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New hyperelastic orthotropic models are proposed for the simulation of textile membranes used in civil engineering applications. In contrast to published models, part of the new models is polyconvex and ensures thereby a physically meaningful and mathematically sound formulation. The models are adjusted to uniaxial tension tests performed in warp and fill direction, where not only the stress-strain response in tension direction is accounted for but also the lateral contraction. Thereby, the crosswise interaction between the warp and fill direction is captured. In a series of different boundary value problems the new models as well as a competitive formulation given in literature are compared with respect to the accuracy to represent the experimental data, the mathematical properties as well as the numerical robustness. As it turns out, most formulations including the model from the literature show a loss of material stability and non-converging Newton iterations in structural simulations. Only one of the proposed polyconvex formulations works robustly in numerical simulations of realistic structural engineering problems. Thereby, this new orthotropic model enables realistic simulations of textile membranes in a fully geometrically nonlinear setting, which does not require simplifications based on linearized strains, which are currently used as standard in engineering practice. © 2019 Elsevier Ltd