Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory

It has been known since the pioneering works by Piola, Cosserat, Mindlin, Toupin, Eringen, Green, Rivlin and Germain that many micro-structural effects in mechanical systems can be still modeled by means of continuum theories. When needed, the displacement field must be complemented by additional kinematical descriptors, called sometimes microstructural fields. In this paper, a technologically important class of fibrous composite reinforcements is considered and their mechanical behavior is described at finite strains by means of a second-gradient, hyperelastic, orthotropic continuum theory which is obtained as the limit case of a micromorphic theory. Following Mindlin and Eringen, we consider a micromorphic continuum theory based on an enriched kinematics constituted by the displacement field u and a second-order tensor field psi describing microscopic deformations. The governing equations in weak form are used to perform numerical simulations in which a bias extension test is reproduced. We show that second-gradient energy terms allow for an effective prediction of the onset of internal shear boundary layers which are transition zones between two different shear deformation modes. The existence of these boundary layers cannot be described by a simple first-gradient model, and its features are related to second-gradient material coefficients. The obtained numerical results, together with the available experimental evidences, allow us to estimate the order of magnitude of the introduced second-gradient coefficients by inverse approach. This justifies the need of a novel measurement campaign aimed to estimate the value of the introduced second-gradient parameters for a wide class of fibrous materials.