effective moduli of continuous fiber-reinforced lamina
TRANSCRIPT
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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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+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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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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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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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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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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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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%% 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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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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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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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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% %
% %
% %
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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[ ]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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' % % '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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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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& &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 = + + +