ABSTRACT
To the best of our knowledge this chain length statistics has not so far been applied to rubber elasticity or softening models. For example, it appears to be very advantageous for full network rubber models. These models assume a continuous directional distribution of chains. Within this concept, three-dimensional models result from the integration of single polymer chain energies over the unit sphere. Both isotropic and anisotropic spatial distribution of chains can be taken into account (see, e.g. (Wu & Giessen 1993)). In the case of initially isotropic directional chain distribution the integration can be carried out analytically provided the entropic energy of a single polymer chain can be developed in a Taylor series of the stretch square (Itskov, Ehret, & Dargazany 2010). Alternatively, various integration schemes can be applied. For example, James-Guth 3-chain (James & Guth 1943) and Arruda-Boyce 8-chain models (Arruda & Boyce 1993) are based on a 3 and 8-point integration scheme, respectively (see also (Beatty 2003)). In the present contribution, we are thus going to implement this distribution to a full network rubber elasticity and softening model. Applying the analytical integration over the unit sphere (Itskov, Ehret, & Dargazany 2010) an analytical full network rubber elasticity model is formulated for the case of initially isotropic directional distribution of chains. A softening model is based on an assumption that the minimal number of chain segments available in the
1 INTRODUCTION
According to the classical statistical theory of polymerization the probability that a linear polymer molecule is composed of exactly k segments is given by [Flory 1953]
P k p kk( ) ( )p= , , , ,−1 1= 2 … (1)
where 0 1p denotes the probability of the chain propagation while 1− p represents then the probability of the chain termination. This probability function is illustrated in Figure 1 for various values of p.
