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15.11.*
625.0
6.1
1
16
9
45
48
1
1
45
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16/9 When
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formula quadraticby solving,0)169(1816
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16
9
)2)(5.2)(1(16
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:0.5 withCase ributionCrack Dist Consistent-Self Isotropic
2
2
2
2
2
22
22
==
+
=
+
=
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−
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E
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15.11.* Continued
MATLAB Plots of effective moduli ratios and comparison with isotropic dilute case :
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
0.1
0.2
0.3
0.8
0.9
1
Crack Density Parameter,
Effective Poissons Ratio,
Self-Consistent
Isotropic Dilute
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Crack Density Parameter,
Effective Youngs Modulus Ratio, E/E
Self-Consistent
Isotropic Dilute
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
0.1
0.2
1
Crack Density Parameter,
Effective Shear Modulus Ratio,
15.12.
1
1
1
,,
2
1
with,,,,
:Theory Stress Couple lDimensiona-TwoFor
2
3
2
3222
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2
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15.13.
checks,00
checks,00
checks,00
(14.4.7) relations equlibrium thein forms theseUsing
,
,,,
:on Formulati FunctionStress Couple
2
22
2
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15.14.
( ) ( )
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text in 393 page of top theon ationsdifferenti suggested theFollowing
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22
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2222
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44
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=−
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15.15.
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where
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6
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66
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2
:Solution Stress Hoop
Case Micropolar -HoleCircular Aroundion Concentrat Stress
max
11
1
1
22
1
2
2
1
11
1
1
2
1
2
2
1
2
2
max
2
1
2
1
2
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2
1
2
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−
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15.16.
)(
2
2
22
)(
2
2
)(
)(
where,2)(
2
2
:gives results theseCombining
2
2
2
)(
2
2
2
2
2
)(
2
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2
2
)(
for relation theUsing
0,2
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2)(
2)(
:Condtions Stress eUnder Plan
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Voids with Materials ElasticIsotropic for Linear EquationsveConstituti
22
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+=++
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===
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15.17.
0
,function stressAiry usual thegIntroducin
6,-7 Exercisein given
results elasticity classical original theas form identical in thecast becan relations These
2
2
)(22
)(4
2
2
)(22
)(4
2
2
2
2
2
)(4
2
2
2
)(
2
2
16-14 Exercisein developed relationsstrain -stress theFrom
2
(7.2.6)by given 7,Chapter in developed
originally that as same theisrelation ity compatibil stress plane thes, with voidelasticityFor
24
2
2
2
2
2
2
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=
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15.18.*
3
2
1
/)2/,(
isresult ingcorrespond the,elasticity classical From
31
)(
)(
1
2
1
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)(
2
)(
2
)(
2
)(
1
)()(
)(
2
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: Functions Besselof properties theFrom
)2(
)/2(46
2
)(,
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2
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2
2,
2
1
When
)/(
)/(26
)(
)/(
)/(22
)(
)/(
)/(21
1)(
31
)(
)(
2cos1
2
voids withelasticityfor solution stress Hoop
4
4
2
2
4
4
2
2
2
2
2
21102
212
2
4
2
4
2
2
2
2
2
2
2
3
2
4
2
2
3
2
3
2
2
3
2
2
2
4
4
2
2
2
++=
++
−
+=
+
+++=
−−=
+=
++=
====
+=
+−=
++=
+−
+
+=
a
a
Tr
r
a
aF
rF
a
r
a
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15.19.*
MATLAB Contour Plots of Doublet Mechanics Solution –
p1 and p2 Microstresses for Flamant Problem
p1 – Contours
p2 – Contours
+
-
+
-
