A Simplified Nonlinear Generalized Maxwell Model for Predicting the Time Dependent Behavior of Viscoelastic Materials

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1 Wrld Jural f Mechaics, 20,, di:0.4236/wj Published Olie Jue 20 ( A Siplified Nliear Geeralized Maxwell Mdel fr Predictig the Tie Depedet Behavir f Viscelastic Materials Abstract Marc Delphi Msia Départeet de Physique, Uiversité d Abey-Calavi, Abey-Calavi, Béi E-ail: siadelphi@yah.fr Received April 3, 20; revised May 2, 20; accepted May 23, 20 I this paper, a siple liear Maxwell del csistig f a liear sprig cected i series with a liear dashpt beyig a pwer-law with cstat aterial paraeters, fr represetig successfully the tie-depedet prperties f a variety f viscelastic aterials, is prpsed. Nuerical exaples are perfred t illustrate the sesitivity f the del t aterial paraeters. Keywrds: Hyperlgistic-Type Fucti, Maxwell Mdel, Nliear Stress-Tie Relatiship, Riccati Equati, Viscelasticity. Itrducti T better uderstadig echaical respses f aterials subjected t defratis r frces, theretical dels are required. These dels prvide iprtat aalytical tls fr predictig ad siulatig aterial fuctis. Whe a aterial is subjected t defratis, the resultig stress is related t the strai by a atheatical relatiship kw as the cstitutive law. This cstitutive equati ca, thus, prvide usefuless i deteriig rhelgical prperties f aterials. Sice viscelastic aterials exhibit bth cbied viscus ad elastic aterial behavirs, their cstitutive equati ust atheatically relate stress, strai ad their tie derivatives. Fr this purpse, these cstitutive equatis are fte deteried fr cbiatis f sprigs ad dashpts arraged i series ad/r parallel []. Sice echaical respses f viscelastic aterials are i geeral liear, the well-kw established liear thery f viscelasticity ust be reasably replaced by liear theries. But, liear dels are re difficult t frulate tha liear theries, because these dels lead fte t slve liear differetial equatis that are geerally -itegrable. I this perspective, several dels f differet cplexities have bee prpsed t describe viscelastic aterial fuctis [2]. T take it accut liear viscelastic aterial prperties, it is eeded t dify the siple classical Maxwell ad Vigt dels r their differet cbiatis fr icludig liear ters. May successful predictive dels are shw t be based the extesi f classical liear rhelgical dels t fiite defratis [3] ad i press [4-8]. The dificatis csist t itrduce liear elastic sprigs ad/r liear dashpts i the classical liear dels. Ather way is t csider that the aterials fuctis deped the agitude f the stress, strai r the strai rate. The resultig del accrdig t Alfrey ad Dty [], is iterestig sice, it evaluates the aterial prperties i ters f differetial equatis that ca be slved fr a wide variety f trasiet cditis. I viscelasticity thery, there are ly a few theretical dels frulated with cstat-value aterial cefficiets [3]. Thus, cstat cefficiets liear rhelgical dels are required. Fllwig this viewpit, Crr et al. [3] extedig the Maxwell fluid del t fiite defratis, cstructed a Riccati differetial equati that is useful t describe the strai stiffeig ad sfteig respse f se viscelastic aterials. Recetly, Msia i press [4] utilizig a pwer series expasi ethd that csisted i a exteded Vigt del t large defratis takig it csiderati the iertia ter, develped a hyperlgistic equati which represets successfully the tie- depedet echaical prperties, that is t say, the strai stiffeig ad sfteig behavir, f a variety f viscelastic aterials. Recetly agai, Msia i press [5] usig a secd-rder elastic Cpyright 20 SciRes.

2 M. D. MONSIA 59 sprig i series with a classical Vigt eleet, which is a exteded fr f the stadard liear slid t fiite strais, frulated a hyperlgistic-type equati t reprduce the liear tie-depedet stress respse f se viscelastic aterials. Mre recetly, Msia i press [6] develped a sigle differetial cstitutive equati derived fr a stadard liear slid del csistig f a plyial elastic sprig i series with a classical Vigt eleet fr the predicti f tiedepedet liear stress f a class f viscelastic aterials. Mre recetly agai, Msia i press [7] frulated a liear fur-paraeter rhelgical Vigt del csistig f a liear Vigt eleet i series with a classical liear Vigt eleet with cstat aterial cefficiets fr represetig the liear stiffeig respse f the iitial lw-lad prti ad the sfteig, that is t say, the S-shaped echaical behavir f se viscelastic aterials. Very lately, Msia i press [8] geeralized successfully the previus del i press [7] by replacig the secd-rder elastic sprig preset i the liear Vigt eleet by a plyial elastic sprig. The del i press [8] was shw t be able t predict accurately the liear stiffeig respse f the iitial lw-lad prti ad the sfteig behavir f a variety f viscelastic aterials. I this study, a siple liear Maxwell del (Figure ) csistig f a liear sprig cected i series with a liear dashpt beyig a pwer-law with cstat aterial paraeters, fr represetig successfully the tie-depedet prperties f a variety f viscelastic aterials, is prpsed. Uder a liear strai-path ctrl the cstitutive law gives a atheatical descripti f the stress versus tie relatiship as a hyperlgistic fucti, which appears pwerful t repre- set ay S-shaped curve. The del ca the crrectly reprduce the strai stiffeig ad sfteig respses ted i se viscelastic aterials at large strais. Nuerical exaples are perfred t illustrate the sesitivity f the del t aterial cefficiets ad the validity f the del. I particular the del is shw t be very sesitive t the agitude f the rate f applicati f the strai. Figure. The prpsed rhelgical del. 2. Mechaical Mdel 2.. Theretical Frulati I this part we describe the theretical rhelgical del ad derive the gverig differetial equati icludig the liear restrig frce ad dapig effects. Mst viscelastic aterials are highly iflueced by the liear elastic ad viscus dapig ters s that, their rhelgical aterial prperties are liear tiedepedet. Fr this, a best descripti f these aterials ust prceed fr the use f liear theries. T build ur prpsed viscelastic del, we start fr the classical liear Maxwell del [] i which we replace the liear elastic sprig with a liear elastic sprig (with stiffess E) beyig a pwer-law ad als the liear dashpt with a liear dashpt (with viscsity ) beyig a pwer-law as shw i Figure. Thus, the echaical prperties f the csidered aterial are divided it tw parts: a liear elastic eleet which captures the liear pure elastic behavir f the aterial at equilibriu, actig i series with a liear dapig eleet capturig the tie depedet histry respse f the aterial. Fr the atheatical pit f view, the liear stiffess ad the liear dapig ters are icluded i a del i rder t lss the liearity i the differetial cstitutive equati that represets the dyaic prperties f the echaical syste studied. Due t the fact that the eleets are i series the ttal stress σ ad the ttal strai ε ca be writte as σ = Eε σ = ε& 2 () ε = ε + ε 2 where ε ad ε 2 are the strais f the liear sprig ad the liear dashpt, respectively. is the viscsity dule, ad E is the elasticity dule. The dt detes the tie derivative ad, ad are liearity paraeters. By differetiatis with respect t tie ad akig apprpriate substitutis, e ca deduce fr Equati () the cstitutive differetial equati E / σ σ σ& + = ε& (2) Equati (2) represets atheatically i the sigle differetial fr the relati betwee the ttal stress σ iduced i the aterial uder a strai histry ε. This equati is a first-rder liear rdiary differetial equati i σ fr a give strai histry ε whe the ubers ad are idetically differet fr the uber Cpyright 20 SciRes.

3 60 M. D. MONSIA e Diesializati If M, L ad T dete the ass, legth ad tie diesi, respectively, the diesi f the stress varies 2 as ML T. The strai ε is a diesiless quatity. Therefre, i Equati (2) the cefficiet E pssesses the sae diesi with the stress σ, that f varies as 2 ML T Slutis Usig a Liear Strai-Path Ctrl We derive i this secti the hyperlgistic-type sluti allwig the descripti f the tie-depedet stress iduced i the aterial studied. Fr this, we csider that the aterial uder csiderati is subjected t a liear strai-path ctrl, that is t say ε ( t) = αt (3) where α is the rate f applicati f the strai. Thus, Equati (2) beces E / σ σ σ& + = α (4) I rder t slve Equati (4) we prceed t the fllwig chage f variable r y σ = (5) σ = y (6) Differetiatig Equati (5) with respect t tie yields y& = σ σ & (7) Substitutig these relatiships (Equati (6) ad (7)) it Equati (4), the resultig equati beces E y+ y = α E & (8) Equati (8) is als a first-rder rdiary differetial equati i y, which ca be slved aalytically with the suitable budary cditis f the echaical prble csidered i hyper-expetial r hyperlgistic-type fucti fr special values f the expet Case A: = I this particular case where =, Equati (8) beces a siple liear first-rder rdiary differetial equati E y& + y = α E (9) which ca be easily slved aalytically usig the iitial cditi 0 y t = y t =, ( ) Thus, we ca btai as sluti E y() t = α + ( y α )exp t (0) Fr the Equati (6) we ay deduce takig it accut the Equati (0) the stress versus tie as E σ () t = α + ( y α ) exp t () Equati () gives the tie variati f the stress i the viscelastic aterial studied. It dels the tie- depedet stress as a hyper-expetial fucti shwig that the iitial stress is differet fr zer. Mrever, Equati () predicts a stress that asypttically appraches a axiu value with icreasig tie Case B: = 2 2 I this particular case where =, Equati (8) beces a first-rder Riccati liear rdiary differetial equati [3] ad i press [4-8]. /2 E 2 /2 y& + y = α E (2) which ca be easily slved aalytically usig the iitial cditi 0 y t = y t =, ( ) Therefre, we ca btai as sluti y t = ( ) E y + α + y α exp 2 α t α E y α y α + exp 2 α t (3) We ca deduce fr Equati (6) takig it csiderati the abve Equati (3) the stress versus tie σ t = ( ) α 2/ E y + α + ( y α ) exp 2 α t E y + α ( y α ) exp 2 α t (4) Cpyright 20 SciRes.

4 M. D. MONSIA 6 Equati (4) describes the tie variati f the stress i the viscelastic aterial studied as a hyperlgistic-type fucti, which is pwerful t reprduce ay S-shaped curve [9,0]. 3. Nuerical Results ad Discussi I this part se uerical exaples ccerig the tie-depedet stress are preseted t illustrate the ability f the del t reprduce the echaical respse f the viscelastic aterial studied. The depedece f the stress versus tie curve the aterial paraeters is als discussed. 3.. Case A: = Figure 2 exhibits the typical tie-depedet stress curve with a icreasig util a peak asypttical value, btaied fr Equati () with the value f cefficietsα =, =, E =, = 3, y = 0.0. It ca be see fr Figure 2 that the del is capable t represet atheatically ad accurately the typical expetial stiffeig f se viscelastic aterials, fr exaple, sft livig tissues ad sils, as shw i [3] ad i press [4-8]. The del predicts a echaical respse i which the slpe, after reachig its axiu value at the iflexi pit, declies gradually with icrease tie util the failure pit at which the slpe reduces t zer. Figure 3(a), (b), (c), (d) ad (e) shws the effect f aterial paraeters the tie-stress respse. The effects f these paraeters are studied by varyig e cefficiet while keepig the ther fur cstat. Figure 3(a) illustrates hw the rate f applicati f the strai α affects the axiu value f the stress. The graph shws that a icreasig α, icreases the axiu stress ad the slpe, ad has sigificat effect the tie required t reach the axiu stress ad the iitial value f the stress. The red clr crrespds t α =, the blue t α = 2, ad the gree t α = 3. The ther paraeters are =, E =, = 3, y = 0.0. I Figure 3(b) is shw the depedece f the stress the viscsity cefficiet. A icrease, icreases the peak stress ad the tie eeded t reach it. The slpe icreases als. But a icreasig, has iprtat effect the iitial value f the stress. The red clr crrespds t =, the blue t = 2, ad the gree t = 3. The ther paraeters are α =, E =, = 3, y = 0.0. The stress curves at varius values f the elasticity dule E fr the aterial uder study are shw i Figure 3(c). A icreasig E, greatly ad fast icreases the value f the stress the tie perid csidered. The slpe icreases with icrease E. The red clr crrespds t E =, the blue t E = 2, ad the gree t E = 3. The ther paraeters are α =, =, = 3, y = 0.0. Figure 3(d) shws the sesitivity f the stress-tie curve t the liearity paraeter. A icreasig liearity paraeter, has a high effect the stress value i the tie perid csidered. Ideed, a icreasig, sigificatly ad fast icreases the value f the stress ad als the iitial value f the stress icreases with icrease. The red clr crrespds t = 3, the blue t = 5, ad the gree t = 7. The ther paraeters are α =, =, E =, y = 0.0. We bserve fr Figure 3(e) that a icreasig iitial value y, has a sigificat effect the stress value i the tie perid csidered. I fact, a icrease y, icreases sigificatly ad fast the stress value ad als the iitial value f the stress. A icreasig y, reduces the slpe. The stress beces cstat, that is t say, σ = α = cste r Figure 2. Typical stress-tie plttig exhibitig a axiu asypttical value. Cpyright 20 SciRes.

5 62 M. D. MONSIA (a) (b) (c) Cpyright 20 SciRes.

6 M. D. MONSIA 63 (d) (e) Figure 3. (a). Stress versus tie curves at varius values f the strai rate; (b). Stress-tie curves with differet values f the viscsity dule; (c). Stress versus tie curves shwig the effect f the elasticity dule E; (d). Stress versus tie curves with differet values f the liearity paraeter ; (e). Stress-tie curves fr three differet values f y. with σ = y = y = α cste ad the slpe reduces t zer. The red clr crrespds t y = 0.0, the blue t y = 0., ad the gree t y = 0.5. The ther paraeters are α =, =, E =, = Case B: = 2 Figure 4 exhibits the typical tie-depedet stress curve with a icreasig util a peak asypttical value, btaied fr Equati (4) with the value f cefficiets α =, =, E =, =, y = 0.0. It ca be bserved fr Figure 4 that the del is able t represet accurately the typical tie-depedet stress curve f a variety f viscelastic aterials, as etied abve, fr exaple, sft livig tissues ad sils, as shw i [3] ad i press [4-8]. The stress versus tie curve is liear, with a liear begiig iitial prti, ad illustrates the the sigid echaical behavir f the viscelastic aterial csidered. The plttig shwig a liear sigid behavir idicates, csequetly, the aterial stiffeig fllwed by sfteig. The del predicts a echaical respse i which the slpe, after reachig its axiu value at the iflexi pit, declies gradually with icrease tie util the failure pit at which the slpe reduces t zer. I Figure 5(a) is shw the depedece f the stress versus tie curve the strai rate α. The graph idicates that the peak stress icreases with icreasig α. The slpe icreases als with icrease the rate f appli- Cpyright 20 SciRes.

7 64 M. D. MONSIA ad the gree t α = 3. The ther paraeters are =, E =, = 3, y = 0.0. Figure 5(b) illustrates hw the viscsity cefficiet cati f the strai. But, a icreasigα, has iprtat effect the tie required t attai the peak stress. The red clr crrespds t α =, the blue t α = 2, Figure 4. Typical stress versus tie curve exhibitig a axiu asypttical value. (a) (b) Cpyright 20 SciRes.

8 M. D. MONSIA 65 (c) (d) (e) Figure 5. (a). Stress-tie curves at varius values f the rate f applicati f the strai α. (b). Stress versus tie curves at three differet values f the cefficiet f viscsity. (c). Stress-tie curves fr varius values f the elasticity dule E. (d). Stress-tie curves shwig the effect f the liearity paraeter ; (e). Stress versus tie curves fr three differet values y. Cpyright 20 SciRes.

9 66 M. D. MONSIA affects the axiu value f the stress. The graph shws that a icreasig, icreases the axiu stress ad icreases als the tie eeded t attai the axiu stress. The red clr crrespds t =, the blue t = 2, ad the gree t = 3. The ther paraeters are α =, E =, = 3, y = 0.0. It ca be bserved fr Figure 5(c) the depedece f the stress the elasticity dule E. A icrease E, has a great ifluece the stress value. Ideed, it icreases fast i the early perids f tie. The, this ifluece decreases as tie teds t ifiity, ad the curves ted twards the sae asypttic value f the stress. The slpe icreases with icrease E. The red clr crrespds t E =, the blue t E = 2, ad the gree t E = 3. The ther paraeters are α =, =, = 3, y = 0.0. The stress curves at varius values f the liearity paraeter fr the aterial uder csiderati are shw i Figure 5(d). A icreasig, has a high effect the stress value. I fact, the stress icreases fast i the early perids f tie. The, this effect decreases as tie teds t ifiity, ad the curves ted twards the sae asypttic value f the stress. The liearity f the iitial prti f curves beces less iprtat with icreasig. The red clr crrespds t =, the blue t = 3, ad the gree t = 5. The ther paraeters are α =, =, E =, y = 0.0. We bserve fr Figure 5(e) that a icreasig iitial value y, has a iprtat effect the stress value. Ideed, the stress icreases fast i the early perids f tie. The, this ifluece decreases as tie teds t ifiity, ad the curves ted twards the sae asypttic value f the stress. Mrever, a icreasig y, icreases the iitial value f the stress. The red clr crrespds t y = 0.0, the blue t y = 0., ad the gree t y = 0.5. The ther paraeters are α =, =, E =, = 3. The previus uerical exaples shw that theretical dels are iprtat tls fr the predicti ad siulati f viscelastic behavir f aterials. I this wrk, a siple liear viscelastic del is preseted. The preset del has bee develped fllwig tw wrkig hypthesis. The first pstulates that the aterial prperties ca be divided it a liear pure elastic cpet beyig a pwer-law ad actig i series with a liear dapig eleet beyig a pwer-law ad capturig the tie-depedet deviati fr the equilibriu state. The secd hypthesis assues that the aterial is subjected t a liear strai-path ctrl. Uder these restrictis, the del predicted the tiedepedet stress iduced i the aterial as a hyperlgistic-type fucti, which is able t reprduce ay S-shaped curve as shw by uerical exaples. These predicted results by the prpsed del are i very agreeet with thse published i the literature. The prpsed del is a extesi f the classical Maxwell del t large defratis by eas f tw paraeters ad. It appeared evidet that fr = ad =, the preset del reduces t the well-kw Maxwell del. Csequetly, these paraeters assure the rle f liearity cefficiets. 4. Cclusis A cplete characterizati f viscelastic aterials is very difficult t perfr, due t the fact that the echaical respse f these aterials is tie-depedet ad histry-depedet, ad rever, their stress-strai curve is liear. Fllwig this viewpit, liear theretical dels are ecessary t better predict ad uderstad the tie-depedet behavir f aterials. Fr this purpse, a liear geeralized Maxwell del has bee develped. The del allwed, accrdig t the btaied results, describig atheatically ad accurately the liear tie-depedet stress i se viscelastic aterials, as a hyperlgistic-type fucti, that is pwerful t represet ay sigid curve. The preset del, i particular, is shw t be very sesitive t the agitude f the strai rate. Altgether, re experietal results ad practical tests are eeded t further validate the feasibility f this del. 5. Refereces [] T. Alfrey ad P. Dty, The Methds f Specifyig the Prperties f Viscelastic Materials, Jural f Applied Physics, Vl. 6, N., 945, pp di:0.063/ [2] R. Chtard-Ghdsia ad C. Verdier, Rhelgy f Livig Materials, I: F. Mllica, L. Prezisi ad K. R. Rajagpal, Eds., Mdelig f Bilgical Materials, Spriger, New Yrk, 2007, pp. -3. di:0.007/ _ [3] D. T. Crr, M. J. Starr, R. Vaderby, Jr ad T. M. Best, A Nliear Geeralized Maxwell Fluid Mdel fr Viscelastic Materials, Jural f Applied Mechaics, Vl. 68, N. 5, 200, pp di:0.5/ [4] M. D. Msia, Labert ad Hyperlgistic Equatis Mdels fr Viscelastic Materials: Tie-Depedet Aalysis, Iteratial Jural f Mechaical Egieerig, Serials Publicatis, New Delhi, Idia, Jauary-Jue 20. [5] M. D. Msia, A Hyperlgistic-Type Mdel fr the Predicti f Tie-Depedet Nliear Behavir f Viscelastic Materials, Iteratial Jural f Mechaical Egieerig, Serials Publicatis, New Delhi, Idia, Jauary-Jue 20. [6] M. D. Msia, A Nliear Geeralized Stadard Slid Cpyright 20 SciRes.

10 M. D. MONSIA 67 Mdel fr Viscelastic Materials, Iteratial Jural f Mechaical Egieerig, Serials Publicatis, New Delhi, Idia, Jauary-Jue 20. [7] M. D. Msia, A Mdified Vigt Mdel fr Nliear Viscelastic Materials, Iteratial Jural f Mechaical Egieerig, Serials Publicatis, New Delhi, Idia, Jauary-Jue 20. [8] M. D. Msia, A Nliear Geeralized Fur-paraeter Vigt Mdel fr Viscelastic Materials, Iteratial Jural f Mechaical Egieerig, Serials Publicatis, New Delhi, Idia, July-Deceber 20. [9] C. Debuche, Présetati Crdée de Différets Mdèles de Crissace, Revue de Statistique Appliquée, Vl. 27, N. 4, 979, pp [0] O. Garcia, Uifyig Sigid Uivariate Grwth Equatis, Frest Bietry, Mdellig ad Ifrati Scieces, Vl., 2005, pp Cpyright 20 SciRes.