This document is partially blurred.
Unlock all pages and 1 million more documents.
Get Access
12.1.
( )
( )
ijoijkk
ijoijokkijij
okkkkokkkk
ijoijkkij
T
ij
M
ijij
ijkkij
M
ij
ijo
T
ij
TTee
TTTTee
E
TTe
E
TT
E
e
TT
EE
eee
EE
e
TTe
−+−+=
−−−−
−
+
+
=
−−
−
=−+
−
=
−+
−
+
=+=
−
+
=
−=
)()23(2
)()(3
211
)(3
21
)(3
21
)(
1
1
:Strains Mechanical
)( :Strains Thermal
)()(
)(
)(
12.2.
)
1
1
(
1)1(
1
gives results theseCombining 1
2
Setting
)(
111
1
)(
1
)(
1
setting and relationsity compatibil strain theinlaw s Hooke'Using
)(
1
: LawsHooke'
0 :ityCompatibil Strain
,,
,,
,,
,,,,,
jij,
,,,,,,,,
,,,,
,,,,
kk
ij
ij
ijkk,kkij
kkkkmm
ijijkkijkkmmijkkkkij
jkikikjkkkijijkkjkikmmikjkmmkkijmmijkkmm
ikjkjkikijkkkkij
ijoijkkijij
ikjljlikijklklij
T
T
E
T
E
ji
TT
E
TTTT
E
lk
TT
EE
e
eeee
−
+
+
+
−=
+
+
−
−==
+
+
−
+
=
+
+
−−+
+
−−−+
+
=−−+
=
−+
−
+
=
=−−+
12.3.
dczbyaxT
dczGc
dz
dG
c
z
T
zGbyFb
y
F
b
y
T
zyFaxTa
x
T
xz
T
xz
e
x
e
z
e
y
e
yxz
e
zy
T
zy
e
z
e
y
e
x
e
xzy
e
z
T
y
T
x
T
z
T
x
T
z
e
x
e
xz
e
z
e
x
e
y
T
z
T
y
e
z
e
zy
e
y
e
z
e
x
T
y
T
x
e
y
e
yx
e
x
e
y
e
eeeTeee
zyxTe
yyzxy
zx
y
x
xy
zx
yz
x
xzzxxz
z
yyz
z
y
y
x
xyy
x
zxyzxyzyx
ijij
+++=
+===
+==
=
+==
=
=
+
+
−
=
=
=
+
+
−
=
=
=
=
=
+
=
+
=
+
=
+
=
+
=
+
=
+
=
+
=
+
======
=
)(
),(
00
00
0 relations threeabove The
002
002
002
relationsity compatibilstrain Using
0,
),,( :expansion thermaledUnrestrict
2
22
222
2
2
2
2
2
2
2
2
2
2
2
2
2
22
2
2
2
2
2
2
2
2
2
2
2
22
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
12.4.
s
n
yyox
s
n
yyyxxy
n
y
s
n
xyxo
s
n
xyxyxx
n
x
xyxy
oxyy
oyxx
xyxyoxyyoyxx
TnTT
E
u
vE
n
v
uE
TnnT
Tn
x
v
y
uE
nTT
E
y
v
x
uE
TnnT
e
E
TT
E
ee
E
TT
E
ee
EE
eTT
E
eTT
E
e
)()(
)(
)(
)(
)1(2
)(
1
)(
1
)(
1
)(
1
)(
1
)(
1
)(
1
1
,)()(
1
,)()(
1
:Stress PlaneFor
2
2
2
2
=
−
−
+
+
+
=+=
=
+
+
+
−
−
−
+
−
=+=
+
=
−
−
−+
−
=
−
−
−+
−
=
+
=−+−=−+−=
12.5.
( ) ( )
0)( becomes stress planefor relation ity compatibil the thusand
1
)21(
1-12 Tableation transform theusesimply result, stress plane ingcorrespond thedetermine To
0
1
)(
toreducesstatement ity compatibil previous the thusand02
00,00
forces,body zero with equations mequilibriu fromBut
2)()1(
2)]()1[()]()1[(
2relationity compatibil in the strains theseUsing
1
,)]()1[(
1
,)]()1[(
1
strains for the Solving
2,))(23(2)(,))(23(2)(
:Strain Plane
22
2
22
2
2
2
2
2
2
2
22
2
2
2
2
2
2
2
22
2
2
2
2
2
2
2
2
2
2
=++
+
+
=
−
++
=
+
+
=
+
=
+
=
+
=
+
+
+
=++−
=−+−−
+−+−−
=
+
+
=−+−−
+
=−+−−
+
=
=−+−++=−+−++=
TE
EE
T
E
yxyx
xyyyxyxxyx
yxyx
TE
yx
TTE
x
TTE
y
yx
e
x
e
y
e
E
eTTE
E
eTTE
E
e
eTTeeeTTeee
yx
yx
y
x
xy
xyyyxyxy
x
xy
x
y
x
xy
yx
xy
oxyoyx
xyy
x
xyxyoxyyoyxx
xyxyoyyxyoxyxx
12.6.
( )
( ) ( )
:)2,1( PlotsMATLAB
coshsinh
sinh
,
coshsinh
sinhcosh
constants twofor the Solving
0)sinh
1
cosh(sinh
/sinhcosh
0),(),( :conditionsboundary free Stress
cos)sinh
1
cosh(sinh
sin)cosh
2
sinh(cosh
sinsinsinhcosh
by given werestresses the1,-12 ExampleFrom
3
2
2
32
2
32
32
2
32
2
32
2
==
+
=
+
+
−=
=
++
−=+
==
++−=
++=
−+−=
a
aaa
aTE
C
aaa
aaaTE
C
aaaCaC
TEaaCaC
yaya
yxxxCxC
yxxxCxC
yTEyxxCxC
oo
o
xyx
xy
y
ox
siny = 1
y/ETo
12.7.
( )
( )
( )
2
3
2
1
2
3
2
21
2
21
2
1
2
2
24
4
2
2
2
)1log2(
4
log
log
1
1
timeseeresult thr thisgIntegratin
111
1
since and,0
111
0 equation governing stress plane the
11
,
11
case, symmetricradially For the
r
C
Cr
C
Tdr
r
E
CrCrdrrCTdrEr
CrCTEr
dr
d
r
C
dr
dT
rEr
dr
d
rdr
d
r
dr
dT
r
dr
d
r
Er
dr
d
rdr
d
r
dr
d
r
dr
d
rdr
dT
r
dr
d
r
E
dr
d
r
dr
d
rdr
d
r
dr
d
r
TE
dr
d
r
dr
d
rdr
d
r
dr
d
rdr
d
r
dr
d
rdr
d
rdr
d
r
r
r
r
r
r
++−+
−=
+++−=
++−=
+−=
−=
==
+
=+
=
=+=
12.8.
+−+
−
=
+−+
−
=
T
dr
du
r
uE
T
r
u
dr
duE
r
])1([
1
])1([
1
:forms stress icaxisymmetr Using
2
2
12.9.
( )
0,
)1(16
3
)1log2(
4
)(
)1(16
)1log2(
4
timeshreeequation t governing thegIntegratin
11
1
0
1
strain, planefor equation Governing
)(
4
4
)(
0boundedremain must )0(
:ConditionsBoundary 4
log gIntegratin
1
0
1
2
2
3
2
1
2
2
2
2
3
2
1
2
424
22
2
2
1
2
21
2
2
r
hE
r
C
Cr
C
r
dr
d
dr
d
r
hE
r
C
Cr
C
hE
r
d
d
r
d
hE
T
E
ra
h
TT
a
h
TCTaT
CT
r
h
CrCT
h
dr
dT
r
dr
d
r
h
r
T
rr
T
r
o
r
o
r
o
r
o
o
o
o
oo
o
oo
=
−
+−++==
=
−
+++−=
=
−
==
−
+
−+=
+==
=
−+=
−=
=+
+
12.10.
0)(
0)()1(
1
)]()1([
1
0
22
1
)1(
)1()1(
: wherecase For the
)1()1(
)1(
0)(
00at ntsdisplaceme Bounded
)1(
(12.7.6),by given solution general theFrom
2
2
22
2
2
2
2
2
2
1
2
2
1
==
=
=
−+−+
−
=−+−+
−
=
=
−+=
+
−
+
=
=
+
−
+
=
+
−==
==
+
++=
r
oorr
o
a
o
r
o
o
ar
a
r
r
dr
d
dr
d
TT
r
u
r
uE
TTee
E
a
a
rr
r
Td
a
rT
d
r
T
u
TT
dT
a
r
dT
r
u
dT
a
Aau
Ar
dT
rr
A
rAu
12.11.
2
22
321
4
2
3
2
21
43
2
1
2
223
2
2
2
1
4
2
3
2
21
24
1log4 where
020
)(
00)(
(12.5.9) conditionsboundary general , and on tractionszero With
1
2
)3log2()(
2
)1log2(
1
becomes thenfield stress resulting theand
1
loglog
give tointegrated be canresult This
0
111
toreducesy axisymmetr with0 equation governing stress plane The
−
−
=
=++=
=+=
==
−−+++−==
=
−+++=
=
−+
+
+=
=
+
=+
a
b
a
b
a
b
N
AAA
dr
ad
a
a
a
a
AAa
brar
Trdr
TE
A
r
A
A
r
d
d
Trdr
r
E
a
A
a
r
a
A
r
A
dr
d
r
ddTEA
a
r
A
a
r
a
r
A
a
r
A
dr
dT
r
dr
d
r
E
dr
d
r
dr
d
rdr
d
r
dr
d
r
TE
a a
r
r
a
r
r
a a
12.12.
−
−
−
+
−+
=
−+−
++=
−+−+++==
−
−
−
+
−
−+
=
−+−+
+=
−+−+++=
=====
=
)(
)21()21)(1(
)()23(2)2(
)()23(2)(
Likewise,
)(
)21(
2)1(
)21)(1(
)()23(2)2(
)()23(2)(
0,,
:Symmetry Spherical withsCoordinate Spherical
TT
E
R
u
R
u
E
TT
R
u
R
u
TTeeee
TT
E
R
u
R
u
E
TT
R
u
R
u
TTeeee
eee
R
u
ee
R
u
e
o
RR
o
RR
oR
o
RR
o
RR
oRRR
RR
RR
R
12.13.
))(1(2
)1(
,)2(
))(1(3
)21(
constants, for the Solving
0
1
1
2
213
1
2
1
1
2
0)(
0
1
1
2
216
1
1
2
0)(
3
1
2
1
)(1
,0)(,)( :Conditions Boundary eTemperatur
00
:roblemSymmetry P Spherical
33
33
2
22
33
1
3
21
3
21
32
0
22
21
2
1
22
−−
+
−=−+
−−
−
=
=
+
−
−
+
−
−−
−=
=
+
−
−
+
−−
−=
−
−
=−
−
=
−
−
=
−
−=
−
−===
+−==
=
ii
i
R
i
R
i
R
i
R
i
ii
i
ab
baT
Caabb
ab
aT
C
a
ECEC
a
b
ab
aT
E
a
b
ECEC
ab
aT
E
b
RbR
ab
aT
db
ab
aT
dT
R
b
ab
aT
T
ab
aT
C
ab
abT
CbTTaT
C
R
C
T
dR
dT
R
dR
d
T
12.14.
=
+
+
+
+
=
=
=
=+=
+
+
=
=+++−
+++−+
+
+
=
+++−
+
+
=
+
++−
+
+
=
+
+
++−
+
+
=
+−
+
+
=
+
=+
=
=
=
1
1
21
1
thatshow to1-12 Tablein constants elastic of einterchang theusesimply case, stress plane For the
and , case for this thusand,)1(
)(2
)23(
ifoccur willThis
vanish. will termsre temperatuthe,0)(2)23(factor theif Now
)](2)23([2
)(2)23(2
22)23(2
2)()23(2
)23(2
strain planefor Law sHooke'in definitionion decomposit thisUsing
determined be oconstant t a is where,,
fieldnt displaceme theofion decomposit thesuggested thisand ,
that determined it wasfunction re temperatuintegrated andcomplex theof sdefinition Using
xyxyyyxx
R
RR
RIR
x
IR
IR
T
x
u
y
v
x
u
x
t
T
x
u
y
v
x
u
x
t
x
t
T
x
u
y
v
x
u
x
t
y
t
x
t
T
x
u
y
v
x
u
T
x
u
y
v
x
u
tvvtuu
y
t
x
t
T
12.15.
222
)(
2222
)(
at Likewise
0
)(
222
)(
2222
)(
At
)(
2122121
)]()([)()(
0)]()([)()()(
:conditionsboundary free Stress
)(
log)(,log)(
(12.8.15) relations From
2
3
22
322
22
3
2
2222
22
2
22
3
22
322
22
3
2
2222
322
22
222
2
2222
2
______
,
2
______
,
222
22
22
2
+
−−
+
++
+
−+
+
−=−
=
=
+
−
+
−=
+
−−
+
++
+
−+
+
−=−
=
+
−−
+
++
+
−+
+
−=
+
−
+
=
+
−
+
=−
+
−=
+
−=
−
−−−
−−
=
−
−−−
−−
=
=
=
i
i
o
ooi
i
oi
o
i
oi
i
o
o
i
i
oi
i
o
o
i
oi
i
o
o
i
rrrr
i
o
i
oii
o
oi
i
i
ioi
i
oi
i
i
oi
i
i
i
i
i
oi
i
i
i
i
oi
i
i
i
i
rrrr
i
i
oi
oi
oi
i
oioi
i
rrr
i
rrrrr
oi
oi
oi
r
r
rrr
err
r
e
rr
er
r
e
e
rr
er
r
e
rr
er
r
e
i
erz
e
rrr
r
rr
r
r
rrr
err
r
e
rr
er
r
e
e
rr
er
r
e
rr
er
r
e
i
erz
zrr
rr
zrr
z
z
z
e
rr
z
zrr
z
z
zzzezz
zzzezzi
rrz
rr
zAz
rr
z
zAz
o
i
oi
oi
12.16.
+=
==
==−=
+=
+=
==
+=
=+
=+
+
==
−=
+
=
+
+
=
=
+
+
=
sin
),( :Infinityat Conditions
00
),(
:holeon condition boundary Insulated
and in constant theabsorbed have wewhere,sin issolution eTemperatur
)( isequation - oSolution t
sin)( 1with offunction odd be should field aturebut temper
,cossin)( isequation - oSolution t
0)()( and0)()(
1
)(
constant
)(
)(
)(
)(
1
)(
0
11
)()(),(let , variablesof separation Using
0
11
0
:Equation Conduction
2
1
1
2
2
2
2
1
21
2
1
2
1
2
2
2
2
2
2
2
2
22
2
2
r
a
r
k
q
T
k
q
Cy
k
q
T
CaC
a
C
C
r
aT
CCA
r
C
rCT
r
C
rCrfr
Agc
cBcAg
gcgrf
r
c
rf
r
rf
c
g
g
rf
rf
r
rfr
gf
r
gf
r
gf
grfrT
T
rr
T
rr
T
T
12.17.
( )
( )
+
−=
−
−=
−
=
−=−
+−
+
−=
+−−=
−+−=
+
=+−
−=+
+
=+
+
−
=
+
−=
=
=
+
=+
+
−=
++−==
−−−
sin
2
1
,sin
2
1
stresses normal individual for the Solving
cos
2
1
andsin
partsimaginary and real Separating
sincoscos
)1(
8
121
2
1
22)()(22
sin
1
4
1
1
3
stress, planeFor
,sin
)1(
8
4)(4)()(2
)1(
2
where,1log)(,log)(
by given where4-12 Examplefrom potentials The
3
3
3
3
3
3
3
3
2
2
2
22
3
3
2
32
3
2
2
22
2
2
______
2
2
2
r
a
r
a
k
qaE
r
a
r
a
k
qaE
r
a
r
a
k
qaE
kr
qaE
r
a
i
r
a
kr
iqa
e
r
e
r
a
e
r
Ae
zz
a
z
z
Aezzzei
kr
qaE
kr
qa
z
A
RezRezz
k
qai
Az
z
a
AzzAz
r
rr
iiii
ii
rr
r
r
12.18*.
:problem eachfor isotherms of PlotsMATLAB
sin andcos
withregion mapped thein scoordinate theare , where
,sin :5-12 efor Exampl FieldeTemperatur
sin :4-12 efor Exampl FieldeTemperatur
2
2
+=
+=
+=
+=
+=
− m
Ry
m
Rxe
m
eRz
a
k
q
T
r
a
r
k
q
T
ii
Example 12-4: Circular Hole Case
Example 12-5: Elliptical Hole Case
12.19.
−
=
−
=
++−−
=
+
−=
+
−=
−++++
−=
−
−=
−
−=
++−+
−=
===
===
cos
2
cos)(
cos])1()[(
)(2
sin
2
sin)(
2
}3sin2sin])1(){[(
)(2
sin
2
sin)(
2
sin])1()[(
)(2
and0let 5,-12 Examplefrom case holecircular eextract th To
3
3
242
8
22242
3
3
242
8
222242
3
3
242
8
22242
r
a
r
a
k
qaE
qaE
mmm
kh
qaE
r
a
r
a
k
qaE
k
qaE
mmmm
kh
qaE
r
a
r
a
k
qaE
k
qaE
mmm
kh
qaEa
r
eareaz
aRmba
ii
12.20*.
:1,2/1,0for Plots MATLAB
)1/(1)2/( :Note
)2cos21(
]3sinsin)1)[(1(
)2cos21(2
}3sin2sin])1(1){[(1(
/
1 with (12.8.32)equation
bygvien isboundary hole around stress hoop ldimensiona-non the5,-12 Example From
22
2
22
22
2
=
+−=
+−
−+++
−=
+−
−++++
−=
=
=
m
m
mm
mmmm
mm
mmmm
kqaE
m = 1
12.21.
yxzzFReyxFyxFyxT
yxFyxFyxT
k
kkk
i
k
k
kkk
kkkk
k
kkk
TkkkyxTT
y
T
k
yx
T
k
x
T
k
Tk
Tkq
q
yy
xyyyxx
yy
xy
xyyyxx
yyxxxyxy
yy
yyxyxx
yyxyxx
yyxyxx
ijij
jiji
ii
+==+++=
+++=
−
−=
−−=
=++
=
+++=
=
+
+
=
−=
=
* where,*)}({2)()(),(
real bemust re temperatu thesincebut
)()(),( :becomessolution general theSo
,
:pairs conjugatecomplex be willroots thewith
442
2
1
:solutionwith
02 :equation sticcharacteri get the we
0)2()( :solutionsfor Looking
02
gives dimensions-for twowhich ,0 relations two theseCombining
:law conductionheat cAnisotropi
0 :case statesteady for equation energy Uncoupled
21
2
2
2
2
2
2
22
2
2
,
,
,
Trusted by Thousands of
Students
Here are what students say about us.
Resources
Company
Copyright ©2022 All rights reserved. | CoursePaper is not sponsored or endorsed by any college or university.