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Coupling of Elastic or Elastic-Plastic Solids withCompressible Two-Phase Fluids for the Numerical

Investigation of Cavitation Damaging

Christian Dickopp

Siegfried Muller, Roman Gartz, Mathieu Bachmann

Institut fur 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 DamagingUniversity Pierre and Marie Curie, Paris 1

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1 Motivation

2 Model Problem and Numerical Setup

3 Mathematical Models and Numerical Methods

4 Numerical Results

5 Conclusion


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Outline

1 Motivation



4 Numerical Results

5 Conclusion


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Material Damage

(Published under the ShareAlike Licence v. 2.5)

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

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


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Experiments with Laser-Induced Cavitation Bubbles 2

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

(Co

urt

esy

of

H.

So

hn

ho

lz)

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

2H. Sohnholz: Temperatureffekte bei der laserinduzierten Kavitation. Dissertation, Universitat Gottingen, to be submitted).


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Experiments with Laser-Induced Cavitation Bubbles3

Damage on aluminium specimens caused by 100 cavitation bubbles

(γ = 1.91)

(Co

urt

esy

of

A.

Ph

ilip

p)

(γ = 1.41)

γ =distance between bubble origin and wall

maximal bubble radius3

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


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Experimental Investigations of Cavitation Bubbles

Experimental findings:

Bubble oscillation, splitting and coalescingFormation of liquid jets/counter jetsPenetration of bubble and emission of pressure waves into liquidFormation of a vortex ring and its breakup into a swarm of tiny bubblesLight emission


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Short comings:

Dynamics of flow field can only be visualized by discrete snapshots orprobes at local positionsMeasurements of gas states inside bubble not possible


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Short comings:

Dynamics of flow field can only be visualized by discrete snapshots orprobes at local positionsMeasurements of gas states inside bubble not possible

=⇒ Need numerical simulations

Improve understanding of the phenomenaGain new insight in the causal connections with material damage

Our numerical simulations show a reasonable explanation !


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Outline

1 Motivation



4 Numerical Results

5 Conclusion


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Numerical Setup

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

Material parameters and homogeneous initial conditions:Steel Water Air

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

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


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Outline

1 Motivation



4 Numerical Results

5 Conclusion


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Solid: Linear Elastic or Linear Elastic - Linear Plastic

Most important assumption: small displacement velocities(⇒ Df

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

(∇ (v) + (∇ (v))T

)σ: stress tensor, λ,µ: Lame-coefficients, v: displacement velocityIN PROGRESS:Linear elastic - linear plastic stress-strain relation:elastoplastic tangent modulus as the slope of the stress–strain relationis switched between Youngs modulus and a modulus of regidity(plastic modulus) depending on the yield criterion following von MisesRadial Return mapping onto the yield surface if a trial stress stateafter 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


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Motivation of the elastic-plastic modelling

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

General stress-strain relation: Brittle material: sudden failure


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Linear elastic-plastic model

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


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Return mapping in 1D

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

Calculation of one time step using the pure elastic approximationmethod leading to trial stress states σtrial

n+1

Check the yield criterion f (σtrialn+1) = |σtrial

n+1| − (Y + Kαn) for everyobtained stress state σtrial

n+1 using hardening variables α on time level nIf f (σtrial

n+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 σtrial

n+1 onto the yield curve⇒ σn+1, εn+1, ε

pn+1


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

Most appropriate (pressure-insensitive) yield functions for metalsin 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

2s(σ) : s(σ) s(σ) := σ − 1

3trace (σ) (4)

HERE: von Mises yield criterion is usedChristian Dickopp Cavitation Damaging

University Pierre and Marie Curie, Paris 15/ 34

Yield criteria and return mapping in 2D - 1

Tresca and von Misesyield criteria in 2D:

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


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Yield criteria and return mapping in 2D - 2

Same treatmemt of stress states on the yield surface indicating that plasticeffects have already taken place:

A more complicated model for plasticity including nonlinear,isotropic/kinematic hardening based on the J2-stress invariant (von Misescriterion) and Prandtl-Reuss equations as plastic flow rule is justimplemented


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Modelling of the Two-phase Flow

Compressible Euler equations for gas and liquidLevel set: sharp interface approach for bubbles wallStiffened gas law for liquid and gas, material parameters chosen bysign of the level set functionReal ghost fluid method at the gas-liquid interface:

Fluid A Ghost Fluid A

Pi

u i

iRρ

Ghost Fluid B Fluid B

i−2 i−1 i i+1 i+2

i−2 i−1 i i+1 i+2

UR=Ui+1

UL=Ui

Pi

iLρ

u iN

Phase boundary

N


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WEIGHTED Fluid-Structure Coupling

Weak, Weighted Fluid-Structure Coupling:Alternating calculations by both solvers (partitioned, Gauss-Seidel orDirichlet-Neumann approach) using additional weighting factors κn

f

and κns for error damping within the boundary conditions obtained by

a splitting of the transition conditions at the interface Γ:

Solid

Un+1

= Un

+ ∆tRs(Un)

nTΓσ nΓ = −κn

f p(Un+1) on Γ

p←−

vT nΓ−→

Fluid

Un+1 = Un + ∆ tRf (Un)

vT nΓ = κns vT (U

n)nΓ on Γ

Errors concerning kinematic and dynamic transiton conditions:

εn+1kin := vT (Un+1)nΓ − vT (U

n+1) nΓ ≈ 0 (5)

εn+1dyn := nT

Γσ(U

n+1) nΓ + p(Un+1) ≈ 0 (6)

Idea: Error reduction by choice of the weighting factors κnf and κn

s

Stable, Fast and Accurate!Christian Dickopp Cavitation Damaging

University Pierre and Marie Curie, Paris 19/ 34

Outline

1 Motivation



4 Numerical Results

5 Conclusion


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Numerical Setup


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Numerical Setup

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

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


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Numerical Setup

Solid Liquid-GasΩ [−0.04, 0.01]× [0, 0.05] m2 [0.01, 0.06]× [0, 0.05] m2

l = 0 4× 4 16× 16L = 8 2048× 2048 6400× 6400

∆t 10−10 s 10−10 sN 100000 100000T 10 µs 10 µs


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Numerical Results

t = 0.50 µs

Pressure gradient magnitude

higher pressure in liquid: rarefaction wave in liquid shock wave inside bubble shrinking of bubble


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Numerical Results

t = 0.50 µs

Normal stress gradient magnitude Pressure gradient magnitude


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Numerical Results

t = 1.50 µs


liquid:rarefaction wave is reflected at bubbleand structure interface with decreasingstrength

gas bubble:shock wave focusses in center and isreflected at interfacebubble continues shrinking at increasingspeed

reflection at structure interface: von Schmidt wave due to dilatationwaves (solid) propagating faster thansound waves (liquid)


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Numerical Results

t = 1.50 µs

Normal stress gradient magnitude Pressure gradient magnitude


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Waves in a Coupled Solid-Fluid System


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Numerical Results

t = 3.50 µs

Fluid pressure

gas bubble:low pressure region causes displacementof the bubbles center towards thestructure due to a kind of Bjerknesforce

liquid:focussing of waves takes place in theliquid at about the original position ofbubble center due to an inertia effect


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Numerical Results

t = 3.50 µs

Normal stress component Fluid pressure


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Numerical Results

t = 3.80 µs

Fluid pressure

liquid:equation of state behaves essentiallystiffer for the liquid than for the gas focussing wave leads to a strong,concentric pressure wave generating ashock wave


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Numerical Results

t = 3.80 µs



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Numerical Results

t = 3.85 µs



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Numerical Results

t = 3.90 µs



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Numerical Results

t = 4.00 µs



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Numerical Results

t = 4.10 µs



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Numerical Results

t = 4.50 µs



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Numerical Results

t = 4.80 µs



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Numerical Results

t = 5.00 µs

Fluid pressure

liquid:shock wave hits structure interfacecausing very high stresses inside thestructure material damage inside structure


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Numerical Results

t = 5.00 µs



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Numerical Results

t = 6.00 µs



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Numerical Results

t = 10.00 µs

Fluid pressure

fluid:penetration of the bubble by the liquidjetformation of a vortex ringimpact of the jet onto the solid

solid:little effects on stress states of thestructure NO contributions to damaging


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Numerical Results

t = 10.00 µs



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Material Damage

Linear elastic model: no plastic deformation!

Post-processing step: Analyze in a whether pure elastic range of thesolid material has been exceeded.

Hypothesis: Material yields irreversibly and breaks if yield stressexceeds certain tolerances!

Ansatz: Comparison of von Mises stress with constant yield strength4

5

Φ(σ) := σvM − Y0

Y0 = 260 MPa (structural steel), Y0 = 502 MPa (stainless steel)

4A.S. Khan, S. Huang, Continuum Theory of Plasticity, John Wiley & Sons., 1995.

5J. Lemaitre, A Course on Damage Mechanics, Second Edition, Springer Verlag, 1996.


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Transient development of the von Mises yield criterion

Indication to plastic effects within the solid in the case of example of steelAISI 316 (Φ(σ) := σvonMises − Y0,Y0 = 260 MPa 6 7 ) as solid material:

t = 4.8µs: t = 5.0µs: t = 5.2µs: t = 5.4µs:

t = 5.6µs: t = 5.8µs: t = 6.0µs: t = 6.5µs:

6A.S. Khan, S. Huang, Continuum Theory of Plasticity, John Wiley & Sons., 1995.

7J. Lemaitre, A Course on Damage Mechanics, Second Edition, Springer Verlag, 1996.


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First Results using ELASTIC-PLASTIC Solid Modelling

t = 4.50 µscoarser spatial discretization than before (L = 8→ L = 6)

perfect plasticity - no hardening is modelled


fluid:generated shock wave hits thesolid surface and is reflectedthere

solid:surface loads cause stress wavesreaching the yield bound of thesolid material


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Normal stress gradient magnitude Pressure gradient magnitudeELASTIC solid Modelling:


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Normal stress gradient magnitude Pressure gradient magnitudeELASTIC-PERFECT-PLASTIC solid Modelling:


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Outline

1 Motivation



4 Numerical Results

5 Conclusion


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Summary & Outlook

Summary:Observed effects (bubble collapse, shock wave) in fluid that arequalitatively similar to experiment.

Observed significant effects (von Schmidt waves) due to coupling inthe solid as well as the fluid.Pressure wave develops within the liquid as a consequence of afocussing effect due to a displacement of the bubble center andBjerknes force characterizing an asymmetric bubble collapsePressure wave causes stress waves in solid that exceed the yieldstrength of structural steel.Formation of the liquid jet, its impact onto the solid surface and thevortex ring are not important for damaging.Modelling of plastic effects changes the wave processes

Outlook:

Improved description of plastic effects using a plastic flow ruleNeed multilevel time stepping at interface allowing for different timesteps in solid and fluid, respectively.


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Summary & Outlook

Summary:Observed effects (bubble collapse, shock wave) in fluid that arequalitatively similar to experiment.Observed significant effects (von Schmidt waves) due to coupling inthe solid as well as the fluid.

Pressure wave develops within the liquid as a consequence of afocussing effect due to a displacement of the bubble center andBjerknes force characterizing an asymmetric bubble collapsePressure wave causes stress waves in solid that exceed the yieldstrength of structural steel.Formation of the liquid jet, its impact onto the solid surface and thevortex ring are not important for damaging.Modelling of plastic effects changes the wave processes

Outlook:



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Summary & Outlook

Summary:Observed effects (bubble collapse, shock wave) in fluid that arequalitatively similar to experiment.Observed significant effects (von Schmidt waves) due to coupling inthe solid as well as the fluid.Pressure wave develops within the liquid as a consequence of afocussing effect due to a displacement of the bubble center andBjerknes force characterizing an asymmetric bubble collapse

Pressure wave causes stress waves in solid that exceed the yieldstrength of structural steel.Formation of the liquid jet, its impact onto the solid surface and thevortex ring are not important for damaging.Modelling of plastic effects changes the wave processes

Outlook:



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Summary & Outlook

Summary:Observed effects (bubble collapse, shock wave) in fluid that arequalitatively similar to experiment.Observed significant effects (von Schmidt waves) due to coupling inthe solid as well as the fluid.Pressure wave develops within the liquid as a consequence of afocussing effect due to a displacement of the bubble center andBjerknes force characterizing an asymmetric bubble collapsePressure wave causes stress waves in solid that exceed the yieldstrength of structural steel.

Formation of the liquid jet, its impact onto the solid surface and thevortex ring are not important for damaging.Modelling of plastic effects changes the wave processes

Outlook:



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Summary & Outlook

Summary:Observed effects (bubble collapse, shock wave) in fluid that arequalitatively similar to experiment.Observed significant effects (von Schmidt waves) due to coupling inthe solid as well as the fluid.Pressure wave develops within the liquid as a consequence of afocussing effect due to a displacement of the bubble center andBjerknes force characterizing an asymmetric bubble collapsePressure wave causes stress waves in solid that exceed the yieldstrength of structural steel.Formation of the liquid jet, its impact onto the solid surface and thevortex ring are not important for damaging.

Modelling of plastic effects changes the wave processes

Outlook:



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Summary & Outlook

Summary:Observed effects (bubble collapse, shock wave) in fluid that arequalitatively similar to experiment.Observed significant effects (von Schmidt waves) due to coupling inthe solid as well as the fluid.Pressure wave develops within the liquid as a consequence of afocussing effect due to a displacement of the bubble center andBjerknes force characterizing an asymmetric bubble collapsePressure wave causes stress waves in solid that exceed the yieldstrength of structural steel.Formation of the liquid jet, its impact onto the solid surface and thevortex ring are not important for damaging.Modelling of plastic effects changes the wave processes

Outlook:



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Summary & Outlook


Outlook:Improved description of plastic effects using a plastic flow rule

Need multilevel time stepping at interface allowing for different timesteps in solid and fluid, respectively.


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Summary & Outlook


Outlook:Improved description of plastic effects using a plastic flow ruleNeed multilevel time stepping at interface allowing for different timesteps in solid and fluid, respectively.


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Literature

S. Andreae.

Wave Interactions with Material Interfaces.Dissertation, RWTH Aachen, Shaker-Verlag, 2008.

S. Muller, Ph. Helluy, J. Ballmann.

Numerical Simulation of Cavitation Bubbles by Compressible Two-Phase Fluids.Int. Journal for Numerical Methods in Fluids, 62(6), 591-631, 2010.

S. Muller, M. Bachmann, D. Kroninger, Th. Kurz, Ph. Helluy.

Comparison and validation of compressible flow simulations of laser-induced cavitation bubbles.Computers & Fluids, 38, 1850-1862, 2009.

M. Bachmann.

Dynamics of cavitation bubbles in compressible two-phase fluid flow.Dissertation, RWTH Aachen, 2012.

Ch. Dickopp, R. Gartz, S. Muller.

Coupling of elastic solids with compressible two-phase fluids for the numerical investigation of cavitation damaging.International Journal on Finite Volumes, 10, 1-39, 2013.

Thank you for your attention!


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