Part I deals with material modelling of woven fabric membranes. Due to their structure of crossed yarns embedded in coating, woven fabric membranes are characterised by a highly nonlinear stress-strain behaviour. In order to determine an accurate structural response of membrane structures, a suitable description of the material behaviour is required. A linear elastic orthotropic model approach, which is current practice, only allows a relative coarse approximation of the material behaviour. The present work focuses on two different material approaches: A first approach becomes evident by focusing on the meso-scale. The inhomogeneous, however periodic structure of woven fabrics motivates for microstructural modelling. An established microstructural model is considered and enhanced with regard to the coating stiffness. Secondly, an anisotropic hyperelastic material model for woven fabric membranes is considered. By performing inverse processes of parameter identification, fits of the two different material models w.r.t. measured data from a common biaxial test are shown. The results of the inversely parametrised material models are compared and discussed.