COMPRSSION BHAVIOUR AND ASTIC RCOVRY OF HIGHOFT MATRIAS (KVIN-MAXW MOD) Jana Přívraská *Kaarina Zelová Technical Universiy of iberec Faculy of Science, Humaniies and ducaion Sudenská, 46 7, iberec, Czech Republic janaprivraska@ulcz * Technical Universiy of iberec Faculy of Texile ngineering, Sudenská, 46 7, iberec, Czech Republic kaarinazelova@ulcz Absrac The behavior in compression and he elasic recovery of highslof maerials can be described by a Kelvin-Maxwell rheological model The proposed model is a serial combinaion of he Kelvin model (parallel connecion of Newonian viscous fluids and elasic maerials) and he Maxwell model (serial combinaion of Newonian viscous fluids and elasic maerials) This combinaion is able o cover he plasic deformaion and relaxaion behavior In his paper an algorihm for he deerminaion of he inpu parameers of he proposed rheological model based on experimenal daa on condiion ha he load phase is carried ou a consan sress for he ime will be presened Inroducion I was shown in [,, 3] ha he compression resisance and he elasic recovery (Fig ) of highlof nonwovens (low densiy fibrous nework srucures characerised by a high raio of hickness o weigh per uni area) Fig Behavior of a highlof maerial in loading-recovery es can be described by a rheological model composed of Kelvin and Maxwell models arranged in series (K-M model), (Fig ) 89
Fig Kelvin-Maxwell model Model Descripion The resuling srain ε of his model is he sum of he srain ε of he Kelvin model and ( ε + ε 3 ) of he Maxwell one, where ε describes he srain of is elasic par and ε 3 he srain of is plasic par Boh pars, Maxwell and Kelvin, are under he same sress [4] 3 ε 3 ε, () where, are Young modules of springs (elasic elemens) and, are viscosiies of viscosiy elemens The sress-srain relaion is deermined by he differenial equaion [] 3 d d d ε () oading Modus The maerial is compressed a ime by a consan sress and kep for some ime d d As for, equaion () will be simpler d 3 ε Is soluion under he iniial condiions ε (a jump of srain of he elasic par in Kelvin model) and 3 (velociy of plasic srain of he firs and he hird par of M-K model) is for ε e 3 (3) (4) (5) (6) 9
where is he ime when maximal compression is reached (srain ε ) For ε he compression srain is consan, lasic Recovery Regime A he ime, he sress is removed and ha is followed by a jump of srain ε ε This represens he elasic recovery of he maerial The plasic or enacious srain is equal o ε 3 As he sress for, he lef side of equaion () is zero and we ge d ε The soluion of he differenial equaion (7) for elasic recovery regime is under he iniial condiions ε ε3 and ε ε3, (7) (8) ε ε ε ε 3 3 ε ε ε ε 3 3 e e,, (9) () Deerminaion of Model Parameers The analysis of measured sress-srain and recovery curves (Fig ) makes i possible o find he inpu parameers,,,3 for he M-K model In experimens we can measure, ε, ε and ε 3 The parameer is deermined from he equaion () ε ε ε The parameer 3 is deermined from he equaions () and (6), as ε 3 represens residual plasic srain 3 () ε 3 The rae x can be deermined from he elasic recovery curve () ε x ln ε ε 3 ε 3 () (3) 9
Comparing he equaions (6) and (9) i is possible o find ε ε e x 3 or ε ε e x Than 3 for (4) for (5) (6) x Conclusion The deerminaion of inpu parameers for he Kelvin-Maxwell model from experimens enables o find a se of consans characerizing highlof maerials Wih hese resuls i is possible o perform compuer simulaions of oher heoreical experimens in order o propose heir opimum design ieraure [] BHARANITHARAN, R; PŘÍVRATSKÁ, J; JIRSÁK, O:Modelling of Compressional Properies of Highlof Texiles In Proceedings of STRUTX, iberec 3, pp 47 43 ISBN 8-783-769- [] PŘÍVRATSKÁ, J; JIRSÁK, O; BHARANITHARAN, R: Maxwell-Kelvin model for highlof maerials In Proceedings of Programs and Algorihms of Numerical Mahemaics, Prague 4, pp96-99 ISBN 8-8583-53-5 [3] DONG, X; ZHANG, J; ZHANG, Y; YAO, M: A sudy on he relaxaion behavior of fabric's crease recovery angle, Inernaional Journal of Clohing Science and Technology, Vol 5, No, 3, pp 47-55 [4] SOBOTKA, Z: Reologie hmo a konsrukcí, Academia, Praha 98 Prof RNDr Jana Přívraská, CSc PhD, Ing Kaarina Zelová 9
CHOVÁNÍ VYSOC OBJMNÝCH MATRIÁŮ PŘI KOMPRSI A ASTICKÉM ZOTAVNÍ (KVINŮV-MAXWŮV MOD) Chování při kompresi a pružné zoavení vysoce objemných maeriálů lze popsa pomocí Kelvinova-Maxwellova reologického modelu Jde o sériové spojení Kelvinova modelu (paralelní spojení newonovské viskózní kapaliny a hookovské elasické láky) a Maxwellova modelu (sériová kombinace newonovské viskózní kapaliny a hookovské elasické láky) Tao kombinace je schopna obsáhnou plasickou deformaci i relaxační chování Uvádíme algorimus, jak urči vsupní paramery pro eno reologický model pomocí experimenálních da v případě, že záěžová fáze probíhá při konsanním napěí po dobu VRHATN VON HIGHOFT MATRIAN UNTR DRUCKBANSPRUCHUNG UND DAS RÜCKSTVRMÖGN (KVIN-MAXW-MOD) Das Verhalen von hochflorigen (highlof) Maerialien uner Druckbelasung und deren elasische rholung kann durch ein Kelvin-Maxwell Modell beschrieben werden s is dies die Reihenschalung eines Kelvin-Modells (Parallelschalung von Newonschen viskosen Flüssigkeien und elasischen Maerialien) und eines Maxwell-Modells (serielle Kombinaion von Newonschen viskosen Flüssigkeien und elasischen Maerialien) Diese Kombinaion is in der age, die plasische Verformung und das Relaxaionsverhalen abzubilden s wird ein Algorihmus vorgesell, mi dem die relevanen Parameer für das beschriebene rheologische Modell aus experimenellen Daen besimm werden können, wenn die Belasungsphase der Zei uner konsaner Spannung erfolg ZACHOWANI WYSOKO OBJĘTOŚCIOWYCH MATRIAŁÓW PRZY KOMPRSJI I ASTYCZNYM ODKSZTAŁCNIU (MOD KVINA-MAXWA) Zachowanie przy kompresji i elasycznym odkszałceniu wysoko objęościowych maeriałów można opisać przy wykorzysaniu reologicznego modelu Kelvina-Maxwella Jes o szeregowe połączenie modelu Kelvina (równoległe połączenie lepkiej cieczy Newona i maeriału elasycznego Hooka) oraz modelu Maxwella (szeregowe połączenie lepkiej cieczy Newona i maeriału elasycznego Hooka) To połączenie jes w sanie objąć zniekszałcenie plasyczne oraz odkszałcenie (zachowanie "relaksu") Prezenujemy algorym w celu określenia paramerów wejściowych dla ego modelu reologicznego przy pomocy danych doświadczalnych, gdy faza obciążenia odbywa się przy sałym napięciu przez czas 93