Page 309 - Fiber Fracture

FRACTURE OF SYNTHETIC POLYMER FIBERS 29 I

Fig. 2. Model representation of an unoriented polymer network. The dots represent entanglement loci and
the dashed lines denote the van der Waals bonds.

rubber elasticity (Treloar, 1958. According to this theory, the stress on a stretched chain
strand having a vector length r is given by
cr = crkTL-'(r/nz) (3)

In Eq. 3, n denotes the number of statistical chain segments of length 1 in the strand.
Also, L-' is the inverse Langevin function and ct is a proportionality constant.
Eqs. 1-2 for the vdW bond breaking process are implemented on the computer with
the help of a Monte-Carlo lottery which breaks a bond i according to a probability

Pi = ui /urn,, (4)
in which ui is obtained from Eq. 1, whereas urn, is the rate of breakage of the most
strained bond in the array. After each visit of a bond, the time t is incremented by
l/[u,,n(t)] where n(t) is the total number of intact bonds at time t. The simulation
of chain slippage through entanglements is executed using a similar technique. For that
process, n(t) now denotes the total number of entanglements left at time t.
After a very small time interval 6t has elapsed, the vdW bond breaking, chain
slippage and fracture processes are halted and the network is elongated along the y-axis
by a small constant amount that is determined by the rate of elongation E.. Subsequently,
the network is relaxed to its minimum energy configuration. This relaxation is executed
using a series of fast computer algorithms, described in Termonia et al. (1985), which
steadily reduce the net residual force acting on each entanglement point. After these