On the asymptotic properties of a canonical diffraction integral.

We introduce and study a new canonical integral, denoted I + - ε , depending on two complex parameters α 1 and α 2. It arises from the problem of wave diffraction by a quarter-plane and is heuristically constructed to capture the complex field near the tip and edges. We establish some region of analyticity of this integral in C 2 , and derive its rich asymptotic behaviour as |α 1 | and |α 2 | tend to infinity. We also study the decay properties of the function obtained from applying a specific double Cauchy integral operator to this integral. These results allow us to show that this integral shares all of the asymptotic properties expected from the key unknown function G +- arising when the quarter-plane diffraction problem is studied via a two-complex-variables Wiener-Hopf technique (see Assier & Abrahams, SIAM J. Appl. Math., in press). As a result, the integral I + - ε can be used to mimic the unknown function G +- and to build an efficient 'educated' approximation to the quarter-plane problem

An Efficient Semi-Analytical Scheme for Determining the Reflection of Lamb Waves in a Semi-Infinite Elastic Waveguide

The classical problem of reflection of Lamb waves from a free edge perpendicular to the centre line of an elastodynamic plate is studied. It is known that Lamb wave expansions for the displacement and stress fields poorly represent the irregular behaviour near corners, leading to the slow convergence of a series of such waves. The form of the irregularity for an elastodynamic corner is derived asymptotically, and a new solution method, which incorporates this corner behaviour analytically, is then implemented. Results are presented showing that this new approach represents the near-field and far-field behaviour very accurately, requiring very modest numbers of Lamb wave and corner modes. Further, it is revealed that the method can recover the trapped-mode phenomenon encountered in this configuration at the Lamé frequency and a specific Poisson’s ratio that we find to be approximately 0.224798

High-frequency homogenization for periodic dispersive media

High-frequency homogenization is used to study dispersive media, containing inclusions placed periodically, for which the properties of the material depend on the frequency (Lorentz or Drude model with damping, for example). Effective properties are obtained near a given point of the dispersion diagram in frequency-wavenumber space. The asymptotic approximations of the dispersion diagrams, and the wavefields, so obtained are then cross-validated via detailed comparison with finite element method simulations in both one and two dimensions

High-order homogenisation of the time-modulated wave equation: non-reciprocity for a single varying parameter

Laminated media with material properties modulated in space and time in the form of travelling waves have long been known to exhibit non-reciprocity. However, when using the method of low frequency homogenisation, it was so far only possible to obtain non-reciprocal effective media when both material properties are modulated in time, in the form of a Willis-coupling (or bi-anisotropy in electromagnetism) model. If only one of the two properties is modulated in time, while the other is kept constant, it was thought impossible for the method of homogenisation to recover the expected non-reciprocity since this Willis-coupling coefficient then vanishes. Contrary to this belief, we show that effective media with a single time-modulated parameter are non-reciprocal, provided homogenization is pushed to the second order. This is illustrated by numerical experiments (dispersion diagrams and time-domain simulations) for a bilayered modulated medium

Effective dynamics for low-amplitude transient elastic waves in a 1D periodic array of non-linear interfaces

This article focuses on the time-domain propagation of elastic waves through a 1D periodic medium that contains non-linear imperfect interfaces. The array considered is generated by a, possibly heterogeneous, cell repeated periodically and bonded by interfaces that are associated with transmission conditions of non-linear "spring-mass" type. More precisely, the imperfect interfaces are characterized by a linear dynamics but a non-linear elasticity law. The latter is not specified at first and only key theoretical assumptions are required. In this context, we investigate transient waves with both low-amplitude and long-wavelength, and aim at deriving homogenized models that describe their effective motion. To do so, the two-scale asymptotic homogenization method is deployed, up to the first-order. To begin, an effective model is obtained for the leading zeroth-order contribution to the microstructured wavefield. It amounts to a wave equation with a non-linear constitutive stress-strain relation that is inherited from the behavior of the imperfect interfaces at the microscale. The next first-order corrector term is then shown to be expressed in terms of a cell function and the solution of a linear elastic wave equation. Without further hypothesis, the constitutive relation and the source term of the latter depend non-linearly on the zeroth-order field, as does the cell function. Combining these zeroth-and first-order models leads to approximation of both the macroscopic behavior of the microstructured wavefield and its small-scale fluctuations within the periodic array. Finally, particularizing for a prototypical non-linear interface law and in the cases of a homogeneous periodic cell and a bilaminated one, the behavior of the obtained models are then illustrated on a set of numerical examples and compared with full-field simulations. Both the influence of the dominant wavelength and of the wavefield amplitude are investigated numerically, as well as the characteristic features related to non-linear phenomena

High-frequency homogenisation in periodic media with imperfect interfaces

International audienceIn this work, the concept of high-frequency homoge-nisation is extended to the case of one-dimensional periodic media with imperfect interfaces of the spring-mass type. In other words, when considering the propagation of elastic waves in such media, displacement and stress discon-tinuities are allowed across the borders of the periodic cell. As is customary in high-frequency homogenisation, the homogenisation is carried out about the periodic and antiperiodic solutions corresponding to the edges of the Brillouin zone. Comparisons are made with the exact solutions obtained by the Bloch-Floquet approach for the particular examples of monolayered and bilayered materials. Asymptotic approximations are provided for both the higher branches of the dispersion diagram (second-order) and the resulting wave field (leading-order). In these two cases, convergence measurements are carried out to validate the approach. The special case of two branches of the dispersion diagram intersecting with a non-zero slope at an edge of the Brillouin zone (occurrence of a so-called Dirac point) is also considered in detail and illustrated numerically

Manipulating thermal fields with inhomogeneous heat spreaders

We design a class of spatially inhomogeneous heat spreaders in the context of steady-state thermal conduction leading to spatially uniform thermal fields across a large convective surface. Each spreader has a funnel-shaped design, either in the form of a trapezoidal prism or truncated cone, and is forced by a thermal source at its base. We employ transformation-based techniques, commonly used to study metamaterials, to determine the require thermal conductivity for the spreaders. The obtained materials, although strongly anisotropic and inhomogeneous, can be accurately approximated by assembling isotropic, homogeneous layers, rendering them realisable. An alternative approach is then considered for the conical and trapezoidal spreaders by dividing them into two or three isotropic, homogeneous components respectively. We refer to these simple configurations as neutral layers. All designs are validated numerically both with and without the effects of thermal contact resistance between interfaces. Such novel designs pave the way for future materials that can manipulate and control the flow of heat, helping to solve traditional heat transfer problems such as controlling the temperature of an object and energy harvesting

High-order homogenisation of the time-modulated wave equation: non-reciprocity for a single varying parameter

Laminated media with material properties modulated in space and time in the form of travelling waves have long been known to exhibit non-reciprocity. However, when using the method of low frequency homogenisation, it was so far only possible to obtain non-reciprocal effective media when both material properties are modulated in time, in the form of a Willis-coupling (or bi-anisotropy in electromagnetism) model. If only one of the two properties is modulated in time, while the other is kept constant, it was thought impossible for the method of homogenisation to recover the expected non-reciprocity since this Willis-coupling coefficient then vanishes. Contrary to this belief, we show that effective media with a single time-modulated parameter are non-reciprocal, provided homogenization is pushed to the second order. This is illustrated by numerical experiments (dispersion diagrams and time-domain simulations) for a bilayered modulated medium