Materials for engineering, 3rd Edition - (Malestrom)

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Materials for engineering
by a phase angle δ. The peak values of the stress and strain, σ 0 and ε 0 , are
related by the complex modulus E*, where E* = E′ + i E″, E′ is known as the
storage modulus and E″ as the loss modulus.
One can also define a complex compliance, D* = 1/E* = D′ – i D″, where
D′ and D″ represent the storage compliance and loss compliance, respectively.
From a consideration of the energy dissipated in a given cycle and by
neglecting heat losses to the surrounding environment, the temperature rise
per unit time (∆T) may be described by the equation:
π fD′′ ( f , T ) σ 0 2
∆ T =
ρc
p
where ρ is the density and c p the specific heat of the polymer. As the temperature
of the specimen rises, its elastic modulus will decrease, resulting in larger
deflections for a given applied stress amplitude. These larger deflections will
in turn give even greater hysteretic energy losses until a point is reached
when the specimen can no longer support the applied load. In Fig. 5.9, ∆σ –
N f curves show three distinct regions, marked I, II and III. Region III represents
an endurance limit, below which heat is dissipated to the surroundings as
rapidly as it is generated, and the temperature of the specimen stabilizes at
a value which is insufficient to cause thermal failure after 10 7 cycles.
Thermal fatigue may be reduced by decreasing the test frequency, cooling
the test sample, or increasing the surface-to-volume ratio of the testpiece.
The introduction of intermittent rest periods allows the specimen to cool and
is another means of achieving an increased fatigue life.
Mechanical fatigue
The existence of region I (Fig. 5.9) depends on whether crazes form at high
values of ∆σ, and whether the crazes cause microscopic cracks to nucleate.
∆σ
I II III
log N f
5.9 Schematic S–N curve for polymers.