Math 2.0

Mechanics of deformable solids Definition

Contact mechanics

Builds on Solid mechanicsCauchy stress tensorSignorini problem

Definition

Contact mechanics is the part of solid mechanics that studies deformable bodies which may touch along part of their boundaries. On the possible contact surface Γc\Gamma_c let gg be the gap between the bodies measured along the normal nn. Let p=−n⋅σnp = -n \cdot \sigma n be the contact pressure, where σ\sigma is the Cauchy stress. A contact problem is a boundary value problem of solid mechanics with the following conditions on Γc\Gamma_c:

  1. non-penetration: g≥0g \ge 0,
  2. no adhesion: p≥0p \ge 0,
  3. complementarity: g p=0g \, p = 0, so pressure acts only where the gap is closed,
  4. a law for the tangential traction, for example zero traction (frictionless contact) or a friction law such as Coulomb's.

Remarks

Example: a linear elastic body pressed against a rigid flat obstacle without friction. Its equilibrium equations together with clauses 1 to 4, with zero tangential traction, form the Signorini problem.

Non-example: a body glued to a rigid support. There the displacement is prescribed on the whole surface and the traction may pull as well as push. So clauses 1 to 3 are replaced by an equality, and the problem is an ordinary boundary value problem.

Clauses 1 to 3 are the Signorini conditions, also called the Hertz–Signorini–Moreau conditions. Adhesive contact, as in the JKR model, relaxes clause 2.

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Cauchy stress tensor, concept, no evidence levelCauchy stress tensorconceptClosed set, concept, no evidence levelClosed setconceptCoercive bilinear form, concept, no evidence levelCoercive bilinear formconceptContinuous dual space, concept, no evidence levelContinuous dual spaceconceptConvex set, concept, no evidence levelConvex setconceptHilbert space, concept, no evidence levelHilbert spaceconceptLamé equations of elasticity, concept, no evidence levelLamé equations of ela…conceptMonotone operator, concept, no evidence levelMonotone operatorconceptSobolev space, concept, no evidence levelSobolev spaceconceptSolid mechanics, concept, no evidence levelSolid mechanicsconceptLions–Stampacchia theorem, statement, no evidence levelLions–Stampacchia the…statementVariational inequality, problem, no evidence levelVariational inequalityproblemSignorini problem, problem, no evidence levelSignorini problemproblemContact mechanics, definition, no evidence levelContact mechanicsdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cauchy stress tensor concept
  2. Closed set concept
  3. Coercive bilinear form concept
  4. Continuous dual space concept
  5. Convex set concept
  6. Hilbert space concept
  7. Lamé equations of elasticity concept
  8. Monotone operator concept
  9. Sobolev space concept
  10. Solid mechanics concept
  11. Lions–Stampacchia theorem statement
  12. Variational inequality problem
  13. Signorini problem problem
  14. Contact mechanics definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Contact_mechanics
About this page
TagC68BDNFQ7N
URIm2://C68BDNFQ7N
Kinddefinition
WritingW0 stub
MSC74M15, 74M10, 74B05
DefinesContact mechanics
UsesSolid mechanics, Cauchy stress tensor, Signorini problem
Changed3 November 2026

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