effective moduli of continuous fiber-reinforced lamina

8/21/2019 Effective Moduli of Continuous Fiber-Reinforced Lamina

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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 1

Effective Moduli of a Continuous Fiber-Reinforced Lamina

Review of Linear Constitutive Relations

General anisotropic material behavior is iven b! a relationship

"herein all # tensor stress components are related to all # tensor

strain components$

11 1% 1& 1' 1( 1#

%1 %% %& %' %( %#

&1 &% && &' &(

'1 '% '& '' '( '#

(1 (% (& (' (( (#

#1 #% #& #' #( ##

xx xx

yy yy

zz zz

yz yz

zx zx

xy xy

C C C C C C

C C C C C C

C C C C C C

C C C C C C C C C C C C

C C C C C C

=

or

) * + ,) *C =


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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina %

+C, is s!mmetric but the matri. is full/ he material constants in

+C, are called the stiffness or elastic constants or moduli2/

3nvertin the last relation ives

) * + ,) *S =

+4, is usuall! called the compliance matri./ 5ote that

1

+ , + ,S C

= /

he determination of the material constants for a eneral

anisotropic material is e.tremel! difficult since the material has

mechanical properties 6oun7s modulus oisson7s ratio etc/2 that

var! "ith the direction in "hich the! are measured and all stressesare coupled "ith all strains/


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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina &

Orthotropic Material (3-D

A material that has mechanical properties that can be associated

"ith an orthoonal principal material coordinate s!stem is calledorthotropic/ A t!pical e.ample is a unidirectional composite

lamina sho"n belo"$

9rthotropic lamina

"ith principal

material 1%&2 andnon-principal .!:2

coordinates/ 5ote

that 1 is enerall!

ta;en as the fiber

direction and % is

transverse to the

fiber but in the plane

of a fiber la!er/

his unidirectional composite lamina has three mutuall!

orthoonal planes of material propert! s!mmetr! and is called an


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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina '

orthotropic material/ 3n the above fiure the 1%& coordinates a.es

are referred to as the principal material coordinates since the! are

associated "ith the reinforcement directions/ 9ne can sho" thatfor speciall! orthotropic materials "herein the 1%& a.es are

principal material directions the compliance matri. has the form$

11 1% 1&

%1 %% %&

&1 &% &&

''

((

##

0 0 0

0 0 0

0 0 0+ ,

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

S S S

S S S

S S SS

S

S

S

=

5ote that shear stresses and strains2 are no" uncoupled from

normal stresses and strains2/


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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina (

the compliance terms can be "ritten in terms of enineerin

material constants so that "e have the follo"in relation bet"een

enineerin strains and stress/ +Recall that enineerin shear strainis t"ice the tensor shear strain i/e/ 1% 1%% = /,

11 1 %1 % &1 & 11

%% 1% 1 % &% & %%

&& 1& 1 &% % & &&

%& %& %&

&1 &1 &1

1% 1% 1%

1> > > 0 0 0

> 1> > 0 0 0

> > 1> 0 0 0

0 0 0 1> 0 0

0 0 0 0 1> 0

0 0 0 0 0 1>

E E E

E E E

E E E

G

G

G

=

1 % & E E E are 6oun7s moduli in the 1 % & directions>ij jj ii = ? oisson7s ratio for transverse normal strain in the @

direction "hen a normal stress is applied

in the i direction and

ijG are shear moduli in the i-@ plane/


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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina #

4ince the compliance matri. +4, must be s!mmetric "e see that

ij ji

i jE E

=

ence the strain-stress relation could also be "ritten as$

11 1 1% 1 1& 1 11

%% %1 % % %& % %%

&& &1 & &% & & &&

%& %& %&

&1 &1 &1

1% 1% 1%

1> > > 0 0 0

> 1> > 0 0 0

> > 1> 0 0 0

0 0 0 1> 0 00 0 0 0 1> 0

0 0 0 0 0 1>

E E E

E E E

E E E

GG

G

=

5ote that for the speciall! orthotropic material there are B

enineerin constants$ 1 % & 1% 1& %& 1% 1& %& E E E G G G /


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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina

Orthotropic Lamina in !lane "tress

For a sinle laminae the lamina is often assumed to be in a simple

t"o-dimensional state of plane stress in the 1-% plane2 such that&& &% &1 0 = = = / he lamina compliance relation simplifies to

11 11 1% 11

%% %1 %% %%

1% ## 1%

0

0

0 0

S S

S S

S

=

or

11 1 1% 1 11

%% %1 % % %%

1% 1% 1%

1> > 0

> 1> 0

0 0 1>

E E

E E

G

=

ence there are onl! ( non-:ero compliances onl! are ' are

independent since +4, is s!mmetric2 and ' independent material

constants 1 % 1% 1% E E G 2/


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he last e=uation can be inverted to obtain the lamina stiffness

relation but is "ritten in terms of tensor strains as$

11 11 1% 11

%% %1 %% %%

1% ## 1% 1%

0

0

0 0 % > %

Q Q

Q Q

Q

= =

or

11 11

%% %%

1% 1% 1%

+ ,

> %

Q

= =

"here the ijQ are components of the lamina stiffness matri. and

are iven b!$


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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina B

%% 111 %

1% %111 %% 1%

11 %%% %

1% %111 %% 1%

1% 1% % %1 11% %1%

1% %1 1% %111 %% 1%

## 1%##

1

1

1 1

1

S EQ

S S S

S EQS S S

S E EQ Q

S S S

Q GS

= =

= =

= = = =

= =


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4ome t!pical values of orthotropic lamina enineerin constants$

Material 1 2E Msi

% 2E Msi

1% 2G Msi

1%

fv

4cotchpl! 100%

E-lass>epos!

(/# 1/% 0/# 0/%# 0/'(

Devlar 'B>B&'

Aramid>epo.!

11/0 0/8 0/&& 0/&' 0/#(

A4>&(01Graphite>epo.!

%0/0 1/& 1/0 0/& 0/#(

oron>((0(

oron>epo.!

%B/# %/#8 0/81 0/%& 0/(

fv ? olume fraction ? ratio of volume of fibers

to total volume of composite


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#ransformation of Material !roperties ($-% to &-'

or the Generall! 9rthotropic Lamina2

3n order to anal!:e laminates havin multiple laminae "ith fibers

in different directions it is necessar! to determine material

properties in an arbitrar! .-! coordinate s!stem in terms of

material properties in the 1-% principal material directions/ his is

a simple transformation similar to stress transformation done in

E5GR %1' from "hich Mohr7s Circle is obtained2/

Consider a lamina that has the principal 1 material a.es at anle to the . a.is countercloc;"ise2 as sho"n belo"/ He can

transform forces from .-! to 1-% coordinates usin the simplerelationship$

1 I

%

x x

y y

F FF c sT

F FF s c

= =

cos sinc s = =


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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 1%

1% is the local

material2

coordinates!stem

.! is the

lobal

coordinate

s!stem

he stress transforms as a second order tensor i/e/

[ ] [ ]11 1%

%1 %%

I Ixx xy T

yx yy

T T

=


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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 1&

his can be e.panded to ive the familiar set of e=uations seen in

E5GR %1' and>or AER9 &0# i/e/

% %11 %% 1%

% %11 %% 1%

% %11 %% 1%

%

%

2

xx

yy

xy

c s sc

s c sc

sc sc c s

= +

= + +

= +

or in matri. notation as

% %

11% %

%%% %

1%

cos sin %sin cos

sin cos %sin cossin cos sin cos cos sin

xx

yy

xy

=

he strain transforms the same as stress if "e use tensor strain

enineerin shear strain is not a tensor =uantit!2/ Recall that


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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 1'

enineerin shear strain is t"ice the tensor shear strain i/e/

1% 1%% = /

% %11 %% 1%

% %11 %% 1%

% %11 %% 1%

%

%

2

xx

yy

xy

c s sc

s c sc

sc sc c s

= +

= + +

= +

his last relation can be "ritten in matri. notation as

% %

11% %

%%% %

1% 1%

cos sin %sin cos

sin cos %sin cos> %> % sin cos sin cos cos sin

xx

yy

xy xy

=

==

5ote that the s=uare matrices in and are identical/

Jefine the transformation matri. +, as


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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 1(

% %

% %

% %

cos sin %sin cos

+ , sin cos %sin cos

sin cos sin cos cos sin

T

=

5ote that +, is not the s=uare matri. in and but is similar/

3nvertin this matri. "e obtain

% %

1 % %

% %

cos sin %sin cos

+ , sin cos %sin cos

sin cos sin cos cos sin

T

=

Comparin e=uation and "e see that can be "ritten in terms of1+ ,T $


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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 1#

[ ]11

1%%

1%

xx

yy

xy

T

=

Li;e"ise for the strain

[ ]

111

%%

1% 1% > %> %

xx

yy

xy xy

T

= ==

5ote that e=uation can be inverted to obtain$

[ ]11

%%

1% 1% > % > %

xx

yy

xy xy

T

= = =


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5o" "e are read! to transform the compliance from material 1%2

directions to lobal .!2 directions/ First substitute e=uation into

e=uation to obtain$

[ ]11 11

%% %%

1% 1% 1%

+ , + ,

> % > %

xx

yy

xy xy

Q Q T

= = = =

5o" substitute e=uation into e=uation to obtain

[ ] [ ] [ ]

11

1 1%%

1%

+ ,

> %

xx xx

yy yy

xy xy xy

T T Q T

= =

=

4o "e have no" have the stress-strain relation in .-! directions but

"ritten in terms of the stiffness in 1-% material directions$


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[ ] [ ]1+ ,

> %

xx xx

yy yy

xy xy xy

T Q T

=

=

he triple matri. product must then be the transformed lamina

stiffness matri.in .-! lobal directions/ ence "e define the

transformed lamina stiffness matrixin .-! lobal directions b!$

[ ] [ ]1

+ , + ,Q T Q T

=

and becomes

11 1% 1#

%1 %% %#

#1 #% ##

+ ,

> % > %

xx xx xx

yy yy yy

xy xy xy xy xy

Q Q Q

Q Q Q Q

Q Q Q

= =

= =

Carr!in out the matri. multiplication ives$


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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 1B

' % % '11 11 1% ## %%cos % % 2sin cos sinQ Q Q Q Q = + + +

% % ' '

1% 11 %% ## 1% ' 2sin cos sin cos 2Q Q Q Q Q = + + +

' % % '%% 11 1% ## %%sin % % 2sin cos cosQ Q Q Q Q = + + +

& &1# 11 1% ## 1% %% ## % 2sin cos % 2sin cosQ Q Q Q Q Q Q = + +

& &%# 11 1% ## 1% %% ## % 2sin cos % 2sin cosQ Q Q Q Q Q Q = + +% % ' '## 11 %% 1% ## ## % % 2sin cos sin cos 2Q Q Q Q Q Q = + + +

5ote that the stiffness matri. + ,Q no" loo;s li;e an anisotropic

material since the &.& has nine non-:ero terms/ o"ever the

material is still orthotropic because the stiffness matri. can be

e.pressed in terms of ' independent lamina stiffness terms

11 1% %% ## Q Q Q Q 2/

he compliance matri. in can be similarl! "ritten/ From


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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina %0

11 11

%% %%

1% 1%

+ ,S

=

ransform to lobal direction similarl! to that done for K2 to

obtain$

[ ] [ ]+ , + ,

> %

xx xx xxTyy yy yy

xy xy xy xy

T S T S

= =

= "here

' % % '

11 11 1% ## %%

cos % 2sin cos sinS S S S S = + + +

% % ' '1% 11 %% ## 1% 2sin cos sin cos 2S S S S S = + + +

' % % '%% 11 1% ## %%sin % 2sin cos cosS S S S S = + + +


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A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina %1

& &1# 11 1% ## %% 1% ##% % 2sin cos % % 2sin cosS S S S S S S =

& &%# 11 1% ## %% 1% ##% % 2sin cos % % 2sin cosS S S S S S S =

% % ' '## 11 %% 1% ## ##%% ' 2sin cos sin cos 2S S S S S S = + + +

effective moduli of continuous fiber-reinforced lamina

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