3 11 Mechanics of Materials Recitation 10 November 18 2003 TA Kristin Domike Rubber 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 are linearly related at a given time and temperature Elastic Solid E Hooke s law Fluid viscous dv dy Newtonian Viscosity oexp Q RT Temperature dependence Velocity gradient dv dy is related to a strain rate gamma dot gamma t l t Two 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 time constant stress changing strain Strain 2 recoverable over time 10 5 stress stress 4 3 2 1 1 0 0 20 40 time 60 2 1 0 0 20 40 time Strain 1 elastic 3 60 Strain 3 Irrecoverable strain Newtonian flow Two 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 strain changing stress 10 strain stress 10 x 0 0 0 20 40 time 60 0 20 40 time 60 Temp 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 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 slide Relaxation modulus vs Temp 10 9 GPa Glassy regime Viscoelastic regime 10 5 GPa Rubbery regime Tg Temp Spring dashpot models for viscoelasticity Spring A spring is E Dashpot d dt d dt is the strain drain rate E d d 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 spring dashpot total spring dashpot total Relaxation Response const strain Stress 0 E exp Et exponential decay Physically not very realistic most polymers require more than 1 relaxation term to describe relaxation wouldn t decay to 0 Creep Response const d dt 0 E t elastic dashpot Poor representation of real polymers because wouldn t recover strain from dashpot Model 2 Voigt or Kelvin Model in parallel spring dashpot total spring dashpot total E d dt Relaxation Response Strain strain rate 0 stress Not physically realistic Creep Response strain vs time graph 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 viscoelastic behavior Viscoelastic Problems Answers to Problem 2 2 Solution Problem 3 3
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