THE THEORY OF MULTIPLE PEELING. Nicola M. Pugno

Transcription

1 HE HEOR O MULIPLE PEELING Ncoa M. Pgno Dept. of Strctra Engneerng an Geotechncs, Potecnco orno, orso Dca eg brzz 4, 9, orno, IL Laboratory of Bo-nspre Nanomechancs Gseppe Mara Pgno e: ; ax: ; Mobe: ; Ema: ncoa.pgno@poto.t; Skype: ncoa.pgno; Webpage: Natona Insttte of Ncear Physcs (INN, Natona Laboratores of rascat, Va E. erm 4, 44, rascat, IL Natona Insttte of Metroogca Research (INRIM, Straa ee acce 9, I- 35, orno, IL onsorzo Nazonae Internerstaro per e Scenze sche ea Matera (NISM, a ea Vasca Naae 84, 46, Roma, IL bstract In ths paper we soe the mtpe peeng probem by appyng a fractre mechancs approach to a compex system of fms, aherng to the sbstrate an hang a common hnge, where the png force s appe. he smpest V- shape system, consstng of two entca peeng tapes s consere as a case sty (to be soe copng sx nonnear eqatons; an optma peeng ange, at whch aheson s maxma, s scoere.. Introcton In spte of the nterest of the fractre mechancs commnty on peeng, the Kena (975 moe remans the nersay aopte theory for snge peeng. Its extenson to mtpe peeng has neer been formate an s the am of the present paper.. he theory of mtpe peeng Let s conser a three-mensona compex system compose by N ahese tapes conergng to a common pont P, where an externa force s appe.

2 Each tape has cross-secton area, ong mos, ength an orentaton efne by the ntary ector n, see gre. he eastc spacement δ η (assme to be sma,.e. tape orentatons o not change sgnfcanty of the pont P can be cacate as foows. he eongaton of each tape s δ δη n, ths the tape tenson (f negate, the corresponng tape oes not work, an the externa oa s spporte by the other tapes s δ n kδη nn, where k s the tape stffness. he eqbrm of the matera pont (hnge P, where the oa s appe, mposes N or eqaenty [ K ] δ η, where [ ] K s the known (by comparng the ast two eqatons stffness matrx of the system. he eastc spacement δ η s ths cacate as: [ K] δ η (a from whch the tape eongatons δ, tensons an strans can be eaate: η δ δ n, δ k, δ,,,n (b Imagne to mpose a fnte (the tape orentatons change sgnfcanty spacement Δ η at the pont P, to be accommoate by mtpe rta eamnatons Δ an eastc eongatons of the tapes. new goba confgraton, enote by the symbo prme, takes pace, see gre. rom the scheme reporte n g. we ece the aty of the foowng eqatons: ( Δ Δη (, n, ( Δ n,,,n ( he strans are known an ther crrent aes can be ere, accorng to eq. (, as a fncton of the nknown orentatons n. ccorngy, copng eqs. (b an (, we can wrte 4N scaar eqatons n 4N nknowns: the N amptes of the rta eamnatons Δ (ther rectons are known a pror from the confgraton of aherng tapes, the N crrent strans an the N sgnfcant components of the new tape orentatons n ( n. Inertng the preos probem, assmng as known three eamnaton amptes n eq. (, we co ere the other compatbe eamnatons as we the spacement Δ η of the pont P. hs means that ony three rta eamnatons can be consere as nepenent.

3 he rta forces reqre for the eamnaton of the th tape can be cacate by the Grffth s energy baance. ccorngy, the eamnaton takes pace when: Π γ w, Π E W,,,N (3a where Π s the tota potenta energy, E s the eastc energy, W s the externa work, γ s the srface energy of the th tapesbstrate nterface an w s ts wth. he eastc energy araton can be cacate as: ΔE he araton of the externa work s: he rea crtca force s: N ( (3b ΔW Δη (3c { } j mn (3 an correspons to the eamnaton of the j th tape. he agebrac system s nonnear bt can be nearze conserng the fferentas nstea of the fnte fferences (e.g. Δ η η. Howeer note that the physca system remans ntrnscay geometrcay nonnear e to the exstence of the orentaton aratons. Moreoer, the energy baance remans non near n the force. 3. Dobe peeng he eeope treatment s here appe to sty a obe peeng system, gre 3. rom eq. ( we ere: ( ( sn, sn sn(, ( sn( > he preos eqatons are a for, ths for < < π. If a tenson s negate ony the other tape sstans the entre oa an ths we hae a cassca snge peeng (f both the tensons are negate the oa cannot be n eqbrm. rom eq. ( we hae: 3

4 Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ sn sn sn sn sn sn sn sn where an Δ Δ are the horzonta an ertca components of the spacement η Δ. Note that the cassca snge peeng ony reqres one eqaton, snce no ange an stran aratons occr rng eamnaton. onserng an song the preos system n the mt of sma aratons (.e. sbstttng the fnte fferences wth the fferentas, yes: Δ [ ] sn sn sn sn sn sn sn sn sn sn [ ] sn b, [ ], [ ] sn b x [ ] [ ] [ ] [ ] b b x Eq. (3b n the mt of sma aratons, ges: E as we as eq. (3c poses: W sn 4

5 ccorng to eq. (3a an (3 the eamnaton force can now be easy obtane. or exampe, conserng the symmetrc case (,, π, an conseqenty an we fn the foowng sotons: ( ( sn sn [( ] ( sn sn ( sn ( sn he preos eqatons hae been nearze n. ccorngy, the energy baance s sef-consstenty wrtten conserng terms p to the secon power of. he rest yes: ( 4 λ, where λ γ t an t w s the tape thckness. Song ths eqaton, the crtca ae of the stran for eamnaton s obtane. he preos eqaton s srprsng: t s entca to that of the snge peeng probem. Howeer the force reqre for eamnaton s fferent snce here we hae: sn ths ony for π the precton s that of the snge peeng tape oae by a force, for whch, as t mst physcay be. he eamnaton force s ths: ( sn 4λ he behaor s epcte n gre 4. n ange for optma aheson opt ceary emerges as a fncton of the parameter λ. 5

6 3. oncson Herewth we hae soe the mtpe peeng probem. he system consstng of two peeng tapes has been consere as a case sty. or sch a case, we hae srprsngy obsere ( a goernng eqaton for the stran entca to that of the snge peeng an ( an optma peeng ange, at whch aheson s maxma. he ast rest s of a great mportance for the expanaton of the fnctona mechansm of boogca aheses an for ahese technoogy as we. References Kena, K. 975, hn-fm peeng-the eastc term. J. Phys. D: pp. Phys. 8, IGURES n P gre. Dagram of the mtpe peeng system consere n ths sty. Δ ( ( P Δη P gre. nte eamnaton of the th tape. 6

7 gre 3. Dagram of the obe peeng system consere n ths sty f gre 4. Dmensoness force ( ( π f erss ange by aryng the mensoness aheson strength λ ; π 4λ. 7