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8.39.
)cos31(
2
cos
2
)cos1(
2
sin
2
)
2
sin
3
5
2
3
sin()
2
cos5
2
3
cos(
1
4
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2
sin
3
1
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2
cos
2
3
(cos
1
4
3
)
2
cos
2
3
(cos)
2
sin3
2
3
(sin
1
4
3
)
2
cos
3
5
2
3
(cos)
2
sin5
2
3
(sin
1
4
3
gives relations stress general theinto results thesengSubstituti
3
)2(
,,
2
3
gave problem theon condtionsBoundary
])2sin()2()2cos()2(sincos[)1(
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cos)1(sin)1([
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2
2
22
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−
++
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−=−=
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−=
++
+−=
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−=−==
−−−−−+−−−=
−+−++−=
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−+−−=
−+−++=
−
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r
r
BA
r
BA
r
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r
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8.40*.
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r
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r
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r
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sin
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3
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cos
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sin2
2
3
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cos
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2222
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2222
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max - Contours – Mode I
max - Contours – Mode II
8.41.
2
sin
2
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2
cos
2
,
2
sin
become stresses and ntsdisplaceme theso and2/110
10at StressesSingular
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:surfacescrack on conditionsboundary stress Zero
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11
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r
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r
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8.42.*
% Solution Contours for Antiplane Strain Crack
Fields
clc; clear all; clf
[x,y]=meshgrid(-0.5:0.05:1,-1:0.01:1);
z Contours
8.43*.
2/)(,
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2222
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baR
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2
8.44.
generated. becan
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:case for this ConditionsBoundary Check
sin
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1
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cos
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22
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8.45*.
:Plots MATLAB
1
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42
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)/(1
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4
42
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are 0on stresses normal two the35,-8 Figurein shown problemdisk For
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8.46*.
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and of einterchang simple usingshown as problems two theofion superposit Use
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4
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8.46*. Continued
MATLAB Comparison Plot:
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8.47.
0,lim
, as case limiting for the so and Load Total
,16,8,4,0, (c) case general for the Thus
0,
8
twooffactor aby stresses theincrease thusandamount same theaddsimply will(b) casein
shown as loadings morefour ofaddition the).,(isotropic chydrostati are case(a)for stresses theSince
(b) Case
0
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42)()()()(2
42)()()()(2
:bygiven arecenter sdisk' at the stresses the8.46, Exercise From (a) Case
0,
4
2
2
4
1
2
4
2
2
4
1
2
0,
4
2
2
4
1
2
4
2
3
4
1
3
0,
4
2
3
4
1
3
4
2
2
4
1
2
=−=
−==
→
===
==
−==
=
−==
=
+
−
−
+
+
−
−
=
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−
+
+
−
+
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−
+
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−
+
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→
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=
=
xy
N
yx
xyyx
xyyx
yx
xy
yx
y
yx
x
p
D
NP
N
D
NP
pNPDp
N
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P
r
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r
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r
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r
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P
D
r
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r
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r
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r
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D
P
D
r
xR
r
xR
r
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r
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8.48.
disk solid theof that twiceisdisk annular thein stress maximum the
)(,
8
3
)0()0(
11-8 ExampleFrom
8
)3(
)(, when
8
)3(
)(
8
)3(
8
31
)()(
8
)3(
)(
8
)3(
8
31
8
)3(
,)(
4
)3(
gives and for relations conditionboundary theSolving
0
28
)3(
0)(
0
28
)3(
0)(
28
31
28
)3(
22
max
22
max
2222222
max
222
2
22222
222
2
222
1
21
2
21
22
2
21
22
2
21
22
2
21
22
+
===
+
+
++
+
+
+
−==
+
++
+
+
+
−=
+
−=+
+
=
=++
+
−=
=++
+
−=
−+
+
−=
++
+
−=
abb
bab
bbaaa
ba
r
bar
baCbaC
CC
b
CC
bb
a
CC
aa
r
CC
r
r
CC
r
r
r
r
r
8.49.
)()(
11
)(
1
)(
1
:Law sHooke'
)(
1
:surface On the
)(
1
(8.5.4)1
xpp
E
p
E
p
EE
e
xp
Edx
ud
e
xp
Edx
ud
yxx
xyxx
x
x
x
−==+=
−
−
+=−=
−
−==
−
−=
8.50.
% Calculate and Plot Stress Contours Under Surface Loading to a Half Space
% Numerically Evaluate Integrals Using quadv(....) Function in Eqn.(8.5.13)
% Elliptical Normal Loading Case: p(s)=(2P/pi*a)sqrt(1-(s/a)^2)
function elliptical
