Code_Aster
®
Version
5.0
Titrate:
SSLV112 - Calculation of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. DURAND
Key
:
V3.04.112-B
Page:
1/8
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
Organization (S):
EDF/AMA
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
Document: V3.04.112
SSLV112 - Calculation 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 Lagrangian 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 mesh 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.
Code_Aster
®
Version
5.0
Titrate:
SSLV112 - Calculation of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. DURAND
Key
:
V3.04.112-B
Page:
2/8
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
1
Problem of reference
1.1 Geometry
H
= 1.
Y
G
E
B
With
F
I
C
D
X
O
Z
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 = 2.10
5
MPa
Poisson's ratio:
= 0.3
1.3
Boundary conditions and loadings
Face OEGH: U
X
= 0
Face OCIH: U
y
= 0
Face ABEDC: U
Z
= 0
Face IFGH: uniform stress of traction
Z
= 1 MPa
Code_Aster
®
Version
5.0
Titrate:
SSLV112 - Calculation of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. DURAND
Key
:
V3.04.112-B
Page:
3/8
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
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
:
(
)
G
E
has
= -
1
4
2
2
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 mesh, (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
in any point
G
NR
mm
=
-
2.433 10
4
.
.
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 K
I
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
Code_Aster
®
Version
5.0
Titrate:
SSLV112 - Calculation of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. DURAND
Key
:
V3.04.112-B
Page:
4/8
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
3
Modeling a: method
THETA_LAGRANGE
3.1
Characteristics of modeling
Y
X
Z
3.2
Characteristics of the mesh
A number of nodes: 1754
A number of meshs and types: 304 PENTA 15 and 131 HEXA 20
3.3 Functionalities
tested
Controls
MECHANICAL AFFE_MODELE
3D
ALL
CALC_MATR_ELEM OPTION
“RIGI_MECA_LAGR”
CALC_G_LOCAL_T' THETA_LAGRANGE”
PROPAGATION: 1
DEGREE: 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.
Code_Aster
®
Version
5.0
Titrate:
SSLV112 - Calculation of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. DURAND
Key
:
V3.04.112-B
Page:
5/8
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
4
Results of modeling A
4.1 Values
tested
The number between brackets indicates it nor
ème
position of the node on the bottom
Identification Reference
Aster %
difference
G local Node A (1)
1.2165 10
- 4
1.2406
10
- 4
1.98
G local Node (5)
1.2165 10
- 4
1.1268
10
- 4
7.96
G local Node (10)
1.2165 10
- 4
1.1406
10
- 4
6.65
G local Node (15)
1.2165 10
- 4
1.1892
10
- 4
2.30
G local Node (20)
1.2165 10
- 4
1.2013
10
- 4
1.26
G local Node (25)
1.2165 10
- 4
1.1825
10
- 4
2.88
G local Node B (33)
1.2165 10
- 4
1.3042
10
- 4
7.21
4.2 Notice
In calculation Aster, G local corresponds to the virtual extension of only one lip of the fissure
(half-crown), the value obtained is thus to compare with
G
NR
mm
ref.
2
12165
=.
Code_Aster
®
Version
5.0
Titrate:
SSLV112 - Calculation of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. DURAND
Key
:
V3.04.112-B
Page:
6/8
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
5
Modeling b: method
THETA_LEGENDRE
5.1
Characteristics of modeling
Y
X
Z
5.2
Characteristics of the mesh
A number of nodes: 1754
A number of meshs and types: 304 PENTA 15 and 131 HEXA 20
5.3 Functionalities
tested
Controls
MECHANICAL AFFE_MODELE
3D
ALL
CALC_MATR_ELEM OPTION
“RIGI_MECA_LAGR”
CALC_THETA THETA_3D
CALC_G_LOCAL_T' THETA_LEGENDRE”
PROPAGATION: 1
DEGREE: 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.
Code_Aster
®
Version
5.0
Titrate:
SSLV112 - Calculation of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. DURAND
Key
:
V3.04.112-B
Page:
7/8
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
6
Results of modeling B
6.1 Values
tested
The number between brackets indicates it nor
ème
position of the node on the bottom
Identification Reference
Aster %
difference
G local Node A (1)
1.2165 10
- 4
1.1455
10
- 4
6.20
G local Node (5)
1.2165 10
- 4
1.1258
10
- 4
8.06
G local Node (10)
1.2165 10
- 4
1.1476
10
- 4
6.00
G local Node (15)
1.2165 10
- 4
1.1797
10
- 4
3.12
G local Node (20)
1.2165 10
- 4
1.1974
10
- 4
1.60
G local Node (25)
1.2165 10
- 4
1.1960
10
- 4
1.71
G local Node B (33)
1.2165 10
- 4
1.1929
10
- 4
1.98
Code_Aster
®
Version
5.0
Titrate:
SSLV112 - Calculation of G local by a Lagrangian method
Date:
17/02/02
Author (S):
G. DEBRUYNE, C. DURAND
Key
:
V3.04.112-B
Page:
8/8
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
7
Summaries of the results
Calculation of G local:
·
2 methods (
THETA_LEGENDRE
and
THETA_LAGRANGE
) the same ones give 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.