Nonlinear waves and solitons in complex solids

Abstract

The problem of the propagation of nonlinear waves in complex solids, namely bodies with different internal microstructures, is analyzed. In the first part, we make use of a general model of microstructured solids as introduced by Engelbrechet and Pastrone (Acc Sc Torino Mem Sc Fis 2011; 35: 23–36) and study two particular relevant models: one-dimensional solid with hierarchical microstructure and with concurrent microstructures. As expected, the hierarchical microstructure leads, with a particular but meaningful choice of the strain energy function, to a sixth-order partial differential equation (PDE) with a characteristic hierarchical structure. Hence, the case of two concurrent microstructures, as introduced by Berezovski, Engelbrecht and Berezovski (Acta Mech 2011; 220(1–4): 349–363), is studied and again for suitable explicit forms of the energy function we can obtain a fourth-order PDE and actually prove the possibility of propagation of solitary and cnoidal waves.