Metals and Polymers: StressStrain and ViscoElastic Responses

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Metals and Polymers Stress-Strain and
Visco-Elastic Responses
Polymers Stress-Strain Response
YS stress-max (necking) TS fracture point
PMA polymethyl methacrylate

• Type of polymer matters
• Strain-rate dependent
• Temperature dependent.

Calister, Ed. 6 (2003)
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Polymers Visco-Elastic Deformations
Semicrystalline polymer
Load vs. Time ta time of applied stress tr
time of released stress
Calister, Ed. 6 (2003)
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Polymers Visco-Elastic Deformations
Relaxation Modulus Er(t) ?(t)/?0
Er(t) vs T
log-Er(t) vs log-t
Melting Tc
Glass Tg
Calister, Ed. 6 (2003)
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Stress-Strain and Anelastic (t and T dependent)
Responses

• Adiabatic specimen is loaded so quickly that
  there is no time for it to absorb thermal energy
  from surroundings -
• Temperature of specimen will drop below
  surroundings by the time peak stress is reached!

Aadiabatic pt. I isothermal pt. O-I loaded
so slowly that specimen remains isothermal with
surroundings. A-I If adiabatically stressed
speciment is held at constant stress, it will,
with passage of time, warm up and elongate by
thermal expansion.
Hayden, Moffat and Wulff (1965)
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Stress-Strain and Anelastic Work
Elastic Work Dissipated Work Done - Work
Recovered
Hayden, Moffat and Wulff (1965)
Although elastic hysteresis loop may be very
small, the elastic hysteresis effect is important
if material is vibrated rapidly total work
cycles hysteresis-area/cycle!
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Energy Dissipation versus Frequency
(un)load intermediate Large dissipation!
(un)load quickly Purely adiabatic
(un)load slowly Purely isothermal
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Metals Anelastic Snoek Effect (measure by
friction) thermoelastic effect by stress-induced
diffusion

• Carbon interstitials occupy octahedral sites
  randomly. They slightly distort unit cell.
• Under stress, the unit cell elongates in
  direction of applied stress and, by the Poisson
  Effect, the edges of the unit cell perpendicular
  to stress contract, ?z - ? ?x.
• Contraction is more difficult at an edge ? to
  force until that atom jumps into edge to
  force.
• If stress is applied slowly, C has time to
  diffuse, and, if released slowly, it has time to
  reassume random distribution.
• If stress is applied rapidly, C atom diffuses
  out of unfavorable site with passage of time,
  allowing a Poisson contraction and corresponding
  elongation with time.
• ?? os like OAI, or hysteresis loop under
  cycling.

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Polymers Visco-Elastic Strain vs. Time
Hayden, Moffat and Wulff (1965)
Fraction of strain that lags

• Relaxation time directly measured
• rise to 1/e of final value on loading.
• decrease to 1/e of its initial value on
  unloading.

Loading Unloading
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Damping Capacity

• Alternative to direct measurement of relaxation
  time is damping capacity as function of frequency
  of stressing.
• Forced vibration (beam or torsion pendulum.
• free vibration (decay of amplitude is measured).

Energy Dissipated/Cycle
Total Stored Elastic Energy
Dissipation Loop
Measure of damping (relative loss)

• ? vs. f, fixed T
• ? vs. T, fixed f

For free oscillation, amplitude decreases with
time track logarithmic decrement, strain in one
cycle to next cycle
Hayden, Moffat and Wulff (1965)