Creep in Viscoelasic Subsances Numerical mehods o calculae he coefficiens of he Prony equaion using creep es daa and Herediary Inegrals Mehod Navnee Saini, Mayank Goyal, Vishal Bansal (23); Term Projec AML3; Indian Insiue of Technology Delhi Absrac Sudy of Viscoelasic subsance in an enclosed boundary volume. This erm paper aims a he developmen of a model for sudying and modelling a creep es in a viscoelasic maerial. The basic aim of our projec is narrowed down o he applicaion of recurring numerical mehods o he soluion of coefficiens of he Prony equaion/model of he represenaion of Viscoelasic maerials. We have used he creep es daa from NASA on a maerial and used he daa for he calculaion of he maerial properies namely he above coefficiens. Lieraure Survey Viscoelasic fluids (subsances o be exac) exhibi resemblance o an elasic solid under some condiions and a viscous liquid under oher. There are he following models amongs many for he sudy of Viscoelasic subsances: MODEL MAXWELL MODEL KELVIN-VOIGT MODEL MAIN ASSUMPTION This is represened by a purely viscous damper and a purely elasic spring conneced in series. This model employs a Newonian Damper and a Hookean elasic spring. CHARACTERISTIC EQUATION dε oal d = dε D d + dε S d = σ η + dσ E d σ() = Eε() + η dε() d SYMBOLIC MODEL STANDARD LINEAR SOLID MODEL I combines a Hookean Spring and he Maxwell Model in parallel E 2 dε ( η dσ η d = E 2 d + σ E ε) E + E 2 GENERALISED MAXWELL MODEL PRONY SERIES I considers ha he relaxaion does no occur a a single ime, bu wih a disribuion of ime inervals (i.e. a variable ime disribuion). I has muliple spring-dashpos in parallel as required for he accurae represenaion of he model Takes ino accoun he linear model The equaion for creep: only wih a ε() = σ combinaion of. J() + ( ξ) dσ(ξ) dashpo and spring.
P a g e 2 Linear viscoelasic consiuive models are represened by simple physical models composed of springs and dashpos. The spring is he linear-elasic componen, and is consiuive equaion is σ = E. ε. The dashpo is he viscous componen, and is consiuive equaion is σ = η. ε where η is he viscosiy consan. Mahemaical Modelling (Prony Series) The consiuive equaion of a viscoelasic maerial characerises dependence upon he boh he srain and he srain rae. The equaion for creep: ε() = σ. J() + ( ξ) dσ(ξ) The coupling of σ = E. ε and σ = η. ε gives σ() = Y(). ε where Y() = E. ( p i is he i h Prony Consan τ i is he i h Prony reardaion ime consan E is he insananeous modulus of he maerial n i= p i. ( e τ i In a creep es, a creep compliance funcion, J() is defined as ε() = J(). σ n Y() = E. ( p i i=. ( e τ i)) For ime =, Y() = E and for =, y( ) = E (-Σpi). The compliance funcion is hen deermined by procedures analogous o hose described To deermine he sress sae in a viscoelasic maerial a a given ime, he deformaion hisory mus be considered. For linear viscoelasic maerials, a superposiion of herediary inegrals describes he ime dependen response a. If a specimen is load free prior o he ime = O, a which a sress, σ + σ(), is applied he srain for ime > can be represened as follows. ε() = ε. J() + J( ξ) dσ(ξ) Where J() is he compliance funcion of he maerial and dσ(ξ) is he sress rae. A similar equaion can be used for he relaxaion model o obain he sress funcion inroduced by an arbirary srain funcion σ() = ε. Y() + Y( ξ) dε(ξ) where Y() is he relaxaion funcion (Equaion 4) and dε(ξ) _is he srain rae. An example of applying herediary inegrals for a muliple loading segmen process is shown nex. )) Herediary inegrals for a muliple loading process Herediary inegrals wih Prony series kernels can be applied o model a loading process such as he one shown in Figure 2. The process is divided ino four segmens for which srain and srain rae funcions are defined. The funcions are: Navnee Saini, Mayank Goyal, Vishal Bansal
P a g e 3 ε. ( ) < ε() = { ε ε. ( 3 2 ) < 2 ; 2 < 3 2 < 4 ε dε ( ) < d = { < 2 ; 2 < 4 ε ( 3 2 ) 2 < 3 where ε = ε() = and =. And we have used ε =.; ε 2 =; =; =5; 2 =55; 3 =6 For a maerial wih a relaxaion funcion in he form of a Prony series (Equaion 4), he sress funcions of he loading process can be derived as follows: Sep. ( < ) Subsiue Equaions 4 and firs srain and srain rae funcions of Equaion 8 ino Equaion 7 and obain: σ() = ε. Y() + Y( ξ) dε(ξ) ] = E. ε [ξ p i. ξ + p i. τ i. e ( ξ) τi = E. ε [ p i. + p i. τ i. e τ i ] where n is he number of erms in he Prony series. To simplify he expression, n will no be shown in following equaions. Pi and "ciare he consans in he i-h erm of he Prony series. Sep 2. ( < 2 ) Using he second srain rae funcion obain: σ() = ε. Y() + Y( ξ) dε(ξ) + Y( ξ) dε(ξ) = E. ε [ p i. + p i. τ i. e ( ) τi + p i. τ i. e τ i ] Sep 3. ( 2 < 3 ) The hird srain rae funcion yields: σ() = E. ε [ p i. + p i. τ i. e ( ) τi + p i. τ i. e τ i ] + E. ε ( 3 2 ) [ p i. + p i. τ i 2 + p i. 2 + p i. τ i. e ( 2) τi Noe, he firs porion of his equaion is equal o he equaion of sep 2. Sep 4. ( 3 < 4 ) Similarly, he equaion for he fourh sep can be wrien as follows: σ() = σ 2 () E. ε ( 3 2 ) [ p i. + p i. τ i 2 + p i. 2 + p i. τ i. e ( 2) τi ] Mehod o Find he Consans of Prony Series Weighed non-linear regression: The Prony series coefficiens and reardaion ime are deermined using regression analysis. Regression analysis is basically a mehod used o deermine relaion beween he wo variables using experimenal resuls. In his mehod one independen variable is varied while keeping oher consan and he variable wih which relaion is o be found ou is ploed, and hen from graph we use differen echniques o find he relaion. One mosly used echnique is error mehod. ] Navnee Saini, Mayank Goyal, Vishal Bansal
P a g e 4 In his error funcion (f) wih respec o unknown consans is defined as, n F(a) = { y i y(x i, a) } 2 σ i Where xi and yi are experimenal daa, funcion y (xi, a) is he model o be fied, and σi is he sandard deviaion of measuremen error of i h daa poin. A se of unknown consans (a) will be deermined ha minimize he error funcion approximaed by is Taylor series wih he quadraic form: a. D. a F (a) = c d. a + 2 Where c is a consan and d is he gradien of F wih respec o he parameers a, which will be zero where F is minimum. Marix D is he second parial derivaive of F wih respec o he parameers a. Since each Prony erm includes wo variables (pi and ci) and since he insananeous modulus (E) mus be deermined, he oal number of variables in he regression is 2n+l. Since viscoelasic effecs are mos significan a he beginning of he relaxaion period, he fiing error in his region is significan. Since he percenage of he number of daa poins a he beginning of he relaxaion period is less, he error funcion χ 2 is dominaed by a long uniform ail region of he relaxaion period. To reduce he error and improve he fi a beginning of he relaxaion period, he weigh funcion [w = ; < σ <] is used. σ Numerical Soluions Working Non Linear for Regression Mehod The specimen (2 in. x 2 in. x.768 in.) was loaded in 22 seconds o a maximum deflecion of.3 in. a he middle of he span. Then he deflecion was held for,7 seconds (Figure 3). Since no analyical mehod exiss o form he weigh funcion, a rial-and-error mehod based on maerial properies was used o obain a fi curve. The viscoelasic sress decays exponenially in he relaxaion es. The viscoelasic sress decays exponenially in he relaxaion es, herefore he viscoelasic effec is more significan during he loading period and a he beginning (< 3seconds) of he holding period. The number of daa poins in hese periods (<) is much less han he number of daa poins in ail region of he relaxaion period (>5). The error funcion (F) will be dominaed by along uniform ail region of he relaxaion period if a uniform weigh funcion is applied. Therefore, a piecewise weigh funcion was used o obain beer fis for hese periods and improve he accuracy of he regression. Figure4 shows he load relaxaion a he beginning of he process. The dos represen he es daa. Three regression resuls are shown. The dash-do curve is he resul of a regression analysis wihou he weigh funcion (w/o WF) for a wo erm Prony series. Navnee Saini, Mayank Goyal, Vishal Bansal
P a g e 5 The long-dash curve is he resul for a wo-erm Prony series wih weigh funcion number (WF) shown below: 5 < 22 w = { 22 5 5 < 7 The shor-dash curve is he resul for a hree-erm Prony series wih weigh funcion 2 (WF2) as follows:. < 22 w = { 6 ( 2 )22 < 7 As is well known, he regression daa fis beer in a paricular region if he relaive weigh facor (weigh facor / sum of weigh facors) in ha region is greaer han he average value. In Figure 4, he curves wih weighed funcions were closer o he daa poins near he beginning of he relaxaion period. Since he weigh facor a = 22 seconds of funcion 2 (=266) is greaer han he value of he weigh funcion (=), he curve of funcion 2 is a beer mach o he daa han curve of funcion a he beginning of relaxaion. However, as shown in he Figure3, weigh funcion2 does no fi he daa as well as he oher curves afer seconds. I appears ha funcion2 was over weighed a he beginning of he relaxaion and he relaive weigh facor a he oher region was oo small. The Prony consans of he regression are shown in Table. The resuls showed ha a weigh funcion should be seleced based on he maerial properies and es daa disribuion. Properly selecing he weigh funcion in he mos significan region can improve accuracy of he regression. Navnee Saini, Mayank Goyal, Vishal Bansal
P a g e 6 Conclusions In his we came wih echnique o deermine he Prony coefficiens ha can represen viscoelasic sress and srain equaion. The herediary inegral mehod was employed o obain an analyical represenaion of maerial response when i is subjeced o rae dependen loading. The analyical represenaion was used in a nonlinear regression analysis, wih measured daa, o evaluae he Prony series consans. Several regression analyses were performed using differen weigh funcions. Noe, he daa of similar lengh in ime had been analyzed for loading and relaxaion regions. Oher weighing funcions may be needed for differen loading schedules. The mehod could be used o provide highly accurae represenaion of he maerial behavior in he rae dependen loading region. Since once afer consans are deermined, he response of a viscoelasic maerial for oher loading schedules can also be calculaed. I can also be used o deermine equaion for oher loading schedules where we can keep relaxaion ime from our choice. References Deermining a Prony Series for a Viscoelasic Maerial from Time Varying Srain Daa (26); Tzikang Chen; U.S. Army Research Laboraory, Vehicle Technology Direcorae, Langley Research Cener, Virginia; Naional Aeronauics and Space Adminisraion. Microfluidic Flows of Viscoelasic Fluids; Monica S. N. Oliveira, Manuel A. Alves, Fernando T. Pinho; Transpor and Mixing in Laminar Flows: From Microfluidics o Oceanic Currens. Engineering Viscoelasiciy (2); David Roylance; Deparmen of Maerials Science and Engineering, Massachuses Insiue of Technology. Aachmens. Sress.m 2. Srain.m 3. Presenaion.pp Navnee Saini, Mayank Goyal, Vishal Bansal