A conventional rubber elasticity equation based on the inverse Langevin function, combined with a yield stress (Y0) has been used for the purpose of studying published tensile stress–strain curves for thermoplastic elastomers. In order to simplify the calculation a Pade approximation has been employed [Cohen A. Rheol. Acta. 1991;30:270] for the inverse Langevin function which leads to the following equation, relating f, the nominal or engineering stress, to the extension ratio l : f Y0 =l 1 Cr=3l 3 2 l2 =n= 1 2 l2 =n 2 1=l2 3 2 1=ln= 1 2 1=ln: When suitable values of the parameters are selected the equation may be used to model nominal stress–strain curves for the new ultralow density polyethylenes, ethylene vinyl acetate copolymers and SBS block copolymers. The parameters Cr and n selected in this way represent identifiable physical entities; Cr the initial modulus and n 1/2 the limit of extensibility. However Cr does not increase with temperature as with a conventional rubber, but declines as the temperature is raised. With the polyethylenes this may be related to the gradual melting of the crystals which are believed to act as cross links [Bensason S, Stepanov EV, Chum S, Hiltner A, Baer E. Macromols 1997;30:2436]. However, with an SBS block copolymer the reason for the fall in Cr and the rise in n are not clear. Generally, for instance when the temperature is reduced and the materials become stiff, Cr will increase and n decrease. However when it is plotted against crystallinity with the ultralow density polyethylenes, n does not follow Cr but shows a minimum at a crystallinity of 30% after which it appears to increase. With polyethylenes n is more sensitive to molecular weight than Cr and gives a linear Flory plot for n 1/2 against 1/T at 08C. At 258C the values of n obtained are very high and when the molecular weight falls to 32 000 and the stress–strain curve is found to follow a Gaussian equation. This supports the mathematical requirement that the equation reduces to a Gaussian form when n is very large. The same result can be predicted from a series approximation suggested by Treloar.
Посилання на статтю:
The application of non-Gaussian chain statistics to ultralow density polyethylenes and other thermoplastic elastomers / R.N. Haward // Polymer. – 1999. – N 40. – P. 5821–5832.