On the post-buckling of elastic beams on gradient foundation

The post-buckling of an axially loaded elastic beam resting on linearly elastic medium is investigated in this paper from a geometrically exact analysis. It is known that the elastic foundation increases the bifurcation limit, but it may have a destabilizing effect on the post-buckling behavior associated to imperfection sensitivity. This unstable nature of the post-buckling behavior may lead to drastic softening phenomena, as already investigated for plasticity or Continuum Damage Mechanics media. It is suggested in this paper to study the influence of gradient terms in the interaction foundation model on the post-buckling behavior of this structural system. The gradient elasticity foundation model of Pasternak is used and introduced by variational arguments in a geometrically exact framework. A general nonlinear fourth-order differential equation is obtained, and numerically solved with a nonlinear boundary value solver. The post-buckling behavior is analyzed from an asymptotic method. The gradient elasticity constitutive law significantly affects the post-localization process.