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.

The Elastic Field for Spherical Hertzian Contact Including Sliding Friction for Transverse Isotropy

1992 ◽  
Vol 114 (3) ◽  
pp. 606-611 ◽  
Author(s):  
M. T. Hanson
Keyword(s):  
Coulomb Friction ◽  
Sliding Friction ◽  
Contact Region ◽  
Transverse Isotropy ◽  
Elastic Field ◽  
Hertzian Contact ◽  
Elementary Functions ◽  
Elastic Bodies ◽  
Title Problem ◽  
Equivalent Expressions

This paper gives closed-form expressions in terms of elementary functions for the title problem of spherical Hertzian contact of elastic bodies possessing transverse isotropy. Traction in the contact region is also included in the form of Coulomb friction; thus the shear stress is proportional to the contact pressure. The present expressions derived here by integration of the point force Green’s functions are simpler and easier to apply than equivalent expressions which have previously been given.

The Elastic Field for Conical Indentation Including Sliding Friction for Transverse Isotropy

1992 ◽  
Vol 59 (2S) ◽  
pp. S123-S130 ◽  
Author(s):  
M. T. Hanson
Keyword(s):  
Shear Stress ◽  
Coulomb Friction ◽  
Sliding Friction ◽  
Contact Region ◽  
Transverse Isotropy ◽  
Elastic Field ◽  
Elementary Functions ◽  
Elastic Bodies ◽  
Title Problem ◽  
Conical Indentation

This paper gives closed-form expressions in terms of elementary functions for the title problem of conical indentation of elastic bodies possessing transverse isotropy. Traction in the contact region is also included in the form of Coulomb friction; thus, the shear stress is taken proportional to the contact pressure. The present expressions are derived here by integration of the point force Green’s functions.

The elastic contact influences on passive walking gaits

Robotica ◽  
2010 ◽  
Vol 29 (5) ◽  
pp. 787-796 ◽  
Author(s):  
Feng Qi ◽  
Tianshu Wang ◽  
Junfeng Li
Keyword(s):  
Coulomb Friction ◽  
Dynamic Models ◽  
Contact Stiffness ◽  
Contact Forces ◽  
Hertz Contact ◽  
Walking Gait ◽  
Routes To Chaos ◽  
Coulomb Friction Law ◽  
Contact Parameters ◽  
Passive Model

SUMMARYThis paper presents a new planar passive dynamic model with contact between the feet and the ground. The Hertz contact law and the approximate Coulomb friction law were introduced into this human-like model. In contrast to McGeer's passive dynamic models, contact stiffness, contact damping, and coefficients of friction were added to characterize the walking model. Through numerical simulation, stable period-one gait and period-two gait cycles were found, and the contact forces were derived from the results. After investigating the effects of the contact parameters on walking gaits, we found that changes in contact stiffness led to changes in the global characteristics of the walking gait, but not in contact damping. The coefficients of friction related to whether the model could walk or not. For the simulation of the routes to chaos, we found that a small contact stiffness value will lead to a delayed point of bifurcation, meaning that a less rigid surface is easier for a passive model to walk on. The effects of contact damping and friction coefficients on routes to chaos were quite small.

The Elastic Field Resulting From Elliptical Hertzian Contact of Transversely Isotropic Bodies: Closed-Form Solutions for Normal and Shear Loading

1997 ◽  
Vol 64 (3) ◽  
pp. 457-465 ◽  
Author(s):  
M. T. Hanson ◽  
I. W. Puja
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Elastic Field ◽  
Transversely Isotropic ◽  
Normal Pressure ◽  
Shear Loading ◽  
Potential Functions ◽  
Hertzian Contact ◽  
Elementary Functions ◽  
Shear Traction

This analysis presents the elastic field in a half-space caused by an ellipsoidal variation of normal traction on the surface. Coulomb friction is assumed and thus the shear traction on the surface is taken as a friction coefficient multiplied by the normal pressure. Hence the shear traction is also of an ellipsoidal variation. The half-space is transversely isotropic, where the planes of isotropy are parallel to the surface. A potential function method is used where the elastic field is written in three harmonic functions. The known point force potential functions are utilized to find the solution for ellipsoidal loading by quadrature. The integrals for the derivatives of the potential functions resulting from ellipsoidal loading are evaluated in terms of elementary functions and incomplete elliptic integrals of the first and second kinds. The elastic field is given in closed-form expressions for both normal and shear loading.

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).

Some Consequences of Coulomb Friction in Modeling Longitudinal Traction

1990 ◽  
Vol 18 (1) ◽  
pp. 13-65 ◽  
Author(s):  
W. W. Klingbeil ◽  
H. W. H. Witt
Keyword(s):  
Shear Force ◽  
Coulomb Friction ◽  
Interfacial Shear ◽  
Force Generation ◽  
Behavior Patterns ◽  
Inflation Pressure ◽  
Friction Law ◽  
The Coefficient Of Friction ◽  
Coulomb Friction Law ◽  
Perfect Adhesion

Abstract A three-component model for a belted radial tire, previously developed by the authors for free rolling without slip, is generalized to include longitudinal forces and deformations associated with driving and braking. Surface tractions at the tire-road interface are governed by a Coulomb friction law in which the coefficient of friction is assumed to be constant. After a brief review of the model, the mechanism of interfacial shear force generation is delineated and explored under traction with perfect adhesion. Addition of the friction law then leads to the inception of slide zones, which propagate through the footprint with increasing severity of maneuvers. Different behavior patterns under driving and braking are emphasized, with comparisons being given of sliding displacements, sliding velocities, and frictional work at the tire-road interface. As a further application of the model, the effect of friction coefficient and of test variables such as load, deflection, and inflation pressure on braking stiffness are computed and compared to analogous predictions on the braking spring rate.

Motion of a Ball in a Ball Bearing

1973 ◽  
Vol 95 (1) ◽  
pp. 263-268
Author(s):  
H. Portig ◽  
H. G. Rylander
Keyword(s):  
Deformation Theory ◽  
Coulomb Friction ◽  
Ball Bearing ◽  
Digital Simulation ◽  
Equations Of Motion ◽  
Unsteady Motion ◽  
Hertzian Contact ◽  
Contact Deformation ◽  
Normal Forces ◽  
Simulation Language

A method is developed which allows the digital simulation of the unsteady motion of a single ball constrained only by two moving bearing races. Any desired motion of the races can be simulated. Normal forces acting on the ball are calculated by Hertzian contact deformation theory. If there is slippage between ball and races, Coulomb friction is assumed to occur. Solutions to the differential equations of motion were obtained on a computer with the digital simulation language MIMIC. The phenomenon of ball control as well as the behavior of the ball as it reached a controlled state from rest were observed. This analysis can produce more realistic results than methods that assume that the ball is controlled at all times, especially when the races are radially or angularly displaced with respect to each other.

scholarly journals Thermomechanical Couplings in Aircraft Tire Rolling/Sliding Modeling

2011 ◽  
Vol 274 ◽  
pp. 81-90 ◽  
Author(s):  
Ange Kongo Kondé ◽  
Iulian Rosu ◽  
F. Lebon ◽  
L. Seguin ◽  
Olivier Brardo ◽  
...  
Keyword(s):  
Friction Coefficient ◽  
Thermal Effects ◽  
Coulomb Friction ◽  
Large Deformations ◽  
Element Model ◽  
Slip Angle ◽  
Tire Model ◽  
Vertical Load ◽  
Coulomb Friction Law ◽  
Lateral Friction

This paper presents a finite element model for the simulation of aircraft tire rolling. Large deformations, material incompressibility, heterogeneities of the material, unilateral contact with Coulomb friction law are taken into account. The numerical model will allow estimating the forces in the contact patch - even in critical and extreme conditions for the aircraft safety and security. We show the influence of loading parameters (vertical load, velocity, inflating pressure) and slip angle on the Self Aligning torque and on the lateral friction coefficient. A friction coefficient law corresponding to Chichinadze model is considered to take into account thermal effects in the aircraft tire model behaviour.

Analysis of a dynamic frictional contact problem for hyperviscoelastic material with non-convex energy density

2017 ◽  
Vol 23 (3) ◽  
pp. 359-391 ◽  
Author(s):  
Mikaël Barboteu ◽  
Leszek Gasiński ◽  
Piotr Kalita
Keyword(s):  
Contact Problem ◽  
Coulomb Friction ◽  
Frictional Contact ◽  
Tangential Velocity ◽  
Dynamic Contact ◽  
Contact Boundary ◽  
Dynamic Contact Problem ◽  
Monotone Dependence ◽  
Coulomb Friction Law ◽  
Stored Elastic Energy

Using the time approximation method we obtain the existence of a weak solution for the dynamic contact problem with damping and a non-convex stored elastic energy function. On the contact boundary we assume the normal compliance law and the generalization of the Coulomb friction law which allows for non-monotone dependence of the friction force on the tangential velocity. The existence result is accompanied by two numerical examples, one of them showing lack of uniqueness for the numerical solution.

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