Deformation and failure of a film/substrate system subjected to spherical indentation: Part 1. Experimental validation of stresses and strains derived using Hankel transform technique in an elastic film/substrate system

Our concern here is to rationalize experimental observations of failure modes brought about by indentation of hard thin ceramic films deposited on metallic substrates. By undertaking this exercise, we would like to evolve an analytical framework that can be used for designs of coatings. In Part I of the paper we develop an algorithm and test it for a model system. Using this analytical framework we address the issue of failure of columnar TiN films in Part II [J. Mater. Res. 21, 783 (2006)] of the paper. In this part, we used a previously derived Hankel transform procedure to derive stress and strain in a birefringent polymer film glued to a strong substrate and subjected to spherical indentation. We measure surface radial strains using strain gauges and bulk film stresses using photo elastic technique (stress freezing). For a boundary condition based on Hertzian traction with no film interface constraint and assuming the substrate constraint to be a function of the imposed strain, the theory describes the stress distributions well. The variation in peak stresses also demonstrates the usefulness of depositing even a soft film to protect an underlying substrate.

Hankel Transform in Quantum Calculus and Applications

Mathematics Subject Classification: 44A15, 33D15, 81Q99

On Hankel Transform of Generalized Mathieu Series

Mathematics Subject Classification: Primary 33E20, 44A10; Secondary 33C10, 33C20, 44A20

Exact solution for the unsteady motion of a viscous-fluid in a porous annulus

An exact solution to the problem of time-dependent motion of a viscous fluid in an annulus with porous walls is obtained under the assumption that the rate of suction at one wall is equal to the rate of injection at the other. Finite Hankel transform is used to obtain a closed-form solution for the axial velocity. The average axial velocity profiles are depicted graphically.

Torsional impact of transversely isotropic solid with functionally graded shear moduli and a penny-shaped crack

The torsional impact response of a penny-shaped crack in an unbounded transversely isotropic solid is considered. The shear moduli are assumed to be functionally graded such that the mathematics is tractable. Laplace transform and Hankel transform are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress fields are obtained. Investigated are the influence of material nonhomogeneity and orthotropy on the dynamic stress intensity factor. The peak value of the dynamic stress intensity factor can be suppressed by increasing the shear moduli's gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface.

任意横向载荷下弹性圆形及圆环形薄板的弯曲

<正> 一.前言 波兰科学院通讯院士W.诺瓦茨基教授不久以前在北京所作的关于他在弹性薄板方面的工作的介绍报告,引起我们的兴趣与注意。在讨论的过程中,本文著者曾试从有限变换式(fnite transform)的角度上来说明诺瓦茨基教授从物理观点导出的一个普遍的方法。 下面我们将讨论边缘为夹住的或自由支承的弹性圆及圆环形薄板在任意横向载荷分布下的弯曲问题,说明有限汉克尔变换式(fnite Hankel transform)的应用。

Exercise sheet 5

Exercises and solutions in LaTex

Exercise sheet 5 pdf

Exercises and solutions in PDF

Desenvolvimento de antenas de microfita com Patch em anel utilizando materiais ferrimagnéticos e metamateriais

In general, the materials used as substrates in the project of microstrip antennas are: isotropic, anisotropic dielectrics and ferrimagnetic materials (magnetic anisotropy). The use of ferrimagnetic materials as substrates in microstrip patch antennas has been concentrated on the analysis of antennas with circular and rectangular patches. However, a new class of materials, called metamaterials, has been currently the focus of a great deal of interest. These materials exhibit bianisotropic characteristics, with permittivity and permeability tensors. The main objective of this work is to develop a theoretical and numerical analysis for the radiation characteristics of annular ring microstrip antennas, using ferrites and metamaterials as substrates. The full wave analysis is performed in the Hankel transform domain through the application of the Hertz vector potentials. Considering the definition of the Hertz potentials and imposing the boundary conditions, the dyadic Green s function components are obtained relating the surface current density components at the plane of the patch to the electric field tangential components. Then, Galerkin s method is used to obtain a system of matrix equations, whose solution gives the antenna resonant frequency. From this modeling, it is possible to obtain numerical results for the resonant frequency, radiation pattern, return loss, and antenna bandwidth as a function of the annular ring physical parameters, for different configurations and substrates. The theoretical analysis was developed for annular ring microstrip antennas on a double ferrimagnetic/isotropic dielectric substrate or metamaterial/isotropic dielectric substrate. Also, the analysis for annular ring microstrip antennas on a single ferrimagnetic or metamaterial layer and for suspended antennas can be performed as particular cases

Antenas de microfita com patch em anel e múltiplas camadas dielétricas anisotrópicas uniaxiais

This work presents an analysis of the annular ring microstrip antennas printed on uniaxial anisotropic substrates and with superstrate.The analysis uses the full-wave formulation by means of the Hertz vector potentials method, in the Hankel transform domain. The definition of the Hertz vector potentials and the application of the appropriate boundary conditions to the structure allow determining the dyadic Green functions, relating the current densities in the conducting patch to the transforms of the tangential electric field components. Galerkin s method is then used to obtain the matrix equation whose nontrivial solution gives the complex resonant frequency of the antenna. From the modeling, it is possible to obtain results for the resonant frequency, bandwidth and quality factor, as a function of several parameters of the antenna, for different configurations. We have considered annular ring microstrip antennas on a single dielectric layer, antennas with two anisotropic dielectric layers, and annular ring microstrip antennas on suspended substrates. Numerical results for the resonant frequency of the these structures printed on isotropic substrates are also presented and compared with those published by other authors, showing a good agreement

Análise de antenas de microfita com patches circulares sobre substratos anisotrópicos usando o método dos potenciais de Hertz

This work consists on the theoretical and numerical analysis of some properties of circular microstrip patch antennas on isotropic and uniaxial anisotropic substrates. For this purpose, a full wave analysis is performed, using Hertz Vector Potentials method in the Hankel Transform domain. In the numerical analysis, the moment method is also used in order to determine some characteristics of the antenna, such as: resonant frequency and radiation pattern. The definition of Hertz potentials in the Hankel domain is used in association with Maxwell´s equations and the boundary conditions of the structures to obtain the Green´s functions, relating the components of the current density on the patch and the tangential electric field components. Then, the Galerkin method is used to generate a matrix equation whose nontrivial solution is the complex resonant frequency of the structure. In the analysis, a microstrip antenna with only one isotropic dielectric layer is initially considered. For this structure, the effect of using superconductor patches is also analyzed. An analysis of a circular microstrip antenna on an uniaxial anisotropic dielectric layer is performed, using the Hertz vector potentials oriented along the optical axis of the material, that is perpendicular to the microstrip ground plane. Afterwards, the circular microstrip antenna using two uniaxial anisotropic dielectric layers is investigated, considering the particular case in which the inferior layer is filled by air. In this study, numerical results for resonant frequency and radiation pattern for circular microstrip antennas on isotropic and uniaxial anisotropic substrates are presented and compared with measured and calculated results found in the literature

Hankel operators on Fock spaces

We study Hankel operators on the weighted Fock spaces Fp. The boundedness and compactness of these operators are characterized in terms of BMO and VMO, respectively. Along the way, we also study Berezin transform and harmonic conjugates on the plane. Our results are analogous to Zhu's characterization of bounded and compact Hankel operators on Bergman spaces of the unit disk.

Generalized Convolution Transforms and Toeplitz Plus Hankel Integral Equations

Mathematics Subject Classification: 44A05, 44A35

On the fast Lanczos method for computation of eigenvalues of Hankel matrices using multiprecision arithmetics

The use of the fast Fourier transform (FFT) accelerates Lanczos tridiagonalisation method for Hankel and Toeplitz matrices by reducing the complexity of matrix-vector multiplication. In multiprecision arithmetics, the FFT has overheads that make it less competitive compared with alternative methods when the accuracy is over 10000 decimal places. We studied two alternative Hankel matrix-vector multiplication methods based on multiprecision number decomposition and recursive Karatsuba-like multiplication, respectively. The first method was uncompetitive because of huge precision losses, while the second turned out to be five to 14 times faster than FFT in the ranges of matrix sizes up to n = 8192 and working precision of b = 32768 bits we were interested in. We successfully applied our approach to eigenvalues calculations to studies of spectra of matrices that arise in research on Riemann zeta function. The recursive matrix-vector multiplication significantly outperformed both the FFT and the traditional multiplication in these studies.