Also see Scribd's Privacy Policy. For Scribd Unlimited payment and billing information, cancellation information, and restrictions, see. The twin paradox. Addition of velocities. The relativistic Doppler effect. 7.3 Relativistic dynamics. The momentum-energy four-vector. Conservation of energy. Mass and energy. Example inelastic collision. The principle of equivalence. Lagrangian and Hamiltonian formulations. 7.4 Accelerated Systems.
Stresses in a contact area loaded simultaneously with a normal and a tangential force. Stresses were made visible using. Contact mechanics is the study of the of that touch each other at one or more points. A central distinction in contact mechanics is between acting to the contacting bodies' surfaces (known as the ) and stresses acting between the surfaces. This page focuses mainly on the normal direction, i.e.
On frictionless contact mechanics. Is discussed separately. Normal stresses are caused by applied forces and by the present on surfaces in close contact even if they are clean and dry.
Contact mechanics is part of mechanical. The physical and mathematical formulation of the subject is built upon the and and focuses on computations involving, and bodies in or contact. Contact mechanics provides necessary information for the safe and energy efficient design of technical systems and for the study of,. Principles of contacts mechanics are implemented towards applications such as locomotive wheel-rail contact, devices, systems, seals, metal forming, and many others. Current challenges faced in the field may include of contact and coupling members and the influence of and material on.
Applications of contact mechanics further extend into the - and realm. The original work in contact mechanics dates back to 1882 with the publication of the paper 'On the contact of elastic solids'. Hertz was attempting to understand how the optical properties of multiple, stacked might change with the 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 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 life in bearings, gears, and any other bodies where two surfaces are in contact. When a sphere is pressed against an elastic material, the contact area increases. 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 and, 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, Kendall, and Roberts found a similar solution for the case of contact. This theory was rejected by 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 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. 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. Contact between two spheres. 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 are highly concentrated in this area. Such a contact is called concentrated, otherwise it is called diversified.
A common approach in is to 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, the 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.