A micro–macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity

A micromechanically based non-affine network model for finite rubber elasticity incorporating topological constraints was discussed in Part I (2004. J. Mech. Phys. Solids 52, 2617–2660) of this work. In this follow-up contribution we extend the non-affine micro-sphere model towards the description of time-dependent viscoelastic effects. The viscoelastic network model is constructed by an additive split of the overall response into elastic equilibrium-stress and viscoelastic overstress contributions. The equilibrium response of the network is understood to be related to results obtained from an infinite relaxation process and modeled by our above mentioned elasticity formulation. Inspired by (2004. J. Mech. Phys. Solids 52, 2617–2660), the rate-dependent overstress response is assumed to be driven by two micro-kinematical mechanisms related to the stretch and the area contraction of a tube containing a prototype chain. Firstly, a retraction of fictitiously unconstrained dangling chains is explained by diffusive reptile motions. Secondly, a release of constraint effects due to surrounding chains is modeled by a time-dependence of a tube cross-section area. The latter contribution is considered to be a result of the retraction of forest chains. We outline a distinct micromechanical model for the viscous overstress in terms of the above outlined two micro-kinematic mechanisms and discuss its numerical implementation in context of an affine homogenization procedure of space orientations. The characteristics and modeling capabilities of the proposed micro-sphere model of finite rubber viscoelasticity are reported for a broad spectrum of experimentally-based benchmark simulations. They demonstrate an excellent performance of the model in simulating rate and hystereses effects of rubbery polymers.