This thesis aims to give the macroscopic behaviors for the small deformations of textile structures, as well as mathematical explanations of phenomena observed in real experiments. The textile structure is presented as a woven canvas made of long, thin fibers, crossing each other in a periodic pattern. The square cloth is partially clamped, and the fibers can slide in the in-plane directions up to a specific grade of freedom to simulate friction.





Due to the problem's complexity, the first part is devoted to extending the theoretical results of the periodic unfolding to sequences that are anisotropically bounded and defined on periodic lattices. In the second part, the linear elasticity problem is taken into account. The partial clamp and strength of contact that holds the fibers together (from glued to strong to loose) led to different cases to study. Through the homogenization and dimension reduction, each asymptotic behavior as period and thickness go to zero is treated. Among the obtained results, of great interest are the in-plane rotation effects of the fibers in the not-clamped areas, and the maximum rotation allowed with respect to the contact strength.