Rarefaction pulses on tensegrity lattices are just $$\text {sech}^2$$-solitary (dark) waves

Abstract

This study investigates the propagation of rarefaction solitary waves in one-dimensional, tensegrity-like mass-spring lattices that are subject to an initial state of pre-compression. The analyzed systems exhibit a cubic interaction potential between adjacent masses that accurately captures the constitutive response of tensegrity prisms with elastically softening behavior. Analytical results are presented for the propagation of rarefaction solitary waves that produce a reduction of the initial prestress exhibited by the system. It is known in the literature that the use of cubic interaction potentials in one-dimensional lattices enables the prediction of the propagation of solitary waves with sech$$^2$$ 2 profile. Investigating the particular case of pre-compressed, softening-type tensegrity lattices, this study shows that such a noticeable result can be derived using both the classical and the improved Boussinesq equation. The given results reveal the presence of rarefaction solitary waves in a suitable range of wave speeds, and offer an explicit formula for the upper bound of the rarefaction wave speed that leaves the system in a compressed state. The outcomes of the present work pave the way to the development of analytic models for the design of radically new, metamaterial-type impact protection systems. Numerical simulations show the ability of the tensegrity-like model in predicting the propagation of rarefaction solitary waves in a physical model of a tensegrity mass-spring chain.