Materials for engineering, 3rd Edition - (Malestrom)

Organic polymeric materials 167
seen in Fig. 5.4, curve 1, and Treloar extended his single-chain model to the
case of a cross-linked network to give an expression of the form:
F = NkT(λ 2 – λ –1 )
where N is the number of separate chains per unit volume, and λ = L/L o . The
measured stress–strain curves of elastomers are in good agreement with this
relationship for strains of up to 400%.
In simple shear, the shear strain γ is equal to (λ – λ –1 ), and the force
deformation relationship is
F = NkTγ
so that Hooke’s Law is obeyed in simple shear and the shear modulus in the
unstrained state is:
G = Nkγ
showing that the shear modulus of elastomers will increase as the number of
cross-links between the polymer chains is increased.
Polymers with higher tensile modulus
The tensile modulus of polymers is increased if the molecular chains are
cross-linked together. For example, cross-linked rigid thermosets were among
the first synthetic polymers to be manufactured. Such materials exhibit the
highest tensile moduli (and lowest breaking strains) of common polymers,
typified by curve 2 in Fig. 5.4.
5.4.3 Yielding
Drawing
Many partially crystalline polymers, such as polyethylene, polypropylene
and polyamide show a tensile response illustrated in curve 4, Fig. 5.4. A nonlinear
pre-yield behaviour is followed by a yield point. The stress then drops
and stabilizes under further straining until a final rise in stress and failure.
During the region of constant stress a stable neck is formed which extends
progressively throughout the gauge length. This is known as cold drawing
and, during this process, the molecules in the polymer are being extended in
the direction of straining, leading to a highly preferred orientation in the
test-piece.
The result is a necked region which is much stronger than the unnecked
material; that is why the neck spreads instead of simply causing failure.
When the drawing is complete, the stress–strain curve rises steeply to final
fracture.