissure in an infinite medium


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2
Reference solution

2.1
Method of calculation used for the reference solution

The reference solution is an analytical solution resulting from SIH [bib1] and [bib2].


It is noted that the angle indicates here the parametric angle of the point M (angle compared to axis OX of
projected M on the circle of radius b) and not the polar co-ordinate of this point.

1/4
1/2
K
1
b2


b2
I
=
2
sin +
2
cos
with K = 1



B
E (K)
a2


has


2

/2
1/2
E (K) = (1 - k2 2
sin) D
0


Here: = 25 mm B = 6 mm have, therefore K = 0,9707728

The values of the elliptic integrals E (K) are tabulées in [bib3], according to asin (K) which is worth
here 76,11°. One finds then: E (K) = 1,0672.
1/4

B ²

From where the factor of intensity of the constraints in MPa.mm: K () =
0680
,
4
sin ² +
cos ²

I



² has


1
(- ²)
Then, starting from the formula of Irwin (plane deformation): G () =
K I () 2
E

The total rate of refund of Gref energy is calculated by integration of G (): Gref = 5,76.10-3
J/Misters.

Derived from G (modeling C):
For the derivative of G compared to the Young modulus E, one can write:

G

G
G =
(with =
,
302 4) thus
= -
E
E

E
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Titrate:
SSLV110 - Elliptic Fissure in an infinite medium


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:
V3.04.110-C Page:
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In addition, while varying the Fz loading, one finds:


2
G
G = F
with =
9
,
2276 thus
= F

2

Z
Z
F
Z

2.2 Bibliography

[1]
G.C. SIH: Mathematical Theories off Brittle Fracture - FRACTURE, flight II - Academic Press -
1968
[2]
Mr. K. KASSIN and G.C. SIH: Three-dimensional stress distribution around year elliptical ace
under arbitrary loadings J. Appl. Mech., 88, 601-611, 1966.
[3]
H. TADA, P. PARIS, G. IRWIN: The Stress Analysis off Cracks Handbook - Third Edition -
International ASM - 2000
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Titrate:
SSLV110 - Elliptic Fissure in an infinite medium


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:
V3.04.110-C Page:
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3 Modeling
With

3.1
Characteristics of modeling

With = N01099 (S = 0.)
B = N01259 (S = 26.68)
C = N01179 (S = 17.8; =/4)

C1
C2
C3
With
C
Z
Y
B
X


Loading: Unit pressure distributed on the face of the block opposed to the plan of the lip:

P = 1.MPa in the plan Z = 1250.mm.

3.2
Characteristics of the grid

A number of nodes: 1716
A number of meshs and types: 304 PENTA15 and 123 HEXA20

3.3 Functionalities
tested

Commands



DEFI_FOND_FISS LEVRE_SUP
GROUP_MA ALL

CALC_THETA FOND_3D



THETA_3D



CALC_G_THETA_T RESULT
TOUT_ORDRE



NUME_ORDRE



LIST_ORDRE


CALC_G_LOCAL_T
“THETA_LEGENDRE”
DEGRE = 7


“G_LEGENDRE”




R_INF_FO/R_SUP_FO



AFFE_CHAR_MECA FORCE_FACE




3.4 Remarks

The degree of the polynomials of LEGENDRE used to calculate G (S) is 7 (maximum value Aster).

For the 3 crowns of integration, radii R_INF and R_SUP vary linearly along the bottom of
fissure.
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SSLV110 - Elliptic Fissure in an infinite medium


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4
Results of modeling A

4.1 Values
tested

The values tested are:

·
the total rate of refund of energy G,
·
the rate of refund of energy room G in all the nodes of the bottom of fissure.

The grid includes/understands only one of the lips of the fissure, it is thus necessary to use key word “SYME_CHAR”
automatically to multiply by 2 in Aster calculation the rate of refund of energy calculated
by virtual extension of the single lip.

In the same way, G total calculated here corresponds to the quarter of G of reference defined previously, only one
eighth of parallelepiped being represented.

Identification Reference
Aster %
difference




G Crown C1
1.44 10­3 1.410
10­3
- 2.1
G Crown C2
1.44 10­3 1.451
10­3
0.8
G Crown C3
1.44 10­3 1.424
10­3
- 1.1




G (A) crowns C1 7.171
10-5
6.829 10­5
- 4.8
G (A) crowns C2 7.171
10-5
7.239 10­5
0.95
G (A) crowns C3 7.171
10-5
6.864 10­5
- 4.3




G (B) crowns C1 1.721
10-5
1.48 10­5
- 13.8
G (B) crowns C2 1.721
10-5
1.57 10­5
- 8.7
G (B) crowns C3 1.721
10-5
1.90 10­5
- 6.9




G (C) crown C1 5.215
10-5
4.992 10­5
- 4.3
G (C) crown C2 5.215
10-5
5.124 10­5
- 1.7
G (C) crown C3 5.215
10-5
5.013 10­5
- 3.9

4.2 Notice

The results are rather stable between the crowns safe at the point B where the variation of G (S) is more
large and results far away from the reference solution. One can explain this variation by the grid
of poor quality.

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5 Modeling
B

5.1
Characteristics of modeling

With = N01099 (S = 0.)
B = N01259 (S = 26.68)
C = N01179 (S = 17.8)

C1
C2
C3
With
C
Z
Y
B
X


Loading: Unit pressure distributed on the face of the block opposed to the plan of the lip:

P = 1 MPa in the plan Z = 1250 Misters.

5.2
Characteristics of the grid

A number of nodes: 1716
A number of meshs and types: 304 PENTA15 and 123 HEXA20

5.3 Functionalities
tested

Commands


DEFI_FOND_FISS LEVRE_SUP
GROUP_MA


CALC_THETA FOND_3D


THETA_3D



CALC_G_THETA_T



CALC_G_LOCAL_T' THETA_LAGRANGE”



“G_LEGENDRE”
DEGRE = 4



R_INF_FO/R_SUP_FO



AFFE_CHAR_MECA FORCE_FACE




5.4 Remarks

“THETA_LAGRANGE”: the field is discretized starting from the functions of forms of the nodes of the bottom
of fissure, but G (S) is always discretized starting from the polynomials of LEGENDRE.

The degree of the polynomials of LEGENDRE used to calculate G (S) is 4 [R7.02.01].

For the 3 crowns of integration, radii R_INF and R_SUP vary linearly along the bottom of
fissure.
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SSLV110 - Elliptic Fissure in an infinite medium


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6
Results of modeling B

6.1 Values
tested

The values tested are:

·
the total rate of refund of energy G,
·
the rate of refund of energy room G in all the nodes of the bottom of fissure.

The grid includes/understands only one of the lips of the fissure, it is thus necessary to use key word “SYME_CHAR”
automatically to multiply by 2 in Aster calculation the rate of refund of energy calculated
by virtual extension of the single lip.

In the same way, G total calculated here corresponds to the quarter of G of reference defined previously, only one
eighth of parallelepiped being represented.

Identification Reference
Aster %
difference




G Crown C1
1.44 10­3 1.410
10­3
- 2.1
G Crown C2
1.44 10­3 1.451
10­3
0.8
G Crown C3
1.44 10­3 1.424
10­3
- 1.1




G (A) crowns C1 7.171
10-5 7.120
10-5 - 0.7
G (A) crowns C2 7.171
10-5 7.452
10-5 3.9
G (A) crowns C3 7.171
10-5 7.431
10-5 3.6




G (B) crowns C1 1.721
10-5 1.608
10-5 - 6.6
G (B) crowns C2 1.721
10-5 1.662
10-5 - 3.4
G (B) crowns C3 1.721
10-5 1.706
10-5 - 0.9




G (C) crown C1 5.215
10-5 4.978
10-5 - 4.5
G (C) crown C2 5.215
10-5 5.096
10-5 - 2.3
G (C) crown C3 5.215
10-5 5.014
10-5 - 3.9

6.2 Notice

The results are better than in modeling A at the point B, but the disparity between
crowns remains strong.

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SSLV110 - Elliptic Fissure in an infinite medium


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7 Modeling
C

7.1
Characteristics of modeling

With = N01099 (S = 0.)
B = N01259 (S = 26.68)
C = N01179 (S = 17.8)

C1
C2
C3
With
C
Z
Y
B
X


Loading: Voluminal force FZ equivalent to a unit pressure on the face of the block opposed to
plan of the lip:

FORCE_INTERN: FZ = 8.10­4 NR/mm3 on all the elements of volume.

7.2
Characteristics of the grid

A number of nodes: 1716
A number of meshs and types: 304 PENTA15 and 123 HEXA20

7.3 Functionalities
tested

Commands



DEFI_FOND_FISS LEVRE_SUP
GROUP_MA

CALC_THETA FOND_3D



THETA_3D



CALC_G_THETA_T SENSITIVITY



CALC_G_LOCAL_T
“THETA_LEGENDRE”
DEGRE = 7


“G_LEGENDRE”




R_INF_FO/R_SUP_FO



AFFE_CHAR_MECA FORCE_INTERN




7.4 Remarks

The degree of the polynomials of LEGENDRE used to calculate G (S) is 7 (maximum value Aster).

For the 3 crowns of integration, radii R_INF and R_SUP vary linearly along the bottom of
fissure.
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SSLV110 - Elliptic Fissure in an infinite medium


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V3.04.110-C Page:
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8
Results of modeling C

8.1 Values
tested

The values tested are:

·
the total rate of refund of energy G,
·
the rate of refund of energy room G in all the nodes of the bottom of fissure,
·
the derivative of G compared to E and to the loading in voluminal force Fz.

The grid includes/understands only one of the lips of the fissure, it is thus necessary to use key word “SYME_CHAR”
automatically to multiply by 2 in Aster calculation the rate of refund of energy calculated
by virtual extension of the single lip.

In the same way, G total calculated here corresponds to the quarter of G of reference defined previously, only one
eighth of parallelepiped being represented.

Identification Reference
Aster %
difference




G Crown C1
1.44 10­3 1.437
10­3
- 0.2
G Crown C2
1.44 10­3 1.479
10­3
2.7
G Crown C3
1.44 10­3 1.450
10­3
0.7




G (A) crowns C1 7.171
10-5 6.962
10-5 ­ 2.9
G (A) crowns C2 7.171
10-5 7.379
10-5 +2.9
G (A) crowns C3 7.171
10-5 6.997
10-5 ­ 2.4




G (B) crowns C1 1.721
10-5 1.509
10-5 ­ 12.2
G (B) crowns C2 1.721
10-5 1.598
10-5 ­ 7.1
G (B) crowns C3 1.721
10-5 1.629
10-5 ­ 5.2




G (C) crown C1 5.215
10-5 5.085
10-5 ­ 2.5
G (C) crown C2 5.215
10-5 5.219
10-5 0.1
G (C) crown C3 5.215
10-5 5.107
10-5 ­ 2.1




DG/of C1 crown
- 6.8610 10­9 - 6.842
10­9
­ 0.2
DG/of C2 crown
- 6.8610 10­9 - 7.041
10­9
2.7
DG/of C3 crown
- 6.8610 10­9 - 6.907
10­9
0.7




DG/dFz C1 crown 3.599 3.592
- 0.1
DG/dFz C2 crown 3.599 3.697
2.7
DG/dFz C3 crown 3.599 3.629
0.9

8.2 Notice

The results are rather stable between the crowns. One always notes worse results with
node B.
The errors on the derivative of G are comparable with those on G.
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V3.04.110-C Page:
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9 Modeling
D

9.1
Characteristics of modeling

With = N01099 (S = 0.)
B = N01259 (S = 26.68)
C = N01179 (S = 17.8)

C1
C2
C3
With
C
Z
Y
B
X


Loading: Voluminal force Fz equivalent to a unit pressure distributed on the face of the block
opposed to the plan of the lip:

FORCE_INTERN: FZ = 8.10­4 NR/mm3 on all the elements of volume.

9.2
Characteristics of the grid

A number of nodes: 1716
A number of meshs and types: 304 PENTA15 and 123 HEXA20

9.3 Functionalities
tested

Commands



DEFI_FOND_FISS LEVRE_SUP
GROUP_MA
CALC_THETA FOND_3D



THETA_3D



CALC_G_THETA_T


CALC_G_LOCAL_T' THETA_LAGRANGE”



“G_LEGENDRE”
DEGRE = 7



R_INF_FO/R_SUP_FO



AFFE_CHAR_MECA FORCE_INTERN




9.4 Remarks

“THETA_LAGRANGE”: the field is discretized starting from the functions of forms of the nodes of the bottom
of fissure, but G (S) is always discretized starting from the polynomials of LEGENDRE.

The degree of the polynomials of LEGENDRE used to calculate G (S) is 7.

For the 3 crowns of integration, radii R_INF and R_SUP are supposed to vary linearly on
bottom of fissure.
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SSLV110 - Elliptic Fissure in an infinite medium


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V3.04.110-C Page:
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10 Results of modeling D

10.1 Values
tested

The values tested are:

·
the total rate of refund of energy G,
·
the rate of refund of energy room G in all the nodes of the bottom of fissure.

The grid includes/understands only one of the lips of the fissure, it is thus necessary to use key word “SYME_CHAR”
automatically to multiply by 2 in Aster calculation the rate of refund of energy calculated
by virtual extension of the single lip.

In the same way, G total calculated here corresponds to the quarter of G of reference defined previously, only one
eighth of parallelepiped being represented.

Identification Reference
Aster %
difference




G Crown C1
1.44 10­3 1.437
10­3
- 0.2
G Crown C2
1.44 10­3 1.479
10­3
2.7
G Crown C3
1.44 10­3 1.450
10­3
0.7




G (A) crowns C1 7.171
10-5 7.259
10-5 1.2
G (A) crowns C2 7.171
10-5 7.597
10-5 5.9
G (A) crowns C3 7.171
10-5 7.575
10-5 5.7




G (B) crowns C1 1.721
10-5 1.636
10-5 ­ 4.9
G (B) crowns C2 1.721
10-5 1.992
10-5 ­ 1.7
G (B) crowns C3 1.721
10-5 1.734
10-5 0.7




G (C) crown C1 5.215
10-5 5.071
10-5 ­ 2.7
G (C) crown C2 5.215
10-5 5.192
10-5 0.4
G (C) crown C3 5.215
10-5 5.108
10-5 ­ 2.1

10.2 Notice

The results are better than in modeling C at the point B.

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11 Summary of the results

Calculation of G local:

·
2 methods (LEGENDRE and LAGRANGE) give the same results appreciably
(less than 5% of error compared to the analytical solution) except at the point B (not end
ellipse on the large axis) where the Lagrange method is most precise,
·
loading case: the values obtained with the voluminal loading are slightly
higher than those obtained with imposed constraints (including for the values of G).
The differences are tiny and due to numerical integrations different on the term from
volume and the term of edge.

Calculation of derived from G:

·
the errors on the derivative of G are weak and comparable with those on G.

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SSLV110 - Elliptic Fissure in an infinite medium


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