A Shear Lag Theory Integrated with Second Strain Gradient Continuum Model for the Composite Reinforced with Extensible Nano-Fibers

A continuum model is formulated to obtain the mechanical response of an elastic solid reinforced with randomly distributed nanofibers. The kinematics of embedded nanofibers are formulated by incorporating the shear lag parameter and Krenchel Orientation factor into the second strain-gradient model. The Euler equilibrium equations and associated boundary conditions are obtained by means of variational principles and iterated integration by parts. In particular, we obtain the rate of change in curvature via the computation of third gradient of deformations, and this results in an additional set of boundary conditions from which the expressions of Piola-type triple stress and its coupled triple forces are formulated. The numerical solutions of the resulting systems of nonlinear coupled differential equations are obtained via the custom-built finite element analysis scheme. Finally, we simulate the uniaxial extension test with respect to various fiber aspect ratios and triple stress parameters to obtain the mechanical response of the nanofiber composite system. It is observed that, among other interesting features, the fiber aspect ratio and the third gradient constitutive parameter are related to the effective Young’s modulus of the third gradient continua.