Answers to Problems - Elsevier BA=r2 0 000 2 4 3 5. 2 Answers to Problems. 2.72 div u آ¼ 3A. 2.73 آ½آ¼ru

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Text of Answers to Problems - Elsevier BA=r2 0 000 2 4 3 5. 2 Answers to Problems. 2.72 div u آ¼ 3A. 2.73...

  • Answers to Problems

    CHAPTER 2 2.1 (b) SijSij ¼ 28, (c) SjiSji ¼ SijSij ¼ 28, (d) SjkSkj ¼ 23, (g) Snmaman ¼ Smnaman ¼ 59. 2.3 (a) b1 ¼ 2; b2 ¼ 2; b3 ¼ 2. (b) s ¼ 6. 2.4 (c) Eij ¼ BmiCmkFkj.

    2.7 i ¼ 1 ! a1 ¼ @v1 @t

    þ v1 @v1 @x1

    þ v2 @v1 @x2

    þ v3 @v1 @x3

    ; etc:

    2.10 d1 ¼ 6; d2 ¼ �3; d3 ¼ 2.

    2.12 (2) For i ¼ k; LS ¼ RS ¼ 0 if j 6¼ l 0 if j ¼ l ¼ i 1 if j ¼ l 6¼ i

    8< :

    2.20 (b) T½ � ¼ 0 0 1

    0 0 �1 �1 1 0

    2 4

    3 5.

    2.21 (c) T aþ bð Þ ¼ 10e1.

    2.22 T½ � ¼ 2 0 �1 0 1 3

    1 3 0

    2 4

    3 5.

    2.23 T½ � ¼ �1=2 0 1=2 �1=2 0 1=2 0 0 0

    2 4

    3 5.

    2.24 (a) T½ � ¼ 1 0 0

    0 �1 0 0 0 1

    2 4

    3 5, (b) T½ � ¼ 1 0 00 1 0

    0 0 �1

    2 4

    3 5.

    2.25 (a) R½ � ¼ 1 0 0

    0 cos y �sin y 0 sin y cos y

    2 4

    3 5, (b) R½ � ¼ cos y 0 sin y0 1 0

    �sin y 0 cos y

    2 4

    3 5.

    2.26 (b) T½ � ¼ 1 3

    1 �2 �2 �2 1 �2 �2 �2 1

    2 4

    3 5, (c) Ta ¼ � 3e1 þ 2e2 þ e3ð Þ.

    2.27 ½T� ¼ 1 3

    1 �2 �2 �2 1 �2 �2 �2 1

    2 4

    3 5.

    Copyright © 2010, Elsevier Ltd. All rights reserved.

  • 2.28 (b) n ¼ e1 þ e2 þ e3ð Þ= ffiffiffi 3

    p .

    2.29 T½ � ¼ 1 3

    1þ 2 cos y 1� cos yð Þ � ffiffiffi3p sin y 1� cos yð Þ þ ffiffiffi3p sin y 1� cos yð Þ þ ffiffiffi3p sin y 1þ 2 cos yð Þ 1� cos yð Þ � ffiffiffi3p sin y 1� cos yð Þ � ffiffiffiffi3p sin y 1� cos yð Þ þ ffiffiffi3p sin y 1þ 2 cos yð Þ

    2 4

    3 5.

    2.30 (b) RA ¼ sin yE.

    2.31 (a) S½ � ¼ 1 0 0

    0 �1= ffiffiffi2p �1= ffiffiffi2p 0 �1= ffiffiffi2p 1= ffiffiffi2p

    2 4

    3 5; (b) T½ � ¼ 1 0 00 �1= ffiffiffi2p 1= ffiffiffi2p

    0 1= ffiffiffi 2

    p 1=

    ffiffiffi 2

    p

    2 4

    3 5, (d) c½ � ¼ 11= ffiffiffi2p

    5= ffiffiffi 2

    p

    2 4

    3 5.

    2.37 a ¼ 2e 01. 2.38 (b) a ¼ e 01 þ

    ffiffiffi 3

    p e 02.

    2.39 T 011 ¼ 4=5; T 012 ¼ �15= ffiffiffi 5

    p ; T 031 ¼ 2=5.

    2.40 (a) T 0ij h i

    ¼ T½ � 0 ¼ 0 �5 0 �5 1 5 0 5 1

    2 4

    3 5.

    2.42 (b) TijTij ¼ 45, (c) T½ � 0 ¼ 2 5 1

    2 3 1

    0 0 1

    2 4

    3 5.

    2.48 (a) TS � � ¼ 1 3 53 5 7

    5 7 9

    2 4

    3 5, TA� � ¼ 0 �1 �21 0 �1

    2 1 0

    2 4

    3 5, (b) tA ¼ e1 � 2e2 þ e3.

    2.50 (d) For l ¼ 1; n ¼ a1e1 þ a2e2 � a1 þ a2ð Þe3½ �= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a21 þ a22 þ a23

    q .

    2.55 y ¼ 120o. 2.56 (c) For l ¼ 1; n ¼ �e3. (d) For l ¼ �1; n ¼ a1e1 þ a2e2; a21 þ a22 ¼ 1, (e) y ¼ p. 2.59 (a) For l1 ¼ 3; n1 ¼ �e3. For l2 ¼ �3; n2 ¼ � e1 � 2e2ð Þ=

    ffiffiffi 5

    p .

    2.60 (a) For l1 ¼ 3; n1 ¼ �e1. For l2 ¼ 4; n2 ¼ � e2 þ e3ð Þ= ffiffiffi 2

    p .

    2.61 For l1 ¼ 0; n1 ¼ � e1 � e2ð Þ= ffiffiffi 2

    p . For l2 ¼ l3 ¼ 2; n ¼ � ae1 þ ae2 þ a3e3ð Þ; 2a2 þ a23 ¼ 1.

    2.65 (b) At ð0; 0; 0Þ; ðdf=drÞmax ¼ jrfj ¼ 2 in the direction of n ¼ e3. At ð1; 0; 1Þ; ðdf=drÞmax ¼ jrfj ¼ 17 in the direction of n ¼ 2e1 þ 3e2 þ 2e3ð Þ=

    ffiffiffiffiffi 17

    p .

    2.67 (a) q ¼ �3k e1 þ e2ð Þ; (b) q ¼ � 3ke1 þ 6ke2ð Þ. 2.69 (a) ½rv�ð1;1;0Þ ¼ 2½I�, (b) rvð Þv ¼ 2e1, (c) div v ¼ 2; curl v ¼ 2e1, (d) dv ¼ 2ds e1 þ e3ð Þ.

    2.71 ru½ � ¼ �A=r2 �B 0

    B A=r2 0 0 0 0

    2 4

    3 5.

    2 Answers to Problems

  • 2.72 div u ¼ 3A.

    2.73 ru½ � ¼ A� 2B=r3 0 0

    0 Aþ B=r3 0 0 0 Aþ B=r3

    2 4

    3 5.

    2.77 div Tð Þr ¼ div Tð Þy ¼ div Tð Þz ¼ 0.

    CHAPTER 3

    3.1 (b) v1 ¼ kx1 1þ kt ; v2 ¼ 0; v3 ¼ 0.

    3.2 (a) v1 ¼ a; v2 ¼ v3 ¼ 0; a1 ¼ a2 ¼ a3 ¼ 0; (b) y ¼ A atþ X1ð Þ. Dy=Dt ¼ Aa, (c) y ¼ BX2; Dy=Dt ¼ 0.

    3.3 (b) v1 ¼ 0; v2 ¼ 2bX21t; v3 ¼ 0 and a1 ¼ 0; a2 ¼ 2bX21; v3 ¼ 0, (c) v1 ¼ 0; v2 ¼ 2bx21t; v3 ¼ 0 and a1 ¼ 0; a2 ¼ 2bx21; a3 ¼ 0.

    3.4 (b) v1 ¼ 2bX22t; v2 ¼ kX2; v3 ¼ 0 and a1 ¼ 2bX22; a2 ¼ 0; a3 ¼ 0, (c) v1 ¼ 2bx22t=ð1þ ktÞ2; v2 ¼ kx2=ð1þ ktÞ; v3 ¼ 0; a1 ¼ 2bx22= 1þ ktð Þ2; a2 ¼ a3 ¼ 0.

    3.5 (b) v1 ¼ k sþ X1ð Þ; v2 ¼ 0; v3 ¼ 0 and a1 ¼ 0; a2 ¼ 0; a3 ¼ 0, (c) v1 ¼ k sþ x1ð Þ= 1þ ktð Þ; v2 ¼ 0; v3 ¼ 0 and a1 ¼ 0; a2 ¼ 0; a3 ¼ 0.

    3.6 (b) For X1;X2;X3ð Þ ¼ 1; 3; 1ð Þ and t ¼ 2; v1 ¼ �4 3ð Þ2 2ð Þ ¼ �72; v2 ¼ �1; v3 ¼ 0: (c) For x1; x2; x3ð Þ ¼ 1; 3; 1ð Þ and t ¼ 2; v1 ¼ �200; v2 ¼ �1; v3 ¼ 0:

    3.7 (a) For X1;X2;X3ð Þ ¼ 1; 1; 0ð Þ and t ¼ 2; v1 ¼ 2k; v2 ¼ 2k; v3 ¼ 0: (b) For x1; x2; x3ð Þ ¼ 1; 1; 0ð Þ and t ¼ 2; v1 ¼ 2k= 1þ 4kð Þ; v3 ¼ 0:

    3.8 (a) t ¼ 2 ! x1 ¼ 5; x2 ¼ 3; x3 ¼ 0, (b) X1 ¼ �3; X2 ¼ 1; X3 ¼ 2, (c) a1 ¼ 18; a2 ¼ 0; a3 ¼ 0, (d) a1 ¼ 2; a2 ¼ 0; a3 ¼ 0.

    3.9 (b) ai ¼ 0. 3.10 (a) a ¼ �4xex � 4yey, (b) x2 þ y2 ¼ constant ¼ X2 þ Y2.

    Or, x ¼ �Y sin 2tþ X cos 2t and y ¼ Y cos 2tþ X sin 2t. 3.11 (a) a ¼ k2 xex þ yey

    � � , (b) x ¼ Xekt; y ¼ Ye�kt. Or xy ¼ XY.

    3.12 Material description: a ¼ 2k2 x2 þ y2� � xex þ yey� �. 3.14 (b) a1 ¼ 0; a2 ¼ �p2 sin ptð Þ sin p X1ð Þ; a3 ¼ 0. 3.15 (b) a ¼ �ða2

    ffiffiffi 2

    p =4Þer; DY=Dt ¼ 2ak.

    3.16 (b) a ¼ �ða2 ffiffiffi 2

    p =4Þer; DY=Dt ¼ 0.

    3.17 (b) ds1=dS1 ¼ ð1= ffiffiffi 2

    p Þ

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ kð Þ2 þ 1

    q ¼ ds2=dS2,

    cos p=2� gð Þ ¼ sin g ¼ f� 1þ kð Þ2 þ 1g=f 1þ kð Þ2 þ 1g.

    Answers to Problems 3

  • (c) For k ¼ 1; ds1=dS1 ¼ ds2=dS2 ¼ ffiffiffiffiffiffiffiffi 5=2

    p ; sin g ¼ �3=5.

    For k ¼ 10�2; ds1=dS1 ¼ ds2=dS2 � 1:005; g ¼ �0:0099 rad: (d) 2E 012 ¼ �0:01.

    3.19 (a) E½ � ¼ 0 0 k=2 0 k 0 k=2 0 0

    " # , (b) 10�5=2.

    3.20 (a) E11 ¼ 5k ¼ 5� 10�4; E22 ¼ 2k ¼ 2� 10�4; 2E12 ¼ k ¼ 10�4rad. 3.21 (a) E 011 ¼ 10�4=3. 3.22 (a) E 011 ¼ ð58=9Þ � 10�4, (b) 2E 012 ¼ ð32=

    ffiffiffiffiffi 45

    p Þ � 10�4rad:

    3.23 (a) E 011 ¼ ð37=25Þ � 10�4, (b) 2E 0012 ¼ ð72=25Þ � 10�4rad: 3.24 (a) I1 ¼ 11� 10�4; I2 ¼ 31� 10�8; I3 ¼ 17� 10�12. 3.25 I1 ¼ 0; I2 ¼ �t2; I3 ¼ 0. 3.26 At 1; 0; 0ð Þ; lmax ¼ 3k ¼ 3� 10�6. 3.27 (a) D dVð Þ=dV ¼ 0, (b) k1 ¼ 2k2. 3.28 (b) At 1; 2; 1ð Þ; E 011 ¼ k; (c) max elongation ¼ 4k; (d) DV ¼ k. 3.32 E11 ¼ a; E22 ¼ c; E12 ¼ b� aþ cð Þ=2. 3.33 (a) E12 ¼ �100� 10�6: (b) For l1 ¼ 261:8� 10�6; y ¼ �31:7o, or

    n ¼ 0:851e1 � 0:525e2. For l2 ¼ 38:2� 10�6; y ¼ 58:3o, or n ¼ 0:525e1 þ 0:851e2. 3.34 (a) E12 ¼ 0, (b) Prin. strains are 10�3 in any direction lying on the plane of e1 and e2. 3.35 E11 ¼ a; E22 ¼ 2bþ 2c� að Þ=3; E12 ¼ b� cð Þ=

    ffiffiffi 3

    p .

    3.36 E11 ¼ 2� 10�6; E22 ¼ 1� 10�6; E12 ¼ ½1=ð2 ffiffiffi 3

    p Þ� � 10�6.

    3.37 E11 ¼ 2� 10�3; E22 ¼ 2� 10�3; E12 ¼ 0.

    3.38 (a) D½ � ¼ 0 kx2 0 kx2 0 0 0 0 0

    2 4

    3 5; W½ � ¼ 0 kx2 0�kx2 0 0

    0 0 0

    2 4

    3 5, (b) D nð Þ nð Þ ¼ 3k.

    3.39 D11 ¼ �a 1þ kð Þ; D 011 ¼ 1þ kð Þ=2. 3.40 (a) D12 ¼ p cos t cos px1ð Þ=2; W12 ¼ �W21 ¼ � p cos t cos px1ð Þ=2.

    (b) D11 ¼ 0, D22 ¼ 0, D 011 ¼ p=2. 3.42 (a) Drr ¼ �1=r2; Dyy ¼ 1=r2; other Dij ¼ 0; W½ � ¼ 0½ �. (b) Drr ¼ �1=r2.

    3.43 At r ¼ 2; ar ¼ �18; ay ¼ 0; (b) D½ � ¼ 0 �1�1 0 � �

    .

    3.44 (a) ar ¼ �ðAr þ B=r2Þ2sin2y=r; ay ¼ �cos y sin yðAr þ B=r2Þ2=r; af ¼ 0. (b) Drf ¼ �ð3B=2r3Þsiny; Dyf ¼ 0.

    4 Answers to Problems

  • 3.45 W½ � ¼ 0: 3.49 k ¼ 1. 3.50 f ¼ g yð Þ=r. 3.51 vy ¼ � k=2ð Þsin y=

    ffiffi r

    p .

    3.53 v1 ¼ f x2ð Þ; v2 ¼ 0: 3.54 (a) r ¼ ro 1þ ktð Þ�a=k; (b) r ¼ r*xo=x1. 3.55 r ¼ roe�at

    2

    .

    3.60 (b) 2kX1X2 ¼ f X2;X3ð Þ þ g X1;X3ð Þ.

    3.62 @r @t

    þ vr @r @r

    þ vy r

    @r @y

    þ vz @r

    @z

    � � þ r @vr

    @r þ vr

    r þ 1

    r

    @vy @y

    þ @vz @z

    ¼ 0.

    3.63 (b) U½ � ¼ 1 0 0

    0 2 0

    0 0 3

    2 4

    3 5; (c) B½ � ¼ 9 0 00 1 0

    0 0 4

    2 4

    3 5, (d) R½ � ¼ 0 0 1�1 0 0

    0 �1 0

    2 4

    3 5,

    (e) E* � � ¼ 0 0 00 3=2 0

    0 0 4

    2 4

    3 5, (f) e*� � ¼ 4=9 0 00 0 0

    0 0 3=8

    2 4

    3 5, (g) DV

    DVo ¼ 6, (h) dA ¼ �3e3.

    3.64 (b) U½ � ¼ 1 0 0

    0 2 0

    0 0 3

    2 4

    3 5, (c) B½ � ¼ 4 0 00 9 0

    0 0 1

    2 4

    3 5, (d) R½ � ¼ 0 1 00 0 1

    1 0 0

    2 4

    3 5,

    (e) E* � � ¼ 0 0 00 3=2 0

    0 0 4

    2 4

    3 5, (f) e*� � ¼ 3=8 0 00 4=9 0

    0 0 0

    2 4

    3 5, (g) DV

    DVo ¼ 6, (h) dA ¼ 3e1.

    3.65 (b) U½ � ¼ 1 0 0

    0 2 0

    0 0 3

    2 4

    3 5, (c) B½ � ¼ 1 0 00 9 0

    0 0 4

    2 4

    3 5, (d) R½ �