On rough Marcinkiewicz integrals along surfaces

In this paper, the authors establish the LP-mapping properties of certain classes of Marcinkiewicz integral operators along surfaces with rough kernels. The results in this paper essentially extend as well as improve previously known results.     

Scattering from infinite rough tubular surfaces

We study the Helmholtz equation in the exterior of an infinite perturbed cylinder with a Dirichlet boundary condition. Existence and uniqueness of solutions are established using the variational technique introduced (SIAM J. Math. Anal. 2005; 37 (2):598–618). We also provide stability estimates with explicit dependence of the constants in terms of the frequency and the perturbed cylinder thickness. Copyright © 2006 John Wiley & Sons, Ltd.     

Contact of ropes and orthotropic rough surfaces

Contact between arbitrary curved ropes and arbitrary curved rough orthotropic surfaces has been revised from the geometrical point of view. Variational equations for the equilibrium of ropes on orthotropic rough surfaces are derived, first, using the consistent variational inclusion of frictional contact constraints via Karush-Kuhn-Tucker conditions expressed in Darboux basis. Then, the systems of differential equations are derived for both statics and dynamics of ropes on a rough surface depending on the sticking-sliding condition for orthotropic Coulomb's friction. Three criteria are found to be fulfilled during the static equilibrium of a rope on a rough surface: “no separation”, condition for dragging coefficient of friction and inequality for tangential forces at the end of the rope. The limit tangential loads still preserve the famous “Euler view” T = T₀e^ωs for the curves and surfaces of constant curvature. It is shown that the curve of the maximum tension of a rough orthotropic surface is geodesic. Equations of motion are derived in the case if the sliding criteria is fulfilled and there is “no separation”. Various cases possessing analytical solutions of the derived system, including Euler case and a spiral rope on a cylinder are shown as examples of application of the derived theory. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of friction phenomena between randomly rough elastic surfaces

Friction induced vibrations are a widely studied field in which the friction coefficient is one of the most important parameters. Measurements show that the friction coefficient underlies stochastic fluctuations. To gain more knowledge about the friction coefficient a finite element study is carried out in order to simulate the friction forces. The Bowden-Tabor model is implemented which calculates the friction force as the force which is needed to shear apart contact areas hold together by welding or adhesion. The dependency of the friction value on sliding velocity and normal pressure can be determined with this model. Different realization are studied and the stochastic properties of the friction value such as mean value, standard deviation, amplitude spectrum and correlation coefficient can be calculated depending on the roughness of the surface. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Frictional contact of elastomer materials on rough rigid surfaces

The frictional behavior of elastomer materials is still unexplored, but very important for many industrial applications. Special attention is turned to rubber friction on rough road tracks. Due to the non‐rigid material characteristics of the rubber it is not sufficient to use a constant friction coefficient like Coulombs law. The frictional qualities depend on many different influences like sliding velocity, applied normal stress, surface roughness, material properties and the temperature in the contact zone. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Holomorphic differentials and Laguerre deformation of surfaces

A Laguerre geometric local characterization is given of L-minimal surfaces and Laguerre deformations (T-transforms) of L-minimal isothermic surfaces in terms of the holomorphicity of a quartic and a quadratic differential. This is used to prove that, via their L-Gauss maps, the T-transforms of L-minimal isothermic surfaces have constant mean curvature \(H=r\) in some translate of hyperbolic 3-space \({\mathbb {H}}^3(-r^2)\subset \mathbb {R}^4_1\), de Sitter 3-space \({\mathbb {S}}^3_1(r^2)\subset \mathbb {R}^4_1\), or have mean curvature \(H=0\) in some translate of a time-oriented lightcone in \(\mathbb {R}^4_1\). As an application, we show that various instances of the Lawson isometric correspondence can be viewed as special cases of the T-transformation of L-isothermic surfaces with holomorphic quartic differential.     

Plastic deformation of bicrystals within continuum dislocation theory

We analyze the evolution of the plastic distortion and the nucleation and accumulation of dislocations within a model bicrystal with one active slip system in each single crystal (symmetric with respect to the interface), which is subject to prescribed displacements of plane–strain shear and extension, and we present closed–form analytical solutions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integrable deformation of critical surfaces in spaceforms

We investigate a Goursat-type transformation for Bryant surfaces and a few of its geometric features. The basic result is that, when regarded as a non-isometric group action on the moduli space of CMC1 immersions of a fixed surface, the Lawson correspondence is equivariant with respect to the more familiar Goursat transform on minimal immersions. This generalizes certain well-known deformations of critical surfaces and enlarges the number of explicitly computable cousin pairs, visualized here in a quaternionic upper-half space.     

Integral equation methods for scattering by infinite rough surfaces

In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane. These boundary value problems arise in a study of time‐harmonic acoustic scattering of an incident field by a sound‐soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double‐ and single‐layer potential and a Dirichlet half‐plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half‐plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single‐layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd.     

Conformal deformation of metrics on subdomains of surfaces

We consider prescribing Gaussian curvature on subdomains of a surface. We employ thedistribution of mass principle (Theorem 3.3) to smooth subdomains of a Riemannian manifold to obtain that for critical and supercritical cases, a function can be the Gaussian curvature of some pointwise conformal metric, provided it satisfies certain conditions.     

Existence of solution in elastic wave scattering by unbounded rough surfaces

We consider the two‐dimensional problem of the scattering of a time‐harmonic wave, propagating in an homogeneous, isotropic elastic medium, by a rough surface on which the displacement is assumed to vanish. This surface is assumed to be given as the graph of a function ?∈C^1,1(?). Following up on earlier work establishing uniqueness of solution to this problem, existence of solution is studied via the boundary integral equation method. This requires a novel approach to the study of solvability of integral equations on the real line. The paper establishes the existence of a unique solution to the boundary integral equation formulation in the space of bounded and continuous functions as well as in all L^p spaces, p∈[1, ∞] and hence existence of solution to the elastic wave scattering problem. Copyright © 2002 John Wiley & Sons, Ltd.     

Triebel-Lizorkin space boundedness of rough singular integrals associated to surfaces of revolution

We consider the boundedness of the rough singular integral operator T_(?,ψ,h) along a surface of revolution on the Triebel-Lizorkin space F~α_( p,q)(R~n) for ? ∈ H~1(~(Sn-1)) and ? ∈ Llog~+L(S~(n-1)) ∪_1q∞(B~((0,0))_q(S~(n-1))), respectively.     

Plastic deformation of models of a crystalline polymer in hydrostatic tension

The strength of a rhombic model of a defective polymer crystal in hydrostatic tension has been calculated for the set of configurations assumed by the model in the process of plastic deformation. The dislocation centers and twinning curves are calculated for models with one and two coordination polygons in hydrostatic tension. It is shown that in hydrostatic tension the twinning limit of the models is considerably reduced.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 20–25, January–February, 1976.     

Reflection of plane waves by rough surfaces in the sense of Born approximation

The topic of the present paper is the reflection of electromagnetic plane waves by rough surfaces, that is, by smooth and bounded perturbations of planar faces. Moreover, the contrast between the cover material and the substrate beneath the rough surface is supposed to be low. In this case, a modification of Stearns’ formula based on Born approximation and Fourier techniques is derived for a special class of surfaces. This class contains the graphs of functions where the interface function is a radially modulated almost periodic function. For the Born formula to converge, a sufficient and almost necessary condition is given. A further technical condition is defined, which guarantees the existence of the corresponding far field of the Born approximation. This far field contains plane waves, far‐field terms such as those for bounded scatterers, and, additionally, a new type of terms. The derived formulas can be used for the fast numerical computations of far fields and for the statistics of random rough surfaces. Copyright © 2013 John Wiley & Sons, Ltd.     

A model describing pressure buildup in rough surfaces with partially filled gaps

Because of an ecological and economical need to reduce the application of lubricants, the tribological regime of starved lubrication in the field of mechanical engineering has gained importance during the last decades. In order to describe the respective processes properly, models should consider that the gap between the bodies is not fully but only partially filled with a fluid. Recently, the authors introduced a model which explicitly describes the fluid flow and its interaction with the buildup of pressures under these conditions [1]. This paper points out that the correlation between the filling ratio and a time- and space-averaged pressure value is highly nonlinear. This is due to the fact that the system's load bearing character changes from being a local effect to a global effect, in particular near the fully filled regime. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bianchi surfaces in R3 and deformation of hyperelliptic curves

New solutions of the problem of describing hyperbolic surfaces of specified negative Gaussian curvature are obtained. The answer is given in terms of 9 functions.Translated from Matematicheskie Zametki, Vol. 52, No. 3, pp. 78–88, September, 1992.The first author (D. A. K.) wishes to express his appreciation to A. I. Bobenko for useful discussion.     

Plastic deformation of models of a crystalline polymer with increased lattice defectivity

Conclusions 1. On the basis of two-dimensional models, the structure of a polymeric crystal containing more than one lattice defect has been determined on an electronic computer. The types of defects are: a) double vacancy, model I; b) two isolated vacancies, model II; c) a combination of isolated vacancies and double vacancies, model III. Plastic deformation curves have been calculated for all three models.2. It has been observed that increase in the degree of defectivity in a crystalline polymer significantly reduces the elastic limit and reduces the strength limit of the structure very little. The elastic limit falls particularly strongly with increase of the defectivity in a state of hydrostatic stress.3. Increase in the density of crystallographic defects causes significant mutual rearrangement of the coordination polygons of the crystal lattice, which occurs as the result of plastic deformation.Report presented at the Third All-Union Conference on Polymer Mechanics, Riga, November 10–12, 1976.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 397–402, May–June, 1977.