Formation Response of High Frequency Electromagnetic Waves

Core samples from rock formations respond to electromagnetic radiation based on an effective permittivity, which depends on the conductivity and permittivity of the constituent components of the rock, as well as the geometric structure of these constituents and the frequency of the radiation. This study analyzes the effect, for radiation of 1 to 100 Mhz, of discrete inclusions having a different permittivity from the surrounding medium. The focus is on the effect of certain geometric features, namely, the individual size of the inclusions, their overall volume fraction, the presence of sharp edges, and their aspect ratio. It is found that the volume fraction has the strongest impact on the effective permittivity, linear at first but higher order at higher volume fractions. The aspect ratio of the inclusions has a moderate effect, which is exaggerated in the extreme case of needle-like inclusions, and which can also be seen in a stronger nonlinearity. There is also a possibility that some features in the shape of the inclusion boundaries may influence the frequency dependence of the effective permittivity. Inclusion size and sharp edges have negligible effect

Spectral properties of elliptic operator with double-contrast coefficients near a hyperplane

In this paper we study the asymptotic behaviour as $\varepsilon\to 0$ of the spectrum of the elliptic operator $\mathcal{A}^\varepsilon=-{1\over b^\varepsilon}\mathrm{div}(a^\varepsilon\nabla)$ posed in a bounded domain $\Omega\subset\mathbb{R}^n$ $(n \geq 2)$ subject to Dirichlet boundary conditions on $\partial\Omega$. When $\varepsilon\to 0$ both coefficients $a^\varepsilon$ and $b^\varepsilon$ become high contrast in a small neighborhood of a hyperplane $\Gamma$ intersecting $\Omega$. We prove that the spectrum of $\mathcal{A}^\varepsilon$ converges to the spectrum of an operator acting in $L^2(\Omega)\oplus L^2(\Gamma)$ and generated by the operation $-\Delta$ in $\Omega\setminus\Gamma$, the Dirichlet boundary conditions on $\partial\Omega$ and certain interface conditions on $\Gamma$ containing the spectral parameter in a nonlinear manner. The eigenvalues of this operator may accumulate at a finite point. Then we study the same problem, when $\Omega$ is an infinite straight strip ("waveguide") and $\Gamma$ is parallel to its boundary. We show that $\mathcal{A}^\varepsilon$ has at least one gap in the spectrum when $\varepsilon$ is small enough and describe the asymptotic behaviour of this gap as $\varepsilon\to 0$. The proofs are based on methods of homogenization theory.Comment: In the second version we added the case r=0, also the presentation is essentially improved. The manuscript is submitted to a journa

Free and forced propagation of Bloch waves in viscoelastic beam lattices

Beam lattice materials can be characterized by a periodic microstructure realizing a geometrically regular pattern of elementary cells. Within this framework, governing the free and forced wave propagation by means of spectral design techniques and/or energy dissipation mechanisms is a major issue of theoretical interest with applications in aerospace, chemical, naval, biomedical engineering. The first part of the Thesis addresses the free propagation of Bloch waves in non-dissipative microstructured cellular materials. Focus is on the alternative formulations suited to describe the wave propagation in the bidimensional infinite material domain, according to the classic canons of linear solid or structural mechanics. Adopting the centrosymmetric tetrachiral cell as prototypical periodic topology, the frequency dispersion spectrum is obtained by applying the Floquet-Bloch theory. The dispersion spectrum resulting from a synthetic Lagrangian beam lattice formulation is compared with its counterpart derived from different continuous models (high-fidelity first-order heterogeneous and equivalent homogenized micropolar continua). Asymptotic perturbation-based approximations and numerical spectral solutions are compared and cross-validated. Adopting the low-frequency band gaps of the dispersion spectrum as functional targets, parametric analyses are carried out to highlight the descriptive limits of the synthetic models and to explore the enlarged parameter space described by high-fidelity models. The microstructural design or tuning of the mechanical properties of the cellular microstructure is employed to successfully verify the wave filtering functionality of the tetrachiral material. Alternatively, band gaps in the material spectrum can be opened at target center frequencies by using metamaterials with inertial resonators. Based on these motivations, in the second part of the Thesis, a general dynamic formulation is presented for determining the dispersion properties of viscoelastic metamaterials, equipped with local dissipative resonators. The linear mechanism of local resonance is realized by tuning periodic auxiliary masses, viscoelastically coupled with the beam lattice microstructure. As peculiar aspect, the viscoelastic coupling is derived by a mechanical formulation based on the Boltzmann superposition integral, whose kernel is approximated by a Prony series. Consequently, the free propagation of damped Bloch waves is governed by a linear homogeneous system of integro-differential equations of motion. Therefore, differential equations of motion with frequency-dependent coefficients are obtained by applying the bilateral Laplace transform. The corresponding complex-valued branches characterizing the dispersion spectrum are determined and parametrically analyzed. Particularly, the spectra corresponding to Taylor series approximations of the equation coefficients are investigated. The standard dynamic equations with linear viscous damping are recovered at the first order approximation. Increasing approximation orders determine non-negligible spectral effects, including the occurrence of pure damping spectral branches. Finally, the forced response to harmonic single frequency external forces in the frequency and the time domains is investigated. The response in the time domain is obtained by applying the inverse bilateral Laplace transform. The metamaterial responses to non-resonant, resonant and quasi-resonant external forces are compared and discussed from a qualitative and quantitative viewpoint