Bishop nanorods with stress-driven nonlocal elasticity and arbitrary boundary conditions

One-dimensional structural elements, such as thick rods, are essential in the development of nanoscale electromechanical systems. Their design and mechanical behavior at small scales can be effectively described using nonlocal continuum mechanics, which captures size-dependent effects and long-range interactions while reducing computational costs compared to atomistic methods. This study analyzes small-scale thick rods with realistic boundary conditions using Bishop’s model, incorporating a nonlocal integral constitutive relation. Usually, the nonlocal relation is derived from Eringen’s integral model, in which the nonlocal stress at a point depends on the entire strain field through a Fredholm-type convolution integral. However, this strain-driven formulation may lead to inconsistencies when applied to bounded domains. To address these issues, a stress-driven nonlocal integral formulation is adopted, providing a well-posed mechanical model that admits analytical solutions. A parametric study is conducted to evaluate the influence of nonlocal effects under various boundary conditions. Theoretical and numerical results contribute to a better understanding of the mechanical behavior of nanorods and offer valuable insights for the design of small-scale components.