continuum mechanics
DESCRIPTION
handout of continuum mechanicsTRANSCRIPT
http://arak-master-mechanics88.persianblog.ir 1
C O N T IN U U M M E C H A N IC S
-
)(
:
http://arak-master-mechanics88.persianblog.ir 2
.
.
m nA:
A =Aij =
mnmmm
n
n
AAAA
AAAA
AAAA
.....
.....
.....
.....
.....
321
2232221
1131211
Aij ij.
I=1000m
J=1000n range convention Aijij
3×31×3
s ector .A B C
a bc.
Transpose matrix
.A :
A = AT Aij =Aji (2.2)
:
Antisyy metric matrix A= -AT Aij = -Aji (2.3)
Skiew syymmetric
unit matrix
I = ij , I,j =1,2,3 (2.4)
.
http://arak-master-mechanics88.persianblog.ir 3
0...
1
31231312
332211 (2.5)
:
I =100
010
001
Kronecker -Delta
1
0ij ji
ji
:Substitution rulen
.
3
1332211
jIKKIKIKIJKij AAAAA
(2.6)
ikkj
ikikijkij 3
3
132211
3
1332211
jkikikikikjij AAAAA
Trace :
Trace .
A )3×3 (:
Exp:
A =
333231
232221
131211
AAA
AAA
AAA
Trace
A + A11 +A22+A33=
3
1iiiA
tr (A)=3
1iijA (2.7)
http://arak-master-mechanics88.persianblog.ir 4
3
1
3)(i
ii Itr
(2.8)
3×3 A
) A(det .:
det (A)= KTJSirrstijktsrkjI
AAAee3
1
3
1
3
1
3
1
3
1
3
16
1(2.9)
eijk permutation symbel Alternative symbel
eijk = ejki = = 1
ejik = ekji = =-1
eiik =ejjk = = 0
det A 0A .A
A-1 :
AA-1=A-1A=I (2.10)
Q )Orthogonal (:
QT =Q-1 (2.11)
Q :
Q.QT =QT.Q =I (2.12)
:
det Q = 1 (2.13)
det Q
= 1.
:
detT
QQ =det 1
det Q .det Q T =1 {det Q T }2=1
det Q = 1
Q1 Q2 Q1.Q2 .
http://arak-master-mechanics88.persianblog.ir 5
: Summation Convention
.
)(
123 ..
)2.7 ()I (
)I (:
Tr A=3
1iiiA
Tr A = Aii =A11+A22+A33
)I (no summation on i
2.6:
3
1iikJKij AA
ikjkij AA
2.8:
ii =3
)2.9 (.
det A =6
1eijk ers +Air Ajs Akt (2.14)
:
ifij
ij
B
A
B
A (a
i
JA,B :
A.B=3
1ijkijjkij BABA Aik Bkj
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(b a B=AT Bij =Aji ijA.AT
:
A.AT = Aik Ajk
A Q )2.12 (:
Q.QT=I
ij
ij
kj
jk
ki
ik
Q
Q
Q
Q (2.15)
(cx y :
X=AY (2.16)
A .:
XI =Aij yj (2.17)
{X}=(A){Y}
(d Trace A.B I
k a :
AB=Aij Bjk
Tr(AB) = Aij Bji (2.18)
ABC =
ABC = Aij Bjk ckm
Tr (ABC) = Aij Bjk cki (2.19)
ij e :
eijp eijq = 2 pq
eijp ersp = ir js
is jr (2.20)
ktkskr
jtjsjr
itisir
= eijk .erst (2.21)
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:
iA eA i
, jB eB j
jj
ii
jiJi BA
BAeeBAij
BA. sub.rule
eI ej=eijk ek j I
k
A B AI eI Bj ej =AIBj eijk ek
B C . D
= (BI eI CJ ej ).DL eL =Bicjeijk ek .DLeL
=Bicjeijk kLDL
=BiCjDLeijk Kl
=BiCjDkeijk
=BiCjDLeijL
:
B C D321
321
321
DDD
CCC
BBB
=333
222
111
DCB
DCB
DCB
=BiLjDk eijk
:: |A|=eijk A1iA2jA3K=eijk Ai1Aj2Ak3)3×3 (
:
|A|=
333231
232221
131211
AAA
AAA
AAA
kk
jj
ii
DA
CA
BA
3
2
1
|A|=
321
321
321
DDD
CCC
BBB
A =eijk BiCjDk =eijkA1I A2j A3k Kl DL
BiCjDL eijk Kl =BiCjDkeijk
http://arak-master-mechanics88.persianblog.ir 8
:det[Aij] =
333231
232221
131211
AAA
AAA
AAA
A11(A22 A33
A32 A23)-A21(A12A33
A32A13)+ A31(A12A23
A22A13)
=A11(e1jk Aj2AK3) A21(- e2jkAJ2 Ak3)+A31(e3jk Aj2Ak3)
=Ai1 eijk AJ2 AK3 = eijk Ai1 Aj2 Ak3
: 321
321
321
kkk
jjj
iii
eijk = that Show
:
iti
isi
jri
A
A
A
3
2
1
321
321
321
ttt
sss
rrr
=eijk ri sj tk =erst
:
erst |A|=eijk Ari Asj Atk
)2.14 ()2.21 (:
erst|A|=eijk Ari Asj Atk empq detA=eijk Aim Ajp Akq (2.22)
: Dummy Inde
D. I .
Aii=Ajj
.
Free Index
.
:
AX = X (2.23)
AX
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.A 3×3.
:
( A - I )X=0 (2.24)
I
.)2.24 (man trivial solution:
det ( A - I )X=0 (2.25)
characteristic.
)2.25 (
)123 ,,((eigen value) A .
.
:
A=402
050
207
|A- I|=402
050
207
[(7- I)(5- I)(4- I) ]
(4-(5- I)) =0
- 3I3 +16 2I2 - 79 I +120 =0
( A - I )X=0
=
8402
0850
2087
3
2
1
X
X
X
=0
n (14
2,0,
14
1)
-X1 2X3 =0 X1=-2X3
3
2
1
X
X
X
3
3
0
2
X
X
-X3 =0 X2 =0 -2X1 4X3 =0
http://arak-master-mechanics88.persianblog.ir 10
5=
5402
0550
2057
3
2
1
X
X
X
= 0
n(5
1, 0,
5
2)
-2x1
2x3=0
-2x1
x3 = 0 x3 =-2x1
3=
3402
0350
2037
3
2
1
X
X
X
= 0
n(5
1, 0,
5
2)
4x1 2x3 =0 x1=L
x3
2x2 =0 X2=0 -2x1+ x3 =0
1 )2.24 (: ( A - 1I )X=0
x(1) ..
x(1)
eigen vector A eigen vector1
.
E.Vec x(2)x(3) 32
.
32 1 )2.25 (3)1- ()2.25 (
:
det ( A - I ) =( 1- )( 2- )( 3- ) (2.26)
= 0 :
detA = 1 2 3 (2.27)
:
http://arak-master-mechanics88.persianblog.ir 11
A E.Va E.Ve
.
:
x(1)
1
)1(X:
AX (1) = 1 x(1) (2.28)
)2.28 (:
)1(x A = 1 X
T)1(
(2.29)
)2.28 (X(1)T)2.29 (X(1) :
( 1- 1 ) X(1)T. X
(1)= 0 (2.30)
X(1)2.240
X(1).X
(1)T
11.
E.Va E.Va .
)2.28 (t)2(x :
x(2)T AX(1) = 1 x(2)T X(1)
(2.31)
X
X(1)A x(2) = 2 x(1)T x(2) (2.32)
2.31:
x(1)TA x(2) = 1x(1)T x(2)
2.32:
( 1 - 2) x(1)T x(2) (2.33)
1
2E.Va
x(1)T. x(2)=0 :
x(r)T x(s) =0 r s (2.34)
E.Ve :
x(r)T. x(r) =1 (2.35)
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E.VeNormalize .
3×3 P E.Ve Normalizex(3) x(2)
x(1) :
P=
x(3)
x(2)
x(1)
PT=(x(1) x(2) x(3) ) (2.36)
)2.34 ()2.35 (:
PPT = 1
P orthagonad .)2.28 (X(2)
X(3) :
APT=(AX(1) ,AX(2) ,AX(3) )
=( 1 X(1) ,
2 X(2) , 3 X
(3)) (2.37)
)2.35 ()2.36 ()2.37 (:
PAPT =
3
2
1
00
00
00
(2.38)
PAPT . )2.23 (:
AX= X A2X= AX , 2X (2.39)
A 2
A2
X .n
Anx
det A =0
.n .
: The Cayley Hamilton Theorem
)2.38 (:
Tr (PAPT)= 1+ 2+ 3 :
Tr(PAPT)2 =23
22
21
http://arak-master-mechanics88.persianblog.ir 13
P artogonal)2.15 (:
Tr (PAPT)=Tr (PijAjkPkm)=PijAjkPki =PijPkiAjk
= jk Ajk = Ajj = Tr A
E.Va .
Tr (PAPT)2 =Tr(PAPT.PAPT) =Tr (PA2PT)
Pij Ajp Apk Pki = kj Ajp Apk =Akp Apk= Tr A2
:
223
22
21
321
TrA
TrA
(2.40)
)2.25 ()2.26 (:
3
( 1+ 2+ 3) 2+( 2 3+ 3 1+ 1 2) -
Adet
321 =0
2.272.40:
3
Tr A 2 +2
1{(Tr A)2
Tr A2}
det A =0 (2.41)
:
:
A3
(Tr A)A2 + 2
1A[(Tr A)2
Tr A2]
I det A =0 (2.42)
.
: The Palar De Composition Ther
APositive definite )P.d (X
XTAX.A )P.d( E.VaA
.Fnon
singular )(
:
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F =R . U , F=V .R (2.43)
R orthogonalV,U )P.d (
3×3:
3×3 F C =FT.F , X = FX
X .C .
C = FT .F
Cik =Bij Ajk jiij
T
jkjk
FBFB
FAFA
Cik =Fji Fjk
I k
C.
X TCX = XTFTX = X
T X
X
.TXX
XTCXC )P.d (E.Va
21
22
23
123 .
)2.28 (PT E.Ve
normalizeCP orthogonal:
PCPT =23
22
21
00
00
00
U:
U =PT
3
2
1
00
00
00
P (2.44)
U 123 )P.d (P
http://arak-master-mechanics88.persianblog.ir 15
Orthogonal:
U2 = PT
23
22
2
00
00
00
P= C (2.45)
R R= F U-1 R
Orthoogonal)2.43()2.45 (:
RTR = U-1FT. FU-1 =U-1C U-1 =U-1U2U-1 =I
R
Orthoogonal.
V:
V = RURT
RUV
F
3×3
.
0e1
e2
e3
(Bace Vector)
.
eI :
a =aieI (3.1)
a = iiaa =23
22
21 aaa
(3.2)
a .b =aibI (3.3)
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ei.ej= ij (3.4)
ab ab
a b sin
:
a b =321
321
321
bbb
aaa
eee
permutation syn :
a b= aibjeijkek ei ej =eijkek (3.6)
:
(a b).c=
321
321
321
ccc
bbb
aaa
(a b).c=eijk aIbjek .em em =eijk aIbjcmekem =eijkaibjck (3.7)
: Coordinate TransFormation
.
.
.
0 0
P 0:
0
0
X
X
X
X
X
X
e1
e2
e3
.aI
http://arak-master-mechanics88.persianblog.ir 17
a.
0e1
e2
e3
.aaI
ia.:
a=aiei= iaie
(3.8)
ieej mij :
m ij= ie
. ej (3.9)
ej:
ie = mij ej (3.10)
mijOrthogonal
)3.4 (
:
ie
ie = ij
)3.10:
ij = ie je
=mir er . mjs es = mir mjs er es = mir mjs rs
= mir mjr= ij = ie je
(3.11)
ijmjr mir m
Orthogonal.
)3.11 (m ij )3
(:
m.mT =1
e j3.10:
http://arak-master-mechanics88.persianblog.ir 18
ej = m ij . ie
(3.13)
(3.10)
m ij ie = m ij m ij ej Tr
a orthogonal
(3.13)
)8-3 ()3.13 (:
ia
ie = aieI =ajej = ajmij ie ia
= jmij (3.14)
aia
orthogonalm . )3.8 ()3.10 (:
I =mji ja
(3.15)
ax
P 0 :
ix =mij xj
XI=mji jx
(3.16)
)ba. (
.:
a.b = Ia Ib
=ajmijbkmik=ajbkmijmik=ajbj (3.17)
a b= Ia Ib eijk ke = eijk(Mipap)(Mjpbq)([krer)= eijk(MipMjqMkr)
)3.18( eijk ie je ke =eijkeiejek
Invariant form .
: TheDyadic Product
.
a b
a =aiei
b = bjej
http://arak-master-mechanics88.persianblog.ir 19
a b = aiei
bjej = aibjei ej = aibjei ej (3.19)
:
:
( a) b =a ( b)= (a b)
a (b+c)=a b+a c (3.20)
(b+c) a=b a + c a
:
a b=aiei bjej=aibjei ej (3.21)
)3.21(I nvariant farn.
:
a b = ia jb ie je
=Mip ap Mjq bq Mirer Mjses
=Mip Mir Mjq Mjs ap bq er es
=ar bser es (3.22)
)3.20 (:
cbacba
cbacba
).().(
).().( (3.23)
:
cbacba
cbacba
).(.
).(. (3.24)
)3.24 (:
a b.c =aiei bjej .ck ek = aibjckei ej.ek=aibjcjeI (3.25)
:
.
http://arak-master-mechanics88.persianblog.ir 20
:
A dyadic :
A = Aijei ej (3.26)
Aij A .
.
ieA ijA.:
A =Aijei ej = ijA ie je
(3.27)
)3.13 (:
ijA ei ej= ijA Mpi pe Mqj qe =
pqA
MMA qipiij pe qe
= pqA pe qe = ijA ie je
pqA = Mpi Mqj ijA
ijA =Mip Mjq Apq (3.28)
:
A =TMAM
)3.28 ()Trans formation (.
n:
A=Aij .meI ej em (3.29)
)3.29 (:
pqA t......... =Mpi Mqj Mtm Aij m (3.30)
http://arak-master-mechanics88.persianblog.ir 21
) (.
)3.28(:
pqjqip
qjpiij
AMMij
pqMMA
A
A (3.31)
)3.30 (:
Aij ..m =Mpi Mqj Mtm pqA t........ (3.32)
: Rule Quotient
Cijk
bIeI b= )3.25.(
A=Aijei ej pqA pe qe
A )Aij=Aji ()3.28 (:
qpA =Mqi Mpj Aij =Mpj Mqi Aji = pqA
(3.33)
trans formation.
:
(Aij)T = Aij (3.34)
A=Aijei ej , AT =Ajiei ej =Aijej eI
:
)(2
1)(
2
1 TT AAAAA
(3.35)
) : (
ejeI iij : .N Isotropicn :e
http://arak-master-mechanics88.persianblog.ir 22
13 .3 :
ieMijej (3-13)
)103(Mijejie
jiijsrrs
srsirisrsjriij
ssjrriijjiij
eeee
eeMMeeMM
eMeMeCI
Iij
.
.PIP .
.
:
kjiijk eekee
kjiijk eekePe
P
.
:
1. lkjiklij eeee
2. lkjijeik eeee
3. lkjijkil eeee
:
)27.3()( lkjijkiljlikklij eeee
.
pqijkpqijk ee ,
iAijpqCjpqi eA
http://arak-master-mechanics88.persianblog.ir 23
.ijpqCij
.
:
jpiqjqippqijijpq cbaC
:
kpqijkpqijijpq eeC )3)2)1
:
ii eaa .jiij eeBBie .
ijij eeM
a) (B____
, iij aB :
rsjsirij
pipi
BMMB
aMa__
__
jkiijk BaC :
kjiijk eeeCC
C_
ieijkC__
. :
____
)38.3( jkirspksjripprsksjripijk BaBaMMMCMMMC
CaB
.
Ba
) : (
Contraction :
kjiijk eeeCCijkC
http://arak-master-mechanics88.persianblog.ir 24
.
rstktjsirijk CMMMC_
k,j .
)39.3(__
rssirrststirrstjsirijj CMCMCMMC
rssC .
rsqpDij .........
n ).
(rsqpDij .........
)n-2 ( .
ijAiiA .
:
kjkikijjki
kjjkii
eBaeBa
eeBeaba ...1(3.40)
kiikijjkkkjiij
iikjjk
aeBBeaeeB
eaeeBaB ...2
B :
a.B = B.a
jijiiiijT
fliilfijlkkljiij
LjkkLijLkjikLijLkkljiij
eeAeeAA
eeaeBAeeBBeeAA
eeBAeeeeBAeeBeeABA
.,.
..)).((..3
ljilij
ljikklijlkijklijT
eeBA
eeBAeeeeBABA ...4(3.41)
.
0ikmik e)
immksiksee 2)
http://arak-master-mechanics88.persianblog.ir 25
tiskijtsikjiistijk eeeeee)
0kjijk AAe)
)..( A)
)..( P
)
AAA **)().()
1A :
(3.42) IAAIAA ..,. 11
1AA
TAA 1
AOrthogonal .
:
(3.44) RVF
URURF kjikij
.
.
:
ijAe i
__
ijA
__
ie
:
jiji eMe__
.E.Va__
A :
0)det(__
IA
A :
(*) TMM
MAMAT
T
http://arak-master-mechanics88.persianblog.ir 26
* :
TTT MIAMMIMMAM )(det)(det
)det(
)det()det()det()det()det( 2
IA
IAMMIAM T
,.VaEE.Va .
AE.Va321 ,, AAA
.E.VainitePositveDef
.
E.VaA321 ,, AAA .E.Ve
E.Va)3()2()1( ,, XXXNormalize
.
3,2,11)( iX i
P :
),,( )3()2()1( XXXPT
)1(jij XP
:
)46.3(
00
00
00
3
2
1__
A
A
A
PAPA T
E.Ve
) . :
. (
E.VaAA
.
:
http://arak-master-mechanics88.persianblog.ir 27
33
32
31
23
22
21
321
AAA
AAA
AAA
47.3
.321 ,, AAA
A .
)3.46 ( :
321
__
)( AAAATr
trAAAAAPPA
A
A
A
PAPA iirrrssrsisirijT
_
3
2
1__
00
00
00
)48.3(
Tr(A).
:
22_
2
__2_23
22
21
)()(
)49.3(
.
ATrATr
TrAAAAAPPPP
APPAPPAAATrAAA
rpprrspqkrkqisip
rsiskrpqkqipkiik
)49.3(
:
33_
TrAATr
47-3 :
)50.3(33 TrATrATrA
123 ,, AAA :
3213
3231212
3211
AAAI
AAAAAAI
AAAI
)51.3(
123 ,, AAA2I :
http://arak-master-mechanics88.persianblog.ir 28
22
2__
23
22
21
23212
)(2
1
)((2
1
)()(2
1
TrATrA
ATrATr
AAAAAAI
3I )3.46 ( :
)det(det_
3TPAPAI
APAP T det)det()det()det(
321 ,, III :
AI
TrATrAI
TrAI
det
)(2
1
3
222
1
)3.42 ( :
)53.3(0322
13 IIAIAIA
Tr :
)54.3(3
12
21
33 TrAITrAITrAI
:
Deviatoric Tensor :
A :
)55.3(3
1ITrAAA
ATranceTranceDeviatoric
ADeviatorieA .
Deviatoric .
http://arak-master-mechanics88.persianblog.ir 29
)56.3(3
1ITrAAA
23 , II :
33233
2222
3
1)(
2
1
2
3
3
12
1)((
2
1
)57.3(ATrATrATrATrATrI
ATrATrATrI
:
Vecter & Tensor Calculuonse :
)321 ,, XXX ( :
nex
ex
gread
ex
ex
ex
grad
ii
ii
)58.3(33
22
11
constont )=321 ,, XXX (
Ctexxx ),,( 321
.
iieea :
3
3
2
2
1
1
..
x
a
x
a
x
a
x
a
iji
x
ja
je
ja
ie
ix
aadiv
i
i
Curl a :
)60.3(*
321
321
321
aaa
xxx
eee
eeaaCurl ixj
akijk
Problem Set (I)
::
http://arak-master-mechanics88.persianblog.ir 30
(
()(
: .
Ctex
u
x
uE
EET
i
j
j
iij
ijijkkij
),(,2
1
2
:
kk
iiijjj
EIe
jujjuiIe,),(
F tU 1
.
fC
2
21
12
221
2
x
uC
t
u
:rRr .
22
2
)1()()
)1(.)
0)(*)
)3().()
nn
nn
n
nn
RnnRd
RnnRc
rRb
RnrRa
:
xx,_
:
http://arak-master-mechanics88.persianblog.ir 31
ijiji bxMx
B :
kiki BMB
jx :
M
kmjik
j
m
m
kik
j
i
x
BMM
x
x
x
BM
x
B.
j
i
x
B .
.
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http://arak-master-mechanics88.persianblog.ir 32
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http://arak-master-mechanics88.persianblog.ir 35
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http://arak-master-mechanics88.persianblog.ir 36
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http://arak-master-mechanics88.persianblog.ir 38
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http://arak-master-mechanics88.persianblog.ir 39
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http://arak-master-mechanics88.persianblog.ir 40
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http://arak-master-mechanics88.persianblog.ir 41
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http://arak-master-mechanics88.persianblog.ir 42
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http://arak-master-mechanics88.persianblog.ir 43
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http://arak-master-mechanics88.persianblog.ir 44
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http://arak-master-mechanics88.persianblog.ir 45
)5.6(
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http://arak-master-mechanics88.persianblog.ir 46
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http://arak-master-mechanics88.persianblog.ir 47
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http://arak-master-mechanics88.persianblog.ir 48
:
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http://arak-master-mechanics88.persianblog.ir 49
3213
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http://arak-master-mechanics88.persianblog.ir 50
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http://arak-master-mechanics88.persianblog.ir 51
(6.15) :
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http://arak-master-mechanics88.persianblog.ir 52
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http://arak-master-mechanics88.persianblog.ir 53
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http://arak-master-mechanics88.persianblog.ir 54
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http://arak-master-mechanics88.persianblog.ir 55
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http://arak-master-mechanics88.persianblog.ir 56
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http://arak-master-mechanics88.persianblog.ir 57
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http://arak-master-mechanics88.persianblog.ir 58
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http://arak-master-mechanics88.persianblog.ir 59
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