Closed form solution for a nonlocal strain gradient rod in tension

A size-dependent integral elasticity model is developed for a small-scaled rod in tension based on the nonlocal strain gradient theory. The integral rod model contains a nonlocal parameter and a material length scale parameter to incorporate the scaling effects of nonlocal stress and microstructure-dependent strain gradient. In comparison to size-dependent differential models, the developed integral rod model is both self-consistent and well-posed. The governing equations and boundary conditions for the nonlocal strain gradient rod in tension are derived by employing the principle of virtual work. In addition to the classical natural and essential boundary conditions, non-classical natural and essential boundary conditions are present for the integral rod model. The closed-form solutions for predicting the displacement and reduced Young's modulus are derived for four types of boundary conditions. It is shown explicitly that the integral rod model can exert stiffness-softening and stiffness-hardening effects by considering various values of size-dependent parameters and boundary conditions. It is found that, the developed rods with four different boundary conditions can predict the scaling effects of the Young's modulus of single-walled carbon nanotube, and the scaling effects are more sensitive to the size-dependent parameters (the material length scale parameter and the nonlocal parameter) in comparison with the non-classical boundary conditions.