Page 308 - Fiber Fracture

290 Y. Termonia

Stretching

(High T)
I
I”.
Low Stiff ness

High Stiffness

Fig. 1. (a) Network of polymer chains after polymerization. (b) Same network after drawing.

coordination number of an entanglement is assumed to be 4, the actual 3-dimensional
network has been for convenience given a planar x-y configuration. The y-axis is
chosen as the direction of draw.
The network of Fig. 2 is deformed at a constant temperature T and a rate of
elongation i.. This leads to straining of the van der Waals (vdW) bonds which are broken
according to the Eyring kinetic theory of fracture (Kausch, 1987), at a rate

v=texp[-(U-/3a)/kT] (1)
In Eq. 1, t is the thermal vibration frequency, U and @ are, respectively, the activation
energy and volume whereas
O=KE (2)
is the local stress. In Eq. 2, E is the local strain and K is the elastic constant for the
bond. These vdW bond breakages lead to a release of the chain strands, which are now
to support the external load. Once broken, vdW bonds are assumed not to reform.
As the stress of the ‘freed’ chain strands increases, slippage through entanglements is
assumed to set in at a rate that has the same functional form as that for vdW breakings
(Eq. 1) but, with different values for the activation energy U and volume @. In Eq. 1, a
now denotes the difference in stress in two consecutive chain strands that are separated
by an entanglement. This stress difference is calculated using the classical treatment of