h 406 3 orthotropiclamina

8/10/2019 H 406 3 OrthotropicLamina

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Principalstresses

e

pr nc pa

rec ons

are

so u ons

o

ress nvar an s:

,

thestresstensor is diagonal:

Stressinvariantsinprincipalcoordinates3


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Generalized Hookes law

4th order tensor ofelastic constants

Because ofthesymmetry

ofthe

strain tensor

Because ofthesymmetry

ofthe

stress

tensor

Equation:

Because ofthe3symmetry relationships,thenumber

ofindependent elastic constantsis reduced

from34=81to21inthemost general anisotropic material

Theorder of

partial

differentiation

Maybe changed

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Changeofcoordinates intheelastic constants

Lettwo coordinate systemsxandxrelated bytherotationmatrixA=aij

Change of coordinates of thestress tensor:

(same rule forthestrain tensor)

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Tensor ofelastic constant:

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r o rop ccompos e:

axes

o

symme ry

Theelastic constantsdonotchangeunder coordinate transformationsthat preserve symmetry

x x laneDirectioncosines:ofsymmetry

ne n s:

Mustbe =0

Similarly,8constantsmustbe equal to0:

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Directioncosines:(x2,x3)plane

ofsymmetry

Thefollowing contants

Mustalso be equal to0:

e e s noa ona con oncom ng om e p aneo symme y x1,x3 .

Overall,there are2112=9independent elastic constantsforanorthotropic material.

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Orthotropic materials:9independent elastic constants

oo e s aw may e w en nma r x o m

Vector ofengineering En ineerin stresscomponents

strain componentsStiffness matrix

[theaxes

1,2,3

coincide with the

natural (orthotropy)

axes

of

the

material]

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Stiffness matrix

Compliance matrix

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Stressstrain relationsandengineeringconstantsfororthotropic lamina

column bycolumn byconsidering 3load cases:

1.

Onegets thefirstcolumn

ofthecompliance matrix

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2.

2nd column ofthe

Compliance matrix

3.

3rd column ofthe

Comp ance matr x

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Orthotro ic laminainnatural axes

1.Compliance matrix

Foranisotropic lamina,

EL=ET,G=E/2(1+)

2.Stiffness matrix

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Changeofreference frame

We seek thetransformationmatrix [T]

Changeofcoordinates

Fora2nd order tensor:

[T]is nota

rotation matrix !!

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Stressstrain relationship

InLTaxes:

Inarbitrary axes:

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Stiffness matrix inarbitrary axes

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Compliance matrix inarbitrary axes:

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Example:find thestrains inthelamina=60

Elastic constants:Stresses:

ep :compu e es esses no o opy axes

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Step 2:compute thestrains innatural axes:

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Step 3:compute thestrains intheaxes(x,y)

LT

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Engineeringconstants

Thecompliance matrix in

ar rary axesmay e wr en:

terms of

the

4

constants:

EL,

ET,

GLT,

LT.21


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Graphiteepoxy system Boronepoxy system

Ex


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Balancedlamina:ET=EL,LT=TL

Notisotropic !!

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h 406 3 orthotropiclamina

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