Contact Mechanics and Elements of Tribology
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1 Contact Mechanics and Elements of Tribology Foreword Vladislav A. Yastrebov MINES ParisTech, PSL Research University, Centre des Matériaux, CNRS UMR 7633, Evry, Centre des Matériaux February 8, 2016
2 Outline Acquaintance Questionnaire Course content Historical remark Complexity Notations
3 Questionnaire V.A. Yastrebov 3/10 Please fill in the questionnaire Message to DMS students : Rank all suggested topics 1 for your report in the order of decreasing attractiveness Before the lunch one topic will be assigned to everybody On Friday (12/02) you will have 10 minutes to present briefly your topic to other students If you are not a DMS student but would like to follow, you are welcome Basava, Andrei and Paolo will also present rapidly their research projects on 12/02 By Friday (19/02) you will need to send me your report on your topic ( 10 pages in Français/English) Criteria of evaluation : no plagiarism, scientifically sound report 1 or suggest a different one but not related to your master project
4 Course content V.A. Yastrebov 4/10 Lectures : 1 Motivation : industrial applications 2 Contact Mechanics I & II 3 Contact at small scales : surface roughness 4 Computational Contact Mechanics 5 Contact rheology & friction laws 6 Elements of tribology : friction, adhesion, wear 7 Wear and fretting (by Henry Proudhon) 8 Multiphysical problems in contact 9 Your presentations
5 Course content V.A. Yastrebov 5/10 Lectures : 1 Motivation : industrial applications 2 Contact Mechanics I & II 3 Contact at small scales : surface roughness 4 Computational Contact Mechanics 5 Contact rheology & friction laws 6 Elements of tribology : friction, adhesion, wear 7 Wear and fretting (by Henry Proudhon) 8 Multiphysical problems in contact 9 Your presentations Practice : 1 Chalk and blackboard : Boussinesq, Cerruti, Flamant 2 Chalk and blackboard : Hertz contact 3 Computer : rough contact 4 Computer : frictionless and frictional cylindrical contact 5 Computer : elasto-plastic spherical indentation
6 Historical remark V.A. Yastrebov 6/10 Use of friction and overcoming of friction Frictional heat - lighting of fire > years ago. Ancient Egypt - lubrication of surfaces with oil > years ago.
![friction laws and many other tribological topics Isaac Newton [1687] Newton s third law for bodies](/docs-images/81/82610294/images/7-2.jpg "interaction From Leonardo da Vinci s notebook Guillaume Amontons [1699] rediscovered friction laws")
7 Historical remark V.A. Yastrebov 7/10 First scientific studies on contact and friction Leonardo da Vinci [ ] first friction laws and many other tribological topics Isaac Newton [1687] Newton s third law for bodies interaction From Leonardo da Vinci s notebook Guillaume Amontons [1699] rediscovered friction laws Leonhard Euler [ ] roughness theory of friction Roughness theory of friction
![[1789] friction independence on sliding velocity and roughness, the influence of the time](/docs-images/81/82610294/images/8-1.jpg "of repose Heinrich Hertz [1881-1882] the first study on contact of deformable solids")
8 Historical remark V.A. Yastrebov 8/10 First scientific studies on contact and friction Charles-Augustin de Coulomb [1789] friction independence on sliding velocity and roughness, the influence of the time of repose Heinrich Hertz [ ] the first study on contact of deformable solids Photoelasticity analysis of Hertz contact problem (fringes correspond to max shear stresses) Holm [1938], Ernst and Merchant [1940], Bowden and Tabor [1942] difference between apparent and real contact areas, adhesion theory. Apparent and real areas of contact
9 Complexity V.A. Yastrebov 9/10 Zoom on the contact interface Between different heterogeneous materials with specific rough surfaces Under various loads
10 Involved physical phenomena V.A. Yastrebov 10/10 Primary phenomena (atomic / electronic interactions) 1 Contact (non-penetration) 2 Friction (tangential resistance) 3 Adhesion (resistance to tension) 4 Surface energy 5 Quantum effects (tunneling, Casimir effect) Secondary phenomena 1 Wear 2 Lubrication (fluid-structure interaction) 3 Environmental effects (oxidation) 4 Aging, diffusion 5 Heat production 6 Heat and electricity transfer
11 Contact complexity : physics and mathematics Particular difficulties related to contact problems : multiphysical aspects, mathematical aspects Fractality of surfaces Interface chemistry Hardly accessible contact interface Generation and diffusion of heat Multiscale and multiphysical nature of friction V.A. Yastrebov 11/10
12 Contact complexity : physics and mathematics V.A. Yastrebov 12/10 Particular difficulties related to contact problems : multiphysical aspects, mathematical aspects One of the most hard problems in mechanics Lack of standard optimization problem Non-convexity and non-differentiability Non-continuous character Bad scalability
13 Notations V.A. Yastrebov 13/10 Vectors and tensors a, α scalars a b = c scalar (dot) product b vectors a b = c vector (cross) product C, β 2nd order tensors a b = C tensor product = = = 4 D = 4th order tensors A = T transposition a = B = gradient operator a = c divergence operator a = B = rotor operator I = = e i e i unitary 2nd order tensor Mechanics σ = Cauchy stress tensor ε = Small strain tensor g, g n gap, normal gap ξ position vector in parent space ɛ penalty parameter n outward unit normal vector λ, λ n, λ t lagrange multipliers ϱ, ϱ surface tangent vectors ξ 1 ξ 2 σ n =(σ n) n contact pressure f Coefficient of friction =
14 Welcome to CMET course!
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