Code_Aster
®
Version
6.0
Titrate:
HPLA310 - Biblio_49 Fissures radial external in a circular bar
Date:
22/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.310-A
Page:
1/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
Organization (S):
EDF-R & D/AMA, CS IF
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear of the structures
axisymmetric
V7.01.310 document
HPLA310 - Biblio_49 Fissures radial external in
a circular bar subjected to a thermal shock
Summary:
This test results from the validation independent of version 3 in breaking process.
It is about a basic static test into axisymmetric under non stationary thermal loading.
behavior of the structure is thermoelastic linear isotropic.
It includes/understands only one axisymmetric modeling.
Code_Aster
®
Version
6.0
Titrate:
HPLA310 - Biblio_49 Fissures radial external in a circular bar
Date:
22/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.310-A
Page:
2/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
1
Problem of reference
1.1 Geometry
has
R
Z
R
O
With
B
C
D
H
External annular fissure in a semi-infinite cylindrical bar
One will take has/R = 0,5 and H/R 5.
has = 1 m,
R = 2 m,
H = 10 Mr.
1.2
Properties of material
The material is thermoelastic linear isotropic.
Young modulus
E = 2e11 AP
Poisson's ratio
= 0,3
Linear expansion factor
= 1E-5 C°1
Thermal conductivity
= 50 W/Mr. C°
Thermal diffusivity
=
/
CP
=
0,5 m2/S
Coefficient of heat exchange
H = 250 W/m2 C°
One will choose H such as Bi = hR/
= 10.
Code_Aster
®
Version
6.0
Titrate:
HPLA310 - Biblio_49 Fissures radial external in a circular bar
Date:
22/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.310-A
Page:
3/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
1.3
Boundary conditions and loading
Boundary conditions mechanical
UX = Ur = 0 on the axis of revolution R = 0
UY = Uz = 0 on the ligament 0 R has
Null resulting thrust load at the higher edge; one will translate this boundary condition by a unit
of (n-1) linear relations UY (1) = UY (2) =… = UY (N) between longitudinal displacements of N
nodes of the edge higher (free axial dilation, conservation of the flatness of the cross section
bar).
Conditions of unilateral contact on the lip of the fissure in order to manage the closing of the aforementioned.
Boundary conditions thermal
Heat flux no one on the axis of revolution AB (by symmetry)
Heat flux no one on ligament OA (by symmetry) and on the fissure AC.
Flow of convection
(
)
T
R
H T
T
ext.
=
-
at the edge R = R,
T
ext.
indicating the temperature of the external medium.
Thermal loading
The temperature of the external medium undergoes an instantaneous level Text = T0 * H (T) where H (T) is the function
level-unit of Heaviside. Taking into account the boundary conditions the temperature does not vary in
function of Z. One will take T0 = 100 °C in order to obtain the closing of the lip, in the vicinity of the skin
part, at the beginning of the thermal shock.
1.4 Conditions
initial
Mechanical initial conditions
Null displacements, strains and stresses in all points.
Thermal initial conditions
Null initial temperature in any point.
Code_Aster
®
Version
6.0
Titrate:
HPLA310 - Biblio_49 Fissures radial external in a circular bar
Date:
22/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.310-A
Page:
4/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
2
Reference solution
2.1
Method of calculation used for the reference solution
Field of temperature:
exact analytical calculation.
Thermomechanical calculation: thermoelastic stress field in the bar not fissured given
by an exact analytical expression
displacement of the lips of the fissure calculated starting from functions
of influence determined numerically by finite elements
factor of intensity of the stresses calculated starting from the surface stresses
released along the fissure, by using functions weight of the solid
unlimited for a distribution of pressure on the lips constant by
interval along the radius.
2.2
Results of reference
Numbers of Fourier:
Fo
T
R
=
2
(adimensional time)
Numbers of Biot:
Bi
hR
=
(coefficient of exchange without dimension)
Expression of the temperature according to R and T:
(
)
(
)
()
(
)
()
()
T
T
BiJ
R R
Bi J
Fo
BiJ
J
N
N
N
N
N
N
N
N
=
-
+
·
-
=
=
0
0
2
2
0
1
2
0
1
1 2
µ
µ
µ
µ
µ
µ
µ
/
exp
where
eigenvalues
µ
N are the solutions of the equation above in which J0 and J1 are them
functions of Bessel of first species of command 0 and 1.
The tables below summarize the values of the temperatures (°C) for three particular radii and
for three numbers of Fourier:
Code_Aster
®
Version
6.0
Titrate:
HPLA310 - Biblio_49 Fissures radial external in a circular bar
Date:
22/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.310-A
Page:
5/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
F0 = 0,001
Ref. (10000 terms)
Aster %
variation
R = 0
3,9968E-12
2,06844E-18
-
R = 1
2,2204E-13
8,94917E-12
-
R = 2
2,79689E+1
2,80013E+1
0,116
F0 = 0,4
Ref. (900 terms)
Aster %
variation
R = 0
1,6230E-1
1,78593E-1
10,04
R = 1
6,2391E+0
6,2555E+0
0,262
R = 2
7,7365E+1
7,73319E+1
0,044
F0 = 1
Ref. (900 terms)
Aster %
variation
R = 0
9,8644E+1
9,8637E+1
0,006
R = 1
9,9018E+1
9,9013E+1
0,005
R = 2
9,9835E+1
9,9834E+1
0,001
Expression of axial stress in the bar not fissured according to R and of T:
(
)
µ
µ
µ
µ
µ
µ
zz
AND
Bi
N
Bi
J
N
N Fo
N
J
N
R
R
Bi
N
J
N
=
-
+
-
=
-
2
0
1
2
2
0
2
1
0
2
2 0
exp
The table below summarizes the values of the stresses
zz (AP) for R = has (bottom of fissure) and for
three numbers of Fourier:
Ref. (900 terms)
Aster %
variation
F0 = 0,001
4,584029E+6
4,58508E+6
0,017
F0 = 0,4
6,397099E+7
6,38746E+7
0,151
F0 = 1
8,200300E+5
8,23974E+5
0,481
Code_Aster
®
Version
6.0
Titrate:
HPLA310 - Biblio_49 Fissures radial external in a circular bar
Date:
22/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.310-A
Page:
6/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
Factor of intensity of the stresses (adimensional) according to the number of Fourier
2.3
Uncertainty on the solution
Lower than 5%.
2.4 References
bibliographical
[1]
J.M. ZHOU, T. TAKASE and Y. IMAI: Opening and closing behavior off year external circular
ace due to axisymmetrical heating. Engng.Fract.Mechs., 47, n°4, 559-568, 1994.
Code_Aster
®
Version
6.0
Titrate:
HPLA310 - Biblio_49 Fissures radial external in a circular bar
Date:
22/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.310-A
Page:
7/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
3 Modeling
With
3.1
Characteristics of modeling
Non stationary thermal calculation precedes mechanical calculation. Two calculations are done using
even mesh to avoid the phenomena of smoothing.
Complete mesh
Zoom on the point of fissure
Code_Aster
®
Version
6.0
Titrate:
HPLA310 - Biblio_49 Fissures radial external in a circular bar
Date:
22/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.310-A
Page:
8/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
3.2
Characteristics of the mesh
The mesh consists of 8651 nodes and 2772 elements, including 2732 elements QUA8 and 40 elements
TRI6.
The radial density of the mesh is determined by successive tests in order to reduce to 1% the variation
between the theoretical solution and the numerical solution, as well from the thermal point of view as
thermomechanics, in the case of the bar not fissured.
The height of the half-model is fixed arbitrarily at 5 times radius R. One supposes a priori that the effect
limitation of size of the mesh in direction Z on the factor of intensity of the stresses is
lower than 1%.
An indeformable block, located under the lip, was with a grid in order to manage the contact without induced friction
by the closing of the lip.
3.3
Functionalities tested
Controls
MECHANICAL AFFE_MODELE AXIS ALL
AFFE_CHAR_MECA TEMP_CALCULEE
AFFE_CHAR_MECA CONTACT
NORMAL MAIT_ESCL
MECA_STATIQUE
CALC_THETA THETA_2D
CALC_G_THETA OPTION
CALC_G
Code_Aster
®
Version
6.0
Titrate:
HPLA310 - Biblio_49 Fissures radial external in a circular bar
Date:
22/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.310-A
Page:
9/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
4
Results of modeling A
4.1 Values
tested
Identification Reference
Aster %
difference
G (Fo = 0,001) (J.m
2
)
2,3E+2 2,9449E+2 *
30
KI (Fo = 0,001) (Pa.m
0,5
)
7,0E+6 8,0438E6 **
14
G (Fo = 0,04) (J.m
2
)
1,0E+4 1,25016E+4 *
19
KI (Fo = 0,04) (Pa.m
0,5
)
4,8E+7 5,24175E7 **
9
G (Fo = 1) (J.m
2
)
1,0 1,2104864 * 15
KI (Fo = 1) (Pa.m
0,5
)
4,8E+5 5,1579E+5 **
7
*
In the case of axisymmetric calculations, to obtain the total rate of refund, it is necessary to divide it
rate of refund obtained with ASTER by R
fissure
= has (cf Reference material
[R7.02.01] - page18).
** Values obtained with the formula of IRWIN in plane deformations, by supposing that K
II
= 0, and
by taking G calculated by ASTER, which does not allow the automatic calculation of KI in
axisymmetric.
4.2 Remarks
To calculate G
ref.
, one uses the formulas of IRWIN in plane deformations:
(
)
Gref
E
K
K
I
II
= -
+
1
2
2
2
, KII = 0
The raised maximum change is 30% on G (Fo = 1), of 14% on KI (Fo = 1).
The maximum relative variation on the temperature in bottom of fissure, compared to the analytical solution
(summoned on 900 terms), is lower A 1%.
The maximum relative variation on
zz in the bar before cracking, compared to the analytical solution
summoned on 900 terms, with the site of the later bottom of fissure, is lower than 0,5%.
With ASTER, in axisymmetric mode, the stress field obtained is following form:
SIXX
SIYY
SIZZ
SIXY
and the associated stresses are:
SIRR SIZZ SITT SIRZ
To calculate the values of reference, we use the curve in Log/Log (page 5). Precision with
reading of the values not being very good, we can estimate that the results on the rate of
restitution of energy G are not too far away from the reference.
It should be noted that the rate of refund of energy G is invariable on the crowns of calculation.
Code_Aster
®
Version
6.0
Titrate:
HPLA310 - Biblio_49 Fissures radial external in a circular bar
Date:
22/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.310-A
Page:
10/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
5
Summary of the results
With regard to the bar not fissured, the Aster results in temperature and stress are very
close relations of the reference (less than 1% maximum for the temperature and less than 0,5% maximum
for the stresses). On the other hand, for the rate of refund of energy, the Aster results are
moved away from the reference since we raise a maximum change of 30% for FO = 0,001, with one
precision announced of 5% on the reference solution.