Long time stress relaxation of amorphous networks under uniaxial tension:Â The Dynamic Constrained Junction Model

The constrained junction model that represents the stress-strain relations of amorphous networks in equilibrium is modified to analyze stress relaxation. Deviation of stress from equilibrium when a network is stretched suddenly is represented by a time dependent constraint contribution that is of the same form as that of the equilibrium theory. The time dependent motions of the junctions are assumed to obey the Langevin equation. The only new term in the model is a time dependent κ parameter that vanishes at long times. Results of the model are compared with uniaxial stress relaxation experiments on polyisoprene networks with different degrees of cross-linking. Experiments show that the time dependent κ parameter obeys a stretched exponential form, κ(t)=κ0exp[−(t/τ)β] with β = 0.4 and τ = 40 s, both of which are the same for all extensions and cross-link densities studied. The front factor κ0 depends on cross-link density in the same way as in the equilibrium case. Comparison with stress relaxation experiments shows satisfactory agreement at a wide range of extensions and for different degrees of cross-linking. The relatively low value of the stretched exponent parameter, β = 0.4, is interpreted in terms of a molecular picture where entanglements contribute to relaxation at a wide spectrum of time scales.