Hertz contact stress mechanical engineering general. The following assumptions are made in determining the solutions of hertzian contact problems. The theoretical contact area of two spheres is a point 0dimensional the theoretical contact area of two parallel cylinders is a line 1dimensional as a result, the pressure between two curved surfaces should be infinite the infinite pressure at the contact should cause immediate yielding of both surfaces. Ciavarellab,1 adepartment of mechanical engineering and applied mechanics, university of michigan, ann arbor, mi 481092125, usa bdepartment of engineering science, university of oxford, parks road, oxford, ox1 3pj, uk abstract contact problems are central to solid mechanics, because contact is the principal method of applying loads to a. Having determined how a bearing carries load, we can determine. Normal contact of elastic solids hertz theory chapter 4. A new theory in contact mechanics for modeling of soft fingers is proposed to define the relationship between the normal force and the radius of contact for soft fingers by considering general softfinger materials, including linearly and nonlinearly elastic materials. Contact mechanics fully elastic models hertz analyzed the stresses at the contact of two elastic solids, and thereby assumed small strains within the elastic limit. It is tempting to accept the predictions regarding indentation depth and radius of circle of contact between two elastic bodies in contact given by the wellknown hertz equations at face value. Assessment of the applicability of the hertzian contact theory to edgeloaded prosthetic hip bearings. It will be shown in part ii that the ball and the large object will deform and.
I the principal reference for this chapter is contact mechanics by johnson, 1985. Pdf contact mechanics amir rostami and xianzhang wang. Introduction to contact mechanics, second edition is a gentle introduction to the mechanics of solid bodies in contact for graduate students, post doctoral individuals, and the beginning researcher. A general method for the numerical analysis of frictionless nonconformable non hertzian contact of bodies of arbitrary shape is developed. Modeling of contact mechanics and friction limit surfaces for. A central distinction in contact mechanics is between stresses acting perpendicular to the contacting bodies surfaces known as the normal direction and frictional stresses acting tangentially between the surfaces. Pdf assessment of the applicability of the hertzian. Apr 22, 20 step by step procedure of how to do static structural analysis of hertzian contact analysis ansys workbench. For the great majority of cases, an application of hertz theory requires little more than selecting the correct formula, and this end could be served by a simple handbook compilation. Although hertzian contact theory has been applied in the case of concentric ceramiconceramic hip contact mak and jin, 2002, its merits and shortcomings have not been assessed in abnormal states and multiple material couples. Tables of hertzian contact stress coefficients duane h. Hertzian theory of nonadhesive elastic contact in hertzs classical theory of contact, he focused primarily on nonadhesive contact where no tension force is allowed to occur within the contact area. Even though the derivation of the theory is relatively difficult, the final solution is a set of simple analytical equations relating the properties of the system to the developed stress. Elastic compression of spheres and cylinders at point and line contact by m.
However, it is nevertheless of interest to examine these predictions either by experiment or by independent computation. Contact mechanics has been analyzed in industry for a long time. Cooper abstract formulations are given for the coefficients x, u, u defined by h. Solid mechanics part i kelly 32 illustrated in fig. It can be used to calculate mechanical point and line contact dimensions and material stresses. Pdf assessment of the applicability of the hertzian contact. In the end of the 19th century hertz presented a formula for calculating the contact stresses in two infinite elastic half spaces magi and melkersson, 2009, which became the general. These contact stresses are called hertz contact stresses. Stress concentration in periodically rough hertzian. One of the methods that adams uses is extrapolated from hertzian contact theory. However, experimental observations suggest that as the contact radius approaches the cone crack radius at the surface, interaction effects lead to a non hertzian pressure distribution.
Zhou y, wang a and muser mh 2019 how thermal fluctuations affect hardwall repulsion and thereby hertzian contact mechanics. The limits normally adopted for hertz s contact stresses are 4 times the allowable stress for linear contact and 5. Numerical difficulties arise because the solution is extremely sensitive to the manner in which one discretizes the governing integral equation. We investigate the normal contact stiffness in a contact of a rough sphere with an elastic halfspace using 3d boundary element calculations. Contact mechanics an overview sciencedirect topics. Knowing the deformation at the center of tne contact and. It is therefore certainly desirable to model the effect of thermal 31. The unit of frequency, cycle per second, was named the hertz in his honor.
The output from a contact mechanics study generally includes contact stresses, both at the bearing surfaces also known as the contact pressure and. As a preparatory step, we will summarize a few results of the theory of elasticity that have a direct application to contact mechanics. Bearing l 10 life, defined as number of revolutions for which 90% of bearings survive at a given load, is a strong function of peak hertzian contact pressure and the peak orthogonal subsurface shear stress. The following video demonstrates steps to mesh the geometry using edge sizing. The contact radius a is considered significantly smaller than the radius of curvature r, and the two contacting surfaces, as depicted in figure 1, assumed to be nonconformal. Elastic compression of spheres and cylinders at point and. We perform molecular dynamics simulations on the detachment of nanoparticles from a substrate. Pdf rigorous treatment of contact problems hertzian contact. Frontiers how thermal fluctuations affect hardwall.
Stress concentration in periodically rough hertzian contact. Indentation depth may be readily compared using standard experimental. The contact area will change depending on the magnitude of the contact stress. Contact mechanics of nanoparticles university of akron. As with the block, the force will actually be distributed over a contact region. Contact mechanics is the study of the deformation of solids that touch each other at one or more points. Sliding hertzian point contact tests phoenix tribology. Assessment of the applicability of hertzian contact theory. Lecture 28 antennas and radiation and the hertzian dipole in this lecture you will learn. For small normal forces, it is found that the stiffness behaves according to the law of pohrtpopov for nominally flat selfaffine surfaces, while for higher normal forces, there is a transition to hertzian behavior. The material at the contact plane wants to expand outward due the compressive contact force and poissons effect. Numerical modeling of accretionary wedge mechanics. This page focuses mainly on the normal direction, i. Kita, hideki hirai, takene iizuka, tateoki and osumi, kazuo 2003.
Hertz in terms of the solution of a transcendental equation involving elliptic integrals and used by him to describe the deformation of bodies subjected to contact stresses. Next he discusses the influence of friction and the topographical roughness of surfaces, and this is incorporated into the theory of contact mechanics. Hertz contact stress calculations and graphs with hertzwin. Equations 7 and 8 can be used to determine the contact parameters between a cylinder and an internal spherical surface or a flat surface.
Ciavarellab,1 adepartment of mechanical engineering and applied mechanics, university of michigan, ann arbor, mi 481092125, usa bdepartment of engineering science, university of oxford, parks road, oxford, ox1 3pj, uk. For an electronic copy pdf of the relevant report, click here consider a simple problem that illustrates a comparison between the analytical and numerical results. Pdf how thermal fluctuations affect hardwall repulsion and. Journal of the mechanics and physics of solids, vol. The company was founded in 2011 and is majority owned by fouriertransform ab vicura, 20. The influence of strain hardening on cumulative plastic deformation in rolling and sliding line contact. How thermal fluctuations affect hardwall repulsion and thereby hertzian contact mechanics article pdf available in frontiers in mechanical engineering 5. We consider the deformations in an elastic halfspace, which are caused by a given stress acting upon its. The magnitude of the load carried by an individual ball depends on the internal geometry of the bear. These elevated stresses are beyond the hertzian field.
Lecture 28 antennas and radiation and the hertzian dipole. We can also do wear analysis due to frictional rubbing between the contact surfaces. Tutorial on hertz contact stress university of arizona. The stress distribution for the hertzian contact is given in fig. Original article kl johnson and contact mechanics da hills1, d nowell1 and jr barber2 abstract in 2008, the journal of mechanical engineering science was approaching its 50th anniversary and the editorial board arranged some personal contributions from those who had been material in.
The hertz contact theory the hertz contact theory consists of a series of mathematical equations to calculate normal, hoop, radial, and shear stress components on and within a soft flat surface or halfspace, known as the specimen, when pressed by a. It applies to normal contact between two elastic solids that are smooth and. Contact mechanics was first studied by hertz in the 1880s with reference to optical lenses hertz, 1896. Contact mechanics, developed based on the continuum mechanics and mechanics of materials, is a theory to describe. The common approach to the study of contact mechanics is either. Zhou et al thermal contact mechanics 30 phase diagram of hertzian spheres 10. Effect of grain size on hertzian contact damage in alumina. A detailed description of the impact function will follow in chapter 2. Cambridge core solid mechanics and materials contact mechanics by k. Jul 18, 20 we investigate the normal contact stiffness in a contact of a rough sphere with an elastic halfspace using 3d boundary element calculations. W, new algorithms for calculating hertzian stresses, deformations, and contact zone parameters, amp journal of technology, 1992. This interaction or contact can occur on many different scales, ranging from. For analyses of adverse loading, hertzian theory may be useful and applicable provided that the following key assumptions of the theory are approximately. Introduction to contact mechanics nanohubtechtalks.
Jan 21, 2014 2014w engr380 lecture18 spur gear design lewis equation for tooth bending stress, part i duration. Generation of radiation by oscillating charges and currents hertzian dipole antenna ece 303 fall 2005 farhan rana cornell university maxwells equations and radiation maxwells equation predict outgoing radiation from sinusoidally. There are no real life applications in which this type of contact occurs. The mechanics of surfaces in contact is called contact mechanics johnson, 1985. Another contact case entails friction that causes shear stresses on the contact zone. Contact mechanics of hertzian cone cracking sciencedirect. Detailed finite element contact mechanics of the actual cracked body are used to show that the contact pressure is singular at the edge of contact once the. However, material further away has lower compressive stress along the zaxis and does not want to expand outward as much so it constrains the material below it pushing inward. Hertzian theory of nonadhesive elastic contact in hertz s classical theory of contact, he focused primarily on nonadhesive contact where no tension force is allowed to occur within the contact area.
Contact stresses and deformations university of utah. Numerical solution of nonhertzian elastic contact problems. Dobrynin polymer program, institute of materials science, and department of physics, university of connecticut, storrs, connecticut 06268, united states abstract. Frictional and mechanical properties of fe5si3particledispersed si3n4 fabricated by solutioninfiltration method. Pdf hertz contact stress analysis and validation using.
Therefore, it is very imporant to calculate the actual stress at the point of contact, the socalled contact stress by theory of elasticity, developed by hertz, in 1881, when he was 23 years old. Hertzian contact equations for elliptical, spherical and. In essence, many of these tests were invented before we understood either surface chemistry or contact mechanics. A theoretical background to the hertz contact theory can be found here. However, real surfaces are not smooth and the contact mechanics of two real bodies can bitterly deviate from the hertzian predictions since only. The application performs all calculations automatically.
Traditional hertzian contact solution for determining subsurface elastic stress fields for two bodies in contact is applicable only for homogeneous materials. You could divide the hertz stresses by the same factors before comparing them to your allowables. Contact mechanics in general contact is the study of deformable bodies that touch each other at least one point in the space. In vivo comparison of hip mechanics for minimally invasive versus traditional total hip arthroplasty. Contact stress causes deformation, plastic or elastic. Extended hertz theory of contact mechanics for casehardened. Boussinesq made a classical approach to finding the elastic stresses and their spatial distributions. Introduction when there are two nonconforming surfaces that come into contact the point or line of contact can be a location of very high stress values. The output from a contact mechanics study generally includes contact stresses, both at the bearing surfaces also known as the contact pressure and within the component, and the contact area. This second edition maintains the introductory character of the first with a focus on materials science as distinct from straight solid mechanics.
Contact mechanics is a fundamental field of tribology and generally refers to the interaction of solid surfaces. Vicura ab is an engineering company that delivers products which are used in different mechanical transmission systems and electrical drivelines. Ball bearing mechanics the loads carried by ball bearings are transmitted through the balls from one ring to the other. Pdf contact mechanics is a fundamental field of tribology and generally. Thwaite summary the purpose of this paper is primarily to present in a convenient form the formulae and data for the calculation of the compression effects which occur in the measurement and use of spheres and cylinders in dimensional metrology. Hertzian contact theory is a classical theory of contact mechanics and is a very useful tool for engineers and researchers. A 3d object may contact on a or a series of shared points or surfaces.