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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
4
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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:
7
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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.
8
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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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