FRACTIONAL THERMOVISCOELASTIC MATERIALS *G. Alba, *M. Di Paola

Transcription

1 Meccanica dei Maeriali e delle Sruure Vol. 6 (26), no.2, pp. - ISSN: X Diparimeno di Ingegneria Civile, Ambienale, Aerospaziale, dei Maeriali - DICAM FRACIONAL HERMOVISCOELASIC MAERIALS *G. Alba, *M. Di Paola *Diparimeno di Ingegneria Civile, Ambienale, Aerospaziale, dei Maeriali (DISAG) Universià di Palermo Viale delle Scienze, 928 Palermo, Ialy giovanni.alba@communiy.unipa.i, mario.dipaola@unipa.i (Ricevuo 7 agoso 26, Acceao seembre 26) Key words: maser-curve, fracional calculus, linear viscoelasiciy, emperaure effecs, ime-emperaure superposiion, viscoelasic maerials Absrac. his paper aims o invesigae he effecs of he change of emperaure in he fracional consiuive law of viscoelasic maerial. I is shown ha he well known William Landell and Ferry (WLF) (or he Harrenius) principle is no valid for real fracional viscoelasic maerials unless he order of fracional operaors is indipenden on he emperaure. Wih he aid of some experimenal dae obained by performing Creep es a differen emperaure he order of he fracional operaors is no consan and hen hey may be considered hermoreologically simple. his modificaion of he WLF is here proposed in order o fully invesigae he effecs of he emperaure in he fracional consiuive law. INRODUCION Creep and Relaxaion funcion of real maerials are well fied by power law []. As a consequence, in he linear hypohesis, he Bolzmann superposiion principle holds and he convoluion inegrals are he so called fracional operaors [3,4,5]. Experimenal evidence shows ha he parameers are very sensiive o he change of emperaure. As his effec will be aken ino accoun, he various parameers will depend on he emperaure and hen implicily on he ime if he emperaure is ime dependen is i happens in many engineering applicaions. Many papers have been devoed o his subjec [7,8,,3,4]. hermoviscoelasic heory has been also invesigaed in many papers Lord and Shulman Dhaliwal and Sherif [5,6]. How saed before he parameers sricly depend on he emperaure and in lieraure many sudies have been devoed o his subjec. In paricular he so-called ime emperaure superposiion is used. Such a principle saes ha as he emperaure increase (or decrease) as he emporal scale is properly modified he Creep funcion will remain he same a he various emperaure. his principle sricly holds only for hermoreologically simple maerials. his principle however does no holds rue for real maerials. his is due o he fac ha Creep and/or relaxaion funcion are Power-law [] raher han exponenial like in he classical models of viscoelasiciy (Kelvin-Voig, Maxwell, Zener, Burger,...). his produce wo effecs: firs of all he consiuive laws are ruled by fracional operaors (Riemann-Liouville Meccanica dei Maeriali e delle Sruure 6 (26), 2, PP. -

2 and Capuo s fracional derivaive) ruher han classical derivaives and inegrals; he second effec is ha he parameers of he fracional viscoelasic model are srongly influenced by he emperaure. I follows ha in he case in which he emperaure depends on he ime (daynigh, summer-winer) also he parameers depend on he ime and he Creep and/or relaxaion funcion will depend no only on he sress (or he srain) bu also on he emperaure ha in urn depend on he ime. In his paper i is shown ha he S for he fracional model is sill applicable only if he order of fracional derivaive (or inegral) is indipenden on he emperaure. ha is he fracional model hermoreologically simple only if he order of he fracional operaor does no depends on he emperaure. wo real maerials have been aken ino accoun he Polypropylene and Ehylene Vinyl Aceae (EVA). Creep es are such a maerials performed a differen emperaures show ha boh are no hermoreologically simple because he order of fracional operaors are no consans. In order o work wih such a maerial a modified William Landell and Ferry law is proposed ha is no only he emporal scale is modified bu also he shif on he order of fracional operaors is examined. 2 PRELIMINARIES IN FRACIONAL VISCOELASICIY Nuing [], on he basis of a wide experimenal campaign observed ha real maerials like rubber, polymers seal..., saes ha he for an assigned sress hisory σ ( ) = σ U ( ) ( U ( ) being he uni sep funcion) he corrisponding srain is given in he form: ε ( ) = a σ ;( a, µ R ); () µ wih a, and µ characerisic parameers depending on he maerial a hands. In paricular if σ = eq.() becomes he so called Creep funcion in he following denoed C (). he conseguence of eq.() is ha as we assume ha σ is small, hen he creep funcion assumes he form: C ()= ; (2) C Γ ( ) where Γ( ) is he Euler Gamma funcion and C is an anomalous coefficien ( Pas ). As soon as he Creep funcion is assumed in he form expressed in eq.(2) he Bolzmann superposiion principle, for an assigned sress hysory σ ( ), is given as: ε ( ) = σ ( τ )( τ ) dτ C ( ) Γ & (3) Such an equaion is valid if he sysem is quiescen a = ( ε () = ) and σ () =. If σ () we have o add a he r.h.s of eq.(3) he erm C () σ (). Inegraion by par eq.(3) leads o: ε ( ) = C ( I σ )( ) (4) Meccanica dei Maeriali e delle Sruure 6 (26), 2, PP. - 2

3 where he symbol ( I )( ) means Riemann Liouville fracional inegral, ha is: σ I f τ f τ dτ Γ( ) (5) ( )( ) = ( ) ( ) Le us now denoe as R ( ) he relaxaion funcion ha is he sress hysory o an imposed srain hisory of he kind ε ( ) = U ( ). Since he following general law of he viscoelasiciy holds rue in Laplace domain (Flugge) ˆ (s) ˆ (s)=/s 2 C R (6) where C ˆ (s) and R ˆ (s) are he Laplace ransform of C () and R (), respecively and s is he (complex) parameer of he Laplace operaor. From eq.(6), afer some algebra, i is recognised ha R (), corresponding o he creep funcion in eq.(2) is given as: R C ( ) = Γ ( (7) ) I follows ha he Bolzmann superposiion principle gives he sress hisory corresponding o a general srain hisory ε ( ) as: C σ ( ) = ε ( τ )( τ ) dτ Γ( ) & (8) or in equivalen form: σ ( ) = C ( C D ε )( ) (9) where he symbol ( )( ) is he so called Capuo s fracional derivaive defined as: D C ε ( D f )( C ) = ( ) ( ) ; ( ) τ f τ d & τ Γ () he consiuive law expressed in eq.(9) is valid provided ε () =. In his case he Capuo s fracional derivaive coalesces wih he Riemann Liouville fracional derivaive. Eq.(4) and (9) represen he consiuive laws of fracional viscoelasiciy. A his sage some remarks are necessary: i) As = he elasic law is recovered and for = he Newon- Perof law is recovered, i follows ha he inermediae value < < gives an inermediae behaviour of he wo exreme (idealized) cases; ii) for a quiescen sysem (a = ) he operaors (5) and () are inverse each anoher; iii) operaors (5) and () are linear ones and for such operaors all he rules of classical derivaive and inegral hold rue (lineariy, inegraion by pars, Leibniz rule,...), moreover in Fourier and in Laplace domain hey Meccanica dei Maeriali e delle Sruure 6 (26), 2, PP. - 3

4 exacly behaves like for he classical derivaive and inegrals; iv) hese operaors are valid for R or even for C (wih Re( ) > ). In he nex secion he effecs of he emperaure on he fracional consiuive law is presened. 3 FRACIONAL ORDER HERMOVISCOELASICIY In his secion we assume ha he body is hysoropic, and he componen of sress σ and he corresponding srain are composed by only one componen as i happens for he purely angenial sress or purely volumeric componen. Such an example for he purely angenial sress τ eq.(4) is rewrien in he form γ ( ) = G ( I τ )( ), γ being he angenial srain corresponding o he angenial sress τ ; or for he volumeric sress ε v = 3 k ( I σ )( ) v and C has been assumed as G and 3k σ v i corresponds o, because hey are he shear and he Bulk s modulus (anomalous) for hese wo cases. In he following we shall assume he general law in eq.(4) and in eq.(9). Now i has been bserved ha as he emperaure varies, he parameers and C varie ha is = ( ) ; C = C ( ) () he variaion of hese coefficiens is general very large. Such an example for he waer a o C he liquid phase disappear (glass) and ;, while a > o C he solid phase disappears and ; (pure Newonian fluid). hese variaions may be used in order o reduce imes of experimenal resuls. Such an example for a resin a a emperaure of 25 o C we need of 5 hours o deermine he experimenal Creep or Relaxaion funcion, bu if we performe he same experimen a 35 o C we may use a reduced ime of order 2 minus. Anoher reason o invesigae in his field is ha afer many maerials commonly used in srucural elemens like pulruded beams in srucural engineering or biumens in he railways engineering suffer of cyclic change of emperaure (day-nigh and/or summer-winer) and in hese circumsances he evaluaion of he response o given loads in erms of deformaion depend on hese changes i follows ha we have o ake ino accoun for hese variaion, in he equaion of moion. Usually empirical models have been proposed in lieraure in order o ake ino accoun he effecs of he change of emperaure he so-called shif facor ha is valid for he so-called hermoreologically simple maerials is applied.his principle is formally proposed in he following form: R (, ) = R (, ) (2) ha reads: he relaxaion funcion a a given ime a he emperaure equals o he relaxaion funcion a ime corresponding o a reference emperaure. he general form of is: = a (3) Meccanica dei Maeriali e delle Sruure 6 (26), 2, PP. - 4

5 where a is he so-called shif facor. Differen expression of he shif facor have been proposed in lieraure. he mos popular expression is he William-Landel and Ferry (WLF), ha is: C ( ) log a = C ( ) 2 (4) where C and C 2 are coefficiens obanied by bes fiing on experimenal daa. he WLF expression is valid for > g, being g is he glass ransiion emperaure. For < g he oher model is ha proposed by Arrhenius, ha is: H a log a = ( ) (5) R where R is he universal consan of gasses and H a is he acivaion energy. For hermoreologically simple maerials he ime emperaure Superposiion (S) holds ha basically consiss in assuming ha he ime scale of observaion a he given emperaure, if properly changed, produce idenical resul of ha obained a a reference experimen by changing he emporal scale according o he eq.(3). Now he quesion is: wha are he condiions for which a fracional viscoelasic maerial is hermoreologically simple? In order o answer o his fundamenal quesion le us assume ha he maerial is hermorologically simple hen, according o eq.(2) we may wrie: C ( ) C ( ) = Γ( ( )) Γ( ( )) ( ) ( ) (6) From his equaion i is eviden ha a maerial is no hermoreologically simple unless ( ) = ( ). Unforunaely his is no he case for many real maerials, such an example from he experimenal daa in he laboraory of Palermo [8] performed on he propilene for a o o o o given load ( 55MPa ) and for differen emperaure ( C,23 C,37 C,5 C and 65 o C ) he Creep funcions of he maerial a differen emperaure are represened in fig. Meccanica dei Maeriali e delle Sruure 6 (26), 2, PP. - 5

6 Fig. Creep funcion a differen emperaures (experimens) he bes fiing deermine he parameers C and for he differen emperaures and he resul are depiced in fig(2) and (3), respecively Fig.2 Bes fiing of C ( ) Fig.3 Bes fiing of ( ) Meccanica dei Maeriali e delle Sruure 6 (26), 2, PP. - 6

7 From figs.(2) and (3) we may saes ha C ( ) is well suied by he law: where: C ( ) = C C C C (7) C =.99344, C =.43893, C =.36, C = 8.547* ( ) = A A A A a >, b > (8) where: 6 A =.6429, A =.3322, A2 =.73832, A3 =.93437* and equaion (8) reveals ha since ( ) is no a consan he polyvinile is no hermoreologically simple. Analogous resul is found for he Ehylen Vinil aceae (EVA) commonly used for a solar cells, glass cover and so-on. A recen sudy [] give he resul of and C based by he bes fiing performed a differen emperaure. hese resuls are repored in able [ o C ] C [ Pa* s ] able resul by bes fiing based on experimenal campaign a differen emperaures (resul repored in []) From his able is apparen ha ( ) may no be considered consan wih respec o he emperaure, so also EVA may no be considered hermoreologically simple maerials. In he nex secion a Modified S (MS) is proposed o define he behavior of maerials no hermoreologically simple. 4 MODIFIED IME EMPERAURE SUPERPOSIION PRINCIPLE We inroduce wo shif facors one for he ime a and he oher one for he variaion of ( ) ha will labeled as b. Eq.(2) for a hermoreologically simple maerial read as: Meccanica dei Maeriali e delle Sruure 6 (26), 2, PP. - 7

8 R (,, ) = R ( a *,, ) (9) for he maerials for which is no consan we propose: R (,, ) = R ( a *,, b * ) (2) I is obvious ha his modificaion reproduce he S for he case in which is indipenden on he emperaure ( b = ). Such an example for he EVA he coefficien b is ploed in fig.4. he bes fiing is performed by assuming = 8o C, =,5 and a hird order polynomial of he form b = b b b b mach quie well all he experimenal daa (doed line) wih he value b = , b =.47932, b =.76426, b = 6.372* Fig.4 b for differen value of he emperaure; coninuos line bes fiing Once b is obained he value of he shif facor for he emperaure may be easily found. In fig.(5) he log a is ploed versus for he EVA. In doed line are repored he log a and in solid line. he resul of he bes fiing wih a polynomial of hird order 2 3 log a = a a a2 a3. For he EVA he coefficien are a = , a = , a =.5735, a = Meccanica dei Maeriali e delle Sruure 6 (26), 2, PP. - 8

9 Fig.5 log a for differen value of he emperaure; 5 CONCLUSION In his paper he effec of he emperaure on he parameers of he fracional consiuive law (order : ) has been invesigaed. I has been shown ha for he fracional model he ime-emperaure shif mehod may no be applied unless he exponen of he power law in he Creep funcion is independen on he emperaure. I is also shown ha from he experimenal daa obained for polypropylene and on he EVA ah differen emperaures he order of derivaive is no consan. hen a modificaion of he WLF is proposed by shifing boh he ime and he order. REFERENCES [] NUING, P. G., A New General Law of Deformaion, Journal of he Franklin Insiue, Volume 9, pagg , Maggio 92. [2] FLUGGE, W., Viscoelasiciy, Blaisdell Publishing Company, Walham, 967. [3] DI PAOLA, M., PIRROA, A., Fracional Calculus - Applicaion o Visco-Elasic Solids, Meccanica dei Maeriali e delle Sruure, Volume, n. 2, pagg , 29. [4] DI PAOLA, M., ZINGALES, M., An Exac Mechanical Model for Fracional Viscoelasic Maerial, Journal of Rheology, aricle under review, 2. [5] DI PAOLA, M., ZINGALES, M., An Exac Mechanical Model for Fracional Viscoelasic Maerial, Journal of Rheology, aricle under review, 22. [6] BAGLEY, R. L., Power Law and Fracional Calculus Model of Viscoelasiciy, AIAA Journal, Volume 27, n., pagg , 989. [7] M. GERGESOVA, B. ZUPAN c ( I c (, I. SAPRUNOV, AND I. EMRI, he closed form -- P shifing (CFS) algorihm,2 by he Sociey of Rheology, Inc. J. Rheol. 55(), -6 January/February (2) Meccanica dei Maeriali e delle Sruure 6 (26), 2, PP. - 9

10 [8] MENG DENG, JACK ZHOU, Effecs of emperaure and Sress Level on Creep and ensile Propery of Polypropylene Suures, Journal of Applied Polymer Science, Vol. 9, (23) Wiley Periodicals, Inc. [9] N.W. SCHOEGL,WOLFGANG G. KNAUSS,IGOR EMRI,he Effec of emperaure and Pressure on he Mechanical Properies of hermo- and/or Piezorheologically Simple Polymeric Maerials in hermodynamic Equilibrium A Criical Review,Mechanics of ime-dependen Maerials 6: 53 99, 22. [] PAGGI, MARCO; SAPORA, ALBERO. An Accurae hermoviscoelasic Rheological Model for Ehylene Vinyl Aceae Based on Fracional Calculus. In: INERNAIONAL JOURNAL OF PHOOENERGY, vol. 25 n. Aricl, pp ISSN -662X [] DANON GUIERREZ-LEMINI, Engineering Viscoelasiciy, Springer New York Heidelberg Dordrech London, ISBN [2] HAL F. BRINSON, L. CAHERINE BRINSON ; Polymer Engineering Science and Viscoelasiciy, An Inroducion, Springer New York Heidelberg Dordrech London,, ISBN [3] EMMANUEL CHAILLEUX,GUY RAMOND, CHRISIAN SUCH CHANAL DE LA ROCHE, A mahemaical-based maser-curve consrucion mehod applied o complex modulus of biuminous maerials, Road Maerials and Pavemen Design. EAA 26, pages 75 o 92 [4] NICOLE HEYMANS, Consiuive equaions for polymer viscoelasiciy derived from hierarchical models in cases of failure of ime emperaure superposiion,physique des Maeriaux de Synhese 259, BE-5 Brussels, Belgium [5] LORD, H.W. AND SHULMAN, Y., A Generalized Dynamical heory of hermoelasiciy, J. Mech. Phys. Solids, Vol. 5, pp , 967. [6] SHERIEF, H.H. AND DHALIWAL, R.S., A Uniqueness heorem and Variaional Principle for Generalized hermoelasiciy, J. herm. Sresses, Vol. 3, No. 4, pp , 98. Meccanica dei Maeriali e delle Sruure 6 (26), 2, PP. -