Strain gradient viscoelasticity theory of polymer networks

Abstract

A physically-based strain gradient viscoelasticity theory is proposed specially for polymer networks, accounting for the strain gradient effect and the history-dependent behavior, where the microstructure-dependence and history-dependence of stress and hyperstress are interpreted physically and quantitatively. The chain representation of the Helmholtz free energy density is transferred to the strain gradient continuum field representation through the geometric (spatial) connection between the chain stretch ratio and the continuum deformation measure. The necessity of using strain gradients to characterize asymmetric deformation of the microstructure of polymer networks serves as the origin of microstructure-dependent hyperstress, which is work-conjugated to the strain gradient field. The free energy per chain is assumed history-dependent to account for chain-environment interactions (e.g., friction between segments), which ultimately results in the history-dependent behavior of polymeric solids. A general constitutive relation is provided, which together with the assumed network structure, gives a concrete one. It is shown that independent of the assumed network structure, the stress, and the hyperstress share the same dimensionless relaxation function. A strain gradient viscoelastic constitutive relation based on the eight-chain network is applied to the analysis of polymer nanocomposites, where the complex moduli are rationally defined and calculated numerically. It is found that the strain gradient effect can be viewed as an amplifier that transforms the strain-induced stress into the effective stress.