Study On Elastic Wave Transport Properties Of Square Lattice Topological Metamaterial

Elastic topological metamaterial is a kind of man-made composite materials with periodic distribution,which show new physical properties such as directional transmission,nonreciprocity,negative refraction,etc.It has great academic value and application prospect in mechanical engineering,aerospace,civil engineering and other fields.At present,the great progress has been made in the study of elastic wave transport characteristics of topological metamaterials.However,most of them are confined to the study of elastic wave edge states,and there is little work on new high-order topological states.At the same time,there are few reports on the coupling of edge and corner states,topological rainbow capture and fractal higher-order topological states in elastic wave systems.Therefore,this paper carries out the research on the elastic wave transport characteristics of the cubic lattice elastic wave topological metamaterial.The specific work is as follows:(1)Based on k·p perturbation theory,a new high-order topological metamaterial of elastic wave is proposed,and the Hamiltonian matrix is derived to obtain the two-dimensional Dirac mass topological quantity.In this system,the multi-dimensional topological transitions can be developed for physical transfer among the corner,edge,and bulk elastic wave modes by tuning the rotation angle of the L-shaped pillars.Different from the four degenerate corner states in the symmorphic space system,these corner states in this metamaterial evolve into two doublets because of the lack of mirror symmetries,whose quality factor is up to 2800.Besides,the coupling effect between corner and edge states is analyzed in the multi-dimensional topological system.The edge-corner coupling effect could be applied to relevant fields,such as highsensitive detection,elastic energy harvesting,and optomechanical sensing.(2)Based on the equivalent tight-binding theory,a C4 v high-order topological metamaterial is constructed,meanwhile,nontrivial topological insulators and trivial topological insulators are determined by bulk polarization theory analysis.The edge state and the corner state are observed in the supercell system,which composed of two different topological properties.Compared with the trivial corner state,the nontrivial corner state is immune to defects.Combined with functional gradient design,a metamaterial of one and cross type gradient topologies is constructed,and unidirectional and multidirectional spatial frequency division rainbow phenomenon is realized.A two-dimensional fractal elastic wave high order topological metamaterial is constructed by using fractal geometry model,and 36 nontrivial corner states are observed.(3)Based on the C4 v model and the band folding mechanism,a spatial symmetric topological metamaterial model of C2 v is proposed.By tuning the radius of the cylinder,two kinds of non-trivial topological insulators can be obtained,and the multi-dimensional topological transitions could be realized.The boundary states of heterogeneous supercells are analyzed,and two wide band gaps are found in the boundary states.What’s more,the topological corner states are observed in both bands.Different from the four degenerate nontrivial corner states in the C4 v system,there are six nontrivial corner states in the C2 v system.In addition,a two-dimensional C2 v fractal high-order topological metamaterial is constructed,and 54 nontrivial corner states are observed.It is further proved that there are also highly localized topological corner states in the fractal metamaterial.