An Extension of Hertz’s Theory in Contact Mechanics

2006 ◽  
Vol 74 (2) ◽  
pp. 373-374 ◽  
Author(s):  
Guanghui Fu
Keyword(s):  
Contact Area ◽  
General Procedure ◽  
Contact Problems ◽  
The Body ◽  
Polynomial Functions ◽  
Contact Pressure Distribution ◽  
Total Load ◽  
Normal Contact ◽  
Elastic Bodies ◽  
Hertz's Solution

Hertz’s theory, developed in 1881, remains the foundation for the analysis of most contact problems. In this paper, we consider the axisymmetric normal contact of two elastic bodies, and the body profiles are described by polynomial functions of integer and noninteger positive powers. It is an extension of Hertz’s solution, which concerns the contact of two elastic spheres. A general procedure on how to solve this kind of problem is presented. As an example, we consider the contact between a cone and a sphere. The relations among the radius of the contact area, the depth of the indentation, the total load, and the contact pressure distribution are derived.

A General Three-Dimensional Numerical Technique for Determining the Contact Area of an Arbitrary Punch on an Elastic Half-Space

2016 ◽  
Vol 08 (01) ◽  
pp. 1650005 ◽  
Author(s):  
Jing Jin Shen ◽  
Feng Yu Xu ◽  
Guo Ping Jiang
Keyword(s):  
Numerical Method ◽  
Contact Area ◽  
Numerical Technique ◽  
Three Dimensional ◽  
Contact Problems ◽  
Linear Interpolation ◽  
Contact Boundary ◽  
Reciprocal Theorem ◽  
Normal Contact ◽  
Indentation Force

The paper presents a numerical method for determining the contact area in three-dimensional elastostatic normal contact without friction. The method makes use of the theorem developed by Barber, the contact area is that over which the total indentation force achieves its maximum value. By approximating the punch by linear interpolation, the analytical expression for the indentation force is derived by virtue of the reciprocal theorem. The physical meaning of the parameter which determines the contact boundary is discussed, and its feasible range corresponding to the contact area is found. Then, the numerical algorithm for determining the parameter is developed and applied to solve several normal contact problems. The results show that the proposed numerical method possesses a good property on accuracy and convergency.

scholarly journals Penetration factor of the surfaces in fast algorithm of solution of contact problems for wheel and rail.

2015 ◽  
Vol 2015 (1) ◽  
pp. 46-51
Author(s):  
Владимир Сакало ◽  
Vladimir Sakalo ◽  
Алексей Сакало ◽  
Aleksey Sakalo
Keyword(s):  
Contact Area ◽  
Fast Algorithm ◽  
Contact Problems ◽  
Normal Contact ◽  
Contact Patch ◽  
Penetration Factor ◽  
Rolling Surface

Values of penetration factor of wheel and rail surfaces that is used on application of fast algorithm for solution of normal contact problems are defined. Three cases had been considered: contact area has not strongly expressed spatial character; contact patch is situated nearby middle of rail rolling surface, radii of curvature of wheel and rail profiles have close values; contact patch is situated on fillet sections of the profiles.

Development and Experimental Validation of a Point Contact Model for Dynamical Problems With Friction

2001 ◽  
Author(s):  
Walter Sextro
Keyword(s):  
Pressure Distribution ◽  
Relative Motion ◽  
Contact Area ◽  
Point Contact ◽  
Contact Problems ◽  
Normal Pressure ◽  
Tangential Direction ◽  
Contact Model ◽  
Impact Oscillator ◽  
Normal Contact

Abstract The contact forces are dependent on many parameters, such as contact stiffnesses, surface profiles, material parameters, temperature distribution, relative motion and normal pressure distribution. These parameters can change within the contact area and from here, it is impossible to derive a general force law. The only possibility to overcome this problem is to discretize the contact areas, since in general the relative motion and the contact parameters are not constantly distributed within the contact surface. This leads to a point contact model, which has to include all main physical effects as described above, which are important, when simulating dynamical contact problems with friction. The friction model includes the main parameters such as the roughness of the contact surfaces, the nonlinear friction law, the contact stiffnesses in normal and tangential direction. The decreasing characteristic of the friction coefficient with respect to the relative velocity has to be modeled in a sufficient way. With respect to the dissipation of energy, the hysteretic behavior is studied with respect to the normal and tangential direction. Separation of the contact is included. This point contact model is be applied to real dynamical contact problems. In the first example, a simple impact oscillator with an elastic contact is used to check the overall modeling with respect to the elastic normal contact. Then, a self excited friction oscillator is investigated with respect to the tangential contact. Here, the modeling of surface waviness leads to high periodic solutions, which is also observed within the experiments. In both examples, the comparison of measurements and calculated phase plots is good. Furthermore, the influence of wear on to the surface profile, contact area and normal pressure distribution is investigated. From here, it follows, that friction leads to time dependent systems.

scholarly journals Topology optimization of quasistatic contact problems

2012 ◽  
Vol 22 (2) ◽  
pp. 269-280 ◽  
Author(s):  
Andrzej Myśliński
Keyword(s):  
Topology Optimization ◽  
Optimality Condition ◽  
Contact Stress ◽  
Optimization Problem ◽  
Contact Problems ◽  
The Body ◽  
Necessary Optimality Condition ◽  
Normal Contact Stress ◽  
Normal Contact ◽  
Topology Optimization Problem

Topology optimization of quasistatic contact problemsThis paper deals with the formulation of a necessary optimality condition for a topology optimization problem for an elastic contact problem with Tresca friction. In the paper a quasistatic contact model is considered, rather than a stationary one used in the literature. The functional approximating the normal contact stress is chosen as the shape functional. The aim of the topology optimization problem considered is to find the optimal material distribution inside a design domain occupied by the body in unilateral contact with the rigid foundation to obtain the optimally shaped domain for which the normal contact stress along the contact boundary is minimized. The volume of the body is assumed to be bounded. Using the material derivative and asymptotic expansion methods as well as the results concerning the differentiability of solutions to quasistatic variational inequalities, the topological derivative of the shape functional is calculated and a necessary optimality condition is formulated.

Measurement and Visualization of the Contact Pressure Distribution of Rubber Disks and Tires

1995 ◽  
Vol 23 (4) ◽  
pp. 238-255 ◽  
Author(s):  
E. H. Sakai
Keyword(s):  
Pressure Distribution ◽  
Contact Pressure ◽  
Contact Area ◽  
Light Absorption ◽  
Lateral Force ◽  
Contact Pressure Distribution ◽  
High Definition ◽  
The Road ◽  
Close Relationship ◽  
Processing Techniques

Abstract The contact conditions of a tire with the road surface have a close relationship to various properties of the tire and are among the most important characteristics in evaluating the performance of the tire. In this research, a new measurement device was developed that allows the contact stress distribution to be quantified and visualized. The measuring principle of this device is that the light absorption at the interface between an optical prism and an evenly ground or worn rubber surface is a function of contact pressure. The light absorption can be measured at a number of points on the surface to obtain the pressure distribution. Using this device, the contact pressure distribution of a rubber disk loaded against a plate was measured. It was found that the pressure distribution was not flat but varied greatly depending upon the height and diameter of the rubber disk. The variation can be explained by a “spring” effect, a “liquid” effect, and an “edge” effect of the rubber disk. Next, the measurement and image processing techniques were applied to a loaded tire. A very high definition image was obtained that displayed the true contact area, the shape of the area, and the pressure distribution from which irregular wear was easily detected. Finally, the deformation of the contact area and changes in the pressure distribution in the tread rubber block were measured when a lateral force was applied to the loaded tire.

scholarly journals CONTACT PROBLEMS FOR NONLINEARLY ELASTIC MATERIALS: WEAK SOLVABILITY INVOLVING DUAL LAGRANGE MULTIPLIERS

2010 ◽  
Vol 52 (2) ◽  
pp. 160-178 ◽  
Author(s):  
A. MATEI ◽  
R. CIURCEA
Keyword(s):  
Lagrange Multipliers ◽  
Variational Equation ◽  
Small Deformation ◽  
Contact Problems ◽  
Point Theory ◽  
The Body ◽  
Elastic Materials ◽  
Weak Formulation ◽  
Nonlinearly Elastic ◽  
Weak Solvability

AbstractA class of problems modelling the contact between nonlinearly elastic materials and rigid foundations is analysed for static processes under the small deformation hypothesis. In the present paper, the contact between the body and the foundation can be frictional bilateral or frictionless unilateral. For every mechanical problem in the class considered, we derive a weak formulation consisting of a nonlinear variational equation and a variational inequality involving dual Lagrange multipliers. The weak solvability of the models is established by using saddle-point theory and a fixed-point technique. This approach is useful for the development of efficient algorithms for approximating weak solutions.

Interaction Model for “Shelled Particles” and Small-Strain Modulus of Granular Materials

2018 ◽  
Vol 85 (10) ◽  
Author(s):  
Shun-hua Zhou ◽  
Peijun Guo ◽  
Dieter F. E. Stolle
Keyword(s):  
Elastic Modulus ◽  
Contact Stiffness ◽  
Scaling Law ◽  
Interaction Model ◽  
Contact Problems ◽  
Hertzian Contact ◽  
Spherical Particles ◽  
Thin Coating ◽  
Normal Contact ◽  
Laboratory Test Results

The elastic modulus of a granular assembly composed of spherical particles in Hertzian contact usually has a scaling law with the mean effective pressure p as K∼G∼p1/3. Laboratory test results, however, reveal that the value of the exponent is generally around 1/2 for most sands and gravels, but it is much higher for reclaimed asphalt concrete composed of particles coated by a thin layer of asphalt binder and even approaching unity for aggregates consisting of crushed stone. By assuming that a particle is coated with a thin soft deteriorated layer, an energy-based simple approach is proposed for thin-coating contact problems. Based on the features of the surface layer, the normal contact stiffness between two spheres varies with the contact force following kn∼Fnm and m∈[1/3,  1], with m=1/3 for Hertzian contact, m=1/2 soft thin-coating contact, m=2/3 for incompressible soft thin-coating, and m=1 for spheres with rough surfaces. Correspondingly, the elastic modulus of a random granular packing is proportional to pm with m∈[1/3,  1].

Stresses Due to Tangential and Normal Loads on an Elastic Solid With Application to Some Contact Stress Problems

1953 ◽  
Vol 20 (2) ◽  
pp. 157-166
Author(s):  
J. O. Smith ◽  
Chang Keng Liu
Keyword(s):  
Plane Strain ◽  
Coefficient Of Friction ◽  
Contact Area ◽  
Normal Load ◽  
Elastic Solid ◽  
Real Variable ◽  
The Body ◽  
Shearing Stress ◽  
Tangential Load ◽  
The Coefficient Of Friction

Abstract The results of two-dimensional approach using real variable method to Hertz’s problem of contact of elastic bodies are presented. Both normal and tangential loads are assumed to be distributed in Hertzian fashion over the area of contact. The magnitude of the intensity of the tangential load is assumed to be linearly proportional to that of the normal load when sliding motion of the body is impending. The stresses in the elastic body due to the application of these loads on its boundary are presented in closed form for both plane-stress and plane-strain cases. A numerical value of f = 1/3 is assumed for the linear proportionality (coefficient of friction) between the tangential and normal loads in order that the distribution of stresses may be illustrated. The significance of the stress distribution, across the contact area and in the body, is also discussed. It is shown that when the combination of loads considered in the paper are applied at the contact area of bodies in contact the maximum shearing stress may be at the surface instead of beneath the surface. For example, for plane strain, if the coefficient of friction is f = 1/3, the maximum shearing stress is at the surface and is 43 per cent larger than the maximum shearing stress, which would be below the surface, that occurs when the normal force acts alone. The effect of range of normal stress and of shearing stress on the plane of maximum shear and on the plane of maximum octahedral shear on failure by progressive fracture (fatigue) is discussed.

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