Torsion of a Rigid Smooth Elliptic Insert in an Infinite Elastic Plane

1991 ◽  
Vol 58 (2) ◽  
pp. 370-375 ◽  
Author(s):  
Robert S. Gross ◽  
James G. Goree
Keyword(s):  
Differential Equation ◽  
Contact Region ◽  
Hertzian Contact ◽  
Torsional Stiffness ◽  
Elastic Plane ◽  
Tangential Stresses ◽  
Present Solution ◽  
Hertzian Contact Stress ◽  
Rigid Smooth ◽  
Weakly Dependent

A solution is presented for the non-Hertzian contact stress problem developed during torsion of a smooth rigid elliptic insert in an infinite, linearly elastic plane. The problem is reduced to a singular integro-differential equation which is solved numerically. Results are presented for the contact zones, and for the normal and tangential stresses at the plane-insert interface. The contact region is independent of the magnitude of the applied moment, but depends strongly on the shape of the ellipse, and is weakly dependent on Poisson’s ratio of the plane. Results are also given for the reduction in torsional stiffness between a fully bonded insert and the present solution, (i.e., a completely failed bond).

The Elastic Field for Spherical Hertzian Contact of Isotropic Bodies Revisited: Some Alternative Expressions

1993 ◽  
Vol 115 (2) ◽  
pp. 327-332 ◽  
Author(s):  
M. T. Hanson ◽  
T. Johnson
Keyword(s):  
Coulomb Friction ◽  
Contact Region ◽  
Complete Solution ◽  
Elastic Field ◽  
Hertzian Contact ◽  
Hertz Contact ◽  
Elementary Functions ◽  
Present Solution ◽  
Coulomb Friction Law ◽  
Isotropic Bodies

Closed-form expressions in terms of elementary functions are derived for the elastic field resulting from spherical Hertz contact of isotropic bodies. Shear traction is also included using a Coulomb friction law; thus the shear stress in the contact region is equal to the contact pressure multiplied by a friction coefficient. This paper provides alternative expressions to those recently given by Hamilton (1983) and Sackfield and Hills (1983a). Two methods are outlined for obtaining the present solution and the complete solution for displacements and stresses are given for both normal and tangential loading in terms of just two distorted length parameters. The elastic field is written in a complex notation allowing the expressions to be put in a compact form. This also allows the expressions for sliding in two directions to be written as simply as for sliding in one direction.

Surface Durability of Spur Gears at Hertzian Stresses Over Shakedown Limit

1974 ◽  
Vol 96 (2) ◽  
pp. 359-372 ◽  
Author(s):  
Akira Ishibashi ◽  
Taku Ueno ◽  
Shigetada Tanaka
Keyword(s):  
Fatigue Testing ◽  
Testing Machine ◽  
Carbon Steels ◽  
Rolling Contact ◽  
Hertzian Contact ◽  
Load Testing ◽  
Short Period ◽  
Hertzian Contact Stress ◽  
Surface Durability ◽  
Shakedown Limit

Using a new type of gear-load testing machine and a disk-type rolling fatigue testing machine designed and made by the authors, the upper limits of Hertzian contact stress allowable on rolling contact surfaces were investigated. It was shown conclusively that gears and rollers made of soft carbon steels could be rotated beyond 108 revolutions at Hertzian stresses over shakedown limit (≈ 0.4 HB). In the case of gears, pits having a pitting area ratio of 0.04 percent occurred during 1.16 × 108 rotations at a Hertzian stress of 0.50 HB. However, no pitting occurred on the roller rotated through 1.20 × 108 revolutions at a Hertzian stress of 0.71 HB, although appreciable changes in texture were observed at the subsurface. In order to rotate gears or rollers at Hertzian stresses over shakedown limit, their surface must either be very smooth initially or after a short period of running, and an oil film must be formed between contacting surfaces.

Effect of Lubricant on Surface Rolling Contact Fatigue Cracks

2010 ◽  
Vol 97-101 ◽  
pp. 793-796 ◽  
Author(s):  
Khalil Farhangdoost ◽  
Mohammad Kavoosi
Keyword(s):  
Fatigue Crack ◽  
Fatigue Cracks ◽  
Rolling Contact Fatigue ◽  
Contact Fatigue ◽  
Rolling Contact ◽  
Hertzian Contact ◽  
Element Analysis ◽  
Crack Face ◽  
Hertzian Contact Stress ◽  
Model Of Friction

This study performed the finite element analysis of the cycle of stress intensity factors at the surface initiated rolling contact fatigue crack tip under Hertzian contact stress including an accurate model of friction between the faces of the crack and the effect of fluid inside the crack. A two-dimensional model of a rolling contact fatigue crack has been developed with FRANC-2D software. The model includes the effect of Coulomb friction between the faces of the crack. The fluid in the crack was assumed not only to lubricate the crack faces and reduce the crack face friction coefficient but also to generate a pressure.

A Study of the Effect of m and n Coefficients of the Hertzian Contact Theory on Railroad Vehicle Dynamics

2007 ◽  
Author(s):  
J. P. Pascal ◽  
Khaled E. Zaazaa
Keyword(s):  
Contact Area ◽  
Contact Region ◽  
Local Deformation ◽  
Hertzian Contact ◽  
Vehicle Dynamic ◽  
Railroad Vehicle Dynamics ◽  
Original Table ◽  
Data Points ◽  
Railroad Vehicle ◽  
Contact Ellipse

For the wheel/rail contact problem, the Hertz theory for two elastic bodies in contact is commonly used to determine the shape and dimensions of the contact area and the local deformation of the wheel and rail surfaces at the contact region. The shape of the contact area is assumed to be elliptical. The ratio of the contact ellipse semi-axes is equal to the ratio of two non-dimensional contact area coefficients, known as m and n coefficients. Hertz presented a table of these two coefficients, determined as a function of an angular parameter, θ. Most railroad vehicle dynamic codes use this table with online interpolation to determine the contact ellipse semi-axes. Recently, it was found that this original table may be too coarse, and that more data points are needed within the table for solving the wheel/rail contact accurately. This paper discusses the effect of the accuracy of the m and n coefficients in solving for wheel/rail contact, and demonstrates this effect with two numerical examples that show the resulting differences in the dynamic behavior of railroad vehicles dependent on this accuracy. A new table with more data points is presented that is recommended for use in railroad vehicle dynamic codes that employ the Hertzian contact for solving the wheel/rail contact interaction. This modified table was originally derived by Jean-Pierre Pascal as a part of collaborative research between the Federal Railroad Administration (FRA) and the French Ministry of Transportation.

Energy Losses of Balls Rolling on Plates

1959 ◽  
Vol 81 (2) ◽  
pp. 233-238 ◽  
Author(s):  
R. C. Drutowski
Keyword(s):  
Surface Roughness ◽  
Rolling Force ◽  
Primary Source ◽  
Sample Surface ◽  
Hertzian Contact ◽  
Energy Losses ◽  
Cyclic Process ◽  
Damping Capacity ◽  
Hertzian Contact Stress ◽  
Point To Point

The apparatus for measuring the rolling force of a ball supported between two plates is described. The rolling force is an extremely small quantity compared to the normal force. Instantaneous values of the rolling force vary greatly from point to point on the sample surface and this variation is explained in terms of surface roughness and material homogeneity. The energy losses of balls rolling on plates are shown as functions of load, material, and surface roughness. The rolling of a ball on a plate is examined as a cyclic process in which elastic hysteresis losses appear to be the primary source of energy dissipation. An analysis involving the Hertzian contact stress field is used to derive an equation relating the rolling force and the material damping capacity.

Contact Fracture Behaviors of Ti3SiC2 Synthesized by Hot Pressing Using TiCx and Si Powder Mixture

2005 ◽  
Vol 486-487 ◽  
pp. 217-220
Author(s):  
Sung Sic Hwang ◽  
Sang Whan Park ◽  
Seong Jai Cho ◽  
Dong Bok Lee
Keyword(s):  
Plastic Deformation ◽  
Fracture Behavior ◽  
Hertzian Contact ◽  
Coarse Grained ◽  
Vickers Indentation ◽  
Hertzian Indentation ◽  
Fine Grained ◽  
Fracture Behaviors ◽  
Hertzian Contact Stress ◽  
Contact Fracture

The contact fracture behaviors of fine-grained Ti3SiC2 and coarse-grained high purity Ti3SiC2 are examined by the Hertzian indentation and Vickers indentation technique. The Vickers hardness of bulk Ti3SiC2 is as low as 5.3~6.3 Gpa, and the Hertzian contact stress-strain curves for Ti3SiC2 deviate much from linearity, which resembles the fracture behavior of a ductile metal rather than a brittle ceramic. The contact damages by both Vickers indentation and Hertzian indentation reveal a fairly good plastic deformation nature of Ti3SiC2. Un-reacted TiCx in fine-grained Ti3SiC2 may impede the plastic deformation by slip along basal plan inside Ti3SiC2 grain, making Ti3SiC2 less plastic under loading.

Planar Tooth Profile Synthesis for Relative Curvature

2017 ◽  
Author(s):  
Zhiyuan Yu ◽  
Kwun-Lon Ting
Keyword(s):  
Contact Stress ◽  
Tooth Profile ◽  
Hertzian Contact ◽  
Rolling Motion ◽  
Special Cases ◽  
Systematic Methodology ◽  
Hertzian Contact Stress ◽  
Curvature Theory ◽  
The Given ◽  
Tooth Pair

This paper is the first that uses the new conjugation curvature theory [1] to directly synthesize conjugate tooth profiles with the given relative curvature that determines the Hertzian contact stress. Conjugation curvature theory offers a systematic methodology to synthesize the relative curvature for a tooth pair. For any given relative curvature between the contact tooth profiles, a generating point can be located on an auxiliary body. Under the rolling motion among the pinion pitch, the gear pitch and the pitch on the auxiliary body, the generating point will trace fully conjugate profiles on the pinion and gear bodies with the given relative curvature at the instant of the contact. Full conjugation throughout the contact of the profiles is guaranteed with the three instant centers remaining coincident [1]. The methodology is demonstrated with a planar tooth profile synthesis with given relative curvature. One may find that the Wildhaber-Novikov tooth profile, which is known to have low relative curvature and Hertzian contact stress, and its variations become special cases under such methodology.

A Unified Treatment of Axisymmetric Adhesive Contact on a Power-Law Graded Elastic Half-Space

2013 ◽  
Vol 80 (6) ◽  
Author(s):  
Fan Jin ◽  
Xu Guo ◽  
Wei Zhang
Keyword(s):  
Half Space ◽  
Power Law ◽  
Contact Region ◽  
Adhesive Contact ◽  
Elastic Half Space ◽  
Hertzian Contact ◽  
Contact Radius ◽  
Abel Transformation ◽  
Numerical Solution Methods ◽  
Special Cases

In the present paper, axisymmetric frictionless adhesive contact between a rigid punch and a power-law graded elastic half-space is analytically investigated with use of Betti's reciprocity theorem and the generalized Abel transformation, a set of general closed-form solutions are derived to the Hertzian contact and Johnson–Kendall–Roberts (JKR)-type adhesive contact problems for an arbitrary punch profile within a circular contact region. These solutions provide analytical expressions of the surface stress, deformation fields, and equilibrium relations among the applied load, indentation depth, and contact radius. Based on these results, we then examine the combined effects of material inhomogeneities and punch surface morphologies on the adhesion behaviors of the considered contact system. The analytical results obtained in this paper include the corresponding solutions for homogeneous isotropic materials and the Gibson soil as special cases and, therefore, can also serve as the benchmarks for checking the validity of the numerical solution methods.

Hertzian Contact-Stress Deformation Coefficients

1969 ◽  
Vol 36 (2) ◽  
pp. 296-303 ◽  
Author(s):  
Duane H. Cooper
Keyword(s):  
Contact Stress ◽  
Digital Computer ◽  
Approximate Calculation ◽  
Transcendental Equation ◽  
Contact Stresses ◽  
Elliptic Integrals ◽  
Hertzian Contact ◽  
Stress Deformation ◽  
Hertzian Contact Stress

Formulations are given for the coefficients λ, μ, ν defined by 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. Methods of approximate calculation are explained and errors in the tables prepared by Hertz are noted. For the purpose of providing a more extensive and more accurate tabulation, using an automatic digital computer, these coefficients are reformulated so that a large part of the variation is expressed by means of easily interpreted elementary formulas. The remainder of the variation is tabulated to 6 places for 100 values of the argument. Graphs of the coefficients are also provided.

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