Numerical Investigation of Abradable Coating Wear Through Plastic Constitutive Law Application to Aircraft Engines Mathias Legrand & Christophe Pierre McGill University IDETC/CIE 2009 - DETC2009-87669 San Diego, USA September 1, 2009
Introduction Solution Method Results Conclusion Outline 1 Introduction Problem statement Wear or material removal? General strategy 2 Solution Method Equations of motion Abradable constitutive law: plasticity Proposed algorithm 3 Results Model convergence Modal interactions Final wear profile maps Animations 4 Conclusion Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 2 / 13
Introduction Solution Method Results Conclusion Problem statement Problem statement Technologies for improved efficiency ∙ light and composite materials ∙ few turbine and compressor stages with higher conicity ∙ aerodynamically improved blade designs ∙ higher operating temperatures ■ CFM56- CFM International New strategy: use of abradable coatings ∙ minimal operating clearance for improved compression rates ∙ capable of undergoing structural contacts through wear ∙ unexpected diverging behaviors? Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 3 / 13
Introduction Solution Method Results Conclusion Problem statement Problem statement Technologies for improved efficiency ∙ light and composite materials ∙ few turbine and compressor stages with higher conicity ∙ aerodynamically improved blade designs ∙ higher operating temperatures ■ CFM56- CFM International New strategy: use of abradable coatings ∙ minimal operating clearance for improved compression rates ∙ capable of undergoing structural contacts through wear ∙ unexpected diverging behaviors? modeling of abradable coating wear in aircraft engines Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 3 / 13
Introduction Solution Method Results Conclusion Wear or material removal? Wear or material removal? Referenced mechanisms ∙ abrasion , erosion, corrosion, adhesion, fretting ∙ drilling, grinding , turning, milling ∙ high speed tangential velocities Abradable properties ∙ strength to withstand gas turbine environment ∙ wear characteristics to minimise incurring blade wear Current knowledge and modeling ∙ Wear: experimental and empirical → Archard’s law ∙ Machining : kinematic relationships ∙ CPU-intensive numerical approaches in the FEM framework Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 4 / 13
Introduction Solution Method Results Conclusion General strategy General strategy casing Assumptions abradable ∙ A single blade in contact ∙ Statically distorted casing on interface nodes Ω a 2-nodal diameter shape ∙ No friction Ω ∙ No thermomechanical coupling Solution method ∙ Blade dynamics reduction through Ω Craig-Bampton technique ∙ Abradable coating wear through quasi-static plastic constitutive law fixed boundaries ∙ Time-marching solution method ■ chosen configuration Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 5 / 13
Introduction Solution Method Results Conclusion Equations of motion Equations of motion Virtual Work Principal � � � σ 1 ( E 1 ) : δ ¯ ¯ ρ 1 ¨ u 1 δ u 1 dV + ǫ 1 dV = − ← blade ¯ ¯ t N δ g dS Ω 1 Ω 1 Γ c � � σ 2 ( E 2 , E T , σ Y ) : δ ¯ ¯ ǫ 2 dV = t N δ g dS ← abradable coating ¯ ¯ Ω 2 Γ c Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 6 / 13
Introduction Solution Method Results Conclusion Equations of motion Equations of motion Virtual Work Principal � � � σ 1 ( E 1 ) : δ ¯ ¯ ρ 1 ¨ u 1 δ u 1 dV + ǫ 1 dV = − ← blade ¯ ¯ t N δ g dS Ω 1 Ω 1 Γ c � � σ 2 ( E 2 , E T , σ Y ) : δ ¯ ¯ ǫ 2 dV = t N δ g dS ← abradable coating ¯ ¯ Ω 2 Γ c Finite element discretization M ¨ u + D ˙ u + Ku + λ = 0 ← blade λ = F int ← abradable Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 6 / 13
Introduction Solution Method Results Conclusion Equations of motion Equations of motion Virtual Work Principal � � � σ 1 ( E 1 ) : δ ¯ ¯ ρ 1 ¨ u 1 δ u 1 dV + ǫ 1 dV = − ← blade ¯ ¯ t N δ g dS Ω 1 Ω 1 Γ c � � σ 2 ( E 2 , E T , σ Y ) : δ ¯ ¯ ǫ 2 dV = t N δ g dS ← abradable coating ¯ ¯ Ω 2 Γ c Finite element discretization Contact conditions M ¨ u + D ˙ u + Ku + λ = 0 ← blade λ ⩾ 0 , g ⩾ 0 , λ g = 0 λ = F int ← abradable Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 6 / 13
Introduction Solution Method Results Conclusion Equations of motion Equations of motion Virtual Work Principal � � � σ 1 ( E 1 ) : δ ¯ ¯ ρ 1 ¨ u 1 δ u 1 dV + ǫ 1 dV = − ← blade ¯ ¯ t N δ g dS Ω 1 Ω 1 Γ c � � σ 2 ( E 2 , E T , σ Y ) : δ ¯ ¯ ǫ 2 dV = t N δ g dS ← abradable coating ¯ ¯ Ω 2 Γ c Finite element discretization Contact conditions M ¨ u + D ˙ u + Ku + λ = 0 ← blade λ ⩾ 0 , g ⩾ 0 , λ g = 0 λ = F int ← abradable Explicit time stepping technique: central finite differences u n = u n + 1 − 2 u n + u n − 1 u n = u n + 1 − u n − 1 ¨ ˙ , h 2 2 h Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 6 / 13
Introduction Solution Method Results Conclusion Abradable constitutive law: plasticity Abradable constitutive law: plasticity Assumptions ∙ Uni-axial plasticity in compression only σ ∙ Isotrope hardening: K σ Y f ( σ, α ) = σ − ( σ Y + K α ) ∙ Linear additive law: ε = ε e + ε p E ε σ = E ε e ε p ε e Integration ■ plastic constitutive law ∙ Kuhn & Tucker conditions ∙ Prediction-correction incremental approach: Return Map algorithm ▶ if f p ⩽ 0, admissible prediction ▶ if f p > 0, correction of prediction Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 7 / 13
Introduction Solution Method Results Conclusion Proposed algorithm Proposed algorithm 1. Displacement prediction by neglecting contact constraints � D � − 1 �� 2 M � M h 2 + D � 2 h − M � � u n + 1 , p = h 2 − K u n + u n − 1 h 2 2 h 2. Determination of g n + 1 , p by identifying all abradable elements i ∈ I being penetrated by the blade 3. Abradable internal forces computation u n + 1 , p → ε i ∈ I → ( σ i ∈ I , α i ∈ I ) → F int ε p i ∈ I → abradable profil update 4. Displacements correction consistent with abradable internal forces ( ∼ contact forces) � − 1 � M h 2 + D F c u n + 1 = u n + 1 , p − 2 h Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 8 / 13
Introduction Solution Method Results Conclusion Model convergence Model convergence Modal convergence ∙ Craig-Bampton reduction technique ∙ frequency deviation Δ f = | f f − f r | < 0 . 1 % f f ▶ initial size : ∼ 80 , 000 dof ▶ final size : 109 dof ■ First five modes of the blade Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 9 / 13
Introduction Solution Method Results Conclusion Model convergence Model convergence Modal convergence ∙ Craig-Bampton reduction technique ∙ frequency deviation Δ f = | f f − f r | < 0 . 1 % f f ▶ initial size : ∼ 80 , 000 dof ▶ final size : 109 dof ■ CB componant modes ■ First five modes of the blade ■ CB static modes Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 9 / 13
Introduction Solution Method Results Conclusion Model convergence Model convergence Time-step convergence ∙ Conditionally stable explicit time-stepping scheme: very small time-step ∙ Blade displacement convergence ∙ Abradable wear profile convergence Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 9 / 13
Introduction Solution Method Results Conclusion Model convergence Model convergence Time-step convergence ∙ Conditionally stable explicit time-stepping scheme: very small time-step ∙ Blade displacement convergence ∙ Abradable wear profile convergence 1 1 vibratory level 0 wear level -1 0.5 -2 -3 0 0 T f 0 2 π angular position on the casing (rad) time ■ Final abradable profiles for ■ Vibratory level of the blade for different densities interface node 1 with different abradable densities Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 9 / 13
Introduction Solution Method Results Conclusion Modal interactions Modal interactions 1,5 | FFT ( u ) | (mm) 1 k=6 k=4 k=8 0,5 k=2 0 0,4 1,5 0,3 1 0,2 0,5 0,1 freq. Ω 0 0 ■ No wear Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 10 / 13
Introduction Solution Method Results Conclusion Modal interactions Modal interactions 1,5 | FFT ( u ) | (mm) 1 0,5 0 0,4 1,5 0,3 1 0,2 0,5 0,1 freq. Ω 0 0 ■ Low wear Mathias Legrand IDETC-CIE 2009: Numerical Investigation of Abradable Coating Wear 2009-10-01 10 / 13
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