Moving layers and interfaces : a thermodynamical approach of Wear Contact

Transcription

1 Moving layers and interfaces : a thermodynamical approach of Wear Contact C. Stolz M. Peigney, M. Dragon-Louiset LMS, CNRS-UMR7649, Ecole polytechnique, Palaiseau LaMSID, CNRS-UMR2832, EdF R&D, Clamart 1

2 Sxh= Ω 3 Ω 1 V Ω 2 Loss of sound material is due to wear. Friction - contact between bodies - interface propagation 2

3 Essai de video essai de vidéo 3

4 Ω 1 Ω 1 Γ 13 Γ 13 Sxh= Ω 3 Ω 3 Ω 3 Γ 23 Γ 23 Ω 1 V Ω 2 Ω 2 Ω 2 Model of damage - Dissipation - Global behaviour of the interface Ω 3 Ω 3 = Sxh : detached particles, damaged media, fluids : complex behaviour Integration over the thickness : constitutive equation for the interface S 4

5 Moving interface / moving layer Moving interface / moving layer Material characteristics change : Brutal transition : moving interface and discontinuities Continuous Transition : moving layer and continuous damage laws. Symi-2010-Keer symposium 5

6 Moving interface and Transformation of material Moving interface and Transformation of material Material 1 is changed in material 2 along a moving interface the mass flux is the loss of sound material. Symi-2010-Keer symposium 6

7 Preliminaries Preliminaries φ n Ω 2 Ω 1 Γ F = b a f(x, t) dx Γ(t) b = a f(x, t) dx + Γ(t) f(x, t) dx Symi-2010-Keer symposium 7

8 Preliminaries φ n Ω 2 Ω 1 Γ F = b a f dx + ( f(γ +, t) f(γ, t) ) Γ [ f ] Γ = f(γ +, t) f(γ, t) Propagation of interface jump conditions. Symi-2010-Keer symposium 8

9 Preliminaries φ n Ω 2 Ω 1 Γ Each medium is linear elastic : x proportion of phase 2 Equilibrium : Energy : Energy release rate σ = E 1 ε 1 = E 2 ε 2 = Σ, u(0) = 0 W(σ, x) = 1 2 σ2 ( x E x E 1 ) G = W x = 1 2 σ2 ( 1 E 2 1 E 1 ); D m = Gẋ Symi-2010-Keer symposium 9

10 Preliminaries Quasicrack : antiplane shear conditions τ α > 0 α = 0 σ = K/ rf(θ) as r e 2 A O R o α < 0 R m R h σ = 0 Ω d Γ c e 1 [ K 2 = h τ2 o ( R m ) α+1 + α 1 ] α + 1 R o 2 A Neuber (1969), Bui-Ehrlacher (1978), Stolz (2010). Symi-2010-Keer symposium 10

11 Wear description- Dissipation Wear description- Dissipation Ω 1 Ω 1 Γ 13 Γ 13 Sxh= Ω 3 Ω 3 Ω 3 Γ 23 Γ 23 Ω 1 V Ω 2 Ω 2 Ω 2 D m = S (G 13 φ 1 + G 23 φ 2 + h d 3 dz) ds G = [ w ] Γ σ : [ ε ] Γ Symi-2010-Keer symposium 11

12 Interpretation Interpretation D Γ = G φ, m = ρφ Then m = 0, no loss of sound material no wear friction D 3 = G(X, t) < G c, φ = 0 G(X, t) = G c, φ 0 h d m dz = h (σ : ε w 3 ) dz Symi-2010-Keer symposium 12

13 Moving Layer Moving Layer The material is described with continuous damage In Ω free energy : w(ε, d, α) = (1 d)w o (ε, α) + H(α) d varies from 0 to 1, no discontinuities. Y = w d, A = w α d m = Y d + A α Symi-2010-Keer symposium 13

14 Dissipation Dissipation d(x) φ Ω 2 Ω 3 Ω 1 but in the moving frame D m = h 2h Y d + A α dz f(x, t) = f(x φt n, t) f = φ f.n + D φ f Symi-2010-Keer symposium 14

15 Dissipation 1-D Example : Free energy : W = 1 2 E(d)ε2 2h d(x) φ Ω 2 Ω 3 Ω 1 Matching Conditions E(0) = E 1, E(1) = E 2 Total Dissipation under steady-state condition ( d + φ d.n = 0) 1 0 D m = φ 2 σ2 ( E E ) d dz = φ1 2 2 σ2 ( E )dd = Gφ E2 h 1 Symi-2010-Keer symposium 15

16 Dissipation with moving layer Dissipation with moving layer D m = φ h (Y d + A α)dz + h (Y D φ d + AD φ α) dz that is D m = Ḡφ + h d φ dz Steady-state motion : the last term is zero. Ḡ = h (Y d + A α)dz hy c Symi-2010-Keer symposium 16

17 Dissipation with moving layer Geometry of the layer, Curvature tensor : n.a α = b β αa β h a α X S h n Γ 2 Γ Γ 1 B ds i = det(b ± i h I) = j i ds f dω = ( f j(z)dz) ds = Γ h Γ f S ds Symi-2010-Keer symposium 17

18 Dissipation with moving layer Eshelby momentum tensor P = w I σ. u Then div P T = Y d A α Introducing the relative coordinate Γ x = X S + zn, D φ x = φn + z φ Ḡφ(s) ds = P : (D φ x) dω + n.p.d φ ( x)j i ds B Γ i Symi-2010-Keer symposium 18

19 Dissipation with moving layer Ω 1 Ω 1 Γ 13 Γ 13 Sxh= Ω 3 Ω 3 Ω 3 Γ 23 Γ 23 Ω 1 V Ω 2 Ω 2 Ω 2 D m = S (Ḡ1φ 1 + Ḡ2φ 2 + h 3 d 3 dz) ds Ḡ i (s) = ( (Y d + A α)j ds).n(s) h i Symi-2010-Keer symposium 19

20 Dissipation with moving layer Ω 1 Γ 13 Sxh= Ω 3 Ω 3 Γ 23 Ω 1 V Ω 2 Ω 2 ψ s ρ s = D s ρ s = h h w(ε, α) z ρ z j(z) dz D( ε, α) z ρ z j(z) dz Symi-2010-Keer symposium 20

21 Dissipation associated with the interface Dissipation associated with the interface Total potential energy : Ω wρ dω + Γ ψ sρ s ds D m = n.σ.v dω d dt Ω ( w i ρ i dω + i = + Ω Γ Γ ( σ i : [ ε i ] Γ + ρ i (w i ψ S ))j i φ i ds D S v i.d φ u i + D S α.d φα + Q.D φ ρ s ds The mass density is an internal parameter for the interface. Γ ρ s ψ S ds) Symi-2010-Keer symposium 21

22 The behaviour of the layer The behaviour of the layer Main idea : h z h Continuity of displacement Then u 3 (X S, z) = u o (X S ) + zu 1 (X S ) +... u 1 = u 3 (X s, h), u 2 = u 3 (X s, h) w(x s ) = u 1 u 2 = 2h u.n, u o = 1 2 (u 1 + u 2 ) + o(h 2 ) Introducing these asymptotic expansion in global potential for the layer Symi-2010-Keer symposium 22

23 Order 0 Order 0 Free global energy : ρ s ψ s (u 1, u 2, α) = Thermodynamical forces h w 3 (ε(x + zn), α)ρ 3 j(z)dz Global potential of dissipation Then Equilibrium on interface T r ψ s ψ s i = ρ s, A = ρ s u i α, G ψ s h = ρ s h ρ s D s (v 1, v 2, α) = T ir i h = ρ s D s v i,... n.σj i = T r i + T ir i ρ 3 d 3 ( ε, α)dz Symi-2010-Keer symposium 23

24 More informations : order 1 More informations : order 1 The energy depends on u o, w, so ρ s ψ s = f(u i, u i ), ρ s D s = g(v i, v i ) ρ s = ρ 3 dz, Conservation The equilibrium is then obtained as h n.σj i = ρ s ψ s u i (ρ s ψ s u i ) + ρ s D s v i (ρ s D s v i ) Symi-2010-Keer symposium 24

25 More informations : order 1 Example of constitutive laws : f volume fraction of defects ψ s = 1 2 k(f)(w n) µ(f)(w t α) 2 d(ẇ, α) = 1 2 η t(f)ẇt η n(f)ẇn H(f) α2 2 n.σ = (k n w n +η n ẇ n )n+(µ(w t α)+η t ẇ t )τ; A = k t (w t α) = H α Strömberg, IJSS 1996, Del Piero, Eur. J. Mech (2010), Stupkiewicz, Wear (1990),... Symi-2010-Keer symposium 25

26 Moving punch on a half-space Moving punch on a half-space V Rigid Punch on an elastic half space. (Dragon-Louiset,2000) Symi-2010-Keer symposium 26

27 Moving punch on a half-space The interface 3 : fluid + particles. Suspension of particles in a viscous fluid : κ(c), η(c) Integral equations on an half elastic space : Galin s Equation Displacement on the boundary due to stresses on the boundary. ψ s = 1 2 κ(c)w2 n, D s = 1 2 η(c)ẇ2 t Symi-2010-Keer symposium 27

28 Moving punch on a half-space Behavior of the interface (3) viscosity in shear σ xy = η(c)( u 1 x u 2 x), σ yy = k(c)(u 1 y u 2 y) Galin s Equations for the elastic half space c 1 u x,x (x) = c 2 σ yy (x) + V p 1 π c 1 u y,x (x) = c 2 σ xy (x) + V p 1 π a a a σ xy (s) s x ds a σ yy (s) s x ds c 1 = E 2(1 ν 2 ), c 2 = (1 2ν) 2(1 ν) Symi-2010-Keer symposium 28

29 Moving punch on a half-space Asymptotic solution for small concentration of particles c Law η(c), κ(c). order 0 : Hertz contact solution order 1 : depends upon criterion of wear : φ = λσ 2 yy Symi-2010-Keer symposium 29

30 Effect of viscosity in shear Effect of viscosity in shear Equivalent to answer with friction law. Symi-2010-Keer symposium 30

31 One more example One more example Cyclic accommodation for a Cyclic Horizontal motion of a rigid punch L poinçon translation périodique Symi-2010-Keer symposium 31

32 One more example The solution of an half space with a boundary η(x) In Galin s Equation, the kernel is changed at order 1 in η. u 1 (x) = A N o (x, y)t 1 (y)dy + A N 1 (x, y)t o (y)dy The applied loading follows the boundary, using of D φ (σ.n) leads to D φ (σ.n) = σ.n + d (ησ.τ) + γφσ.n ds Symi-2010-Keer symposium 32

33 Asymptotic Respons 1 Asymptotic Respons Final State : η(x) Control Variable η Cost function (dissipation) J J = η dx + a Γ T Γ < G(η, x, t) G c > 2 + dxdt Symi-2010-Keer symposium 33

34 Results Results Lost of matter : Γ η(x) dx G/10G c p/10max(p) x/a Symi-2010-Keer symposium 34

35 Results G/10G c p/10max(p) x/a Symi-2010-Keer symposium 35

36 Results Possibility of using homogeneization technics to describe wear contact and to propose wear laws. The mass density in the interface is a important internal parameter. Symi-2010-Keer symposium 36

37 Results Peigney M : Int. J. Solids and Structures, 2004,41, Dragon-Louiset, Stolz : Comptes Rendus Acad. Sci. Paris, 1999, 327, Dragon-Louiset : Int. J. Solids Structures, 2001, 38, C. Stolz : Entropy 12, , Bui H.D, D-L, C. S : On Prand l lifting equation arising in wear mechanics Arch. Mech., 2000, 52, Galin, Goryatcheva, Mroz, Zmitrowicz, Stupkiewicz,... Symi-2010-Keer symposium 37