Characterization of the pressure wave emitted from implosion of submerged cylindrical shell structures
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ABSTRACT: A nonlinear, large deflection, elasto-plastic finite element code (EPSA) has been developed for the analysis of shells in an acoustic medium subjected to dynamic loadings. The nonlinear equations of shells are discretized with the aid of a finite difference/finite element method based upon the principle of virtual work. The resulting system of equations contains the nodal displacements as the generalized co-ordinates of the problem. The integration in time of the equations of motion is done explicitly via a central difference scheme.Shell strain-displacement relations are established by a two-dimensional finite difference scheme. The shell constitutive equations are formulated in terms of the shell stress resultants and the shell strains and curvatures. The fluid-structure interaction is accounted for by means of the doubly asymptotic approximation (DAA) expressed in terms of orthogonal fluid expansion functions. The analytically produced results satisfactorily reproduce available experimental data for dynamically loaded shells.International Journal for Numerical Methods in Engineering 05/1983; 19(6):811 - 824. · 2.06 Impact Factor
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ABSTRACT: A technique is presented for determining mechanical properties of materials under dynamic tensile loads. A Dynapak metalworking machine was modified into a test fixture capable of producing the required dynamic loads for uniaxial and certain biaxial tensile tests. Results from uniaxial dynamic tests on 6061-T6 aluminum alloy are presented and compared to static data obtained from a universal testing machine. The dependence of tensile strength on strain rate and the augmenting effect of temperature on this dependence can be seen. The results of biaxial tests are described in terms of a modified form of the distortion-energy failure theory.Experimental Mechanics 01/1966; 6(4):204-211. · 1.55 Impact Factor
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ABSTRACT: The treatment of zero energy modes which arise due to one-point integration of first-order isoparametric finite elements is addressed. A method for precisely isolating these modes for arbitrary geometry is developed. Two hourglass control schemes, viscous and elastic, are presented and examined. In addition, a convenient one-point integration scheme which analytically integrates the element volume and uniform strain modes is presented.International Journal for Numerical Methods in Engineering 04/1981; 17(5):679 - 706. · 2.06 Impact Factor
1 Copyright © 2012 by ASME
Proceedings of the Internoise 2012/ASME NCAD meeting
August 19-22, 2012, New York City, NY, USA
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
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
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. , ), 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  although an
externally pressurized closed volume of any shape will be
vulnerable to implosion . 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
IMPLOSION BEHAVIOR FOR CYLINDERS
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 :
2 Copyright © 2012 by ASME
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.
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 .
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.
3 Copyright © 2012 by ASME
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.
The cylinders and fluid were modeled using Weidlinger
Associates, Inc. (WAI) explicit dynamic finite element software
EPSA . The structure is modeled using a thick shell element
formulation with a strain-displacement matrix derived
according to Flanagan and Belytschko . 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
(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
4 Copyright © 2012 by ASME
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 . Rate enhancement is
added based on work performed by Hoge . 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 . 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
Figure 6. Rate-dependent uniaxial stress-strain relation for Al
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
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
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
Figure 10. Impulse histories for Model A (blue) and Model B
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
Figure 10 shows the impulse histories at the sensor
???= −0.211 and
???= −0.203. This means that there is very little difference
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
6 Copyright © 2012 by ASME
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.
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
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