In this talk, we lift this restriction and generalize linear elasticity to active media with non-conservative microscopic interactions that violate mechanical reciprocity.

The number of coefficients of proportionality between stress and strain, i.e., the elastic moduli, is constrained by energy conservation. Hooke's law states that the forces or stresses experienced by an elastic object are proportional to the applied deformations or strains. In this talk I will demonstrate with plenty of examples how this principle works for the traditional classes of topological meta-materials and how one can easily extend the principle in new settings, such those of quasi-periodic and quasi-crystalline meta-materials. This statement is applied as it follows: Since the boundary invariant is computed from the boundary modes, a non-zero value of the bulk invariant automatically implies the existence of the boundary modes, because otherwise the boundary invariant, hence also the bulk one, will be zero. It goes as follows: For each class of the periodic table of topological phases, one can explicitly define a bulk as well as a boundary topological invariant and the two are equal. Surprisingly, not many are aware that such formulation of the bulk-boundary principle is in fact available. A principle, however, becomes useful when it contains the mechanism that explains the observations made in the statement. Quite often, the bulk-boundary principle is formulated as the emergence of topological boundary modes when a topological meta-material is cut in half.

Body swap metamorphose