Abstract Summary
Periodic wave propagation in a Euler-Bernoulli beam resting on a bilinear elastic foundation has been subject of recent analytical investigations. To realistically model embedding media with different reactions in compression and tensile conditions, different stiffnesses of the foundation for positive and negative values of the displacements have been considered. Several interesting behaviors have revealed by the analyses, such as veering-like phenomena with changes from single wave to multiple wave solution, and cusp points in the wave path. If interest is moved from macro- to micro- and nano-scale, materials and structures have been shown to display peculiar performances which are strongly related to their very small size. Indeed, the mechanical properties undergo a significant size effect, which in turn influences the static and dynamic behavior on the model. To account for the size effect in modelling the material behavior at the nanoscale, classical continuum elasticity, which is a scale free theory, cannot be resorted, so that nonlocal model have to be adopted. In this framework, a nonlocal model of Euler-Bernoulli beam resting on a bilinear elastic foundation is proposed, in order to possibly describe the behavior of carbon nanotubes embedded in elastic medium with different compression and tensile stiffnesses. The variational formulation is developed by resorting to a generic quadratic form of the elastic potential energy density. The ensuing equation of motion is governed by the sixth order spatial derivative, instead of the fourth order one obtained from the classical elasticity theory. The wave propagation is analytically investigated in order to describe the behavior of periodic waves with single and multi periodicity. The influence of scale length material parameters as well as foundation stiffnesses on the beam dynamical properties is discussed.
