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Contact mechanics is the study of the deformation of solids that touch each other at one or more points. 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. Central aspects in contact mechanics are the pressures and adhesion acting perpendicular to the contacting bodies' surfaces (known as the normal direction) and the frictional stresses acting tangentially between the surfaces. This page focuses mainly on the normal direction, i.e. on frictionless contact mechanics. Frictional contact mechanics is discussed separately.
Contents
- History
- Classical solutions for non adhesive elastic contact
- Contact between a sphere and a half space
- Contact between two spheres
- Contact between two crossed cylinders of equal radius R displaystyle R
- Contact between a rigid cylinder with flat ended and an elastic half space
- Contact between a rigid conical indenter and an elastic half space
- Contact between two cylinders with parallel axes
- Bearing contact
- The Method of Dimensionality Reduction
- Hertzian theory of non adhesive elastic contact
- Assumptions in Hertzian theory
- Analytical solution techniques
- Point contact on a 2D half plane
- Line contact on a 2D half plane
- Point contact on a 3D half space
- Numerical solution techniques
- Contact between rough surfaces
- Adhesive contact between elastic bodies
- Bradley model of rigid contact
- Johnson Kendall Roberts JKR model of elastic contact
- Derjaguin Muller Toporov DMT model of elastic contact
- Tabor parameter
- Maugis Dugdale model of elastic contact
- Carpick Ogletree Salmeron COS model
- References
Contact mechanics is part of Mechanical engineering; it provides necessary information for the safe and energy efficient design of technical systems and for the study of tribology and indentation hardness. Principles of contacts mechanics can be applied in areas 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 1882 with the publication of the paper "On the contact of elastic solids" ("Ueber die Berührung fester elastischer Körper") by Heinrich Hertz. Hertz was attempting 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. 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 Johnson, Kendall, and 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 DMT (after Derjaguin, Muller and Toporov) model, and the Johnson et al. model came to be known as the JKR (after Johnson, Kendall and Roberts) model for adhesive elastic contact. This rejection proved to be instrumental in the development of the Tabor and later 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 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 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 Greenwood and Williamson (1966), Bush (1975), and 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 (i.e., pressure, 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.
Contact between a sphere and a half-space
An elastic sphere of radius
The applied force
where
and
The distribution of normal pressure in the contact area as a function of distance from the center of the circle is
where
The radius of the circle is related to the applied load
The depth of indentation
The maximum shear stress occurs in the interior at
Contact between two spheres
For contact between two spheres of radii
Contact between two crossed cylinders of equal radius R {displaystyle R}
This is equivalent to contact between a sphere of radius
Contact between a rigid cylinder with flat-ended and an elastic half-space
If a rigid cylinder is pressed into an elastic half-space, it creates a pressure distribution described by
where
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
with
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 indentation depth:
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).
The Method of Dimensionality Reduction
Some contact problems can be solved with the Method of Dimensionality Reduction. 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. 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
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:
Additional complications arise when some or all these assumptions are violated and such contact problems are usually called non-Hertzian.
Analytical solution techniques
Analytical solution methods for non-adhesive contact problem can be classified into two types based on the geometry of the area of contact. A conforming contact is one in which the two bodies touch at multiple points before any deformation takes place (i.e., they just "fit together"). A non-conforming contact is one in which the shapes of the bodies are dissimilar enough that, under zero load, they only touch at a point (or possibly along a line). In the non-conforming case, the contact area is small compared to the sizes of the objects and the stresses are highly concentrated in this area. Such a contact is called concentrated, otherwise it is called diversified.
A common approach in linear elasticity is to superpose a number of solutions each of which corresponds to a point load acting over the area of contact. For example, in the case of loading of a half-plane, the Flamant solution is often used as a starting point and then generalized to various shapes of the area of contact. The force and moment balances between the two bodies in contact act as additional constraints to the solution.
Point contact on a (2D) half-plane
A starting point for solving contact problems is to understand the effect of a "point-load" applied to an isotropic, homogeneous, and linear elastic half-plane, shown in the figure to the right. The problem may be either plane stress or plane strain. This is a boundary value problem of linear elasticity subject to the traction boundary conditions:
where
for some point,
Line contact on a (2D) half-plane
Normal loading over a region ( a , b )
Suppose, rather than a point load
Shear loading over a region ( a , b )
The same principle applies for loading on the surface in the plane of the surface. These kinds of tractions would tend to arise as a result of friction. The solution is similar the above (for both singular loads
These results may themselves be superposed onto those given above for normal loading to deal with more complex loads.
Point contact on a (3D) half-space
Analogously to the Flamant solution for the 2D half-plane, fundamental solutions are known for the linearly elastic 3D half-space as well. These were found by Boussinesq for a concentrated normal load and by Cerruti for a tangential load. See the section on this in Linear elasticity.
Numerical solution techniques
Distinctions between conforming and non-conforming contact do not have to be made when numerical solution schemes are employed to solve contact problems. These methods do not rely on further assumptions within the solution process since they base solely on the general formulation of the underlying equations . Besides the standard equations describing the deformation and motion of bodies two additional inequalities can be formulated. The first simply restricts the motion and deformation of the bodies by the assumption that no penetration can occur. Hence the gap
where
At locations where there is contact between the surfaces the gap is zero, i.e.
These conditions are valid in a general way. The mathematical formulation of the gap depends upon the kinematics of the underlying theory of the solid (e.g., linear or nonlinear solid in two- or three dimensions, beam or shell model). By restating the normal stress
After discretization the linear elastic contact mechanics problem can be stated in standard Linear Complementarity Problem (LCP) form.
Contact between rough surfaces
When two bodies with rough surfaces are pressed into each other, the true contact area
In contact between a "random rough" surface and an elastic half-space, the true contact area is related to the normal force
with
can be reasonably estimated as half of the effective elastic modulus
For the situation where the asperities on the two surfaces have a Gaussian height distribution and the peaks can be assumed to be spherical, the average contact pressure is sufficient to cause yield when
The Greenwood-Williamson model requires knowledge of two statistically dependent quantities; the standard deviation of the surface roughness and the curvature of the asperity peaks. An alternative definition of the plasticity index has been given by Mikic. Yield occurs when the pressure is greater than the uniaxial yield stress. Since the yield stress is proportional to the indentation hardness
In this definition
In both the Greenwood-Williamson and Mikic models the load is assumed to be proportional to the deformed area. Hence, whether the system behaves plastically or elastically is independent of the applied normal force.
Adhesive contact between elastic bodies
When two solid surfaces are brought into close proximity, they experience attractive van der Waals forces. Bradley's van der Waals model provides a means of calculating the tensile force between two rigid spheres with perfectly smooth surfaces. The Hertzian model of contact does not consider adhesion possible. However, in the late 1960s, several contradictions were observed when the Hertz theory was compared with experiments involving contact between rubber and glass spheres.
It was observed that, though Hertz theory applied at large loads, at low loads
This indicated that adhesive forces were at work. The Johnson-Kendall-Roberts (JKR) model and the Derjaguin-Muller-Toporov (DMT) models were the first to incorporate adhesion into Hertzian contact.
Bradley model of rigid contact
It is commonly assumed that the surface force between two atomic planes at a distance
where
The Bradley model applied the Lennard-Jones potential to find the force of adhesion between two rigid spheres. The total force between the spheres is found to be
where
The two spheres separate completely when the pull-off force is achieved at
Johnson-Kendall-Roberts (JKR) model of elastic contact
To incorporate the effect of adhesion in Hertzian contact, Johnson, Kendall, and Roberts formulated the JKR theory of adhesive contact using a balance between the stored elastic energy and the loss in surface energy. The JKR model considers the effect of contact pressure and adhesion only inside the area of contact. The general solution for the pressure distribution in the contact area in the JKR model is
Note that in the original Hertz theory, the term containing
where
The approach distance between the two spheres is given by
The Hertz equation for the area of contact between two spheres, modified to take into account the surface energy, has the form
When the surface energy is zero,
The tensile load at which the spheres are separated, i.e.,
This force is also called the pull-off force. Note that this force is independent of the moduli of the two spheres. However, there is another possible solution for the value of
If we define the work of adhesion as
where
The tensile load at separation is
and the critical contact radius is given by
The critical depth of penetration is
Derjaguin-Muller-Toporov (DMT) model of elastic contact
The Derjaguin-Muller-Toporov (DMT) model is an alternative model for adhesive contact which assumes that the contact profile remains the same as in Hertzian contact but with additional attractive interactions outside the area of contact.
The radius of contact between two spheres from DMT theory is
and the pull-off force is
When the pull-off force is achieved the contact area becomes zero and there is no singularity in the contact stresses at the edge of the contact area.
In terms of the work of adhesion
and
Tabor parameter
In 1977, Tabor showed that the apparent contradiction between the JKR and DMT theories could be resolved by noting that the two theories were the extreme limits of a single theory parametrized by the Tabor parameter (
where
Subsequently Derjaguin and his collaborators by applying Bradley's surface force law to an elastic half space, confirmed that as the Tabor parameter increases, the pull-off force falls from the Bradley value
Maugis-Dugdale model of elastic contact
Further improvement to the Tabor idea was provided by Maugis who represented the surface force in terms of a Dugdale cohesive zone approximation such that the work of adhesion is given by
where
In the Maugis-Dugdale 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. Hertz contact is assumed in the region
where the Hertz contact force
The penetration due to elastic compression is
The vertical displacement at
and the separation between the two surfaces at
The surface traction distribution due to the adhesive Dugdale stress is
The total adhesive force is then given by
The compression due to Dugdale adhesion is
and the gap at
The net traction on the contact area is then given by
Non-dimensionalized values of
In addition, Maugis proposed a parameter
where the step cohesive stress
Zheng and Yu suggested another value for the step cohesive stress
to match the Lennard-Jones potential, which leads to
Then the net contact force may be expressed as
and the elastic compression as
The equation for the cohesive gap between the two bodies takes the form
This equation can be solved to obtain values of
Carpick-Ogletree-Salmeron (COS) model
The Maugis-Dugdale model can only be solved iteratively if the value of
where
The case