Contact mechanics I: basics
Georges Cailletaud1 St´ephanie Basseville1,2 Vladislav A. Yastrebov1
1Centre des Mat´eriaux, MINES ParisTech, CNRS UMR 7633 2Laboratoire d’Ing´enierie des Syst`emesde Versailles, UVSQ
WEMESURF short course on contact mechanics and tribology Paris, France, 21-24 June 2010 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Table of contents
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 2/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Plan
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 3/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Short historical sketch
Use and opposition to friction Frictional heat - lighting of fire - more than [40 000 years ago]. Ancient Egypt -lubrication of surfaces with oil [5 000 years ago].
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 4/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Short historical sketch
First studies on contact and friction
Leonardo da Vinci [1452-1519] first friction laws and many other trobological topics;
From Leonardo da Vinci’s notebook Issak Newton [1687] Newton’s third law for bodies interaction;
Guillaume Amontons [1699] rediscovered firction laws;
Leonhard Euler [1707-1783] roughness theory of friction; Roughness theory of friction
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 5/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Short historical sketch
First studies on contact and friction
Charles-Augustin de Coulomb [1789] friction independence on sliding velocity and roughness; the influence of the time of repose.
Photoelasticity analysis of Hertz Heinrich Hertz [1881-1882] contact problem (shear stresses) the first study on contact of deformable solids;
Holm [1938], Ernst and Merchant [1940], Bowden and Tabon [1942] difference between apparent and real contact areas, adhesion theory. Apparent and real areas of contact
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 6/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Practice VS theory
1900: Theory is several steps behind the practice
Theory Practice
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 7/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Practice VS theory
1940: Theory is behind the practice
Theory Practice
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 7/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Practice VS theory
1960: Theory catchs up with practice
Practice and Theory
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 7/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Practice VS theory
1990: The trial-and-error testing becomming more and more difficult. Theory leads practice.
Practice Theory
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 7/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Plan
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 8/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Surface interaction properties
Surface properties: Coefficient of friction Adhesion Wear parameters
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 9/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Surface interaction properties
Surface properties are not fundamental Coefficient of friction Adhesion / Wear parameters/ /
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 9/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Surface interaction properties
Surface properties are not fundamental Fundamental properties: Coefficient of friction Volume: Adhesion / Young’s modulus; Wear parameters/ Poisson’s ratio; / shear modulus; yield stress; elastic energy; thermal properties. Surface: chemical reactivity; absorbtion capabilities; surface energy; compatibility of surfaces;
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 9/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Surface interaction properties
Surface properties are not fundamental Fundamental properties: Coefficient of friction Volume: Adhesion / Young’s modulus; Wear parameters/ Poisson’s ratio; / shear modulus; yield stress; elastic energy; thermal properties. Surface: chemical reactivity; absorbtion capabilities; surface energy; compatibility of surfaces;
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 9/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Surface interaction properties
Surface properties are not fundamental Fundamental properties are interdependent Coefficient of friction Volume: / Adhesion / Young’s modulus; Wear parameters/ Poisson’s ratio; / shear modulus; yield stress; elastic energy; thermal properties. Surface: chemical reactivity; absorbtion capabilities; surface energy; compatibility of surfaces;
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 9/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Surface interaction properties
Surface properties are not fundamental Fundamental properties are interdependent Coefficient of friction Volume: / Adhesion / Young’s modulus; Wear parameters/ Poisson’s ratio; / shear modulus; yield stress; More fundamental properties elastic energy; thermal properties. solids are made of atoms; Surface: atoms are linked by bonds; chemical reactivity; many of the volume and surface absorbtion properties are the properties of the capabilities; bonds. surface energy; compatibility of surfaces;
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 9/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Surface interaction properties
Surface properties are not fundamental Fundamental properties are interdependent Coefficient of friction Volume: / Adhesion / Young’s modulus; Wear parameters/ Poisson’s ratio; / shear modulus; yield stress; More fundamental properties elastic energy; thermal properties. solids are made of atoms; Surface: atoms are linked by bonds; chemical reactivity; many of the volume and surface absorbtion properties are the properties of the capabilities; bonds. surface energy; compatibility of surfaces;
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 9/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Material properties interdependence
Young’s modulus and yield strength interdependence [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 10/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Material properties interdependence
Penetration hardness and yield Young’s modulus and melting temperature stress interdependence interdependence [Rabinowicz, ] [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 11/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Material properties interdependence
Thermal coefficient of expansion and Young’s modulus Surface energy and hardness interdependence interdependence [Rabinowicz, ] [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 12/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real area of contact
Real area of contact depends on normal load: real area of contact is proportional to the normal load; coefficient of proportionality is inverse of the material hardness;
sliding distance: contact area might be 3(!) times as great as the value before shear forces were first applied;
time: (for creeping materials) real area of contact increases with time;
surface energy: the higher the surface energy, the greater the area of contact.
[Ref: Course of Julian Durand on surface roughness]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 13/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real area of contact
Real area of contact depends on normal load: Ar ∼ F real area of contact is proportional to the normal load; coefficient of proportionality is inverse of the material hardness; Ar - real contact area, F - applied load sliding distance: contact area might be 3(!) times as great as the value before shear forces were first applied;
time: (for creeping materials) real area of contact increases with time;
surface energy: the higher the surface energy, the greater the area of contact.
[Ref: Course of Julian Durand on surface roughness]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 13/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real area of contact
Real area of contact depends on F normal load: Ar = real area of contact is proportional to the p normal load; coefficient of proportionality is inverse of the material hardness; Ar - real contact area, F - applied load; p - hardness. sliding distance: contact area might be 3(!) times as great as the value before shear forces were first applied;
time: (for creeping materials) real area of contact increases with time;
surface energy: the higher the surface energy, the greater the area of contact.
[Ref: Course of Julian Durand on surface roughness]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 13/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real area of contact
Real area of contact depends on F normal load: Ar = real area of contact is proportional to the p normal load; coefficient of proportionality is inverse of the material hardness;
sliding distance: contact area might be 3(!) times as great as the value before shear forces were first applied;
time: (for creeping materials) real area of contact increases with time;
surface energy: the higher the surface energy, the greater the area of contact.
[Ref: Course of Julian Durand on surface roughness]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 13/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real area of contact
Real area of contact depends on F normal load: Ar = real area of contact is proportional to the p normal load; coefficient of proportionality is inverse of the material hardness;
sliding distance: contact area might be 3(!) times as great as the value before shear forces were first applied;
time: (for creeping materials) real area of contact increases with time;
surface energy: the higher the surface energy, the greater the area of contact.
[Ref: Course of Julian Durand on surface roughness]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 13/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real area of contact
Real area of contact depends on F normal load: Ar = real area of contact is proportional to the p normal load; coefficient of proportionality is inverse of the material hardness;
sliding distance: contact area might be 3(!) times as great as the value before shear forces were first applied;
time: (for creeping materials) real area of contact increases with time;
surface energy: the higher the surface energy, the greater the area of contact.
[Ref: Course of Julian Durand on surface roughness]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 13/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Engineering friction
First approximations: friction coefficient does not depend on normal load apparent area of contact velocity sliding surface roughness time Friction force direction is opposite to the sliding
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 14/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Engineering friction
First approximations: friction coefficient does not depend on normal load apparent area, of contact velocity , sliding surface/ roughness / time / / , Friction/ force, direction is opposite to the sliding ,
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 14/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: normal load
First approximation: Exceptions: friction coefficient does at micro scale for small slidings (fig. 1); not depend on normal for very large normal loads (metal forming) load. friction force is limited; for very hard (diamond) or very soft (teflon) materials: α ˆ 2 ˜ generally T = cF , α ∈ 3 ; 1 ; thin hard coating and a softer substrate (fig.2).
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 15/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: normal load
First approximation: Exceptions: friction coefficient does at micro scale for small slidings (fig. 1); not depend on normal for very large normal loads (metal forming) load. friction force is limited; for very hard (diamond) or very soft (teflon) materials: α ˆ 2 ˜ generally T = cF , α ∈ 3 ; 1 ; thin hard coating and a softer substrate (fig.2).
Fig. 1. For very small sliding, the force of friction is not proportional to the normal force [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 15/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: normal load
First approximation: Exceptions: friction coefficient does at micro scale for small slidings (fig. 1); not depend on normal for very large normal loads (metal forming) load. friction force is limited; for very hard (diamond) or very soft (teflon) materials: α ˆ 2 ˜ generally T = cF , α ∈ 3 ; 1 ; thin hard coating and a softer substrate (fig.2).
Fig. 1. For very small sliding, the force of friction is not proportional to the normal force [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 15/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: normal load
First approximation: Exceptions: friction coefficient does at micro scale for small slidings (fig. 1); not depend on normal for very large normal loads (metal forming) load. friction force is limited; for very hard (diamond) or very soft (teflon) materials: α ˆ 2 ˜ generally T = cF , α ∈ 3 ; 1 ; thin hard coating and a softer substrate (fig.2).
Fig. 1. For very small sliding, the force of friction is not proportional to the normal force [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 15/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: normal load
First approximation: Exceptions: friction coefficient does at micro scale for small slidings (fig. 1); not depend on normal for very large normal loads (metal forming) load. friction force is limited; for very hard (diamond) or very soft (teflon) materials: α ˆ 2 ˜ generally T = cF , α ∈ 3 ; 1 ; thin hard coating and a softer substrate (fig.2).
Fig. 1. For very small sliding, the force of friction is not proportional to the normal force [Rabinowicz, ]
Fig. 2. In case of hard surface layer on a softer substrate, at moderate loads friction is determined by the hard surface, higher load brakes the coating and softer material begins to define the frictional properties [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 15/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: normal force
Friction coefficient versus tangential movement; experiments from [Courtney-Pratt and Eisner, 1957]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 16/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: friction direction
First approximation: Exceptions: friction force direction is the direction of the friction force remains within opposite to the sliding. [178; 182] degrees to sliding direction (fig. 1); the difference is higher for oriented surface roughnesses.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 17/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: friction direction
First approximation: Exceptions: friction force direction is the direction of the friction force remains within opposite to the sliding. [178; 182] degrees to sliding direction (fig. 1); the difference is higher for oriented surface roughnesses.
Fig. 1. Change of the direction of friction force with sliding [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 17/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: friction direction
First approximation: Exceptions: friction force direction is the direction of the friction force remains within opposite to the sliding. [178; 182] degrees to sliding direction (fig. 1); the difference is higher for oriented surface roughnesses.
Fig. 1. Change of the direction of friction force with sliding [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 17/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: apparent area and roughness
First approximation: First approximation: Friction coefficient does not depend Friction coefficient does not depend on the apparent area of contact. on sliding surface roughness. Exceptions: Exceptions: very smooth and very clean surfaces. very smooth or very rough surfaces (fig. 1).
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 18/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: apparent area and roughness
First approximation: First approximation: Friction coefficient does not depend Friction coefficient does not depend on the apparent area of contact. on sliding surface roughness. Exceptions: Exceptions: very smooth and very clean surfaces. very smooth or very rough surfaces (fig. 1).
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 18/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: apparent area and roughness
First approximation: First approximation: Friction coefficient does not depend Friction coefficient does not depend on the apparent area of contact. on sliding surface roughness. Exceptions: Exceptions: very smooth and very clean surfaces. very smooth or very rough surfaces (fig. 1).
Fig. 1. Friction roughness influences the coefficient of friction [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 18/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: time and velocity
First approximation: First approximation: Friction coefficient does not depend Friction coefficient does not depend on time. on sliding velocity. Exceptions: Exceptions: creeping materials. if material behaves differently at different loading rate, then the friction depends on the sliding velocity;
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 19/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: time and velocity
First approximation: First approximation: Friction coefficient does not depend Friction coefficient does not depend on time. on sliding velocity. Exceptions: Exceptions: creeping materials. if material behaves differently at different loading rate, then the friction depends on the sliding velocity;
Static coefficient of friction evolution with time
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 19/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: time and velocity
First approximation: First approximation: Friction coefficient does not depend Friction coefficient does not depend on time. on sliding velocity. Exceptions: Exceptions: creeping materials. if material behaves differently at different loading rate, then the friction depends on the sliding velocity;
Static coefficient of friction evolution with time Kinetic friction decreases with increasing sliding velosity G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 19/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: velocity
First approximation: Friction coefficient does not depend on sliding velocity. Exceptions: if material behaves differently at different loading rate, then the friction depends on the sliding velocity;
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 20/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: velocity
First approximation: Friction coefficient does not depend on sliding velocity. Exceptions: if material behaves differently at different loading rate, then the friction depends on the sliding velocity;
Friction coefficient slightly decreses with increasing velocity of sliding, titanium on titanium [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 20/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: velocity
First approximation: Friction coefficient does not depend on sliding velocity. Exceptions: if material behaves differently at different loading rate, then the friction depends on the sliding velocity;
Friction coefficient slightly decreses with increasing velocity of sliding, titanium on titanium [Rabinowicz, ]
Friction coefficient dependence on velocity of sliding for lubricated surfaces [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 20/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Real friction :: velocity
First approximation: Friction coefficient does not depend on sliding velocity. Exceptions: if material behaves differently at different loading rate, then the friction depends on the sliding velocity;
Friction coefficient increases and decreases with increasing velocity of sliding, hard on soft (steel on lead, steel on indium) [Rabinowicz, ]
Friction coefficient dependence on velocity of sliding for lubricated surfaces [Rabinowicz, ]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 20/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Three scales of contact study
Nanoscale: Study of molecular junctions, van des Waals forces and Casimir effect.
Microscale: Roughness and microstructure study
Macroscale: Stress-strain state of contacting solids
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 21/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Plan
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 22/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Macroscopic contact
Signorini contact law (1933) r n Fn ≤ 0
u n un ≤ 0 Fnun = 0
Compliance contact law r n [Kragelsky, 1982], [Oden-Martins, 1985] mn −Fn = Cn(un)+
u n [Song, Yovanovich, 1987] 2 c2u −Fn = C1e n
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 23/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Hertz theory (1882)
Geometry of smooth, non-conforming surface in contact Expression of the profile of each surface
1 2 1 2 P z1 = 0 x1 + 00 y1 2R1 2R1 δ 2 „ « 1 2 1 2 z2 = − 0 x1 + 00 y2 2R2 2R2 S2 uz δ 1 0 00 δ 2 where Ri and Ri are the principal radii of curvature 0 x−y plane of the surface i. δ2 uz2 Separation between the two surfaces 2 a S1 z 2 2 h = z1 − z2 = Ax + By δ1 P Displacement
uz1 + uz2 + h = δ1 + δ2 [Johnson, 1996]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 24/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Hertz theory (1882)
Assumptions in the Hertz theory: The surface are continuous and non-conforming, a << R The strains are small, a << R
Each solid can be considered as an elastic half-space, a << R1,2, a << l
The surfaces are frictionless, q − x = qy = 0 Applications 1 Solids of revolution 2 Two-dimensional contact of cylindrical bodies Note 1 1 − ν1 1 − ν2 = + ∗ E E1 E2
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 25/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Hertz theory : Solids of revolution
Simple case of solids of revolution Principal radii of curvature 0 00 Ri = Ri = Ri , i = 1, 2 Boundary conditions for the displacement
2 uz1 + uz2 = δ − (1/2R)r
Pressure distribution 2 1/2 p = p0{1 − (r/a) } Consequences Pressure !1/3 6PE ∗2 p0 = P3R2
Radius of the contact circle „ 3PR «1/3 a = 4E ∗2 Displacement !1/3 9P2 δ = 16RE ∗2
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 26/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Hertz theory : Solids of revolution
Distributions of stresses
σr (x = 0, z) = σθ (x = 0, z) p0 p0
= −(1 + ν){1 − (z/a)tan−1(a/z)}
1 2 2 −1 + 2 (1 + z /a )
σz (x = 0, z) = −(1 + z2/a2)−1 p0
1 Maximum shear stress τ1 = 2 |σr − σθ | (τ1)max = 0.31p0 at the deph of 0.48a (for ν = 0.3)
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 27/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites 2D contact of cylindrical bodies
a a p0
1 1−ν1 1−ν2 ∗ = + E E1 E2 p(x) 1 = 1 + 1 R R1 R2
O x 2 −1/2 q(x) p(x) = p0(1 − (x/a) )
M(x, y) ¡ σxx q 4PR a = πE∗ τxz σzz q PE∗ p0 = πR
z
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 28/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites 2D contact of cylindrical bodies
Example : cylinder/plate
Distributions of normal pressure (Hertz) and tangential stress
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p0 2 2 2 2 −1/2 σxx (x = 0, z) = − a {(a + 2z )(a + z ) − 2z}
2 2 −1/2 σzz (x = 0, z) = −p0a(a + z )
2 2 2 −1/2 τmax (x = 0, z) = p0a{z − z (a − z ) }
σxz = σxy = σyz = 0
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 29/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Macroscopic friction 1/2
Tresca 8 |Ft | ≤ g > > <> If |Ft | < g, then Vslide = 0 > > :> If |Ft | = g, ∃λ > 0 such Vslide = −λFt
Coulomb
8 |Ft | ≤ µ|Fn| > > <> If |Ft | < µ|Fn|, then Vslide = 0 stick > > :> If |Ft | = µ|Fn|, ∃λ > 0 such Vslide = −λFt slip
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 30/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Macroscopic friction 2/2
Regularized Coulomb [Oden, Pires, 1983], [Raous, 1999]
i |Ft | = −µφ (Vslide ) |Fn| φ1 = √ Vslide 2 2 Vslide +ε 2 Vslide φ = tanh ε
Variable friction
mt 8 |Ft | ≤ Ct (ut ) > > <> mt If |Ft | < Ct (ut ) , then Vslide = 0 > > :> mt If |Ft | = Ct (ut ) , ∃λ > 0 such Vslide = −λRt
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 31/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Transition toward the slip
Definition of sliding Relative peripheral velocity of the surfaces at their point of contact Sliding of non-conforming elastic bodies Question Fixed slider P The tangential traction due to the friction at the S 2 contact surface influences the size and shape of the Q contact area or the distribution of normal pressure ? x S Evaluation of the elastic stresses and displacements 1 Q Basic premise of the Hertz theory V 2a Relationship between the tangential traction and the z normal pressure
|q(x, y)| |Q| = = µ Coulomb’s law Sliding contact p(x, y) P Application Cylinder sliding perpendicular to its axis [Johnson, 1996, Goryacheva, 1998]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 32/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Cylinder sliding perpendicular to its axis
Distributions of normal pressure (Hertz) and tangential traction q 2P x 2 p(x) = πa 1 − ( a ) (1) q 2P x 2
q(x) = ±µp(x) = ±µ πa 1 − ( a )
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G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 33/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Partial slip
Relation between slip zone (c) and contact zone (a)
+ = glissement a c +c +a − − a +a a c +c +a − − − zone coll´ee
x2 1/2 c x2 1/2 q (x) = µ p 1 q (x) = µ p 1 q(x) = q1(x) q2(x) 1 0 − a2 2 a 0 − c2 −
s c Q = 1 − a µP
If x < c : stick condition. The local contact shear stress is
r x c r x 2 2 τxz = µp0 1 − ( ) − µp0 1 − ( ) a a c
If c < x < a: slip condition. The local contact shear stress is
r x 2 τxz = µp0 1 − ( ) a
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 34/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Friction instability : “stick-slip” 1/4
Definiton of the stick-slip :
Intermittent relative motion between the contact surfaces, alternation of slip and stick. F F Fs
Fd
time time time Phenomenon occurs at various scales:
Macroscopic : discontinuities in the gravity center displacement of contact body and loads. Microscopic : location of the phenomenon at the interface
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 35/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Friction instability : “stick-slip” 2/4
The stick-slip is a coupling result : The dynamic response of the friction system stiffness, damping, inertia
Friction dynamic at the interface Difference between static (µs ) and dynamic (µd ) friction coefficient µs and µd dependence on the sliding velocity and time
A simple stick-slip model Friction law
F ¡
¢¡¢F Fs
¡
¢¡¢ ¡
¢¡¢ k
¡
¢¡¢ ¡
¢¡¢ X ¡
¢¡¢ F ¡
¢¡¢ d ¡
¢¡¢ m
¡ ¢¡¢ v v
Fig : Plot of frictional force vs. time.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 36/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Friction instability : “stick-slip” 3/4
During sliding, the problem is:
Fs mx¨ − Fd = −kx x(0) = x˙ (0) = v k
and the solution is
1 v r k x(t) = {(Fs − Fd )cos(ωt) + Fd } + sin(ωt), ω = k ω m
or the velocity v is negligible compared to dx/dt:
1 x(t) ≈ {(Fs − Fd )cos(ωt) + Fd } k
Tinertia F F Characteristic time of sliding s
r m Tinertia = 2π k
The force F oscillates between Fs and 2Fd − Fs . 2Fd − Fs
t G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 37/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Friction instability : “stick-slip” 4/4
Fig : Regions of stable and stick-slip motion.
The red curve in the parameters plane at the other parameters being fixed, demarcates the regions of stable and unstable motion.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 38/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Plan
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 39/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Examples
(a) Vickers indenta- (b) Contact zone under tion test, palladium Vickers indenter, zirconium glasses glasses
(c) Sracth resistance of soda-Lime Silica Glasses
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 40/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Hill’s theory: Elastic-plastic indentation
Assumptions Within the core: Cavity model of an Hydrostatic component of stress p elastic-plastic indentation cone da da Outside the core: a a Radial symmetry for stresses and displacement β dh c core r At the interface ( between core and plastic zone) da dc du(r) Hydrostatic stress (in the core)= radial component of Plastic stress (in the external zone)
Elastic The radial displacement on r=a during an increment dh must accommodate the volume of material.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 41/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Characteristic
In the plastic zone: a ≤ r ≤ c σr = −2ln(c/r) − 2/3 In the elastic zone: r ≥ c Y σr 2 ` c ´3 Y = − 3 r σθ = −2ln(c/r) + 1/3 Y σθ 1 ` c ´3 Y = 3 r where Y denotes the value of the yield strees of material in simple shear and simple compression.
Core pressure 9 > > p σr 2 > Y = −[ Y ]r=a = 3 + 2ln(c/a) > > > Pressure in the core, Radial displacement > => for an incompressible material „ «ff du(r) T 2 2 > p 2 Etan(β) dc = E {3(1 − ν)(c /r ) − 2(1 − 2ν)(r/c)} > = 1 + ln > Y 3 3Y > Conservation volume > > > > 2πa2du(a) = πa2dh = πa2tan(β)da ;
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 42/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Unloading indentation: elastic strain energy
Example: spherical indenter
R R’ R ρ
a a a a Before loading Under loading After unloading
Residual depth δ − δ0: Estimation of the energy dissipated ∆W in one cycle of the load ∆W = α R Pdδ where α is the hysteresis-loss factor. (α = 0, 4% for hard bearing steel) „ ∗2 5 « 9 W = 2 9E P > 5 16R > > > 3 “ 1 1 ” 3P > 4a R − ρ = E∗ > 0 9πPpm = δ = 2 with pm = 0.38Y in fully plastic state 16E0 P 0 2 “ ”1/3 = 0, 38(δ /δY ) a = 3P > PY 4E0 > > > 2 “ 2 ”1/3 > a 9P > δ = R = 3 ; 16RE0
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 43/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Sharp indentation
Characterisation of P-h reponse During the loading,
P = Ch2 Kick’s law During the unloading, ˛ dPu ˛ dh initial slope ˛hm
hr Residual indentation depth after complete unloading
Three independent quantities Schematic illustration of a typical P-h reponse of an elasto-plastic material [?]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 44/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Plastic behavior
Model 8 Eε for σ ≤ σ <> y σ = > n : Rε for σ ≥ σy
E Young’s modolus R a strength coefficient with n the strain hardening exponent σy the initial yield stress
Assumption : The power law elasto-plastic stress-strain The theory of plasticity with the von Mises effect stress. behavior Parameters for an elasto-plastic behavior E, ν, σy , n
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 45/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Dimensional analysis
Objective Prediction of the P − h reponse from elasto-plastic properties
Application of the universal dimensionless functions : the Π theorem
Material parameter set
(E, ν, σy , n) or (E, ν, σr , n) or (E, ν, σy , σr )
Load P
∗ ∗ P = P(h, E , σy , n) or P = P(h, E , σr , n) or P = P(E, ν, σy , σr ) with
!−1 1 − ν2 1 − ν2 E ∗ = + i E Ei
Unload
∗ Pu = Pu (h, hm, E, ν, Ei , νi , σr , n) or Pu = Pu (h, hm, E , σr , n)
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 46/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Determination of hm
Application of the dimensional analysis during the load
Load Applying the Π theorem in dimensional analysis ∗ P “ E∗ ” P = P(h, E , σy , n) C = = σr Π1 , n h2 σr
∗ P A “ E∗ σr ” P = P(h, E , σr , n) C = = σy Π , h2 1 σy σy
P B “ E∗ σy ” P = P(E, ν, σy , σr ) C = = σr Π , h2 1 σr σr
with Π are dimensionless functions.
And then hm !
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 47/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Determination of hr
Application of the dimensional analysis during the unload
dPu dPu ∗ = (h, hm, E , σr , n) dh dh thus dPu „ hm σr « = E ∗hΠ0 , , n dh 2 h E ∗ Consequently, ˛ dPu „ σr « „ σr « ˛ = E ∗h Π0 1, , n = E ∗h Π , n ˛ m 2 ∗ m 2 ∗ dh ˛h=hm E E Or „ « ∗ ∗ hm σr Pu = Pu (h, hm, E , σr , n) = E Πu , , n h E ∗ Finaly,
„ hm σr « hr „ σr « P = 0 implies 0 = Π , , n whether = Π , n u u ∗ 3 ∗ hr E hm E
and then hr ! Πi ,i =1,2,3 can be used to relate the indentaion reponse to mechanical properties.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 48/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites FE Vickers indentation test
max Maximum penetration hs 3.11 µm
Parameters
Elastic E=81600MPa, ν = 0, 36
Elasto-plastic model E=81600MPa, ν = 0, 36, σy = 1610MPa
Drucker-Prager model E=81600MPa, ν = 0, 36, t c σy = 1600MPa, σy = 1800MPa
[Laniel, 2004]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 49/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Computation results
Penetration depth
von Misesmax Residual stress
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 50/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Computation results
Elasto-plastic contact reponse
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 51/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Computation results
Elasto-plastic contact reponse
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 52/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Spherical indentation on a single crystal
Hypothesis Sphere radius : R=100µm Copper and zinc single crystals : crystal plasticity Silicon substrate : isotropic elastic max Maximum penetration hs : 3.5 µ m
[Casal and Forest, 2009]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 53/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Contact reponse
(d) Elastic anisotropic contact reponse (e) Elasto-plastic anisotropic contact reponse
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 54/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites von Mises stress fields
Figure: (a) f.c.c and (b) h.c.p crystals. Penetration depth: hs = 1.25µm
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 55/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Plastic zone morphology
(a) f.c.c copper crystals (b) h.c.p zinc crystals
Figure: Penetration depth: hs = 3.5µm
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 56/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Plan
1 Introduction
2 Basic knowledges
3 Contact mechanics of elastic solids
4 Normal contact of inelastic solids
5 Contact of inhomogeneous bodies
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 57/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Bounds for the global coefficient of friction
Lubricant µ = 0.2 9 1 PiP P Coating µ2 = 0.8
q(x) R µ(x)p(x)dS µ(x) = µ = p(x) R p(x)dS P X µi ci fi Uniform stress ≡ µi fi ≤ µ ≤ P ≡ Uniform strain ci fi
2 Ei (1−νi ) with ci = 2 (1−νi ) (1−2νi ) [Dick and Cailletaud, 2006]
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 58/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Bounds for the global coefficient of friction
Cont A ν E (GPa) C (GPa) µ Cont B ν E (GPa) C (GPa) µ Comp 1 0.32 8 11.45 0.1 Comp 1 0.15 55 58.08 0.1 Comp 2 0.15 55 58.08 0.5 Comp 2 0.32 08 11.45 0.5
0.5 0.5 0.45 0.45 0.4 0.4 0.35 0.35 0.3 0.3
µ 0.25 µ 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 P =P 0.05 P =P 1 2 ε 1 ε 2 εy1=εy2 y1= y2 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Component 2 (%) Component 2 (%)
Case A: µ1 = 0.1, E1 = 11.45GPa Case B: µ1 = 0.5, E1 = 58GPa µ2 = 0.5, E2 = 58GPa µ2 = 0.1, E2 = 11.45GPa
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 59/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites FE computations VS analytic estimation
0.5 0
0.45 -1
0.4 -2
0.35 -3
0.3 -4 comp.2 comp.1 comp.2 COF
0.25 -5
0.2 normalized contact pressure -6 analytic CComp.1 0.15 analytic -7 CComp.2 P1-10 P1-10: pComp.1, pComp.2 P2-10 P2-10: pComp.1, pComp.2 0.1 -8 0 0.2 0.4 0.6 0.8 1 -0.01 -0.005 0 0.005 0.01 Component 2 (%) x (mm)
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 60/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Different CSL geometries 1/2
Bulk material Component 1 Component 2 E (GPa) 119 8 55 ν 0.29 0.32 0.15 R0 ( MPa) - 200 500
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 61/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Different CSL geometries 2/2
0.5
0.45
0.4
0.35
0.3 COF
0.25
analytic 0.2 analytic S20_el 0.15 R20_el L20_el Lxx_el 0.1 0 0.2 0.4 0.6 0.8 1 Component 2 (%)
0 0 -1 -0.5 comp.2 comp.1 comp.2 -2 -1 -3 -1.5 -4 -2 -5 -2.5 -6 comp.2 comp.1 comp.2 -3 -7 -3.5
-8 CComp.1 -4 CComp.1 normalized contact pressure CComp.2 normalized contact pressure CComp.2 -9 S20_el: pComp.1, pComp.2 -4.5 S20_el: pComp.1, pComp.2 R20_el: pComp.1, pComp.2 R20_el: pComp.1, pComp.2 -10 L20_el: pComp.1, pComp.2 -5 L20_el: pComp.1, pComp.2 Lxx_el: pComp.1, pComp.2 Lxx_el: pComp.1, pComp.2 -11 -5.5 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 x (mm) x (mm)
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 62/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Influence of the number of dimples
10% comp 2 70% comp 2 ESD type 2a (µm) lESD (µm) nb.ESD lESD (µm) nb.ESD R10 360 11.1 32.4 33.3 10.8 R20 360 22.2 16.2 66.6 5.4 R40 360 44.4 8.1 133.3 2.7
0.5
0.45
0.4
0.35
0.3 COF
0.25
0.2 R10_el R20_el 0.15 R40_el S20_el S40_el 0.1 0 0.2 0.4 0.6 0.8 1 Component 2 (%)
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 63/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Influence of plastic deformations
0.5 -0.5
-1 0.45 -1.5 0.4 -2 0.35 -2.5
0.3 -3 comp.2 comp.1 comp.2 COF -3.5 0.25
analytic -4 0.2 analytic normalized contact pressure -4.5 R40_el CComp.1 0.15 R40_pl CComp.2 -5 S40_el R40_el: pComp.1, pComp.2 S40_pl R40_pl: pComp.1, pComp.2 0.1 -5.5 0 0.2 0.4 0.6 0.8 1 -0.04 -0.02 0 0.02 0.04 Component 2 (%) x (mm)
0.0038 0.011 0.019 0.027 0.035 0.042 0.05 0 0.0076 0.015 0.023 0.031 0.038 0.046
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 64/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Conclusion
Estimation of the upper and lower bound. The friction coefficient depend on the CSL geometry the dissimilarity of the CSL component materials the compliance of substrate and counter body
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 65/68 Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites Summary
Contact and friction complicated phenomena; depend on many material properties; not yet well elaborated. Analytical solutions hertzian contact; nonlinear material; friction; stick-slip instabilities. Numerical analysis examples of indendation tests; analysis of heterogeneous friction.
G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 66/68 Thank you for your attention!
Georges Cailletaud
Casal, O. and Forest, S. (2009). Finite element crystal plasticity analysis of spherical indentation. Computational Materials Science, 45:774–782.
Dick, T. and Cailletaud, G. (2006). Analytic and FE based estimations of the coefficient of friction of composite surfaces. Wear, 260:1305–1316.
Goryacheva, I. (1998). Contact mechanics in tribology. London. Johnson, K. (1996). Contact mechanics. Cambridge.
Laniel, R. (2004). Simulation des proc¨ı¿½d¨ı¿½s d’indentation et de rayage par ¨ı¿½l¨ı¿½ments finis et ¨ı¿½l¨ı¿½ments disctincts.
Rabinowicz, E. Friction and Wear. G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 68/68