Jump conditions with memory for the dynamic homogenization of a thin row of elastic inclusions in a viscoelastic matrix

We study the propagation of anti-plane shear waves across a thin periodic row of elastic inclusions embedded in a Kelvin–Voigt viscoelastic matrix. In the long-wavelength regime, a matched asymptotic expansion in the Laplace variable formally yields an effective interface model with jump conditions with memory: the jumps of the displacement and of the normal stress combine instantaneous averages with time convolutions against two causal kernels determined by cell problems. The cell solution depends on the Laplace parameter only through the complex modulus contrast, from which two distinct memory mechanisms are proved: the normal kernel originates from the flux bypassing the inclusions through the matrix and vanishes for a full layer, whereas the transverse kernel then retains a closed-form expression, active only at oblique incidence. An exact Herglotz identity, which extends the envelope formulas to the complex contrast plane, proves the positivity of the Stieltjes measure of the interface coefficients and, with it, the passivity of both kernels on the whole half-plane. Numerical validation against the scattering by the actual row, solved by finite elements whose accuracy floor is itself measured, shows errors consistent with second-order convergence at normal and oblique incidence—where the transverse coefficient is indispensable—and identifies the spectral regime in which memory becomes decisive.