Force and w ork

Th e great est d efo rm at io n , wh ich is eq u al t o t h e su m o f t h e t wo o t h er d efo rm at io n s, is d esign at ed t h e p rin cip le d efo rm at io n j g : Th e p rin cip le d efo rm at io n m u st be a kn o wn q u an t it y, as it fo rm s t h e basis fo r every calcu lat io n , fo r exam p le o f d efo rm at io n fo rce. It is easy t o d et erm in e, as it carries a d ifferen t sign t o t h e o t h er t wo . In t h e co m - p ressio n o f a cu bic bo d y, fo r exam p le, t h e in crease o f wid t h b 1 b an d len gt h l 1 l resu lt s in a p o sit ive sign , wh ile t h e d ecrease o f h eigh t h 1 h p ro d u ces a n egat ive sign Fig. 2.2.1 . Acco rd in gly, t h e abso lu t e great est d efo rm at io n is alo n g t h e vert ical axis j 1 . Sim ilar t o t h e su m o f d efo rm at io n s, t h e su m o f d efo rm at io n rat es j m u st always be eq u al t o zero : Th e flo w law ap p lies ap p ro xim at ely: wit h t h e m ean st ress s m given by Th e m at erial flo w alo n g t h e d irect io n o f t h e st ress wh ich lies bet ween t h e largest st ress s m ax an d t h e sm allest st ress s m in will t h erefo re be sm all an d will be zero in cases o f p lan e st rain m at erial flo w, wh ere d efo rm at io n is o n ly in o n e p lan e.

2.2.3 Force and w ork

In calcu lat in g t h e fo rces req u ired fo r fo rm in g o p erat io n s, a d ist in ct io n m u st be m ad e bet ween o p erat io n s in wh ich fo rces are ap p lied d irect ly an d t h o se in wh ich t h ey are ap p lied in d irect ly. Direct ap p licat io n o f fo rce m ean s t h at t h e m at erial is in d u ced t o flo w u n d er t h e d irect ap p li- 28 Basic principles of metal forming ϕ ϕ ϕ ϕ 1 2 3 = − + = g . ˙ ϕ 1 + ˙ j 2 + ˙ ϕ 3 = ϕ ϕ ϕ σ σ σ σ σ σ 1 2 3 1 2 3 : : : : , = − − − m m m σ σ σ σ m = + + 1 2 3 3 Metal Forming Handbook Schuler c Springer-Verlag Berlin Heidelberg 1998 cation of an exterior force. Th is req u ires su rfaces to m ove d irectly again st on e an oth er u n d er p ressu re, for exam p le wh en u p settin g an d rollin g. In d irect ap p lication of force, in con trast, in volves th e exertion of a force som e d istan ce from th e actu al form in g zon e, as for exam p le wh en th e m aterial is d rawn or forced th rou gh a n ozzle or a clearan ce. Ad d ition al stresses are gen erated d u rin g th is p rocess wh ich in d u ce th e m aterial to flow. Exam p les of th is m eth od in clu d e wire d rawin g or d eep d rawin g. In t h e d irect ap p licat io n o f fo rce, t h e fo rce F is given by: wh ere A is t h e area u n d er co m p ressio n an d k w is t h e d efo rm at io n resis- t an ce. Th e d efo rm at io n resist an ce is calcu lat ed fro m t h e flo w st ress k f aft er t akin g in t o acco u n t t h e lo sses arisin g, u su ally t h ro u gh frict io n . Th e lo sses are co m bin ed in t h e fo rm in g efficien cy fact o r η F : Th e fo rce ap p lied in in d irect fo rm in g o p erat io n s is given by: wh ere A rep resen ts th e tran sverse section area th rou gh wh ich th e force is tran sm itted to th e form in g zon e, k wm is th e m ean d eform ation resistan ce an d k fm th e m ean stability factor, both of wh ich are given by th e in tegral m ean of th e flow stress at th e en try an d exit of th e d eform ation zon e. Th e arith m etic m ean can u su ally be u sed in p lace of th e in tegral valu e. Th e referen ced d eform ation work w id is th e work n ecessary to d eform a volu m e elem en t of 1 m m 3 by a certain volu m e of d isp lacem en t: Th e sp ecific fo rm in g wo rk can be o bt ain ed by grap h ic o r n u m erical in t egrat io n u sin g available flo w cu rves, an d in exact ly t h e sam e way as t h e flo w st ress, sp ecified as a fu n ct io n o f t h e d efo rm at io n j g . Figure 2.2.2 illu st rat es t h e flo w cu rves an d relat ed wo rk cu rves fo r d if- feren t m at erials. 29 Basic terms F A k A k A w wm g fm F g i d F = ⋅ ⋅ = ⋅ ⋅ = ⋅ ϕ η ϕ η w k d k i d f fm g g = ⋅ ≅ ⋅ ∫ ϕ ϕ ϕ η F = k k f w F A k w = ⋅ Metal Forming Handbook Schuler c Springer-Verlag Berlin Heidelberg 1998 If t h ere is n o flo w cu rve available fo r a p art icu lar m at erial, it can be d et erm in ed by exp erim en t at io n . A t en sile, co m p ressive o r h yd rau lic in d en t at io n t est wo u ld be a co n ceivable m et h o d fo r t h is. If t h e sp ecific d efo rm at io n wo rk w id an d t h e en t ire vo lu m e V o r t h e d isp laced vo lu m e V d are kn o wn q u an t it ies, t h e t o t al d efo rm at io n wo rk W is calcu lat ed o n t h e fo llo win g basis:

2.2.4 Formability