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ResearchCite this article: Chahine GL, Hsiao C-T. 2015

Modelling cavitation erosion using

fluid – material interaction simulations.

Interface Focus 5: 20150016.

http://dx.doi.org/10.1098/rsfs.2015.0016

One contribution of 13 to a theme issue

‘Amazing (cavitation) bubbles: great potentials

and challenges’.

Subject Areas:biomedical engineering, mathematical physics

Keywords:cavitation, bubble dynamics, erosion,

fluid and material interaction

Author for correspondence:Georges L. Chahine

e-mail: [email protected]

& 2015 The Author(s) Published by the Royal Society. All rights reserved.

Modelling cavitation erosion usingfluid – material interaction simulations

Georges L. Chahine and Chao-Tsung Hsiao

Dynaflow, Inc., 10621-J Iron Bridge Road, Jessup, MD, USA

Material deformation and pitting from cavitation bubble collapse is investi-

gated using fluid and material dynamics and their interaction. In the fluid, a

novel hybrid approach, which links a boundary element method and a com-

pressible finite difference method, is used to capture non-spherical bubble

dynamics and resulting liquid pressures efficiently and accurately. The

bubble dynamics is intimately coupled with a finite-element structure

model to enable fluid/structure interaction simulations. Bubble collapse

loads the material with high impulsive pressures, which result from shock

waves and bubble re-entrant jet direct impact on the material surface. The

shock wave loading can be from the re-entrant jet impact on the opposite

side of the bubble, the fast primary collapse of the bubble, and/or the

collapse of the remaining bubble ring. This produces high stress waves,

which propagate inside the material, cause deformation, and eventually fail-

ure. A permanent deformation or pit is formed when the local equivalent

stresses exceed the material yield stress. The pressure loading depends on

bubble dynamics parameters such as the size of the bubble at its maximum

volume, the bubble standoff distance from the material wall and the pressure

driving the bubble collapse. The effects of standoff and material type on the

pressure loading and resulting pit formation are highlighted and the effects

of bubble interaction on pressure loading and material deformation are

preliminarily discussed.

1. IntroductionCavitation is the explosive growth and intense collapse of bubble nuclei in a liquid

when exposed to large pressure variations. It has been long known by engineers

for its deleterious effects in a variety of applications involving fluid machinery,

pumps, propellers, valves, sluice gates, lifting surfaces, etc. Unwanted effects

of cavitation include material erosion, performance degradation, and noise

and vibrations.

Over the past century, since the early works of Besant [1] and Rayleigh [2], a

significant amount of studies have been dedicated to seeking out measures to

mitigate, if not eliminate, cavitation. Cavitation erosion has attracted wide

attention from the materials and fluids engineering, and scientific communities

not only because significant damage to the machinery is frequently observed,

but also because the underlying physics of the erosion process itself involves

interesting and complex fluid–structure interaction dynamics, which need to

be addressed.

Conversely, cavitation can be generated for useful purposes and many studies

have been conducted to explore cavitation merits and enhance its intensity for

efficient underwater cleaning; e.g. [3–6], ultrasonic cleaning [7–9], chemical com-

pounds oxidation, e.g. [10,11], microorganisms disinfection, e.g. [12], algae oil

extraction, e.g. [13,14], etc.

Recent developments in utilization of ultrasound cavitation for biomedical

applications reveal a need for delicate control of the cavitation intensity, because

the boundaries between meritorious effects (e.g. tumour removal) and deleter-

ious effects (e.g. kill of nearby healthy cells and tissue) depend on some

critical irradiation set-up conditions. For example, use of cavitation to produce

cell sonoporation produces therapeutic effects only when the detrimental ‘side

effects’ on cells due to over dosage does not occur [15–17]. Numerous studies

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have also observed that cavitation activity could result in dele-

terious bio-effects such as haemolysis [16,18], haemorrhage

[19–21], kidney failure [22] when patients are treated with

ultrasound-based medical treatments such as shock wave

lithotripsy or high-intensity focused ultrasound.

Even though different mechanisms can cause cavitation

and different cavitation types exist, the underlying physics

of materials erosion and of cell membranes poration is the

same from the microscopic bubble dynamics point of view.

It has been known for more than a century [1,2] for cavitation

and underwater explosion bubbles that volume implosion of

the bubble can generate very high-pressure pulses and shock

waves. Many pioneering studies have also shown, exper-

imentally as well as analytically, that the collapse of these

bubbles near rigid boundaries results in high-speed re-

entrant liquid jets, which penetrate the highly deformed

bubbles to strike a nearby rigid boundary generating water

hammer like impact pressures [23,24]. The resulting re-

entrant jet has been found a key element in pit formation

on the material surface [25–27] as well as to induce the

erosion of endothelia [28] and haemorrhage [29].

Re-entrant jets can also be induced through bubble/

bubble interaction. For example, re-entrant jets induced by

tandem bubble interactions have recently been introduced

and gained attention for its precise control on cell membrane

poration [30,31]. Also, multi-bubble interactions in a bubble

cloud can result in collective synergetic behaviour, which

can result in very high pressures and loading on nearby

structures [6,32–36].

Cavitation bubble collapse near boundaries has been

extensively investigated numerically by assuming flow

incompressibility and using the boundary element method

(BEM) [37–41]. The BEM enables to accurately describe the

re-entrant jet formation, development, advance through the

bubble and impact on nearby boundaries. It can provide

the jet geometric and dynamic characteristics as functions

of time and space as the bubble wall velocities (including

the re-entrant jet velocity) are most often small relative to

the speed of sound in water. On the other hand, during the

bubble explosive growth, rebound, and at jet impact, com-

pressible effects can be non-negligible and need to be

included. These dynamic stages may lead to shock wave for-

mation and shock propagation and impact on a nearby

material. Compressible flow solvers have been developed

for this purpose [42–44]. However, such flow solvers often

use a finite difference method which requires very fine spatial

resolution and small time-step sizes especially to resolve the

re-entrant jet accurately [45,46]. This makes them not very

efficient for simulating the relatively long duration bubble

period but excellent at describing the shock phase.

To overcome this, this work applies a novel hybrid

numerical procedure involving a time domain decomposition

into incompressible and compressible stages in order to cap-

ture the full bubble dynamics period as well as the shock

phase occurring during re-entrant jet impact, bubble collapse

and bubble rebound. This numerical procedure takes advan-

tage of an accurate shock-capturing method and of a BEM

shown to be both very efficient in modelling cavitation

bubble dynamics problems and very accurate in capturing

the re-entrant jet and its dynamics.

In proximity of a deforming boundary, the complex flow

phenomena and the bubble dynamics itself can be significantly

altered [27,47–51]. In this study, a finite-element structure code

is coupled with the hybrid fluid flow solver to investigate

material response to the pressure field generated by the

bubble dynamics.

This paper presents first the methods used, then describes

the pressure loading and material deformation resulting from

a single bubble collapse near a material. The effect of bubble/

bubble interaction on pressure loading and on material defor-

mation resulting for tandem bubbles and bubble clouds are

then considered. Although the actual case studies shown

are for cavitation erosion on a flat material surface, the key

findings and the numerical approaches apply as well to

complex geometries (with appropriate corrections) and to

biomedical applications such as for the study of sonoporation

and breakage and removal of calculi.

2. Numerical approachThe numerical approach applied to model material pitting in

this paper is part of a general hybrid approach which was

developed by the authors to simulate fluid–structure inter-

action (FSI) problems involving shock and bubble pressure

pulses [52,53]. As illustrated in figure 1, for a highly inertial

bubble such as a spark-generated bubble, an underwater

explosion bubble (UNDEX), or a laser-generated bubble

[54–57], a compressible–incompressible link is required at

the beginning to handle the emitted shock wave and the

flow field generated by the exploding bubble. Cavitation

bubbles on the other hand, generate a small pressure peak

and no shock wave during the growth phase. As a result,

no initial shock phase compressible solution is required. An

incompressible BEM code can then be used to simulate

most of the bubble period until the end of the bubble collapse

where, due to high liquid speeds or to the bubble re-entrant

jet impacting on the liquid or on the structure, compressible

flow effects prevail again.

In this study, the solution of the incompressible BEM

code is passed to a fully compressible code capable of

shock capturing to simulate re-entrant jet impact and bubble

ring collapse.

2.1. Incompressible flow modellingThe potential flow code used in this study, 3DYNAFS-BEM

q, is

based on a BEM [58–60]. The code solves the Laplace

equation for the velocity potential, f,

r2f ¼ 0, ð2:1Þ

with the velocity vector defined as u ¼ rf. A boundary inte-

gral method is used to solve the Laplace equation based on

Green’s theoremðV

ðfr2G� Gr2fÞ dV ¼ð

Sn � [frG� Grf]dS: ð2:2Þ

In this expression, V is the domain of integration having

elementary volume dV. The boundary surface of V is S,

which includes the surfaces of the bubble and the nearby

boundaries with elementary surface element dS. n is the

local normal unit vector. G ¼ �1=jx� yj is Green’s function,

where x corresponds to a fixed point in V and y is a point

on the boundary surface S. Equation (2.1) reduces to

Green’s formula with ap being the solid angle under which


fluid codes FSC structure codes

shock phase compressible code

DynaNParadynSalinasEPSA

etc.

compressible to incompressible linkafter initial shock moves out

bubble phase incompressible code

incompressible to compressible link just prior to reentrant jet touchdown

rebound phase compressible code

time

pressures

positions,velocities

mapping,interpolation

Figure 1. Schematic diagram of the numerical approach used to simulate the interaction between a highly inertial bubble or a cavitation bubble and a structure.A time domain decomposition approach is used and the liquid and structure codes communicate through the FSC interface. (Online version in colour.)


Focus5:20150016

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x sees the domain, V:

apfðxÞ ¼ð

SfðyÞ @G

@nðx, yÞ � Gðx, yÞ @f

@nðyÞ

� �dS, ð2:3Þ

where ap is the solid angle. To solve (2.3) numerically, the

BEM, which discretizes the surface of all objects in the

computational domain into panel elements, is applied.

Equation (2.3) provides a relationship between f and

@f/@n at the boundary surface S. Thus, if either of these

two variables (e.g. f ) is known everywhere on the surface,

the other variable (e.g. @f/@n) can be obtained.

To advance the solution in time, the coordinates of the

bubble and any free surface nodes, x, are advanced according

to

dx

dt¼ rf: ð2:4Þ

The velocity potential f on the bubble and free surface

nodes is obtained through the time integration of the material

derivative of f, i.e. df/dt, which can be written as

df

dt¼ @f@tþrf:rf, ð2:5Þ

where @f/@t can be determined from the Bernoulli equation

r@f

@tþ 1

2rf:rfþ gz

� �þ pl ¼ p1, ð2:6Þ

p1 is the hydrostatic pressure at infinity at z ¼ 0, where z is

the vertical coordinate. pl is the liquid pressure at the

bubble surface, which balances the internal pressure and

surface tension,

pl ¼ pv þ pg � sC, ð2:7Þ

where pv is the vapour pressure, s is the surface tension, and

C is the local bubble wall curvature. pg is the gas pressure

inside bubble and is assumed to follow a polytropic law

with a compression constant, k, which relates the gas pressure

to the gas volume, V , and reference (often initial) value,

pg0, and V0.

pg ¼ pg0V0

V

� �k

: ð2:8Þ

2.2. Compressible flow modellingThe multi-material compressible Euler equation solver used

here is based on a finite difference method [46,61]. The

code solves continuity and momentum equations for a com-

pressible inviscid liquid in Cartesian coordinates. These can

be written in the following format:

@Q@tþ @E@xþ @F@yþ @G@z¼ S, ð2:9Þ

and

Q ¼

r

rurvrwret

26666664

37777775

, E ¼

ruru2 þ pruvruw(ret þ p)u

26666664

37777775

,

F ¼

rvrvurv2 þ prvw(ret þ p)v

26666664

37777775

, G ¼

rwrwurwvrw2 þ p(ret þ p)w

26666664

37777775

and

S ¼

0

0

0

rgrgw

26666664

37777775

, ð2:10Þ

where r is the fluid density, p is the pressure, u, v and w are

the velocity components in the x, y, z directions, respectively

(z is vertical), g is the acceleration of gravity, and et ¼ e þ0.5(u2 þ v2 þ w2) is the total energy with e being the internal

energy. The system is closed by using the equation of state of

each material, which provides for this material the pres-

sure as a function of the specific internal energy and the

density. In this study, a g-law (with g ¼ 1.4) is used for

the gas–vapour mixture [62]

p ¼ ðg� 1Þre, ð2:11Þ



Focus5:20150016

4


and the Tillotson equation is used for water [63]

p ¼ p0 þ vrðe� e0Þ þ Amþ Bm2 þ Cm3,

m ¼ r

r0

� 1, ð2:12Þ

where v, A, B and C are constants, and p0, e0 and r0 are the

reference pressure, reference specific internal energy and

reference density, respectively: v ¼ 0.28, A ¼ 2.20 � 109 Pa,

B ¼ 9.54 � 109 Pa, e0 ¼ 3.54 � 105 m2 s22, p0 ¼ 1.0 � 105 Pa,

r0 ¼ 1000 kg m23.

The compressible flow solver uses a high-order Godunov

scheme and employs the Riemann problem to construct a

local flow solution between adjacent cells. The numerical

method is based on a higher-order MUSCL scheme and

tracks each material. To improve efficiency, an approximate

Riemann problem solution replaces the full problem. The

MUSCL scheme is augmented with a mixed cell approach

[64] to handle shock wave interactions with fluid or material

interfaces. This approach uses a Lagrangian treatment for the

cells including an interface and an Eulerian treatment for cells

away from interfaces. A re-map procedure is employed to

map the Lagrangian solution back to the Eulerian grid

[46,61] The code has been extensively validated against

experiments [45,49].

2.3. Compressible – incompressible link procedureBoth incompressible and compressible flow solvers are able

to model the full bubble dynamics on their own. However,

each method has its shortcomings when it comes to specific

parts of the bubble history. The BEM-based incompressible

flow solver is efficient, reduces the dimension of the problem

by one (line integrals for an axisymmetric problem, and sur-

face integrals for a three-dimensional problem) and thus

allows very fine gridding and increased accuracy with

reasonable computations times. It has been shown to provide

re-entrant jet parameters and speed accurately [39,58,60,65].

However, it has difficulty pursuing the computations

beyond surface impacts (liquid–liquid and liquid–solid).

On the other hand, the compressible flow solvers used here

(GEMINI developed by the US Navy at NSWCIH [46] and

3DYNAFS-COMP [61]) are most adequate to model shock wave

emission and propagation, liquid–liquid, and liquid–solid

impacts. The method requires, however, very fine grids and

very small time steps to resolve shock wavefronts. This

makes it appropriate to model time portions of the bubble

dynamics. Concerning bubble–liquid interface and the re-

entrant jet dynamics, the procedure is diffusive as the interface

is not directly modelled, and re-entrant jet characteristics are

usually less accurate than obtained with the BEM approach.

Hence, we employ here a general novel approach, which

combines the advantages of both methods and consists in

executing the following steps:

(1) Set-up the initial flow field using the Eulerian compressi-

ble flow solver and run the simulation until the initial

shock fronts go by and the remnant flow field can be

assumed to be incompressible with bubble dynamics

independent of initial treatments.

(2) Transfer at that instant to the Lagrangian BEM potential

flow solver, all the flow field variables needed for the sol-

ution to proceed: geometry, bubble pressure and moving

boundary normal velocities, @f=@n.

(3) Solve for bubble growth and collapse using fine BEM

grids to obtain a good description of the re-entrant jet

until the point where the jet is very close to the opposite

side of the bubble.

(4) Transfer the solution back to the compressible flow solver

with the required flow variables. To do so, compute

using the Green equation all flow field quantities on

the Eulerian grid.

(5) Continue advancing the solution with the compressible

code to obtain pressures due to jet impact and to the col-

lapse of the remnant bubble ring.

2.4. Structure dynamics modellingTo model the dynamics of the material, the finite-element

model DYNA3D is used. DYNA3D is a nonlinear explicit struc-

ture dynamics code developed by the Laurence Livermore

National Laboratory [66]. Here, it computes the material

deformation when the loading is provided by the fluid

solution. DYNA3D uses a lumped mass formulation for effi-

ciency. This produces a diagonal mass matrix M, to express

the momentum equation as

Md2x

dt2¼ Fext � Fint, ð2:13Þ

where Fext represents the applied external forces, and Fint the

internal forces. The acceleration, a ¼ d2x=d2t for each

element, is obtained through an explicit temporal central

difference method. Additional details on the general formu-

lation can be found in [66].

2.5. Material modelsIn this study, two metal alloys: Aluminium Al7075 and Stain-

less Steel A2205, two rubbers: a Neoprene synthetic rubber

and a Polyurethane, and a Versalink-based polyurea are con-

sidered. The metals are modelled using elastic-plastic models

with linear slopes (moduli); one for the initial elastic regime

and the second, a tangent modulus, for the plastic regime.

The material parameters required for the current elastic-plastic

model are specified as follows:

— Aluminium Al7075: Young modulus 71.7 GPa, tangent

modulus 680 MPa, yield stress 503 MPa and density

2.81 g cm23.

— Stainless Steel A2205: Young modulus 190 GPa, tangent

modulus 705 MPa, yield stress 515 MPa and density

7.88 g cm23.

Concerning the rubbers, a Blatz-Ko’s hyper-elastic

model is used. This model is appropriate for materials

undergoing moderately large strains and is based on

the implementation in [67]. The material motion satisfies

the formal equation

rsd2x

dt2¼ r � s, ð2:14Þ

where rs is the material density, x is the position vector of the

material and s is the Cauchy stress tensor.

If we denote F the deformation gradient tensor, then the

Cauchy stress tensor is computed by

s ¼ 1

JFSFT, ð2:15Þ


(a)

Pamb = P(t)

R0 = 50 µm

X = 1.5 mm

(b)

00 10–4

10–3

10–2

10–1

1

10

102

0.5

1.0

1.5

2.0

Req

(m

m)

Rmax

P amb

(MPa

)2.5

3.0

3.5

4.0

1

Dt

Pd

time (ms)

imposed pressureRayleigh–Plesset solution

2 3

Figure 2. (a) Illustration of the problem of a bubble growth and collapse near a wall. Initial spherical bubble radius 50 mm at 1.5 mm distance from the wall.(b) Imposed time-varying pressure field and resulting time history of the bubble radius obtained by solving the Rayleigh – Plesset equation [2,70]. (Online version incolour.)


Focus5:20150016

5


where J is the determinant of F, and S is the second Piola–

Kirchoff stress tensor and is computed by

Sij ¼ GCij

V

� �� V�1=ð1�2nÞdij

� �: ð2:16Þ

In the above equation, G is the shear modules, V is the rela-

tive volume to original which is related to excess

compression and density, Cij is the right Cauchy–Green

strain tensor, and n is Poisson’s ratio and is set to 0.463. In

this study, the shear module and density are specified as

99.7 MPa and 1.18 g cm23 for the Neoprene synthetic

rubber (Rubber no. 1) and 34.2 MPa and 1.20 g cm23 for

Polyurethane (Rubber no. 2).

The Versalink-based polyurea was modelled as a visco-

elastic material with the following time-dependent values

for the shear modulus, G:

GðtÞ ¼ G1 þ ðGo � G1Þe�bt ð2:17Þ

with

G1 ¼ 41:3 MPa, Go ¼ 79:1 MPa,

K ¼ 4:948 GPa and b ¼ 15 600 s�1Þ ð2:18Þ

While the time dependence of G is considered in this

model, the bulk modulus, K, was assumed to be constant.

The material parameters in (2.18) are derived from simplifica-

tion of fourth-order Prony series in [68] taking the second

term as the major contributor.

2.6. Fluid – structure interaction couplingFluid–structure interaction effects are captured in the

simulations by coupling the fluid codes (GEMINI or

3DYNAFS-COMPq) with DYNA3D through a fluid–structure

coupler interface (FSC). The coupling is achieved through

the following steps:

— The fluid code solves the flow field and deduces the press-

ures at the structure surface using the positions and

normal velocities of the wetted body nodes.

— In response, the structure code computes material stresses,

strains and deformations and velocities of the wetted inter-

face in response to this loading.

— The new coordinates and velocities of the structure surface

nodes become the new boundary conditions for the fluid

code at the next time step.

Additional details on the procedure can be found in

[41,45,51,59]. This FSI coupling procedure has only a first-

order time accuracy. A predictor–corrector approach is also

implemented in the coupling to iterate and improve the sol-

ution but was not used here. This is because the numerical

error due the time lag is negligible thanks to the very small

time steps used. These are controlled by the steep pressure

waves, which have a time scale that is two orders of magni-

tude shorter than the time response of the material. This

method has been shown in UNDEX studies to correlate very

well with experiments [69].

3. Single cavitation bubble collapse near wall3.1. Problem descriptionWe consider an initially spherical bubble of radius 50 mm,

located at a distance of X ¼ 1.5 mm from a flat material sur-

face and subject it to a time-varying pressure field as

represented in figure 2 and expressed as follows:

pðtÞ ¼105 Pa; t , 0,103 Pa; 0 � t � 2:415 ms,107 Pa; t . 2:415 ms:

8<: ð2:19Þ

This imposed pressure variation is different from that

used in many classical studies on bubble collapse near a

wall and where a bubble with a maximum radius is suddenly

subjected to a pressure higher than the internal pressure such

as in [71–74]. Here, bubble growth is included (this allows

one to include standoff distances smaller than the bubble

maximum radius and covers a large range of applications),

and the time-varying pressure field represents for example

the pressure encountered by a bubble nucleus captured in

the shear layer of a cavitating jet. In the considered

expression, the bubble nucleus which is in equilibrium at

atmospheric pressure is entrained in a jet low pressure

region, travels to the wall and enters in stagnation region of


(a)

1.5 mm

(b)

X Y

Z

X Y

Z

time

time

Figure 3. Bubble shape outlines at different times showing (a) bubble growth (0 , t , 2.415 ms) and (b) collapse (2.415 ms , t , 2.435 ms). Resultsobtained by the 3DYNAFS-BEM simulations for R0 ¼ 50 mm, Rmax ¼ 2 mm, Pd ¼ 10 MPa, �X ¼ 0:75 and p(t) described by equation (2.19). (Online version incolour.)


Focus5:20150016

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the cavitating jet impacting the wall. The bubble transport in

the flow is not included here. The stagnation pressure at the

wall, which is a portion of the nozzle pressure can exceed

10 MPa as 104 psi jets (70 MPa) are commonly used in practice.

Similarly, a bubble nucleus travelling near a propeller blade

can encounter a pressure, which drops to a value close to the

vapour pressure and then, in the pressure recovery region,

can be impacted by a shock wave emitted from the collapse

of a nearby cloud collapse, which can exceed 10 MPa.

The duration of the pressure drop is selected to corre-

spond to practical distances of travel of the bubble in the

above two cases. The exact value is selected to achieve a nor-

malized bubble standoff from the wall of 0.75 in order to

compare between cases. The nucleus size is chosen arbitrarily

to be 50 mm. A complete study requires varying this size over

the range of bubble nuclei distribution in water. However, we

expect the results to depend more on Rmax. Figure 2b shows

an approximate time history of the bubble radius in response

to the imposed pressure field obtained by solving the

Rayleigh–Plesset equation ignoring for now the presence of

the wall. It is seen that the bubble starts an explosive grow

as soon as the pressure suddenly drops to a low pressure

below the critical pressure. The bubble continues to grow

asymptotically until the pressure rises back to the bubble

‘collapse driving pressure’, Pd, here of 107 Pa.

3.2. Non-spherical bubble dynamics and re-entrant jetdevelopment

To simulate the bubble dynamics near the wall, the incom-

pressible BEM solver is first applied with a total of 400

nodes and 800 panels used to discretize the bubble surface.

This corresponds to a grid density, which provides grid inde-

pendent solution [41]. Figure 3 shows the variations of the

bubble outer contours as time advances. As the bubble

grows between t ¼ 0 and t � 2.4 ms, it behaves almost spheri-

cally on its portion away from the wall, while the side close to

the material flattens and expands in the direction parallel to

the wall actually never touching the wall as a layer of

liquid remains between the bubble and the wall. Such a

behaviour has been confirmed experimentally by Chahine

et al. [41]. Note that at maximum bubble volume, the non-

dimensional standoff, �X ¼ X=Rmax, is less than one (here�X ¼ 0:75), where Rmax is the maximum equivalent bubble

radius deduced from its volume.

The bubble will continue expanding following pressure

reversal due to the inertia of the outward flow of the

liquid. Due to the asymmetry of the flow, the pressures at

the bubble interface on the side away from the wall are

much higher than those near the material; thus, the collapse

proceeds with the far side moving towards the material

wall. The resulting acceleration of the liquid flow perpendicu-

lar to the bubble-free surface develops a Rayleigh–Taylor

instability [75–78] at the axis of symmetry, which results

into a re-entrant jet that penetrates the bubble and moves

much faster than the rest of the bubble surface to impact

the opposite side of the bubble and the material boundary.

In the present approach, the simulation of the bubble

dynamics is switched from the BEM to the compressible

flow solver right before the jet touches the opposite side of

the bubble (dubbed touchdown). Ideally, the time of the

‘link’ between incompressible and compressible approaches

should be at the time the bubble becomes multi-connected.

However, to avoid increased errors/fluctuations in the BEM

solution when the distance between a jet panel and the oppo-

site bubble side panels continues to decrease as the jet

advances, the ‘link’ time is selected to be when the distance

between the jet front and the opposite bubble surface

becomes less than or equal to 1.5 times the local panel size.

This results in an underestimate of tlink by less than 1%.

Figure 4a shows the pressure contours and velocity vec-

tors at the selected time, tlink, when the compressible flow

solver starts its computations and FSI effects become non-

negligible [79]. Figure 4b shows the corresponding velocity

vectors and velocity magnitude contour levels. Note that,

for this bubble collapse condition, prior to jet impact on the

opposite side of the bubble, the liquid velocities near the

tip of the jet exceeds 1400 m s21. The maximum liquid vel-

ocity of the jet exceeds the sound speed after the ‘link’ time

when the computation was continued with the compressible

code. The peak value reaches about 1600 m s21 at the time

when the jet touches down the opposite side of the bubble.


–1.00

0.5

1.0

1.5

2.0(a)

–0.5 0r (mm)

z (m

m)

0

0.5

1.0

1.5

2.0(b)

z (m

m)

0.5 1.0

–200–150–100–50050100150200250

pressure(MPa)

–1.0 –0.5 0r (mm)

0.5 1.0

0100200300400500600700800900100011001200130013501400

|V|, m s–1

Figure 4. (a) Pressure contour levels with velocity vectors and (b) velocity vectors and magnitude contour levels at t ¼ 2.435 ms, time at which the incompressible –compressible link procedure is applied. These serve as initial conditions for the compressible flow and structure solvers for R0 ¼ 50 mm, Pd ¼ 10 MPa, �X ¼ 0:75 andp(t) described by equation (2.19). (Online version in colour.)

4002000

200

400

z(m

m)

600

(a) (b)

600 800 0

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6

re-entrantjet

x (mm)0.8 1.0

r (mm)

Figure 5. Axisymmetric computational domain used for the computation of the bubble dynamics by the compressible flow solver: (a) full domain, (b) zoom on thebubble/wall region. The blue region is the inside of the bubble after it formed a re-entrant jet on the axis of symmetry, which is the Z-axis here. (Online versionin colour.)


Focus5:20150016

7


This value is much higher than values reported in the litera-

ture for bubble collapse under atmospheric conditions, such

as in ‘shallow’ underwater explosions or for laser or spark-

generated bubbles where the reported values are of the

order of 100 m s21. This is because the re-entrant jet velocity

is proportional to the square root of the collapse driving

pressure, Pd. Detailed discussions of the effect of Pd on the

re-entrant jet velocity can be found in [25,26].

3.3. Re-entrant jet impact and bubble ring collapseThe compressible flow solver is used to continue the simulation

after re-entrant jet touchdown. An axisymmetric domain with

a total of 220 � 1470 grid points in a 1 m � 1 m domain was

used with stretched grids concentrated in the immediate

region surrounding the bubble. The grids were distributed

such that there was a uniform fine mesh with a size of 10 mm

in the area of interest where the interaction between bubble

and plate is important as shown in figure 5. The axisymmetric

computational domain ended at the large distance of 1 m

(20 000 R0) in the far field (radial direction and away from the

wall) and at the plate material wall. A reflection boundary

condition was imposed on the axis of symmetry, i.e. all phys-

ical variables such as density, pressure, velocities and energy

are reflected from the axis, while transmission non-reflective

boundary conditions (i.e. the flow variables are extrapolated

along the characteristic wave direction) were imposed at the

far-field boundaries.

Figure 6 shows the bubble shapes and corresponding

pressure contours computed at six time instances after tlink ¼

2.435 ms. It is seen that at t2tlink ¼ 0.05 ms, the jet has comple-

tely penetrated the bubble and touched the opposite side. The

liquid–liquid impact event generates a localized high-pressure

region which then expands quasi-spherically to reach the

material liquid interface at t2tlink ¼ 0.2 ms. The volume of

the bubble ring remaining after the jet touchdown shrinks

and reaches a minimum at t2tlink ¼ 0.7 ms. The collapse of

the bubble ring generates another high-pressure wave, which

then propagates towards the axis of the cylindrical domain

and reaches the wall at t2tlink ¼ 0.9 ms.

To display better the pressure field dynamics during the

bubble collapse impulsive loads period, a zoom of the

pressure contours in the region between the re-entrant jet

impact on the opposite side of the bubble and the rigid


t – tlink = 0.05 µs t – tlink = 0.2 µs t – tlink = 0.4 µs

pressure(MPa)

pressure(MPa)

1500

–400

–210

–20

170

360

550

740

930

1120

1310

1500

–400–210–2017036055074093011201310

t – tlink = 0.7 µs t – tlink = 0.9 µs

(a) (b) (c)

(d) (e) ( f )

1.5

1.0

0.5

0

z(m

m)

–1.0 –0.5 0r (mm)

0.5 1.0 –1.0 –0.5 0r (mm)

0.5 1.0

t – tlink = 0.6 µs 1.5

1.0

0.5

0–1.0 –0.5 0

r (mm)0.5 1.0

z(m

m)

Figure 6. Pressure contours and bubble outlines at different instances following re-entrant jet touchdown. Initial bubble radius R0 ¼ 50 mm, Rmax ¼ 2 mm,d0 ¼ 1.5 mm, the initial normalized standoff �X ¼ 0:75, and collapse driving pressure Pd ¼ 10 MPa. (Online version in colour.)

bubble

jet axis

jet impact



0.3

0.2

0.1

0

0.3

0.2

0.1

0

z(m

m)

z(m

m)

pressure(MPa)

1000

–400–260–12020160300440580720860

pressure(MPa)

1000

–400–260–12020160300440580720860

–0.2 –0.1 0r (mm)

0.1 0.2 –0.2 –0.1 0r (mm)

0.1 0.2 –0.2 –0.1 0r (mm)

0.1 0.2

(a) (b) (c)

(d) (e) ( f )

Figure 7. Zoom on the region between the re-entrant jet impact on the opposite side of the bubble (frame a) and the rigid wall at z¼ 0. Pressure contours at six differenttimes showing the resulting shock wave reaching the wall (between frames d and e) and reflecting from it (frames e and f ) �X ¼ 0:75, R0 ¼ 50 mm, Rmax ¼ 2 mmand Pd ¼ 10 MPa: (Online version in colour.)


Focus5:20150016

8


wall is shown in figure 7. Eight time steps are selected

between the jet impact and the shock wave reflection from

the wall. The re-entrant jet impact in figure 7a forms a

strong quasi-ellipsoidal shock wave, which later propagates

outwards in all directions losing intensity from reflections

into the bubble-free surface but remaining strong in the

axial region of the bubble (figure 7b,c). The shock front

advances towards the solid wall at the fluid sound speed

and impacts in between frames (d ) and (e). It then reflects

as a reinforced shock wave between frames (e) and (f ). One


direction ofpropagation of the

wave

t – tlink = 0.182 µs t – tlink = 0.189 µs t – tlink = 0.196 µs t – tlink = 0.206 µs

0

00

500pres

sure

(M

Pa)

1000

0.02 0.04

z (mm)

0.06 0.08 0.10 0.12

Figure 8. Pressure distribution along the axis of symmetry at four times beforeand after the shock wave from jet-bubble wall impact reaches the rigid wall.These times correspond to frames c – f in figure 7. R0 ¼ 50 mm, Rmax ¼

2 mm, Pd ¼ 10 MPa and �X ¼ 0:75. (Online version in colour.)

00

0.5

1.0

1.5

impa

ct p

ress

ure

(GPa

)

Req

, dis

tanc

e (m

m)

2.0

2.5

3.0

3.5

0.5 1.0

jet touchdown

bubble ringcollapse

peak due tobubble ring collapsepeak due to

jet impact

t – tlink (µs)1.5 2.0 2.5

0

0.2

0.4

0.6

impact pressurebubble equivalent radiusdistance between bubble poles

0.8

Figure 9. Zoom at bubble collapse of the equivalent bubble radius evolutionversus time (red dashed line), and the pressure recorded at the rigid wall onthe axis. R0 ¼ 50 mm, Rmax ¼ 2 mm, Pd ¼ 10 MPa and �X ¼ 0:75. (Onlineversion in colour.)

z

Or

Figure 10. Finite-element axisymmetric grid used in DYNA3D to study thematerial response to loads due to collapsing bubbles. (Online version incolour.)


Focus5:20150016

9


can also observe the generation of a small bubble ring (dark

blue spots on either side of the axis) in frames (e) and ( f ) due

to the generated local high tensile stresses. This bubble ring

continues to grow and becomes a clear two-phase interface

as seen in figure 6b–e, while the main larger bubble ring con-

tinues to shrink. This is associated with the vortical flow

induced in the re-entrant jet shear layer after the jet pierces

the bubble.

The corresponding pressure distribution along the axis of

symmetry at the last four time steps is shown in figure 8. It

can be seen that the incoming pressure of maximum ampli-

tude 700 MPa (red curve) is almost doubled following

reflection at the rigid wall to about 1.3 GPa (green curve).

The magnitude of this high-pressure loading level has been

extracted or deduced from both numerical studies and from

experimental measurements [26,80,81]. Figure 9 shows the

time history of the bubble equivalent radius and the pressure

at the axis of symmetry on the wall surface (z ¼ 0). One can

observe the generation by the bubble collapse of two distinct

pressure peaks, one resulting from the re-entrant jet impact at

the wall and the other occurring right after the remainder

ring bubble reaches its minimum size. In addition to these

two peaks, many other pressure peaks are observed due to

pressure or shock waves bouncing back and forth between

the target wall, the bubble surface, and any other daughter

bubbles in the near wall flow field.

3.4. Dynamics response of material due to pressureloadings

The pressures generated during bubble collapse and rebound

are at least two orders of magnitude higher than those gener-

ated during the bubble growth period when the bubble

dynamics is that due to a driving pressure as in figure 2

[25]. Also, material deformation during bubble dynamics

up to re-entrant jet impact has very little influence on the

re-entrant jet impact and bubble ring collapse phases. Conse-

quently, FSI simulations in this study are only carried out

after the incompressible–compressible link has occurred

and when the compressible code is used to simulate the

flow field.

For the structure computations, a circular plate with a

radius of 1 m and a thickness of 0.01 m is discretized using

rectangular brick elements. As shown in figure 10, a stretched

grid with 220 elements in the radial direction and 446

elements in the axial direction are used to discretize the

plate. The elements are distributed such that a uniform fine

10 mm mesh size exists near the centre of the plate where

the high-pressure loading occurs. The mesh sizes were

tested to establish convergence and grid independence of

the solution. The motion of the nodes at the plate bottom

was restricted in all directions. The nodes along the vertical

axis were only allowed to move in the vertical direction.

Figure 11 shows a time sequence of the contours of the

von Mises equivalent stresses in the material for one of the

two metallic alloys considered here, Al7075. It is seen that

high stresses appear at the plate centre near the surface

when the re-entrant jet impact pressure reaches the wall

first at t2tlink ¼ 0.2 ms. The high stress wave is observed to

propagate and move radially away from the impact location.

As the first high stress wave starts to attenuate, another high

stress is observed initiating from the top centre of the plate at

t2tlink ¼ 0.9 ms, the time at which the high-pressure wave

generated by the collapse of the remaining bubble ring

reaches the wall (figure 6f).All high stresses due to the bubble dynamics eventually

attenuate. However, residual stresses remain in the material

below the surface due to the plastic deformations of some

regions of the plate. In the conditions of figure 11, these

have their highest value occurring at a depth of 0.2 mm


5 .0 × 102

4.7 × 102

4.3 × 102

4.0 × 102

3.7 × 102

3.3 × 102

3.0 × 102

2.7 × 102

2.3 × 102

2.0 × 102

1.7 × 102

1.3 × 102

1.0 × 102

5 .0 × 102

4.7 × 102

4.3 × 102

4.0 × 102

3.7 × 102

3.3 × 102

3.0 × 102

2.7 × 102

2.3 × 102

2.0 × 102

1.7 × 102

1.3 × 102

1.0 × 102

5 .0 × 102

4.7 × 102

4.3 × 102

4.0 × 102

3.7 × 102

3.3 × 102

3.0 × 102

2.7 × 102

2.3 × 102

2.0 × 102

1.7 × 102

1.3 × 102

1.0 × 102

5 .0 × 102

4.7 × 102

4.3 × 102

4.0 × 102

3.7 × 102

3.3 × 102

3.0 × 102

2.7 × 102

2.3 × 102

2.0 × 102

1.7 × 102

1.3 × 102

1.0 × 102

5 .0 × 102

4.7 × 102

4.3 × 102

4.0 × 102

3.7 × 102

3.3 × 102

3.0 × 102

2.7 × 102

2.3 × 102

2.0 × 102

1.7 × 102

1.3 × 102

1.0 × 102

stress (MPa) stress (MPa) stress (MPa)

stress (MPa) stress (MPa) stress (MPa)

5 .0 × 102

4.7 × 102

4.3 × 102

4.0 × 102

3.7 × 102

3.3 × 102

3.0 × 102

2.7 × 102

2.3 × 102

2.0 × 102

1.7 × 102

1.3 × 102

1.0 × 102



–1.0

–1.5

–1.0

z(m

m)

–0.5

0

–1.5

–1.0

z(m

m)

–0.5

0

–0.5 0r (mm)

0.5 1.0 –1.0 –0.5 0r (cm)

0.5 1.0 –1.0 –0.5 0r (cm)

0.5 1.0

(a) (b) (c)

(d) (e) ( f )

Figure 11. Time sequence of the equivalent stress contours in the Al7075 plate for R0 ¼ 50 mm, Rmax ¼ 2 mm, Pd ¼ 10 MPa and �X ¼ 0:75: (Online version incolour.)

t – tlink (µs) 0

–50

–45

–40

–35

–30

–25

–20

–15

–10

–5

0

0.5

1.0

1.5

2.0

2.5

5 10 15 20 25

impact pressure

impa

ct p

ress

ure

(GPa

)

vertical displacement

vert

ical

dis

plac

emen

t (µm

)

Figure 12. Time history of the pressure and vertical displacement at the topsurface centre of an Al7075 plate following the collapse of a cavitation bubble.R0 ¼ 50 mm, Rmax ¼ 2 mm, Pd ¼ 10 MPa and �X ¼ 0:75: (Online versionin colour.)

–40–400 –400 0 400 400

–35

–30

–25

–20

–15

–10

–5

0

5

10

z(µ

m)

r (µm)

pit depth

pit radius

Dz (mm)

1.200.35–0.50–1.35–2.20–3.05–3.90–4.75–5.60–6.45–7.30–8.15–9.00

Figure 13. Profile of the permanent deformation of an Al7075 platefollowing the collapse of a cavitation bubble. R0 ¼ 50 mm, Rmax ¼ 2 mm,Pd ¼ 10 MPa and �X ¼ 0:75: (Online version in colour.)


Focus5:20150016

10


below the surface as shown in figure 11f. As the material is

modelled as elastic-plastic, permanent deformation should

occur wherever the local equivalent stresses exceed the

material yield point. With the yield stress of Al7075 being

503 MPa, all regions that have seen red stress contour levels

as shown in figure 11, experience permanent deformation

due to loads from either re-entrant jet impact or bubble

ring collapse.

To quantitatively examine the material response to the

pressure loading, the time histories of the liquid pressure

and the vertical displacement of the material surface at the

centre of the Al7075 plate/liquid interface are shown together

in figure 12. The material starts to get compressed as the

high-pressure loading due to the re-entrant jet impact reaches

it, and the plate surface centre point starts to move into the

material direction at t2tlink ¼ 0.45 ms. The maximum defor-

mation occurs when the highest pressure loading peak

due to the bubble ring collapse reaches the centre of the

plate at time t2tlink ¼ 1.15 ms. Once the pressure loading

due to the full bubble dynamics has virtually vanished at

t2tlink ¼ 4 ms, the surface elevation continues to oscillate

due to stress waves propagating back and forth through the

metal alloy thickness and lack of damping in the model.




1.0(a) (b) (c)

(d) (e) ( f )

0.5

z(m

m)

1.0

0.5

z(m

m)

pressure(MPa)

pressure(MPa)

1500

–400–210–2017036055074093011201310

1500

–400–1.0 –0.5 0

r (mm)0.5 1.0 –1.0 –0.5 0

r (mm)0.5 1.0 –1.0 –0.5 0

r (mm)0.5 1.0 –210

–2017036055074093011201310

Figure 14. Pressure contours at different instances during bubble collapse near the wall for an initial normalized standoff distance �X ¼ 0:5, R0 ¼ 50 mm, Rmax ¼

2 mm, Pd ¼ 10 MPa. (Online version in colour.)

impa

ct p

ress

ure

(GPa

)

1 2 30

0

1

2

3

4X– = 0.75X– = 0.50

t – tlink (µs)

Figure 15. Pressure versus time at the centre of the plate for two initial non-dimensional bubble plate standoff distances. R0 ¼ 50 mm, Rmax ¼ 2 mm,Pd ¼ 10 MPa. (Online version in colour.)

X– = 0.75X– = 0.50

5

0

–5

–10

–15disp

lace

men

t(µm

)

–20

–25–400 –200

r (µm)0 200 400

Figure 16. Comparison of pit shapes between �X ¼ 0:5 and �X ¼ 0:75 forAl7075. R0 ¼ 50 mm, Rmax ¼ 2 mm, Pd ¼ 10 MPa. (Online version incolour.)

1.5

1.0

0.5

0

0.2 0.3

rigidAI7075

0.4

impa

ct p

ress

ure

(GPa

)

t – tlink (µs)

Figure 17. Comparison of the time history of the impact pressure betweenthe case where the plate was considered as rigid and no FSI was allowedand an Al7075 deformable plate. �X ¼ 0:75, R0 ¼ 50 mm, Rmax ¼ 2 mm,Pd ¼ 10 MPa. (Online version in colour.)


Focus5:20150016

11


Finally, a permanent deformation in the form of a pit remains

as a result of the high-pressure loading causing local stresses

that exceed the Al7075 elastic limit. The vertical displacement

of the monitored location eventually converges to a non-zero

value of about 9 mm.

The radial extent of the permanent deformation is shown

in figure 13. The permanent deformation generated on the

plate surface shows a profile, which is qualitatively similar

to those observed in previous experimental studies such as

in [80,82,83].

3.5. Effect of normalized standoff distanceIt is known from previous studies on an explosion bubble

near a rigid wall [41,65,79,84] that the impact pressure due

to the re-entrant jet attains a maximum at non-dimensional

distances between half and three quarters of the maximum

bubble radius, i.e. 0:5 , �X , 0:75: In those studies, the jet

hits the wall almost at the same time when re-entrant jet

touchdown occurs even though a small liquid film always


impa

ct p

ress

ure

(GPa

)

t – tlink (µs)1050

20

40

60

60

40

20

0

80(a) (b)

jet impact

ring collapse

versalink platerubber plateAL7075 plateA2205 platerigid plate

15 20 25t – tlink (µs)

10.09.59.0 10.5 11.0 11.5

Figure 18. (a) Overall time history of the pressure at the plate centre and (b) zoom near the pressure peaks due to ring collapse, for a rigid wall, two metallicalloys, and two compliant materials. �X ¼ 0:75, R0 ¼ 50 mm, Rmax ¼ 2 mm, Pd ¼ 0.1 MPa. (Online version in colour.)

0.6 rigid wallrubber

0.5

0.4

0.3

0.2

0.1

2 4 6 8 10 12

Req

(mm

)

t – tlink (µs)

Figure 19. Comparison of the time history of the equivalent bubble radiusfor a bubble collapse near a rigid wall and near a rubber wall. Zoom at theend of the collapse phase. �X ¼ 0:75, R0 ¼ 50 mm, Rmax ¼ 2 mm, Pd ¼

0.1 MPa. (Online version in colour.)

R0 = 50 µm standoff X

spacing D

Figure 20. Illustration of problem set-up for two bubbles near a wall.(Online version in colour.)


Focus5:20150016

12


exists between bubble and wall. However, for this study, as

shown in figure 6, the jet touchdown occurs at a small dis-

tance away from the wall. The shock generated by the jet

touchdown needs to travel a distance and thus attenuates

before reaching the wall. To show the effect of a direct jet

impact, a smaller standoff at �X ¼ 0:5 illustrated in figure 14

with the pressure contours at six selected time. One can see

clearly that the re-entrant jet directly impacts the wall when

the jet touchdown occurs at t 2 tlink ¼ 0.05 ms. Similar to

the �X ¼ 0:75 case, the volume of the bubble ring remaining

after the jet touchdown continues to shrink and reaches a

minimum at about t 2 tlink ¼ 0.92 ms. However, the high-

pressure shock wave generated by the bubble ring collapse

originates much closer to the wall as compared with the�X ¼ 0:75 case. This results in a higher concentrated pressure

loading at the plate centre when the shock wave propagates

towards and reaches the axis at t 2 tlink ¼ 1.1 ms.

Figure 15 shows the pressure versus time monitored at the

plate centre for different standoff distances. It is seen that

under the present conditions, the pressure loading due to

the jet impact is much higher for �X ¼ 0:5 because the jet

directly impacts on the wall when it penetrates the other

side of the bubble. The higher pressure loading due to the

direct jet impact and later more concentrated pressure loading

due to the ring collapse is expected to results in a different pit

shape on material surface. As shown in figure 16, the pit

radius is smaller with �X ¼ 0:5 than with �X ¼ 0:75, while the

pit depth is larger with �X ¼ 0:5 than with �X ¼ 0:75 because

of the higher magnitude and concentrated pressure loadings.

3.6. Effect of material complianceAs seen in figures 9, 12 and 15, two main pressure peaks can

be attributed to the jet impact and bubble ring collapse. As

shown below, the magnitudes of these pressure peaks

depend on the level of deformation of the material when

the pressure loading on the material is a result of full inter-

action between the collapsing bubble and the responding

material. Figure 17 shows a comparison of the time histories

of the collapse impulsive load between a non-deformable

wall (where FSI was not exercised and a rigid boundary con-

dition was used) and an Al7075 deformable plate (FSI used)

at the moment when the initial shock wave reaches the wall.

The figure shows that the pressure peaks felt at the plate

centre are notably smaller when the solid boundary deforms

and absorbs part of the energy.

To further study the effect of material compliance on the

pressure loading, we consider compliant materials with the


22°jet

43°jet

68°

jet

Figure 21. Shape of the bubbles and jetting directions at times close to re-entrant jet touchdown for each of the three normalized values of the�D ¼ 0:5, 1:5 and 2:5, and �X ¼ 0:75 (Darker areas are the flat jets penetrating the rest of the bubble). (Online version in colour.)

2000

1000

1500

mom

entu

m a

vera

ged

jet v

eloc

ity (

ms–1

)

500

13 14time (ms)

single bubbleD = 0.5 RmaxD = 1.5 RmaxD = 2.5 Rmax

15 16

Figure 22. Comparison of the re-entrant jet speed versus time between thesingle bubble and the three cases of tandem bubbles for �X ¼ 0:75 and�D ¼ 0:5, 1:5 and 2:5: (Online version in colour.)

1800single bubble jet speed 1640 m s–1

1600

1400

1200

1000

800

600

2 4 6 8 10D–

12 14 16 18 20

mom

entu

m a

vera

ged

jet v

eloc

ity (

ms–1

)

Figure 23. Effect of normalized bubble spacing on the re-entrant jet velocityat touchdown. (Online version in colour.)

0.4

0.3

0.2

0.1

0

–0.2 0y (cm)

0.2 –0.2 0y (cm)

0.2 –0.2 0y (cm)

0.2

z(c

m)

600p (MPa) p (MPa)

53847641435229022816610442

–20

600p (MPa)

53847641435229022816610442

–20–20122264406548690832974

12581400

1116

t = 14.49 µs t = 15.05 µs t = 15.67 µs

(a) (b) (c)

Figure 24. Pressure contours at different instances during the collapse of the two bubbles separated by �D ¼ 0:5 near a wall at �X ¼ 0:75. (Online versionin colour.)

0.4

0.3

0.2

0.1

0

–0.2 0y (cm)

0.2 –0.2 0y (cm)

0.2 –0.2 0y (cm)

0.2

z(c

m)

600p (MPa)

53847641435229022816610442–20

600p (MPa)

53847641435229022816610442–20

140012581116974832690548406264122–20

p (MPa)t = 16.32 µs t = 16.56 µs t = 16.79 µs

(a) (b) (c)

Figure 25. Pressure contours at different instances during the collapse of the two bubbles separated by �D ¼ 1:5 near a wall at �X ¼ 0:75. (Online versionin colour.)


Focus5:20150016

13



t =16.31 µs60053847641435229022816610442

–20

p (MPa)t = 15.69 µs0.4

0.3

0.2

0.1

0

–0.2 0y (cm)

z(c

m)

0.2 –0.2 0y (cm)

0.2 –0.2 0y (cm)

0.2

600

p (MPa)

53847641435229022816610442

–20

t =16.84 µs140012581116974832690548406264122

–20

p (MPa)

(a) (b) (c)

Figure 26. Pressure contours at different instances during the collapse of the two bubbles separated by �D ¼ 2:5 near a wall at �X ¼ 0:75. (Online version in colour.)

14

0

0.5

1.0

1.5

2.0

impa

ct p

ress

ure

(GPa

) 2.5

3.0

3.5

15 16time (µs)

D = 2.5 Rmax

D = 1.5 Rmax

D = 0.5 Rmax

single bubble

17 18

Figure 27. Comparison of time history of the pressure at the plate centrebetween the single bubble and the three cases of tandem bubbles for�X ¼ 0:75 and �D ¼ 0:5, 1:5 and 2:5: (Online version in colour.)

D = 2.5 Rmax

D = 1.5 Rmax

D = 0.5 Rmax

single bubble

–3

–0.20

–0.15

–0.10

–0.05

0

–2 –1 0y (mm)

z (m

m)

1 2 3

Figure 28. Comparison of the resulting pit shape in Al7075 between thesingle bubble and the three cases of tandem bubbles for �X ¼ 0:75 and�D ¼ 0:5, 1:5 and 2:5: (Online version in colour.)

0.2

0.4

0 0.5 1.0

sum of pit volume from two single bubble = 0.3 mm3

1.5 2.0 2.5 3.0D–

pit v

olum

e (m

m3 )

Figure 29. Comparison of the resulting pit volume in Al7075 for the threecases of tandem bubbles for �X ¼ 0:75 and �D ¼ 0:5, 1:5 and 2:5: (Onlineversion in colour.)


Focus5:20150016

14


properties described in §2.5. Due to the low compressive and

shear strength of the considered compliant materials, a smaller

collapse driving pressure, Pd ¼ 0.1 MPa, is chosen as these

materials cannot handle the large Pd value used with the met-

allic alloys. Figure 18a shows a comparison of the time histories

of the pressures resulting from the bubble dynamics for the

same base case shown in the previous section. The figure

shows for illustration the results on a rigid fixed plate, on

A2205, Al7075, Rubber no.2 and Versalink.

Figure 18b shows a zoom on the pressure peaks due to the

bubble ring collapse. This figure clearly shows that the press-

ures felt by the metallic alloys and the rigid plate are very

close. On the other hand, due to absorption of energy

through deformation, lower magnitude of the impact press-

ures are felt by the softer materials. This was already

observed experimentally by Chahine & Kalumuck [85].

Also, a delay in the peak timing indicates that a soft material

tends to elongate the bubble period as compared with the

rigid material as shown in figure 19.

4. Second bubble effect on bubble collapse nearwall

As in the previous sections of this paper, studies of the inter-

action of inertial bubbles with nearby boundaries have

mostly concentrated on isolated bubbles. However, bubbles

are seldom isolated and their interactions with each other


XY

Z

XY

Z

Figure 30. Shapes and pressure contours at the time of incompressible – compressible link for two bubbles separated by �D ¼ 1:64 with the left-hand side bubblehaving R0 ¼ 142 mm, Pg0 ¼ 115 GPa and �X 1 ¼ 0:565, while the right-hand side bubble has R0 ¼ 50 mm, Pg0 ¼ 1,500 GPa and �X 2 ¼ 0:665: (Online version incolour.)

t = 14.65 µs t = 15.07 µs t = 15.55 µs

t = 16.27 µs t = 17.24 µs t = 17.92 µs

1700 MPa

1500 Mpa

(a) (b) (c)

(d) (e) ( f )

–0.4–0.1

0

0.1

0.2

0.3

0.4

0.5

–0.2 0y (cm)

z(c

m)

–0.1

0

0.1

0.2

0.3

0.4

0.5

z(c

m)

P (MPa)

60053847641435229022816610442–20

P (MPa)

60053847641435229022816610442–20

P (MPa)

60053847641435229022816610442–20

P (MPa)

60053847641435229022816610442–20

P (MPa)

1500134811961044892740588438284132–20

P (MPa)

17001528135611841012840688496324152–20

0.2 –0.4 –0.2 0y (cm)

0.2 –0.4 –0.2 0y (cm)

0.2

Figure 31. Pressure contours at different instances during the collapse of two bubbles separated by �D ¼ 1:64 with the left-hand side bubble having R0 ¼

142 mm, Pg0 ¼ 0.115 GPa and �X 1 ¼ 0:565, while the right-hand side bubble has R0 ¼ 50 mm, Pg0 ¼ 1.5 GPa and �X 2 ¼ 0:665: (Online version in colour.)


Focus5:20150016

15


affect their dynamics and the way by which they load a

nearby boundary. Also, there are configurations, such as for

controlled cell sonoporation [30,31], or for underwater

explosion applications [86,87], where the interactive

dynamics of two or more bubbles is purposely sought.

In this section, we consider for illustration a simple

example of two identical inertial bubbles initially located as

illustrated in figure 20. The two bubbles are initially spherical

and at the same normalized standoff distance, �X ¼ 0:75,

from the plate. Their centres are separated from each other

by a distance, D. The initial bubble radius for each of

the bubbles is R0 ¼ 50 mm and the initial gas pressure

(Pg0 ¼ 1500 GPa) is specified such that the bubble will grow

to a maximum radius Rmax of 2 mm if it was isolated in a

free field when the ambient pressure (bubble collapse

pressure) is Pd ¼ 100 MPa, for example for a bubble gener-

ated by energy deposition in a the high stagnation pressure

of a cavitating jet. The very high initial pressure was obtained

by applying the Rayleigh–Plesset equation to obtain an

initial bubble radius of 50 mm and a maximum value of

2 mm. This would correspond to a laser-generated bubble

where actually the value of R0 and Pg0 do not really corre-

spond to a physical configuration but are selected to

represent mathematically the energy deposit in the laser

beam, which generates a 2-mm bubble.

Here, we consider the effects of the normalized bubble spa-

cing, �D ¼ D=Rmax, on the pressure loading and the pit shapes

for the same �X ¼ 0:75: Figure 21 presents for three different


–3

–0.20

–0.15

–0.10

–0.05

0

–2 –1 0

bubble no. 1 alonebubble no. 2 alonebubble no. 1 and bubble no. 2

y (mm)

z (m

m)

1 2 3

Figure 32. Comparison of the resulting pit shapes in Al7075 between thetandem bubble and two single bubble cases. (Online version in colour.)


Focus5:20150016

16


values of the normalized bubble spacing, �D ¼ 0:5, 1.5 and 2.5,

the bubble shapes at a time when the re-entrant jet (appearing

as a darkened area of the grid in figure 21) is about to impact the

opposite side of the bubble. The jet angle is obtained from the

velocity vector at the fastest node on the bubble re-entrant jet

surface. While a single bubble collapsing near a rigid wall

forms a re-entrant jet perpendicular to the wall, in the two

bubbles problem the re-entrant jet direction is controlled by

both the presence of the second bubble and the wall. Therefore,

the jet forms a shallower angle, between 208 and 708 for the

various bubbles spacing.

To compare quantitatively the effect of the spacing, �D, on

the re-entrant jet behaviour, we introduce as a jet character-

istic the momentum averaged jet velocity vector, Vmom,

defined as the average of the liquid velocity integrated in

the whole re-entrant jet volume (region of the bubble surface

with negative curvature) as follows:

Vmom ¼1

V

ðVd V , ð4:1Þ

where V is the liquid velocity at any point inside the jet, and V

is the jet volume.

The jet speed versus time for the three values of �D can be

seen in figure 22. We can observe in the figure that the pres-

ence of other nearby dynamically behaving bubbles does not

necessarily enhance the cavitation dynamic effects and can

negatively affect the strength of each bubble re-entrant jet

impact at the wall. These effects depend strongly on the

bubble and wall spacing as well as on the timing of the

two bubbles and deserve a full study.

Figure 22 shows that the single bubble re-entrant jet is

the shortest lived and the fastest, reaching sound speed

under the present conditions. The other cases are subsonic,

with the �D ¼ 0:5 condition resulting in the lowest jet speed

with almost one-third of the isolated bubble jet speed.

Figure 23 analyses further the effect of the bubble spacing�D on the jet velocity at touchdown. It can be seen from the

figure that the jet speed of the tandem bubbles at touchdown

approaches that of a single bubble when �D increases. Vmom is

only 5.5% less than the single bubble value for �D ¼ 6 and

2.5% for �D ¼ 10: This is consistent with previous work on

the limit of influence of a rigid wall (where the image plays

the role of the second bubble), which corresponds to a

standoff from the wall of about 3 [88].

Figures 24, 25 and 26 show the pressure contours at differ-

ent instances during the bubble collapse near the wall for the

three normalized standoffs �D ¼ 0:5, 1:5 and 2:5, respectively.

Each time sequence is shown with its own pressure scale in

order to be able to highlight the important high-pressure

regions locations and the displacement of these ‘hot spots’

with time.

When the two bubbles are very close, �D ¼ 0:5 (figure 24),

the re-entrant jets are so much deviated from the direction of

the wall that they impact on each other (and not on the

bottom wall) in the plane of symmetry of the problem.

A high impact pressure is then generated away from the

bottom wall. Figure 24a shows the initiation of the shock

when the two re-entrant jets impact each other. The pressure

then reaches in figure 24b its highest peak when the top

halves of the two bubbles complete their collapse. A high-

pressure wave is then observed to propagate radially away

from the highest pressure spot with a pressure front moving

towards the wall. The pressure amplitude is however

significantly attenuated before it reaches the bottom wall in

figure 24c.

For �D ¼ 1:5, each of the two bubbles is equidistant from

the rigid wall and the plane of symmetry. This equal attrac-

tion results in two very wide re-entrant jets both moving

towards the centre of the plate. Figure 25a shows the complex

shape of each bubble before touchdown on both the plane of

symmetry and the bottom plate as seen in figure 25b. This

generates high-pressure spots at the three impact locations:

two on the bottom plate and one in the plane of symmetry.

As the re-entrant jet front continues moving towards the

plate centre, the residuals of bubbles near the centre even-

tually collapse completely and result in a highly focused

pressure loading at the centre of the plate.

In the other extreme case, �D ¼ 2:5 (figure 26), the inter-

action between the bubbles is much weaker, the re-entrant

jets remain close to perpendicular to the wall, and two

shock centres are generated first when each jet impacts the

wall. These shocks then move along the wall surface towards

the plane of symmetry. The pressure loading on the wall is

then much higher than for the �D ¼ 0:5 case.

A quantitative comparison of the time history of the

pressure at the plate centre is shown in figure 27. It is seen

that �D ¼ 1:5 results here in a highest pressure loading

among the three tandem bubble cases at the centre of plate.

However, when compared with the single bubble case, the

pressure is only about 88% of the isolated bubble case.

Figure 28 shows a comparison of the shape of the final pit

in Al 7075 for the three bubble spacing cases. It is seen that

the D ¼ 1.5 Rmax results in the deepest pit among the three

tandem bubble cases because of the highest concentrated

loading at the centre of plate. However, the �D ¼ 2:5 case

results in a wider pit because of the two separate high-

pressure impacts from each of the bubbles. The single

bubble case is however the most erosive for the present con-

ditions. Figure 29 shows a comparison of the volume of the

final pit in Al 7075 for the three bubble spacing cases. It is

seen that the total volume of the pit, as for the depth,

increases as the spacing is increased and it surpasses the

sum of the pit volumes from the two isolated single bubbles

for �D ¼ 2:5 case (figure 29).


excitationsource

plan

e of

sym

met

ry

no r

efle

ctio

n B

C

no reflection BC

polyurea

bubble cloud

50 mm

100 mm

rigid bodyx

z

20 mm

15 mm

6 mm

Figure 33. Bubble cloud material interaction problem set-up. A side view of the half domain is shown. (Online version in colour.)

ZY

X

Figure 34. Illustration of the distribution of the 360-bubble cloud shown inthe quarter domain (90 bubbles). For the computations shown below allinitial bubble radii are 100 mm, and the finest mesh size near the bubblesis 25 mm. A total of 4.7 million cells are used in the 1/4th domain. (Onlineversion in colour.)


Focus5:20150016

17


Figure 30 investigates another tandem bubble configur-

ation to illustrate conditions where the two bubble sizes and

the two standoffs are not the same. The first bubble is selected

to have an initial radius R0 ¼ 142 mm, and initial gas pressure

Pg0 ¼ 115 GPa, and is located at a normalized standoff,�X1 ¼ 0:565, while the second bubble has R0 ¼ 50 mm, Pg0 ¼

1500 GPa and �X2 ¼ 0:665: The initial pressures are again

selected such that each bubble would grow, if isolated and

in an infinite medium, to a maximum radius Rmax ¼ 2 mm

when the ambient pressure is Pd ¼ 100 MPa The normalized

spacing between the two bubbles is �D ¼ 1:64. As the second

bubble, located in the right-hand side of the figure, collapses

first, the incompressible–compressible link in this case was

initiated right before the re-entrant jet of the second bubble

touches down, as shown in figure 30.

Figure 31 shows pressure contours at different instances

during the collapse of the two bubbles. It is seen that the

right-hand side bubble collapses first at t ¼ 15.55 ms and

results in a high-pressure loading of about 1700 MPa on the

material surface as seen in figure 31c. The left-hand side

bubble collapses next at t ¼ 17.92 ms and results in a pressure

loading of about 1500 MPa as seen in figure 31f.Figure 32 compares the resulting permanent deformation

outlines in the plane of symmetry of the problem for Al 7075.

The figure compares the tandem bubble results with each of

the two bubbles when alone. It is seen that each of the two iso-

lated bubbles results in about the same pit depth with the

second bubble producing a slightly wider pit. Both single

bubble pits are, however, much larger than when the tandem

bubbles act together. The tandem bubbles generate two pits

with the right bubble, which collapses first and generates a

higher pressure loading, resulting in a wider and deeper pit

on the material than the left bubble. The sum of the volumes

of the two pits is about the same as that due to two of the

isolated single bubbles. A systematic study on the various com-

binations of tandem bubbles sizes and position would need to

be conducted in the future to further understand the various

effects of bubble/bubble interaction (constructive versus

destructive) on the permanent deformation.

5. Bubble cloud effectsIn this section, we discuss, as an illustration of multi-bubble

dynamics, the behaviour of an inertial cloud of bubbles

interacting with a soft material, Polyurea. The selected three-

dimensional computational domain is 100 � 100 � 50 mm

and possesses two planes of symmetry (the XOY and YOZplanes) as shown in figure 33. The bubble ‘cloud’ behaviour

is triggered by the dynamic of a bubble of initial radius

3.15 mm and initial internal pressure 25 atm placed in the

domain which has an ambient pressure of 1 atm. The driving

bubble is located on the Z-axis 20 mm away from the material

surface whose initial interface with the liquid is located in the

plane XOY. The material is a 6 mm thick block of polyurea

(Versalink). The bubble cloud is composed for ease of analysis


t = 10 µs t = 19 µs t = 21 µs

0.050 0.10 0.15 0.20

x (cm)

0.25 0.30 0.351 × 1062 × 1063 × 1064 × 1065 × 1066 × 1067 × 1068 × 1069 × 1061 × 1071.1 × 1071.2 × 1071.3 × 1071.4 × 1071.5 × 1071.6 × 1071.7 × 1071.8 × 1071.9 × 1072.0 × 1072.1 × 1072.2 × 1072.3 × 1072.4 × 107

p (dyne cm–2)p (dyne cm–2)p (dyne cm–2)

0.050 0.10 0.15 0.20

x (cm)

0.25 0.30 0.350.0500.45

0.50

0.55

0.60

0.65

0.70

0.75(a) (b) (c)

0.10 0.15 0.20

x (cm)

z(c

m)

0.25 0.30 0.35

Figure 35. Pressure field in the bubble cloud region at three different instances: (a) t ¼ 10 ms, (b) t ¼ 19 ms and (c) 21 ms (at the end of the first cloud collapse). Thebubble cloud shields the wall below it from the incoming high pressure first before it collapses in its turn and raises the pressure. (Online version in colour.)

0

50

100

radi

i of

bubb

les

in th

e cl

oud

(µm

)

150

200

50

R0 = 100 µm

R0 = 150 µm

Pd

time (µs)100 150

0

5

10

driv

ing

pres

sure

(at

m)

15

20

25

Figure 36. Time history of the radii of several randomly selected bubbles inthe cloud. Two cases are shown one where all initial bubble radii were 100mm and the other where they were 150 mm. (Online version in colour.)

0

5 × 106

1.0 × 107

1.5 × 107

2.0 × 107

5 × 10–5

time (s)

pres

sure

(Pa

)

0.00010.0001

R0 = 100 µmR0 = 150 µm

Figure 37. Time history of the pressure at the centre on the material surface.Two cases are shown one where all initial bubble radii were 100 mm and theother where they were 150 mm. (Online version in colour.)


Focus5:20150016

18


of bubbles with the same radius initially at equilibrium with

the 1 atm ambient pressure.

Figure 34 illustrates such a cloud configuration in the one-

fourth volume (because of double symmetry). The bubbles in

the cloud are located within the projected square [22 mm ,

x , 2 mm, 22 mm , y , 2 mm] and are equally spaced in

the vertical direction within a 1 mm height [z , 1 mm]. The

bubbles in the cloud are distributed in four layers: the

bottom layer (closest to the Versalink) has 10 � 10 bubbles,

the second layer has 8 � 10 bubbles, the third layer has

10 � 10 bubbles and the top layer has 8 � 10 bubbles in the

quarter domain. Bubble centres in each layer are staggered

by half bubble spacing from the adjacent layers. The total

number of bubbles is 360 in the full domain. The finest

mesh size of 25 mm is used near these bubbles. A total of

4.7 million cells are used to model the problem. In the far

field, no reflection boundary conditions are applied.

The simulations in this section were obtained from the

compressible code only without resorting to the

compressible–incompressible link.

Figure 35 shows the pressure field near the bubble cloud 10,

19 and 21 ms after the initiation of the driving bubble growth

and as the high-pressure wave reaches the bubble cloud.

The cloud is seen to first act as a shield, preventing the high

pressure generated by the source bubble from reaching the

wall. This is seen in figure 35a,b where the high pressure (red

region) impacts the wall outside the bubble cloud while the

pressure under the bubble cloud remains close to the initial

1 atm. As first observed in [89], the bubbles on the outer edge

of the cloud begin to collapse first, and then the phenomenon

propagates inward to the centre of the bubble cloud. The

bubbles in the cloud then collapse strongly (figure 35c) and

then go through multiple collapse–rebound cycles, which

generate multiple pressure pulses.

Figure 36 shows the time history of the radii of a few

selected bubbles in the cloud. Overall, the bubbles in the

cloud practically collapse and rebound almost simul-

taneously. The first collapse is the strongest as this brings

the bubbles to their smallest volume during their history

(figure 35c). Figure 36 also shows the internal pressure of

the driving bubble, and the radii of the bubbles in the

cloud when they all have initial radii of 150 mm. Even

though the outside contour of the cloud has not changed

from the previous case, the cloud period is increased–in

the same way as the individual bubble radii following the

Rayleigh collapse time—and is almost doubled. Here again

all bubbles collapse and rebound almost in unison resulting

in a very high pressure at collapse. Figure 37 shows the

corresponding pressure time history at the origin of coordi-

nate, i.e. centre of the material surface. The first pressure

peak from the collapse of the 150 mm bubbles is more than

four times higher than that of the 100 mm bubbles.


4 × 10–6

5 × 107

pres

sure

(Pa

)

1.5 × 108

1.0 × 108

6 × 10–6 8 × 10–6 1.0 × 10–5 1.2 × 10–5 1.40

time (s)

R0 = 100 µm, 100 atm air bubble source

1 bubble in cloud, rigid wall1 bubbles in cloud, Versalink10 bubbles in cloud, rigid wall10 bubbles in cloud, Versalink34 bubbles in cloud, rigid wall34 bubbles in cloud, Versalink74 bubbles in cloud, rigid wall74 bubbles in cloud, Versalink

Figure 38. Effect of the number of bubbles in the cloud and of the wallmaterial response on the pressure generated by the bubble cloud collapse.(Online version in colour.)

1.2 × 10–5 1.4

5 × 107

pres

sure

(Pa)

1.5 × 108

1.0 × 108

04 × 10–6 6 × 10–6 8 × 10–6 1.0 × 10–5

time (s)

R0 = 100 µm, 100 atm air bubble source

1 bubble in cloud, rigid wall1 bubbles in cloud, Versalink74 bubbles in cloud, rigid wall74 bubbles in cloud, Versalink

Figure 39. Comparison of the effects of wall compliance on wall pressuredue to one bubble and 74 bubbles. (Online version in colour.)


Focus5:20150016

19


Several computations with a different number of bubbles

in the cloud are illustrated in figures 38 and 39. Figure 38

shows the time histories of the pressure at the centre on the

material obtained with different number of bubbles in the

cloud both near a rigid wall and when the wall is responsive

and made of Versalink. It can be observed that the period of

the bubble cloud collapse and the pressure peak both increase

as the number of bubbles in the cloud increases. This is the

case for both material responses: one with a rigid wall and

the other with a compliant Versalink wall. The bubble

cloud works as an energy accumulator; the energy from the

incoming pressure wave is absorbed by the bubble cloud,

and then re-emitted as the bubble cloud collapses at a later

time. A larger cloud with a larger number of bubbles

accumulates more energy and then generates later a larger

pressure peak at the cloud collapse.

Figure 39 shows four curves selected from figure 38. The

pressure peaks with the rigid wall are observed to be two to

three times higher than those with the compliant wall. As the

high pressure reaches the compliant wall, the wall responds

by compressing first and absorbing energy through

deformation. This fluid–structure interaction and energy

absorption by deformation reduces the pressure peaks.

6. ConclusionIn this contribution, the material pitting due to cavitation

bubble collapse is studied by modelling the dynamics of

growing and collapsing bubbles near responding and

deforming materials. The pressure loading on the material

surface during the bubble collapse is shown to be due to

the re-entrant jet impact and to the collapse of the remaining

bubble ring. The high-pressure loading results in high stress

waves, which propagate radially from the loading location

into the material and cause material deformation. Permanent

deformation in the shape of a pit is formed when the local

equivalent stresses exceed the material yield stress.

The impulsive pressure loading due to the bubble collapse is

highly dependent on the initial standoff distance between the

bubble and the nearby boundary. This standoff distance affects

the jet characteristics in a non-monotonic fashion. Higher jet vel-

ocities occur at the larger standoff distances. However, a higher

jet velocity does not necessarily result in a higher impact pressure,

as the impact pressure also depends on the distance the jet

front has to travel to impact the wall after touching down the

opposite side of the bubble. The energy transferred to the wall

is maximum at a normalized standoff distance close to�X ¼ 0:75. A more concentrated pressure loading on the material

surface is obtained for smaller standoffs where both the jet

touches downs and the bubble ring collapses very close to the

wall. Such concentrated pressure loadings result in deeper but

narrower pits. As a result, the shape of the pit, i.e. the ratio of

pit radius and depth does not vary monotonically with standoff.

The magnitude of the pressure peak felt by the material

depends on the response and amount of deformation of the

solid. Fluid–structure interaction simulations show that the

load on the material is damped with material deformation.

The load reduction with wall response increases when the

solid boundary deformation increases due to increased

energy absorption. Impact pressures for metallic alloys are

very close to those on a rigid plate while compliant materials

deform and absorb energy. This results in lower magnitude

of the impact pressures for the coatings and delays in peak

occurrence due to lengthening of the bubble period.

Interaction between bubbles significantly influences the

pressure loading on the material surface and the resulting

pit shape. This interaction requires extensive study as both

enhancement and negative interference of the interaction on

the resulting damage can be seen. A classification of these

effects deserves further investigations. Similarly, the work

needs to be extended to the study of actual damage and

material loss, which requires fracture and damage models

as opposed to the present relatively simple elastic-plastic

model used here. Such models would involve various criteria

including tension and shear failure such as for hardened and

brittle materials and heating effects such as for coatings.

Competing interests. We declare we have no competing interests.

Funding. This work was conducted under support from Dynaflow, Inc.internal IR&D and partial support from the Office of Naval Researchunder contract N00014-12-M-0238, monitored by Dr Ki-Han Kim.

Acknowledgements. We thank Dr Kim for his support. We thankMr Gregory Harris from the Naval Surface Warfare Center, IndianHead, for allowing us to access the GEMINI code and giving us theopportunity to contribute to the coupling within DYSMAS of ourcode 3DYNAFS and the DYNA3D Structure code. We are also gratefulto many colleagues at DYNAFLOW, who have contributed to severalaspects of this study, most particularly, Dr Jin-Keun Choi and DrAnil Kapahi for their contributions.


20


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