Séminarie du groupe de travail Méthodes Numériques du Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie, Paris 6

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1 Séminarie du groupe de travail Méthodes Numériques du Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie, Paris 6 MODÉLISATION ET SIMULATION NUMÉRIQUE DU DURCISSEMENT INDUSTRIEL DE L ACIER PAR INDUCTION-CONDUCTION J. M. Díaz Moreno, C. García Vázquez M. T. González Montesinos and F. Ortegón Gallego Departamento de Matemáticas, Universidad de Cádiz Departamento de Matemática Aplicada I, Universidad de Sevilla ESPAGNE Paris, le 7 mai 2012 This research was partially supported by Ministerio de Educación y Ciencia under grant MTM with the participation of FEDER, and Consejería de Educación y Ciencia de la Junta de Andalucía, research group FQM-315

2 Steel hardening In the automotive industry, many important moving (rotating/translating) pieces are in close contact in order to transmit the desired rotation/translation movement: gear wheels, toothed rings, bevel gears, rack and pinion, etc.

3 Steel hardening One common example is the steering rack and pinion.

4 Steel hardening

5 Steel hardening The rack and pinion are very important workpieces inside an automobile. It is part of the steering system of the vehicle. The lifetime of the rack and pinion kit must be of at least 20 years!

6 Steel hardening Wear and abrasion: drill chuck and key.

7 Steel hardening These workpieces are made of steel. Prior to any hardening treatment, steel is a ductile material. Rotating/translating workpieces in close contact are subject to stresses during its lifetime. Hardening treatment is necessary in order to avoid wear and abrasion.

8 Steel hardening A convenient hardening treatment is then applied in order to produce: a hard boundary layer to hinder wear and abrasion, and a soft inner part to reduce fatigue. hard soft

9 Some facts on steel Steel is an iron based alloy. Iron may appear in two type of crystal lattices: face centered cubic (fcc) body centered cubic (bcc) Different solid phases in steel: Austenite: Solution of C in fcc iron. Only possible if concentration of C up to 2.11%; if so, only possible at a high temperature range. Ferrite: Nearly pure bcc iron. Pearlite: Lamellar structure of ferrite and cementite (Fe 3 C). Martensite: Tetragonally bbc iron crystal distorted by C atoms. It can only stem from austenite.

10 Iron Carbide Phase Diagram Liquid Temperature ( C) C γ+liquid γ 1148 C A (Austenite) A + C 912 C 727 C 0.77 (eutectoid) 600 α (Fe) α + C Fe hypereutectoid % C Fe 3 C (C) hypoeutectoid

11 Iron Carbide Phase Transitions ferrite heating cooling ferrite pearlite bainite martensite austenite pearlite bainite martensite Phase transitions in hypo/hyper/eutectoid steel Austenite pearlite, bainite (slow cooling down temperature rate) Austenite martensite (very rapid cooling down temperature rate) Phases have different physical properties Pearlite: soft and ductile. Martensite: hard and brittle.

12 TTT Diagrams Temperatura ( C) 800 A c3 A c1 700 A M s A + M 200 M f 100 A + F A + F + C A + F + C P 1 P 2 P 3 P Hardness (Rockwell) P 1 : Martensite P 2 : Martensite and nodular pearlite P 3 : Fine pearlite P 4 : Pearlite 1s 2s 5s 10s 20s 50s 100s 200s 500s 10 3 s 10 4 s 10 5 s Time in seconds 1mn 2mn 15mn 1h 2h 4h 8h 24h 62

13 The industrial procedure: 1. Heating An inductor (copper) is put in contact with the workpiece. A high frequency electric current (83 kh) is supplied. This induces eddy currents. Joule s effect heats up the boundary layer till austenization is reached.

14 The industrial procedure: 2. Cooling The power supply is switched off, and the workpiece is then quenched (aquaquenching). Martensite is produced just where it is needed.

15 The industrial procedure: 2. Cooling

16 Thermomechanical phenomena thermal strain temperature dependent parameters stress σ displacement u mechanical dissipation temperature θ transformation kinetics transformation kinetics transformation strain and plasticity phase fraction dependent parameters latent heat phase fraction dependent parameters phase volume fractions z = (z 0, z 1, z 2 ) z 0 + z 1 + z 2 = 1

17 Steel hardening: Maxwell equations

18 Steel hardening

19 Steel hardening This monument was unveiled on November 25th 2008, George Street, Edinburgh, Scotland, UK.

20 Thermomechanical + electromagnetics modeling D. Hömberg, W. Weiss, K. Chełminski, D. Kern (WIAS, Berlin): Electromagnetics production (main heat source) (b(θ) φ) = 0 b 0 (θ)a t + ( 1 µ A)+b 0(θ) φ = 0 A = 0 Viscoelasticity { σ = f σ = K(ε(u) β(θ, z)i t 0 γ(θ, z, z τ)s dτ) Phase { fractions z1,t = max((a eq (θ) z 1 )/τ a (θ), 0)H(θ A s3 ) z 2,t z 2,t = 1.4(1 z 2 )H(M s θ)h( θ (t)) Temperature { α(θ,σ, z)θt (k θ)+3κβ θ θ u t = b(θ) A t + φ 2 +(ρl+trσ z β + 9κθβ θ z β)z t +γ(θ, z, z t ) S 2 whereα(θ,σ, z) = ρc+trσβ θ + 9κβ 2 θ θ, κ is the bulk modulus and L is the latent heat.

21 Electromagnetics + thermal + phase fractions modeling Simplified model by neglecting mechanical effects. Ω = Ω c Ω s S D Ω c Γ S Ω s

22 EM + thermal + phase fractions modeling Heating stage (Joule s heating): (0, T h ) (b(θ) φ)=0 in Ω (0, T h ), b(θ) φ [ n ] = 0 on Ω (0, T h), = j S onγ (0, T h ), b(θ) φ n b 0 (θ)a t + ( 1 µ A) δ ( A)+b 0(θ) φ = 0 in D (0, T h ), A = 0 on D (0, T h ), A(0)=A 0 in Ω, z t = F(θ, z) in Ω s (0, T h ), z(0)=z 0 in Ω s, ρc ǫ θ t (k(θ) θ) = b(θ) A t + φ 2 +ρlz t + G in Ω (0, T h ), θ n = 0 on Ω (0, T h), θ(0)=θ 0 in Ω.

23 EM + thermal + phase fractions modeling Cooling stage (aquaquenching): (T h, T c ) z t = F(θ, z) in Ω s (T h, T c ), z(t h )=z Th in Ω s, ρc ǫ θ t (k(θ) θ) = ρlz t + G in Ω (T h, T c ), k(θ) θ n =η(θ θ e) on Ω (T h, T c ), θ(t h )=θ Th in Ω. η: heat transfer coefficient.

24 EM + thermal + phase fractions modeling The harmonic regime Electromagnetic fields generated by high frequency currents are sinusoidal in time. Then, we may introduce the complex-valued fieldsϕ, A and j as φ = Re[e iωt ϕ(x, t)], A = Re[e iωt A(x, t)], j S = Re[e iωt j(x, t)]. and approximate 1 t+ω A t + φ 2 1 ω t 2 iωa+ ϕ 2.

25 EM + thermal + phase fractions modeling Heating stage in the harmonic regime ( 1 iωb 0 (θ)a+ µ A (b(θ) ϕ) = 0 in Ω (0, T h ), ϕ n = 0 on Ω (0, T h), [ b(θ) ϕ ] = j onγ (0, T h ), n Γ ) δ ( A) = b 0 (θ) ϕ in D (0, T h ), A = 0 on D (0, T h ), z t = F(θ, z) in Ω s (0, T h ), z(0) = z 0 in Ω s, ρc ǫ θ t (k(θ) θ) = 1 2 b(θ) iωa+ ϕ 2 +ρlz t + G in Ω (0, T h ), θ n = 0 on Ω (0, T h), θ(, 0) = θ 0 in Ω, An existence result by M. T. González Montesinos and FOG.

26 Numerical simulation Finite elements in space (P2-Lagrange) + finite difference in time (Crank-Nicolson): elements, vertices

27 Numerical simulation Dh

28 Numerical simulation (Rockwell) Temperature: heating stage, Th = 5.5 seconds P24 c3 2 Ms P21 IsoValue J. M. Díaz, C. García, M. T. González and F. Ortegón t=0 t=1 t=3 t=5 t = 5.5 Simulation numérique du durcissement de l acier

29 Numerical simulation (Rockwell) Temperature: cooling stage, Tc = 11 seconds P24 c3 2 Ms P21 IsoValue J. M. Díaz, C. García, M. T. González and F. Ortegón t = 5.57 t = 5.97 t = 6.23 t = 7.08 t = 11 Simulation numérique du durcissement de l acier

30 Numerical simulation Austenite transformation at the end of the heating stage T h = 5.5 seconds IsoValue

31 Numerical simulation Martensite transformation at the end of the cooling stage T c = 11 seconds Martensita final IsoValue