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 r · = v 8 2 1 c Displacement compatibility 2 c 1 2 " = ( u + u ) f = 2 r r u Constitutive equation σ = W0(") = = Two bodies in contact Boundary conditions 0 Dirichlet: u = u ; x Γu 8 2 0 Neumann: n σ = t ; x Γf · = 8 2 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 r · = v 8 2 1 c Displacement compatibility 2 c 1 2 " = ( u + u ) f = 2 r r u Constitutive equation σ = W0(") = = Two bodies in contact Boundary conditions Include contact conditions 0 ... Dirichlet: u = u ; x Γu 8 2 0 Neumann: n σ = t ; x Γf · = 8 2 V.A. Yastrebov 4/18 Intuitive conditions 1 No penetration Ω1(t) Ω2(t) = \ ; f 2 No adhesion 1 i n σ n 0; x Γc u · = · ≤ 8 2 1 c 3 No shear stress 2 c i n σ (I n n) = 0; x Γc 2 · = · − ⊗ 8 2 f Twou bodies in contact Intuitive contact conditions for frictionless and nonadhesive contact V.A. Yastrebov 5/18 Intuitive conditions 1 No penetration Ω1(t) Ω2(t) = \ ; f 2 No adhesion 1 i n σ n 0; x Γc u · = · ≤ 8 2 1 c 3 No shear stress 2 c i n σ (I n n) = 0; x Γc 2 · = · − ⊗ 8 2 f Twou 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 p = n σ n 0 − · = · ≥ 1 2 f which being applied at Γc and Γc results in x1 = x2; x1 Γ1; x2 Γ2 1 8 2 c 2 c and evidently u 1 Ω1(t) Ω2(t) = c \ ; 2 c Two bodies in contact 2 1 f Unfortunately, we do not know Γc in advance, u 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 p = n σ n 0 − · = · ≥ 1 2 f which being applied at Γc and Γc results in x1 = x2; x1 Γ1; x2 Γ2 1 8 2 c 2 c and evidently u 1 Ω1(t) Ω2(t) = c \ ; 2 c Two bodies in contact 2 1 f Unfortunately, we do not know Γc in advance, u 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 j j 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 j j 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