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  • Coupling of Elastic or Elastic-Plastic Solids with Compressible Two-Phase Fluids for the Numerical

    Investigation of Cavitation Damaging

    Christian Dickopp

    Siegfried Müller, Roman Gartz, Mathieu Bachmann

    Institut für Geometrie und Praktische Mathematik

    RWTH Aachen University

    University Pierre and Marie Curie, Paris

    DFG-CNRS-Research Group FOR 563 ”Micro-Macro Modelling and Simulation of Liquid-Vapor Flows”

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 1

    / 34

  • 1 Motivation

    2 Model Problem and Numerical Setup

    3 Mathematical Models and Numerical Methods

    4 Numerical Results

    5 Conclusion

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 2

    / 34

  • Outline

    1 Motivation

    2 Model Problem and Numerical Setup

    3 Mathematical Models and Numerical Methods

    4 Numerical Results

    5 Conclusion

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 3

    / 34

  • Material Damage

    (Published under the ShareAlike Licence v. 2.5)

    Lord Rayleigh1: ”Hydraulic blows” (pressure waves) emitted by collapsing vapor bubbles near ship propellers

    1 Lord Rayleigh, On the pressure developed in a liquid during the collapse of a spherical cavity, Phil. Mag., 34 (1917), 94–98.

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 4

    / 34

  • Experiments with Laser-Induced Cavitation Bubbles 2

    Shock wave from bubble collapse near solid wall, von Schmidt wave and counter jet

    (C o

    u rt

    es y

    o f

    H .

    S ö

    h n

    h o

    lz )

    (d = Rmax = 0.7mm, image section: 7.5mm × 4.0mm)

    2 H. Söhnholz: Temperatureffekte bei der laserinduzierten Kavitation. Dissertation, Universität Göttingen, to be submitted).

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 5

    / 34

  • Experiments with Laser-Induced Cavitation Bubbles3

    Damage on aluminium specimens caused by 100 cavitation bubbles

    (γ = 1.91)

    (C o

    u rt

    es y

    o f

    A .

    P h

    il ip

    p )

    (γ = 1.41)

    γ = distance between bubble origin and wall

    maximal bubble radius 3

    A. Philipp, W. Lauterborn, Cavitation erosion by single laser-produced bubbles, J. Fluid Mech., 361 (1998), 75–116.

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 6

    / 34

  • Experimental Investigations of Cavitation Bubbles

    Experimental findings: Bubble oscillation, splitting and coalescing Formation of liquid jets/counter jets Penetration of bubble and emission of pressure waves into liquid Formation of a vortex ring and its breakup into a swarm of tiny bubbles Light emission

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 7

    / 34

  • Experimental Investigations of Cavitation Bubbles

    Experimental findings: Bubble oscillation, splitting and coalescing Formation of liquid jets/counter jets Penetration of bubble and emission of pressure waves into liquid Formation of a vortex ring and its breakup into a swarm of tiny bubbles Light emission

    Short comings: Dynamics of flow field can only be visualized by discrete snapshots or probes at local positions Measurements of gas states inside bubble not possible

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 7

    / 34

  • Experimental Investigations of Cavitation Bubbles

    Experimental findings: Bubble oscillation, splitting and coalescing Formation of liquid jets/counter jets Penetration of bubble and emission of pressure waves into liquid Formation of a vortex ring and its breakup into a swarm of tiny bubbles Light emission

    Short comings: Dynamics of flow field can only be visualized by discrete snapshots or probes at local positions Measurements of gas states inside bubble not possible

    =⇒ Need numerical simulations Improve understanding of the phenomena Gain new insight in the causal connections with material damage

    Our numerical simulations show a reasonable explanation !

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 7

    / 34

  • Outline

    1 Motivation

    2 Model Problem and Numerical Setup

    3 Mathematical Models and Numerical Methods

    4 Numerical Results

    5 Conclusion

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 8

    / 34

  • Numerical Setup

    Model problem: A single bubble collapses near the surface of an elastic or elastic–plastic structure:

    Material parameters and homogeneous initial conditions: Steel Water Air

    ρ kg/m3 7800 ρ kg/m3 1000 0.0266 E Pa 210× 109 p Pa 5× 107 2118 v m/s 0 v m/s 0 0

    c1 (c2) m/s 5990 (3458) c m/s 1480 340 λ kg/m s2 9.3× 1010 γ 7.15 1.4 µ kg/m s2 9.3× 1010 π Pa 3× 108 0

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 9

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  • Outline

    1 Motivation

    2 Model Problem and Numerical Setup

    3 Mathematical Models and Numerical Methods

    4 Numerical Results

    5 Conclusion

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 10

    / 34

  • Solid: Linear Elastic or Linear Elastic - Linear Plastic

    Most important assumption: small displacement velocities (⇒ DfDt := ∂t f + v

    T · ∇f ≈ ∂t f ) Starting point: Newton’s law Differentiation of Hooke’s law w.r.t. time: ∂σ ∂ t = λ div (v) I + µ

    ( ∇ (v) + (∇ (v))T

    ) σ: stress tensor, λ,µ: Lamé-coefficients, v: displacement velocity IN PROGRESS: Linear elastic - linear plastic stress-strain relation: elastoplastic tangent modulus as the slope of the stress–strain relation is switched between Youngs modulus and a modulus of regidity (plastic modulus) depending on the yield criterion following von Mises Radial Return mapping onto the yield surface if a trial stress state after update within a time step fullfills the yield criterion (Neuber’s method) - orthogonal to yield surface (Drucker-hypothesis) Plastic multiplier h as additional dependent variable ⇒ Hyperbolic (sub)system of equations for v, σ and h

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 11

    / 34

  • Motivation of the elastic-plastic modelling

    First motivation: 1D-experiment with a solid speciment like metal:

    General stress-strain relation: Brittle material: sudden failure

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 12

    / 34

  • Linear elastic-plastic model

    Approximation by two linear stress-strain relations using as slopes Young’s modulus E within the elastic range and the plastic modulus K characterizing strain hardening; switch/intersection at the yield stress Y → elastoplastic tangent modulus mathematical/numerical aspect: additional nonlinearity within the system of equations needs special numerical treatment → return mapping algorithms

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 13

    / 34

  • Return mapping in 1D

    Basic steps of return mapping algorithms in 1D for isotropic hardening materials for time stepping n→ n + 1:

    Calculation of one time step using the pure elastic approximation method leading to trial stress states σtrialn+1 Check the yield criterion f (σtrialn+1) = |σtrialn+1| − (Y + Kαn) for every obtained stress state σtrialn+1 using hardening variables α on time level n If f (σtrialn+1) < 0⇒ pure elastic stress state⇒ set σn+1 = σtrialn+1 READY If f (σtrialn+1) ≥ 0 ⇒ plastic parts of the stress state ⇒ RETURN MAPPING

    (different strategies !) of σtrialn+1 onto the yield curve

    ⇒ σn+1, �n+1, �pn+1

    Christian Dickopp Cavitation Damaging University Pierre and Marie Curie, Paris 14

    / 34

  • Concepts for elasto-plastic modelling in 2D and 3D

    Most appropriate (pressure-insensitive) yield functions for metals in two or three space dimensions:

    f (σ) := σcompar (σ)− σy (α) (1)

    α: hardening internal variable, for example equivalent, plastic stress (kind of integral meassure for the plastic strains)

    Comparison stress σcompar maps the stress tensor to a real value (1D):

    Tresca yield criterion: based on the maximum shear stress τmax (σ) (σ1, σ2, σ3: principial stresses):

    σmin := min (σ1, σ2, σ3) , σmax := max (σ1, σ2, σ3) (2)

    σcompar (σ) := τmax (σ) := 1

    2 (σmax − σmin) (3)

    von Mises yield criterion: based on the distortional strain-energy, described by the invariant J2(σ) of the stress deviator s(σ):

    σcompar (σ) := J2(σ) := 1

    2 s(σ) : s(σ) s(σ) := σ − 1

    3 trace (σ) (4)

    HERE: von Mises yield criterion is used Christian Dickopp Cavitation Damaging

    University Pierre and Marie Curie, Paris 15 / 34

  • Yield criteria and return mapping in 2D - 1

    Tresca and von Mises yield criteria in 2D:

    Return mapping in 2D (radial or othogonal version):

    Christian Dickopp Cavitation Damaging