Code Aster Aniso

Code_Aster Version defaultTitre : Elasticit anisotrope Date : 02/02/2012 Page : 1/14Responsable : Jean-Michel PROIX Cl : R4.01.02 Rvision : 8448

Anisotropic elasticity

Rsum

This document treats anisotropic elasticity, used for the modelizations of continuums 3D and 2D (C_PLAN, D_PLAN, AXIS), or the layers of the composite shells.

The springy medium can be anisotropic according to the 3 directions (orthotropic elasticity is spoken), or in isotropic in two directions (one speaks about transverse isotropic elasticity).

Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and is provided as a convenience.

Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)


Contents1 Introduction ........................................................................................................................... 3 2 Topologie of the matrixes of Hooke ...................................................................................... 3.2.1

transverse ............................................................................................................................. orthotropy 3.2.2 Isotropy ........................................................................................................................ 4.2.3 Isotropy ................................................................................................................................ 4

3 Matrix of Hooke and flexibility ............................................................................................... 4.3.1 Notations .............................................................................................................................. 4.3.2 Cas 3D .................................................................................................................................

6.3.2.1 0rthotropie .......................................................................................................... 6 3.2.1.1 Stamp flexibility ......................................................................................... 6 3.2.1.2 Matrice of Hooke .......................................................................................

6.3.2.2 Isotropy transverse ............................................................................................. 7 3.2.2.1 Matrice flexibility ........................................................................................ 7 3.2.2.2 Matrice of Hooke .......................................................................................

8.3.2.3 lasticit cubic .................................................................................................... 9.3.2.4 Isotropy .............................................................................................................. 10

3.2.4.1 Matrice flexibility according to E and ..................................................... 10 3.2.4.2 Matrice of Hooke according to E and .................................................... 10 3.2.4.3 Matrice flexibility according to the coefficients of Lam and ............... 11 3.2.4.4 Matrice of Hooke according to the coefficients of Lam and .............. 11.3.3

orthotropic in plane strains and axisymmetric Cas 2D .......................................................... 11.3.3.1 Matrix of flexibility ............................................................................................. 11.3.3.2 Matrice of Hooke .............................................................................................. 12.3.4

Cas 2D orthotropic in plane stresses .................................................................................... 12.3.4.1 Matrice flexibility ............................................................................................... 12.3.4.2 Matrice of Hooke .............................................................................................. 12

4 Utilisation in Code_Aster ...................................................................................................... 13 5 Bibliographie ......................................................................................................................... 14 6 Description of the versions of the document ......................................................................... 14




1 Introductionthe objective of this document is to give the form of the matrixes of flexibility and Hooke for elastic materials orthotropic, isotropic transverse and isotropic in the cases 3D, 2D-forced, 2D - plane strains and axisymetry.

We speak about matrixes of Hooke because, by preoccupation with a simplification, we did not adopt the notation of a tensor of order 4.

In any rigour, for the linear elastic materials, the stresses are linear functions of the strains.

One writes: ij=H ijklkl

The symmetric nature of { } and [ ] the adoption for these tensors of order 2 of a vectorial form make it possible to write:

{ }= [H ] { }

where { } and [ ] are the vectorial representation of the tensors of order 2 { } and [ ] where [H ] is a matrix 66 .

2 Topology of the matrixes of Hooke2.1 the orthotropy

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

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

[H ]=H 11 H 12 H 13 H 14 H 15 H 16

H 22 H 23 H 24 H 25 H 26H 33 H 34 H 35 H 36

SYM H 44 H 45 H 46H 55 H 56

H 66

An orthotropic material has two orthogonal planes 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 nine independent components.




In the axes of orthotropy:

[H ]=H 11 H 12 H 13 0 0 0

H 22 H 23 0 0 0H 33 0 0 0

SYM H 44 0 0H 55 0

H 66

9 coefficients thus should be provided.

2.2 Transverse isotropythe transverse isotropy is a restriction of the orthotropy in where one has the isotropy in one of the two orthogonal planes of elastic symmetry.

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

2.3 Isotropy the material is isotropic if [H ] remains invariant in any change of reference.

Two coefficients are enough to determine [H ] .

3 Stamp of Hooke and from flexibility3.1 Notations

Au lieu d' to use indices 1.2 and 3 to identify the axes, one will use the corresponding indices L , T and N :

1. L for longitudinal2. T for transverse3. N for normal

N T L




Les coefficients which intervene are the following:

Key word

Notation meaning

E_L EL longitudinal Modulus YoungE_T ET transverse Modulus YoungE_N EN normal Modulus YoungG_LT GLT Shear modulus in plane L ,T G_TN GTN Shear modulus in plane T , N G_LN GLN Shear modulus in plane L , N NU_LT LT Poisson's ratio in plane L ,T NU_TN TN Poisson's ratio in plane T , N NU_LN LN Poisson's ratio in the Remarque L , N

plane very important:LT is different from TL :

If one applies a tension according to L

LL=LLEL

(Hookes law according to a direction).

This tension is accompanied, proportionally, of a contraction according to T ,LT .LLEL

and of a contraction according to N ,LN .LLEL

.

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

Then one exerts a tension according to T , then a tension according to N ; one obtains:

LL=LLE L

TLTTET

NL NNEN

TT= LT LLE L

TTET

NTNNE N

NN=LN LLE L

TNTTET

NNE N

} S The matrix of flexibility [H ]1 being symmetric; one of deducted:

LTEL

=TLET

LNE L

=NLE N




TNET

=NTE N

3.2 Case 3D3.2.1 0rthotropie

3.2.1.1 Matrix of flexibility

[LLTTNN2LT2LN2TN

]=[1EL

TLET

NLEN

0 0 0

LTEL

1ET

NTEN

0 0 0

LNEL

TNET

1EN

0 0 0

1G LT

0 0

SYM 1GLN0

1GTN

][LLTTNNLT LNTN ] [H ]1 - Orthotropy

3.2.1.2 Stamps of Hooke

[LLLLNNLT LNTN

]= 1 [1TN NT

ET E N TLNL TN

ET . E N NLTL NT

ET . E N0 0 0

LT LN NT E LE N

1NL LN E L . E N

NTNL . LT E L . EN

0 0 0

LNLT .TN E L . ET

TNTL . LN E L . ET

1LT .TL E L . ET

0 0 0

GLT 0 0SYM GLN 0

GTN

] [ LLTTNN2 LT2LN2TN ] [H ] - Orthotropy with:

TLET

=LTE L

; NLE N

=LNE L

; NTEN

=TNET




1=

E LET E N

1TN NTNL LNLT TL2TN NL LT

3.2.2 Transverse isotropy

3.2.2.1 Matrice flexibility

N

T L

the matrix [H ]1 can be deducted directly of the matrix [H ]1 - Orthotropy by using the properties of the transverse isotropy. In the plane L ,T :

EL=ETTL=LT

GLT=EL

2 1LT

In the planes L , N and T , N :

NT=NLLN=TNGTN=G LN

N

T L




EL=ETLT=TL

GLT=EL

2 1LT NT=NLLN=TNGTN=G LNNTEN

=LNEL

[LL TTNN2LT2LN2TN

]=[1E L

LTE L

NLEN

0 0 0

TLE L

1E L

NTEN

0 0 0

LNE L

TNE L

1EN

0 0 0

2 1LT E L

0 0

SYM 1GLN

0

1GTN

][ LLTT NN LT LN TN ] [H ]1 - Transverse Isotropy

3.2.2.2 Matrice of Hooke

the matrix [H ] has same symmetries as [H ]1 .

N

T L




[ LL TT NN LT LNTN

]= 1 ' [1NL . LNE L . E N

LTNL LNE L . EN

NLLT NLE L . EN

0 0 0

TLNL LNE L . E N

1 NL. LNE L . EN

LNLT LNE L . EN

0 0 0

LN LT . LNE L

2

TN LT .TNE L

2

1 LT2

E L2 0 0 0

EL . '2 1LT

GLN . 'GLN . '

][ LLTTNN2LT2LN2TN ] [H ] - Transverse Isotropy

1'=

E L2 .EN

[12NL .LN LT22NL LN LT ]

3.2.3 cubic lasticitcubic elasticity corresponds to a matrix of elasticity of the form :

[y1111 y1122 y1122y1122 y1111 y1122y1122 y1122 y1111

y1212y1212

y1212]

Being given cubic symmetry, it remains to determine 3 coefficients:E L=EN=ET=E ,GLT=GLN=GTN =G ,LN=LT= LN=

To reproduce cubic elasticity with ELAS_ORTH, it is enough to calculate the coefficients of the orthotropy such that the matrix of elasticity obtained is form above:

y1111=E 12

13223

y1122=E 1

13223y1212=G LT=GLN=GTN

therefore, as long as 132230 (i.e. different from 0.5).




y1122y1111

= 1

what provides = 1

1y1111y1122

then E= y111113223

12

3.2.4 Isotropie

3.2.4.1 Stamps flexibility according to E and

[LLTTNN2LT2LN2TN

]=[1E

E

E

0 0 0

1E

E

0 0 0

1E 0 0 0

1G=

21 E

0 0

SYM 1G=

2 1 E

0

1G=

2 1 E

][ LLTT NN LT LNTN ] [H ]1 - complete Isotropy

3.2.4.2 Matrice of Hooke according to E and

[ LLTT NN LT

LN

TN]= E1 12 [

1 0 0 01 0 0 0

1 0 0 0

SYM 122

0 0

122

0

122

][ LLTTNN2LT

2LN

2TN

] [H ] - complete Isotropy




3.2.4.3 Matrice flexibility according to the coefficients of Lam and

the Hookes law takes the following form with the coefficients of Lam and .

ij=kkij2 ij

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

[ LLTT NNLT ]= 11LT . TL [EL TL .ET 0 0

LT .EL ET 0 00 0 0 00 0 0 G LT

][ LLTTNN2LT ] [H ] - Plane Orthotropy in plane stresses

3.2.4.4 Stamps of Hooke according to the coefficients of Lam and

[ LLTT NNLN LTTN

]=[ 2 0 0 02 0 0 02 0 0 0SYM 0 0 0][

LLTTNN2LN2LT2TN

] [H ] - complete Isotropy with the coefficients of Lam

3.3 Cas 2D orthotropic in plane strains and axisymmetric3.3.1 Matrice of flexibility

[ LLTT02LT

]=[1E L

TLET

NLE N

0

LTE L

1ET

NLE N

0

0 0 0 0

0 0 0 1GLT

] [ LL TT NN LT ] [H ]1 - plane Orthotropy in plane strains and Matrice




3.3.2 axisymetry of Hooke

[ LL TT NN LT ]= 1 [1TN NT

ET E NTLNL TN

ET . E N NLTL NT

ET . EN0

LTLN NT EL EN

1NL LN E L . EN

NT NL. LT E L .E N

0

LN LT .TN E L .ET

TNTL . LN E L .ET

1 LT .TL EL .ET

0

0 0 0 GLT] [ LLTT0 LT ]

[H ] - plane Orthotropy in plane strains and axisymetry

1=

E LET EN

1TN NTNL LN LT TL2TN NL LT

3.4 orthotropic Cas 2D in Matrice

3.4.1 plane stresses flexibility

[ LLTT

NN2LT

]=[1E L

TLET

0 0

LTE L

1ET

0 0

0 0 0 0

0 0 0 1G LT

][ LLTT NN LT ] [H ]1 - plane Orthotropy in Matrice

3.4.2 plane stresses of Hooke

[ LLTT0 LT ]= 11 LT .TL [E L TL ET 0 0

LT E L ET 0 00 0 0 00 0 0 GLT

][ LLTTNN2LT ] [H ] - Orthotropy in Utilisation




4 plane stresses in Code_Aster Dans Aster , the definition of the constant orthotropic elastic characteristics or functions of the temperature are carried out by command DEFI_MATERIAU , key words ELAS_ORTH, ELAS_ISTR, ELAS_ISTR_FO or ELAS_ORTH_FO for the isoparametric shell elements and solid elements or layers constitutive of a composite (see command DEFI_COMPOSITE ).

To define the reference of orthotropy L ,T , N related to the elements, one can refer to documentations [U4.42.03] DEFI_COMPOSITE and [U4.42.01] AFFE_CARA_ELEM .

NT L

L, T et N : directions d'orthotropie longitudinale, transversale et normale

/ELAS_ORTH = _F ( E_L = ygl longitudinal Modulus Young. E_T = ygt transverse Modulus Young. E_N = ygn normal Modulus Young. GL_T = glt Shear modulus in the plane LT . G_TN = gtn Shear modulus in the plane TN . G_LN = gln Shear modulus in the plane LN . NU_LT = nult Poisson's ratio in the plane LT . NU_TN = nutn Poisson's ratio in the plane TN . NU_LN = nuln Poisson's ratio in the plane LN .

Notice important:

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 in the following way in Aster:

if one exerts a tension according to the axis L giving place to a strain according to this axis

equalizes with ygl

lL

= , one has a strain according to the axis T equalizes with:

ygl*-nult lt

= .




5 Bibliography1) J.C. MASSON: Stamp of Hooke for the orthotropic materials, internal Rapport Applications in

Mechanics, n79-018, CiSi, 1979.

2) D. GAY: Composites, Hermes Edition, 1987

3) J.L. BATOZ, G. DHATT: Modelization of structures by finite elements, Volume 1, Edition Hermes

6 Description of the versions of the documentVersion Aster Auteur (S)

Organisme (S)Description of amendments

6.4 A. ASSIRE, EDF-R&D/AMA

initial Texte

8.4 A. ASSIRE, X. DESROCHES, J.M. PROIX EDF-R&D/Tiny

AMA Corrections



1Introduction2Topology of the matrixes of Hooke2.1the orthotropy2.2Transverse isotropy2.3Isotropy

3Stamp of Hooke and from flexibility3.1Notations3.2Case 3D3.2.10rthotropie3.2.1.1Matrix of flexibility3.2.1.2Stamps of Hooke

3.2.2Transverse isotropy3.2.2.1Matrice flexibility3.2.2.2Matrice of Hooke

3.2.3cubic lasticit3.2.4Isotropie 3.2.4.1Stamps flexibility according to and 3.2.4.2Matrice of Hooke according to and 3.2.4.3Matrice flexibility according to the coefficients of Lam and 3.2.4.4Stamps of Hooke according to the coefficients of Lam and

3.3Cas 2D orthotropic in plane strains and axisymmetric3.3.1Matrice of flexibility3.3.2axisymetry of Hooke

3.4orthotropic Cas 2D in Matrice3.4.1plane stresses flexibility3.4.2plane stresses of Hooke

4plane stresses in Code_Aster 5Bibliography6Description of the versions of the document

Documents

Code Aster Aniso