Topological mechanics in rigid mechanical metamaterials

Mass-spring systems have been widely studied in recent years, where isostatic systems stand out. In these systems there is a delicate balance between degrees of freedom and constraints, however the materials are usually more rigid, so it is important to study the case where the constraints exceed the degrees of freedom, or hyperstatic systems. In this work, two problems in hyperstatic systems are addressed, taking as reference systems and results previously found in isostatic systems. In the first place, the topological protection of states at finite frequency, of systems composed of isostatic and hyperstatic parts, is studied, based on topological characterizations previously described in the literature. In this work, the predominance between invariants in multiple systems is compared. On the other hand, in the nonlinear regime, the existence of a soliton in a one-dimensional system is shown, previously studied in the isostatic case. It is shown that said soliton can prevail in the hyperstatic case under certain requirements to the new restrictions. In both problems, numerical and experimental results are presented that show the possibility of extending some results from isostatic systems to hyperstatic systems.