3.11 Mechanics of MaterialsRecitation #10Rubber ElasticityRubber elasticity, cont…ViscoelasticityTwo common tests used to characterize viscoelasticityTwo common tests used to characterize viscoelasticityTemp. effect on amorphous polymersGraph to demonstrate last slideSpring-dashpot models for viscoelasticitySpring-dashpot Model #1Model #2Viscoelastic ProblemsAnswers to Problem 2Solution Problem 33.11 Mechanics of MaterialsRecitation #10November 18, 2003TA: Kristin DomikeRubber Elasticity• An extra sample problem – Just a little review of what we’ve just covered– Problem #1• If a single polymer chain has a Helmholtz free energy of 1.543*10^-18 J when its end-to-end distance is 2.25um, what is the chain stiffness k? In that configuration, what is the restoring force?• At what end-to-end distance can a single polymer chain have no restoring force acting on it?Rubber elasticity, cont…Viscoelasticity• Time dependent behavior can be modeled by spring-dashpot models• A linear-viscoelastic(applications:polymers, biomatls)– Behavior between that of elastic solid and viscous fluid– Time and temperature dependent behaviour–Linear: σ & ε are linearly related at a given time and temperature• Elastic Solid: σ = E ε (Hooke’s law)• Fluid (viscous): τ = µ dv/dy (Newtonian)– Viscosity, µ = µoexp (Q/RT) Temperature dependence– Velocity gradient, dv/dy is related to a strain rate gamma dot = gamma/t = (δ/l)/tTwo common tests used to characterize viscoelasticity•1st: CREEP TEST–Apply constant stress, measure strain as a function of time– If material is linear viscoelastic, then strain at any time is proportional to the stress. • i.e. double stress, then double strain at a given timeconstant stress0123450204060timestresschanging strain0100204060timestressStrain 1: elastic Strain 2: recoverable over timeStrain 3: Irrecoverable strain (Newtonian flow)1123Two common tests used to characterize viscoelasticity•2nd: RELAXATION TEST–Apply constant strain, measure stress as a func. of time– Keep in mind this is occuring at a constant temperature.constant strain0100204060timestrainchanging stress0100204060timestressxTemp. effect on amorphous polymers• Consider the relaxation modulus at a given time, as a function of temperature– At low temp.• Hard, brittle glassy behavior (high Er (modulus 10^9GPa)• Modulus depends on change in energy with straining associated with stretching or compressing bonds– As temp. increases• Expansion of material – somewhat more open structure & segments of polymer chains can begin to slide voer one another– Tg, secondary bonds melt (middle of viscoelastic regime)• Much more open structure– Increase in temp Æ rubbery regime– Modulus depends on entropic changes (rubbery regime depends on change in entropy with strain (E about 10^5GPa)– If no covalent x-links, get viscous fluid (initial physical entanglement of chains causes viscous flow to occur over some temperature range)Graph to demonstrate last slide10^9GPa10^5GPaRelaxation modulus vs. Temp.Temp.TgGlassy regimeViscoelastic regimeRubbery regimeSpring-dashpot models for viscoelasticity•Spring– A spring is E = σ/ε• Dashpot– η = σ/(dε/dt) Æ dε/dt is the strain drain rate•E = (dσ/dε)– Like a piston (think that you are trying to close a door that has hinges acting against your force)Spring-dashpot Model #1• Maxwell Model – In series– σspring= σdashpot= σtotal– εspring+ εdashpot = εtotal• Relaxation Response ε const.– σ = E εexp (-Et/ η) exponential decay– Physically not very realistic Æ most polymers require more than 1 relaxation term to describe relaxation & σwouldn’t decay to 0.• Creep Response σ const.–dσ/dt =0, ε = σ/E + (σ/η)t (elastic + dashpot)– Poor representation of real polymers because wouldn’t recover strain from dashpot.strainStressÆ0Model #2• Voigt or Kelvin Model (in parallel)– σspring+ σdashpot= σtotal– εspring= εdashpot = εtotal– σ = Eε + η (dε/dt)• Relaxation Response– Not physically realistic• Creep Response– These are all very simplistic b/c doesn’t give exponential response for creep and relaxation. To really model a polymer, you need multiple springs & dashpots to model the flow.– To get exponential response for both, you need to sum voigt and maxwell models to get a better approximation of viscoelasticbehaviorStrainÆ strain rate = 0stressstrain vs. time graphViscoelastic ProblemsAnswers to Problem 22.Solution Problem
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