Mechanics of third-gradient continua reinforced with fibers resistant to flexure in finite plane elastostatics

Abstract

A third-gradient continuum model is developed for the deformation analysis of an elastic solid, reinforced with fibers resistant to flexure. This is framed in the second strain gradient elasticity theory within which the kinematics of fibers are formulated, and subsequently integrated into the models of deformations. By means of variational principles and iterated integrations by parts, the Euler equilibrium equation is obtained which, together with the constraints of bulk incompressibility, compose the system of the coupled nonlinear partial differential equations. In particular, a rigorous derivation of the admissible boundary conditions arising in the third gradient of virtual displacement is presented from which the expressions of the triple forces are derived. The resulting triple forces are, in turn, coupled with the Piola-type triple stress and are necessary to determine a unique deformation map. The proposed model predicts smooth and dilatational shear angle distributions, as opposed to those obtained from the first- and second-gradient theory where the resulting shear zones are either non-dilatational or non-smooth.