Analysis of a gradient-elastic beam on Winkler foundation and applications to nano-structure modelling

• A simplified strain gradient elastic beam model on Winkler foundation is analyzed.
• Higher-continuity Hermite finite elements have been developed for the solution.
• The Winkler foundation and gradient elasticity parameter are studied parametrically.
• In the case of the static problem, an analytical solution has also been obtained.
• The cases of a cantilever and a simply supported beam have been examined.

An equation of motion governing the response of a first strain gradient beam, including the effect of a Winkler elastic foundation, is derived from the Hamilton–Lagrange principle. The model is based on Mindlin's gradient elasticity theory, while the Euler-Bernoulli assumption for slender beams is adopted. Higher-continuity Hermite Finite Elements are presented for the numerical solution of related Initial-Boundary Value (IBV) problems. In the static case an analytical solution is derived and the convergence characteristics of the proposed Finite Element formulation are validated against the exact response of the configuration. Several examples are presented using “equivalent beam” data for Carbon Nanotubes (CNT's) and the effect on the Winkler foundation is studied. Finally, applicability of the derived model for the simulation of micro-structures, as for example CNT's or Microtubules, is discussed.