Solutions Manual for Introduction to Continuum Mechanics (4th Edition) by Lai, Rubin & Krempl - Complete, All Chapters

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SOLUTION MANUAL
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Laig etg al,g Introductiong tog ContinuumgMechanics



CHAPTER g 2,g PARTg A

2.1 Given
1 0 2 1
Sijggg 0 1 2 andg a i ggg 2
  g 
  g 
3 0 3 3
Evaluateg (a)g S iig,g (b)g SijgS ijg ,g (c)g SgjigS gjig ,g (d)g S gjkgSkj (e) a m a mg,g (f)g Smngamang ,g (g)g Snmama n

Ans.g (a)g Siig gS11g gS22 g gS 33g g1gg1gg3gg5g.
(b) S ijgS ijg gS g2 g gSg2g gSg2 g gSg2g gSg2g gSg2 g gS g2 g gSg2g gS g2 g 
11 12 13 21 22 23 31 32 33
1gg0gg4gg0gg1gg4gg9gg0gg9gg28g.
(c) S gjigS gjig=gSijgS ijg =28.
(d) S gjkgS kjg gS 1kgSk1 g gS2kgSkg2g gS 3kgS kg3
gS 11 S 11g gS12 gS 21g gS 13S 31g gS 21S 12g gS22gS22 g gS 23S 32g gS 31S 13g gS32 gS23g gS33S33
g 11gg  0  0  gg  2  3  gg  0  0  g g 11g g  2  0  gg  3  2  gg  0  2  g g 33 g g 23g .
(e) a m a mg ga12g ga22 g ga32 g g1gg4gg9gg14g .
(f) S mn gamang g S1na1 ang gS2na2 ang gS3na3 ang 
S 11 a 1 a1g gS 12a 1a2 g gS13a1 a3g gS21a2 a1g gS22 a2a 2g gS23a2 a3g gS31a3 a1g gS32 a3a 2g gS33 a3a3
g 111gg 01 2 gg 21 3 g g 02 1 g g 1 2 2g g  2  2  3  g g 33 1 
  0  3  2  gg 3 3  3g g 1gg 0g g 6g g 0g g 4g g 12g g 9g g 0g g 27g g 59.
(g) S nm ama ng =gSmngaman g =59.

2.2 Determine g whichg ofg thesegequationsg havegangidenticalg meaningg withg a gQg a 'g .
i ijg g j
(a)ga gQ a ' ,g (b)g a gQg g a 'g ,g (c)g a g a 'g Q .
p pmg m p m n mn
qp g
q


Ans.g (a)g andg(c)

2.3 Giveng theg followingg matrices
 
1 2 3 0
aiggg0g ,gBijggg0 5 1
2 0 2 1
Demonstrate g theg equivalence g ofg theg subscriptedg equationsg andg correspondingg matrixg equationsg ing t
heg followingg twog problems.
(a) b g g Bg a g andg bgg Ba ,g (b)g sggBg aga andgsgga Ba 
Tg

i ijg g j ijg ig g j


Ans.g (a)
b i g gBij agj ggb1 ggB1gj a gjggB11 a1 ggB12a 2ggB13a 3gg21gg30gg02gg2
b 2 g g B2gja gjg gB21a1 ggB22a2 g gB23 a3g g2, b 3 g g B3 gj a gjg gB31a 1g gB32a2 g gB33a 3g g2g.

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2 3 0g1 2
bggBag g 0 5 1gg0g gg2g g.g Thus,g big gBijagjg givesgthegsamegresultsgasgbggBa


 0 2 1g2 2
(b)
sggBijga iagjg gB11a 1a1 ggB12a 1a2 g gB13a1 a3g gB21a 2a1 ggB22a 2a2 g gB23a2a3
B31 a 3 a 1gg B32 a3a 2g gB33a3 a3 g g2(1)(1)gg3(1)(0)gg0(1)(2)gg0(0)(1)
5(0)(0)gg1(0)(2)gg0(2)(1)gg2(2)(0)gg1(2)(2)gg2gg4gg6.
2 3 0g1 2
andg sgga Ba gg1g 0 2 0 5 1 g0g  g1g g 0 2g2g g 2gg4gg6g.
  g  g
Tg

0 2g 1g2 2


Writegingindicialgnotationgthegmatrixgequationg(a)g gAggBC,g(b)g DggB C  and (c)
Tg
2.4 g g g



 E   B C F  .
g g
Tg
g g g




Ans.g(a)gA
g  gg B CgggAg gBg C ,g (b)gDggB C  A
Tg
g g g gBg C .
ij igm g g mg j ij mig g mj
(c) EggB C F  E
Tg
g g g g g gBg C Fg .
ij mig g mkg g kj


2 2 2  2  2 2
2.5 Writeg ing indicialg notationg theg equationg (a)g sg g A1 g g A2 g g A3 g andg (b)g 2g g 2g g 2g g0g.
x1 x2 x3

2 2 2  2   2  2  2 
Ans.g (a)g sggA1 g gA2 g gA3g g AigAig . (b)g 2 g g 2g g 2 g g0gg g0g.
x1 x2 x3 xigxig

2.6 Givengthatg Si g j g =ai aj gandgSij g=aiaj g ,gwhereg ai =Qmi gam gandg aj g=Qn gj ang ,gan QikgQjkg g ijg .
d
Showgthatg Si i g=Sii g .

Ans.g Si j g =Qmi amQn g j an g=Qmi Qn g jam an g gSi ig =Qmi Qni aman g= mn am ang =am amg gSmm g gSii g .

vig
2.7 Writeg ai g gv vi ing longg form.
t g x gj
j




Ans.
v
igg1gga g g 1g gv v1 g v1 g v v1 g v
g gvg 1g gv gv3gg 1g .
1
t j
g x gj t 1
g x1
2
g x 2 x3
v2g v2g v2 g v2g v2g v
ig g 2ggag g gvg g gvg gvg gvg 2g .
2
t j
g x gj t 1
g x1 2
g x 2
3g
x3
v3g v3 g v3g v3g v3 g v3g
igg3gga g g gvg g gvg gvg gvg .

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Laig etg al,g Introductiong tog Continuumg Mechanics
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t j
g xgj t 1
g x1 2
g x2
3g
x3




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