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The classical models used for describing the behavior of woven fabrics do notfully account for the whole set of phenomena that occur during the testing ofsuch materials. This lack of precision is mainly due to the absence of energyterms related to the microstructural properties of the fabric and, inparticular, to the bending stiffness of the yarns. In this paper it is shownthat in the unbalanced fabrics the different bending stiffnesses of the warpand weft yarns produce macroscopic effects that are extremely visible as, forexample, the asymmetric S-shape during a Bias Extension Test (BET).  We propose to introduce a constrained micromorphic model and a discrete modelthat are able to account for i) the angle variation between warp and weft tows,ii) the unbalance in the bending stiffness of the yarns and iii) the relativeslipping of the tows.  The constrained micromorphic model is framed in the spirit of the Principleof Virtual Powers for the equilibrium of continuum bodies. A suitableconstraint is introduced by means of Lagrange multipliers in the strainenergydens ity and the resulting constrained model tends a particular secondgradientone. The main advantage of using such constrained micromorphic model isthat the kinematical and traction boundary conditions that can be imposed onthe boundary of the considered body take a natural and unique meaning.  The discrete model is set up by opportunely interconnecting Euler-Bernoullibeams with different bending stiffnesses in the two directions by means ofrotational and translational elastic springs. The main advantage of suchdiscrete model is that the slipping of the tows is described in a ratherrealistic way. Suitable numerical simulations are presented for both thecontinuum and the discrete models and a comparison between the simulations andthe experimental results is made showing a definitely good agreement.

Continuum and discrete models for unbalanced woven fabrics