SSLV131 - Orthotropism in an unspecified reference mark

Code_Aster

Version

5.0

Titrate:

SSLV131 - Orthotropism in an unspecified reference mark

Date:

16/11/01

Author (S):

C. DURAND

Key

V3.04.131-A

Page:

1/10

Manual of Validation

V3.04 booklet: Linear statics of the voluminal structures

HI-75/01/010/A

Organization (S):

EDF/MTI/MN

Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
Document: V3.04.131

SSLV131 - Orthotropism 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.

Code_Aster

Version

5.0

Titrate:

SSLV131 - Orthotropism in an unspecified reference mark

Date:

16/11/01

Author (S):

C. DURAND

Key

V3.04.131-A

Page:

3/10

Manual of Validation

V3.04 booklet: Linear statics of the voluminal structures

HI-75/01/010/A

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, Transverse and Normal.

The units will not be specified.

13000

7000

10500

8000

;

5000

;

11000

8000

5000

11000

AND

(It is known that

AND

that is to say

06875

020

0.396,

For the transverse isotropy, one keeps the same values while knowing as:

)

2 (1

and

AND

It is reminded the meeting 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.

Code_Aster

Version

5.0

Titrate:

SSLV131 - Orthotropism in an unspecified reference mark

Date:

16/11/01

Author (S):

C. DURAND

Key

V3.04.131-A

Page:

4/10

Manual of Validation

V3.04 booklet: Linear statics of the voluminal structures

HI-75/01/010/A

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

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

in the total reference mark is thus constant and equal to:

2 3 4
3 5 6
4 6 7

Code_Aster

Version

5.0

Titrate:

SSLV131 - Orthotropism in an unspecified reference mark

Date:

16/11/01

Author (S):

C. DURAND

Key

V3.04.131-A

Page:

5/10

Manual of Validation

V3.04 booklet: Linear statics of the voluminal structures

HI-75/01/010/A

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
identify local (A, L, NR, T).

That is to say

the tensor of deformation in the local reference mark. One a:

= P

The tensor of Hooke

is known in the local reference mark, that is to say

the tensor of the stresses in it

identify. One a:

The tensor is obtained

stresses in the total reference mark by:

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, stresses and plane deformations, the selected reference was:
`Matrix of Hooke for orthotropic materials `. Report/ratio interns applications in Mechanics
n° 79-018 of Jean-Claude Masson CISI.

Code_Aster

Version

5.0

Titrate:

SSLV131 - Orthotropism in an unspecified reference mark

Date:

16/11/01

Author (S):

C. DURAND

Key

V3.04.131-A

Page:

6/10

Manual of Validation

V3.04 booklet: Linear statics of the voluminal structures

HI-75/01/010/A

3 Modeling

With

3.1

Characteristics of modeling

Following modelings are implemented:

2D:
- axisymmetric
- stresses

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 stresses because this case corresponds to

isotropy.

b) For 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 reminded the meeting 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 stresses:
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 mesh

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

Controls

Key word

DEFI_MATERIAU ELAS_ORTH

DEFI_MATERIAU ELAS_ISTR

MASSIVE AFFE_CARA_ELEM

Code_Aster

Version

5.0

Titrate:

SSLV131 - Orthotropism in an unspecified reference mark

Date:

16/11/01

Author (S):

C. DURAND

Key

V3.04.131-A

Page:

7/10

Manual of Validation

V3.04 booklet: Linear statics of the voluminal structures

HI-75/01/010/A

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)

field epsielgadepl

Epxy 3

Epxz 4

Epyz 6

field sief.elga.depl

If xx

50461,97

If yy

80136,037

If zz

68682,137

If xy

39559,096

If xz

30622,542

If yz

84027,579

field sigmelnodepl

If xx

50461,971

field emelelga Ep

1.23652.10

Field emelelnoelga Ep

1.23652.10

Case of the orthotropism 3D

name of the result: Mest2

field depl

Dy (c)

field epsielgadepl

Epxy 3

Epxz 4

Epyz 6

field siefelgadepl

If xx

23170,539

If yy

78600,676

If zz

78692,318

If xy

86435,100

If xz

16449,622

If yz

125577,226

field sigmelnodepl

if xx

2370,539

field enelelga Ep

1.55286.10

field enelelnoelga Ep

1.55286.10

Case of the transverse isotropy in
axisymmetric

name of the result: Mest3

field depl

Dy (c)

field epsielgadepl

Exxy 4

field siefolgadepl

If xx

42930,079

If yy

52252,113

If xy

37288,135

field enelelga Ep

4.15741.10

Code_Aster

Version

5.0

Titrate:

SSLV131 - Orthotropism in an unspecified reference mark

Date:

16/11/01

Author (S):

C. DURAND

Key

V3.04.131-A

Page:

8/10

Manual of Validation

V3.04 booklet: Linear statics of the voluminal structures

HI-75/01/010/A

field enelcluoelga Ep

4.15741.10

Case of the orthotropism into axisymmetric

name of the result: Mest4

field depl Dy (c)

field epsielgadepl

Epxy 4

field siefelgadepl

If xx

19438,248

If yy

75231,714

If xy

53867?974

field siefelgaelga Ep

4,9131710

field enelchroelga Ep

4.9131710

Case of the transverse isotropy in
plane deformations

name of the result: Mest5

field depl Dy (c)

field epsielgadepl Epxy

field siefelgadepl

If xx

31612,684

If yz

40934,718

If xy

37288,135

field sigmelnodepl

if xx

31612,684

field enelelga Ep

2.42167.10

field enelelnoelga Ep

2.42167.10

Case of the orthotropism in deformations
plane

name of the result: Mest6

field depl Dy (c)

field epsielgadepl Epxy

field siefelgadepl

If xx

9931,422

If yy

68733,870

If xy

51262,119

field sigmelnodepl

if xx

9931,422

field Epenelelga Ep

3.180807.10

field enelelnoelga

3.180807.10

Case of the orthotropism in stresses
plane

name of the result: Mest7

field depl Dy (c)

field epsielgadepl Epxy

field siefelgadepl

If xx

7454,007

field emelelga Eo

3.10347.10

field emelelnoelga Ep

3.10347.10

In the asymmetrical case, the values of the field of the deformations and field of the stresses 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

Code_Aster

Version

5.0

Titrate:

SSLV131 - Orthotropism in an unspecified reference mark

Date:

16/11/01

Author (S):

C. DURAND

Key

V3.04.131-A

Page:

9/10

Manual of Validation

V3.04 booklet: Linear statics of the voluminal structures

HI-75/01/010/A

Summary of the results

The results provided by Mathématica and Aster are identical for all modelings usable
with materials isotropic transverse and orthotropic.