DYNAMIC INSTABILITY OF COMPOSITE CYLINDERS IN UNDERWATER CONFINING ENVIRONMENTS

A comprehensive series of experiments was conducted to understand the effect of confining environments on the mechanics of implosion of carbon/epoxy composite cylinders. In the case of implosion, a confining environment can be defined as any environment which limits, restricts, or otherwise manipulates particle motion towards an implodable volume during an implosion event. As such, the implosion of composite structures within two different types of confining environments was investigated: implosion within full confinement, and implosion within partial confinement. It was found that a fully-confining environment considerably limits the energy available to drive implosion, and thus two implosion phenomena can result: full implosion and partial implosion. Full implosion occurs when the energy contained in the compressed water bounded by the confining structure is sufficient to cause wall contact in the implodable. This resulted in water-hammer pressure spikes at the ends of the confining chamber due to the formation and subsequent collapse of large cavitation bubbles. Partial implosion occurs when the energy contained in the compressed volume of water bounded by the confining structure is not sufficient to cause wall contact in the implodable structure, causing an arrest in the implosion process and resulting in structural oscillation of the walls of the impolodable. This resulted in pressure oscillations of the same frequency throughout the confining chamber, with oscillations increasing in amplitude with distance from the axial center. In partially-confining environments, it was found that the implosion of composite structures resulted in pressure oscillations which behaved as a damped harmonic oscillator of frequency f, amplitude ∆Pmax, and damping ratio ξ. Pressure oscillations were experimentally characterized, and an analytical investigation was conducted in which expressions for f and ∆Pmax were derived, showing good agreement with experimental results. Finally, it was experimentally shown that by decreasing the energy stored in the compressed volume of water bounded by the partially-confining structure, implosion with dwell can be achieved, in which there is a short pause in the implosion process. This phenomenon is analogous to partial implosion within full confinement.


Abstract
A fundamental experimental study was conducted to understand the physical phenomena resulting from the dynamic instability of carbon/epoxy composite tubes in an underwater pressurized tubular confining environment. The confining nature of the environment limits the potential energy available to drive instability, resulting in a decrease in hydrostatic pressure with the onset of instability and allowing the carbon/epoxy composite tubes to recover. Unsupported tube length and tube diameter were varied in order to determine the effect of tube geometry on the failure mechanisms of the tube and pressure waves emitted throughout the confining chamber during the instability event. High-Speed photography coupled with Digital Image Correlation techniques were employed alongside the acquisition of pressure-history data from each experiment to relate specimen displacement behavior to resulting pressure pulses. Tubes of 55 mm diameter experienced partial implosion, in which the walls of the specimen oscillated radially with no wall contact. This resulted in pressure oscillations of the same frequency throughout the confining chamber, with oscillations increasing in amplitude with distance from the axial center. Amplitude of pressure and radial structural oscillations were found to be dependent on pressure just prior to instability. Tubes of 35 mm diameter experienced full implosion, which resulted in water-hammer pressure spikes at the ends of the confining chamber due to the formation and subsequent collapse of large cavitation bubbles. Longer tubes were observed to undergo significantly more damage during full implosion, reducing their ability to recover radially and thus effectively reducing the strength of hammer pulses.

Introduction
A fundamental experimental study was conducted to investigate the instability of anisotropic filament wound carbon fiber cylindrical shells initiated in a tubular confining chamber. The experiments were conducted to examine the failure mechanisms of carbon fiber shells in a pressurized limited energy environment and understand how their instability affects the local surroundings. In confined conditions, the potential energy available to drive instability is limited and thus begins to decrease with any decrease in specimen volume. This effect gives rise to two very different instability cases: partial implosion and full implosion. Both cases are carefully examined and discussed in this study.
The use of composite materials for marine applications has recently been the subject of extensive research due to the various advantages they offer. Compared with metals, composite materials offer reduced weight, greater corrosion resistance, and greater potential operating depths per unit weight for submerged structures. Furthermore, their low thermal, magnetic, and acoustic signatures make them hard to detect, making them ideal for military applications. For these reasons, composite materials have already been incorporated in the design of various unmanned underwater vehicles (UUV's), submarine bow domes, and ship masts [1]. However, the lack of complete understanding of these materials' behavior, especially under extreme and complex loading environments, remains a hindrance to their widespread application. In naval applications such environments could include the main ballast tanks of a submarine.
These are flooded, confined spaces between the pressure hull and outer hull, and often house sensitive equipment related to navigation and weapons systems. For this reason, the present study investigates the problems of partial and full implosion occurring in a confining, limited energy environment.
An implosion event occurs when a closed, hollow structure of lower internal pressure is submerged in a medium of higher pressure such that the differential results in instability in the structure walls, causing it to collapse inwards on itself. When this occurs underwater, the sudden increase and subsequent arrest in fluid motion emits a pressure pulse into the fluid which can be damaging to and even initiate the implosion of nearby structures [2,3]. The problem of implosion itself has been a topic of study for many decades, with the first equations for the critical buckling pressure of a hydrostatically loaded cylindrical shell having been derived by von Mises in the early 1900's [4,5].
Early works focusing on the pressure waves released into the surrounding fluid by an imploding volume were conducted by imploding glass structures as a source of underwater acoustic signals [6,7]. The strength of these pressure waves depends in part on the geometry and material of the collapsing volume. In 2001, at Japan's Super-Kamiokande neutrino observatory, the implosion of a single photomultiplier tube released a pressure wave powerful enough to trigger a chain reaction of implosion. This accident resulted in the implosion of nearly 7000 nearby tubes, causing $20-$30 million in damage [8]. The danger posed to surrounding structures by implodable volumes has thus sparked investigations from researchers in the naval community.
The implosion of aluminum tubes has been widely investigated, with Turner and Ambrico having identified the key stages of the implosion process in a free-field environment with respect to the local pressure about the collapsing volume [2]. This work also resulted in the development of robust and accurate fluid-structure computational models. Farhat et al. furthered this work by studying 2 and 4 mode implosion events in aluminum cylindrical shells and using these experiments to verify existing computational models [9]. Gupta et al. then showed that the presence of nearby structures can also have a dramatic impact on the implosion process and resulting pressure histories due to their effect on the fluid motion in the system [10,11].
Compared to metals, the implosion of composite structures has received little attention.
The first analytical equations to predict critical buckling pressure of a composite tube were derived from von Mises' original equations as recently as 1993 [12].
Experimentally, Moon et al. conducted studies on the implosion of filament-wound carbon/epoxy composite cylindrical shells to determine the effect of winding angle on collapse mode critical buckling pressure [13]. The critical buckling pressure and collapse modes of carbon/E-glass composite tubes have also been studied by Smith and Ross, who used experimental results to create design tables to aid in the implementation of composites [14,15]. Most recently [16 -18], studies designed to capture full field displacement data together with pressure data showed that the complex failure mechanisms behind the implosion of composite tubes resulted in pressure profiles which differed dramatically from those resulting from the implosion of metal tubes. Namely, the brittle failure of carbon/epoxy composite tubes resulted in stronger pressure pulses than aluminum tubes, while the energy intensive damage processes involved in the failure of glass/PE composite tubes resulted in weaker pressure pulses [16]. Similar experiments also showed that the architecture of the composite structure also had a significant impact on resulting pressure profiles [17,18]. These studies have been conducted to examine the physics of an imploding composite structure in a free-field environment, where the hydrostatic pressure acting on the tube does not change over the course of the implosion. However, it remains unclear how a confining, limited energy environment will affect the implosion process of a composite structure. It was therefore the aim of the present study to examine this phenomenon by employing underwater digital image correlation techniques together with the use of pressure data obtained from strategically-placed pressure sensors.
Experimental results from the present study showed that initiating hydrostatic instability in carbon/epoxy composite tubes within a limited energy confining environment can result in two very different instability events: partial implosion and full implosion.
Partial implosion occurred when the potential energy in the system decreased with specimen volume too rapidly to continue the implosion process. This caused the specimen to radially oscillate in a harmonic manner. These oscillations directly affect the resulting pressure-time history, with stronger oscillations seen at the ends of the confining chamber. Full implosion occurred when the potential energy in the system is sufficient to force wall contact in the specimen and result in large pressure spikes at the end of the confining chamber due to the water-hammer effect, and due to the formation and subsequent collapse of small cavitation bubbles on the chamber walls. Full implosion specimens with a greater unsupported length resulted in smaller waterhammer pulses due to more extensive damage incurred in the specimen.

Experimental Procedure
A series of experiments was conducted using state of the art facilities to understand partial and full collapse of composite cylindrical structures in an underwater fully confined environment. A brief description of the material and specimen geometry, experimental apparatus and sensor calibration is presented below.

Composite Tube Specimens
The specimen geometries used in this study were selected such that one would exhibit full implosion and the other would exhibit partial implosion. . At the initiation of instability, the potential energy driving the collapse of the specimen decreases with change in volume of the specimen. Thus, to ensure that these tubes would exhibit partial implosion behavior, the geometry was selected such that, upon instability, the potential energy in the confined system would decrease fast enough to prevent a full implosion. This requires that the buckling pressure of the specimen be low, and that the radius of the specimen be relatively large in order to increase the ratio of specimen external volume ( ) to pressure vessel volume ( ). As such, the partial implosion specimens studied had an outer diameter (d) of 55 mm, a wall thickness of (h) 1.9 mm, and an unsupported length (l) of 381 mm. Three experiments were conducted with this specimen geometry, one of which had lower ovality and wall eccentricity parameters than the other two, resulting in a 13% increase in critical buckling pressure Pc.
Specimens for the full implosion case were fabricated with the same material as the partial implosion case, however the thickness and layup were different. In order to ensure that the critical buckling pressure of full implosion specimens did not exceed the safe operating pressures of the pressure vessel used, the wall thickness of the full implosion specimens was required to be less than that of partial implosion specimens.
According to the filament winding process employed by the supplier, wall thickness is governed by the composite architecture, and thus to achieve the required wall thickness a slightly different layup schedule was selected. In all cases, specimens were sealed at both ends with 12.7 mm protruding aluminum endcaps outfitted with circumferential O-rings, and sealed with a thin layer of epoxy to ensure that the tube remains airtight. It has been shown that initial imperfections, namely ovality and variation in wall-thickness, can play a role in critical buckling pressure of tubes [19]. To quantify these imperfections, the ovality (∆ 0 = ( − min )/( + )) and wall eccentricity ( 0 = (ℎ − ℎ min )/(ℎ + ℎ )) were determined for each specimen prior to experiments. Prior to experiments, a high-contrast random speckle pattern was applied to the surface of the specimen facing the cameras using flat black and white paint. Using wall thickness measurements, special care was taken to ensure that this surface will experience the maximum inward radial deformation. This allows the high-speed images captured during the event to be analyzed using commercially available digital image correlation (DIC) software to provide full-field displacement measurements across the speckled surface of the specimen.

Sensor Calibration: Methodology and Results
The Piezotronic pressure transducers used in this study have a sensitivity of around 1mV/psi, however, the exact sensitivity varies from sensor to sensor, and thus the sensors must be calibrated to determine their exact respective sensitivities. To calibrate pressure sensors, the vessel was pressurized to a value near a given goal pressure, held at this value to ensure that the signal had settled, and the current hydrostatic pressure was recorded. The vessel was then quickly depressurized by opening a valve located on the pressure vessel. Data capture is triggered manually in unison with the release of pressure, and the resulting voltage drop in each sensor was used to determine respective sensitivity, according to the following equation: To verify repeatability over a range of pressures, this procedure was done three times for each goal pressure of 300, 400, 500, and 600 psi. Therefore, it follows that , denotes the sensitivity of a certain sensor given from trial . is the hydrostatic pressure just prior to depressurization and is the resulting voltage drop given by:

Where
, is the average signal prior to depressurization and , is the average signal after depressurization. The sensitivity for each sensor was then taken as the average of all trials conducted over all goal pressures : Table 2 gives the sensitivities of each sensor as determined by the above process: To illustrate this process, the deviation between sensor voltage outputs for a 600 psi pressure drop case is shown in figure 2a. There was a notable deviation in response although all the sensors measured the same pressure drop. This deviation was removed by applying the sensitivities shown in Table 2 such that all the sensors measure the same pressure drop, as shown in figure 2b.

Results and Discussion
The results of the two instability cases are presented and discussed below. In the case of partial implosion, the potential energy in the system was not sufficient to drive a full collapse, and thus wall contact was not achieved. Full implosion cases involved a relatively high amount of potential energy, causing wall contact in the specimen and causing significant pressure spikes (water hammer pulses) at the ends of the confining vessel. 3D-DIC displacement data, pressure data, and frequency information are discussed in depth.

Partial Implosion
Partial implosion occurs when the hydrostatic pressure in the system is enough to cause instability in the specimen, however the total available energy is insufficient to drive a full collapse. This results in an oscillating radial displacement behavior which causes harmonic pressure pulses throughout the confining chamber. After the initial instability event, trigger delay allowed further addition of water in the confining chamber causing progressive damage and resulting in subsequent instability events. For the sake of consistency, this study focuses on the initial instability event only. Since the visible damage on the specimen includes the damage that occurred due to subsequent instability events, the post-mortem analysis of partial implosion specimens has been omitted. It is important to note here that all time measurements are taken such that time t = 0.00 ms corresponds to the specimen experiencing its first peak inward radial deflection. The pressure at instability is referred to as PI, and is taken as the hydrostatic pressure the instant prior to instability. Deviations in ovality ∆ 0 and wall eccentricity 0 between partial implosion specimens resulted in specimens with instability pressures of 2.34 MPa and 2.06 MPa. These will be respectively referred to as case (a) and case (b) for the following discussion.

DIC Results
Digital image correlation technique was used to obtain position and velocity data from the instability event to provide additional insight into the failure mechanisms at play. To characterize specific phenomena that occur during the instability event, images   The maximum inward radial deflection at this point was 5.3 mm. From this point, the This moment marked the beginning of the second oscillation, during which the specimen experienced a maximum inward deflection of 5.4 mm at time t = 3.93 ms.
Note that in case (a), maximum inward radial deflection occurred in the first oscillation and did not increase in any successive oscillation. However, case (b) showed increasing inward radial deflection with each oscillation. Furthermore, it can be seen in figure 4B that this deflection begins to accumulate about the axial center of the tube, with larger portions showing greater inward radial deflections with each oscillation. This is due to the increased average pressure acting on the tube for the duration of the event which, as opposed to case (a), was relatively high due to the lower initial decrease in specimen volume. This phenomenon will be further discussed in the following section. Figure 4B also shows, along with figure 4C, the harmonic behavior of the specimen as it oscillates at a frequency of 259 Hz.

Pressure History
The normalized pressure histories at various points throughout the pressure vessel resulting from the partial implosion for the case (a) and case (b) are shown in figure 5A and 5B, respectively. Radial deformation histories extracted from the axial center of the specimens are superimposed over the respective pressure-time histories to illustrate the effect of specimen deformation. The pressure for both cases oscillated harmonically at a frequency equal to that of their respective radial oscillations obtained from DIC data, indicating that the frequency of oscillation of the pressure profile is governed by specimen deformation. The initial drop in normalized pressure ( ) at the axial center in case (a) was 2 times that of case (b). This difference in initial pressure drop is reflected in the difference in initial inward deflection (8.6 mm for case (a), 5.3 mm for case (b)) between the two cases and can be attributed to the difference in instability pressure, since a higher critical buckling pressure provides more driving energy for a longer time. However, experienced at the ends of the pressure vessel in case (a) was 2.4 times that of in case (b). This is then reflected in the amplitude of subsequent oscillations, which were considerably greater than those of case (b). This result suggests that while the pressure change at the axial center is directly related to the change in radius of the specimen, the pressure change will increase with longitudinal distance from the specimen as fluid motion begins to play a bigger role in the pressure profiles. It was noted in the previous section that case (b) experienced progressive inward radial deformation with each oscillation, while case (a) did not. To understand the mechanism behind this phenomenon, it is important to understand the pressure acting on the specimen itself. As the pressure-time history given by Channel 4 is taken very close to the specimen, it can be assumed to be the pressure acting on the specimen. For case (a), it is shown in figure 5A that the normalized pressure near the specimen dropped to 0.6 and oscillated in unison with specimen deformation. For case (b), figure 5B shows that similar pressure oscillations occurred but with lower amplitude. However, because the normalized pressure dropped to 0.8, higher magnitude pressures were imposed on case

Full Implosion
Full implosion occurs in a limited-energy environment when the specimen is subjected to a critical hydrostatic buckling pressure Pc such that instability is initiated, and the energy in the system is sufficient to continue radial inward deformation until wall contact is reached. Full implosion experiments were conducted for carbon/epoxy tubes with unsupported lengths of 381 mm, 330.2 mm, and 279.4 mm. To ensure repeatability three experiments were conducted for each length case. It is important to note that for all cases presented, time measurements are taken such that time t = 0.00 ms at wall contact.

DIC Results
A typical sequence of events seen in all full implosion cases is shown in figure   6A. The specimen experienced a certain amount of pre-deformation prior to collapse, as shown at time t = -1.80 ms. Due to the steady increase in hydrostatic pressure, deformation lobes began to form on the tube. Once the critical hydrostatic pressure was reached, the specimen began to deform radially inward, as shown at time t = -0.96 ms.
At time t = 0.00 ms, the specimen achieves wall contact, and the buckle began to propagate longitudinally across the specimen. While the buckle propagated, shown at time t = 0.67 ms, the center of the tube began to recover, and once the buckle had propagated as far as it will, the rest of the tube showed recovery. This behavior is shown at time t = 4.47 ms, when the buckle front is 'retreating' back to the center of the tube.
Finally, at time t = 6.00 ms the buckle was seen propagating and then retreating as the specimen continues to recover. Note that it was the decrease in potential energy with the onset of instability that allowed the specimen to recover, as otherwise the damaged material would be unable to recover with its critical buckling pressure still acting on it after implosion. This recovery behavior continued in an oscillating manner, recovering more and more with each oscillation. Taking the Fast Fourier Transform of radial displacement curves shows that the dominant frequency of this oscillation is 315 Hz for all specimens. It should be kept in mind that the post-collapse behavior of the specimen will be affected by many factors such as loss in stiffness and reflections from hammerpulses at the ends of the confining chamber, and thus further work will aim to expand on this frequency analysis and identify the physical mechanisms responsible for these oscillations. along the length of the tube. Furthermore, greater radial velocities resulted in more extensive damage throughout the specimen, which impaired its ability to recover after the implosion process. This in turn had a strong effect on the resulting pressure histories at the ends of the confining chamber, as will be discussed in the following section. The average change in radius (dR) over the entire viewable surface of the deforming specimen is included in these figures to allow for comparisons between global specimen deformation behavior and the resulting pressure profiles. At the axial center, a sudden decrease in pressure was observed as the specimen began to deform. Once the specimen achieves wall contact at time t = 0.00 ms, fluid motion was suddenly arrested, causing a pressure spike. This pressure spike was registered at all sensors throughout the confining chamber, except for those located at the ends of the chamber due to the absence of water at these locations. For all cases, the implosion event drew water from the ends of the confining tube, creating a vacuum at these locations. This is apparent in the pressure-time history for every case, as the presence of the vacuum resulted in periods of near-zero pressure at the ends of the chamber, which will be referred to as endplate cavitation. Note that the duration of endplate cavitation differed with specimen length according to the amount of water displaced by the implosion event.

Pressure history
For 279.4 mm long specimens, endplate cavitation lasted for 1.3 ms while for the 381 mm long specimens, endplate cavitation lasted for 2.2 ms. Just prior to the formation of the vacuum, liquid in tension was observed as the water separated from the confining chamber walls, given by a drop in pressure below the line denoting endplate cavitation.
Once the tube collapse was complete and momentum of water towards the axial center of the confining chamber had arrested, the pressure differential at the ends of the chamber caused this displaced water to accelerate back towards the ends of the confining chamber, creating a high amplitude pressure spike which acted on the endplate for about 2 ms. Meanwhile, as water began to move very rapidly away from the walls of the confining chamber, small cavitation bubbles formed and subsequently collapsed. In most full implosion events, these cavitation bubbles formed directly on the sensor surface. When these cavitation bubbles collapse, they resulted in very high amplitude pressure spikes seen in all plots. While the time duration of these pressure spikes was always very small and thus represented relatively low-energy events, they were seen to reach peaks up to 34.5 MPa. The full amplitudes of these spikes are not shown in Figure 7.
When comparing the 279.4 mm and 381 mm length cases, it can be seen that the pressure profiles at the axial center of the confining chamber were nearly identical. However, the longer tube resulted in smaller water hammer pressure spikes while the shorter tube length resulted in significantly larger water hammer pressure spikes. Post-mortem analysis reveals that longer specimens experienced significantly more damage throughout the implosion process, with increased bifurcation and intersection of major cracks which isolated large portions of material from the main structure of the tube.
These damage mechanisms will be further discussed in the following section. Increased damage in the 381 mm long tube resulted in a lower post-implosion stiffness, visible the partial implosion case that the dynamic pressure measured throughout the pressure vessel was heavily dependent on the amplitude of structural oscillation in the deforming specimen, especially at the ends of the confining chamber. In full implosion, the magnitude of the hammer pulse is dictated by the amount of water displaced at the ends of the confining chamber and the velocity at which that water returns. Much like the partial implosion case, the recovering tube acts like a spring which, coupled with the pressure differential between the cavitation/liquid interface, forces water towards the ends of the confining chamber at a greater velocity. This is evident also in the subsequent pressure oscillations seen in figure 7A between time t = 6 ms and time t = 8 ms, and in figure 7B between time t = 6 ms and time t = 10 ms. In these regions, a greater amplitude in pressure is still observed in 279.4 mm tube, corresponding with its greater velocity of radial recovery.

Post-Mortem Analysis
An in-depth post-mortem analysis was conducted to understand the failure mechanisms at play in the implosion of carbon/epoxy tubes, and to identify how they vary with tube length and critical buckling pressure. Post-mortem images of 330. The fact that the second outer-most layup orientation plays a larger role in the behavior of the major valley cracks has a significant effect on the failure mechanics of the tube.  propagation causes the major valley crack to approach the major lobe crack, causing the two to intersect (G). If these two cracks are through cracks, as shown in Figure 10, this intersection can separate a large portion of material from the tube structure (H), which can also lead to fiber pullout (I). Looking closely at this separated piece of material, it can be seen that the specimen also undergoes inter-lamina delamination (J).
When comparing specimens of differing length, it can be seen from figures 9 and 10 that the 381 mm long tubes experience more extensive through-thickness cracking along the lobes, and the major valley crack branches off to more surface areas. This allows it to intersect with the major lobe crack in more locations, in turn isolating larger amounts of material from the tube structure. These more extensive damage mechanisms at the valley can be attributed to the higher inward radial velocities experienced in this region during the implosion process, shown in figures 7D and 7E. The 279.4 mm long tubes also experience branching along the major valley crack, but to a lesser degree. Because of branching, the major valley crack does intersect with the major lobe crack in some areas, however because of the reduced amount of through-thickness cracking, these intersections fail to isolate any large portion of material and thus the tube remains relatively intact.

Conclusions
A fundamental experimental study was conducted to understand the effects of partial and full implosion of carbon/epoxy composite tube specimens occurring in a limited-energy environment. During experiments, high-speed images were captured which, together with the application of 3-D DIC techniques, provided full-field displacement data for the surface of the specimens. Furthermore, dynamic pressure data Finally, post-mortem analysis of full implosion specimens was conducted to further understand the failure mechanisms at play in the implosion of carbon composite tubes.
Key findings from this study are presented as follows: (1) By increasing the volume and decreasing the critical buckling pressure of a composite tube subject to hydrostatic load in a limited energy environment, the energy available to drive implosion was decreased. As a result, dynamic instability events transitioned from a full implosion event, in which wall contact is achieved, to a partial implosion event, in which the specimen oscillates radially without achieving wall contact.
(2) Partial implosions resulted in multiple pressure surges throughout the vessel, occurring at the exact same frequency at which the specimen oscillated. The amplitude of these oscillations increased considerably near the ends of the confining chamber.
(3) The amplitudes of pressure oscillations in the partial implosion case were heavily dependent on instability pressure, with a 13% increase in critical buckling pressure causing a 400% increase in peak amplitude of pressure oscillations.
(4) Full implosion resulted in pressure spikes at the end caps of the confining vessel due to the water hammer effect. The rapid motion of water at the ends of the confining chamber also lead to the formation and collapse of small cavitation bubbles, which were low energy, high amplitude events.
(5) Full implosion results in a significant decrease in hydrostatic pressure in the system which allows the imploded specimen to recover in an oscillating manner.
(6) For full implosion, increased radial velocities during the implosion process increased damage in longer specimens, thereby greatly reducing the velocity at which the specimen could recover. This reduced velocity of recovery resulted in lower amplitude hammer pulses and subsequent pressure oscillations. , frequency of oscillation , and damping ratio ξ. It is shown that decreases with increasing implodable volume to confinement volume ratio / , and is theoretically determined with excellent correlation to experimental results. Finally, amplitude of oscillation ∆ is theoretically determined and used in conjunction with theoretically determined and experimentally determined ξ to fully define the hammer wave oscillations resulting from partially-confined implosion.

Nomenclature
The

Introduction
The results, findings, and analytical investigation of an experimental study conducted on the hydrostatic implosion of filament wound carbon fiber epoxy composite tubes within partial confinement. The term "partial confinement" refers to a tubular confining structure that is closed on one end and open to free-field pressurized environment on the other. This differs from full confinement which is closed on both ends, thus limiting the energy available to drive implosion [1,2]. Implosion experiments are conducted using filament wound carbon/epoxy composite tubes of varying implodable volume to confinement volume ratios.
An implosion event begins when a closed, hollow structure of lower internal pressure is submerged in a medium of higher pressure such that the differential results in instability tubes and causing $20-$30 million in damage [5]. The problem of implosion has been a topic of study for many decades, with the first closed-form expressions for the critical buckling pressure, or of a cylindrical shell under uniform radial and axial compression having been derived by von Mises in the early 1900's [6,7].
The mechanics and fluid-structure interaction of implosion in free-field conditions have been well-established for metallic tubes [3,[8][9][10] and composite shell and double-hull structures [11][12][13]. In one study [3], Turner and Ambrico divided the implosion process of metallic cylinders into 4 main stages, summarized as follows: (1)  implosion within a free-field environment due to the limited energy available to drive collapse [1]. Then, studies conducted by Matos et al. [15] and Gupta et al. [16] showed that the hydrostatic and shock-initiated implosions of metallic tubes within partial confinement resulted in a large hammer pulse of significant impulse which oscillated throughout confinement at a frequency close to that predicted by water hammer theory. Since partial implosion can be initiated in composite implodables within confining structures, some of the implodable structures in the present study were designed to implode with minimal energy available within the volume of water bounded by partial confinement. This resulted in a phenomenon known as implosion with dwell, in which the structure began to implode, paused due to lack of sufficient driving energy available in the immediate surroundings, then continued as energy was made available from the open end of confinement. All implosion events detailed in the current study resulted in large pressure pulses due to the water hammer effect. It is observed that the water hammer pulse behaves as a damped harmonic oscillator defined by amplitude ∆ , frequency of oscillation , and damping ratio ξ. As such, the dynamic pressure histories resulting from implosion are characterized by experimentally identifying these parameters. Finally, theoretical expressions which define ∆ and are developed and presented, and correlate well with the experimental results.

Experimental Procedure
Implosion experiments were conducted in a state-of-the art pressure vessel facility equipped with dynamic pressure transducers and optically-clear windows which allow for the use of 3D digital image correlation (DIC). This section details the facilities used to create partial confinement conditions within a hydrostatic loading environment, and presents tabulated geometric data which fully describe the filament wound carbon/epoxy tubes used. Finally, a brief discussion is given which validates the use of DIC in the complex underwater environment described, and quantifies any expected error.

Pressure Vessel Facility with Partial Confinement
Implosion events were initiated within a 1.  To conduct implosion experiments, implodable structures were placed concentrically within the confining structure such that the axial midpoint of the implodable was in-line with the axial midpoint of the acrylic section. The hydrostatic pressure of the pressure vessel was slowly increased by pressurizing a small void at the top of the vessel with nitrogen gas until the critical buckling pressure of the implodable structure was reached. High-speed photographs of the resulting structural deformations, as well as pressure data throughout the confining structure were recorded by the facilities described above.

Composite Tube Specimens
The mm OD will be referred to as Case A and Case B, respectively. Furthermore, each specimen geometry will be referred to as geometry or tube A or B, as per its OD, followed immediately by 28, 33, or 38, which refers to the length of the tube rounded to the nearest centimeter. Thus, a tube of 41.3 mm OD that is 330.2 mm in length will be referred to as to as geometry B33 or tube B33. Average measured geometric values, including implodable radius , unsupported length , implodable wall thickness ℎ , and implodable to confinement volume ratio / given as a percentage are given for each geometry in table 1. Note that the term refers to the volume of air bounded by the implodable structure's walls, and does not account for wall thickness. Due to insignificant variations in / between tubes of the same geometry, no standard deviation is given for this value. Also given are the experimentally-obtained critical buckling pressure and confinement energy , which represents the strain energy stored in the volume of water bounded by the confining structure at the moment of implosion. The confinement energy is calculated according to the following equation: Where is the compressibility of water and is the bulk modulus of water.

Validation of DIC Methods in a Complex Underwater Environment
It has been well-established that 3D DIC is valid for submerged objects viewed through a window so long these additional media (water and window) are included in the calibration process [9]. The experiments conducted for this study required a third media, a curved acrylic viewing window (see Fig. 1) in which lay the possibility for distortion of the object being viewed by high-speed cameras. A process was developed by Senol et al [17] to minimize the error induced by a curved viewing window surrounded by air and filled with water. The resulting errors were no greater than 4%.
Thus, a similar scheme was adopted in this work in which the calibration procedure described in [9] was conducted without the curved acrylic window in place, and then DIC measurements of the object in question taken with the inclusion of the curved window. To validate this procedure over a range of tube radii, a rigid cylinder of sequentially varying radii was manufactured, painted with a high-contrast speckle pattern, and the DIC-measured radius of each section compared to its true radius. The cylinder is shown behind the acrylic window in Fig. 2A indicates that on average, DIC-measured values were slightly smaller than true values.
As a final precaution, radius measurements of the implodable structure itself were taken prior to every experiment and compared to true radius values. Of these measurements, none deviated more than 5% of the true radius value of the tube implodable structure.

Experimental Results
Partially-confined implosion experiments were conducted in the state-of-the-art displacement data and velocity data taken at the surface of the imploding structure were made available from 3d DIC. Note that dynamic pressure, or ∆ ( ), is given as the deviation from . Thus the hydrostatic pressure is equal to when ∆ ( ) is zero, and the hydrostatic pressure is equal to absolute zero when ∆ ( ) is equal to − . Note further that in this manuscript, radial deformation towards the axial center of the implodable is denoted as negative (-), while radial deformation away from the axial center of the implodable is denoted as positive (+). These will be referred to as inward and outward , respectively.

Water Hammer Behavior
In all cases, implosion within partial confinement resulted in an initial drop in hydrostatic pressure at all locations within confinement which will be referred to the initial under pressure region. This is due to the decrease in implodable volume and

Under Pressure Oscillations
Over Pressure Oscillations   Fig. 3 Water hammer pressure oscillations taken from channel 2 resulting from the partial implosion of implodable geometry B28. Water hammer oscillation is seen to behave as a damped harmonic oscillator, defined by amplitude ∆ , frequency , and damping ratio ξ structure achieves wall contact, fluid motion stopped abruptly and kinetic energy was suddenly converted into strain energy in the water, resulting in a large pressure spike resulting in an over pressure region which will be referred to as the hammer pulse. With analogy to pipe flow mechanics, this behavior is akin to opening a valve long enough for a volume of water equal to the volume of the implodable and with hydraulic head equal to to pass through the duct, then closing the valve suddenly. In all pressure time histories shown in this section, time = 0 at the moment the maximum pressure is observed anywhere in the confining chamber. This maximum over pressure, due to the water hammer effect, will be referred to as ∆ . It was noted that all dynamic pressure histories observed at channels 1-4 behave generally as a damped harmonic oscillator, described by the equation: Where ξ is the damping ratio and is the radial frequency of oscillation (equal to 2 , where is the frequency of oscillation). To illustrate the manner in which these parameters are present in a typical water hammer pulse resulting from implosion, Fig.   3 gives the dynamic pressure taken at channel 2 from implodable geometry B28, annotated with all parameters required to define ∆ ( ) from equation (2).
While six implodable geometries were used in this study, there were no fundamental differences observed between geometries of the same case aside from variations in ∆ , ξ, and . As such, this discussion will focus mainly on comparing and contrasting dynamic pressure history and structural deformation phenomena seen in the implosion of geometries A38 and B28, which are the largest and smallest tubes used in this study. All representative dynamic pressure histories from each geometry can be found in Appendix C, attached to the end of this manuscript.
The oscillatory behavior of the hammer pulse seen in Figs. 4A and 4B shows significant differences between the pressure histories resulting from the largest tube in this study, tube A38, and the smallest tube in this study, tube B28. Note that the y-axis scales and time scales differ from Fig. 4A to Fig. 4B so that the finer features of pressure behavior can be seen. Furthermore, note that a black dotted line is plotted at ∆ = − to indicate absolute pressure of 0 MPa. Upon initial inspection it can be immediately noted that there is a dramatic difference in the frequency of oscillation between the two tubes.
In fact, the average of the dynamic pressure history resulting from the implosion of geometry A38 was about 115 Hz, less than half of that observed in geometry B28, which was about 250 Hz. Generally, it was seen that the frequency of oscillation decreased as implodable volume ratio / increased, and it will be shown in section 4 that is in fact a function of both and / . This phenomenon was not seen to any degree in implodable geometries of Case B.
Finally, it should be noted that during the first under pressure region, all implodable geometries of Case A experienced a brief "dwell" period, in which inward slows, causing a brief decrease in the rate of pressure drop. This phenomenon is explained further in section 3.2.

Partially-Confined Implosion with Dwell
The term "implosion with dwell" is given to partially-confined implosion events in which the implodable structure is subject to such that structural instability is initiated, however the energy contained in the fluid immediately surrounding the structure is insufficient to continue inward radial deformation, , and thus there is a brief delay in the collapse process. This delay only lasts until energy from the open end of the confining chamber arrives to the implodable structure, at which point inward radial deformation accelerates until wall-contact is achieved and the implosion process is completed. Note that all implodable structures of Case A exhibit this behavior. The energy available to drive collapse from the initiation of instability until the brief moment of dwell is equal to the strain energy in the pressurized volume of water contained by the confining chamber at the moment of implosion. This value, given as , was defined by equation (1) in section 2 but is repeated here for the reader's convenience: Where is the compressibility of water and is the bulk modulus of water. is calculated for all implodable volumes and tabulated alongside geometric data in table 1. Note that geometries of Case A had significantly less immediate energy available to drive collapse than geometries of case B. This is largely due to the fact that, all other factors held constant, an increase in diameter of a tubular implodable structure will decrease , which is the variable that contributes most to . This decreased resulted in a "dwell" effect that can be broken up into 3 primary stages: partial collapse, dwell, and buckle propagation. stage, however, is also distinguishable from dynamic pressure data, especially at the locations measured by channels 1, 2, and 3. From the perspective of pressure data, the partial collapse stage is given by a gradual drop in pressure, the dwell stage is given by a leveling off at some hydrostatic pressure below , and then the buckle propagation stage is given by a sudden degrease and subsequent high-magnitude spike in pressure.
To illustrate this, a magnified view of Fig 4A is given by Fig. 5B showing only data taken at channels 1, 2, and 3, and divided into the corresponding sections much like Fig.   5A.

Model of ∆ ( ) using Experimentally-Obtained Values
It can be seen from the pressure-time histories given by Fig. 3 that the water hammer pressure pulse generated from an implosion event behaves very similarly to a damped harmonic oscillator with amplitude equal to experimental ∆ , , frequency of oscillation , and damping coefficient . Note that refers to the specific channel or location at which the pressure history in question was measured. This is important especially with regards to the damping coefficient and maximum over pressure because, as will be shown, and ∆ , are dependent on location. Thus, if these parameters are extracted from experimental data, the pressure history at any location measured by channels 1-4 can be modeled to take the following form: Note that this model only describes the pressure from time = 0 onwards, as the under pressure region does not conform to the behavior of a damped harmonic oscillator.
Furthermore, it is seen that the pressure history measured at channel 5 behaves much differently than those measured at channels 1-4, and thus will not be modeled according to equation (2).
To determine the value of ∆ , , one need simply to take the first maximum value from each channel . The first step in determining the damping coefficient of ∆ ( ) from experimental data is to determine the average logarithmic decrement, , given as: Where ∆ 1 and ∆ 2 are the values of two successive peaks in the decaying pressure signal. From , the damping coefficient can be given as: For each sensor location, the decay constant is determined for each oscillation and then these values are averaged to give a value of for the entire signal. Using equations (4) and (5), was determined for all dynamic pressure measured at channels 1-4 and for all geometries, and is tabulated in tables 2 and 3 alongside corresponding ∆ , values.
Where is the period of oscillation , and is the total number of full, distinguishable frequency, however, will not be considered for use in equation (3) as it is the aim of equation (3) to model only pressure histories at locations within confinement. After this low-frequency contribution, the frequency which contributes most to the pressure signal the frequency marked by the white dotted line in Fig. 7. As this value does not vary with position within confinement, it will be considered that = = for use in equation (3). Thus was determined for all geometries and is plotted against , given as a percentage, for all experiments in Fig. 6.
Finally, using ∆ , , , and obtained from experimental data, equation (3) can be used to model the dynamic pressure oscillations resulting from a partially confined implosion event. To demonstrate the ability of this method to recreate the pressure signals obtained from channels 1 and 3, the results from equation 3 are plotted over their respective dynamic pressure histories for implodable geometries A38 and B28 in Figs. show that the entire initial overpressure region conforms well the proposed model.
As can be seen from Fig.8, the experimental model given by equation (3) correlates well with experimental data, and shows that the dynamic pressure history resulting a partially confined implosion event can indeed be modeled as a damped harmonic oscillator primarily dependent on the contribution of a single frequency. That is to say, that while it is clear from Fig. 6 that ∆ ( ) is the result of the superposition of various phenomena occurring at frequencies other than , these contributions are largely negligible and can be neglected for modeling purposes. This is especially important for the following discussion, in which ∆ and are theoretically derived.  Tables 2 and 3 are given for dynamic pressure histories taken at channels 1 and 3 for (A) implodable geometry B28 and (B) implodable geometry A38

Discussion
As evident from the previous section, the pressure oscillations resulting from the implosion of a tube within partial confinement can be accurately modeled using experimentally-determined parameters when it is assumed that the dynamic pressure history ∆ ( ) behaves as a damped harmonic oscillator, given by equation (2) which is repeated here: Where ∆ is the maximum change in pressure, ξ is the damping coefficient, is the angular frequency of oscillation (equal to 2 , where is the frequency of oscillation), and is time. While ∆ , ξ, , and can be determined easily from experimental data, conducting experiments is an extremely costly method of determining constants which can be resolved from a more fundamental, theoretical understanding of the problem. Of these parameters, those of most significance are ∆ and , as these two parameters provide sufficient information to fully describe the first and strongest overpressure oscillation resulting from implosion. As such, it is the purpose of this section is to determine and ∆ from the fundamentals of water hammer theory and by equations which describe the energy partition in a shockwave travelling through water. Theoretical values are then compared to experimental results. For reference, all variables and constants mentioned in this section are defined in the nomenclature located at the beginning of this manuscript.

Preliminary Considerations: The Water Hammer Effect
A water hammer event occurs in typical pipeline or duct of a given length when fluid passing through the duct at initial velocity is brought to rest by sudden valve closure. For application to the case presented in this study, it will be considered that the confining chamber is the duct in question with length . Furthermore, it can be considered that the flow of water into the confining chamber begins with the initiation of implosion at and occurs at an average velocity of . One compete cycle of a dynamic pressure oscillation due to the resulting hammer wave in a frictionless case is described [18].
The completion of the implosion event is analogous to sudden valve closure, and for the sake of this discussion will be considered to occur at time  9A shows the process at time 0 < ≤ / , Fig. 9B shows the process at time / < ≤ 2 / , Fig. 9C shows the process at time 2 / < ≤ 3 / , and Fig. 9D shows the process at time 3 / < ≤ 4 / .

Theoretical Determination of Frequency of Oscillation
From standard water hammer theory, the pressure pulse resulting from a water hammer event must travel a length of 4 times the confinement length, or 4 , in order to complete a full under pressure and overpressure cycle [18]. Thus, the frequency of oscillation of a wave travelling at the speed of sound in the fluid can be given by equation (3): By definition, is a function of the bulk modulus of the fluid and the and the density of the fluid , given by equation (4): It is commonly known that the speed of sound in water is = 1440 m/s, however using equation (3) to determine from the values of plotted in Fig. 6 shows that the wave speed observed in partial implosion experiments ranged from about 1100 m/s to as low as about 500 m/s, with wave speed decreasing with increasing implodable volume. According to work done by Kabori et al. and Pearsall [19,20], such a decrease in wave speed is to be expected in any fluid-air mixture. According to their work, the wave speed in such a mixture can defined simply by equation 2 when the bulk modulus of the mixture and density of the mixture are defined by the properties and volume content of the mixture's constituents. By considering the volume contained within confinement to be a mixture of water and the air contained by the impolodable , the same method can be applied here. Beginning first with the definition of bulk modulus, can be written by equation (5), substituting in the confinement volume and for the change in pressure and so that: Where ∆ can be expressed as the sum of the change in volume of each constituent, or: The change in volume of water can be determined simply from the definition of bulk modulus, such that: Where is the bulk modulus of water, given to be 2.2 GPa, and is equal simply to − . To determine , it is assumed that during the implosion process, all of the air escapes the implodable and is subsequently pressurized to . This is a valid assumption for the implosion of composite structures as the damage sustained by the structure is often sufficient to allow the air contained within to escape. The volume of air within the confining chamber at can thus be defined by the ideal gas law according to equation (8): Where is the total mass of the air given by = , is the gas constant, is the temperature in K, and is the molar mass of air. Plugging equation (8) into the definition of bulk modulus gives the solution for ∆ , given by equation (8): For a gas volume under hydrostatic compression, it can be shown that the bulk modulus is equal to the hydrostatic pressure. Thus, is replaced with , yielding equation (10): Thus, plugging in equations (6), (7), and (10) into equation (5) yields: Now that the bulk modulus of the fluid-air mixture contained within confinement has been determined, the only variable necessary to determine the wave speed is , which is given simply by the rule of mixtures: Thus, the wave speed of the water-air mixture is given by: The wave speed defined by equation (13) is valid for any water-air mixture contained by which is pressurized to and contains an unpressurized volume of air equal to . However, the effect of the confining structure itself on wave speed has still not been considered. According to Tijsseling et al. [21], the wave speed of a fluid contained within a pipe made of segments of differing materials can be approximated by equation (14), which is a weighted average between coupled wave speeds in the aluminum section and the acrylic section. Thus, the final expression for the wave speed during a partially-confined implosion event, , is given as: Where the length of the aluminum and acrylic sections are given by and , respectively, the elastic moduli of aluminum and acrylic are given by and , respectively, the diameter of the confining chamber is given by , and the thickness of

Determination of Maximum Overpressure ∆
Now that has been determined, the only remaining parameter required to fully define the first overpressure period is ∆ , which, as will be shown, is dependent on . In the derivation of the following expression for ∆ , it is important to note the following assumptions:  When an implodable volume is subject to hydrostatic pressure from a liquid medium of unlimited driving energy, the hydrostatic potential energy, ℎ , is limited only by the work that can be done on the volume by the hydrostatic pressure being applied. In the case of implosion, the hydrostatic pressure is , the implodable volume is , and thus ℎ can be given by the equation: Note that equation (15) requires the first assumption stated above to be true, as otherwise this term would be ∫ ( implosion event, and is the maximum energy available to create the hammer pulse and cause fluid motion within confinement. According to Urik [22], the energy passing through one unit of surface area from time 1 to time 2 due to a one-dimensional planar compression wave, , can be defined as: It is also noted by Chamberlin et al. [10] that a portion of the hydrostatic potential energy is converted to work in compressing the air bounded by the implodable volume, denoted by . Assuming that all the air contained with the implodable volume is compressed by pressure (shown to be a valid assumption due to the accuracy of theoretical values of , shown in Fig. 10) and assuming adiabatic compression, can be given by the equation: Thus, multiplying the term by the surface area of the shock front , adding , and considering the sum equal to ℎ yields the energy balance resulting from the implosion of a partially-confined composite structure: According to assumptions 3 and 4, the shock front is planar and thus for the case of partially-confined implosion within a tubular confining structure, is defined as: For the determination of ∆ in a water hammer wave resulting from a partiallyconfined implosion event, it is convenient to modify equation (2) by eliminating the damping term and considering a phase shift of 2 for algebraic simplification. Thus the first overpressure region takes the form: Where is known from section 4.1. Furthermore, the period of oscillation can be easily determined from , as = 1 . Thus, the limits of integration for can be given as the duration of the overpressure region, bounded by 1 = 0 and 2 = /2. Also, it must be considered that the fluid through which the hammer wave is travelling is in fact a water-air mixture contained by a duct composed of aluminum and acrylic segments.
Thus, the terms and in equations (16) and (17) become and , which vary from implodable to implodable and are defined in section 4.1. Considering all of these substitutions and inserting equations (19) and (20) into equation (18) yields the relationship between ∆ and ℎ : Finally, evaluating the integrals from 0 to 2 and solving for ∆ yields the closedform expression: Which, because = 1 , reduces further to: Note that the value of ∆ determined by equation (23) is the maximum value of ∆ ( ) possible in the confining chamber at time = 0, and does not specify where this dynamic pressure will occur. However, it is seen consistently that for the experimental setup used for this study ∆ occurs at channel 2 (25.4 mm from the confinement floor), and thus can be assumed to occur within this vicinity. The ability to express ∆ in this way is a very powerful tool in the design of any structure which will be subject to hydrostatic loading and surrounded by partial confinement, as it allows for an array of parametric studies to easily be conducted. Such studies could determine optimum implodable geometries which meet the design requirements of the structure while minimizing the risk posed to surrounding structures. Also necessary for such a study would be an accurate formulation for , which can be determined for composite and metallic implodable structures based on material properties and geometries [6,7,23].
Using experimentally-obtained and carrying out the entire process defined by equations  for each implodable structure used in this study to theoretically determine ∆ gives good agreement with experimental results, as seen in Fig. 11.
Note, however, that while excellent agreement is seen for implodable volumes with below 2 MPa, theoretical values begin to deviate from experiments values as increases. This is likely due to two reasons. The first is that implodable volumes of high (namely, all the volumes of 41.3 mm OD) experience much larger radial velocities than implodable volumes of lower (all tubes of 63.5 mm OD). Unless cavitation forms between the surface of the implodable and the water, water can be assumed to be moving at the same velocity as the implodable. Thus, due to greater fluid velocities, turbulent Fig. 11 Experimentally-obtained ∆ given alongside ∆ determined from equation (23), showing excellent agreement for low , however deviating for higher flow and thus energy dissipated through turbulent losses are more likely. Second, it was noted in section 3 that implodable volumes of high resulted in the formation of cavitation at the closed end of the confining chamber. Thus it is likely that the energy required for the change in phase necessary for cavitation was dissipated through this means, further increasing the deviation between experimental and theoretical . This is to say that as increases, partially confined implosion events exhibit more energydissipating phenomena which are not accounted for by equation (23). It should also be noted that losses due to fracture and delamination of the implodable structure have been neglected in equation (23), which also explains why theoretical ∆ is always greater than experimental ∆ .
Using the ∆ and values defined in this section in equation (1) Finally, it should be noted that the case derived in this section is identical to a case in which a closed valve of a piping duct containing a volume of gas is suddenly opened for enough time to allow a volume of liquid pressurized at pressure to enter the duct, and then suddenly closed again. The only difference in this case is that would be equal to for the determination of (equations 2-13) and in equation (23), and would be equal to . Thus the range of applicability for the equations derived in this section expand beyond the case of implosion within partial confinement, and into the realm of pipeline analysis.

Conclusions
A comprehensive study of the failure mechanics and dynamic pressure histories resulting from the implosion of filament wound carbon/epoxy composite structures within an open-ended confining structure is presented. It was found that dynamic pressure oscillations can be described according to water hammer theory, and that the frequency of oscillation decreased as implodable to confinement volume ratio / 3. Implosion with dwell can be divided into three main stages: partial collapse, dwell, and buckle propagation.

The wave speed resulting from a partially-confined implosion event is in fact equal
to the wave speed in a fluid-air mixture defined by the amount of air contained within the implodable volume, the volume of water bounded by the confining structure, and critical buckling pressure . This allows for an accurate determination of frequency of oscillation .
5. Using theoretically-determined frequency of oscillation , the limits of integration necessary to define amplitude ∆ and be determined, and thus an energy balance can be used to determine ∆ with good agreement to experimental results

Chapter 3: Topics for Future Study
It is of the nature of scientific research that the process of defining, investigating, and drawing conclusions from a problem results in just as many questions as answers. The investigations detailed in Chapter 1 and Chapter 2 of this thesis are no exception to this, and thus it is the purpose of this third and final chapter to identify some of the questions or topics brought to light by these investigations and suggest methods as to their solution. The topics discussed in this chapter are closely related to the topics of fully and partially-confined implosion, and as such will be divided into two sections: Topics for Fully-Confined Implosion and Topics for Partially-Confined Implosion.

Topics for Fully-Confined Implosion
Problem Statement #1: Determination of for Partial and Full Implosion events In Chapter 2, it was shown that the behavior of the dynamic pressure history resulting from implosion events occurring within partial confinement was heavily dependent on the term , which is the bulk modulus of the gas-fluid mixture bounded by the confining structure. The gas in this mixture is the gas bounded by the implodable structure at critical buckling pressure and fluid is the fluid bound by confinement by the time of implosion. By determining the theoretical and resulting wave speed , it should be possible to predict the frequency of oscillation of dynamic pressure pulses resulting from implosion within full confinement, and by following a similar energy balance method shown in Section 4.3 of Chapter 2, it may be possible to fully define pressure oscillations.

Suggested Solution #1:
It was shown in section 4.2 of Chapter 2 that can be determined from the following equation: Where is the critical buckling pressure, is the total volume bounded by the confining structure, and ∆ is equal to the total change in volume of the fluid-gas mixture. In the case presented in Chapter 2, catastrophic failure of the implodable structure was assumed and thus it was considered that the entire volume of air bounded by the implodable was compressed to by . In the case of partial implosion, however, catastrophic failure is not seen, and thus it would be of interest to experimentally determine (from experimental wave speed ) how this affects the value of . This can be compared to any theoretical predictions, which could assume no compression of the air bounded by the implodable, or could attempt to predict the compression of the air. Furthermore, the process defined in section of 4.2 of Chapter 2 could be applied to a full implosion case, and any discrepancies could be determined and analyzed.

Problem Statement #1: Investigation of Water Hammer Oscillations Little to no
Interference from an Implodable Structure It was noted in Chapter 2 that the amplitude of oscillation of dynamic pressure histories resulting from the implosion of composite structures in partial confinement can be estimated using the following energy balance: Where ℎ is the hydrostatic potential energy, is the radiated energy, and is the energy required to compress the air bounded by the implodable structure. It was further stated in Chapter 2 that the energy balance does not consider losses, , which are shown to increase with . These losses can include turbulent losses, losses due to the formation of cavitation, and losses due to fracture in the implodable structure, . While can be crudely estimated based on the energy release rate of the composite material, the energy lost due to fracture could be better estimated experimentally, according to the following two methods.

Suggested Solution #1a:
Instead of using a composite implodable to initiate the water hammer oscillations, it would be beneficial to use a very brittle material such as glass, as this would ensure sudden and total loss of stiffness of the implodable upon implosion. As such, there should be very little to negligible energy lost to fracture in the event, and the pressure pulses resulting could be analyzed and compared to an implosion event using a composite structure at similar to determine the energy lost in the composite structure.

Suggested Solution #1b:
This solution would require skilled signal processing, but would have the advantage of completely eliminating a structure which could absorb energy during implosion. Instead of using an implodable structure to create a change in volume at and initiate a water hammer event, a small explosive could be used to generate a bubble of some volume , which could be determined from the radius of the bubble using DIC. This would be primarily for validating the theory explained in Section 4 of Chapter 2, as large hydrostatic pressures would be required to maintain a bubble size small enough to be unaffected by the confining structure. This technique would require skilled signal processing to filter out the incident and reflected pressure pulses resulting from the UNDEX event itself, as only the collapse of the bubble is of interest. This should not be impossible as the frequency of the hammer pulse resulting from the collapse of the bubble should be governed by water hammer physics, and thus can be predicted and used as an estimate for any filtering cutoffs. to 0%, 40%, and 80% of the structure's internal volume. As the volume of each implodable structure was 0.290 liters, each structure was filled with 0 liters, 0.116, and 0.232 liters of water, respectively. The volume of the water was determined from the mass of the water prior to filling each implodable structure. In the following discussions, the implodable structures will be referred to as V0, V40, and V80, respectively.

Experimental Details
Prior to being filled with water, each structure was painted with a high-contrast speckle pattern to enable the use of 3D digital image correlation (DIC), which provided real-time structural deformation data during the implosion process. Once filled with its respective volume of water and sealed with aluminum endcaps, each implodable volume was placed concentrically within the confining pressure vessel, such that the axial center of the implodable structure aligned with the center of the acrylic viewing window. The confining chamber was then filled with water, and it was ensured that no air was trapped in the system. Hydrostatic implosion was initiated by increasing the pressure in the closed system using a hydrostatic test pump. Implosion triggered the capture of highspeed images from two Photron SA1 high speed cameras, recording at 30,000 frames per second, as well as the capture of dynamic pressure data from pressure transducers located at various points throughout the confining chamber. Of particular interest were the pressure histories taken at the axial center of the confining structure and the pressure history taken at the end of the confining chamber, where

Experimental Results
The dynamic pressure histories taken at channel 1 (left end of confining chamber) and channel 4 (axial midpoint of the confining chamber) are presented alongside radial displacement taken at the axial midpoint of the imploding structure, for each case. It is important to note that in the case of V0, dynamic pressure data was sampled at a rate of 25 kHz, while in the cases V40 and V80 the sampling rate was 1 MHz. Note further that in this section, radial deformation towards the axial center of the implodable is denoted as negative (-), while radial deformation away from the axial center of the implodable is denoted as positive (+). These will be referred to as inward and outward , respectively, and for all plots, time will be given as time = 0 when has reached a minimum value. Finally, it is important to note that all dynamic pressure histories are normalized to the critical buckling pressure of the structure, , according to the equation: Where is the normalized pressure and ∆ is the dynamic pressure. As such, the absolute pressure is equal to when = 1. and is equal to 0 MPa when = 0.

a) Unfilled tube, V0
The normalized dynamic pressure data taken at channels 1 and 4 are shown alongside center point displacement during the implosion of the unfilled tube is shown in Fig. A1. It is important to note that the implodable structure never quite achieves wall contact. This is evident from both the radial displacement data shown as well as from examination of the imploded structure after it was removed from confinement, in which a gap is seen between the walls of the deformed structure. Note that there is relatively no spike in pressure seen in the vicinity of the implodable structure (channel 4), due to the fact that wall contact was never made. The slight recovery in seen just after minimum is reached is due to elastic recovery in the deformed material after being plastically strained. As is common in implosion events occurring within a fully-confined

b) Tube filled with 40% of water, V40
It was stated in the previous section that the unfilled implodable structure V0 did not achieve wall contact during implosion. This resulted in a large amount of the implodable volume still bounded by the structure after the implosion event. This remaining volume was greater than 40% of the original implodable volume, and due to this fact filling the tube with 40% of water had no effect on the resulting dynamic pressure histories taken at channel 1 or channel 4. Because no notable differences were seen between the implosion of structures V0 and V40, discussion of V40 will be omitted.

c) Tube filled with 80% of water, V80
For the case of V80, in which eighty percent of the implodable structure volume was filled with water, it is important to note two phenomena seen in Fig. A2 which distinguish this case from the implosion of V0 and V40. First, the addition of water in the implodable volume was able to prevent wall contact, as evident from the fact that center point displacement reaches a minimum of -12.2 mm. Second, it should be noted that no cavitation is seen at the left end (channel 1), as evident from the fact that the normalized pressure never reaches 0. However, a water hammer pulse is still seen at this location, and is seen to oscillate with decreasing amplitude. It should be noted that the frequency of these oscillations is less than that of case V0, and is likely due to the fact that the bulk modulus of the fluid-air mixture bounded by the confining structure is greater for the case of V80, therefore increasing wave speed . Finally, note also that

Fig. A2
Change in radius measured at the axial midpoint of the implodable structure is shown alongside normalized pressure resulting from the implosion of structure V80 taken at left end (channel 1) and axial center (channel 4) of confinement the maximum radial velocity has been decreased to 9 m/s, as opposed to 19 m/s in case V0.

Conclusions
A series of three implosion experiments was conducted within full confinement in which implodable structures were filled with varying amounts of water equal to 0%, 40%, and 80% of the total volume bounded by the implodable structure. 3D digital image correlation was used to provide structural deformation data, while dynamic pressure transducers captured the pressure-time histories at various points throughout the confining chamber. Key conclusions from this study are as follows: 1. By filling an implodable structure with 80% water, both wall contact and cavitation at the ends of confinement can be avoided.
2. The frequency of oscillation of water hammer pulses resulting from implosion is greater for implodable structures filled with 80% water. This is likely due to the increased bulk modulus of the water-air mixture bounded by confinement, , which increases as the volume of air bounded by the implodable structure decreases.
surrounding the pre-deformed structure excites vibration in the structure, which is measured using DIC. Note that the amount of water transferred to the brass section is not exactly equal to , as there will be a certain amount of air trapped in the brass section which will be compressed as pressurized water enters. Note further that Valve 2 MUST be closed before opening Valve 1 or else water will spray out of the assembly uncontrolled. Due to the fact that air is trapped in the brass section when Valve 1 is opened, the change in pressure cannot be calculated simply from change in volume of water using the bulk modulus of water, and thus a series of calibration tests were conducted to identify the expected pressure drop resulting from the opening of Valve 1.
Calibration tests were conducted by pressurizing the fluid in the pressure vessel to a pressure 1 , quickly opening Valve 1, and then recording the final pressure of the fluid, 2 , which is the pressure at which structural vibrations are excited. This was done for many values of 1 . Initial Pressure 1 is plotted against final pressure 2 and a best fit line is determined, as shown in Fig. B2. Using Fig. 2B, the pressure 1 necessary to excite structural oscillations about any desired pressure 2 can be determined.

Speckle Pattern and Calibration Technique
The composite implodable structure used for vibration experiments was a filament wound carbon fiber/epoxy tube of 20.7 mm outer radius, 1.67 mm wall thickness, and 280 mm unsupported length. The layup schedule of the tube was [±15/~90/±45/±15]. A 51 mm long section of the camera-facing side of the tube was painted with flat white paint, and a very fine black speckle pattern was applied using a "misting" technique. The term "misting" refers to the application of very fine black dots to a surface by very lightly spraying paint at the surface from about 305 mm (~1 foot) away from the target surface. Once properly mastered, this technique is an effective and efficient way of applying very small, random speckle patterns to a surface suitable for the use of 3d DIC.
Once the implodable structure was painted and speckled, it was placed in the fullyconfining pressure vessel such that the center of the speckled portion of the structure water. Then, two high-speed cameras with 100 mm lenses were arranged such that the optical center of each camera's frame of view was aligned with the center of the speckled portion of the implodable structure. The frame rate was set to 67,500 frames per second, and the resolution to 256 x 256 pixels. The cameras were then further adjusted to ensure that the diameter of the implodable tube fit perfectly within the 256pixel viewable frame. Once the cameras were positioned, the position of the implodable structure within the pressure vessel was measured to ensure that it could later be returned to the same position. The water was then removed and the implodable pushed off to the side to allow for in-air calibration.
Calibration of high speed cameras for DIC purposes was conducted in air using a custom-designed 12 x 5 -2 mm calibration grid with an offset in X of 2, an offset in Y of 2, a length in X of 9, and a length in Y of 2. The calibration grid is shown from the perspective of the camera in Fig. B3a. The calibration score resulting from calibrations conducted for the experiments detailed in this appendix was 0.15. Note that because vibration experiments were conducted underwater, calibrations conducted in air will not

Experimental Procedure
Once the calibration process described in section 2 had been completed, the implodable structure was placed in its original position, such that the optical center of each camera was aligned with the center of the implodable region of interest. After ensuring that Valve 1 and Valve 2 are closed, the pressure vessel was then filled with water, with special care taken to ensure that no air remained in the vessel. At this point, the experimental procedure was as follows: 1. Ensure that cameras are set to the "start" recording option, and begin recording. into a bucket or other container. The compressed air within the assembly will expel the water quickly, and ensure that no water remains in the brass section.
7. Save first 1000 or so frames taken from high speed cameras. Save more or fewer frames if necessary, depending on the expected frequency of oscillation.

Experimental Results
The experimental procedure detailed in the previous section was conducted for the implodable structure described in section 2 for 2 values of 253 psi, 133 psi, 54 psi, and 23. Note that these values correspond to 55%, 29%, 12%, and 5% of the implodable structure's experimentally-determined critical buckling pressure of 457 psi, respectively. Radial deformation values were determined from high-speed photos, and then a fast Fourier transform (FFt) was conducted on time-deformation curves to determine the frequency content of oscillations. This process is shown in Fig. B4, which gives a segment of the radial deformation curve and its corresponding FFt plot. From the FFt plot, it can be seen that two dominant frequencies were excited in the structural behavior of the composite structure. The low-frequency contribution seen in Fig. B4 is referred to as , and is the frequency at which the structure was loaded (the frequency at which Valve 1 was opened). The low frequency contribution is also seen in the radial deformation curve in Fig. B4, given by the upwards trend of the structural oscillations.
The high frequency contribution, is also seen from the FFt plot at around 500 Hz.
This frequency is the value which corresponds to the natural frequency of oscillation of the structure while under pressure 2 . Because the signal processing techniques used in the determination of resulted in frequency measurements with a discrimination of 32.9 Hz, the value of determined from the FFt is only approximate and thus was used to divide the deformation curve into time windows of 1 in length, and a peak value was determined for each window. Using the time difference between peaks, the frequency of each oscillation was determined, and the average frequency was determined for the entire radial deformation curve. The average frequencies of structural oscillation are given for all 2 values in table B1. Note that with the exception of 2 = 54 psi, decreases with increasing pressure 2 .This is in agreement with accepted principles of added mass effect, which states that the frequency of oscillation approaches zero as 2 approaches the critical buckling pressure.

Conclusions
A novel experimental technique was developed to measure the frequency of vibration of an implodable structure under hydrostatic pressure, and was verified by conducting a series of 4 experiments. Major conclusions from this study are as follows: 1. 3D digital image correlation can be used to successfully measure small vibrations in implodable structures excited by a sudden drop in hydrostatic pressure.
2. The "misting" speckling technique is an ideal method for measuring small displacements with 3D DIC.
3. Generally, the frequency of vibration of structures submerged in a pressurized fluid can be shown experimentally to decrease with increasing hydrostatic pressure