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Converging shocks in water and material effects

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Converging shocks in water and material effects

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Content CONVERGING SHOCKS IN WATER AND MATERIAL EFFECTS by Chuanxi Wang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) December 2013 Copyright 2013 Chuanxi Wang To my family ii Acknowledgments First, I wish to thank my advisor, Veronica Eliasson, for her tolerance about my procrastination in many cases and advices throughout the years. I have learned a lot about shock wave physics, experimental instrumentation and simulation tech- niquesunderherguidance. OneimportantthingIlearnedfromheristhatsentence she repeated over so many circumstances, “Do whatever makes you happy”. I would also like to thank my committee members, Dr. Stephan Haas, Dr. Werner D¨ appen and Dr. Thompson Richard from the Department of Physics and Astronomy, and Dr. Larry Redekopp from the Department of Aerospace and Mechanical Engineering, for their suggestions and support. Special thanks to Dr. Lessa Grunenfelder and Dr. Steven Nutt for their generous help on fabricating the carbon fiber samples for my research, and Dr. Bill Henshaw for his help on the Overture package. I am very thankful for all the help and fun brought by my fellow graduate students in the shock wave lab, James Tamashiro, Daniel Gally, Gauri Khanolkar, Orlando Delpino Gonz´ ales, Stylianos Koumlis, Shi Qiu. I am putting their names in an order following the time they joined the lab so that I can recall even years later. It was also my pleasure to work with so many undergraduate and master students, Felipe Figueroa, Martina Troesch, Catherina Ticksay, Sayeed Ahmed, iii Fredrick Krafft. Being part of such a young and dynamic team endows me a lot of great ideas and precious skills. I have met and made many friends since I started graduate school at USC. Although it will be a lengthy list to mention one by one, I would like to thank them all for numerous encouraging conversations and cheerful activities, which make graduate school not always that stressful. Last but not the least, this dissertation is dedicated to my family for their unconditional love and support. I hope my work could convince my parents that it is worth the six years Ph.D. study. Without my wife Yarong, I could not reach this far in my graduate study. Thank you for taking care of me and your love. My son Andrew was born towards my defense which added tremendous joy to my life. I hope in the future he could see this and be proud. iv Table of Contents Dedication ii Acknowledgments iii Abstract xix Chapter 1: Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Shock wave and shock reflection . . . . . . . . . . . . . . . . 2 1.2.2 Shock focusing . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Underwater explosion and structural dynamics . . . . . . . . 6 1.2.4 Shock wave generation . . . . . . . . . . . . . . . . . . . . . 10 1.3 Outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter 2: Theoretical considerations 12 Chapter 3: Experimental setup and methods 16 3.1 Gas gun and shock wave generation . . . . . . . . . . . . . . . . . . 16 3.1.1 Gas gun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.2 Hugoniot relationship and x−t diagram . . . . . . . . . . . 18 3.2 Schlieren optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 v 3.2.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.2 Single-shot photography . . . . . . . . . . . . . . . . . . . . 24 3.2.3 High-speed photography . . . . . . . . . . . . . . . . . . . . 25 3.3 Background oriented schlieren . . . . . . . . . . . . . . . . . . . . . 26 3.3.1 Principles of background oriented schlieren . . . . . . . . . . 27 3.3.2 Optical system . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.3 Image background . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.4 Image correlation and density calculation . . . . . . . . . . . 31 3.3.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . 32 3.3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Strain gauge installation and data acquisition . . . . . . . . . . . . 37 Chapter 4: Shock dynamics and weak coupling 39 4.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1.1 Specimen geometry and assembly . . . . . . . . . . . . . . . 39 4.1.2 Polycarbonate specimen . . . . . . . . . . . . . . . . . . . . 39 4.1.3 Aluminum specimen . . . . . . . . . . . . . . . . . . . . . . 42 4.1.4 PMMA specimen . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2.1 Two-dimensional rigid boundary simulation . . . . . . . . . 50 4.2.2 Two-dimensional fluid-structure interaction simulation . . . 54 4.2.3 Axisymmetric simulation . . . . . . . . . . . . . . . . . . . . 58 4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Chapter 5: Shock dynamics and strong coupling 63 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 vi 5.2.1 Steel specimens . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2.2 Aluminum specimens . . . . . . . . . . . . . . . . . . . . . . 71 5.2.3 Composite specimens . . . . . . . . . . . . . . . . . . . . . . 73 5.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.3.1 Physical model . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.3.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . 80 5.3.3 Material properties . . . . . . . . . . . . . . . . . . . . . . . 81 5.3.4 Numerical considerations . . . . . . . . . . . . . . . . . . . . 83 5.3.5 Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3.6 Parametric study of impact speed . . . . . . . . . . . . . . . 85 5.3.7 Steel specimens . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.3.8 Aluminum specimens . . . . . . . . . . . . . . . . . . . . . . 94 5.3.9 Composite specimens . . . . . . . . . . . . . . . . . . . . . . 95 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Chapter 6: Conclusions 104 Appendix A 108 Appendix B 110 Appendix C 113 Bibliography 115 vii List of Tables 3.1 Material properties. Here, ρ is density, a is speed of sound, C 0 and S are parameters for the linear U s -U p equation. For Delrin C 0 is picked to be the same as the speed of sound and S is set to be 0.. . 18 4.1 Initial conditions in region 1 ahead of the shock wave, and region 2 behind the shock wave. . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Shock wave speed comparison between the aluminum specimen and thetwo-dimensionalfluid-structureinteractionsimulation. Thesim- ulationresultshaveanaveraged3.8%deviationfromtheexperimen- tal values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.1 Material properties of Delrin and Polycarbonate. . . . . . . . . . . . 81 5.2 Material properties of Aluminum 6061-T651. " p in the table stands for the logarithmic plastic strain. . . . . . . . . . . . . . . . . . . . 82 5.3 Material properties and parameters for 1018 low-carbon steel. . . . 82 5.4 Material properties and parameters for water. . . . . . . . . . . ..83 viii 5.5 Materialpropertiesofthefive-harness-satincarbonfiberfabric. The direction 1 corresponds to the warp direction of the fibers. The direction 2 corresponds to the fill direction. The direction 3 corre- sponds to the direction normal to the plane of carbon fibers. . . . . 96 5.6 Russell errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 ix List of Figures 1.1 One dimensional shock discontinuity. . . . . . . . . . . . . . . . . . 3 1.2 Shock reflection types. a) Regular reflection. I is the incoming shock wave, and R is the reflected wave. b) Mach reflection. M is the Mach stem, T is the triple point and S is the slipstream. Image taken from Skews et al. [1]. . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Geometrical shock dynamics. Solid lines are shock front at different time instances. Dashed lines are the separations of the ray tubes..6 1.4 Pressure pulse (Top) and pulsation of the gas products (Bottom) from an UNDEX. Image taken from Hart [2]. . . . . . . . . . . . . . 7 1.5 An example of contact explosion. Resulting in a 12 m×18 m hole in the ship’s hull and severe internal damage. Image from Webster [3]. 8 1.6 Schematics for a sandwich structure. . . . . . . . . . . . . . . . . . 10 2.1 Logarithmic spiral given by Equation (2.1). L is the length of the logarithmic spiral duct, the angle χ determines the overall shape and also the local curvature. θ and r are polar coordinates with the origin at the tip of the duct. . . . . . . . . . . . . . . . . . . . . . . 12 x 3.1 Experimentalsetup. (1)pressurechamber, (2)gunbarrel,(3)veloc- ity sensors, (4) light source, (5) spherical mirror, (6) flat mirror,(7) smallflatmirror,(8)schlieren knifeedge, (9)lenses, (10)high-speed camera,and(11)experimentalspecimeninthetestregion. Thelight beamisshowntoillustratetheopticalpathoftheZ-foldedschlieren system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 An x−t graph of incident and reflected waves. u p , u pb and u bw are thevelocities oftheprojectile, theparticlevelocity attheprojectile- blocker interfaceandtheparticlevelocity attheblocker-water inter- face. U p , U b and U w are the shock wave velocities in the projectile, polycarbonate blocker and water, respectively. U r is the velocity of the reflected rarefaction wave front in the blocker. . . . . . . . . . . 19 3.3 Hugoniotequations onP−u graph. 1)Left-goingshock wave inthe projectile. 2) Right-going shock wave in the blocker. 3) Left-going rarefaction wave head in the blocker. 4) Right-going shock wave in the water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4 Shock velocity in the water as a function of the projectile impact velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.5 Demonstration of the BOS principle and the key quantities. Z a is the distance between the background and sample, and it is 300 mm in our experiments. Z b is the thickness of the sample. y is the displacement of the background observed at the image plane. y ! is the projected displacement at the background plane. . . . . . . . . 28 xi 3.6 Example of a background plane for BOS. Random white dots on a black background are printed onto a transparency. . . . . . . . . . . 31 3.7 Schematic drawing of the experimental test sample assembly for BOS visualization. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.8 The displacement vectors plots corresponding to the raw data. a) t=38.45 µs,b) t=46.14 µs,c) t=53.83 µs,d) t=61.52 µs and e) t=69.21 µs.............................. 34 3.9 ThedensityplotscalculatedfromthecorrespondingsubplotsinFig- ure 3.8. Plotted along the center line y=25. a) t=38.45 µs,b) t=46.14 µs,c) t=53.83 µs,d) t=61.52 µs and e) t=69.21 µs.. 35 3.10 Thepressureplots. a)t=38.45µs,b)t=46.14µs,c)t=53.83µs, d) t=61.52 µs and e) t=69.21 µs.. . . . . . . . . . . . . . . . . . 36 4.1 Specimengeometryandassembly. (a)Experimentalcorewithdimen- sionsinmillimeter. Thecorethickness is25.4mmforthealuminum and polycarbonate samples. For the PMMA samples, this thickness is 6.3 mm. (b) Specimen assembly with (1) window (only one is depicted for clarity), (2) core, (3) water-filled section, and (4) blocker. 40 4.2 A series of high-speed schlieren images using a polycarbonate spec- imen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3 Three single-shot schlieren experiments using a polycarbonate spec- imen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 A series of high-speed schlieren images using an aluminum specimen. 44 xii 4.5 A single-shot schlieren experiments using an aluminum specimen. . 45 4.6 Experiment of PMMA sample using BOS technique. (a) The initial background with no displacement. The white dashed line in (a) is the center line. The white arrow in (b) points at the location of the shock front. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.7 The displacement of the background at different locations. The spatial grid points are predetermined by the interrogation window size of the image correlation analysis. The size and direction of the vectors indicate the relative magnitude and orientation of the displacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.8 The reconstructed pressure measurement along the center line. . . . 48 4.9 Grid sensitivity study by investigating peak pressures at different initial grid sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.10 Grid setup for the two-dimensional rigid boundary simulation. (a) An overview of the composite grid used for two-dimensional rigid boundarysimulations. (b) A blowup of the overlapping griddetails. (1) background grid; (2) inlet grid; (3) logarithmic spiral boundary grid. Interpolation points are marked as dots. . . . . . . . . . . . . 51 4.11 Initial pressure distribution at time t=0. . . . . . . . . . . . . . . 52 4.12 The shape of the shock front in the two-dimensional rigid boundary simulation. (a) A snapshot of pressure distribution at time 40.5 µs. (b) The shock front is normal to the wall as a consequence of apply- ing slip-wall condition at the boundary. . . . . . . . . . . . . . . . . 53 xiii 4.13 Pressure distributions at different times and Mach number of the shock front at the boundary. The arrow points in the direction of the wave propagation. . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.14 Gridsetup forthetwo-dimensional FSIsimulation. (a)Anoverview of the composite grid for the FSI simulation. The solid colored blocks are finer grids at the interface. (b) A blowup of the marked region of the interface from (a). (1) solid domain; (2) boundary grid for the solid domain; (3) boundary grid for the fluid domain; (4) fluid domain. Interpolation points are marked as black dots. The boundary condition between grid (1) and grid (2) is given by a traction boundary condition. . . . . . . . . . . . . . . . . . . . . . . 55 4.15 Initial configuration for the FSI simulation. Region 1 and 2 repre- sents the low and high pressure regions in the fluid domain. The solid part is plotted using the x-component of the stress tensor, σ ij , which is initially zero for all the grid points. . . . . . . . . . . . . . 56 4.16 Theshape oftheshock front inthetwo-dimensional FSIsimulation. (a) A snapshot of the FSI simulation at time t=43.5 µs. (b) The shock front is not perpendicular to the interface. Instead, the shock angleα is88.2 ◦ forthis particular timeinstance. Inthesolidregion, thedivergenceofdisplacementfieldisexpressedbythecontourlines. In the fluid region, pressure change is described by the contour lines. 57 4.17 Pressureplotsalongthefluid-structureboundary. (a)Pressureplots along the interface at times t=19.6, 37.7, 55.8 µs. (b) Pressure plots along the interface at times t=59.6, 62.3, 64.3 µs. . . . . . . 58 xiv 4.18 Composite grid for axisymmetric simulation. . . . . . . . . . . . . . 59 4.19 Comparisonbetweentwo-dimensionalandaxisymmetricrigidbound- ary simulations of the time evolution of the pressure profile along the rigid boundary. Time is given in µs. (a) The incoming shock wave. (b) The reflected shock wave. All the time instances are consistent with the ones in Figure 4.17. . . . . . . . . . . . . . . . . 60 5.1 The specimen assembly for the samples in 5.8 mm thickness. a) Blocker; b) test sample; c) polycarbonate window. The location of the strain gauges are also marked. . . . . . . . . . . . . . . . . . . . 65 5.2 A series of high-speed schlieren images using an 5.8 mm thick steel specimen. The location of strain gauges marked in (a) is used as a convention throughout all the experiments. . . . . . . . . . . . . . . 66 5.3 Strain measurement at two locations on the outer surface of the 5.8 mm thick steel specimen. The dashed lines corresponds to the time instances when Figure 5.2(e)–(g) were taken. . . . . . . . . . . 67 5.4 A series of high-speed schlieren images using an 1.3 mm thick steel specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.5 Strain measurement at two locations on the outer surface of the 1.3mmthicksteel specimen. Threeregionsofinterest areseparated by dashed lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.6 Strain measurement at two locations on the outer surface of the 5.8 mm thick aluminum specimen. . . . . . . . . . . . . . . . . . . . 71 xv 5.7 Strain measurement at two locations on the outer surface of the 1.3 mm thick aluminum specimen. . . . . . . . . . . . . . . . . . . . 72 5.8 Schematic of the layup for composite fabrications. 1) Aluminum base plate. 2) Teflon release film. 3) Breathable edge dam. 4) Prepreg plies. 5) Porous release film. 6) Breather cloth. . . . . . . . 73 5.9 A series of high-speed schlieren images using an carbon fiber com- posite specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.10 Motion of the interface near the tip of the composite sample. . ... 76 5.11 Strain measurement at two locations on the outer surface of the carbon fiber composite specimen. Dashed lines correspond frame (c), (f) and (h) in Figure 5.9. . . . . . . . . . . . . . . . . . . . . . 77 5.12 Delaminationatthetipregionofthecompositesample. a)Overview. b) Scanning Electronic Microscope (SEM) image of the dashed line region in a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.13 Assembly of the Finite Element (FE) model. a) Projectile; b) Blocker; c) Domainfor fluid motion; d) Test sample; e) Rigid plane. The compass in the top right corner shows the global coordinate system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.14 Meshing the FE model. (a) Mesh configuration for the solid parts. (b) Mesh distribution of the Eulerian meshes. . . . . . . . . . . . . 84 5.15 Comparison of the strain result from a single element and the aver- aged one among six elements. . . . . . . . . . . . . . . . . . . . . . 86 xvi 5.16 Parametricstudyofimpactspeed. (a)StrainmeasurementsatGauge 1 and (b) pressure measurements at the tip region of the water domain for three impact speeds. . . . . . . . . . . . . . . . . . . . . 86 5.17 Timehistoryoftruestrainfromboththeexperiment andsimulation for the 5.8 mm thick steel sample. . . . . . . . . . . . . . . . . . . . 87 5.18 Time history of pressure measured at locations 15 mm apart along the centerline. 0 mm is the location of the focal point. . . . . . . . 88 5.19 Effect of the fluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.20 Strain rate dependence for the the 5.8 mm thick steel sample. Solid lines are for the case with the strain rate term. Dashed lines are for the case with no such term. . . . . . . . . . . . . . . . . . . . . . . 90 5.21 Timehistoryoftruestrainfromboththeexperiment andsimulation for the 1.3 mm thick steel sample. . . . . . . . . . . . . . . . . . . . 91 5.22 Origin of the precursor wave for the 1.3 mm thick steel sample. a) Flexural waves in the solid. b) Precursor pressure waves. . . . . . . 92 5.23 Deformationpattern on the 1.3 mm thick steel specimen. a) Exper- imental result. b) Numerical result. . . . . . . . . . . . . . . . . . . 92 5.24 Locations of maximum displacement. Data points are measured at 20 µs time interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.25 Plastic energy dissipation for the 1.3 mm thick and 5.8 mm thick steel samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 xvii 5.26 Timehistoryoftruestrainfromboththeexperiment andsimulation for the 5.8 mm thick aluminum sample. . . . . . . . . . . . . . . . . 94 5.27 Timehistoryoftruestrainfromboththeexperiment andsimulation for the 1.3 mm thick aluminum sample. . . . . . . . . . . . . . . . . 95 5.28 Global plastic energy dissipation for the 1.3 mm thick and 5.8 mm thick aluminum samples. . . . . . . . . . . . . . . . . . . . . . . . . 96 5.29 Material coordination for the outermost lamina. . . . . . . . . . . . 97 5.30 Timehistoryoftruestrainfromboththeexperiment andsimulation for the carbon fiber composite sample. . . . . . . . . . . . . . . . . 97 5.31 Deformationplotsatthetipofthecarbonfibercompositespecimen. a) Results from the numerical simulations showing deformation in the composite sample. b) Results from the numerical simulations showing strain levels in the composite sample. . . . . . . . . . . . . 99 5.32 Results from the numerical simulations showing pressure wave pat- terns. a) 5.8 mm steel sample; b)carbon fiber composite sample. The grey areas in the plots are regions having pressure higher than the color contour’s upper limit. . . . . . . . . . . . . . . . . . . . . 100 A.1 Schematic for evaluating the shock wave speed in water. . . . . . . 108 B.1 Two-part alignment system. a) Female part. b) Male part and the sample.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 B.2 Photography of the alignment assembly before testing. (1) Gun barrel. (2) Male part. (3) Female part. (4) Dowel pin. (5) Sample. . 112 xviii Abstract During events of underwater explosions, the resulting shock wavesposesextreme mechanical loadings to the nearby naval structures. Therefore this dissertation studiestheshockwavedynamicsofwaterinsideasolidstructureandthedynamical response of such a structure. The goal is to understand the effect of underwater shock wave focusing on materials possessing various mechanical properties with an emphasis on lightweight materials. This research has a direct impactonthe material selection and hydrodynamic considerations in naval architecture. Two steps are taken towards the research goal. First, a special geometry of the confinement structure called logarithmic spiral is chosen. This type of shape will help to focus the shock wave, which will yield the maximum energy at thefocal region. Numerical simulations areconducted toconfirm thederived characteristics of the geometry. When including weak material coupling with the liquid, both experiments and simulations demonstrate that the shock dynamics in water is sensitive to the coupling. Secondly, significant coupling effect is introduced by reducing the thickness of thesolidstructure. Experimentalandnumericalinvestigationsarebothcarriedout to shed light onto the details of the fluid-structure coupling. The results revealed that the thickness ofthe materialhas the most significant impact onboththefluid xix dynamics and the deformation mechanisms of the structure. Lightweight carbon fiber composite structures are also studied under the same framework. Alongtheway,thisdissertationalsoproposesanddesignsexperimentalmethod- ologies to enable the study of highly dynamic underwater events. xx Chapter 1 Introduction 1.1 Motivation Underwater explosions (UNDEX) have been a main factor of concern for engi- neers when designing naval structures [4, 5]. A significant portion oftheenergy generated by UNDEX is carried by the resulting shock wave in water [6]. When this energetic wave interacts with marine structures, it could lead to deformation, or even structural failure. Besides the damage caused to the marine structures, the transmitted waves either through the structures or inside the structures could be harmful to crew and passengers on board. Thus, knowledge of both shock wave dynamics in water, and the interaction between the shock wave in water and the naval structure will facilitate the design for marine structuressubjectedto UNDEX. In particular, during UNDEX, when the shock wave propagates into structures of convergent geometries, such as where propeller shaft merging from the ship’s hullorthebowthruster,focusingoftheshockwavecanoccur[7]. Thus, knowledge of the strength of a converging shock wave and its interaction with surrounding structuresinanUNDEXscenarioiscrucialforimprovingthesurvivabilityofstruc- tures during the explosion events. Another aspect of this problem that is worth exploring is the application of lightweight composite materials. The attraction of composites is the high strength to mass ratio, corrosion resistance and reduction 1 of fuel costs etc. Therefore the motivation of this dissertation is to study the per- formance of modern materials under the condition of UNDEX events, especially converging shock waves. By visualizing the dynamics of the shock wave in water while it is traveling inside a convergent structure and at the same time monitor- ing the dynamic response of the surrounding solids, this dissertation aims to shed light on such a complex coupling process and can contribute to the future naval structure design. 1.2 Background Beforediscussingthedetailsofthepresentinvestigation,areviewoftheshockwave phenomenon, shock focusing and shock loading on solid structures, which are all in relevance to the research investigations in later chapters, will be presented. 1.2.1 Shock wave and shock reflection A shock wave is a nonlinear wave propagating with supersonic speed compared to the sound speed of the medium it propagates in. It is a strong disturbance, which leads to discontinuity in thermodynamic state variables such as density, pressure and temperature. For example, the governing equations of a shock wave in fluids are the continuity equation, the momentum equation and the energy equation [8]. The one-dimensional integral form for the three equations are d dt ! x 1 x 2 ρdx+[ρu] x 1 x 2 =0, (1.1) 2 d dt ! x 1 x 2 ρudx+[ρu 2 +p] x 1 x 2 =0, (1.2) d dt ! x 1 x 2 ( 1 2 ρu 2 +ρe)dx+[( 1 2 ρu 2 +p)u+pu] x 1 x 2 =0. (1.3) The convention for subscription 1 and 2 is depicted in Figure 1.1. Quantities ρ, u, p and e are the density, particle velocity, pressure and internal energy ofthe fluid. In Equations 1.2–1.3, the velocity gradients and heat addition terms are not included. The equations can be closed by the equation of state, which changes its form for different substances. x Figure 1.1: One dimensional shock discontinuity. When the shock wave interacts with other shocks or solid boundaries, shock wave reflections will occur. Because of the nonlinear nature of shock waves, the reflection angles are not always equal to the incident angles. There are two basic categories of shock reflection for a traveling shock of moderate strength, regular reflection in Figure 1.2(a) and Mach reflection (a type of irregular reflections) in Figure 1.2(b). A regular reflection resulting from a large incident angle will transition into a Mach reflection when decreasing the incident angle [1]. Regular 3 reflection is well described by the planar shock relationships, see Equation 1.1– 1.3. However a three-shock theory is derived [9] to describe the flow quantities near the triple point (T) for a Mach reflection, see Figure 1.2(b). The triple point is sometimes also referred as a ”shock-shock”, because there exists a jump in the orientation of the shock front at this point. For weak shock waves and small incident angles, the three-shock theory may not have a standard solution or even no solution will exist. Therefore new reflection types, such as von Neumann reflection, belonging to irregular reflection category emerge. (a) (b) Figure 1.2: Shock reflection types. a) Regular reflection. I is the incoming shock wave, and R is the reflected wave. b) Mach reflection. M is the Mach stem, T is the triple point and S is the slipstream. Image taken from Skews et al. [1]. The shock wave propagation in water attracts research interest in many areas. Inthermodynamics,dynamiccompressionofwaterbyusingshockwaveexperiment allowstostudythekineticsoffastphasetransitions[10]. Inengineeringdisciplines, a shock wave can be used as a method to test the dynamic response and failure modesofmaterials,suchasunderwaterexplosiontests, orinexptracorporealshock wave lithotripsy [11], where shock waves are used to destroy kidney stones. The 4 most commonly used equation of states for water is the Tammann-Tait type [12]. The parameters for the Tait equation of states can be readily determined by using the shock Hugoniot curves of water [13]. 1.2.2 Shock focusing Shock focusing typically occurs because of the reflection from complex boundaries surrounding the shock focusing media or inhomogeneities of the flow field [14]. For example, a planar shock front becomes curved as it propagates inside a shock wave lithotripter, a reflector in a particular shape used to generateafocused shock wave to destroy kidney stones [15]. Shock focusing is a fundamental tool to generate extreme conditions at the region where the shock implodes. It is utilized in both basic research and various applications such as in the study of shock stability, implosion fusion reactors and medical treatment for kidney stones [16, 17, 18]. Many analytical, experimental and numerical studies have been conducted on shock focusing since the 1940’s. Most of the work focused on the generation and stability of cylindrical and spherical converging shock waves in gases[19,20,21,22,23]. Inrecent years, theuseofshockwave focusinginbiomed- ical applications have increased. Examples are shock wave lithotripsy [15, 24, 25] and drug delivery by shock waves [26, 27]. The working medium for bothofthese applications is water or a water solution. In general, shock focusing in water gen- erates much higher pressures than shock focusing in air, due to the increase in density and speed of sound in water as compared to air [12, 28]. Thedifficultyindescribingshockfocusinginsideawater-filledconvergentstruc- ture analytically is that it combines two different effects. On one hand,theshock wave interacts with the complex geometry of the solid boundary. On the other hand, the flow behind the shock front may have a nonlinear effect on the shock 5 wave [8]. For moderate to strong shock waves, Whitham’s Geometrical Shock Dynamics (GSD), an analytical approximation, yields accurate results [22, 29]. In thisapproach,analogiestogeometricopticshavebeenmade. Conceptualraytubes orthogonal to the shock wave front are constructed, see Figure 1.3. The sections of the shock wave transmit locally inside the ray tubes. The instantaneous shock Mach number is directly related to the local cross section of the ray tube. Shock front Shock front Figure 1.3: Geometrical shock dynamics. Solid lines are shock front at different time instances. Dashed lines are the separations of the ray tubes. 1.2.3 Underwater explosion and structural dynamics When UNDEX happens, the explosive material undergoes a rapid chemical reac- tion on the order of nanoseconds. The product of this chemical reaction is gases at high pressure and temperature. The temperature in the resultant gases is on the order of 3000 ◦ C and the pressure is around 5 GPa [6]. This detonation pro- cess is carried by a detonation wave traveling inside the explosive media. Once the detonation wave reaches the boundary between the explosive and water, the surrounding water willbecompressed andastrong shock disturbance isgenerated, propagating away from the boundary in a spherical fashion. The first dominant 6 0 0 Figure 1.4: Pressure pulse (Top) and pulsation of the gas products (Bottom) from an UNDEX. Image taken from Hart [2]. pressure peaknear time0inFigure1.4corresponds tothis initialrelease ofenergy. The high pressure gaseous bubble formed from the explosion will continue to grow in size until the boundary decelerates to zero velocity. At this point,thepressure insidethebubbleislower thanthehydrodynamicpressureoutside, thusthebubble starts to collapse inward. The bubble will then reach to a minimum radiusand a successive expansion will start. This oscillation dissipates the residual energy inside the gas cavity. The shock waves generated from the initial detonation and the oscillations of the gas bubble are the main threats for any nearby solid marine structures. Although full scale ship shock testing are typically mandatory for the lead ship of a new class of ships [30], the logistic effort and expenses are tremendous. Instead, numerical consideration about the survivability of the whole ship [31] or ship-like structures [32] were conducted. In general, damage resulting from the UNDEX to ships are expected to be occurring on the bottom or side panels, see 7 Figure 1.5: An example of contact explosion. Resulting in a 12 m×18 m hole in the ship’s hull and severe internal damage. Image from Webster [3]. Figure 1.5. Thus the study of the whole ship can be reasonably reduced to a simpler problem which considers an air-backed plate under the shock loading from water. Topics such as fluid-plate interaction, reloading effect on the plate, elas- tic and plastic deformation and contact explosion are all extensively studied [33]. For non-contact explosions the thin plates’ permanent deflection is predictable. To generate a shock wave of equivalent strength as that from an UNDEX event, experimental approaches utilize either a direct charge explosion [34]orgasgun impact with careful scaling [35]. In the light of advanced test facilities and analyt- icalpredictions available, theperformanceofmonolithicsteelplatesareengineered to have a significant improvement in the yield strength, the uniform ductility and fracture strain in the context of underwater shock loading [36]. There are two main distinctions between shock loading on a planar surface and shock loading on a convergent structure. First, when compared toanormalshock impact, a shock focusing event can generate a higher transient pressure in local regions of the fluid, which consequently yields high stress inside the nearby solid structure[37,38]. Second,whentheshockwavetravelsintoaconvergent structure, 8 the interaction time between the shock wave and the structure is in general much longer than that of aninstantaneous loading, which suggests that more energy can be transferred into the solid structure. These two factors indicate that underwater shock focusing in convergent structures will have complex dynamics and failure modes which are different from the panels subjected to planar shock loadings. There is an increasing trend in using advanced lightweight composite materials on the ship’s hull due to the improved maneuverability resulting from the reduced mass [39]. Thus investigations on the performance of composite structures, usu- ally plates, are carried out [40, 41]. Two types of composite structures are studied the most, laminates and sandwich structures. They are essentially two different kinds of architectures to combine materials. The laminated composites are typi- cally layered resin materials, such as epoxy or polyester. Each layer of the epoxy is reinforced by fiber fabrics such asglass fiber, carbonfiber etc. After shock impacts the composite laminates, bending occurs on the composite which leadstocrack formation at the rear surface of the laminates and fiber fractures [42]. A major difference between sandwich structures and monolithic panels is that sandwich structures have a core material made of foam or other topological structural ele- ments [43]. Sandwich structures are composed of two face sheets and a core made from a plastic compressible solid, see Figure 1.6. When considering the interaction between UNDEX and sandwich structure materials, there are threestages: • Fluid-structure interaction phase. In this phase, momentum transfers from fluid to structure. At the same time, fluid flow gets changed by this interac- tion, which might result in cavitation on site. • Core compression phase. Energy deposits onto the compressible core of the sandwich structure. 9 • Beam deforming phase. Excessive energy in the structure converts to defor- mation of the beam, which might result in structure failure. Explosion Core Face sheets Figure 1.6: Schematics for a sandwich structure. 1.2.4 Shock wave generation To conduct research on shock focusing in water, a controllable way of generating the shock wave is always preferable. There are several different ways in generating shock waves in water [44], including: • Explosive. This is the most direct way, and requires a high degree of con- trol and safety protocols. Therefore, this technique is rarely used at higher education institutions but often in national research laboratories. • Spark gap. Spark gap is often used by medical research in treating kidney stones, because of the temporal sharpness of the pressure pulse. However, this feature defeats our main goal is to simulate underwater explosion, in which case pressure pulse decay is exponential. • Impact water by a striker. A solid striker hitting a water column showed results in an exponentially decaying pressure pulse [45]. This is in good 10 agreement with the scenario of interest in this dissertation. This method has also been extensively used in other research investigations, such as liquid drop impact on solid [46], and supersonic liquid jet formation [47]. 1.3 Outline The chapters of the dissertation are organized in the following order. Chapter 2 introduces the idea of a “worst-case scenario” for a shock focusing event. Based on the theoretical derivations from Archer and Milton [48], a loga- rithmic spiral geometry for shock implosions in air is extended to shock implosions in water. Chapter 3 describes in detail the experimental facilities and techniques. To visualizetheshockwaveinwater, bothqualitativeschlieren andquantitativeback- ground oriented schlieren methods are applied together with high-speed photogra- phy. The installation of strain gauges and data acquisition will also be explained. Chapter 4 considers the case of weak coupling between the shock wave inwater andthesurroundingstructures. Thesampleconfigurationandexperimentalresults will be presented first, followed by numerical investigations. In Chapter 5 the case of strong coupling between the shock wave in water and the surrounding structure will be presented. 11 Chapter 2 Theoretical considerations Shock focusing in a logarithmic spiral is being investigated due to its ability to minimize reflections offthesurrounding boundary. Alogarithmicspiralisdepicted in Figure 2.1. shock wave L θ r χ Figure 2.1: Logarithmic spiral given by Equation (2.1). L is the length of the logarithmic spiral duct, the angle χ determines the overall shape and also the local curvature. θ and r are polar coordinates with the origin at the tip of the duct. The derivation of the logarithmic spiral is based on Whitham’s ray-shock the- ory, or so-called Geometrical Shock Dynamics (GSD) theory [8]. The form of the curve can be expressed as following r = L cosχ e χ−θ tanχ , (2.1) where the variables r, L, χ and θ are shown in Figure 2.1. Simulations and experiments have both shown that a two-dimensional logarithmic spiral geometry produces shock focusing in air without observable reflections off the surrounding boundary [49, 50]. 12 Given the properties of a logarithmic spiral duct in air, an extension ofthe theory follows in order to apply the concept to shock propagation and focusing in water. To account for water instead of air, the GSD theory has to be modified by changing the equation of state (EOS) of the shock focusing medium. Here, a stiffened EOS, also referred to as Tammann-Tait EOS [12] in the literatures, has been chosen to account for water as the shock medium. The stiffened EOS is given by e = p+γ ∞ p ∞ ρ(γ ∞ −1) . (2.2) The variable e is the internal energy, p is the pressure, ρ is the density; and the constants γ ∞ =7.415,p ∞ =2.962× 10 8 Pa are chosen to match the speed of sound of water in laboratory experiments. This EOS has been used in previous works related to shock waves in water, such as underwater explosions [51], and the collapse of shock-induced bubbles in water [52]. Rewriting theRankine-HugoniotjumpconditionsusingthestiffenedEOSgives the following equations u 2 =a 1 2(M 2 −1) (γ ∞ +1)M , (2.3) ρ 2 = ρ 1 (γ ∞ +1)M 2 (γ ∞ −1)M 2 +2 , (2.4) p 2 = ρ 1 a 2 1 2(M 2 −1) γ ∞ +1 +p 1 . (2.5) Here, M, is the shock Mach number, u, is the particle velocity and a is the speed of sound. Quantities with a subscript 1 denote variables in the undisturbed region ahead of the shock wave and quantities with a subscript 2 denotes variables in the region behind the shock. This convention will remain the same through this 13 chapter. The Rankine-Hugoniot conditions, Equations (2.3)–(2.5),cannowbe substituted into the following equation for the C + characteristic, dp 2 dx +a 2 ρ 2 du 2 dx + ρ 2 a 2 2 u 2 u 2 +a 2 1 A dA dx =0, (2.6) where A = A(x) denotes the cross-sectional area of a ray tube where the C + characteristic is being considered. The C + characteristic belongs to the family of characteristics having positive slope on the time-position graph, when considering flow properties behind a shock wave moving in a quasi one-dimensionaltube. By substituting Equations (2.3)–(2.5) into Equation (2.6), the result is M M 2 −1 λ(M) dM dx + 1 A dA dx =0, (2.7) where λ(M)=(1+ 2 γ ∞ +1 1−µ 2 µ )(1+2µ+ 1 M ), (2.8) µ 2 = (γ ∞ −1)M 2 +2 2γ ∞ M 2 −(γ ∞ −1) . (2.9) Equation (2.7) can be rewritten in the form A MA ! =− M 2 −1 λ(M)M 2 . (2.10) 14 Now, by using Whitham’s GSD theory, purely from geometric considerations the characteristic angle is given by tanχ = " − A MA ! #1 2 . (2.11) Finally, Equation (2.11) can be rewritten in terms of the shock Mach number M, and γ ∞ , the parameter from the stiffened EOS, tanχ = " M 2 −1 λ(M)M 2 #1 2 . (2.12) where λ(M) is given by Equation (2.8). With χ known, we are able to construct the shape of a logarithmic spiral in the case of shock wave focusing in water. As seen in Equation (2.12), the characteristic angle χ depends only on the Mach number of the shock wave and γ ∞ from the equation of state. However, the Mach number depends largely on the nature of the problem understudy. In this paper, we focus on the impact of shock waves from UNDEX. The value of the incident shock wave is determined considering a 135 kilogram Trinitrotoluene (TNT) charge. Thepeakpressure behindasphericalshockwave isdecreasing with time. Thus, the shock front will experience a deceleration, and the Mach number of the shock wave will decay from M=1.2 at the close proximity of the charge, to M = 1, at a distance far away [6]. Since we are considering the structures close to where UNDEX happens, we chose an initial Mach number of M=1.1 for all the simulations. This Mach number represents a shock wave 3 meters away from the 135 kilogram TNT charge. 15 Chapter 3 Experimental setup and methods In the following sections the method for generating shock waves in water, visual- ization methods and strain measurement procedures are discussed. 3.1 Gas gun and shock wave generation 3.1.1 Gas gun Alltheexperimentalstudiesinthisdissertationarefulfilledbyanimpacttechnique where a projectile from a gas gun impacts onto the experimental specimens. A single-stage gas gun is installed in Dr. Eliasson’s research lab, see Figure 3.1. The gun barrel is a 2.1 m long steel tube with 50.8 mm inner diameter. The gun barrel is connected to a pressure chamber filled with compressed air. The projectile is a 75 mm long cylinder with 50 mm diameter made out of PTFE- filled Delrin. During the experiments, the chamber pressure of the gas gun is measured by a digital pressure gauge (Druck DPI 104, absolute type, 6.9× 10 5 Pa (100 psi)). The velocity of the projectile is measured by two optical sensors (Avago Technology HFBR-1505CZ), which are placed close to the exit of the gun barrel. The optical sensor receives light from a light emitting diode (LED). When the moving projectile blocks the light from the LED, a signal will be generated. By this means, two optical sensors placed a known distance apart will give the velocityoftheprojectile. Foralltheexperiments, thepressurechamberisoperated 16 1 2 3 4 5 5 6 6 7 8 9 10 11 Figure 3.1: Experimental setup. (1) pressure chamber, (2) gun barrel, (3) velocity sensors, (4) light source, (5) spherical mirror, (6) flat mirror, (7) small flat mirror, (8) schlieren knife edge, (9) lenses, (10) high-speed camera, and (11) experimental specimen in the test region. The light beam is shown to illustrate the optical path of the Z-folded schlieren system. at 2.75×10 5 Pa (40 psi), which corresponds to a projectile velocity of 48 ±1m/s at the barrel exit. 17 3.1.2 Hugoniot relationship and x−t diagram Before performing the experiments, it is necessary to understandhowtheshock wave propagates into the sample assembly and how strong the shock wave will be given a projectile impact speed. This section will elucidate the processofshock wave propagation and discuss a systematic way of determining the resulting shock strength in water. During the laboratory experiments, a moving projectile will strike normally onto one surface of a polycarbonate blocker, resulting in a weak shock wave trans- mittedthroughtheblocker andfinallyintothewater, seethebottomofFigure3.2. Adetailedx−tschematicoftheincidentandreflectedwavesisshowninFigure3.2. For the following discussion we will use subscript p for the Delrin projectile, b for the blocker, w for water, pb for the projectile-blocker interface and bw for the blocker-water interface. The blocker made from polycarbonate is for the purpose of increasing the efficiency of shock wave transmission and sealing water. The projectile is moving from left to right in Figure 3.2. At time t 1 ,the projectile impacts onto the polycarbonate blocker which is initially is at rest. The impact generates a left-going shock wave in the projectile and a right-going shock wave in the blocker. Physical properties [53, 54] related to wave transmission and reflection are listed in Table 3.1. Delrin Polycarbonate Water ρ [kg/m 3 ] 1420 1190 1000 a [m/s] 1531 2270 1482 C 0 [m/s] 1531 2330 1490 S 0 1.57 1.92 Table 3.1: Material properties. Here, ρ is density, a is speed of sound, C 0 and S are parameters for the linear U s -U p equation. For Delrin C 0 is picked to be the same as the speed of sound and S is set to be 0. 18 Time Distance up u pb Uw u bw Up Ur U b Solid wall Water Water Projectile Projectile Blocker Blocker t 2 t 1 pb bw Figure 3.2: An x−t graph of incident and reflected waves. u p , u pb and u bw are the velocities of the projectile, the particle velocity at the projectile-blocker interface andtheparticlevelocityattheblocker-waterinterface. U p ,U b andU w aretheshock wave velocities in the projectile, polycarbonate blocker and water, respectively. U r is the velocity of the reflected rarefaction wave front in the blocker. The shock wave in the projectile propagates with a lower velocity than that in the blocker, which is indicated by a steeper slope on the x−t graph. Once the shock wave in the blocker is formed, it will propagate to the right until reaching the interface between the block and the water domain at time t 2 . The behavior of 19 the shock wave at the interface depends on the impedance mismatchbetweenthe media on both sides. The impedance of polycarbonate (subscript pc)andwater are given by Z pc =a pc ρ pc =2.7×10 6 kg/m 2 s, (3.1) Z w =a w ρ w =1.5×10 6 kg/m 2 s. (3.2) Because the impedance in the polycarbonate blocker is larger than that of the water, Z pc >Z w , a shock wave will be transmitted into the water. At the same time, a rarefaction wave will be reflected back because the blocker-water interface moves to the right faster than the projectile-blocker interface does, resulting in an expansion of the block. The change in curvature of the shock wave path on x−t graph indicates that in general the shock wave experiences an acceleration when moving toward the tip of a converging structure. To quantitatively determine the speed of the resulting shock wave in the water, pressure and particle velocity, P − u, Hugoniot equations are utilized [55], see Figure3.3. Attheprojectile-blockerinterface,thepressureandtheparticlevelocity u 1 on each side have to match respectively to maintain a contact interface. The velocity of the left-going shock wave in the projectile is described by U p =C p +S p (u 1 −u p ), (3.3) in which, U p and u p are the shock velocity and the projectile velocity; C p and S p are C 0 and S in Table 3.1 for the projectile material. Because C p and S p are 20 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 0.05 0.1 0.15 0.2 0.25 u (km/s) P (GPa) 1 2 3 4 up uw pw Figure 3.3: Hugoniot equations on P−u graph. 1) Left-going shock wave in the projectile. 2) Right-going shock wave in the blocker. 3) Left-going rarefaction wave head in the blocker. 4) Right-going shock wave in the water. not available for PTEF-filled Delrin, and the experiments are performed in a low- speed regime, U p is close to the sound speed, a p .Thus U p = a p is a reasonable approximation in our experiments. The hydrodynamic pressure of the projectile material behind the shock wave p p is given by p p = ρ p a p (u 1 −u p ), (3.4) For the right-going shock wave in blocker, the hydrodynamic pressure, p b , is p b = ρ b u 1 (C b +S b u 1 ). (3.5) 21 Since the hydrodynamic pressures are the same at the contact interface, p p = p b , Equations (3.4) and (3.5) can be solved with only two unknown variables. The same technique can be applied to the blocker-water interface to get the particle velocity u 2 in the water. Once u 2 is obtained, the following equation U w =C w +S w u 2 , (3.6) can be used to determine the shock velocity in water. This procedure enables a systematic way to relate the impact velocity of the projectile and the resulting shock velocity in the water, as plotted in Figure 3.4. 0 50 100 150 200 250 300 1450 1500 1550 1600 1650 1700 1750 1800 u p (m/s) U w (m/s) Figure 3.4: Shock velocity in the water as a function of the projectile impact velocity. 22 3.2 Schlieren optics The schlieren technique is an optical method to visualize the density gradient in fluid flows. It is a powerful tool to capture even a slight change in index of refrac- tion in the fluid, which ultimately is related to the density change. Although only qualitative information about density can be obtained from such a method, basic flow characteristics such as the wave speed or the wavelength are readily measur- able. In Section 3.3, a novel method based on schlieren optics will be explored to quantitatively determine thedensity or pressure ina flow field, especially for shock waves in water. 3.2.1 Principles A Z-folded schlieren optical system is shown in Figure 3.1. The light source (4) is approximately a point source. By shedding light on a concave spherical reflective mirror (5) (in some cases a converging lens), a collimated light beam is generated. The parallel light beam is going through a transparent test region (11) that may havevariationsoftheindexofrefraction,inourcaseduetocompressibleflows. The light beam deflection according to the spacial gradient of the index of refraction is " = L n 0 ∂n ∂s . (3.7) In Equation (3.7), " is the deflection angle; n is the index of refraction as a function of spacial coordinates; s is the spacial coordinate along the direction of deflection; L is the extent of change in distance along the optical axis; n 0 is the index of refraction of the undisturbed media. These deflections and the main light beam will be focused by another concave mirror (5) to where the schlieren knife edge (8) is placed. The spacial location of the focused deflection light is 23 different from that of the main parallel light beam on the focal plane. By partially blocking the focused deflection light using the knife edge, the regionswherethe lightoriginallygotdeflectedwillappeardarkonthecamera. Inthisway, a contrast is created between the regions where there are changes in the index of refraction andtheregions where there isno change. Since thedensity is linearly proportional totheindex ofrefractioninatransparent media, thesamecontrast willbedirectly related to the qualitative density distributions. By controlling the shape and the location of the schlieren knife edge, the contrast type and sensitivity of the image can be fine-tuned [56]. 3.2.2 Single-shot photography High-resolution single-shot schlieren photographsaretaken with a Nikon D90SLR camera withaAF-SNikkor 18-105mmzoomlens using an18nssparklightsource (Nanolite,High-SpeedPhoto-Systeme). Theadvantageofthistechniqueisthehigh resolution, but only one photograph per test can be obtained. In the single-shot experimentspresentedinthisdissertation,thecameraisoperatingat100mmfocal length with 2848×4288 pixel resolution, which corresponds to a 163 mm×245 mm regionof the sample being visualized. During the experiments, thelaboratoryis in a dark room setting and the camera shutter will be open for two seconds waiting for a flash from the spark light source. When the projectile breaks a laser beam just ahead of the experimental specimen assembly, a trigger signal is sent out to a pulse/delay generator unit (BNC Model 575). After an appropriate time delay, thedelayunit sends out asignaltotriggerthesparksource sothatthephotograph can be recorded. 24 3.2.3 High-speed photography A Phantom V711 camera, equipped with an AI-S Nikkor 50mm f/1.4 lens, is used for obtaining high-speed schlieren photos. All high-speed photographs have been processed in MATLAB using the Control Point Selection Tool in the Imaging Processing Toolbox to remove any optical distortions. Before each experiment, a transparent sheet, with control points in a square grid pattern, was placed in the test section where the experimental specimen normally is located. The image of the grid with control points can then be used to remove any optical distortions in the experimental photographs. The size of the images and framing rate are chosen according to different experimental setups. For example, if the resolution of each imageis320×128pixels, thetimebetweensubsequent framesisthensetto8.38µs. The corresponding measurement size of thespecimen is 118.5mm×47.4 mm. Dur- ing the 1 µs minimum exposure time that the high-speed camera has, the shock wave inwater willtravelapproximately 1.4mm, whichcanyieldblurryimages. To overcome this intrinsic limitation of the camera, a synchronized pulsed laser light source (SONY SLD1332VLaser diodecombined with PicoLAS LDP-V03-100UF3 Driver module) is used to provide intensive lighting within tens of nanosecond time scale. A typical pulse duration for the experiments is 40 ns, whichmeansthe exposure time is reduced to significantly minimize the blur. During experiments, when the event of interest takes place, a control signal is sent to a pulse/delay generator (BNC Model 575). The pulse generator receives the signal and outputs a sequence of TTL pulses to both the light source and the camera to synchronize the two. A series of high-speed schlieren images will be recorded by the camera. 25 3.3 Background oriented schlieren It is just until recent years that researchers started to use the same basic principle as schlieren optics and developed a quantitative technique referredtoasBack- ground Oriented Schlieren (BOS) [57, 58, 59]. There exists only a few published research papers on this subject that elaborate on the details of their protocols, making this technology in general daunting to utilize. The aim of this section is to provide a step-by-step description on the procedures of performing BOS mea- surements. At the same time, another motivation is to prove that BOS is capable of measuring density variation in water due to shock impacts, and thereby the pressure magnitudes can be inferred from the density information. For studies of shock wave propagation in water, the surface pressure on the boundary walls or sometimes the whole pressure field in the test samples is of crit- ical importance. Due to the small scale of laboratory fluid samples, non-intrusive measurements are always favorable when considering finite-size disturbances that instrumentation probes may introduce. There are other non-intrusive techniques such aspressure-sensitive paintsavailableforpressure measurements. However the response time, at least in hundreds of microseconds [60], is much greater than the time window of the shock dynamics in water. Therefore BOS combined with high- speedphotographyprovidesaplatformforperformingdetailedmeasurements. The requirements for a functional BOS system are easy to obtain in most laboratory settings, in particular if there is already a schlieren system present. The following components are necessary: a schlieren optical system, a regular or random back- ground pattern, and image post-processing algorithms, see more in Section 3.3.3– 3.3.4. 26 Experimental results in Section 3.3.5 show that the BOS technique success- fully captures the pressure increase in water when a shock wave propagates into a convergent geometry. 3.3.1 Principles of background oriented schlieren As suggested by Meier, [61], BOS is suitable for quantifying a density change in transparent media. To evaluate the pressure variation in water, a direct relation- ship between the density and pressure in water will be needed, while assuming temperature remains constant. Here the International Association for the Proper- ties of Water and Steam (IAPWS) formulation for the thermodynamic properties ofordinarywaterhasbeenutilized[62]. Usingsucharelationshipbetweentheden- sity and pressure, pressure measurements can be readily obtained by measuring density variations. TheprincipleofBOSisillustratedinFigure3.5. Thecameraisfocusedontothe background plane that has certain prescribed patterns [63]. The sample, placed in between the background plane and the camera, serves as a transfer channel function [61]. The variation of the sample’s index of refraction causes a deflection angleoftheoriginallyparallellight, which leads tothedisplacement oftheoriginal background on the image plane of the camera, see the enlarged view in Figure 3.5. By comparing the images obtained with and without the sample, the information aboutthetransfer channel functionorthechange ofindexof refractioninthesam- ple can be extracted. The deflection angle α is a key quantity, which connects the measurabledisplacement y ! andtheunknownn,theindexofrefraction. Here, only the displacement in the y-direction is considered for simplicity. Two relationships 27 α α y " y Za Z b higher density lower density Z b /2 Background plane Sample Lens Image plane Principal optical axis Figure 3.5: Demonstration of the BOS principle and the key quantities. Z a is the distancebetweenthebackgroundandsample, anditis300mminourexperiments. Z b isthethickness ofthesample. y isthedisplacement ofthebackgroundobserved at the image plane. y ! is the projected displacement at the background plane. can be obtained for the deflection angle α. First, by assuming that the angle α is small, α can be expressed by y ! and additional experimental parameters as α = y ! Z a +Z b /2 . (3.8) InEquation(3.8),Z a isthedistancebetweenthebackgroundplaneandthesample, and Z b is the thickness of the sample. By increasing Z a , the error in determining 28 the angle α can be reduced. Second, based on Snell’s law, α and the index of refraction n have the following relationship, α = 1 n 0 ! Za+Z b Za ∂n ∂y dz. (3.9) By combining Equation (3.8) and (3.9) and assuming the width of the sample Z b is much smaller than Z a , Z b #Z a , the following can be obtained, ∂n ∂y = n 0 y ! Z b (Z a +Z b /2) . (3.10) The assumptions made above transform the problem into two-dimensions, which means that the density is only a function of the x and y coordinates but remains constant along the z direction. The medium considered in this work is water, and a direct relationship between the density ρ and index of refraction n of water can be given by [64], n =n 0 +κ(ρ−0.99824), (3.11) in which n 0 equals to 1.332 and κ equals to 0.322. Equation (3.11) is also referred as Gladstone-Dale relation [65]. Equation (3.11) can then be substituted into Equation (3.10) to yield a relationship between ρ and y ! given by ∂ρ ∂y = n 0 y ! κZ b (Z a +Z b /2) . (3.12) Since y ! is directly measurable, it is evaluated through the displacement of the background on the image plane y. The magnification factor M is defined as M =y/y ! . (3.13) 29 Equation (3.12) can be rewritten using M and y as ∂ρ ∂y = n 0 y MκZ b (Z a +Z b /2) . (3.14) AsseeninEquation(3.14),theunknownρisdirectlyrelatedtoy,whichisobtained through the analysis of experimental image data, see Section 3.3.4. To solve the complete two-dimensional problem instead of the simplified one-dimensional ver- sion, a Poisson Equation of the density distribution can be formulatedas ∂ 2 ρ(x,y) ∂ 2 x + ∂ 2 ρ(x,y) ∂ 2 y =k " ∂u(x,y) ∂x + ∂v(x,y) ∂y # , (3.15) k = n 0 MκZ b (Z a +Z b /2) . (3.16) In Equation (3.15), u(x,y)and v(x,y) are the displacements of the background in x andy directions at location (x,y) on the image. A systematic numerical method has been developed to solve this partial differential equation, see Section 3.3.4. 3.3.2 Optical system Because BOS is an extension of schlieren method, the optical system shown in Figure 3.1 is also suitable for BOS technique. However no schlieren knife edge is needed for BOS. 3.3.3 Image background The pattern for the background is generated using PIVMat Toolbox, an open- source Matlab code [66]. The background is designed to have random white dots on a black background. The distribution density and size of the white dots are controlled to yield an optimum resolution of the image on the high-speedcamera, 30 so that the analysis error for post-processing could be minimized [67, 68], see Figure 3.6. The image generated from the program is printed onto a black and white transparency by using a standard inkjet printer. The optical quality of the transparency ensures that the scattering of the incoming collimated light is minimum. This property is critical to eliminate the error associated withthe mapping of spacial coordinates between the background and the test sample. Figure 3.6: Example of a background plane for BOS. Random white dotsona black background are printed onto a transparency. 3.3.4 Image correlation and density calculation A Digital Particle Image Velocimetry (DPIV) tool for Matlab [69] is utilized to extract the displacements of the location of the dots. A three point Gaussian sub- pixel estimator is selected when performing analysis for sub-pixel accuracy [68]. The maximum displacement for the type of application under consideration is no more than 2 pixels, thus an initial 8×8 pixel 2 interrogation window is adopted. 31 Through a multiple vector validation, the final resolution of the displacements from the image correlation step is 4 pixels in either direction, which corresponds to 1.5 mm in the physical measurement. Once the displacement information is extracted, the data on the meshes is fed into a Poisson solver according to Equa- tion (3.15). A second-order finite volume method combined with the successive over relaxation method [70] is the main component of the solver. The solver has been validated by problems with known solutions. To solve the problem presented here, a Neumann boundary condition is enforced on all four edges oftherectangu- larmeshdomain. Thevaluesattheboundaryarecalculatedfromthedisplacement field. The initial condition for the density field is simply the density of water at rest, 998.24 kg/m 3 . This code converges within 50 iterations for a typical mesh size of 50×13. 3.3.5 Results and discussion Windows Blocker Core sample Water-filled region Figure 3.7: Schematic drawing of the experimental test sample assembly for BOS visualization. 32 The test sample is shown in Figure 3.7. The blocker and the core sample, 6.35mmthick, areallmadeofpolycarbonate. The173mm×192mmcorecontains a22 ◦ symmetric wedge cavity [7]. During the experiments, the cavity is filled with water. Water will stay well sealed inside the cavity with the help of the blocker, two optically transparent polycarbonate windows on both sides of the core and a silicone sealant. Once the projectile impacts the blocker, a pressure wave will propagate through the blocker and reach the water region in the form of a shock wave. A series of images were recorded by the high-speed camera. When applying the DPIV tool explained in Section 3.3.4 to the image pairs, the displacement at different mesh locations are revealed and plotted by arrows, see Figure 3.8. The dashed lines in Figure 3.8(a)–(e) indicate the progression of the shock wave in water withtime. Thearrowstotheleftandrightofthedashedlines allpoint away from it at each time instance, which indicates that the density at the location of the lines is higher than the surroundings. The average wave speed of this marked peak is determined through dividing the marching distance of the peakbetween two consecutive frames by the corresponding time interval. The calculated shock speeds based on frames in Figure 3.8 (a)–(e) are 1555 ± 100 m/s, 1494 ± 100 m/s, 1713 ±100m/sand1961 ±100m/s. These valuesarecomparablewiththevalues found in the literature [7] using traditional schlieren technique. The shock wave is acceleratedwhenapproachingthetipofthewedgeduetotheconvergent geometry. After performing the density calculation by the Poisson solver, the two- dimensional density distributions at different time instances are reconstructed. Figure 3.9 shows the line plots of density along y = 25 for five time instances. The density peaks marked by the arrows match the locations of the dashed lines on the left of Figure 3.8, which confirms with the previous observation of the high density 33 20 40 60 80 100 120 140 160 180 10 20 30 40 50 location(pixel) location(pixel) 20 40 60 80 100 120 140 160 180 10 20 30 40 50 location(pixel) location(pixel) 20 40 60 80 100 120 140 160 180 10 20 30 40 50 location(pixel) location(pixel) 20 40 60 80 100 120 140 160 180 10 20 30 40 50 location(pixel) location(pixel) 20 40 60 80 100 120 140 160 180 10 20 30 40 50 location(pixel) location(pixel) a) b) c) d) e) Figure 3.8: The displacement vectors plots corresponding to the rawdata. a) t=38.45µs,b) t=46.14µs,c) t=53.83µs,d)t=61.52µs and e) t=69.21µs. 34 0 20 40 60 80 100 120 140 160 180 200 1 2 3 4 5 999.5 1000 1000.5 1001 1001.5 1002 1002.5 1003 1003.5 1004 1004.5 location(pixel) time step density(kg/m 3 ) a b c d e Figure 3.9: The density plots calculated from the corresponding subplots in Fig- ure 3.8. Plotted along the center line y=25. a) t=38.45 µs,b) t=46.14 µs,c) t=53.83 µs,d) t=61.52 µs and e) t=69.21 µs. region. However the magnitude of the peak decreases with time until it reaches the tip of the wedge, due to the energy dissipation of the wave. At the last time instance t=69.21 µs, the density peak rises, since a convergent geometry, in this case the wedge shape [7], has the ability to focus the shock wave. The range of density variation observed through the density reconstruction is only 4 kg/m 3 , which is only 0.4% change of the overall density. However, when calculating the pressure inside water through theIAPWS formula, see Figure3.10, the maximum pressure can reach 8×10 6 Pa, which is 80 times greater than the atmospheric pressure of water. This transient high pressure may cause failure of a surrounding solid structure [71]. 35 0 20 40 60 80 100 120 140 160 180 200 1 2 3 4 5 −2 0 2 4 6 8 10 x 10 6 location(pixel) time step pressure(Pa) a b c d e Figure 3.10: The pressure plots. a) t=38.45 µs,b) t=46.14 µs,c) t=53.83 µs, d) t=61.52 µs and e) t=69.21 µs. 3.3.6 Conclusion The Background Oriented Schlieren method has been developed and fully inte- grated with the in-house high-speed schlieren optical system. The reconstructed results show matching wave speeds with previous results using schlieren optics. At the same time quantitative density and pressure values for the flow field are also made available. By examining the pressure profile along the centerline or even the whole area of interest for the test sample, focusing effects from the convergent wedge-shaped sample is confirmed, which otherwise requires an underwater sensor sensitive to high dynamic pressure. Moreover, detailed information of the setup and algorithms used is provided, so that this technique can serve more research laboratories as a standard non- intrusivequantitativemethodtostudyfluidflows, especiallywaterflows,withhigh resolution. The boundary conditions used in the Poisson solver may besubjectto changes for different types of fluid flows. The resolution of the images for a given 36 optical setup can be controlled by modifying the interrogation window size, but then the particle density on the background has to be changed correspondingly. FilteredBackProjectionTechnique(FBPT)[57]issuitableforaxisymmetricthree- dimensional test sample. However, for a fully three-dimensional reconstruction, a more sophisticated optical setup will be needed, which is out of the scope of this dissertation. 3.4 Strain gauge installation and data acquisi- tion General purpose manganin strain gauges (Micro-measurements EP-08-125AD- 120) are utilized to perform the strain measurements. The strain gauges are installed onto the clean sample surface following the instruction bulletin (Micro- measurements B-127-14). After soldering the electrical wire on the gauge elec- trodes, rosin solvent is applied to remove rosin and solder residuals. In the end, because all the experiments are in contact with water, the gauges are covered by a protective layer of silicone rubber (Micro-measurements M-coat C). During the measurements, strain gauges are connected into a quarter Wheat- stone bridge provided by the signal conditioning amplifier (Vishay 2310B). The excitation voltage of the bridge is 2.7 V. The amplifier is operated at no filtering and a gain of 60. The output from the amplifier is recorded by an oscilloscope (LeCroy WaveSurfer 24Xs-A) with up to 500 MHz sampling rate. After acquiring the data, the strain values are calculated from the voltage reading through " = 4V out V ext ×G×K . (3.17) 37 TheV out isthevoltagerecordedbytheoscilloscope,V ext isthebridgeexcitation voltage, G is the gain applied and K is the gauge factor of the strain gauge. Due to the noise level associated with the measurements, all the strain data are post- smoothed in MATLAB by a moving average filter with a span of 11 points. 38 Chapter 4 Shock dynamics and weak coupling By weak coupling, we assume the amount of deformation of the solid structure is small enough so that only linear elasticity effect need to be considered. 4.1 Experiments 4.1.1 Specimen geometry and assembly The experimental specimen is shown in Figure 4.1. The core material with a con- vergent water-filled section is sandwiched between two transparent optical quality polycarbonate windows measuring 182×205×12.7 mm. A polycarbonate blocker is placed at the entrance of the convergent section to prevent the water from leak- ing out. The core materials are made of polycarbonate, aluminum or PMMA, and the windows are made of polycarbonate. The edges between the windows and the core are sealed with silicone to prevent water leakage from the specimen. Special careistakentoremoveanybubblesinthewater-filledregionbeforeanexperiment. 4.1.2 Polycarbonate specimen A series of high-speed schlieren images with a projectile impact speed of 50.6 ± 1.0 m/s using the polycarbonate specimen is shown in Figure 4.2. Figure 4.2(a) 39 38 L=114 173 192 192 (a) Core 1 2 3 4 (b) Assembly Figure 4.1: Specimen geometry and assembly. (a) Experimental core with dimen- sions in millimeter. The core thickness is 25.4 mm for the aluminum and polycar- bonate samples. For the PMMA samples, this thickness is 6.3 mm. (b) Specimen assembly with (1) window (only one is depicted for clarity), (2) core, (3) water- filled section, and (4) blocker. showsthespecimenbeforeimpactwiththeundisturbedwater-filledregion(marked A in the figure) and the blocker (marked B in the figure). The pressure wave front is marked by white arrows. Time 0 µs is when the pressure wave enters the water region. The wave speed along the center line for frames (c)-(h) was measured using the distance between the wave front positions, and the timing betweensuccessiveframes. Thewavespeedbetweenframes(c)-(d)is1370±44m/s, (d)-(e) is 1326 ±44 m/s, (e)-(f) is 1458 ±44 m/s, (f)-(g) is 1547 ±44 m/s, and (g)-(h)1458 ±44 m/s. The average wave speed is 1432 ± 44 m/s, which gives a Mach number M a =u a /c water =(1432±44)/(1491±2.8)= 0.96±0.03. However the maximum Mach number reaches M s =u s /c water =(1547±44)/(1491±2.8)= 1.04±0.03. 40 In Figure 4.2(g) the angle between the interface of the convergent section and the shear waves in the polycarbonate core was measured to be 34 ◦ ± 1 ◦ on the upper edge and lower edge. From such an angle, one can estimate thewavespeed in water by assuming that the shear wave remains constant in the surrounding solid [7]. In later frames, Figure 4.2(h), this angle has decreased to 29 ◦ ± 1 ◦ . This decrease is due to the curvature change of the fluid-solid interface, and if the horizontalcomponent ofthe shear wave iscomputed itis clear that thewave speed in water remains constant, see Appendix A for calculation details. Three single-shot schlieren experiments are presented in Figure 4.3. Fig- ures 4.3(a), (c) and (e) show the configuration before the experiment, and (b), (d) and (f) during the experiment. The arrow in Figure 4.3(a) shows the location of two holes in the catcher box side panels where the laser beam to trigger the spark light source enters and exits the catcher box. Figure 4.3(b) shows the result ofaslightlyobliqueimpact. Thearrowspointattheshockwave(1),thetransverse waves in the surrounding polycarbonate core (2), and the longitudinal wave in the polycarbonatecoretraveling aheadofthe shock wave inthewater-filled region(3). Figure 4.3(d) was taken at a longer delay time compared with Figure 4.3(b). The arrow in Figure 4.3(d) points at the curved shear wave in the polycarbonate core. This curvature is formed due to the fact that the interface between the fluid and the solid has a nonlinear geometry. As shown in Figure 4.3(e), a large air bubble, marked by an arrow, was introduced into the water before the experiment. Since the wave speed is much slower in the air, the shock front in Figure 4.3(f) is skewed by the bubble, thus indicating the importance of removing any air bubbles if a planar shock wave is to be formed. 41 A B (a) Before test (b) t=0 µs (c) t=8.38 µs (d) t=16.76 µs (e) t=25.14 µs (f) t=33.52 µs (g) t=41.90 µs (h) t=50.28 µs (i) t=58.66 µs (j) t=67.04 µs Figure4.2: Aseriesofhigh-speedschlierenimagesusingapolycarbonatespecimen. 4.1.3 Aluminum specimen A series of high-speed schlieren images with a projectile impact speed of 50.7 ± 1.0 m/s using an aluminum specimen is shown in Figure 4.4. Figure 4.4(a) shows 42 A B (a) Before test 1 2 2 3 (b) t=86.7 µs (c) Before test (d) t=111.4 µs (e) Before test (f) t=85.8 µs Figure 4.3: Three single-shot schlieren experiments using a polycarbonate speci- men. the specimen before impact with the undisturbed water-filled region (marked A in the figure) and the blocker (marked B in the figure). The dark area in region A of Figure 4.4(a) is caused by the interference of light from the light source. 43 A B (a) Before test (b)Δt=0 µs (c)Δt=8.38 µs (d)Δt=16.76 µs (e)Δt=25.15 µs (f)Δt=33.53 µs (g)Δt=41.91 µs (h)Δt=50.29 µs (i)Δt=58.68 µs (j)Δt=67.06 µs Figure 4.4: A series of high-speed schlieren images using an aluminum specimen. The shock speed along the center line for frames (b)-(i) in Figure 4.4 was measured using the distance between the shock front positions, and the timing between successive frames. The shock speed between frames (b)-(c) is 1724 ± 44 44 m/s, (c)-(d) is 1591±44 m/s, (d)-(e) is 1591±44 m/s, (e)-(f) is 1503±44 m/s, (f)-(g)is1503±44m/s,(g)-(h)is1547±44m/s,andfinally(h)-(i)is1503±44m/s. The average shock speed is 1566 ±44 m/s, which gives the shock Mach number M s =u s /c water =(1566±44)/(1491±2.8)=1.05±0.03. Figure4.5showsasingle-shotschlierenexperimentwithaluminumassurround- ing core material. The projectile impact speed of this test is at 79±1m/s. The arrows point at the shock wave in water (1), the oblique precursor wave from the shear wave in the aluminum core material (2) and the pressure wave in the poly- carbonatewindows. Bubbles formbetween the precursor wave andthemain shock front. (a) Before test 1 2 2 3 (b) t=39.4 µs Figure 4.5: A single-shot schlieren experiments using an aluminum specimen. 45 4.1.4 PMMA specimen The purpose of using PMMA samples is to test BOS and obtain quantitative pressuremeasurementsduringtheexperiments. Asmentionedbefore,thethickness of the PMMA samples is only a quarter of the thickness of other type of samples. ThethicknessisreducedtorealizeanoptimumresultforBOSvisualizations. Other dimensions of the PMMA samples stay the same as the rest of the test samples. (a) Before test (b) t=54.5 µs Figure 4.6: Experiment of PMMA sample using BOS technique. (a) The initial background with no displacement. The white dashed line in (a) is the center line. The white arrow in (b) points at the location of the shock front. Two snapshots selected from a series of high-speed BOS images are shown in Figure 4.6. The projectile impact speed during the test is 50.0 ±1.0m/s. By performingtheimagecorrelationanalysisofthetwoimages,aspatialdisplacement of the original background has been constructed, see Figure 4.7. 46 18.75 37.5 56.25 75 3.75 7.5 11.25 15 18.75 22.5 26.25 location(mm) location(mm) Figure 4.7: The displacement of the background at different locations. The spa- tial grid points are predetermined by the interrogation window size of the image correlation analysis. The size and direction of the vectors indicate the relative magnitude and orientation of the displacement. Two-dimensional pressure distributionattime54.5µsisreconstructed byfeed- ing the displacement information into a solver, see [72] for details. The resulting pressure profile along the center axis of the logarithmic spiral geometry is plotted in Figure 4.8. The peak pressure behind the shock is 10 MPa. There is a 4 mm spatial span of the pressure rise as seen in Figure 4.8, which is due to the finite window size of the image analysis. Behind the pressure peak, the pressure drops to 3.9 MPa, which corresponds to a long trailing expansion wave. 47 0 15 30 45 60 75 90 0 2 4 6 8 10 12 x 10 6 Location(mm) Pressure(Pa) Figure 4.8: The reconstructed pressure measurement along the center line. 4.2 Numerical simulations Inthissection,resultsfromthreedifferenttypesofsimulationsofconvergingshocks in water are presented; (i) a two-dimensional rigid confinement, (ii) an axisym- metricrigidconfinement, and(iii) atwo-dimensionalmulti-physicssimulationthat accounts for the fluid-structure interaction. The Overture package, a finite difference code for partial differential equations, is utilized to perform all the simulations [73]. The inviscid and adiabatic flows under consideration are governed by the Euler equations of gas dynamics, i.e. conservation of mass, momentum and energy, ∂ρ ∂t +∇·(ρu)=0, (4.1) ∂(ρu) ∂t +∇·(ρuu)+∇p=0, (4.2) ∂(ρE) ∂t +∇·(u(ρE +p)) = 0, (4.3) 48 E =e+ 1 2 (u 2 +v 2 +w 2 ). (4.4) where ρ,u,p are the density, velocity vector and pressure. In Equation (4.4), E and e are the total energy and the internal energy per unit mass, andu,v,w are the velocity components in three directions. A stiffened equation of state (some- times referred to as Tammann-Tait EOS [12]) is assumed, see Equation (2.2). The Rankine-Hugoniot jump conditions using the stiffened EOS are given asEqua- tions (2.3)–(2.5) in Chapter 2. Theconservationequations,Equations(4.1)–(4.3),aresolvednumericallyusing a shock-capturing high-order Godunov method [74, 75]. Composite grids are gen- erated by overlapping structured grids [75]. The main reason for using composite grids is to save memory because the main part of all grids are rectangular Carte- sian grids which are more efficient in terms of memory usage thancurvilinear grids [73]. Ameasureofinterestisthepeakpressureatthetipofthelogarithmicspiral,i.e. the focal region. This peak pressure is an indicator of the destructive potential of the converging shock wave to the surrounding structure. Transient high pressures will induce strong tensile and shear stresses inside the surrounding solid. The stress fieldcaninturncauseinitiationandgrowthofanydefectsalreadypresent in the structure, which eventually can lead to crack formation [76] with devastating results. This parameter is also used for a grid sensitivity study. In Figure 4.9, the peak pressure converges to about 4 GPa when reducing the initial grid size below 0.1 mm. This indicates that the simulation results will be independent of the initial grid size, when using a grid size smaller than 0.1 mm. An Adaptive Mesh Refinement (AMR) algorithm in Overture is also utilized to capturethe fine structures of the shock focusing process. The AMR grids are two levels of 49 refinement factor of 4, which will yield a minimum grid size of 0.0125 mm in our simulations. 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 2.5 3 3.5 4 4.5 Initial mesh size(mm) Peak pressure (GPa) Figure4.9: Gridsensitivitystudybyinvestigatingpeakpressuresatdifferentinitial grid sizes. 4.2.1 Two-dimensional rigid boundary simulation This type of simulation, in which only the liquid part of the problem is considered without any influence from the surrounding solid structure, is informative in two aspects. First, it enables the possibility to investigate how the geometry influences the shock focusing event. Second, this case serves as a base case for comparison with the other types of simulations. To perform the simulation, a composite grid made of three different grids over- lapping each other is prepared, Figure 4.10(a). A rectangular Cartesian grid is first generated as a background grid. All other grid components cut or replace part of the background grid. The reflector grid giving the outer boundary of the logarithmic spiral is obtained by marching from a spline whose points are given by the equation for a logarithmic spiral. The reflector grid cuts out the geometry and shares interpolation points with background grid, Figure 4.10(b). 50 (a) (1) (2) (3) (b) Figure 4.10: Grid setup for the two-dimensional rigid boundary simulation. (a) An overview of the composite grid used for two-dimensional rigid boundary sim- ulations. (b) A blowup of the overlapping grid details. (1) background grid; (2) inlet grid; (3) logarithmic spiral boundary grid. Interpolation points are marked as dots. Step functions of pressure, p, velocity, u, and density, ρ, are used as initial conditions to represent an initially planar incident shock wave. The conditions in the undisturbed region ahead of the shock wave are given by room temperature (21 ◦ C), atmospheric pressure (101,325 kPa) and zero velocity. The conditions behindtheshockwavearecomputedusingshockjumpconditions,Equations(2.3)– (2.5),basedonthechosen shockMachnumber, M=1.1,andtheconditionsahead of the shock. The initial conditions are summarized in Table 4.1. The initial 51 configuration of the simulation is shown in Figure 4.11 which shows the simulation at time t=0. Region 1 Region 2 p [MPa] 0.1 109.8 u [m/s] 067 ρ [kg/m 3 ] 1000 1043 Table 4.1: Initial conditions in region 1 ahead of the shock wave, and region 2 behind the shock wave. Region 2 Region 1 Figure 4.11: Initial pressure distribution at time t=0. At the left boundary of the inlet grid, a constant inflow condition (thesame as region 2 in Table 4.1) is maintained to sustain the shock wave strength. The boundaryconditionatthewallprofileissettobeaslip-wallcondition, whichkeeps the normal velocity of the fluid zero at the wall boundary. As a consequence of this slip-wall boundary condition, the shock front is always perpendicular to the wall, as shown in Figure 4.12. FoursnapshotsatvarioustimeinstancesareshowninFigure4.13. Astheshock wave propagates, the Mach number of the shock front at the boundary increases due to the slope of the boundary. The observed peak pressure at the tip is 4 GPa. It is 36.9 times the initial pressure, p 2 , behind the shock wave. However, in order 52 (a) (b) Figure 4.12: The shape of the shock front in the two-dimensional rigid boundary simulation. (a) A snapshot of pressure distribution at time 40.5µs. (b) The shock front is normal to the wall as a consequence of applying slip-wall condition at the boundary. (a) t=20.2µ s, M=1.13 (b) t=33.7µ s, M=1.26 (c) t=54.0µ s, M=1.32 (d) t=63.4µ s, M=1.60 Figure4.13: PressuredistributionsatdifferenttimesandMachnumberoftheshock front at the boundary. The arrow points in the direction of the wave propagation. to quantify what shock focusing in water does to the surrounding structure, it is necessary to include the solid structure. 53 4.2.2 Two-dimensional fluid-structure interaction simula- tion To determine the loads on the surrounding structure, a two-dimensional fluid- structure interaction simulation is performed. This is a coupled problem because theshockgeneratedinthewater results inbothlongitudinalandtransversal waves in the surrounding material. In turn, the longitudinal and transversal waves in the solid interact with the water. Experimental results from Eliasson et al. [7] showed clear evidence of precursor waves in water ahead of the moving shock wave when a convergent aluminum confinement was used. The precursor wavesoccurreddue to faster longitudinal and transversal wave propagation in the surrounding solid compared to the shock wave speed in water. The governing equations for thefluid part is thesame as inthe first case with a rigid confinement. Linear elasticity is assumed in the solid part and the dynamics of the displacement is given by ρ ∂ 2 u i ∂t 2 = ∂σ ij ∂x j +ρf i , (4.5) σ ij = λ(" kk )δ ij +2µ" ij . (4.6) In Equation (4.5), u i , σ ij and f i are the displacement function of material points in direction i, the Cauchy stress tensor, and the body force in direction i. And in Equation (4.6), " ij is the strain tensor, while λ and µ are the two Lam´ e constants. Material properties of aluminum are utilized in order to be able to compare results with experimental data. The density, ρ, is 2.7×10 3 kg/m 3 .TheLam´econstants, λ and µ, are 54 GPa and 27.8 GPa respectively. 54 The composite grid for this type of simulation consists four parts including a Cartesian grid for the fluid domain, a Cartesian grid for the solid domain, and two curvilinear grids for communication between the fluid and the solid, Figure 4.14. The boundary condition between the two curvilinear boundary grids is a traction boundarycondition, whichmeansthatthenormalstress atthefluid-solidinterface is matched. For initial conditions, a shock jump is set in the fluid domain as in the previous case with the rigid boundary. The solid part is left undisturbed. The initial configuration is shown in Figure 4.15. (a) (1) (2) (3) (4) (b) Figure 4.14: Grid setup for the two-dimensional FSI simulation. (a) An overview of the composite grid for the FSI simulation. The solid colored blocks are finer grids at the interface. (b) A blowup of the marked region of the interface from (a). (1) solid domain; (2) boundary grid for the solid domain; (3) boundary grid for the fluid domain; (4) fluid domain. Interpolation points are marked as black dots. The boundary condition between grid (1) and grid (2) is given by a traction boundary condition. As the shock wave starts to propagate into the logarithmic spiral duct, it will deform the interface between the fluid and the solid region. Because the fluid will be moving together with the deforming boundary to maintain contact, the overall velocity of the fluid behind the shock wave will be moving toward the undisturbed boundary. As aconsequence, the shock fronthas tobeatanangle less than 90 ◦ to 55 Solid Region 1 Region 2 Figure4.15: InitialconfigurationfortheFSIsimulation. Region1and2represents the low and high pressure regions in the fluid domain. The solid part is plotted using the x-component of the stress tensor, σ ij , which is initially zero for all the grid points. the boundary to balance this flow condition [77], Figure 4.16. Energy carried by the shock wave is dampened by the transmission of energy into solid, Figure 4.16. This results in a lower peak pressure of 880 MPa at the focal region ascompared to the previous result from the rigid boundary simulation with a peak pressure of 4GPa. Detailedplotsofpressurealongtheinterfacefortheincomingandreflected shock waves are given in Figure 4.17. The pressure in front of the incoming shock wave is lower than zero, which indicates the possibility of cavitation formation. This feature is not observed in the two dimensional rigid boundary simulation. Acomparisonwiththeresultsfromthistypeofsimulationandtheexperiments performedonthealuminumspecimenshowsimilartrendsfortheshockwavespeed during the focusing phase. The simulated shock wave speeds are calculated at the same spatial locations as the high-speed visualization results shown in Figure 4.4, summarized in Table 4.2. The general trends observed for the shockwavefront 56 (a) α (b) Figure 4.16: The shape of the shock front in the two-dimensional FSI simulation. (a) A snapshot of the FSI simulation at time t=43.5 µs. (b) The shock front is not perpendicular to the interface. Instead, the shock angle α is 88.2 ◦ for this particular time instance. In the solid region, the divergence of displacement field is expressed by the contour lines. In the fluid region, pressure change is described by the contour lines. Experiment (m/s) Simulation (m/s) (b)-(c) 1724 1628 (c)-(d) 1591 1642 (d)-(e) 1591 1640 (e)-(f) 1503 1639 (f)-(g) 1503 1515 (g)-(h) 1547 1480 (h)-(i) 1503 1515 Table 4.2: Shock wave speed comparison between the aluminum specimen and the two-dimensional fluid-structure interaction simulation. The simulation results have an averaged 3.8% deviation from the experimental values. are a deceleration of the shock wave speed and also a dispersion in space, which is in agreement with the experimental observation. 57                       (a)                         (b) Figure 4.17: Pressure plots along the fluid-structure boundary. (a) Pressure plots along the interface at times t=19.6, 37.7, 55.8 µs. (b) Pressure plots along the interface at times t=59.6, 62.3, 64.3 µs. The present results take the fluid-solid interaction into account butdonot take three-dimensional effects into account. In order to understand how three- dimensionaleffectsinfluencethepeakpressureatthefocalregionanaxi-symmetric simulation has been performed. 4.2.3 Axisymmetric simulation To study three-dimensional effects of the shock focusing, a cylindrical symmetric or so-called axisymmetric coordinate system is used to perform the simulation. 58 The axis of rotationalsymmetry is along the line where the characteristic length L is defined, Figure 4.1. Accordingly, the governing equations are also expressed in axisymmetric coordinates. Because the solution does not depend on the rotational angle (there is no swirl in the solution, [78]), the equations can be discretized and solved on a two-dimensional mesh, as shown in Figure 4.18. The horizontal line in the figure is the axis of symmetry. On this axis, the boundary conditions are set based on symmetry considerations. Slip-wall boundary conditions are again applied to the wall boundary, and constant inflow conditions are assigned to the left in-flow boundary. Figure 4.18: Composite grid for axisymmetric simulation. A quantitative representation of the difference is shown in Figure 4.19. At early times,t< 19.4 µs, the pressure profiles along the boundary are identical in the two cases. As the shock front propagates closer to the focal region, higher pressure starts to build up behind the shock wave in the axisymmetric simulation than in the rigid simulation. As a consequence, the shock front accelerates. At time t=55.6 µs, the peak pressure in the axisymmetric simulation is 2.5 times higher than that in the rigid boundary simulation, and the Mach numbers have ratio1.2. Afterreflection, thedistinctionbetweenthetwo casesismoresignificant, see Figure 4.19(b). The resulting peak pressure at the focal region in the axisymmetric simulation is20.5GPa,whichishigherthanthatofthebothpreviouslydescribedsimulations. The mainreasonfor thishighpeak pressure is dueto thefact thata larger amount 59 0 10 20 30 40 50 60 70 80 90 100 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 x P t=19.4 0 10 20 30 40 50 60 70 80 90 100 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 x P t=37.5 0 10 20 30 40 50 60 70 80 90 100 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 x P Axisymmetric case 2D case t=55.6 (a)                                                                      (b) Figure 4.19: Comparison between two-dimensional and axisymmetric rigid bound- ary simulations of the time evolution of the pressure profile along the rigid bound- ary. Time is given in µs. (a) The incoming shock wave. (b) The reflected shock wave. All the time instances are consistent with the ones in Figure 4.17. of energy carried by the shock wave is brought to the focal region as compared with the other two previous cases. 60 From the discussion above, much higher peak pressure can be achieved by using a three-dimensional logarithmic spiral shape, which suggests the significance of including the three-dimensional effects. 4.3 Summary In this section, basic fluid dynamics features of the shock focusing process inwater insidelogarithmicspiralstructures arediscovered throughsingle-shotphotography as well as high-speed photography. Quantitative measurement of the pressure distributions is also obtained through BOS technique. The peak pressure along the center line can be readily compared with the numerical simulations. Complex fluiddynamics such ascavitationoccur bothinfront andbehindtheshock wave in water. Cavitation in front of the shock wave is due to the interactionbetweenthe fastertravelingstresswavesinsidethesurroundingstructure, andthisisseeninthe aluminum test case. Cavitation behind the shock wave is due to bubble nucleation sites in the water-filled region that get excited when the shock wavespassesthem. More complex fluid dynamics models would be necessary in the future in order to correctly model the cavitation bubble events and possible interaction with the surrounding solid surfaces. Computer simulations arealso conducted using theOverture package. The two dimensional rigid boundary simulation confirms that the shock wave undergoes a focusing process, and reaches a peak pressure at the focal region of the loga- rithmic spiral shape. The fluid-structure simulation, which includes the material effects from the surrounding boundary, indicates that the coupling between fluid and structure changes the shock wave dynamics. As a result, the peak pressure magnitude at the focal region is lower in the two dimensional FSI simulation as 61 comparedtotherigidboundarysimulation. Bycontrast, arigidboundaryaxisym- metric simulation demonstrates a much higher peak pressure when compared to thetwodimensionalrigidboundarycase,indicatingathree-dimensionalsimulation would be necessary to infer the correct response due to three-dimensional effects. When comparing the experiments with the simulations, the overall dynamics of the shock front agrees. However the experimental reconstructed peak pressure from BOS is an order lower than the inflow pressure assigned in the simulation. This is attributed to the spatialaveraging with the fixed size interrogation window used in BOS, and thus the shock wave is smeared over several pixels.Another reason that accounts for the differences observed between the experiments and simulations is the estimation of shock strength by using Hugoniot datatakenfrom literature. Exact material properties of each component used in the experiments might provide more accurate data to use for shock Hugoniot matching techniques. In short, we conclude that a shock wave propagating in water is efficiently focused by the use of a logarithmic spiral geometry. When including the coupling betweenthewaterandthesolidsample,boththeexperimentsandsimulationsshow a reduction of the shock strength comparing with the ideal case, which suggests an evident fluid-structure interaction. The degree of the coupling strongly depends on the surrounding material. 62 Chapter 5 Shock dynamics and strong coupling In Chapter 4, we discussed the possibility of using the logarithmic spiral shape to focus a shock wave in water. Both experimental and numerical investigations emphasized the effect of the geometric shape and the weak coupling between the fluid and the bulk solid structure. In this chapter, we will study solid structures with the same kind of convergent geometry but of finite thicknesses. By reducing thicknesses of the structures, a strong coupling between the fluid and solid is expected. 5.1 Introduction Transient dynamic loading on thin solid structures during an UNDEX event has been extensively studied. On one hand, direct loading on plates from UNDEXs is a topic receiving wide attentions in modeling, experiments and numerical simula- tions, because of the simplicity and fundamental importance in such ideal config- uration [79]. Aspects such as added mass effect [80], strain rate dependance [81] and failure [40] etc. are investigated. On the other hand, axial impacts on water- filledtubesrepresents anotherclass offluid-structure interactionproblems[82,83]. 63 The shock wave in water travels perpendicular to the tube’s wall, therefore cre- ates a lateral loading on the solid structure. The pressure wave speed in water is significantly altered when reducing the wall thickness of the tubes. By studying dynamic loading on a plate of convergent geometry like the loga- rithmic spiral shape, it is a combination of direct and lateral loading. When the shock wave first enters the cavity, the shock loading is lateral with regard to the fluid-structure interface. However when approaching the tip region, a fraction of the shock wave can impact directly onto the structure. From this point of view, our work will fill in the gap between direct loading on plates and loading ofwater- filled tubes. At the same time, the logarithmic spiral shape focuses the shock wave therefore it will also allow us to test the solid structure under extreme pressure conditions. In this chapter, experiments and finite element simulations on under- water shock focusing in convergent structures of finite thicknesses are performed on three different types of materials; steel, aluminum and carbon fiber composites. 5.2 Experiments 5.2.1 Steel specimens The specimens are made of type 1018 steel. Gas gun experiments on two sets of the samples were performed under the same impact conditions. Specimen geometry and preparation. The steel samples were fabricated by a wire cut machine at the Caltech Aero shop. This machining process has a high accuracy in the resulting geometry and also leaves a minimum residual stress in the test sample. Two sets of the steel samples in distinct thicknesses were tested. One type of sample is 5.8 mm in thickness and the other type is 1.3 mm 64 thick. The two distinct thicknesses are chosen in order to compare with the carbon fiber composite sample in Section 5.2.3. The 5.8 mm thick steel sample has the same thickness as the composite test sample, while the 1.3 mm thick steel sample has the same weight as the composite test sample. The test assembly is depicted in Figure 5.1. The test sample is securely connected to the gas gun to makesureevery singleimpactisnormalwithout anoblique angle,seeAppendix B. a b c Gauge 1 Gauge 2 Figure5.1: Thespecimenassemblyforthesamplesin5.8mmthickness. a)Blocker; b) test sample; c) polycarbonate window. The location of the strain gauges are also marked. Strain measurement. The installation of strain gauges and data acquisition are explained in details in Section 3.4. The strain gauges applied for the following experiments are all installed on the outer surface of the samples. The locations of the strain gauges are indicated by arrows in Figure 5.1. During the experiments, the strain gauges are measuring the strain on the surface along the direction per- pendicular to the span direction. 65 The 5.8 mm thick specimen. A series of high-speed schlieren images with a A B Gauge 1 Gauge 2 (a) Before test (b)Δt=0 µs (c)Δt=6.38 µs (d)Δt=12.76 µs (e)Δt=19.15 µs (f)Δt=25.53 µs (g)Δt=31.91 µs (h)Δt=38.29 µs (i)Δt=44.68 µs (j)Δt=51.06 µs Figure 5.2: A series of high-speed schlieren images using an 5.8 mm thick steel specimen. The location of strain gauges marked in (a) is used as a convention throughout all the experiments. projectile impact speed of 45.7 ± 1.0 m/s using the 5.8 mm thick steel specimen is shown in Figure 5.2. Figure 5.2(a) demonstrates the sample before the impact 66 happened with the water filled region (marked A in the figure) and the polycar- bonate blocker (marked B in the figure). The shock front in each frame is traced by an white triangle. The time interval between the frames is 6.38 µs with a 40 ns exposure time dedicated by the pulsed light source. At time 0µsthe shock wave is transmitted into thewater region. The shock speed was measured along the center line for frames (b)–(g). The shock speed between frames (b)–(c) is 1489±68 m/s, (c)–(d) is 1557±68 m/s, (d)–(e) is 1557±68 m/s, (e)–(f) is 1557±68 m/s, (f)–(g) is 1692±68 m/s. The average shock speed is 1570±68 m/s, which gives the shock Mach number M s =u s /c water =(1570±68)/(1491±2.8)=1.05±0.04. The white arrow in Figure 5.2(d) points at the precursor wave formed in front of the main shock wave. The speed of the source is around 2900 m/s, thus this is the feedback wave resulting from the shear wave traveling in the solid material. 0 10 20 30 40 50 60 -20 -15 -10 -5 0 5 x 10 -4 Time (μs) Strain Gauge1 (experiment) Gauge2 (experiment) (e) (f) (g) Figure5.3: Strainmeasurement attwolocationsontheoutersurfaceofthe5.8mm thick steel specimen. The dashed lines corresponds to the time instances when Figure 5.2(e)–(g) were taken. 67 Simultaneous strain measurements are taken to assist the understanding of the steel specimen. The locations of the strain gauges are indicated in Figure 5.2(a). When strain gauge 1 registered a rise, it was just after the shear wave passed the location of its location as seen in Figure 5.2(e). However when the shear wave passed the location of gauge 2 (after dashed line (g) in Figure 5.3), the rise of the signal in strain gauge 2 was more retarded than the previous one. This might be caused by the increase of curvature along the logarithmic spiral shape, which will lead to an increase in the bending stiffness. In other words, the closer to the focal region of the structure, the more reluctantly it moves. After the main shock wave in water passed the location of gauge 1 in Figure 5.2(f), gauge 1 experienced a large negative strain which is due to the bending moment exerted by the high pressure right behind the shock wave. The 1.3 mm thick sample. A series of high-speed schlieren images with a projectile impact speed of 56 ± 1.0 m/s using the 1.3 mm thick specimen is shown in Figure 5.4. Figure 5.4(a) demonstrates the sample before the impact happened withthewaterfilledregion(markedAinthefigure) andthepolycarbonateblocker (marked B in the figure). The pressure wave front in each frame is traced by an white triangle. The time interval between the frames is 7.1 µs with a 40 ns exposure time. Around time 0 µs the pressure wave is transmitted into the water region. The wave speed was measured along the center line for frames (b)–(h). The wave speed between frames (b)–(c) is 1301±81 m/s, (c)–(d) is 1219±81 m/s, (d)–(e) is 1463±81 m/s, (e)–(f) is 1463±81 m/s, (f)–(g) is 1626±81 m/s, (g)–(h) is 1626±81 m/s. The average wave speed is 1450±81 m/s, which gives the Mach number M s =u s /c water =(1450±81)/(1491±2.8)= 0.97±0.05. In Figure5.4(b), thearrowpointsattheprecursorwave(1)duetotheshearwaveinthesolid. There are successive precursor waves (2) generated in Figure 5.4(c). From the curvature 68 of these waves, it indicates that the source is decelerating. We speculate the formation of the curved precursor waves is due to the traveling of flexural waves in thesolid,whichinnaturewillattenuateanddeceleratewhilepropagating. Another evidence isthatinFigure5.4(g)these waves startto broadenwhich might alsodue to the dispersive characteristic of the flexural waves [84]. This will be discussed more in Section 5.3.7. A B (a) Before test 1 (b)Δt=0 µs 2 (c)Δt=7.09 µs (d)Δt=14.19 µs (e)Δt=21.3 µs (f)Δt=28.4 µs (g)Δt=35.5 µs (h)Δt=42.59 µs (i)Δt=49.69 µs (j)Δt=56.8 µs Figure 5.4: A series of high-speed schlieren images using an 1.3 mm thick steel specimen. 69 The above schlieren images well demonstrated the flow features of a 1.3 mm thick sample interacting with a converging shock wave in water. No simultaneous strain measurement was recorded as the high-speed photographywastaken. How- everthestrainmeasurements areobtainedthroughaseparatetestwithaprojectile impact speed of 48.4 ± 1.0 m/s. In Figure 5.5, after the initial disturbance from the shear wave and the flexural wave (1), both strain gauge channels recorded a negative-going strain(2), which iscaused bythemainshockwave. Thisisfollowed by a violent ringing period (3) with high frequency signals. 0 10 20 30 40 50 60 70 80 90 100 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 x 10 -3 Time (μs) Strain Gauge1 (experiment) Gauge2 (experiment) (1) (2) (3) Figure5.5: Strainmeasurement attwolocationsontheoutersurfaceofthe1.3mm thick steel specimen. Three regions of interest are separated by dashed lines. To summarize experiments on steel specimens, the visualization results reveal that in the two sets of experiments, the formation of precursor waves has different origins. The measured shock wave velocity tends to increase while theshockwave is approaching the focal region, going from subsonic to supersonic. Strain gauge 70 measurements identify the regions of motion for the solid structures. Counterin- tuitively, the minimum strain of the 1.3 mm thick steel sample measured at gauge 1 is greater than that of the 5.8 mm thick steel sample, which implies that the 1.3 mm thick steel sample deforms less at the locations of the strain gauges for the 100 µs of the strain recording. 5.2.2 Aluminum specimens The type 6061 aluminum serves as a comparison with the steel samples, but with an absence of strain rate dependence [85]. Only strain measurements were taken for the aluminum test specimens to investigate the different dynamicstheymay have. The strain gauge locations follow the convention used in Section 5.2.1. 0 10 20 30 40 50 60 70 80 90 100 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 x 10 -3 Time (μs) Strain Gauge1 (experiment) Gauge2 (experiment) Figure5.6: Strainmeasurement attwolocationsontheoutersurfaceofthe5.8mm thick aluminum specimen. The5.8mmthickspecimen. Strainmeasurements weretakenwithaprojectile impact speed of 47.1 ± 1.0 m/s using the 5.8 mm thick aluminum specimen. The 71 plot in Figure 5.6 shows similar trend at both strain gauge locations comparing with the plot for the steel specimen of the same thickness. Only the level of strain is quantitatively different. 0 10 20 30 40 50 60 70 80 90 100 -1 0 1 2 3 4 5 x 10 -3 Time (μs) Strain Gauge1 (experiment) Gauge2 (experiment) Figure5.7: Strainmeasurement attwolocationsontheoutersurfaceofthe1.3mm thick aluminum specimen. The1.3mmthickspecimen. Strainmeasurements weretakenwithaprojectile impact speed of 48.2 ± 1.0 m/s using the 1.3 mm thick aluminum specimen. The plot in Figure 5.7 shows qualitative deviations from the plot for the steel specimen of the same thickness at both strain gauge locations. The strain signals obtained from the 1.3 mm thick aluminum sample show less oscillation. More about this comparison will be discussed in Section 5.3.8. 72 5.2.3 Composite specimens Sample fabrication. Composite test samples were fabricated from layers of prepreg (carbon fiber fabric pre-impregnated with epoxy resin). The prepreg used in this study was formulated for cure in an oven, under vacuum pressure. The material was composed of a five-harness-satin carbon fiber fabric and a toughened epoxy resin (CYCOM 5320,Cytec Engineered Materials, USA). Test samples were constructed from 16 layers of prepreg, laid-up in a [0/ ±45/90] 2s quasi-isotropic orientation. 1 2 3 4 5 6 Figure 5.8: Schematic of the layup for composite fabrications. 1) Aluminum base plate. 2) Teflon release film. 3) Breathable edge dam. 4) Prepreg plies. 5) Porous release film. 6) Breather cloth. Samples were bagged according to a standard vacuum bagging assembly. Prepreg was laid-up on a metal tool, over a layer of Teflon release film. Breathable edge dams were created from vacuum sealant tape wrapped with dry fiberglass cloth. Edge dams were placed around the perimeter of the prepreg.Aperforated release film was placed over the prepreg plies, followed by a breather cloth. 73 Lastly, the lay-up was vacuum bagged. After bagging, samples were cured using a two-dwell cure profile, as dictated by the manufacturer, see Figure 5.8. Vacuum was pulled continuously during cure. A B (a) Before test (b)Δt=0 µs (c)Δt=6.88 µs (1) (d)Δt=13.78 µs (2) (e)Δt=20.66 µs (f)Δt=27.56 µs (g)Δt=34.44 µs (h)Δt=41.34 µs (i)Δt=48.22 µs (j)Δt=55.11 µs Figure5.9: Aseries ofhigh-speed schlieren imagesusing ancarbonfiber composite specimen. 74 Results. A series of high-speed schlieren images with a projectile impact speed of 49.1 ± 1.0 m/s using the carbon fiber composite specimen is shown in Fig- ure 5.9. As can be seen from the images, the shock wave is planar and this indicates a planar impact. Figure 5.9(a) demonstrates the sample before the impact happened with the water filled region (marked A in the figure) and the polycarbonate blocker (marked B in the figure). The shock front in each frame is traced by an white triangle. The time interval between the frames is 6.9 µs with a 50 ns exposure time. Around time 0 µs the shock wave is transmitted into the water region. The shock speed was measured along the center line for frames (b)–(g). The shock speed between frames (b)–(c) is 1863±69 m/s, (c)– (d) is 2214±69 m/s, (d)–(e) is 1661±69 m/s, (e)–(f) is 1937±69 m/s, (f)–(g) is 1937±69 m/s. The average shock speed is 1907±69 m/s, which gives the shock MachnumberM s =u s /c water =(1907±69)/(1491±2.8)=1.28±0.05. Whileread- ing out the shock wave locations, the density gradient across the shock wave may interfere this process, thus causing the anomalous measured shock speed between frames (c)–(d) and (d)–(e). If averaging the two shock wave speed calculations, an averagevalueof1937m/sisobtained. Att=20.68µs,aback-propagatingwave(1) in Figure 5.9(d) appeared at the tip region of the water domain. As this acoustic wave propagated more, cavitation bubbles (2) in Figure 5.9(e) started to form. The formation of the bubbles is due to the propagation of the stress waves in the carbon fibers to the tip region [86]. The stress waves in the fibers travel with a speed three times faster than the shock wave in the water region. The image correlation tool used in Section 3.3.4 has been applied to analyze the motion of the water-solid interface near the focal region. Result shows an inward (towards water) displacement, see Figure 5.10, of the solid boundary in Figure 5.9(d) and (e). The displacement is estimated to be around 0.07 mm. This contraction is 75 Water Composite Figure 5.10: Motion of the interface near the tip of the composite sample. believed to be the origin of the back-propagating acoustic wave. Expansion wave following the acoustic wave created lower pressures thus inducing bubbles to grow in water. Simultaneous strain measurements, Figure 5.11, are taken to monitor the dynamics of the composite sample. As soon as the shock wave being transmit- ted into water, there are already changes in strain at both gauge locations. This is a direct detection of the stress waves traveling in the carbon fiber, which propa- gates at a speed of 6200 m/s [86]. After the main shock wave in water passes the location of gauge 1 (indicated by (f) in Figure 5.11), the strain gauge1records negative-going signal which is caused by the bending moment exerted by the high pressure behind the shock wave. Afterperformingtheexperiment,thesamplewasexaminedfordamage. Delam- ination was found at the location of the tip of the solid structure, see Figure 5.12. 76 0 10 20 30 40 50 60 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 x 10 -3 Time (μs) Strain Gauge1 (experiment) Gauge2 (experiment) (c) (f) (h) Figure 5.11: Strain measurement at two locations on the outer surface of the carbon fiber composite specimen. Dashed lines correspond frame (c), (f) and (h) in Figure 5.9. 5mm (a) 500 µm (b) Figure 5.12: Delaminationat the tipregion of the composite sample. a) Overview. b) Scanning Electronic Microscope (SEM) image of the dashed line region in a). 77 Visible delamination zone and micro-cracks were both presented. Further numer- ical investigation in Section 5.3.9 will consider the mechanism for the occurrence of the delamination. 5.3 Numerical simulations AsdescribedintheforegoingexperimentalresultsinSection5.2,itisobservedthat the interaction between shock waves in water and materials of finite thicknesses involves high-speed dynamics, large geometrical changes and transient material responses. The commercial Abaqus/Explicit version 6.12 is applied to simulate such events. Abaqus/Explicit has been proved to be quite robust and reliable to predict dynamic behaviors of material and the solid-solid or fluid-solid interaction, in applications such as bird strikes onto aircraft [87] and hydrodynamic ram phe- nomenon[88]. Oneadvantageoftheexplicit methodisthatglobaltangentstiffness matrix is not involved, thus no iterations or tolerances are needed [89]. 5.3.1 Physical model The explicit dynamics procedures are in general implemented through an explicit time-integration rule together with a diagonal element mass matrix [90]. The equation of motion for the simulated subjects are integrated in time by using second order central difference method, ˙ u (i+ 1 2 ) = ˙ u (i− 1 2 ) + Δt (i+1) +Δt (i) 2 ¨ u (i) , (5.1) u (i+1) =u (i) +Δt (i+1) ˙ u (i+ 1 2 ) . (5.2) 78 ˙ u and ¨ u stand for the velocity and acceleration vectors, i is the increment step number, and non-integer superscripts label intermediate results.Δt is the time increment value. The acceleration in Equation (5.1) is obtained as following, ¨ u (i) =M −1 (F (i) −I (i) ). (5.3) M, F and I are the mass matrix, the applied load vector and the internal force vector. The central difference method is conditionally stable. To have a stable explicit integration, the time increments need to be less than a critical value, Δt≤ 2 ω max , (5.4) where ω max is the maximum frequency of the system under investigation. The problem discussed in this Chapter is not only highly dynamic, but also has interaction between the liquid and the solid. For solid structure interactions with moderate deformations, Lagrangian meshes will be sufficient to capture the correctdynamics. HoweverifdescribingthemotionofliquidinLagrangianmeshes, severe mesh distortion is inevitable. Thus Eulerian meshes are necessary to be implemented for the fluid part of the simulation, while solid structures are still represented by the Lagrangian meshes. Abaqus/Explicit has the capability to performcoupled Eulerian-Lagrangian(CEL) calculations. The general idea is that proper boundaryconditions areassigned when theLagrangianmesh interacts with the Eulerian one. Pressure boundary condition is applied to the Lagrangian mesh from the results calculated in the Eulerian mesh. At the same time, thenormal velocity to the mesh boundaries obtained from the Lagrangian mesh is enforced onto the Eulerian mesh [91]. 79 5.3.2 Model description A realistic replication of most of the test components are modeled with the help of Abaqus/CAE. a b c c d e Constraint Constraint Figure5.13: AssemblyoftheFiniteElement (FE)model. a)Projectile; b)Blocker; c) Domain for fluid motion; d) Test sample; e) Rigid plane. The compass in the top right corner shows the global coordinate system. There is a initial gap between the projectile (green) and the blocker (grey). The projectile will travel in the positive x direction with a prescribed velocity. In order to constrain fluid flow inthe z direction, rigid planes aredefined on each side of the fluiddomain. Only a fractionof the volume inthe fluid domainis filled with material in the beginning of the simulations, see the light blue part of region (c) in Figure 5.13. The test specimen in the simulations is constraint in bothyand z directions at the top and bottom edges, see Figure 5.13. All other components are subjected to no constrain and interact with each other throughgeneral contact method. 80 5.3.3 Material properties Asdescribed inEquation (5.3), tocalculate theacceleration at each materialpoint the mass matrix and the internal force vector are required. It is accomplished by user-supplied material properties. The material properties of each part instance in the finite element (FE) model assembly are presented in Tables 5.1 through 5.3. Delrin Polycarbonate Density ρ [kg/m 3 ]1420 1190 Young’s Modulus E [GPa] 2.9 2.32 Poisson’s Ratio ν 0.3 0.3912 Table 5.1: Material properties of Delrin and Polycarbonate. Theprojectileandtheblocker areassignedasDelrinandpolycarbonaterespec- tively, Table 5.1. They are modeled by the linear elastic material model because they only serve as the media to generate and transmit stress waves where their plastic deformation or failure mode is out of the current research interest. The yield stress of type 6061-T6 aluminum is known to have little strain rate dependence [85]. Thus only work hardening effect, or in other words the depen- dence of the yield stress on plastic strains, is included for such material, Table 5.2. For type 1081 steel, both work hardening and strain rate dependence are modeled in the simulations. The empirical Johnson-Cook plasticity model with a von Mises yield criterion is utilized to model this type of steel under high strain rate defor- mation. It considers the strain hardening effect and the strain ratedependenceas two independent terms with no coupling. The flow stress is calculated as σ=[A+B(¯ " pl ) n ][1+Cln( ˙ ¯ " pl ˙ " 0 )](1−θ), (5.5) 81 in which θ =              0,T<T t (T−Tt) (T−Tm) T t ≤T ≤T m 1,T>T m . The ¯ " pl and ˙ ¯ " pl aretheequivalent plasticstrainandtheequivalent plastic strain rate respectively. T t and T m are the transition temperature and the melting tem- perature of the material. Density ρ [kg/m 3 ]2784 Young’s Modulus E[GPa] 75.6 Poisson’s Ratio ν 0.33 Yield Stress@(" p =0) Y 1 [GPa] 0.3 Yield Stress@(" p =1) Y 2 [GPa] 0.33 Table 5.2: Material properties of Aluminum 6061-T651. " p in the table stands for the logarithmic plastic strain. Johnson-Cook plasticity Others A [GPa] 0.35 ρ [kg/m 3 ]7700 B [GPa] 0.275 E[GPa] 210 n 0.36 ν 0.28 T m [K] 1811 Specific Heat C [J/(kg·K)] 452 T t [K] 293 Inelastic Heat Fraction 0.9 C 0.022 ˙ " 0 [s −1 ]1 Table 5.3: Material properties and parameters for 1018 low-carbon steel. ThedynamicsofwaterismodeledbyNavier-Stokesequationsfornearlyincom- pressible fluid together with a Mie-Gr¨ uneisen EOS. The Mie-Gr¨ uneisen EOS is simplified by using a linear U s −U p Hugoniot form, 82 p = ρ 0 c 2 0 η (1−sη) 2 (1− Γ 0 η 2 )+Γ 0 ρ 0 E m , (5.6) U s =c 0 +sU p . (5.7) η in Equation (5.6) is (1− ρ ρ 0 ). The c 0 , s and Γ 0 in Equation (5.6) and (5.7) are empirical parameters for Hugoniot data of water. Initial valuesfor E m and p equal to 0, see Table 5.4. The quantities ρ 0 and µ are the reference density and dynamic viscosity of water respectively. ρ 0 [kg/m 3 ] c 0 [m/s] s Γ 0 µ [Pa·s] 1000 1490 1.92 0.1 0.001 Table 5.4: Material properties and parameters for water. 5.3.4 Numerical considerations Allthesolidcomponentsofthemodeledsystemincludingtheprojectile,theblocker and the test specimen are discretized by eight-node 3D stress elements (C3D8R) with reduced integration formula. Since the test sample is the main subject of interest, it is meshed with much finer elements comparing with the other two part instances, see Figure 5.14(a). The mesh size of the blocker and the projectile is 2.5 mm in all three directions, and the element size selection of the curved plate is discussedfurthernext. Theaspectratiooftheelementswasheldclosetounityand the virtual topology feature of Abaqus/CAE is utilized to avoid severe skewness of the elements. The Eulerian domain is discretized by linear hexahedron elements (EC3D8R), which is the Eulerian version of C3D8R. To save computational cost, 83 only the region initially filled with water and its vicinity are meshed with fine unstructured meshes, see Figure 5.14(b). (a) (b) Figure 5.14: Meshing the FE model. (a) Mesh configuration for the solid parts. (b) Mesh distribution of the Eulerian meshes. Mesh convergence is studied by varying the size of elements for the test sample and the Eulerian mesh. Three mesh seed sizes were investigated, 1 mm, 0.8 mm and 0.6 mm. By evaluating strain history at locations on the sample and the peak pressure at the focalregion, all the results converges to values obtained using 0.6 mm mesh seeding. Thus 0.6 mm mesh seed is required for all the numerical models. The resulting model contains around 715K elements and 771Knodes. 84 5.3.5 Postprocessing Because in experiments strain gauges are attached to the curved surface of the sample, they are measuring strain values along the local material orientation. In simulations, results are given with regard to the global coordinate. In order to compare simulation results with experimental ones, transforming the global coor- dinate to the local coordinate at location of interest is always necessary. Local coordinate systems are defined by using 3 nodes on an element surface. When plottingthestrainmeasurement inFEsimulations, itismoreconvenient to plot result on a specific element. However in experiments all the strain gauges have finite sizes, thus certain degree of averaging occurs. In Figure 5.15, the plot compares the numerical result from a single element and an averaged result at the location of strain gauge 1 for the 5.8 thick steel sample. The averaged result is obtained from six elements which covers 3.6 mm corresponding to the length of the active area on a strain gauge. There is no significant difference between the two. Thusitissufficient enoughtouseresultfromsingleelementswhencomparing with measurements in experiments. 5.3.6 Parametric study of impact speed During experiments the projectile is not always shot at the same speed, although controlledprocedures were taken. Thisislargelyduethefactthattherearealways slight changes in the test environments such as room temperature, atmospheric pressure and projectile mass etc. Thus it is necessary to investigate the effect of the variation of projectile speed. The initial gap between the projectile and the blocker is changed for each case to keep the impact happening at the same time during the simulation. The 5.8 mm thick steel specimen is investigated for this parametric study. For all the 85 0 50 100 150 200 250 300 -4 -2 0 2 4 6 8 10 x 10 -3 Time (μs) Strain Single element Average Figure5.15: Comparisonofthestrainresultfromasingleelementandtheaveraged one among six elements. 0 10 20 30 40 50 60 70 80 90 100 -2 0 2 4 6 8 10 12 14 16 x 10 7 Time (μs) Pressure (Pa) v impact =46 m/s v impact =48 m/s v impact =50 m/s (a) 0 10 20 30 40 50 60 70 80 90 100 -2 -1.5 -1 -0.5 0 0.5 1 x 10 -3 Time (μs) Strain v impact =46 m/s v impact =48 m/s v impact =50 m/s (b) Figure 5.16: Parametric study of impact speed. (a)Strain measurements at Gauge 1 and (b) pressure measurements at the tip region of the water domain for three impact speeds. experiments discussed in Section 5.2, an average projectile speed of48m/swas used, thus three velocities, 46 m/s, 48 m/s and 50 m/s were numerically studied to understand the influence of impact speed variation. In Figure 5.16, both the strain measurement at the location of gauge 1 and the pressure in water at the 86 focalregionareplotted. The error ofthe peak pressure is within 5%difference and the error of minimum strain measurement at the location of gauge 1 is within 3% difference. Thus the slight variation of impact speed introduced by environments will change the quantities of interest in less than 5%. 5.3.7 Steel specimens The 5.8 mm thick specimen. To compare with the simulations, time histories 0 50 100 150 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x 10 -3 Time (μs) Strain Gauge1 (experiment) Gauge2 (experiment) Gauge1 (simulation) Gauge2 (simulation) Figure 5.17: Time history of true strain from both the experiment and simulation for the 5.8 mm thick steel sample. of the true strain are plotted at the same locations as strain gauge1and2used in the experiments, see Figure 5.17. The time origin 0 µs is different from the one defined in Section 5.2.1. It is shifted to the time when the projectile impact happened for the ease of comparison between the experimental and numerical results. The numerical simulation is carried out at an impact speed of 48 m/s. The total duration of simulated time is 300 µs. In Figure 5.17 only the first 150 µs results are compared due to the limitation of the acquired experimental data. A general observation is that the numerical result at gauge 1 agrees well 87 with the experimental data, whereas there is certain deviation at gauge 2. An error measurement provided by Russell [92] can be utilized here to quantify the degree of agreement between the experiment and simulation, see Appendix C for more details. The calculated errors are presented in Table 5.6. 0 50 100 150 -2 0 2 4 6 8 10 12 14 16 18 x 10 7 Time (μs) Pressure (Pa) 75 mm 60 mm 45 mm 30 mm 15 mm 0 mm Figure 5.18: Time history of pressure measured at locations 15 mm apart along the centerline. 0 mm is the location of the focal point. During the focusing phase of the shock wave, the strength of the shock wave along the centerline stays constant until 15 mm away from the focal point. At the focal point, an almost five times enhancement of the shock strength is observed. The arrow in Figure 5.18 points at the slight rise of pressure about 20 µs ahead of the main shock wave. This is the pressure wave originated from the fast-going shear wave in the solid. Tofurtherunderstandtheeffectofcouplingbetweenwaterandthesolidsample, anumerical experiment wasconducted withonlythesolidcomponents. Theinitial water filled in the logarithmic spiral cavity is removed. This type of simulation can also help to identify the contribution to the strain measurements purely from the mechanical contacts between the blocker and the test sample. Results in 88 0 50 100 150 200 250 300 -4 -2 0 2 4 6 8 10 x 10 -3 Time (μs) Strain Gauge1 (water) Gauge2 (water) Gauge1 (no water) Gauge2 (no water) Figure 5.19: Effect of the fluid. Figure 5.19 shows that for the first 100 µs the mechanical vibration from loadings other than the shock wave in water can reach up to 50% of the total strain at the location of gauge 2. At later time, as more energy transferring from water to the sample, the strain curve ramps at both gauges, while for the case without water only oscillations in small amplitude remain. This suggests that the critical interaction time between the water and the solid structure stays much longer than the time duration of the shock focusing process. To identify the role that the strain rate dependence is playing in this case, the strain rate dependence term, see Equation (5.5), is removed and the whole simulation was repeated. Plastic strains in the material direction normal to the span is plotted in Figure 5.20. For both of the gauges, plastic deformations occur around the same time instance no matter if the strain rate term is included or not. This suggests that in the two separate simulations, the test sample is subjected to similar external load from the water. However, the magnitudes of the plastic strain show a variation. In the simulation with the strain rate term, bytheendof 89 the simulation the plastic strain at gauge 1 is about 7.6% higher in absolute value than that in the simulation without the strain rate term. At gauge 2 this value is 30%. Therefore the effect from strain rate must be included when simulating this type of steel. 0 50 100 150 200 250 300 -1 0 1 2 3 4 5 6 7 x 10 -3 Time (μs) Plastic strain Gauge1 (with strain rate term) Gauge2 (with strain rate term) Gauge1 (no strain rate term) Gauge2 (no strain rate term) Figure 5.20: Strain rate dependence for the the 5.8 mm thick steel sample. Solid lines are for the case with the strain rate term. Dashed lines are for the case with no such term. The 1.3 mm thick specimen. Numerical simulations with 48 m/s projectile speed was performed on the 1.3 mm thick steel sample. The numerical results are compared against the experimental strain data on the same type of sample, Fig- ure5.21. Thetimeorigin0µsisdefinedasthetimeinstanceofimpact. Thenumer- ical simulation captures the three stages of interaction during the shock focusing processmentionedinSection5.2.1. Thecorrelationbetween theexperimentaldata and simulation results is again measured by Russell error, see Table 5.6. From the point of view of fluid dynamics, one distinct feature of the visual- ization result on the 1.3 mm thick steel sample is the multiple curved precursor 90 0 10 20 30 40 50 60 70 80 90 100 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 x 10 -3 Time (μs) Strain Gauge1 (experiment) Gauge2 (experiment) Gauge1 (simulation) Gauge2 (simulation) Figure 5.21: Time history of true strain from both the experiment and simulation for the 1.3 mm thick steel sample. waves. Simulation results are also utilized to explain such phenomenon. A partial view of the 1.3 mm thick steel sample at 40.8 µs after the impact is shown in Fig- ure 5.22(a). It is a snapshot of the deformed mesh plotted in a 1000 deformation scalefactor. Thecontourofthevelocityfieldinydirectionandthewavy patternof the deformation all suggest a traveling flexural wave in the solid sample. A direct measurement of the separation between the pressure waves inside the fluid gives an average around 7 mm, see Figure 5.22(b), which agrees with valuesmeasured in the experimental visualization. We have previously speculated the possible root of such wave patterns in Section 5.2.1. With the help of simulation, the cause of precursor waves is confirmed to be flexural waves in the solid specimen. The deformation contour of the 1.3 mm thick steel specimen after 160 µs of simulation is shown in Figure 5.23(b). Until this point, the location and strength 91 (a) (b) Figure 5.22: Origin of the precursor wave for the 1.3 mm thick steel sample. a) Flexural waves in the solid. b) Precursor pressure waves. Original Deformed (a) (b) Figure 5.23: Deformation pattern on the 1.3 mm thick steel specimen. a) Experi- mental result. b) Numerical result. of the deformation of the sample matches with the result from the experiment, Figure 5.23(a),which indicates that the deformationfully develops. The dynamics of the displacement is demonstrated by plotting the location of the maximum deformation on the sample with the magnitude of the maximum deformation, see Figure5.23(b). Themaximumdeformationisinitiallyoccurringat107.2mmaway 92 Figure 5.24: Locations of maximum displacement. Data points are measured at 20 µs time interval. from the sample tip. At later time, the location of maximum deformationmoves towards its final location. 0 10 20 30 40 50 60 70 80 90 100 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Time (μs) Energy 5.8 mm steel 1.3 mm steel Figure5.25: Plastic energydissipation forthe1.3mmthickand5.8mmthick steel samples. The global plastic dissipation for the steel samples is plotted in Figure 5.25. Because of the larger inertia the 5.8 mm thick sample has, plastic deformation of such sample starts at 40 µs, while the 1.3 mm thick sample starts to deform at 93 20 µs. However there is more energy dissipated through plasticity in the 5.8 mm thick sample, which means the 5.8 mm thick sample has an overall larger defor- mation. When plotting the plastic strain at both gauge locations for the 1.3 mm thick sample, no plastic strains are presented throughout the simulation. Further investigation shows that plastic deformation develops only at certain locations for the1.3 mm thick steel sample. Inother words, theplastic deformation islocalized. This behavior of the 1.3 mm thick sample is again different from the observation for the 5.8 mm thick sample. 5.3.8 Aluminum specimens 0 10 20 30 40 50 60 70 80 90 100 -5 -4 -3 -2 -1 0 1 2 3 x 10 -3 Time (μs) Strain Gauge1 (experiment) Gauge2 (experiment) Gauge1 (simulation) Gauge2 (simulation) Figure 5.26: Time history of true strain from both the experiment and simulation for the 5.8 mm thick aluminum sample. Numerical simulations with 48 m/s projectile speed was performed on the 5.8 mm and 1.3 mm thick aluminum samples. The numerical results are compared against the experimental strain data on the same type of samples, see Figure 5.26 and 5.27. The simulations successfully capture the trend of the strain variation at 94 0 10 20 30 40 50 60 70 80 90 100 -1 0 1 2 3 4 5 x 10 -3 Time (μs) Strain Gauge1 (experiment) Gauge2 (experiment) Gauge1 (simulation) Gauge2 (simulation) Figure 5.27: Time history of true strain from both the experiment and simulation for the 1.3 mm thick aluminum sample. both locations of strain gauges. However, the magnitude of the numerical results are dampened for the 5.8 mm thick aluminum sample, whereas in the 1.3 mm thick sample case the simulation overpredicts the strain values. The correlation between theexperimentaldataandsimulationresultsismeasuredbyRussellerror, see Table 5.6. The global plastic energy dissipation for the aluminum samples of two thick- nesses is plotted in Figure 5.28. The effect of thickness to the plastic energy dissipation is similar to the steel samples, see Figure 5.25 as a comparison. 5.3.9 Composite specimens The mechanical properties for the five-harness-satin carbon fiber fabric are not available through the supplier andtherefore all the material properties of the com- posite used in the numerical simulations are either estimated from properties of unidirectional tapes or obtained from open literature [93]. The averaged material 95 0 10 20 30 40 50 60 70 80 90 100 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Time (μs) Energy 5.8 mm Aluminum 1.3 mm Aluminum Figure 5.28: Global plastic energy dissipation for the 1.3 mm thick and 5.8 mm thick aluminum samples. properties for a single lamina are listed in Table 5.5. The epoxy matrix andthe carbon fiber fabrics are not modeled separately. In current numerical model, the composite is modeled as an anisotropic elastic material, with neither a detailed plasticity model nor failure criteria about fiber failure, matrix failure and delaminations. This is regarded as the first step of a full understanding for the dynamics of such type of composite materialundershock focusing events. ρ [kg/m 3 ] E 11 [GPa] E 22 [GPa] E 33 [GPa] 1660 77 75 11 ν 12 ν 13 = ν 23 G 12 [GPa] G 13 =G 23 [GPa] 0.07 0.336 6.5 6.9 Table 5.5: Material properties of the five-harness-satin carbon fiber fabric. The direction 1 corresponds to the warp direction of the fibers. The direction 2 corre- sponds to the fill direction. The direction 3 corresponds to the direction normal to the plane of carbon fibers. 96 2 1 1 3 2 2 2 2 2 2 1 1 1 3 3 21 3 1 2 1 1 3 3 1 1 3 2 2 1 3 3 2 2 1 3 1 3 2 2 2 3 1 3 1 3 2 2 1 1 1 1 3 3 3 2 3 3 3 2 2 3 1 1 3 2 3 2 2 1 2 3 3 1 2 2 1 2 1 1 2 3 3 2 1 1 1 1 2 1 3 3 1 3 1 3 3 1 3 3 2 3 1 3 2 1 2 1 1 3 3 3 1 3 2 1 2 1 3 3 3 3 3 1 1 3 3 X Y Z Figure 5.29: Material coordination for the outermost lamina. Thefiniteelement modelofthecompositesampleispartitionedfromthemodel used for the 5.8 thick steel sample. Sixteen layer are created, and each layer contains a lamina. The local material coordinate is defined based on the geometry of surfaces at each layer, see Figure 5.29. The orientation of the plies in each layer is following the experimental layup sequence ([0/±45/90] 2s ). 0 10 20 30 40 50 60 70 80 90 100 -5 -4 -3 -2 -1 0 1 2 3 x 10 -3 Time (μs) Strain Gauge1 (experiment) Gauge2 (experiment) Gauge1 (simulation) Gauge2 (simulation) Figure 5.30: Time history of true strain from both the experiment and simulation for the carbon fiber composite sample. 97 Numerical simulations with 48 m/s projectile speed was performed on the car- bon fiber composite sample. The numerical results are plotted in Figure 5.21 at the location of the two strain gauges in order to compare with experimental strain measurements. The time origin 0 µs is defined as the time instance of impact. The initial disturbance from the fast-going stress wave traveling in the carbon fiber tows are missing in the simulation results because the material modeling does not differentiate the fibers from the matrix. For the numerical results, the overall trend of changes in strain at both gauge locations are qualitatively similar with the experimental data, but the amplitude is half or less of experimental val- ues. The correlation between the experimental data and simulation results is also measured by Russell error, see Table 5.6. Because an area of delamination at the tip region is found after performing the experiments, one benefit of numerical simulation is to explain the cause of such phenomenon. Since there is no interlaminar modeling in the current simulations, sothelocationswheredelaminationislikelytooccurareinferredfromdeformation and strain measurements, but cannot be observed directly. Two plots at a time instance 98.2 µs are shown in Figure 5.31 (a) and (b). In Figure 5.31 (a), the original undeformed structure is shown in grey color and the deformed shape is demonstrated in green color using a magnified deformation scale of eight times. A clear stretching of the inner surface and shrinkage of the outer surface close to the tip region of the composite structure can be observed. Such deformation suggests an internal stress accumulation. By plotting the normal strain component in the horizontal direction, Fig. 8, this behavior can be confirmed. Positive values of the strain shown on the plot (red color close to the inner surface) indicate a tensile stress, which can initiate interlaminar cracks causing delaminations [94]. 98 (a) (b) Figure 5.31: Deformation plots at the tip of the carbon fiber composite specimen. a) Results from the numerical simulations showing deformation in the composite sample. b) Results from the numerical simulations showing strain levels in the composite sample. In the experimental visualizations, there is no precursor wave observed for the carbon fiber samples, because the shear wave in the sample is much slower than its metallic counterparts. The pressure field displayed for the water-filled region of both specimens is plotted using the same scale in both Figure 5.32(a) and (b) at a time instance 46.4 µs after the projectile impacts onto the specimens. As can be seen in Figure 5.32(a), there is a distinct pressure wave pattern showing up in the case of a 5.8 mm thick steel structure. This is also directly comparable with the experimental visualization inFigure5.2(f). When plottingthe pressure field inthe water for the composite sample, no precursor waves are visible in the simulations, see Figure 5.32(b). The simulation of the carbon fiber composite can predict basic features in the water andof thecomposite sample. Further modeling effort isnecessary to explain the bubble formation ahead of the main shock and damage to the sample. 99 (a) (b) Figure 5.32: Results from the numerical simulations showing pressurewavepat- terns. a) 5.8 mm steel sample; b)carbon fiber composite sample. The grey areas in the plots are regions having pressure higher than the color contour’s upper limit. Magnitude error Phase error Comprehensive error Gauge 1 (5.8 mm steel) 0.29 0.25 0.34 Gauge 2 (5.8 mm steel) 0.25 0.34 0.37 Gauge 1 (1.3 mm steel) -0.35 0.70 0.70 Gauge 2 (1.3 mm steel) -0.09 0.48 0.44 Gauge 1 (5.8 mm aluminum) 0.31 0.07 0.28 Gauge 2 (5.8 mm aluminum) 0.56 0.32 0.57 Gauge 1 (1.3 mm aluminum) -0.29 0.15 0.29 Gauge 2 (1.3 mm aluminum) 0.62 0.50 0.71 Gauge 1 (composite) 0.18 0.11 0.19 Gauge 2 (composite) 0.43 0.49 0.58 Table 5.6: Russell errors. 100 A list of Russell errors for different specimen materials and gauge locations is presented in Table 5.6. For comprehensive error (CE) lower than 0.15, the correla- tion between the experimental and numerical results is considered to be excellent. If0.15<CE≤0.28,thecorrelationisacceptable,whereCE>0.28isregardedaspoor correlation [41]. In the case of steel specimens, both gauges for the 1.3 mm thick specimen have a negative magnitude error because the experimental data are in general greater than the simulation results, while for the 5.8 mm thick specimen it is the opposite. Although results for the 5.8 mm thick specimen show better correlation than the 1.3 mm thick specimen, the agreement between the experi- mental and numerical results is in general poor under the criteria stated before. In the case of aluminum and carbon fiber composite specimens, the simulation result agrees well with the experiment data at gauge 1, but has a great deviation at gauge 2. 5.4 Summary Previous experiments studying impacts of UNDEX on thin solid structures rarely consider the complex fluid dynamics but mainly look at the dynamics and defor- mation of the solid structures. In our investigation, an advanced high-speed pho- tography technique has been applied to visualize the fluid dynamics together with simultaneous strain diagnostics to monitor the response of the solid structure. In this combined approach, insights of the characteristics in the fluid-structure inter- action process were gained. For all the samples in reduced thicknesses, the shock wave in water starts to accelerate earlier than the bulk samples discussed in Chapter 4. This suggests that when the shock wave reaches the focal region, the shell structures might be 101 subjected to higher transient pressures. For samples made from the same mate- rial, there is a general trend that the 1.3 mm thick samples oscillate at a higher frequency than the 5.8 mm samples and the precursor wave patterns are not the same between the two thicknesses. When looking at the effect of materials, the aluminum samples has larger strain and deformation when comparing with their steelcounterparts. Thestrainmeasurementresultofthecompositesampleiscloser to the 5.8 mm thick aluminum sample. Composite samples exhibit a fast cavita- tion event ahead of the main shock wave due to the high stress wave speed in the carbon fibers. Computer simulations are carried out by using Abaqus/Explicit 6.12. The numerical results are compared with the experimental strain data at the same locations. A good agreement between the experimental and numerical results was obtained. The shock wave in water reaches a peak pressure of 180 MPa when approaching the focal point. This suggests that even with a strong coupling between the structure and the fluid, logarithmic-spiral shape keeps the ability to focus the shock wave. For type 1018 steel, the strain rate term in Johnson-Cook model must be included to have a correct description of the plastic deformation. The numerical simulations help to elucidate some speculations in the analysis of the experimental results. The origin of curved precursor waves in the 1.3 mm thicksteelsamplewasthoughttobeflexuralwaves. Thisisconfirmedbynumerical results through plotting the deformation of the sample. After the impact test, the carbon fiber composite sample shows delaminations at the tip region. The high tensile strain level at the same region in the simulation results explains the root of the delamination. In conclusion, the logarithmic spiral shaped samples of finite thicknesses can maintain the ability to focus the shock wave. The strong coupling between the 102 solid structure and the fluid gives rise to intense precursor waves in front of the shockwaveforthemetallicsamplesandthepost-testmetallicsamplesexhibitlarge plasticdeformations. Whileforthelightweight compositesamples, faststresswave traveling in the carbon fibers induces cavitation in water at the tip region before the main shock wave arrives. The composite samples show less ductilitythanthe metallicsamplesbuthavecracksanddelaminationsformingclosetothetipregion. 103 Chapter 6 Conclusions Throughout this dissertation, we study the fluid dynamics of the shock focusing process inside a logarithmic-spiral-shaped solid structure and its interaction with the surrounding solid structures. The main contributions and conclusions will be summarized here. • Experimental setup for high-speed photography is built and tested in a laboratory environment. Quantitative BOS method is developed for two- dimensional shock flows in water. • Inorder tostudy navalstructures under extreme loadingconditions, the idea of using a logarithmic spiral to focus the shock wave [48] was utilized. The derivation of the logarithmic spiral shape has been modified to incorporate the Tait equation of state for water. • Bothexperimentalandnumericalstudiesarecarriedout,forthecaseofweak coupling between the fluid and the bulk solid structure. – Minimum reflection was observed during the experiments when com- pared to the study on convergent wedge-shaped duct [7]. This is an expected behavior of the shock wave traveling inside a logarithmic- spiral-shaped structure. – After the shock wave enters the water-filled region, its speed stays con- stant orevendecelerates byasmuchas5%duringthefirst 50µs. Itwill 104 accelerate again when approaching the focal region. Both experiments and simulations confirm this behavior. – The pressure distribution for the shock focusing event has been esti- mated with the BOS technique. It is the first time that BOS has been applied to pseudo-steady shock propagation in water. – When including the interaction between the water and its surrounding materials in computer simulations, it was found out that the peak pres- sure achieved at the focal region is lower than that obtained from the ideal rigid boundary simulation, due to the energy dissipation through the surrounding materials. • Both experimental and numerical studies are carried out, for the case of strong coupling between the fluid and the solid structure with finite thick- nesses. – For all the samples of finite thicknesses, the shock wave in water starts to accelerate during the first 50 µs after entering the convergent water- filled cavity. – For the 5.8 mm thick steel sample, the precursor wave ahead of the shock wave is caused by the shear wave in the surrounding solid. When the thickness is further reduced to 1.3 mm, a flexural wave is excited and generates a precursor wave. The carbon fiber composite sample does not show any precursor wave but instead a back-propagatingwave is formed in front of the shock wave due to the high stress wave velocity inside the carbon fibers. – Russell error was utilized to correlate the numerical results with the experimental ones. It shows that the numerical calculations of strain 105 changes are in good agreement with the experimental results for the carbon fiber composite samples. But for the steel and aluminum sam- ples, the agreement between experiments and simulations are not as good. This might be due to the material model used for the two types of materials are empirical and obtained from literature. Further work is necessary to understand the difference. – The numerical simulation results help to explain the cause of delamina- tion in the carbon fiber composite samples. Conclusionsofthisresearchindicatethatconvergentsectionsonamarinestruc- turesmadefromlightweightcompositelaminatesmaybesubjectedtocrackgrowth and delamination at the focal region due to cavitation formation and structure motion. Therefore, enhancement inthetipregionisnecessary tokeep theintegrity of the composite structure. In contrast, if the convergent section is composed of ductile metallic materials, side walls are more likely to experience large deforma- tion during underwater explosions. Strengthening the side walls by increasing the thickness or strength of the metallic material would be preferable. Logarithmic- spiral-shaped sections should always be avoided where shocks might impinge and focus. This dissertation is farfroma complete investigation of shock focusing inwater and the associated material effects, but serves as a first step. In the future, devel- opment in the following areas may greatly help to improve the understanding of UNDEX and its material effects. • The BOS method described in Chapter 3 is highly suitable for samples with small span along the optical axis. This limitation requires reduction of the span of the sample to be much less than the distance from the background 106 plane to the sample. To further develop BOS to overcome this constraint and allow samples with larger span length will be useful for many future experiments. • Although for the strong coupling case, experiments and numerical simula- tions agree to some degree, a general observation is that all the strain mea- surementsfromthesimulationsaremoredampenedthanthoseobtainedfrom experiments. A careful bottom-up material modeling process could provide better predictions. The current model for the carbon fiber composite only considers the material as a anisotropic, homogeneous material andhasno yield or failure criteria. Improvements, such as including the modeling for each constituent or the modeling of fiber and matrix failure, will be of great importance. • Modeling of complex fluid phenomena such as cavitation is absent in cur- rent study. In order to correctly predict the location and size of cavitation, numerical simulations have to be extended. • Asdiscussed,advancedlightweightcompositestructureshavepotentialappli- cations in naval vessels. In current investigation only composite laminates of a complex geometry is studied. Sandwich composite structures, which are knowntohavesuperiorenergyabsorptioncapability,arealsoworthexploring in the future. 107 Appendix A There areseveral ways of calculating the speed of the shock wave. The most direct way is to measure the spatial locationof the shock wave at different time instances and the averaged velocity of the shock wave is readily obtained by u a = d t . (A.1) v a is the averaged shock velocity, d is the displacement of the shock wave between two frames and t is the time difference. θ α u s Solid Water Interface Shear wave front Figure A.1: Schematic for evaluating the shock wave speed in water. For the case that the shock wave in water interacts with a bulk polycarbonate surrounding material, thereisamoreaccurateway ofevaluatingtheinstantaneous shock wave speed. The shear wave in polycarbonate travels slower than the shock wave in water. A direct consequence will be that a oblique shear wave front forms 108 in the polycarbonate, see Figure 4.3(b). This characteristic can be utilized to calculated the instantaneous shock wave speed given the shear wave speed in poly- carbonate is known. The angle between the shear wave front and the interface is defined as θ andtheangle between thetangentialof theinterface and thedirection of the shock propagation is defined as α, see Figure A.1. The surface speed of the shock wave u ss can be expressed in the angle θ and the shear wave speed in solid, c s ,as u ss = c s sin(θ) . (A.2) The surface speed of the shock wave is related to the shock speed by u s =u ss cos(α). (A.3) Combining Equations (A.1)–(A.2), the shock wave speed in water is related to the shear wave speed in the solid as u s = c s cos(α) sin(θ) . (A.4) Thus, the shock speed in water can be determined from measuring the angles θ and α for interfaces of any geometric shape. 109 Appendix B Toensureeveryimpacttestwillgenerateanormalshockwithnoobliqueanglewith respect to the sample-water interface, a two-part alignment system was designed and machined, see Figure B.1. Both parts are made from a cast iron washer set, which have a large impact strength. The female part stays onthe gas gun close to the exit. The sample are securely fastened onto the male part. The two parts can be easily connectedbyfourdowel pins (6.35 mm diameter). The assembly of this alignment system is shown in Figure B.2. The female part can also stop the projectile from its further impact to the sample. 110 6.35mm 61.20mm 35.56mm 35.56mm (a) Male part Test sample (b) Figure B.1: Two-part alignment system. a) Female part. b) Male part and the sample. 111 1 2 3 4 5 FigureB.2: Photographyofthealignmentassembly beforetesting. (1)Gunbarrel. (2) Male part. (3) Female part. (4) Dowel pin. (5) Sample. 112 Appendix C In order to quantify the correlation between simulation results and experimental data, Russell error is one option [92]. The error measure utilizes two sets of time historyinputdatawiththesametimessteps. Eachsetoftheinputdataisregarded as a vector. The magnitude error considers the magnitude difference between the two vectors of the input data and the phase error is measuring the phase shift of one vector regarding to the other. The comprehensive error factor combines the magnitude error and phase error to evaluate the overall difference. The magnitude error is defined as " m = sign(m)log 10 (1+|m|), (C.1) in which m = (Σs 2 i −Σe 2 i ) ( Σe 2 i Σs 2 i . (C.2) 113 sandearetwosetsofdatafromthesimulationandexperiment respectively. Index i labels each data point in the data set and the summations are over index i.The phase error is defined as " p = 1 π cos −1 ) Σe i s i ( Σe 2 i Σs 2 i * . (C.3) The overall comprehensive error factor can be expressed as " c = π 4 + " 2 m +" 2 p . (C.4) 114 Bibliography [1] B. W. Skews and J. T. Ashworth, “The physical nature of weak shock wave reflection,” Journal of Fluid Mechanics, vol. 542, pp. 105–114, 2005. [2] D. T. Hart, “Ship shock trial simulation of USS Winston S. Churchill (ddg- 81): Surrounding fluid effect,” Master’s thesis, Naval Postgraduate School, 2003. [3] K. G. 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Abstract (if available)

Abstract During events of underwater explosions, the resulting shock waves poses extreme mechanical loadings to the nearby naval structures. Therefore this dissertation studies the shock wave dynamics of water inside a solid structure and the dynamical response of such a structure. The goal is to understand the effect of underwater shock wave focusing on materials possessing various mechanical properties with an emphasis on lightweight materials. This research has a direct impact on the material selection and hydrodynamic considerations in naval architecture. ❧ Two steps are taken towards the research goal. First, a special geometry of the confinement structure called logarithmic spiral is chosen. This type of shape will help to focus the shock wave, which will yield the maximum energy at the focal region. Numerical simulations are conducted to confirm the derived characteristics of the geometry. When including weak material coupling with the liquid, both experiments and simulations demonstrate that the shock dynamics in water is sensitive to the coupling. ❧ Secondly, significant coupling effect is introduced by reducing the thickness of the solid structure. Experimental and numerical investigations are both carried out to shed light onto the details of the fluid-structure coupling. The results revealed that the thickness of the material has the most significant impact on both the fluid dynamics and the deformation mechanisms of the structure. Lightweight carbon fiber composite structures are also studied under the same framework. ❧ Along the way, this dissertation also proposes and designs experimental methodologies to enable the study of highly dynamic underwater events.

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Creator Wang, Chuanxi (author)

Core Title Converging shocks in water and material effects

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 10/14/2013

Defense Date 09/06/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag fluid-structure interaction,OAI-PMH Harvest,Schlieren optics,shock focusing,underwater explosion

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Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Eliasson, Veronica (committee chair), Daeppen, Werner (committee member), Däppen, Werner (committee member), Haas, Stephan W. (committee member), Redekopp, Larry G. (committee member), Thompson, Richard S. (committee member)

Creator Email chuanxiw@usc.edu,chuanxiwang@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-338110

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Identifier etd-WangChuanx-2103.pdf (filename),usctheses-c3-338110 (legacy record id)

Legacy Identifier etd-WangChuanx-2103.pdf

Dmrecord 338110

Document Type Dissertation

Format application/pdf (imt)

Rights Wang, Chuanxi

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA