Analytical solutions of curvilinear nano-beams: An application to nano-circular ring problems and gradient initial stresses

This study investigates the analytical solutions for homogeneous, not only isotropic but also anisotropic curved nano-beams with axial symmetry and extends classical elasticity (CE) to gradient elasticity (GE). The stress fields are determined using a gradient Airy stress function, which corresponds to the classical Airy stress potential. For both cases, the gradient Airy stress functions are derived from analytical solutions of the governing differential equations, which are written in the form of the (classical) Airy stress function. The corresponding stress in GE is determined for different cases with displacement fields derived from CE. The analytical solutions show that GE stresses and displacement fields contain expressions with Bessel and hypergeometric functions. These solutions make it possible to compare the CE and GE stress and displacement fields. As an application, the stress and displacement fields for a nano-circular ring representing a multiply connected body are solved analytically for classical and nano-scale cases. In addition, the initial stresses are extended to GE for nested rings, where the GE initial stresses are introduced. Finally, it is shown analytically and numerically that the GE solutions for all derived stress and displacement fields, including the gradient Airy stress functions, approach CE when the gradient coefficient c converges to zero.