Correlation between beam on Winkler-Pasternak foundation and beam on elastic substrate medium with inclusion of microstructure and surface effects

Abstract

A novel beam-elastic substrate element with inclusion of microstructure and surface energy effects is proposed in this paper. The modified couple stress theory is employed to account for the microstructure-dependent effect of the beam bulk material while Gurtin-Murdoch surface theory is used to capture the surface energy-dependent size effect. Interaction mechanism between the beam and the surrounding substrate medium is represented by the Winkler foundation model. The governing differential equilibrium and compatibility equations of the beam-elastic substrate system are consistently derived based on virtual displacement and virtual force principles, respectively. Both essential and natural boundary conditions of the system are also obtained. Two modified Tonti's diagrams are presented to provide the big picture of both displacement-based and force-based formulations of the system. Due to similarity between the current problem and the one related to the beam on Winkler-Pasternak foundation, the so-called "natural" beam-Winkler-Pasternak foundation element coined by the authors is employed to perform two numerical simulations to study the characteristics and behaviors of a beam-substrate system with inclusion of microstructure and surface effects.