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Titrate:
HSNV100 - Thermoplasticity in simple traction


Date:
03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
1/20

Organization (S): EDF-R & D/AMA

Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
V7.22.100 document

HSNV100 - Thermoplasticity in simple traction

Summary:

This test treats the thermo plasticity of Von Mises with isotropic work hardening on a three-dimensional problem
(modeling A into axisymmetric) and two-dimensional (modeling B in plane constraints). Interest of the test
holds with the dependence of the elastic limit with the temperature. It also makes it possible to test the orthotropism in
thermo elasticity because it applies to an isotropic material then with an isotropic material declared orthotropic.
This makes it possible to test the functionalities of the orthotropism. One tests there also the calculation of the deformation energy.

Two modelings (C with element TUYAU, D with element TUYAU_6M) are added to test
thermoplasticity in these elements.

A modeling (E) makes it possible to test the good taking into account of the variation of the coefficients of
behavior VMIS_CINE_LINE with the temperature.

A modeling (F) makes it possible to test the calculation of the thermoelastic deformation energy in the beams.

Modeling (G) makes it possible to test the same functionalities as modelings A and B, but in 3D.

The solution is analytical.

Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HT-66/03/008/A

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Titrate:
HSNV100 - Thermoplasticity in simple traction


Date:
03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
2/20

1
Problem of reference

1.1 Geometry

Axisymmetric cylinder (modeling A) or plates rectangular (modeling B) or right pipe
(modelings C and D)

Z or y
D
C
H
B
has
R or X
0
With
B

Appear 1.1-a: Géométrie of the structure

Interior radius: has = 1 mm

external radius: B = 2 mm (width AB: 1 mm)

height: H = 4 mm

1.2
Property of materials

E = 200.000 MPa modulus Young
AND = 50.000 MPa modulates tangent
= 0 3
.
(T)
y
= 0 (1 - S (T - T0) elastic limit
= 400 MPa =
0
y (T0)

S =
-
10 2 °C-1
= -
10 5 °C-1 thermal dilation coefficient
C p = 0
J/(mm3°C) heat voluminal
= -
10 3
W/(mm°C)
thermal conductivity

For isotropic material declared orthotropic, it comes:

E_L = E_T = E_N = E

Nu_LT = Nu_LN = Nu_TN = Naked =

E
G_LT = G_LN = G_TN =
(
= 76923,077
2 1+)

ALPHA_L = ALPHA_T ALPHA_N = ALPHA =
Handbook of Validation
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HT-66/03/008/A

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Titrate:
HSNV100 - Thermoplasticity in simple traction


Date:
03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
3/20



AND
0

AND
y T
()
E


Appear 1.2-a: Courbe of traction of material

1.3
Boundary conditions and loadings

Modeling A into axisymmetric: uz = 0 on the sides AB and CD (Axe OZ fixes)

Modeling B in plane constraints: uy = 0 on sides AB and CD, ux = 0 in A

T (T) = T + T0 = 1°C/S T0 = 0°C.

Modelings C and D: embedding in A, Dy = 0 out of C

2
Reference solution

2.1
Method of calculation used for the reference solution

Axisymmetric case (2D)

déplacemen

of

Fields
T: U = ur (R) er
(
in

blocking
Z)
U '0
0

R
R


déformatio

of

Fields
N:
(U)


= 0
0
0


according to Z

ur



0 0



R


0 0 0

R




constraint

of

Fields
S:
= L0 1 0
limits)
with

conditions

(cf.


according to Z




0 0 0



Parallelepipedic case

déplacemen

of

Fields
T: U = ux (X) E + U
X
y (y) E y
(
in

blocking
Z)
U '0
0

X
X
déformatio

of

Fields
N:
(U)



= 0
0
0


according to Z





0
0 U y '

y
0 0 0

X




constraint

of

Fields
S:
= L0 1 0
limits)
with

conditions

(cf.


according to Z




0 0 0

y

The case could be studied in plane constraints and 3D.
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HT-66/03/008/A

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Titrate:
HSNV100 - Thermoplasticity in simple traction


Date:
03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
4/20


AND

E
y
2µ = 1+
E
3K =
E
1 - 2



The law of behavior is written (variable scalar intern p):

1
1


=
tr Id +
D + p +
O

(T - T) Id
9K


with:
1
D
= - tr Id
(diverter of the constraints)


3

3
D
3

P
&
=
p&
, with
=
D D

2


éq
2
éq
p & =

0 if F (, p) =
-
éq
R (p) < 0
p &

0 if F (, p) = 0



R (p) indicates the function of work hardening:

E E
R (p)
T
= +
p
y

E - AND

The rate &p can be expressed, when F (, p) = 0. Indeed, from &p F identically no one, one draws:
&p &f+ &p F = 0. Thus, when one is on the criterion (F =)
0, necessarily &f = 0. I.e.:

3D &
D

- R,
T
&T - R, p &p = 0
2
éq

3D &
D
E E

+ O
T
y S &
T -
&p = 0
2
E -

E
éq
T
Handbook of Validation
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HT-66/03/008/A

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Titrate:
HSNV100 - Thermoplasticity in simple traction


Date:
03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
5/20

From where:

E - E
D
D

T
3 &

&p
O
=

+ S
y
&T

if &p 0, for
=
éq
R (p)
E AND 2 éq


(criterion reached, in “load”)

The stress field being uniaxial, one a:

- 1 0 0


D
L
=
0 2
0
3 0 0 - 1

As follows:


=
éq
L

and:

- 1 0 0


P
&p
&
=
sgn (L) 0 2 0
2

0 0 - 1

The relation of behavior leads to:



&p
&rr = & = -
&
- sgn
L
(L) + &T (= &xx = &yy for the case of the parallelepiped)

E
2



1
&zz = 0 =
&
L + &p sgn (L) + &T

E

From where:



3
1 -
2
&rr = & = &T +
&L
2
2nd




&p = sgn (
L
L) - &
&
T -
0
if
L R (p)



E =
<




D
D
E - E


T
3 &

= max 0;

+ O

y
&
St

if not

E E
2


T
éq






I.e., in the case L = R (p) (criterion reached):


E - E

&p
Max 0;
T (sgn (
O
=
L) & + S
L
y
&T)
E E

T


Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HT-66/03/008/A

Code_Aster ®
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Titrate:
HSNV100 - Thermoplasticity in simple traction


Date:
03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
6/20

2.1.1 Phase
rubber band

At the beginning of the thermal loading, L being lower than y, &p is null.

From where:

&

= - E
L
&T; &rr = & = &T (1+).

As follows:




= - E
L
T
(compressionL <) 0



=

rr

= (1+) T

Validity of the elastic solution

The criterion is:

() - () =
= - O
T
T
E
T
(1 - S
L
Y
y
T) 0

The criterion is not crossed for T = [0, ty], with:

O

T
y
y
=

(E
O
+ S
y
)
y - L
OY
T
T y


At the moment ty:

E O

y
L (ty) =
-
E + OY S

1
The density of deformation energy is worth: (
W T =

y)
E (T) 2
2

The total deformation energy is worth in the parallelepipedic case:
1
W (T
2
=
.(-).
y)
E (T) X
X
H
B
With
2

1
(R 2 -
2
R 2).
The total deformation energy is worth in the axisymmetric case: W (T =
.
y)
E (T)
B
With
H
2
2

(for 1 radian)

Handbook of Validation
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HT-66/03/008/A

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Titrate:
HSNV100 - Thermoplasticity in simple traction


Date:
03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
7/20

2.1.2 Phase
elastoplastic

T ty. One is on the criterion. Then:


E - E

&p
Max 0;
T (& sgn
O
=
L
(L) + S
y
&T)
E E

T


By admitting that one is “charges some” (&p >)
0, then one eliminate &p to have:


E - E

&

= - E &T + sgn ()
T

S O
L
T
L
y

E AND


then:
E - E

S O

y
&p
T
=
&T- sgn (


L) +

E
E




With T = ty, = - E
L
ty < 0; one integrates then these expressions for T T (T
y & =):



E - E


(T) = - E
T
O
L
T
(T - ty) -
S -
y
L (ty)

E E

T



(
E - E
p T)
T
=


2
[E +soy] (t-ty)

E

Maybe, after rearrangement, (T ty):



E
T

(T) = bone
T
1
1

L
y
T - +
-


E




T
y



O


y (E - AND)
(
T
p T) =

-
1

E 2

T

y



Validity of this elastoplastic solution

It should be made sure that ()
L T remains negative. Knowing that S T < 1, and that T > T y, the preceding result
confirm that ()
L T < 0.

Lastly, it is noticed that:

1 -
2
sgn (L)
p +
= (1+)
& &rr
&T
2

from where:

1 -
(
2
T) = (T)
rr

= (1+) T +
(
p T), T [
ty, T fine]
2

(since ()
L T < 0).
Handbook of Validation
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HT-66/03/008/A

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Titrate:
HSNV100 - Thermoplasticity in simple traction


Date:
03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
8/20

2.2
Results of reference

,
rr
xx
zz
or and p
ty

in
and with beyond:

Elastic phase: for T < ty

= - E T =
L
rr
= (1+) T
into axisymmetric
= (

1+)
xx
T
in plane constraints.


The yield stress is reached in T
0
y =
66,666 S from where
(
=
E + S
0)


L (ty) = - 1+ S



0

E

Elastoplastic phase: for T ty


E
T
(T) = S
T
1
1

L
T - +
-
0
E



T
y
0 (E - E

T)
(
T
p T) =

-
1
E 2

T

y


1 -
2
=
rr
= (1+) T +
(
p T) into axisymmetric
2
1 -
2
or
=
xx
= (1+) T +
(
p T) in plane constraints
2

E = 200.000 MPa; = 0 3
,
; =
-
10 5 °C-1; = 10
. s-1

O

= 400 MPa; To = 0 °C; S = -
10 2 °C-1; T
< 100s
y
end


E
= 50.000 MPa
T

From where:

T

= 66 6666
.
S
y





133 333
.

L

(ty) = -
MPa


elastic phase

-
rr
(ty) = (ty) = 0866666
.
10 3
.





w=4.44410- ²
W=0.17778 (PLAN or 3D)
W=0.26666 (axi)
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Titrate:
HSNV100 - Thermoplasticity in simple traction


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03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
9/20

Then, elastoplastic phase:

with T =
S
80:
()
80
= -
L
100 0
. MPa
(
p
)
80
=
-
0 3000
.
10 3
.
()
80
= ()
80
=
- 3
rr

1100
.
10
.

with T =
S
90:
()
90
= -
L
75 00
.
MPa
(
p
)
90
=
-
0 5250
.
10 3
.
()
90
= ()
90
=
3
rr

1275
.
10
.

2.3
Uncertainty on the solution

Analytical solution.
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Titrate:
HSNV100 - Thermoplasticity in simple traction


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03/11/03
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J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
10/20

3 Modeling
With

3.1
Characteristics of modeling

QUAD4 - Axisymétrique

GRN03
Z
N4
N3
D
C
GRN04
GRN02
With
B
N1
N2
GRN01

Appear 3.1-a: Modeling A

3.2
Characteristics of the grid

A number of nodes: 4
A number of meshs and types: 1 QUAD4, 4 SEG2

3.3 Functionalities
tested

Commands




DEFI_MATERIAU ELAS_ORTH



DEFI_MATERIAU TRACTION
SIGM


AFFE_CHAR_MECA DDL_IMPO
TEMP_CALCULEE


STAT_NON_LINE COMP_INCR
RELATION
VMIS_ISOT_TRAC

CALC_ELEM OPTION
EPSI_ELNO_DEPL


CALC_ELEM OPTION
EPOT_ELEM_DEPL


CALC_ELEM OPTION
ENEL_ELGA

POST_ELEM ENER_TOTALE




3.4 Remarks

Functionality AFFE_CARTE is also tested but it is not documented in the test.
Handbook of Validation
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Code_Aster ®
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Titrate:
HSNV100 - Thermoplasticity in simple traction


Date:
03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
11/20

4
Results of modeling A

4.1 Values
tested

Variables Moments
(S) Référence
Aster %
error
Tolerance
relative

T = 66.666
8.6666 10­4 8.66658
10­4
0 0.1


rr =
T = 80
1.1000 10­3 1.10029
10­3
0.026 0.1

T = 90
1.2750 10­3 1.27529
10­3
0.023 0.1

T = 66.666
0
0
0
0.1
p
T = 80
3.0000 10­4
3.0000 10­4
0 0.1

T = 90
5.2500 10­4
5.2500 10­4
0 0.1

T = 66.666
­ 133.333
­ 133.332
­ 0.001
0.1
zz
T = 80
­ 100.000
­ 100.00
0
0.1

T = 90
­ 75.000
­ 75.000
0
0.1
ENEL_ELGA
T = 66.666
4.444. 10-2 4.444.
10-2 0.00 0.1






ENER_TOTALE T = 66.666
0.2666
0.2666
­ 0.00
0.1
ENER_POT
T = 66.666
0.2666
0.2666
­ 0.00
0.1

4.2 Notice

One obtains well the same results with isotropic material declared orthotropic as with material
isotropic in thermo elasticity, i.e. for the sequence number 1 with T = 66.666 S.

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Titrate:
HSNV100 - Thermoplasticity in simple traction


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03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
12/20

5 Modeling
B

5.1
Characteristics of modeling

QUAD4 - Contraintes plane

y
N4
N3
D
C
With
B
N1
N2

Appear 5.1-a: Modeling B

5.2
Characteristics of the grid

A number of nodes: 4
A number of meshs and types: 1 QUAD4, 4 SEG2

5.3 Functionalities
tested

Commands




DEFI_MATERIAU TRACTION
SIGM


AFFE_CHAR_MECA DDL_IMPO
TEMP_CALCULEE


STAT_NON_LINE COMP_INCR
RELATION
VMIS_ISOT_TRAC

CALC_ELEM OPTION
EPSI_ELNO_DEPL


CALC_ELEM OPTION
EPOT_ELEM_DEPL


CALC_ELEM OPTION
ENEL_ELGA

POST_ELEM ENER_TOTALE



POST_ELEM ENER_POT


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Titrate:
HSNV100 - Thermoplasticity in simple traction


Date:
03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
13/20

6
Results of modeling B

6.1 Values
tested

Variables Moments
(S) Référence
Aster %
error
Tolerance
relative

T = 66.666
8.6666 10­4 8.66658
10­4
0 0.1
xx
T = 80
1.1000 10­3
1.1000 10­3
0 0.1

T = 90
1.2750 10­3
1.2750 10­3
0 0.1

T = 66.666
0
0
0
0.1
p
T = 80
3.0000 10­4
3.0000 10­4
0 0.1

T = 90
5.2500 10­4
5.2500 10­4
0 0.1

T = 66.666
­ 133.333
­ 133.332
­ 0.001
0.1
yy
T = 80
­ 100.
­ 100.00
0
0.1

T = 90
­ 75.000
­ 75.00
0.001
0.1
ENEL_ELGA
T = 66.666
4.444. 10-2 4.444.
10-2 0.00 0.1






ENER_TOTALE T = 66.666
0.17777
0.17777
­ 0.00
0.1
ENER_POT
T = 66.666
0.17777
0.17777
­ 0.00
0.1

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Titrate:
HSNV100 - Thermoplasticity in simple traction


Date:
03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
14/20

7 Modeling
C

7.1
Characteristics of modeling

1 element TUYAU

C
With

7.2
Characteristics of the grid

1 element TUYAU

7.3 Functionalities
tested

Commands




AFFE_MODELE
MODELING PIPE

STAT_NON_LINE COMP_INCR
RELATION
VMIS_ISOT_TRAC


TUYAU_NCOU
1


TUYAU_NSEC
16


8
Results of modeling C

8.1 Values
tested

Variables Moments
(S) Référence
Aster %
difference

T = 66.666
0
0
0
p
T = 80
3. 10­4
3.003 10­4 0.1

T = 90
5.25 10­4 5.2526
0.05

T = 66.666
­ 1.333
­ 1.3313
­ 0.16
yy
T = 80
­ 100
­ 99.82
­ 0.18

T = 90
­ 75
­ 74.85
­ 0.2

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Titrate:
HSNV100 - Thermoplasticity in simple traction


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03/11/03
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:
V7.22.100-C Page:
15/20

9 Modeling
D

9.1
Characteristics of modeling

1 element TUYAU 6M

C
With

9.2
Characteristics of the grid

1 element TUYAU

9.3 Functionalities
tested

Commands




AFFE_MODELE
MODELISATION TUYAU_6M

STAT_NON_LINE COMP_INCR
RELATION
VMIS_ISOT_TRAC


TUYAU_NCOU
1


TUYAU_NSEC
16


10 Results of modeling D
10.1 Values
tested

Variables Moments
(S) Référence
Aster %
difference

T = 66.666
0
0
0
p
T = 80
3. 10­4
3.003 10­4 0.1

T = 90
5.25 10­4 5.2526
0.05

T = 66.666
­ 1.333
­ 1.3313
­ 0.16
yy
T = 80
­ 100
­ 99.82
­ 0.18

T = 90
­ 75
­ 74.85
­ 0.2

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Titrate:
HSNV100 - Thermoplasticity in simple traction


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03/11/03
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:
V7.22.100-C Page:
16/20

11 Modeling
E

11.1 Characteristics of modeling

QUAD4 - Axisymétrique. Test of the variation of the coefficients of VMIS_CINE_LINE according to
temperature, in this case AND (given by D_SIGM_EPSI) varies like: AND = 105 (1­10­2 (TT0)).
2nd E
constant of Prager is worth: C
T
=
.
3rd - AND

GRN03
Z
N4
N3
D
C
GRN04
GRN02
With
B
N1
N2
GRN01

Appear 3.1-a: Modeling E

11.2 Characteristics of the grid

A number of nodes: 4
A number of meshs and types: 1 QUAD4, 4 SEG2

11.3 Functionalities
tested

Commands




DEFI_MATERIAU ECRO_LINE_FO D_SIGM_EPSI


DEFI_MATERIAU PRAGER_FO
C


DEFI_MATERIAU TRACTION
SIGM


AFFE_CHAR_MECA DDL_IMPO
TEMP_CALCULEE


STAT_NON_LINE COMP_INCR
RELATION
VMIS_ECMI_TRAC

CALC_ELEM OPTION
EPSI_ELNO_DEPL


STAT_NON_LINE COMP_INCR
RELATION
VMIS_CINE_LINE


11.4 Notice

One tests the variation of AND (D_SIGM_EPSI) with the temperature per comparison with
behavior VMIS_ECMI_TRAC where C (constant of Prager) varies with the temperature in way
similar (not of analytical solution).
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HT-66/03/008/A

Code_Aster ®
Version
6.4
Titrate:
HSNV100 - Thermoplasticity in simple traction


Date:
03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
17/20

12 Results of modeling E

12.1 Values
tested

Variables
Moments (S)
Reference (Aster)
Aster
% error
Tolerance
(VMIS_ECMI_TRAC) (VMIS_CINE_LINE)
relative

T = 66.666
8.6666 10­4 8.66658
10­4 0
0.1


rr =
T = 80
1.112 10­3 1.112
10­3 0 0.1

T = 90
1.303 10­3 1.303
10­3 0 0.1

T = 66.666
­ 133.333
­ 133.332
0
0.1
zz
T = 80
­ 88
­ 88
0
0.1

T = 90
­ 47
­ 47
0
0.1

12.2 Notice

One obtains well the same results with behavior VMIS_CINE_LINE as with
behavior VMIS_ECMI_TRAC what validates the taking into account of the temperature in this model.

Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HT-66/03/008/A

Code_Aster ®
Version
6.4
Titrate:
HSNV100 - Thermoplasticity in simple traction


Date:
03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
18/20

13 Modeling
F

13.1 Characteristics of modeling

1 element POU_D_T

C
With

13.2 Characteristics of the grid

1 mesh SEG2

13.3 Functionalities
tested

Commands




AFFE_MODELE
MODELISATION TUYAU_6M

STAT_NON_LINE COMP_INCR
RELATION
ELAS

CALC_ELEM OPTION
EPOT_ELEM_DEPL


POST_ELEM ENER_POT



14 Results of modeling D
14.1 Values
tested

Variables Moments
(S) Référence
Aster %
difference
yy
T = 66.666
­ 1.333
­ 1.3313
­ 0.16
ENER_POT
T = 66.666
0.3555
0.3555
0.00

Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HT-66/03/008/A

Code_Aster ®
Version
6.4
Titrate:
HSNV100 - Thermoplasticity in simple traction


Date:
03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
19/20

15 Modeling
G

15.1 Characteristics of modeling

3D, H=1

y
N4
N3
D
C
With
B
N1
N2

Appear 5.1-a: Modeling G

15.2 Characteristics of the grid

A number of nodes: 8
A number of meshs and types: 1 HEXA8

15.3 Functionalities
tested

Commands




DEFI_MATERIAU TRACTION
SIGM


AFFE_CHAR_MECA DDL_IMPO
TEMP_CALCULEE


STAT_NON_LINE COMP_INCR
RELATION
VMIS_ISOT_TRAC

CALC_ELEM OPTION
EPSI_ELNO_DEPL


CALC_ELEM OPTION
EPOT_ELEM_DEPL


CALC_ELEM OPTION
ENEL_ELGA

POST_ELEM ENER_TOTALE



POST_ELEM ENER_POT


Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HT-66/03/008/A

Code_Aster ®
Version
6.4
Titrate:
HSNV100 - Thermoplasticity in simple traction


Date:
03/11/03
Author (S):
J.M. PROIX, I. DEBOST-EYMARD, F.VOLDOIRE Key
:
V7.22.100-C Page:
20/20

16 Results of modeling G

16.1 Values
tested

Variables Moments
(S) Référence
Aster %
error
Tolerance
relative

T = 66.666
8.6666 10­4 8.66658
10­4
0 0.1
xx
T = 80
1.1000 10­3
1.1000 10­3
0 0.1

T = 90
1.2750 10­3
1.2750 10­3
0 0.1

T = 66.666
0
0
0
0.1
p
T = 80
3.0000 10­4
3.0000 10­4
0 0.1

T = 90
5.2500 10­4
5.2500 10­4
0 0.1

T = 66.666
­ 133.333
­ 133.332
­ 0.001
0.1
yy
T = 80
­ 100.
­ 100.00
0
0.1

T = 90
­ 75.000
­ 75.00
0.001
0.1
ENEL_ELGA
T = 66.666
4.444. 10-2 4.444.
10-2 0.00 0.1






ENER_TOTALE T = 66.666
4.444. 10-2 4.444.
10-2 ­ 0.00 0.1
ENER_POT
T = 66.666
4.444. 10-2 4.444.
10-2 ­ 0.00 0.1

17 Summary of the results

The results are satisfactory and validate the behaviors thermoplastic of Von Mises with
isotropic work hardening and linear kinematics. The finite elements used are the elements 2D
(quadrilaterals in plane constraints or axisymetry) and elements TUYAU.

One notes in particular a good modeling of the variation of the elastic limit and
constant of Prager with the temperature.
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HT-66/03/008/A

Outline document