ME 612 Metal Forming and Theory of Plasticity. 6. Strain


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1 Mtal Forming and Thory of Plasticity mail: Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü
2 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv. Lo is th original lngth and ΔL th lngth chang aftr th load application. Enginring strain: 0 0 Tru or logarithmic strain: d d (6.1) (6.)
3 6.1. Uniaxial Strain In addition to th abov normal nominal strain, on can dfin th nginring shar strain as th chang of angl as shown in Fig. 6.. For small angl chang, w can writ: a b (6.3) Figur 6.. Shar strains ar usd to dfin chang of angls upon application of forcs. 3
4 6.. TwoDimnsional Enginring Strain Figur 6.3. Plan strain involving small distortions To simplify th prsntation w only discuss th dfinition of th twodimnsional strain componnts but an xtnsion to 3D will b apparnt. In Figur 6.3 an infinitsimally small cub is givn (bfor and aftr dformation) with dg lngths dx and dz. Hr th dformation is only in xz plan and th dformation is a function of x and z. 4
5 6.. TwoDimnsional Enginring Strain At point A small strain componnts ar dfind. With th assumption of small dformation; A PA C and tan(pa C ) is thn PA C. From Figur 6.3 u A dx u u u P 1 x AC dx u x dx 1 (6.4) (6.5) With a similar analysis w (6.6) zz z If 3D cas is analyzd: v yy (6.7) y Hr u,v,w ar diplacmnts in x,y,z dirctions. 5
6 6.. TwoDimnsional Enginring Strain Shar strains is associatd with angular distortions shown as angls RA B and PA C. Again with small dformations, PA C angl As u x <<1; arctan w x PA C angl. w dx x AP w dx arctan x u dx dx x (6.8) (6.9) (6.10) A similar analysis for RA B angl: u RA B angl. z (6.11) 6
7 6.. TwoDimnsional Enginring Strain Total shar strain is th sum of ths angls or xz w x u z (6.1) For 3D cas; xy u y v x (6.13) yz v z w y (6.14) It is important to raliz that th g form of shar strains givn in Eq. 6.1, 6.13 and 6.14 is quivalnt to simpl shar strain as masurd in a torsion tst. 7
8 6.3. Th Strain Tnsor Lik strss tnsor a similar form can b usd for strains: ij yx xz xy yy yz xz yz zz (6.15) Th tnsor shar strain is qual to half of shar strain givn abov; xy = 1 xy 1 u y v x (6.16) 8
9 6.3. Th Strain Tnsor Figur 6.4. Illustration showing that pur shar, (a) and (b) is rlatd to simpl shar (c) by a rotation (d). 9
10 6.3. Th Strain Tnsor Avrag rotation of infinitsimally small cub is dfind by w j. For j=x,y,z th qualitis ar givn as; w x w yz 1 w y v z (6.17) w y w xz 1 u z w x (6.18) w z w xy 1 v x u y (6.19) 10
11 6.3. Th Strain Tnsor Figur 6.5. Exampls of strain stats (a) Uniaxial tnsion for an isotropic matrial (b) qual hydrostatic tnsion in th thr Cartsian axs and (c) shar. 11
12 6.3. Th Strain Tnsor Th 9 componnts of strain tnsor ar ncssary to dfin th dformation status of th cub. Strain tnsor is symmtric. Ex: xy = yx. Gnrally x is usd instad of. Principal strain indics ar shown by 1,,3 and thus principal strains ar 1,, 3. Always; yy zz 1 3 (6.0) Trm is corrct. 1
13 6.4. Rlativ Volum Chang in Trms of Strain Componnts Considr a unit cub (dimnsions 1 1 1)) along th principal strain dirctions. Undr loading, th cub will dform to anothr cub of dimnsions (1 + 1 ) (1 + ) (1 + 3 ). dilation; V V is dfind as th rlativ volum chang (1)(1)(1) V 1 3 V x 3 1 y 1 z 3 (6.1) (6.) Not that if th dformation prsrvs volum (incomprssibl dformation), thn; ( 1 )(1 )(1 3) 1 1 (6.3) 1 3 x y z 0 (6.4) 13
14 6.5. Transformation of Strain Componnts in Plan Strain Conditions Figur 6.6. Dformation of a small lmnt with sids originally paralll to x and y axs. u and ν ar hr th displacmnts of point O in th dirctions of th axs x and y, rspctivly. 14
15 6.5. Transformation of Strain Componnts in Plan Strain Conditions Similarly to th transformation quations drivd for th strss componnts, w can driv transformation quations for th strain componnts. Using th notation of Fig. 6.6, w dfin th strains as follows: x x u x (6.5) y y v y (6.6) xy u y v x (6.7) 15
16 6.5. Transformation of Strain Componnts in Plan Strain Conditions Th final strain transformation quations hav th following form: yy yy yy cos xy sin (6.8) yy yy cos xy sin yy xy sin xy cos (6.9) (6.30) Th principal strain dirctions (whr xy = 0) ar found from: tan p xy yy (6.31) 16
17 6.5. Transformation of Strain Componnts in Plan Strain Conditions Similarly, th magnituds of th principal strains ar: 1, yy yy xy (6.3) Th maximum sharing strains ar found on plans 45 rlativ to th principal plans and ar givn by: max yy xy 1 (6.33) Not that th abov transformation quations ar only valid for small strain. W will not nd th transformation quations for th logarithmic strain as w will always try to work on principal strain axs!! 17
18 6.6. Mohr s Circl for Small Strain Figur 6.7. Th Mohr circl for plan strain problms. 18
19 6.6. Mohr s Circl for Small Strain Bcaus w hav concludd that th transformation proprtis of strss and strain ar idntical, it is apparnt that a Mohr s circl for strain may b drawn and that th construction tchniqu dos not diffr from that of Mohr s circl for strss(figur 6.7). In Mohr s circl for strain, th normal strains ar plottd on th horizontal axis, positiv to th right. Whn th shar strain is positiv, th point rprsnting th x axis strains is plottd a distanc γ/ blow th lin, and th y axis points a distanc γ/ abov th lin, and vic vrsa whn th shar strain is ngativ. 19
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