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11.16*.
:cases isotropic and corthotropifor PlotsMATLAB
0,
1 case, isotropic For the
0
)(
cos
)(
cos
)]()sin)(coscos(sin
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0
sincos
sincos
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)(
sin
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sincos
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sincos
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)(
sin
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)]()cos(sin)()cosRe[(sin2
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21
21
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Isotropic Case
11.17*.
materials) corthotropifor regions stresshigher (Note : PlotMATLAB
]cos)1(cossin)12(
cossin)2(sin)[(
)cos)(sincos(sin
Re
withcase corthotropi For the
]cos)(cossin)12(
cossin)2(sin)[(
)cos)(sincos(sin
Re
(11.5.47)by given wasproblem holecircular dpressurize for the solution stress hoop The
3
21
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21
3
2121
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2,11,2
3
21
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21
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i
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11.18*.
: PlotsMATLAB
3
2
3
1
2
1
2
),2/(),0(
(8.4.15) from comes case isotropic for the Solution
1
)(
)1(
1
)(
)1()(
),0(
case holecircular for the (11.5.51) solution From
4
4
2
2
4
4
2
2
222
2
2
2
2
2
2
222
1
2
1
1
2
1
21
++=
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−
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−
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−
−
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y
a
y
aS
S
y
aS
y
aS
yy
aya
y
aya
yS
Sy
x
x
x(0,y)/S
11.19.
)O(1/order of crack tip at the stresses theof naturesingular theNote
01
)(
1
)(
1
)(
1
)(
1
)(
,components stressother for the likewise and
2
1
)(
1
)(
and, and,0for Now
1
)(
1
)(
1
))((
1
))((
,0 with (11.5.50)solution From
22
2
2
2
2
21
22
1
2
1
1
21
22
21
22
2
2
2
2
221
22
1
2
1
1
121
22
2222
21
2
22
21
1
2211
22
2
2
2
2
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22
1
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1
21
1
22
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1
r
bz
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bi
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=
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+
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−
−
=
−
−
−
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−
−
−
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−
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−
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−
−
−
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−
−
−
−+=
===
−
−
−
+
−
−
−
−+=
−
−
−
+
−
−
−
−+=
→
11.20.
()()
()()
()()
()()
()()
0
1
)(2
1
)(2
2
)]()([2
1
)(2
1
)(2
2
)]()([2
1
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1
)(2
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1
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)(
)(2
)(
1
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)(
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)(
22
22
21
2
22
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21
2
11
222111
22
22
22
2
21
1
2
2
22
11
22
1
21
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1
2211
22
22
22
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21
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2
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22
11
22
1
21
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12
2
2
11
22
2
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2
1
22
22
22
2
21
1
2
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1
22
22
21
1
2
2222
22
11
22
1
21
2
2
111
1
22
11
21
2
2
1111
=
−+−
−
−+
−+−
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+
−=
−+−
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=
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azzaz
Sa
A
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zzRe
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Sa
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xy
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x
11.20. Continued
()
( )
()
−
−
−
−
=
−
−
−
−
=
−
−
−
−
=
+
−
+
−
=
−
−
=
−+−
++==−
+
−+−
++==−
sincos
1
sincos
1
2
sincossincos
1
2
toreduce components stressother thecrack tip, at the Likewise
sincossincos2
sincossincos)(2
ˆˆ
)(2
ˆ
2
11
2,,sincos
ˆ
ˆ
2
1
ˆ
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ˆ
2
11
2,,sincos
ˆ
:TipCrack At
21
21
21
1
2
2
1
21
1
1
2
2
21
21
1
1
2
2
21
21
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1
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21
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zaa
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11.21.*
Anisotropic Crack Tip y/KI Contours (1=0.753i , 2=2.870i)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-1
-0.8
-0.6
-0.4
-0.2
0.6
0.8
11.22.
0
21
1
2
),( Note
sincos
1
2sincos2
1
*)}(){(2
sincos2sincos2
1
*)}(){(2
sincos
2
sincos
2
}sincos{2*)}({2
*2/*)()/(2 where,**)( Choosing
)( :equation sticcharacteri From
*)}(){(2,*)}(){(2,*)}({2
:15-11 Exercise From
33
34544
4544
35545
5545
4445
3
4555
3
45553
44455545
45445545
=−=
−
=
+
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11.23.
( )
( ) ( )
121
22
1
121
22
1
22
21
2
22
1
1
1
2
1
1
11
11
222
2
2
2
2
2
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222
2
,
,)(,0)(
:ConditionsBoundary theapplying and : as form stress radial theRewriting
)(
1
)(
11
)(
1
)(
11
isSolution ,0)1(
:form theof solutionsfor Looking
/ where,0
1
0
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1
11
1
0 :Eqn mEquilibriu
,
1
,
1
−−−
+
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11.24.
R
R
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RR
RR
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R
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+
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−
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+
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2
2
22
1211
222312232212
1211
2,2
1
)1(
,
)1(
where
gives stressesshear for the (4) relations Solving
, stepssimilar following Likewise
)( lines twoprevious Combine
2
1
2
)(
1
)3()2(
)()1(
)4( . . . . 2/,2/,
1
2/
)3( . . . .
1
)2( . . . .
1
)1( . . . . )(
1
11.25.
2 where,0
2
)(2
2
2
)(,2 Using
0)(
2
0,)(,)(With
0)3cot2(
1
sin
11
0]3cot)[(
1
sin
11
0)cot2(
1
sin
11
Eqns Equilib. General
11
122322
2
2
2
2
2
23221212111211
2322121211
R
R
R
R
R
R
R
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k
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−+
==−+
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=−+
======
=++
+
+
=+−+
+
+
=+−−+
+
+
11.26.
32
21
23
21
2
32
2
2
2
2
2
2
)]/(9[
)1(
)]/(9[
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try solution, Particular
, form theofsolution shomogeneoufor Look
type.ODE shomogeneou-nonEuler -Cauchy . . . .
)1(
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0)74.4.8(
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−=
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−
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+
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11.27.
( )
( )
( )
)(
9
3
)3()3(
1
)(
)3(
0)3()(0)( :ConditionBoundary
)3()(
1
3
11
)]/(9[
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,0disk solidFor
2132
2
213
3
1
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−−
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−
−
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=+++−
−
=
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+++
−
=
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−
=
+
−
=
−
−
−=+===
11.28.
case isotropic as stresses same0 :Case cOrthotropi
equation governing satisfies0)44(
3
4
3
2
3
2
02)2(2
0
4
3
4
4
3
)124(
6
2
4
3
;00
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63
4
3
2616
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16
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3
11
3
16
4
4
11
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4
16
22
4
6612
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4
26
4
4
22
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3
11
16
3
22
11
16
3
422
11
16
3
2
3
==
=−=
−−−
=
+
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++
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=−=
−=
−+−===
−+−=
SS
P
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S
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