  The classical continuum models used for the woven fabrics do not fully
describe the whole set of phenomena that occur during the testing of those
materials. This incompleteness is partially due to the absence of energy terms
related to some micro-structural properties of the fabric and, in particular,
to the bending stiffness of the yarns. To account for the most fundamental
microstructure-related deformation mechanisms occurring in unbalanced
interlocks, a second-gradient, hyperelastic, initially orthotropic continuum
model is proposed.
  A constitutive expression for the strain energy density is introduced to
account for i) in-plane shear deformations, ii) highly different bending
stiffnesses in the warp and weft directions and iii) fictive elongations in the
warp and weft directions which eventually describe the relative sliding of the
yarns. Numerical simulations which are able to reproduce the experimental
behavior of unbalanced carbon interlocks subjected to a Bias Extension Test are
presented. In particular, the proposed model captures the macroscopic
asymmetric S-shaped deformation of the specimen, as well as the main features
of the associated deformation patterns of the yarns at the mesoscopic scale.
