Numerical Modeling and Simulation of the Movable Contact Tool-Worpiece and Application in Technological Processes

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1 Numrcal Mdlng and Smulan f h Mvabl Cnac l-wrpc and Applcan n chnlgcal Prcsss Ln KUKIELKA Agnszka KULAKOWSKA Radslaw PAYK Facul f Mchancal Engnrng Kszaln Unvrs f chnlg Śnadckch Kszaln Pland ABSRAC Grndng mbssng burnshng hrad rllng drawng cung urnng ar vr cmplcad chnlgcal prcsss. ncras h qual f h prduc and mnmz h cs f prcss w shuld knw h phscal phnmna whch xs durng h prcss. h phnmna n a pcal ncrmnal sp wr dscrbd usng a sp-b-sp ncrmnal prcdur wh an updad Lagrang s frmulan. h chnlgcal prcsss ar cnsdrd as gmrcal and phscal nn-lnar nal and bundar prblms. h fn lmn mhd (FEM) and h dnamc xplc mhd (DEM) wr usd ban h slun. h applcan was dvlpd n h ANSYS/LS-DYNA ssm whch maks pssbl a cmplx m analss f h phscal phnmna: sas f dsplacmns srans and srsss. Numrcal cmpuans f h sran and srss hav bn cnducd wh h us f mhdlg whch rqurs a prpr dfnn f h cnac zn whu h ncss nrduc bundar cndns. Exampls f calculans ar prsnd. Kwrds: chnlgcal Prcsss Updad Lagrangan Frmulan FEM DEM Numrcal Mdlng Numrcal Smulan Gmrcal Cnac Cndns.. INRODUCION h chnlgcal prcsss as grndng mbssng burnshng hrad rllng drawng cung ar vr cmplcad. ncras h qual f h prduc and mnmz h cs f prcss w shuld knw h phscal phnmna whch xs durng h prcss. hs papr prsns h mdlng and smulan f a cnac prblm n h pran f chnlgcal prducn f bjcs. h prcsss ar cnsdrd as a gmrcal and phscal nn-lnar nal and bundar valu prblms. h mahmacal mdl n a pcal ncrmnal sp m wr dscrbd usng sp-b-sp prcdur wh updad J.L. Lagrang s frmulan []. A nw ncrmnal maral mdl f lasc (dman rvrsbl) and vsc-plasc (dman nn-rvrsbl) wh mxd hardnng ncludng hgh sran ras and gmrcal and phscal nnlnars s usd. h mdl aks n accun h hsr f dfrman. h dnfcan f cnsuv paramrs n h mdl f ld srss s mad usng undrcnal s n h sudd dffrn marals. An ncrmnal mdl f h cnac prblm fr mvabl las/vsc-plasc bd fr spaal sas (D) s bng cnsdrd. Gmrcal cnac cndns (GCC) fr h cas f a dfrmd bjc and a rgd r lasc l wh a ran and ranslan f h bds ar nrducd. A GCC frm usd n numrc calculans s drmnd. Dpndncs bwn ncrmns f un frcs n h cnac ara f bds s nrducd. Basc ncrmnal quans f h dg dsplacmn n h rvrsbl and nn rvrsbl zn ar dfnd. h dscrpn f a gmrcal cnac cndns and frcn cndns n h rangs f sck-slp ar cnsdrd. h mdls band ar usd a varanal frmulan f quan f mn and dfrman n hr dmnsns fr hs cas. hn h fn lmn mhd (FEM) and dnamc xplc mhd (DEM) wr usd ban h slun. h prcdur has bn mplmnd n h fn lmn cmpur prgram ANSYS whch maks pssbl a cmplx m analss f h phscal phnmna: sas f dsplacmns srans and srsss. Numrcal cmpuans f h sran hav bn cnducd wh h us f mhdlg a prpr dfnn f h cnac zn whu h ncss nrduc bundar cndns. Exampls f numrcal analss f cnac bds (l-bjc) n dffrn chnlgcal prans as grndng and hrad rllng prcsss ar shwn. h nflunc f h a sngl abrasv gran gmr and h cung angl n h sas f sran and srss n h surfac lar durng machnng s xpland. Exampls f smulan f h nflunc a varus hrad rllng prcss cndns n h sas f sran and srss wr prsnd.. MAHEMAICAL MODEL OF PROCESS A mahmacal mdl f h chnlgcal prcsss s frmulad n ncrmns and cnans h fllwng: a maral mdl a cnac mdl an quan f mn and dfrman wh nal and bundar cndns. Maral mdl Yld srss: Yld srss s h ms mpran paramr characrzng h rssanc f a vsc-plasc dfrman. h ncrmnal mdl f h ld srss fr a pcal sp m was dfnd as []: (P) (P) (P) F ( ) F ( ) () Y (P) whr Δε and Δε ar h ncrmn f ffcv vscplasc sran and sran ra rspcvl F ( ) s h (P ) cmpnn f chang n h mprar ld srss Y wh (P) chang f h vsc-plasc sran whr F ( ) Y ( )/ s mprar hardnng paramr fr cnsan accumulad ffcv vsc-plasc sran ra a m ( ε (P) cns ) (P ) F ( ) s h cmpnn f chang n h mprar ld srss Y wh chang f h vsc-plasc sran ra whr (P) F ( ) Y ( )/ s mprar hardnng paramr fr cnsan accumulad ffcv vsc-plasc sran a m (ε (P) cns ). Elasc/vsc-plasc maral mdl: A nw mdl f mxd hardnng fr srpc maral whch ncluds h cmbnd ffcs f lasc (rvrsbl dman) vscplasc (nn-rvrsbl dman) (E/P) s usd. h mdl aks n accun h hsr f h maral. h cnsuv quan f ncrmn cmpnns f a al sran nsr aks frm: Δεj (Dj Δσ A Sj ) () S and f ncrmn cmpnns f h al srss nsr: Δσ j Cj Δε ψ Sj ( Sj Cj Δε A) ()

whr: s a psv scalar varabl Sj S C j S Sj Cjmn Smn (4) j S j Sj σ Y s a cmpnn f a srss nsr Y (P) (C E ) (5) σ (P) A σ Y Δε (6) ε s a psv scalar varabl j s h ncrmn cmpnn f h scnd Pla-Krchhff srss nsr D

2 whr: s a psv scalar varabl Sj S C j S Sj Cjmn Smn (4) j S j Sj σ Y s a cmpnn f a srss nsr Y (P) (C E ) (5) σ (P) A σ Y Δε (6) ε s a psv scalar varabl j s h ncrmn cmpnn f h scnd Pla-Krchhff srss nsr D j ar h cmpnns f nsr D [ C ] n m Δε j s h ncrmn cmpnn f Grn-Lagrang sran nsr C j ar h cmpnns f lasc cnsuv nsr C. Mdl f cnac l-bjc h qualfcan f h ara ral shap f h bds cnac zns s cmbnd wh h drmnan n hs aras f h sas f ladng mchancs (prssurs and frcs f frcn) and h sa f h dfrman f h bjc maral and h pps. In praccal cnsdrans hs sas ar uncupld n h wa ha h frs n drmns h shap and h fld f h cnac pn ara f bds and hn lads h rsul fr hs cndns. h abv cas f h cnac prblm has an ssnal manng: h cnac frcs cnac sffnss shap and fld f h cnac ara f bds cnac bundar cndns and frcn cndns n hs ara. Frcs n h cnac zns: In h m ncrmn ncrmn f un frc pn acd n a prpndcular drcn h cnac surfac hwvr ncrmns f un frcs ncrmns f un frcs pj and j j= ar angnal hs surfac. q q p p (7) N whr Δq s h ncrmn f h rsulan un frc wh cmpnns q. h cmpnns f ncrmn frcs pj and j add up ldng cmpnns qj f rsulng frc q acng angnal h surfac f cnac: q p j. (8) j j Cnac sffnss: Cnac frcs caus h dsplacmn f h dg f bds n cnac. h valu f hs dsplacmn s dpndn f h cnac sffnss whch s dfnd b h rlan f acng frc n h surfac h valu f h dsplacmn surfac f h cnac n h drcn f h frc wrkng. Cnac sffnss ccurs n h nrmal and angnal drcns. Dpndnc f un frc-dsplacmn ( p u ) can b nrducd wh h hlp f w lns (Fg. ). h frs n cncrns h rang f h lnar rvrsbl dsplacmn ld srss p (rang E) hwvr h scnd n h nn-lnar nn-rvrsbl dsplacmn (rang P). Δp p p E Fgur : p u dagram fr cnac l-bjc j Fr mvmn f maral Δu k τ u τ u P Δu Δu k (P) τ τ u Blckng h mvmn f maral u Δq z Δp Δ Δp n Δq An ncrmn f h rsulng dsplacmn n h drcn a a pcal m ncrmn s calculad wh h us f h fllwng quan: Δu Δu Δu (9) Δp Fgur: Incrmns f un frcs n an pn f cnac zn Frm Fg. h fllwng dpndncs ncrmns f un frcs: q Δq Δp p q p p p p N p Δq q Δp N Δ Δq z p Δ Δq z rsul amng whr u s h ncrmn f an lasc dsplacmn s h ncrmn f a vsc-plasc dsplacmn. Frm p u dagram w ban: Δp k Δu Δu (P) Δp k Δu (0) (P) whr k and k ar mprar sffnss cffcns n drcn fr rang E and P rspcvl. hs cffcns ar gvn b: k p fr u p k fr u 0 fr (P) p k fr u k (P) p p p p p p p p ()

3 whr u s h accumulad cmpnn f lasc dsplacmn p s h accumulad cmpnn f h nrmal un frc a m. Frm Eqs. (0) and () w ban h rlans bwn h ncrmn f rsulng dsplacmn u and h ncrmn f prssur p : Δu [k ] Δp fr p p k -k Δu Δu Δp fr (P) k k (P) p p. () I rsuls frm rlanshps Eq. () ha h ncrmn f h rsulng un frc s h funcn f h ncrmn f h dg dsplacmn. h qualfcan f hs rlanshps dmands h knwldg f xprmnal curv p u fr ral cndns f h cnac. I s fn vr dffcul ralz h drmnan f such dpndnc r s unfasbl. In h prsn papr hs dffcul s lmnad b a varanal frmulan f mvmn quans and h us f rav mhds f slun. Assumng ha h sa f h ncrmn f h prssur and frcns frc s knwn frm h prvus ran mprar cffcns f cnac sffnss k and k and dpndnc p u ar drmnd analcall. (P) Bundar cndns n h cnac zn: h gmrcal cndn f h cnac dfns currn dsanc g bwn pns n h dg f bds alng h nrmal drcn.. prpndcular h angnal plan bh bds (Fg. ). A gmrcal cndn f h cnac wll bcm frmulad n ncrmns n a gnral frm.. fr spaal sas a fundan ha bh bjcs as h ls undrg ranslan and urn a whch l hav much largr sffnss n cmparsn wh h bjc. h bds rman undr nflunc f frcs and mmns. mprar dsanc τ g(z) h dg f h bjc frm h acv surfac f h l n a nrmal drcn fllwng h dpndnc bwn h cmpnn vcrs: τ () () g(z;τ) g(z;) ΔK Δu (z;δ) Δu (z;δ) 0 () () whr ΔK s a al nflunc f h ranslan ncrmn and rans f ls and h bjc n h dsplacmn ncrmn f h bjc s dg Δu () (z; Δ) and l Δu () (z; Δ). z z z τ R τ r 0 F(z) τ g R Δu () r Δu () Δw () 0 f(zτ) Δw () M () ω () ω () B g m τ A Fgur : Illusran f gmrcal cndns f cnac Frm cndn Eq. () hr rsul h fllwng cass: a) f g( ) 0 and g( ) 0 hn h pn n qusn ls bnd ara f cnac l τ O (p) M () Δl (p) ΔΦ () Δl () m τ τ O () O (p) ΔΦ () () O () bjc F () F ( ) 0 f(z) m b) f g( ) 0 and g( ) 0 hn n h cnsdrd pn a cnac fllwd c) f g( ) 0 and g( ) 0 hn cnsdrd pn n ara f cnac sll sas n d) f g( ) 0 and g( ) 0 hn n h cnsdrd pn a lss f cnac ccurrd. Cndn Eq. () s usd n numrcal calculans. h applcan f an rav prcdur dfns h dsplacmn cndns n h cnac ara. hn aks r n h ran prcss ha dsanc g( ) 0 and g( ) 0 (funcns ar knwn frm fundan) ncrmn f dsplacmn f l () dg u as a rsul f dfrman s knwn frm prvus ran hwvr sks slf an ncrmn f h dg () dsplacmn f bjc u frm ransfrmd Eq. () frm: () [] () [] () [] Δu (z;δ) ΔK ( ) Δu (z; Δ). (4) Bundar cndns fr dsplacmn Eq. (4) ar appld n numrcal analss f h cnac prblm n cnsdran. Incrmnal mdl f mn and dfrman aranal frmulan: h quan f mn and dfrman f h bjc s dvlpd n h updad Lagrang s frmulan. Assumng ha numrcal sluns ar band a dscr m h slun fr m s b band. A hs cas a funcnal ncrmn s frmulad fr ncrmn dsplacmn F [ u u u ] F ( ) whr u u u ar h h ncrmn cmpnns f h dsplacmn vlc and acclran vcrs rspcvl. Usng h cndns f sanar f funcnal F () w ban a varanal quan f mn and dfrman: [ F( )] u C j j ( ) C ( j (u u ) ( u ) d C j j ( C j (E) j C j j ) C ( ( j j ( u j) d ( ( j j ( ) d ) d ) d d ) d ) d j ( j ) ( ) d C C ( ) C ( C j (E) j ( ) d r u ( u ) d j j j j C ) C j j j ( ) d ( ) d j j d d ( ) d j ) ( ) d ( u ) d ( f f ) ( u ) d (qˆ qˆ ) ( u ) dk k 0 (5) whr j s h cmpnn f Cauch s srss nsr and ar cnsans ( b drmnd frm w gvn dampng ras ha crrspnd w unqual frquncs f vbran) ar lnar and nn-lnar ncrmn cmpnns f j j

4 Grn-Lagrang s sran ra nsr j 05 ( u j u j ) and j 05 ( u k u jk ) ar h lnar and nn-lnar ncrmns cmpnns f Grn-Lagrang s sran nsr rspcvl s h mass dns a m j s a accumulad cmpnn f al sran nsr a m (dpnd n h hsr f dfrman) f f ar h cmpnns f h nrnal frc and ncrmn frc vcrs rspcvl q q ar h cmpnns f h xrnall appld surfac frc and surfac ncrmn frc vcrs n h cnac bd zns rspcvl j s h cmpnn f h gr nsr. h ngrans ar prfrmd vr h vlum and surfac f h bd rspcvl. Implmnan f h fn lmns mhd: Assum ha h cmpl bd undr cnsdran has bn dalzd as an assmblag f fn lmns w hav a pcal sp m fr lmn n h lcal crdna {x}: u u u N N N w w w ε B w ε B w B w ε w B w ε w B w σ ( S C w B ) w (6) whr w w w ar ncrmns vcrs f dsplacmn vlc and acclran n h all W ndal pns f lmn rspcvl w s h marx f dsplacmns ncrmns h N s dsplacmn nrplan marx B B ar lnar and nn-lnar ncrmnal sran - ncrmnal dsplacmn ransfrman marcs S nw dfn h ncrmnal srss whn lmn as a funcn f h ndal pn ncrmnal dsplacmn. h varans n h Eq. (5) n h lcal Carsan crdna {x} s: ( u ) N ( w ) ( ε ) B ( w ) ( ε ) w B ( w ) ( ε ) B ( w ) ( ε ) w B ( w ). (7) Usng h Eqs. (6) and (7) and subsung n h varanal Eq. (5) w ban h dscrzd quans f mn fr an assmblag f lmns n h glbal crdna {z}: k u M r C( w w w ) CG ( ) r ( S ) K K ( w w ) K w w) K ( w ) K ( w w ) K ( ) K ( ) 4 ( 5 C K ( ) K ( w w w w ) r F( f w ) F( ) whr: A R( qˆ) { F( f ) R( qˆ) w Wx WxN A WxN WxW r Nx WxN (8) A A (9) A s h ransfrman marx rla h bass f lcal ssm {x} and glbal ssm {z} A s h Bl s marx (lgc marx) w s h ndal pn ncrmn dsplacmn vcr f lmn n h lcal crdna {x} r s h ndal pn ncrmn dsplacmn vcr f ssm n h glbal crdna {z} N s h numbr f all ndal pns f ssm. Inrducd h fllwng nan: K K C C( w w w ) CG ( ) k u ( S ) K K ( ) K( ) KC( ) K ( ) ( S ) K 4( ) K 5( ) K ( ) K ( ) F F( f w ) F( ) R(ˆ) q F F( f ) R R( qˆ) w can wr h Eq. (8) n h frm: r K K ) (0) M r C ( r R F F () whr mass marx M dampng marx C sffnss marx K and xrnal and nrnal frc vcr F ar knwn a m. Hwvr ncrmn sffnss marx K xrnal ncrmnal lad vcr R nrnal ncrmnal frcs vcr F ncrmnal vcrs f dsplacmn r vlc r and acclran r f fn lmn assmbl a a pcal sp m ar n knwn. In rdr slv hs prblm w appl h ngran mhds - cnral dffrnc mhd (DEM) whch s n f mhds f drc ngran h Eq. ().. DEM SOLUION Assumng ha an ncrmn f mprar sp s vr small s pssbl xcu a lnarzan f Eq. () and usng h ncrmnal dcmpsn w ban an quan fr m : M r C r K r F Q. () hn usng h cnral dffrnc mhd (DEM) n whch s assumd ha: r r r r r r r () and subsung h Eqs. () n Eq. () w ban: whr: s ffcv mass marx and Q F QK s ffcv lads. M r (4) Q M C M r r r r M C (5) h ngran mhd rqurs ha h m sp s smallr han crcal valu kr whch can b calculad frm h mass and sffnss prprs f h cmpl lmn assmblag: kr N / whr N s h smalls prd f h fn lmn assmblag wh N dgrs f frdm.

4. RESULS OF NUMERICAL CALCULAIONS Rllng prcss f h rund hrad [] h man am f h smulan was dfn h nflunc f frcn cffcn n h sa f dfrman (dsplacmns and sran) and srss n h surfac lar

h l s cnsdrd as rgd E r lasc bd hwvr h maral mdl as an ls/vsc-plasc bd wh nn-lnar hardnng. h mdl has dscrzd b fn lmn PLANE8 wh nn-lnar funcn f h shap.

Analzng h dsrbun f dfrman f h fn lmn grd and sa f ffcv srans and srsss whr h nflunc f h lubrcan cndn s bsrvd. Fr 0 n h cnac zn l wrk pc (Fg.

Ohr sd ncras h frcn cffcn causs ncras brakng f h maral. Fr hgh valu f h frcn cffcn (Fg. 4b) ccurs srng brakng f maral n h cnac zn. Frm als h adhsn zn f maral.

a) b) = 0 M MX Fgur 4: h dfrman f grd and h maps f ffcv srsss n a lngudnal cung plan fr varus valu f frcns cffcn h frcn cffcn has hgh nflunc n valu and dsrbun f sran.

In hs zn h valu f sran s vr small. Fr 0 9 srans ar clsr h cnac surfac and gng smallr valu 0 006 (lasc srans) (MN Fg. 5b). Whra h lcal maxmum f srans (MX) mvng dwn n surfac lar.

5a) valu 0 54 fr 0 9 (MX Fg.5b). Nx n lcal maxmum (MX) s lcad n dph f maral n smmr axs pass hrugh p f h MX = 0.9 hrad.

a) b) MX Fgur 5: h maps f ffcv srans n a lngudnal cung plan fr varus valu f frcn cffcn Grndng prcss [5] Fr h crrc mdllng and analss f h grndng prcss h knwldg f h curs f h

h mdl f abrasv gran (Fg. 6) spcfd n papr [4] s cnsdrd as rgd r lasc bd. h bjc s cnsdrd as h lasc/vsc plasc bd and s rang wh angular vlc ω arund wn ax.

5 4. RESULS OF NUMERICAL CALCULAIONS Rllng prcss f h rund hrad [] h man am f h smulan was dfn h nflunc f frcn cffcn n h sa f dfrman (dsplacmns and sran) and srss n h surfac lar f h bjc. h numrcal analss fr D sas f dfrman and D sas f srss was appld n h xampl f sl C55. h l s cnsdrd as rgd E r lasc bd hwvr h maral mdl as an ls/vsc-plasc bd wh nn-lnar hardnng. h mdl has dscrzd b fn lmn PLANE8 wh nn-lnar funcn f h shap. h cnac l wh wrk pcs was mdlng b lmns ARGE69 and CONA7. Exmplar rsuls f h numrcal smulan ar prsn n Fgs 4 and 5. Analzng h dsrbun f dfrman f h fn lmn grd and sa f ffcv srans and srsss whr h nflunc f h lubrcan cndn s bsrvd. Fr 0 n h cnac zn l wrk pc (Fg. 4a) durng h frmng h uln f h hrad maral sn brakng b l and sld hrugh h cnac surfac. h curvng f vrcal ln f h fn lmn grd s nvsbl. Ohr sd ncras h frcn cffcn causs ncras brakng f h maral. Fr hgh valu f h frcn cffcn (Fg. 4b) ccurs srng brakng f maral n h cnac zn. Frm als h adhsn zn f maral. ha caus hghr dsplacmns f maral n h zn placd fahr frm h cnac zn. hn h ln f h fn lmn grd ar srngr curvd. a) b) = 0 M MX Fgur 4: h dfrman f grd and h maps f ffcv srsss n a lngudnal cung plan fr varus valu f frcns cffcn h frcn cffcn has hgh nflunc n valu and dsrbun f sran. Fr 0 h maxmum f ffcv sran 078 s lcad n h bm f h hrad nar h cnac surfac (MX Fg. 5a). Fr 0 appar an adhsn zn f maral n h bm f h hrad whch ak characrsc shap f a wdg. In hs zn h valu f sran s vr small. Fr 0 9 srans ar clsr h cnac surfac and gng smallr valu (lasc srans) (MN Fg. 5b). Whra h lcal maxmum f srans (MX) mvng dwn n surfac lar. hn appar addnal w lcal maxmums f h ffcv srans. Scnd maxmum (MX) s placd nar h cnac zn f h sd f h hrad whr hghr valu f frcn cffcn ncras srans valu frm 076 fr 0 (Fg.5a) valu 0 54 fr 0 9 (MX Fg.5b). Nx n lcal maxmum (MX) s lcad n dph f maral n smmr axs pass hrugh p f h MX = 0.9 hrad. Hr srans ar gng smallr ghr wh ncrasng f frcn cffcn frm valu 0 5 fr 0 (Fg. 5a) 0 4 fr 0 9 (Fg. 5b). a) b) MX Fgur 5: h maps f ffcv srans n a lngudnal cung plan fr varus valu f frcn cffcn Grndng prcss [5] Fr h crrc mdllng and analss f h grndng prcss h knwldg f h curs f h phscal phnmna ccurrng n h machnng zn n ral cndns (.. gmr f gran and chnlgcal paramrs) prvs b ncssar. Fr hs purps an analss f h prcss f cung wh a sngl abrasv gran was cnducd. h mdl f abrasv gran (Fg. 6) spcfd n papr [4] s cnsdrd as rgd r lasc bd. h bjc s cnsdrd as h lasc/vsc plasc bd and s rang wh angular vlc ω arund wn ax. An abrasv gran wh h apx angl f 80 0 and h crnr radus r 0 00 m s ld n rlan h fundan b l cung dg angl [6]. h dph f cu was g 0 0 m. h valu f h ral dph f cu f h maral rmvd as a rsul f lasc dsplacmn was smallr and was ca. g r 0009 m. x g Φ = 0 v c r MX chp ω lasc/vsc-plasc bd Fgur 6: h schma f cnsdrd prcss f cung wh n abrasv gran h lasc/vsc-plasc bd: g dph f cu g r ral dph f cu g s lasc dfrman f maral r crnr radus v c chp vlc ω angular vlc l rak angl Φ shar angl Numrcal smulan n h ANSYS ssm was cnducd fr dffrn angls and f h abrasv gran. h bjc machnd and h abrasv gran wr dgzd b lmns f PLANE6 p wh a nn-lnar funcn f shap. h cnac gran wh bd was mdlng b Sngl Surfac Au MX abrasv gran g s g r MX MN MX = 0.9

6 D (ASSD). h n f fnshd lmns was cncnrad n h cnac ara. Sampl smulan rsuls ar prsnd n Fgs. 7 and 8. Whl analsng h rsuls band was fund ha ghr wh h chang f h angls and h valus f srans and srsss ar subjc chang. Abrup ncrass f srsss ar h rsul f h chp cran phnmnn. ghr wh h ncras f h l cung dg angl h shar angl Ф f h maral sparad frm h fundan ncrass as wll. I was fund ha bh angls hav a sgnfcan nflunc n h chp shap. Fr h l cung dg angl 45 w bsrv fas dsurbancs f h chsn f h maral bwn h nghburng chp lmns. hs rsuls n h fac ha h chp drps ff frm h cung dg n h frm f spara lmns a sgmnal chp (Fg. 7). NODAL SOLUION.0E SEP= SUB =8 LS-DYNA usr npu Fgur 7: Map f ffcv srans n h chp cran phas fr 0 45 r 0 00 m Fr angl 65 hr ccurs h phnmnn f chp curlng (Fg. 8) n h drcn f h fundan machnd a sppd chp. hs s h rsul f h fac ha h chp ln frm h sd f h cung dg acn surfac s lngr han h chp ln n s pps sd. Fr angl 55 h chps crad ar sgmn chps. Fas crackng f h chp lmns s bsrvd. NODAL SOLUION dsplacmn sran srss c. durng h prcss wh nwadas chnqu f a masurmn s mpssbl. Abu hr curs w culd cnclud n h prpr f h prduc. An applcan f mdrn mahmacal mdllng numrcal mhds and cmpung ssms allws an analss f cmplx phscal phnmna ccurrng n h prcss undr nvsgan. h applcan dvlpd n h ANSYS ssm nabls a m analss f h prcss wh h cnsdran f h changabl f h lubrcans cndns. On h curs f phscal phnmna n h wrkng zn w can frcas a chnlgcal qual f h prduc. h band rsuls f h cmpur smulan f h hrad rllng prcss shw ha h frcn cffcn nflunc n h sas f dsplacmns srans and srsss n h surfac lar f h hrad als s n f h facrs dcdng abu h chnlgcal and h xplan qual. h bs pranal qual f h hrad s rcvd durng h rllng prcss n gra lubrcan cndns ( 0 ). h smulan rsuls fr cndn f lubrcan can b us f whl dsgnng h rund hrad rllng prcss: makng a slcn f h prcss cndn and knd f h lubrcan facr n h aspc f h chnlgcal qual f h hrad. h band rsuls f h cmpur smulan f h cung prcss wh a sngl abrasv gran wh a gmr f 0 and cung dg angl 45 cncd wh h rsuls band b Ka and Id []. h maral flashs band bfr h gran cung dg and s shaps smlar h rsuls f xprnal nvsgans cnfrm h jusfabl f h us f cmpur smulans and hr rlabl. h dsrbuns f srsss and srans band fr dffrn gran gmrs and acn angls n parcular phass f h dfrman prcss can b mad us f whl dsgnng machnng: makng a slcn f h machnng cndns and s pmsng n h aspc f h chnlgcal qual f h prduc. h dsrbuns f srsss and srans band fr dffrn gran gmrs and acn angls n parcular phass f h dfrman prcss can b mad us f whl dsgnng machnng: makng a slcn f h machnng cndns and s pmsng n h aspc f h chnlgcal qual f h prduc E E+09.E+0.75E+0.8E+0.6E+0.06E+0.50E+0.9E+0 SEP= SUB =9 LS-DYNA usr npu Fgur 8: Map f ffcv srsss n h chp cran phas fr r 0 00 m 6. CONCLUSIONS h chnlgcal prcsss ar gmrcal and phscal nnlnar nal and bundar prblm. Bundar cndns n h cnac zn l-bjc ar n drmnd. Masurmn f a prcss paramrs dcd n h chnlgcal qual such as: MN MX 7. REFERENCES [] K.J. Bah Fn Elmn Prcdurs n Engnrng Analss Prnc Hall Englwd Clffs N.J. 98. [] Y. Ka M. Id h mchansm f mal rmval b abrasv l War N. 978 pp [] K. Kuklka L. Kuklka Mdlng And Numrcal Analss Of h hrad Rllng Prcss WILEY-CH rlag GmbH & C. KGaA l. 6 Issu 006 pp [4] L. Kuklka J. Kusra Numrcal analss f hrmal phnmna and dfrmans n prcssng zn n h cnrlss cnnuus grndng prcss Cmpuan Mhds and Exprmnal Masurmns fr Surfac ramn Effcs WIPRESS Suhampn Bsn 00 pp [5] L. Kukłka J. Chdór Numrcal analss f chp frman durng machnng fr dffrn valu f falur sran Jurnal PAMM l. 7 Issu 008 pp [6] W. Lrz A mdl f h cung mchansm n grndng War N pp. 5-8.

10.5 Linear Viscoelasticity and the Laplace Transform

10.5 Linear Viscoelasticity and the Laplace Transform Scn.5.5 Lnar Vclacy and h Lalac ranfrm h Lalac ranfrm vry uful n cnrucng and analyng lnar vclac mdl..5. h Lalac ranfrm h frmula fr h Lalac ranfrm f h drvav f a funcn : L f f L f f f f f c..5. whr h ranfrm