  Topological defects with symmetry-breaking phase transitions have captured
much attention. Vortex generated by topological defects exhibits exotic
properties and its flow direction can be switched by altering the spin
configurations. Contrary to electromagnetic and acoustic domains, the
topological transport of elastic waves in periodic structures with topological
defects is not well explored due to the mode conversion between the
longitudinal and transverse modes. Here, we propose an elastic topological
insulator with spontaneously broken symmetry based on the topological theory of
defects and homotopy theory. Multiple topological transitions for elastic waves
are achieved by topologically modifying the ellipse orientation in a triangular
lattice of elliptical cylinders. The solid system, independent of the number of
molecules in order parameter space, breaks through the limit of the point-group
symmetry to emulate elastic pseudospin-orbit coupling. The transport robustness
of the edge states is experimentally demonstrated. Our approach provides new
possibilities for controlling and transporting elastic waves.
