The thermoelastic Hertzian contact problem

A numerical solution is obtained for the steady-state thermoelastic contact problem in which heat is conducted between two elastic bodies of dissimilar materials at different temperatures with arbitrary quadratic profiles. Thermoelastic deformation causes the initially elliptical contact area to be reduced in size and to become more nearly circular as the temperature difference is increased. There is also a small but identifiable deviation from exact ellipticity at intermediate temperature differences. An approximate analytical solution is obtained, based on approximating the contact area by an ellipse.     

Frictionless indentation by an elastic punch: a dynamic Hertzian contact problem

Frictionless indentation of an elastic half-plane by a relatively blunt, symmetric elastic punch at an ar: bitrary speed is analyzed by treating the more general problem of frictionless Hertzian contact between elastic solids. As in the quasi-static problem, the analysis assumes that the solid surface contours are approximately flat. In addition, the contact strip expands at a constant rate and the imposed rigid body motions and surface contours are represented by polynomial curves. Homogeneous function techniques allow analytic solutions to the basic mathematical problem. As an example, the general results are then applied to the uniformly accelerating parabolic punch on a half-plane.     

Rolling contact problem involving surface roughness

Rolling contact problems for two elastic rollers are investigated. The surface micro-roughness is taken into account. For modelling of boundary roughness a new model is proposed. The problem is solved by the boundary integral method. Presented results show the effect of the roughness on the shearing traction and on the creepforce–creep relation.     

Two-dimensional third- and fifth-order nonlinear evolution equations for shallow water waves with surface tension

Nonlinear Dynamics - We derive two new two-dimensional third- and fifth-order nonlinear evolution equations that model a unidirectional wave motion in shallow water waves with surface tension....     

Two-dimensional thermoelastic contact problem of functionally graded materials involving frictional heating

The two-dimensional thermoelastic sliding frictional contact of functionally graded material (FGM) coated half-plane under the plane strain deformation is investigated in this paper. A rigid punch is sliding over the surface of the FGM coating with a constant velocity. Frictional heating, with its value proportional to contact pressure, friction coefficient and sliding velocity, is generated at the interface between the punch and the FGM coating. The material properties of the coating vary exponentially along the thickness direction. In order to solve the heat conduction equation analytically, the homogeneous multi-layered model is adopted for treating the graded thermal diffusivity coefficient with other thermomechanical properties being kept as the given exponential forms. The transfer matrix method and Fourier integral transform technique are employed to convert the problem into a Cauchy singular integral equation which is then solved numerically to obtain the unknown contact pressure and the in-plane component of the surface stresses. The effects of the gradient index, Peclet number and friction coefficient on the thermoelastic contact characteristics are discussed in detail. Numerical results show that the distribution of the contact stress can be altered and therefore the thermoelastic contact damage can be modified by adjusting the gradient index, Peclet number and friction coefficient.     

Experimental study of an impact oscillator with viscoelastic and Hertzian contact

Modeling an impact event is often related to the desired outcome of an impact oscillator study. If the only intent is to study the dynamic behavior of the system, numerous researchers have shown that simpler impact models will often suffice. However, when the geometric contours and material properties of the two colliding surfaces are known, it is often of interest to model the contact event at a greater level of complexity. This paper investigates the application of a finite time impact model to the study of a parametrically excited planar pendulum subjected to a motion-dependent discontinuity. Experimental and numerical studies demonstrate the presence of multiple periodic attractors, subharmonics, quasi-periodic motions, and chaotic oscillations.     

The elliptical Hertzian contact of transversely isotropic magnetoelectroelastic bodies

First, the general solution for transversely isotropic magnetoelectroelastic media is given concisely in form of five harmonic functions. Second, the extended Boussinesq and Cerruti solutions for the magnetoelectroelastic half-space are obtained in terms of elementary functions by utilizing this general solution. Third, the coupled fields for elliptical Hertzian contact of magnetoelectroelastic bodies are solved in smooth and frictional cases. At last, the graphic results are presented.     

Frictional Hertzian contact problems under cyclic loading using static reduction

In this study we investigate an axisymmetric Hertzian contact problem of a rigid sphere pressing into an elastic half-space under cyclic loading. A numerical solution is sought to obtain a steady state, which demands a large amount of computer memory and computing speed. To achieve a tractable problem, the current numerical model uses a “static reduction” technique, which employs only a contact stiffness matrix rather than the entire stiffness of the problem and is more accurate than the approach used by most finite element codes. Investigation of the tendency of contact behavior in the transient and steady states confirms that a steady state exists, showing converged energy dissipation. The dependence of dissipation on load amplitude shows a power law of load amplitude less than 3, which may explain some deviations in the experimental findings.     

Axisymmetric contact problem for a biphasic cartilage layer with allowance for tangential displacements on the contact surface

A unilateral axisymmetric contact problem for a biphasic cartilage layer indented by a rigid punch is considered. The refined linearized kinetic relationship which takes into account both the radial and tangential displacements of the boundary points of the biphasis cartilage layer is imposed. The obtained analytical solution is valid over long-time periods and can be used for increasing loading conditions. It can be used as it is and as a benchmark for verification of FEM model accuracy.     

On the undamped free vibration of a mass interacting with a Hertzian contact stiffness

Three methods are used to determine the natural frequency of undamped free vibration of a mass interacting with a Hertzian contact stiffness. The exact value is determined using the first integral of motion. The harmonic balance method is used on a transformed equation for an approximate solution, and the multiple scales method is used on an approximate equation. The maximum initial displacement avoiding contact loss is also determined, and the corresponding exact natural frequency is also obtained analytically. The methods are evaluated by studying the free vibration of an elastic sphere on a flat rigid surface.     

Control of vibroimpact dynamics of a single-sided Hertzian contact forced oscillator

The control of vibroimpact dynamics of a single-sided Hertzian contact forced oscillator is investigated analytically and numerically in this paper. The control strategy is introduced via a fast excitation and attention is focused on the response near the primary resonance. The fast excitation is added to the basic harmonic force, either through a harmonic force applied from above, or via a harmonic base displacement added from bellow, or by considering the stiffness of the oscillator as a periodically and rapidly varying in time. The results reveal that the threshold of vibroimpact response initiated by jump phenomenon near the primary resonance can be shifted toward lower or higher frequencies of the slow dynamic system depending on the fast excitation taken into consideration. It was also shown that the most realistic and practical way for controlling the vibroimpact dynamics is the introduction of a fast harmonic base displacement.     

Analytical and experimental study of a circular membrane in Hertzian contact with a rigid substrate

The problem that is addressed here is that of a pressurized circular membrane in contact with a rigid substrate. A closed-form membrane analysis with Hertz-type contact is developed to describe the relationship between pressure, contact radius and contact force. Both the variation in the slope of the deflection profile of the portion of the membrane outside the contact zone and the contact radius itself are measured by an apparatus based on moiré deflectometry. Contact experiments with a 3 μm PET film and a glass substrate show that this analysis predicts both the slope field and contact radius well.     

Effect of changes of surface tension and contact angle on normal force measurement with the Weissenberg rheogoniometer

Summary In experiments with water and dilute aqueous solutions of polyacrylamide spurious normal forces were observed. This phenomenon was traced to changes in contact angle between the test liquid and the plates brought about by starting and stopping shear. There are circumstances in which contact angle and surface tension changes with shear could be indistinguishable from true normal forces.

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Zusammenfassung In Versuchen mit Wasser und verdünnten wässerigen Lösungen von Polyacrylamid wurden falsche Normal-Kräfte beobachtet. Diese Erscheinung wird durch die Änderung des Kontaktwinkels zwischen der Versuchsflüssigkeit und den Platten erklärt. Diese Änderungen werden an sich durch den Anfang und die Beendigung der Bewegung erzeugt. Unter Umständen können die Änderungen des Kontaktwinkels und der Oberflächenspannung mit der Scherung nicht von den Normal-Kräften zu unterscheiden sein.

     

Simulating surface tension with smoothed particle hydrodynamics

A method for simulating two‐phase flows including surface tension is presented. The approach is based upon smoothed particle hydrodynamics (SPH). The fully Lagrangian nature of SPH maintains sharp fluid–fluid interfaces without employing high‐order advection schemes or explicit interface reconstruction. Several possible implementations of surface tension force are suggested and compared. The numerical stability of the method is investigated and optimal choices for numerical parameters are identified. Comparisons with a grid‐based volume of fluid method for two‐dimensional flows are excellent. The methods presented here apply to problems involving interfaces of arbitrary shape undergoing fragmentation and coalescence within a two‐phase system and readily extend to three‐dimensional problems. Boundary conditions at a solid surface, high viscosity and density ratios, and the simulation of free‐surface flows are not addressed. Copyright © 2000 John Wiley & Sons, Ltd.     

The hertz contact problem with finite friction

The indentation of an elastic half-space by an axisymmetric punch under a monotonically applied normal force is formulated as a mixed boundary value problem under the assumption of Coulomb friction with coefficient in the region of contact. Within an inner circle the contact is adhesive, while in the surrounding annulus the surface moves inwards with increasing load. The slip boundary between the two regions depends on and the Poisson ratio v, and is found uniquely as an eigenvalue of a certain integral equation.For power law indentors of the form zr ⁿ , a group property of the integral operator connecting stresses and displacements makes it possible to derive the contact stress distributions from those under a flat punch by a simple quadrature, and shows that the slip radius is the same in all such cases.An iterative numerical solution using a dual system of Volterra equations is described, and calculated distributions of surface stress presented for the cases of indentation by a flat punch and by a sphere.

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Zusammenfassung Das Eindringen eines axial-symetrischen Stempels in einen elastischen unendlichen Halbraum, unter Einwirkung einer monotonischen senkrechten Kraft, wird dargestellt als ein gemischtes Grenzwert problem, wobei ein Coulombscher Reibungskoeffizient im Kontaktvolumen angenommen wird. Innerhalb eines inneren Kreises der Kontakt is haftend während in dem umgebenden Kreisring eine nach innen gerichtete Bewegung der Ebene stattfindet, die mit der Kraft wächst. Die Gleitgrenze zwischen diesen beiden Gebieten hängt von und dem Poisson Verhältnis ab und ist ein eindeutiger Eigenwert eines Integralgleichung.Für Stempel, deren Form zr ⁿ gehorcht, wird gezeigt, dass eine Gruppeneigenschaft des Integral Operators, welche Spannungen und Verschiebungen verknüpft, ermöglicht, die Kontaktspannungsverteilung in der Umgebung eines flachen Stempels durch einfache Quadratur abzuleiten und zu zeigen, dass der Gleitradius in allen Fällen der gleiche ist.Es wird eine iterative numerische Lösung beschrieben, die Volterra-Gleichungen benutzt. Die Berechnungen der Oberflächenspannungsverteilung für das Eindringen eines ebenen Stempels in eine Kugel wird gegeben.