Code_Aster
®
Version
6.1
Titrate:
SSNV152 - Elastic traction. Calculation of the stresses of Cauchy
Date
:
19/08/02
Author (S):
P. MASSIN, G. BERTRAND
Key
:
V6.04.152-A
Page:
1/6
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal elements
HT-66/02/001/A
Organization (S):
EDF/AMA, CS IF
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal elements
Document: V6.04.152
Elastic SSNV152- Traction. Calculation of the stresses
of Cauchy
Summary
The goal of this test is to validate the calculation of the stresses of Cauchy in the control
CALC_ELEM
by the option
SIGM_ELNO_COQUE
.
Code_Aster
®
Version
6.1
Titrate:
SSNV152 - Elastic traction. Calculation of the stresses of Cauchy
Date
:
19/08/02
Author (S):
P. MASSIN, G. BERTRAND
Key
:
V6.04.152-A
Page:
2/6
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal elements
HT-66/02/001/A
1
Problem of reference
1.1 Geometry
The geometry of this test is a square plate in the plan (X, y) round of 30° compared to X around
of Z.
room
X
total
X
One calls
L
the length of the deformed plate, one will note X, y, Z, the co-ordinates of the configuration
deformation and X, Y, Z, co-ordinates of the initial configuration
1.2
Properties of materials
One takes
0
and
.
000
200
=
=
MPa
E
1.3
Boundary conditions and loadings mechanical
One locks the nodes No1, NO5 and No2 so that DX=DY=DZ=DRX=DRY=DRZ=0,
and one imposes a local displacement Dx=100. on nodes NO3, No4 and No7.
.
1000
0
=
L
No2
No1
NO5
No4
No7
NO3
M1
Z
y
X
Code_Aster
®
Version
6.1
Titrate:
SSNV152 - Elastic traction. Calculation of the stresses of Cauchy
Date
:
19/08/02
Author (S):
P. MASSIN, G. BERTRAND
Key
:
V6.04.152-A
Page:
3/6
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal elements
HT-66/02/001/A
2
Reference solution
2.1
Method of calculation used for the reference solution
The reference solution is analytical.
Passage of the initial state in a deformed state:
Z
B
B
Z
Y
has
has
y
X
L
X
0
0
0
,
,
1
=
=
=
where
A is the length of the deformation of the plate following Y,
has
0
is the initial length of the plate,
B is the thickness of the deformed plate,
B
0
is the initial thickness of the plate.
Owing to the fact that
0
=
and of the assumptions of hull, one has
0
0
,
B
B
has
has
=
=
Green-Lagrange tensor:
By definition of the tensor of Green-Lagrange, one has
+
+
=
J
K
I
K
I
J
J
I
ij
X
U
X
U
X
U
X
U
E
2
1
With
X
L
L
L
X
X
U
0
0
-
=
-
=
, one thus has
(
)
-
=
-
+
-
+
-
=
2
0
2
0
2
2
0
2
0
0
0
11
2
1
2
1
L
L
L
L
L
L
L
L
L
L
L
L
E
While replacing, one has
105
.
0
1000
1000
1100
2
1
2
2
2
11
=
-
=
E
Gradient of deformation:
By definition:
=
=
1
0
0
0
1
0
0
0
0
L
L
dZ
dz
Dy
dz
dX
dz
dZ
Dy
Dy
Dy
dX
Dy
dZ
dx
Dy
dx
dX
dx
F
That is to say
0
det
L
L
F
J
=
=
Code_Aster
®
Version
6.1
Titrate:
SSNV152 - Elastic traction. Calculation of the stresses of Cauchy
Date
:
19/08/02
Author (S):
P. MASSIN, G. BERTRAND
Key
:
V6.04.152-A
Page:
4/6
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal elements
HT-66/02/001/A
Stresses of Piola-Kirchhoff of second species:
That is to say S the stress of PK2, in our case,
21000
105
.
0
200000
.
11
11
=
×
=
=
E
E
S
Stress of Cauchy
That is to say S the tensor of stresses of Cauchy, one with the relation
(
)
T
F
S
F
F
S
.
.
det
1
=
, one deduces some then
that
23100
21000
.
1000
1100
.
.
.
1
11
0
0
11
0
0
=
=
=
=
S
L
L
L
L
S
L
L
L
L
S
xx
2.2
Results of reference
One calculates displacements DX and DY with node NO3, the stresses of PK2 and the stresses of
Cauchy on the M1 mesh.
2.3
Uncertainty on the solution
Analytical result.
2.4 References
bibliographical
Nothing.
Code_Aster
®
Version
6.1
Titrate:
SSNV152 - Elastic traction. Calculation of the stresses of Cauchy
Date
:
19/08/02
Author (S):
P. MASSIN, G. BERTRAND
Key
:
V6.04.152-A
Page:
5/6
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal elements
HT-66/02/001/A
3 Modeling
With
3.1
Characteristics of modeling
Elements COQUE_3D are used
3.2
Characteristics of the mesh
Co-ordinates of the main nodes:
Coor_x node
Coor_y
Coor_z
N01 500
866.025
0.
N02
0 0 0.
N03 866.025
500
0.
N04 366.025
1366.025
0.
The meshs used are:
1 mesh QUAD9
2 meshs TRIA7
3.3
Functionalities tested
Controls
Option
CALC_ELEM
SIGM_ELNO_COQUE
No2
No1
No4
NO3
Code_Aster
®
Version
6.1
Titrate:
SSNV152 - Elastic traction. Calculation of the stresses of Cauchy
Date
:
19/08/02
Author (S):
P. MASSIN, G. BERTRAND
Key
:
V6.04.152-A
Page:
6/6
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal elements
HT-66/02/001/A
4
Results of modeling A
4.1 Values
tested
Identification Reference
Aster
Difference
DX (No4) 8.66025
E+01
8.66025 E+01
4.66 E-05%
DY (No4) 50.0
50.0
0%
PK2-SIXX (M1) 21000.
21000.
2.04
E-08%
Cauchy-SIXX (M1) 23100.
23100.
2.14
E-08%
5
Summary of the results
The found results are in agreement with the analytical solution.