The paper is dedicated to the modeling of the elasticity problem for a textile structure. The textile is made of long and thin fibers, crossing each other in a periodic pattern, forming a woven canvas of a square domain. The textile is partially clamped. The fibers cannot penetrate each other but can slide with respect to each other in the in‐plane directions. The sliding is bounded by a contact function, which is chosen loose. The partial clamp and the loose contact lead to a domain partitioning, with different expected behaviors on each of the four subdomains. The homogenization is made via the periodic unfolding method, with an additional dimension reduction. The macroscopic limit problem results in a Leray–Lions problem with only macroconstraints in the plane.