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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!
A=adiabatic pt.
I = isothermal pt.
O-I: loaded so slowlythatspecimen remains isothermal
with surroundings.
A-I: If adiabatically stressedspeciment is held at constant
stress, it will, with passage oftime, warm up and elongate
by thermal expansion .
Thermally expand
with time
Thermally contract
with time
(un)loaded in
continuous cycle
Hayden, Moffat and Wulff (1965)
Stress-Strain and Anelastic Work
Elastic WorkDissipated= Work Done- WorkRecovered
Although elastic hysteresis loop may be very small, the
elastic
hysteresis effectis important if material is vibrated
rapidly:
total work = cycles * hysteresis-area/cycle!
Hayden, Moffat and Wulff (1965)
Energy Dissipation versus Frequency
(un)load quickly:
Purely adiabatic(un)load slowly:
Purely isothermal
(un)load intermediate:
Large dissipation!
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Metals: Anelastic SnoekEffect (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 ofthe unit
cell perpendicular to stress contract, "z= - #"x.
Contraction is more difficult at an edge $to forceuntil that
atom jumps into edge || to force.
If stress is applied slowly, C has time to diffuse, and,
if released slowly, it has time to reassume random
distribution.
If stress is applied rapidly, C atom diffuses out of
unfavorable site with passage of time, allowing a
Poisson contraction and corresponding elongation with
time.
!"os like OAI, or hysteresis loopunder cycling.
Polymers: Visco-Elastic Strain vs. Time
a="r# "
u
"r
Fraction of strain that lags
"="r(1#ae#t/$ )
"=a"re#(t#t1)/ $
Loading
Unloading
Relaxation time directly measured:
rise to 1/e of final value on loading.
decrease to 1/e of its initial value on
unloading.
Hayden, Moffat and Wulff (1965)
"="0 sin#t
$=$0 sin(#t% &)
Damping Capacity
Alternative to direct measurement of relaxation time is
damping capacityasfunction of frequency of stressing.
Forced vibration (beam or torsion pendulum.
free vibration (decay of amplitude is measured).
Dissipation Loop
"U= #d$%#0
2
E& 'sin(
M )E$0
2'sin(
U=
"0
2
2E=
E#0
2
2
Energy Dissipated/Cycle Total Stored
Elastic Energy
"U
U# 2$sin%
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:
"= ln#1
#2
$1
2
%u
u
Hayden, Moffat and Wulff (1965)
