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Dynamic materials are materials whose constitutive properties vary at a rate comparable to the frequency of waves traveling within. Also referred to as spatio-temporal composites or modulated phononic crystals, they constitue a playground for remarkable non-standard wave phenomena. Of most interest in the currently trending context of non-reciprocal and non-time-reversible wave motion, is their ability to block wave forms traveling in a given direction while transmitting the same wave forms if incident in the opposite direction. This asymmetry between left and right is visible in the dispersion diagram where aligned left and right gaps, blocking left- and right-going waves, get tilted and become misaligned, thus blocking either left- or right-going waves.
Band tilting in dynamic materials is expected to be a function of the time profiles of the constitutive properties and of their rate of change. Interestingly, it is proven that the ratio tilt to rate only depends on a number of well-identified qualitative parameters and is insensitive to the detail of said time profiles. Specifically, the tilt-to-rate ratio turns out to be a robust (topological) perturbation-immune quantity quantized in units of length of the unit-cell. The proof makes use of two quantum mechanical tools, namely the adiabatic theorem and Berry’s phase, generalized and adapted to the case of elastic waves. The example of a modulated 3-periodic spring-mass lattice whose spring constants are being slowly and periodically modulated in time is thoroughly treated so as to illustrate theoretical findings.
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