Nonlocal effective medium theory for phononic temporal metmaterials.

We have developed a nonlocal effective medium theory (EMT) for phononic temporal metamaterials using the multiscale technique. Our EMT yields closed-form expressions for effective constitutive parameters and reveals these materials as reciprocal media with symmetric band dispersion. Even without spatial symmetry breaking, nonzero Willis coupling coefficients emerge with time modulation and broken time-reversal symmetry, when the nonlocal effect is taken into account. Compared to the local EMT, our nonlocal version is more accurate for calculating the bulk band at high wavenumbers and essential for understanding nonlocal effects at temporal boundaries. This nonlocal EMT can be a valuable tool for studying and designing phononic temporal metamaterials beyond the long-wavelength limit.