Technical University of Lublin, Poland. The New Concept of Reversibility for Viscoelastic Maxwell Class Models

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1 Absrac Sławmir KARAŚ, Kaarzyna OSINA Tchnical Univrsiy f Lublin, Pland Th Nw Cncp f Rvrsibiliy fr Visclasic Maxwll Class Mdls Assuming Blzmann s suprpsiin rul and nn-aging marials h rvrsibiliy f linar phnmnlgical visclasic mdls ar invsigad wih spcial anin Maxwll-Burgrs bdis lik. On h basis f Trzaghi s prus grund and hysrsis phnmnn, h nw cncp f rvrsibiliy is prsnd. Th abv ida is illusrad in h cass f Maxwll and Klvin-Vig mdls. Finally, h ask f simpl bam vibrain is analyzd, whil is marial prpris ar fund as adqua Maxwll s mdl. Th rsul ha has bn baind is prving ha Maxwll mdl, and gnralizing Maxwll Class mdls, ar rvrsibl.. Inrducin N frqunly, bu i happns ha in chnical liraur, spcially in highway nginring, hr appars incrrc classificain f Burgrs mdl [], [], [3]. As i is prsnd a Fig.., Burgrs crp funcin is an adqua apprximain curv fr dscribing prsnd labrary rsuls, bu h sraigh applicain yild rahr larg las squars mhd rrr []. Th cmpnns f Burgrs mdl can b scind ff in fllwing lmns, B linar v xpnnal v, () xpnnial B v whr - sands fr h immdia lasic srain, linar v - linar viscus flw prcss f srain, xpnnial v - xpnnial viscus flw; h arrws rinain is assciad wih - - applid lad prid and - dsrssing. Th abv dscripin srngly shws h absnc f linar v cmpnn in dsrsing prcss. Th lack f his lmn nails raing h rsidual valu in Burgrs crp funcin as an irrvrsibl par and as a cnsqunc classify Burgrs mdl as irrvrsibl. Our aim is shw h cnradicin f such cncp and ry frmula h ida f rvrsibiliy. Wih rfrnc sraighfrward applying f Burgrs mdl is br us h xpandd Burgrs mdl by sris cnncin wih Sain-Vnan lmn [5] () Σ B S V whr Σ - summary srain, B - Burgrs mdl srain, S V - plasic srain., in ha cas w bar in mind ha h invlvd mdl blngs Bingham Class bdis. In such cas, n can avid h inaccuracy f apprximain rrrs. In furhr cnsidrains Maxwll Class mdl is undrsd as - h Maxwll as wll as Gubanw and Burgrs and similar hr mdls.

2 ,,9,8,7,6,5,,3, ()/σ Cnsan mpraur: C, [σ][mpa] σ. σ.35 σ.5 σ.6 Fig.. An xampl f ypical rad pavmn asphal-mrar crp s graph. Th rdinas rlad h las abscissa valu, in chnical sns, ar rad as masurs f plasic dfrmains. 5 im [sc]. Visclasiciy Visclsiciy culd b rad as a naural xnsin f lasiciy. Th lasiciy is prcisly dfind by mahmaical rlains and physical cncps. In h cas f visclasiciy h dfiniin is n s clar. W can dfin visclasiciy as a rlain bwn srss and srain nsrs and hir im drivaivs f diffrn ranks Φ ( ) ( σ ) ( ) ( σ) ( ) ( ) ( ) ( I,I,I,I &,,I && σ,&& I,...,) F( σ,,σ, & &,σ,, && &,...,) &, (3) m n m n m n σ whr I m, m,, 3 - invarians f h srss nsrσ, I n, n,,3 - invarians f h srain nsr, ds which can b nicd vr h symbls mans parial drivaivs accrding im. Th scnd lmn f visclasisiy dfiniin cms frm Blzmann s suprpsiin, which rfrs h lad hisry. W cnclud ha: - Each lad is a cmpnn f h lad hisry, - Th al final srain is a sum f dfrmains causd by all cmpnns f h lad hisry. Alhugh h xampls f h Blzmann s principl can b fund in many wrks, w illusra hm blw by aking in accun h lad () acing Klvin-Vig and Burgrs mdls σ σ H( ) H( )] σ [H( ) H( )] () [ 3 ( ) whr: (5) ( (.) ) H( (.) ) - Havisida s sp funcin, ( (.) ) < σ, σ ; sig[ σ ] sig[ σ] - diffrn srss valus, < < 3 < - im mmns.

3 a) b) c) σ( ) (, σ ) (, σ ) σ 3 3 σ( ) σ (, σ ) (, σ ) σ( ) σ σ σ Fig.. Blzmann s suprpsiin. a) Lads. b) Klvina-Vig mdl racin. c) Burgrs bdy rspns. In h cas b) h unlading causs ha h dfrmain fads afr lngnugh im, s w g lim lim, (6) whras fr Burgrs marial c), accrding h sam lad hisry a), w bsrv prmann srain valu whn : lim, lim, lim. (7) Accrding Fig.., fr : Klvina-Vig mdl rurn h iniial null srain valu whil Burgrs mdl achiv a rsidual (nn-iniial) srain valu and lks lik b irrvrsibl n, as has bn nicd in h bginning f h papr. All h im w hav in mind ha bh f hs mdls ar linarly visclasic. L us als discuss h cncp f Przyna [6]. In spi f h fac ha h rgards viscplasiciy, his mhdlgy is vry usful fr adping ur prblm. W can rad h mhds f viscplasiciy blng nighr rhlgy nr plasiciy. On ha basis h inrducs w marial classs: - Elasic - viscplasic fr which h marial shws viscus prpris in bh lasic and plasic rgins. - Elasic / viscplasic - rsrvd fr marials shwing viscus prpris in plasic rgin nly. In bh cass abv h viscus flw dfrmain culd b furhr dividd in irrvrsibl and rvrsibl pars 3

4 , (8) i i v i p i v p sinc i, i, i, i - ar as i fllws: al, lasic, viscus and plasic srain nsr cmpnns. Cnmprary bh prcsss ar classifid as Bingham mdl. Adping Przyna s cncp w can inrduc w yps f visclasiciy: - Visc - lasiciy whn viscus and lasic cmpnns dvlp in paralll and - Visc / lasiciy fr h prvius immdia lasic dfrmain and afr ha viscus dfrmain prcss, bu bh f hm ar rvrsibl. Accrding h im scal f rad pavmn asphal-mrar crp s, i culd b includd in visc / lasiciy class, which is idnical wih visclasiciy in furhr cnsidrains. Rlain () can b rad as a cmpl cnsiuiv rlain and, in ur cnsidrain, w cnfin urslvs invarian linar prcsss i.. saisfying h rlain ( σ,) k m m,,3 σ m f [, ( )] m, (9) whr -srain, σ(,) -srss in h n dimnsinal cas and is a im paramr. W wuld lik mphasiz ha h crp prblm is rducd h cas f n dimnsinal bdy and h analysis f an iniial sag ransin crp. W disrgard sady sa crp and crp failur. Addiinally, w assum ha iniial cndiins ar always hmgnus. 3. Maxwll Class Mdls L us cmpar h diffrnial cnsiuiv rlains f svral lmnary visclasic mdls: - Maxwll σ & a σa _ & b, (.) - Burgrs σ & a σ& a σa _ & b & b, (.) - Klvina-Vig σ a b & b, (.) - Znr σ & a σa b & b; (.) whr a,a,a,b,b, b - ar h marial characrisics. In spi f h sam nain hy can hav diffrn valus fr diffrn mdls. Nwacki [7] cmmns h fllwing frmula P(D) σ () Q(D) () by snncs Th ranks f Q(D) and P(D) prars shuld b qual (mn) and I is admissibl assum h cas whn h rank f Q(D)is n rank highr hn h P(D) rank.

5 Als Rzanicyn analyss h prblm [8] wriing fllwing rmarks Th rank f diffrnial quain is qual h numbr f dashps in h mdl and In cas f lacks f cnsan lasic cnncin bwn dfrming pins, hn insad f w hav is im drivaiv. Th lng-sanding srss yilds h incras up infiniy. Fr (.-) and (.-) h cndiins frmulad by Nwacki and Rzanicyn ar fulfilld bu w can als nic n discussd faur i.. h absnc f n squnc lmn in Maxwll and Budgrs mdls, nifid by _ in (.-) frmula. Nw w can als dfin h Maxwll Class as mdls fr which hirs cnsruciv quains ar f yp (6) and in gnral as h marials dscribd in fllwing way ( n ) ( n ) σ a n _ σ a n... σ&& a _ & b && b _ σa... ( m 3) ( m ) ( m) b m 3 _ b m b m () In mahmaical sns, h absnc f b lmn in rlain (.) r (.) is h rasn f s-calld irrvrsibiliy f hs mdls. Having in mind ha iniial cndiins ar hmgnus, w can slv h diffrnial quain baining zr valus fr all cnsans. L us assum ha h lad funcin fr a Maxwll mdl is f h frm σ σ [ H( ) H( )], <, (3) hn w arriv a fllwing sluin ( ) η( ) ( ) η( ) ( ) (s ()) E η E η whr fr > w bain ( ) ( ) - prmann valu. () η W us prmann valu disinc abv rsul, which is h sluin diffrnial quain ( ) χ & sinc χ R and >, (5) whr h valu f diffrnial quain cnsan C cming frm iniial cndiins. Cncluding prmann valu is causd by h lad funcin and h mdl has n b irrvrsibl. Ending his par w rpa sm infrmain f Maxwll mdl. W us h classical nain. Visclasic bhavir can b simulad by spring lasic rsisanc E and viscus dashp characrizd by η and which ar cnncd ghr in sris way. Th cusmary diffrnial dfiniin frm is as fllws 5

6 d dσ σ d E d η r σ& σ &. (6) E η Applying h Carsna-Havisid a ransfrmain [9] w bain - ( p σ ) p E pη Invlving Dirac s impuls df. θ θ δ( )f ()d H( )f ( ) Subsqunly, w g [ δ( ) δ( )] σ& ( θ) ( θ) dθ. (7) E η d d df., (3) [ H( )] δ( ). (8) σ& ( ) σ, (9) ( ) η( ) ( ) η( ) ( ). () E η E η Dividing im abscissa in scrs, xrm pins ccur if lim () ( ) ( ), (.) E η E η lim () ( E η, and > ) ( E η ) ( η ) > lim (), (.) diffrs infinily frm apprprialy appraching frm lf and righ sid >. Fig. 3. is an illusrain f crp funcin accrding Maxwll mdl. (, σ ) E η (- ) _ η ( - ) 8 /E σ( ) σ Fig. 3. Th dfrmain prcss in h cass f Maxwll mdl and lad funcin (3) 6

7 . Th nw cncp f rvrsibiliy fr Maxwll Class marials In h bginning, l us rcall w simpl xampls f physical prcsss. Firs is hanks s-calld Trzaghi s prus grund [] als knwn as Trzaghi s fam. H usd succssfully h fllwing squnc sima slmn f h buildings: - Th uppr surfac f grund in naural cndiin is ladd by rcd building; a firs w can bsrv immdia dflcin causd b lasic grund srucur rspns, nx - Th rarding and dcaying flw is bsrvd du war disldging, which was filling grund prus. L us lnga his cncp, adding w nw sags - Th building srucur is disassmbld which invlvs immdia rspns f grund lasic srucur - Afr lngnugh im w can als nic h rcvring flw which culd b causd by graviy frcs. In such prcss, war can fill dsrssd grund prus vlums again. If human liv is prprly lng, n can prbably prv abv sry. Th scnd xampl is xrmly rivial. W cnsidr hr cnfigurains f ypical car shck absrbr h srucur f which is pracically idnical wih Nwnian lmn in Maxwll mdl (Fig..). Assuming an arbirary cnfigurain as iniial - K w push h dvic wih P frc. Tha dfrms h absrbr acual cnfigurain K a (Fig..a.). Squnially, applying frc P w pull h absrbr ( P culd b qual r n P ) achiving h K a cnfigurain (Fig..b.). a) p K p K a P L L b) K a K a P L L Fig.. Th squnc f Nwn bdy mdl cnfigurains Nw w hav answr h arusd qusin: ar h cnfigurains K a and K a qual r n? In gmrical maning hy ar qual wihu dub. In physical sns h ss {dash p srucur, wrk, nrgy} f K a and K a can b rad in many siuains as cnvrgn r vn qual. Fulfilling h suprpsiin rul, rvrsibiliy f Maxwll Class mdls can b br undrsd wih h hlp f hysrsis. W can fully adp all faurs which cm as a rsul f his nin i..: hardning, sfning. Y, h ms craiv lmn is h advn f h alrna squnc f lads and ani-lads which gnra h lp frm iniial and back iniial cnfigurain. 7

8 W shuld als rmmbr ha ur nars cncp Bauschingr hysrsis is alms always prsnd as a rlain f σ crdinas. On h basis f abv analysis and invlvd assumpins, w can prps h fllwing nw dfiniin f visclasic rvrsibiliy - Sinc h squnc f lads and ani-lads is dfind (i culd b calld fundamnal) and - If afr applying h squnc, h rsidual dfrmain is bsrvd h marial blngs Bingham Class, - If cnrary h marial is visclasc and is characrizd as nirly rvrsibl. L us nw build h fundamnal squnc fr Maxwll and Klvin-Vig mdls assuming { H( ) H( )] [H( ) H( )]} σ σ, () [ 3 sinc < < 3 <. Fig. 5. shws h graph f mdls rvrsibiliy in dails accrding (). (, σ ) Maxwll Mdl 3 (, σ ) Klv in-vig Mdl 3 σ( ) Fundamnal lad squnc σ 3 σ Fig. 5. Rvrsibiliy f visclasic mdls, including Maxwll Class. 5. Th Vibrain f Simply Supprd Maxwll Bam Tha subc was invsigad by many auhrs, fr xampl Nwacki [5] lkd fr cmmn ffcs cmparing lasiciy and visclasiciy in Maxwll and Klvin-Vig mdls. Th pic f ur cnsidrain is xamin h vibrain prcss a h infini im mmn. 8

9 Th dynamical quilibrium quain fr lasic infinisimal bam lmn is as fllws γa EJ w(x, ) q(x, ) x g, (3) whr: w-bam dflcin, A-cnsan bam crss-scin, EJ-bnding rigidiy, q -lad linar dnsiy. Dning bam span by L, w inrduc dimnsinlss crdinas x ξl, w ωl, τ ; () whr - psiiv cnsan wih im unis, ha invlvs EJ IV γal ω ( ξ, τ) ω& ( ξ, τ) q( ξ, τ), (5) 3 g ( L) ( ) IV sinc: γ - marial wigh dnsiy, g graviy acclrain, ω - furh rdr drivaiv accrding ξ, ω& & - h scnd rank dimnsinlss im paramr drivaiv. Assuming: ( ) w bain ( ) γa L and J g ( L) 3 q q (6.-) J E ω IV ω& q. (7) Wih h hlp f Carsn-Havisid ransfrmain accrding τ, and invlving Alfry s analgy w can urn in visclasic prblm E ω IV, (8) ( p) ω q sinc h iniial cndiins ar hmgnus. In h cas f Maxwll mdl w bain E p, (9) E η whr E is n Yung mdulus (E) ransfrmain. L h lad funcin b Dirac s impuls q ha yild ( τ τ ) qξ qτ δ ξ δ (3) 9

10 q q q ξ τ. (3) W lk fr h sluin xpanding h unknwns and lad funcin in Furir sin sris accrding ξ - ω ( ξ,p) ω ( p) sin πξ ( q ),,..., q( ξ,p) q ( p) ( q ) sin πξ τ ξ,,..., (3.-) π ξ sin πξ dξ sin δ ξ. (33) By viru f sris prpris, w hav,,..., ω Th rs f f ( p) q τ ( q ) ( p) ( π) E ξ ( p) ar as fllws q p τ ( q ) ( π) ( π) E ξ η f q τ ( q ) ( p) ( p p )( p p ) ξ. (3) ( π) E p, m ( ) ( ) m. (35) E η π Applying falung hrm w can find Furir cfficins whr [ ] ω τ p ω ( ) δ( θ τ ) ( )( ) ( τ θ) θ qξ C d (36) p p p p p ( q ) ( ) ( )( ) ( ) ξ H τ τ C τ τ p p p p, C is h symbl f Carsn-Havisida rransfrmain. Bh rs (35) ar funcin. Having in mind ha E and η ar psiiv w arriv a E lim lim, (37) η ( π) and in cnsqunc fr larg nugh valu h rs ar ral and qual p, ( π) p. (38.-) E W hav analyz hr pnial varians I. E > > h rs ar ral and ngaiv, p < p, η ( π)

11 II. III. E < < h rs ar cnugaiv cmplx p η ( π) E w hav dual ral r p η ( π) n indx valu. p whn ( p ) ( π) R <, p, i culd b nly fr E Th cmplxiy f h prblm cnsiss in simulanus ccurrnc f all (I-III) varians. Simplifying, l us assum ha w fund gr. by slving III and gr. N. gr. is dividing dmain in w pars whr - - Varian I is valid fr < gr. and - Varian II whn > gr.. Addiinally, w can sa gr. is n larg and w can nglc h cndiin (38.-) which bys. Varian I is assciad wih hard viscus damping. Varian II dscribs dcaying bam vibrain. In ur prblm h Jrdan s lmma is fulfilld and w can apply rsidual hrm. Th riginal fr h varian I has h frm - C p Rs ( p p )( ) p p [ p ( )] [ p ( )] pτ τ sh ( τ). (39) Fr varian II, wih h hlp f Eulr s frmula, w arriv a C p ( )( ) C p p [ ( )] [ ( )] p p p i p i pτ sin( τ) τ Rs [ p ( i )] [ p ( i )] p, () whr i. In cas f larg valus w hav lim sin τ ( ) and lim On h basis f (6) w g τ. () and ω sin π H ( τ τ ) ( τ τ ) sh [ ( τ τ )] ()

12 ω π ( τ τ ) [ ( τ τ )] sin H( τ τ ) sin. (3) Th sluin f h prblm has h fllwing sris frm In( ) ω,,,... In( ),... gr. ( ξ τ) ω sin( πξ) ω sin( πξ) gr.. () Nw w can find h limi f (5) fr ξ and τ limω, τ lim τ τ In(gr. ),,... In(gr. ),... sin sin π π ( τ τ ) ( τ τ ) sin [ ( τ τ )] sh [ ( τ τ )]. (5) Th rsul ha has bn baind is clarly prving ha Maxwll mdl, and gnralizing Maxwll Cass mdls, ar rvrsibl. 6. Bibligraphy []. Gawł I., Kalabinśka M., Piła J.: Mariały drgw i nawirzchni asfalw, Oficyna Wydawnicwa Kmunikaci i Łącznści, Warszawa,. []. Chałabis J, Kukiłka J. Srukura, wyrzymałść i dkszałcalnść bnów asfalwcmnwych, Prac Naukw Plichniki Lublski, sr. 55. [3]. Gldn J.M., Graham G.A.C. - Bundary Valu Prblms in Linar Visclasiciy, Springr, Brlin-Hidlbrg-Nw Yrk-Lndyn-Paris-Tki, 988., p. 7 h. []. Mackiwicz P. Ocna dfrmaci miszank minraln-asfalwych na pdsawi paramrów rlgicznych, Drgwnicw,, 5. [5]. Baak M., Mazurk K.: Właściwści bnu asfalw-cmnwg (BAC) wdług mdlu Burgrsa, Drgwnicw, 5, 5, 59. [6]. Przyna P. Fundamnal Prblms in Viscplasiciy, Advancs in Applid Mchanics, vl. 9, 966. [7]. Nwacki W. - Tria płzania, Arkady, Warszawa, 963. [8]. Rzanicyn A.R. - Th Crp Thry (in Russian), Mscw [9]. Brnszn I.N., Simindiaw K. A. Mamayka. Pradnik ncyklpdyczny, PWN, Warszawa, 976. []. Trzaghi K. Erdbaumchanik auf bdnphysikalischr Grundlag, Duick, Lipzig, 95.