Strain hardening in glassy polymers

An alternative form of eq. (8.41a) representing the entanglement density in terms of the material density p and the molecular weight Me of the chain length between molecular entanglements is (Treloar 1975) 

Incorporation of the hardening component of the flow stress into the plastic-resistance component then gives the total stress strain relation beyond the flow state for which hardening by molecular alignment becomes prominent. This gives for uniaxial behavior of tension and compression rc. through the use of eqs. (8.34a), the following expressions  

In eqs. (8.42) we recall that the dilatancy factor depends on f, the liquid-like-material fraction, which increases with increasing plastic strain -f (or e ) from a low value cp- of around 0.05 at -f= 0 in the annealed structure to = 0.5 in the flow state through the strain-softening range and remains stationary after that. In the stress-strain relationship all plastic strains are represented as the true equivalent plastic strain sP, where we recall that 

Finally, the plastic resistance in shear where there is no interaction energy between the dilatational transformation strain and a mean normal stress since the latter is not present, becomes 

It is important to note that the plastic-resistance component has a very different temperature dependence that is based on an Arrhenian expression 

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