Bending of thin periodic plates

We show that nonlinearly elastic plates of thickness $h\to 0$ with an $\varepsilon$-periodic structure such that $\varepsilon^{-2}h\to 0$ exhibit non-standard behaviour in the asymptotic two-dimensional reduction from three-dimensional elasticity: in general, their effective stored-energy density is "discontinuously anisotropic" in all directions. The proof relies on a new result concerning an additional isometric constraint that deformation fields must satisfy on the microscale.Comment: 35 page

High contrast homogenisation in nonlinear elasticity under small loads

We study the homogenisation of geometrically nonlinear elastic composites with high contrast. The composites we analyse consist of a perforated matrix material, which we call the "stiff" material, and a "soft" material that fills the pores. We assume that the pores are of size $0<\varepsilon\ll 1$ and are periodically distributed with period $\varepsilon$. We also assume that the stiffness of the soft material degenerates with rate $\varepsilon^{2\gamma},$ $\gamma>0$, so that the contrast between the two materials becomes infinite as $\varepsilon\to 0$. We study the homogenisation limit $\varepsilon\to 0$ in a low energy regime, where the displacement of the stiff component is infinitesimally small. We derive an effective two-scale model, which, depending on the scaling of the energy, is either a quadratic functional or a partially quadratic functional that still allows for large strains in the soft inclusions. In the latter case, averaging out the small scale-term justifies a single-scale model for high-contrast materials, which features a non-linear and non-monotone effect describing a coupling between microscopic and the effective macroscopic displacements.Comment: 31 page

An investigation of film wavy structure in annular flow using two simultaneous LIF approaches

The paper is devoted to development and validation of film thickness measurement techniques in interfacial gas-liquid flows. The specific flow investigated here is that of downwards (co-flowing) annular flow in a vertical pipe, however, many of the observations and findings are transferable to similar flow geometries. Two advanced spatially resolved techniques, namely planar laser-induced fluorescence and brightness-based laser-induced fluorescence , are used simultaneously in the same area of interrogation. A single laser sheet is used to excite fluorescence along one longitudinal section of the pipe, and two cameras (one for each method) are placed at different angles to the plane of the laser sheet in order to independently recover the shape of the interface along this section. This allows us to perform a cross-validation of the two techniques and to analyse their respective characteristics, advantages and shortcomings

The Impact of Inflation Hedge Assets on Portfolio Optimizations for US and Canadian Investors

The research is based on “Gold: Inflation Hedge and Long-Term Strategic Asset.” paper by Dempster and Artigas (2010). Authors used basic portfolio for the US investor, which includes Corporate Bonds, US Treasuries, Equity US and Equity Ex-US. By adding, alternatively, the four potential inflation-hedges, researchers showed Gold as the most appropriate Long-Term Strategic Asset. In our research, we constructed basic investment portfolio for US and Canadian investors. For each case, alternatively, four potential Inflation Hedges, which are Gold, S&amp;P GSCI Index, REITs and TIPS, were added to the basic portfolio. The optimization results are based on the post-crisis period from 2009 to 2016. The final results for the US suggest that Gold should be considered as a strong long-term strategic asset. For the Canadian case, Gold, and S&amp;P GSCI tend to be appropriate long-term strategic assets, which should be added to the basic portfolio. Canadian REITs get allocation under base case assumptions but sensitivity analysis indicates that the results are not robust

Asymptotic analysis of some spectral problems in high contrast homogenisation and in thin domains

We study the spectral properties of two problems involving small parameters. The first one is an eigenvalue problem for a divergence form elliptic operator Aε with high contrast periodic coefficients of period ε in each coordinate, where ε is a small parameter. The coefficients are perturbed on a bounded domain of 'order one' size. The local perturbation of coefficients for such operator could result in emergence of localised waves in the gaps of the Floquet-Bloch spectrum. We prove that, for the so-called double porosity type scaling, the eigenfunctions decay exponentially at in infinity, uniformly in ε. Then, using the tools of twoscale convergence for high contrast homogenisation, we prove the strong twoscale convergence of the eigenfunctions of Aε to the eigenfunctions of a two-scale limit homogenised operator A₀ , consequently establishing 'asymptotic one-to-one correspondence' between the eigenvalues and the eigenfunctions of these two operators. We also prove by direct means the stability of the essential spectrum of the homogenised operator with respect to the local perturbation of its coefficients. That allows us to establish not only the strong two-scale resolvent convergence of Aε to A₀ but also the Hausdor convergence of the spectra of Aε to the spectrum of A₀ , preserving the multiplicity of the isolated eigenvalues. As the second problem we consider the eigenvalue problem for the Laplacian in a network of thin domains with Dirichlet boundary conditions. We construct an asymptotic solution to the problem using the method of matched asymptotic expansions to obtain appropriate boundary conditions for the terms of the asymptotics near the junctions of thin domains. We justify the asymptotics and prove the error bound of order h3=2 , where h is the width of thin domains.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Spectral convergence for high-contrast elliptic periodic problems with a defect via homogenization

We consider an eigenvalue problem for a divergence-form elliptic operator Aε that has high-contrast periodic coefficients with period ε in each coordinate, where ε is a small parameter. The coefficients are perturbed on a bounded domain of “order one” size. The local perturbation of coefficients for such an operator could result in the emergence of localized waves—eigenfunctions whose corresponding eigenvalues lie in the gaps of the Floquet–Bloch spectrum. For the so-called double porosity-type scaling, we prove that the eigenfunctions decay exponentially at infinity, uniformly in ε Then, using the tools of twoscale convergence for high-contrast homogenization, we prove the strong two-scale compactness of the eigenfunctions of Aε. This implies that the eigenfunctions converge in the sense of strong two-scale convergence to the eigenfunctions of a two-scale limit homogenized operator A0, consequently establishing “asymptotic one-to-one correspondence” between the eigenvalues and the eigenfunctions of the operators Aε and A0. We also prove, by direct means, the stability of the essential spectrum of the homogenized operator with respect to local perturbation of its coefficients. This allows us to establish not only the strong two-scale resolvent convergence of Aε to A0 but also the Hausdorff convergence of the spectra of Aε to the spectrum of A0, preserving the multiplicity of the isolated eigenvalues

Two-scale Γ-convergence of integral functionals and its application to homogenisation of nonlinear high-contrast periodic composites

An analytical framework is developed for passing to the homogenisation limit in (not necessarily convex) variational problems for composites whose material properties oscillate with a small period ε and that exhibit high contrast of order 1/ε between the constitutive, “stress-strain”, response on different parts of the period cell. The approach of this article is based on the concept of “two-scale Γ-convergence”, which is a kind of “hybrid” of the classical Γ-convergence (De Giorgi and Franzoni in Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur (8)58:842–850, 1975) and the more recent two-scale convergence (Nguetseng in SIAM J Math Anal 20:608–623, 1989). The present study focuses on a basic high-contrast model, where “soft” inclusions are embedded in a “stiff” matrix. It is shown that the standard Γ-convergence in the L^p -space fails to yield the correct limit problem as ε tends to 0, due to the underlying lack of L^p -compactness for minimising sequences. Using an appropriate two-scale compactness statement as an alternative starting point, the two-scale Γ-limit of the original family of functionals is determined via a combination of techniques from classical homogenisation, the theory of quasiconvex functions and multiscale analysis. The related result can be thought of as a “non-classical” two-scale extension of the well-known theorem by Müller (Arch Rational Mech Anal 99:189–212, 1987)

Comparison of disturbance wave parameters with flow orientation in vertical annular gas-liquid flows in a small pipe

The interfacial wave structure of the liquid film in both upward and downward annular gas-liquid flows in an 11.7 mm pipe were investigated using the Brightness Based Laser Induced Fluorescence technique (BBLIF). Film thickness measurements were carried out with high spatial and temporal resolution between 330 and 430 mm from the inlet, where the properties of disturbance waves are almost stabilised. Using a tracking algorithm to detect disturbance waves, a full characterisation in terms of their velocity, frequency, longitudinal size and spacing was carried out. Direct comparison between both flow orientations while testing the same flow conditions shows that although the flow orientation does not affect the velocity of disturbance waves, the fraction of film surface occupied by the disturbance waves is smaller in upwards flow. Thus, more liquid travels in the base film in upwards flow, which is consistent with the base film thickness measurements. These observations, together with qualitatively different behaviour of ripple wave velocity in upwards and downwards flows, studied using 2D Fourier analysis, indicate that the role of gravity is much more important on the base film than on disturbance waves. This supposedly occurs due to a local decrease in the interfacial shear stress on the base film surface because of the resistance of the disturbance waves to the gas stream in upward flow