Contact Mechanics By K L Johnson.pdf

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Li, Q., Argatov, I.I., Popov, V.L., 2018, Onset of detachment in adhesive contact of an elastic half-space and flat-ended punches with noncircular shape: Analytic estimations and comparison with numeric analysis, Journal of Physics D: Applied Physics, 51(14), 145601.

[10], which defines a general relation between the adhesive solution and that without adhesion. Strictly, the argument requires that the contact area be such as to give a uniform stress intensity factor around the perimeter as in axisymmetric problems, but it might reasonably be expected to give good approximations in other cases.

Sketch of traction distributions in DMT and JKR type of solution. In DMT, the contact area coincides with that obtained from the non-adhesive problem, the adhesive tractions are thus confined in the separated zone. In a JKR solution, both compressive and adhesive tractions are exchanged within the contact area. (Online version in colour.)

by assuming that unloading is approximately defined by the Hertzian analysis from load P0 and contact radius a0. Using this value in the JKR solution (2.3) he then obtained an increased pull-off force

In discussing figure 8, we noted that solutions even of the smooth contact problem are very sensitive to numerical perturbations in and just outside the unstable range. This effect is of course even more pronounced if the surfaces have random roughness, provided this is not of such large amplitude as to suppress the instability. The patterns developed analogous to those of figure 7 are now irregular and significant differences are observed between different realizations of the same roughness statistics.

Although contact splitting provides an explanation for adhesion enhancement in some bio-applications, we are still far for a complete understanding of how adhesion is effectively controlled at the interface. While Artz et al. [95] showed a strong correlation in flies, beetles, spiders and lizards between the areal density of attachment hairs and the body mass, later Peattie & Full [96] surveyed 81 species with hierarchical fibrillar structures and found no such correlation when the data are analysed within the same taxa (see also [97]). Bartlett et al. [98] proposed a more general criterion based on total energy minimization, which predicted that the maximum adhesion force would scale according to the relationship

Barreau and co-authors [105] have shown that one strategy for better adaptation to surface roughness is to use a small pillar diameter to take advantage of the contact splitting effect, but not smaller than the mean spacing between local peaks on the substrate, insofar as this can be defined for a multi-scale surface. The problem is that when diameter is too small compared with this criterion, bending and buckling events occur, storing strain energy, which effectively reduce adhesion.

An alternative strategy is to design the contact geometry to include mushroom or funnel-shaped tips. These shapes can increase the pull-off stress even by an order of magnitude because of the shift from the severe singular edge stresses of the flat punch to the more uniform stresses with almost no singularity at the edge of the mushroom flaps [106]. More precisely, the singular stress multiplier is determined by the thickness of the flaps, rather than the diameter of the fibril.

Classical contact mechanics models for adhesion and friction interaction date back to the seminal work of Savkoor & Briggs [115] who extended the JKR solution for a smooth sphere to friction. They assumed a singular stress field also in tangential direction (mode II), and combined the energy release rate G as

Contact mechanics is the study of the deformation of solids that touch each other at one or more points.[1][2] 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\"[3] (\"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.

Classical contact mechanics is most notably associated with Heinrich Hertz.[3][4] 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.[5] This theory was rejected by Boris Derjaguin and co-workers[6] who proposed a different theory of adhesion[7] in the 1970s. The Derjaguin model came to be known as the DMT (after Derjaguin, Muller and Toporov) model,[7] 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[8] and later Maugis[6][9] 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.[10][11] 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)[12] 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),[13] Bush (1975),[14] and Persson (2002).[15] 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.

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 [16]

For contact between two spheres of radii R 1 {\\displaystyle R_{1}} and R 2 {\\displaystyle R_{2}} , the area of contact is a circle of radius a {\\displaystyle a} . The equations are the same as for a sphere in contact with a half plane except that the effective radius R {\\displaystyle R} is defined as [4]

In the case of indentation of an elastic half-space of Young's modulus E {\\displaystyle E} using a rigid conical indenter, the depth of the contact region ϵ {\\displaystyle \\epsilon } and contact radius a {\\displaystyle a} are related by[17]

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.[19][20] MDR is based on the solution to axisymmetric contact problems first obtained by Ludwig Föppl (1941) and Gerhard Schubert (1942)[21] 153554b96e

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