Frictional Contact of Rubber on rough Surfaces

The aim of this work is to investigate the frictional behaviour of rubber on rough rigid surfaces. Rubber friction is composed of hysteretic and adhesional effects. The main focus lies on the hysteresis theory which motivates rubber friction with the internal damping inside the viscoelastic material. It is necessary to perform a multiscale analysis to regard the roughness of the surface on all important length scales. The fractal surface is approximated by a superposition of harmonic functions. The simulations will start at the lowest wavelength and proceed to larger scales by replacing the smaller roughness with the ascertained friction law. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Frictional contact of a nonlinear spring

We describe and analyze a frictional problem for a system with a compressed spring which behaves as if it has a spring constant that is negative over a part of its extension range. As a result, the problem has three critical points. The friction is modeled by the Coulomb law. We show that there are three separate stick regions for some values of the parameters, centered on the critical points. We model three other versions of the process. Then we describe a numerical scheme for the models and present a number of computer simulations.     

Plastic deformation of rough surfaces

Friction forces are only transferred within the the real area of contact A_real, which is usually smaller than the apparent area of contact A_o. The maximum friction stress τ_fric is therefore determined by the shear limit τ_max in the area of real contact and the fraction of the real area of contact (τ_fric = τ_max (A_real/A_o)). For rough surfaces the size of A_real is governed o by the plastic deformation of the surface roughness. We present a fully elasto-plastic halfspace contact formulation based on the work of Jacq et al. [1]. Linear elastic-plastic material behavior is modeled based on v. Mises plasticity with isotropic hardening. The algorithm gives the residual stress as well as the full plastic deformation field due to a frictionless normal contact. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Locally homogeneous rigid geometric structures on surfaces

We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection ${\nabla}$ on a two dimensional torus is complete and, up to a finite cover, homogeneous. Let ${\nabla}$ be a unimodular real analytic affine connection on a real analytic compact connected surface M. If ${\nabla}$ is locally homogeneous on a nontrivial open set in M, we prove that ${\nabla}$ is locally homogeneous on all of M.     

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.     

Square-tiled surfaces and rigid curves on moduli spaces

We study the algebro-geometric aspects of Teichmüller curves parameterizing square-tiled surfaces with two applications.(a) There exist infinitely many rigid curves on the moduli space of hyperelliptic curves. They span the same extremal ray of the cone of moving curves. Their union is a Zariski dense subset. Hence they yield infinitely many rigid curves with the same properties on the moduli space of stable n-pointed rational curves for even n.(b) The limit of slopes of Teichmüller curves and the sum of Lyapunov exponents for the Teichmüller geodesic flow determine each other, which yields information about the cone of effective divisors on the moduli space of curves.     

A note on the rigid core of affine surfaces

During recent decades, $$\mathbb {G}_a$$-actions, especially certain invariants of $$\mathbb {G}_a$$-actions, have been important tools in the study of affine varieties. The $$\mathbb {G}_a$$-actions are usually studied through locally nilpotent derivations in characteristic zero and exponential maps (see Definition 1.1) in arbitrary characteristic. The “Makar-Limanov invariant” of locally nilpotent derivations played a pivotal role in solving the linearization conjecture in the 1990s, while invariants of exponential maps were central to N. Gupta’s resolution of the Zariski cancellation problem in positive characteristic. In the study of locally nilpotent derivations on commutative algebras containing $$\mathbb {Q}$$, Freudenburg and Moser-Jauslin (Mich Math J 62:227–258, (2013), Theorem 6.1) have introduced a new invariant called “rigid core” and used it to formulate an alternative version of Mason’s theorem and to prove a well-known analogue of Fermat’s last theorem for rational functions (Freudenburg and Moser-Jauslin (2013), Corollary 6.1). In this note, we consider the concept of the rigid core in the framework of exponential maps on commutative algebras over an algebraically closed field k of arbitrary characteristic. We observe that for any factorial k-domain B with $${\text {tr.deg}}_k(B)=2$$, the concept of rigid core coincides with the Makar-Limanov invariant. We also show that over any affine two-dimensional normal k-domain B, its rigid core is a stable invariant.     

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)     

Homogenization for contact problems with friction on rough interface

We consider a contact problem between a macroscopic solid with a smooth boundary and a technical textile, while the textile has a periodic microscopic structure and microscopically rough surface. Two–scale homogenization approach is applied to the problem. The microscopic solution is approximated in terms of macroscopic solution and some concentration factor, given as a solution of auxiliary boundary value or contact problems of elasticity on the periodicity cell. Local friction condition is represented as a continuous non–linear functional over the stress field. Two–scale convergence is used to prove the convergence of friction functional. The macroscopic initial frictional limit is found. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Logarithmic transformations of rigid analytic elliptic surfaces

We give new examples of algebraic elliptic surfaces and non-algebraic rigid analytic elliptic surfaces by means of logarithmic transformations. In the complex analytic case, it is known that all multiple fibers of elliptic surfaces are obtained by logarithmic transformations. Using rigid analytic geometry, we construct similar transformations of elliptic surfaces over complete non-Archimedean valuation base fields. These operations yield rigid analytic elliptic fibrations with multiple fibers. When the resulting surface admits an ample line bundle, we may algebraize the surface. In the positive characteristic case, we obtain new types of algebraic elliptic surfaces. We also obtain a non-algebraic rigid analytic surface the combination of whose invariants appears neither in the algebraic case nor in the complex analytic case.     

On the structure of rigid semistable sheaves on algebraic surfaces

LetS be a smooth projective surface, letK be the canonical class ofS and letH be an ample divisor such thatH • K < 0. We prove that for any rigid sheafF (Ext¹ (F, F) = 0) that is Mumford-Takemoto semistable with respect toH there exists an exceptional set (E ₁ ,..., E _n ) of sheaves onS such thatF can be constructed from {E _i } by means of a finite sequence of extensions. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 692–700, November, 1998. The author wishes to express his gratitude to S. A. Kuleshov for useful discussions and to A. N. Rudakov and A. L. Gorodentsev for their attention to the present work. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01323 and by the INTAS Foundation.     

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)     

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.