Code_Aster ®
Version
5.0
Titrate:
SSLV112 - Calcul of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. Key DURAND
:
V3.04.112-B Page:
1/8

Organization (S): EDF/AMA

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
Document: V3.04.112

SSLV112 - Calcul of G by the method
Lagrangian for a circular fissure

Summary

It is about a test in statics for a three-dimensional problem. This test allows the calculation of the rate of refund of
energy room by the Lagrangienne method of propagation for an initial fissure quasi-circular plunged
in a presumedly infinite medium. One transforms it into circular fissure of more important radius.

The interest of the test is to study the validity of the calculation of the rate of refund of energy room after extension of
fissure. It is also to be able to calculate the rate of refund of energy starting from a grid fixed on one
fissure variable geometry (in elasticity). Methods of calculation of G_LOCAL, THETA_LAGRANGE and of
THETA_LEGENDRE are used.

The test includes/understands two modelings.

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV112 - Calcul of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. Key DURAND
:
V3.04.112-B Page:
2/8

1
Problem of reference

1.1 Geometry

Z
= 1.
G
H
I
F
O
B
E
Y
With
C
X
D


Large initial axis: OA = 35 mm

Small initial axis: OB = 33.95 mm

SupX = Face OEGH

SupY = Face OCIH

Supfissz: Face ABEDC

mailpress: Face IFGH

1.2
Material properties

Young modulus: E = 2.105 MPa

Poisson's ratio: = 0.3

1.3
Boundary conditions and loadings

Face OEGH: ux = 0

Face OCIH: uy = 0

Face ABEDC: uz = 0

Face IFGH: uniform constraint of traction Z = 1 MPa

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV112 - Calcul of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. Key DURAND
:
V3.04.112-B Page:
3/8

2
Reference solution

2.1
Method of calculation used for the reference solution


has



For a circular fissure of radius has in an infinite medium, the rate of refund of energy G is equal to
:

(1 - 2)
G =
4 2
has


E

Numerical application:

Initially, the fissure is not strictly circular (OA = 35 mm, OB = 33.95 mm).

One transforms it into circular fissure of radius has = 42 mm without touching with the grid, (it is the goal of
this method) but while forming on the modules of the field theta in each node of the bottom. One has then
NR
in any point G =
-
2.433 10 4
.
.
mm

2.2
Results of reference

Values of G local in bottom of fissure.

The solutions given in the “handbook” of SIH give the value of KI divided by by report/ratio
with the traditional definition [bib1].

2.3 References
bibliographical

[1]
Solution of Sneddon (1946) in G.C. SIH: Handbook off stress-intensity factors Institute off
Fracture and Solid Mechanics - Lehigh University Bethlehem, Pennsylvania
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV112 - Calcul of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. Key DURAND
:
V3.04.112-B Page:
4/8

3
Modeling a: method THETA_LAGRANGE

3.1
Characteristics of modeling

Z
Y
X


3.2
Characteristics of the grid

A number of nodes: 1754

A number of meshs and types: 304 PENTA 15 and 131 HEXA 20

3.3 Functionalities
tested

Commands



MECHANICAL AFFE_MODELE
3D
TOUT

CALC_MATR_ELEM OPTION
“RIGI_MECA_LAGR”


CALC_G_LOCAL_T' THETA_LAGRANGE”




PROPAGATION: 1



DEGRE: 4




3.4 Notice

The initial fissure is not circular (OA = 35 mm, OB = 33.95 mm) but the transformation
Lagrangian makes it circular thanks to the field theta of module different from 1 in each node from
melts of fissure (OA = OB = 42 mm in the final configuration).

·
The degree of the polynomials of LEGENDRE used to calculate G (S) is 4.
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV112 - Calcul of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. Key DURAND
:
V3.04.112-B Page:
5/8

4
Results of modeling A

4.1 Values
tested

The number between brackets indicates the nth position of the node on the bottom

Identification Reference
Aster %
difference
G local Noeud A (1)
1.2165 10-4 1.2406
10-4
1.98
G local Noeud (5)
1.2165 10-4 1.1268
10-4
7.96
G local Noeud (10)
1.2165 10-4 1.1406
10-4
6.65
G local Noeud (15)
1.2165 10-4 1.1892
10-4
2.30
G local Noeud (20)
1.2165 10-4 1.2013
10-4
1.26
G local Noeud (25)
1.2165 10-4 1.1825
10-4
2.88
G local Noeud B (33)
1.2165 10-4 1.3042
10-4
7.21

4.2 Notice

In Aster calculation, G local corresponds to the virtual extension of only one lip of the fissure
Gréf
NR
(half-crown), the value obtained is thus to compare with
= 12165
.

2
mm

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV112 - Calcul of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. Key DURAND
:
V3.04.112-B Page:
6/8

5
Modeling b: method THETA_LEGENDRE

5.1
Characteristics of modeling

Z
Y
X


5.2
Characteristics of the grid

A number of nodes: 1754

A number of meshs and types: 304 PENTA 15 and 131 HEXA 20

5.3 Functionalities
tested

Commands


MECHANICAL AFFE_MODELE
3D
TOUT
CALC_MATR_ELEM OPTION
“RIGI_MECA_LAGR”


CALC_THETA THETA_3D



CALC_G_LOCAL_T' THETA_LEGENDRE”




PROPAGATION: 1




DEGRE: 4




5.4 Notice

The initial fissure is not circular (OA = 35 mm, OB = 33.95 mm) but the transformation
Lagrangian makes it circular thanks to the field theta of module different from 1 in each node from
melts of fissure (OA = OB = 42 mm in the final configuration).

·
The degree of the polynomials of LEGENDRE used to calculate G (S) is 4.
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV112 - Calcul of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. Key DURAND
:
V3.04.112-B Page:
7/8

6
Results of modeling B

6.1 Values
tested

The number between brackets indicates the nth position of the node on the bottom

Identification Reference
Aster %
difference
G local Noeud A (1)
1.2165 10-4 1.1455
10-4
6.20
G local Noeud (5)
1.2165 10-4 1.1258
10-4
8.06
G local Noeud (10)
1.2165 10-4 1.1476
10-4
6.00
G local Noeud (15)
1.2165 10-4 1.1797
10-4
3.12
G local Noeud (20)
1.2165 10-4 1.1974
10-4
1.60
G local Noeud (25)
1.2165 10-4 1.1960
10-4
1.71
G local Noeud B (33)
1.2165 10-4 1.1929
10-4
1.98

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV112 - Calcul of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. Key DURAND
:
V3.04.112-B Page:
8/8

7
Summaries of the results

Calculation of G local:

·
2 methods (THETA_LEGENDRE and THETA_LAGRANGE) give the same ones appreciably
results (8% of error to the maximum compared to the analytical solution),
·
the precision of the results is average because the extension of the fissure is approximately 1.2, which
is close to the maximum of extension reasonable for this method for a fissure 3D,
·
method LEGENDRE is less expensive in time CPU.
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

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