Constitutive modelling of fibre networks with stretch distributions. Part I: Theory and illustration

In this contribution, we propose a novel continuum mechanical concept to determine the homogenised mechanical response of random fibre networks. Their free energy is calculated as an average of the fibre free energy over the distribution of stretch, which forms the core of the new theory. The fibre-scale kinematic information contained in this distribution is intrinsic to the model and available for any state of macroscopic deformation in addition to the homogenised macroscopic response. In contrast to the various approaches that use directional averaging over the unit sphere, the presented model does not rely on a relation between fibre deformation and orientation, and therefore applies to highly non-affine networks as well as affine ones. In the present Part I of the work, the new concept is formally introduced, put into perspective with current approaches, and elaborated for the case of networks of elastic fibres with uniform orientation distribution, which generate a macroscopically isotropic hyperelastic response. The essential constitutive assumption of the theory establishes a relation between the macroscopic deformation of the network and the microscopic stretch distribution within. A phenomenological approach is used to illustrate the new concept, expressing statistical moments of the stretch distribution in terms of the macroscopic principal strain invariants. Application of the concept to 2D and 3D Voronoi networks with fibres of different properties finally exemplifies the accurate agreement of the model with discrete network simulations in terms of both the macroscopic and microscopic response. While the here presented phenomenological variant of the approach therefore represents an advance in the analytical multiscale modelling of network materials itself, the work also provides the basis for further developments, where the relation between the stretch distribution and macroscopic strain is derived from alternative principles.