Gradient elasticity theory for fiber composites with fibers resistant to extension and flexure

A model for the mechanics of an elastic solid, reinforced with bidirectional fibers is presented in finite plane elastostatics. The fiber's resistance to stretch and flexure are accounted for with variational computations of first and second gradient of deformations, respectively. Within the framework of strain-gradient elasticity, the Euler equation and necessary boundary conditions are formulated. A rigorous derivation of the corresponding linear theory is also developed from which, a complete analytical solution is obtained for small deformations superposed on large. In particular, we assimilate plane bias extension test results indicating that the proposed model successfully predicts smooth transitions of shear strain fields unlike those depicted by the first gradient theory where significant discontinuities are present. The proposed model can serve as an alternative 2D Cosserat theory of non-linear elasticity.