Contact Mechanics and Elements of Tribology Lectures 2-3. Mechanical Contact

Vladislav A. Yastrebov

MINES ParisTech, PSL Research University, Centre des Mat´eriaux,CNRS UMR 7633, Evry, France

@ Centre des Materiaux´ February 8-9, 2016 Outline

Lecture 2 1 Balance equations 2 Intuitive notions 3 Formalization of frictionless contact 4 Evidence friction 5 Contact types 6 Analogy with boundary conditions

Lecture 3 1 Flamant, Boussinesq, Cerruti 2 Displacements and tractions 3 Classical elastic problems Boundary value problem in elasticity

Reference and current configurations

x = X + u f Balance equation (strong form) 1 σ + ρf = 0, x Ωi u ∇ · = v ∀ ∈ 1 c Displacement compatibility 2 c

1 2 ε = ( u + u ) f = 2 ∇ ∇ u Constitutive equation

σ = W0(ε) = = Two bodies in contact Boundary conditions

0 Dirichlet: u = u , x Γu ∀ ∈ 0 Neumann: n σ = t , x Γf · = ∀ ∈ V.A. Yastrebov 3/18 Boundary value problem in elasticity

Reference and current configurations

x = X + u f Balance equation (strong form) 1 σ + ρf = 0, x Ωi u ∇ · = v ∀ ∈ 1 c Displacement compatibility 2 c

1 2 ε = ( u + u ) f = 2 ∇ ∇ u Constitutive equation

σ = W0(ε) = = Two bodies in contact

Boundary conditions Include contact conditions 0 ... Dirichlet: u = u , x Γu ∀ ∈ 0 Neumann: n σ = t , x Γf · = ∀ ∈ V.A. Yastrebov 4/18 Intuitive conditions

1 No penetration

f Ω1(t) Ω2(t) = ∩ ∅ 1

u 2 No adhesion 1 c i 2 n σ n 0, x Γ c = c · · ≤ ∀ ∈ 2 f u 3 No shear stress

n σ (I n n) = 0, x Γi · = · − ⊗ ∀ ∈ c Two bodies in contact

Intuitive contact conditions for frictionless and nonadhesive contact

V.A. Yastrebov 5/18 Intuitive conditions

1 No penetration

f Ω1(t) Ω2(t) = ∩ ∅ 1

u 2 No adhesion 1 c i 2 n σ n 0, x Γ c = c · · ≤ ∀ ∈ 2 f u 3 No shear stress

n σ (I n n) = 0, x Γi · = · − ⊗ ∀ ∈ c Two bodies in contact

Intuitive contact conditions for frictionless and nonadhesive contact

V.A. Yastrebov 6/18 Gap function

Gap function g g>0 gap = – penetration non-contact asymmetric function n n g=0 defined for n contact separation g > 0 • contact g = 0 • g<0 penetration g < 0 • penetration governs normal contact Gap between a slave point and a master surface Master and slave split Gap function is determined for all slave points with respect to the master surface Gap function

Gap function g g>0 gap = – penetration non-contact asymmetric function n n g=0 defined for n contact separation g > 0 • contact g = 0 • g<0 penetration g < 0 • penetration governs normal contact Gap between a slave point and a master surface Master and slave split rs Gap function is determined for all slave points with respect to the n master surface ( ( Normal gap h i gn = n r ρ(ξ ) , · s − π Definition of the normal gap n is a unit normal vector, rs slave point, ρ(ξπ) projection point at master surface Frictionless or normal contact conditions

n No penetration 0 Always non-negative gap g g 0 ≥ No adhesion Always non-positive contact pressure

σ∗ 0 n ≤ Complementary condition Either zero gap and non-zero pressure, or Scheme explaining normal non-zero gap and zero pressure contact conditions g σn = 0 No shear transfer (automatically)

σt∗∗ = 0

σn∗ = (σ n) n = σ :(n n) = · · = ⊗   σ = σ n σnn = n σ I n n t∗∗ = · − · = · =− ⊗

V.A. Yastrebov 9/18 Frictionless or normal contact conditions

n No penetration 0 Always non-negative gap non-contact g g > 0, n = 0 0

g 0

≥ < t n c

No adhesion a

t , n Always non-positive contact pressure 0 restricted

o c = regions σ∗ 0 g n ≤ Complementary condition Either zero gap and non-zero pressure, or Improved scheme explaining non-zero gap and zero pressure normal contact conditions g σn = 0 No shear transfer (automatically)

σt∗∗ = 0

σn∗ = (σ n) n = σ :(n n) = · · = ⊗   σ = σ n σnn = n σ I n n t∗∗ = · − · = · =− ⊗

V.A. Yastrebov 10/18 Frictionless or normal contact conditions

In mechanics: n 0 Normal contact conditions non-contact g g > 0, n = 0

≡ 0 Frictionless contact conditions < t n c

a

t ≡ ,

1 [2] n

0 restricted

Hertz -Signorini conditions o c = regions ≡ g Hertz1 -Signorini [2]-Moreau [3] conditions also known in optimization theory as Improved scheme explaining Karush [4]-Kuhn [5]-Tucker [6] conditions normal contact conditions

g 0, σn 0, gσn = 0 ≥ ≤

1Heinrich Rudolf Hertz (1857–1894) a German physicist who first formulated and solved the frictionless contact problem between elastic ellipsoidal bodies. 2Antonio Signorini (1888–1963) an Italian mathematical physicist who gave a general and rigorous mathematical formulation of contact constraints. 3Jean Jacques Moreau (1923) a French mathematician who formulated a non-convex optimization problem based on these conditions and introduced pseudo-potentials in contact mechanics. 4William Karush (1917–1997), 5Harold William Kuhn (1925) American mathematicians, 6Albert William Tucker (1905–1995) a Canadian mathematician.

V.A. Yastrebov 11/18 Contact problem

Problem ≈

Find such contact pressure f p = n σ n 0 1 − · = · ≥ u 1 2 1 which being applied at Γc and Γc results in c 2 1 2 1 1 2 2 c x = x , x Γc , x Γc ∀ ∈ ∈ 2 f and evidently u Ω1(t) Ω2(t) = ∩ ∅ Two bodies in contact

1 Unfortunately, we do not know Γc in advance, it is also an unknown of the problem. Related problem Suppose that we know p on Γc Then what is the corresponding displacement field u in Ωi?

V.A. Yastrebov 12/18 Contact problem

Problem ≈

Find such contact pressure f p = n σ n 0 1 − · = · ≥ u 1 2 1 which being applied at Γc and Γc results in c 2 1 2 1 1 2 2 c x = x , x Γc , x Γc ∀ ∈ ∈ 2 f and evidently u Ω1(t) Ω2(t) = ∩ ∅ Two bodies in contact

1 Unfortunately, we do not know Γc in advance, it is also an unknown of the problem. Related problem Suppose that we know p on Γc Then what is the corresponding displacement field u in Ωi?

This afternoon

V.A. Yastrebov 13/18 Evidence of friction

Existence of frictional resistance is evident Independence of the nominal contact area

Think about adhesion and introduce a threshold in the interface τc Globally:

- stick: T < Tc(N) Rectangle on a flat surface - slip: T = Tc(N) From experiments:

- Threshold Tc N ∼ - Friction coefficient f = N/Tc | | Locally

- stick: στ < τc(σn)

- slip: στ = f σn

V.A. Yastrebov 14/18 Evidence of friction

Existence of frictional resistance is evident Independence of the nominal contact area

Think about adhesion and introduce a threshold in the interface τc Globally:

- stick: T < Tc(N) Rectangle on a flat surface - slip: T = Tc(N) From experiments:

- Threshold Tc N ∼ - Friction coefficient f = N/Tc | | Locally

- stick: στ < τc(σn)

- slip: στ = f σn

Torque V.A. Yastrebov 15/18 Types of contact

Known contact zone conformal geometry flat-to-flat, cylinder in a hole initially non-conformal geometry but huge pressure resulting in full contact Unknown contact zone general case Point and line contact Frictionless conservative, energy minimization problem Frictional path-dependent solution, from the first touch to the current moment

Example

V.A. Yastrebov 16/18 Types of contact

Known contact zone conformal geometry flat-to-flat, cylinder in a hole initially non-conformal geometry but huge pressure resulting in full contact Unknown contact zone general case Point and line contact Frictionless conservative, energy minimization problem Frictional path-dependent solution, from the first touch to the current moment

Example

V.A. Yastrebov 17/18 Types of contact

Known contact zone conformal geometry flat-to-flat, cylinder in a hole initially non-conformal geometry but huge pressure resulting in full contact Unknown contact zone general case Point and line contact Frictionless conservative, energy minimization problem Frictional path-dependent solution, from the first touch to the current moment

Example

V.A. Yastrebov 18/18 Analogy with boundary conditions

Flat geometry Compression of a cylinder

Frictionless uz = u0

Full stick conditions u = u0ez

Rigid flat indenter uz = u0

V.A. Yastrebov 19/18 Analogy with boundary conditions

Flat geometry Compression of a cylinder

Frictionless uz = u0

Full stick conditions u = u0ez

Rigid flat indenter uz = u0 Curved geometry Polar/spherical coordinates ur = u0 If frictionless contact on rigid surface y = f (x) is retained by high pressure (X + u) e = f ((X + u) e ) · y · x

V.A. Yastrebov 20/18 Analogy with boundary conditions

Flat geometry Compression of a cylinder

Frictionless uz = u0

Full stick conditions u = u0ez

Rigid flat indenter uz = u0 Curved geometry Polar/spherical coordinates ur = u0 If frictionless contact on rigid surface y = f (x) is retained by high pressure (X + u) e = f ((X + u) e ) · y · x Transition to finite friction

From full stick, decrease f ≈ by keeping uz = 0 and by replacing in-plane Dirichlet BC by in-plane Neumann BC

V.A. Yastrebov 21/18 Analogy with boundary conditions II

In general

Type I: prescribed tractions

p(x, y), τx(x, y), τy(x, y) Type II: prescribed displacements u(x, y) Type III: tractions and displacements

uz(x, y), τx(x, y), τy(x, y) or

p(x, y), ux(x, y), uy(x, y) Type IV: displacements and relation between tractions

uz(x, y), τx(x, y) = fp(x, y) ±

V.A. Yastrebov 22/18 To be continued. . .  Concentrated forces

Normal force: in-plane stresses and displacements (plane strain) 2 3 2 2N cos(θ) 2N x y 2N y 2N xy σr = or σx = , σy = , σxy = − π r − π (x2+y2)2 − π (x2+y2)2 − π (x2+y2)2 1 + ν ur = N cos(θ) [2(1 ν) ln(r) (1 2ν)θ tan(θ)] + C cos(θ) πE − − − 1 + ν u = N sin(θ) [2(1 ν) ln(r) 2ν + (1 2ν)(1 2θ ctan(θ))] C sin(θ) θ πE − − − − − Tangential force 2 2 2T sin(θ) 2T x3 2T xy 2T x y σr = or σx = , σy = , σxy = π r − π (x2+y2)2 − π (x2+y2)2 − π (x2+y2)2 1 + ν ur = T sin(θ) [2(1 ν) ln(r) (1 2ν)θctan(θ)] C sin(θ) − πE − − − − 1 + ν u = T cos(θ) [2(1 ν) ln(r) 2ν + (1 2ν)(1 + 2θ tan(θ))] + C cos(θ) θ πE − − − Distributed load

Distributed tractions p(x)dx = dN(x), τ(x)dx = dT(x) Use superposition principle for the stress state and for displacements

Tractions on the surface

Za Za 2y p(s)(x s)2 ds 2 τ(s)(x s)3 ds σx(x, y) = − − − π ((x s)2 + y2)2 − π ((x s)2 + y2)2 b − b − − − Za Za 2y3 p(s) ds 2y2 τ(s)(x s) ds σy(x, y) = − − π ((x s)2 + y2)2 − π ((x s)2 + y2)2 b − b − − − Za Za 2y2 p(s)(x s) ds 2y τ(s)(x s)2 ds σxy(x, y) = − − − π ((x s)2 + y2)2 − π ((x s)2 + y2)2 b − b − − − Distributed load

Distributed tractions p(x)dx = dN(x), τ(x)dx = dT(x) Use superposition principle for the stress state and for displacements

Tractions on the surface

Za Za 2y p(s)(x s)2 ds 2 τ(s)(x s)3 ds σx(x, y) = − − − π ((x s)2 + y2)2 − π ((x s)2 + y2)2 b − b − − − Za Za 2y3 p(s) ds 2y2 τ(s)(x s) ds σy(x, y) = − − π ((x s)2 + y2)2 − π ((x s)2 + y2)2 b − b − − − Za Za 2y2 p(s)(x s) ds 2y τ(s)(x s)2 ds σxy(x, y) = − − − π ((x s)2 + y2)2 − π ((x s)2 + y2)2 b − b − − − Distributed load

Distributed tractions p(x)dx = dN(x), τ(x)dx = dT(x) Use superposition principle for the stress state and for displacements Consider displacements on the surface

Tractions on the surface

 x a  a Z Z  2 Z (1 2ν)(1 + ν)   2(1 ν ) ux(x, 0) = −  p(s) ds p(s) ds − τ(s) ln x s ds+C1 − 2E  − − πE | − | b x b − − Distributed load

Distributed tractions p(x)dx = dN(x), τ(x)dx = dT(x) Use superposition principle for the stress state and for displacements Consider displacements on the surface Or rather their derivatives along the surface

Tractions on the surface

 x a  a Z Z  2 Z (1 2ν)(1 + ν)   2(1 ν ) ux(x, 0) = −  p(s) ds p(s) ds − τ(s) ln x s ds+C1 − 2E  − − πE | − | b x b − − Za (1 2ν)(1 + ν) 2(1 ν2) τ(s) ux x(x, 0) = − p(x) − ds , − E − πE x s b − − Near-surface stress state Distributed load

Distributed tractions p(x)dx = dN(x), τ(x)dx = dT(x) Use superposition principle for the stress state and for displacements Consider displacements on the surface Or rather their derivatives along the surface

Tractions on the surface

 x a  a Z Z  2 Z (1 2ν)(1 + ν)   2(1 ν ) uy(x, 0) = −  τ(s) ds τ(s) ds − p(s) ln x s ds+C2 2E  − − πE | − | b x b − − Distributed load

Distributed tractions p(x)dx = dN(x), τ(x)dx = dT(x) Use superposition principle for the stress state and for displacements Consider displacements on the surface Or rather their derivatives along the surface

Tractions on the surface

 x a  a Z Z  2 Z (1 2ν)(1 + ν)   2(1 ν ) uy(x, 0) = −  τ(s) ds τ(s) ds − p(s) ln x s ds+C2 2E  − − πE | − | b x b − − Za (1 2ν)(1 + ν) 2(1 ν2) p(s) uy x(x, 0) = − τ(x) − ds , E − πE x s b − − Rigid stamp problem

Link displacement derivatives with tractions

Za τ(s) π(1 2ν) πE ds = − p(x) ux x(x, 0) x s − 2(1 ν) − 2(1 ν2) , b − − − − Za p(s) π(1 2ν) πE ds = − τ(x) uy x(x, 0) x s 2(1 ν) − 2(1 ν2) , b − − − − If in contact interface we can prescribe p, ux,x or τ, uy,x, then the problem reduces to Za (x) F ds = (x) x s U b − The general solution (case a− = b):

Za Za 1 √a2 x2 (s) ds C (x) = − U + , C = (s)ds F π2 √a2 x2 x s π √a2 x2 F a − a − − − −

V.A. Yastrebov 31/18 Rigid stamp problem

Link displacement derivatives with tractions

Za τ(s) π(1 2ν) πE ds = − p(x) ux x(x, 0) x s − 2(1 ν) − 2(1 ν2) , b − − − − Za p(s) π(1 2ν) πE ds = − τ(x) uy x(x, 0) x s 2(1 ν) − 2(1 ν2) , b − − − − If in contact interface we can prescribe p, ux,x or τ, uy,x, then the problem reduces to Za (x) F ds = (x) x s U b − The general solution (case a− = b):

Za Za 1 √a2 x2 (s) ds C (x) = − U + , C = (s)ds F π2 √a2 x2 x s π √a2 x2 F a − a − − − −

flat frictionless punch, consider P.V.

V.A. Yastrebov 32/18 Three-dimensional problem

Analogy to Flamant’s problem Potential functions of Boussinesq Boussinesq problem concentrated normal force Cerruti problem concentrated tangential force 1 Displacements decay as r− ∼ 1 2ν N ur(x, y, 0) = − p − 4πG x2 + y2 1 ν N uz(x, y, 0) = − p 4πG x2 + y2 2 Stress decay as r− ∼ Superposition principle

V.A. Yastrebov 33/18 Classical contact problems

Various problems with rigid flat stamps: circular, elliptic, frictionless, full-stick, finite friction Hertz theory normal frictionless contact of elastic solids 2 2 Ei, νi and zi = Aix + Biy + Cixy, i = 1, 2 K.L. Johnson Wedges (coin) and cones Circular inclusion in a conforming hole Steuermann, 1939,Goodman, Keer, 1965 Frictional indentation z xn Incremental approach Mossakovski,∼ 1954 self-similar solution Spence, 1968, 1975 I.G. Goryacheva Adhesive contact Johnson et al, 1971, 1976 Contact with layered materials (coatings) Elastic-plastic and viscoelastic materials Sliding/rolling of non-conforming bodies Cattaneo, 1938,Mindlin, 1949,Galin, 1953,Goryacheva, 1998 Note: ur (1 2ν)/G, so if (1 2ν )/G = (1 2ν )/G tangential ∼ − − 1 1 − 2 2 tractions do not change normal ones V.L. Popov V.A. Yastrebov 34/18 Next. . .