Code_Aster ®
Version
3
Titrate:
HSNV120 - Hyperelastic Traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A Page:
1/12
Organization (S): EDF/IMA/MN
Handbook of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
Document: V7.22.120
HSNV120 - Hyperelastic Traction
of a bar under thermal loading
Summary:
This quasi-static thermomechanical test consists in heating a parallelepipedic bar uniformly, it
to subject to an important traction for finally letting it return in a discharged state. One validates thus
kinematics of the great hyperelastic deformations (command STAT_NON_LINE, key word COMP_ELAS)
for a relation of elastic behavior non-linear (ELAS_VMIS_LINE and ELAS_VMIS_TRAC) with
thermal loading.
The bar is modelled by a voluminal element (HEXA20, modeling A) or quadrangular (QUAD8,
assumption of the plane constraints, modeling B).
The results obtained by Aster do not differ from the theoretical solution.
Handbook of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A

Code_Aster ®
Version
3
Titrate:
HSNV120 - Hyperelastic Traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A Page:
2/12
1
Problem of reference
1.1 Geometry
y
1.000 (mm)
1
4
2
3
Z
1.000 (mm)
X
1.2
Material properties
The material obeys a law of isotropic nonlinear behavior hyperelastic to work hardening
linear isotropic.
S
E
= 2.105 MPa
AND = 2.103 MPa
y
AND
y = 103 MPa
E

= 0,3

= 10­4 K1
E
Handbook of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A

Code_Aster ®
Version
3
Titrate:
HSNV120 - Hyperelastic Traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A Page:
3/12
1.3
Boundary conditions and loadings
The bar blocked in direction OX on the face [1,2] is subjected to a uniform temperature T and
a tractive effort F distributed on the face [3,4]. The sequences of loading are as follows:
1
4
Tunif
F
2
3
T °
(C)
F (MPa)
120
1298
20
T (S)
0
1
2
3
T (S)
0
1
2
3
Temperature of reference: Tréf = 20°C.
Handbook of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A

Code_Aster ®
Version
3
Titrate:
HSNV120 - Hyperelastic Traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A Page:
4/12
2
Reference solution
2.1
Method of calculation used for the reference solution
One seeks the field of displacement U in the form:
ux


U (X, y, Z) =
vy

vz

The gradient of the transformation, the deformation and its mechanical share are then:
1+ U
0
0


F =
0
1+

v
0

0
0
1+ v
(
U U + 2)


0
0


2
1
v v
T
+

2
E =
(F F)
(
)
1
=
0
0

2

2


v (v + 2)
0
0


2


0 have

0


EM = E - T
1 = 0 B 0

0 0 B
with:

U (U + 2)
has

=
- T


2

v (v + 2)
B

=
- T



2
Note:
(EM) = a-b = a-b (one supposes that >b) has
eq
Handbook of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A

Code_Aster ®
Version
3
Titrate:
HSNV120 - Hyperelastic Traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A Page:
5/12
The relation of behavior is written:

2
S
= K

xx
(+ B has
2) + G (has - b)

3


1
S
= S
= K
yy
zz
(+ B has
2) - G (has - b)


3
with:
E
3K = 1-2
modulate compressibility
To determine G by taking account of linear work hardening, one introduces:
E
· the modulus of rigidity: 2µ = 1+
E E
· the module of work hardening: R
T
'= E -,
AND
The “pseudo variable interns” p is worth then:
2µ (EM)
y
-
y
eq
2µ (has - b) -
p =
=
R' + 3µ
R' + 3µ
Finally, G is written:
y
+ R' p
G =
- B has
By taking account of the boundary conditions:

F
Sxx =
(dead load)

1+ U
Syy =


0 (free edge)
The system to be solved is written:

2
has - B
y
-
F
K (has + B
2)
(
)
y
+ + R'
=
3
R


'+ 3µ
1+ U





1
has - B
y
-
K (has + B
2)
(
)
y
-

+ R'
= 0
3
R


'+




Handbook of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A

Code_Aster ®
Version
3
Titrate:
HSNV120 - Hyperelastic Traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A Page:
6/12
It is also written:

F
3K (has + 2b) =


1+ U

F

3

µ
2 (has - b) =
1
y
+
µ

1

+ U

R' -
R'
WITH F fixed, it is thus about a nonlinear system out of U and v, since A is quadratic out of U and B
quadratic in v.
Nevertheless, one can choose to fix U (thus has) and to solve a linear system out of F and B (of which one
deduced p and v):
(
U U + 2)
·
has =
- T

2
1 F - 6 K B = 3

K has
1+ U
·



1
3

µ
1+
F + 2µ B = 2
y




µ has
R'
+
1

+ U
R'
2µ (has - b)
y
-
·
p =
R' + 3µ
·
v =
1+ (
2 B + has T
) - 1
It then remains to express the constraint of Cauchy:
=
1
F S FT
Det (F)
That is to say here:

1+ U

xx =

(
S
1+ v) 2 xx

yy = zz =

0
As for the force exerted on the face [3,4], because of assumption of died loads, she is written
simply:
F
X = F So
where So: initial surface of the face [
3,4]
F
y = 0
F
Z = 0
Handbook of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A

Code_Aster ®
Version
3
Titrate:
HSNV120 - Hyperelastic Traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A Page:
7/12
2.2
Results of reference
One will adopt like results of reference displacements, the constraint of Cauchy and the force
exerted on the face [3,4] (in 3D only):
At time T = 2 S (T = 100°C, traction F)
In fact, one seeks F such as lengthening:
U = 0 1
,
·
K = 166.666 MPa µ = 76.923 MPa R' = 2.020 MPa
·
= 0.095 have

0.90909

F - 106 B = 47 500
104.76 F +153.85 103 B = 128.85 103


·
F = 1.298 MPa
B = -.

0 046
·
p = 8.91 10 ­ 2
·
v = - 3.70 10 ­ 2

9
xx
= 1 399.66 MPa
xy = 0
Fx = 1.298 10 NR
·
yy = 0
xz = 0
Fy = 0
zz = 0
yz = 0
Fz = 0
At time T = 3 S (T = 0, F =)
0
The bar returned in its initial state:
U

= 0

= 0
p =

0
2.3
Uncertainty on the solution
The solution is analytical. With the round-off errors near, one can consider it exact.
2.4 References
bibliographical
One will be able to refer to:
[1]
E. LORENTZ: A nonlinear relation of behavior hyperelastic - internal Note EDF
DER HI-74/95/011/0
Handbook of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A

Code_Aster ®
Version
3
Titrate:
HSNV120 - Hyperelastic Traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A Page:
8/12
3 Modeling
With
3.1
Characteristics of modeling
Voluminal modeling:
1 mesh HEXA20
1 mesh QUAD8
Z
5
20
8
17
19
18
7
6
16
13
y
15
1
12
4
1.000 (mm)
9
11
2
10
3
X
Boundary conditions:
N2:
U = U = U
X
y
Z = 0
N9, N13, N14, N5, N17: U X = 0
N1:
U = U
X
Z = 0
N6:
U = U
X
y = 0
Charge: Traction on the face [3 4 8 7 11 16 19 15]
3.2
Characteristics of the grid
A number of nodes: 20
A number of meshs: 2
1 HEXA20
1 QUAD8
3.3 Functionalities
tested
Commands
Keys
STAT_NON_LINE
COMP_ELAS:
DEFORMATION:
“GREEN”
[U4.32.01]
RELATION:
“ELAS_VMIS_LINE”
“ELAS_VMIS_TRAC”
EXCIT:
CHARGE:
THERMIQUE
CALC_NO
OPTION:
“FORC_NODA”
[U4.61.03]
GEOMETRIE:
“DEFORMEE”
Handbook of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A

Code_Aster ®
Version
3
Titrate:
HSNV120 - Hyperelastic Traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A Page:
9/12
4
Results of modeling A
4.1 Values
tested
Identification
Reference
Aster
% difference
T = 2 Déplacement DX (N8)
100
100.
0.
T = 2 Déplacement DY (N8)
­ 37
­ 37.005
0.013
T = 2 Déplacement DZ (N8)
­ 37
­ 37.005
0.013
T = 2 Contraintes SIGXX (PG1)
1399.66
1399.67
0.001
T = 2 Contraintes SIGYY (PG1)
11013.986
10­10
/
T = 2 Contraintes SIGZZ (PG1)
0
10­10
/
T = 2 Contraintes SIGXY (PG1)
0
10­12
/
T = 2 Contraintes SIGXZ (PG1)
0
10­12
/
T = 2 Contraintes SIGYZ (PG1)
0
10­11
/
T = 2 Variable p VARI (PG1)
8.9110­2
8.91 10­2
0.
T = 3 Déplacement DX (N8)
0
10­13
/
T = 3 Déplacement DY (N8)
0
10­13
/
T = 3 Déplacement DZ (N8)
0
10­14
/
T = 3 Contraintes SIGXX (PG1)
0
10­10
/
T = 3 Contraintes SIGYY (PG1)
0
10­11
/
T = 3 Contraintes SIGZZ (PG1)
0
10­11
/
T = 3 Contraintes SIGXY (PG1)
0
10­11
/
T = 3 Contraintes SIGXZ (PG1)
0
10­11
/
T = 3 Contraintes SIGYZ (PG1)
0
10­11
/
T = 3 Variable p VARI (PG1)
0
0
/
T = 2 Force nodal DX (N8)
­ 1.0817108
­ 1.0817 108
­ 0.003
T = 2 Force nodal DY (N8)
0
10­5
/
T = 2 Force nodal DZ (N8)
0
10­6
/
4.2 Remarks
Calculation of the nodal force:
The force applied to the face [3,4], Fx, is distributed between the various nodes according to weighting
following:
· nodes nodes: ­ 1/12Fx
· nodes mediums: 4/12Fx
4/12
­ 1/12
­ 1/12
4/12
4/12
­ 1/12
­ 1/12
4/12
4.3 Parameters
of execution
Version: NEW 3.03.15
Machine: CRAY C90
Obstruction memory:
8 MW
Time CPU To use:
47.2 seconds
Handbook of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A

Code_Aster ®
Version
3
Titrate:
HSNV120 - Hyperelastic Traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A Page:
10/12
5 Modeling
B
5.1
Characteristics of modeling
Modeling 2D forced plane:
1 mesh QUAD8
1 mesh SEG3
y
8
1
4
5
7
2
X
6
3
Boundary conditions:
N2:
U X = 0
U y = 0
N1:
U X = 0
N5:
U X = 0
Loading:
Traction on the face [3 4 7] (mesh SEG3)
5.2
Characteristics of the grid
A number of nodes: 8
A number of meshs: 2
1 QUAD8
1 SEG3
5.3 Functionalities
tested
Commands
Keys
STAT_NON_LINE
COMP_ELAS:
DEFORMATION:
“GREEN”
[U4.32.01]
RELATION:
“ELAS_VMIS_LINE”
“ELAS_VMIS_TRAC”
EXCIT:
CHARGE:
THERMIQUE
Handbook of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A

Code_Aster ®
Version
3
Titrate:
HSNV120 - Hyperelastic Traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A Page:
11/12
6
Results of modeling B
6.1 Values
tested
Identification
Reference
Aster
% difference
T = 2 Déplacement DX (N4)
100
100
0
T = 2 Déplacement DY (N4)
­ 37
­ 37.004
0.013
T = 2 Contraintes SIGXX (PG1)
1399.66
1399.67
0.001
T = 2 Contraintes SIGYY (PG1)
0
10­12
/
T = 2 Contraintes SIGXY (PG1)
0
10­12
/
T = 2 Variable p VARI (PG1)
8.9110­2
8.91 10­2
0
T = 3 Déplacement DX (N4)
0
10­14
/
T = 3 Déplacement DY (N4)
0
10­13
/
T = 3 Contraintes SIGXX (PG1)
0
10­10
/
T = 3 Contraintes SIGYY (PG1)
0
10­10
/
T = 3 Contraintes SIGXY (PG1)
0
10­10
/
T = 3 Variable p VARI (PG1)
0
0
/
6.2 Parameters
of execution
Version: 3.03.13
Machine: CRAY C90
Obstruction memory:
8 MW
Time CPU To use:
123,8 seconds
Handbook of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A

Code_Aster ®
Version
3
Titrate:
HSNV120 - Hyperelastic Traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A Page:
12/12
7
Summary of the results
The numerical and analytical results coincide remarkably. One can however be astonished by
execution time manifestly longer for modeling in plane constraints (123,8 S) that for
the 3D (47,2 S). The difference is explained by a discretization in time much finer for
plane constraints, related to problems of convergence (the algorithm of resolution of the equation
nonlinear scalar out of p is still rudimentary).
Handbook of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A

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