Modeling the Nonlinear Rheological Behavior of Materials with a Hyper-Exponential Type Function

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1 Mechanical Engineering Research Vl. 1, N. 1; December 011 Mdeling the Nnlinear Rhelgical Behavir f Materials with a Hyper-Expnential Type Functin Marc Delphin Mnsia Département de Physique, Université d Abmey-Calavi Abmey-Calavi, 09 B.P. 305, Ctnu, Bénin Tel: mnsiadelphin@yah.fr Received: Nvember 8, 011 Accepted: December 11, 011 Published: December 31, 011 di: /mer.v1n1p103 URL: Abstract This paper describes a nnlinear rhelgical mdel cnsisting f a mdified and extended classical Vigt mdel fr predicting the time dependent defrmatin f a variety f viscelastic materials exhibiting elastic, viscus and inertial nnlinearities simultaneusly. The usefulness f the mdel is illustrated by numerical examples. Keywrds: Hyper-expnential functin, Lambert-type equatin, Mathematical mdeling, Nnlinear time dependent defrmatin, Viscelasticity, Vigt mdel 1. Intrductin In viscelasticity, mathematical mdels are required fr studying the time dependent prperties f materials under varius lading cnditins. In the characterizatin f materials, the well knwn established linear thery f viscelasticity is nly valid fr small defrmatins r lw stresses. When the material underges large defrmatins the linear thery becmes inapplicable, and nnlinear mdels are needed. Cntrary t the linear thery f viscelasticity that is usually described in the Bltzmann single integral r in the differential frm, a standard framewrk des nt exist in nnlinear viscelasticity. Therefre, nnlinear mathematical rhelgical mdels are ften cnstructed by thrugh mdificatins and extensins t higher rder stress r strain terms f the linear thery. Frm a mathematics pint f view, the integral representatin f viscelastic cnstitutive equatin is mre difficult t perfrm than the differential frm. Thus, several mdels with varius cmplexities have been develped fr describing the nnlinear behavir f these materials that are characterized by elastic, viscus and inertial nnlinear cntributins (Bauer et al. 1979; Bauer 1984). Hwever, in these mdels, due t the mathematical cmplicatins, nly the elastic r viscus nnlinearity is ften taken int accunt (Mnsia 011a,b,c) and the inertial cntributin is ignred. Mrever, there are nly a few theretical mdels frmulated with cnstant-value rhelgical material parameters. Therefre, nnlinear mdels with cnstant rhelgical cefficients are required. Fllwing this viewpint, by using a secnd-rder elastic spring in series with a classical Vigt element, that is an extended frm f the standard linear slid t finite strains, Mnsia (011a) frmulated a hyperlgistic-type equatin t reprduce the nnlinear time dependent stress respnse f sme viscelastic materials. Recently, in Mnsia (011b), a single differential cnstitutive equatin derived frm a standard nnlinear slid mdel cnsisting f a plynmial elastic spring in series with a classical Vigt element fr the predictin f time dependent nnlinear stress f a class f viscelastic materials is develped. Mre recently, in Mnsia (011c), a nnlinear fur-parameter rhelgical Vigt mdel cnsisting f a nnlinear Vigt mdel in series with a classical linear Vigt element with cnstant material cefficients fr representing the nnlinear stiffening respnse f the initial lw-lad prtin and the sftening behavir f sme viscelastic materials is frmulated. In Mnsia (011d) the elastic and viscus nnlinearities are taken simultaneusly int cnsideratin thrugh a simple nnlinear generalized Maxwell fluid mdel cnsisting f a nnlinear spring cnnected in series with a nnlinear dashpt beying a pwer law with cnstant material cefficients. Accrding t Bauer (1984), suitable cnstitutive equatins f viscelastic materials must relate stress, strain and their higher time derivatives, t say, must take int cnsideratin the elastic, viscus and inertial nnlinearities simultaneusly. T vercme the mathematical cmplexities in viscelastic mdeling, Bauer (1984) develped, fr a cmplete characterizatin f rhelgical prperties f arterial walls, a thery based n the classical Vigt mdel. The Bauer s thery (1984) is aimed t give satisfactrily and simultaneusly accunt f strng elastic, viscus and inertial nnlinearities characterizing a viscelastic material. The methd cnsists essentially t Published by Canadian Center f Science and Educatin 103

2 Mechanical Engineering Research Vl. 1, N. 1; December 011 decmpse the ttal stress exciting the material as the sum f three cmpnents, that is t say, the elastic, viscus and inertial stresses and t express the pure elastic stress as a nnlinear functin f defrmatin. The pure viscus and inertial stresses are then frmulated as a first and secnd time derivatives f a similar functin f defrmatin t the nnlinear elastic functin, respectively. A fundamental theretical difficulty in the use f the Bauer s thery (1984) cnsists f the determinatin f apprpriate nnlinear elastic restring frce functin that tends twards the expected linear elastic behavir fr small defrmatins. In the Bauer s study (1984), the pure elastic stress is expanded in a pwer series f strain, the pure viscus stress is develped as a first time derivative f a similar pwer series f strain, and the pure inertial stress is expressed as a secnd time derivative f a similar pwer series f strain. The Bauer s stress decmpsitin methd (1984), cnsisting t express the stress as a sum f three elementary stresses, has been after used by many authrs (Armentan et al. 1995; Gamer et al. 001) fr a cmplete characterizatin f arterial behavir. Recently, Mnsia (011e), using the Bauer s methd (1984), develped a hyperlgistic equatin that is useful fr representing the time dependent behavir f sme viscelastic materials. In Mnsia (011e), fllwing the Bauer s thery (1984), the pure elastic stress is develped in an asympttic expansins in pwers f defrmatin, the pure viscus stress is expressed as a first time derivative f a similar asympttic expansins in pwers f defrmatin, and the inertial stress is frmulated as a secnd time derivative f a similar asympttic expansins in pwers f defrmatin. In this wrk, using a hyperblic functin f defrmatin fr the pure elastic cnstitutive relatinship, we develped fllwing the Bauer s thery (1984) a ne-dimensinal nnlinear theretical rhelgical mdel with cnstant material cefficients taking int accunt all tgether elastic, viscus and inertial nnlinearities characterizing viscelastic materials. The theretical btained results shw that the mdel can be successfully applied t represent the nnlinear time defrmatin f sme viscelastic materials. Numerical examples are perfrmed t illustrate the effects f rhelgical parameters actin n the material respnse.. Frmulatin f the Mathematical Mdel.1 Theretical cnsideratins In this part we develp the gverning equatins including the elastic, viscus and inertial nnlinearities with material cnstant cefficients. Mst viscelastic materials exhibit elastic, viscus and inertial nnlinearities, s that their mechanical respnses are nnlinear time dependent, and require advanced mathematical mdels fr their descriptin. T that end, the use f nnlinear viscelasticity is suggested (Bauer 1984). T cnstruct then ur prpsed ne-dimensinal rhelgical mdel, in which the stresses and strain are scalar functins, we will use the Bauer s thery (1984) that brings significant mdificatins and extensins t the classical mechanical Kelvin-Vigt mdel fr vercming the preceding mentined difficulties. The first significant mdificatin herein cnsists t intrduce a nnlinear restring spring frce functin ( ) f defrmatin fr capturing the pure elastic cmpnent f the stress induced in the material studied. Then, fr a general frmulatin, the pure elastic cnstitutive equatin may be written in the frm a() e (1) where a is a stiffness cefficient. The secnd imprtant mdificatin invlves the fact that the pure nnlinear viscus cnstitutive equatin is then directly given by d b( ) dt v () where b is a viscsity mdule. The third significant mdificatin cnsists t derive frm ( ) the nnlinear inertial stress cmpnent as d c ( ) dt i (3) in which c is an inertia mdule. The cefficients a, b and c are time independent material parameters. Thus, nting σ t the ttal stress due t the external exciting frce acting n the material, the superpsitin f elastic stress, viscus stress and inertial stress is given by 104 ISSN E-ISSN

3 Mechanical Engineering Research Vl. 1, N. 1; December 011 d d b d a 1 ( ) t d d c d c c (4) where the dt ver the symbl dentes a differentiatin with respect t time and the inertial mdule c is different frm zer. Equatin (4) determines the differential cnstitutive relatinship between the ttal external stress σ t and the resulting strain (t) fr a given nnlinear functin ( ). It is required, at this stage f the mdel-building, t specify the nnlinear functin ( ). In this paper, the nnlinear spring frce functin ( ) is chsen as the fllwing hyperblic law Thus, using Equatin (5), Equatin (4) may be written as ( ) 1 (5) b a 1 ( 1) ( 1) t 1 c c c Equatin (6) dentes a secnd-rder nnlinear rdinary differential equatin in (t) fr a given ttal stress σ t. (6). Dimensinalizatin Nting that the strain (t) is a dimensinless quantity, the abve material parameters used in Equatin (6) have then the fllwing dimensins. Nting als M, L and T the mass, length and time dimensin respectively, the dimensin f the stress becmes ML -1 T -. Therefre, the dimensin f a is given by ML -1 T -, that f b varies as ML -1 T -1, and that f c varies as ML -1 (mass per unit length)..3 Slutin using a stress t Evlutin equatin f the defrmatin (t) In the absence f external exciting stress ( t 0 ), the internal dynamics f the viscelastic material studied can be represented by the fllwing nnlinear rdinary differential equatin r b a where, and. c c b a ( 1) 0 1 c c ( 1) 0 (7) 1 Equatin (7) represents the time dynamics equatin f the strain (t) induced in the material under the external exciting stress fixed t zer. By intrducing the auxiliary variable Equatin (7) transfrms, after sme algebraic manipulatins, int x 1 (8) x x x x x 0 (9) x In Equatin (9), the first term is prprtinal t the basic inertial stress, the secnd t a nnlinear quadratic viscus stress, the third term t the linear viscus stress, the furth term t the classical linear elastic stress and the last term t a quadratic nnlinear elastic stress. Equatin (9) is a Lambert-type differential equatin which can be analytically slved by using an apprpriate change f variable and suitable bundary and initial cnditins that Published by Canadian Center f Science and Educatin 105

4 Mechanical Engineering Research Vl. 1, N. 1; December 011 satisfy the time dynamics f the viscelastic material cnsidered..3. Slving time dependent defrmatin equatin Fr slving Equatin (9), a nvel change f variable is required. Making the fllwing substitutin 1 x y (10) with y 0, Equatin (9) transfrms, after a few algebraic manipulatins as y y (11) Equatin (11) is the well-knwn secnd-rder linear rdinary differential equatin with a right-hand member different frm zer. Integratin yields fr y (t) the fllwing slutin where y y t) A exp( rt) A exp( r t) 1 (1) ( 1 1 and r (1 1 ) r (1 ) are the tw negative real rts f the characteristic equatin r r 0 with 1 4 A 1 and A are tw integratin cnstants determined by the initial cnditins. Thus, using the suitable initial cnditins and t 0, ( t) 0 t 0, ( t) f where the cefficient f dentes the initial strain rate, and taking int cnsideratin Equatin (8) and Equatin (10), the desired strain versus time relatinship can be written as ( t) 1 f exp( ( ) t) exp( (1 1 4 ) t) Equatin (13) describes mathematically the time dependent strain in the viscelastic material under study. It predicts the strain versus time relatinship f the material studied as a hyper-expnential type functin that asympttically appraches a maximum value with increasing time. (13) 106 ISSN E-ISSN

5 Mechanical Engineering Research Vl. 1, N. 1; December Numerical Results and Discussin In this sectin sme numerical examples are perfrmed t illustrate the predictive quality f the mdel fr representing the mechanical behavir f the material cnsidered and the influence f rhelgical cefficients actin n the material respnse. Since viscelastic materials are characterized by high elastic, viscus and inertial nnlinearities, advanced analytical frmulatins are needed fr determining and predicting accurately their prperties. In this regard the Bauer s thery (1984) becmes an imprtant mathematical tl in viscelastic mdeling. Fr the present mdel, the nnlinear restring spring frce functin is expressed as a hyperblic law. The hyperblic law has been extensively used in sils cnstitutive mdeling. The hyperblic functin prpsed here is inspired by the wrk (Prevst and Keane 1990). This law gives the advantage t be an exact analytical expressin, cmpared with the asympttic series used in Mnsia (011e). Mrever, the chice f Equatin (5) prceeds frm the fact that when 1, Equatin (5) reduces t a gemetric series frmula, s that fr small defrmatins, ( ) shws linear behavir as expected. In this respect, the prpsed hyperblic law agrees well with the pwer series f defrmatin used by Bauer (1984). Anther feature that invlves the advantage f Equatin (5) is that ( ) becmes infinite frce fr a finite defrmatin, that is, fr 1, in ther wrds, fr a strain range f abut 100 %, like the nnlinear elastic FENE (Finitely Extensible Nnlinear Elastic) spring frce, cntrary t plynmial restring frce in which the defrmatin shuld tend als twards infinity. Fr, that is, fr 1, ( ) tends twards a finite value. By applying the Bauer s thery (1984) and using the preceding hyperblic restring frce functin, the gverning equatin btained is a Lambert-type nnlinear differential equatin. By cnsidering then suitable change f variable and initial cnditins, this equatin led t btain the time versus strain relatinship as a hyper-expnential type functin, demnstrating that the present mdel is applicable t represent mathematically the mechanical behavir f a variety f viscelastic materials. Hwever, mre experimental results are required t further verify the feasibility f the mdel. 4. Cnclusins A nnlinear rhelgical mdel with cnstant material cefficients, taking int accunt all tgether elastic, viscus and inertial nnlinearities, is develped. The theretical btained results shwed that the mdel can be successfully applied t represent the nnlinear time defrmatin behavir f sme viscelastic materials. Numerical results illustrated als the effects f rhelgical parameters actin n the defrmatin respnse f the material studied, and particularly the sensitivity f the mdel t the initial rate f applicatin f the strain. The present uniaxial mdel can act as a starting pint t a mre general three dimensinal mdel, but this study will be dne as future wrk. References Armentan R. L., Barra J. G., Levensn J., Simn A., & Pichel R. H. (1995). Arterial wall mechanics in cnscius dgs. Assessment f viscus, inertial, and elastic mduli t characterize artic wall behaviur. Circ Res. 76, Bauer R. D., Busse R., Schabert A., Summa Y., & Wetterer E. (1979). Separate determinatin f the pulsatile elastic and viscus frces develped in the arterial wall in viv. Pflügers. Arch. 380, Bauer R. D. (1984). Rhelgical appraches f arteries. Birhelgy. Suppl. I, Gamer L. G., Armentan R. L., Barra J. G., Simn A., & Levensn J. (001). Identificatin f Arterial Wall Dynamics in Cnscius Dgs. Exp. Physil. 86, Lesecq S., Baudin V., Kister J., Pyart C., & Pagnier J. (1997). Influence f the A Helix Structure n the Plymerizatin f Hemglbin S*. The Jurnal f Bilgical Chemistry 7, Mnsia M. D. (011a). A Hyperlgistic-type Mdel fr the Predictin f Time-dependent Nnlinear Behavir f Viscelastic Materials. Int. J. Mech. Eng. 4, 1-4. Mnsia M. D. (011b). A Nnlinear Generalized Standard Slid Mdel fr Viscelastic Materials. Int. J. Mech. Eng. 4, Mnsia M. D. (011c). A Mdified Vigt Mdel fr Nnlinear Viscelastic Materials. Int. J. Mech. Eng. 4, Mnsia M. D. (011d). A Simplified Nnlinear Generalized Maxwell Mdel fr Predicting the Time Dependent Behavir f Viscelastic Materials. Wrld Jurnal f Mechanics. 1, Published by Canadian Center f Science and Educatin 107

6 Mechanical Engineering Research Vl. 1, N. 1; December Mnsia M. D. (011e). Lambert and Hyperlgistic Equatins Mdels fr Viscelastic Materials: Time-dependent Analysis. Int. J. Mech. Eng, 4, Prevst J. H., & Keane C. M. (1990). Shear stress-strain curve generatin frm simple material parameters. Jurnal f Getechnical Engineering, 116, Figure 1. Typical strain versus time curve Figure 1 shws the typical time versus strain variatin with an increase until a maximum value, btained frm Equatin (13) with the fixed value f parameters at, 0. 5, f 1. It can be seen frm Figure 1 that the prpsed mdel is able t reprduce mathematically and accurately the typical expnential defrmatin respnse f a variety f viscelastic materials, fr example, sft living tissues (Lesecq et al., 1997). The mdel predicts a time dependent respnse in which the slpe declines gradually with increasing time until the failure pint at which the slpe reduces t zer. Figure. Strain versus time curves shwing the effect f the cefficient Figure, 3 and 4, illustrates the effects f material cefficients n the time dependent strain respnse f the material studied. These effects are studied by varying ne cefficient while the ther tw are kept cnstant. Figure exhibits the effect f the damping cefficient λ change n the strain-time relatinship. The graph shws that an increasing λ, reduces the value f the strain n the time perid cnsidered. The slpe als decreases with increase λ. The red line crrespnds t, the blue line t 3, and the green line t 4. The ther parameters are 0.5, f ISSN E-ISSN

7 Mechanical Engineering Research Vl. 1, N. 1; December 011 Figure 3. Strain-time curves at three varius values f cefficient As shwn in Figure 3, an increase f the natural frequency cefficient, decreases als the value f the strain n the time perid cnsidered. The slpe decreases slwly in the early perids f time with increase. The red line crrespnds t 0. 5, the blue line t 1, and the green line t The ther parameters are 4, f 1. Figure 4. Strain-time curves fr three different values f the parameter Figure 4 exhibits hw the initial rate f applicatin f the strain f affects the time dependent respnse f the material studied. The graph shws that the curves becme mre linear with decreasing f. The slpe decreases als with decreasing f. The red line crrespnds t f 0. 01, the blue line t f 0. 1, and the green line t f 1. The ther parameters are, f Published by Canadian Center f Science and Educatin 109