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1 Copyright © 2012 by ASME

Proceedings of the Internoise 2012/ASME NCAD meeting

August 19-22, 2012, New York City, NY, USA

IN12-586

CHARACTERIZATION OF THE PRESSURE WAVE EMITTED FROM IMPLOSION OF

SUBMERGED CYLINDRICAL SHELL STRUCTURES

Michael D. Shields Ph.D.

Weidlinger Associates, Inc.

New York, NY, USA

shields@wai.com woelke@wai.com

ABSTRACT

Buckling of submerged cylindrical shells is a sudden and

rapid implosion which emits a high pressure pulse that may be

damaging to nearby structures. The characteristics of this

pressure pulse are dictated by various parameters defining the

shell structure such as the length to diameter ratio, shell

thickness, material, and the existence and configuration of

internal stiffeners. This study examines, through the use of high

fidelity coupled fluid-structure finite element computations, the

impact of various structural parameters on the resulting

pressure wave emanating from the implosion. The results

demonstrate that certain structural configurations produce

pressure waves with higher peak pressure and impulse thereby

enhancing the potential for damage to nearby structures.

Pawel Woelke Ph.D.

Weidlinger Associates, Inc.

New York, NY, USA

Najib N. Abboud Ph.D.

Weidlinger Associates, Inc.

New York, NY, USA

abboud@wai.com

INTRODUCTION

The implosion phenomenon is the result of structural

instability (either elastic or inelastic) of an enclosed volume

with relatively low internal pressure subjected to large external

hydrostatic pressure. This instability, when it occurs, very rapid

and can result in a complete inward collapse of the structure.

Engineers must be concerned with implosion when designing

or analyzing any enclosed structure that is expected to

withstand large external pressure. This environment may arise

for example in deep sea applications such as off-shore oil

drilling, sea exploration, or deep sea naval exercises as well as

in certain hydraulic systems. Although design standards are in

place to aid designers in avoiding such a catastrophic failure

(e.g. [1], [2]), these codes do not necessarily cover some of the

extreme environments (such as underwater shock) that may be

encountered in the applications listed above.

The problem of implosion extends beyond the structural

collapse itself (and the potential payload loss that may result)

because an implosion event may also result in a violent shock-

type pressure wave emanating from the implosion in the

surrounding fluid. This pressure wave has the potential to

damage nearby structures. The objective of this work is to

begin characterizing this emitted pressure wave in terms of

specific structural characteristics using high fidelity fluid-

structure finite element calculations. The most common

implodable volumes are cylinders and spheres [3] although an

externally pressurized closed volume of any shape will be

vulnerable to implosion [4]. The study presented in this work

focuses specifically on the characteristics of the fluid response

resulting from implosion of two unstiffened cylindrical shell

structures with differing length, diameter, and shell thickness

and represents a small sample of a much larger effort to

characterize the fluid response for a generalized set of structural

parameters.

IMPLOSION BEHAVIOR FOR CYLINDERS

Structural Behavior

Implosion of an unstiffened cylindrical structure generally

results from one of two possible failure mechanisms: elastic

buckling or plastic buckling. The elastic buckling pressure, ???,

for unstiffened cylinders has been studied exhaustively and can

be predicted with reasonable accuracy using the equation

presented by Timoshenko and Gere [5]:

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2

2

2

2

2

2

2

2

2

2

2

1 12

1

1

2

1

1

l

a

n

a

h

a

l

n

l

a

n

a

Eh

qcr

abrupt halt in the flow of fluid filling the void. This results in a

high pressure wave being emitted in all directions in the fluid

(Figure 1b). The initial contact typically occurs at the center-

span of the cylinder and then contact propagates both

circumferentially and along the length of the cylinder. The

cylinder fully closes first circumferentially (Figure 1c) and then

longitudinally. As this is occurring, a continual high pressure

wave emanates from the vicinity of the cylinder until contact

reaches the end of the cylinder. At this point, void regions are

no longer being created in the fluid and a final high pressure

wave is emitted into the fluid (Figure 1d). This behavior has

been repeatedly observed both experimentally and numerically.

The fluid response from implosion can also be observed

by considering a pressure gauge in the fluid located adjacent to

the longitudinal center of the cylinder and separated from the

cylinder center by a distance ? as shown in Figure 2.

(1)

where ? is the elastic modulus of the material, ? is Poisson’s

ratio, ℎ is the cylinder shell thickness, ? is the cylinder radius,

and ? is the length of the cylinder. Equation (1) is minimized for

integer values of ? to determine the elastic buckling pressure

and respective Eigen mode. Plastic buckling results from

yielding of the cylinder causing significant plastic deformations

and ultimately a loss of stability. For design purposes, failure is

assumed to occur instantaneously at the moment of yield

although this is generally not true because collapse is only

initiated after sufficient deformation has occurred.

Internally ring stiffened cylinders, on the other hand, are

more complicated but implosion is generally a result of the

same two mechanisms with some limited exceptions.

Additional factors which may affect the collapse pressure for a

cylinder and the mode of collapse include various forms of

structural imperfections which may be present including out-of-

roundness, thickness variations, specification tolerances, and

corrosion. For a general review of the design and analysis of

ring stiffened cylinders and cylinders with imperfections, the

reader is referred to a concise review work published by

MacKay from Defense R&D of Canada [6].

Fluid Behavior

Implosion is an inherently coupled problem involving

complex interactions between the highly deforming structure

and the surrounding fluid flow. In particular, as the pressure in

the fluid increases, the structure deforms elastically until it

reaches ???, at which point a bifurcation occurs and the

structure becomes unstable. As the structure collapses it creates

a void into which the adjacent high pressure fluid flows. This

serves to temporarily reduce the pressure in the surrounding

fluid (Figure 1a). The structure continues to collapse until the

opposite sides make contact. This initial contact causes an

Figure 2. Hypothetical pressure sensor (red dot): end view

(left), side view (right)

At the location of this pressure sensor, a typical pressure history

is shown in Figure 3. This history can be conveniently broken

into five distinct domains which highlight the evolving fluid

behavior during implosion. The specific details of the pressure

histories will vary depending on the cylinder design (and

particularly the circumferential failure mode), but the overall

behavior is generally valid. The first domain (denoted by a (1.)

Figure 1. Qualitative depiction of the various stages of a mode 2 implosion. Blue coloration indicates low pressure while red

indicates high pressure and green intermediate pressure. (a.) Collapse initiation. (b.) First contact. (c.) Full circumferential closure

at mid-span. (d.) Full longitudinal closure.

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in the figure) is the so-called “under-pressure” domain wherein

the fluid pressure is reduced during the period of time where

the faces of the cylinder are collapsing but have not yet made

contact. The second domain corresponds to the initiation of

contact and the circumferential flattening of the cylinder center.

During this time domain, two peaks occur. The first peak

corresponds to the initial contact between cylinder faces.

Shortly thereafter, the peak pressure pulse arrives. This is

followed by a high pressure plateau in domain (3.) during

which time the collapse is propagating longitudinally down the

cylinder. Once the collapse propagates to the end of the

cylinder, the full volume has closed and a final pressure wave is

emitted as shown in domain (4.). Finally, the structure and

fluid return to equilibrium as shown in domain (5.).

Figure 3. Representative implosion pressure time history.

STUDY CYLINDERS AND COMPUTATIONAL MODELS

Two specific cylindrical structures are considered in this

study and will be referred to as Model A and Model B. The

design parameters for each of the cylinders are presented in

Table 1. Model A is a shorter cylinder with a larger diameter

having length/diameter ratio (L/D) equal to 5.36 while Model B

is a longer and more slender cylinder with L/D = 10.9.

Furthermore, the volumes of Model A and B are approximately

equal with ??= 297 ??3 and ??= 289 ??3. Using Eqn. (1)

Model A has an analytical hydrostatic collapse pressure

???≅ 1050 ??? and Model B has ???≅ 1400 ???. Both

models buckle in mode ? = 2.

Table 1. Design parameters for studied cylinders.

Length

Diameter

Thickness

Model A

22.16 in.

4.13 in.

0.14 in.

Model B

35.23 in.

3.23 in.

0.133 in.

The cylinders and fluid were modeled using Weidlinger

Associates, Inc. (WAI) explicit dynamic finite element software

EPSA [7]. The structure is modeled using a thick shell element

formulation with a strain-displacement matrix derived

according to Flanagan and Belytschko [8]. The fluid is modeled

using Lagrangian hexahedral elements. Three planes of

symmetry are utilized to reduce model size as shown in Figure

4 for Model A. The structure mesh for both Model A and Model

B have 120 elements around the circumference (30 elements

through ¼ of circumference modeled) and the fluid hexahedra

match the structure mesh at the interface and slowly grow in

size radially to a silent boundary. The fluid mesh for Model A

extends 32 in. radially and 28 in. axially with a total of

approximately 650,000 elements (see Figure 4). The fluid

mesh for Model B extends 25 in. radially and 35 in. axially

with a total of approximately 940,000 elements (not shown).

Figure 4. FE representation of Model A showing the structure

(blue) and fluid (gray) along with symmetry planes and

boundary conditions.

(OOR) imperfection has been added to each structure model.

This is done for two reasons. First, it will ensure collapse in the

desired direction relative to pressure recording locations in the

fluid mesh. Second, it allows collapse to be initiated in both

cylinders at identical hydrostatic pressure for direct comparison

of the resulting fluid pressure curves. This is an alternative to

other methods of inducing implosion such as modeling a rigid

indenter similar to one that is used in a controlled test. In both

cases, collapse is initiated at a fluid pressure of 1000 psi.

Table 2. Basic rate-independent material properties for

two aluminum alloys.

Additionally, a small mode ? = 2 out-of-roundness

Al. 6061

10,000 ksi

0.3

38 ksi

48 ksi

Al 5086-H32

10,000 ksi

0.3

30 ksi

42 ksi

?

?

??

??

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different aluminum alloys. These cylinders have been tested

experimentally and the results presented herein have been

validated against these experiments although this will not be

explicitly discussed. Model A is machined from Al 6061 while

Model B is machined from Al 5086-H32. The basic rate-

independent material properties used for these alloys are given

in Table 2.

For Al 6061, the material model is elastic-plastic with

strain hardening based on uni-axial tension tests performed at

the University of Texas at Austin [9]. Rate enhancement is

added based on work performed by Hoge [10]. Figure 5 shows

single-element uniaxial tension stress-strain relations at various

strain rates for this material model. Current material data for Al

5086-H32 were not available for calibration of the material

model for Model B. Consequently, the material was fit using a

typical uniaxial tensile stress-strain curve found in [11]. Rate

enhancement was assumed to be the same as for Al 6061.

Figure 6 shows single-element uniaxial tension stress-strain

relations at various strain rates for the Al 5086-H32 material

model. Material failure is not considered.

Model A and B are based on physical models made of

Figure 5. Rate-dependent uniaxial stress-strain relation for Al

6061.

Figure 6. Rate-dependent uniaxial stress-strain relation for Al

5086-H32.

shear strength where each element represents an incompressible

volume. The water is endowed with a bilinear material model

that accounts for cavitation.

Water is modeled using a material idealization with zero

RESULTS

Both models collapse in mode ? = 2 general instability at

1000 psi. The collapsed shapes for these models are shown in

Figure 7. The failure shapes are nearly identical with the

cylinders flattening nearly completely.

Figure 7. Collapsed shape for Model A (top) and Model B

(bottom) - not to scale.

To observe fluid pressure, several sensors were placed in the

fluid in a radial arrangement at various lengths along the

cylinders. For both Model A and Model B, the fluid response in

this study is evaluated at a point at the mid-span of the cylinder

at a distance of 6” from the center of the cylinder. That is, the

sensor is oriented as shown in Figure 2 with ? = 6".

Figure 8 shows the implosion pressure histories for both

models at the specified sensor location. Numerous interesting

features are observed in these curves. First, Model B – which

has approximately 25% smaller diameter – implodes faster. In

particular, the elapsed time from collapse initiation to first

contact is shorter (as observed from the duration of the under-

pressure phase of the pressure history). This may be expected

considering it has smaller diameter. However, Figure 9, which

shows the collapse velocity of the node at the center of the

collapsing face, demonstrates that Model B also collapses with

greater velocity. Model A has a peak velocity of 2080 in/sec.

while Model B has peak velocity of 2750 in/sec. Additionally,

it is observed that the peak negative pressure is larger for

Model B than for Model A. The greater collapse velocity

therefore causes the pressure reduction in the near-field fluid to

be enhanced.

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5 Copyright © 2012 by ASME

Figure 8. Fluid pressure time histories for Model A (blue) and

Model B (red).

Figure 9. Structural collapse velocities at center span for

Model A (blue) and Model B (red).

that the pressure pulse from first contact in Model B has

approximately twice the magnitude of Model A and, more

importantly, the peak pressure at the fluid sensor location from

Model B, ????= 598 ???, is approximate 50% larger than

from Model A where ????= 429 ???. Note that these are

gauge pressures and therefore represent an increase in pressure

above the 1000 ??? ambient hydrostatic pressure. This, once

again, is a result of the higher velocity of collapse in Model B.

After the initial high pressure wave passes, there is a high

pressure plateau in the fluid as the collapse propagates along

the cylinder length. This plateau is much more pronounced in

Model B – extending from time ? = 0.0021 ???. to ? =

0.0032 ???. – than in Model A where the plateau extends only

from ? = 0.0022 ???. to ? = 0.0027 ???. This is because

Model B is significantly longer and more slender than Model A.

Hence, the collapse has a longer distance to propagate.

Finally, the collapse reaches the cylinder ends and a final

pressure pulse propagates through the fluid. This pressure pulse

is much more noticeable in Model A than in Model B for two

reasons. First, Model A is a shorter cylinder. This means that

the end cap is located closer to the pressure sensor in the fluid

so the wave has not diminished by simple spreading loss. In

Next, it is observed from the pressure histories in Figure 8

addition, the end cap of Model A is larger and will abruptly halt

a larger quantity of flowing fluid. These two factors together

account for the larger final pressure pulse arriving at the sensor

from collapse of Model A. In fact, the final pressure pulse from

Model B is hardly distinguishable from the high pressure

plateau and is only noticeable because the pressure plateau is,

in fact, decreasing slightly with time.

The impulse delivered by the implosion pressure pulse

can be computed by time integration of the pressure history.

This impulse is a measure of energy delivered to the sensor

location. Of particular interest is the difference between the

maximum and minimum impulse values because, for shock

loading, it correlates strongly with damage potential. For

hydrostatic implosion, there is very little positive impulse so

this metric of interest reduces to the absolute value of the

minimum impulse.

Figure 10. Impulse histories for Model A (blue) and Model B

(red).

location for Model A and Model B. There are only minor

differences between the two curves. Most importantly, the

minimum impulses are very similar with ??

??

in energy delivered to the sensor location between the two

cylinders.

Figure 10 shows the impulse histories at the sensor

???= −0.211 and

???= −0.203. This means that there is very little difference

DAMAGE POTENTIAL

The potential for damage to nearby structures as a result

of an implosion event is a function of both the energy delivered

by the implosion pulse (i.e. impulse) and the rate at which that

energy is delivered (i.e. peak pressure). There has been much

debate over which quantity is more important but both have

been shown to correlate with damage.

Based on the analyses in the previous section, it appears

that implosion of Model B poses a larger damage risk to nearby

structures. Although the impulses delivered by implosion of

Model A and Model B are approximately equal, the implosion

of Model B delivers that energy at an appreciably higher

pressure.

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CONCLUSIONS

This work has presented a comparison of computational

models to capture the fluid response from implosion of two

different unstiffened cylindrical shell structures of comparable

volume at equivalent hydrostatic collapse pressure. The general

characteristics of the pressure wave emitted by an implosion

event have been described and the differences in the particular

fluid response characteristics for the two studied cylinders have

been linked to the geometric properties of the cylinder.

Furthermore, the potential for the implosion of these two

cylinders to cause damage to a nearby structure has been

assessed and it appears that the longer and more slender

cylinder poses a larger threat.

ACKNOWLEDGMENTS

The authors acknowledge the support of the Office of

Naval Research Future Naval Capabilities program under

Contract number N00014-08-C-0242 with Dr. Louise

Couchman and Dr. Stephen Turner as program managers. The

authors are also grateful for the collaboration of Naval

researchers at the Naval Surface Warfare Center – Carderock

Division and the Naval UnderSea Warfare Center – Newport

Division.

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