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Titrate:
SSLV131 - Orthotropie in an unspecified reference mark


Date:
16/11/01
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C. Key DURAND
:
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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
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Titrate:
SSLV131 - Orthotropie in an unspecified reference mark


Date:
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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:
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Author (S):
C. Key DURAND
:
V3.04.131-A Page:
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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

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SSLV131 - Orthotropie in an unspecified reference mark


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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 ®
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SSLV131 - Orthotropie in an unspecified reference mark


Date:
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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

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SSLV131 - Orthotropie in an unspecified reference mark


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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
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Titrate:
SSLV131 - Orthotropie in an unspecified reference mark


Date:
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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 ®
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SSLV131 - Orthotropie in an unspecified reference mark


Date:
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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 ®
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Titrate:
SSLV131 - Orthotropie in an unspecified reference mark


Date:
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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 ®
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Titrate:
SSLV131 - Orthotropie in an unspecified reference mark


Date:
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