Contact mechanics is the study of the deformation of solids that touch each other at one or more points. This can be divided into compressive and adhesive forces in the direction perpendicular to the interface, and frictional forces in the tangential direction. Frictional contact mechanics is the study of the deformation of bodies in the presence of frictional effects, whereas frictionless contact mechanics assumes the absence of such effects.
Frictional contact mechanics is concerned with a large range of different scales. At the macroscopic scale, it is applied for the investigation of the motion of contacting bodies (see Contact dynamics). For instance the bouncing of a rubber ball on a surface depends on the frictional interaction at the contact interface. Here the total force versus indentation and lateral displacement are of main concern. At the intermediate scale, one is interested in the local stresses, strains and deformations of the contacting bodies in and near the contact area. For instance to derive or validate contact models at the macroscopic scale, or to investigate wear and damage of the contacting bodies' surfaces. Application areas of this scale are tire-pavement interaction, railway wheel-rail interaction, roller bearing analysis, etc. Finally, at the microscopic and nano-scales, contact mechanics is used to increase our understanding of tribological systems (e.g., investigate the origin of friction) and for the engineering of advanced devices like atomic force microscopes and MEMS devices. This page is mainly concerned with the second scale: getting basic insight in the stresses and deformations in and near the contact patch, without paying too much attention to the detailed mechanisms by which they come about.
History Several famous scientists, engineers and mathematicians contributed to our understanding of friction. They include Leonardo da Vinci, Guillaume Amontons, John Theophilus Desaguliers, Leonhard Euler, and Charles-Augustin de Coulomb. Later, Nikolai Pavlovich Petrov, Osborne Reynolds and Richard Stribeck supplemented this understanding with theories of lubrication.
Deformation of solid materials was investigated in the 17th and 18th centuries by Robert Hooke, Joseph Louis Lagrange, and in the 19th and 20th centuries by d’Alembert and Timoshenko. With respect to contact mechanics the classical contribution by Heinrich Hertz stands out. Further the fundamental solutions by Boussinesq and Cerruti are of primary importance for the investigation of frictional contact problems in the (linearly) elastic regime.
Classical results for a true frictional contact problem concern the papers by F.W. Carter (1926) and H. Fromm (1927). They independently presented the creep versus creep force relation for a cylinder on a plane or for two cylinders in steady rolling contact using Coulomb’s dry friction law (see below). These are applied to railway locomotive traction, and for understanding the hunting oscillation of railway vehicles. With respect to sliding, the classical solutions are due to C. Cattaneo (1938) and R.D. Mindlin (1949), who considered the tangential shifting of a sphere on a plane (see below).
In the 1950s, interest in the rolling contact of railway wheels grew. In 1958, Kenneth L. Johnson presented an approximate approach for the 3D frictional problem with Hertzian geometry, with either lateral or spin creepage. Among others he found that spin creepage, which is symmetric about the center of the contact patch, leads to a net lateral force in rolling conditions. This is due to the fore-aft differences in the distribution of tractions in the contact patch.
In 1967, Joost Jacques Kalker published his milestone PhD thesis on the linear theory for rolling contact. This theory is exact for the situation of an infinite friction coefficient in which case the slip area vanishes, and is approximative for non-vanishing creepages. It does assume Coulomb's friction law, which more or less requires (scrupulously) clean surfaces. This theory is for massive bodies such as the railway wheel-rail contact. With respect to road-tire interaction, an important contribution concerns the so-called magic tire formula by Hans Pacejka.
In the 1970s, many numerical models were devised. Particularly variational approaches, such as those relying on Duvaut and Lion’s existence and uniqueness theories. Over time, these grew into finite element approaches for contact problems with general material models and geometries, and into half-space based approaches for so-called smooth-edged contact problems for linearly elastic materials. Models of the first category were presented by LaursenLaursen, T.A., 2002, Computational Contact and Impact Mechanics, Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis, Springer, Berlin and by Wriggers.Wriggers, P., 2006, Computational Contact Mechanics, 2nd ed., Springer, Heidelberg An example of the latter category is Kalker’s CONTACT model.
A drawback of the well-founded variational approaches is their large computation times. Therefore, many different approximate approaches were devised as well. Several well-known approximate theories for the rolling contact problem are Kalker’s FASTSIM approach, the Shen-Hedrick-Elkins formula, and Polach’s approach.
More information on the history of the wheel/rail contact problem is provided in Knothe's paper. Further Johnson collected in his book a tremendous amount of information on contact mechanics and related subjects. With respect to rolling contact mechanics an overview of various theories is presented by Kalker as well. Finally the proceedings of a CISM course are of interest, which provide an introduction to more advanced aspects of rolling contact theory.
Problem formulation Central in the analysis of frictional contact problems is the understanding that the stresses at the surface of each body are spatially varying. Consequently, the strains and deformations of the bodies are varying with position too. And the motion of particles of the contacting bodies can be different at different locations: in part of the contact patch particles of the opposing bodies may adhere (stick) to each other, whereas in other parts of the contact patch relative movement occurs. This local relative sliding is called micro-slip.
This subdivision of the contact area into stick (adhesion) and slip areas manifests itself a.o. in fretting wear. Note that wear occurs only where power is dissipated, which requires stress and local relative displacement (slip) between the two surfaces.
The size and shape of the contact patch itself and of its adhesion and slip areas are generally unknown in advance. If these were known, then the elastic fields in the two bodies could be solved independently from each other and the problem would not be a contact problem anymore.
Three different components can be distinguished in a contact problem. First of all, there is the deformation of the separate bodies in reaction to loads applied on their surfaces. This is the subject of general continuum mechanics. It depends largely on the geometry of the bodies and on their (constitutive) material behavior (e.g. elastic vs. plastic response, homogeneous vs. layered structure etc.). Secondly, there is the overall motion of the bodies relative to each other. For instance the bodies can be at rest (statics) or approaching each other quickly (impact), and can be shifted (sliding) or rotated (rolling) over each other. These overall motions are generally studied in classical mechanics, see for instance multibody dynamics. Finally there are the processes at the contact interface: compression and adhesion in the direction perpendicular to the interface, and friction and micro-slip in the tangential directions.
The last aspect is the primary concern of contact mechanics. It is described in terms of so-called contact conditions. For the direction perpendicular to the interface, the normal contact problem, adhesion effects are usually small (at larger spatial scales) and the following conditions are typically employed: The gap between the two surfaces must be zero (contact) or strictly positive (separation, ); The normal stress acting on each body is zero (separation) or compressive ( in contact). Mathematically: . Here are functions that vary with the position along the bodies' surfaces.
In the tangential directions the following conditions are often used: The local (tangential) shear stress (assuming the normal direction parallel to the -axis) cannot exceed a certain position-dependent maximum, the so-called traction bound ; Where the magnitude of tangential traction falls below the traction bound , the opposing surfaces adhere together and micro-slip vanishes, ; Micro-slip occurs where the tangential tractions are at the traction bound; the direction of the tangential traction is then opposite to the direction of micro-slip .
The precise form of the traction bound is the so-called local friction law. For this Coulomb's (global) friction law is often applied locally: , with the friction coefficient. More detailed formulae are also possible, for instance with depending on temperature , local sliding velocity , etc.
Solutions for static cases
Rope on a bollard, the capstan equation
Consider a rope where equal forces (e.g., ) are exerted on both sides. By this the rope is stretched a bit and an internal tension is induced ( on every position along the rope). The rope is wrapped around a fixed item such as a bollard; it is bent and makes contact to the item's surface over a contact angle (e.g., ). Normal pressure comes into being between the rope and bollard, but no friction occurs yet. Next the force on one side of the bollard is increased to a higher value (e.g., ). This does cause frictional shear stresses in the contact area. In the final situation the bollard exercises a friction force on the rope such that a static situation occurs.
The tension distribution in the rope in this final situation is described by the capstan equation, with solution:
The tension increases from on the slack side () to on the high side . When viewed from the high side, the tension drops exponentially, until it reaches the lower load at . From there on it is constant at this value. The transition point is determined by the ratio of the two loads and the friction coefficient. Here the tensions are in Newtons and the angles in radians.
The tension in the rope in the final situation is increased with respect to the initial state. Therefore, the rope is elongated a bit. This means that not all surface particles of the rope can have held their initial position on the bollard surface. During the loading process, the rope slipped a little bit along the bollard surface in the slip area . This slip is precisely large enough to get to the elongation that occurs in the final state. Note that there is no slipping going on in the final state; the term slip area refers to the slippage that occurred during the loading process. Note further that the location of the slip area depends on the initial state and the loading process. If the initial tension is and the tension is reduced to at the slack side, then the slip area occurs at the slack side of the contact area. For initial tensions between and , there can be slip areas on both sides with a stick area in between.
Generalization for a rope lying on an arbitrary orthotropic surface If a rope is laying in equilibrium under tangential forces on a rough orthotropic surface then three following conditions (all of them) are satisfied:
This generalization has been obtained by Konyukhov A.,
Sphere on a plane, the (3D) Cattaneo problem