Polymer networks with slip-links: 2. Constitutive equations for a cross-linked network

Stress-strain relations are derived for the mechanical response of elastomers at arbitrary three-dimensional deformations with finite strains. An elastomer is treated as an incompressible network of chains bridged by permanent (chemical cross-links and physical cross-links whose lifetime exceeds the characteristic time of deformation) and temporary (entanglements modeled as slip-links) junctions. Two types of chains are introduced in the network to distinguish between permanent and temporary nodes. Type-I chains have free ends, and their motion at the micro-level is constrained by a random number of slip-links. Type-II chains are Gaussian chains permanently connected to the network. Concentration of type-I chains is fixed, while the number of type-II chains per unit volume can change under deformation. The governing equations involve two (networks with constant concentrations of type-II chains) or three (networks where the content of type-II chains is affected by mechanical factors) material parameters. These parameters are found by fitting observations on rubbers, thermoplastic-elastomers, and thermoplastic-elastomer composites. Good agreement is demonstrated between the experimental data in uniaxial tensile tests and the results of numerical simulation at elongations up to 1,000%. It is shown that the adjustable parameters are affected by chemical composition and molecular architecture of polymers in a physically plausible way.