The basic mathematical concepts and tools for differentiable manifolds are provided as needed in the following, with emphasis on symplectic manifolds.

Various types of fibers encountered in Network materials are presented and classified in this chapter. Their mechanical behavior is of primary concern here. The first section describes the structure and mechanical behavior of cellulose fibers, polymeric fibers used in nonwovens, and collagen fibers forming connective tissue. The remainder of the chapter is divided into three parts presenting the mechanical behavior of athermal fibers, thermal filaments, and of fiber bundles. The linear, nonlinear, and rupture characteristics of athermal fibers are presented. Thermal filaments, which form molecular networks such as elastomers and gels, are described by the Gaussian, Langevin, and self-avoiding random walk models. Models describing the mechanics of semiflexible filaments are presented. The section on the mechanics of fiber bundles presents a number of results relevant for bundles of continuous and discontinuous (staple) fibers, including the effect of bundle twisting and of packing on the axial stiffness and strength of the bundle. These results apply to many networks of practical importance which are composed from fiber bundles.

This chapter presents results related to the interaction of fibers. In networks, fibers form crosslinks and develop contacts. The mechanical behavior of the crosslinks between athermal fibers and between thermal filaments is discussed, with emphasis on crosslink failure and strength. Contacts between fibers form either along the fiber length, within bundles of parallel fibers, or at specific sites where fibers cross at an angle. The mechanics of elastic contacts with and without cohesive interactions between fibers is reviewed. Sliding of fibers in contact is often encountered in networks and friction has strong effects on network mechanics. Several results related to friction at elastic contacts are reviewed.