Code_Aster ®
Version
6.4
Titrate:
Anisotropic elasticity

Date
:

28/10/03
Author (S):
A. Key ASSIRE
:
R4.01.02-A Page
: 1/14

Organization (S): EDF-R & D/AMA
Handbook of Référence
R4.01 booklet: Composite materials
Document: R4.01.02
Anisotropic elasticity

Summary

This document treats anisotropic elasticity.


Handbook of Référence
R4.01 booklet: Composite materials
HT-66/03/005/A

Code_Aster ®
Version
6.4
Titrate:
Anisotropic elasticity

Date
:

28/10/03
Author (S):
A. Key ASSIRE
:
R4.01.02-A Page
: 2/14

Count

matters

1 Introduction ............................................................................................................................................ 3
2 Topology of the matrices of Hooke ......................................................................................................... 3
2.1 The Orthotropism .................................................................................................................................... 3
2.2 Transverse isotropy ......................................................................................................................... 4
2.3 Isotropy ........................................................................................................................................... 4
3 Matrix of Hooke and flexibility ........................................................................................................ 4
3.1 Notations .......................................................................................................................................... 4
3.2 Case 3D ............................................................................................................................................ 6
3.2.1 0rthotropie .............................................................................................................................. 6
3.2.1.1 Stamp flexibility ................................................................................................. 6
3.2.1.2 Stamp of Hooke ....................................................................................................... 6
3.2.2 Transverse isotropy ................................................................................................................ 7
3.2.2.1 Stamp flexibility ................................................................................................. 7
3.2.2.2 Stamp of Hooke ....................................................................................................... 9
3.2.3 Isotropy ................................................................................................................................ 10
3.2.3.1 Stamp flexibility according to E and ............................................................. 10
3.2.3.2 Stamp of Hooke according to E and ................................................................... 10
3.2.3.3 Stamp flexibility according to the coefficients of Lamé and µ ......................... 11
3.2.3.4 Stamp of Hooke according to the coefficients of Lamé and µ ............................... 11
3.3 Orthotropic in plane deformations and axisymmetric case 2D ...................................................... 11
3.3.1 Stamp flexibility ............................................................................................................ 11
3.3.2 Stamp of Hooke ................................................................................................................. 12
3.4 Orthotropic case 2D in plane constraints ..................................................................................... 12
3.4.1 Stamp flexibility ............................................................................................................ 12
3.4.2 Stamp of Hooke ................................................................................................................. 12
4 Use in Code_Aster ................................................................................................................. 13
5 Bibliography ........................................................................................................................................ 14

Handbook of Référence
R4.01 booklet: Composite materials
HT-66/03/005/A

Code_Aster ®
Version
6.4
Titrate:
Anisotropic elasticity

Date
:

28/10/03
Author (S):
A. Key ASSIRE
:
R4.01.02-A Page
: 3/14

1 Introduction

The objective of this document is to give the form of the matrices of flexibility and Hooke for
elastic materials orthotropic, isotropic transverse and isotropic in the cases 3Dn 2D-constraints,
plane 2D-deformations and axisymetry.

We speak about “matrices” of Hooke because, by preoccupation with a simplification, we did not adopt
notation of a tensor of command 4.

In any rigor, for linear elastic materials, the constraints are linear functions
deformations.

One writes: ij = Hijkl. kl

The symmetrical nature of [] and [] and adoption for these tensors of command 2d' a vectorial form
allows to write:

{} = [H] {}

or {} and {} are the vectorial representation of the tensors of a nature 2 {} and [] and where [H] is one
stamp 6 X 6.

2
Topology of the matrices of Hooke

2.1 Orthotropism

One can show the symmetry of the matrix of Hooke H.

We thus have twenty and one independent components in the case 3D.

H11
H12 H13 H14 H15 H16
H 22 H23 H24 H25 H26
[H]
H33 H34 H35 H36
=

SYM
H 44 H45 H46
H55 H56
H 66

An orthotropic material has two orthogonal plans of elastic symmetry.

This wants to say that if one calls [H'] the matrix [H] after symmetry (S)
[H'] = [H].

The relations obtained between the coefficients make it possible to write that [H] is defined by new
independent components.
Handbook of Référence
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Code_Aster ®
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Titrate:
Anisotropic elasticity

Date
:

28/10/03
Author (S):
A. Key ASSIRE
:
R4.01.02-A Page
: 4/14

In the axes of orthotropism:

H
H
H
0
0
0
11
12
13
H
H
0
0
0
22
23
[H]
H
0
0
0
33
=

SYM
H
0
0
44
H
0
55
H66

9 coefficients thus should be provided.

2.2 Isotropy
transverse

The transverse isotropy is a restriction of the orthotropism in where one has the isotropy in one of both
orthogonal plans of elastic symmetry.

The matrix [H] will have the same form as for the orthotropism but with additional relations
between the components.
5 components are enough to determine [H].

2.3 Isotropy


The material is isotropic if [H] remains invariant in any change of reference mark.

Two coefficients are enough to determine [­ H].


3
Stamp of Hooke and flexibility

3.1 Notations

Instead of using indices 1, 2 and 3 to locate the axes, one will use the corresponding indices L,
T and NR:

L for longitudinal
T for transverse
NR for normal


NR
T
L

Handbook of Référence
R4.01 booklet: Composite materials
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Code_Aster ®
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Titrate:
Anisotropic elasticity

Date
:

28/10/03
Author (S):
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:
R4.01.02-A Page
: 5/14

The coefficients which intervene are as follows:

E_L
: Longitudinal Young modulus
E_T
: Transverse Young modulus
E_N
: Normal Young modulus
G_LT
: Modulus of rigidity in the plan (L, T)
G_TN
: Modulus of rigidity in the plan (T, NR)
G_LN
: Modulus of rigidity in the plan (L, NR)
NU_LT: Poisson's ratio dasn the plan (L, T)
NU_TN: Poisson's ratio in the plan (T, NR)
NU_LN: Poisson's ratio in the plan (L, NR)

Very important remark:

Naked _ LT is different from Naked _ TL:
If one applies a traction according to L
L
L =
(law of Hooke following a direction).
EL


This traction is accompanied, proportionally, of a contraction according to
L
T, - Naked _ LT.

EL

and of a contraction according to
L
NR, - Naked _ LN
.
EL

The first index indicates the axis where the effect of the loading is exerted and the second index indicates
direction of the loading.

Then one exerts a traction according to T, then a traction according to NR; one obtains:





L

=
- Naked _ LT TT - Naked _ LN NR
L

E
E
E
L
T
NR






= - Naked _ TL L
TT
+
- Naked _ TN NR
TT
(
S)
E
E
E
L
T
NR






= - Naked _ NL L - Naked _ NT TT
NR
NR
+

E
E
E
L
T
NR


The matrix of flexibility [H] ­ 1 is symmetrical; one deduces some:

U _ LT
Naked _ TL
=

E
E
L
T

Naked _ LN
Naked _ NL
=

E
E
L
NR

Naked _ TN
Naked _ NT
=

E
E
T
NR

In all that follows NAKED will be noted.
Handbook of Référence
R4.01 booklet: Composite materials
HT-66/03/005/A

Code_Aster ®
Version
6.4
Titrate:
Anisotropic elasticity

Date
:

28/10/03
Author (S):
A. Key ASSIRE
:
R4.01.02-A Page
: 6/14

3.2
Case 3D

3.2.1 0rthotropie
3.2.1.1 Stamp flexibility








1
-
-
L
LT
LN






0
0
0



E
E
E
L

L
NR






-
1
-
TL
TN

TT
0
0
0


E
E
E




L
T
NR
TT




-
-
1


NR

NL
NT
0
0
0





E
E
E

L
T
NR


=
NR


1







0
0

LT


GLT



LT

1



SYM
0


LN




G



LN



LN

1






G
TN




TN


TN





H1 ­ Orthotropie

3.2.1.2 Stamp of Hooke





1

L
- TN NT


(+


LT
LN
NT)
(
+
LN
LT
TN)
L






0
0
0


E E
E E
.
E .E




T
NR
T
NR
T
NR

L
(+
1

.
TL
TN
NL)
(- NL LN)
(
+
TN
LN
TL)

TT


0
0
0






E E
E .E
E E
.
L
NR
L
NR
L
NR




(+.

.
1
.

NL
NT
TL)
(
+
NT
LT
NL)
(- LT TL)

NR




E E E
0
0
0
NR
=

L
T
NR

E E
.
E .E
E E
.



L
T
L
T
L
T


1 -
TN
NT




GLT




LT


0
0


-
LT

NL
LN

*





-

GLN


LT
TL


LN


SYM
0


-
2

*
LN

TN
NL
LT






GTN

1


TN =


* TN









H ­ Orthotropie with TL
LT
NL
LN
NT
TN
=
;
=
;
=

E
E
E
E
E
E
L
T
L
NR
T
N
Handbook of Référence
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Code_Aster ®
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Titrate:
Anisotropic elasticity

Date
:

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Author (S):
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:
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3.2.2 Isotropy
transverse

3.2.2.1 Stamp flexibility

NR
T
L


The H1 matrix can be deduced directly from the H1-Orthotropie matrix by using the properties
transverse isotropy.

In the plan (L, T):

E = E
L
T
=

TL
TL EL
G
=
LT
(
2 1+ LT)

In the plans (L, NR) and (T, NR):


=
NT
NL

=

LN
TN
G
= G
TN
LN
Handbook of Référence
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Code_Aster ®
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Titrate:
Anisotropic elasticity

Date
:

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:
R4.01.02-A Page
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NR
T
L


E = E
L
T
=
LT
TL
EL
G
=
LT
(
2 1+ LT)

=
NT
NL


=
LN
TN
G
= G
TN
LN


NT
LN
=
E
E
L
NR







1
-
-
L
LT
LN






0
0
0



E
E
E
L

L
L
NR






-
1
-
TL
LN

TT
0
0
0


E
E
E




L
L
NR
TT




-
-
1


NR

NL
NT
0
0
0





E
E
E

L
L
NR


=
NR

(

2 1+ LT)









0
0

LT


EL



LT

1



SYM
0


LN




G



LN



LN

1







G

TN
TN


TN





H1 - Isotropie transverse
Handbook of Référence
R4.01 booklet: Composite materials
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Code_Aster ®
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Titrate:
Anisotropic elasticity

Date
:

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Author (S):
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:
R4.01.02-A Page
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3.2.2.2 Stamp of Hooke

The matrix [H] has same symmetries as [H] ­ 1


NR
T
L








L


1 -.
+

+

NL
LN
LT
NL
LN
LN
LT
LN



0
0
0
L




E .E
E .E
E E
.
L
NR
L
NR
L
NR


TT




E 2.E
+
1 -
.

+


L
NR
TL
NL
LN
NL
LN
LN
LT
LN



0
0
0
TT

1

2
.
E .E
E .E
E E
.


-
NL
LN
L
NR
L
NR
L
NR




2

NR
2
+.
+.
1 -




- LT
NL
LT
NL
NL
LT
NL
LT


=

0
0
0
2
2
2
NR

E
E
E


-
2



NL
LN
LT
L
L
L


LT




E
. '
1
L





=





2 (1 + LT)
LT




LN



G

. '

LN
LN



TN
G

. '

LN



TN









H ­ Isotropie transversal
Handbook of Référence
R4.01 booklet: Composite materials
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Code_Aster ®
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Titrate:
Anisotropic elasticity

Date
:

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Author (S):
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: 10/14

3.2.3 Isotropy


3.2.3.1 Stamp flexibility according to E and










L
1
-
-
L


0
0
0


E
E
E



1
-




TT

0
0
0
TT


E
E




1



0
0
0

NR


E
NR
=




1
1
(
2 +)



LT
=
0
0
LT


G
E


1
(
2 1+)




SYM
=
0




LN

G
E
LN


1
(
2 1+)



=




G
E



TN
TN









Complete H1 ­ Isotropie

3.2.3.2 Stamp of Hooke according to E and



L
1


0
0
0
L











TT
1

0
0
0
TT




NR

1
0
0
0


NR



E
1


2

LT
SYM
0
0
LT
=


(1+) (1 -
2)

2






1 -
2





0

LN
LN

2






1 -
2








2

TN




TN

H ­ Isotropie complete
Handbook of Référence
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Code_Aster ®
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Titrate:
Anisotropic elasticity

Date
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3.2.3.3 Stamp flexibility according to the coefficients of Lamé and µ

The law of Hooke takes the following form with the coefficients of Lamé and µ.

ij = kk
ij + 2µij

By using the system of equations (S), one obtains:


E
E
.
0
0

L

L
TL
T
L






1
E
.
E
0
0

TT
LT L
T
TT
=


1 -
.
0
0
0
0

NR
LT
TL
NR

0
0
0 G

LT

LT LT

H ­ Orthotropie planes in plane constraints

3.2.3.4 Stamp of Hooke according to the coefficients of Lamé and µ




0
0
0
L
+
L







0
0
0
TT

+
TT

+ 2µ 0 0 0
NR =
NR

SYM
µ 0 0
LN

LN

µ 0
LT
LT


µ
TN


TN

H ­ Isotropie supplements with the coefficients of Lamé

3.3
Orthotropic in plane deformations and axisymmetric case 2D

3.3.1 Stamp flexibility

1
1

1
.

.
0
0

L
(- NL LN)
-
(+
TL
TN
NL)


L

E
E
L
L




TT
- 1 (+
1
.
1
.
0
0
LT
LN
NT)
(- TN NT)
TT

= E
E




T
T


0
0
0
0
0 NR

1

0
0
0

LT



LT


GLT

H-1 ­ Orthotropie planes in plane deformations and axisymetry
Handbook of Référence
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Code_Aster ®
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Titrate:
Anisotropic elasticity

Date
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3.3.2 Stamp of Hooke



E
E
E

1
.

.



0

L
L (
TN
NT)

-
(+
LT
LN
)
NR
NT
(+
+
NL
NT
TL)
L




'
'
'







TT
E
E
E
T (+
+
1
.
1


0
TL
TN
)
T
NL
(- NL
)
T
LN
(-
+
+
NT
LT
NL)
TT


=


'
'
'







E
E

0
NR
NR (
+.



0
0
NL
NT
)
NR
TL
(
+
+
NT
LT
NL)




'
'

LT
LT

0
0
0
GLT

'=1 -.
-.
-. -
2

TN
NT
NL
LN
LT
TL
TN
NL
LT
H ­ Orthotropie planes in plane deformations and axisymetry

3.4
Orthotropic case 2D in plane constraints

3.4.1 Stamp flexibility

1
LT


0
0

L

-
L

E
E
L
T




TT

1
TL
TT
-

=
0
0
E
E




L
T



0
0
0
0

NR

NR




1
0
0
0

LT
LT


GLT
H1 ­ Orthotropie planes in plane constraints

3.4.2 Stamp of Hooke


E
E
0
0

L

L
TL
T
L






1
E
E
0
0

TT
LT L
T
TT
=


0 1 -
.
0
0
0
0



LT
TL
NR

0
0
0 G

LT

LT LT
H ­ Orthotropie in plane constraints

Handbook of Référence
R4.01 booklet: Composite materials
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Code_Aster ®
Version
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Titrate:
Anisotropic elasticity

Date
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4 Use
in
Code_Aster

In Aster, the definition of the constant orthotropic elastic characteristics or functions of
temperature are carried out by the command DEFI_MATERIAU, key word ELAS_ORTH or ELAS_ORTH_FO
for the elements of hull and the solid elements isoparametric or the constitutive layers
of a composite (see command DEFI_COQU_MULT).

To define the reference mark of orthotropism (L, T, NR) related to the elements, one can refer to documentations
[U4.42.03] DEFI_COQU_MULT and [U4.42.01] AFFE_CARA_ELEM.

NR
T
L
L, T and NR: directions of orthotropism
longitudinal, transverse and normal


/ELAS_ORTH = _F
(E_L = ygl Module of longitudinal Young.








E_T = ygt Module of transverse Young.








E_N = ygn Module of normal Young.








GL_T = glt Module of shearing in plan LT.








G_TN = gtn Module of shearing in plan TN.








G_LN = gln Module of shearing in plan LN.








NU_LT = nult Coefficient of Poisson in plan LT.








NU_TN = nutn Coefficient of Poisson in plan TN.








NU_LN = nuln Coefficient of Poisson in plan LN.

Important remark:

The talk of this note of reference is based on the convention of the books of J.L.Batoz and D. Gay.
Documentation U of DEFI_MATERIAU describes these choices, and coefficient NU_LT is interpreted
the following way in Aster:
if one exerts a traction according to the axis L giving place to a deformation according to this axis equalizes with
L


L
L =
, there is a deformation according to the axis T equalizes with: T = - nult *
.
ygl
ygl

Handbook of Référence
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Code_Aster ®
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Titrate:
Anisotropic elasticity

Date
:

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5 Bibliography

[1]
J.C. MASSON: Stamp of Hooke for orthotropic materials, internal Rapport
Applications in Mécanique, n°79-018, CiSi, 1979.
[2]
D. GAY: Composite materials, Edition Hermes, 1987
[3]
J.L. BATOZ, G. DHATT: Modeling of stuctures by finite elements, Volume 1, Edition
Hermes

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