11.1.
exist moduli elastict independen 21only that implies thus
and symmetric ismatrix stiffness canisotropi 66 general the thusand
s,Hooke‘ From
=
ij
kl
kl
ij
ijkl
kl
ij
ee
C
e
11.2.
relations. newany producenot wouldplane, about the reflection (third) final theEmploying
constants. elastict independen nine with materials corthotropifor form desired thegiving
0
00
000
000
000
ations transform twoThese
0
0
0
0
tensor,elasticity for the lawtion Transforma
100
010
001
Q:plane, about the Reflection
0
0
0
0
0
0
0
0
tensor,elasticity for the lawtion Transforma
100
010
001
Q:plane, about the Reflection
s.reflection three theChoose planes. coordinate theofeach about sreflection three
or axes coordinate about the rotations threedoeither toequivalent isIt
13
66
55
44
33
2322
131211
4523311332223145
3633122133331236
2622122122221226
1611122111111216
ij32
3533311333333135
5631122113311256
4623122132231246
3433233233332334
2522311322223125
2422233222222324
1511311311113115
1411233211112314
ij21
−
=
=−===
=−===
=−===
=−===
=
−
=−
=−===
=−===
=−===
=−===
=−===
=−===
=−===
=−===
=
−
=−
xx
C
C
C
C
CC
CCC
C
CCCQQQQCC
CCCQQQQCC
CCCQQQQCC
CCCQQQQCC
CQQQQC
xx
CCCQQQQCC
CCCQQQQCC
CCCQQQQCC
CCCQQQQCC
CCCQQQQCC
CCCQQQQCC
CCCQQQQCC
CCCQQQQCC
CQQQQC
xx
ij
mnpqqpnm
mnpqqpnm
mnpqqpnm
mnpqqpnm
mnpqlqkpjnimijkl
mnpqqpnm
mnpqqpnm
mnpqqpnm
mnpqqpnm
mnpqqpnm
mnpqqpnm
mnpqqpnm
mnpqqpnm
mnpqlqkpjnimijkl
11.3.
22
222
22
4224
22
4
22
22
1211
22
12
4
11
22
4
66
22
12
22
11
4
2222
4
222211
2
21
2
222121
2
22
2
212112
2
22
2
21
1221
2
22
2
211212
2
22
2
211122
2
22
2
211111
4
21
2222222222
22
5555
22
44
2
55
2
32321212323111123132121131311111
3311331331331313313155
55
2211665544
311256231246233145331236
333135332334221226223125
222324111216113115112314
)cossin()coscossin2(sin
coscossin)(2cossin2sin
coscossin4cossin2sin
C
:Check
)sin(cossincos
C
:Check
100
0cossin
0sincos
:bygiven isrotation particular The
2/)(, :and
0
0
0
:zero are moduli following thecase, For this
:Symmetry Material
CCC
CCCCC
CCCC
CQCQQCQQCQQ
CQQCQQCQQCQ
CQQQQC
C
CCCC
CQQCQQCQQCQQ
CQQCQQQQCQQQQC
C
Q
CCCCC
CCCCCCCC
CCCCCCCC
CCCCCCCC
CQQQQC
mnpqqpnm
qnqnqnqnmnpqqpnm
ij
mnpqlqkpjnimijkl
=+=++=
+−++=
+++=
++++
+++=
=
=
=+=+=
+++=
===
=
−
=
−==
========
========
========
=
11.4.
0
3
2
0)23(40
2
2
2
0
2
2
0:Case Isotropic
2)(00
00
00
:Case Isotropicly Transverse
2
00
00
00
00
:Case cOrthotropi
0 are elements diagonal all and
0 minors principal all so and definite positive bemust
2
4
2
13121133
1313
1112
13
3313
1312
12
3313
1311
114
12
11
1112
3
2
133311
3313
1311
21
2
1233
2
1322
2
2311312312332211
2313
2212
13
3313
2312
12
3323
2322
114
2
122211
2212
1211
3
2
133311
3313
1311
2
2
233322
3323
2322
1
++
+
+
+
+
+
++−
=
+++
+−
p
CCCC
CC
CC
C
CC
CC
C
CC
CC
Cp
CC
CC
CC
p
CCC
CC
CC
pp
CCCCCCCCCCCC
CC
CC
C
CC
CC
C
CC
CC
Cp
CCC
CC
CC
p
CCC
CC
CC
p
CCC
CC
CC
p
pC iij
11.5.
11
0
11
0
11
0
positive be ofminor principaleach that function energy strain definite Positive
0
00
000
000
000
, materials corthotropiFor
2
1
3
2
31
2
3
31
13
2
131133
1113
1333
2
3
2
32
2
3
32
32
2
233322
3323
2322
1
2
2
21
2
2
21
21
2
122211
2212
1211
66
55
44
33
2322
131211
−
−
−
=
E
E
E
E
EEE
SSS
SS
SS
E
E
EEE
SSS
SS
SS
E
E
EEE
SSS
SS
SS
S
S
S
S
S
SS
SSS
S
i
ij
ij
ij
11.6.
11.7.
0
02
as writtenbe can equation aldifferenti original The
,2
roots for the quadratic theSolving
equation sticcharacteri02
0)2(
parametera is where,)(),( form theof solutionsfor Look
02
:equation sHomogeneou
21
22
2
2
22
55
45
2
2
55
44
55
44
21
55
45
21
55
5544
2
4545
2,1
4445
2
55
2
554544
554544
=
−
−
=
+
−
==+
−
=
=+−
=
+−
+=
=+−
xyxy
yyxx
yyxS
S
xS
S
S
S
S
S
S
SSSS
SSS
fSSS
yxfyx
SSS
i
yyxyxx
11.8.
xyyx
xyyxxyyxxy
xyyx
xyyxxyyxy
xyyxx
z
xyyxxyyxx
S
SSS
S
SSSS
S
SSSS
BBBe
SSS
S
SSSS
S
SSS
S
SSSS
BBBe
SSSe
SS
S
SSSS
S
SSSS
S
SSS
BBBe
S
SSSS
B
S
SSS
B
S
SSSS
B
S
SSS
B
S
SSSS
B
S
SSS
B
−
+
−
+
−
=++=
++=
−
+
−
+
−
=++=
++=
===
−
+
−
+
−
=++=
−
=
−
=
−
=
−
=
−
=
−
=
662616
33
2
363366
33
36233326
33
36133316
662616
262212
33
36233326
33
2
233322
33
23133312
262212
161211
2313
33
36133316
33
23133312
33
2
133311
161211
33
36233326
26
33
2
363366
66
33
36133316
16
33
2
233322
22
33
23133312
12
33
2
133311
11
2
00 withrelations stress plane develop to wishNow we
strain planefor law s Hooke‘into relations theseUsing
,,
,,
(11.5.3) Relations
11.9.
form the yieldsbefore as gIntegratin
, with
0 : EquationGoverning
,:2Case
4321
4321
4321
−
==
−
==
=
==+===
xy
DD
xy
DD
DDDD
i
11.10.
( )
ii
E
S
SS
E
E
S
S
S
S
S
SS
S
SS
SSSS
SS
SSSSSS
47.2,69.0
12.6,48.0)94.2(46.66.6
2
1
6.6
7
50
)27.0(22
2
,94.2
17
50
:materialy Glass/Epox–SFor
22
2
1
:for equation quadratica as Solving
0)2(
0 materials corthotropiFor
02)2(2
2,1
2
2
2,1
12
1
12
11
6612
2
1
11
22
11
22
2
11
6612
11
6612
2
2,1
2
22
2
6612
4
11
2616
2226
2
6612
3
16
4
11
=
−−=
−−=
=+−=
+−=
+
===
+
+
=+++
==
=+−++−
11.11.
)]()([2
)()()()(
)()()()(
)]()([2
)()()()(
)]()([2
)()()()(
)()()()(
)()()()(
222222111111
222222111111
2
2211
22221111
2
22
2
211
2
1
22
2
222
2
211
2
111
2
1
22
2
222
2
211
2
111
2
1
2
2
________
2222
________
1111
zFzFRe
zFzFzFzF
zFzFzFzF
yx
zFzFRe
zFzFzFzF
x
zFzFRe
zFzFzFzF
zFzFzFzF
y
zFzFzFzF
xy
x
+
−=
−
−
−
−=
−
−
−
−=
−=
+
=
+
+
+
=
+
=
+
+
+
=
+
+
+
=
=
+++=
11.12.
strainshear for the expression thein results theseUsing
)()]()([2
)()]()([2
relationsnt displaceme–stain thegIntegratin
/)(and where
)]()()()[(22
)]()([2
)]()([2
)]()([2)]()([2)]()([
stress, planeFor
)]()([2
)]()([2
)]()([2
222111
222111
2226121216
2
11
2222211111662616
22221111262212
222111
2221111622111222
2
211
2
111
161211
222111
2211
22
2
211
2
1
v
u
xgzqzqRev
yfzpzpReu
SSSqSSSp
x
v
y
u
zqpzqpReSSSe
y
v
zqzqReSSSe
x
u
zpzpRe
zzReSzzReSzzReS
SSSe
zzRe
zzRe
zzRe
iiiiii
xyyxxy
xyyxy
xyyxx
xy
y
x
+
+
=
+
+
=
+−=+−=
+
=
++
+=++=
=
+
=++=
=
+
=
+
−
+
+
+
=
++=
+
−=
+
=
+
=
11.13.
)]()sincos()()sincos[(2
)]()sincoscossincossin(
)()sincoscossincossin[(2
)sin(coscossincossin
)]()sin(cos)()sin[(cos2
)]()cossin2cossin()()cossin2cossin[(2
cossin2cossin
)]()cos(sin)()cos[(sin2
)]()cossin2sincos()()cossin2sincos[(2
cossin2sincos
theory,ation transformstress From
)]()([2
)]()([2
)]()([2
:Stresses
22221111
22221111
22
2
2
2
2
2
2
11
2
1
2
1
2
1
22
22
2
211
2
1
222
222
2111
222
1
22
22
2
211
2
1
222
222
2111
222
1
22
222111
2211
22
2
211
2
1
zpqzpqRe
z
zRe
zzRe
zzRe
zzRe
zzRe
zzRe
zzRe
zzRe
xyyxr
xyyx
xyyxr
xy
y
x
−+−=
+−+−+
+−+−=
−++−=
++
+=
+++
−+=
−+=
−+
−=
−++
−+=
++=
+
−=
+
=
+
=
11.14*.
 
784.5,643.0 :poxyKevlar49/E
545.5,577.0 :yBoron/Epox
474.2,693.0 :yGlass/Epox–S
: MATLABUsing
1422
2
1
2
2
,1 case, isotropic For the
Defining
4
22
2
1
:for equation quadratica as Solving
0)2(
0 materials corthotropiFor
02)2(2
:equation sticcharacteri General
2,1
2,1
2,1
2,1
22
2,1
11
6612
11
22
2
2,1
2
2,12,11,2
11
22
2
11
6612
11
6612
2
2,1
2
22
2
6612
4
11
2616
2226
2
6612
3
16
4
11
=
=
=
=−−−==
+
=
−==
−
+
+
−=
=+++
==
=+−++−
S
SS
S
S
i
S
S
S
SS
S
SS
SSSS
SS
SSSSSS
11.15.
*)}(){(2
*)}(){(2
*)}({2*)}({2*)(*)( real bemust Since
* where,*)(*)()()(
as written becan equation governing osolution t general so and
),( pairs conjugatecomplex are rootsshown previously As
–
:gives roots twofor the Solving
02
equation sticcharacteri quadratic theproduces termcommon theCancelling
0)2(gives)( solutionsfor Looking
020 Eqns. Equilbrium
22
22
0
0
2
2
0
0
0
0
0
00
00
00
becomes materials monclinicfor law sHooke‘ thereforeand
,0),(,0With
4544
5545
111
2121
2
44
5544
2
4545
5545
2
44
2
444555
2
2
44
2
45
2
2
55
45444544
55455545
66
55
4544
3633
262322
16131211
zFCCRe
zFCCRe
zFRezFRezFzFww
yxzzFzFyxFyxFw
CCC
CCCC
CCC
F
FCCCyxFw
w
C
w
C
w
C
x
w
C
y
w
CeCeC
x
w
C
y
w
CeCeC
e
e
C
C
CC
CC
CCC
CCCC
eeeeyxwwvu
yz
xz
yz
xz
xzyzyz
xzyzxz
xyzyx
zx
yz
xy
zx
yz
z
y
x
xyzyx
+=
+=
==+=
+=+=+++=
−
=
=++
=
+++=
=
+
+
=
+
+
=+=
+
=+=
====
=
=======