Effective Resonant Model and Simulations in the Time-Domain of Wave Scattering from a Periodic Row of Highly-Contrasted Inclusions

Abstract

The time-domain propagation of scalar waves across a periodic row of inclusions is considered in 2D. As the typical wavelength within the background medium is assumed to be much larger than the spacing between inclusions and the row width, the physical configuration considered is in the low-frequency homogenization regime. Furthermore, a high contrast between one of the constitutive moduli of the inclusions and of the background medium is also assumed. So the wavelength within the inclusions is of the order of their typical size, which can further induce local resonances within the microstructure. In (20), two-scale homogenization techniques and matched-asymptotic expansions have been employed to derive, in the harmonic regime, effective jump conditions on an equivalent interface. This homogenized model is frequency-dependent due to the resonant behavior of the inclusions. In this context, the present article aims at investigating, directly in the time-domain, the scattering of waves by such a periodic row of resonant scatterers. Its effective behavior is first derived in the time-domain and some energy properties of the resulting homogenized model are analyzed. Time-domain numerical simulations are then performed to illustrate the main features of the effective interface model obtained and to assess its relevance in comparison with full-field simulations.

Revisiting imperfect interface laws for two-dimensional elastodynamics

“…The case of highly contrasted inclusions able to produce internal, low-frequency, resonances is more challenging. Already considered in the scalar, antiplane, case by Pham et al [26] and Touboul et al [27], it is difficult to anticipate how the effect of these resonances translates in two-or three-dimensional elasticity. From a practical point of view, the study of guided waves by such imperfect interfaces and their link with Rayleigh, Stoneley and Love waves in realistic configurations is of interest in a geophysical context.…”

Wave propagation across a functionally graded interphase between soft and hard solids: Insight from a dynamic surface elasticity model

“…In the case of resonant meta-interfaces, an alternative approach was to obtain effective jump conditions by applying a suitable homogenization process [33,34]. Moreover, some studies on elastic wave propagation showed that when the interphase is located between two surrounding media with microstructure, inertial properties also play a pivotal role in the modeling of the equivalent interface [35][36][37].…”

Damping in a row of locally-resonant inclusions: Dynamic homogenization and scattering of transient shear waves

“…In elasticity, works on interface homogenization focused first on rows of non-resonant inclusions [9,28,8,27], and then on resonant inclusions [31,29,41]. In [31,41], two-scale asymptotic method has been combined with matched-asymptotic expansions to yield effective jump conditions, both in the frequency domain [31] and in the time domain [41]; in the latter case, the jump conditions turn out to be non-local. Second-order accuracy in terms of the small ratio k m h has been reached, where k m is the wavenumber in the background medium and h is the typical size of the inclusions.…”

“…As is usual in structural dynamics, a phenomenological model is therefore considered. In this framework, the results presented in [41,40] are modified and need to be re-examined on their main features: (i) effective jump conditions, (ii) energy balance, (iii) auxiliary fields, and (iv) discretization of the interfaces. These four points are addressed here.…”

“…These four points are addressed here. For the sake of brevity, the reader is referred to [41,40] for details about unchanged features. Moreover, the correspondence principle is used to take shortcuts when the whole proof does not bring anything new.…”