Localized buckling of an elastic strut in a visco-elastic medium
Certain types of long, axially compressed structures have the potential to buckle locally in one or more regions rather than uniformly along their length. Here, the potential for localized buckle patterns in an elastic layer embedded in a visco-elastic medium is investigated using a strut-on-foundation model. Applications of this model include the growth of geological folds and other time-dependent instability processes. The model consists of an elastic strut of uniform flexural stiffness supported by a Winkler-type foundation made up of discrete Maxwell elements. Mathematically, this model corresponds to a nonlinear partial differential equation which is fourth-order in space and first-order in time. The nature of the buckling process is characterized by an initial period of elastic deformation followed by an evolutionary phase in which both elasticity and viscosity have a role to play. Two different formulations are studied: the first combines linear strut theory with a nonlinear foundation and is valid for small, but finite, deflections; the other incorporates the exact expression for curvature of the strut resulting in geometrical nonlinearities and is capable of modelling large deflections. The evolution of non-periodic buckle patterns in each system is examined under the constraint of controlled end displacement. Two independent methods are used to approximate the solution of the governing equations. Modal solutions, based on the method of weighted residuals, complement accurate numerical solutions obtained with a boundary-value solver. In either case, the results suggest that for the perfect system, localized solutions follow naturally from the inclusion of nonlinear elasticity with softening characteristics. Emphasis throughout is on the qualitative features displayed by the phenomenon of localization rather than specific applications. Nevertheless, the ideas and results are a step towards accounting for the rich variety of deformed shapes exhibited by nature.