# 02 - CHAPTER 2 STRAIN Displacement f time=0 5‘ reference...

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Unformatted text preview: CHAPTER 2 STRAIN Displacement _ f; time=0 5‘ reference conﬁguratlon The reference conﬁguration is the conﬁguration at time 0. The current conﬁguration is the conﬁguration at time t. X : material coordinate. » ’\ -—v r ~ I') \l xzspatial coordinate. 7g w> *iumt‘i‘m‘f‘ a7 .4; yr} u: displacement. u=x—X=x(X,t)-—X h I I l "' Ti? Err: rim/{MAL 7 L, 2-1 (1) General Deformation ~/ A/ W159“) Carr" mo n N = dX/|dx| (2) # n = dx/|dx[ (3) dx=F-dX (4) F: deformation gradient tensor. dxi = Fijde (5) F._ = axi(X’t) :9 J3"; & {kw/Xi) 0} V' . —— I ’« ax: (6) ax}. NJ Define / / dso =lXm . I (7) dS =[dx[ (8) x1 = dS / dS0 = stretch ratio 1 . - . , , , , ,r r .L to, / \. < ., k .V ~- n fit/9Q f‘3”5{:'-'V’J2"”" ﬁf/‘Q ’C' ' '9 > I" 3 ’ r' 3‘ r '1“ch -' F“ A ,, I : ., '. t. v v - . '. ' ' /. , r /{ O A alum/r. , V. #4., _ a j .H‘ , :) I, V/ .— Infinitesimal gSmalh Strain Tensor Elementar Deﬁnition of Strains 2-D _ I ______ —-' A112 l I ~ X2 5 AXZ l L—J L_ X1 A)(l A111 Aul - / ; . -" 8 =-—- A .» ~, (9) 11 AXI : w ;_, i A » V v' ” 3 , ‘ Au2 ‘ ' 522 = (10) AX 2 12 1 2 AXI AXZ a» \V ‘ \il-‘w 1 812 = 5712 (12) As AX 1 , AX 2 —> 0 a ’ ~’ ” H - J7» 17/- ~ :/)— if"? “#537; 611 = 5%- T“ T>KLY‘I‘{'!-7J2 ‘ " VIVJ’". ’-€ 7 "‘f (/il-J/ ~ - I 1 A yk raw-afw far” Fret/v15, auz 8 = (14) 22 3X2 V 1 au2 aul 1 5 = _ — + = _ 15 12 2 6X2 2 9/12 ( ) 812 : mathematipglﬁgﬁgiﬂtﬁn of thgvs‘he‘ar strain 712 {engiﬂﬁring‘dgﬁnition of thashgargtrain t): T- !— L ’ ’ 5 3? r ’1 7 ’ 2-3 o 9' X NZ; {5* ﬁrm 0,9 .T0g(w exfmmﬂan. ( aux AZ /p€ISPlﬂLC€W1/\t MA‘CMb AMA}: ) ’ WA“! .9le 3% ’ . . . <1 a 5 1'14: lgf’jfrr/ . —_ H 4-. V7 ' InfimteSImal Straln Tensor 3-D F I I A > C a 7’6 "327’ '5 ,1“:- ‘ 77*0‘1" X3 u” = u.Pe. ‘ (16) ug =uerJ. (17) ( >\$wd§ 5.2:,macla ‘/ s“: <’ (“m aui ~ But. \J ' ‘ ‘ " " ' ’ u. . c c: ' ' (18) "’ —3x: = F77/v “if :41, j »: displayergent gradieptfa u? —uip =ufj(xjg —xf) ' - (19) J J Au;7=u?—uf,Ax.=x?—x’.’ (20) Aui : rélative displacements Aui = um.ij (21) 1 1 = (uhj + it”) + -2— (ui‘j — MN _ M WA 7 ‘vf/ » , / if“! = (517+ cub.)ij (23) y Cf)?" 7‘3 07‘ 1:) 5‘ 5) L i",/ {Jyf ¢ / 2—4 8i]. : inﬁnitesimal strain tensor. my. : rotational tensor. 1 8i]. =—2-(ui,j +14”) 1 2-2-(ujj =8-. ‘9 Jl PM» jumpy/fife”. 81.}. are symmetric. #81 1 812 813 [8]: 821 £22 £23 831 855" €33 is has 6 components. a)” -E(ulj —-uj,i) 1 =_—i(u“_ui,j) —_ r? _ £0}, 7:, :68, 0 £012 [(0] = _ [012 0 . " (013 _ 5023 ‘ to has 3 components. h \ 2-5 (24) (25) (26) (27) (28) (29) (30) (31) 22; JTsFlMemw-t growmQI/x‘t 3X: 'tV‘amsla-‘tTOrx 0-9 r}wa Loo/Law rcmoves 13")“ hot FOfa-tToh, rm 0 {To n W“ Move staf‘kn I ,\ Rotational Tensor Deﬁne g = iei = £191 + 5262 + §3e3 where (012 = ‘8}!2353 e i 7f ' (Mr/.3,” 17' . Therefore, (0.. = —8.. 1,” 4kg" Au; : my A23 Aui = (6,). my.)ij 'r‘T-C‘: 1’»? r }/C V_ _, (Ant. =60”.ij =—8ijk§kij ' = gikjgkij M" m ﬂ m 4 6}— Au’ =Aui’ei / = gikjgkijei : gkjigkijei = f x Ax "Y A Y : I? on «V; A; /‘._,, N 1 — :1: A 3 A2”; .3) 2-6 .7“, U ‘ #9! (32) r (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) _ (43) (44) (45) (52> 6”“ W6'W”T“8 F m“ 9—wme 021/- ii)— 5/14 » dk/XDQEQ, i; I I 7C; : A\ ‘8 T/ J 9y 02b 2*) 01% /L—\7Q [2/]: [OZJEEJCOQTJ “(3 I 752 ﬁx” Feasa ﬁndfx3< Izzjfosa *SM& (03\$; 7 0 45w Strain Transformation 13-D! 311 512 513 [3] = 521 822 623 (46) 831 532 533 [a]: aZ'l a'2’2 a'2’3 (47) I 5i; = ai’kaj’lgkl y (48) 01' [8’] = [a][a][a]’ <49) Strain Transformation 12-DL [£]=[811 812:] (50) 821 822 ' [ ]_ cos6 sin6 (51 a _ —sin6 00819 ’ ) 2—7 So 01‘ = [—2 ii ll |——‘I [ 811 821 8116 + Ens — 811s + 6126‘ 812 C 822 S .65] 812C + 8223 C ‘S —812s+822c s c 81ch + (522s2 +2£lzsc 2 2 —(£11—822)SC+812(C —s ) 81'1 = 8“ cos2 6 + £22 sin2 6 + 2612 sin 6cos 6 51,2 = ‘(8n - 822 ) sin 6 cos 6 + 8,2 (cos2 — sin2 6) 6:2 = 611 sin2 6 + £22 cos2 6— 2512 sin 60056 a. 8 +8 8 _8 11 22 + g c0529+glzsln26 2 2 I./~/r q A Vi; M0170” ad" mgr; age. ‘ nah ,- a 8 . WW __ 11 22 Sln2€+ 812 (:0326 ———( 811—822 2 jcos 26 — 812 sin 26 2-8 ‘ 2 2 —(£11—822)sc+612(c —s ) SHSZ + 82202 — 2812SC y; (52) (53) (54) (55> _ (56) (57) (58) (59) (60) (61> ’ (62) (63) Mohr’s Circle / 1.. L9! ~ ’xv. ". / (2/ ,x {/77 Ex ‘ "if: ; / " « V, Tim" L \ ‘Cw’ w . * U / 63k \ I JP, f m 3, {WW writ: MW '" ﬂ . <i"\ a 1 Ir, "L E» 4 <3??? 3‘ . H ,. ( . _. 1 8' —€ 2 E R: (———“ 22] +8122 (64) The directions denoted by 1 and 2 are the principal directions where only the normal strains exist. 2 1 811 " 522 2 EmaXJnin '—_'2‘(£11""5‘22)i 2 +512 28 tan 249p = —12— 511 _ 822 Example 2-1 Three strain gages a, b and c in the three directions as shown 7 ‘I {j Gage a = ill/in = 8a = 611 A V inﬁgwwwﬁ ‘E:MWM02' ,x‘ Gage b = 0.0025 in/in = 8b = 81,1 W a ‘ ,9. ‘ Emmy” . - I I .21" ' ﬂea/2’ Gage c /= 0.0005 1n/ 1n 2 EC = 822’ L -.:. I} - / ,, ‘ 7 “If, r k )- Find 8'12 and the principal strains in-the x-y plane. a +8 8 —8 I” ‘ . 51,1 = ——'H 22 + —“ 22 cos 26 4— 812/‘s1n26 2 2 \J 812 = 0.00101 in/in 2-10 (67) (68) . , . V .. 1 E —8 _ u 22 Emaxmin — “(811+ 822) i— [ 2 2 = 0.00251 or 0 Mohr’s Circle (0.001250) ‘ (0.0025, _) A‘(0.00251,0) 81.1 60°=20 (0.002,-0.00101) . V En ineerin Strain and True SMn 1-D model Tensile bar . . . V L— L0 englneermg stam: e = y’xv' ') 1 1 “ -.' 3 ¥‘(. ‘ v E I.» /v‘ . :._‘-> k_» _ 2 +29122 , L0 is the reference gage length. in .r r/ -«" 2-11 i", 5.; ' ‘ “I ’ ' ' ‘ - dl . true straln: deﬁne the d1fferent1al true stram as d8 = 7 Where I IS the current length (69) (70) 37/ 3/1 9-0 r rrr 5 x A 4 Z , dl 8 = d8 = — 71 f l I ( ) = In I = In L — 111 L0 / d}'_,’,£ R4 7'2- 55 r" Ingmar : " .‘ , 3r :ﬁmﬁi I (72) L I _1 73 8— n[ j ' r ( ) ~ L' dln[_lll;] 4W, A, VIP/,0” ’ro/ _ 0 _1n[L0]|L=LD + dL |L0 tL—Lo) . A}. F (av-{Vijij 1 d2 / V p Y 0 2 + — —-—— L — + . . . 74 2 w I“ ( L°) - ( ) L— 1 L— 2 = Lo___( 210) +... (75) i9 2 1° ; ' 1 2 . 6:6—‘e + 7‘: 3 lax/j: * P7 1 S 7- 9‘ e 9 2 ’ ase—>O,8:—: yan" ‘ a L4 1 > I ’ IJ C A /{“:f: x: |/_ "in \ J #7174 A I'y / k0”? Example 2-2 75v Amie, Ari} 90 m €970 sf’émfn C rubber If L=2L0, 8:? :13?" '/ 2L ' e =/ 1n{——°—] =1n 2 = 0.693 (77) 0 e = 2% = 100% (inappropriate from the small strain approach) ' (78) e isonly a good approximation for strain up to a few percents. 1,. i I ‘ ii I?! N V :_ V. To induce the same magnitude of strain in compression: (3,: o 1&5,ny , 3M ‘ 5: “23/ 2 J 2-12 T / _ g 8 =1n££~J = —1n 2 = —0.693 L0 L = 0.5L0 Example 2-3 L0=50mm, L=60mm, e=?, 8:? e= 60—50 =0.2=20% so a =1n i =ln(éj = 0.182 = 18.2% 10 5 1 (a e —Ee2 = 0.2 —o.02 = 0.18) |e| > Isl in tension. Example 2-4 L0 =50mm, 8=—0.182, L=? 8 = —0.182 = ln(£) = ln(§) 50 6 L = 41.67mm L—L0 L0 = —O.167 = —16.7% e: lel < l8| in compression. 2-13 (79) (80) (81) (82) (83) (84) (85) (86) ...
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