Derivation of full-network models with chain length distribution

The majority of network models for rubber-like materials are derived for homogeneous materials: all chains have the same length. So, these models are not defined to predict the mechanical response of materials composed by chains of different lengths. Recently, some specific constitutive equations for inhomogeneous rubber-like materials have been proposed: most of them consist in assemblies of homogeneous networks with different chain lengths assuming equal stretch of chains. Only few papers adopt an equal stress assumption. Here, we propose a full-network model for materials with chain length distribution. In each direction of the unit sphere, chains of different lengths are supposed to bear the same force. More precisely, the affine deformation of the unit sphere containing non-Gaussian freely jointed chains is considered. It is demonstrated that: (i) the equal force assumption leads to the equality of relative stretches in chains of different length, and (ii) the response of an inhomogeneous network is similar to the one of a homogeneous network, characteristics of which being defined from the chain length distribution.