Code_Aster ®
Version
5.0
Titrate:
SSLV131 - Orthotropie in an unspecified reference mark
Date:
16/11/01
Author (S):
C. Key DURAND
:
V3.04.131-A Page:
1/10
Organization (S): EDF/MTI/MN
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
Document: V3.04.131
SSLV131 - Orthotropie in an unspecified reference mark
Summary
This case test validates modelings relating to linear elasticity which implement materials
orthotropic whose properties are known in a reference mark defined by the user different from the total reference mark.
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/01/010/A
Code_Aster ®
Version
5.0
Titrate:
SSLV131 - Orthotropie in an unspecified reference mark
Date:
16/11/01
Author (S):
C. Key DURAND
:
V3.04.131-A Page:
2/10
Count
matters
1 Problem of reference ........................................................................................................................... 3
1.1 Geometry ........................................................................................................................................ 3
1.2 Properties of the material .................................................................................................................... 3
1.3 Boundary conditions and loadings ............................................................................................ 4
2 Reference solution ............................................................................................................................. 4
2.1 Method of calculation ............................................................................................................................ 4
2.2 Results of reference ..................................................................................................................... 5
2.3 Uncertainties on the solution ............................................................................................................... 5
2.4 Bibliographical references ........................................................................................................... 5
3 Modeling A ........................................................................................................................................ 6
3.1 Characteristics of modeling ................................................................................................. 6
3.2 Characteristics of the grid ........................................................................................................... 6
3.3 Functionalities tested .................................................................................................................... 6
4 Result of modeling A ................................................................................................................. 7
4.1 Values tested ................................................................................................................................ 7
5 Summary of the results ........................................................................................................................... 9
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/01/010/A
Code_Aster ®
Version
5.0
Titrate:
SSLV131 - Orthotropie in an unspecified reference mark
Date:
16/11/01
Author (S):
C. Key DURAND
:
V3.04.131-A Page:
3/10
1
Problem of reference
1.1 Geometry
The total reference mark is reference mark (A, X, Y, Z). In this reference mark the co-ordinates of the nodes are:
To (0., 0., 0.)
B (3., 1., 0.)
C (2., 3., 0.)
D (3.1, - 1)
For the 2D, one will study the behavior of the triangle ABC whose material properties are defined
in the total reference mark (A, X, y) represented on the figure; this reference mark is turned of an angle of 30° around
of Z compared to the total reference mark.
For the 3D, one will study the behavior of the tetrahedron ABCD whose material properties are
defined in a local reference mark (A, X, y, Z) obtained by rotation of the total reference mark according to angles'
nautical (= 30°, = 20°, = 10°).
This reference mark is not represented on the figure.
1.2
Properties of material
The materials used are orthotropic and isotropic transverse.
One adopts the convention of terminology used in ASTER, i.e the suffixes L, T and NR means
Longitudinal, Transversal and Normal.
The units will not be specified.
EL =
,
11000 AND =
,
5000 IN =
,
8000
= 0. ,
18
= 0. ,
15
= 0.11
LT
LN
TN
EL = 11000; AND = 5000; IN = 8000
= 0. ,
18
= 0. ,
15
= 0. ,
11
LT
LT
TN
G =
,
10500 G =
,
7000 G = 13000
LT
LT
TN
EL
EL
AND
(It is known that
=
X
,
=
X
,
=
X,
TL
LT
NL
LN
NT
TN
AND
IN
IN
that is to say =
0.396,
=
,
62
.
020
= 06875
.
0
TL
NL
NT
For the transverse isotropy, one keeps the same values while knowing as:
AND = EL,
=
G
and
TL
LN
EL
LT= 2 (1
+
)
LT
It is pointed out that these coefficients are defined in a local reference mark (A, L, T, NR) turned of 30° in
plan (L, T) compared to the reference mark total for the 2D and turned with the nautical angles (30°, 20°, 10°)
compared to the total reference mark for the 3D.
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/01/010/A
Code_Aster ®
Version
5.0
Titrate:
SSLV131 - Orthotropie in an unspecified reference mark
Date:
16/11/01
Author (S):
C. Key DURAND
:
V3.04.131-A Page:
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1.3
Boundary conditions and loadings
The boundary conditions are of Dirichlet type. One makes the assumption of a field of displacement
linear in X and y so that the field of deformation is constant.
For the 2D one takes
dX = 2x + 4y
Dy = 4x + 3y
For the 3D one takes
dX = 2x + 3y + 4z
Dy = 3x + 5y + 6z
dZ = 4x + 6y + 7z
For the 2D, one thus will impose:
·
for node A
dX = 0, Dy = 0
·
for the node B
dX = 10, Dy = 15
·
for the node C
dX = 16, Dy = 17
and for the 3D:
·
for node A
dX = 0, Dy = 0, dZ = 0
·
for the node B
dX = 9, Dy = 14, dZ = 18
·
for the node C
dX = 13, Dy = 21, dZ = 26
·
for the node D
dX = 5, Dy = 8, dZ = 11
2
Reference solution
2.1
Method of calculation
Calculation is analytical.
One used the formal calculation programme Mathématica to carry it out.
One exposes of it the principle only for the 3D.
It is known that the field of displacement is:
dX = 2x + 3y + 4z
Dy = 3x + 5y + 6z
dZ = 4x + 6y + 7z
The field of deformations G in the total reference mark is thus constant and equal to:
2 3 4
G = 3 5 6
4 6 7
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/01/010/A
Code_Aster ®
Version
5.0
Titrate:
SSLV131 - Orthotropie in an unspecified reference mark
Date:
16/11/01
Author (S):
C. Key DURAND
:
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That is to say P the matrix of passage allowing to make pass a vector of the total reference mark (A, X, Y, Z) to
locate local (A, L, NR, T).
That is to say
T
L the tensor of deformation in the local reference mark. One a:
.
L = P
. P
G
The tensor of Hooke H L is known in the local reference mark, that is to say L the tensor of the constraints in it
locate. One a:
= H.
L
L
L
The tensor is obtained
T
G of the constraints in the total reference mark by:
= P.
G
L. P
2.2
Results of reference
They are obtained by carrying out the operations described above with Mathematica.
2.3
Uncertainties on the solution
Uncertainty is null because the solution is analytical.
2.4 References
bibliographical
For the description of the matrices of Hooke for materials isotropic transverse and orthotropic for
plane modelings 3D, constraints and plane deformations, the selected reference was:
`Matrix of Hooke for orthotropic materials `. Report/ratio interns applications in Mécanique
n° 79-018 of Jean-Claude Masson CISI.
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/01/010/A
Code_Aster ®
Version
5.0
Titrate:
SSLV131 - Orthotropie in an unspecified reference mark
Date:
16/11/01
Author (S):
C. Key DURAND
:
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3 Modeling
With
3.1
Characteristics of modeling
Following modelings are implemented:
·
2D:
- axisymmetric
- constraints
plane
- deformations
plane
·
3D.
For each one of these modelings, one tests materials isotropic transverse and orthotropic.
Note:
has) The transverse isotropy is not tested for the plane constraints because this case corresponds to
isotropy.
b) Pour the axisymmetric case the stress field depends on the point of calculation.
This point is selected at the point of integration of the triangle (i.e it is the center of gravity of the triangle).
c) It is pointed out that the orthtropie in an unspecified reference mark is not available for modeling
as a Fourier because there is then coupling of all the components of the tensor of constraints:
Implementation the current makes it possible to use only the symmetrical components to leave
which one can find the antisymmetric components but so that it is possible, it
is not necessary that the slips induce tensile stresses.
3.2
Characteristics of the grid
For the 2D, there is an element triangle with 3 nodes ABC.
For the 3D, there is an element tetrahedron with 4 nodes ABCD.
3.3 Functionalities
tested
Commands
Key word
DEFI_MATERIAU ELAS_ORTH
DEFI_MATERIAU ELAS_ISTR
MASSIVE AFFE_CARA_ELEM
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/01/010/A
Code_Aster ®
Version
5.0
Titrate:
SSLV131 - Orthotropie in an unspecified reference mark
Date:
16/11/01
Author (S):
C. Key DURAND
:
V3.04.131-A Page:
7/10
4
Result of modeling A
4.1 Values
tested
Identification Reference
Aster %
difference
Case of the transverse isotropy 3D
name of the result: Mest1
field depl
Dy (c)
21
21
0
field epsielgadepl
Epxy 3
3
0
Epxz 4
4
0
Epyz 6
6
0
field sief.elga.depl
If xx
50461,97
50461,97
0
If yy
80136,037
80136,037
0
If zz
68682,137
68682,137
0
If xy
39559,096
39559,096
0
If xz
30622,542
30622,542
0
If yz
84027,579
84027,579
0
field sigmelnodepl
If xx
50461,971
50461,971
0
field emelelga Ep
1.23652.106 1.23652.106
Field emelelnoelga Ep
1.23652.106 1.23652.106
Case of the orthotropism 3D
name of the result: Mest2
field depl
Dy (c)
21
21
0
field epsielgadepl
Epxy 3
3
0
Epxz 4
4
0
Epyz 6
6
0
field siefelgadepl
If xx
23170,539
23170,539
0
If yy
78600,676
78600,676
0
If zz
78692,318
78692,318
0
If xy
86435,100
86435,100
0
If xz
16449,622
16449,622
0
If yz
125577,226
125577,226
0
field sigmelnodepl
if xx
2370,539
2370,539
0
field enelelga Ep
1.55286.106 1.55286.106 0
field enelelnoelga Ep
1.55286.106 1.55286.106 0
Case of the transverse isotropy in
axisymmetric
name of the result: Mest3
field depl
Dy (c)
17
17
0
field epsielgadepl
Exxy 4
4
0
field siefolgadepl
If xx
42930,079
42930,079
0
If yy
52252,113
52252,113
0
If xy
37288,135
37288,135
0
field enelelga Ep
4.15741.105 4.15741.105 0
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/01/010/A
Code_Aster ®
Version
5.0
Titrate:
SSLV131 - Orthotropie in an unspecified reference mark
Date:
16/11/01
Author (S):
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field enelcluoelga Ep
4.15741.105 4.15741.105
0
Case of the orthotropism into axisymmetric
name of the result: Mest4
field depl Dy (c)
17
17
0
field epsielgadepl
Epxy 4
4
0
field siefelgadepl
If xx
19438,248
19438,248
0
If yy
75231,714
75231,714
0
If xy
53867?974
53867?974
0
field siefelgaelga Ep
4,91317105 4,91317105 0
field enelchroelga Ep
4.91317105 4.91317105 0
Case of the transverse isotropy in
plane deformations
name of the result: Mest5
field depl Dy (c)
17
17
0
field epsielgadepl Epxy
4
4
0
field siefelgadepl
If xx
31612,684
31612,684
0
If yz
40934,718
8
0
If xy
37288,135
37288,135
0
field sigmelnodepl
if xx
31612,684
31612,684
0
field enelelga Ep
2.42167.105 2.42167.105 0
field enelelnoelga Ep
2.42167.105 2.42167.105 0
Case of the orthotropism in deformations
plane
name of the result: Mest6
field depl Dy (c)
17
17
0
field epsielgadepl Epxy
4
4
0
field siefelgadepl
If xx
9931,422
9931,422
0
If yy
68733,870
68733,870
0
If xy
51262,119
51262,119
0
field sigmelnodepl
if xx
9931,422
9931,422
0
field Epenelelga Ep
3.180807.105 3.180807.105 0
field enelelnoelga
3.180807.105 3.180807.105 0
Case of the orthotropism in constraints
plane
name of the result: Mest7
field depl Dy (c)
17
17
0
field epsielgadepl Epxy
4
4
0
field siefelgadepl
If xx
7454,007
7454,007
0
field emelelga Eo
3.10347.105 3.10347.105 0
field emelelnoelga Ep
3.10347.105 3.10347.105 0
In the asymmetrical case, the values of the field of the deformations and field of the constraints are
given to the point of integration of the triangle (i.e its center of gravity) whose co-ordinates are:
X = 1.666667
Y = 1.333334
Z = 0
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/01/010/A
Code_Aster ®
Version
5.0
Titrate:
SSLV131 - Orthotropie in an unspecified reference mark
Date:
16/11/01
Author (S):
C. Key DURAND
:
V3.04.131-A Page:
9/10
5
Summary of the results
The results provided by Mathématica and Aster are identical for all modelings usable
with materials isotropic transverse and orthotropic.
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HI-75/01/010/A
Code_Aster ®
Version
5.0
Titrate:
SSLV131 - Orthotropie in an unspecified reference mark
Date:
16/11/01
Author (S):
C. Key DURAND
:
V3.04.131-A Page:
10/10
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Handbook of Validation
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