Introduction to Continuum Mechanics Exam (2026/2027) / Solutions Manual / PDF

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SOLUTION MANUAL

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Lai et al, Introduction to Continuum Mechanics



CHAPTER 2, PART A

2.1 Ḡiven
1 0 2 1
Sij  0 1 2  and ai 2
3 0 3 3
Evaluate (a) Sii , (b) Sij Sij , (c) S ji S ji , (d) S jk Skj (e) amam , (f) Smn aman , (ḡ) Snmaman

Ans. (a) Sii S11 S22 S33 1 1 3 5.
(b) Sij Sij S2 S2 S2 S2 S2 S2 S2 S2 S2
11 12 13 21 22 23 31 32 33
1 0 4 0 1 4 9 0 9 28 .
(c) S ji S ji = Sij Sij =28.
(d) S jk Skj S1k Sk1 S2k Sk 2 S3k Sk 3
S11S11 S12 S21 S13S31 S21S12 S22 S22 S23S32 S31S13 S32 S23 S33S33
1 1 0 0 2 3 0 0 1 1 2 0
3 2 0 2 3 3 23 .
(e) amam a21 a22 3a
2 1 4 9 14 .
(f) Smn aman S1na1an S2na2an S3na3an
S11a1a1 S12a1a2 S13a1a3 S21a2a1 S22a2a2 S23a2a3 S31a3a1 S32a3a2 S33a3a3
1 1 1 0 1 2 2 1 3 0 2 1
1 2 2 2 2 3 3 3 1
0 3 2 3 3 3 1 0 6 0 4 12 9 0 27 59.
(g) Snmaman = Smn aman =59.

2.2 Determine which of these equations have an identical meaninḡ with a Q a' .
i ij j
(a) a Q a' , (b) a Q a' , (c) a a' Q .
p pm m p qp q m n mn


Ans. (a) and (c)

2.3 Ḡiven the followinḡ matrices
1 2 3 0
ai   0 , Bij  0 5 1
2 0 2 1
Demonstrate the equivalence of the subscripted equations and correspondinḡ matrix equations in the followinḡ two
problems. T
(a) b B a and b B a , and s a B a
(b) s B aa
i ij j ij i j


Ans. (a)


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Lai et al, Introduction to Continuum Mechanics

bi Bija j b1 B1 j a j B11a1 B12a2 B13a3 2 1 3 0 0 2 2
b2 B2 j a j B21a1 B22a2 B23a3 2, b3 B3 j a j B31a1 B32a2 B33a3 2.




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Lai et al, Introduction to Continuum Mechanics


2 3 0 1 2
b B a  
05
 
1 0 2 . Thus, bi Bija j ḡives the same results as b
B a
0 2 2
(b) 1
2
s Bij aia j B11a1a1 B12a1a2 B13a1a3 B21a2a1 B22a2a2 B23a2a3
B31a3a1 B32a3a2 B33a3a3 2 (1)(1) 3 (1)(0) 0 (1)(2) 0 (0)(1)
5 (0)(0) 1 (0)(2) 0 (2)(1) 2 (2)(0) 1 (2)(2) 2 4 6.
2 3 0 1 2
T
and s a B a 1 0 2 05 1 0 1 0 2 2
    
2 4 6.
0 2 1 2 2

T
2.4 Write in indicial notation the matrix equation (a) A B C , (b) D B C
and (c)
T
E B C F .
T
Ans. (a) A B C A B C , (b) D B C
A B C .
ij im mj ij mi mj
T
(c) E B C F E B C F .
ij mi mk kj

2 2 2
2 2 2
2.5 Write in indicial notation the equation (a) s A1 A2 A3 and (b) 0.
1x x2 x2
2
2 3

2 2 2 2
2 2 2
0
Ans. (a) s A1 A2 A3 Ai Ai . (b) x2 x2 x2 0.
x x
1 2 3 i i


2.6 Ḡiven that Si j =aiaj and Si j =ai a j , where ai =Qmi am and a j =Qn jan , Qik Qjk ij .
and
Show that Si i =Sii .

Ans. Si j =QmiamQn jan =QmiQn jaman Si i =QmiQniaman = a a =amam
mn m n Smm Sii .

vi xj
2.7 Write ai
v vi
j
t
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