Code_Aster ®
Version
6.1
Titrate:
SSNV152 - Elastic Traction. Calculation of the constraints of Cauchy Date
:

19/08/02
Author (S):
P. MASSIN, G. BERTRAND Clé
:
V6.04.152-A Page:
1/6

Organization (S): EDF/AMA, CS IF
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal elements
Document: V6.04.152

Elastic SSNV152- Traction. Calculation of the constraints
of Cauchy

Summary

The goal of this test is to validate the calculation of the constraints of Cauchy in command CALC_ELEM by the option
SIGM_ELNO_COQUE.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal elements
HT-66/02/001/A

Code_Aster ®
Version
6.1
Titrate:
SSNV152 - Elastic Traction. Calculation of the constraints of Cauchy Date
:

19/08/02
Author (S):
P. MASSIN, G. BERTRAND Clé
:
V6.04.152-A Page:
2/6

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.

L
= 1000.
0

No1

No4

M1

NO5
No7
Z

NO3
y
No2
X


room

X
X

total

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 E = 200
.
000 MPa and = 0

1.3
Boundary conditions and loadings mechanical

One blocks 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.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal elements
HT-66/02/001/A

Code_Aster ®
Version
6.1
Titrate:
SSNV152 - Elastic Traction. Calculation of the constraints of Cauchy Date
:

19/08/02
Author (S):
P. MASSIN, G. BERTRAND Clé
:
V6.04.152-A Page:
3/6

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:

1
has
B
X =
X, y =
Y, Z =
Z
L
has
B
0
0
0

where
A is the length of the deformation of the plate following Y,
a0 is the initial length of the plate,
B is the thickness of the deformed plate,
b0 is the initial thickness of the plate.

Owing to the fact that = 0 and of the assumptions of hull, one have has = has, B = B
0
0

Green-Lagrange tensor:
1 U
U
U
U
I
J



K

K


By definition of the tensor of Green-Lagrange, there are E

ij =

+
+







2


X
X
X
X
J
I I J



L - L
1 L - L
L L
L L
1 L
L
0
- 0 (-) 2
0


2 - 2
With U = X - X
0
=
X, one thus has E

11 =

+
+
2
=
0
L


2


2 L
L
L
2
L
0

0



0

1 11002 - 10002
While replacing, there are E =
= 0.105
11
2
10002




Gradient of deformation:

By definition:
dx
dx
dx L



dX
Dy
dZ

0 0


Dy
Dy
Dy
l0

F =
= 0 1 0
dX
Dy
dZ
dz
dz
dz
0
0 1






dX
Dy
dZ

L
That is to say J = det F =

l0
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal elements
HT-66/02/001/A

Code_Aster ®
Version
6.1
Titrate:
SSNV152 - Elastic Traction. Calculation of the constraints of Cauchy Date
:

19/08/02
Author (S):
P. MASSIN, G. BERTRAND Clé
:
V6.04.152-A Page:
4/6

Constraints of Piola-Kirchhoff of second species:

That is to say S the constraint of PK2, in our case, S = E.E = 200000 × 0.105 = 21000
11
11

Constraint of Cauchy

1
That is to say S the tensor of constraints of Cauchy, one with the relation S =
(
T
F S
. F
.
), one deduces some then
det F




1 L
L
L
1100
that S

xx =
.S.
= .S =
21000
.
= 23100
11
11
L L
L
L
1000
0
0
0


l0

2.2
Results of reference

One calculates displacements DX and DY with node NO3, the constraints of PK2 and the constraints of
Cauchy on the M1 mesh.

2.3
Uncertainty on the solution

Analytical result.

2.4 References
bibliographical

Nothing.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal elements
HT-66/02/001/A

Code_Aster ®
Version
6.1
Titrate:
SSNV152 - Elastic Traction. Calculation of the constraints of Cauchy Date
:

19/08/02
Author (S):
P. MASSIN, G. BERTRAND Clé
:
V6.04.152-A Page:
5/6

3 Modeling
With

3.1
Characteristics of modeling

Elements COQUE_3D are used

3.2
Characteristics of the grid

No1

No4

No2
NO3

Co-ordinates of the principal 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

Commands
Option



CALC_ELEM
SIGM_ELNO_COQUE

Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal elements
HT-66/02/001/A

Code_Aster ®
Version
6.1
Titrate:
SSNV152 - Elastic Traction. Calculation of the constraints of Cauchy Date
:

19/08/02
Author (S):
P. MASSIN, G. BERTRAND Clé
:
V6.04.152-A Page:
6/6

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.

Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal elements
HT-66/02/001/A

Outline document