Contact mechanics is the study of the deformation of solids that touch each other at one or more points. A central distinction in contact mechanics is between stresses acting perpendicular to the contacting bodies' surfaces (known as normal stress) and frictional stresses acting tangentially between the surfaces (shear stress). Normal contact mechanics or frictionless contact mechanics focuses on normal stresses caused by applied normal forces and by the adhesion present on surfaces in close contact, even if they are clean and dry. Frictional contact mechanics emphasizes the effect of friction forces.
Contact mechanics is part of mechanical engineering. The physical and mathematical formulation of the subject is built upon the mechanics of materials and continuum mechanics and focuses on computations involving elastic, viscoelastic, and plastic bodies in static or dynamic contact. Contact mechanics provides necessary information for the safe and energy efficient design of technical systems and for the study of tribology, contact stiffness, electrical contact resistance and indentation hardness. Principles of contacts mechanics are implemented towards applications such as locomotive wheel-rail contact, coupling devices, braking systems, tires, bearings, combustion engines, mechanical linkages, gasket seals, metalworking, metal forming, ultrasonic welding, electrical contacts, and many others. Current challenges faced in the field may include stress analysis of contact and coupling members and the influence of lubrication and material design on friction and wear. Applications of contact mechanics further extend into the micro- and nanotechnological realm.
The original work in contact mechanics dates back to 1881 with the publication of the paper "On the contact of elastic solids"H. Hertz, 1881, Über die berührung fester elastischer Körper, Journal für die reine und angewandte Mathematik 92, pp.156-171. (For English version, see: Hertz, H., 1896. On the contact of elastic solids, In: Miscellaneous Papers, Chapter V, pp.146-162. by Hertz, H. and Lenard P., translated by Jones, D. E. and Schott G.A., London: Macmillan. "Über die Berührung fester elastischer Körper" by Heinrich Hertz. Hertz attempted to understand how the optical properties of multiple, stacked lenses might change with the force holding them together. Hertzian contact stress refers to the localized stresses that develop as two curved surfaces come in contact and deform slightly under the imposed loads. This amount of deformation is dependent on the modulus of elasticity of the material in contact. It gives the contact stress as a function of the normal contact force, the radii of curvature of both bodies and the modulus of elasticity of both bodies. Hertzian contact stress forms the foundation for the equations for load bearing capabilities and fatigue life in bearings, gears, and any other bodies where two surfaces are in contact.
History
Classical contact mechanics is most notably associated with Heinrich Hertz.Hertz, H. R., 1882, Über die Berührung fester elastischer Körper und Über die Härte, Verhandlungen des Vereins zur Beförderung des Gewerbefleisscs, Berlin: Verein zur Beförderung des Gewerbefleisses, pp.449-463 (For English version, see: Hertz, H., 1896. On the contact of rigid elastic solids and on hardness, In: Miscellaneous Papers, Chapter VI, pp.163-183. by Hertz, H. and Lenard P., translated by Jones, D. E. and Schott G.A., London: Macmillan. In 1882, Hertz solved the contact problem of two elastic bodies with curved surfaces. This still-relevant classical solution provides a foundation for modern problems in contact mechanics. For example, in mechanical engineering and tribology, Hertzian contact stress is a description of the stress within mating parts. The Hertzian contact stress usually refers to the stress close to the area of contact between two spheres of different radii.
It was not until nearly one hundred years later that Kenneth L. Johnson, Kevin Kendall, and Alan D. Roberts found a similar solution for the case of adhesive contact. This theory was rejected by Boris Derjaguin and co-workers who proposed a different theory of adhesion in the 1970s. The Derjaguin model came to be known as the Derjaguin–Muller–Toporov (DMT) model (after Derjaguin, M. V. Muller and Yu. P. Toporov), and the Johnson et al. model came to be known as the Johnson–Kendall–Roberts (JKR) model for adhesive elastic contact. This rejection proved to be instrumental in the development of the David Tabor and later Daniel Maugis parameters that quantify which contact model (of the JKR and DMT models) represent adhesive contact better for specific materials.
Further advancement in the field of contact mechanics in the mid-twentieth century may be attributed to names such as Frank Philip Bowden and Tabor. Bowden and Tabor were the first to emphasize the importance of surface roughness for bodies in contact. Through investigation of the surface roughness, the true contact area between friction partners is found to be less than the apparent contact area. Such understanding also drastically changed the direction of undertakings in tribology. The works of Bowden and Tabor yielded several theories in contact mechanics of rough surfaces.
The contributions of J. F. Archard (1957) must also be mentioned in discussion of pioneering works in this field. Archard concluded that, even for rough elastic surfaces, the contact area is approximately proportional to the normal force. Further important insights along these lines were provided by James A. Greenwood and J. B. P. Williamson (1966), A. W. Bush (1975), and Bo N. J. Persson (2002). The main findings of these works were that the true contact surface in rough materials is generally proportional to the normal force, while the parameters of individual micro-contacts (pressure and size of the micro-contact) are only weakly dependent upon the load.
Classical solutions for non-adhesive elastic contact The theory of contact between elastic bodies can be used to find contact areas and indentation depths for simple geometries. Some commonly used solutions are listed below. The theory used to compute these solutions is discussed later in the article. Solutions for multitude of other technically relevant shapes, e.g. the truncated cone, the worn sphere, rough profiles, hollow cylinders, etc. can be found in
Contact between a sphere and a half-space
An elastic sphere of radius indents an elastic half-space where total deformation is , causing a contact area of radius
The applied force is related to the displacement by
where
and , are the elastic moduli and , the Poisson's ratios associated with each body.
The distribution of normal pressure in the contact area as a function of distance from the center of the circle is
where is the maximum contact pressure given by
The radius of the circle is related to the applied load by the equation
The total deformation is related to the maximum contact pressure by
The maximum shear stress occurs in the interior at for .
Contact between two spheres
For contact between two spheres of radii and , the area of contact is a circle of radius . The equations are the same as for a sphere in contact with a half plane except that the effective radius is defined as
Contact between two crossed cylinders of equal radius This is equivalent to contact between a sphere of radius and a plane.
Contact between a rigid cylinder with flat end and an elastic half-space
If a rigid cylinder is pressed into an elastic half-space, it creates a pressure distribution described by
where is the radius of the cylinder and
The relationship between the indentation depth and the normal force is given by
Contact between a rigid conical indenter and an elastic half-space
In the case of indentation of an elastic half-space of Young's modulus using a rigid conical indenter, the depth of the contact region and contact radius are related by
with defined as the angle between the plane and the side surface of the cone. The total indentation depth is given by:
The total force is
The pressure distribution is given by
The stress has a logarithmic singularity at the tip of the cone.
Contact between two cylinders with parallel axes
In contact between two cylinders with parallel axes, the force is linearly proportional to the length of cylinders L and to the indentation depth d:
The radii of curvature are entirely absent from this relationship. The contact radius is described through the usual relationship
with
as in contact between two spheres. The maximum pressure is equal to
Bearing contact
The contact in the case of bearings is often a contact between a convex surface (male cylinder or sphere) and a concave surface (female cylinder or sphere: bore or hemispherical cup).
Method of dimensionality reduction
Some contact problems can be solved with the method of dimensionality reduction (MDR). In this method, the initial three-dimensional system is replaced with a contact of a body with a linear elastic or viscoelastic foundation (see fig.). The properties of one-dimensional systems coincide exactly with those of the original three-dimensional system, if the form of the bodies is modified and the elements of the foundation are defined according to the rules of the MDR. MDR is based on the solution to axisymmetric contact problems first obtained by Ludwig Föppl (1941) and Gerhard Schubert (1942)
However, for exact analytical results, it is required that the contact problem is axisymmetric and the contacts are compact.
Hertzian theory of non-adhesive elastic contact The classical theory of contact focused primarily on non-adhesive contact where no tension force is allowed to occur within the contact area, i.e., contacting bodies can be separated without adhesion forces. Several analytical and numerical approaches have been used to solve contact problems that satisfy the no-adhesion condition. Complex forces and moments are transmitted between the bodies where they touch, so problems in contact mechanics can become quite sophisticated. In addition, the contact stresses are usually a nonlinear function of the deformation. To simplify the solution procedure, a frame of reference is usually defined in which the objects (possibly in motion relative to one another) are static. They interact through surface tractions (or pressures/stresses) at their interface.
As an example, consider two objects which meet at some surface in the (,)-plane with the -axis assumed normal to the surface. One of the bodies will experience a normally-directed pressure distribution and in-plane surface traction distributions and over the region . In terms of a Newtonian force balance, the forces:
must be equal and opposite to the forces established in the other body. The moments corresponding to these forces:
are also required to cancel between bodies so that they are kinematically immobile.
Assumptions in Hertzian theory The following assumptions are made in determining the solutions of Hertzian contact problems: The strains are small and within the elastic limit. The surfaces are continuous and non-conforming (implying that the area of contact is much smaller than the characteristic dimensions of the contacting bodies). Each body can be considered an elastic half-space. The surfaces are frictionless. Additional complications arise when some or all these assumptions are violated and such contact problems are usually called non-Hertzian.
Analytical solution techniques