Chapter 3

 

Constitutive Models MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzaiaeaa aaaaaaa8qacaWFtacaaa@3788@  Relations between Stress and Strain

 

 

 

3.6 Linear Viscoelastic Materials MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzaiaeaa aaaaaaa8qacaWFtacaaa@3788@  time-dependent behavior of polymers at small strains

 

Amorphous polymers show complex time-dependent behavior when subjected to a history of stress or strain.  Viscoelasticity theory was developed to approximate this behavior in polymers that are subjected to modest strains (less than 0.5%).  A typical application might be to model the energy dissipation during cyclic loading of a polymeric vibration damper, or to model human tissue responding to an electric shaver.

 

 

3.6.1 Features of the small-strain rate dependent response of polymers

 

The principal features of polymers (and some biological tissue) are summarized below

 

1.      Polymers strongly resist volume changes at all temperatures.  The bulk modulus (the ratio of volume change to hydrostatic component of stress) is comparable to that of metals or covalently bonded solids;

2.      The shear response of a polymer is strongly temperature dependent.  At low temperatures (the glassy regime), the shear modulus is high, and comparable to that of metals.   At a critical temperature (the glass transition) the modulus drops. At temperatures well above the glass transition temperature (the rubbery regime), the shear modulus can be as low as 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaGymaiaaicdadaahaaWcbeqaaiabgk HiTiaaiwdaaaaaaa@341B@  times that of most metals.

3.      At temperatures near the glass transition, the shear modulus is strongly time (and load history) dependent MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  this behavior is discussed in more detail below. The time dependent shear response can be measured in two ways: (i) by applying a step load to a sample; or (ii) by applying a harmonic (sinusoidal) load to the specimen.

4.      The time dependent modulus of polymers is also temperature dependent.  Reducing the temperature is qualitatively equivalent to increasing the strain rate.   The equivalence of temperature and strain rate is discussed in more detail below.

5.      Most amorphous polymers are isotropic MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  their stress-strain response is independent of material orientation.

 

 

Time dependent response to step loading

 

The time dependent shear response can be measured in one of two ways:

1.      Take a specimen that is free of stress at time t=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamiDaiabg2da9iaaicdaaaa@3386@ , apply a constant shear stress Δτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeuiLdqKaeqiXdqhaaa@33F8@  for t>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamiDaiabg6da+iaaicdaaaa@3388@  and measuring the resulting shear strain ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqyTdugaaa@3274@  as a function of time.  The results are generally presented by plotting the `creep compliance’ J(t)=ε(t)/Δτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOsaiaacIcacaWG0bGaaiykaiabg2 da9iabew7aLjaacIcacaWG0bGaaiykaiaac+cacqqHuoarcqaHepaD aaa@3CCB@  as a function of time.

2.      Take a specimen that is free of stress at time t=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamiDaiabg2da9iaaicdaaaa@3386@ , apply a constant shear strain ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqyTdugaaa@3274@  for t>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamiDaiabg6da+iaaicdaaaa@3388@  and measuring the resulting shear stress τ(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiXdqNaaiikaiaadshacaGGPaaaaa@34E4@  as a function of time. In this case the results are presented by plotting the Relaxation Modulus: G(t)=τ(t)/Δε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4raiaacIcacaWG0bGaaiykaiabg2 da9iabes8a0jaacIcacaWG0bGaaiykaiaac+cacqqHuoarcqaH1oqz aaa@3CC8@

 

The results of such a test depend on the degree of cross-linking in the polymer.  Heavily cross-linked materials show `retarded elastic’ behavior, while un-cross-linked materials show steady-state creep.  A detailed description of each type of behavior follows.

 

 

  Retarded Elastic Behavior (observed in strongly crosslinked polymers):

 

The notable features of this behavior are:

1.      There is always an instantaneous strain Δε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabew7aLbaa@3375@  in response to a step change in stress Δτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabes8a0baa@3393@ .  The instantaneous compliance J g =Δε/Δτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOsamaaBaaaleaacaWGNbaabeaaki abg2da9iabfs5aejabew7aLjaac+cacqqHuoarcqaHepaDaaa@3AB1@  is low,  and only weakly dependent on temperature.

2.      At temperatures significantly below the glass transition temperature the solid is essentially elastic (there may be a very slow increase in compliance with time). At low temperatures the compliance is low, comparable to J g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOsamaaBaaaleaacaWGNbaabeaaaa a@32B6@ .

3.      At temperatures significantly above the glass transition temperature, the solid is very compliant, and the compliance is a function of temperature.  The specimen will show an initial transient response, but will quite quickly settle to a constant strain.  The time taken to reach steady state decreases with increasing temperature, and for some materials the transient may be short enough to be neglected.  In this case the material can be modeled using the hyperelastic constitutive law described in the preceding section.

4.      For a range of temperatures both above and below the glass transition temperature, the solid shows a slow transient response.

5.      The deformation is reversible MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  if the loading is removed, the specimen will eventually return to its original configuration, although in the transition regime this may take a very long time.

 

 

  Steady-state creep behavior (observed in uncrosslinked polymers and polymer melts):

 

The notable features of this behavior are:

1.      There is always an instantaneous strain in response to a step change in stress, exactly as in crosslinked polymers.

2.      At low temperatures (well below the glass transition temperature) the solid is essentially elastic (there may be a very slow rate of creep), and has a very low compliance, comparable to J g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOsamaaBaaaleaacaWGNbaabeaaaa a@32B6@ .

3.      At temperatures above the glass transition temperature, the solid is very compliant.  It may show rubbery behavior for very low stresses, but for most practical ranges of loading the compliance will increase more-or-less linearly with time (especially for short time intervals).  The rate of change of compliance is strongly temperature dependent, as discussed below.

4.      Above the glass transition temperature, the deformation is irrreversible MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  if the loading is removed, the specimen will not return to its original shape.

 

Response to harmonic loading

 

In addition to measuring the response of a material to a step change in load, one can subject it to cyclic strain, e.g. with strains that vary sinusoidally with time ε(t)= ε 0 cos(ωt)= ε 0 Re[ exp(iωt) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqyTduMaaiikaiaadshacaGGPaGaey ypa0JaeqyTdu2aaSbaaSqaaiaaicdaaeqaaOGaci4yaiaac+gacaGG ZbGaaiikaiabeM8a3jaadshacaGGPaGaeyypa0JaeqyTdu2aaSbaaS qaaiaaicdaaeqaaOGaciOuaiaacwgadaWadaqaaiGacwgacaGG4bGa aiiCaiaacIcacaWGPbGaeqyYdCNaamiDaiaacMcaaiaawUfacaGLDb aaaaa@4E8F@ , where Re(z) denotes the real part of a complex number z.  The stress history will also be harmonic, and could be expressed as τ= τ 0 ε 0 Re[ exp(iωt)exp(iδ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiXdqNaeyypa0JaeqiXdq3aaSbaaS qaaiaaicdaaeqaaOGaeqyTdu2aaSbaaSqaaiaaicdaaeqaaOGaciOu aiaacwgadaWadaqaaiGacwgacaGG4bGaaiiCaiaacIcacaWGPbGaeq yYdCNaamiDaiaacMcaciGGLbGaaiiEaiaacchacaGGOaGaamyAaiab es7aKjaacMcaaiaawUfacaGLDbaaaaa@4B48@ , where τ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes8a0naaBaaaleaacaaIWaaabeaaaa a@3313@  is the stress amplitude, and δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKbaa@320D@  is a phase shift. Both τ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes8a0naaBaaaleaacaaIWaaabeaaaa a@3313@  and δ depend on ω.  One can define a complex modulus as

G * (ω,T)= τ 0 exp(iδ)/ ε 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4ramaaCaaaleqabaGaaiOkaaaaki aacIcacqaHjpWDcaGGSaGaamivaiaacMcacqGH9aqpcqaHepaDdaWg aaWcbaGaaGimaaqabaGcciGGLbGaaiiEaiaacchacaGGOaGaamyAai abes7aKjaacMcacaGGVaGaeqyTdu2aaSbaaSqaaiaaicdaaeqaaaaa @44F1@

Experimental data is usually presented by plotting the real part G'(ω,T) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGNaGaaiikaiabeM8a3jaacY cacaWGubGaaiykaaaa@368E@  of the complex modulus G * (ω,T) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeadaahaaWcbeqaaiaacQcaaaGcca GGOaGaeqyYdCNaaiilaiaadsfacaGGPaaaaa@36C8@  against the inverse of frequency, where

G * =G'+iG''G'= τ 0 cos(δ)/ ε 0 G''= τ 0 sin(δ)/ ε 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeadaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWGhbGaai4jaiabgUcaRiaadMgacaWGhbGaai4jaiaacEca caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caWGhbGaai4jaiabg2da9iabes8a0naaBaaaleaacaaIWaaa beaakiGacogacaGGVbGaai4CaiaacIcacqaH0oazcaGGPaGaai4lai abew7aLnaaBaaaleaacaaIWaaabeaakiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaadEeacaGGNaGaai4jaiabg2da9iabes8a0naaBaaaleaacaaI WaaabeaakiGacohacaGGPbGaaiOBaiaacIcacqaH0oazcaGGPaGaai 4laiabew7aLnaaBaaaleaacaaIWaaabeaaaaa@762F@

The variation of the modulus with frequency is illustrated in the picture.

 

 

 

Williams-Landell-Ferry (WLF) Time/temperature equivalence

 

You may have noticed that the figures showing the variation of modulus with temperature and frequency are remarkably similar. Of course these are just sketches, but in fact the connection between temperature and loading rate is more than just a qualitative trend.  This can be demonstrated by means of the following experiment:

1.      At temperature T 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaGymaaqabaaaaa@3228@ , subject a specimen to a step change  in shear strain Δε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabew7aLbaa@3375@  and measure the relaxation modulus G(t, T 1 )=τ(t)/( 2Δε ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacYcacaWGub WaaSbaaSqaaiaaigdaaeqaaOGaaiykaiabg2da9iabes8a0jaacIca caWG0bGaaiykaiaac+cadaqadaqaaiaaikdacqqHuoarcqaH1oqzai aawIcacaGLPaaaaaa@4122@

2.      Repeat the experiment at several progressively higher temperatures T 2 , T 3 ... T n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaGOmaaqabaGcca GGSaGaamivamaaBaaaleaacaaIZaaabeaakiaac6cacaGGUaGaaiOl aiaadsfadaWgaaWcbaGaamOBaaqabaaaaa@38BD@  to obtain a series of relaxation modulus curves G(t, T 2 ),G(t, T 3 )... MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacYcacaWGub WaaSbaaSqaaiaaikdaaeqaaOGaaiykaiaacYcacaWGhbGaaiikaiaa dshacaGGSaGaamivamaaBaaaleaacaaIZaaabeaakiaacMcacaGGUa GaaiOlaiaac6caaaa@3E61@

3.      Plot log(G(t)) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacYgacaGGVbGaai4zaiaacIcacaWGhb GaaiikaiaadshacaGGPaGaaiykaaaa@37AF@  -v- log(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacYgacaGGVbGaai4zaiaacIcacaWG0b Gaaiykaaaa@358A@  for the raw data. The results will look like a complicated mess MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  something like the picture on the right.

4.      However, you will find that if you simply shift the modulus curves for the higher temperatures to the right, you can make the data collapse onto a single master-curve, as shown.

5.      This observation can be expressed mathematically as log(G)=f{ log(t)+log[ A(T; T 1 ) ] } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacYgacaGGVbGaai4zaiaacIcacaWGhb Gaaiykaiabg2da9iaadAgadaGadaqaaiGacYgacaGGVbGaai4zaiaa cIcacaWG0bGaaiykaiabgUcaRiGacYgacaGGVbGaai4zamaadmaaba GaamyqaiaacIcacaWGubGaai4oaiaadsfadaWgaaWcbaGaaGymaaqa baGccaGGPaaacaGLBbGaayzxaaaacaGL7bGaayzFaaaaaa@49C6@  where the function f represents the master curve, and logA(T; T 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacYgacaGGVbGaai4zaiaadgeacaGGOa GaamivaiaacUdacaWGubWaaSbaaSqaaiaaigdaaeqaaOGaaiykaaaa @38B9@  represents the horizontal shift from temperature T1 to T2 A(T; T 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeacaGGOaGaamivaiaacUdacaWGub WaaSbaaSqaaiaaigdaaeqaaOGaaiykaaaa@35E9@  is known as the WLF shift function.

6.      If you measure A(T; T 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeacaGGOaGaamivaiaacUdacaWGub WaaSbaaSqaaiaaigdaaeqaaOGaaiykaaaa@35E9@  at a series of temperatures, and plot log(A(T; T 1 )) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacYgacaGGVbGaai4zaiaacIcacaWGbb GaaiikaiaadsfacaGG7aGaamivamaaBaaaleaacaaIXaaabeaakiaa cMcacaGGPaaaaa@3A12@  as a function of temperature T, you will find that the data can be well approximated by a function of the form log[A(T; T 1 )]= C 1 (T T 1 )/{ C 2 +(T T 1 ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaciiBaiaac+gacaGGNbGaai4waiaadg eacaGGOaGaamivaiaacUdacaWGubWaaSbaaSqaaiaaigdaaeqaaOGa aiykaiaac2facqGH9aqpcqGHsislcaWGdbWaaSbaaSqaaiaaigdaae qaaOGaaiikaiaadsfacqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqa aOGaaiykaiaac+cadaGadaqaaiaadoeadaWgaaWcbaGaaGOmaaqaba GccqGHRaWkcaGGOaGaamivaiabgkHiTiaadsfadaWgaaWcbaGaaGym aaqabaGccaGGPaaacaGL7bGaayzFaaaaaa@4DDE@ . The scaling holds for any two temperatures, but of course C 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaBaaaleaacaaIXaaabeaaaa a@328C@  and C 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaBaaaleaacaaIYaaabeaaaa a@328D@  must depend on the choice of T 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamivamaaBaaaleaacaaIXaaabeaaaa a@329D@ . In practice it is convenient (and conventional) to use the glass transition temperature as the reference temperature.  The scaling law can then be written as

A(T; T g )=exp{ C 1 g (T T g ) C 2 g +(T T g ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyqaiaacIcacaWGubGaai4oaiaads fadaWgaaWcbaGaam4zaaqabaGccaGGPaGaeyypa0JaciyzaiaacIha caGGWbWaaiWaaeaacqGHsisldaWcaaqaaiaadoeadaqhaaWcbaGaaG ymaaqaaiaadEgaaaGccaGGOaGaamivaiabgkHiTiaadsfadaWgaaWc baGaam4zaaqabaGccaGGPaaabaGaam4qamaaDaaaleaacaaIYaaaba Gaam4zaaaakiabgUcaRiaacIcacaWGubGaeyOeI0IaamivamaaBaaa leaacaWGNbaabeaakiaacMcaaaaacaGL7bGaayzFaaaaaa@4DF3@

The values of C 1 g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaDaaaleaacaaIXaaabaGaam 4zaaaaaaa@336B@  and C 2 g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaDaaaleaacaaIYaaabaGaam 4zaaaaaaa@336C@  vary slightly (but surprisingly little) from one polymer to another: typical ranges are   C 1 g 1040 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaDaaaleaacaaIXaaabaGaam 4zaaaakiabgIKi7kaaigdacaaIWaGaeyOeI0IaaGinaiaaicdaaaa@3900@  and C 2 g 50100Kelvin MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaDaaaleaacaaIYaaabaGaam 4zaaaakiabgIKi7kaaiwdacaaIWaGaaGPaVlabgkHiTiaaigdacaaI WaGaaGimaiaaykW7caqGlbGaaeyzaiaabYgacaqG2bGaaeyAaiaab6 gaaaa@424D@ . The expression works (again surprisingly) for T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamivaaaa@31A6@  both above and below T g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamivamaaBaaaleaacaWGNbaabeaaaa a@32C0@  - but of course the expression blows up if T< T g C 2 g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamivaiabgYda8iaadsfadaWgaaWcba Gaam4zaaqabaGccqGHsislcaWGdbWaa0baaSqaaiaaikdaaeaacaWG Nbaaaaaa@3831@ .  For temperatures below this critical value, the material is perfectly elastic (with constant elastic moduli).

7.      Note that because A(T; T 1 )=A( T g ; T 1 )A(T; T g ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyqaiaacIcacaWGubGaai4oaiaads fadaWgaaWcbaGaaGymaaqabaGccaGGPaGaeyypa0JaamyqaiaacIca caWGubWaaSbaaSqaaiaadEgaaeqaaOGaai4oaiaadsfadaWgaaWcba GaaGymaaqabaGccaGGPaGaamyqaiaacIcacaWGubGaai4oaiaadsfa daWgaaWcbaGaam4zaaqabaGccaGGPaaaaa@43AB@ , the constants C 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaBaaaleaacaaIXaaabeaaaa a@328C@ , C 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaBaaaleaacaaIYaaabeaaaa a@328D@ , T 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamivamaaBaaaleaacaaIXaaabeaaaa a@329D@  and C 1 g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaDaaaleaacaaIXaaabaGaam 4zaaaaaaa@336B@ , C 2 g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaDaaaleaacaaIYaaabaGaam 4zaaaaaaa@336C@ , T g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamivamaaBaaaleaacaWGNbaabeaaaa a@32C0@  are related by C 1 = C 1 g C 2 g /( C 2 g + T 1 T g ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWGdbWaa0baaSqaaiaaigdaaeaacaWGNbaaaOGaam4qamaa DaaaleaacaaIYaaabaGaam4zaaaakiaac+cacaGGOaGaam4qamaaDa aaleaacaaIYaaabaGaam4zaaaakiabgUcaRiaadsfadaWgaaWcbaGa aGymaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaadEgaaeqaaOGaai ykaaaa@42BB@ , C 2 = C 2 g + T 1 T g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaaGOmaaqabaGccq GH9aqpcaWGdbWaa0baaSqaaiaaikdaaeaacaWGNbaaaOGaey4kaSIa amivamaaBaaaleaacaaIXaaabeaakiabgkHiTiaadsfadaWgaaWcba Gaam4zaaqabaaaaa@3B59@ .  This means that if you measure a time dependent modulus G(t, T 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacYcacaWGub WaaSbaaSqaaiaaigdaaeqaaOGaaiykaaaa@3600@  at temperature T 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaGymaaqabaaaaa@3228@ , and know the values of C 1 g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaDaaaleaacaaIXaaabaGaam 4zaaaaaaa@336B@ , C 2 g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaDaaaleaacaaIYaaabaGaam 4zaaaaaaa@336C@ , T g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamivamaaBaaaleaacaWGNbaabeaaaa a@32C0@  for the material, you can immediately calculate the modulus at any other temperature as G(t,T)=G(A(T; T 1 )t, T 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacYcacaWGub Gaaiykaiabg2da9iaadEeacaGGOaGaamyqaiaacIcacaWGubGaai4o aiaadsfadaWgaaWcbaGaaGymaaqabaGccaGGPaGaamiDaiaacYcaca WGubWaaSbaaSqaaiaaigdaaeqaaOGaaiykaaaa@412E@ , where

A(T; T 1 )=exp( C 1 g C 2 g (T T 1 ) [ C 2 g + T 1 T g ][ C 2 g +T T g ] ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeacaGGOaGaamivaiaacUdacaWGub WaaSbaaSqaaiaaigdaaeqaaOGaaiykaiabg2da9iGacwgacaGG4bGa aiiCamaabmaabaWaaSaaaeaacqGHsislcaWGdbWaa0baaSqaaiaaig daaeaacaWGNbaaaOGaam4qamaaDaaaleaacaaIYaaabaGaam4zaaaa kiaacIcacaWGubGaeyOeI0IaamivamaaBaaaleaacaaIXaaabeaaki aacMcaaeaadaWadaqaaiaadoeadaqhaaWcbaGaaGOmaaqaaiaadEga aaGccqGHRaWkcaWGubWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0Iaam ivamaaBaaaleaacaWGNbaabeaaaOGaay5waiaaw2faamaadmaabaGa am4qamaaDaaaleaacaaIYaaabaGaam4zaaaakiabgUcaRiaadsfacq GHsislcaWGubWaaSbaaSqaaiaadEgaaeqaaaGccaGLBbGaayzxaaaa aaGaayjkaiaawMcaaaaa@59EF@

 

 

 

3.6.2 General constitutive equations for linear viscoelastic solids

 

The general stress-strain law for a linear viscoelastic solid is constructed as follows:

* Assume that the material experiences small shape changes and rotations.  The deformation can then be characterized using the infinitesimal strain tensor ε ij =( u i / x j + u j / x i )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaqadaqaaiabgkGi2kaadwhadaWgaaWcbaGaamyA aaqabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaki abgUcaRiabgkGi2kaadwhadaWgaaWcbaGaamOAaaqabaGccaGGVaGa eyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaai aac+cacaaIYaaaaa@4880@  defined in Section 2.1.7.

 For small strains, all stress measures are equal.  We can use the Cauchy stress σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3434@  as the stress measure.

 Assume that for time t<0, the solid is stress free, and ε ij =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcaaIWaaaaa@35E2@ .

 For small strains/stresses, we can assume that the stress and strain are related through linear equations.  (This doesn’t mean that stress is proportional to strain, of course MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  instead stress, strain and their rates are related by a time dependent linear ODE, as discussed below)

 Assume that the material is isotropic.

 In most practical applications we can assume that material response to a pure volumetric strain ( ε 11 = ε 22 = ε 33 =(V V 0 )/(3V) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaH1oqzdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGa eyypa0JaeqyTdu2aaSbaaSqaaiaaiodacaaIZaaabeaakiabg2da9i aacIcacaWGwbGaeyOeI0IaamOvamaaBaaaleaacaaIWaaabeaakiaa cMcacaGGVaGaaiikaiaaiodacaWGwbGaaiykaaaa@4609@  with all other ε ij =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcaaIWaaaaa@35E2@  ) is perfectly elastic (with no time dependent behavior).  The volumetric strain will induce a state of hydrostatic tension σ 11 = σ 22 = σ 33 =σ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGa eyypa0Jaeq4Wdm3aaSbaaSqaaiaaiodacaaIZaaabeaakiabg2da9i abeo8aZbaa@3F90@  with all other σ ij =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcaaIWaaaaa@35FE@ .  The stress is related to the strain by σ=3KδV/V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZjabg2da9iaaiodacaWGlbGaeq iTdqMaamOvaiaac+cacaWGwbaaaa@38CC@  where K is the bulk modulus.

 Viscoelastic response most commonly characterized by the shear relaxation modulus G(t, T 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacYcacaWGub WaaSbaaSqaaiaaigdaaeqaaOGaaiykaaaa@3600@  measured at some reference temperature T 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaGymaaqabaaaaa@3228@ .  (Recall that the shear relaxation modulus can be measured by subjecting a specimen to a step increase in shear strain Δε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabew7aLbaa@3375@ , and measuring the resulting shear stress τ(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes8a0jaacIcacaWG0bGaaiykaaaa@347F@ .  The relaxation modulus follows as  G(t, T 1 )=τ(t)/(2Δε) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacYcacaWGub WaaSbaaSqaaiaaigdaaeqaaOGaaiykaiabg2da9iabes8a0jaacIca caWG0bGaaiykaiaac+cacaGGOaGaaGOmaiabfs5aejabew7aLjaacM caaaa@40F2@  )

 The temperature dependence of the modulus is characterized by the WLF constants C 1 g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaDaaaleaacaaIXaaabaGaam 4zaaaaaaa@336B@ , C 2 g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaDaaaleaacaaIYaaabaGaam 4zaaaaaaa@336C@  and the glass transition temperature T g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamivamaaBaaaleaacaWGNbaabeaaaa a@32C0@ , through the WLF shift function defined in the preceding section.

 Since the stress is linearly related to strain, the stress history σ ij (t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccaGGOaGaamiDaiaacMcaaaa@3690@  resulting from an arbitrary strain history ε ij (t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccaGGOaGaamiDaiaacMcaaaa@3674@  can be computed by appropriately superposing the step response.  The result is

σ ij (t)= 0 t 2G( A(T; T 1 )(tξ), T 1 )[ ε ˙ ij (ξ) ε ˙ kk (ξ) δ ij ]dξ +K ε kk δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccaGGOaGaamiDaiaacMcacqGH9aqpdaWdXbqaaiaaikdacaWG hbWaaeWaaeaacaWGbbGaaiikaiaadsfacaGG7aGaamivamaaBaaale aacaaIXaaabeaakiaacMcacaGGOaGaamiDaiabgkHiTiabe67a4jaa cMcacaGGSaGaamivamaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawM caamaadmaabaGafqyTduMbaiaadaWgaaWcbaGaamyAaiaadQgaaeqa aOGaaiikaiabe67a4jaacMcacqGHsislcuaH1oqzgaGaamaaBaaale aacaWGRbGaam4AaaqabaGccaGGOaGaeqOVdGNaaiykaiabes7aKnaa BaaaleaacaWGPbGaamOAaaqabaaakiaawUfacaGLDbaacaWGKbGaeq OVdGhaleaacaaIWaaabaGaamiDaaqdcqGHRiI8aOGaey4kaSIaam4s aiabew7aLnaaBaaaleaacaWGRbGaam4AaaqabaGccqaH0oazdaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@6BEB@

Here, the temperature T is assumed to be constant up to time t.   It is not hard to extend the formula to account for time varying temperatures but the result looks messy and is difficult to visualize.

 

To apply this stress-strain relation in practice, it is necessary to find a convenient way to fit the relaxation modulus G(t, T 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacYcacaWGub WaaSbaaSqaaiaaigdaaeqaaOGaaiykaaaa@3600@ .  Various approaches to doing this are described in the next two sections.

 

 

 

3.6.3 Spring-Damper approximations to the relaxation modulus

 

Spring-damper models are often used as a simple, approximate model of the behavior of a viscoelastic solid.  The figures illustrate the general idea: in each case the force applied to the spring-dashpot system represents shear stress, while the extension represents shear strain.  It is straightforward to show that they are related by

kσ+η dσ dt =kη dε dt Maxwell σ=kε+η dε dt Kelvin-Voigt k 1 σ+η dσ dt = k 1 k 2 ε+( k 1 + k 2 )η dε dt 3Parameter MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaqabeaacaWGRbGaeq4WdmNaey4kaSIaeq 4TdG2aaSaaaeaacaWGKbGaeq4WdmhabaGaamizaiaadshaaaGaeyyp a0Jaam4AaiabeE7aOnaalaaabaGaamizaiabew7aLbqaaiaadsgaca WG0baaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caqGnbGaae yyaiaabIhacaqG3bGaaeyzaiaabYgacaqGSbaabaGaeq4WdmNaeyyp a0Jaam4Aaiabew7aLjabgUcaRiabeE7aOnaalaaabaGaamizaiabew 7aLbqaaiaadsgacaWG0baaaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaabUeacaqGLbGaaeiBaiaabAhaca qGPbGaaeOBaiaab2cacaqGwbGaae4BaiaabMgacaqGNbGaaeiDaaqa aiaadUgadaWgaaWcbaGaaGymaaqabaGccqaHdpWCcqGHRaWkcqaH3o aAdaWcaaqaaiaadsgacqaHdpWCaeaacaWGKbGaamiDaaaacqGH9aqp caWGRbWaaSbaaSqaaiaaigdaaeqaaOGaam4AamaaBaaaleaacaaIYa aabeaakiabew7aLjabgUcaRiaacIcacaWGRbWaaSbaaSqaaiaaigda aeqaaOGaey4kaSIaam4AamaaBaaaleaacaaIYaaabeaakiaacMcacq aH3oaAdaWcaaqaaiaadsgacqaH1oqzaeaacaWGKbGaamiDaaaacaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaIZaGaaGPaVlaaykW7caqGqbGaaeyyaiaabkhacaqGHbGaaeyB aiaabwgacaqG0bGaaeyzaiaabkhaaaaa@37A9@

For a material with time independent bulk modulus K, these can be generalized to multi-axial loading as

e ij = ε ij ε kk δ ij σ ij = S ij +K ε kk δ ij k S ij +η d S ij dt =kη d e ij dt Maxwell S ij =k e ij +η d e ij dt Kelvin-Voigt k 1 S ij +η d S ij dt = k 1 k 2 e ij +( k 1 + k 2 )η d e ij dt 3Parameter MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaqabeaacaWGLbWaaSbaaSqaaiaadMgaca WGQbaabeaakiabg2da9iabew7aLnaaBaaaleaacaWGPbGaamOAaaqa baGccqGHsislcqaH1oqzdaWgaaWcbaGaam4AaiaadUgaaeqaaOGaeq iTdq2aaSbaaSqaaiaadMgacaWGQbaabeaakiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8Uaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaa beaakiabg2da9iaadofadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaey 4kaSIaam4saiabew7aLnaaBaaaleaacaWGRbGaam4AaaqabaGccqaH 0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVdqaaiaadUgacaWGtbWaaS baaSqaaiaadMgacaWGQbaabeaakiabgUcaRiabeE7aOnaalaaabaGa amizaiaadofadaWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaamizai aadshaaaGaeyypa0Jaam4AaiabeE7aOnaalaaabaGaamizaiaadwga daWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaamizaiaadshaaaGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaab2eacaqGHbGaaeiEaiaabEhacaqGLbGa aeiBaiaabYgaaeaacaWGtbWaaSbaaSqaaiaadMgacaWGQbaabeaaki abg2da9iaadUgacaWGLbWaaSbaaSqaaiaadMgacaWGQbaabeaakiab gUcaRiabeE7aOnaalaaabaGaamizaiaadwgadaWgaaWcbaGaamyAai aadQgaaeqaaaGcbaGaamizaiaadshaaaGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8Uaae4saiaabwgacaqGSbGaaeODaiaabMga caqGUbGaaeylaiaabAfacaqGVbGaaeyAaiaabEgacaqG0baabaGaam 4AamaaBaaaleaacaaIXaaabeaakiaadofadaWgaaWcbaGaamyAaiaa dQgaaeqaaOGaey4kaSIaeq4TdG2aaSaaaeaacaWGKbGaam4uamaaBa aaleaacaWGPbGaamOAaaqabaaakeaacaWGKbGaamiDaaaacqGH9aqp caWGRbWaaSbaaSqaaiaaigdaaeqaaOGaam4AamaaBaaaleaacaaIYa aabeaakiaadwgadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaey4kaSIa aiikaiaadUgadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGRbWaaS baaSqaaiaaikdaaeqaaOGaaiykaiabeE7aOnaalaaabaGaamizaiaa dwgadaWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaamizaiaadshaaa GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaG4maiaaykW7caaMc8Uaaeiuai aabggacaqGYbGaaeyyaiaab2gacaqGLbGaaeiDaiaabwgacaqGYbaa aaa@AA9C@

Qualitatively, these models describe the behavior of a typical polymer.  The Kelvin-Voigt model gives retarded elastic behavior, and represents a crosslinked polymer.  The Maxwell model gives steady state creep, and would represent an uncrosslinked polymer.  With an appropriate choice of k 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4AamaaBaaaleaacaaIXaaabeaaaa a@32A6@  and k 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4AamaaBaaaleaacaaIYaaabeaaaa a@32A7@ , the 3 parameter model can describe both types of behavior.

 

For hand calculations it is often more convenient to use the differential equations relating stress to strain than the integral integral form given in the preceding section.  However, it is straigthforward to calculate the relaxation modulus for the Maxwell and 3 parameter models

G(t)=k e kt/η Maxwell G(t)= k 2 + k 1 e k 1 t/η 3Parameter MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaqabeaacaWGhbGaaiikaiaadshacaGGPa Gaeyypa0Jaam4AaiaadwgadaahaaWcbeqaaiabgkHiTiaadUgacaWG 0bGaai4laiabeE7aObaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aeytaiaabggacaqG4bGaae4DaiaabwgacaqGSbGaaeiBaaqaaiaadE eacaGGOaGaamiDaiaacMcacqGH9aqpcaWGRbWaaSbaaSqaaiaaikda aeqaaOGaey4kaSIaam4AamaaBaaaleaacaaIXaaabeaakiaadwgada ahaaWcbeqaaiabgkHiTiaadUgadaWgaaadbaGaaGymaaqabaWccaWG 0bGaai4laiabeE7aObaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaIZaGaaGPaVlaaykW7 caqGqbGaaeyyaiaabkhacaqGHbGaaeyBaiaabwgacaqG0bGaaeyzai aabkhaaaaa@9B1D@

The Kelvin-Voigt model does not have a well defined relaxation modulus.

 

 

 

 

3.6.4 Prony series representation for the relaxation modulus

 

The models described in the preceding section are too simple to give a good quantitative fit to any polymer over an extended period of time.  We can make a more versatile model by connecting a bunch of Maxwell elements in series, and adding a spring in parallel with the whole array.  The relaxation modulus for  this material has the form

G(t)= G + i=1 N G i e t/ t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4raiaacIcacaWG0bGaaiykaiabg2 da9iaadEeadaWgaaWcbaGaeyOhIukabeaakiabgUcaRmaaqahabaGa am4ramaaBaaaleaacaWGPbaabeaakiaadwgadaahaaWcbeqaaiabgk HiTiaadshacaGGVaGaamiDamaaBaaameaacaWGPbaabeaaaaaaleaa caWGPbGaeyypa0JaaGymaaqaaiaad6eaa0GaeyyeIuoaaaa@45E0@

where G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4ramaaBaaaleaacqGHEisPaeqaaa aa@3346@  is the steady-state stiffness (represented by the parallel spring), and G i , t i i=1...N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4ramaaBaaaleaacaWGPbaabeaaki aacYcacaaMc8UaaGPaVlaaykW7caWG0bWaaSbaaSqaaiaadMgaaeqa aOGaaGPaVlaaykW7caaMc8UaamyAaiabg2da9iaaigdacaGGUaGaai Olaiaac6cacaWGobaaaa@4474@  are the time constants and stiffnesses of the Maxwell elements.  These parameters are used directly as the properties of the material. The sum of exponentials is known as the `Prony series.’  

 

 

 

 

3.6.5 Calibrating the constitutive laws for linear viscoelastic solids

 

Experimental data for the time dependent behavior of polymers can be presented in several different ways:

1.      The Young’s modulus E(t, T i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweacaGGOaGaamiDaiaacYcacaWGub WaaSbaaSqaaiaadMgaaeqaaOGaaiykaaaa@3631@  or shear modulus G(t, T i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacYcacaWGub WaaSbaaSqaaiaadMgaaeqaaOGaaiykaaaa@3633@  as a function of time t,at various temperatures T i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaamyAaaqabaaaaa@325B@  

2.      The tensile compliance C(t, T i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeacaGGOaGaamiDaiaacYcacaWGub WaaSbaaSqaaiaadMgaaeqaaOGaaiykaaaa@362F@  or shear compliance J(t, T i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacaGGOaGaamiDaiaacYcacaWGub WaaSbaaSqaaiaadMgaaeqaaOGaaiykaaaa@3636@  as a function of time, at various temperatures;

3.      The complex modulus G*(ω, T i )=G'(ω, T i )+iG''(ω, T i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGQaGaaiikaiabeM8a3jaacY cacaWGubWaaSbaaSqaaiaadMgaaeqaaOGaaiykaiabg2da9iaadEea caGGNaGaaiikaiabeM8a3jaacYcacaWGubWaaSbaaSqaaiaadMgaae qaaOGaaiykaiabgUcaRiaadMgacaWGhbGaai4jaiaacEcacaGGOaGa eqyYdCNaaiilaiaadsfadaWgaaWcbaGaamyAaaqabaGccaGGPaaaaa@49CA@ , or, more usually, just the real part of the complex modulus G'(ω, T i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGNaGaaiikaiabeM8a3jaacY cacaWGubWaaSbaaSqaaiaadMgaaeqaaOGaaiykaaaa@37B2@  as a function of frequency ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3baa@3235@  and temperature. 

4.      The complex compliance J*(ω,T) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacaGGQaGaaiikaiabeM8a3jaacY cacaWGubGaaiykaaaa@3694@  or the real part of the complex compliance J'(ω,T) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacaGGNaGaaiikaiabeM8a3jaacY cacaWGubGaaiykaaaa@3691@  as a function of frequency and temperature.

 

The material parameters G , G i , t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeadaWgaaWcbaGaeyOhIukabeaaki aacYcacaWGhbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaadshadaWg aaWcbaGaamyAaaqabaaaaa@383E@  must be fit to this data.  For each data set, the first step is to combine data from tests at various temperatures into a master-curve of G(t,T) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacYcacaWGub Gaaiykaaaa@350F@ , G'(ω,T) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGNaGaaiikaiabeM8a3jaacY cacaWGubGaaiykaaaa@368E@  or J(t,T) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacaGGOaGaamiDaiaacYcacaWGub Gaaiykaaaa@3512@ J'(ω,T) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacaGGNaGaaiikaiabeM8a3jaacY cacaWGubGaaiykaaaa@3691@  at a single reference temperature, using the WLF scaling procedure described in 3.6.1.  The parameters should then be chosen to give the best fit to this master curve. (A simple way to fit the parameters is to choose t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshadaWgaaWcbaGaamyAaaqabaaaaa@327B@  to be spaced at exponentially increasing time intervals, and then choose G , G i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeadaWgaaWcbaGaeyOhIukabeaaki aacYcacaWGhbWaaSbaaSqaaiaadMgaaeqaaaaa@3571@   to minimize the square of the difference between the predicted and measured values log(G(t)) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacYgacaGGVbGaai4zaiaacIcacaWGhb GaaiikaiaadshacaGGPaGaaiykaaaa@37AF@  ).

 

To do the fit, it is helpful to find formulas for G(t,T) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacYcacaWGub Gaaiykaaaa@350F@  or G'(ω,T) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGNaGaaiikaiabeM8a3jaacY cacaWGubGaaiykaaaa@368E@  in terms of material properties. It is straightforward to show that

G(t,T)= G + i=1 N G i exp(t/ t i ) G'(ω,T)= G + i=1 N G i ω 2 t i 2 1+ ω 2 t i 2 G''(ω,T)= i=1 N G i ω t i 1+ ω 2 t i 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4raiaacIcacaWG0bGaaiilai aadsfacaGGPaGaeyypa0Jaam4ramaaBaaaleaacqGHEisPaeqaaOGa ey4kaSYaaabCaeaacaWGhbWaaSbaaSqaaiaadMgaaeqaaOGaciyzai aacIhacaGGWbGaaiikaiabgkHiTiaadshacaGGVaGaamiDamaaBaaa leaacaWGPbaabeaakiaacMcaaSqaaiaadMgacqGH9aqpcaaIXaaaba GaamOtaaqdcqGHris5aaGcbaGaam4raiaacEcacaGGOaGaeqyYdCNa aiilaiaadsfacaGGPaGaeyypa0Jaam4ramaaBaaaleaacqGHEisPae qaaOGaey4kaSYaaabCaeaadaWcaaqaaiaadEeadaWgaaWcbaGaamyA aaqabaGccqaHjpWDdaahaaWcbeqaaiaaikdaaaGccaWG0bWaa0baaS qaaiaadMgaaeaacaaIYaaaaaGcbaGaaGymaiabgUcaRiabeM8a3naa CaaaleqabaGaaGOmaaaakiaadshadaqhaaWcbaGaamyAaaqaaiaaik daaaaaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOtaaqdcqGHris5 aOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caWGhbGaai4jaiaacEcacaGGOa GaeqyYdCNaaiilaiaadsfacaGGPaGaeyypa0ZaaabCaeaadaWcaaqa aiaadEeadaWgaaWcbaGaamyAaaqabaGccqaHjpWDcaWG0bWaaSbaaS qaaiaadMgaaeqaaaGcbaGaaGymaiabgUcaRiabeM8a3naaCaaaleqa baGaaGOmaaaakiaadshadaqhaaWcbaGaamyAaaqaaiaaikdaaaaaaa qaaiaadMgacqGH9aqpcaaIXaaabaGaamOtaaqdcqGHris5aaaaaa@A2A7@

It is slightly more cumbersome to fit the Prony series parameters to compliance measurements.  The compliances can be expressed in terms of  G , G i , t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeadaWgaaWcbaGaeyOhIukabeaaki aacYcacaWGhbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaadshadaWg aaWcbaGaamyAaaqabaaaaa@383E@  as follows

J(t,T)= L 1 { 1 s 2 ( G s + i=1 N t i G i t i s+1 ) 1 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacaGGOaGaamiDaiaacYcacaWGub Gaaiykaiabg2da9iaadYeadaahaaWcbeqaaiabgkHiTiaaigdaaaGc daGadaqaamaalaaabaGaaGymaaqaaiaadohadaahaaWcbeqaaiaaik daaaaaaOWaaeWaaeaadaWcaaqaaiaadEeadaWgaaWcbaGaeyOhIuka beaaaOqaaiaadohaaaGaey4kaSYaaabCaeaadaWcaaqaaiaadshada WgaaWcbaGaamyAaaqabaGccaWGhbWaaSbaaSqaaiaadMgaaeqaaaGc baGaamiDamaaBaaaleaacaWGPbaabeaakiaadohacqGHRaWkcaaIXa aaaaWcbaGaamyAaiabg2da9iaaigdaaeaacaWGobaaniabggHiLdaa kiaawIcacaGLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaaakiaawU hacaGL9baaaaa@5411@

J'(ω,T)= G'(ω,T) G' (ω,T) 2 +G'' (ω,T) 2 J''(ω,T)= G''(ω,T) G' (ω,T) 2 +G'' (ω,T) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacaGGNaGaaiikaiabeM8a3jaacY cacaWGubGaaiykaiabg2da9maalaaabaGaam4raiaacEcacaGGOaGa eqyYdCNaaiilaiaadsfacaGGPaaabaGaam4raiaacEcacaGGOaGaeq yYdCNaaiilaiaadsfacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaey4k aSIaam4raiaacEcacaGGNaGaaiikaiabeM8a3jaacYcacaWGubGaai ykamaaCaaaleqabaGaaGOmaaaaaaGccaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaamOsaiaacEcacaGGNaGaaiik aiabeM8a3jaacYcacaWGubGaaiykaiabg2da9maalaaabaGaam4rai aacEcacaGGNaGaaiikaiabeM8a3jaacYcacaWGubGaaiykaaqaaiaa dEeacaGGNaGaaiikaiabeM8a3jaacYcacaWGubGaaiykamaaCaaale qabaGaaGOmaaaakiabgUcaRiaadEeacaGGNaGaai4jaiaacIcacqaH jpWDcaGGSaGaamivaiaacMcadaahaaWcbeqaaiaaikdaaaaaaaaa@84AC@

where L 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeadaahaaWcbeqaaiabgkHiTiaaig daaaaaaa@330E@  denotes an inverse Laplace transform (which can be calculated using a symbolic manipulation program), and G',G'' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGNaGaaiilaiaadEeacaGGNa Gaai4jaaaa@34B1@  were defined above. 

 

If you are given experimental data for Young’s modulus E(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweacaGGOaGaamiDaiaacMcaaaa@3384@  or tensile compliance C(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeacaGGOaGaamiDaiaacMcaaaa@3382@ , you will need to estimate G(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacMcaaaa@3386@  or J(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacaGGOaGaamiDaiaacMcaaaa@3389@ .  Precise values can’t be found without knowing the bulk modulus or Poisson’s ratio of the material, but for most practical applications you can assume that the bulk modulus is very large, in which case G(t)=E(t)/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacMcacqGH9a qpcaWGfbGaaiikaiaadshacaGGPaGaai4laiaaiodaaaa@3918@  and J(t)=3C(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacaGGOaGaamiDaiaacMcacqGH9a qpcaaIZaGaam4qaiaacIcacaWG0bGaaiykaaaa@3866@ .

 

 

 

 

3.6.6 Representative values for viscoelastic properties of polymers

 

The properties of polymers are very sensitive to their molecular structure, so for accurate predictions you will need to obtain data for the particular material you intend to use.  As a rough guide to typical values, data for the Young’s modulus of polyisobutylene (from McCrum, Buckley, Bucknall “Principles of polymer engineering” OUP New York 1988, p. 112) is shown in the picture on the right.  They give the glass transition temperature for this material as T g =193K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaam4zaaqabaGccq GH9aqpcaaIXaGaaGyoaiaaiodacaWGlbaaaa@3674@ .

 

The master-curve of G(t, T g ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacYcacaWGub WaaSbaaSqaaiaadEgaaeqaaOGaaiykaaaa@3631@  and the WLF shift function A(T, T g ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeacaGGOaGaamivaiaacYcacaWGub WaaSbaaSqaaiaadEgaaeqaaOGaaiykaaaa@360B@  can be deduced from their data, following the procedure discussed in Section 3.5.1.  The results  are plotted below.   The resulting WLF parameters, together with moduli and time constants for a 7-term Prony series fit to the data are listed in the table.   The shear modulus predicted by the Prony series is shown on the modulus-v-time plot for comparison with the experimental data.  

 

   

 

 

Viscoelastic properties of polyisobytylene

WLF parameters: C 1 g =36.9, C 2 g =57.6Kelvin  T g =193Kelvin MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaqhaaWcbaGaaGymaaqaaiaadE gaaaGccqGH9aqpcaaIZaGaaGOnaiaac6cacaaI5aGaaiilaiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam4qam aaDaaaleaacaaIYaaabaGaam4zaaaakiabg2da9iaaiwdacaaI3aGa aiOlaiaaiAdacaaMc8UaaGPaVlaabUeacaqGLbGaaeiBaiaabAhaca qGPbGaaeOBaiaabccacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGubWaaS baaSqaaiaadEgaaeqaaOGaeyypa0JaaGymaiaaiMdacaaIZaGaaGPa VlaaykW7caqGlbGaaeyzaiaabYgacaqG2bGaaeyAaiaab6gaaaa@7431@

G =0.143MPa  G 1 =7.3GPa  t 1 =100sec G 2 =4.9GPa  t 2 =2000sec G 3 =0.48GPa  t 3 =2× 10 4 sec G 4 =57MPa  t 4 =2× 10 5 sec G 5 =6.4MPa  t 5 =2× 10 6 sec G 6 =1.3MPa  t 6 =2× 10 7 sec G 7 =0.1MPa t 7 =4× 10 8 sec  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4ramaaBaaaleaacqGHEisPae qaaOGaeyypa0JaaGimaiaac6cacaaIXaGaaGinaiaaiodacaqGnbGa aeiuaiaabggacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaeiiaiaadEeadaWgaaWcbaGaaGymaaqabaGccqGH9a qpcaaI3aGaaiOlaiaaiodacaqGhbGaaeiuaiaabggacaqGGaGaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG0bWaaSbaaSqaaiaa igdaaeqaaOGaeyypa0JaaGymaiaaicdacaaIWaGaci4Caiaacwgaca GGJbaabaGaam4ramaaBaaaleaacaaIYaaabeaakiabg2da9iaaisda caGGUaGaaGyoaiaabEeacaqGqbGaaeyyaiaabccacaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaadshadaWgaaWcbaGaaGOmaaqaba GccqGH9aqpcaaIYaGaaGimaiaaicdacaaIWaGaci4CaiaacwgacaGG JbGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam4r amaaBaaaleaacaaIZaaabeaakiabg2da9iaaicdacaGGUaGaaGinai aaiIdacaqGhbGaaeiuaiaabggacaqGGaGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caWG0bWaaSbaaSqaaiaaiodaaeqaaOGaeyyp a0JaaGOmaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI0aaaaO Gaci4CaiaacwgacaGGJbaabaGaam4ramaaBaaaleaacaaI0aaabeaa kiabg2da9iaaiwdacaaI3aGaaeytaiaabcfacaqGHbGaaeiiaiaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamiDamaaBaaaleaaca aI0aaabeaakiabg2da9iaaikdacqGHxdaTcaaIXaGaaGimamaaCaaa leqabaGaaGynaaaakiGacohacaGGLbGaai4yaiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaadEeadaWgaaWcbaGaaGynaa qabaGccqGH9aqpcaaI2aGaaiOlaiaaisdacaqGnbGaaeiuaiaabgga caqGGaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG0bWa aSbaaSqaaiaaiwdaaeqaaOGaeyypa0JaaGOmaiabgEna0kaaigdaca aIWaWaaWbaaSqabeaacaaI2aaaaOGaci4CaiaacwgacaGGJbaabaGa am4ramaaBaaaleaacaaI2aaabeaakiabg2da9iaaigdacaGGUaGaaG 4maiaab2eacaqGqbGaaeyyaiaabccacaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaadshadaWgaaWcbaGaaGOnaaqabaGccqGH9aqp caaIYaGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaaiEdaaaGcci GGZbGaaiyzaiaacogacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caWGhbWaaSbaaSqaaiaaiEdaaeqaaOGaeyypa0JaaGim aiaac6cacaaIXaGaaeytaiaabcfacaqGHbGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaamiDamaaBaaaleaacaaI3aaabe aakiabg2da9iaaisdacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGa aGioaaaakiGacohacaGGLbGaai4yaiaabccaaaaa@B1DD@