Frequency analysis and continued fractions

The frequency analysis problem is solving by the positive Perron-Carathéodory continued fraction, associated with monic Szegö polynomials.     

Dynamic characteristic analysis of an electrostatically-actuated circular nanoplate subject to surface effects

This study investigates the influence of surface effect on the nonlinear behavior of an electrostatically actuated circular nanoplate. The Casimir force, surface effects, and the electrostatic force are modelled. In performing the analysis, the nonlinear governing equation of a circular nanoplate is solved using a hybrid computational scheme combining a differential transformation and finite differences. The method is used to model systems found in previous literature using different methods, producing consistent results, thus verifying that it is suitable for treatment of the nonlinear electrostatic coupling phenomenon. The obtained results from numerical methods demonstrated that the relationship between the thickness, radius, and gap size of a circular nanoplate, and its pull-in voltage, is scale-dependent. The model exhibits size-dependent behavior, showing that surface effects significantly influence the dynamic response of circular nanoplate actuators. Moreover, the influence of surface stress on the pull-in voltage of circular nanoplate is found to be more significant than the influence of surface elastic modulus. Finally, the effects of actuation voltage, excitation frequency, and surface effects on the dynamic behavior of the nanoplate are examined through use of phase portraits. Overall, the results show that the using hybrid method here presented is a suitable technique for analyzing nonlinear behavior characteristic of circular nanoplates.     

Earth surface effects on active faults: An eigenvalue asymptotic analysis

We study in this paper an eigenvalue problem (of Steklov type), modeling slow slip events (such as silent earthquakes, or earthquake nucleation phases) occurring on geological faults. We focus here on a half space formulation with traction free boundary condition: this simulates the earth surface where displacements take place and can be picked up by GPS measurements. We construct an appropriate functional framework attached to a formulation suitable for the half space setting. We perform an asymptotic analysis of the solution with respect to the depth of the fault. Starting from an integral representation for the displacement field, we prove that the differences between the eigenvalues and eigenfunctions attached to the half space problem and those attached to the free space problem, is of the order of d^-2, where d is a depth parameter: intuitively, this was expected as this is also the order of decay of the derivative of the Green's function for our problem. We actually prove faster decay in case of symmetric faults. For all faults, we rigorously obtain a very useful asymptotic formula for the surface displacement, whose dominant part involves a so called seismic moment. We also provide results pertaining to the analysis of the multiplicity of the first eigenvalue in the line segment fault case. Finally we explain how we derived our numerical method for solving for dislocations on faults in the half plane. It involves integral equations combining regular and Hadamard's hypersingular integration kernels.     

Effect of silicon carbide coatings on the strength of carbon fibers

Conclusion The character of the effect of silicon carbide coatings on the strength of carbon fibers is a function of the strength of the coating, which decreases with an increase in its thickness. The results obtained permit hoping for a significant increase in the strength of fibers by application of high-modulus coatings. It can be attained as a result of increasing the critical value of the coating thickness due to optimization of the conditions of preparation, which will improve the structure of the articles.Translated from Mekhanika Kompozitnykh Materialov, No. 4, pp. 604–608, July–August, 1988.     

A stefan-like problem with a kinetic condition and surface tension effects

A two-dimensional, one phase Stefan-type problem is described as a model for an industrial erosion/deposition process, which includes surface tension effects and a kinetic condition at the free boundary. Special solutions (similarity and ‘traveling wave’) are considered. The stability of the free boundary of these special solutions is proved within the class of planar solutions, as is their linear stability as solutions of the full problem. The role of surface energy and the interaction rate in stabilizing solutions corresponding to deposition is discussed.     

Mathematical modelling of surface vibration with volume constraint and its analysis

A hyperbolic problem with volume constraint is investigated by means of a minimizing method called the discrete Morse flow. A weak solution is introduced in a constructive way and results of numerical computations are shown.     

Finite-length mask effects in the isolation oxidation of silicon

Author to whom all correspondence should be addressed A moving-boundary problem modelling the two-dimensional isolationoxidation of silicon is analysed in the limit of reaction-controlledoxidation for a finite-length nitride mask. Encroachment underthe mask caused by silicon oxidation then occurs from both sidesto produce two ‘bird's beaks’, and it is the interactionbetween these beaks on which attention is focused. This effect,termed ‘bird's beak punchthrough’, is currentlyof interest in submicron silicon-isolation technologies.     

Size dependent forced vibration of nanoplates with consideration of surface effects

In this article, an analytical method is presented to study the size dependent forced vibration of rectangular nanoplates under general external loading using a generalized form of Kirchhoff plate model. The effects of surface properties including surface elasticity, surface residual stresses and surface mass density are considered which are bases for size dependent behaviors due to increase in surface to volume ratios at smaller scales. At first, a complete discussion is given for size dependent natural frequencies which are then used in forced vibration analyses. It is shown that the surface properties compact the frequency spectrums of the nanoplates. Saving generality and using the superposition principle, closed form solution is derived for time response of nanoplates under general harmonic loads. As a result, some elliptic curves are obtained for which the surface properties will not change the time response of nanoplates when a point load is applied on any point of these curves. It is observed that, for actuations inside these ellipses, the surface effects reduce the vibration amplitude while increase it for actuations outside the curves. Sensitivity of the problem to the excitation frequencies is also studied and various examples are given to illustrate the trend of size dependencies.     

Local smoothing effects for the water-wave problem with surface tension

The water-wave problem with a one-dimensional free surface of infinite depth is considered, based on the formulation as a second-order nonlinear dispersive equation. The local smoothing effects are established under the influence of surface tension, stating that on average in time solutions acquire locally 1/4 derivative of smoothness as compared to the initial state. The analysis combines energy methods with techniques of Fourier integral operators. To cite this article: H. Christianson et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Frequency response analysis of laminated composite beams

Conclusions The modeling of laminated composite beams has been derived systematically from the three-dimensional elasticity relations. The correctness of the solution found by using the present finite element model is verified by comparison with the results obtained by analytical solutions and other results presented in the literature. Numerical results indicate that the present technique can given accurate results for frequency response analysis for laminated composite beams. Loss factors of structures obtained by the method of complex eigenvalues and the direct frequency response method exhibit very good agreement. Optimum design of a laminated composite beam by the finite element method and the method of experiment planning has been successfully presented.Published in Mekhanika Kompozitnykh Materialov, Vol. 30, No. 5, pp. 664–674, September–October, 1994.     

Phenomenological theory of surface effects in ferroelectrics and its relationship with transverse Ising model

The relationship between the transverse field Ising model and the Landau phenomenological theory for ferroelectrics is analyzed, and the Landau free energy expression for ferroelectrics having surfaces is derived. It is pointed out that the traditional expression in which the surface integral has only a term of the square polarization is valid only for special cases, in general a term of the polarization to the four should be included as well. By use of the newly derived free energy expression, the thickness-dependence of the spontaneous polarization and Curie temperature of ferroelectric films is calculated; thereby some experimental results incompatible with the traditional phenomenological theory are successfully explained.     

Effect of surface treatment of carbon fibers on the strength properties of carbon fiber plastics

Conclusions The strength characteristics of composites based on carbon fibers having a coating of silicon carbide are in direct dependence on the coating thickness and on the porosity, which makes it possible to assume the possibility of increasing the degree of realization of the strength characteristics of fibers having a coating in a composite by increasing the degree of impregnation of the carbon cord with the binder. The latter finds confirmation also in the fact that at a small coating thickness on the carbon fiber (of the order of 5 nm) the porosity of the composite obtained is equal to the porosity of the material based on the carbon fiber without coating. Moreover, as is evident from Fig. 3c, the casing of silicon carbide does not form a continuous coatting over the whole perimeter of the cord. The presence of these prerequisites, and also the high resistance of carbon fibers having a silicon carbide coating to oxidation [9], open up wide prospects for creating new composite materials based on them.Translated from Mekhanika Kompozitnykh Materialov, No. 4, pp. 603–606, July–August, 1979.     

A geometrically nonlinear finite element model of nanomaterials with consideration of surface effects

In conventional continuum mechanics, the surface energy is usually small and negligible. But at nano-length scale, it becomes a significant part of the total elastic energy due to the high specific surface area of nanomaterials. A geometrically nonlinear finite element (FE) model of nanomaterials with considering surface effects is developed in this paper. The aim is to extend the conventional finite element method (FEM) to analyze the size-dependent mechanical properties of nanomaterials. A numerical example, analysis of InAs quantum dot (QD) on GaAs (0 0 1) substrate, is given in this paper to verify the validity of the method and demonstrate surface effects on the stress fields of QDs.     

Mathematical analysis of carbon nanotube model

In this paper, we present a recently developed mathematical model for a short double-wall carbon nanotube. The model is governed by a system of two coupled hyperbolic equations and is reduced to an evolution equation. This equation defines a dissipative semi-group. We show that the semi-group generator is an unbounded nonselfadjoint operator with compact resolvent. Moreover, this operator is a relatively compact perturbation of a certain selfadjoint operator.     

Isogeometric analysis of surface elasticity: a comparison with isoparametric FEM

We report here on the isogeometric treatment of surface-elastic effects and compare the results obtained from Finite Element simulations and Isogeometric Analysis. As expected, the latter delivers smoother solutions than a standard finite element approach. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An analysis of the traffic on highways with changing surface conditions

In this paper we present an application of the theory of traffic flow formulated by Lighthill and Whitham to problems where a section of a road imposes different conditions on the flow of cars. To represent these situations mathematically, we use a different flow-density relationship which contains implicitly the changes in the properties of this special section of road.With this approach we analyze first some simplified problems of this type, the solutions of which are of use in more complicated problems. Also, we obtain the corresponding steady-state solutions and discuss their stability. Finally, to illustrate these ideas on more realistic situations, we obtain the solutions of the flow produced by heavy rain on a section of a highway and by the slowing down of a convoy of trucks moving on one lane of a three-lane highway.     

Numerical analysis of longitudinal elastic-plastic waves with radial effects

Summary The two-dimensional problem of longitudinal elastic-plastic waves in circular rods taking radial-inertia effects into account is solved based on the finite-element method and an explicit integration algorithm. The elastic-plastic constitutive equations are the yield criterion of von Mises with isotropic work hardening and the Prandtl-Reuss flow rule. Numerical results are shown for the region near the impact end of a semi-infinite rod for two sets of boundary conditions, namely prescribed longitudinal velocity and prescribed longitudinal stress at the bar end. The lateral motion of the struck end is assumed to be unrestrained (zero shear stress).The numerical results show response characteristics which deviate from the one-dimensional solution and which are in a good qualitative agreement with a number of experimental observations reported in the literature. These impact test results have been examined with respect to the predictions in order to separate radial- and strain-rate effects. Several specific calculations for the various test conditions have been performed to obtain quantitative agreement with experimental observations.

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Zusammenfassung Das zweidimensionale Problem der elastisch-plastischen Längswellen in kreisförmigen Stäben unter Berücksichtigung der Querschwingungen wird mit Hilfe der finiten Elemente und eines expliziten Integrationsalgorithmus gelöst. Als Materialgleichungen wurden die von Misessche Fliebedingung und das Fliegesetz nach Prandtl-Reuss verwendet. Für den halbunendlichen Stab werden numerische Resultate fur den Bereich des belasteten Endes gezeigt, wobei entweder die Geschwindigkeit oder die Spannung am belasteten Ende vorgegeben ist. Zusätzlich wird angenommen, da sich das Stabende in der Querrichtung frei bewegen kann.Die numerischen Resultate zeigen abweichend von der eindimensionalen Lösung Details, die gut mit einigen in der Literatur beschriebenen experimentellen Beobachtungen übereinstimmen. Diese Versuchsergebnisse wurden den numerischen Resultaten gegenübergestellt, um so den Einflu der Querschwingungen vom Einflu der Dehnungsgeschwindigkeit zu trennen. Um eine quantitative Übereinstimmung mit den experimentellen Beobachtungen zu erzielen, wurden einige Rechnungen mit den Versuchbedingungen als Eingabewerte durchgeführt.

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This paper is based on the doctoral dissertation of first author's dissertation at the Swiss Federal Institue of Technology (ETH), Zürich, Switzerland.