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4: Non-Linear Viscoelasticity

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4: Non-Linear Viscoelasticity Normal stress effects (Weissenberg et al. 1947) 4: non-linear viscoelasticity Main goals Ranges of viscoelasticity: the Deborah number - To develop constitutive models that can describe non-linear phenomena such as rod climbing - To use the equations in practical applications such as polymer processing and soft tissue mechanics A dissappointing point of view …… Three major topics: - Non-linear phenomena (time dependent) - Normal stress difference - Shear thinning - Extensional thickening - Most simple non-linear models - More accurate models Normal stress differences in shear Shear thinning - Data for a LDPE melt - Lines from the Kaye-Bernstein, Kearsley, Zapas (K-BKZ) model For small shear rates: Normal stress coefficients Shear thinning & time dependent viscosity Interrelations between shear functions For small shear rates: Cox-Merz rule Gleissle mirror rule Lodge-Meissner; after step shear: LDPE Steady shear (solid line) Cox Merz (open symbols) Gleissle mirro rule (solid points) Extensional thickening Extensional thickening Time-dependent, uniaxial uniaxial extensional viscosity extensional viscosity versus shear viscosity Non-linear behavior in uniaxial extensional viscosity is more sensitive to molecular archtecture than non-linear behavior in shear viscosity PS, HDPE: no branches 2 LDPE’s: branched, tree-like Stressing viscosities Second order fluid Simplest constitutive model that predicts a first normal stress difference: Upper convective derivative (definition): m = ½ uniaxial extension m = 1 biaxial extension Finger tensor: m =0 planar extension with the substantial or material derivative: Extremely rare results Second order fluid in simple shear Second order fluid in uniaxial extension For: extensional thickening for low extension rates Useful for non-uniform complex flow More complex model (Criminale-Ericksen-Filby): Can not predict neither stress Growth or stress relaxation Upper Convected Maxwell model Upper Convected Maxwell model: shear flow Start-up flow 1-D Maxwell model (linear viscoelasticity): Homogenous flow = 0 - Non-linear viscoelastic model (product of and ) Symmetry: - Small strain: non-linear terms dissapear, material derivative - Steady flow, small strain rate: Newtonian flow - Transient flow, high strain rates: Neo-Hookean. Upper Convected Maxwell model: shear flow Upper Convected Maxwell model: shear flow Start-up flow Stress growth, non-zero components: Steady state results (time derivatives are zero) Viscosity and first normal stress coefficient are constant Upper Convected Maxwell model:extension Upper Convected Maxwell model Steady state uniaxial extension Integral form How to use integral models ? Different shear histories . Extremely extension thickening, viscosity rises to infinity when : → : strain accumulated between t an t’ Upper Convected Maxwell model Upper Convected Maxwell model Step shear strain Start-up uniaxial extension Start-up steady shear Upper Convected Maxwell model Upper Convected Maxwell model Start-up uniaxial extension Start-up uniaxial extension Upper Convected Maxwell model Upper Convected Maxwell model Start-up uniaxial extension Integral form; multi mode Corresponding differential forms Upper Convected Maxwell model Upper Convected Maxwell model Summarizing: Oldroyd-B constitutive equation (1950) Including the (viscous) solvent contribution Pro’s - Recaptures all of the linear viscoelastic modelling - Newtonian / Neo-Hookean behavior for the limiting case (slow / fast flow) - Predicts first normal stress difference and extensional thickening HWM polyisobutylene in Con’s poly(1-butene) / kerosene - No second normal stress difference - No shear rate dependence of viscosity and first normal stress difference __ UCM equation (i.e no shear thinning) --- Oldroyd-B - extensional thickening is too severe Works for very dilute solutions (< 0.5% concentration) and dilute solutions with very high solvent viscosities (Boger fluids) More accurate constitutive models More accurate constitutive models Integral constitutive models Integral constitutive models Lodge: How to obtain the right energy function: Lodge, step strain Two invariants IB, IIB and time t: - lots of experiments required (problem!) - or guidance from molecular theory Use general elastic solid: Time-dependent elastic First, restrict tot simple shear flow energy function Special case: Lodge: More accurate constitutive models More accurate constitutive models Integral constitutive models: simple shear Integral constitutive models: simple shear / step strain The function can be obtained by taking the tome derivative of the relaxing shear stress after step shear strain With the expression: one can obtain and More accurate constitutive models More accurate constitutive models Time-strain factorabilibty Simple shear / step strain - M(t-t’) from linear viscoelastic measurements - Non-linear measurements for U(IB,II B) -For τ12 and N1 the so-called damping function h(γ) needs to be measured Time-strain factorability!! (works also for other than shear) More accurate constitutive models More accurate constitutive models Stress growth Simple shear / step strain /damping function h(γ) With Gi, λi and h(γ) known predictions for various shear flows can be made Steady state viscosity & first Wagner (1976) (--) normal stress coefficient Laun (1978) (__) Khan & Larson (1987) Stress relaxation Notice that the Lodge-Meisner relation is obeyed ( ) A single damping function captures a wealth of non-linear shear data for many polymer melts More accurate constitutive models More accurate constitutive models Factorization doesn’t work always Concentrated polystyrene solution More general expression: Also problems with strain reversal (complete failure) Doesn’’t work for other type of flows (extensional) Based on a molecular theory, limited applicability More accurate constitutive models Maxwell-type differential equations Recent integral models: - Wagner stress function - Pom-Pom model (Larson-McLeish) Modifies the rate of stress build up Modifies the rate of stress decay Multi-modes required to describe experimental data Larson model Phan Thien-Tanner model Step shear Uniaxial extension Step biaxal extension Maxwell-type differential equations Notice: mostly only one Maxwell-type differential equations non-linear parameter!! Maxwell-type differential equations Maxwell-type differential equations Recent differential models Example: viscosity and normal stress - Pom-Pom differential approximation coefficients for Johson-Segelman model - eXtended Pom-Pom (XPP) - Rolie-Poly Steady state: Some algebra → Summary.
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