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SSLV135 Endommagement by fatigue under biaxial loading alternate Date:
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J. Key ANGLES
:
V3.04.135-B Page:
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Organization (S): EDF-R & D/AMA
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal systems
V3.04.135 document
SSLV135 Endommagement by fatigue under
alternate biaxial loading
Summary:
One presents a test here having an analytical reference [bib1]. The geometry treated here is a cube without defect
with which one carries out a linear elastic mechanical calculation followed calculation of the plan of shearing criticizes in
each point of Gauss and in each node.
Each of four modelings tests a criterion:
·
modeling a: criterion MATAKE;
·
modeling b: criterion DANG_VAN_MODI_AC;
·
modeling C: criterion DOMM_MAXI;
·
modeling D: criterion DANG_VAN_MODI_AV,
·
modeling E: criterion FATEMI_SOCIE.
The first two criteria are said “to plan of critical shearing”, they are adapted to the loadings
periodicals. The last two criteria can be qualified criteria “in plan of critical damage”, they
can be used when the loading is not periodical.
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:
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1
Problem of reference
1.1 Geometry
has
Face 1
has = 10
has =
m
10 m
Face 1
(above)
P1
P2
P
P5
P6
Face 4
(on the right-hand side)
Face 5
F
(behind)
Face 3
F
(side ga
G uche)
P3
P4
Face 6
F
(in front of)
(Dev.
Y
X
P7
P8
F
F this 2
(below)
Z
The cube has 10 mm on side.
1.2
Material properties
Young modulus: E = 200000 MPa
Poisson's ratio: = 0.3
Ultimate constraint: =
0
.
850 MPa
U
Curve of Wöhler (alternate traction and compression controlled in constraint):
Half amplitude of constraint 138.0 152.0 165.0 180.0 200.0 250.0 295.0
(MPa)
A number of cycles
1.0E+6 0.5E+
0.2E+
0.1E+
0.05E+
0.02E+
12.0E+
6
6
6
6
6
4
Half amplitude of constraint 305.0 340.0 430.0 540.0 690.0 930.0 1210.0
(MPa)
A number of cycles
10.0E+
5.0E+
2.0E+
1.0E+
5.0E+2 2.0E+2 1.0E+2
4
3
3
3
Half amplitude of constraint 1590.0 2210.0 2900.0
(MPa)
A number of cycles
50.0
20.0
10.0
Table 1.2-1: Curve of Wöhler
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1.3
Boundary conditions and loadings
·
Displacements according to axis X of face 3 are blocked (DX=0.0).
·
Displacements according to the axis Y of face 2 are blocked (DY=0.0).
·
Displacements of the P3 point are blocked according to axis Z (DZ=0.0).
·
We apply an alternate biaxial loading (traction and compression) according to axes X and Y.
Fx (T) represents the alternate efforts applied to face 4 according to axis X and Fy (T) represents them
alternate efforts applied to face 1 according to axis Y.
Loading for modelings A and b:
200N
20
Fx (
Fx T)
100N
10
T
0
1s
2s
- 100N
- 200N
Fy (T)
Loading for modelings C and D:
1.4 Conditions
initial
Without object for a static analysis.
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:
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2
Reference solution
2.1
Method of calculation used for the reference solution
In the case of an alternate biaxial loading where the pressures applied are such as: =,
X
y
with > 1 and < 0, one show [bib1] that it half amplitude of maximum shearing
2 = (
+
, where
2 and
2 represent the half amplitudes of constraints
X
y) 4
X
y
applied according to axes X and Y. Moreover, there are two critical plans in which shearing
is maximum:
Y
n1
n1
n22
X
Z
L S two pl
S two p years of cis
years
has
of cis illement
maximum men
max badly
2.2
Results of reference for modelings A and B
See the references [bib2] and [R7.04.01].
Half amplitude of maximum shearing:
2 (MPa)
2 (MPa)
2 (MPa)
X
y
100.200.150
Note:
The half amplitude of maximum shearing is identical for the two critical plans.
Normal vectors in the two critical plans:
n1
N2
Component X
- 1 2
1 2
Component y
1 2
1 2
Component Z 0
0
Normal maximum constraints in the fields of the normals n1 and N2:
NR
(N) = 50 MPa
max
1
and NR
(N) = 50 MPa
max
2
.
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Hydrostatic pressure maximum, independent with respect to the plans of normals n1 and N2:
P =
33333
,
33
MPa.
Normal average constraints in the fields of the normals n1 and N2:
NR (N)
MPa
m
0
1 =
and NR (N)
MPa
m
0
2 =
.
Normal maximum deformations in the fields of the normals n1 and N2:
4
max (
-
n1) = 75
,
1
10 and
4
max (
-
N2) = 75
,
1
10
Normal average deformations in the fields of the normals n1 and N2:
(N) 0
m
1 =
and (N)
0
m
2 =
.
Criterion of MATAKE
(N)
I
+ NR has
(N) B, I =,
1 2
2
max
I
where has = 1 and B = 2.
Equivalent constraints within the meaning of MATAKE in the fields of the normals n1 and N2:
(N)
I
F
(N) =
N
eq
I
+ NR has
()
I
, I =,
1 2
2
max
T
where F and T represent, respectively, the limit of endurance in alternating bending and the limit
of endurance in alternate torsion. Here F T is equal to 5
,
1. Consequently we have:
(N)
MPa
eq
300
1 =
and
(N)
MPa
eq
300
2 =
.
Numbers of cycles to the rupture in the fields of the normals n1 and N2:
From the curve of Wöhler, cf [Tableau 1.2-1], and equivalent constraints within the meaning of
MATAKE, we obtain:
Nb (N) = Nb (N) = 10946
Cr
1
Cr
2
cycles.
Damage in the fields of the normals n1 and N2:
5
ENDO (
-
n1) = ENDO (N2) = 913565
,
10.
Criterion of Dang Van adapted to the periodic loadings:
DANG_VAN_MODI_AC
(N)
I
+ has P B, I =,
1 2
2
where has = 1 and B = 2.
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Equivalent constraints within the meaning of DANG VAN in the fields of the normals n1 and N2:
(N)
I
C
(N) =
eq
I
+ has P, I =,
1 2
2
T
where C and T represent, respectively, the limit of endurance in alternate shearing and the limit
of endurance in alternate traction and compression. Here C T is equal to 5
,
1. Consequently we have:
(N) = 275 MPa and (N) = 275 MPa.
eq
1
eq
2
Numbers of cycles to the rupture in the fields of the normals n1 and N2
From the curve of Wöhler, cf [Tableau 1.2-1], and equivalent constraints within the meaning of DANG
VAN, we obtain:
Nb (N) = Nb (N) = 14903cycles.
Cr
1
Cr
2
Damage in the fields of the normals n1 and N2:
- 5
ENDO (N) = ENDO (N) = 709959
,
6
10.
1
2
2.3
Results of reference for modelings C and D
See the references [bib2] and [R7.04.01].
Half amplitude of constraint:
2 (MPa)
2 (MPa)
X
y
100 200
Criterion of MATAKE adapted to the nonperiodic loadings: DOMM_MAXI
For this criterion there are no analytical results.
Criterion of Dang Van adapted to the nonperiodic loadings:
DANG_VAN_MODI_AV
For this criterion there are no analytical results.
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3 Modeling
With
3.1
Characteristics of modeling
Modeling 3D: 125 quadratic elements of volume:HEXA8.
Grid of the cube made with GIBI 2000
Appear of the grid of the cube
Test criterion MATAKE.
3.2
Characteristics of the grid
The grid of the cube was obtained starting from the version 2000 of maillor GIBI.
A number of nodes: 216
A number of meshs: 465
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3.3 Functionalities
tested
Commands Options
LIRE_MAILLAGE
DEFI_GROUP CREA_GROUP_NO
CREA_GROUP_MA
DEFI_FONCTION NOM_PARA
“SIGM”
Interpol
“LOG”
DEFI_FONCTION NOM_PARA
“INST”
DEFI_MATERIAU ELAS
FATIGUE
CISA_PLAN_CRIT
DEFI_LIST_REEL
“MECHANICAL” AFFE_MODELE
“3D”
AFFE_MATERIAU ALL
AFFE_CHAR_MECA DDL_IMPO
GROUP_NO
FACE_IMPO
GROUP_MA
FORCE_FACE
GROUP_MA
STAT_NON_LINE
MECA_STATIQUE
CALC_FATIGUE TYPE_CALCUL
“FATIGUE_MULTI”
OPTION
“DOMA_ELGA”
“DOMA_NOEUD”
TYPE_CHARGE
“PERIODIQUE”
RESULTAT
SOL
CHAM_MATER
MAT
GROUP_MA
“FACE1”, “FACE2”, “FACE3”
“FACE4”, “FACE5”, “FACE6”
GROUP_NO
“FACE4”, “FACE5”, “FACE6”
MAILLAGE
CUBE
CRITERE
“MATAKE”
METHODE
“CERCLE_EXACT”
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4
Results of modeling A
4.1 Values
tested
Identification
Type of value
Reference
Aster Variation
(%)
Nodes:N1;N206;
(Moment: 3)
1.00000E+02
1.00000E+02
0.0
xx
Net: M60,
Not Gauss: 3
,
(Moment: 3)
2.00000E+02
2.00000E+02
0.0
yy
,
(N) 2
1.500000E+02
1.500000E+02
0.0
1
,
component X of n1 7.071068E01 7.071068E01
0.0
,
component there of n1
7.071068E01 7.071068E01
0.0
,
component Z of n1
0.0 6.123234E17
0.0
,
NR
()
max n1
5.000000E+01 5.000000E+01
0.0
,
NR (N)
m
1
0.0 4.235754E14
0.0
,
()
max n1
1.750000E04 1.750000E04
0.0
,
(N)
m
1
0.0 1.564995E19
0.0
,
(N)
eq
1
3.000000E+02 3.000000E+02
0.0
,
(N)
Cr
Nb
1
1.094600E+04 1.094600E+04
0.0
,
ENDO (N)
1
9.135647E05 9.135647E05
0.0
,
(N) 2
1.500000E+02 1.500000E+02 0.0
2
,
component X of N2 7.071068E01 7.071068E01 0.0
,
N2 component there 7.071068E01 7.071068E01 0.0
,
component Z of N2
0.0 6.123234E17
0.0
,
NR
(
)
max N2
5.000000E+01 5.000000E+01
0.0
,
NR (N)
m
2
0.0 4.235754E14
0.0
,
(
)
max N2
1.750000E04 1.750000E04
0.0
,
(N)
m
2
0.0 1.564995E19
0.0
,
(N)
eq
2
3.000000E+02 3.000000E+02
0.0
Nodes:N1;N206;
(N)
Cr
Nb
2
1.094600E+04 1.094600E+04
0.0
Net: M60,
Not Gauss: 7
,
ENDO (N)
2
9.135647E05 9.135647E05
0.0
The variations being lower than 1.0E-08 we put zero in the table above.
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5 Modeling
B
5.1
Characteristics of modeling
Except the criterion of fatigue tested, modeling B is identical to modeling A.
Test criterion DANG_VAN_MODI_AC.
5.2
Characteristics of the grid
Identical to modeling A.
5.3 Functionalities
tested
The functionalities tested are identical to modeling A. only option CRITERE of
order CALC_FATIGUE is different:
Commands Options
CALC_FATIGUE CRITERION
“DANG_VAN_MODI_AC”
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6
Results of modeling B
6.1 Values
tested
Identification
Type of value
Reference
Aster Variation
(%)
Nodes:N1;N206;
(Moment: 3)
1.00000E+02
1.00000E+02
0.0
xx
Net: M60,
Not Gauss: 3
,
(Moment: 3)
2.00000E+02
2.00000E+02
0.0
yy
,
(N)
1
1.500000E+02
1.500000E+02
0.0
,
component X of n1
7.071068E01 7.071068E01
0.0
,
component there of n1
7.071068E01 7.071068E01
0.0
,
component Z of n1
0.0 6.123234E17
0.0
,
NR
()
max n1
5.000000E+01 5.000000E+01
0.0
,
NR (N)
m
1
0.0 4.235754E14
0.0
,
()
max n1
1.750000E04 1.750000E04
0.0
,
(N)
m
1
0.0 1.564995E19
0.0
,
(N)
eq
1
2.750000E+02 2.750000E+02
0.0
,
(N)
Cr
Nb
1
1.490300E+04 1.490300E+04
0.0
,
ENDO (N)
1
6.709959E05 6.709959E05
0.0
,
(N)
2
1.500000E+02 1.500000E+02 0.0
,
component X of N2 7.071068E01 7.071068E01
0.0
,
N2 component there 7.071068E01 7.071068E01 0.0
,
component Z of N2
0.0 6.123234E17
0.0
,
NR
(
)
max N2
5.000000E+01 5.000000E+01
0.0
,
NR (N)
m
2
0.0 4.235754E14
0.0
,
(
)
max N2
1.750000E04 1.750000E04
0.0
,
(N)
m
2
0.0 1.564995E19
0.0
,
(N)
eq
2
2.750000E+02 2.750000E+02
0.0
Nodes:N1;N206;
(N)
Cr
Nb
2
1.490300E+04 1.490300E+04
0.0
Net: M60,
Not Gauss: 7
,
ENDO (N)
2
6.709959E05 6.709959E05
0.0
The variations being lower than 1.0E-08 we put zero in the table above.
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7 Modeling
C
7.1
Characteristics of modeling
Except the criterion of fatigue tested and the loading, cf [§ 1.3], modeling C is identical to
modeling A.
Test criterion DOMM_MAXI.
7.2
Characteristics of the grid
Identical to modeling A.
7.3 Functionalities
tested
The functionalities tested are identical to modeling A except the following options: CRITERE
is modified, METHODE is not used any more and PROJECTION is added:
Commands Options
CALC_FATIGUE CRITERION
“DOMM_MAXI”
POJECTION
“UN_AXE”
“DEUX_AXES”
8
Results of modeling C
8.1 Values
tested
Identification
Type of value
Reference
Aster Variation
(%)
Nodes:N1;N206;
(Moment: 3)
1.00000E+02
1.00000E+02
0.0
xx
Net: M60,
Not Gauss: 3
,
(Moment: 3)
2.00000E+02
2.00000E+02
0.0
yy
,
component X of N
_ _
1
_ _ 3.746066E01
3.907311E01
,
component there of N
_ _
1
_ _ 9.271839E01
9.205049E01
,
component Z of n1
_ _ 6.123234E17
_ _
,
ENDO (N)
1
_ _ 7.049845E05
_ _
In the table above, components X and of n1 have two values there because there are two
vectors which correspond to the same value of damage ENDO (N)
1.
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9 Modeling
D
9.1
Characteristics of modeling
Except the criterion of fatigue tested and the loading, cf [§ 1.3], modeling D is identical to
modeling A.
Test criterion DANG_VAN_MODI_AV.
9.2
Characteristics of the grid
Identical to modeling A.
9.3 Functionalities
tested
The functionalities tested are identical to modeling A except the following options: CRITERE
is modified, METHODE is not used any more and PROJECTION is added:
Commands Options
CALC_FATIGUE CRITERION
“DANG_VAN_MODI_AV”
POJECTION
“UN_AXE”
“DEUX_AXES”
10 Results of modeling D
10.1 Values
tested
Identification
Type of value
Reference
Aster Variation
(%)
Nodes:N1;N206;
(Moment: 3)
1.00000E+02
1.00000E+02
0.0
xx
Net: M60,
Not Gauss: 3
,
(Moment: 3)
2.00000E+02
2.00000E+02
0.0
yy
,
component X of n1
_ _ 7.071068E01
_ _
,
component there of n1
_ _ 7.071068E01
_ _
,
component Z of n1
_ _ 6.123234E17
_ _
,
ENDO (N)
1
_ _ 1.341992E04
_ _
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11 Modeling
E
11.1 Characteristics of modeling
Except the criterion of fatigue tested and the loading, cf [§ 1.3], modeling E is identical to
modeling A.
Test criterion FATEMI_SOCIE.
11.2 Characteristics of the grid
Identical to modeling A.
11.3 Functionalities
tested
The functionalities tested are identical to modeling A except the following options: CRITERE
is modified, METHODE and MECA_STATIQUE are not used any more and PROJECTION is added:
Commands Options
CALC_FATIGUE CRITERION
“FATEMI_SOCIE”
POJECTION
“UN_AXE”
“DEUX_AXES”
12 Results of modeling E
12.1 Values
tested
Identification
Type of value
Reference
Aster Variation
(%)
Nodes:N1;N206;
(Moment: 3)
1.00000E+02
1.00000E+02
0.0
xx
Net: M60,
Not Gauss: 3
,
(Moment: 3)
2.00000E+02
2.00000E+02
0.0
yy
,
- 8.00000E-04
- 8.00000E-04
0.0
xx (Instant: 3)
,
1.15000E-03
1.15000E-03
0.0
yy (Instant: 3)
,
- 1.50000E-04
- 1.50000E-04
0.0
zz (Instant: 3)
,
component X of n1
_ _ ±4.383711E01
_ _
,
component there of n1
_ _ 8.987940E01
_ _
,
component Z of n1
_ _ 6.123234E17
_ _
,
ENDO (N)
1
_ _ 1.682346E01
_ _
13 Summary of the results
The results obtained are in perfect agreement with the reference solution for modelings A
and B. Les modeling C, D and E do not have reference solutions associated with the criteria.
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