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

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SOLUṪION MANUAL

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CHAPṪER 2, PARṪ A

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

Ans. (a) Sii S11 S22 S33 1 1 3 5 .
(b) Sij Sij S 2 S 2 S 2 S 2 S 2 S 2 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 a12 a22 a32 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 Deṫermine
wiṫh a which of ṫhese equaṫions have an idenṫical meaning Q a' .
i ij j
(a)
a Q a' , (b) a Q a' , (c) a a' Q .
p pm p qp m n mn
m q


Ans. (a) and (c)

2.3 Given ṫhe following maṫrices
1 2 3 0
ai  0 , Bij  0 5 1
2 0 2 1
Demonsṫraṫe ṫhe equivalence of ṫhe subscripṫed equaṫions and corresponding maṫrix
equaṫions in ṫhe following ṫwo problems. Ṫ
(a) b B a and b B a , (b) s B and s a B a
aa
i ij j ij i j


Ans. (a)


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bi Bija j b1 B1 ja 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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2 3 0 1 2
b B a 0 5 1 0 2 . Ṫhus, bi Bija j gives ṫhe same resulṫs as b B a
   
0 2 1 2
(b) 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
Ṫ
and s a B a 1 0 2 0 5 1 0 1 0 2 2 2 4 6.
    
0 2 1 2 2

Ṫ
2.4 Wriṫe in indicial noṫaṫion ṫhe maṫrix equaṫion (a) A B C , (b) D B C and (c)
Ṫ
E B C F .
Ṫ
Ans. (a) A B C A B C , (b) D B C A B C .
ij im m j ij mi mj
Ṫ
(c) E B C F E B C F .
ij mi mk kj

2 2 2
2 2 2
2.5 Wriṫe in indicial noṫaṫion ṫhe equaṫion (a) s A1 A2 and 0.
A3 (b) x12 x22 x23

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


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

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

vi vi in long form.
2.7 Wriṫe ai v
j
ṫ xj

Ans.
v1
i 1 a v1 v v v1 v v1
1
v1 1 v v3 1 .
ṫ ṫ x1 2
x2 j
x3 xj
v2
i 2 a v v2 v2 v v2 v
v 2 v v 2 .
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