MARINE COMPOSITE PANELS UNDER BLAST LOADING
A Dissertation
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
Dushyanth Sirivolu
August, 2016 MARINE COMPOSITE PANELS UNDER BLAST LOADING
Dushyanth Sirivolu
Dissertation
Approved: Accepted:
______Advisor Department Chair Dr. Michelle S. Hoo Fatt Dr. Sergio Felicelli
______Committee Member Interim Dean of College Dr. Graham Kelly Dr. Eric Amis
______Committee Member Dean of the Graduate School Dr. Kwek-Tze Tan Dr. Chand Midha
______Committee Member Date Dr. Atef Saleeb
______Committee Member Dr. Dmitry Golovaty
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ABSTRACT
Composite and composite sandwich panels are being used and considered as an alternative to metal panels in ship structures due to their high stiffness- and strength- to weight ratio, improved corrosion resistance and low radar and magnetic signatures. In such applications they may be subjected to both in-air and under-water blast loading and it is important to understand the response of composite panels to blast loading.
Analytical solutions were developed to elucidate the deformation and damage initiation of composite and composite sandwich panels under blast. Three different problems were considered in this study: composite shells subjected to external pressure pulse, composite sandwich shells subjected to blast loading and composite sandwich plates subjected to air and blast loading.
The response of a double-curvature, composite shell under external blast was examined using Novozhilov non-linear shell theory and Lagrange’s equations of motion.
The predicted stable response of the shell was shown to compare well with FEA results from ABAQUS Explicit. The Budiansky-Roth criterion was used to examine the instability of the shell. It was shown that the dynamic pulse buckling strength of the shell could be increased by reducing the radius of curvature of the shell for a fixed span, or by increasing angular extent for fixed radius of curvature. Both these parameters triggered instability at higher buckling modes and were responsible for higher buckling strength.
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A multi-layered, analytical model was developed to study the blast response of double-curvature, composite sandwich panels with crushable, elastic-plastic PVC foam core. Plastic core crushing and energy absorption are important core properties for blast mitigation. The PVC foam core was modeled with isotropic and transversely isotropic properties. Predicted solutions using isotropic foam core was shown to be in good agreement with FEA results from ABAQUS Explicit. For sandwich shells with higher curvature and in-plane membrane resistance, lower blast resistance was found with transversely-isotropic foam core than with an isotropic foam core. The study suggested that modeling the sandwich core as an isotropic material, as is commonly done in practice, leads to non-conservative estimates in the structure’s ability to resist blast loads.
A fluid-solid interaction model was developed to examine the dynamic response of a composite sandwich plate subjected to air blast/air back, water blast/water back, and water blast/water back conditions. Reflected and radiated surface traction vectors were introduced to account for the plate motions at the interface of fluid and solid. The fluid damping terms resulted in substantially reducing and slowing down the deformation of the sandwich plate. This caused higher pressure loads to induce damage in the water blast/air back and water blast/water back panels when compared to air blast/air back panel. For the thick composite sandwich plates, the panels with the high density foam was more blast resistant in air blast/air back condition, while sandwich panels with lower density foam were more blast resistant in the water blast/air back and water blast/water back conditions. For thin composite sandwich plates, sandwich panels with lower density foam were most blast resistant in all blast loading conditions. It was also shown that the core plasticity in the air blast/air back panel was primarily due to transverse shear.
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However, in the water blast/air back, it was due to combined transverse compression and shear, and in the water blast/water back case, plasticity in the core was due to hydrostatic pressure and transverse shear.
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ACKNOWLEDGEMENTS
I want to convey my sincere appreciation and gratitude to my advisor Dr.
Michelle S. Hoo Fatt. During my studies at The University of Akron, she encouraged me to work on various research problems, provided me with an unlimited access at all times and more importantly, valuable guidance in completing this research. The accomplishment of this degree would be impossible without her consistent help and support.
I would like to thank Dr. Yapa Rajapakse, Solids Mechanics Program Manager at the Office of Naval Research, for financially supporting this project under the Grant
N00014-11-1-0485.
I would like to thank my committee members: Dr. Graham Kelly, Dr. Kwek-Tze
Tan from the Department of Mechanical Engineering, Dr. Atef Saleeb from the
Department of Civil Engineering and Dr. Dmitry Golovaty from the Department of
Theoretical and Applied Mathematics, for agreeing to be on my committee and providing me with valuable suggestions during the dissertation proposal. Their comments and suggestions have improved the quality of my research work.
I would like to thank the staff and faculty in the Department of Mechanical
Engineering for their help during my study at The University of Akron. A special thanks to Mr. Cliff Bailey and Ms. Bayaan Jundi for allowing me to use the Mechanical
Engineering computer lab and patiently helping me with computer/software related
vi issues. Also, I would like to thank Ms. Cortney Castleman for helping me with various office-related matters.
I would also like to thank my friends and colleagues both in the Department and outside The University. I would like to thank Preethi, Isaac and Ahamed for their help during my work. Outside the lab, Kranthi, Kalyan, Sandeep, Mani Harsha, Bharat, and many others have been very supportive and made my stay enjoyable.
I would like to thank my cousin Shravan Sirivolu, his wife, Roshni Sirivolu and their children for their love and support. Last but not the least, special thanks to my parents, Hari Kishan Sirivolu and Bharathi Sirivolu, and my sister Hithaswi Sirivolu.
There is no way that I could have done it without their love, encouragement, patience and understanding.
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TABLE OF CONTENTS
Page
LIST OF TABLES ...... xi
LIST OF FIGURES ...... xii
CHAPTER
I. INTRODUCTION ...... 1
II. RESPONSE OF DOUBLE CURVATURE SHELLS UNDER EXTERNAL BLAST LOADS …...... 9 2.1 Background ...... ……………………………………………...... 9
2.2 Problem Formulation ...…………………...... 13
2.2.1 Novozhilov non-linear shell theory …...... 13
2.3 Nonlinear Equations of Motion …………...... 16
2.4 Stable Forced Response ……………...... 18
2.4.1 Solution of Lagrange’s equations of motion ...... 19
2.4.2 Finite element analysis ………………………………..……...... 20
2.5 Dynamic Instability ………………………………………………...... 24
2.5.1 Critical buckling curves .…………………………………………...... 25
2.5.2 Influence of shell geometry ...……………………………………...... 27
2.6 Summary ….……………………………………………………………...... 32
III. BLAST RESPONSE OF DOUBLE-CURVATURE SANDWICH PANELS ...... 33
3.1 Background ...... ……………………………………………...... 33
3.2 Problem Formulation ...…………………...... 37 viii
3.2.1 Facesheet kinematics …………………...... 40
3.2.2 Core kinematics ………………………...... 41
3.2.3 Transient response of shell ……………...... 43
3.3 Predicted Response with Isotropic and Transversely-isotropic Core ...... 51
3.3.1 Isotropic foam core ...…………………...... 53
3.3.2 Transversely isotropic foam core ….…...... 57
3.4 Failure of Sandwich Shell …………...... 60
3.4.1 Facesheet failure …………………...…...... 60
3.4.2 Core failure ………………………...…...... 61
3.5 Influence of Transverse Isotropy ...………………………………...... 62
3.6 Summary …….…………………………………………………………...... 65
IV. WATER BLAST RESPONSE OF COMPOSITE SANDWICH PLATES ……...... 66
4.1 Background ………...... 66
4.2 Problem Formulation ...... 72
4.2.1 Acoustic pressure loading ...... 72
4.3 Structural Model ...... 77
4.3.1 Facesheet kinematics ...... 77
4.3.2 Core kinematics ...... 78
4.4 Equations of Motion ...... 80
4.5 Finite Element Analysis ………...... 86
4.6 An Example ...... 89
4.6.1 Structural response ...... 90
4.6.2 Cavitation ……...... 92
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4.7 Blast Resistance and Energy Absorption ...... 95
4.7.1 Effect of foam type ...... 98
4.7.2 Effect of plate aspect ratio ...... 99
4.8 Summary ….………...... 101
V. CONCLUDING REMARKS …….……………………...... 103
BIBLIOGRAPHY ...... 106
APPENDICES ...... 115
APPENDIX A. NOVOZHILOV NON-LINEAR SHELL THEORY ...... 116
APPENDIX B. CONVERGENCE OF DEFLECTION AND STRESSES WITH DOUBLE FOURIER SERIES ……………………………………...... 119
APPENDIX C. FINITE ELEMENT ANALYSIS FOR SHELL WITH ISOTROPIC FOAM CRUSHING ...... 122
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LIST OF TABLES
Table Page
2.1 Material properties of 0/90 Woven Roving E-glass/Vinyl Ester ...... 19
3.1 Material properties of 0/90 Woven Roving E-glass/Vinyl Ester ...... 51
3.2 Assumed isotropic properties of PVC H250 …………...... 52
3.3 Transversely isotropic properties of PVC H250 ……...... 52
4.1 E-Glass/Vinyl Ester Woven Roving facesheet properties ...... 89
4.2 Various foam properties ………………………………...... 89
4.3 Acoustic properties of air and water …………………...... 90
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LIST OF FIGURES
Figure Page
1.1 US Raven, Osprey class coastal minehunter, monocoque hull made of glass reinforced plastic material. Courtesy of Wikipedia.org ………………………….... 2
1.2 Swedish Visby-Class Corvette with hull made of Carbon/Vinyl laminate and PVC core. Courtesy of Wikipedia.org ..…………………………………..……….. 3
1.3 US DDG 1000 Zumwalt Class, guided-missile destroyer with composite superstructure section. Courtesy of www.naval-technology.com ...……………..... 3
1.4 Shockwave and oscillation of bubble pulse generated by an underwater blast [12] ………………………………………………………....…… 7
1.5 Destruction of an Australian warship by a torpedo 295 kg of high explosive. Courtesy of maritimequest.com .……………………………………….……..…… 7
2.1 Double curvature shell subjected to an external pressure pulse loading ……….... 14
2.2 Shell response for p0=0.9 MPa and 1 ms with various number Fourier components: (a) shell center deflection and (b) deflection profile at t=0.9 ms along y=b0/2 ………………………………………………………...... …….... 20
2.3 Finite element model of double curvature shell ………………………..……….... 21
2.4 Load-deflection response of shell under uniformly-distributed, quasi-static pressure load (post-buckling analysis) ..…………………………...... 22
2.5 Comparison of predicted and FEA stable vibration response with p0=0.9 MPa and τ=1ms: (a) shell center deflection and (b) deflection profile at t=0.9 ms along y=b/2 ..….………………………….….... 23
2.6 Determination of pcr when τ=1ms: (a) shell response with varying peak pressures and (b) stability curve indicating snap-through buckling …………………………….. 25
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2.7 Dynamic stability of Woven Roving E-Glass/Vinyl Ester shell: (a) critical buckling curve and (b) stability curves at 0.1, 1, 10 and 100 ms ...... 26
2.8 Woven Roving E-Glass/Vinyl Ester shell with fixed radius of curvature and various span (angular extent changes with span): (a) shell geometry and (b) critical buckling curves ……………………………………………………..... 27
2.9 Woven Roving E-Glass/Vinyl Ester shell with fixed span and various radius of curvature (angular extent changes with radius of curvature): (a) shell geometry and (b) critical buckling curves ..……………………..…….... 28
2.10 Deformation and buckling modes of shells (R/h=158.4) with a pulse decay constant of 1 ms: (a) a/h=55.3, 20 deg, (b) a/h=110.6, 40 deg, and (c) a/h=221.2, 80 deg ………...... 30
2.11 Deformation and buckling modes of shells (a/h=110.6) with a pulse decay constant of 1 ms: (a) R/h=316.8, 20 deg, (b) R/h=158.4, 40 deg, and (c) R/h=79.2, 80 deg ……………………………………………………….…….. 31
3.1 Geometry and loading of double curvature sandwich shell …………………….... 39
3.2 Elastic-plastic stress-strain response; strain energy in plastic region is area under OACD ………………………………………………………………….….. 48
3.3 Elastic and plastic regions in core mid-surface at t=0.25 ms ……………….…..... 54
3.4 Transient deflections at mid-surface of core along y=b0/2 assuming isotropic core ……………………………………………………………………... 55
3.5 Stress distribution in isotropic core mid-surface along y=b0/2 at t=0.16 ms: (a) in-plane stress components and (b) out-of-plane stress components ……….... 56
3.6 Equivalent stress distribution in isotropic core mid-surface along y=b0/2, at various times ..…………………………………………….………………….... 56
3.7 Stress distribution in facesheets of sandwich shell with isotropic core along y1=b0/2 or y2=b0/2 and at t=0.25 ms (maximum defection): (a) outer facesheet, top side, (b) outer facesheet, bottom side, (c) inner facesheet, top side, and (d) inner facesheet, bottom side ...…………….. 58
3.8 Comparison of panel center deflections at core mid-surface assuming isotropic and transversely isotropic core ..…………………………………..….... 59
3.9 Distribution of failure index in core mid-surface along y=b0/2 (F=1 denotes yielding and plastic zone): (a) isotropic crushable foam and (b) transversely isotropic Tsai-Wu foam ..………………………………….….... 59
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3.10 Failure modes in sandwich shell …………………………………………….….... 60
3.11 Variation of failure pressure (blast resistance) with sandwich shell curvature ratio (shell becomes flat panel as radius-to-thickness ratio increases) ..…….….... 64
4.1 Sandwich panel subjected to air/water blast conditions: (a) panel geometry and (b) acoustic loading on differential surface elements on front and back facesheets ……………………………………………………………………….... 73
4.2 Deformation of sandwich plate in: (a) elastic-plastic regime and (b) overlap regime ...... … 85
4.3 Assembly of (a) Air blast/air back condition ...……………………………….… 87 (b) Water blast/water back condition and (c) Water blast/water back condition .... 88
4.4 Transient deflections at the center and mid-surface of core for air blast/air back, water blast/air back, and water blast/water back panels …..………………...….... 91
4.5 Cavitation process in the water blast/air back condition: (a) incident wave at t<0 sec, (b) incident (blue line) and absolute pressure wave (red line) at t=0.05 ms, (c) incident and absolute pressure wave at t=0.07 ms, (d) incident and absolute pressure wave at t=0.3 ms and (e) incident and absolute pressure wave at t=0.6 ms ………………………...…. 93
4.6 Cavitation process in the water blast/water back condition: (a) incident wave at t<0, (b) radiated wave from back facesheet at t<0, (c) incident (blue line) and absolute pressure (red line) wave near front facesheet at t=0.25 ms, (d) radiated wave from back facesheet at t=0.25 ms, (e) incident and absolute pressure wave near front facesheet at t=0.56 ms, (f) radiated wave from back facesheet at t=0.56 ms, (g) incident and absolute pressure wave near front facesheet at t=1.6 ms, (h) radiated wave from back facesheet at t=1.6 ms …………………..………... 94
4.7 Predicted (a) failure pressure and (b) energy absorption for PVC H250 foam sandwich panel assuming an exponential decay time constant of 1 ms .……….... 96
4.8 Core mid-surface stresses at the onset of yielding (a) in-plane stresses and (b) out-of-plane stresses in air blast/air back condition ..………....…………….... 97
4.9 Core mid-surface stresses at the onset of yielding (a) in-plane stresses and (b) out-of-plane stresses in water blast/air back condition .…………..…..…….... 97
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4.10 Core mid-surface stresses at the onset of yielding (a) in-plane stresses and (b) out-of-plane stresses in water blast/water back condition ..………...... …….... 97
4.11 Predicted (a) blast resistance and (b) energy absorption of various sandwich panels for air blast/air back, water blast/air back, and water blast/water back conditions ..……………………. 99
4.12 Blast resistance per unit areal weight density for various sandwich panels for air blast/air back, water blast/air back, and water blast/water back conditions ….. 99
4.13 Predicted (a) Blast resistance per unit areal weight density and (b) energy absorption for sandwich panels with aspect ratio = 6.97 for air blast/air back, water blast/air back, and water blast/water back conditions .... 100
4.14 Predicted (a) Blast resistance per unit areal weight density and (b) energy absorption for sandwich panels with aspect ratio = 10.72 for air blast/air back, water blast/air back, and water blast/water back conditions .... 101
4.15 Predicted (a) Blast resistance per unit areal weight density and (b) energy absorption for sandwich panels with aspect ratio = 17.86 for air blast/air back, water blast/air back, and water blast/water back conditions .... 101
B1. Convergence of Fourier series for transverse deflection: (a) transient deflection at center of core mid-surface plane and (b) transverse deflections at mid-surface of core along y=b0/2 when t=0.25 ms .. 120
B2. Convergence of Fourier series for in-plane stress (1 or S11) at the bottom side of the inner facesheet: (a) stress history at clamped edge x2=a0, y2=b0/2 and (b) stress distribution along y2=b0/2 when t=0.25 ms ...... …….. 121
C1. FEA model of the sandwich shell ……………………………………...... …….. 122
C2. Mesh convergence for (a) transient deflection at center of core mid-surface plane and (b) S11 history at clamped edge x2=a0, y2=b0/2 of the bottom side of the inner facesheet ………………………………………………………………...... …….. 123
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CHAPTER I
INTRODUCTION
Composite panels are being increasingly used in marine applications and ship structures. Composite structures may consist of fiber-reinforced polymers (FRP) panels or can appear in sandwich configurations with FRP facesheets and crushable polymeric foam cores. For naval applications, E-glass/Vinyl Ester and carbon fiber/Vinyl Ester based facesheets and PVC foam core are extensively used. Vinyl Ester resins are used in ship structures because of their low water absorption properties, environmental resistance, relatively low cost and their suitability for vacuum infusion [1]. Lightweight
PVC foams are semi-rigid, closed-cell and cross-linked foams [2]. They have low water absorption and high deformation energy absorption capacity [3, 4]. The composite sandwich panels not only offer high strength - and stiffness-weight ratio and improved environmental resistance when compared to their traditional counterparts, but they can dissipate energy when subjected to blast loads. The cellular microstructure of the polymer foam allows it to attenuate forces and absorb energy as it crushes at almost constant flow stress. The collapse of the foam is associated with viscoelastic/viscoplastic cell wall buckling, fracture, friction and viscous air/fluid flow.
In the past, use of composite panels was restricted to river boats and minesweepers due to their low magnetic signatures. Figure 1.1 shows a USS Raven,
Osprey Class Coastal Minehunter. The monocoque hull is made of glass reinforced
1 plastic material. Nowadays, due to improved manufacturing techniques, reduced life maintenance costs, and because of the above-mentioned properties, composite materials are being used in several surveillance and battle warships. Figure 1.2 shows a Swedish
Visby-class corvette, a stealth patrol boat. The hull is constructed using a PVC core and carbon/vinyl laminate, thus reducing the structural weight and increasing the payload capacity. Reducing the structural weight resulted in high navigating speeds and reduced fuel consumption costs. Figure 1.3 shows the USS Zumwalt (DDG 1000) guided-missile destroyer. The superstructure of the ship which contains the radar system and control center is made of carbon fiber/vinyl ester skins and balsa core. Balsa core is cost- effective and less flammable when compared to PVC foam core materials. Reducing the weight of the superstructure provides increased weapons payload and better sea-keeping of these new class of destroyers. However, recent sequestrian of the US Government has led to the cancellation of building these ships.
Figure 1.1 USS Raven, Osprey class coastal minehunter, monocoque hull made of glass reinforced plastic material. Courtesy of Wikipedia.org
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Figure 1.2 Swedish Visby-Class Corvette with hull made of Carbon/Vinyl laminate and PVC core. Courtesy of Wikipedia.org
Figure 1.3 US DDG 1000 Zumwalt Class, guided-missile destroyer with composite superstructure section. Courtesy of naval-technology.com
During wartime/military drills, composite naval warships may undergo both air- blast loading and/or underwater explosions. They undergo bending/membrane deformations, core shear and display various failure modes. During an underwater blast, fluid-solid interactions and cavitation also play an important role along with the structural response. Although composite structures offer desirable properties, it is important to
3 understand the response of composite panels under blast loading for effective design and survivability of these panels.
The objective of this dissertation is to develop analytical models for the structural response and survivability of the composite shell and composite sandwich structures under blast loading. Such analytical models are benchmarked against more refined finite element analysis (FEA) using ABAQUS. The analytical models elucidate physical mechanisms that are responsible for blast resistance and mitigation during explosions in air and water. Three specific problems are of interest in this study: a. double-curvature, composite shells subjected to external pressure pulse, b. double-curvature, composite sandwich shells subjected to external pressure pulse and c. flat composite sandwich panels subjected to air and water blast. These three problems will be dealt with specifically in Chapters II, III and IV, respectively. Each chapter is organized in the same way. Each chapter will explain the previous work done in that area, provide a guide to problem formulation and the analytical methods involved. Finally, results will be discussed.
In Chapter II, double-curvature composite shells under external blast are studied using Novozhilov nonlinear shell theory and Lagrange’s equation of motion. Composite shells are commonly used in ship hulls, fuselages of airplanes, pressure vessel design and many other engineering structures. An important aspect in the design of shell structure is to reduce the thickness and weight of the structure. Composite shells when subjected to external pressure loading may result in excessive deformation of the structure. Due to in- plane membrane compression, composite shells can buckle or fracture, which in-turn will
4 lead to collapse of the structure. This kind of situation is undesirable and it is necessary to predict the critical buckling load.
Buckling is classified into static and dynamic buckling. Static buckling due to quasi-static loads involves lowest fundamental mode. When exposed to blast loading, composite shells may undergo dynamic instability. This specific type of dynamic instability is called dynamic pulse buckling and usually involves many higher modes [5].
Dynamic pulse buckling is characterized by pressure pulse loading and is different from vibration buckling, which is characterized by oscillatory or periodic loading. In vibration buckling, parametric resonance occurs when loading frequency is equal to one of the natural frequency of the structure. In dynamic pulse buckling, the structure undergoes large deformation with a small increase in the load [6-8]. These curved structures when subjected to transverse loads above some critical values will cause a change in the curvature of the shell structure and this reversal of curvature is called snap-through buckling [9] and is often undesirable.
Critical buckling curves for a given shell geometry is investigated using
Budiansky-Roth criterion [10] in Chapter II. The shell’s stable transient response is compared to the finite element analysis results using ABAQUS Explicit. A parametric study is performed to determine the influence of pulse duration, shell aspect ratio and angular extent on the buckling resistance of the shell.
Expanding on the work done in Chapter II, the blast response of a double- curvature, composite sandwich panel with polymeric foam core is investigated in Chapter
III. When the sandwich panel is subjected to dynamic loading, work due to the applied loads is transferred to kinetic and strain energy of the facesheets as well as the foam. The
5 foam crushes due to relative motion of the facesheets, undergoes significant deformation and absorbs impact energy. Thin shell theory is used in facesheet kinematics and thick shell theory for the core. Structural deformation of foam core includes elastic and elastic- plastic behavior. Conventional sandwich theory is extended to include elastic-plastic crushing of the core by incorporating it as a non-linear elastic behavior and plastic damping. Since the properties of foam core in the through-thickness direction vary from in-plane properties [11], two different core models are considered in this research: a. isotropic foam core and b. transversely isotropic foam core. The predicted transient response using isotropic foam core is compared to FEA results from ABAQUS Explicit.
Chapter IV describes the effect of fluid medium on the composite sandwich plate during underwater explosions. In underwater shock wave loadings and/or hull plummeting against waves, the inertial loading of the fluid on the structure is an important component of the analysis. Fluid-structure interactions take place with the fluid exerting pressure on the structure and thereby cause excessive deformation of the structure which in turn alters the flow of the fluid. These interactions may be stable or vibratory and failing to consider the effects of these interactions may not be conservative and in some cases, may be even devastating to the structures. Depending on the depth of the explosion, the bubble moves towards the surface during its oscillating stage and forms a plume or large bulk cavitation area, as shown in Fig. 1.4. This cavitated region causes a significant damage to the nearby structure. As shown in Fig. 1.5, an Australian warship was destroyed using a 295 kg high explosive well below its keel and a plume is caused by the collapse of a large gas bubble sucking sea water upward in a powerful jet.
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Figure 1.4 Shockwave and oscillation of bubble pulse generated by an underwater blast [12].
Figure 1.5 Destruction of an Australian warship by a torpedo 295 kg of high explosive. Courtesy of Maritimequest.com.
Underwater explosions include many variables and it is extremely difficult to validate all the variables with experiments. In Chapter IV, the response of a composite sandwich panel to a pressure pulse propagating in air or fluid environments is examined.
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Three different cases are analyzed: a. Air blast-Air backed (A-A), b. Water blast-Air backed (W-A) and c. Water blast-Water backed (W-W) panels. The fluid-structure interactions between facesheet and fluid are studied and the transient response of the composite sandwich panel is compared with ABAQUS Explicit results. Cavitation phenomenon is analyzed at the interface of sandwich panel and water, and also in the surrounding water. Chapter V concludes this work and insights are provided for future research.
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CHAPTER II
RESPONSE OF DOUBLE CURVATURE SHELLS UNDER EXTERNAL BLAST
LOADS
2.1. Background
Thin-walled composite shells are commonly used to construct lightweight aerospace, military, transportation and civilian structures. When exposed to blast, these shells may undergo dynamic instability instead of stable transient and vibratory response.
The specific type of dynamic instability referenced herein is dynamic pulse buckling [5].
This type of dynamic buckling is characterized by pressure pulse loading and is distinct from vibration buckling, which is characterized by oscillatory or periodic loading.
Simitses [13] discusses various approaches for determining the dynamic stability of structures under sudden loading and a more recent review on the topic, including experimental studies, can be found in Singer et al. [14]. In general the dynamic stability of structures can be addressed using direct equations of motion or the Budiansky-Roth approach [10], a total energy-phase plane or the Hsu approach [15], and the total potential energy or Simitses approach [13]. In the last two approaches, lower and upper bounds of critical stability conditions are established. The Budiansky-Roth criterion simply states that instability occurs when there is a large increase in deformation response from little or no increase of load, and it involves numerical solution of non-linear coupled equations of
9
motion. This is perhaps the most commonly used dynamic stability criterion for highly nonlinear systems which lack analytical stability criteria.
Dynamic pulse buckling due to lateral pressure pulse has been addressed for isotropic shell structures since the early 60's [16-19] and more recently for laminated composite shells, including composite sandwich shells [20-24]. In Ref. [22] the equations of motion governing transient motion of a thin composite cylinder under external pressure pulse loading resulted in a set of Mathieu equations for which the instability conditions are well-known. These Mathieu equations are due to parametric resonance when the hoop mode couples with specific bending modes of the cylinder. Clamped arches and single curvature shells exhibit more complicated vibration modes, and it is difficult to develop closed-form solutions for the stability of them. Numerical solutions in combinations with one of the aforementioned stability criteria are used instead. In Gao and Hoo Fatt [23], nonlinear equations of motion were derived for clamped, single- curvature laminated composite shells exposed to external pressure pulse loading. The buckling pressures were determined using the Budiansky-Roth criterion. It was found that thicker shells were more likely to fail by fracture during stable vibratory response rather than buckling, while thinner shells were susceptible to this mode of failure. The buckling strength could also be enhanced by adjusting shell layup. Gao and Hoo Fatt later extended this study to examine local facesheet buckling or wrinkling of a single- curvature, sandwich shell [24]. A sandwich shell, consisting of two thin facesheets and a compressible core, deforms globally like a thick shell while the thin facesheet may deform locally and undergo wrinkling.
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In this study, a more general case of pulse buckling of a thin, double-curvature composite shell that is fully clamped along its edges is examined. Solutions are given for shells of any angular extent, i.e., both shallow and deep shells, and for pulse durations ranging from impulsive to step loading. Novozhilov nonlinear shell theory [25] is used to describe the nonlinear strain-displacement relationships, which are expanded to quadratic terms, of a double-curvature shell. The quadratic or nonlinear terms must be included in buckling analysis because they are the only means by which destabilizing effects are introduced into the analysis. In Gao and Hoo Fatt [24], mid-surface strains and changes of curvature of a single-curvature shell were expanded to quadratic terms and this allowed for very good comparison of the transient shell response with ABAQUS Explicit results. It is believed that the same approach may be used to predict the transient response and dynamic stability of the double-curvature shell.
Alijani and Amabili [26] recently reviewed geometrically nonlinear shell theories as they apply to nonlinear vibrations and stability. In fact one may find an improved version of Novozhilov nonlinear shell theory in Amabili [27], although quadratic terms for the changes in curvature and twisting of the mid-surface are not given in this reference. As mentioned in the introduction, vibration buckling is another form of dynamic buckling, and it should be distinguished from pulse buckling, which is the topic of this study.
An interesting study on static buckling estimates of thin shells of revolution was done by Teng and Hong [28]. Teng and Hong proposed generalized nonlinear strain- displacement relations to garner numerically-predicted static buckling loads of thin shells of revolution and showed, by comparison, the differences in results from the use of some
11 traditional nonlinear shell theories [29-30]. Examples of traditional nonlinear shell theories include Maguerre [29] and Donnell [30] shell theories, which are specialized or limited to shallow shells, and Sanders [31] and Koiter [32] shell theories, in which only linear terms in changes in curvature and twisting of the mid-surface are retained. Teng and Hong concluded in their study that the omission of certain important terms was a major flaw in applying many of the traditional shell theories to buckling problems. A more recent comparison on the performance of nonlinear, thin shell theories to predict dynamic buckling of spherical shells under step loading was given by Shams and Porfiri
[33]. They found that the Sanders–Koiter shell theory to be accurate for all the considered shell thicknesses, while the Teng–Hong and Donnell shell theories only gave accurate results for the very thin shells. Dynamic stability under step loading was also examined by Ganapathi et al. [34] for a clamped, composite spherical cap. Their formulation was based on first-order shear deformation theory and geometric nonlinearity was introduced using von Karman strain–displacement relations. An implicit numerical scheme, employing the Newmark’s integration technique coupled with a modified
Newton–Raphson iteration, was used to solve the governing equations of motion.
In this work, the dynamic stability of a double- curvature shell under general pulse loading, which is described by an exponentially-decaying function of time. Critical buckling curves, combination of peak pressures and impulses, are specifically produced for a doubly-curved, fully fixed, thin 0/90 Woven Roving E-glass/Vinyl Ester shell following this solution. The transient shell response is found using Lagrange's equation of motion. This response is then compared to finite element analysis results using
ABAQUS Explicit only when the shell undergoes stable transient response. Dynamic
12 stability of the composite shell is investigated with the Budiansky-Roth criterion. A parametric study is performed to determine the influence of pulse duration, shell aspect ratio and angular extent on the buckling resistance of the shell.
2.2. Problem Formulation
Consider a doubly-curved laminated composite shell of radius R1, R2, subtended
angles 00 ,, and thickness h , subjected to uniformly-distributed pressure pulse loading as shown in Fig. 1. The external pressure pulse loading is defined as
t 0ep)t(p (2.1)
where p0 is the peak pressure, is the decay time constant and t is the time. The shell
deforms with mid-surface transverse displacement 21 t,,(w ), and tangential displacements 21 )t,,(u and 21 t,,(v ), where , 21 are Gaussian curvature coordinates defined at the mid-surface of the shell in Fig. 2.1.
2.2.1 Novozhilov non-linear shell theory
Novozhilov’s non-linear shallow shell theory for thin shells [25] is used to obtain the strain-displacement relations. His shell theory is valid when strains are small and rotations are arbitrary. The shell strains are given by
z 1m11 (2.2)
z 2m22 (2.3)
1212 m z 12 (2.4)
13 where m2m1 ,, 12m are the mid-surface strains and 21 ,, 12 are changes in curvature.
In obtaining mid-surface strains and changes in curvatures and twisting of the mid- surface, Novozhilov assumed 21 1Rz,Rz by virtue of the thinness of the shell.
Generalized equations for them are given in Appendix A.
t 0eptp
Figure 2.1 Double curvature shell subjected to external pressure pulse loading.
Novozhilov generalized strain-displacement expressions are expanded to quadratic or second-order terms in deformation as follows:
2 2 2 1 wu 1 1 wu 1 1u uw (2.5) m1 R1 R1 2 R1 R1 R 2 R1 R1
2 2 2 1 wv 1 1 wv 1 1v vw (2.6) m2 R 2 R 2 2 R 2 R 2 R1 R 2 R 2
14
1 1u 1v 1v wu 1 1u wv 12m R 2 R1 R1 R1 R R 21 R 2 R 2 (2.7) 1 uw 1 vw R1 R R 21 R 2
1 1 2 1w 1u 1 wu 1 1 wv 1 2 R R11 R1 R R11 R R R 211 R 2 1 2 1w 1u v RR w 1 2 1u vw 21 2 2 R1 R1 R 2 R R RR 2112 R 2 R 2 2 1 2 1u uw 1 3u 2v 2u 1 w 2 2 2 (2.8) R1 R1 R1 R1 R 2 R1 R R11 1 1 1v uu 3 uw 2 R R11 R 2 R1 R1 R1 w RR3 1 2u v RR2 w 21 21 2 R1 R 2 R1 R 2 R 2 R1
1 1 2 1w 1v 1 wv 1 1 wu 2 2 R R 22 R 2 R R 22 R R R122 R1 1 2 1w 1v u RR w 1 2 1v uw 21 2 2 R 2 R 2 R1 R R RR 2121 R1 R1 2 1 2 1v vw 1 3v 2u 2v 1 w 2 2 2 (2.9) R 2 R 2 R 2 R 2 R1 R 2 R R 22 1 1 1v vu 3 vw 2 R R12 R 2 R 2 R 2 R 2 w RR3 1 2v u R2R w 21 21 2 R 2 R1 R 2 R1 R1 R 2
15
2 2 wRR 2 1w 2vu 1w 1u v 21 12 RR 21 RR 21 RR 21 R1 R 2 RR 21 1 2 1u 2 1v vw 1 2 1v 2 1u uw 2 2 2 2 R 2 RR 21 R 2 R 2 R1 RR 21 R1 R1
RR 21 1 1u 1v 1u v RR2 21 1 1w w (2.10) RR R121 R 2 R 2 R1 RR 21 R 2 R1 2 RR 21 w 1 1u v R2R 21 1 uw RR RR R 22121 R1 RR 21 R 2 R1
RR2 21 1 vw RR 21 u v RR 21 R1 R 2 RR R R 2121
Amabili [27] has proposed a variant of the above mid-surface strains based on by
2 2 retaining Rz,Rz 21 and neglecting 1 Rz,Rz 2 instead. These have been applied with only the linear curvature and twisting retained to solve nonlinear vibrations of double-curvature shells [35], including shells with shear deformations [35]. Both strain- displacement forms are acceptable provided the shell is very thin and deflections are a few times the shell thickness. However, for large deflection analysis involving snap through buckling, the quadratic or second-order terms describing changes in curvature and twisting become necessary [24].
2.3. Nonlinear Equations of Motion
The boundary conditions for a shell that is clamped at the four edges are
w w w 0vu at ,0 and w 0vu at .,0 In order to be 0 0 consistent with the above clamped boundary conditions, deflections are expressed in by the following double Fourier series:
16
N M 2 2 m n mn cos1aw cos1 cos cos (2.11) 0n 0m 0 0 0 0
N M m n mn sinbu sin (2.12) 1n 1m 0 0
N M m n mn sincv sin (2.13) 1n 1m 0 0
Geometric imperfections are not considered in the above series. It should be mentioned that power law functions have also been used to describe displacement of a clamped shell by Abe et al. [37].
The dynamic response of the sandwich shell is found by satisfying Lagrange’s equations of motion: d T T U Q (2.14) dt q q q mn mn mn mn where T and U are the kinetic and strain energy of the shell, q mn are generalized
coordinates ( mnmn ,b,a and cmn), and q mnare generalized velocities, and Qmn are generalized forces.
The kinetic energy is
2/h 1 0 0 222 T 21 dRdRwvu dz 2 0 0 2/h (2.15) where is the mass density of the shell. Rotary inertia is neglected also because of the
thinness assumption, 21 1Rz,Rz .
17
For an orthotropic shell, the elastic strain energy is given by
1 0 0 2 2 2 U 11 m1 22 m2 A2AA 12 m2m1 A 1266 m 2 0 0 2 2 D2DD 2 dRdRD 11 1 22 2 12 21 1266 21 (2.16)
where Aij and Dij are the membrane and bending stiffness of the shell.
Generalized forces Qmn are obtained from virtual work W :
)W( Qmn (2.17) mn)q(
For a uniformly distributed external pressure pulse,
0 0 21 21 dRdRw)t,,(pW (2.18) 0 0
Hence,
0 0 2 2 m n cos1)t(pQ cos1 cos cos dRdR (2.19) mn 21 0 0 0 0 0 0 Note that the function for transverse deflection w is not orthogonal and this will lead to coupled, nonlinear equations of motion upon satisfying Lagrange’s equations of motion. The following section describes the forced response of the shell subjected to a pressure pulse that causes stable vibratory response.
2.4. Stable Forced Response
An orthotropic shell made of 0/90 Woven Roving E-Glass/Vinyl Ester with
thickness 5.2h mm, radii RR 21 396 mm and subtended angles 00 40 deg is considered. Laminate properties for the 0/90 Woven Roving E-Glass/Vinyl Ester
18 were taken from Boh et al. [38] and are shown in Table 2.1. This shell is subjected to external pressure pulse loading with peak pressure 0 9.0p MPa and time decay constant 1ms.
Table 2.1 Material properties of 0/90 Woven Roving E-glass/Vinyl Ester [38].
0/90 Woven Roving Material E-Glass/Vinyl Ester Density (kg/m3) 1,391
E11 (GPa) 17
E22 (GPa) 17
E33 (GPa) 7.48
12 0.13
23 0.28
13 0.28
31 0.12
G12=G21 (GPa) 4.0
G23=G32 (GPa) 1.73
G13=G31 (GPa) 1.73
2.4.1 Solution of Lagrange’s equations of motion
The coupled equations of motion were solved with MATLAB using ode23tb solver and the MATLAB parallel computing tool box. The MATLAB solver ode23tb is based on an implementation of TR-BDF2, an implicit Runge-Kutta formula with a first stage that is a trapezoidal rule step and a second stage that is a backward differentiation formula of order 2, and has no numerical damping. The absolute and relative tolerances
19 in the MATLAB program were chosen to be 1e-3 and 1e-3, respectively. The program ran for a step time of 3t ms with various numbers of terms in the double Fourier series.
Figure 2.2(a) shows the transverse deflection in the middle of shell when ,4MN
6MN , and 8MN . The corresponding deflection profile of the shell at 0.9 ms along 2by is shown Fig. 2.2(b). Thus it was concluded from these results that
6MN in the double Fourier series should provide adequate accuracy.
(a) (b)
Figure 2.2 Shell response for 0 MPa9.0p and 1 ms with various number Fourier components: (a) shell center deflection and (b) deflection profile at t=0.9 ms along .2by
2.4.2 Finite element analysis
Finite element analysis using ABAQUS Explicit Version 6.13 was used to corroborate results from the analytical model. The FEA model for the double curvature shell is shown in Fig. 2.3. The dimensions of the shell are the same as in the above example. All sides of the shell were encastre or fully fixed. Three-dimensional, continuum eight node (C3D8) or brick elements were chosen. Three solid elements were used in the thickness direction, and a total of 314,928 elements were needed to assure
20 accurate (converged) FEA results. Solid elements were used instead of shell elements in order to obtain the most accurate FEA prediction. Conventional and continuum shell elements were not used in this particular application because their nonlinear behaviors are very much dependent on the kinematics and constraints from which these elements were derived.
Figure 2.3 Finite element model of double curvature shell.
To benchmark the FEA model, the shell is subjected to a uniformly distributed, quasi-static pressure load. The pressure load is increased from zero to 0.6 MPa at a loading rate of 2.4 MPa/s. Equations of motion were solved in MATLAB using ode23tb solver with N=M=6. Figure 2.4 shows the comparison of the center deflection of the plate between the predicted and FEA. The shell loses the stability at approximately the same pressure load (0.45 MPa) and deflection (0.9 mm) in both cases.
21
Figure 2.4 Load-deflection response of shell under uniformly-distributed, quasi-static pressure load (post-buckling analysis).
The shell material was assumed to be orthotropic, linear elastic with properties of
E-Glass/Vinyl Ester Woven Roving. A uniformly distributed pressure, expressed by Eq.
(2.1) where 0 9.0p MPa and 1ms , was applied to the shell. A Dynamic, Explicit step with nonlinear geometry was run, and the default values for the linear and quadratic bulk viscosity of 0.06 and 1.2 were used.
Solutions for the mid-surface deflections of panel center are shown in Fig. 2.5(a), and these are compared to predictions using Lagrange’s equations of motion over a period of 3 ms. Overall the stable forced response from FEA and Lagrange’s equations of motion were found to be consistent in terms of center panel deflections and vibrations up to 1.5 ms. The maximum deflection predicted from nonlinear shell theory is 11.4 mm, which is over four times the shell thickness, and it is of the same magnitude as that obtained from FEA. One should not expect an exact correlation between the transient response using two-dimensional, nonlinear shell theory and three-dimensional
22
FEA with ABAQUS Explicit because of differences in formulations. In particular, the
3D solid continuum elements (C3D8) are not subjected to the plane stress assumption that is inherent in Novozhilov thin shell theory. Transverse or through-thickness properties and their effect on dynamic behavior are taken into consideration in the FEA, while they are ignored in the thin shell of the analytical model.
The deflection profiles at 0.9 ms along the panel ( 2by ) are also compared with respect to FEA and the Lagrange’s equations of motion methods in Fig. 2.5(b). At
0.9 ms, both FEA and Lagrange’s equations of motion give the same panel center deflection and appear to be the most synchronized with respect to dynamic mode so this is a good time to compare deflection shapes. Since the deflections are of similar shapes
(capturing the similar vibration modes), one can conclude from this comparison that
Novozhilov nonlinear shell theory and the associated equations of motion were accurate in predicting the transient shell response.
(a) (b)
Figure 2.5 Comparison of predicted and FEA stable vibration response with
0 MPa9.0p and 1 ms : (a) shell center deflection and (b) deflection profile at t=0.9 ms along .2by
23
2.5. Dynamic Instability
Dynamic instability of the shell was assessed using the Budiansky-Roth buckling criterion [10]. The application of this criterion requires solving the equations of motion for different values of peak pressure p0 , monitoring the significant change in shell responses and determining the critical peak pressure pcr . The time variation of center deflection of the shell with various peak pressures and a decay constant 1ms, are shown in Fig. 2.6(a). One may observe from this plot that there is a sudden change in vibratory behaviour between pressure amplitudes 1.5 MPa and 1.75 MPa, where the maximum deflection changes from 18 mm to 55.5 mm.
A plot of the peak pressure and the maximum deflection during the forced vibration response produces a stability curve for the 1 ms decay constant as shown in Fig.
2.6(b). Now it is clear to see from Fig. 2.6(b) that the shell has experience a sudden change in vibratory response when 0 MPa5.1p and 1 ms. This sudden “jump” in deflection is the onset of dynamic buckling and the pressure load at which this takes place is the buckling load. For pressure amplitudes above cr,p the shell experiences snap-through instability. After snap-through, the shell experiences stable response because of membrane stiffening.
24
(a) (b)
Figure 2.6 Determination of pcr when 1 ms : (a) shell response with varying peak pressures and (b) stability curve indicating snap-through buckling.
2.5.1 Critical buckling curves
The critical buckling curve allows one to examine the buckling strength of the shell for any load that is characterized by an exponential pulse. The exponential decay constant determines whether the pressure pulse loading is impulsive, dynamic or quasi-dynamic. Again consider the 0/90 Woven Roving E-Glass/Vinyl Ester shell thickness 5.2h mm, radii 21 RRR 396 mm and subtended angles
00 40 deg. The combination of critical peak pressures and impulses for decay constant 0.001, 0.01, 0.1, 1, 10 and 100 ms are shown in Fig. 2.7(a), where both axes are on logarithmic scale. The insert graph of Fig. 2.7(a) indicates how the load transients change with increasing decaying constant, . Notice that a general trend of this curve is that (a) the critical buckling pressure asymptotically approaches a limiting value when the pressure pulse decay constant is very large, i.e, step loading response and (b) critical buckling impulse asymptotically approaches a limiting value when the decay constant is very small, i.e., impulse response.
25
Stability curves used to determine the critical buckling curves at four different load decay constants are shown in Fig. 2.7(b). The longer duration decay time constant pressure pulses give a sharper indication of shell instability. As the decay constant decreases, the critical buckling pressure rises in value and the difference in deflections at the bifurcation point diminishes. At a decay constant of 0.1 ms, there is more of a change in the slope or stiffness the instability point. Results for decay constants less than 0.1 ms are not shown in this graph because the pressure loads are too high.
(a) (b)
Figure 2.7 Dynamic stability of Woven Roving E-Glass/Vinyl Ester shell: (a) critical buckling curve and (b) stability curves at 0.1, 1, 10 and 100 ms.
26
2.5.2 Influence of shell geometry
The buckling strength of the double curvature shell is controlled by two geometric parameters: span and curvature. A parametric study was done to examine both. In all
cases 21 RRR and 00 for simplicity. In the first study, the shell radius was fixed at R 396 mm,i.e. hR 158 4. . Three different spans were then considered by making 0 20,40,80 .deg A diagram explaining this parametric study is given in Fig.
2.8(a). It should be noted that a 20-deg shell is shallow, while an 80-deg shell is rather deep shell. Figure 2.8(b) shows critical buckling curves for the various span-to-thickness ratios or angular extent. Clearly buckling strength increases with increasing angular extent or depth of the shell.
(a) (b)
Figure 2.8 Woven Roving E-Glass/Vinyl Ester shell with fixed radius of curvature and various span (angular extent changes with span): (a) shell geometry and (b) critical buckling curves.
27
In the second parametric study, the span was fixed at a 276 5. mm, i.e.
ha 110 .6. Three different shell radius-to-thickness ratios were then considered. As indicated in Fig. 2.9(a), the shell angular extent changes in order to keep the same span.
Now the shell with the smallest radius of curvature is the deepest shell, and the shell with the largest radius curvature is the shallowest shell. Critical buckling curves for the three geometries are shown in Fig. 2.9(b), where it is clearly seen that buckling strength increases with curvature or decreasing radius of curvature for a fixed span. The 80-deg shell is a deep shell, and it resists more of the pressure from in-plane membrane action than the 40- and 20-deg shells. The above conclusion was experimentally verified by
Kumar et al. [39], when they conducted shock tube tests on carbon-epoxy shells.
(a) (b)
Figure 2.9 Woven Roving E-Glass/Vinyl Ester shell with fixed span and various radius of curvature (angular extent changes with radius of curvature): (a) shell geometry and (b) critical buckling curves.
An explanation for the increased buckling strength with increasing angular extent can be given by examining the deformation and buckling modes for each shell when they are subjected to a pulse with 1ms-decay in Figs. 2.10(a)-(c) and Figs. 2.11(a)-(c). The shells in Figs. 2.10(a)-(c) are taken from the first parametric study (varying a/h), while
28 the shells in Figs. 2.11(a)-(c) are from the second parametric study (varying R/h). The 20 deg shells show a definite snap-through buckling mode very similar to quasi-static buckling modes. However, the 40 and 80 deg shells buckle with higher mode numbers.
The 80 deg shell with a/h=110.6 and R/h=79.2 (Fig. 2.11(c)) actually buckles with Mode
4, while the 80 deg shell with a/h=221.2and R/h=158.4 (Fig. 2.10(c)) buckles with Mode
2 owing to longer span and higher radius of curvature to maintain the same 80 deg angle.
Critical buckling curves for the 80 deg shell with a/h=110.6 and R/h=79.2 are higher than the critical buckling curves for the 80 deg shell with a/h=221.2and R/h=158.4 regardless of the exponential decay constant of the pressure pulse.
The parametric study in Figs. 2.10(a)-(c) and Figs. 2.11(a)-(c) reveals that higher buckling modes are excited as the angular extent of the shell increases. Shams and
Porfiri [33] have looked at thin spherical shells subjected to step loading and found shell thickness can have an influence on dynamic buckling modes. They noticed that as the shell thickness increases, the wavelength of the buckling pattern decreases so that the shell buckles with fewer modes. Buckling with higher mode numbers is a type of wrinkling instability that is often observed in cylindrical shells under pressure pulse loading [5, 22]. These higher modes are responsible for the increased buckling strength because they are associated with higher bending resistance of the shell. The parametric study suggests that the dynamic pulse buckling resistance of a curved panel may be increased by increasing its angular extent or making the shell deeper, whether by increasing span for a fixed radius of curvature or decreasing the radius of curvature of the shell for a fixed span.
29
Mode 0, Snap-through 0 10 20
(a)
20 Mode 2 0 40
(b)
Mode 2 40
0 80
(c) Figure 2.10 Deformation and buckling modes of shells (R/h=158.4) with a pulse decay constant of 1 ms: (a) a/h=55.3, 20 deg, (b) a/h=110.6, 40 deg, and (c) a/h=221.2, 80 deg.
30
Mode 0, snap-through buckling
(a)
Mode 2 20 0 40
(b)
Mode 4 40
0 80
(c) Figure 2.11 Deformation and buckling modes of shells (a/h=110.6) with a pulse decay constant of 1 ms: (a) R/h=316.8, 20 deg, (b) R/h=158.4, 40 deg, and (c) R/h=79.2, 80 deg.
31
2.6. Summary
Dynamic pulse buckling of a fully clamped, double-curvature composite shell was examined in this work. Novozhilov nonlinear shell theory was used to capture the large deformation response of the shell under an exponentially decaying pressure pulse.
Equations of motion were derived using a Lagrangian approach. The clamped boundary conditions imposed on the shell resulted in coupling of nonlinear differential equations because of non-orthogonal modes in the transverse deflections. Predictions of the shell’s stable vibratory response were shown to compare relatively well with finite element analysis using ABAQUS Explicit.
Critical buckling curves were computed for a given shell geometry using the
Budiansky-Roth criterion. It was shown that the dynamic pulse buckling strength of a curved panel may be increased by increasing its angular extent or making the shell deeper, whether by increasing span for a fixed radius of curvature or decreasing the radius of curvature of the shell for a fixed span. Higher buckling modes are induced by increasing the angular extent of the shell or making the shell deeper, and these higher buckling modes are responsible for the increasing the buckling strength because they are associated with higher bending resistance of the shell.
32
CHAPTER III
BLAST RESPONSE OF DOUBLE CURVATURE SANDWICH PANELS
3.1. Background
Naval composite sandwich panels, consisting of fiber-reinforced polymer skins and crushable polymeric foam cores, may be exposed to blast or pressure pulse loading.
The performance of a sandwich panel under blast loading is primarily dependent on the geometry and material properties of facesheet and core. This literature review is divided into two categories: a. experimental research and b. analytical models.
In the past few years, several blast experiments were conducted on flat composite sandwich panels [40-48] and curved metallic sandwich panels [48-49]. However, there is no experimental work available in the open literature for the curved composite sandwich panels subjected to air blast loading. Hence, only experimental research on flat composite panels and curved metallic sandwich panels will be discussed. Langdon et al.
[40] conducted blast load experiments on circular clamped marine sandwich panels.
Composite facesheets were made of E-Glass/Vinyl Ester and two different foam core materials, Divinycell H80 and H200, were considered in the experiments. With increasing impulse, the following failure pattern was observed in the sandwich panels: delamination, fibre fracture, matrix failure, debonding, core damage, and facesheet fracture. The major energy absorption modes in the sandwich panel subjected to blast loading included delamination, core compression and fiber fracture. Tekalur et al. [41] 33 conducted shock loading experiments on composite sandwich panels with 3-D woven E- glass skins and stitched foam cores. The front facesheet deformation was independent of back facesheet in unstitched foam core sandwich panel, while there was little difference in the front and back facesheet deformation of the stitched foam sandwich panel. It was also found that stitched foam core sandwich panel performed better in delaying damage initiation. Wang et al. [42] investigated the performance of composite sandwich panels under simultaneous in-plane loading and blast loading. It was found that lower blast loads were transmitted into back facesheet in sandwich panels with foam cores having longer plastic-densification regions. This indicated that the back facesheet deformation depended on the core plasticity property.
Experimental studies in the field by Arora et al. [43-44], have indicated that front facesheet damage and core shear are the primary damage modes in the composite sandwich panels subjected to blast loads. Also, an elastic-plastic core is more representative of the blast behaviour of foams used nowadays in composite sandwich construction. While the current design of a sandwich panel under quasi-static loads prohibits plastic crushing of the core, plastic core crushing is inevitable under very high intensity loading, such as one due to a nearby explosion. Plastic core crushing is a desirable feature in blast mitigation. The importance of plastic core behaviour in the sandwich panels with metallic facesheets and PVC foam core subjected to blast loading was also confirmed by Hassan et al. [45]. Wang et al. [46] have conducted shock-tube tests on step-wise graded cores with different foam core crushing abilities to show that the blast resistance of a composite sandwich panel can be improved by staggering the foams such that the softest core, which would experience significant core crushing and
34 plasticity, is the first incident layer during transmission of the through-thickness shock wave from a blast. The blast resistance and energy absorption of the composite sandwich panel with step-wise graded cores can be even further enhanced with polyurea interlayers
[47].
In curved metallic sandwich panels, Shen et al. [48] analyzed the effect of curvature and observed three main differences between curved composite sandwich panels and flat panels subjected to blast loading: the reduction of impulse on the front facesheet of the sandwich panel, wrinkling of back facesheet as new failure pattern and three different deformation regimes against two regimes in flat sandwich panels. Qi et al.
[49] performed numerical analysis on curved metallic sandwich panels subjected to blast loading using LS-DYNA. The radius of curvature was found to affect the deformation response and the energy absorption capacity of the sandwich panel.
Analytical models were developed for both flat composite sandwich panels and curved composite sandwich panels. Several multi-layered sandwich shell theories that incorporate higher-order shell kinematics to address transverse core compressibility and shear deformations have been proposed to address blast loading on composite sandwich panels [50-53]. However, these solutions are restricted to elastic core behavior. In this study, the blast response of composite sandwich shells with crushable foam cores that exhibit elastic-plastic response is examined. Hoo Fatt and Palla [54] developed an analytical model to describe the response of circular clamped sandwich panels with PVC foam core subjected to blast loading. An elastic-plastic wave propagation model was used for core compression. The transient response and damage initiation of the sandwich panel compared well with the results from ABAQUS Explicit. Sandwich panels with
35 high dense foam were shown to perform better when measured with critical impulse to failure criterion. Hoo Fatt [55-57] developed analytical models to describe the deformation response of cylindrical, flat and single curvature composite sandwich panels.
A multi-layered sandwich model with crushable, elastic-plastic foam core was considered to develop analytical solutions for single-curvature sandwich shell under blast loading
[57]. Facesheet wrinkling was found to primary failure mode in sandwich panels with thin facesheets and strong foam cores.
The current work is concerned with the blast response of a shallow composite sandwich shell with elastic-plastic core. Specifically, an analytical model for the transient response and failure of a fully-clamped, double-curvature, composite sandwich panel with a crushable, elastic-plastic polymeric foam core when it is subjected to uniformly-distributed pressure pulse loading is developed. Finite element analysis using
ABAQUS Explicit is performed to compare model predictions with more refined three- dimensional numerical predictions. This analytical study shall elucidate the effect of core crushing as the sandwich shell undergoes transient deformation under blast loading.
Although many foams are transversely isotropic, they are often assumed to behave in an isotropic manner. In a flat sandwich panel subjected to lateral pressure loading, the foam core resists primarily transverse shear and compression. Modeling the core as an isotropic material with transverse or out-of-plane compression and shear properties is adequate for analysis of flat sandwich panels and often produces accurate solutions. However, in a curved sandwich panel or shell subjected to outer lateral pressure loading, the foam core must resist in-plane compression in addition to transverse shear and compression because of shell curvature. Such in-plane or membrane
36 compression may be responsible for local facesheet buckling if the sandwich shell has thin facesheets and a strong core [24]. In Ref. [58], it was shown that both in-plane and out-of-plane compressive normal stresses are about the same magnitude in the elastic core of a composite sandwich shell that is externally loaded by a pressure pulse. Since most structural foams are transversely isotropic [2, 11], this indicates that the in-plane foam properties would be needed to accurately determine the response of a composite sandwich shell, which carries much of the lateral blast load in membrane compression.
Solutions for the blast response of a composite sandwich shell with isotropic core crushing as well as a transversely isotropic core crushing are given and compared in this work.
3.2. Problem Formulation
The double-curvature, shallow composite sandwich shell is defined in Fig. 3.1 with facesheet thickness h and core thickness .H The mid-surfaces of the composite facesheets are defined with radius R,R,R and .R Curvilinear coordinates xyx 211 y2
z,y,x 111 and z,y,x 222 are defined with respect to the outer and inner facesheets, respectively. The sandwich shell is fully clamped along all edges, and is considered to be shallow; i.e., a shell with a rise-to-span ratio of less than approximately 1/5 [59-61]. Lim
[61] compared the accuracy of deep shell theories and shallow shell theories and found that the shallow shell theory is accurate for the shells with a subtended angle of not more than 400. This sandwich shell is subjected to uniformly-distributed pressure pulse of amplitude po and duration ,T which is given by
37
t 0 1p Tt0, )t(p T (3.1) otherwise,0
The above triangular pulse load is a simplification of the blast loading on a structure. In an actual explosion, charge weight, stand-off distance and reflected pressure waves affect the actual pressure distribution on the surface of the structure. An exponentially-decaying pressure time history is usually what is measured in blast experiments, but the triangular pulse load is a very good approximation of the initial stages of this loading history. It should be noted that the current work does not model an actual explosion, but rather gives a solution methodology that one can use to model blast response of a sandwich shell with an elastic-plastic core.
The composite sandwich shells considered in this work are composed of fiber- reinforced polymeric facesheets and PVC foam cores. The mass density of facesheets and core are f and c , respectively. The facesheets are considered to be orthotropic, linear elastic-brittle material, while the core is idealized as an elastic, perfectly-plastic material. The following assumptions concerning sandwich material behaviors are made in this study:
1. Foam cores do not experience appreciable crushing (core compression) before
facesheet fracture so that they can be described as an elastic, perfectly-plastic
material. This assumption is generally applicable to uniformly-loaded composite
sandwich panels with fiber-reinforced polymeric facesheets that are brittle
(fracture strains less than 5% [62]) because the facesheet would fracture long
before core densification. It may not be true for more localized loading, however.
38
2. There is perfect bonding between facesheet and core. The validity of this
assumption depends on adhesion between facesheet and core. In many cases a
good choice of adhesive and bonding practice ensures that there is perfect
adhesion between core and facesheets. Gdoutos and Daniel [63] have examined
failure modes of composite sandwich beam that were perfectly bonded and
subjected to four-point bend tests. They have documented core failure due to
transverse shear cracking along lines approximately 45 degrees to a transverse
cross-sectional plane; this was followed by cracking just under and parallel to the
bond line between facesheet and core [63]. Post-mortem sectioning of fully
clamped composite sandwich panels subjected to air-blast in Ref. [44] also
indicates similar shear cracking patterns in the core.
3. Strain rate effects in both facesheet and core material behaviour are neglected.
Under blast loading, facesheet and core materials experience high strain rates.
Fiber-reinforced polymer and polymer foam materials are best described by
dynamic constitutive relations since experiments indicate that they exhibit rate-
dependent behaviors [64, 65].
Figure 3.1 Geometry and loading of double curvature sandwich shell.
39
3.2.1 Facesheet kinematics
Donnell’s nonlinear shallow shell theory is used to obtain the strain-displacement relations in the outer facesheet ( 1i ) and inner facesheet ( 2i ):
z (3.2) ii xim,xx i
z (3.3) ii yim,yy i
z (3.4) iiii yxim,yxyx ii where the mid-surface strain and the change in curvature in the outer and inner facesheets are
2 u w 1 w i i i (3.5) i m,x x R 2 x i xi i
2 v w 1 w ii i (3.6) i m,y y R 2 y yi
u v w w i i i i (3.7) ii m,yx yi xi xi yi
2 w i (3.8) xi 2 xi
2 w i (3.9) yi 2 yi
2w 2 i yx ii (3.10) yx ii
40
This non-linear shell theory was chosen to describe facesheet strain-displacement relations because of its simplicity and accuracy for shallow shell [66].
3.2.2 Core kinematics
As indicated in Figure 3.1, the core mid-surface is located at mean radii:
1 R RR (3.11) x 2 1 xx 2
1 R RR (3.12) y 2 1 yy 2 and it is defined with a set of curvilinear coordinates .z,y,x Core deformations, u,w 0o and o ,v are assumed to be compatible with the facesheet deflections so that
ww w 21 (3.13) 0 2
uu u 21 (3.14) 0 2
vv v 21 (3.15) 0 2
The in-plane strain-displacement relations for the core are
z xm,xx (3.16)
z ym,yy (3.17)
xyxy m, z xy (3.18)
41 where the mid-surface strain and the change in curvature are
2 u w00 1 w0 ,mx (3.19) x Rx 2 x
2 v w00 1 w0 ,my (3.20) y Ry 2 y
u v w w 0000 (3.21) xy,m y x x y
y (3.22) x x
x (3.23) y y
y 1 11 v u x 00 (3.24) xy 2 RRxy xy x y
and y and x are rotations of plane sections about the x- and y-axes, respectively. The nonlinear terms may be neglected for thick cores if the sandwich shell deflections are small compared to the full thickness of sandwich shell. Transverse shear strains in the core are given by
w u 00 xz y (3.25) x R x
w v00 yz x (3.26) y R y
42
The following approximations are made for the rotations of plane sections and the radial compressive strains:
uu 21 y (3.27) H
vv 21 x (3.28) H
ww 21 z (3.29) H
Equations (3.27)-(3.29) are first-order approximations for rotations and compressive strains in the core. The first two assumptions regarding rotations of plane sections ignore deformation due to bending strains in the facesheets and suggest that plane cross-sections of the core rotate with a constant angle. The last assumption is equivalent to assuming a linear variation in radial deformations of the core. As a result of these assumptions, transverse shear and normal strains within the core are assumed to be uniformly- distributed through the thickness, i.e. they do not depend on z-coordinate. While there are more accurate, higher-order theories to describe core deformations [54-55], they were not chosen because of the resulting complexities that would have resulted in the setting up the following equations of motion.
3.2.3 Transient response of shell
Out-of-plane and in-plane deflections of the facesheets are expressed in double Fourier series as follows:
x2 1 y2 1 xm 1 yn 1 (3.30) w1 mn cos1a cos1 cos cos 0n 0m a0 b0 a0 b0
43
xm 1 yn 1 1 mnsinbu sin (3.31) 1n 1m a0 b0
xm 1 yn 1 1 mn sincv sin (3.32) 1n 1m a 0 b0
x2 2 y2 2 xm 2 yn 2 (3.33) w2 mn cos1d cos1 cos cos 0n 0m a0 b0 a0 b0
xm 2 yn 2 2 mn sineu sin (3.34) 1n 1m a 0 b0
xm 2 yn 2 2 mn sinfv sin (3.35) 1n 1m a 0 b0
where the outer and inner shell surface areas are projected on a rectangular plane a xb 00 , which is area associated with the mid-surface of the core. The Fourier series selected above satisfy the deflection and slope boundary conditions for a fully clamped sandwich panel. Unlike the case of pinned boundary conditions which is more commonly found in the literature, radial deformations in Eqs. (3.30) and (3.33) are non-orthogonal functions and this will be later found to lead to mode coupling in the equations of motion. The sandwich shell was considered to be fully clamped because it is not practical to pin a soft core whether in real life experiments or in FEA. Applying roller supports to simulate pinned boundary conditions may be possible for flat sandwich plates but membrane forces in a curved sandwich plate causes the facesheets to slide instead of rotate at the rollers.
44
The dynamic response of the sandwich shell is found by satisfying Lagrange’s equations of motion: d T T U Q (3.36) mn dt q mn q mn q mn where T and U are the total kinetic and strain energy in the sandwich shell and q mn are generalized coordinates, e,d,c,b,a mnmnmnmnmn and mn,f and Qmn are generalized forces.
Plastic work dissipated during core crushing is accounted for in the strain energy expression.
The kinetic energy of the two facesheets is
1 2 2 2 1 2 2 2 T 1ff 1 wvuh 1 dydx 2f11 2 wvuh 2 dydx 22 2 S 2 S 1 2 (3.37) where S1 and S2 are outer and inner facesheet mid-surface areas. The kinetic energy of the core is
1 2 2 2 1 3 2 2 T 0cc 0 0 dxdywvuH cH y x dxdy (3.38) 2 S0 24 S0
where S0 is the core mid-surface area. Since w1 and w 2 are non-orthogonal functions, the kinetic energy expressions involve coupling of Fourier modes after integration.
Generalized forces Qmn are obtained from virtual work W :
)W( Qmn (3.39) mn)q(
45
For uniformly distributed external pressure pulse,
w)t(pW 1 dydx 11 aQ mnmn (3.40) S1 0n 0m
Hence,
a 0b 0 x2 1 y2 1 xm 1 yn 1 (3.41) mn cos1)t(pQ cos1 cos cos dydx 11 0 0 a0 b0 a0 b0
The facesheets remain linear elastic even though the core may undergo inelastic deformation during crushing. The elastic facesheet strain energy is
1 U A 2 2 A2A A 2 2 2 D2DD D 2 dydx f 11 1 m,x 22 1 m,y 12 11 m,ym,x 66 11 m,yx 11 x1 22 y1 12 yx 11 66 yx 11 11 2 S1 1 A 2 A 2 A2 A 2 2 2 D2DD D 2 dydx 11 2 m,x 22 2 m,y 12 22 m,ym,x 66 22 m,yx 11 x 2 22 y2 12 yx 22 66 yx 22 22 2 S 2
(3.42)
Again it should be emphasized that use of non-orthogonal functions for facesheet radial deformations to satisfy the clamped boundary conditions leads to mode coupling after integration in the strain energy. Special cases of purely elastic and elastic-plastic response of the core are considered in the following sections.
46
Core elastic response
During fully elastic response, the stress-strain relation for an orthotropic core is
x 131211 000CCC x y 232212 000CCC y z 332313 000CCC z (3.43) yz 44 00C000 yz 0C0000 zx 55 zx C00000 xy 66 xy
where Cij are the elastic stiffness components. The elastic strain energy is therefore given by
1 2 2 2 2 2 2 Uc 11 x 22 y 33 z 12 yx 23 zy 13 xz 44 yz 55 zx CCCC2C2C2CCC xy66 dydxdz (3.44) 2 V0
where V0 is the volume of the core enclosed between the two facesheets. If the foam is
transversely isotropic with respect to the 1-2 plane, 1122 , 2313 , 4455 ,CCCCCC and
CCC 121166 .2 If the foam is assumed to be isotropic, then 112233 ,CCC CCC 121323
and 1211445566 .2CCCCC
Core elastic-plastic response
Cellular foam exhibits elastic-plastic behavior under both compression and shear
[2, 11, 67]. Yield criteria for an isotropic foam and transversely isotropic foam are described in this section. After initial yield, it is assumed that continued plastic flow or deformations can be represented by a nonlinear elastic stress-strain response.
Deformation theories of plasticity have been used extensively in practice because of their simplicity and are accurate as long as unloading does not occur. In addition to this, non-
47 hardening or perfectly plastic behavior is assumed to take place after initial yield. A representative stress-strain behavior for one component of the generalized three- dimensional stress-strain state of the foam is shown in Fig. 3.2 where ij0 and ij0 are the strain and stress components at the initial yield. The stress and strain components at initial yield are found by satisfying the yield criterion, as will be explained later. The physical soundness of plasticity deformation theory for loading paths other than proportional loading has been studied on the basis of Drucker’s postulate by Budiansky
[68]. These assumptions are made in order to avoid mathematical complexities associated with satisfying a three-dimensional plasticity flow rule and at the same time allow plastic response to be incorporated into Lagrange’s equation of motion.
B σij
A σ C ij0
Cij
O εij0 D εij
Figure 3.2 Elastic-plastic stress-strain response; strain energy in plastic region is area under OACD.
Isotropic yielding. One of the most commonly used criteria to describe plastic yielding is an isotropic foam yielding [67] of the form
ˆ F 1 (3.45) 0
48 where 0 is the flow stress and the effective stress ˆ is given in terms of the mean stress
: m and von Mises equivalent stress e
1 ˆ 2 2 2 2 (3.46) 2 e p m 1 3 and 23 if the foam plastic Poisson’s ratio is assumed to be zero.
Transversely isotropic yielding. Gdoutos et al. [11] have shown that under biaxial loading, failure of PVC H250 foam is best described by a Tsai-Wu failure criterion.
Foam failure in Ref. [11] is defined as fracture when the foam is subjected to tension and/or shear, while foam failure is considered to be initial yielding when the core is in pure compression and/or shear. Although this study was restricted to biaxial loading (or plane stress), it is assumed that the Tsai-Wu failure criterion could be applied to a general three-dimensional stress state for the PVC H250 foam. The initial yield criterion for the transversely isotropic foam is as follows:
2 2 2 2 2 3311 11 1 33 3 1355 1266 12 1 13 31 1X4X2XX2XX2XX2F (3.47)
2 1 1 1 1 1 1 1 where X1 X, 3 X, 11 X, 33 X, 55 , X Xct Z Zct XX ct ZZ ct S13
2 X 1 11 1 X66 X, 12 , X13 3311 ,XX and Z,X,X tct and Zc are the in-plane and S12 2 2 out-of-plane tensile and compressive strengths (subscripts “t” and “c” denote tension and compression).
49
Plastic response after initial yield. The strain energy in the plastic core regions are approximated by nonlinear elastic strain energy density enclosed in Region OACD of
Fig. 3.2. The area under OACD is equivalent to the area under OBD minus the area of shaded triangle ABC. Hence the core strain energy can be expressed by
1 2 2 2 2 2 2 Uc 11 x 22 y 33 z 12 yx 23 zy 13 xz 44 yz 55 zx CCCC2C2C2CCC xy66 dydxdz 2 Ve 1 2 2 2 2 2 2 11 x 22 y 33 z 12 yx 23 zy 13 xz 44 yz 55 zx CCCC2C2C2CCC xy66 dydxdz (3.48) 2 Vp 1 2 2 2 C110xx C22 0yy C330zz C2 12 0yy0xx C2 23 0zz0yy 2 Vp 2 2 2 C2 13 0xx0zz C44 yzyz 0 C55zxzx 0 C xyxy66 0 dydxdz
where Ve and Vp are the volumes of foam core undergoing purely elastic and elastic- plastic behavior.
The above expression is simplified as follows:
1 2 2 2 2 2 2 Uc 11 x 22 y 33 z 12 yx 23 zy 13 xz 44 yz 55 zx CCCC2C2C2CCC xy66 dydxdz 2 Ve (3.49) C11 0xx C22 0yy C33 0zz C12 0xy0yx C23 0yz0zy C13 0zx0xz Vp
C44 yzyz 0 C55 zxzx 0 C xyxy66 0 dydxdz
1 2 2 2 2 2 2 11 0x 22 0y 33 0z C2CCC 12 0y0x C2 23 0z0y C2 13 0x0z 44 yz0 55 zx 0 CCC xy66 0 dydxdz 2 Vp
When Eq. (3.49) is expanded in Fourier terms and integrated over the shell surface area, reduced stiffness and negative damping forces are introduced into Lagrange’s equations of motion as a consequence of the first and last integral terms. The last integral does not depend on generalized coordinates or Fourier coefficients and is only a function of the size of the plastic zone since it is integrated over p .V
50
3.3 Predicted Response with Isotropic and Transversely-isotropic Core
As an example, consider a composite sandwich shell with .5h 08 mm, H 25 4.
RR 726.44 RR 695.96 ba 248.25 mm, 1 yx 1 mm, 2 yx 2 mm, and 00 mm. The facesheet is E-Glass/Vinyl Ester Woven Roving and the core is Divinycell PVC H250.
Table 3.1 shows properties of the E-Glass Vinyl Ester. Elastic and plastic properties of the Divinycell PVC H250 are taken from data provided in Gdoutos et al. [11] and these are summarized in Tables 3.2 and 3.3 assuming the foam is isotropic and transversely isotropic. In Table 3.3, Xc,/Yc and Zc are the in-plane and out-of-plane compressive yield strengths, while S12 and S13/S23 are the in-plane and out-of-plane shear yield strengths.
The compressive stress-strain response in the through-thickness or transverse directions is used when assuming the foam is isotropic. This panel is subjected to uniformly distributed pressure pulse loading with load duration 1 ms and peak pressure 2.3 MPa.
Table 3.1 Material properties of 0/90 Woven Roving E-glass/Vinyl Ester.
E11=E22 (GPa) 17
E33 (GPa) 7.48
0.13 0.28 G12 (GPa) 4
G13=G23 (GPa) 1.73
Xc /Xt =Yc /Yt (MPa) 200/270
Zc /Zt (MPa) 343/23.2
S13=S23 (MPa) 31.6
S12 (MPa) 40
51
Table 3.2 Assumed isotropic properties of PVC H250.
E (MPa) 403
0.34
G (MPa) 117
(MPa) 6.3
Table 3.3 Transversely isotropic properties of PVC H250 [11].
E11=E22 (MPa) 236
E33 (MPa) 403
0.2
0.34
G12 (MPa) 85
G13=G23 (MPa) 117
Xc /Xt =Yc /Yt (MPa) 4.5/7.3
Zc /Zt (MPa) 6.3/10
S13=S23 (MPa) 5
S12 (MPa) 3.9
52
3.3.1 Isotropic foam core
Equations of motion were solved for the Fourier coefficients described in Eqs.
(3.30)-(3.35) using MATLAB ode45 solver with relative and absolute tolerances set at
1e-4 and 1e-6, respectively. The MATLAB ode45 is a variable time step solver, based on an explicit Runge-Kutta (4,5) formula, the Dormand-Prince pair. The Fourier series was
expanded to 6mn or a total of 242 Fourier terms in e,d,c,b,a mnmnmnmnmn and f mn because it was deemed that having these number of terms would provide the required accuracy for predicting both deformation and stresses in the sandwich shell. It is shown in Appendix B that the assumed double Fourier series converge rapidly for transverse deflection ( 2mn or 34 terms), but it would take 6mn or 242 terms to adequately obtain stresses in the facesheet and subsequently in the core.
The elastic-plastic response of the composite sandwich shell is obtained in two parts: (1) complete linear elastic response up to the point of initial yielding in the core and (2) elastic-plastic response as the plastic zone spreads in the core. The yield criterion, Eq. (3.45), must be evaluated during elastic response until it is met. Yielding in the core mid-surface was found to initiate and spread into the four cross-hatched line regions shown in Fig. 3.3. At initiation εij0 is stored and used to determine elastic-plastic response using Eq. (3.49). The elastic-plastic boundary shown in Fig. 3.3 occurs at 0.25 ms. Prediction of this boundary from FEA (dashed lines), which is discussed in
Appendix C, is also shown in Fig. 3.3 for comparison. The equations of motion during elastic-plastic response incorporated the spread of the plastic zone because it is introduced in the integration limits of the core strain energy in Eq. (3.49).
53
Figure 3.3 Elastic and plastic regions in core mid-surface at t=0.25 ms.
The transient deflections of the mid-surface of the core along 0 2by are shown in Fig. 3.4. The panel reaches maximum deflection at a value of 3 mm at 0.25 ms before rebounding. Solutions from finite element analysis in Appendix C, are also shown in these figures for comparison. The distribution of stresses at the mid-surface of the core along 0 2by were calculated from Eq. (3.43), and are shown at .0t 16 ms in Figs.
3.5 (a) and (b) for the in-plane and out-of-plane stress components, respectively. While the core mid-surface deflections are in very good agreement with FEA, stresses at the core mid-surface and in the facesheets are not captured as well, especially at the clamped boundaries. This is a consequence of the first-order approximations that were made for transverse shear and compressive strains in order to simplify the core kinematics.
54
Figure 3.4 Transient deflections at mid-surface of core along 0 2by , assuming isotropic core.
The magnitudes of the in-plane compressive stresses are greater than that of the out-of-plane compressive stress. This indicates that transversely isotropic foam properties would be needed to accurately determine the foam stress. The maximum out- of-plane or transverse shear stress is about twice the magnitude of the highest normal compressive stress suggesting that foam failure would be predominantly due to transverse shear. Before the foam fails, however, it would yield. Yielding is determined by the foam initial yield criterion. The equivalent stress as defined in Eq. (3.46) is also shown at various times in Fig. 3.6. There is relatively good agreement between predicted effective stresses and FEA. Yielding occurs when this equivalent stress is equal to the yield strength for the PVC H250 foam, which is 6.3 MPa. It was calculated that at t=0.18 ms, plasticity would just initiate at x=27.13 mm, 221.12 mm, and this is approximately the same as was found in the FEA. Plastic zones in the foam core will occur when the equivalent stress exceeds this value. The plastic zones are distributed as shown in Fig.
3.3 throughout the entire mid-surface. A similar plastic region also occurs in core mod- surface from the FEA.
55
(a) (b)
Figure 3.5 Stress distribution in isotropic core mid-surface along 0 2by at t=0.16 ms: (a) in-plane stress components and (b) out-of-plane stress components.
Yielding 0 MPa3.6 ,
Figure 3.6 Equivalent stress distribution in isotropic core mid-surface along , at various times.
Facesheet stresses in the composite sandwich shell at the maximum deflection are shown in Figs. 3.7(a) – (d). The stresses are shown at the top and bottom side of each facesheet where failure would most likely occur. Again the corresponding FEA stresses are shown for comparison, and there is relatively good agreement between the analytical model prediction and FEA. The highest in-plane compressive stress occurs at the edge of the panel on the bottom side of the inner facesheet. Compression failure of the facesheet
56 would be most likely to occur here. Simple failure criteria that can be used to predict the blast resistance of the sandwich shell are introduced in a later section.
3.3.2 Transversely isotropic foam core
To observe the effect of transversely isotropic properties in the foam, transversely isotropic properties of the foam from Table 3.3 were assumed instead. The load was kept the same as in the isotropic case, i.e., a peak pressure of 2.3 MPa and load duration of 1 ms. Again the elastic-plastic response of the composite sandwich shell was obtained for complete linear elastic response up to the point of initial yielding in the core and elastic- plastic response as the plastic zone spreads in the core. The Tsai-Wu yield criterion, Eq.
(3.47), was used to determine core plasticity. Yielding in the core mid-surface was also found to initiate and spread into the four regions similar to those shown in Fig. 3.3.
A comparison of the panel center deflection assuming isotropic and transversely isotropic elastic-plastic core is shown in Fig. 3.8. The panel reaches a maximum deflection of 2.6 mm at 0.23 ms assuming transversely isotropic core, while it was at a maximum deflection of 3 mm at 0.25 ms assuming isotropic core. The maximum deflection is considered to be just before the rebounding in both cases. A reason why the sandwich shell with transversely isotropic core rebounds earlier with a smaller deflection than the shell with the isotropic core has to do with changes in core plasticity when the
Tsai-Wu yield criterion is used instead of the isotropic crushable foam criterion. The failure index used to determine initial yield and the extent of the plastic zone is given in
Figs. 3.9 (a) and (b) for the shell with isotropic core and transversely isotropic core, respectively. Plasticity initiated in the isotropic core at 0.18 ms, while it initiated later at
0.22 ms in the transversely isotropic core. There is actually very little plasticity in the
57 transversely isotropic core before the panel rebounds. In fact, only the last 0.01 ms is spent in elastic-plastic core deformations.
(a) (b)
(c) (d)
Figure 3.7 Stress distribution in facesheets of sandwich shell with isotropic core along
01 2by or 02 2by and at t=0.25 ms (maximum defection): (a) outer facesheet, top side, (b) outer facesheet, bottom side, (c) inner facesheet, top side, and (d) inner facesheet, bottom side.
58
Figure 3.8 Comparison of panel center deflections at core mid-surface assuming isotropic and transversely isotropic core.
(a) (b)
Figure 3.9 Distribution of failure index in core mid-surface along 0 2by (F=1 denotes yielding and plastic zone): (a) isotropic crushable foam and (b) transversely isotropic Tsai-Wu foam.
Even though the sandwich shell with the transversely isotropic foam core has a lower maximum deflection, the inner facesheet stresses at peak deflections are slightly higher than the shell with an isotropic foam core. As in the case of an isotropic core, the highest in-plane compressive stress occurs at the edge of the panel on the bottom side of the inner facesheet. The maximum in-plane compressive stress at the panel edges of the bottom side of the inner facesheet is 187 MPa, while it is 178 MPa in the sandwich shell
59 with the isotropic core. The panel failure pressure or blast resistance would be therefore over-predicted using an assumption of isotropic crushable foam model if the core was indeed transversely isotropic.
3.4 Failure of Sandwich Shell
If the pressure pulse amplitude is high enough, the sandwich shell may fail by foam or facesheet fracture, and these are illustrated in Fig. 3.10. Assumption two in the problem formulation states that the facesheets are rigidly tied to the core, therefore delamination and debonding effects are not considered in this work. Only damage initiation in either the facesheet or core is considered in this work. Simple failure criteria are established for each mode below.
Figure 3.10 Failure modes in sandwich shell.
3.4.1 Facesheet failure
A modified Hashin-Rotem criterion is used to examine lamina failure of the woven roving E-Glass/Vinyl Ester [69]. More refined damage analysis of the facesheets during blast loading is provided in Ref. [70]. For the orthotropic shell, the relationship between the principal stresses and strains are given by
60
x 1211 0QQ x y 2212 0QQ y (3.50) Q00 xy 66 xy
where EQ 1111 /(1 ), EQ 22222112 /(1 2112 ), EQ 221212 /(1 2112 ), and
GQ 1266 .
According to the modified Hashin-Rotem failure criteria, the failure of the composite occurs when
x x 1 if x 0 or 1 if x 0 (3.51) X t Xc
y y 1 if y 0 or 1 if y 0 (3.52) Yt Yc and
x y 1 (3.53) S12 where X t and X c are the tensile and compressive strength in the x-direction; Yt and Yc are the tensile and compressive strength in y-direction; and S12 is the in-plane shear strength. Strengths for the E-Glass Vinyl Ester are given in Table 3.1. For the E-Glass
Vinyl Ester, in-plane compressive stresses were found to be highest at the clamped edges on the on the bottom side of the inner facesheet (see Figs. 3.7(a)-(d)). Thus compressive failure is predicted by either Eqs. (3.51) or (3.52).
3.4.2 Core failure
The stress distributions in the core shown Figs. 3.5 (a) and (b) indicate that transverse shear stresses dominate all other stress components, when and where yielding
61 begins. The points of yielding are also where the foam would tear if the strains exceed the foam ductility. Tearing occurs after yielding [2, 11]. Although core fracture is of a mixed-mode involving not just transverse shear, but also in-plane stress and out-of-plane compression, it is driven mostly by transverse shear deformations. A simple criterion for the onset of core shear failure is to set the transverse shear strain in the core equal to the
core transverse fracture strain, f . which for PVC H250 is 0.45. In other words, core shear failure occurs when
w u 00 xz y f (3.54) x R x and
w v00 yz x f (3.55) y R y
The transverse shear strains in the core were found to be highest approximately 25 mm from the panel edges, as indicated by the plastic zones of Fig. 3.3. Core shear failure occurs when this value reaches the core transverse fracture strain.
3.5 Influence of Transverse Isotropy
The effect of the transversely isotropic core properties would become more apparent when the sandwich shell radius of curvature decreases. A smaller radius of curvature or higher curvature leads to more in-plane membrane compression for the transversely externally loaded shell. A shell with a very large radius of curvature approaches a flat panel in which in-plane membrane stresses would be insignificant compared to out-of-plane bending and shear stresses. A parametric study was done in
62 order to see how the blast resistance of a composite sandwich shell would vary assuming that the core was isotropic and transversely isotropic. Failure could be due to either due to core shear or facesheet fracture, as defined in the previous section.
The parametric study was done on a sandwich shell with E-Glass/Vinyl Ester
Woven Roving facesheets and a Divinycell PVC H250 foam core. In order to isolate the effect of curvature, the facesheet and core thickness were kept at 5h mm and H 25 mm, and the span was also kept constant ba 00 248.25mm. This meant that the
aspect ratio of the sandwich shell, h/a tot0 where tot ,Hh2h would be fixed. The radius of curvature of the facesheets could then be varied accordingly, while still maintaining shallow shell assumptions. A curvature ratio was defined as h/R tot0 , and it was allowed to vary from 10 to 50. This range of curvature ratio translated to shell angular extent that ranged from 7 to 20 deg, thereby ensuring that the shallow shell assumptions were valid.
Figure 3.11 summarizes the results of this parametric study involving blast resistance and curvature ratio for shells with isotropic and transversely isotropic cores. In all cases, compressive failure of the back side of the inner facesheet was found to be the mode of failure. The PVC H250 core did not exhibit transverse shear failure. The failure pressure generally decreases as the curvature ratio increases or as the panel becomes flatter. This is to be expected because shells are designed to carry greater transverse loads through the membrane compression. When h/R tot0 30, there is very little difference in the failure pressure when the core is assumed to isotropic and transversely isotropic. Essentially both sandwich shells are acting like a flat sandwich panel and there
63 is little membrane stress transmitted in the core. However, as h/R tot0 becomes smaller than 20, the blast resistance of the sandwich shell with the isotropic core is seen to be higher than that calculated for the sandwich shell with a transversely isotropic core. One may observe from Table 3.3 that in-plane stiffness and yield strengths of the transversely isotropic foam are lower than in the isotropic foam, and as a result the sandwich panel fails at a lower load under the assumption of transversely isotropic core.
Although the difference is small, about 6%, the results indicate the importance of using the correct in-plane foam properties when addressing blast resistance of applying composite sandwich shells. The sandwich shells considered in this parametric study were geometrically restricted to shallow shell assumptions, but even greater deviation in the blast resistance with isotropic and transversely isotropic core would be expected of a deep sandwich shell which has greater in-plane membrane contributions. From a practical engineering standpoint, this implies that the isotropic core assumption which is widely-used in the literature has the potential to give very non-conservative estimates of blast resistance in composite sandwich shells.
Figure 3.11 Variation of failure pressure (blast resistance) with sandwich shell curvature ratio (shell becomes flat panel as radius-to-thickness ratio increases).
64
3.6 Summary
An analytical model for the blast response of a double-curvature, composite sandwich panel with a crushable foam core was developed. Lagrange’s equations of motion were used to describe the sandwich shell response with an isotropic and a transversely isotropic elastic-plastic core. The predicted transient response, including deformations and stresses in facesheets and core, was found to be in very good agreement with ABAQUS Explicit results assuming an isotropic crushable core.
The elastic-plastic analysis was used to determine the blast resistance of a sandwich shell with E-Glass/Vinyl Ester Woven Roving facesheets and a Divinycell PVC
H250 foam core. Panel failure due to transverse shear cracking of the core and facesheet failure were considered. All the sandwich shells considered in the study were found to fail by compressive failure of the inner facesheet. A parametric study indicated that the blast resistance of the sandwich shells increases as the curvature ratio decreases because shells are designed to carry greater transverse loads through the action of membrane compression. The parametric study also showed that the blast resistance of the sandwich shell with the isotropic core became higher than that calculated for the sandwich shell a transversely isotropic core as the shell radius of curvature decreased. This not only indicated the importance of using the correct in-plane foam properties for some shell configurations but it also implied that modeling the core of the sandwich shell as isotropic would give non-conservative estimates of the structure’s ability to resist blast loading.
65
CHAPTER IV
WATER BLAST RESPONSE OF COMPOSITE SANDWICH PLATES
4.1 Background
The response of submerged structures to blast loading is of great interest to the naval industry. Explosions underwater and near the water surface may cause serious damage to ship structures. It is important to incorporate shock wave propagation and fluid-structure interactions in the design of naval structures against blast loading. Early theoretical work dates back to World War II and most of the research concerning this topic is with metal ship structures [71 -72]. Taylor [73] studied the response of a rigid plate subjected to an exponential decaying shock wave in water and found that the impulse transmitted to the plate is reduced by reducing the mass of the plate. Another important aspect during underwater explosions is the formation of cavitation zones at the fluid-structure interface or in the surrounding water. Kennard [74] studied the cavitation zones in the elastic fluid. Two breaking fronts start propagating in opposite directions when the pressure drops below the cavitation limit in the liquid. The breaking fronts may invert their direction and become closing fronts depending on the pressure of the fluid.
Recently, Liu [75] extended Taylor’s formulation by considering both air backed plates (ABP) and water backed plates (WBP). The WBP experienced less peak pressures and momentum transfer when compared to the ABP. The effect of cavitation was investigated and it was found that the cavitation is more relevant in the ABP than WBP,
66 and if cavitation exists in WBP, it is longer than ABP. Extending this work, Young [76] investigated the response of composite structures during underwater explosions
(UNDEX). A two-dimensional Eulerian-Lagrangian numerical method was used to capture the non-linear FSI and solid equations of motion and constitutive equations were solved using a user-subroutine in ABAQUS Explicit. Different cases were studied to verify Taylor’s FSI effect, including bending/stretching, core compression and boundary effects. Cavitation effect was studied and it was concluded that the influence of cavitation is small when compared to the shock-bubble interaction.
Experimental techniques to study the water-blast loading include: a. explosives and b. shock tubes. Explosives yield spherical waves which are spatially complex and difficult to capture. Considering the complexity in conducting blast experiments, shock tube experiments were built to understand the physical phenomenon of underwater explosions. Ramajeyathilagam [72] performed experiments on rectangular plates under air-backed conditions in a water tank. Numerical analysis was performed using the
MARC software. The effect of strain–rate was studied and it was concluded that the strain-rate plays an important role during underwater explosions. Several authors studied the response of flat metallic plates against water blast loading [77 - 79]. Espinosa [80] conducted experiments on steel plates using shock tube setup. The experimental setup was benchmarked by numerical simulations using ABAQUS Explicit. The Johnson-
Cook plasticity model was considered for steel plates and fluid was modelled using Mie-
Gruneisen Equation of State (EOS) in ABAQUS. The deformation response from
ABAQUS was in good agreement with experimental results, thereby indicating good understanding of cavitation phenomenon in FE simulations.
67
Mouritz [81] conducted experiments to study the flexural properties of GRP laminates due to shock wave produced by underwater explosion. To benchmark the flexural properties, the laminate base material was tested using a four-point bend test.
Following each shock test, flexural properties of the laminate were again measured using a four-point bend test in order to assess damage. The amount of damage in the laminate increased with increase in peak pressure. It was also found that the damage pattern in the blast test was consistent with the damage pattern in the static test. LeBlanc [82] conducted experiments on E-Glass/Vinyl Ester curved composite panels subjected to underwater shock loading. Computational analysis was performed using LS-DYNA.
Delamination damage and failure criteria were included in the numerical modeling, and the results were in close agreement with experimental results. Batra [83] developed a finite element code to study the response of fiber-reinforced composites under water blast loading. Parameters effecting the performance of the laminate such as fiber orientation, delamination, debonding and strain-rate effects were investigated using the code.
Damage initiation process was found to vary with the change in constituent properties of the laminate. The numerical code was restricted to uni-directional laminates and is not valid for woven or stitched composites.
Several studies have shown that sandwich configurations will outperform monolithic plates of same mass under blast loading. Numerous investigations were done to study the response of metallic sandwich panels subjected to underwater blasts [84 -
88]. Despande [85] studied the response of one-dimensional sandwich plate to underwater shock wave. Two different core models were used: a. core as a monolithic plate and b. shock wave propagation through the core. A lumped parameter model was
68 used to predict the fluid-structure interactions accurately at the first cavitation but was not adequate to predict the effect of wave propagation within the cavitated fluid. Xue [86] studied the effect of sandwich plate construction due to air and water blasts. Three different core types were used with metal plates. Quasi-static analysis and dynamic analysis were performed using ABAQUS FEA. Sandwich plates were found to sustain larger blasts loads than the solid plate of equal mass in both air and water blasts.
Latourte [89] conducted experiments on monolithic composite, symmetric and asymmetrical composite sandwich panels using blast simulator. Divinycell PVC H250 foam was used as core material in sandwich panels. The deflections in the sandwich panels were reduced considerably when compared to the solid panel confirming the weight advantages. A linear relationship between central peak deflection versus applied impulse per areal mass was established for both monolithic and composite sandwich panels. Avachat [90] performed tests on sandwich plates consisting of glass fiber reinforced epoxy skins and Divinycell PVC H100 foam core material. It was found that the structures with facesheet-thickness- to-core-thickness ratio varying between 0.15 and
0.4 were most efficient for energy dissipation capacity and structural rigidity.
Hua et al. [91] conducted experimental and numerical analysis on composite sandwich panels subjected to blast loading. Air was modelled as an ideal gas equation and fluid-structure interactions were simulated using the FEA model. Parametric studies revealed that the deformation of back facesheet may be reduced by either decreasing the intensity of the blast or by increasing the facesheet thickness or core thickness. Sprague
[92] used cavitating acoustic finite elements (CAFE) model to examine fluid-structure interactions in a one-dimensional fluid column. A two degree-of-freedom spring-mass
69 system was used to represent the structure. Various mass ratios and discretization parameters like time increment, and element size were considered in the study. While cavitation was found to depend on the discretization parameters, structural response was independent of these discretization parameters. Shin [93] studied the dynamic characteristics of ship at component and sub-component level. Doubly asymptotic approximation (DAA) approach was used to model the underwater fluid-structure interaction. Important parameters during underwater explosions were discussed. A three- dimensional coupled ship and fluid model was developed using LS-DYNA. Simulation results matched well with the test data. Gong [94] performed finite element analysis on the response of floating structures to underwater shock. Benchmark tests was conducted on a plate and a two-dimensional structure involving plane symmetry. Two different hull structures were considered: a. single hull with coating and b. double hull with an interlayer. The two layered structure had significant attenuation on the shock wave response. It was also established that the damping material should have light weight, and high modulus to resist shock loading. Wei [95] simulated fluid-structure interactions using acoustic-Lagrangian based finite element code in ABAQUS Explicit. Composite sandwich panel with Divinycell PVC H250 foam were subjected to shock wave loading and a user-defined subroutine, VUMAT was implemented to include laminate damage criteria.
Makinen [96] studied the transverse response of a one-dimensional sandwich model with water on one side and air on the other to an underwater shock wave. A wave propagation method was used to find the presence of cavitation zones. Two cavitation zones were identified, one at the interface of the sandwich structure and the fluid and
70 other ahead of the structure. The effect of cavitation was found to be more important in the sandwich structures than homogenous structures due to the presence of weak core.
Schiffer [97] studied the response of rigid plate to deep water blast loading. One- dimensional analytical models were developed for two different problems: a. water blast/air back rigid plate and b. water blast/water back rigid plate. The cavitation process was found to influence the structural response to the blast event. Extending the previous work, Schiffer [98 - 101] conducted various experiments on isotropic plates, composite laminated plates, foam core materials and hull specimens. Schiffer [101] conducted water blast experiments on two different composite laminates using shock tube setup.
Three dimensional FE analysis was performed using ABAQUS Explicit. The deformation response of the plate was divided into two phases depending on the time of the flexural wave to reach the center of the plate. The cavitation phenomenon was explained with emergence and propagation of breaking fronts (BF) and closing fronts
(CF). The numerical predictions compared well with the experimental and analytical results.
This work offers analysis that can be used to determine the transient response, blast resistance and energy absorption of composite sandwich panels with crushable polymer foam cores subjected to water blast loading. Prior research has shown that on a per unit weight basis, the blast resistance of a polymer foam-core composite sandwich structures is highest when it undergoes plastic core crushing [56-58]. The current study investigates the foam core plasticity in the case of water blast loading conditions. The study also accounts for the cavitation zones both at the interface of solid and fluid and also in the surrounding water.
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4.2 Problem Formulation
Consider a fully clamped composite sandwich plate with the geometry shown in
Fig. 4.1(a). Coordinates x1,y1,z1 and x2,y2,z2 are defined with respect to the mid-surface of the front and back factsheets, while coordinates x0,y0,z0 are defined with respect to the mid-surface of the core. As described in Fig. 4.1(b), the sandwich plate is surrounded by acoustic media, water or air, on either side of the front and back facesheets. It is subjected to a planar shock wave incident to the front facesheet. Specifically, the incident planar shock wave is described by a uniformly distributed, exponentially- decaying, pressure loading. If the shock wave is transmitted to the front facesheet through water and air it is termed water-blast and air-blast, respectively. Three scenarios are of interest in this study: water blast/air backed panel, water blast/water backed panel, and air blast/air back panel. Lagrange’s equations of motion are developed for the sandwich plate by considering acoustic pressure fields due to the presence of water or air.
These acoustic pressure fields are based on the pioneering work by Taylor [73].
4.2.1 Acoustic pressure loading
Expressions for the acoustic loading on the front and back side of the sandwich plate by considering the differential surface elements, dx1,dy1 and dx2,dy2, on the front and back facesheets shown in Fig. 4.1(b). The acoustic loadings on differential surfaces are later integrated over the entire surface area to give the external virtual work in Lagrange’s equations of motion for the composite sandwich plate.
72
(a) (b)
Figure 4.1: Sandwich panel subjected to air/water blast conditions: (a) panel geometry and (b) acoustic loading on differential surface elements on front and back facesheets.
Front facesheet
Motion of the plate in Taylor’s model was restricted to being in the direction of incident acoustic pressure wave or normal to plate. In this work, tangential or in-plane motions of the plate are allowed implying acoustic pressures waves are radiated in both normal and tangential directions to the wetted surface. Let the surface traction vector on the incident or front side of the facesheet be such that t1 = t1i + t1r (4.1) where t1i is the incident traction vector and t1r is the reflected traction vector. The total surface traction vector on the front facesheet is specified with vector components,
73
ˆˆ ktjtitt ˆ ˆ ˆ kji ˆ zyx 1 1 1 1 1 zyx 1 1 where , , 111 are the unit vectors associated with the ,, 111 axes. The incident traction vector is
ˆ t 1 , IIi 0eppkpt (4.2)
ˆ where p0 is the peak pressure, is the decay time constant, t is time and k1 is the unit normal vector perpendicular to the surface or in the z1-direction. The reflected traction vector t1r is assumed to be of the form
ˆˆ kjipt ˆ r 01 x1 1 y1 1 z1 1 (4.3)
yx where ,, zyx 111 are functionals that vary with , 11 and t at the incident surface.
These functionals lead to radiated traction components not only normal to the incident surface but also parallel to this surface because of in-plane motions of the plate ( ,vu 11 ).
The particle velocity in water is related to the pressure so that the velocity due to the incident pressure is 1i ct where c is the acoustic impedance of water or air.
Continuity of the velocity at the incident surface in the z1-direction gives
1 w pp (4.4) 1 c I 0 z1
In the x1- and y1-directions,
p u 0 (4.5) 1 c x1 and
p v 0 (4.6) 1 c y1
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Equation (4) may be solved for z1 to give
1 wcp z1 I 1 (4.7) p0
Substituting the above expression into Eq. (4.3) and using Eqs. (4.1) and (4.2) give
wcpt 1 2 Iz 1 (4.8)
The other fluid functionals give fluid damping forces from in-plane or tangential motion of the plate:
pt uc x1 0 x1 1 (4.9) and
pt vc y1 0 y1 1 (4.10)
Back facesheet
Acoustic pressure waves radiate from the back facesheet and this affects the response of the plate. Let the total traction vector on the back facesheet be
tt 22 r (4.11) where t2r is the radiated traction vector. The total surface traction vector on the back
ˆˆ ktjtitt ˆ ˆ ˆ kji ˆ facesheet is specified with vector components, 2 2 2 2 2 zyx 2 2 where , , 222 are the unit vectors associated with the ,, zyx 222 axes because the radiated pressure wave propagates in these directions from general motion of the plate. The radiated traction term is of the form
75
ˆˆ kjipt ˆ r 02 x2 2 y2 2 z2 2 (4.12)
yx where ,, zyx 222 are functionals that vary with , 22 and t at the wetted surface. A negative sign is place on radiated traction vector because this wave propagates in the opposite direction on the back facesheet. The particle velocity of the fluid is related to
ˆˆ ˆ the radiated pressure by 2r ct . Denote the velocity R kwjviuw 2222222 .
Continuity of the velocity in the z2-direction gives
pt wc z2 0 z2 2 (4.13)
Continuity of the velocity in the x2 and y2-directions gives
pt uc x2 0 x2 2 (4.14) and
pt vc y2 0 y2 2 (4.15)
Equations (4.13)-(4.15) are recognized as fluid damping forces associated with the motion of the back facesheet.
Cavitation
Water cannot sustain tension or negative pressures, and it will disassociate creating regions of cavitation if the pressure falls below the cavitation pressure.
Cavitation is the formation of vapor cavities, bubbles or voids, and such a process may occur in both in front and at the back of the sandwich panel in water blast/water back scenario because of the negative pressures induced by the motion of the facesheets.
Should cavitation occur the acoustic pressure loading is set to zero. For the sandwich panel with a varying velocity field, some parts of the front facesheet may be under load while other parts may not be loaded and some parts of back facesheet may experience
76 fluid damping forces while others may not. The following equations of motion take this into consideration.
4.3 Structural Model
A multi-layered sandwich panel approach is used to distinguish separate responses of the facesheets and core. The strains in the facesheets and the core are defined with respect to displacements ,, wvu 111 and ,, wvu 222 associated with ,, zyx 111 and zyx 222 ,,, as defined in Fig. 4.1 (a).
4.3.1 Facesheet kinematics
The strain-displacement relations in the front facesheet ( i 1) and back facesheet
(i 2) are given by
z ii ximxx i (4.16)
z ii yimyy i (4.17)
z iiii yximyxyx ii (4.18) where the mid-surface strain and the change in curvature in the front and back facesheets are
2 u 1 w i i (4.19) imx xi 2 xi
2 v 1 w i i (4.20) imy yi 2 yi
77
u v w w i i i i ii ,myx (4.21) yi xi xi yi
2w i (4.22) xi 2 xi
2w i (4.23) yi 2 yi
2w i yx ii 2 (4.24) yx ii
4.3.2 Core kinematics
zyx As indicated in Fig. 4.1(a), the core mid-surface is defined with coordinates .,, Core deformations, u,w 0o and o ,v are assumed to be compatible with the facesheet deflections and are approximated by
ww w 21 (4.25) 0 2
uu u 21 (4.26) 0 2
vv v 21 (4.27) 0 2
The above expressions are the simplest way to relate facesheet deformations with core deformations [57, 102]. More elaborate or higher-order displacement fields in cores are given in Refs. [52-53]. Adopting these instead would add more degrees of freedom and unnecessarily increase the complexity of the solution. The in-plane strain-displacement relations for the core are
78
z x xm x (4.28)
z y ym y (4.29)
z xymxy xy (4.30) where the mid-surface strain and the change in curvature are
u 0 (4.31) xm x
v 0 (4.32) ym y
u v 00 (4.33) xym y x
y (4.34) x x
x (4.35) y y
y x (4.36) xy xy
and y and x are rotations of plane sections about the x- and y-axes, respectively. It is not necessary to carry non-linear terms in strain-displacement relations of the core because it is thick. Transverse shear strains in the core are given by
w 0 (4.37) xz y x
79
w 0 (4.38) yz x y
The following approximations are made for the rotations of plane sections and transverse compressive strains:
uu 21 y (4.39) H
vv 21 x (4.40) H
ww 21 z (4.41) H
Equations (4.39)-(4.41) are first-order approximations for rotations and compressive strains in the core. The first two assumptions regarding rotations of plane sections assumes that plane cross-sections of the core rotate with a constant angle (first-order shear deformation theory). The last assumption is equivalent to assuming a linear variation in transverse deformations of the core. As a result of these assumptions, transverse shear and normal strains within the core are assumed to be uniformly- distributed through the thickness, i.e. they do not depend on z-coordinate.
4.4 Equations of Motion
In order to be consistent with clamped boundary conditions, out-of-plane and in- plane deflections of the facesheets are expressed in double Fourier series as follows:
2x1 2 y1 xm 1 yn 1 1 aw mn cos1 cos1 cos cos (4.42) n0 m0 a b a b
xm 1 yn 1 1 bu mn sinsin (4.43) n1 m1 a b
80
xm 1 yn 1 1 cv mn sinsin (4.44) n1 m1 a b
2x2 2 y2 xm 2 yn 2 w2 dmn cos1 cos1 cos cos (4.45) n0 m0 a b a b
xm 2 yn 2 2 eu mn sinsin (4.46) n1 m1 a b
xm 2 yn 2 2 fv mn sinsin (4.47) n1 m1 a b where the outer and inner plate surface areas are projected on a rectangular plane axb , which is area associated with the mid-surface of the core.
The dynamic response of the sandwich plate is found by satisfying Lagrange’s equations of motion: d T T U Q (4.48) dt q q q mn mn mn mn where T and U are the total kinetic and strain energy in the sandwich plate and qmn are
edcba f Q generalized coordinates, ,,,, mnmnmnmnmn and mn , and mn are generalized forces.
Plastic work dissipated during core crushing is accounted for in the strain energy expression.
The kinetic energy of the two facesheets are
1 2 2 2 1 2 2 2 ff 1 1 wvuhT 1 dydx 11 f 2 2 wvuh 2 dydx 22 (4.49) 2 S1 2 S2
S S where 1 and 2 are outer and inner facesheet mid-surface areas. The kinetic energy of the core is
1 2 2 2 1 223 cc 0 0 0 dxdywvuHT c H xy dxdy2 (4.50) 2 S0 24 S0
81 where S 0 is the core mid-surface area and y and x are rotations of plane sections about the x- and y-axes in the core, respectively.
The facesheets remain linear elastic even though the core may undergo inelastic deformation during core crushing. The elastic facesheet strain energy is
1 2 AAU 2 2A A 2 2 2 2DDD D 2 dydx f 11 1,mx 22 1,my 12 1, 1,mymx 66 11 ,myx 11 x1 22 y1 12 yx 11 66 yx 11 11 2 S1 1 2 AA 2 2A A 2 2 2 2DDD D 2 dydx 11 2,mx 22 2,my 12 2, 2,mymx 66 22 ,myx 11 x2 22 y2 12 yx 22 66 yx 22 22 2 S2 (4.51)
where Aij and Dij are the membrane and bending stiffness of the laminate facesheets (
Bij 0is assumed in this case) and ij ,m and ij are the facesheets midsurface strain and curvatures.
During fully elastic response, the stress-strain relation for an orthotropic core is
x CCC 131211 000 x y CCC y 232212 000 z CCC 332313 000 z (4.52) yz 000 C44 00 yz 0000 C 0 zx 55 zx C xy 00000 66 xy where Cij is the elastic stiffness. The elastic strain energy is therefore given by
1 2 2 2 2 2 2 c 11 x 22 y 33 z CCCCU 12 yx C23 zy 222 13 xz 44 yz 55 zx CCCC 66 xy dydxdz (4.53) 2 V0 whereV0 is the volume of the core enclosed between the two facesheets. The core strains may be determined from facesheet deformations. 82
Cellular foams can exhibit elastic-plastic behavior under both compression and shear when peak pressures are very high. In fact this is an important attribute of a cellular foam in blast mitigation and energy absorption. In order to account for elastic- plastic behavior of the core, one must first describe a foam yield criterion. One of the most commonly used criteria to describe plastic yielding is an isotropic foam yielding
[67] of the form f ˆ 0 0 (4.54) where 0 is the flow stress and the effective stress ˆ is given in terms of the mean stress
: m and von Mises equivalent stress e
2 1 222 ˆ mpe (4.55) 2 1 3 where 23 if the foam plastic Poisson’s ratio is assumed to be zero. This yield criterion is used to separate the boundary between elastic and plastic regions of the core so that the core strain energy may be expressed over elastic and plastic regions by the following expression:
1 2 2 2 2 2 2 c 11 x 22 y 33 z 2CCCCU 12 yx 2C23 zy 2 13 xz 44 yz 55 zx CCCC 66 xy dydxdz 2 Ve
1 2 2 2 2 2 2 11 x 22 y 33 z 2CCCC 12 yx 2C23 zy 2 13 xz 44 yz 55 zx CCCC 66 xy dydxdz 2 V p
1 2 2 2 C11 xx 0 C22 yy 0 C33 zz 0 2C12 0 yyxx 0 2C23 0 zzyy 0 2 V p 2 2 2 2C130 xxzz 0 C44 yzyz 0 C55 zxzx 0 C66 xyxy 0 dydxdz
(4.56)
83 whereVe and V p are the volumes of foam core undergoing purely elastic and elastic-
plastic behavior, and ij 0 and ij 0 are the strain and stress components at the initiation of plasticity.
For the composite sandwich plate, the elastic-plastic response is obtained in the following sequence: (1) complete linear elastic response up to the point of initial yielding in the core, (2) elastic-plastic response as the plastic zone spreads in the core, (3) overlap of plastic zones in the core and (4) total plastic region in the core. The yield criterion,
Eq. (4.54), must be evaluated during elastic response until it is met. Yielding in the core mid-surface was found to initiate and spread into the four regions, as shown in Fig.
4.2(a). As the pressure load increases, the four separate plastic regions grow and start intersecting with other region, as shown in Fig. 4.2(b). During this overlap phase, the integration limits in the elastic and plastic zones of the Eq. (4.56) are changed accordingly. When the core turns completely plastic, the elastic strain energy terms in the
Eq. (4.56) are set to zero.
84
(a) (b)
Figure 4.2: Deformation of sandwich plate in: (a) elastic-plastic regime and (b) overlap regime.
Generalized forces Qmn are obtained from virtual work. Forces on the front facesheet in the z1-direction are given by
W)( 2x1 2 y1 xm 1 yn 1 Qamn 2 I wcp 1 cos1 cos1 coscos dydx 11 (4.57) amn)( S1 a b a b
N M 2x1 2 y1 xi 1 yj 1 where 1 aw ij cos1 cos1 coscos and N and M are the j0 i 0 a b a b number of terms taken in the Fourier series expansion. Each of the above force components depends on all velocity components of amn because w1 is not an orthogonal function. As a result of orthogonality, forces in x1- and y1- direction are
W)( ab Qbmn c bmn (4.58) bmn 4)( and
W )( ab Qcmn c cmn (4.59) cmn 4)(
85
On the back facesheet, forces in the z2-direction are given by
W)( 2x2 2 y2 xm 2 yn 2 Qdmn wc 2 cos1 cos1 cos cos dydx 22 d mn )( S2 a b a b
(4.60)
N M 2x2 2 y2 xi 2 yj 2 where 2 dw ij cos1 cos1 coscos and N and M are the j0 i 0 a b a b number of terms taken in the Fourier series expansion. Again each of the above force
components depends on all d mn because w2 is not an orthogonal function. As a result of orthogonality, forces in x2- and y2- direction are
W)( ab Qemn c emn (4.61) emn 4)( and
W)( ab Q fmn c f mn (4.62) f mn 4)(
The limits of integration of the generalized forces Qmn are adjusted if cavitation occurs.
The pressure at which cavitation is said to take place is zero.
4.5 Finite Element Analysis
Finite element analysis is performed for air blast/air back, water blast/air back and water blast/water back conditions using Dynamic Explicit step available in ABAQUS.
The top and bottom facesheets are rigidly tied to the core. Eight node, continuum brick elements (C3D8) are used to model facesheets and core. In the water blast/air back and water blast/water back conditions, the shock wave propagates along the fluid column, interacts with facesheet and reflects back. This reflected wave reaches the free surface and reflects back into the fluid column. To avoid the interactions of these reflected
86 waves during the structural response, a fluid column of depth 4m is used. The water in the fluid column is modeled using Acoustic elements (AC3D8R). The contact between the fluid column and the facesheet is defined using tie constraint. Water cannot sustain tension and will undergo cavitation when cavitation limit pc = 0. Exponential decay blast load is applied in all the cases. In the air blast/air back panel, the load is applied on the front facesheet. In the water blast/air back and water blast/water back conditions, load is applied at the free end of the fluid column. The assemblies of three models are shown in
Figs. 4.3 (a)-(c). Clamped boundary conditions are assumed and because of the symmetry only quarter model is used in the analysis. Default value is set for quadratic bulk viscosity parameter while linear bulk viscosity parameter is changed to 1.
(a)
87
(b)
(c) Figure 4.3: Assembly of (a) Air blast/air back condition, (b) Water blast/water back condition and (c) Water blast/water back condition.
88
4.6 An Example
As an example, consider the composite sandwich plate with E-Glass/Vinyl Ester
Woven Roving facehsheets and Divinycell PVC H250 core. The sandwich plate has h=5 mm, H=25 mm, and a=b=248.25 mm. Properties of the E-Glass/Vinyl Ester Woven
Roving and Divinycell PVC H250 are given in Tables 4.1 and 4.2, respectively, while acoustic properties for water and air given in Table 4.3. This panel is subjected to planar shock loading, with peak pressure 3.85 MPa and decay time constant 1 ms at the front facesheet.
Table 4.1: E-Glass/Vinyl Ester Woven Roving facesheet properties.
ρf E11 =E22 υ G12 Xt=Yt Xc=Yc S12
3 (kg/m ) (GPa) (GPa) (MPa) (MPa) (MPa)
1391 17 0.13 4 200 210 40
Table 4.2: Various foam properties.
H30 H100 H200 H250 HCP 100
3 ρc (kg/m ) 36 100 200 250 400
E (MPa) 27 105 293 403 650
υ 0.25 0.31 0.3 0.34 0.3
σ0(MPa) 0.3 1.66 4.35 6.3 10.3
γf 0.09 0.4 0.45 0.45 0.35
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Table 4.3: Acoustic properties of air and water.
Air Water
Density, ρ (kg/m3) 1.225 1024
Speed of sound, c (m/s) 340 1492
Acoustic impedance, ρc (kg/m2s) 416.5 1.53e6
4.6.1 Structural response
The transient deflections at the center and mid-surface of the core for three scenarios: air blast/air back; water blast/air back, and water blast/water back is shown in
Fig. 4.4. This plot shows how fluid damping effects from the reflected wave on the incident facesheet and the radiated wave on the back facesheet can affect the amount of deformation in the sandwich panel for the same pressure pulse loading. As shown in
Table 4.3, the acoustic impedance of air is 3,673 times lower than water so that there is very little air damping. In fact, air damping effects are negligible in this case.
At a peak pressure of 3.85 MPa, the PVC core exhibited large scale plasticity in the air blast/air back case would have failed before reaching this deflection. There is an
11% variation between the analytical and FEA prediction. This is due to the approximations in the analytical model for the plastic core. For the water blast/air back case, deflections are reduced by a factor of 6.9 and the maximum deflection occurs 3 times later in comparison to the air blast/air back sandwich plate. There was some plastic core crushing in the PVC foam. Deflections are reduced by more and occur even later, by factors 11 and 4.5, respectively, for the water blast/water back sandwich plate in comparison to the air blast, air back plate. The foam core remained elastic throughout the
90 response. The difference in the deflections in the analytical and FE predictions of water blast/air back and water blast/water back cases may be attributed to the fact that the analytical model has finite number of modes. The facesheets are modelled as 3D continuum elements in FEA studies while shell theory is used in the analytical formulations. Also, through-thickness properties are taken into consideration in the FEA while the analytical model including cavitation phenomenon is in its simple form does not account for this condition and broadly describes the deflection of the sandwich plate.
Figure 4.4: Transient deflections at the center and mid-surface of core for air blast/air back, water blast/air back, and water blast/water back panels.
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4.6.2 Cavitation
Consider the incident pressure wave given by the equation
czt )/( I 0epp (4.63) where p0 is the peak pressure, c is the speed of sound in the water and τ is the decay constant. The interface of the plate and fluid surface is considered as the origin of the z- coordinate. The reflected pressure wave equation is given by
czt )/( R 0epp wc 1 (4.64)
The absolute pressure in the fluid is given by
czt )/( czt )/( RI 0epppp 0ep wc 1 (4.65)
Using Eq. (4.65), the absolute pressure is calculated at various times at the interface of the plate and fluid and also in the fluid column for water blast/air back condition. At any instance t<0 as shown in Fig. 4.5 (a), the incident wave travels with peak pressure (p0). At time t>0, the incident wave hits the sandwich plate causing structural deformation and reflection of the incident pressure wave as shown in Fig. 4.5
(b).
As shown in Fig. 4.5 (c), cavitation starts at t=0.07 ms at the interface of plate and fluid column. At time t=0.3 ms, the cavitation zone propagates along the z-direction in the fluid column. At the interface, there is change in the absolute pressure and therefore arrest of the cavitation zone as shown in Fig. 4.5 (d).
The cavitation process in the water blast/water back condition is similar to the water blast/air back condition except that there is no cavitation zone at the interface of
92 plate and fluid column. The profiles at various times at the front and rear side are shown in Figure 4.6.
Figure 4.5: Cavitation process in the water blast/air back condition: (a) incident wave at t<0 sec, (b) incident (blue line) and absolute pressure wave (red line) at t=0.05 ms, (c) incident and absolute pressure wave at t=0.07 ms, (d) incident and absolute pressure wave at t=0.3 ms and (e) incident and absolute pressure wave at t=0.6 ms.
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(a) (b)
(c) (d)
(e) (f)
(g) (h) Figure 4.6: Cavitation process in the water blast/water back condition: (a) incident wave at t<0, (b) radiated wave from back facesheet at t<0, (c) incident (blue line) and absolute pressure (red line) wave near front facesheet at t=0.25 ms, (d) radiated wave from back facesheet at t=0.25 ms, (e) incident and absolute pressure wave near front facesheet at t=0.56 ms, (f) radiated wave from back facesheet at t=0.56 ms, (g) incident and absolute pressure wave near front facesheet at t=1.6 ms, (h) radiated wave from back facesheet at t=1.6 ms.
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4.7 Blast Resistance and Energy Absorption
The peak pressure with an exponential decay constant of 1 ms that would either initiate facesheet fracture or core shear fracture are found for the E-Glass/Vinyl Ester -
PVC H250 foam core sandwich panel. Figure 4.7 (a) – (b) summarizes the failure pressure pulse amplitudes to initiate damage and energy absorbed by sandwich panel for the three loading cases. For the current sandwich panel, the back facesheet experienced in-plane compressive failure and the PVC H250 core exhibited plasticity near the clamped edges before this failure in the air blast/air back situation when the exponential pressure pulse amplitude is 1.15 MPa and the decay time constant is 1 ms. Because deformations are reduced substantially in the water blast cases, the pressure pulse amplitude increased substantially to get damage initiation by facesheet failure in the sandwich plate with the same 1 ms decay time constant. When compared to the air blast/air back situation, it took significantly higher pressures to cause the same amount of damage in the panel under water blast/air back and a water blast/water back conditions, respectively. Results from Figs. 4.7 (a)-(b) suggest that one can expect a certain degree of conservatism when designing these sandwich plates for underwater explosion loading using air-blast results. In order to understand why plastic work dissipation is higher in the water blast cases, core stresses near yielding were analyzed.
The stress distribution at the mid-surface of the core along y=b0/2 just before the start of plasticity were calculated from Eq. (4.52), and are shown in Figs. 4.8 (a) - Figs.
4.10 (b) for the air blast/air back, water blast/air back and water blast/water back conditions. In these figures, a red circle indicates the position of initial yield in the core.
Yielding initiates at 22mm and 226.25 mm along the foam in the air blast/air back
95 condition, at 20 mm and 228.25 mm in the water blast/air back and at 25 mm and 223.25 mm in the water blast/water back condition. The stress distributions changes with different conditions. In the air blast/air back condition, transverse shear stress (S13) is the most significant component when compared to other stress components so that plasticity is induced primarily due to transverse shear. In the water blast/air back condition, the transverse shear stress (S13) is still dominant but out-of-plane compression
(S33) is also significant. Plasticity is due to combined transverse shear and compression.
In the water blast/water back conditions, the most dominant stress component is the out- of-plane compression (S33), while transverse shear stress (S13) and in-plane stresses
(S11 and S22) are also significant. This change in the stress distribution is due to the fact that the distal fluid column in the water blast/water back condition gives a radiated pressure field to the back facesheet and this in turn causes significant core compression.
Hence in water blast/water back panels, plasticity is due to transverse shear and hydrostatic pressure.
(a)
(b)
Figure 4.7: Predicted (a) failure pressure and (b) energy absorption for PVC H250 foam sandwich panel assuming an exponential decay time constant of 1 ms.
96
(a) (b) Figure 4.8: Core mid-surface stresses at the onset of yielding (a) in-plane stresses and (b) out-of-plane stresses in air blast/air back condition.
(a) (b) Figure 4.9: Core mid-surface stresses at the onset of yielding (a) in-plane stresses and (b) out-of-plane stresses in water blast/air back condition.
(a) (b)
Figure 4.10: Core mid-surface stresses at the onset of yielding (a) in-plane stresses and (b) out-of-plane stresses in water blast/water back condition.
97
4.7.1 Effect of foam type
A parametric study is done to examine the blast resistance and energy absorption of the sandwich panels with various types of foam cores. The dimensions of the facesheet and core are same as in each panel. Table 4.2 lists properties of five different foams, including the PVC H250 foam. The foams are arranged from the least dense to the most dense, with the densest having the highest modulus and crushing resistance. In general, blast resistance increases with increasing foam density whether there is an air blast or water blast as shown in Fig. 4.11(a). Except for the PVC H30 foam, all the panels failed by facesheet fracture. The panel with the PVC H30 foam core failed by core shear failure. Figure 4.11(b) shows the amount of plastic work dissipated by the core when these panels failed. Plastic work is highest in the PVC H200 core and about the same in the PVC H100 and H250 cores.
When the failure loads are normalized by the areal weight density of the panel
(i.e., weight per unit panel surface area), a different conclusion concerning blast resistance is drawn as shown in Fig. 4.12. In the air blast/air back case, panels with PVC
HCP 100 are the most blast resistant, while in water blast/air back and water blast/water back cases, panels with PVC H200 are the most blast resistant. The panels with the PVC
H200 foam clearly absorbed the most energy due to plastic core crushing. Water damping and core plastic damping have a synergistic effect in increasing blast resistance.
Fluid-structure interaction causes the foam plasticity to be influenced by core compression and this makes energy dissipation greater than in air blast/air back panels.
98
(a) (b)
Figure 4.11: Predicted (a) blast resistance and (b) energy absorption of various sandwich panels for air blast/air back, water blast/air back, and water blast/water back conditions.
Figure 4.12: Blast resistance per unit areal weight density for various sandwich panels for air blast/air back, water blast/air back, and water blast/water back conditions.
4.7.2 Effect of plate aspect ratio
The previous parametric study on the effect of foam type was for a fairly thick sandwich plate with an aspect ratio a/htot = 6.97. In the second parametric study, three different aspect ratios a/htot = 6.97, 10.72 and 17.86 are considered for all the five foams under air blast/air back, water blast/air back and water blast/water back conditions as shown in Figs 4.13 (a) – 4.15 (b). Regardless of span, the sandwich panels with PVC
H30 foam core failed by core shear in all the three loading conditions while sandwich panels with all other foams failed by facesheet fracture. For the sandwich panels with
99
PVC H30 foam core, it is observed that with increasing span, there is very little plasticity and the failure pressure loads are very small when compared to other foam core panels.
For sandwich plates with a/htot = 10.72, it is observed that there is no initiation of plasticity in sandwich panel with HCP 100 foam core material. As the aspect ratio increases, there is zero or negligible plasticity in the panels with high density foam cores implying that the failure occurs while the core is still elastic. It is also observed that the sandwich panels with PVC H200 foam core absorbs most energy and were also most blast resistant in the air blast/air back condition while panels with PVC H100 foam core clearly absorbed the most energy due to plastic core crushing and were also most blast resistant in water blast/air back and water blast/water back conditions.
(a) (b)
Figure 4.13: Predicted (a) Blast resistance per unit areal weight density and (b) energy absorption for sandwich panels with a/htot = 6.97 for air blast/air back, water blast/air back, and water blast/water back conditions.
100
(a) (b)
Figure 4.14: Predicted (a) Blast resistance per unit areal weight density and (b) energy absorption for sandwich panels with a/htot = 10.72 for air blast/air back, water blast/air back, and water blast/water back conditions
(a) (b) Figure 4.15: Predicted (a) Blast resistance per unit areal weight density and (b) energy absorption for sandwich panels with a/htot = 17.86 for air blast/air back, water blast/air back, and water blast/water back conditions
4.8 Summary
This work examined the blast response, resistance and energy absorption of PVC foam core composite sandwich panels subjected to air and water blast. Taylor’s formulation of the acoustic pressure loading of a simple one degree-of-freedom elastic plate subjected to underwater blast was extended to a multi-mode, multi-degree-of- freedom, elastic-plastic sandwich plate. It was shown that water-interaction diminished deflection and slowed down the response of the sandwich panel for the same pressure pulse loading. For same pressure pulse loading, two simultaneous cavitation zones were observed in the water blast/air back panels: one at the fluid-solid interface and the other
101 ahead of the plate. In the water blast/water back panels, a cavitation zone was observed ahead of the plate. Because deformations were reduced substantially in the water blast cases, it took significantly higher pressures to cause the same amount of damage in the panel under water blast/air back and water blast/water back conditions than the air blast/air back plate. It was also found that the core plasticity in the air blast/air back panel was primarily due to transverse shear. In the water blast/air back, it was due to combined transverse compression and shear, and in the water blast/water back case, plasticity in the core was due to hydrostatic pressure and transverse shear.
A parametric study on the blast resistance of panels with various foam core types revealed that plastic core deformation is an important foam core property for blast mitigation. For sandwich plates with aspect ratio a/htot = 6.97, it was shown that panels with PVC HCP 100 were the most blast resistant in the air blast/air back case. However, in the cases of water blast/air back and water blast/water back, panels with PVC H200 were the most blast resistant and absorbed the most energy due to plastic core crushing.
For composite sandwich plates with a/htot = 10.72 and 17.86, panels with PVC H200 foam absorbed more energy while panels with PVC H100 foam absorbed most blast energy in water blast/air back and water blast/water back conditions. Panels with PVC
H200 foam core were the most blast resistant in air blast/air back condition while panels with PVC H100 foam were most blast resistant in water blast/air back and water blast/water back conditions. It was also observed that with increase in span, the sandwich panels with high density foam core material exhibited no plastic deformation.
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CHAPTER V
CONCLUDING REMARKS
Analytical models were developed to describe the deformation and damage initiation of composite panels in three different problems: composite shells subjected to external pressure pulse, composite sandwich shells subjected to external pressure pulse, and composite sandwich plates subjected to air and water blast.
Dynamic pulse buckling of double-curvature composite shells under external pressure pulse loading was examined using Novozhilov non-linear shell theory and
Lagrange’s equation of motion. The predicted transient shell response compared well with FEA results from ABAQUS Explicit in the stable regime. Critical buckling loads were determined analytically using the Budiansky-Roth buckling criterion. It was shown that higher buckling loads are obtained by making the shell deeper, i.e., either increasing the angular extent for a fixed radius of curvature or decreasing the radius of curvature for a fixed span. Higher buckling modes were induced by making the shell deeper and they were responsible for improving the dynamic buckling strength of the shell.
The composite laminate shell study was extended by considering double- curvature, composite sandwich shells with crushable, elastic-plastic PVC foam core.
Lagrange’s equations of motion were used to derive the facesheet deformations for sandwich shell with isotropic and transversely isotopic foam core. The predicted solutions from the elastic-plastic model compared well with FEA results with the
103 isotropic foam core. A parametric study indicated that the blast resistance of the sandwich shells increases as the curvature ratio decreases. For the sandwich shells with low radius of curvature (radius-to-thickness ratio below 20), higher loads to initiate failure were obtained for sandwich shells assuming the core was isotropic instead of a transversely isotropic. Although foam core material is transversely isotropic, many analysis is performed on it today using isotropic foam core. The fact that the failure loads are higher when an isotropic core is assumed would lead to non-conservative estimates of the blast resistance of composite sandwich structures. It is recommended that future studies include transversely isotropic properties of the foam core material, especially when analyzing deep sandwich shells subjected to blast loading.
Flat sandwich panels subjected to air and water blast loading were also studied in this work. The panels were analysed under three different conditions: air blast/air back, water blast/air back and water blast/water back conditions. It was found that the deformation of the sandwich plate was reduced and occurred over longer periods due to the fluid damping terms in water blast cases compared to the air blast/air back plates.
The predicted sandwich plate deformations compared fairly well with FEA results from
ABAQUS Explicit. For the same pressure pulse loading, two cavitation zones were found in the water blast/air back panel, and one cavitation zone in the water blast/water back panel.
Higher pressure loads were needed to cause failure in water blast/water back and water blast/air back panel than sandwich plate in air blast/air back condition. For thick sandwich plates, panels with strong, high density foam performed better in the air blast/air back case. However, in the cases of water blast/air back and water blast/water
104 back, panels with low density foam were the most blast resistant and absorbed the most energy due to plastic core crushing. In the case of thin composite sandwich plates, panels with low density foam material performed better in all the three loading conditions. The core plasticity in the air blast/air back panel was primarily due to transverse shear. In the water blast/air back, it was due to combined transverse compression and shear, and in the water blast/water back case, plasticity in the core was due to hydrostatic pressure and transverse shear. The comparative studies were all done assuming the flat sandwich panel had an isotropic foam core. Future fluid-structure interaction models should include multi-axial foam properties for better understanding of the structural response and cavitation phenomenon.
105
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APPENDICES
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APPENDIX A
NOVOZHILOV NON-LINEAR SHELL THEORY
In Chapter VI of Reference [24], Novozhilov presents a theory of deformations for thin shells using Gaussian curvilinear coordinates to describe the mid-surface of the shell. This theory is valid for small strains and arbitrary rotations (arbitrary in the sense that rotations still give small strains). This section summarizes general strain- displacement relations from Novozhilov nonlinear shell theory. A full derivation of
Novozhilov thin shell nonlinear strain-displacement relations may be found in Ref. [24].
The shell coordinate system is described in Fig. 2.1. The shell deforms with mid-
surface radial displacement 21 t,,(w ), and tangential displacements 21 )t,,(u and
21 t,,(v ),where , 21 are Gaussian curvature coordinates defined at the mid-surface of the shell in Fig. 2.1. Novozhilov assumed fibers normal to mid-surface do not elongate, and remain normal before and after deformation. With this assumption, transverse normal and shear strains are neglected, and the remaining strains are represented by
z 1m11 (A1)
z 2m22 (A2)
1212 m z 12 (A3)
where m2m1 ,, 12m are mid-surface normal and shear strains and 21 ,, 12 are variations in curvature and twisting of the mid-surface induced by the deformations.
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Based on the thinness of the shell, Nonvozhilov assumed 1 2 1Rz1Rz1 , he represented the mid-surface strains and variations in curvature and twisting by
1 e eˆ 2 eˆ 2 eˆ 2 (A4) m1 11 2 11 12 13
1 e eˆ 2 eˆ 2 eˆ 2 (A5) m2 22 2 21 22 23
12m eeeeeeee 2313122221112112 (A6)
1 kekeke1 131312121111 (A7)
2 kekeke1 232321212222 (A8)
12 2111 kekekekeke1ke1 23131323221211211222 (A9) where
1 wu e11 (A10) R1 R1
1 wv e22 (A11) R 2 R 2
1 v e21 (A12) R1
1 uw e13 (A13) R1 R1
1 u e12 (A14) R 2
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1 vw e23 (A15) R 2 R 2
1 k11 (A16) 1 RR 1
1 k 22 (A17) 2 RR 2
1 k12 (A18) R1
1 k 21 (A19) R 2
1 k13 (A20) 1 RR 1
1 k 23 (A21) 2 RR 2 and
e1 eˆ eˆ eˆ12231322 (A22)
e1 eˆ eˆ eˆ 21132311 (A23)
eˆ eˆ eˆ eˆ eˆ eˆ 211222112211 (A24)
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APPENDIX B
CONVERGENCE OF DEFLECTION AND STRESSES WITH DOUBLE FOURIER
SERIES
Equations of motion were solved using MATLAB ode45 solver with relative and absolute tolerances set at 1e-4 and 1e-6, respectively, and assuming a finite number of coefficients or degrees of freedom described in double Fourier series of Eqs. (3.30)-
(3.35). The transverse shell deflection taken at the center of the core mid-surface plane (
0 0 2by,2ax ) was first taken as a quantity of interest. Figure B1(a) shows the transient deflection when the Fourier series are expanded up to 6,4,2mn and even 8, while Fig. B1(b) shows the corresponding deflection profile at the mid-surface of core
along 0 2by when t=0.25 ms. The Fourier series converge very rapidly for deflection, and very good approximations of the transient deflection could be achieved when
2mn or 32 Fourier terms in e,d,c,b,a mnmnmnmnmn and f mnare taken.
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(a) (b)
Figure B1 Convergence of Fourier series for transverse deflection: (a) transient deflection at center of core mid-surface plane and (b) transverse deflections at mid-surface of core along 0 2by when t=0.25 ms.
The in-plane stress 1 at the bottom side of the inner facesheet (see Fig. 3.7(d)) was taken as the second quantity of interest since it would contribute to the facesheet failure predictions. This stress is highest at the clamped edges, and the stress history at clamped edge at the bottom side of the inner facesheet ( 0202 2by,ax ) is shown when the Fourier series were expanded up to 6,4,2mn and 8 in Fig. B2(a). The
stress distribution along 02 2by when t=0.25 ms is also shown in Fig. B2(b) for the corresponding Fourier expansions. Unlike deflections, stresses do not converge as rapidly but it can be seen that a good approximation for stress is obtained when
.6mn In other words, the series converge for stresses when 6mn or 242 Fourier terms in e,d,c,b,a mnmnmnmnmn and mn.f
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(a) (b)
Figure B2 Convergence of Fourier series for in-plane stress ( 1 or S11) at the bottom side of the inner facesheet: (a) stress history at clamped edge 0202 2by,ax and (b) stress distribution along 02 2by when t=0.25 ms.
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APPENDIX C
FINITE ELEMENT ANALYSIS FOR SHELL WITH ISOTROPIC FOAM CRUSHING
Finite element analysis using ABAQUS Explicit was used to corroborate results from the analytical model. Dynamic, Explicit analysis was used to obtain the transient response of the sandwich panel. The FEA model for the sandwich shell is shown in Fig.
C1. The dimensions of the sandwich shell are the same as in the example problem described in Section 3.3. The facesheets were rigidly tied to the core. All sides of the sandwich shell were encastre. The facesheets were orthotropic, linear elastic with properties of E-Glass/Vinyl Ester Woven Roving (Table 3.1). The core was modeled with PVC H250 properties assuming crushable foam with isotropic hardening (Table
3.2). A uniformly distributed pressure, expressed by Eq. (3.1) where 0 3.2p MPa and
1T ms, was applied to the outer facesheet of the sandwich shell.
Figure C1 FEA model of the sandwich shell.
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Continuum eight node (C3D8) or brick elements were chosen for both facesheets and core. The mesh density was chosen to give accurate results with respect to quantities of interest such as the transient deflection at center of core mid-surface plane and the stress S11 history at clamped edge 0202 2by,ax of the bottom side of the inner facesheet. Figures C2 (a) and (b) show results for above-mentioned deflection and stress history with 4,375 to 140,000 elements. The mesh convergence study indicated that a total of 140,000 elements were needed to assure accurate FEA results. Solutions for the mid-surface core deflections and stresses along 0 2by at various times from the FEA are given in Figs. 3.4- 3.6. The FEA stresses along axis at the top and bottom sides of each facesheet are shown alongside analytical predictions in Figs. 3.7(a)-(d) at t=0.25 ms, when deflections are at a maximum.
(a) (b)
Figure C2 Mesh convergence for (a) transient deflection at center of core mid-surface plane and (b) S11 history at clamped edge of the bottom side of the inner facesheet.
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