Asymptotics for a thin elastic fiber in contact with a rigid foundation

In this work a 3-D contact elasticity problem for a thin fiber and a rigid foundation is studied. We describe the contact condition by a linear Robin-boundary-condition (by meaning of the penalized and linearized non-penetration and friction conditions). The Robin parameters are scaled differently in the longitudinal and cross-sectional directions. The dimension of the problem is reduced by a standard ([3], [4]) asymptotic approach with an additional expansion suggested to fulfil the contact conditions. The 3-D contact conditions result into 1-D Robin-boundary-conditions for corresponding ODEs. The Robin-coefficients of the 1-D problem depend on the ones from the 3-D statement and on the cross-section of the fiber. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lie Sphere Geometry in Hubert Spaces

We develop Lie sphere geometry for arbitrary real pre-Hilbert spaces of (finite or infinite) dimension at least 2. One of the main results is that a bijection of the set of all Laguerre cycles which preserves contact in one direction must already be a Lie transformation (THEOREM 2). As a first consequence of this theorem we get that a bijection of an arbitrary real pre-Hilbert space of dimension at least 3 which preserves Lorentz-Minkowski distance 0 in one direction must already be a (proper or improper) Lorentz boost up to a dilatation, a translation and an orthogonal mapping (THEOREM 3). This is a generalization of results of A.D. Alexandroff [1], E.M. Schröder [21] and F. Cacciafesta [7]. Another consequence is that a bijection of the set of all Lie cycles which preserves contact in one direction must already be a Lie transformation (THEOREM 4). If we apply this result to the finite dimensional case, we get that the diffeomorphism assumption in the Fundamental Theorem of Lie sphere geometry as stated in Theorem 1.5 in T.E. Cecil [8], p. 33, is not needed for the proof of this theorem (REMARK to THEOREM 4).     

The Vibrational Behavior of Bladed Disks in Consideration of Friction Damping and Contact Elasticity

Rotating turbine blading is subjected to fluctuating gas forces during operation that cause blade vibrations. One of the main tasks in the design of turbomachinery blading is the reduction of the vibration amplitudes of the blades to avoid high resonance stresses that could damage the blading. The vibration amplitudes of the blades can be reduced significantly to a reasonable amount by means of friction damping devices such as underplatform dampers. In the case of blade vibrations, relative displacements between the friction damping devices and the neighboring blades occur and friction forces are generated that provide additional damping to the structure due to the dry friction energy dissipation. In real turbomachinery applications, spatial blade vibrations caused by a complex blade geometry and distributed excitation forces acting on the airfoil accur. Therefore, a three dimensional model including an appropriate spatial contact model to predict the generalized contact forces is necessary to describe the vibrational behavior of the blading with sufficient accuracy, see [1] and [2]. In this paper the contact model presented in [2] is extended to include also local deformations in the contacts between underplatform dampers and the contact surfaces of the adjacent blades. The additional elasticity in the contact influences the resonance frequency of the coupled bladed disk assembly. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The indentation of a punch in the form of an elliptic paraboloid into the plane boundary of an elastic body

The three-dimensional contact problem for an elastic body of arbitrary geometry with a single plane face, into which a punch in the shape of an elliptic paraboloid is indented, is considered. The curvilinear boundary of the body is partially clamped, and the remaining boundary (outside the contact region) is stress-free. It is assumed that the dimensions of the contact area are small compared with the characteristic dimension of the body. Using the method of matched asymptotic expansions a model problem of unilateral contact without friction is derived for the boundary layer, which is solved using the apparatus of Hertz's theory. Asymptotic models of the contact interaction of different degrees of accuracy are constructed, including corrections to the geometry and clamping conditions of the elastic body. The sensitivity of the parameters of the elliptic region of the contact to these factors is investigated.     

A DAE formulation for geared rotor dynamics including frictional contact between the teeth

A DAE approach is presented for geared rotor dynamics simulations with rigid helical evolvent gears. It includes the normal contact force between the teeth as well as tangential components. Given the evolvent tooth flank geometry of gear 1 and gear 2 [1], the contact line and the velocity difference in the contact are found. The requirement of no penetration of the teeth yields a non-holonomic constraint and the contact normal force. The friction caused force and moment are obtained by applying Coulomb's friction model. This approach is used to investigate the dynamics of two ideal rotors with translational DoFs, which are connected by gears to one another. The driving rotor has a given angular speed, while the driven rotates unrestrainedly and is connected to a rotational damper. Because of the periodic friction terms, the solution is periodic. A direct time integration or a harmonic approach can be used for the numerical computation. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The model of dry friction in the problem of the rolling of rigid bodies

The Contensou model of combined dry friction [1] is considered. The problem of integrating the shear stresses over the contact area is solved in terms of elementary functions, unlike the solution in [1], reduced to elliptic quadratures. The problem of the rolling of a homogeneous sphere over a plane with dry friction is investigated.     

Three-dimensional contact problem of the motion of a stamp with friction

There is considered the three-dimensional contact problem of elasticity theory with friction forces collinear to the motion direction. Such a case holds during stamp motion along the boundary of an elastic half-space with anisotropic friction /1/. In the case of an arbitrary friction surface, the mentioned force distribution is satisfied approximately during stamp motion.     

Surface Deformation due to Shear and Ploughing in a Halfspace

Sheet and bulk metal forming are widely used manufacturing methods. The interaction between worktool and workpiece in such a process causes friction which has a remarkable impact on the expended energy of the process. Therefore the influence of friction is important. Friction can be split into shearing and ploughing [1]. Ploughing is the plastic deformation of a soft surface by a hard contact partner. Shear forces are only transferred in the real contact area where material contact occurs. The investigation of the contribution of both ploughing and shearing to the total friction resistance is done with the use of an elasto-plastic halfspace model. The multiscale character of surfaces demands a fine discretization, which results in numerical effort. While a finite element method takes into account both surface and bulk of the contact partners, the halfspace model only regards the contact surfaces and thereby consumes less computing capacity. In order to identify the friction resistance, two rough surfaces get into contact. After full application of the normal load, the surfaces are moved relatively to each other. New asperities of the contact surfaces get into contact and are plastically deformed. These deformations are used to estimate the ploughing effect in dependency on the relative displacement. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A dynamically consistent model of the contact stresses in the plane motion of a rigid body

The problem of determining dry friction forces in the case of the motion of a rigid body with a plane base over a rough surface is discussed. In view of the dependence of the friction forces on the normal load, the solution of this problem involves constructing a model of the contact stresses. The contact conditions impose three independent constraints on the kinematic characteristics, and the model must therefore include three free parameters, which are determined from these conditions at each instant. When the body is supported at three points, these parameters (for which the normal stresses can be taken) completely determine the model, while indeterminacy arises in the case of a larger number of contact points and, in order to remove this, certain physical hypotheses have to be accepted. It is shown that contact models consistent with the dynamics possess certain new qualitative properties compared with the traditional quasi-static models in which the type of motion of the body is not taken into account. In particular, a dependence of the principal vector and principal moment of the friction forces on the direction of sliding or pivoting of the body, as well as on the magnitude of the angular velocity, is possible.     

Dynamic behavior of a moving frictional punch over the surface of anisotropic materials

This paper investigates dynamic, frictional contact of a moving punch over the surface of anisotropic materials. An eigenvalue analysis of the governing equations is performed. The application of the complex function theory produces a singular integral equation exhibiting a non-square-root or unconventional singularity. Numerical tests demonstrate that both the friction coefficient and the moving velocity contribute to the contact behaviors under a moving punch with a flat or cylindrical profile. Furthermore, the present results illustrate that the surface in-plane stress possesses singularity and discontinuation at both edges of the flat punch and has a tensile spike at one edge of the cylindrical punch, which may account for the fatigue and fracture under the contact loading.     

Modelling of lateral oscillations in the interface of brake systems

The research of the dynamics in the boundary layer of brake pads can afford revealing technical expertise concerning different phenomena occurring in brake systems (e.g. squealing, judder, Hot Spots). For manufacturers of brake systems, the phenomenon “squealing” is of particular interest due to the desired driving comfort. The mentioned phenomena have been researched by numerous scientists. Different models have been developed and approved by experimental results. The authors agree upon the fact that the main excitation mechanism of the mentioned effects in brake systems is caused by the variation of the lateral (in-plane) friction force and friction torque respectively [1]. A model describing the dynamical interaction of friction and wear in the contact zone of brake systems has been introduced in [2]. Based on this theory, the brake pad's boundary layer consists of characteristic hard and smooth structures on the mesoscopic length scale. A new concept of the initiation of the squealing mechanism is based on the synchronisation mechanism of these so-called patches emerging from the dynamical interaction of the friction partners. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

裂缝三级摩擦因数及影响因素研究(以砂岩（颗粒胶结体）为例)

讨论摩擦面的摩擦因数模型．认为砂岩的摩擦因数分为砂粒球面摩擦因数、微裂纹平面摩擦因数、凸凹构成的裂缝摩擦因数3个层次,分别代表3类不同的成因,3个层次的耦合是真实岩石摩擦因数的决定因素．岩石摩擦因数是在砂粒球面材料摩擦因数基础上,经过后两种形式的放大而形成岩石的宏观摩擦因数．裂纹表面凸起的平均角度或者分形维数是影响岩石摩擦因数分异的最大影响因素,而颗粒排布模式导致的分异相对小得多．颗粒接触的静摩擦因数大于动摩擦因数的成因与颗粒的平均接触角度有关．     

Evolutional contact with Coulomb's friction on a periodic microstructure

We consider the elasticity problem in a heterogeneous domain with an ε-periodic micro-structure, ε ≪ 1, including a multiple micro-contact in a simply connected matrix domain with inclusions completely surrounded by cracks, which do not connect the boundary, or a textile-like material. The contact is described by the Signorini and Coulomb-friction contact conditions. In the case of the Coulomb friction, the dissipative functional is state dependent, like in [2]. A time discretization scheme from [2] reduces the contact problem to the Tresca one (with prescribed frictional traction or state independent dissipation) on each time-increment. We further look for the spatial homogenization. The limiting energy and the dissipation term in the stability condition were obtained for the contact with Tresca's friction law in [4] for closed cracks and can be extended to textile-like materials. Using these results and the concept of energetic solutions for evolutional quasi-variational problems from [2], for a uniform time-step partition, the existence can be proved for the solution of the continuous problem and a subsequence of incremental solutions weakly converging to the continuous one uniformly in time. Furthermore, the irreversible frictional displacements at micro-level lead to a kind of an evolutional plastic behavior of the homogenized medium. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The interaction of surface topography and friction dynamics in brake systems described by a Cellular Automaton

The friction coefficient μ , which is the quotient of the friction force R and the normal force N is in principal not a stationary material parameter, but also dependent on for instance the relative velocity, the normal load, the temperature, the climate conditions, the location and the event itself. The dynamics in the boundary layer between a brake disc and a brake pad is closely linked with the surface topography dynamics. Growing and destroying processes of hard, thin patches, carrying the friction power, determine the time-dependence of the friction coefficient. This interaction between friction and wear has already been simulated with a set of differential equations [2-4], which give an idea about the equilibrium of flow in the contact zone and which are able to describe the fading effect, for example. Based on this assumption we discretised the boundary layer with a Cellular Automaton [5], which makes it possible to have a more detailed look at the processes in the contact area. This paper will show new conclusions concerning the interdependencies of the friction behaviour and the surface topography. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An investigation of the impact and penetration of solids of revolution into soft earth

The numerical method proposed earlier in [1] is developed to solve problems of the impact and penetration of rigid and deformable bodies of revolution into soft soil, which are described by Grigoryan's model [2]. The effect of the surface Coulomb friction, the bulk compressibility and the shear strength of soft soil on the forces of resistance and contact pressures in the contact zone is analysed. The results of numerical solutions of problems in a coherent formulation are compared with analytical relations and experimental data on the determination of the forces and coefficients of resistance to the penetration of impactors of different shapes into soft soil.     

Asymptotics for thin elastic fibers in unilateral contact

What is the contact condition in a 1D beam-model and is it possible to obtain the frictional moments and forces from the 3D traction? If it is possible does the cross-section of the beams influence these values? These questions motivate to study the dimension reduction of a 3D contact problem for beams. This paper is a continuation of [1]. In [1] the asymptotic dimension reduction of a Robin-type elasticity boundary value problem was presented. In this work the explicit relation between a 3D contact problem and a 3D Robin-type elasticity boundary value problem are established and the 1D equations derived in [1] are interpreted as 1D contact conditions, further some numerical examples are shown. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rolling Contact Problem with the Generalized Coulomb Friction

This paper deals with the numerical solution of the wheel - rail rolling contact problems. The unilateral dynamic contact problem between a rigid wheel and a viscoelastic rail lying on a rigid foundation is considered. The contact with the generalized Coulomb friction law occurs at a portion of the boundary of the contacting bodies. The Coulomb friction model where the friction coefficient is assumed to be Lipschitz continuous function of the sliding velocity is assumed. Moreover Archard's law of wear in the contact zone is assumed. This contact problem is governed by the evolutionary variational inequality of the second order. Finite difference and finite element methods are used to discretize this dynamic contact problem. Numerical examples are provided. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Verification of an anisotropic Coulomb adhesion-friction law for contact surfaces with periodic structure

Sliding friction forces and so-called adhesion forces are the main mechanical characteristics to describe contact interaction. Both together are representing 2D surface constitutive laws in analogy to e.g. elasto-plasticity for 3D continua. The classical model to generalize the Coulomb friction law into anisotropic domains is to introduce an anisotropic friction tensor. Michalowski and Mroz in [1] proposed the structure of the friction tensor considering the sliding of a rigid block on an inclined surface. Zmitrowicz in [2] developed the theoretical basis for the structure of the friction tensor on symmetry groups for the tensor. The current contribution is aimed at verification of this modeling process based on a homogenization procedure for a very fine discretization representing the exact structure of the surface. The validation issue with realistic experiments given in [4] is discussed as well. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)