A model for the second strain gradient continua reinforced with extensible fibers in plane elastostatics

A second strain gradient theory-based continuum model is presented for the mechanics of an elastic solid reinforced with extensible fibers in plane elastostatics. The extension and bending kinematics of fibers are formulated via the second and the third gradient of the continuum deformation. The Euler equations arising in the third gradient of virtual displacement are then formulated by means of iterated integration by parts and variational principles. A rigorous derivation of the associated boundary conditions is also presented from which the expressions of triple forces and stresses are obtained. The obtained triple forces are found to be in conjugation with the Piola-type triple stress and are necessary to determine energy contributions on edges and points of Cauchy cuts. In particular, a complete linear model including admissible boundary conditions is derived within the description of superposed incremental deformations. The obtained analytical solution predicts smooth deformation profiles and, more importantly, assimilate gradual and dilatational shear angle distributions throughout the domain of interest.