First versus second gradient energies for planar sheets with two families of inextensible fibres. Investigation on deformation boundary layers, discontinuities and geometrical instabilities

We present a homogenized model for the analysis of a 2D continuum with two straight families of inextensible fibres embedded in it. The kinematics of the continuum is analyzed and, motivated by phenomenological observations, it is assumed that the strain energy depends on the shear deformation of the fibres and on their bending curvature. It is shown that in order to account for the latter deformation it is necessary to introduce second gradient strains. The problem is formulated as a nonlinear constrained minimization, after introducing a suitable discretization of the domain. Some deformation processes are simulated using different constitutive hypotheses, comparing the predictions obtained assuming the presence of only first gradient or second gradient deformations, or a combination of both. It is found that the first gradient model leads to the presence of discontinuities in the rotation of the fibres, while the second gradient model regularizes these discontinuities by means of boundary layers. In particular in some deformation processes an instability of geometrical nature is observed when the second gradient model is used, that can be suppressed by the first gradient contribution