Continuum and discrete models for unbalanced woven fabrics

The classical models used for describing the mechanical behavior of woven fabrics do not fully account for the whole set of phenomena that occur during the testing of such materials. This lack of precision is mainly due to the absence of energy terms related to the microstructural properties of the fabric and, in particular, to the bending stiffness of the yarns. The importance of the bending stiffness on the overall mechanical behavior of woven reinforcements, if already essential for the complete description of balanced fabrics, becomes even more important in the case of unbalanced ones. In this paper it is shown that the unbalance in the bending stiffnesses of the warp and weft yarns produces macroscopic effects that are extremely visible: we mention, for example, the asymmetric S-shape of a woven interlock subjected to a Bias Extension Test. We propose to introduce a constrained micromorphic model and, simultaneously, a discrete model that are both 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 relative slipping of the tows. The introduced constrained micromorphic model is rigorously framed in the spirit of the Principle of Virtual Work for the study of the equilibrium of continuum bodies. A suitable constraint is introduced in such micromorphic model by means of Lagrange multipliers in the strain energy density and the resulting constrained model is seen to be a particular second gradient one. The main advantage of using such constrained micromorphic model is that the kinematical and traction boundary conditions that can be imposed on some sub-portions of the boundary of the considered body take a natural and unique meaning. The discrete model is set up by opportunely interconnecting Euler Bernoulli beams with different bending stiffnesses in the two directions by means of rotational and translational elastic springs. The main strength of such discrete model is that the slipping of the tows is described in a rather realistic way. Suitable numerical simulations are presented for both the continuum and the discrete models and a comparison between the simulations and the experimental results is performed showing a definitely good agreement. (C) 2016 Elsevier Ltd. All rights reserved.