Revisiting imperfect interface laws for two-dimensional elastodynamics

Abstract

We study the interaction of in-plane elastic waves with imperfect interfaces composed of a periodic array of voids or cracks. An effective model is derived from high-order asymptotic analysis based on two-scale homogenization and matched asymptotic technique. In two-dimensional elasticity, we obtain jump conditions set on the in-plane displacements and normal stresses; the jumps involve in addition effective parameters provided by static, elementary problems being the equivalents of the cell problems in classical two-scale homogenization. The derivation of the model is conducted in the transient regime and its stability is guarantied by the positiveness of the effective interfacial energy. Spring models are envisioned as particular cases. It is shown that massless-spring models are recovered in the limit of small void thicknesses and collinear cracks. By contrast, the use of mass-spring model is justified at normal incidence, otherwise unjustified. We provide quantitative validations of our model and comparison with spring models by means of comparison with direct numerical calculations in the harmonic regime.

Stable GSTC Formulation for Maxwell’s Equations

“…1) Zero-order problem: As for the film, inserting (29) in (1) along with (30) provides equations at the dominant order O( 1 ). In particular, we obtain for h 0 the set of equations 8 > > < > > :…”

“…This could be an incidental remark but it turns out that these approximate problems can be discriminated in terms of their well-posedness or stability. Up to now, this problem has been addressed in the context of metafilms in elastodynamics [29], [30] and in acoustics [31], [32] and it has been shown that unstable formulations may foster numerical instabilities in the time domain. Such instabilities have been reported in [33] for homogeneous thin films, and in [34], [35] for structured thin films.…”

Homogenization of the Helmholtz problem in the presence of a row of viscoelastic inclusions

“…These methods have been used in static elasticity [6,7]. For the propagation of waves, they have been adapted in elasticity, studies have focused on rows of non-resonant inclusions [8,9] then on resonant inclusions [10]. For the propagation of waves in electromagnetism [11][12][13], and in acoustics [5,14].…”

Homogenization of the vibro–acoustic transmission on periodically perforated elastic plates with arrays of resonators

“…Besides the acoustic problems with a standing fluid, similar treatment was employed to study the electromagnetic field [11]. Using higher order approximation involving the correctors at order o(ε 1 ), nontrivial interface conditions capturing acoustic impedance of the thin interfaces have been obtained in [12,13] using an approach based on the so-called inner and outer asymptotic expansions which enable to treat rather general shapes of the perforations, or other heterogeneities.…”