Chapter 1 Fundamentals in Elasticity

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1 Fs s ν

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14 ρ ρ I s s o o o

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17 7.7. Roo of s ss Roo of s w Ooo Ι Dff Oo P z z

18 8 os oo ss os os os ss oo ss

19 9 Ds

20 .. Ds Ds o s s s s s s α s s G s s. α α α

21 s s α α α α s s s s. G s s of s α α α α α α α α α α α α α α α α α α ] [ ] [ ] [ α α α α α α α α α α α α s fs s so foo fo s s << I s s s of s s.

22 R o oo : s s s s fo. fo wo o. Fo s s. s : γ fo oo oo ss D : α α α α

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24 .. o -D Pos I s s f s os q s f oss s? W oos s q o o so? wo so s : f f γ s f o f!!!

25 .. G s Ω Ω Ω Ω Ω Ω Ω Ω Ω F Fo oo F F ] [ ] [

26 6 ] [ ] [. so.- s o qos U U U R R R fos R U U U R U U U R

27 7 os Ros

28 8.. os Lw Gz Hoo s Lw Go qos so s q : ρ -s o : os w : q qos s of s ρ o oos Ds : o o : o o s of s

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30 s I I s II II soo oo I II fo. -O Ro s I s II s I I s II II soo oo I II fo. 9 Roo

31 ssos R. off o N-N N-O.N. - ssos ss s os. oso Roo of ss q q Fo soo q q q q q q q [ [ [ q q [ q q q ] ] ] q q q q q q q ] q q q q q F Ro: wo off λ µ

32 L s os λ µ λ µ D ssos λ λ µ µ µ λ µ λ λ µ µ µ λ λ µ λ λ λ µ µγ µγ µγ λ λ µ q qo s of L os ρ λ µ ρ λ λ µ µ µ µ ρ λ µ µ ρ ρ

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34 τ τ τ ν ν [ ] os γ s γ s γ γ γ τ ν τ γ τ Gγ ν G ν

35 oo

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37 7 Mo Rs M ~ ~ ~ w ~ Q M ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Q M oo fo q Q q q Q q q Q q M ~ Q M Q M

38 8 P Q q M Q M q M W oos so ssf o os o o s oso of??

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40 sso o sss o zz?? ν zz ν zz ow s-s o Posso o so zo w s q o Posso ff. M z I M q I q M I so w fo s o 6 q 8 q 8 6

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42 oso ~ γ G GI Q f G s G γ w G γ M G I G I I G G q

43 6 P Pos s oo s z

44 6.. P ss z sso ss of s s o o o sos of. No z- o o s o sf. o of sss o ss s. o o oo O ± : ± ± ± ± z z z zz sso z z zz q qo

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47 7 6.. P z sso If z-o w so. I o -o No z- o o s o sf. -Ds Ro z zz z z z z z z z os w zz zz ν zz ν γ ν ν ν ν ν

48 8 6. Poo ss Fos o oo w/o o fos ss oos o o os sss ss O

49 9 L sss ss - o o : ss : os o sss s sss Fo o :

50 Mo. sss Mo N. sss Ff o f f :

51 F oo off of oo so s so o o oos ssf. os o sf w s s sf o.f. s o. s f s o s f. - s P I soss f o os o of o s q o so ffs o sss so o os wos s fo s o o s w os of.

52 6.. L o o oos o ± ± ± o sso ss sos o. o oo o ± ± q ss P P P P I P I M I

53 Ds γ P ν I νp ν I P G IG Io P νp f I I f s s P I νp f I P f IG P I P I f νp I f νp I f P IG f P P IG IG P F F IG F F so os. P f νp f P F F I I IG Io of F F P νp P f f 6I 6I 6IG Ds oos P I νp I νp P 6I 6IG P 6I

54 Ds o oos o P I P I νp P 6I 6IG νp P 6I 6IG P I νp 6I P 6IG ow oo s oss. νp P ν νg ν 6I 6IG ν -Ds o νp P νp P I 6I I 6I o P W w sss soo s o ssf!!! W??? Qso o D P w o oss o o oo fo IG o oos so os o o sf q o o os. How o s o oos q o o oos of o sf o oos fo s o. If so w o o s o oos?? w s foo s o oos. D o s so. P 6I P 6I

55 P P P I I I P P I GI P νp P P P I 6I 6IG I GI νp P P P I 6I I I fo of soo νp I P / I ν.% fo. ν. P νp P P P I 6I 6IG I GI P νp P I 6I 6GI soo P P I I P P /.7% GI I fo. ν. P P ν / % 6I I P P P I I I P P I GI P νp P I 6I 6IG P P P νp P 6I I I I GI P GI

56 6.. 6 o o oos. o - q q. o. o -. o oos. so o o o os.. so -s w s o so o fo w. o of fos. Fo M oo. Fo M oo Fo M oo

57 7. Fo M oo Fo M oo. Fo M oo F ssos of ss oos o oos o oo q o oo o sos qos I q q I q q q

58 8 s q oos s q oo : ± I q s q oo : ± I q I q I q f sso fo sss I q I q I q I q s q oos: q ± q I q

59 6.6. s oo 9 X s X os f I s s fo o os fo s s s f s αf o qo fo o s fo. α f α f f G soo ss oos f os α s α os α s α s α[ α α os α α s α α α os α α s α] s α α os α s α[ os α s α os α s α] α osα[ α s α α os α α s α α os α] α os α s α o o oos o s α o s α Io oss s α αos α α s α α os α αs α α s α α αs α α s α α α os α α s α α

60 Fo offs 6 q s q s Rf o. -6 of o of s oso.

61 6 7 P Pos Po oo s

62 7.. Po oo ss 6 ss os s z z os s s os s os os s sfoo os s s os ss os s s os z z z z zz os s s os

63 6 z z z z z z os s os s s os z z s os s os s s os s Ps Do of q qo. F F os os s o s s

64 6 o Ds s os?? os s?? os s?? ss Fo os os?? s???? s s s os os s s s os s s os s os os q qo s s???? os s s os s os s??

65 6 Ds os s s os z os s z z z z z zz s os os s s os - z z z z z z os s os s s os z z s os s os s s os s s os os s os s s Go o of s oos H F H F D D F

66 66 / ' F F F / / o

67 7.. Go qo o oos q ss Fo o 67 oos os Lw Ds o oos s Ro o o oo

68 oos Po oo s o qo I s s fo of o oo of o qo D D ss oos - If o o ss o of oos - sso w s o s- fo.

69 Ds sss 69 ν ν [ ν ν ν ] ν [ ν ν ν ] f ν f ν [ ν ν ν ] f f f f f f f f f f f f f F f H s K os so of o soos o o q.. ν [ ν ν ν ] H s K os H os K s F R o so os s s os K H

70 7 R o oo F

71 7 7.. P Po o O : O : o o o o ss oos o o o o Ds ] [ ν ν o o o

72 7 oos P s o zz zz ν ν P sss o o zz zz ν ν os w R P o o zz zz zz z F ν ν o zz o zz ν o ss : > Ds : ] [ ν ν

73 7 7.. ss w ss: << ] [ ν ν If Ro w ss: / / ] [ / ] [ ν ν ν ν s ν If Ro w L Ro o o o o ] [ ν ν o o Ro o o ν

74 7 o ] [ o o o ν ν ν ] [ ] [ ν ν ν ν ν o o o ] [ o o o ν ν ν o ν ν o ν ν ν o o

75 M M o oo O : O : Mo M ss oo o o Mo oo M D M

76 oo w N ss. M N M N M N 76 M N M N Ds ν [ ν ν ν ] H s K os H os K s F Ds.. w o. / F H o : α o oo 8 α α 8 Mo q o os M α N 8 αn M 6

77 7.7. Ro Ds 77 ω f fo : ρω q qo ss s ρω ν ν ν ν ν ν ν ν q qo s of s ρω - qo G oo ρω [ ν 8 ]

78 ] 8 [ ] 8 8 [ ρω ρω ρω ν ν ν ν ρω ν ρω ν ν ν ν o s 8 8 ρω ρω ρω ρω ρω 8 ρω Ds w o ρω ρω ρω ρω ρω ρω ρω ρω s /.9 / s / f N oo fo o Ds w wo ff owo

79 7.8. P w o 79 ss o o f s os s s os s os s os s os os s s oo fo fo sss o >> oo fo o-fo sss f o : os s ss Fo : f os

80 8 o qo D D f D H K D H K K K H K f f f D f f f D F F F F F F F f f f F f f f os os ss oo s 6 6 os 6 os 6 D D s D D D D s D

81 8 F oo os os s ss oo o os : : osso : s o o - - o Oos os s s os os os os os ±

82 7.9. F soo -D osssq oo 8 o oo o ± q oo / / os P fo. f f f f f f os s s of sss / os / os P / / ss oos Po P os ss oos s P P os os os P s os s P s os s os

83 8 ooo I If ξ f f-ξ Mo s R f ξ ξ ξ R f ξ ξ ξ R Pf Pf P f f f M M f f

84 8 8 Ufo oso. oso

85 8.. ss 8 z Ds os α ϕ osϕ osα osϕ s αs ϕ osϕ s αs ϕ α α s α ϕ s ϕ s α osϕ osαs ϕ s ϕ s α osϕ α α α z : oo W fo z ψ oos zz z z z z ψ z γ γ γ z z z z z z z z z ψ ψ z z ψ ψ z

86 86 ss oos z z zz ψ G ψ G q qo z-o o z z z zz ψ ψ ψ ψ G o f o o sf fo z z z fs wo s o ssf z z z z s s ψ ψ s s ss fo z z ψ G ψ G ψ o os s s G s s os o o.

87 87 s of z z M z z X X Go qo : G o oo : ws o : M

88 oso o so ss fo G G G G G ws G G G M G M M ss M M z z W fo G M G M f G M G M ψ ψ ψ ψ f f M M M M ψ

89 89 ψ M ψ ψ Fo so w os o o

90 8.. R o s oo 9 Go qo G s: o ± ± s so OD -o os Y s o ± ssf. / G G os K K K o s o ± Y ± os 8 Y Y G / 8 / Y Y G fo K / Y s os G ± s os G G / os / / Y / os / G os /

91 9 F oo / os / os os / G K ss / os / os s 6 / G z K / os / s os 6 / G z K M ss 8 7 / os 6 / os 6 / os s 6 / L K K K z G G G G ws 96 7 / 9 / 6 / os / s 6 / os / os os 6 / L K K K K K G G G G G M Now so / G G M L

92 q so.8.79 / M G L G G 9 O so w os ss M G w

93 9 9 Mos

94 9 9.. Po Dfo ν Go qos o oos q qo : os Lw : D : -Ds Ros : Ds o oo : o o o oo : o s Ro o o : o oso Π Π

95 9 Fo s o of s s f o. - 6 of o of s oso o Po Π

96 P of M Po D D D D D D D D D D D D D D Π Π Π Π Π Π D s os f D > fo. fo D > fo D ff.??. q s os o fo Π Π

97 P of Wo If foow q s fo α of wo os. υ Π α Π RR RR ss fo RR D D D D D D D υ α α α α α α α α α α α Π α fo ss If of wo os of o os s ss o qo of of o ss. o so of of wo s fo ss

98 R-Rz Dszo oo L P of M Po Π D q q f K D D D D D D q q q q fo Π P of Wo fo fo f K D D D q q q q q q q q Π Π L

99 99 M Fo Wo sso γ γ γ Ds N M L γ γ γ

100 M L ss-s s Ro D D γ γ γ ν F s qo D N N f K N N D o ss fo

101 9. soo s f of s s ss s 6 ν ν ν ν 6 6 D N ν ν ν ν ν ν ν ν ν ν ν ν ν ν z z K P P f P

102 ν ν ν ν ν P P I P I P ν ν ν 6 P I P I ν ν P I I M I M I P 6 τ ν

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