continuum mechanics

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


.

.

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)

.



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)



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 .



: 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



(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)



:

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



: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



.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



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)

:



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)



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



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 )(

:



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



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)



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



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:



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



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)

:

.



:

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)



) (.

)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



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



.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



.

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)



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



* :

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

.

:



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 :



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 .



)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)

::



(

()(

: .

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,_

:



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 .

.

:

)Gouss ( .

a:DomainR

S :

)61.3(..R s

dsnaduadiv :

dsnAdvA ..

dvdsx .

:R si

i dsnadvx

a)62.3(11

.A

ijA :

)63.3(R s

iiji

ij dsnAdvx

A



:

ds

dxn

dsdyn

dAy

fdsFn

dAx

fdsFn

y

x

Ay

Ax

/

)64.3(

xy nn ,f .

:

s vAdvVAdsn )65.3(**

A* .

VA)* ( )3.65 ( :

dvX

Vdsvn

i

ii

iS

)3.62 ( .

A=T)* ( :

dvx

TdsTn

S vi

ijiji

)3.63 ( .

:

»«

.



»O «ie^

.Bt=0R .0R

0POX .

RX .t=00R

tR .0,0 tPt

P .

:

)1.4(),( tXxx

ixx :

)2.4(),( txxx Rii

Rx Material Coordinate .

ixSpatial Coordinate .

t=0Reference Configurationt=tCorrent Configuration

.

4.2RXiXt .

:

)3.4(),(

),(1.4

txXX

txX

iRR

RXRe. ConfixCo.conf Corft



.

:

),(,

),(

tXXX

tXxx

iRR

Rii

),( tXxx Rii

),( txXX iRRRe. ConfCor. Conf .

Material CoordinateLagrangian Description

Spatial CoordinateEulerion Descsription .

Eulerion

.

:

:

)4.4(XxU

Lagrangian. DisUt,x :

)5.4(),(),( XtXxtXU

Euleriom .DisUt,x :

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11

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:

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13

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