Deformation and material flow

Th e valu e o f t h e flo w st ress d ep en d s o n t h e m at erial, t h e t em p erat u re, t h e d efo rm at io n o r st rain , w , an d t h e sp eed at wh ich d efo rm at io n o r st rain rat e is carried o u t , w . Belo w t h e recryst allisat io n t em p erat u re, t h e flo w st ress gen erally rises wit h in creasin g d efo rm at io n , wh ile t h e t em - p erat u re an d d efo rm at io n rat e exert o n ly a m in im al in flu en ce. Excep - t io n s t o t h is ru le are fo rm in g t ech n iq u es su ch as ro llin g an d fo rgin g, in wh ich ext rem ely h igh d efo rm at io n rat es are u sed . Abo ve t h e recrys- t allisat io n t em p erat u re, t h e flo w st ress is gen erally su bject t o t h e t em - p erat u re an d d efo rm at io n rat e, wh ile a p revio u s d efo rm at io n h ist o ry h as o n ly m in im al in flu en ce. Th e flo w st ress gen erally d ro p s wit h in - creasin g t em p erat u re an d d ecreasin g d efo rm at io n rat e. Acco rd in gly, DIN 8582 d ifferen t iat es bet ween m et al fo rm in g p ro cess- es in vo lvin g a last in g ch an ge in st ren gt h p ro p ert ies an d t h o se in vo lvin g n o ap p reciable ch an ge in st ren gt h p ro p ert ies, p revio u sly d esign at ed as co ld an d h o t fo rm in g. In t h e t em p erat u re ran ge bet ween , d efo rm at io n in vo lves o n ly a t em p o rary ch an ge in t h e st ren gt h p ro p ert ies o f t h e m at erial. In t h is case, t h e d efo rm at io n sp eed is h igh er t h an t h e reco v- ery o r recryst allisat io n rat e. Recryst allisat io n st art s o n ly aft er co m p le- t io n o f t h e fo rm in g p ro cess. Th e ru les o f m et al fo rm in g wit h last in g ch an ge in t h e st ren gt h p ro p ert ies ap p ly in t h is case. Th e DIN 8582 st an d ard also breaks d o wn t h e p ro cess acco rd in g t o fo rm in g wit h o u t h eat in g co ld fo rm in g an d fo rm in g aft er t h e ap p lica- t io n o f h eat h o t fo rm in g. Th ese t erm s sim p ly sp ecify wh et h er h eat in g d evices are n ecessary. Un like t h eir fo rm er m ean in g, t h ese t erm s are n o t p h ysically relat ed t o t h e m at erial co n cern ed . Th e flo w st ress o f t h e in d i- vid u al m at erials is d et erm in ed by exp erim en t s in fu n ct io n o f d efo rm a- t io n o r st rain an d d efo rm at io n rat e o r st rain rat e at t h e vario u s t em - p erat u re ran ges, an d d escribed in flo w cu rves. O n e o f t h e u ses o f flo w cu rves is t o aid t h e calcu lat io n o f p o ssible d efo rm at io n , fo rce, en ergy an d p erfo rm an ce.

2.2.2 Deformation and material flow

Act u al d efo rm at io n w , also called lo garit h m ic o r t ru e st rain , is given by: 26 Basic principles of metal forming . ϕ 1 1 1 = = ∫ dh h h h h h ln Metal Forming Handbook Schuler c Springer-Verlag Berlin Heidelberg 1998 in wh ich w 1 is d efo rm at io n in o n e p rin cip le axis an d w 2 an d w 3 in t h e o t h er t wo p rin cip le axes. Th is eq u at io n will give, fo r exam p le, t h e am o u n t o f co m p ressio n in a bo d y wit h h eigh t h Fig. 2.2.1 . w is calcu - lat ed fro m t h e co m p ressio n relat ive t o t h e st art in g m easu rem en t e o r fro m t h e relat ive d efo rm at io n in wh ich h st an d s fo r t h e h eigh t o f t h e bo d y befo re co m p ressio n an d h 1 t h e fin al h eigh t o f t h e bo d y aft er co m p ressio n : In acco rd an ce wit h t h e law o f vo lu m e co n st an cy, acco rd in g t o wh ich t h e vo lu m e is n o t alt ered by t h e d efo rm at io n p ro cess Fig. 2.2.1 , t h e su m o f all d efo rm at io n valu es is always eq u al t o zero : 27 Basic terms ε 1 1 = = h h h h h – , ∆ ϕ ε 1 1 1 1 = = + ln ln h h ϕ ϕ ϕ 1 2 3 + + = V = l b h = V = l b h 1 1 1 1 F h l b h 1 l 1 b 1 Fig. 2.2.1 Dimensional changes in frictionless upsetting of a cube Metal Forming Handbook Schuler c Springer-Verlag Berlin Heidelberg 1998 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