Deformation Processing - Drawing

Transcription

1 eformation Processing - rawing ver.

2 Overview escription Characteristics Mechanical Analysis Thermal Analysis Tube drawing

3 Geometry b α a F b, σ xb F a, σ xa 3

4 4

5 Equipment 5

6 Cold rawing 6

7 A. urer - Wire rawing Mill (489) (copper wire) 7

8 Characteristics Product sizes: (5µm) to several inches (00-50 mm) Mostly cold (T < 0.4 T melting ) below recrystallization point Small diameter (wire): uses a capstan iameter > inch (5 mm) (rod): bull blocks on a draw bench length up to 40 feet ( m) 8

9 Characteristics Fine wire done through several dies Speeds large diameter: 30 feet per minute (9 m/min) small diameter: 300 feet per minute (90 m/min) fine wires: 5,000 feet per minute (60 mph 00 km/h) 9

10 ie Materials Large diameter high carbon steel high speed steel Moderate diameter tungsten carbide (WC) Small diameter diamond inserts 0

11 Characteristics Lubrication Coatings Oil ie angle (α) typically small: 4-6 o

12 Mechanical analysis (round wire / rod) Reduction in area (RA) b α a F b, σ xb F a, σ xa RA b b a a b ε t ln RA ln b a

13 Slab analysis µp p + d σ x + dσ x dx σ x Assume p, σ x are uniform µp p OK for small α, µ 3

14 Equilibrium π x x σ x 4 π dx π dx p sinα + µ p cosα cosα ( σ + dσ ) ( + d) + π 4 cosα 0 Expanding π x x 4 π dx π dx + p sinα + µ p cosα cosα ( ) ( σ + dσ + d + d ) σ x cosα 0 π 4 4

15 π 4 σ Equilibrium [( ) ( )] σ + σ d + σ d + dσ + dσ d + dσ d x x π 4 x x small small small π dx π dx + p sinα + µ p cosα 0 cosα cosα x x x Eliminating higher order terms, dividing by & π, multiplying by 4 and canceling σ d dx dx x + dσ x + 4 p sinα + 4µ p cosα cosα cosα 0 5

16 Equilibrium d/ α dx Noting d tan α dx dx d tanα sinα cosα σ d d d x + dσ x + 4 p + 4µ p cosα tanα cosα tanα 0 or σ d d x + dσ x + pd + µ p tanα 0 6

17 Equilibrium Finally µ σ x d + d σ x + p + d tanα 0 7

18 Maximum shear stress (Tresca) criterion σ + p τ σ x flow flow µ B tanα p τ flow σ x 8

19 ifferential form d dp 4 τ flow + pb d dσ x Bσ 4τ x flow ( + B) 9

20 Integrating b xb x Bσ 4τ + a d σ σ xa x d σ flow ( B) 0

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22 rawing stress σ + B B xa a xb a flow B b flow τ τ b + σ B where: σ xb back stress (tension) σ xa pulling stress (tension) b σ xb α a σ xa

23 Strain hardening (cold below recrystallization point) For round parts - Tresca τ flow σ flow Y K ε n n + average flow stress: due to shape of element 3

24 Strain rate effect (hot above recrystallization point) τ flow σ flow Y C & ε m For a round part (derived for extrusion) ε & 6v b b 3 b tanα ln average strain rate due to shape of element v b velocity of b side A area 3 a A A b a 4

25 Value for p or p Y σ p τ flow σ p τ flow σ x maximum at entrance 5

26 Effect of back tension die pressure drawing stress without back tension with back tension entry exit 6

27 Maximum RA Solve previous equations with: α 6 o (typical value) µ 0. B σ xb 0 For failure: draw stress material flow {yield} stress σ σ xa Kε n τ flow σ flow Y n Kε n + here, say K 760 MPa, and n 0.9 7

28 8 Maximum RA Yields RA 0.6 must be solved for each µ, α, σ xb + + B b a n n B B n K K ε ε b a

29 Energy / unit volume (u) u F V / A a V σ xa (with no back stress) V volume 9

30 30 Rod/Wire rawing Analysis Ideal deformation External work Work of ideal plastic deformation for ( ) ( ) t t d f f d d u L A u L A t ε σ σ σ ε 0 n t Kε t σ + f f t f t n t d A A Y Y n K 0 ln ε ε ε σ

31 Rod/Wire rawing Analysis Ideal deformation rawing force, F d σ d A f rawing power, P d F d V f Source: S. Kalpakjian & S. Schmidt, 4 th ed

32 rawing Limit Ideal deformation of a perfectly plastic material σ σ d ε o Y ln Y f A o Ao σd σε ln e A f Af Maximum reduction per pass Ao Af % A e o A A 3

33 rawing Limit Ideal deformation of a strain hardening material σ σ d ε A o Kε Y ln A f n + o f ( n+ ) n+ σ σ ε n + d n Kε Maximum reduction per pass A A e A o ε σ ideal+friction σ d n+ ideal σ d σ ε ε 33

34 Example Problem Assuming zero redundant work and frictional work to be 0% of the ideal work, derive an expression for the maximum reduction in area per pass for a wire drawing operation for a material with a true-stress strain curve of σkε n Total work Ideal work + frictional work + redundant work Total work Ideal work + 0. x Ideal work. x Ideal work Or, Total work of deformation. [u x volume] () In drawing, external work of deformation σ d x volume () Equating () and (), we get σ d.u or σ d. ε 0 σ dε t t. ε 0 n Kε dε t t. n+ Kε n + σ d.yε where A ε 0 ln (3) A f 34

35 35 Example Problem Max reduction occurs when total drawing stress, σ d Flow stress of material at die exit, Y max reduction per pass. ln... n f n f f n n n d e A A A e A A n A A n K n K K Y Y ε ε ε ε ε σ

36 rawing - Ex. - etermine power, and plot σ x and p along die length. rawing steel rod from φ 3 mm to φ m/s K 760 MPa, n 0.9 µ 0., α 4 o, σ xb 0 36

37 rawing - Ex. - First, we must see if we can do the process, the limit is σ σ xa max Kε RA -( a / b ) 0.5 5% ε t ln{/(-ra)} ln {/(-0.5)} 0.6 B µ/tanα 0. / tan 4 o.43 n 37

38 38 rawing - Ex. -3 B b a flow xb B b a flow xa B B + + τ σ τ σ + n K Y n flow ε τ n σ Kε max

39 39 rawing - Ex. -4 So, equating the equations (with no back stress) yields + + B b a B B n min a

40 rawing - Ex. -5 Solving gives a-min 8.53 mm, so we can do the process and proceed with the analysis 40

41 rawing - Ex. -6 σ τ xa flow n ( 0.6) 0.9 Kε 760 τ flow Y 446MPa n

42 rawing - Ex. -7 σ xa 0.35 x τ flow 0.35 x 446 MPa 56 MPa F draw σ xa x Area 56 x π(/) 7.6 kn 3938 lbf Power F draw x speed 7.6 kn x.5 m/s 6.4 kw 35.4 hp 4

43 rawing - Ex. -8 imensionless pressures (divided by τflow) sx p diameter (mm) 43

44 Limits on analysis Larger die angles more redundant work σ, p, u will be larger than predicted b α a 44

45 Redundant work d m /L d m ( a + b ) / p Q r σ flow L (contact length) b α a 45

46 Redundant work factor (Backofen) (frictionless) Q r 46

47 Temperature rise (-r)q l o v o / 6 k rq k w, ρ w, c w WIRE IE 47

48 Temperatures θ θ o + θ s + θ f θ o ambient (room) temperature θ s temperature rise in the wire due to plastic shear energy, u s θ f interface temperature rise due to frictional energy, u f 48

49 Specific energies u u s + u f u σ xa u s τ flow ε 49

50 Specific energies From the example above (steel rod): u σ xa 56 MPa u s τ flow ε 446 * MPa u f u - u s MPa 50

51 Shear temperature (θ s ) Since the shear strain is uniform in the wire and all the shear energy remains in the rod as heat Then, we can obtain the shear temperature in the wire: θ s ρ u w s c w 5

52 Material properties For this material: k w 60 W/m-K ρ w 7850 kg/m 3 c w 500 J/kg-K α w.53 x 0-5 m /s For a WC die: k 4 W/m-K 5

53 Shear temperature (θ s ) θ s ρ u w s c w o 8. C 53

54 Frictional heat (Q) Q represents all heat generated by friction Q u f π v 4 v velocity (-r)q goes into the die 54

55 ie and wire temperatures (θ) For the die (steady): θ θ o + ( r) π For the wire (moving): k Q l ln o ref: Carslaw and Jaeger θ θ + θ +.07 r Q α w o s π lk w l v 55

56 Q calculation Q u f π 4 v Q 435 W π

57 imensions mm from o / 6 o 7 mm in this example l contact length reduction in radius / sin α 0.5 mm / sin 4 o 7.7 mm α l dr 57

58 ie temperature θ θ o + ( r) π k Q l ln o θ 0 + ( r) π ln ( r) 58

59 Wire temperature θ θ + θ r Q +.07 π lk α w o s w l v θ π r r

60 Heat flow ratio and Temperature Equating the previous equations yields: r 0.99 hence θ 56 o C 49 K T melt 500 o C 73 K So θ/t melt 0.5, cold (below recrystallization point 60

61 6 Temperature in practice In practice, r all heat goes into wire l v c u c u w w w f w w s o + + α ρ ρ θ θ 0.9

62 Tube drawing 6

63 Tube rawing 63

64 Plane strain / Slab analysis σ τ xa flow B * + B * B * µ die + µ mandrel tanα tan β t t f i B * α semicone angle of die β semicone angle of plug 64

65 Tube rawing Special Cases B * µ die + µ mandrel tanα tan β Fixed mandrel- same friction at both interface (plane tube is modeled as a flat section) B * µ tan α Moving mandrel No friction at interface of mandrel and tube (plane and slab) Moving mandrel with friction towards exit, takes into account motion between mandrel and tube (B may be negative) (plane) Fixed mandrel (slab circular tube) B * B µ tan α µ B * µ tanα * die mandrel µ µ tanα 65

66 Summary escription Characteristics Mechanical analysis Thermal analysis Tube drawing 66

67 67