Polymers- Principles of Viscoelasticity

Question 1

Retardation time for Zener model

Answer 1

τ=Jdη
Jd is for spring in parallel to dashpot
η is for the dashpot

Question 2

Mechanical model for natural rubber

Answer 2

Zener model but with another dashpot in series at the bottom of the parallel bit. The stress applied from top of top spring and bottom of bottom dashpot. Top spring J1 is bond displacement, spring and dashpot J2 and η2 are chain uncoiling, bottom dashpot η3 is chain slippage

Question 3

Compliance for mechanical model for natural rubber

Answer 3

J(t)=J1+J2(1-exp(-t/τ2))+t/η3

Question 4

Distribution of relaxation times

Answer 4

Like Maxwell-Boltzmann distribution curve. Is j(τ) vs τ. Integrating j(t) from 0 to infinity with respect to τ gives JY-JU. Same applies for G using g(τ) which has peak to the right of j.

Question 5

Boltzmann superposition principle

Answer 5

Total response is the sum of the individual responses from each stress increment. Stress vs time goes up in sharp steps. Strain vs time goes up like root x curves adding on top of each other at each stress increase.
e(t)=e1(t)+e2(t)+e3(t)+…=sum 1 to N of J(t-τn)Δ σn
Or the integral from minus infinit to t of J(t-τ)dσsubτ

Question 6

What happens to strain if load suddenly gone?

Answer 6

After the root x curve up (before that vertical up) goes bit vertical down then exponential decrease

Question 7

Origin of temperature dependence and shift factor

Answer 7

See p11 week 8 for details
JsuperT0(t)=JsuperT(αsubT x t)
αsubT is shift factor
Higher T means faster relaxation time so J vs log t graph shifts left.
T is greater than T0 if αsubT is less than 1

Question 8

For viscoelastic materials, what is time equivalent to?

Answer 8

Temperature

Question 9

How to get the master curve for E vs t

Answer 9

There are different curves over the same time period for different temperatures. You want a master curve at one temperature. You use the temperature shift factor for each one to find their translation on the graph and join them together

Question 10

Relaxation time Arrhenius equation

Answer 10

τsubT=τsubinf exp(ΔH/RT)
ΔH is activation enthalpy of the relaxation
R is gas constant

Question 11

Empirical WLF equation

Answer 11

For finding shift factor
log(αT)=-C1(T-Tg)/(C2+(T-Tg))
Tg is glass transition temperature 
C1 is 17.4K
C2 is 51.6K

Question 12

Dielectric thermal analysis

Answer 12

DETA. See p14 week 8

Frequency range 20Hz-100kHz