Pdf new algorithms for calculating hertzian stresses. It applies to normal contact between two elastic solids that are smooth and can be described locally with orthogonal radii of. The problem is linear except the contact condition at the base which is modeled using the contact pair approach. 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. In this case, we can consider a small area of the contact region da, fig. Nov 28, 2014 hertz contact stress analysis and validation using finite element analysis mr. Contact stress causes deformation, plastic or elastic. These stresses were first studied by the german physicist heinrich hertz in 1881. Jan 21, 2014 2014w engr380 lecture18 spur gear design lewis equation for tooth bending stress, part i duration. The typical results we want to see are the peak contact stress, the contact radius and stress variation through depth under peak contact stress.
High contact stress may lead to failures such as cracks, pits and flaking in the surface material. A change in designed surface topology may result from wear that brings in a wide change in contact geometry and stresses. For the simplest model to calculate the stresses, one uses the hertz theory and evaluates the effect of the weight only. One main assumption in the hertzian contact theory is that the surfaces are smooth and quadratic. For 9th and 8thpower hertz stress life exponents for ball and roller bearings, respectively, the lundbergpalmgren model best predicts life. Hertzian contact stress hertzian contact theory is a classical theory of contact mechanics and is a very useful tool for engineers and researchers. This results in a nominal variation of the hertz stresslife exponent. I found a nice paper giving the theoretical solution for a 2d plane strain problem here. Both the loadlife and hertz stresslife relations of weibull, lundberg and palmgren, and ioannides and harris reflect a strong dependence on the weibull slope.
His application of the classical theory of elasticity to this problem forms the basis of stress calculation for machine elements such as ball and rol5. Hertz 1881 considered the stresses and deformations in two perfectly smooth t ellipsoidal, contacting elastic solids much like those shown in figure 2. This edition of technical tidbits discusses the concept of hertz stress. Useful solutions for standard problems dartmouth college. There is plenty of literature in public domain on this topic, but it has not been previously discussed in this forum. C contact mechanicsi the theory developed by hertz in 1880 remains the foundation for most contact problems encountered in engineering. After hertzs work, people do a lot of study on the stresses arising from the contact between two elastic bodies. Pdf hertz contact stress analysis and validation using. For analyses of adverse loading, hertzian theory may be useful and applicable provided that the following key assumptions of. Since wheel movement on rail is accompanied by lateral movement due to wheel profile conisity, so the contact point of wheel and rail is not constant and the contact stresses are therefore. Later extensions of the model add viscosity, shear response including friction and adhesion. Further reading useful solutions for standard problems mike ashby engineering department trumpington street, cambridge cb2 1pz, uk.
Hertz developed a theory to calculate the contact area and pressure between the two surfaces and predict the resulting compression and stress induced in the objects. The stress field under the surface is termed hertzian contact stress reference hertz 1881 has done the hard work to derive the equations we need to calculate the stress when there is contact between two bodies that are conformal or nonconformal. Extended hertz theory of contact mechanics for casehardened. 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 rigid hemisphere called the indentor appendix figure 1 1. I am guessing it is how i am applying the force boundary condition, but not sure how i would come up with 17000 n from n the number i am. Assessment of the applicability of hertzian contact theory. The hertz contact problem see timoshenko and goodier, 1951 provides a classic example for verifying the contact capabilities in abaqus. When this happens, complex stresses are induced under the surface at the contact. The limits normally adopted for hertz s contact stresses are 4 times the allowable stress for linear contact and 5. For linearly graded materials such as casehardened bearing steels, three more design parameters, surface elastic modulus e surface. Oct 23, 2014 the typical results we want to see are the peak contact stress, the contact radius and stress variation through depth under peak contact stress.
The application performs all calculations automatically. If a lubricant film is present, and the sphere slides over the flat, the pressure distribution and surface deformations predicted by hertz s equations must be modified. Spheres in contact vertical stress distribution at center of contact area plot shows material with poissons ratio. These contact stresses are called hertz contact stresses, which was first studies by hertz in 1881. While studying at the gelehrtenschule des johanneums in hamburg, hertz showed an aptitude for sciences as well.
Heinrich rudolf hertz was born in 1857 in hamburg, then a sovereign state of the german confederation, into a prosperous and cultured hanseatic family. Assessment of the applicability of hertzian contact theory to. The previous numerical algorithm is used to treat counterformal contacts, and convergent results can always be obtained. 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 rigid hemisphere called the indentor appendix figure 1 1.
Using trescas criterion, a minimum yield stress of 422 nmm2 will be required for ductile materials. Part i history and definition of stress theory this first body of research led to the first efforts of development of family stress theory burr, 1989 by sociologist earl koos 1946. Contact mechanics is the study of the deformation of solids that touch each other at one or more points. The maximum pressure exists at the center of the contact area 12 point o in. 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. In addition, the problem is analyzed under dynamic conditions in abaqusstandard to illustrate the use of contact. For the derivation of the subsurface stresses for line and point hertzian contact, please refer to refs. This is the case for instance for cylinders crosssection, 2d contact and spheres 3d contact, if the contact region is relatively small to the bodies diameters. Hertzwin is a free hertzian contact stress calculator. Tutorial report on hertzian contact stress theory, opti 521, college of optical sciences, university of arizona, tucson, az. Typically people denote maximum stress according to hertz contact theory as a hertz contact stress.
In 1892, hertz began experimenting and demonstrated that cathode rays could penetrate very thin metal foil such as aluminium. As per hertz theory, these parameters are principal curvatures of two bodies, their material properties, and the normal load acting on the contact. After hertz s work, people do a lot of study on the stresses arising from the contact between two elastic bodies. Spherical contact is a special case of the hertz contact, which is between two spheres, or between a sphere and the surface of a half space. Contrary to most classical fatigue phenomena, rolling con. The hertzian contact hertz 1881, 1882 is referred to the frictionless contact between two bodies. This provides an additional increase in contact stress as the oring deforms to a d shape see figure 3 and the contact area of sealing under pressure increases to almost. International journal for research in applied science. Chapter 6, is expanded, presenting more coverage on electrical strain gages and providing tables of equations.
In this example problem, a steel cylinder with a radius of 5 is pressed against a 2 deep aluminum base. The complete set of assumptions directly or indirectly affecting hertzs contact stress theory is presented. The limits normally adopted for hertzs contact stresses are 4 times the allowable stress for linear contact and 5. Hertzian contact stress usually refers to the stress close to the area of contact between two differentsized spheres. Hertz contact theory is the classical theory to evaluate the contact stress in a contact between parabolic surfaces. He developed a version of the cathode tube and studied the penetration by xrays of various materials. To study the influence of interacting wheel and rail profile topology of standard rail uic60, the standard wheel profile as per. C contact mechanics i mit opencourseware free online.
Ansys workbench contact stress help i am trying to model hertz contact stress for a cylinder on a plane, but my fea pressure is less than a factor of 17 when comparing my calculations to the results. The hertz contact stress usually refers to the stress close to the area of contact between two spheres of different radii. A method for determining the size of this region was first described by heinrich hertz in 1881. The contact area will change depending on the magnitude of the contact stress.
The theory governing the wheelrail contact is hertz theory. It can be used to calculate mechanical point and line contact dimensions and material stresses. Safety and economy of railway traffic is enormously influenced by the contact stress variation caused by wheel rail contact profile changes. As the pressure increases, the oring is forced to the low pressure side of the gland. Hertz contact stress analysis and validation using finite. Contact between two continuous, nonconforming solids is initially a point or line. Hertzian loaddisplacement relation holds for spherical. 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. The stress field created by the contact stresses was first introduced by hendrick hertz in 1881. But yes, in static cases you can often allow for higher stress. It applies to normal contact between two elastic solids that are smooth and can be described locally with orthogonal radii of curvature such as a toroid. Since the stress must continuously go to zero at the edges of the boundary between the contact and the non contact regions like in the hertz theory, it follows that the stress distribution psigma, zeta exhibits a tail extending the whole way down to zero stress as indicated in b. A method permitting systematic characterization and. Hertzs solution became the foundation for contact mechanics and is still relevant today.
Lesson 9 hertz contact problem patran 322 exercise workbook 93 model description. Nov 15, 2008 an example of hertz contact fem simulation. Dec 01, 2012 1 tutorial on hertz contact stress xiaoyin zhu opti 521 december 1, 2012 abstract in mechanical engineering and tribology, hertzian contact stress is a description of the stress within mating parts. Design for limit stresses of orange fruits citrus sinensis. Hertz contact stress calculations and graphs with hertzwin. The contact elliptical area is small so stresses developed will be higher. Tutorial on hertz contact stress university of arizona. If a lubricant film is present, and the sphere slides over the flat, the pressure distribution and surface deformations predicted by hertzs equations must be modified. Pdf calculating the contact stress resulting from lateral.
Usually, hertz contact stresses are judged by the shear stress just below the surface. With the smallest application of load elastic deformation occurs and contact is made over a finite area. Since wheel movement on rail is accompanied by lateral movement due to wheel profile conisity, so the contact point of wheel and rail is not constant and the contact stresses are therefore changeable. Contact of bodies occurs in all interfaces transmitting force or motion, or both.
In hertz stress calculator the data assumed for calculation provided as input for calculating contact stress is shown in fig. Hertzian theory when two curved bodies are brought into contact they initially contact at a single point or along a line. Nov 10, 2011 hardonhard total hip impingement causes extreme contact stress concentrations. For systems that need to survive extreme environments or have no room for failure would need to carefully examine risks and benefits of allowing these high contact stresses. Under the action of a load the solids deform and a contact area is formed as shown in figure 1.
Then the original circular shape is approximated well by a quadratic function. If the block weighs w perpendicular to the surface and the force parallel to the surface f can be applied to the side of the block before it begins moving, then the coefficient of static friction. Contact stress analysis of spur gear for different. You could divide the hertz stresses by the same factors before comparing them to your allowables. This page focuses mainly on the normal direction, i. This kind of stress may not be significant most of the time, but may cause serious problems if not take it into account in some cases. Spur gears, solids mechanics, contact stresses, hertz theory. In 1882, heinrich hertz published a theory concerning circular contact areas and elastic deformations in the case of contact between two spheres1. Koos made the first effort at creating a stress theory with the profile of trouble p. The state of stress in nonconformal contacts where rcf occurs is complex and multiaxial and governed by the hertzian contact theory. The zaretsky model decouples the dependence of the critical shear stresslife relation from the weibull slope. According to the hertz contact theory 11, the distribution of pressure is inhomogeneous in the contact area. In detail study of the contact stress produced in the mating gears is the most important task in design of gears as it is the deciding parameter in finding the dimensions of gear mr. Rolling bearing life prediction, theory, and application.
Hertz contact stress theory allows for the prediction of the resulting contact area, contact pressure, compression of the bodies, and the induced stress in the bodies. Unusually high stresses may occur when a load is applied over a small area of contact. Hertz contact stress mechanical engineering general. Hertzian contact theory is a classical theory of contact mechanics and is a very useful tool for engineers and researchers. Hertz developed a theory to calculate the contact area and pressure between the two surfaces and predicted the resulting compression and stress induced in the objects 32. In the maugisdugdale theory, the surface traction distribution is divided into two parts one due to the hertz contact pressure and the other from the dugdale adhesive stress. Tutorial on hertz contact stress tutorial on hertz contact. For example, butterflies are often associated with high contact stress 2000 to 4000 mpa, as discussed in more detail in the article contact fatigue of hardened steel in this vol ume. According to the widely used elliptical hertz contact theory established by greenwood 15, the contact area formed at the contact point for two elastic bodies under certain contact force is an.
In addition, the problem is analyzed under dynamic conditions in abaqusstandard to illustrate the use of. Hertz contact stress is normally defined as pressurep0sqrt1xa2, where p02ppia, 2a is the contact width, p is load per length that is, units are force per length. 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. This also sets a limit for the maximum contact pressure evaluated with hertz contact stress theory as 0. This results in a nominal variation of the hertz stress life exponent. It also serves as an excellent illustration of the use of substructuring in abaqusstandard for locally nonlinear cases local surface contact. 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. Philipp lenard, a student of heinrich hertz, further researched this ray effect. Hertzian static contact stress acceptance criteria. A new symbolism for solving the hertz contact problem. Pure hertzian model only describes normal deformation of spheres. Patil3 1 2 3head of the department of mechanical engineering, sgdcoe, jalgaon, india. Belt theory friction drives vbelts and vribbed belts the figure to the left illustrates the basic idea of static friction.
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