Plane bias extension test for a continuum with two inextensible families of fibers. A variational treatment with Lagrange multipliers and a perturbation solution

In the present paper, a system constituted by two families of nearly inextensible fibers is studied. The interest towards this kind of structures, which can be suitably modeled by 2D continua with inextensible curves, comes form both theoretical and applicative motivations. Indeed, the advantageous weight/strength ratio and the particularly safe behavior in fractures make them attractive in many engineering fields, while the mechanical properties displayed by them are theoretically interesting and still not well understood in full generality. We focused on dead-loading traction boundary-value problems assuming particular forms for the deformation energy associated to every node of the structure. The main result of the paper consists in the determination of the nonlinear integral equations describing fibers directions for a specific class of deformation energies. A perturbative analysis of a particular solution is also performed, and some physical justification for one of the considered forms for the energy density are provided in the Appendix.