Contact mechanics

Stresses in a contact area under simultaneous loading by a normal force and a tangential force. The stresses were made visible by means of stress optics.

The contact mechanism deals with the calculation of elastic , viscoelastic or plastic bodies in static or dynamic contact.

Contact mechanics is a fundamental engineering discipline that is indispensable for the safe and energy-saving design of technical systems . It is important, for example, in wheel-rail systems, clutches , brakes , tires , plain and roller bearings , combustion engines , joints , seals , forming , material processing , ultrasonic welding , electrical contacts and many others. Your tasks range from proof of the strength of contact and connecting elements to influencing friction and wear through lubrication or material design to applications in micro and nano system technology.

history

Classic contact mechanics is primarily associated with Heinrich Hertz . In 1882, Hertz solved the problem of contact between two elastic bodies with curved surfaces (see also the article Hertzian pressure ). The Hertzian contact theory forms a basis of contact mechanics even today. The Hertz power law is

${\displaystyle F=H\cdot \xi ^{3/2}}$

with the deformation path and the constant , which depends on "the shape of the surfaces and the elasticity conditions in the immediate vicinity of the point of impact". ${\displaystyle \xi }$ ${\displaystyle H}$

Further early analytical work on this subject can be traced back to Joseph Boussinesq and V. Cerruti.

It wasn't until almost a century later that Kenneth L. Johnson , Kevin Kendall and Alan D. Roberts found a solution similar to that of Hertz for an adhesive contact (JKR theory).

Another advance in our knowledge of contact mechanics came in the middle of the 20th century and is associated with the names Bowden and Tabor. You were the first to point out the importance of the roughness of the contacting bodies. Due to the roughness, the true contact area between friction partners is typically orders of magnitude smaller than the apparent area. This insight suddenly changed the direction of many tribological investigations. The work of Bowden and Tabor initiated a number of theories on the contact mechanics of rough surfaces.

The work of John F. Archard (1957) should be mentioned as pioneering work in this area , who came to the conclusion that the contact area is approximately proportional to the normal force even in contact with elastic, rough surfaces . Other important contributions are associated with the names JA Greenwood and JBP Williamson (1966), Bush (1975) and Bo NJ Persson (2002). The main result of this work is that the true contact area on rough surfaces is roughly proportional to the normal force, while the conditions in individual micro-contacts (pressure, size of the micro-contact) depend only slightly on the load.

Today, many tasks in contact mechanics are processed with simulation programs based on the finite element method or the boundary element method. There are a large number of scientific articles on this, some of which can be found in the books by Laursen (2002) and Wriggers (2006) in addition to the fundamentals of numerical contact mechanics.

Classic contact tasks

Contact between a sphere and an elastic half-space

Contact between a sphere and an elastic half-space. The deformation path is indicated here with . ${\displaystyle \xi }$ ${\displaystyle d}$

If an elastic sphere with the radius is pressed into an elastic half-space by the amount ( deformation path or indentation depth ), a contact area with the radius is formed . The force required for this is the same ${\displaystyle R}$ ${\displaystyle \xi }$ ${\displaystyle a={\sqrt {R\xi }}}$

${\displaystyle F={\frac {4}{3}}E^{*}R^{1/2}\xi ^{3/2}}$

in which

${\displaystyle {\frac {1}{E^{*}}}={\frac {1-\nu _{1}^{2}}{E_{1}}}+{\frac {1-\nu _{2}^{2}}{E_{2}}}}$

${\displaystyle E_{1}}$and here are the moduli of elasticity as well as and the Poisson's numbers of both bodies. ${\displaystyle E_{2}}$ ${\displaystyle \nu _{1}}$ ${\displaystyle \nu _{2}}$

Contact between two elastic balls

Contact between two balls

If two spheres with the radii and are in contact, these equations continue to apply with the radius according to ${\displaystyle R_{1}}$ ${\displaystyle R_{2}}$ ${\displaystyle R}$

${\displaystyle {\frac {1}{R}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}}$

The pressure distribution in the contact area is given by

${\displaystyle p=p_{0}\left(1-{\frac {r^{2}}{a^{2}}}\right)^{1/2}}$

With

${\displaystyle p_{0}={\frac {2}{\pi }}E^{*}\left({\frac {\xi }{R}}\right)^{1/2}}$

The maximum shear stress is located inside, for at . ${\displaystyle \nu =0{,}33}$ ${\displaystyle z\approx 0{,}49a}$

Contact between two crossed cylinders with equal radii R

Contact between two crossed cylinders with equal radii

is equivalent to the contact between a sphere with the radius and a plane (see above). ${\displaystyle R}$

Contact between a rigid cylinder and an elastic half-space

Contact between a rigid cylindrical indenter and an elastic half-space

If a rigid, cylindrical punch with the radius is pressed into an elastic half-space , the pressure distribution is through ${\displaystyle a}$

${\displaystyle p=p_{0}\left(1-{\frac {r^{2}}{a^{2}}}\right)^{-1/2}}$

given with

${\displaystyle p_{0}={\frac {1}{\pi }}E^{*}{\frac {\xi }{a}}}$

The relationship between the indentation depth and the normal force is as follows

${\displaystyle F=2aE^{*}\xi }$

Contact between a rigid conical indenter and the elastic half-space

Contact between a cone and an elastic half-space

When an elastic half-space is indented by a rigid conical indenter, the indentation depth and the contact radius are determined by the relationship

${\displaystyle d={\frac {\pi }{2}}a\tan \theta }$

given. is the angle between the plane and the side face of the cone. The pressure distribution has the form ${\displaystyle \theta }$

${\displaystyle p(r)=-{\frac {E\xi }{\pi a\left(1-\nu ^{2}\right)}}\ln \left({\frac {a}{r}}+{\sqrt {\left({\frac {a}{r}}\right)^{2}-1}}\right)}$

The voltage has a logarithmic singularity at the tip of the cone (in the center of the contact area) . The total force is calculated as

${\displaystyle F={\frac {2}{\tan {\theta }\cdot \pi }}E\xi ^{2}}$

Contact between two cylinders with parallel axes

Contact between two cylinders with parallel axes

In the case of contact between two cylinders with parallel axes, the force is linearly proportional to the indentation depth:

${\displaystyle F={\frac {\pi }{4}}E^{*}L\xi }$.

The radius of curvature does not appear at all in this regard. Half the contact width is given by the same relationship

${\displaystyle a={\sqrt {R\xi }}}$

With

${\displaystyle {\frac {1}{R}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}}$

given, as in the contact between two balls. The maximum pressure is the same

${\displaystyle p_{0}=\left({\frac {E^{*}F}{\pi LR}}\right)^{1/2}}$

Contact between rough surfaces

When two bodies with rough surfaces are pressed against one another, the real contact area is initially much smaller than the apparent area . In the case of a contact between a “randomly rough” surface and an elastic half-space, the real contact area is proportional to the normal force and is given by the equation ${\displaystyle A}$ ${\displaystyle A_{0}}$ ${\displaystyle F}$

${\displaystyle A={\frac {\kappa }{E^{*}h'}}F}$

where is the root mean square value of the slope of the surface and . The mean pressure in the true contact area ${\displaystyle h'}$ ${\displaystyle \kappa \approx 2}$

${\displaystyle \sigma ={\frac {F}{A}}\approx {\frac {1}{2}}E^{*}h'}$

is calculated as a good approximation as half of the effective elastic modulus multiplied by the root mean square value of the slope of the surface profile. Is this pressure greater than the hardness of the material and thus ${\displaystyle E^{*}}$ ${\displaystyle h'}$ ${\displaystyle \sigma _{0}}$

${\displaystyle \Psi ={\frac {E^{*}h'}{\sigma _{0}}}>2}$

the micro-roughness is completely in the plastic state. For the surface behaves elastically on contact. The size was introduced by Greenwood and Williamson and is called the plasticity index. The fact whether the system behaves elastically or plastically does not depend on the normal force applied. ${\displaystyle \Psi <{\tfrac {2}{3}}}$ ${\displaystyle \Psi }$

Adhesive contact

The phenomenon of adhesion is most easily observed when a solid body comes into contact with a very soft elastic body, for example a jelly . As a result of the Van der Waals forces, an adhesive neck is created between the bodies. So that the bodies can be taken apart again, it is necessary to apply a minimum force, which is known as the adhesive force. Adhesion can be of technological interest, for example in an adhesive connection, and a disruptive factor, such as when elastomer valves open quickly. Adhesive force between a parabolic rigid body and an elastic half-space was found in 1971 by Johnson, Kendall and Roberts. She is the same

${\displaystyle F_{A}=(3/2)\pi \gamma R}$

where is the separation energy per unit area and the radius of curvature of the body. ${\displaystyle \gamma }$ ${\displaystyle R}$

The adhesive force of a flat rigid punch with the radius was also found in 1971 by Kendall: ${\displaystyle a}$

${\displaystyle F_{A}={\sqrt {8\pi a^{3}E^{*}\gamma }}}$

More complex shapes begin to tear off from the "edges" of the contact.

Method of dimensionality reduction

Contact between a sphere and an elastic half-space and the one-dimensional equivalent model. The deformation path is indicated here with . ${\displaystyle \xi }$ ${\displaystyle d}$

Some contact problems can be solved with the dimensional reduction method. In this method, the original three-dimensional system is replaced by contact with an elastic or viscoelastic Winkler bedding (see picture). The macroscopic contact properties exactly match those of the original system, provided that the parameters of the Winkler bedding and the shape of the bodies are selected according to the rules of the method. The method of dimension reduction provides analytically exact results for axially symmetrical systems whose contact area is compact. The applicability to real, randomly rough surfaces, such as machined metal or road surfaces, is controversial.

literature

• Kenneth L. Johnson : Contact mechanics. Cambridge, 1985, ISBN 978-0-521-25576-9 .
• Valentin L. Popov: Contact Mechanics and Friction. A text and application book from nanotribology to numerical simulation . Springer, 2009, ISBN 978-3-540-88836-9 .
• Ian Sneddon : The Relation between Load and Penetration in the Axisymmetric Boussinesq Problem for a Punch of Arbitrary Profile. Int. J. Eng. Sci. 3, 1965, pp. 47-57, doi: 10.1016 / 0020-7225 (65) 90019-4 .
• Sangil Hyuna, Mark O. Robbins: Elastic contact between rough surfaces: Effect of roughness at large and small wavelengths. Tribology International 40, 2007, pp. 1413-1422, doi: 10.1016 / j.triboint.2007.02.003 .
• Tod A. Laursen: Computational Contact and Impact Mechanics: Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis . Springer, 2003, ISBN 978-3-540-42906-7 .
• Peter Wriggers: Computational Contact Mechanics . Springer, 2006, ISBN 978-3-540-32608-3 .
• Valentin L. Popov: Method of reduction of dimensionality in contact and friction mechanics: A linkage between micro and macro scales . Friction 1, 2013, pp. 1–22, doi: 10.1007 / s40544-013-0005-3 .

Individual evidence

1. ↑ Kristof Heck, Simon Huppertz: Jugend-forscht 2014, Physics: Investigations on the shot put pendulum chain and on the Hertzian contact theory . ( mgm-monschau.de9 [PDF; 3.3 MB ; accessed on December 29, 2016]).
2. ↑ ^{a b c} K. L. Johnson, K. Kendall, AD Roberts: Surface energy and the contact of elastic solids . In: Proc. R. Soc. Lond. A . tape 324 , no. 1558 , September 8, 1971, ISSN  0080-4630 , p. 301–313 , doi : 10.1098 / rspa.1971.0141 ( royalsocietypublishing.org [accessed December 25, 2017]).
3. ↑ K. Kendall: The adhesion and surface energy of elastic solids . In: Journal of Physics D: Applied Physics . tape 4 , no. 8 , 1971, ISSN  0022-3727 , pp. 1186 , doi : 10.1088 / 0022-3727 / 4/8/320 ( iop.org [accessed December 25, 2017]).
4. ↑ Valentin L. Popov, Roman Pohrt, Qiang Li: Strength of adhesive contacts: Influence of contact geometry and material gradients . In: Friction . tape 5 , no. 3 , September 1, 2017, ISSN  2223-7690 , p. 308-325 , doi : 10.1007 / s40544-017-0177-3 ( springer.com [accessed December 25, 2017]).
5. ^ Friction Physics: Science friction: Adhesion of complex shapes. December 6, 2017. Retrieved December 25, 2017 . 
6. ↑ Valentin L. Popov, Markus Heß: Method of dimension reduction in contact mechanics and friction . Springer, 2013, ISBN 978-3-642-32673-8 .
7. ↑ Bo NJ Persson: Contact Mechanics for Randomly Rough Surfaces: On the Validity of the Method of Reduction of Dimensionality . Tribology Letters 58, 2015, doi: 10.1007 / s11249-015-0498-1 .
8. ↑ VL Popov .: Comment on “Contact Mechanics for Randomly Rough Surfaces: On the Validity of the Method of Reduction of Dimensionality” by Bo Persson in Tribology Letters . Tribology Letters 60, 2016, pp. 1-7, doi: 10.1007 / s11249-015-0608-0 .
9. ^ Iakov A. Lyashenko, Lars Pastewka, Bo NJ Persson: On the validity of the method of reduction of dimensionality: area of ​​contact, average interfacial separation and contact stiffness . Tribology Letters 52, 2013, doi: 10.1007 / s11249-013-0208-9 ( arxiv : 1303.0965 ).
10. ↑ Li et al .: A Reply to the Comment by IA Lyashenko et al. Phys Rev Lett 111, 2013, doi: 10.1103 / PhysRevLett.111.189402 .

Web links

• Contact mechanics (English calculator)