Impact-governed dynamics of an axially-incompressible bistable continuous metastructure

Abstract

We study the dynamics of a bistable Single Degree of Freedom mechanism that bifurcates from the trivial straight configuration when subjected to a critical traction force and it is otherwise incompressible. We show that the appearance of “impacts”, in correspondence with the minimum axial extension of the system, merely reflects the adoption of the axial extension as the dependent variable, as opposed to the angular rotation. Within this description, the structure realizes a perfectly elastic obstacle. Next, we construct the continuous limit for a dense chain of such mechanisms and axial strain naturally emerges as the continuous dependent field. Consequently, an unilateral constraint becomes associated with the system. Most importantly, the corresponding Lagrangian problem needs to be supplemented by energy conservation across the impacts to faithfully represent the underlying microstructure. In doing so, we generalize the established procedure to construct the continuous limit of a dense chain of discrete systems to the presence of unilateral constraints. Remarkably, energy conservation allows to apply Hamilton’s principle in the form of a variational equality, in contrast to the inequality format usually encountered when dealing with non-smooth problems. This important result, which greatly simplifies the mathematics, is available provided that variations are extended to accommodate for discontinuities in the variables. Besides, the system dynamics may be now constructed semi-analytically by joining pairs of d’Alembert’s solutions through the conditions obtained from the extended variational principle at the impact time and location (which are obviously unknown). As a result, waves propagating in the system are obtained and they are checked against global energy conservation.