Dynamic Collapse of Double Hull Composite Cylinders: Hydrostatic and Shock-Initiated Implosion

An experimental investigation of double-hull composite cylinders during underwater implosion is performed. Composite materials offer several unique advantages for underwater applications, including greater specific strength and stiffness than metals, as well as better corrosion resistance. Double hull composite structures, which feature a low density core between two facesheets of comparatively higher density, increase these benefits further by adding increased bending strength and acoustic attenuation. Despite these advantages, there exists a large knowledge gap with regards to the underwater implosion behavior of double hull composite structures. To that end, a series of experimental studies is performed to address said knowledge gaps and help provide fundamental understanding of the behavior of these structures during implosion. First, the dynamic collapse of hollow and filled double hull composite cylinders is investigated experimentally. Carbon-fiber/epoxy double cylinders with and without parametrically-graded PVC foam cores in between are collapsed in a large-diameter pressure vessel, and dynamic pressure data is used in conjunction with underwater DIC to determine the effect of the double hull structure on implosion mechanics. Buckling initiation and overall collapse behavior of the specimen are studied, as well as the pressure pulse from the implosion released into the fluid. Incidents of the outer tube imploding but not the inner are reported, in addition to cases where both collapse. Results show heavier foam cores increase collapse pressure dramatically, and this increase in collapse pressure is predictively related to the mechanical and geometric properties of the foam cores themselves. Normalized dynamic pressure emitted from implosions is shown to occur in distinct phases, with an additional underand overpressure region present if the inner tube collapses. When normalized for hydrostatic pressure, fluid impulses from various core densities are shown to remain constant. Energy flux from the implosions, presented as a percentage of available hydrostatic energy, is shown to increase as a function of core density. Increased foam crushing energy at higher core densities, and increased damage observed in post-mortem specimens from heavier cores, are identified as mechanisms responsible for these behaviors. Second, an experimental study is performed which investigates the dynamic collapse of double hull composite cylinders under external hydrostatic pressure and shock loading. All experiments are performed underwater in a 2.1 m diameter semi-spherical pressure vessel that approximates a free-field environment, and Digital Image Correlation (DIC) is used in conjunction with blast transducers to study collapse mechanics. Specimens have carbon-fiber / epoxy facesheets and a PVC foam core that is removed for control in some instances, and are brought to 80% of their natural buckling pressure before being subjected to an underwater explosion (UNDEX) at varying standoff distances. Results show that double hull specimens implode below their natural collapse pressure when subject to explosive loading, but that the addition of the PVC foam core prevents implosion in some cases and substantially increases structural stability in others. The double hull configuration with foam core is also shown to emit significant pressure pulses despite the noisy environment of the pressure vessel.

The dynamic collapse of hollow and filled double hull composite cylinders is investigated experimentally. Carbon-fiber/epoxy double cylinders with and without parametrically-graded PVC foam cores in between are collapsed in a large-diameter pressure vessel, and dynamic pressure data is used in conjunction with underwater DIC to determine the effect of the double hull structure on implosion mechanics. Buckling initiation and overall collapse behavior of the specimen are studied, as well as the pressure pulse from the implosion released into the fluid. Incidents of the outer tube imploding but not the inner are reported, in addition to cases where both collapse. Results show heavier foam cores increase collapse pressure dramatically, and this increase in collapse pressure is predictively related to the mechanical and geometric properties of the foam cores themselves. Normalized dynamic pressure emitted from implosions is shown to occur in distinct phases, with an additional under-and overpressure region present if the inner tube collapses. When normalized for hydrostatic pressure, fluid impulses from various core densities are shown to remain constant.
Energy flux from the implosions, presented as a percentage of available hydrostatic energy, is shown to increase as a function of core density. Increased foam crushing energy at higher core densities, and increased damage observed in post-mortem specimens from heavier cores, are identified as mechanisms responsible for these behaviors.

INTRODUCTION
Implosion is a fluid-structure interaction problem that involves the buckling, collapse, and subsequent pressure pulse emission of an enclosed shell when external fluid pressure around the structure reaches critical value and the shell collapses dynamically. This process happens in distinct stages, with the first phase of instability initiation followed by a second phase in which the walls buckle inward at high velocities and local pressure in the surrounding fluid drops via Bernoulli's principle. After a point, these walls make contact with one another, and abruptly decelerate to zero velocity. As buckle propagation occurs, the sudden change in momentum of the walls simultaneously causes an immense pressure spike in the fluid, and a high pressure wave is released outward [1], [2]. Implosion has been studied for several decades, with early work in the field primarily focused on evaluating the acoustic pulses from deep sea implosions [3]- [5]. Following a series of high profile accidents and developing interest from the naval community, however, implosion research has been reinvigorated in recent years. In the first of these events, a photomultiplier tube at the Super-Kamiokande Neutrino Observatory imploded during filling, causing a chain reaction of pressure pulses that destroyed a total of 7000/11000 such tubes within the facility and cost an estimated $20 million dollars of damage [6]. Later, the AUV Nereus imploded while on a deep-sea research dive off the coast of New Zealand. The vehicle had been expected to reach the deepest depths of the ocean consistently; its failure resulted in a substantial loss of confidence in the subsea engineering community, in addition to the monetary and opportunity cost of the vehicle itself [7]. Jointly, these accidents emphasize the need for increased understanding of the implosion phenomenon, in terms of both the bucklingbased initiation of implosion as well as prediction and mitigation of its associated pressure pulse.

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Several studies from the past decade partially address these gaps. Turner and Ambrico performed experimental and computational investigations of the implosion of both glass spheres and metal tubes, and were able to identify key stages in the implosion process, as well as develop effective computational models that agreed well with their experimental results [1], [2]. Farhat  across the surface of the tube, and related that information to pressure pulse emission [10].
Composite materials have also been studied with regards to the implosion phenomenon. Because of their excellent strength-and stiffness-to-weight ratios, as well as their natural corrosion resistance, they make excellent candidates for marine engineering applications, including those where implosion is a matter for concern. In that vein, the buckling of composite cylinders has been studied by several authors analytically and computationally, in an effort to predict their buckling mode shapes and critical collapse pressures [11]- [16]. Additionally, Pinto et al. studied the implosion behavior of glass-and carbon-fiber composite tubes. This work included the performance of digital image correlation during the collapse process, and investigated both buckling behavior and pressure pulse mechanics [17], [18]. The mitigation of implosion pressure pulses via the application of polyurea coatings was also investigated [19].
That said, the previously mentioned works have only investigated the implosion of isotropic materials and classical composite laminates; there remains little work on the implosion of double hull-structured composite materials, where two facesheets of thin but 5 strong material are concentrically assembled with a thick but comparatively low density core.
Moreover, the work that does exist primarily focuses on the buckling of sandwich structures and doesn't address pressure pulse emission from the implosion process [20]- [26]. None of these studies include experimental verification of their analytical and computational results, and none address the effect of the sandwich construction on pressure pulse emission within the surrounding fluid.
The aim of this study is to experimentally investigate the implosion of double hull composite structures, considering both buckling performance, as well as pressure pulse analysis and mitigation. To most closely approximate the conditions of a submersible or other deep sea vehicle, cylindrical geometry will be investigated. Double hull composites have several unique advantages for this kind of application: in addition to the benefits of composite materials listed previously, double hull structures offer increased buckling resistance, excellent acoustic attenuation, and further increases in specific stiffness over monolayer composites [23].
This study marks the first time that the mechanics unique to a double hull implosion have ever been reported, with several key differences from the behavior of monolayer structures being observed. Double hull structures are found to emit "double pulses" during implosion wherein two distinct pressure pulses are registered, corresponding to the collapse of the outer and inner tubes, respectively. The facesheets are also found to collapse independently of one another, such that the inner tube may or may not survive the collapse of the outer tube, depending on external hydrostatic pressure. The outer tube is found to buckle and flatten against the inner tube, with a time delay sometimes occurring before the collapse of the inner tube, if it collapses at all. 6

MATERIAL SELECTION AND SPECIMEN DESIGN
The specimens in this study use carbon-fiber/epoxy tubes manufactured by Rock West Composites (West Jordan, UT) for their inner and outer facesheets. The outer tube has a [±15/0/±45/±15] layup, with a nominal 60.4 mm ID and 1.7 mm wall thickness. The inner tube has the same layup and nominal thickness, but a nominal ID of 38.6 mm. Prior to assembly, the outer tube is painted by hand with a high-contrast, stochastic, black-and-white speckle pattern that covers half of its circumference and its entire length. During assembly, the tubes are mounted concentrically and sealed using aluminum endcaps such that their unsupported length is 279.4 mm. In addition to being sealed from water, the tubes are also sealed between one another so that water may not enter the inner tube in the event that the outer tube fractures or its seal is otherwise broken.
The PVC foam cores used in the specimens are from the Divinycell H series of foams,

EXPERIMENTAL PROCEDURE
All experiments are performed using a semi-spherical pressure vessel of 2.1 meter diameter, with a maximum static pressure rating of 6.89 MPa, shown in Figure 1. The vessel is designed to maintain constant hydrostatic pressure through the collapse process, and has optical viewports mounted radially around its midsection for photography and lighting during 7 experiments. Two high-speed cameras (Photron SA1, Photron USA, Inc., San Diego, CA) are mounted coaxially with the portholes of the tank, such that a stereoscopic viewing angle of 17° is created toward the speckled surface of the double hull specimen after it is fixed within the vessel. This enables performance of underwater 3-D DIC in post processing, in the manner presented in [10], [17] and [18]. In predicting the collapse pressures of double hull structures, it was decided to focus on the increase in collapse pressure of the tubes on account of the double hull structure, rather than absolute pressures. The increase in collapse pressure of a particular specimen is given by: Where ΔP cr is the increase in collapse pressure of the outer tube due to the double hull structure, P cr is the experimentally measured collapse pressure of that specimen, and P cr,o is the estimated collapse pressure of that same specimen if its foam core were to be removed.

COMPOSITE THICKNESS EFFECTS ON COLLAPSE PRESSURE
In analyzing collapse pressure, it is necessary to account for thickness variation in the outer tubes: small inconsistencies in the manufacturing process of the tubes resulted in variation of average thickness by as much as 0.254 mm, and the collapse pressures of composite tubes are sensitive to variations in thickness.
From literature, the following equation can be used to estimate the collapse pressure of a composite cylinder without a foam core [27]: Where γ is a constant that depends on geometry, E 1 and E 2 are the longitudinal and hoop moduli of the composite cylinder, t is average thickness, μ 12 and μ 21 are the anisotropic Poisson's ratios, L is Length, R c is average cylinder radius, R o is outer cylinder radius, I is moment of inertia about the centerline, and G is shear modulus. Equation (2) shows that critical collapse pressure scales by a factor of t 2.5 . It follows that: Where P cr,1 and P cr,2 are the collapse pressures of two similar tubes with different average thickness, and t 1 and t 2 are those thickness values. For the outer tubes used in this study, Equation (3) predicts the relationship between collapse pressures within 0.5 % for the hollow specimens. Equation (3) can be rearranged to predict the collapse pressure of the outer tube of a given specimen with its core removed: Where P cr,o is the predicted collapse pressure of the outer tube of a given specimen, and t o is that specimen's measured thickness. P cr,ref and t ref are the collapse pressure and thickness of the reference specimens in this study, which had their cores removed when experimented on.
Thus the change in the collapse pressure of a double hull specimen is given by: Careful thickness measurements of the outer tube were needed to make accurate predictions of the collapse pressure of the double hull structure using Eq. (5).
ΔP cr is critical in evaluating the true effect of core material on collapse pressure, as discussed later in Section 1.5.1. This is a very important quantity in the design of subsea vehicles, where avoiding collapse is paramount.

IMPULSE
The area under the pressure-time curve is often of more importance in terms of measuring implosion damage potential than the magnitude of the pressure alone. To that end, the following equation is employed [28]: Where I is the impulse per unit area imparted to the water by the wavefront, dP is dynamic pressure, and t is an arbitrary time of interest. In order to simplify the problem, the total impulse is calculated here for the underpressure regions only, under the assumption that the impulse for the overpressure region will be equal in magnitude, as discussed in [1], [8], [10].
It will be shown in the results section of this paper that some double hull structures release "double pulses" during implosion wherein two distinct pressure pulses are emitted, each corresponding to the collapse of the outer and inner tubes. In such cases, the following formula is used: Where I total is the total impulse per unit area of the double hull structure's collapse, t 1 is the time at which the first underpressure region crosses the t-axis, t 2 is the time at which the second underpressure region begins as the pressure traces crosses below the t-axis again, and t 3 is the time at which the second underpressure region ends as the pressure trace crosses the taxis a third time.
In order to compare the impulse behavior of the various foam cores, it is necessary to account for differences in collapse pressure caused by differing core strengths. The following formula is used, which normalizes the impulse values from Equation (7) with respect to collapse pressure: Here, I norm is the total impulse per unit area of the wavefront, normalized by an impulse term associated with the collapse pressure of the specimen in question. Generally, the impulse of an implosion scales with the ambient hydrostatic pressure at which the implosion occurred.
Equation (8) removes this effect via normalization, and allows for direct comparison of impulse between cases with different collapse pressures.

ENERGY
An acoustic energy flux analysis is also performed to measure implosion damage potential, where the afterflow energy flux through a surface some arbitrary distance from the event is given by [29]: Where E F is acoustic energy flux, ρ 0 is the density of the fluid, and R s is the distance of the sensor from the surface of the collapsing tube. Assuming a spherical pressure wave, the energy flux given by Equation (9) can be multiplied by the surface area of a sphere of radius equivalent to the standoff distance of the sensor to yield the total afterflow energy of the implosion pulse: To compare directly between cases with differing collapse pressures, it is necessary to normalize this value with available hydrostatic energy: Where E H is available hydrostatic energy prior to collapse, given by: and ΔV implosion is the difference in displaced fluid volume before and after the implosion.

PREDICTING COLLAPSE PRESSURE
It is important to be able to predict the buckling behavior of a double hull structure as a function of material and geometrical properties. To that end, a series of experiments were performed to determine the mechanical properties of the foam core materials (Table 1) [30].
Comparting the stress-strain curves from the tests to the maximum compressive strain in the core immediately prior to buckling (Table 1), it can be seen that buckling initiation always occurred in the foam-crushing region of the stress-strain curve. Plotting the measured crushing strengths of the foams against the ΔP cr values measured previously, a linear trend is observed in Figure 2A, intersecting the origin and having a nondimensional slope of 2.6. This implies that for the given core geometry, it would be possible to find an even higher strength 13 core than those studied here, and that applying such a core in the specimen would very likely increase buckling strength along the line in Figure 2A. Note that predeformation prior to buckling is sensitive to small irregularities in the geometry of the cylinders, so a trend is not observed between predeformation and core density in Table 1. Plotting ΔP cr values versus the measured density of each foam core (Table 1) gives the data shown in Figure 2B. The trend is nonlinear for this series of foams. This is noteworthy from an applications perspective because it is often desired to minimize the weight of underwater vehicles. This means that the mass-efficiency of the core increases with density for this series of foams, such that higher density cores are both stronger and more efficient per unit mass.

.2. COLLAPSE BEHAVIOR CATEGORIES
Three overarching behaviors were observed in the collapse of the double hull specimens. In 1), the outer tube collapsed and crushed the foam core but the inner tube did not collapse. In 2), the outer tube collapsed and crushed the foam core, after which there was a brief dwell time before the inner tube collapsed. In 3), the outer tube collapsed, crushed the foam core, and collapsed the inner tube in a singular motion, with no dwell in between.
The outer tubes in this study collapsed in mode 3, whereas the inner tubes collapsed in an asymmetrical mode 2, if they collapsed at all ( Figure 3). In all cases where the inner tube collapsed, the outer tube collapsed first, then remained crushed against the surface of the inner during the latter's collapse, conforming to its shape. In behaviors 1) and 2), the outer tubes formed around the shape of the inner tube, rendering the inner's shape clearly visible; there was not time for this to occur when the inner and outer tubes collapsed in a singular motion. The difference in collapse mode between the inner and outer tubes had no effect on the instability of the inner tube: collapse pressure of the inner tube was the dominant factor.
The addition of a core increased the collapse pressure of the outer tube, sometimes to levels above that of the inner tube. The collapse pressure of the outer tube had to be higher than that of the inner tube (inner tube collapse pressure is approximately 2.4 MPa) for the inner tube to collapse.
Detailed results from each case are presented here, with data from similar collapse behaviors grouped together to enable comparison. propagation, as has been reported in [10], [17], [18]. The same behavior can be seen here,  Total collapse duration for the H60 specimen [ Figure 4 (A)] is 3 ms, greater than that for any other case. This occurs because the ratio between hydrostatic pressure and inner tube collapse pressure is about 1.1 in the H60 case, while being 1.7 for the H100 case. In the former, the ratio is small enough that the pressure exerted on the inner tube during the dwell takes longer to overwhelm it, so it doesn't collapse until about 1.5ms after the implosion of the outer tube. Because the ratio is much higher for the H100 case, the duration of the dwell time is reduced, and the inner tube implosion begins about 1ms after the implosion of the outer tube. For foam cores of lower strength, the outer/inner P cr ratio does not exceed 1, so the inner tube does not collapse. Because the ratio is just over 1 for the H60 specimens, that case features the longest dwell time.
The uniform longitudinal collapse of the inner tube is likely due to material damage in the inner tube causing an asymmetric collapse (visible from post-mortem specimens) that does not comply with the modal symmetry given by structural stability theory [31], [32]. A unique feature of (A) is that maximum collapse velocity is on the order of 60 m/s, at least 50% higher than that seen in other cases. This velocity is located along the midsection of the tube, similar to the inner tube collapse mechanics described in Section 1.5.3.2. These velocities are achievable because, without the dwell observed in other cases, the pressure in the fluid continues to accelerate the structure inward through the duration of the structure's 20 motion. Additionally, uniform high velocities are seen to occur along the length of the structure in time as the inner tube collapses, in much the same manner as described in the previous section. These similarities occur despite the removal of the dwell period. This implosion is also the fastest of all cases, with a total duration of 1.4 ms. to what has been reported in the literature for monolayer composite structures, but the "peakiness" of the behavior, as well as the maximum dynamic overpressure from the implosion, are reduced substantially in this configuration compared to the single-cylinder configuration of similar dimensions reported in [17]. This reduction in overpressure is due to the reduced potential energy of the event, shown with Equation (12): when the inner tube survives, the change in volume ΔV implosion from implosion is reduced, and the less energetic event results in measurable reductions in waveform amplitude. Additionally, the outer tubes in this study collapse entirely in a mode 3 shape, while the tubes in [17] initiate collapse in mode 3 but finish their collapse in mode 2. In this study, the presence of the inner tube prevents the dynamic mode change. Because of the 3-lobe shape here, one of the valleys collapses against the inner tube first, while the other two lobes wrap around the tube.
Displaced fluid volume continues to be reduced by this action, absorbing energy. The result is that the high-frequency pressure pulse observed in the single tube configuration is largely eliminated.

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In the H35 case, shown on the bottom of Figure 7, the presence of the foam core causes energy absorption from cell crushing, and prevents outright wall contact; instead, the foam core densifies to a point where it cannot be crushed more, and a quasi-wall contact is made. This causes a less abrupt change in fluid momentum which, combined with the aforementioned factors, reduces frequency and amplitude in the dynamic overpressure signal as compared with that of the reference. In the H100 case, the amplitude of the first pressure spike is much lower than that of the H60 case, because more energy is absorbed through core compressive strain during implosion. This is because of the higher modulus and crushing strength of the H100 foam.
However, the secondary spike is of much greater amplitude in the H100 case. This is because collapse pressure is substantially higher, for greater E H . The foam core absorbs relatively little energy during the collapse of the inner tube because it is already crushed, so its mitigating effects are minimized during that portion of the collapse. Finally, the time delay between the inner-and outer-tube pulses is greater for the H60 case than any other. This is because of the uniquely long dwell time for the H60 case discussed in Section 1.5.3.2. The pressure pulse of a straight-through collapse appears qualitatively similar to that of the outer tube only collapse, but the underpressure region is of greater duration. This is because the tube collapses a greater distance radially in one motion, so the associated underpressure duration is increased. Additionally, the pressure spike recorded in this case is of greater amplitude than that where only the outer tube collapses. This is due to the greater potential energy of the event: change in volume is increased substantially when the inner tube implodes, so the pulse released is larger.
The peak amplitude of the straight through collapse is less than that reported in literature for composite tubes of similar geometry [17]. In the reference, specimens similar to the outer tube in this study emitted a maximum dynamic overpressure of approximately 0.8, whereas the same value reported here is 0.6. That is despite the substantial increase in  Figure 10A, with foam core density on the horizontal axis. To allow for direct comparison between cases, the integrals from t 2 to t 3 in both the numerator and denominator are taken to be zero for instances where the inner tube did not collapse. The results are constant with foam core density, which implies that the impulse is mainly a function of collapse pressure, and is not directly influenced by core density or other properties. The exception to this trend in the data is the case of the H60 core, which has a lower magnitude normalized impulse. The reason for this is because of the uniquely long time delay between inner and outer tube collapses for these cases, discussed in Section 1.5.3.2. As a result, it can be concluded that foam core properties do not directly affect impulse, with collapse pressure being the dominant factor instead.
A plot of normalized energy values, which present the energy emitted by the implosion event as a function of available hydrostatic energy, is given in Figure 10B. There, normalized energy released increases with core density, with more variation in the H60, -80, and -100 cases. The increase in variation is because the inner tube collapses in the aforementioned cases, and introduces new damage mechanisms. The increase in normalized energy with core density is also significant, as there is a 57% increase in average normalized energy released between H35 and H100 core specimens. This highlights the diminishing returns on energy absorption for higher density cores. Though increasing core density also increases hydrostatic collapse pressure, the energy released by the implosion also increases, and does not scale with increase in hydrostatic energy. Therefore, the amount of energy released in the collapse of double hull tubes increases with hydrostatic energy, and increased core density allows for increased absolute mitigation of implosion energy. These returns do not scale, however, with slightly higher normalized energy released at higher foam core densities. The post-mortem analysis of the specimens yields additional insight into the mechanics of the collapse process, and helps to explain the energy mitigation behavior discussed above. Generally, damage in heavier foam core cases is more extensive because 29 stronger foam cores enable higher collapse pressures: in the H100 case, P cr is approximately triple that of the H35 case, and because the inner tube also collapses, the energy of the event is increased even further. This enables new damage mechanisms, which causes greater overall energy absorption in the heavier foam core cases.  Several important instances of damage should be noted at the tear. First, the fibers along the tear have all failed, with substantial fiber pullout visible. There is also significant delamination at the tear face (C), visible at both the ends of the tube as well as the surface of the damage itself. The tear propagated to the foam core (D). The carbon/epoxy material of the outer and inner tubes has also not recovered to cylindrical shape (E) after being removed from the pressurized environment of the water. This is because the broken fibers from the tubes 31 intertwine with one another where the tear and fold-under occurs, and also cut into the foam core material. This prevents them from returning to their original shape, and implies the presence of residual stress in the material, even after it is removed from water.
Finally, at (F), circumferential cracks from endcap-induced shear are seen. The cracks extend entirely around the circumference of the specimen on one end and largely around it on the other, and the portion of the outer tube that was supported by the endcap is entirely detached on one end.   In cases where the outer tube collapses but the inner does not, an implosion pulse is generated which is similar to a monolayer tube implosion, but of lower frequency and amplitude. This is created by the outer tube collapsing and flattening against the inner.
 In situations where both inner and outer tubes implode, the pulse from the outer tube's collapse is followed by a secondary implosion pulse from the collapse of the inner tube. The impulse and amplitude of the second event may match or exceed that of the first.
 After normalizing for collapse pressure, the impulse of a double hull implosion is constant with core density, regardless of whether the inner tube implodes or not. There are many benefits of composite materials for underwater structural applications: composites generally have higher specific strength and stiffness, improved corrosion resistance, and reduced weight in comparison with metals. These benefits expand further in double hull composite configurations, where a foam core of relatively low density is sandwiched between two facesheets of higher density and stiffness: this configuration allows for increased acoustic attenuation, as well as improved bending strength, making double hull structures particularly good for preventing buckling. This is important because buckling causes the initiation of an implosion, where structural instability results in the walls of an underwater gas-filled structure rapidly moving inward and contacting each another, collapsing the structure entirely and emitting high-pressure waves in the process [2]. While implosions may be initiated from hydrostatic pressure alone, sufficient instability to collapse the structure may also be caused by shock loading at sub-critical pressure, as in the case of a nearby underwater explosion (UNDEX) [3].
Though underwater implosion has been of interest for several decades, there has been a recent resurgence of experimental work in the field. Turner and Ambrico clarified the collapse mechanics of cylindrical metal shells using experiments and simulations, and Farhat et. al did similarly while using high speed cameras to visualize the stages of the collapse [2], 42 [4]. Gupta et al further expanded the literature by using high speed photography in conjunction with digital image correlation (DIC) to study the implosion of metallic tubes [5].
Pinto extended these techniques to the study of composite tubes [6], [7], thus adding to a body of work that had been focused largely on evaluating the collapse pressures of composite cylinders under hydrostatic conditions [8]- [11].
A variety of work has been done on the phenomenon of shock-initiated implosion.
Pegg performed a nonlinear finite element analysis of submerged cylinders with combined hydrostatic and impulse loading, and investigated the effect of varying impulse [12]. Tanov [14], [15]. Fatt studied the underwater pulse loading of GFRP cylinders analytically using Mathieu equations and computationally, and found good agreement between the two [16]. Ikeda studied the implosion of Brass and Aluminum cylinders under hydrostatic pressure and UNDEX loading experimentally using high-speed imaging [17]. Finally, Pinto used high speed imagining and DIC to study the buckling and implosion of carbon-fiber/epoxy composite structures under UNDEX loading [18].
There is a wealth of literature on the behavior of sandwich structures to impulse and blast loading, both in air and water [19]- [35]. However, this body of work focuses on the behavior of plates and beams with few exceptions, and generally does not account for buckling in cases where cylinders are considered. As such, there is a significant knowledge gap with regards to the behavior of sandwich-structured composite cylinders under combined hydrostatic and shock loading. This study seeks to address that gap experimentally, with results showing that the addition of a foam core substantially increases structural stability by extending the duration of collapse, as well as preventing implosion completely in some cases. Solutions Inc., Columbia, SC) to be performed during post-processing in the manner of [5], [6], [18]. Viewports on the side of the tank are used with an additional high speed camera to track UNDEX bubble size. After mounting, the tank is sealed and filled with filtered water, then pressurized by introducing nitrogen gas to a small air pocket at the top of the chamber. The water in the vessel is pressurized to 80% the collapse pressure of the specimen being studied, which is previously measured from identical samples by slowly pressurizing the same setup at a rate of 0.083 MPa/min until natural collapse occurs. When 80% collapse pressure is reached in the tank, a high voltage firing box is triggered alongside the cameras and pressure sensors, such that the charge detonates and imagery and pressure data are saved and matched in time.
All trials are duplicated to ensure repeatability between experiments. Average normalized RMSE of pressure signals is 4.3% for similar trials, with a maximum variation of bubble pulse period of 0.16 ms, and maximum variation in maximum overpressure of 2.18 MPa. Representative data is plotted unless there is significant deviation between trials, in which case differences are highlighted.

Specimen Type Hydrostatic Pressure
During  here, in order to provide complete information about collapse. In each case, plots are taken from a sensor mounted 25mm from the back face of the specimen, as well as standoff distance R s from the charge (Figure 16). This provides an accurate representation of the pressure experienced by the specimen, as the shock wave arrives at both the sensor and the structure simultaneously. Additionally, time 0 is taken by convention to be the time at which the explosive is fired for a particular case, such that the initial shock wave passes the structure shortly thereafter. This convention is also upheld for image data throughout the paper, so pressure and DIC information can be directly compared in time. All pressure and DIC data presented in this study is unfiltered.
Where T is the period of bubble oscillation, Y is total energy available (proportional for a given explosive to charge weight), K is a constant depending on the explosive, and P 0 is the ambient hydrostatic pressure during the event. Assuming constant values of K and Y between experiments, the relationship between bubble pulse periods at differing hydrostatic pressure is given simply by: inward at E and finally making centerpoint wall contact at F. A phase commonly referred to as "buckle propagation" ( [5], [7]) follows, with the walls of the cylinder making contact at increasingly higher longitudinal position, until the walls are completely collapsed at 7.93 ms, marked G. Figure 17: Centerpoint radial displacement data (dR) for a no-core specimen with 102mm standoff Figure 18: High speed imagery with DIC contours superimposed for a no-core specimen with 102 mm standoff. Letter-labels to the left of the images correspond to those in Figure 17.
The pressure-time history recorded from the event provides further insight into collapse mechanics, and is plotted in Figure 19. There, it can be seen that the implosion of the outer tube occurs following the initial shock wave, but before any reflections from the tank walls interact with the specimen. As such, it is clear that the initial shock wave alone possesses enough energy to implode the outer tube at this ambient pressure and standoff distance. As noted previously, the second set of reflections can also be seen to begin at around 2.8 ms, and continue during the time of the rebound labeled C in Figure 17. Interestingly, the first bubble pulse from the UNDEX appears to be concurrent with the buckling initiation of the inner cylinder. Side view photography shows a minimum bubble radius at exactly 4.00 ms, which is the same time as image D in Figure 18, and aligns with the local maximum in pressure marked bubble pulse 1 in Figure 19. From this it may be concluded that the bubble pulse is providing the energy to initiate the collapse of the inner cylinder, even though that structure's collapse pressure is much higher than the ambient hydrostatic pressure in the tank.
After this point, collapse of the inner tube continues until wall contact is made shortly after 7 ms, which also happens to correspond with the second UNDEX bubble pulse, and buckle propagation continues until approximately 8 ms. It is worth noting that, even though an implosion typically releases high-amplitude pressure waves concurrently with buckle propagation, it is not possible to distinguish that feature in the pressure trace here because the environment is already very noisy. This can be seen by cross-referencing with Figure 18, and show that the tubes absorb energy through damage as they collapse.
Also of interest is the back face circumferential cracking in the outer tube that forms a roughly circular shape and can be seen in areas marked B. This cracking is interesting because it only appears on the back face of the tube, in the direction facing the UNDEX.
These circumferential cracks are each about 82 mm from the midplane of the tube, which is itself 102 mm from the center of the UNDEX. Using simple trigonometry, it can be determined that this corresponds to a radius of 130 mm in the initial shock wave, at which point the section of the tube bracketed by these cracks was loaded by the shock but the rest of the tube was not. Given this non-uniform loading and lack of support from a core, the material likely experienced enough stress to fracture, further degrading the stability of the structure and leading to the short duration of collapse. Interestingly, similar cracking can be seen at a corresponding location in the inner tube, marked C in Figure 20.  The collapse of the structure with core removed and a 203 mm standoff completes in 13.8 ms: this is a greater duration than the 102 mm standoff case, but still much shorter than either case where the foam core is included.
DIC information from the event helps explain the key stages and processes of the collapse process: this is shown in Figure 21, which shows centerpoint radial displacement with key stages of the collapse marked, and Figure 22, which shows high-speed imagery from the marked times in the previous figure. Following the firing of the explosive shortly after time 0, the outer tube can be seen to oscillate briefly before collapse initiates at 3.33 ms, marked A.
Interestingly, this occurs after reflections from the tank walls interact with the specimen, as can be seen in Figure 23. From this, it can be concluded that the initial shock wave alone does not possess enough energy to destabilize the outer tube. Following the initiation of collapse at A, the walls of the outer tube move inward and impact the walls of the inner tube at B. The walls of the inner tube immediately continue inward collinearly with those of the outer tube, but eventually stabilize briefly around 6.67 ms, marked C. Shortly afterwards, the second bubble pulse is observed at 6.83 ms from side view imagery, and can be seen clearly in Figure   23. This leads to a substantial amount of cavitation in the tank, which causes brief loss of correlation.
The second bubble pulse provides sufficient energy to collapse the inner tube, such that implosion of that structure initiates at 7.57 ms, marked D. This continues until the structure collapses completely and makes wall contact at centerpoint at E. A distinct buckle propagation phase is not observed following centerpoint contact, and the center of the structure appears to rebound outward briefly at F, while the buckle concurrently propagates in the direction of the endcaps, most clearly visible in Figure 22. Finally, the center of the 55 structure moves back inwards to complete wall contact at 13.77 ms, marked G, and no further changes in the state of the structure are observed.
It should be noted that the structure does not undergo complete collapse, as can be seen from end-state DIC contours in Figure 22G. This implies that the inner tube of the double-hull structure still has some residual stiffness after the collapse event, and is able to resist complete collapse on account of that. It appears that the inner tube only collapsed because the valley of its buckling mode shape was well aligned with that of the outer tube, and that the timing of the second bubble pulse was enough to partially collapse the inner tube in this configuration.  Figure 21.
The pressure history from the no core, 203 mm standoff case is plotted in Figure 23, and investigation of this data can help yield insight into the collapse dynamics seen above.
Because the standoff distance of the explosive is much greater, the amplitude of the shock wave is substantially reduced from the 102 mm configuration, explaining why the outer tube does not collapse until it is subject to additional energy from tank reflections. The first bubble pulse from the UNDEX can also be seen to occur around 4 ms, during which time the walls of the outer tube are collapsing inwards. This adds additional energy to their collapse, and helps drive the inner tube inward after wall contact between inner and outer cylinders is made at 5.23 ms. As noted previously, the second bubble pulse can also be seen to occur shortly before the collapse of the inner tube initiates, which is likely the cause of that event. Finally, it can be seen that the centerpoint rebound in Figure 22F also begins between bubble pulses 3 57 and 4, which are not easily distinguishable in the pressure history but can be observed in side view imagery. Because the rebound occurs between pulses, the local pressure in the fluid would be at a relative minimum, helping to explain why the rebound seen above occurs and why residual stiffness in the outer tube would be better able to initiate the rebound during that time. This is a departure from the 102 mm standoff case, which did not have damage in this area, though it was subject to a stronger blast. The difference is likely because the duration of the collapse was longer in the 203 mm case, as a result of the lower-amplitude shock loading.
Because of this, damage had a longer time to accumulate in the structure, particularly in the outer tube. The inner tube does not have the back face damage present in the 102 mm specimen in Figure 20, showing that the impact between inner and outer facesheets was not as violent in the case of the 203 mm specimen. The addition of a foam core provides substantially increased stability for the double hull cylinders. This is initially obvious from the 30.2 ms collapse duration of the H100 core specimen with 102 mm standoff: this is more than double the duration of the no core, 203 mm case described in the previous section, even though the shock experienced by the specimen is of greater magnitude here. Figure 25 shows centerpoint radial displacement data for the foam core, 102 mm standoff case, with key stages in the collapse marked. Figure 26 shows high speed imagery from these times with DIC contours superimposed, thus providing information about the rest of the cylinder. It is immediately clear from viewing Figure 25 that the qualitative behavior of the specimen is different when the foam core is added: whereas cases with core removed resulted in collapse of the specimen in a roughly singular motion, it can be seen here that the inner tube oscillates for an extended duration after the collapse of the outer tube, before finally losing stability itself.
Several key stages of the collapse are marked in Figure 25 and Figure 26. Following arrival of the initial shock wave shortly after 0 ms, the tube can be seen to bow outward and oscillate for some time, before implosion of the outer tube initiates at about 4.00 ms, marked A. Cross-referencing with Figure 27, it can be seen that this initiation occurs shortly after the second bubble pulse, indicating that the bubble pulse provided the energy to collapse the outer tube. This highlights the reinforcing effect of the foam core, as the initial shock wave alone was able to collapse the outer tube in the no-core case with the same standoff. Here, the outer tube did not collapse until subject to energy from the initial shock wave, two sets of reflections, and two bubble pulses, as well as a much higher ambient hydrostatic pressure, clearly showing the reinforcing effect of the core. Collapse of the outer tube continues until B, at which point complete densification occurs in the foam core and the inner tube stops the inward motion of the walls. Interestingly, there is substantial damage in the outer facesheet at this time, as evidenced by circumferential cracks that appear to extend around much of the circumference of the structure. These cracks are seen in analysis of post-mortem specimens, and highlighted in Figure 28. After the outer tube crushes against the inner at B, the latter tube begins to oscillate at low frequency and an amplitude of about 4 mm for 23.1 ms, until collapse of the outer tube is initiated at E. A local minimum and maximum are marked for 61 reference during this time as C and D, respectively. Following initiation at E, the structure continues to collapse asymmetrically before making centerpoint wall contact at F, after which buckle propagation continues along the longitudinal axis of the structure and completes at 30.2 ms, marked G. Pressure profiles recorded during the event help to explain the behavior seen above, and are plotted in Figure 27 with key features labeled. Of particular note is the substantially reduced time period between bubble pulses. This occurs because the ambient hydrostatic pressure is brought to a much higher level in the foam core cases (2.90 MPa vs 1.0 MPa), resulting in faster oscillations in bubble radius. This means that the specimen is subject to a greater amount of energy in a given time period than in the no core case, though ambient hydrostatic pressure is 80% the natural collapse pressure of the specimen in both cases.
Four bubble pulses are recorded in the first six milliseconds of the collapse, with reflections typically occurring every 1.4 ms. Despite this, the outer tube of the specimen does not initiate collapse until 4.00 ms, which clearly shows the stabilizing effect of the foam core.
After the outer tube implodes, bubble pulses and reflections continue, though individual pulses and reflections lose energy and become more difficult to discern after 7.5 ms. Over time, a Investigation of the post-mortem specimens for this case reveals more information about the failure modes of the structure, with images of the specimen shown in Figure 28. The first thing to note is the circumferential tearing in the vicinity of the endcaps of the outer tube, marked A. This damage of this type also appears in the inner tube, at locations marked E.
Circumferential tearing does not appear in the specimens with core removed. This is because the foam core specimens have a higher collapse pressure, so the experiments on them are performed at a higher hydrostatic pressure. As a result, the cylinders are driven inward more abruptly after losing stability, and undergo substantial tearing from endcap-induced shear.
Large-scale delamination of fibers also occurs near the endcaps in regions marked B, and is enabled by the tearing at A, as well as the increased hydrostatic energy of the foam core cases.
A massive longitudinal tear can be seen in the back of the specimen, marked by C.
This tear extends the full length of the tube, such that the cylinder is fully parted at this Also of note is the crushing of the foam core in an asymmetric mode 2 shape at the regions marked F. This demonstrates that the core is absorbing energy during the collapse of the inner and outer facesheets. Through-thickness longitudinal cracking can be seen in the inner tube at G, and corresponding to the regions that hinged during the collapse of the inner cylinder. This occurs in much the same manner as the no-core specimens; however, the same characteristic cracking does not appear in the outer tube, further implying that the outer cylinder collapsed on account of damage. Finally, it is worth noting that the back-face circumferential cracking seen in the no-core specimens is not present in this case. Here, it 66 appears that the foam core provides enough support for the outer tube to prevent that damage from developing. The case with an H100 foam core and 203 mm standoff is interesting because the specimens either did not implode at all, or only did so after a very long time relative to other cases. For the sake of completeness, the latter trial is focused on in this section, with relevant information included from the no-collapse trial where relevant.
The specimen that imploded did so after 86.9 ms, about triple the duration of the case from Section 2.4.4. Collapse of the outer tube did not initiate until 43.7 ms after the charge was fired, longer than the entire collapse duration of the specimen with 102 mm standoff.
High speed imagery and DIC information help to explain the mechanics of this collapse process, with centerpoint radial displacement data plotted in Figure 29, with key stages of the collapse labeled. High-speed imagery corresponding to these labels is shown in Figure 30, with DIC contours superimposed. Figure 29 contains data from both trials, though labels only apply to the trail that imploded. In the trial that did not implode, steady-state oscillations can be seen about the horizontal axis, beginning around 10 ms. These oscillations continue indefinitely, though only about 64 ms of data is recorded.
Key stages of the collapse are labeled in Figure 29 for the trial that did implode.
Starting around 10 ms, the deflection from the second trial can be seen to diverge from the first, as the specimen bows outward slightly with positive radial displacement. From there, the specimen oscillates radially with a frequency approximately the same as Trial 1. This continues until 43.7 ms, marked A, at which point the outer tube loses stability and begins to collapse. Collapse of the outer tube continues until B, at which point the outer facesheet makes a quasi-wall contact with the inner by crushing the foam core fully.
It is important to understand why the first trial did not collapse and the second trial did. This can be answered in part by image B in Figure 30. There, damage in the form of circumferential cracking can be seen near the endcaps of the specimen. An asymmetric collapse of the outer facesheet is also seen, where the bottom of the cylinder is not deformed in the image, but the top is crushed against the inner tube. Combined, these factors indicate that the outer tube did not collapse from a structural instability, as that would result in a symmetric modal buckling shape. Rather, it is clear that the outer cylinder lost stability on account of material damage which eventually overwhelmed the structure. It is difficult to identify when this damage would have initiated, as cracking can be too small for visible inspection and half the cylinder is not visible in the images.
After the collapse of the outer facesheet at B, the inner cylinder continues to oscillate radially for 27.6 ms, before collapsing entirely at time 83.6 ms, marked D. Meanwhile, damage in the outer cylinder continues to grow, as is apparent from the more extensive cracking in Figure 30C    to the collapse of the outer tube. Interestingly, a definite pulse is registered after the implosion of the inner tube. Following an underpressure region associated with the collapse of the inner tube, a peak in pressure is registered shortly after wall contact at 86.9 ms. This peak is concurrent with buckle propagation along the length of the inner tube, as is typically observed during an implosion. As with the 102 mm standoff case, the pressure pulse from the 70 implosion of the inner tube is registered here because noise in the pressure vessel attenuates over time. Because the foam core specimen survives for a long time following the UNDEX event, and the hydrostatic pressure is higher, a small but significant pressure pulse is detected from the collapse of the inner tube. Figure 31 shows only the pressure trace from the trial where the specimen imploded, as the pressure-time history from the other trial is the same with the exception of the inner tube implosion pulse.

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Post-mortem analysis helps complete the understanding of failure and damage modes, with images of foam core specimens at 203 mm standoff plotted in Figure 32. There, the first thing to note is circumferential tearing around the endcaps at A, as seen in the foam core, 102 mm standoff case. This tearing clearly resulted in loss of stability for the outer tube, though the tearing is less extensive than the 102 mm case because the magnitude of shock loading was reduced. This allowed the outer tube to survive for a relatively long time before collapsing, at which point the pressure in the fluid from the UNDEX had attenuated even further. As a result, the collapse was less violent after stability was lost, and damage was less extensive. Similarly to the 102 mm case, axial cracking and delamination of fibers near the endcaps can also be seen at B.
Another similarity to the 102 mm case is the back-face longitudinal tear extending along the full length of the specimen at C. This tear is again characterized by complete separation of the outer cylinder with itself, as well as massive fiber failure and delamination along the face of the tear. This damage mode occurs where the outer tube folds on the inner as the latter structure collapses. Corresponding damage on the front face is not visible in this case, however, due to the reduced magnitude of shock loading. Large scale delamination of the outer plies of a small grouping of fibers can be seen at D. Again as with the 102 mm standoff case, an asymmetric mode 2 shape can be discerned in the crushed shape of the foam core at E, and characteristic through-thickness cracking corresponding to hinging in a mode 2 buckling shape can be seen in the inner tube at F.
For reference, a side-view image of the specimen that did not collapse is shown in (v) in Figure 32: note the complete lack of any apparent damage in the inner tube, foam core, and outer tube. As stated previously, there was likely some initial damage in the specimen that did implode, which grew over time as the structure was loaded by the UNDEX, secondary bubble 72 pulses, and reflections from the walls of the pressure vessel. The post-mortem images in Figure 32 (i)-(iv) help reinforce this concept. An experimental study was performed to study the shock-initiated implosion behavior of double hull composite cylinders at sub-critical hydrostatic pressures. High-speed photography, underwater digital image correlation, and high-frequency blast transducers were used to study the mechanics of carbon-fiber / epoxy specimens with and without PVC foam cores under these conditions. As a result of this work, the following conclusions can be made:  Double hull composite cylinders can implode at hydrostatic pressures 80% their natural collapse pressure, if they are subject to shock loading.
 These structures may implode from the energy of the initial shock wave, a secondary bubble pulse, or loss of stability due to damage.
 The addition of a foam core substantially increases structural stability: foam cores have been shown to increase the collapse duration of these structures substantially, as well as prevent collapse entirely in some cases.
 The pulse from an implosion in a confined vessel is often insignificant when initiated by an UNDEX. In cases where a foam core is added, the implosion pulse of the inner tube becomes significant, as the core delays collapse until noise in the vessel has attenuated.
 The addition of a foam core may result in a vibrational response of the structure prior to collapse. This can occur in the outer tube, as well as the inner tube after the collapse of the outer.