pdfs.semanticscholar.org...ii analytical modeling of metallic honeycomb for energy absorption and...

ANALYTICAL MODELING OF METALLIC HONEYCOMB FOR ENERGYABSORPTION AND VALIDATION WITH FEA

A Dissertation by

Vinoj Meshach Aaron Jeyasingh

M.S., Wichita State University, USA, 2001

B.E., University of Madras, India, 1997

Submitted to the College of Engineeringand the faculty of the Graduate School of

Wichita State Universityin partial fulfillment of

the requirements for the degree ofDoctor of Philosophy

May 2005

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ANALYTICAL MODELING OF METALLIC HONEYCOMB FOR ENERGYABSORPTION AND VALIDATION WITH FEA

I have examined the final copy of this dissertation for form and content and recommendthat it be accepted in partial fulfillment of the requirements for the degree of Doctor ofPhilosophy with a major in Mechanical Engineering.

______________________________________Dr. Behnam Bahr, Committee Chair

We have read this dissertationAnd recommend its acceptance:

______________________________________Dr. Hamid Lankarani, Committee Member

______________________________________Dr.Dennis Siginer, Committee Member

______________________________________Dr. Gamal Weheba, Committee Member

______________________________________Dr. Jamal Ahmad, Committee Member

Accepted for the College of Engineering

_________________________________Dr. Walter Horn, Dean

Accepted for the Graduate School

_________________________________Dr. Susan Kovar, Dean

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DEDICATION

To my parents, brother, and wife for their for their support, love, and care

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ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Behnam Bahr, for his patient guidance and

constant support during the course of my graduate studies. I would also like to thank my

dissertation committee members Dr. Hamid Lankarani, Dr. Dennis Siginer, Dr. Gamal

Weheba, and Dr. Jamal Ahmad for their valuable suggestions and comments on my

research.

My thanks go to Dr. Amir Adibi for his help and guidance in my research. I

would also like to thank all the faculty and staff members of the Department of

Mechanical Engineering for their help and support.

I am extremely grateful to my parents, Mr. T.Aaron Jeyasingh and Mrs. Janaki

Jeyasingh, for their love, support, and encouragement throughout my academic years. I

also thank my brother, Mr. Manoj Praveen, for his constant encouragement, my wife,

Reja Shining Gold, for her support and encouragement during my studies. I would like to

thank Mr. and Mrs. T. Thankaswamy for their support and prayers. Finally, I would like

to thank all of my friends and other graduate students who helped me in my research.

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ABSTRACT

Honeycomb materials possess high energy absorption characteristics and are

useful for the impact protection of structural members. Various honeycomb

configurations are being developed for a variety of applications. Analytical models are

now available to determine the energy absorption characteristics of the regular hexagonal

type of honeycomb. However, the development a parameterized analytical model that can

determine the energy absorption characteristics of various honeycomb shapes is needed.

In this research, a parameterized analytical model is developed for the typical honeycomb

shape, and is validated using experimental and finite element analysis.

Honeycomb materials exhibit strain-rate effects at impact velocities. They can

have higher energy absorption during dynamic crush than during quasi-static crush. In

order to determine the energy absorption of honeycomb material at higher velocity, the

characterization of it must be made using high-impact testing machines, which are

expensive and time-consuming. Therefore, development of an analytical model that can

predict energy absorption at higher velocities is needed. Also, strain-rate coefficients

must be determined for each particular type of honeycomb since the strain rate depends

on the geometrical properties of the honeycomb. Therefore, strain-rate coefficients were

developed for each honeycomb model in this research. The energy absorption of

honeycombs at higher impact velocities was also determined using the low-velocity test,

which will be useful when only low-velocity machines are available for testing

honeycombs. Finally, a performance analysis was carried out using response surface

methods to maximize energy absorption of the honeycomb.

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TABLE OF CONTENTS

Chapter Page

1. INTRODUCTION ...................................................................................................1

1.1 Motivation ...................................................................................................11.2 Literature Review.........................................................................................21.3 Outline..........................................................................................................71.4 Scope and objectives....................................................................................8

2. REVIEW OF ANALYTICAL MODELING OF METALLIC HONEYCOMB.....9

2.1 Objectives ....................................................................................................92.2 Energy Absorption Properties of Metallic Honeycomb...............................92.3 Various Developments in Analytical Modeling of Honeycomb................12

2.3.1 Euler Buckling of Columns ...........................................................122.3.2 Collapse of Thin Cylindrical Shells...............................................142.3.3 Crushing of Hexagonal Cell Structures .........................................172.3.4 Analytical Model of Honeycomb without Rolling Deformation...202.3.5 Analytical Model of Honeycomb with Rolling Deformation ........24

2.4 Chapter Summary ....................................................................................28

3. ANALYTICAL MODELING OF SYMMETRIC ASYMMETRIC ANDCURVE-SHAPED HONEYCOMB CONFIGURATIONS ..................................29

3.1 Objectives ..................................................................................................293.2 Methodology..............................................................................................293.3 Classification of Honeycomb According to Geometrical Parameters .......29

3.3.1 Symmetrical and Asymmetrical Honeycomb Configuration.........293.3.2 Reinforced Honeycomb Configuration..........................................313.3.3 Curved Edge Honeycomb..............................................................31

3.4 Analytical Modeling of Typical Honeycomb Shapes................................323.4.1 Analytical Modeling of Formgrid Honeycomb .............................32

3.4.1.1 Validation of Analytical Model of FormgridHoneycomb........................................................................48

3.4.2 Analytical Modeling of Half-Hexagonal Honeycomb...................493.4.2.1 Energy Absorption During Buckling .................................513.4.2.2 Energy Due to Rolling Deformation..................................513.4.2.3 Energy Due to Horizontal Hinge Lines .............................533.4.2.4 Energy due to Inclined Hinge Lines ..................................55

3.4.3 Analytical Modeling of Flexible Flexcore Honeycomb ................583.4.3.1 Energy Absorption During Buckling .................................603.4.3.2 Energy Due to Rolling Deformation..................................603.4.3.3 Energy Due to Horizontal Hinge Lines .............................61

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3.4.3.4 Energy Due to Inclined Hinge Lines .................................623.4.3.5 Experimental Validation of Flexcore Analytical Model....64

3.4.4 Analytical Modeling of Double Flex honeycomb .........................663.4.4.1 Energy Absorption During Buckling .................................673.4.4.2 Energy Due to Rolling Deformation..................................683.4.4.3 Energy Due to Horizontal Hinge Lines .............................693.4.4.4 Energy Due to Inclined Hinge Lines .................................69

3.5 Chapter Summary ......................................................................................73

4. PARAMETERIZATION OF HONEYCOMB MODEL.......................................74

4.1 Developments in Parameterization of Honeycomb ...................................744.2 Methodology..............................................................................................774.3 Parameterization of Energy Absorption Parameter E1 Due to Rolling

Radius ........................................................................................................794.4 Energy Absorption Parameter E2 Due to Horizontal Hinge Lines.............824.5 Energy Absorption E3 Due to Inclined Hinge Lines..................................864.6 Validation of the Parameterized Honeycomb Model Using a typical

Honeycomb Configuration.........................................................................914.6.1 Validation using Hexagonal Honeycomb ......................................934.6.2 Validation using Half Hexagonal Honeycomb..............................954.6.3 Validation using Formgrid honeycomb .........................................974.6.4 Validation using Flexcore honeycomb ........................................1004.6.5 Validation using Double Flex honeycomb ..................................102

4.7 Validation of Parameterized Model using New Honeycomb CellConfiguration ...........................................................................................1044.7.1 Diamond-shaped honeycomb model............................................1054.7.2 Triangular-Shaped Honeycomb Model........................................106

4.8 Chapter Summary ....................................................................................108

5. FINITE ELEMENT ANALYSIS OF METALLIC HONEYCOMB ..................109

5.1 Objective ..................................................................................................1095.2 Finite Element Analysis on Buckling of Square tube..............................109

5.2.1 Simulation Stages.........................................................................1115.3 Finite Analysis on Honeycomb Models...................................................115

5.3.1 FEA Analysis on Hexagonal Honeycomb ...................................1165.3.2 FEA Analysis on Half-hexagonal Honeycomb............................1175.3.3 FEA Analysis on Flexcore Honeycomb ......................................1185.3.4 FEA Analysis on Formgrid Honeycomb .....................................1195.3.5 FEA Analysis on Diamond-Shaped Honeycomb ........................1205.3.6 FEA Analysis on Triangular Honeycomb....................................122

5.4 Chapter Summary ....................................................................................123

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6. DYNAMIC CRUSH STRENGTH OF TYPICAL HONEYCOMBS .................124

6.1 Objectives ................................................................................................1246.2 Various Developments in the Dynamic Crush Strength of Thin walled

Materials ..................................................................................................1246.2.1 Cowper-Symond Strain Rate Law ...............................................1246.2.2 Rate-Sensitive Impulse Loaded Structures ..................................1256.2.3 Dynamic Gain of Thin-Walled Material......................................1276.2.4 Dynamic Crush load for Square Tubes........................................1306.2.5 Energy Absorption of Sandwich Honeycomb .............................1316.2.6 Honeycomb Constitutive Model ..................................................132

6.3 Proposed Method of Evaluating Dynamic Crush Strength of TypicalHoneycomb Shapes.................................................................................133

6.4 Evaluation of Dynamic Crush Strength of Flexcore Honeycomb ...........1336.5 Evaluation of Dynamic Experimental Material Constants for Flexcore

Honeycomb..............................................................................................1366.6 Evaluation of Dynamic Crush Strength of Hexagonal honeycomb.........1386.7 Dynamic Crush Strength Equation for Half-Hexagonal Honeycomb .....1416.8 Dynamic Crush strength Equation for Formgrid honeycombs ................1436.9 Dynamic Crush Strength Equation for Double-Flex Honeycombs .........1456.10 Dynamic Crush Strength Equation for Diamond-Shaped Honeycomb ...1476.11 Dynamic Crush Strength Equation for Triangular-Shaped

honeycomb...............................................................................................1486.12 Validation of Dynamic Crush Strength Obtained from Proposed

Analytical Models....................................................................................1486.12.1 Validation of Dynamic Analysis of Honeycombs .......................1496.12.2 Validation of Dynamic Crush Strength Obtained from

Analytical Method .......................................................................1516.12.3 Methodology of Predicting Dynamic Crush Strength from Low

Velocity Material Strain Rate Coefficients Dm and p ..................1556.13 Chapter Summary ....................................................................................160

7. PERFORMANCE ANALYSIS OF HONEYCOMB FOR MAXIMUM ENERGYABSORPTION ....................................................................................................161

7.1 Methodology............................................................................................1617.2 Performance Analysis using Response Surfaces .....................................161

7.2.1 Design for Surface Response Analysis ........................................1627.2.2 Performance Analysis on Formgrid Honeycomb ........................163

7.2.2.1 Model validation and Results...........................................1667.2.3 Performance Analysis on Triangular Shaped Honeycomb..........173

7.2.3.1 Model Validation and Results..........................................1757.3 Chapter Summary ....................................................................................180

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8. CONCLUSIONS AND RECOMMENDATIONS ..............................................181

9. REFERENENCES ...............................................................................................183

APPENDIX..........................................................................................................189Appendix..................................................................................................190

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LIST OF TABLES

Table page

3.1. Analytical and Experimental Values of Crush Strength of Formgrid....................48

3.2. Analytical and Experimental Values of Crush Strength of Half-Hexagonal.........57

4.1. Energy Coefficient R1 for Various Intersecting Angles ........................................81

4.2. Energy Coefficient R3 for Various Intersecting Angles ........................................87

4.3. Validation of Half-Hexagonal Crush Strength Obtained from ParameterizedModel .....................................................................................................................97

5.1. Validation of Experimental and Finite Element Analysis of Square TubeBuckling..............................................................................................................114

5.2. Crush Strength of Formgrid Honeycomb ............................................................119

5.3. Crush Strength of Diamond-Shaped Honeycomb................................................120

5.4. Crush Strength of Triangular- Shaped Honeycomb.............................................122

6.1. Evaluation of Material Strain rate Coefficients for Flexcore Honeycomb..........136

6.2. Evaluation of Material Rate Sensitive Coefficients for HexagonalHoneycomb..........................................................................................................140

6.3. Evaluation of Material Rate Sensitive Coefficients for Half-hexagonalHoneycomb.........................................................................................................142

6.4. Evaluation of Material Rate Sensitive Coefficients for FormgridHoneycomb.........................................................................................................144

6.5. Evaluation of Material Rate Sensitive Coefficients for HexagonalHoneycomb..........................................................................................................151

6.6. Evaluation of Material Rate Sensitive Coefficients for Formgrid .......................153

7.1. 23 Design for the Crush Strength Experiment.....................................................164

7.2. Analysis of Variance Table..................................................................................165

7.3. Reduced Regression Model .................................................................................165

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7.4. 23 Design for the Crush Strength Data for Triangular Shaped Honeycomb .......173

7.5. Analysis of Variance Table..................................................................................174

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LIST OF FIGURES

Figure Page

2.1. Honeycomb cell structure ........................................................................................9

2.2. Flexible Formgrid honeycomb...............................................................................10

2.3. Crushed flexible Formgrid honeycomb .................................................................10

2.4. Schematic diagram of load deflection curve of honeycomb crushed in out-of-plane direction........................................................................................................11

2.5. Schematic diagram of honeycomb crushed in out-of-plane direction ...................12

2.6. Type of end condition in column buckling ............................................................13

2.7. Collapse mode assumed by Alexander ..................................................................15

2.8. Experimental methods for mild steel tube by Alexander.......................................16

2.9. Buckling pattern assumed by McFarland ..............................................................17

2.10. Cell wall rotations during crushing of hexagonal honeycomb assumed byMcFarland ..............................................................................................................18

2.11. Shear mechanism proposed by McFarland ............................................................19

2.12. Out-of-plane buckling of a hexagonal cell.............................................................21

2.13. Plastic buckling of hexagonal honeycomb without rolling deformation ...............22

2.14. Validation of analytical model proposed by Wierzbicki .......................................23

2.15. Buckling pattern proposed by Wierzbicki with rolling deformation .....................24

2.16. Cell wall attachment proposed by Wierzbicki .......................................................25

2.17. Toroidal coordinates ..............................................................................................26

2.18. Analytical results vs. experimental proposed by Wierzbicki considering rollingradius and edge connectivity of the buckling ........................................................28

3.1. Asymmetrical cell configurations - regular Hexagonal honeycomb .....................30

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3.2. Symmetrical cell configurations - Formgrid..........................................................30

3.3. Asymmetrical cell configurations - Half Hexagonal .............................................31

3.4. Curved edge and symmetrical cell configurations – Flexcore...............................31

3.5. Curved edge and symmetrical cell configurations - Double Flex..........................32

3.6. Symmetrical cell configurations-Formgrid............................................................33

3.7. Formgrid honeycomb.............................................................................................33

3.8. Three-dimensional view of the Formgrid cell configuration.................................34

3.9. Three-dimensional view of the Formgrid cell–side view ......................................34

3.10. Three-dimensional view of the Formgrid cell configuration-front view ...............35

3.11. Geometrical representation of the buckling of the Formgridhoneycomb-schematic view...................................................................................35

3.12. Hinge lines influencing energy absorption ............................................................36

3.13. Continuous flow of a thin-walled honeycomb cell wall over a toroidal surface ...37

3.14. Velocity field during the rolling deformation........................................................38

3.15. Buckling angles for Formgrid configuration .........................................................41

3.16. Formgrid honeycomb after fully compressed ........................................................41


3.18. Tributary areas for the half-hexagonal honeycomb ...............................................47

3.19. Validation of analytical modeling of Formgrid honeycomb..................................49

3.20. Half-Hexagonal honeycomb ..................................................................................50

3.21. Hinge lines influencing energy absorption of half-hexagonal honeycomb ...........50

3.22. Buckling of Half-hexagonal honeycomb- schematic view....................................51


3.24. Cell walls influencing the energy absorption.........................................................54

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3.25. Tributary areas for Half-Hexagonal honeycomb ...................................................57

3.26. Flexcore honeycomb..............................................................................................58

3.27. Flexcore honeycomb showing the symmetrical configuration ..............................59

3.28. Single cell of Flexcore honeycomb showing the intersecting wall........................59

3.29. Buckling angles for Flexcore configuration ..........................................................60

3.30. Tributary areas for the Flexcore honeycomb .........................................................64

3.31. Validation of analytical modeling of Flexcore honeycomb F40-5052 ..................64




3.35. Double Flex honeycomb........................................................................................67

3.36. Double Flex honeycomb showing the symmetrical configuration ........................68

3.37. Tributary areas for the Double Flex honeycomb ...................................................72

3.38. Tributary areas for the Double Flex honeycomb ...................................................73

4.1. Validation of experimental and Wierzbicki generalized method ..........................75

4.2. Geometrical shapes of cruciform used for experimental and analytical studies byHayduk and Wierzbicki .........................................................................................76

4.3. Analytical versus experimental data on L-section crushing ..................................77

4.4. Buckling phenomenon of half-hexagonal type honeycomb ..................................78

4.5. Asymmetrical cell configurations - Half-Hexagonal.............................................79

4.6. Symmetrical cell configurations – Formgrid .........................................................80

4.7. Honeycomb wall length associated with energy absorption..................................82

4.8. Honeycomb wall associated with energy absorption.............................................83

4.9. Half-Hexagonal cell configuration showing cell connectivity ..............................85

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4.10. Validation of proposed parameterized model using square tube buckling ............92

4.11. Validation of proposed parameterized model using cruciform section .................93

4.12. Hexagonal cell configurations ...............................................................................94

4.13. Validation of parameterized model using hexagonal honeycomb.........................95

4.14. Half-hexagonal honeycomb cells...........................................................................97

4.15. Formgrid configuration..........................................................................................98

4.16. Experimental validation of Formgrid honeycomb.................................................99

4.17. Flexcore configuration .........................................................................................100

4.18. Validation of Flexcore honeycomb F40-5052 .....................................................101

4.19. Validation of Flexcore honeycomb F40-5052 .....................................................102

4.20. Double Flex configurations..................................................................................103

4.21. Experimental validation of Double Flex..............................................................104

4.22. Diamond shaped honeycomb...............................................................................105

4.23. Triangular shaped honeycomb.............................................................................107

5.1. Square tubes for buckling analysis ......................................................................110

5.2. Buckling of square tube .......................................................................................110

5.3. Validation of quasi-static and dynamic buckling of square tube specimen.........111

5.4. Simulation stages of dynamic buckling of square tube .......................................112

5.5. Simulation stages of quasi-static buckling of square tube ...................................113

5.6. Experimental validation of square tube buckling ................................................114

5.7. Validation of hexagonal honeycomb configuration 1/8-0.001-8.1......................116

5.8. Simulation stages of quasi-static analysis of hexagonal honeycomb ..................116

5.9. Validation of quasi-static analysis of Half-Hexagonal honeycomb ...................117

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5.10. Simulation stages of quasi-static analysis of Half-hexagonal honeycomb..........117

5.11. Simulation stages of quasi-static analysis of Flexcore honeycomb.....................118

5.12. Validation of quasi-static analysis of Flexcore honeycomb ................................118

5.13. Simulation stages of quasi-static analysis of Formgrid honeycomb....................119

5.14. Validation of quasi-static analysis of Formgrid...................................................120

5.15. Simulation stages of quasi-static analysis of diamond shaped honeycomb.........121

5.16. Validation of quasi-static analysis of Diamond-shaped honeycomb...................121

5.17. Simulation stages of quasi-static analysis of Triangular- shaped honeycomb ....122

6.1. Perrone experimental methods for determining rate sensitivity ..........................125

6.2. Bending mechanisms of thin walled structures ...................................................127

6.3. Basic thin plate showing geometry and direction ................................................128

6.4. Crush load versus impact velocity .......................................................................131

6.5. Sandwich plate subjected to indentation by a circular punch ..............................131

6.6. Flexcore honeycomb model.................................................................................134

6.7. Graph for evaluating new material strain rate coefficients ..................................137

6.8. Validation of analytical modeling of square tube performed by Abromowicz &Jones [46] and proposed model............................................................................138

6.9. Hexagonal honeycomb models ............................................................................138

6.10. Graph for evaluating new material strain rate coefficients for hexagonalhoneycomb -1/8-0.001-5052-8.1pcf ....................................................................140

6.11. Graph for evaluating new material strain rate coefficients forHalf-hexagonal.....................................................................................................142

6.12. Formgrid honeycomb model................................................................................143

6.13. Graph for evaluating new material strain rate coefficients for Formgrid ............145

6.14. Graph for evaluating new material strain rate coefficients for Double-Flex .......146

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6.15. Graph for evaluating material strain rate coefficients-Diamond shapedhoneycomb...........................................................................................................147

6.16. Graph for evaluating material strain rate coefficients-Triangularshaped honeycomb...............................................................................................148

6.17. Validation of dynamic crush strength of Flexcore...............................................149

6.18. Validation of dynamic crush strength of Half-hexagonal....................................150

6.19. Validation of dynamic crush strength of Hexagonal honeycomb pcf-22 ............150

6.20. Graph for evaluating material strain rate coefficients for Hexagonalhoneycomb..........................................................................................................151

6.21. Validation of analytical dynamic crush strength of hexagonalhoneycomb -1/8- 0.001-5052-8.1pcf ...................................................................152

6.22. Validation of analytical dynamic crush strength of hexagonalhoneycomb -1/8- 0.001-5052-8.1pcf ...................................................................152

6.23. Validation of analytical dynamic crush strength of Formgridhoneycomb – 40-.0019-pcf-3.1............................................................................153

6.24. Validation of analytical dynamic crush strength of diamond-shapedhoneycomb, strain rate coefficients obtained from 100, 200,300 inches/sec. velocities.....................................................................................154

6.25. Validation of analytical dynamic crush strength of triangular-shapedhoneycomb; strain rate coefficients obtained from 100 and 200 inches/sec.velocities ..............................................................................................................154

6.26. Validation of analytical dynamic crush strength of hexagonal honeycomb-1/8-5052-4.5 pcf; strain rate coefficients obtained from 343,421, 551 and 689 inches/sec. velocities ...............................................................155

6.27. Analytical dynamic crush strength obtained from low velocitymaterial coefficients for half hexagonal- 1/8-0.006-5052-38-pcf;strain rate coefficients obtained from 25 and 50 inches/sec impact velocities ....156

6.28. Analytical dynamic crush strength obtained from low velocity materialcoefficients for hexagonal- 1/8-0.002-5052-8.1pcf; strain ratecoefficients obtained from 25,100 and 200 inches/sec impact velocities ............157

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6.29. Analytical dynamic crush strength obtained from low velocities2.5, 7.5,10,25 inches/sec FEA data for Formgrid honeycomb ............................157

6.30. Analytical dynamic crush strength obtained from low velocitymaterial coefficients for Flexcore- F40-0.0019-5052-3.1pcf, strainrate coefficients obtained from 7.5 and 25 inches/sec impact velocities .............158

6.31. Analytical dynamic crush strength obtained from low velocitymaterial coefficients for Diamond-shaped honeycomb; strain ratecoefficients obtained using impact velocities 100 and 200 inches/sec ...............158

6.32. Analytical dynamic crush strength obtained from low velocitymaterial coefficients for Double Flex; strain rate coefficients obtainedusing impact velocities 50 and 100 inches/sec ....................................................159

6.33. Analytical dynamic crush strength obtained from low velocitymaterial coefficients for Triangular shaped honeycomb; strain ratecoefficients obtained from 100 and 200 inches/sec. velocities............................159

6.34. Analytical dynamic crush strength obtained from low velocitymaterial coefficients for hexagonal honeycomb -1/8-5052-4.5 pcf;strain rate coefficients obtained from 343 and 421 inches/sec. velocities..........160

7.1. Half Normal Plot..................................................................................................164

7.2. Half normal plot for reduced model.....................................................................166

7.3. Normal plot of residuals.......................................................................................167

7.4. Residual vs. Predicted values...............................................................................167

7.5. Outliers.................................................................................................................168

7.6. Residual vs. Edge angle .......................................................................................169

7.7. Residual vs. Gauge Thickness .............................................................................169

7.8. Residual vs. Honeycomb width ...........................................................................170

7.9. Interaction graph for Honeycomb width and gauge thickness.............................171

7.10. Two dimensional contour plots of Response surfaces with edge angle 30

degree..................................................................................................................171

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7.11. Two dimensional contour plots of Response surfaces with edge angle 50degree...................................................................................................................172

7.12 Three dimensional contour plots of response surfaces ........................................172

7.13. Half normal Plot...................................................................................................174

7.14. Normal plot of residuals.......................................................................................175

7.15. Residual vs. Predicted values...............................................................................176

7.16. Outliers.................................................................................................................176

7.17. Residual vs. Edge angle .......................................................................................177

7.18. Residual vs. Gauge Thickness .............................................................................178

7.19. Residual vs. Honeycomb width ...........................................................................178

7.20. Interaction graph for Honeycomb width and gauge thickness.............................179

7.21. Two dimensional contour plots of Response surfaces with edge angle 30degree..................................................................................................................179

7.22. Three dimensional contour plots of Response surfaces .......................................180

1

CHAPTER 1

INTRODUCTION

1.1 Motivation

Metallic honeycomb is widely used as energy absorption material in structural

applications because of its high energy absorption and high strength-to-weight ratio,

compared to other materials. The strength characteristics of metallic honeycomb should

be evaluated before using it in energy absorption devices. Honeycomb applications

depend upon their geometrical configuration. In order to develop different types of

honeycomb configurations according to design needs, crush strength properties must be

evaluated. A parameterized model of honeycomb that can accommodate typical

honeycomb configurations is needed. This will enable the designers to select the

honeycomb and evaluate the crush strength properties. In this dissertation, a

parameterized honeycomb model was developed and validated using experimental

methods and finite element analysis. In the case of a high-impact landing of the space

module, a high crush strength honeycomb material is needed to protect the structural

members of the space module. Therefore, it is necessary to evaluate the crush strength of

the honeycomb material at impact conditions.

An analytical model of dynamic crush strength incorporating the strain rate

material constants extracted from real experimental testing of honeycomb materials is

needed. An analytical model was developed to determine the dynamic crush strength

using the strain rate equation and strain rate material constant obtained from the

experimental testing of honeycombs.

2

1.2 Literature Review

Wierzbicki [1] developed an analytical method for crush strength and energy

dissipation during the crush of hexagonal honeycomb. The mathematical model he

developed essentially dealt with the quasi-static crush strength of the metallic hexagonal

honeycomb. He showed that the cell wall crushes progressively with a buckling

wavelength of 2H. Wierzbicki and Abramowicz [2] developed a basic folding mechanism

of thin-walled structures. In their research, they had shown that the crushing of the

hexagonal axisymmetric cell walls crushes like plastic flow of metal sheet over a toroidal

surface. They also showed that rate of energy dissipation was due to continuous and

discontinuous velocity fields. Abramowicz [3] discussed the crushing distance during the

buckling of metal column and developed the analytical model of effective crushing

distance. Reid et al. [4] performed experiments on the crushing of foam-filled sheet metal

and developed an equation for the mean crushing load of the square tubes. Wierzbicki

and Bhat [5] analyzed the deformation mode of the axisymmetric crushing of tubes and

developed a solution for the mean crushing force. Santosa and Wierzbicki [6] studied the

crash behavior of box columns filled with aluminum honeycomb or foam. Abramowicz

and Jones [7] developed the folding and collapse modes of the circular and square tubes,

and developed an equation for the dynamic progressive buckling of tubes.

Mamalis et al. [8] developed a theoretical model of the plastic collapse of a

conical shell. Langseth and Hopperstad [9] investigated the behavior of square thin-

walled aluminum extrusion, subject to static and dynamic loading. Both static and

dynamic tests were performed and the primary variables were the wall thickness and

temper of the square tubes, and the impact velocity of the projectile. Experimental results

3

also show that the dynamic mean force was significantly higher than the corresponding

static force for the same axial displacement, which indicates a strong inertia effect.

Reid and Reddy [10] studied the deformation modes of tapered sheet metal tubes

with rectangular cross sections. Yasui [11] performed experimental testing on multi-layer

honeycomb and discovered that they absorb more energy than single-layer honeycomb.

This conclusion is useful in sensitivity analysis. Doyoyo and Mohr [12] experimentally

studied the micro-structural response of the aluminum honeycomb.

Aaron [13] developed a honeycomb model for the proposed arresting mechanism

for the drop testing tower. Dynamic analysis was performed to simulate the impact on the

arresting mechanism.

Eskandar and Marzougui [14] performed dynamic simulation using the Ls-dyna

model of honeycomb for the surrogate crash test vehicle impacted with roadside objects.

The compressive properties of honeycomb materials using experimental and theoretical

methods were studied by Paik et al. [15]. Aaron et al. [16] developed experimental

methods, performed finite element analysis to determine the crush strength of Formgrid

honeycomb, and did sensitivity analysis to study the parameters affecting the crush

strength properties. Chou [17] compared various simulation software for modeling

honeycomb properties.

Wierzbicki and Mohr [18] studied the crush responses and energy absorption

properties of double-walled sandwich columns with a honeycomb core. Lee et al. [19]

performed an experimental study on the behavior of aluminum-honeycomb sandwich

panels. Gibson and Ashby [20] discussed the out-of-plane deformation of the hexagonal

4

honeycomb. Johnson et al. [21] studied the in-extensional collapse of thin-walled tubes

under axial compression.

Dynamic shear properties of honeycomb were studied by Adam and Maheri [22]

who developed a setup for measuring the shear properties experimentally. Allan [23]

experimentally performed axial crushing of cylindrical tubes and discussed the

progressive buckling of metal tubes.

Wierzbicki and Abromowicz [24] derived the equation for the crushing of thin-

walled strain rate sensitive materials. They developed the basic bending mechanism for

the dynamic crushing of thin-walled members and developed the concept of the rolling

radius in the deformation mechanism.

Wu [25] experimentally determined the plastic buckling mechanism of metallic

honeycomb. He showed that the half wavelength predicted by Wierzbicki [1] was about

80 percent of the experimental values. He showed that the crush strength was

independent of the number of cells and that cells having a larger width would have lower

crush strength.

Wu and Jiang [26] performed experiments for studying the parameters affecting

dynamic crush strength. They found that the width of the honeycomb plays an important

role in the crush strength values.

Baker et al. [27] developed a method for testing high density metal honeycombs.

They used aluminum and stainless steel material and found out that aluminum behaves

with a strain rate effect. The plastic stress in the case of the dynamic impact testing was

50 percent higher than the static testing. The rate effect was different for static and the

dynamic tests. In contrast, they discovered that there was no difference between the

5

initial and final deformation of the honeycomb tested in the quasi-static and dynamic

environments.

Goldsmith and Sackman [28] performed experimental testing on several the

honeycombs under quasi-static and impact conditions. Their research showed that the

dynamic and static testing of the honeycomb has a variation in the maximum crush load.

Bodner and Symonds [29] studied the Parkes [30] experimental curves on the effect of

rate of strain on yield stress for mild steel and aluminum alloys. They used Cowper-

Symond equations to determine the strain rate. Bandak and Bitzer [35] derived the

equation for designing the energy absorbers in vertical and horizontal impact conditions,

and discussed various applications of the honeycomb. Howell et al. [38] discussed the

drop tests and the methods used to prevent the drop carriage from damaging the load cells

and gears using a crushable aluminum honeycombs. Rawlings [39] studied the response

of the structures to dynamic loads. He discussed the application of the Cowper-Symond

equation for the dynamic region of materials.

Doengi et al. [40] developed lander shock alleviation techniques using aluminum

honeycombs. They performed experimental studies on various hexagonal honeycombs

under quasi-static and impact conditions. Aaron et al. [41] developed the analytical

model for Formgrid honeycomb material and developed the method for analyzing the

honeycomb using half-model configurations. Aaron et al. [42] developed the parameters

affecting the rate sensitivity of the metallic honeycomb and performed the sensitivity

analysis using response surfaces. Zhao and Gray [44] did experimental testing of

metallic honeycomb in both static and dynamic conditions and found the strain rate effect

of aluminum honeycombs. Macaulay and Redwood [45] studied the large-scale

6

deformation of circular tubes and railway coach models, and compared the static and

low- speed crumpling loads.

Santosa and Wierzbicki [50] developed the concept of double-walled sandwich

columns for energy absorption and found that energy absorption is higher and weight

savings of the energy absorption systems is about 30 to 45 percent. They also developed

the combined equation for the double-walled sandwiched column. The LS-DYNA

example manual [51] shows the procedure for developing the buckling analysis of a

square tube. This analysis is performed on both square tubes and honeycomb to develop

finite element models of honeycomb and to study the effect of impact velocities on

honeycombs. Wierzbicki et al. [53] developed a derivation for the impact of sandwich

panel with a crushable core. His analytical model is useful for a impact surface having a

circular cross section.

Hinnerrichs et al. [54], from Sandia National Laboratories, developed the

Honeycomb Constitutive model which can determine the crush strength in bi-axial

loading conditions. This new honeycomb model can be useful in predicting the off-axis

properties. Cote et al. [55] developed the analytical model for the out-of-plane

compressive behavior of metallic honeycomb. They developed the square-shaped

honeycomb and discovered that the energy absorption of stainless steel square-shaped

honeycomb is higher than aluminum hexagonal honeycomb.

7

1.3 Outline

This dissertation report discusses the development of the parameterized model of

metallic honeycomb for typical configurations. Analytical models for dynamic crush

strength for typical honeycomb configurations were derived. Material constants for strain

rate sensitivity of typical configurations were developed, which will be useful in

determining the crush strength at a higher strain rate. A methodology was developed to

determine the crush strength at high velocity using low-velocity test data of the

honeycomb. Performance analysis was performed on the honeycomb geometrical

parameters, which will be beneficial in the design and selection of honeycomb

parameters for energy absorption systems.

A short summary of each chapter follows. Chapter 2 deals with the detailed

review of the important research previously performed on the buckling of honeycomb.

Advantages and disadvantages of the previous research have been discussed and an

alternate solution is proposed in this chapter. Chapter 3 discusses the development of a

methodology for analyzing symmetrical, asymmetrical, curve-shaped and reinforced

honeycombs. Analytical models of Formgrid, Flexcore, Half-Hexagonal, Double Flex,

and two new types triangular and diamond-shaped honeycombs are developed. This

chapter will provide insight for developing a parameterized honeycomb model, which

will be discussed with in Chapter 4. Chapter 5 deals with the finite element model, and

validation of the analytical models of honeycombs that were developed. Chapter 6 is

concerned with the development of the dynamic crush strength and the strain rate

material constants for typical honeycombs and their validations. Chapter 7 discusses the

8

performance analysis of the honeycomb parameters to enhance the energy absorption

properties. Finally, the conclusion and recommendations are provided in Chapter 8.

1.4 Scope and Objectives

A parameterized analytical model of honeycomb was developed to determine

crush strength. Crush strength of new types of honeycomb models can be determined

analytically using parameterized honeycombs models. Since the metallic honeycomb is

used for impact protection, the dynamic crush strength of the honeycomb can be

predicted using the dynamic crush strength equation developed for particular

honeycombs. The methodology proposed to determine the material constant depending

on the geometry of the honeycomb will be useful in evaluating the crush strength at a

higher velocity, which will reduce expensive dynamic testing procedures. A methodology

was developed to predict the dynamic crush strength using crush strength data obtained

from the low-velocity test.

9

CHAPTER 2

REVIEW OF ANALYTICAL MODELING OF METALLIC HONEYCOMB

2.1 Objectives

The objectives of this chapter are to study the energy absorption properties of

metallic honeycomb and to review the analytical models developed for determining crush

strength of honeycomb by various research.

2.2 Energy Absorption Properties of Metallic Honeycomb

The strength of honeycomb is evaluated in three different axes – the T-direction

which is thickness or cell depth, the L-ribbon direction and the W-transverse direction, as

shown in Figure 2.1. Strength characteristics are classified into in-plane properties and

out-of-plane properties. The properties obtained when the load is applied in the L and W

axes are in-plane properties. Energy absorption is higher if the honeycomb is compressed

in the T-direction, which is the out-of-plane direction. Figures 2.2 and 2.3 show the

flexible type of honeycomb in its uncompressed and compressed state.

Figure 2.1. Honeycomb cell structure.

L-direction

W-direction

T-direction

10

Figure 2.2. Flexible Formgrid honeycomb.

Figure 2.3. Crushed flexible Formgrid honeycomb.

The out-of-plane directional axis is the effective buckling or crushing axis, since

the crush strength obtained by other axes is less compared to the out-of-plane axis. In

order to evaluate the crush strength of a honeycomb, it was crushed between the rigid

base and a punch at quasi-static velocities. The Load-deflection curve of the honeycomb

crushed in an out-of-plane direction is shown in Figure 2.4. During the crushing of

11

honeycomb, the peak load occurred due to the breaking up of the bonds between the

inter-connected cells. The strength-to-weight ratio was higher in the case of honeycombs

which more easily absorb higher energy with a lower material weight.

The buckling of the honeycomb was uniform, which can be seen from the mean

load curve. These features enhance the cushioning effect during the impact crush of the

honeycomb. The mean crush load is the average force being absorbed by the honeycomb

during the crush. The crushing of the honeycomb progresses until it attains the

compaction phase which is known as the densification region, and beyond that the load

increases drastically due to the locking up of the cells. The schematic diagram of

honeycomb crushed in out-of-plane direction as shown in Figure 2.5.

Deflection

Loa

d

Figure 2.4. Schematic diagram of load deflection curve of honeycomb crushed in out-of-plane direction.

Mean crush load

Energy Absorbed

Densificationstage

Peak Load

12

Figure 2.5. Schematic diagram of honeycomb crushed in out-of-plane direction.

2.3 Various Developments in Analytical Modeling of Honeycomb

In this section, various methods developed for determining the crush strength of

honeycomb is reviewed. The methodology, advantages, and disadvantages of the crush

strength method developed by previous researchers is discussed.

2.3.1 Euler Buckling of Columns

Basically, when a column is subjected to a compressive load, it fails by crushing

if it is short; otherwise, it buckles. This load is called a buckling load or a crippling load.

In the 1700s, Euler [56] developed the buckling of columns at various end conditions, as

shown in Figure 2.6. The end conditions are as follows: both ends hinged, both ends

fixed, one end fixed and the other end hinged and one end fixed and the other end free.

Euler assumptions are listed as follows

• The column is initially straight and the load is applied along its axis.

• The cross section of the column is uniform.

• The column material is perfectly elastic, homogenous, and isotropic.

• The length of the column is very large as compared to its cross-sectional

area.

Rigid Punch

HoneycombMaterial

Rigid Base

AppliedVelocity

Z

X

13

• Failure takes place only because of buckling.

Figure 2.6 Type of end condition in column buckling

The Euler formula is given as

2

22

2

2

l

EAkC

l

EICP

ππ == (2.1)

2

2

��

��

�=

k

l

EACP

π(2.2)

2AkI = (2.3)

Case 1 Case 2 Case 3 Case 4

P P P P

14

where

P = crippling load

E = modulus of elasticity of column material

A = area of cross section

K = least radius of gyration of cross section

L = length of column

C = constant depending on end condition

C = 1 for both ends of column hinges

C = 2 for both ends of column fixed

C = 3 one end fixed and the other hinged

C = 4 one end fixed and the other end free

The Euler method is useful for the buckling of columns and thin walls where the

end conditions are known. An analysis of tube buckling has been derived by researchers

using the Euler equation. However, his method is not suited for buckling of honeycomb

material which involves adjacent cell wall attachments.

2.3.2 Collapse of Thin Cylindrical Shells

In 1959, Alexander [48] developed the simple mode of collapsing a tube. He

assumed that the collapse of the tube is like a bellows and in the form of a concertina

with straight-sided convolutions, as shown in Figure 2.7. He assumed that the material of

the tube is rigid-plastic. The work done on the circular tube is due to the bending at the

circular joints and the stretching of the metal between the joints. He predicted the total

displacement of a single convolution as 2H.

15

Figure 2.7. Collapse mode assumed by Alexander.

The mean axial load is given by

DtHP 5.1σ= (2.4)

where

P = mean axial load

D = mean diameter of tube

H = 6.08, a constant

t = thickness of the cylindrical shell

� = yield strength of the material

As seen from the graph shown in Figure 2.8, the mean straight line drawn through

the experimental data has a slope H� = 434,000 lb/sq.in and considering � = 70,000 lb/in3

t

H

H�

P

P

D

16

which gives H= 6.2 for experimental methods which is being compared with constant

6.08 from analytical methods.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 0.005 0.01 0.015 0.02

t1.5Sqrt(D) (in)

P(l

b)

D=1.43 inches

D=2.12 inches

D=2.66 inches

Linear (D=1.43 inches)

Figure 2.8. Experimental methods for mild steel tube by Alexander.

This method is simple and is the approximate solution for the crushing of circular

tubes. Also Alexander has explained the convolution of the buckling and predicted the

2H displacement during the crushing of the tube. But this method does not deal with the

edge connectivity and rolling radius of the buckling. These are important aspects of

buckling of honeycomb walls which will be explained later in this chapter.

17

2.3.3 Crushing of Hexagonal Cell Structures

In 1963, McFarland [47] developed a new method for hexagonal structures

subjected to axial loading. He assumed a rigid-plastic material. He also assumed that the

crushing of the hexagonal honeycomb is a function of cell shape, and that, unlike the thin

cylinder shell the collapse mode does not vary with the cell-diameter to wall-thickness

ratio. This is one of the basic concepts in the development of crush strength for

honeycomb materials. He assumed that in-plane deformation of the cell wall occurs

during its collapse. He showed that the energy of deformation consists of energy of

bending deformation and energy of shear deformation. The buckling pattern is shown in

Figure 2.9 and 2.10.

Figure 2.9. Buckling pattern assumed by McFarland.

The energy of bending is calculated using the basic equation

θ∆= ME� (2.5)

]396.12057.2[3

2/1

HMS

E +��

��

�= (2.5 6)

S

D

18

Figure 2.10. Cell wall rotations during crushing of hexagonal honeycomb assumed byMcFarland.

The mean crush stress is calculated by considering the tributary area of the crush

]628.28750.4[

2

2H

S

hH

P ycr +

��

��

�=

σ(2.7)

where

E� = rate of energy of bending

E = energy due to bending

S = size of honeycomb

�y = yield stress

h = gauge thickness

M = plastic bending moment

McFarland [47] calculated the energy of shear deformation which also contributes

to the total crush strength. He considered that the shear deformation is due to the in-plane

P

�

19

deformation of the cell wall and is basically due to the stretching of the thin wall as

shown in Figure 2.11.

The energy due to shear deformation is given as

22 )2sin2( dABhE iy += ψσ (2.8)

The mean crush strength obtained by the energy of shear deformation is given by

��

��

�=S

hqf ypcr 155.1 (2.9)

Figure 2.11. Shear mechanism proposed by McFarland.

The combined mean crushing stress of a hexagonal cell is

[ ]��

��

�+

��

��

�

+=

S

h

S

hH

Hf yp

ycr 155.1

628.28750.4

2

2

σ(2.10)

where

D = width of cell wall

fcr = mean crush stress

P = mean crushing load

qyp = shear yield stress

� ��

2A

d1

d2

20

S = cell minor diameter

l = width of the basic panel element

h = cell wall thickness

E = energy dissipated

H = l/D

The experimental results were obtained for a hexagonal honeycomb configuration

3003 H-19 aluminum alloy with a cell diameter of 0.75 inch. The results show clearly

that the experimental values are less when compared to the analytical results. The shear

energy is less compared to the bending energy in the analytical results when the h/S

values increases.

The in-plane deformation assumed by McFarland is incorrect and is later

explained as a rolling deformation. This theory developed by McFarland lacks the rolling

deformation and plastic hinge traveling energy. Also, this method is based on the

geometrical conditions of the hexagonal cell and does not involve any other typical

shapes.

2.3.4 Analytical Model of Honeycomb without Rolling Deformation

In the 1980s, Wierzbicki [20] predicted that honeycomb cells are compressed in a

progressive manner by a wavelength H, which is approximately equal to the side of the

length of the cell. This is an approximate solution for determining the buckling stress or

the crush strength of the honeycomb. Figure 2.12 and 2.13 show the buckling mechanism

of one cell of the hexagonal honeycomb.

21

Figure 2.12. Out-of-plane buckling of a hexagonal cell.

The total work done per cell wall is given by

)2( DlM +π (2.11)

The total displacement of the cell wall is given by

hH

22

− (2.12)

The applied force is given by

��

��

� − hH

P 22

(2.13)

H

D

θ

b

lh

22

Equating the applied force to the work done on the cell wall

( )DlMhH

P +=��

��

� − 222

π (2.14)

The tributary area of the hexagonal unit cell as shown in Figure 2.12, is

θθ cos)sin(2 lhlAt +=

Figure 2.13. Plastic buckling of hexagonal honeycomb without rolling deformation.

But Force P = (stress �3) * (At area of the cell)

Substituting θθσ cos)sin(2 3 lhlP += in equation (2.14)

Also substituting4

2hM

ysσ=

Considering approximately lH = in equation (2.14)

( )2

2)sin/cos4

)2/(h

llD

lD ys

pl

θθσπσ

++= , (2.15)

P

Plastic HingeH/2

2hBuckingAngle =180

23

Where

ysσ = yield strength

3σ = stress in out-of-plane direction

h = gauge thickness of hexagonal cell

H = wave length of cell buckling

θ = inner edge angle

M - plastic moment

l = side length of hexagonal honeycomb

D = face length of hexagonal honeycomb

plσ = plastic buckling stress or crush strength of hexagonal honeycomb

0

50

100

150

200

250

300

350

400

0 0.0005 0.001 0.0015 0.002 0.0025 0.003

Gauge Thickness (inch)

Cru

shSt

reng

th(p

si)

Experimental Published [10]

Analytical without rolling deformation [20]

Figure 2.14. Validation of analytical model proposed by Wierzbicki.

24

This method can be applied to any typical shapes of honeycomb since the energy

calculation does not depend upon the shape of the honeycomb. Variation as shown in

Figure 2.14 between the experimental and analytical methods is due to the energy

calculation does not include wall edges attachment and their rolling deformation.

2.3.5 Analytical Model of Honeycomb with Rolling Deformation

In 1983, Tomaz Wierzbicki [1] developed a method for determining the crush

strength of honeycomb for the hexagonal honeycomb. He assumed that the plastic hinges

responsible for energy dissipation travel during buckling of the cell wall. He explained

that buckling of the cell wall takes place due to rolling deformation instead of in-plane

deformation. The rolling deformation, as shown in Figure 2.15, is assumed to be thin wall

of honeycomb flowing over the toroidal section.

Figure 2.15. Buckling pattern proposed by Wierzbicki with rolling deformation.

Horizontal hinges

Inclined hinges

Rollingdeformation

25

The total energy dissipated during the buckling process is the sum of energy dissipated

due to the rolling deformation, energy dissipated due to the traveling of horizontal hinge

lines and energy dissipated due to the inclined hinge lines. He also assumed that buckling

takes place in a progressive manner with a constant buckling wavelength 2H. The energy

calculation was calculated by considering the wall attachment, as shown in Figure 2.16.

Figure 2.16. Cell wall attachment proposed by Wierzbicki.

The energy E1 is associated with the energy of the rolling deformation which is a

continuous flow over the toroidal surface. The basic energy flow due to the continuous

plastics deformation is given by

2�o

�

�-2�

H

�

26

� += dsNME )(1 µκ� (2.16)

where

M = circumferential bending moment of buckling of thin plate

= rate of curvature of toroidal element

= rate of extension of toroidal element

N = circumferential membrane force during buckling of thin plate

r = radius of toroidal shell

b = small radius of torus

� = circumferential coordinate

� = meridional coordinate

ds = surface element of toroidal flow as shown in Figure 2.17 is given

by

θϕ bdrdds = (2.17)

Figure 2.17. Toroidal coordinates.

r

�b

27

)(2 LdL

ME θ�� = (2.18)

where L is the length of the horizontal plastic hinges.

While calculating the energy due to horizontal hinges, Wierzbicki assumed a

three-wall attachment of the hexagonal honeycomb, as shown in Figure 2.15. This

assumption was valid only for hexagonal configurations.

Considering eight horizontal hinge lines of length D/2 having gauge thickness h,

and four hinge lines of length D/2 having gauge thickness 2h, since they have a double

layer due to the attachment of the adjacent walls, as shown in Figure 2.15, then

( ) ��

��

� += 222 2

224

228

2h

Dh

DE oo σσπ� (2.19)

Energy dissipated due to buckling of the inclined hinge lines is given as

tLME θ�� =3 (2.20)

where Lt is the total length of the inclined plastic hinge lines.

Total energy absorbed is equal to crush load acting over 2H distance.

3212* EEEKP ++= (2.21)

Using equation (2.21), the crush strength fcr is obtained by considering the tributary area

of the hexagonal honeycomb

3

5

56.16 ��

��

�=s

hf ocr σ (2.22)

28

0

50

100

150

200

250

300

350

400

0 0.0005 0.001 0.0015 0.002 0.0025 0.003

Crush Strength (psi)

Gua

geT

hick

ness

(inc

h)

Analytical [1]


Figure 2.18. Analytical results vs. experimental proposed by Wierzbicki consideringrolling radius and edge connectivity of the buckling.

This type of cell wall analysis can be applied to honeycomb configurations having

asymmetrical walls or three attachment walls, but it cannot be used when there are more

than three interconnecting symmetrical walls, as in the case of Formgrid. No attention

was given to the calculation for the reinforcement wall, as in the case of the Half-

Hexagonal honeycomb, which will be explained by the proposed methodology.

2.4 Chapter Summary

In this chapter, research performed on the crush strength properties of metallic

honeycomb was discussed. Advantages and disadvantages of the method developed

previously have been identified. The next chapter proposes a new parameterized model,

which can accommodate various typical shapes of the honeycomb configuration.

29

CHAPTER 3

ANALYTICAL MODELING OF SYMMETRIC ASYMMETRIC ANDCURVE-SHAPED HONEYCOMB CONFIGURATIONS

3.1 Objectives

The objective of this chapter is to develop analytical modeling of typical

honeycomb shapes: flexible types Formgrid, Flexcore, Double Flex, and reinforcement

type Half-Hexagonal.

3.2 Methodology

Here honeycombs are classified into symmetrical, asymmetrical, curved and

reinforced. Previous methods developed by researchers cannot be used for symmetrical,

curved, and reinforcement types of honeycombs. Flexible honeycombs, namely

Formgrid, having a symmetrical configuration were studied and analyzed to obtain the

crush strength equation. A new methodology of analyzing the honeycomb configuration

and calculating the energy absorption has been proposed. Analytical equations have been

developed for reinforced Half-Hexagonal, Flexcore, and Double Flex honeycombs and

have been validated with experimental methods.

3.3 Classification of Honeycomb According to Geometrical Parameters

Classification of honeycomb is made according to geometrical parameters and

cell connectivity.

3.3.1 Symmetrical and Asymmetrical Honeycomb Configuration

Asymmetrical honeycomb configuration, is shown in Figure 3.1 is defined as such

because symmetry along the cell connectivity does not exist. In the case of symmetrical

configuration, symmetry along the cell connectivity does exist as shown in Figure 3.2.

30

Figure 3.1. Asymmetrical cell configurations - regularHexagonal honeycomb.

Figure 3.2. Symmetrical cell configurations - Formgrid.

60°°°°

2D

31

3.3.2 Reinforced Honeycomb Configuration

In order to increase the crush properties, honeycombs are reinforced with walls as


Figure 3.3 Asymmetrical cell configurations - Half Hexagonal.

3.3.3 Curved Edge Honeycomb

Curved edge honeycomb is shown in Figures 3.4 and 3.5.

Figure 3.4. Curved edge and symmetrical cell configurations - Flexcore.

Asymmetricaltype of cellconnectivity

Re-inforcement wall

32

Figure 3.5. Curved edge and symmetrical cell configurations - Double Flex.

3.4 Analytical Modeling of Typical Honeycomb Shapes

3.4.1 Analytical Modeling of Formgrid Honeycomb

When studying the folding mechanism of the Formgrid configuration, the

interconnecting cell walls are considered the buckling elements. Figure 3.6 shows the

Formgrid cell configuration. The half symmetrical section of the Pi-shaped member, as

shown in Figure 3.6, is analyzed for out-of-plane loading conditions. The basic

assumptions made in this derivation are that the cell buckles in 2H distance and the

material is perfectly plastic.

33

Figure 3.6. Symmetrical cell configurations- Formgrid.

Figure 3.6. Symmetrical cell configurations-Formgrid.

Energy absorption is calculated for only half of the Formgrid honeycomb section

as shown in Figures 3.7 and 3.8. By analyzing this half-section, the energy calculated is

doubled to obtain the total energy absorption of the Formgrid. This type of configuration

is a symmetrical honeycomb type. Analyzing the honeycomb cell gave an insight into

developing a parameterized honeycomb model, which can accommodate most of the

honeycomb configurations.

Figure 3.7. Formgrid honeycomb.

Pi-Shapedconfiguration

60°°°°

2D

34

Figure 3.8. Three-dimensional view of the Formgrid cell configuration.

Figure 3.9. Three-dimensional view of the Formgrid cell–side view.

2�o

2H

35

Figure 3.10. Three-dimensional view of the Formgrid cell configuration-frontview.

Figure 3.11. Geometrical representation of the buckling of the Formgridhoneycomb-schematic view.

�

s

�

�-2�

H

�

s

��-2�

36

The geometrical parameters and the angles are given by the following

D = half cell size

H = half wave-length

2�o = angle between the adjacent walls and is assumed to be constant duringbuckling

� = angle between wall during bending

� = angle of rotations of the walls

s = horizontal displacement during buckling

� = angle between the inclined hinge of wall and the horizontal edge

Figure 3.12. Hinge lines influencing energy absorption.

Total energy dissipated during the buckling process is the sum of energy

dissipated due to the rolling deformation, energy dissipated due to the traveling of

horizontal hinge lines, and energy dissipated due to the incline hinge lines.

Inclined Hinge Line

Rolling Deformation

Horizontal Hinge Lines

37

From Figure 3.12, which shows the partial crushing of the Pi-shaped Formgrid

honeycomb, the compressed distance can be given in equations (3.1) and (3.2)

The vertical crush distanceαδ cos22 HH −= (3.1)

Horizontal distance due to bucklingαsinHs = (3.2)

Therefore, the vertical velocity of the buckling can be written as

ααδ �

� )(sin2H= (3.3)

The horizontal velocity is as follows:

αα �� )(cosHs = (3.3)

The basic folding mechanism for Formgrid honeycomb uses Wierzbicki’s [1]

method for hexagonal honeycomb. According to this method, the plastic buckling of the

hexagonal honeycomb is assumed to be the flow of a thin sheet on a toroidal surface as


Figure 3.13. Continuous flow of a thin-walled honeycomb cell wallover a toroidal surface.

Toroidal surface

38

The tangential velocity should be determined for evaluating the energy dissipated during

the toroidal deformation. The profile of the toroidal deformation has lines BC and MN, an

arc of a circle CD, and a toroidal radius b. The tangential velocity during the toroidal flow

of the material is shown in Figure 3.14

Figure 3.14. Velocity field during the rolling deformation.

The tangential velocity of the plastic flow of the cell wall over the toroidal

surface is given by

ωψ �bVVt == tan/ (3.5)

where

ω� = the angular velocity of the plastic flow of the cell wall

ψ = the angle with respect to 0ψ and β according to the relation below

φπ

ψπψψ oo

2−+=

(3.7)

b - rolling radius

V0

VtV

ϖ

Toroidal flow ofbuckling

B

C M

N

D

39

The meridian � and circumferential � coordinates of the toroidal surface vary as

ψπθψπ +≤≤− 2/2/ (3.7)

βφβ ≤≤− (3.8)

The rate of extension of the toroidal surface is given by

θωλ sin/1 br �

� = (3.9)

The energy flow due to the continuous plastics deformation is given by

� += dsNME )(1 λκ �� (3.10)

where

M = circumferential bending moment of buckling of thin plate

= rate of curvature of toroidal element

λ� = rate of extension of toroidal element

N = circumferential membrane force during the buckling of thin plate

ds = surface element of the toroidal flow given by

θϕ bdrdds = (3.11)

r = radius of the toroidal shell

According to the flow rule, if gauge thickness h is less than four times the larger

radius of the toroidal shell, then it can be concluded that the circumferential bending

moment M can be neglected.

Therefore,

�= dsNE )(1 λ�� (3.12)

40

Substituting equations (3.5), (3.9), and (3.11) in (3.12) and using equations

(3.6), (3.7), and (3.8) integrating the equation (3.13)

θθψ

ααβ

β

ψπ

ψπdbKNE

o

sintan

cos2/

2/1

�

� �+

−�

+

−=

(3.13)

After integrating the equation (3.13) and substituting the limits, then

�

��

��

��

� −−+�

��

��

��

� −−

= βπ

ψπψψβπ

ψπψψψπ

π oooo

oo

HbNE

2cos*coscos

2sinsin

tan)2(

4 01�

�2

0

cos*

π

αα d (3.15)

But 20

4

h

M=σ ; hN 0σ= (3.16)

where

0σ = flow stress

h = gauge thickness

N = plastic membrane force

M = plastic bending moment

41

Figure 3.15. Buckling angles for Formgrid configuration.

Figure 3.16. Formgrid honeycomb after fully compressed.

For the Formgrid honeycomb, � is the angle between the wall during bending can

be determined using the geometrical relations.

p

s�o

2�o

D

�

s

�

�-2�

m

m

42

Figures 3.15 and 3.16 show the relation between buckling as

H

s=αsin (3.17)

p

so =ψsin (3.18)

m

s=αtan (3.19)

p

m=��

��

� − βπ2

tan (3.20)

p

s��

��

�

=��

��

� − αβπ tan2

tan (3.21)

0sin

tantan

ψαβ = (3.22)

l

s=0tanψ (3.23)

l

H=γtan (3.24)

0tanψs

H

= (3.25)

��

��

�=s

H0tantan ψγ (3.26)

43

But sin � = s/H

Therefore,

αψγ

sin

tantan 0= (3.27)

Using the above diagrams, the angle � can be written as

��

��

�= −

oψαβ

sin

tantan 1 (3.28)

where

� - angle between wall during bending

Substituting the � angle in equation 3.14 results in

�

��

��

�

��

��

��

��

� −−

+�

��

��

��

��

��

� −

−=

−

−

2/

01

1

1 cos

sin

tantan

2cos*cos

cossin

tantan

2sinsin

tan)2(

4 παα

ψα

πψπψ

ψψα

πψπψ

ψψππ d

HbNE

o

oo

oo

oo

oo

�

(3.29)

The next step is to integrate the above equation from 0 to 90 degrees since the

crushing of the Formgrid honeycomb occurs when the angle � increases from 0 to 90

degrees. Integrating equation (3.28) with respect to � and substituting �0 = 30 degrees for

the Formgrid configuration, which is the angle between the two intersecting walls then

d

HbM

d

MbHE 6.33

)2(05.1*16(1 == (3.30)

Evaluating the energy dissipated due to the horizontal hinges is as follows.

The rate of dissipation of energy due to the horizontal hinge lines is given by

44

( )( )�=L

DdME α��

2 (3.31)

where � is the rate of rotation of hinges lines.

Integrating with respect to the limit 0-�/2 gives the energy due to

horizontal hinge lines as shown in Figure 3.17.

MDE22

π= (3.32)


Only the half-model of the Formgrid is considered for energy calculations.

Considering the full length D of the cell wall for energy determination, there are eight

hinge lines for the Formgrid with a gauge thickness of h, as shown in the Figure 3.17

therefore

Inclined Hinge line

Rolling deformation

Horizontal hinge lines

45

�

��

= 202 4

1*8

2σπ

DE (3.33)

Substituting 2/40 hM=σ in equation (3.32)

MDE π42 = (3.34)

The energy dissipated due to buckling of the inclined hinge lines in the Formgrid

honeycomb derived from the equation (3.34) is

tLME θ�� =3 (3.35)

where

Lt = total length of the inclined hinge for Formgrid honeycomb

γsin/2HLt =

rate of curvature bVt /=θ�

By substituting Lt, θ� in equation (3.34)

γα

ψψ

sin

cos

tan

243

o

o

b

MHE = (3.36)

Integrating the equation (3.36) using limit 0 to 2/π and considering two angle elements

for Formgrid configuration, the energy dissipation due to inclined hinge lines is

b

MHE

224.383 = (3.37)

The total energy absorbed is the product of the force P acting on the Formgrid

honeycomb to crush a wavelength 2H. Therefore,

46

321)2(* EEEHP ++= (3.38)

b

MHMD

h

MbHHP

212.194

6.332* ++= π (3.39)

H

b

MHMD

h

Mb

P��

��

�

++

=

256.92

8.16 π

(3.40)

In order to obtain the final equation for energy dissipation in terms of known

parameters, gauge thickness h, and edge length D, the unknown parameters H and b

should be eliminated. This can be performed by taking partial differentiation of force P

with respect to H and b. From partial differentiation, the values of K and b can be

determined

3

2

69.4

Dhb = (3.41)

3

2

068.4

hDH = (3.42)

Substituting b and H values in energy equations E1, E2 and E3

01 61.12 DME = (3.43)

02 56.12 MDE = (3.44)

DME 03 52.12= (3.45)

The crush load for a half model of Formgrid is given by

47

H

EP T

2= (3.46)

H

dDP

24

1*69.37 2

0σ= (3.47)

3

1

3

5

052.7 DdP σ= (3.48)

The crush load for a full Formgrid model is given by

3

1

3

5

03.15 DdP ysT σ= (3.49)

The crush strength in terms of per unit area is obtained by considering the tributary area

of the Formgrid cell, as shown in Figure 3.18

Figure 3.18. Tributary areas for the half-hexagonal honeycomb.

( )2sin += θwDAt (3.50)

w�

D

48

The equation for crush strength of Formgrid honeycomb is given as

3

13

5

03.15D

A

df

t

ycr

σ= (3.51)

The total energy absorbed due to the crushing of the metallic Formgrid honeycomb is the

product of crushing strength fcr and maximum crush distance dc

ccrt dfE *= (3.52)

3.4.1.1 Validation of Analytical Model of Formgrid Honeycomb

Validation of the analytical model of Formgrid honeycomb is performed using the

test data obtained from the Formgrid manufacturing company [52].

TABLE 3.1

ANALYTICAL AND EXPERIMENTAL VALUES OF CRUSH STRENGTH OFFORMGRID

Quasi-static Crush Strength

Honeycombconfigurations

Density ofhoneycomb

(pcf)

ExperimentalpublishedData [52]

(psi)

AnalyticalModeling

(psi)

Deviation %

Formgrid-40-0.0019-Al-5052

3.1 170 147 10

Formgrid-40-0.0025 -Al-5052

4.1 255 233 8

Formgrid-40-0.0037-Al-5052

5.7 390 447 14

49

0

50

100

150

200

250

300

350

400

450

500

0.0015 0.002 0.0025 0.003 0.0035 0.004

Guage thickness (in)

Cru

shst

reng

th(p

si)

Analytical model


Figure 3.19. Validation of analytical modeling of Formgrid honeycomb.

3.4.2 Analytical Modeling of Half-Hexagonal Honeycomb

Half-Hexagonal honeycomb is a reinforcement type of honeycomb. This type of

cell configuration is used for high-impact resistance by Sandia Laboratories [34] for

protecting the space module during landing on the earth’s surface. Figure 3.20 shows the

Half-Hexagonal honeycomb model.

50

Figure 3.20. Half-Hexagonal honeycomb.

Figure 3.21. Hinge lines influencing energy absorption of half-hexagonal honeycomb.

Vertical Hinge Lines

RollingDeformation

Horizontal Hinge Lines

ReinforcementWall

51

3.4.2.1 Energy Absorption During Buckling

The total energy absorbed during buckling is given by three types of energy

absorption: E1 (energy dissipated due to the rolling deformation), E2 (energy dissipated

due to the traveling of horizontal hinge lines), and E3 (energy dissipated due to the

inclined hinge lines).

The crushing of the Half-Hexagonal honeycomb occurs with an increase in angle

� from 0 to 90 degrees. Integrating equation (3.14) with respect to � and substituting �o =

30 degrees for the Half-Hexagonal honeycomb configuration, which is the angle between

the two intersecting walls.

3.4.2.2 Energy Due to Rolling Deformation

The equation (3.52) is a general expression for energy dissipation E1 during

rolling deformation of an intersecting honeycomb wall as shown in Figure 3.22.

Figure 3.22. Buckling of Half-hexagonal honeycomb- schematic view.

�

2�o

�

�

H

52

The variables shown in Figure 3.22 is given below

D - half cell size

H - half wave-length

2�o - angle between adjacent walls which is assumed to be

constant during buckling

� - angle between wall during bending

� - angle of rotations of the walls

� - angle between inclined hinge lines of wall and horizontal

edge

��

��

�

��

��

� −

−+�

��

−

−=

2/

01 cos2

cos*

coscos)2

sin(sin

tan)2(

4 παα

βπ

ψπ

ψψβπ

ψπψ

ψψππ d

HbNE

o

ooo

o

oo

�

(3.53)

Integrating equation (3.52) yields energy absorption during rolling deformation

By substituting,

2/40 hM=σ ; hN 0σ= (3.54)

��

��

�= −

oψαβ

sin

tantan 1 (3.56)

( )[ ]h

HbM

h

MbHE 6.33

205.1*161 == (3.57)

53

3.4.2.3 Energy Due to Horizontal Hinge Lines

The rate of energy absorption due to the horizontal hinge lines is given by

( )�=L

DdME α��

2 (3.58)

whereα� is the rate of rotation of hinges.

Integrating with respect to within the limit (0-�/2)

MDE22

π= (3.59)

Full symmetrical model of the Half-Hexagonal honeycomb is considered for the

energy calculations due to horizontal hinge lines. Considering the length D/2 of the cell

wall, there are eight hinge lines having gauge thickness h, four hinge lines of length D/2

having gauge thickness 3h due to reinforcement walls, and four hinge lines of length D

having gauge thickness h, as shown in Figures 3.23 and 3.24.

Therefore, energy absorption due to horizontal hinge line is given by

( ) �

��

++= 20

20202 4

1*43

424

4

1*8

2hDh

DhDE σσσπ

(3.60)

Substituting20

4

h

M=σ in equation (3.58)

MDE π132 = (3.61)

54

E3

E1E2


Figure 3.24. Cell walls influencing the energy absorption.

D/2 D

D/2

55

3.4.2.4 Energy due to Inclined Hinge Lines

Energy absorption during the traveling of inclined hinge lines is derived from the

equation (3.60)

)(3 DdL

ME α�� = (3.62)

where Lt = total length of the inclined hinge

θsin

2KLt =

rate of curvature =b

Vt=θ�


αγα

ψ�

sin

cos

tan

243

ob

MHE = (3.63)

Integrating equation (3.61) using a limit of 0 to 2/π and considering two angle elements

for the Half-Hexagonal configuration, the energy absorption due to inclined hinge lines is

b

HME

2

3

24.38= (3.64)

The total energy absorbed is the product of force P acting on Half-Hexagonal honeycomb

to crush wavelength 2H. Therefore,

tEb

MHMH

d

MbHHP =++=

212.1913

6.332* π (3.65)

b

MHMH

d

MbHP

256.95.6

8.16 ++= π (3.66)

In order to obtain a final equation for energy dissipation in terms of known parameters,

gauge thickness h, and edge length D, unknown parameters H and b should be

56

eliminated. This can be performed by taking a partial differentiation of force P with

respect to H and b. From the partial differentiation, values of H and b can be determined

3

2

45.1

Dhb = (3.67)

3

2

385.0

hDH = (3.68)

Substituting the b and H values in energy equations E1, E2 and E3, total energy is

204

1*49.122 hDET σ= (3.69)

The total crush load is

H

EP T

2= (3.68)

3

1

3

5

0312.11 DhP σ= (3.70)

Crush strength is

tcr A

Pf = (3.71)

where At is the tributary area of the buckling, as shown in Figure 3.25.

3DS = (3.72)

2

43

SAt = (3.73)

Substituting the S and At in equation (3.70), the final crush strength is given as

3

5

24 ��

��

�=s

hf ocr σ (3.74)

Table 32 shows the validation of analytical modeling of Half-Hexagonal honeycomb.

57

Figure 3.25. Tributary areas for Half-Hexagonal honeycomb.

TABLE 3.2

ANALYTICAL AND EXPERIMENTAL VALUES OF CRUSH STRENGTH OFHALF-HEXAGONAL

Quasi-static Crush StrengthHoneycomb

ConfigurationsDensity of

Honeycomb ExperimentalPublished [33]

(psi)

AnalyticalModeling

(psi)

PercentDeviation

Half-Hexagonal-0.006-Al-5052

38 6500 6390 1.6

D

S

58

3.4.3 Analytical Modeling of Flexible Flexcore Honeycomb

Flexcore honeycomb is a flexible type of honeycomb used for impact protection

of curved shapes. Figure 3.26 shows the geometry of the Flexcore honeycomb. In order

to derive the energy equation and to calculate the energy absorption, the configuration of

the Flexcore should be determined. This type of honeycomb is considered to be

symmetrically shaped, as shown in Figure 3.27. Therefore, only half of the symmetrical

shape is used for energy calculation. The total energy absorption will be twice the energy

obtained from half of the symmetrical shape. Figure 3.28 and 3.29 show the

configurations of Flexcore honeycomb.

Figure 3.26. Flexcore honeycomb.

59

Figure 3.27. Flexcore honeycomb showing the symmetrical configuration.

Figure 3.28. Single cell of Flexcore honeycomb showing the intersecting wall.

2�o

D/2

D

h

Symmetricalconfiguration

60

The geometrical parameters of Flexcore honeycomb is given below

D = half cell size.

H = half wave-length

2�o = 50 degrees; angle between adjacent walls

� = angle between wall during bending

� = angle of rotations of walls

� = angle between inclined hinge of wall and horizontal edge


The total energy absorbed during the buckling process is the sum of energy

dissipated due to rolling deformation, energy dissipated due to traveling of horizontal

hinge lines, and energy dissipated due to incline hinge lines.


A general expression for energy absorption E1 during rolling deformation of

an intersecting honeycomb wall, as shown in Figure 3.29, is expressed by equation

(3.75).

Figure 3.29. Buckling angles for Flexcore configuration.

D/2

�

�

�

61

��

��

�

��

��

� −

−+�

��

−

−=

2/

01 cos2

cos*

coscos)2

sin(sin

tan)2(

4 παα

βπ

ψπ

ψψβπ

ψπψ

ψψππ d

HbNE

o

ooo

o

oo

�

(3.75)

After substituting buckling parameters and integrating equation (3.74), the following

equation is obtained for energy absorption due to rolling deformation

h

HbME 0

1

41= (3.76)


The rate of energy absorption due to the horizontal hinge lines is given by

)(2 DdL

ME α�� = (3.77)

where α� is the rate of rotation of hinges

Integrating the Equation (3.76) using limits from 0 to 2/π ,

MDE22

π= (3.78)

Energy due to all horizontal hinge lines is given by

( )��

��

++=

4*4

24

2*2

3*4

4*8

2

20

20

2

2

hDhDhDE

σσσπ (3.79)

02 6 DME π= (3.80)

62

3.4.3.4 Energy Due to Inclined Hinge Lines

Energy absorption due to traveling of inclined hinge lines in Flexcore honeycomb

is obtained by a derivation from equation (3.80)

)(3 DdL

ME α�� = (3.81)

Where Lt is the total length of the inclined hinge for Flexcore honeycomb,

θsin/2HLt = , and θα �

� = ; rate of curvature is given by bVt /=θ� .


αγα

ψ�

sin

cos

tan

243

ob

MHE = (3.82)

Since buckling takes place as � goes from 0 to �/2, integrating with the limits and

substituting �o = 25 degrees for Flexcore configuration

b

MHE

22.133 = (3.83)

The total energy absorbed is the product of the force P acting on the Flexcore honeycomb

to crush a wavelength 2H. Therefore,

tEb

MHMH

d

MbHHP =++=

22.136

412* π (3.84)

b

MHMH

d

MbHP

256.95.6

8.16 ++= π (3.85)

In order to obtain the final equation for total energy absorption in terms of known

parameters, gauge thickness h, and edge length D, the unknown parameters H and b

should be eliminated. This can be done by taking partial differentiation of force P with

63

respect to H and b. From the partial differentiation, the values of H and b can be

determined.

3

2

75.6

Dhb = (3.86)

3

2

52.1

hDH = (3.87)

Substituting b and H in the energy absorption equations and the crush load or force for

one half of symmetry of Flexcore is given as

3

1

3

5

1 13.8 DhPT σ= (3.88)

Total crush load is double the load absorbed from the one half of the Flexcore

honeycomb. Therefore, total crush load is given as

3

1

3

5

26.16 DhPT σ= (3.89)

Crush strength is given by the load absorbed by the tributary area as shown in Figure

3.30. The tributary area for Flexcore is given by

NDAt

12*= (3.90)

where N is the number of cells along the ribbon direction of the Flexcore honeycomb.

The crush strength of Flexcore honeycomb is given by

ND

hfcr

*

12.195

3

2

3

5

σ= (3.91)

64

Figure 3.30. Tributary areas for the Flexcore honeycomb.

3.4.3.5 Experimental Validation of Flexcore Analytical Model

Figures 3.31 to 3.34 show the validation of analytical modeling of Flexcore

honeycomb.

0

50

100

150

200

250

300

350

400

450

500

0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004


Cru

shS

tren

gth

(psi

)

Analytical model


Figure 3.31. Validation of analytical modeling of Flexcore honeycomb F40-5052.

D/2

D

h

12/N

65

0

50

100

150

200

250

300

350

0.001 0.0012 0.0014 0.0016 0.0018 0.002 0.0022 0.0024 0.0026 0.0028


Cru

shst

reng

th(p

si)

Analytical Model


Figure 3.32. Validation of analytical modeling of Flexcore honeycomb F40-5056.

0

100

200

300

400

500

600

700

800

900

1000

0.001 0.0012 0.0014 0.0016 0.0018 0.002 0.0022 0.0024 0.0026 0.0028

Guage Thickness (in)

Cru

shSt

reng

th(p

si)

Analytical Model


Figure 3.33 Validation of analytical modeling of Flexcore honeycomb F80-5056

66

0

100

200

300

400

500

600

700

800

0.001 0.0012 0.0014 0.0016 0.0018 0.002 0.0022 0.0024 0.0026

GuageThickness (in)

Cru

shSt

reng

th(p

si)

Analytical model


Figure 3.34 Validation of analytical modeling of Flexcore honeycomb F80-5052

3.4.4 Analytical Modeling of Double Flex honeycomb

Double Flex honeycomb is a flexible type of honeycomb similar to Flexcore

honeycomb, but it has two-sided flexible curved walls, as shown in Figure 3.35. Double

Flex honeycomb is considered to be a symmetrical configuration honeycomb and,

therefore, only half of the symmetrical section is considered for energy calculation, as

shown in Figure 3.36. The total energy will be twice the energy obtained from one half

of one symmetrical section.

67

Figure 3.35. Double Flex honeycomb.


Total energy absorption during the buckling process is the sum of energy

absorbed due to the rolling deformation, energy absorbed due to the traveling of

horizontal hinge lines, and energy absorbed due to the incline hinge lines.

The crushing of the Double Flex honeycomb occurs when the buckling angle �

increases from 0 to 90 degrees. Integrating equation (3.91) with respect to � and

substituting �o = 45 degree for Double Flex configuration which is half of the angle

between two intersecting walls, the energy absorption is determined.

68

Figure 3.36. Double Flex honeycomb showing the symmetrical configuration.


Equation (3.91) is a general expression for energy dissipation E1 during the rolling

deformation of any intersecting honeycomb wall.

��

��

�

��

��

� −

−+�

��

−

−=

2/

01 cos2

cos*

coscos)2

sin(sin

tan)2(

4 παα

βπ

ψπ

ψψβπ

ψπψ

ψψππ d

HbNE

o

ooo

o

oo

�

(3.92)

SymmetricalConfiguration

2�o

69

After substituting the parameters and integrating equation (3.91), the following equation

is obtained for energy due to rolling:

h

HbME 0

1

8.28= (3.93)


The rate of absorption of energy due to the horizontal hinge lines is given by

)(2 DdL

ME α�� = (3.94)

where α� is the rate of rotation of hinges

Integrating equation (3.93) using the limit (0-�/2)

MDE22

π= (3.95)

The energy absorption due to all horizontal hinges lines is given by

( )��

��

+=

4*4

24

4*2*8

2

20

2

2

hDhDE

σσπ (3.96)

02 4 DME π= (3.97)

3.4.4.4 Energy Due to Inclined Hinge Lines

Energy absorption due to traveling of inclined hinge lines in Double Flex

honeycomb is derived from equation (3.97) as

)(3 DdL

ME α�� = (3.98)

Where Lt = total length of the inclined hinge for Flexcore honeycomb,

70

θsin/2HLt = , θα �

� = and rate of curvature is given by bVt /=θ�

By substituting Lt, θ� , in equation (3.97)

αγα

ψ�

sin

cos

tan

243

ob

MHE = (3.99)

Since buckling takes place as � goes from 0 to �/2, integrating using the limits,

substituting �o= 45 degrees for the Double Flex configuration, and considering two angle

elements, for one half of the symmetric section for the Double Flex configuration, the

energy absorption is obtained as

b

MHE

22.93 = (3.100)

The total energy absorbed is the product of the force P acting on the Double Flex

honeycomb to crush a wavelength 2H. Therefore,

tEb

MHMH

d

MbHHP =++=

22.94

8.282* π (3.101)

b

MHMH

d

MbHP

26.42

4.14 ++= π (3.102)

In order to obtain the final equation for energy absorption in terms of known parameters,

gauge thickness h and edge length D, the unknown parameters H and b should be

eliminated. This can be performed by taking a partial differentiation of force P with

respect to H and b. From the partial differentiation, the values of H and b are obtained as

71

3

2

174.7

Dhb = (3.103)

3

2

6778.1

hDH = (3.104)

Substituting the b and H in the energy equations (3.92) and (3.99)

From this crush load, one half of the symmetry of Double Flex is given as

3

1

3

5

1 6.5 DhPT σ= (3.105)

Total crush load is twice the load absorbed from one half of the Double Flex honeycomb.

Therefore,

3

1

3

5

202.11 DhPT σ= (3.106)

Crush strength is given by the load absorbed by the tributary area, as shown in Figure

3.37. The tributary area is given as

2

12

ND = (3.107)

DDA *= (3.108)

where N is the number of cells along the ribbon direction of the Double Flex honeycomb,

72

Figure 3.37. Tributary areas for the Double Flex honeycomb.

The crush strength of the Double Flex honeycomb is given as

ND

hf cr

*

12.195

3

2

3

5

σ= (3.109)

Validation of the analytical modeling of Double Flex is shown in Figure 3.38.

12/N

D

73

0

50

100

150

200

250

300

350

400

450

500

0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005


Cru

shst

reng

th(p

si)

Analytical model


Figure 3.38 Validation of analytical modeling of Double Flex honeycomb

3.5. Chapter Summary

A method of classifying honeycomb according to its cell connectivity and

configuration is identified. Crush strength equations were developed for Formgrid, Half-

Hexagonal, Flexcore, and Double Flex honeycomb. The next chapter explains how a

parameterized model of the honeycomb was developed by using the methodology for

analyzing the cell structures, which was dealt with in this chapter.

74

CHAPTER 4

PARAMETRIZATION OF HONEYCOMB MODEL

4.1 Developments in Parameterization of Honeycomb

In 1983, Tomaz Wierzbicki [2] developed a general solution for determining the

crush load of thin-walled structural members by studying the energy parameters. He

assumed that the total energy absorbed during the buckling of thin-walled structures was

the sum of energy due to rolling deformation of the thin wall, plus the energy due to

horizontal hinge lines and inclined hinge lines. He obtained the coefficients of the energy

absorptions as

Tm E

b

HA

H

DA

h

bA

M

P =++= 321 (4.1)

333213

h

DAAA

M

Pm = (4.2)

where

Pm = crush load during compression of honeycomb

M = plastic buckling moment

b = rolling radius

h = gauge thickness

D = cell wall size

ET = total energy absorption

A1, A2, and A3 = coefficients of energy absorption

He applied his general form of the equation to the square tube buckling and

obtained the mean crush load of the steel square tube as

75

3

1

3

5

56.9 DhP om σ= (4.3)

The general solution developed was validated with experimental results obtained

from the Abramowicz and Jones [46] experimental studies of axial crushing of square

tubes. Variation between the analytical and experimental methods, as shown in Figure

4.1, is due to a lack of analyzing the square tube by considering symmetrical and

asymmetrical configurations. Also, the extensional lengths of wall and reinforcement

parameters were not introduced in the general solution. This solution can be applied for

known values of energy coefficients and but is not useful for a typical configuration that

involves reinforcement, extensional length of walls, curved structures, and symmetricity

of configurations.

0

5

10

15

20

25

30

35

40

45

50

30 35 40 45 50 55

Square tube size D (mm)

Cru

shlo

adP

(KN

)

Experimental published [46]

Generalized method [2]

Figure 4.1. Validation of experimental and Wierzbicki generalized method.

76

Hayduk and Wierzbicki [49] developed the collapse mode for thin-gauge

cruciform, which is used as an energy-absorbing material. They developed the simple

formula for the L-shaped column, shown in Figure 4.2, which is given as

3 /04.12 hDM

Pm = (4.4)

Figure 4.2. Geometrical shapes of cruciform used for experimental and analyticalstudies by Hayduk and Wierzbicki

They developed the crush load equation for cruciform by considering the L-

shaped equation. Since the L-shaped column equation is doubled to obtain the cruciform

equation, the extensional attachments and reinforcements were not dealt with in this case.

Therefore, the analytical results varied from the experimental results, as shown in Figure

4.3.

D

90°

D

90°

L-Shaped Thin ColumnCruciform

77

80

90

100

110

120

130

140

150

160

0 20 40 60 80 100 120 140 160

D/h ratio

Pm

/M


Analytical model [49]

Figure 4.3. Analytical versus experimental data on L-section crushing.

4.2 Methodology

A parameterization of the analytical model of the honeycomb involves the study

of three types of energy of buckling of walls, and the cell parameters that are associated

with these energies. As a case study, initially the Half-Hexagonal honeycomb was studied

and parameterized. Figure 4.4 shows the three modes of energy absorption during the

buckling of the metallic honeycomb. Total energy absorption is the sum of E1 (energy

absorption due to the rolling deformation), E2 (energy absorption due to the traveling

hinge lines), and E3 (energy absorption due to the inclined hinge lines). A parameterized

equation was derived for these three energy components using the unknown coefficients,

which will be the critical factors for the energy absorption of various honeycombs. The

energy coefficients were derived for the energy due to rolling deformation and the energy

78

E3

E1E2

absorption due to traveling of inclined hinge lines. Cell structures, namely reinforcement

walls, extensional walls, curved walls, and cell attachments layers were included in the

calculation of energy due to horizontal hinges. The symmetrical and asymmetrical

configurations of the honeycomb were clearly defined before completing the energy

calculation. Finally, the parameterized equation for the crush strength was derived by

considering the tributary area of the buckling of a single cell. Validation was performed

to check the parameterized honeycomb model.

Figure 4.4. Buckling phenomenon of half-hexagonal type honeycomb.

Three types of energies, namely E1 (energy absorption due to the rolling

deformation), E2 (energy absorption due to the traveling hinge lines), and E3 (energy

absorption due to the inclined hinge lines), and also cell parameters affecting these

energy should be determined. A generalization of the energy absorption components E1,

E2, and E3 are explained using the Half-Hexagonal honeycomb. Figure 4.4 shows the

partial buckling of the Half-Hexagonal cell.

79

4.3. Parameterization of Energy Absorption Parameter E1 Due to Rolling Radius

First, energy E1 absorbed during the buckling was evaluated. Since the attached

cell walls did not show any symmetry along the centerline of the cell connectivity, then it

is considered to be an asymmetrical model, as shown in Figure 4.5. But in the case of the

Formgrid honeycomb, Figure 4.6 shows symmetry with the centerline of the connectivity,

which is called a symmetrical configuration.

Figure 4.5. Asymmetrical cell configurations - Half-Hexagonal.

AsymmetricalType of CellConnectivity

ReinforcementWall

80

Figure 4.6. Symmetrical cell configurations - Formgrid.

Energy absorption E1 can be parameterized as

11 2tan

32I

h

bMoHE

oo�

��

−=

ψππ

ψ(4.5)

��

��

��

��

��

��

� −−+

��

��

��

��

��

��

� −=

−

−�

o

o

o

ooI

ψα

πψπψψ

ψα

πψπψ

π

sin

tantan

2cos.coscos

sin

tantan

2sinsin

100

12/

0

1

(4.5)

Writing equation (4.5) in the form of coefficients, then

h

HbMmE 01

1 = (4.7)

Half-modelof the cellconnectivity

81

where

11 2tan

32Im

oo�

��

−=

ψππ

ψ(4.8)

The varying parameter in the energy equation is coefficient m1, which depends on

the angle of buckling � that varied between 0 and 90 degrees as the buckling starts and

ends. The other parameter variation in the energy equation is angle �o, which is the angle

between intersecting cell walls. Table 4.1. shows the coefficient R1 for various

intersecting angles of the honeycomb wall.

TABLE 4.1

ENERGY COEFFICIENT R1 FOR VARIOUS INTERSECTING ANGLES

Intersecting Angles 1R

50 41

60 33.6

90 28.8

Therefore, the energy absorption E1 is given in a generalized form as

h

HbMRE o1

1 = (4.9)

where 11 mR = (4.10)

82

4.4 Energy Absorption Parameter E2 Due to Horizontal Hinge Lines

In the evaluation of energy absorption parameter E2, the first step is to check

whether the honeycomb cells are symmetrical or asymmetrical. The Half-Hexagonal

honeycomb is the asymmetrical configuration honeycomb.

Figure 4.7. Honeycomb wall length associated with energy absorption.

Reinforcement layers should be considered before evaluating the energy E2, as

shown in Figure 4.7. Also the presence of extensional walls should be noted for

evaluating the energy. The energy E2 is caused by traveling horizontal hinge lines during

buckling, as shown in Figure 4.8. Eight horizontal hinges of length D/2 each are located

on walls A and B having wall thickness h. However, wall C has four horizontal hinges

having thickness 3h and length D/2. Wall E, which is an extensional or interconnecting

wall, has four horizontal hinges having wall thickness with full length D.

D/2 D

D/2Reinforcement Layer Extensional Wall

Intersecting Walls

83

E3

E1E2

Wall B

Wall A

Wall C

Wall D

Taking all into account, a generalized equation for energy absorption E2 was developed.

Figure 4.8. Honeycomb wall associated with energy absorption.

The rate of energy absorption due to the horizontal hinge lines is

�= αDdME o2 (4.11)

where � increases from 0 to �/2 during the buckling,

D is the length of the horizontal hinge, which is a constant for particular honeycomb.

4

2hM oσ

= (4.12)

Energy E for Half-Hexagonal is given by

4*4

4

)3(*

24

4*

2*8

2

222

2

hD

hDhDE ooo σσσπ ++��

�

��

�= (4.13)

For parameterization consider

84

3212 HHHE ++= (4.14)

where

��

��

�=

4*

2*8

2

2

1

hDH oσπ

for wall A and B (4.15)

4

)3(*

24

2

2

2

hDH oσπ= for wall C (4.16)

4*4

2

2

3

hDH oσπ= for wall E (4.17)

4*4

4

)3(*

24

4*

2*8

2

222

2

hD

hDhDE ooo σσσπ ++��

�

��

�= (4.18)

Now, for generalizing the above energy equation, substituting the unknown parameters

��

��

� ++= rDDknc

DaME o 444*

22

2

π (4.19)

��

��

� ++= rknc

aDME o 222 2

2 π (4.20)

where

a = number of D/c in inter-connecting wall

c = value given either 1 or 2 wall when length D or D/2 is considered according to

the length of the wall involved in energy absorption

k = length of the cell having multi-layer thickness in terms of cell size D

n = number of multi-layers

85

r = length of the extra attachment length in terms of cell size D

��

��

� ++= rknc

aDME o 222 2

2 π (4.21)

Substituting the above parameters a, c, k, n, and r in equation (4.21) to obtain the E2

equation for Half-Hexagonal honeycomb. For half-hexagonal honeycomb, a= 2, c = 2,

k=0.5, r=1

Therefore,

DME oπ132 = (4.22)

Figure 4.9. Half-Hexagonal cell configuration showing cell connectivity.

22 mME o= (4.23)

D/2 D

D/2

a

rK=D/2

86

Therefore, the generalized coefficient for determine out the energy E2 due to horizontal

traveling hinges is given by

��

��

� ++= rknc

aDm 222 2

2 π (4.24)

The next step is to determine the generalized coefficients for finding out the energy E3

due to the inclined hinge lines.

4.5 Energy Absorption E3 Due to Inclined Hinge Lines

The energy absorption component E3 was due to the inclined hinge lines, as

shown in Figure 4.7. While evaluating E3, the attachment mode, namely symmetrical or

asymmetrical, was noted. For an asymmetrical configuration there were two angle

elements, and for an symmetrical configuration there were four angle elements. Energy

absorption E3 due to inclined hinge lines is given by

αγψ

α�

sintan

cos4

2

3o

o b

HME = (4.25)

3

2

3 4 Ib

HME o= (4.26)

where

αγα

ψ

π

dIo�=

2/

0

3 sin

cos

tan

1(4.27)

b

HMmE o

2

33 = (4.28)

33 *4 Im = (4.29)

87

The varying parameter in energy equation E3 is coefficient m3, which is

associated with the angle of buckling � that takes the values from 0 to 90 degrees and the

angle �o, which is the angle between the intersecting cell walls. Then

b

HMRE o

2

33 = (4.30)

Where

33 mR = (4.31)

The energy coefficients are given in table 4.2.

TABLE 4.2

ENERGY COEFFICIENT R3 FOR VARIOUS INTERSECTING ANGLES

Intersecting Angles 3R

50 13.2

60 19.12

90 9.2

where R3 is the energy coefficient for honeycomb

Total energy absorption during the buckling of the metallic honeycomb is given by

ET = E1 + E2 + E3 (4.32)

b

MHRrkn

c

aDM

h

HbMRE o

oo

T

2321 222 +�

�

��

� +++= π (4.34)

The total energy absorption is equivalent to the product of the crush load and the

buckling distance 2H

88

HPE mT 2*= (4.35)

b

HRrkn

c

aD

h

HbR

M

HP

o

m2

321 2222

+��

��

� +++= π (4.36)

b

HRrkn

c

aD

h

bR

M

P

o

m

2222

2321 +�

�

��

� +++= π (4.37)

Taking a partial differentiation of Pm with respect to rolling radius b and half-

wavelength H, equations for rolling radius b and half-wavelength H were derived as

)222(( 2

23

rknc

aD

HRb

++=

π(4.38)

3

21

hR

bRH = (4.39)

Equations (4.37) and (4.38) for b and H can be further reduced to obtain b and H

in terms of measurable geometrical terms, since b and H were difficult to measure during

the experimental process of honeycomb buckling.

3

31

222 )222((

RR

rknc

aDh

H++

=π

(4.40)

89

32

1

322 )222((

R

Rrknc

aDh

b++

=π

(4.41)

By substituting b and H in Equation (4.39),

h

HbMRE 01

1 = (4.42)

3

2

13

1

31

03

2

33

1

3

12

1 *222(

RRhR

MhRhrknc

aDR �

��

��

��

� ++=

π(4.43)

��

��

� ++= rknc

aDME 222 2

01 π (4.44)

By substituting b and H in equation (4.41),

b

MHRE 0

23

3 = (4.45)

��

��

� ++= rknc

aDME o 222 2

3 π (4.46)

The energy parameterized energy absorption E2 is given as

��

��

� ++= rknc

aDME o 222 2

2 π (4.47)

321 EEEET ++= (4.48)

90

The total parameterized energy absorption is given as

��

��

� ++= rknc

aDME oT 2223 2π (4.49)

Parameters which affect energy absorption are D (cell size), n (number of reinforcement

walls), k (length of the reinforcement wall), and r (extensional length of the wall),

therefore,

Tm EHP =2* (4.50)

H

Mrknc

aD

Pm 2

222(3 02

��

��

� ++=

π(4.51)

3

31

222 )222((

RR

rknc

aDh

H++

=π

(4.52)

3

2

23

2

3

1

3

1

33

1

102

222(2

*222(3

��

��

� ++

��

��

� ++=

rknc

aDh

RRMrknc

aD

Pm

π

π(4.53)

( )3

1

210

3

1

2

3

1

3

1

222(

2

3RRMrkn

c

aD

h

Pm ��

��

� ++= π(4.54)

91

The parameterized equation for determining the crush load of a particular honeycomb is

given by

( )3

1

210

3

1

2

3

1

3

1

222(

2

3RRMrkn

c

aD

h

Pm ��

��

� ++= π(4.55)

The parameterized equation for determining the crush strength of a particular honeycomb

is given by

( )

tcr A

RRMrknc

aD

hf

3

1

310

3

1

2

3

122

2(

2

3

��

�

�

��

�

�

��

��

� ++

=

π

(4.56)

where

mP = Crush load

At = tributary area of buckling

fcr = crush strength of buckling

4.6 Validation of the Parameterized Honeycomb Model Using a typicalHoneycomb Configuration

The parameterized equation for crush load of honeycomb is given by

( )3

1

310

3

1

2

3

122

2(

2

3RRMrkn

c

aD

h

Pm��

�

�

��

�

�

��

��

� ++= π (4.57)

Using the above parameterized equation for determining the crush load of honeycomb

shapes, the crush load was determined for square and L-shaped cruciforms, which were

previously determined by Wierzbicki and Hayduk [49] analytical models.

92

From Figure 4.10, it is shown that the proposed parameterized model has a close

match with the experimental data.

The proposed parameterized model, which includes all of the geometrical

configurations, had much closer results than the generalized solution for square tube

buckling given by Wierzbicki. Similarly, in the case of the cruciform section, as shown in

Figure 4.11, the proposed parameterized model shows a closer match with the

experimental data. This model can predict more accurately the crush strength in a better

manner for honeycombs having intersecting walls.

0

5

10

15

20

25

30

35

40

45

50

30 35 40 45 50 55

Square tube size D (mm)

Cru

shlo

adP

(KN

)


Wierzbicki generalized method [2]

Proposed parameterized method

Figure 4.10. Validation of proposed parameterized model using square tube buckling.

93

80

90

100

110

120

130

140

150

160

0 20 40 60 80 100 120 140 160

D/h ratio

P m/M



Proposed parameterized model

Figure 4.11. Validation of proposed parameterized model using cruciform section.

4.6.1 Validation Using Hexagonal Honeycomb

The parameterized model is validated using the Hexagonal honeycomb. The

geometrical parameters are shown in Figure 4.12.

Substituting the parameters a=2, c=2, k=0.5D, n=2, and r=0 in equation (4.56).

The energy coefficients R1=33.6 and R3=19.12 must be considered for the intersecting

angle 60 degrees between hexagonal cell walls.

( )

tcr A

MD

hf

3

1

0

3

1

2

3

112.19*6.330*22*5.0*2

2

2*2(

2

3

��

�

�

��

�

��

��

� ++

=

π

(4.58)

Substituting Mo in equation (4.58)

94

2

4

1hM oo σ= (4.59)

Figure 4.12. Hexagonal cell configurations.

Therefore the crush strength is given as

tcr A

Dhf

3

1

3

5

61.8= (4.60)

where At is the tributary area given by

2

4

3SAt = (4.61)

The validation of the parameterized model using Hexagonal honeycomb is shown in

Figure 4.13.

k=0.5 Dn=2

a=2

D

95

0

50

100

150

200

250

300

350

400

0 0.0005 0.001 0.0015 0.002 0.0025 0.003

Guage Thickness (inch)

Cru

shS

tren

gth

(psi

)

Analytical model [2]Experimental published [33]Proposed parametrized model

Figure 4.13. Validation of parameterized model using Hexagonal honeycomb.

4.6.2 Validation Using Half Hexagonal Honeycomb

Parameters defined in the parameterized equation are as follows:


c = value given either 1 or 2 wall when length D or D/2 according to length of

wall involved in energy absorption

k = length of cell having multi-layer thickness in terms of cell size D


r = length of extra attachment length in terms of cell size D

The geometrical parameters are shown in Figure 4.14, substituting the parameters

a=2, c=2, k=0.5, n=3, and r=1 in equation (4.61) The energy coefficients R1=33.6 and

R3=19.12 must be considered for the intersecting angle 60 degrees between the Half-

Hexagonal cell wall.

96

( )

tcr A

RRMrknc

aD

hf

3

1

310

3

1

2

3

122

2(

2

3

��

�

�

��

�

�

��

��

� ++

=

π

(4.62)

( )

tcr A

MD

hf

3

1

0

3

1

2

3

112.19*6.331*23*5.0*2

2

2*2(

2

3

��

�

�

��

�

��

��

� ++

=

π

(4.62)

By substituting Mo in equation (4.62)

2

4

1hM oo σ= (4.63)

Then the crush strength is given as

tcr A

Dhf

3

1

3

5

14.11= (4.64)

where

2

4

3SAt = (4.65)

3DS = (4.66)

97

Figure 4.14. Half-Hexagonal honeycomb cells.

TABLE 4.3

VALIDATION OF HALF-HEXAGONAL CRUSH STRENGTH OBTAINED FROMPARAMETERIZED MODEL

Quasi-static Crush StrengthHoneycomb

Configurations

Density ofhoneycomb

(pcf)Experimentalpublished [33]

(psi)

Parameterizedmodel(psi)

Deviation%

Half-hexagonal-0.006-Al-5052

38 6500 6390 1.6

4.6.3 Validation Using Formgrid Honeycomb

Parameters defined in the parameterized equation are as follows:


D/2

D

D/2

a=2

k=0.5 D

n=3

r =1 D

98

c = value given length D or D/2 is consider according to the length of wall

involved in the energy absorption

k - length of cell having multi-layer thickness in terms of cell size D

n - number of multi-layers

r - length of extra attachment length in terms of cell size D

Figure 4.15. Formgrid configuration.

Substituting the parameters a = 2, c = 1, k = 0, and n = 0, r = 0 in the equation

(4.56). The energy coefficients R1=33.6 and R3=19.12 must be considered for the

intersecting angle 60 degrees between the Formgrid cell walls.

( )

tcr A

MD

hf

3

1

0

3

1

2

3

112.19*6.330*20*0*2

1

2*2

2

3

��

�

�

��

�

��

��

� ++

=

π

(4.67)

Half-modelof the cellconnectivity

a=2

c=1

99


2

4

1hM oo σ= (4.68)


tcr A

Dhf

3

1

3

5

52.7*2= (4.69)

tcr A

Dhf

3

1

3

5

04.15= (4.70)

The validation of Formgrid Honeycomb is shown in Figure 4.16.

0

50

100

150

200

250

300

350

400

450

500

0.0015 0.002 0.0025 0.003 0.0035 0.004


Cru

shst

reng

th(p

si)

Proposed parametrized model


Figure 4.16. Experimental validation of Formgrid honeycomb.

100

4.6.4 Validation Using Flexcore Honeycomb

The parameters defined in the generalized equation are as follows:


c = value given either 1 or 2 wall when length D or D/2 is according to wall

sharing model or wall non-sharing model configuration

k = length of cell having multi-layer thickness in terms of cell size D


r = length of extra attachment length in terms of cell size D

Figure 4.17. Flexcore configuration.

Substituting the parameters a=2, c=1, k=0.25, n=2, and r=0 in equation (4.56).

Intersecting angles 50 degrees, therefore R1 = 41 and R3 =13.2

D/2

D

h

a=2

n=2

k=0.25 D

101

( )

tcr A

MDD

hf

3

1

0

3

1

2

3

12.13*410*20*25.0*2

1

2*2(

2

3

��

�

�

��

�

��

��

� ++

=

π

(4.71)


2

4

1hM oo σ= (4.72)


tcr A

Dhf

3

1

3

5

13.8*2= (4.73)

Validation of the parameterized model for Flexcore honeycomb is shown in Figures 4.18

and 4.19.

0

50

100

150

200

250

300

350

400

450

500

0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004


Cru

shS

tren

gth

(psi

)



Figure 4.18. Validation of Flexcore honeycomb F40-5052.

102

0

100

200

300

400

500

600

700

800

0.001 0.0012 0.0014 0.0016 0.0018 0.002 0.0022 0.0024 0.0026

GuageThickness (in)

Cru

shS

tren

gth

(psi

)



Figure 4.19. Validation of Flexcore honeycomb F40-5052.

4.6.5 Validation Using Double Flex Honeycomb

The parameters defined in the parameterized Equation are shown in Figure 4.20.

Substituting the parameters a = 2, c = 2, k = 0.25D, n = 2, and r = 0 in the above equation

(4.56). Intersecting between cell wall are angles 90 degrees, therefore R1 = 28.8 and R3 =

9.2 should be considered.

The crush strength is given as:

( )

tcr A

MDD

hf

3

1

0

3

1

2

3

12.9*8.280*20*25.0*2

2

2*2(

2

3

��

�

�

��

�

��

��

� ++

=

π

(4.74)

Substitute Mo in equation (4.74)

103

2

4

1hM oo σ= (4.74)

Figure 4.20. Double Flex configurations.

The crush strength of Double Flex is given as

tcr A

Dhf

3

1

3

5

2.11= (4.75)

Validation of parameterized model for Double Flex honeycomb is shown in Figures 4.21.

SymmetricalConfiguration

a=2

n=2

k=0.25 D

104

0

50

100

150

200

250

300

350

400

450

500

0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005


Cru

shst

reng

th(p

si)



Figure 4.21. Experimental validation of Double Flex.

4.7 Validation of Parameterized Model Using New Honeycomb Cell Configuration

The parameterized model for determining the crush strength of honeycomb was

validated using a new honeycomb cell configuration. Two types of cell configurations

were developed for validating the parameterized model. Since experimental data was not

available for the new honeycomb cell, finite element analysis on these cell configurations

was needed to validate the crush strength properties. The new honeycomb configuration

was checked for compatibility with manufacturing that provides maximum strength. The

cell wall attachments were modeled to include double wall thickness. The two new types

of cell configuration developed for this validation study were diamond-shaped and

triangular-shaped configurations.

105

4.7.1 Diamond-Shaped Honeycomb Model

The crush strength equation of diamond-shaped honeycomb model was derived

from the parameterized honeycomb model. The parameterized honeycomb model is

given as

( )

tcr A

RRMrknc

aD

hf

3

1

310

3

1

2

3

122

2(

2

3

��

�

�

��

�

�

��

��

� ++

=

π

(4.77)

Figure 4.22 shows the schematic diagram of the diamond-shaped honeycomb

model, in which honeycomb, the cells exhibit an asymmetrical honeycomb rather than

symmetrical shape. The connecting wall between cells was doubled to have higher

energy absorption cells and also for the purpose of bonding between cells.

Figure 4.22 Diamond-Shaped Honeycomb

a=D+D

k=0.5Dc=2

106

The parameters defined in the parameterized equation is given below as

a = number of D/c in the inter-connecting wall

c = value given either 1 or 2 wall when length D or D/2 is consider according to

the length of wall involved in energy absorption

k = length of the cell having multi-layer thickness in terms of cell size D


r = length of the extra attachment length in terms of cell size D

Substituting the parameters a=2, c=2, k=0.5, n=2, and r=0 in the above equation

(4.77), intersecting angles 60 degrees, therefore R1 = 33.6 and R3 = 19.12 are considered.

( )

tcr A

MD

hf

3

1

0

3

1

2

3

112.19*6.330*22*5.0*2

2

2*2(

2

3

��

�

�

��

�

��

��

� ++

=

π

(4.78)


2

4

1hM oo σ= (4.79)


tcr A

Dhf

3

1

3

5

47.18= (4.80)

4.7.2 Triangular-Shaped Honeycomb Model

Figure 4.23 shows the schematic diagram of the triangular-shaped honeycomb

model, in which cells exhibit symmetrical rather than asymmetrical shapes. The

connecting walls between cells were doubled to increase energy absorption and also for

the purpose of bonding between cells.

107

Figure 4.23 Triangular shaped honeycomb

Substituting the parameters a = D+0.5D, c = 1, k = 0, n = 0, and r = 0 in equation

(4.77), intersecting angles 60 degrees, therefore R1 = 33.6 and R3 = 19.12.

( )

tcr A

MD

hf

3

1

0

3

1

3

112.19*6.33

1

5.0*2

1

1*2(

2

3

��

�

�

��

�

��

��

� +

=

π

(4.81)

Substitute Mo in equation (4.81)

2

4

1hM oo σ= (4.82)

a=D+0.5Dc=1

108

Therefore, the total crush strength is obtained by multiplying with two since this

triangular-shaped honeycomb is a symmetrical honeycomb

tcr A

Dhf

3

1

3

5

*832.6*2= (4.83)

tcr A

Dhf

3

1

3

5

*66.13= (4.84)

4.8 Chapter Summary

A parameterized model for determining honeycomb crush strength was developed

for typical configuration and validated with experimental results. A crush strength

equation was developed for two new types of honeycombs, triangular and diamond-

shaped. Since experimental data was not available for these two new types of

honeycombs, finite element models were used to validate these new types of

honeycombs.

109

CHAPTER 5

FINITE ELEMENT ANALYSIS OF METALLIC HONEYCOMB

5.1 Objective

This chapter shows the development of a finite element model and analysis of

metallic honeycomb. This is useful if analytical models need to be validated when

experimental data are not available for validation. Buckling of the thin wall structure was

studied using the square tube analysis in order to provide insight into buckling simulation

variables needed to simulate honeycomb models. This chapter explains how finite

element models of hexagonal, half-hexagonal, Formgrid, Flexcore, and Double Flex

honeycomb were developed and validated using experimental methods. New types of

honeycombs, namely, diamond and triangular shapes, were modeled and analyzed to

determine crush strength properties.

5.2 Finite Element Analysis on Buckling of Square Tubes

Buckling analysis was performed on the square tube to study the buckling

behavior of the thin- walled structure. The FEA model of the square tube on which

buckling analysis was performed is shown in Figure 5.1. The square tube was modeled in

FEMB software [57] using nodes and shell elements. FEA analysis was performed using

Ls-dyna software [32]. Simulation variables, material properties and contact definitions

were input in a Ls-dyna solver key file. Quasi-static and impact analysis were performed

on the square tube model. In order to perform a buckling analysis, imperfections were

incorporated on the square tube by introducing boundary conditions which act as an

imperfection to initiate buckling. The imperfections implied were as follows: lower end

nodes of the tube were fully constrained, top end nodes of the tube were free to move

110

only in T-direction, and all the nodal rotations in the top end were fixed. Prescribed

constant velocity was applied to the top end nodes in a negative T direction. These

kinematic features applied to the tube allowed the tube to start buckling, as shown in

Figure 5.2. The sliding of the inner surface to the adjacent surface during the buckling

was prevented by providing sliding contact in LS-DYNA coding.

Figure 5.1. Square tube for buckling analysis.

Quasi-static Dynamic

Figure 5.2. Buckling of square tube.

L axis

T axis

W axis

Top End

Bottom End

111

For dynamic analysis of the square tube, a Cowper-Symond constant for steel

material, namely Dm=40.5, p=5, was used to simulate the rate sensitivity of the material.

The Dm and p are the material constants for the Cowper-Symond model for high rate

sensitivity deformation. As shown in Figure 5.2, the dynamic buckling of the tubes starts

from the bottom end and gradually progresses toward the top end. In the case of quasi-

static buckling, the buckling starts from the top end and progresses toward the bottom

end. These buckling analysis were validated with experimental methods.

Figure 5.3. Validation of quasi-static and dynamic buckling of square tube specimen.

5.2.1 Simulation Stages

Various stages of the quasi-static and dynamic buckling of the square tube are

shown in Figure 5.4 and 5.5. As can be seen, it is clear that dynamic buckling of tubes

starts from the bottom end, and quasi-static buckling starts from the center of the tubes.

112

Figure 5.4. Simulation stages of dynamic buckling of square tube.

113

Figure 5.5. Simulation stages of quasi-static buckling of square tube.

114

Figure 5.6 shows the experimental validation of square tube buckling.

0

5

10

15

20

25

30

35

40

45

50

30 35 40 45 50 55

Width of tube (mm)

Cru

shlo

ad(K

N)

Experimental published [46]FEAProposed parameterized modelAnalytical solution [2]

Figure 5.6. Experimental validation of square tube buckling.

TABLE 5.1

VALIDATION OF EXPERIMENTAL AND FINITE ELEMENT ANALYSIS OFSQUARE TUBE BUCKLING

Size of tubemm

Velocitym/sec

Experimentalmethod [46]

(KN)

Finiteelementanalysis

(KN)

Deviation%

37.11 Quasi-static 17.9 16.32 849.31 10.39 49 44.21 949.31 Quasi-static 35.25 35.7 1.3

115

Using the validated buckling analysis and buckling concepts, honeycomb models

were modeled in FEMB software [57] as regular geometrical models and the finite

analysis was performed.

5.3 Finite Analysis on Honeycomb Models

Quasi-static analysis was performed on Half-Hexagonal, flexible Flexcore,

Hexagonal, and Double-Flex honeycomb models. Quasi-static velocity was applied to the

punch. Honeycomb models were modeled using shell element in FEMB software [57]

which is the preprocessor for LS-DYNA. After completing modeling of honeycomb, LS-

DYNA codes were generated.

Initially LS-DYNA codes consisted of nodes and corresponding elements of the

honeycomb model. Then LS-DYNA codes were input with materials properties,

elemental properties, the defined curve for motion of the impacting punch, surface

contact definition, velocity, and simulation parameters for executing the analysis.

Validation of Hexagonal honeycomb is shown in Figure 5.7. Simulation stages of various

honeycombs are shown in Figure 5.8.

116

5.3.1 FEA Analysis on Hexagonal Honeycomb

0

200

400

600

800

1000

1200

1400

5.00E-04

1.50E-03

2.50E-03

3.50E-03

4.50E-03

5.50E-03

6.50E-03

7.50E-03

8.50E-03

9.50E-03

1.05E-02

Time (sec)

Cru

shS

tren

gth

(psi

)

FEA



Figure 5.7. Validation of hexagonal honeycomb configuration 1/8-0.001-8.1.

Figure 5.8. Simulation stages of quasi-static analysis of Hexagonal honeycomb.

117

5.3.2 FEA Analysis on Half-Hexagonal Honeycomb

0

2000

4000

6000

8000

10000

12000

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Crush Distance/Original Sample Height (in/in)

Cru

shS

tren

gth

(psi

)



Finite element analysis

Figure 5.9. Validation of quasi-static analysis of Half-Hexagonal honeycomb.

Simulation stages of Half-Hexagonal honeycomb and validations are shown in

Figures 5.9 and 5.10.

Figure 5.10. Simulation stages of quasi-static analysis of Half-hexagonal honeycomb.

118

5.3.3 FEA Analysis on Flexcore Honeycomb

Simulation stages of Flexcore honeycomb and validations are shown in Figure

5.11. and 5.12.

Figure 5.11. Simulation stages of quasi-static analysis of Flexcore honeycomb.

0

50

100

150

200

250

300

350

400

5.0E-04 1.5E-03 2.5E-03 3.5E-03 4.5E-03 5.5E-03 6.5E-03 7.5E-03

Time (sec)

Cru

shSt

reng

th(p

si)




Figure 5.12. Validation of quasi-static analysis of Flexcore honeycomb.

119

5.3.4 FEA Analysis on Formgrid Honeycomb

Simulation stage of Formgrid honeycomb is shown in Figure 5.13. Table 5.2 and

Figure 5.14 show the validation of Formgrid honeycomb model.

Figure 5.13. Simulation stages of quasi-static analysis of Formgrid honeycomb.

TABLE 5.2

CRUSH STRENGTH OF FORMGRID HONEYCOMB

Mean crush strength (lbs/sq.inch)Honeycomb type

Velocity ofimpact

(inch/sec) FEM ExperimentalDeviation

%3.1-5052/40 25 188 179 5

3.1-5052/40 10 170 175 2.8

120

0

50

100

150

200

250

300

350

400

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Crush Distance/Original Sample Height (in/in)

Cru

shSt

reng

th(p

si)


Experimental


Figure 5.14. Validation of quasi-static analysis of Formgrid.

5.3.5 FEA Analysis on Diamond-Shaped Honeycomb

Table 5.3 shows the validation of Diamond-Shaped honeycomb model.

TABLE 5.3

CRUSH STRENGTH OF DIAMOND-SHAPED HONEYCOMB


Velocity ofimpact(inch/sec) FEA Analytical

Deviation%

Diamond Quasi-static 1060 984 7

121

Simulation stages of Diamond-shaped honeycomb are shown in Figure 5.15

Figure 5.15. Simulation stages of quasi-static analysis of diamond shaped honeycomb.

0

500

1000

1500

2000

2500

3000

0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 0.0070 0.0080 0.0090

Time (sec)

Cru

shst

reng

th(p

si)

Proposed analytical model


Figure 5.16. Validation of quasi-static analysis of diamond-shaped honeycomb.

122

5.3.6 FEA Analysis on Triangular Honeycomb

Simulation stage of Triangular-shaped honeycomb is shown in Figure 5.17 and

validations are shown in table 5.4.

Figure 5.17. Simulation stages of quasi-static analysis of triangular-shaped honeycomb.

TABLE 5.4

CRUSH STRENGTH OF TRIANGULAR- SHAPED HONEYCOMB


Velocity ofimpact(inch/sec) FEA Analytical

Deviation%

Triangular Quasi-static 576 552 4.2

123

5.4 Chapter Summary

Finite element analysis of the square tube was performed to study the buckling

pattern of thin-walled structures. Finite element analyses of various honeycomb

configurations were performed and validated using experimental data. The analytical

models of the diamond-shaped and triangular-shaped honeycomb were validated using

the finite element analysis. Validated honeycomb models were used to generate crush

strength data for validation purposes.

124

CHAPTER 6

DYNAMIC CRUSH STRENGTH OF TYPICAL HONEYCOMBS

6.1 Objectives

Experimental results showed that a thin-walled metallic structure exhibits strain-

rate sensitivity properties. The metallic honeycomb shows the rate sensitivity properties

at higher speeds. The quasi-static crush strength of the honeycomb is lower than the

dynamic crush strength, which is obtained at higher speeds. Since dynamic testing is a

time consuming and expensive process, an analytical model was needed to evaluate the

dynamic crush strength of metallic honeycomb. This chapter, discusses the development

of analytical model to evaluate the dynamic crush strength of the metallic honeycomb

with typical configurations. Rate sensitivity material coefficients are evaluated for typical

honeycomb configuration which is essential for evaluating the dynamic crush strength.

Also discusses the method to determine the crush strength at higher impact velocities

using the low velocity test.

6. 2 Various Developments in the Dynamic Crush Strength of Thin walledMaterials

6.2.1 Cowper-Symond Strain Rate Law

In 1957, Cowper and Symond, [29] developed an empirical equation for strain

rate sensitivity materials. The relation between the static and dynamic yield stress was

given by their empirical formula

p

omD ��

�

��

�−= 1

σσ

ε� (6.1)

125

where

ε� = strain rate

Dm and p = empirical constants

σ = dynamic yield stress

oσ = static yield stress

6.2.2 Rate-Sensitive Impulse Loaded Structures

In 1965, Perrone [43] developed the method for solving rate-sensitive problems

on the impact of materials. He used the Cowper-Symonds rate-sensitive law [29] to

develop the solution. As shown in Figure 6.1, he used a mass less wire having a constant

cross-section where by a mass is attached to the end of the wire and gravity and wave-

propagation effects were ignored. He assumed a constant strain rate at any instant, and

the wire was rigid and perfectly plastic having a rate-sensitive yield stress.

Figure 6.1. Perrone experimental methods for determining rate sensitivity.

m

Lo

wire

mass

xuo

126

p

mo D

/1

1 ��

��

�+= ε

σσ �

(6.2)

Applying Newton’s second law for the mass shown in the Figure 6.1

( )dx

udm

dt

dumA

2

2==− σ (6.3)

( )dx

udm

DA

p

mo

2/1

21 =

��

��

�

��

��

�+− εσ

�

(6.4)

where

u = Velocity of m

A = area of wire

t = time

ε� = u/Lo - strain rate

x = axial displacement

Written in a dimensionless form

ξµ

d

dff p =+− )1( 2/1 (6.5)

where dimensionless constant are given as

22

2

om LD

uf = (6.6)

oL

x=ξ (6.7)

127

A

LmD

o

om

σµ

2

2

= (6.8)

From this, the final axial strain is given as

� +=

of

pf f

df

02/11

µξ (6.9)

He applied this final axial strain to a simple example assuming the wire material of mild

steel having p = 5 and Dm = 40.4/sec. The final strain of the mild steel rod is given as

µξ 75.364=f (6.10)

Perrone’s methodology of introducing rate-sensitivity to impact problems was used by

later researchers to study the dynamic properties of materials.

6.2.3 Dynamic Gain of Thin-Walled Material

In 1981, Wierzbicki and Abromowicz [24] developed an equation for dynamic

gain, which is the ratio of the dynamic to static stress for thin-walled materials.

Figure 6.2. Bending mechanisms of thin walled structures.

V, velocity

H

2Hb

128

They developed the bending mechanism of thin-walled structures and introduced

the strain rate effect. Before buckling starts, the thin wall is straight with a curvature of

the bent R1 equal to zero, and when the thin wall start to bend, the final curvature R2 is

equal to 1/b, where b is the rolling radius of the thin-walled material, as shown in Figure

6.2.

Figure 6.3. Basic thin plate showing geometry and direction.

The average curvature rate is given by

tRRR ave δ/21 −=� (6.11)

tRR ave δ/2=� (6.12)

tbR ave δ/1=� (6.13)

tHV δ/= (6.14)

y

z

x

h/2

F

129

Hb

VR ave =� (6.15)

The average strain rate is related to the average curvature rate as

zR aveave�

� =ε (6.16)

where z is the average thickness of the thin plate from the mid-surface, as shown in

Figure 6.3, Where z = h/4, Then

Hb

VR ave =� (6.17)

Therefore the average strain rate is given as

hHb

Vave 4

=ε� (6.18)

The dynamic gain can be determined using the Cowper-Symond relation as

p

m

p

mo DHb

Vh

DR

/1/1

411 ��

�

��

�+=��

�

��

�+== ε

σσ �

(6.19)

where

R - dynamic gain

ε� - strain rate

Dm and p - empirical constants

σ - dynamic yield stress

oσ - static yield stress

Also the dynamic load factor is represented as a ratio of impact velocity of two columns

pp

V

VR

/1

2

1

/1

2

1��

��

�=��

�

��

�=

εε�

�

(6.20)

130

Wierzbicki and Abromowicz concluded that the dynamic gain is not dependent

upon any geometrical and material parameters. Their method does not deal with any

geometrical configuration, as in the case of honeycombs. Also, development of new

material constants dependent on the geometrical shape is not indicated. This method does

not show any ways to determine the dynamic crush load when only the static crush load

is available.

6.2.4 Dynamic Crush load for Square Tubes

In 1984, Abromowicz and Jones [46] developed the dynamic crush load for the

axial crushing of square tubes. They used the Cowper-Symonds strain rate equation to

determine the dynamic crush load. The material constants used were Dm=6844 and

p=3.91 from testing steel specimens.

dHb

Vave 4

=ε� (6.21)

6.279.361.43(49.0

1(/3/23/1/1

+��

��

�+��

��

��

��

�+=

h

D

h

D

DD

VMP

p

mm

(6.22)

The analytical model results vary from the experimental data, as shown in Figure

6.4. In their analytical model, material constants Dm and p are taken from the

experimental testing of steel specimens and not from the regular geometry, since the Dm

and p material constants depend upon the geometrical shapes, mainly the width of the

square tubes.

131

40.0

50.0

60.0

70.0

80.0

90.0

100.0

7.5 8 8.5 9 9.5 10 10.5

Impact velocity (m/s)

P mK

N



Figure 6.4. Crush load versus impact velocity.

6.2.5 Energy Absorption of Sandwich Honeycomb

In 1995, Wierzbicki et al. [53] developed an equation for predicting the energy

absorption of sandwich plates with crushable cores, as shown in Figure 6.5.

Figure 6.5. Sandwich plate subjected to indentation by a circular punch.

rp

h

q

132

The energy absorbed by the honeycomb core with sandwich plates is given by

��

��

�

+

��

��

�

−�

�

�

�

��

�

�

��

�

�

��

�

�= 11ln

16

4442

ppo

p

rrh

rqE

ξξσ

π(6.22)

where

rp - radius of punch

h - plate thickness

q - average crush pressure = Pm/At

Pm - Crush load

At - tributary area

�o - yield strength

Using equation (6.23) for energy absorption of the sandwich core, they obtained a

close match with the experimental data. This method is good for assessing the impact of a

circular punch on a sandwich core. However, if the shape is arbitrary, an analytical model

is needed for that shape. Also, measuring the boundary of impact will be difficult and

inaccurate.

6.2.6 Honeycomb Constitutive Model

In 2004, Hinnerichs et al. [54], from Sandia National Laboratories, developed the

Honeycomb Constitutive Model (HCM), which can determine the crush strength in bi-

axial loading conditions. They compared the HCM with the Orthotropic Crush Model

and found that the crush load obtained in bi-axial loading conditions is much closer to the

experimental results in the case of HCM.

133

6.3 Proposed Method of Evaluating Dynamic Crush Strength of Typical

Honeycomb Shapes

This chapter shows the method that was developed to determine the dynamic

crush strength of metallic honeycombs. Analytical equations of average strain rate for

typical honeycomb configurations, namely, Hexagonal, Half-Hexagonal, Formgrid,

Flexcore, and Double Flex were derived using the half-wavelength and rolling radius of

these honeycombs. The dynamic crush strength equations for these honeycomb

configurations were developed using the Cowper-Symond rate-sensitivity law. The new

material strain rate coefficients Dm and p were developed from the dynamic and quasi-

static experimental data of the honeycomb testing. Using these newly developed material

strain rate coefficients, the dynamic crush strength was determined for each particular

honeycomb configuration. Also, the dynamic crush strength of honeycomb was predicted

using the material constant Dm=6500 and p=4 given by Cowper and Symond that was

obtained from the experimental testing of aluminum at dynamic conditions. Validation of

the dynamic crush strength was performed using the experimental data. This

methodology was applied to the new honeycomb configuration, namely, diamond and

triangular-shaped.

The dynamic crush strengths obtained from both of the above materials strain rate

constants were compared. Next, the crush strength data obtained at low speeds were used

to evaluate the crush strength at high speeds. The rate sensitivity material constants

developed at low speeds were used to develop the crush strength at impact velocity.

6.4 Evaluation of Dynamic Crush Strength of Flexcore Honeycomb

The average strain rate of thin-walled material is given as

134

hHb

Vave 4

=ε� (6.22)

The Cowper Symond equation is given by

s

p

m

aved D

σεσ *1/1

�

��

+=

�

(6.24)

Equation (6.25) can be used to determine the dynamic crush strength of metallic

honeycomb, which can written as

cr

p

m

aved

cr fD

f *1/1

�

��

+= ε�

(6.25)

The Flexcore honeycomb is shown in Figure 6.6. The half wavelength H for Flexcore

honeycomb, previously derived in Chapter 3, is given as

3 28697.0 hDH = (6.26)

Figure 6.6. Flexcore honeycomb model.

135

Substituting the half wavelength H in Equation (6.24)

hhDb

Vave 3 28697.0**4

=ε� (6.27)

where b is the rolling radius, which can be given as

3 2529.0 Dhb = (6.28)

By substituting rolling radius, a relation between the average strain rate in terms

of velocity V and the width D of the Flexcore type of honeycomb is obtained.

The average strain rate for Flexcore is given by

D

Vave 84.1

=ε� (6.29)

From the equation (6.30), it can be shown that the average strain rate is influenced

by impact velocity and the width of the honeycomb. Substituting the average strain rate

in the equation (6.26), the new formula for dynamic crush strength is obtained as

cr

p

m

d

cr fDD

Vf *

84.11

/1

�

��

+= (6.30)

The Dm and p are new material strain rate coefficients for the Flexcore

honeycomb. The new material strain rate coefficients were determined for this particular

honeycomb in order to evaluate the crush strength at higher speeds.

136

6.5 Evaluation of Dynamic Experimental Material Constants for FlexcoreHoneycomb

In order to determine material strain rate coefficients Dm and p for this particular

honeycomb material, the above dynamic crush strength equation was written in

logarithmic form as

cr

p

m

d

cr fDD

Vf *

84.11

/1

�

��

+= (6.32)

p

mcr

d

cr

DD

V

f

f/1

84.11 �

��

=

��

�

�

��

�

�− (6.33)

�

��

��

��

�=��

�

�

��

�

�−

DD

V

pf

f

mcr

d

cr 184.1

log1

1log (6.34)

DpD

V

pf

f

mcr

d

cr log1

84.1log

11log −�

��

��

��

�=

��

�

�

��

�

�− (6.35)

Using Equation 6.35, new material strain rate coefficients can be evaluated.

TABLE 6.1

EVALUATION OF MATERIAL STRAIN RATE COEFFICIENTS FOR FLEXCOREHONEYCOMB

HoneycombWidth(inch)

ImpactVelocity(inch/sec

)

AverageStrainRate

aveε�

DynamicCrush

Strength

(d

crf )(psi)

Quasi-StaticCrush

Strength

( crf )(psi)

Log(d

crf /

crf -1)Log( aveε )

0.15 498 1804 269 165 -0.4615 7.490.15 570 2065 285 165 -0.3184 7.630.15 609 2206 290 165 -0.2776 7.690.15 665 2409 307 165 -0.1501 7.78

137

Equation (6.36) is an equation for a straight line with Log (fcrd/fcr-1) plotted

against log (V/1.84*D). The material constant p can be found by taking the slope of the

straight line in graph shown in Figure (6.7), and Dm can be found by taking the intercept

on the ordinate. The strain rate constants Dm and p were found to be 2815 and 0.956

respectively.

y = 1.0451x - 8.3011

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

7.45 7.5 7.55 7.6 7.65 7.7 7.75 7.8 7.85

Log(V/1.84 D)

Log

(fcr

d /fcr

-1)


Figure 6.7. Graph for evaluating new material strain rate coefficients.

3011.80451.1 −= xy (6.36)

Using the above method, the dynamic crush load for the square tube and the new

material constants for square tubes were determined. This method shows a close match

with the experimental methods performed by Abromowicz and Jones [46], as shown in

the Figure 6.8. The material strain rate coefficients for the square tube was found to be

138

Dm =227 and p=1.127, using the proposed model. The material constant used by the

analytical model [46] were Dm=6844 and p=3.91 from testing steel specimens.

0

10

20

30

40

50

60

70

80

90

7 7.5 8 8.5 9 9.5 10 10.5

Impact velocity (inch/sec)

Dyn

amic

crus

hst

reng

th(p

si)


Analytical modeling [46]

Proposed model

Figure 6.8. Validation of analytical modeling of square tube performed byAbromowicz & Jones [46] and proposed model.

6.6. Evaluation of Dynamic Crush Strength of Hexagonal Honeycomb

Figure 6.9. Hexagonal honeycomb models.

139

The average strain rate for thin-walled structure is given as

hHb

Vave 4

=ε� (6.37)

Substituting the half wavelength H for hexagonal honeycomb in Equation (6.36)

hhDb

Vave 3 2821.0**4

=ε� (6.38)

where b is the rolling radius given as

3 2683.0 Dhb = (6.39)

Substituting the values for the rolling radius, a relation between the average strain

rate in terms of velocity V and width D of the hexagonal type of honeycomb is obtained.

The average strain rate for hexagonal type of honeycomb is given as

D

Vave 243.2

=ε� (6.40)

The dynamic crush strength for hexagonal honeycomb is obtained as

fcrDD

Vf

p

m

d

cr *243.2

1/1

�

��

+= (6.41)

The new material strain rate coefficients Dm and p for the above dynamic crush

strength equation should be evaluated using the equation (6.42) and the experimental data

given in table 6.2.

140

mcr

d

cr DpD

V

pf

flog

184.1

log1

1log −�

��

��

��

�=��

�

�

��

�

�− (6.42)

TABLE: 6.2

EVALUATION OF MATERIAL RATE SENSITIVE COEFFICIENTS FORHEXAGONAL HONEYCOMB

Honey-combWidth(inch)

ImpactVelocity(inch/sec)

AverageStrainRate

aveε�

DynamicCrush

Strength

(d

crf )(psi)

Quasi-staticCrush

Strength

( crf )(psi)

Log(d

crf

/ crf -1)

Log(�)

0.125 964 3438.2 1280 830 -0.612 8.140.125 1080 3851.9 1320 830 -0.527 8.250.125 1176 4194.3 1506 830 -0.205 8.340.125 1354 4829.2 1687 830 0.032 8.48

The Dm and p were found to be 4760 and 0.495, respectively using the Figure 6.10.

y = 1.7845x - 14.919

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

8.1 8.15 8.2 8.25 8.3 8.35 8.4 8.45 8.5

log(V/2.242D)

log(

f crd /f

cr-1

)


Figure 6.10. Graph for evaluating new material strain rate coefficients for hexagonalhoneycomb -1/8-0.001-5052-8.1pcf.

141

6.7 Dynamic Crush Strength Equation for Half-Hexagonal Honeycomb

The average strain rate for thin-walled structure is given as

hHb

Vave 4

=ε� (6.43)

Substituting the half wavelength H for half-hexagonal honeycomb in equation (6.43)

hhDb

Vave 3 2375.1**4

=ε� (6.44)

where b is the rolling radius which can be given as

3 28835.0 Dhb = (6.45)

Substituting the values for the rolling radius eventually obtained a relation between the

average strain rate in terms of velocity V and the width D of the Half-Hexagonal.

The average strain rate for Half-Hexagonal type of honeycomb is given as

D

Vave 859.4

=ε� (6.46)

Substituting the average strain rate in the Cowper-Symond equation, the dynamic crush

strength for Half-Hexagonal honeycomb is obtained as

cr

p

m

d

cr fDD

Vf *

859.41

/1

�

��

+= (6.47)

The new material strain rate coefficients Dm and p for the dynamic crush strength

equation (6.47) should be evaluated using the equation (6.48) and the experimental data

given in table 6.3.

142

mcr

d

cr DpD

V

pf

flog

1859.4

log1

1log −�

��

��

��

�=��

�

�

��

�

�− (6.48)

The Dm and p were found to be 26742 and 1, respectively using the Figure 6.11.

TABLE 6.3

EVALUATION OF MATERIAL RATE SENSITIVE COEFFICIENTS FOR HALF-HEXAGONAL HONEYCOMB



AverageStrainRate

aveε�

DynamicCrush

Strength

(d

crf )(psi)

Quasi-Static Crush

Strength

( crf )(psi)

Log(d

crf /

crf -1)

Log(�)

0.125 25 41.1 6510 6500 -6.47 3.710.125 50 82.3 6520 6500 -5.78 4.4

y = x - 10.194

-6.6

-6.5

-6.4

-6.3

-6.2

-6.1

-6

-5.9

-5.8

-5.7

3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5

Log(V/4.859D)

Log

(fcr

d /fcr

-1)


Figure 6.11. Graph for evaluating new material strain rate coefficients forHalf-hexagonal.

143

6.8 Dynamic Crush Strength Equation for Formgrid Honeycombs

Figure 6.12. Formgrid honeycomb model.

The average strain rate for the thin-walled structure is given by

hHb

Vave 4

=ε� (6.49)

Substituting the half wavelength H for Formgrid honeycomb in Equation (6.50)

hhDb

Vave 3 2245.0**4

=ε� (6.50)


3 2213.0 Dhb = (6.51)

144

Substituting the values for the rolling radius, eventually obtained a relation between the

average strain rate in terms of velocity V and the width D of the hexagonal type of

honeycomb.

The average strain rate for Formgrid type of honeycomb is given by

D

Vave 495.1

=ε� (6.52)


strength for hexagonal honeycomb is obtained as

cr

p

m

d

cr fDD

Vf *

495.11

/1

�

��

+= (6.53)

The new material strain rate coefficients Dm and p for the dynamic crush strength

equation (6.54) should be evaluated using the equation (6.48) and the experimental data

given in table 6.4.

mcr

d

cr DpD

V

pf

flog

1495.1

log1

1log −�

��

��

��

�=��

�

�

��

�

�− (6.54)

TABLE 6.4

EVALUATION OF MATERIAL RATE SENSITIVE COEFFICIENTS FORFORMGRID HONEYCOMB



AverageStrainRate

aveε�

DynamicCrush

Strength

(d

crf )(psi)

Quasi-StaticCrush

Strength

( crf )(psi)

Log(d

crf

/ crf -1)

Log(�)

0.15 7.5 33.444 170 165 -3.496 3.5090.15 10 44.593 175 165 -2.803 3.7970.15 25 111.48 178 165 -2.540 4.7130.15 498 2220 271 165 -.44 7.7

145

The Dm and p material strain rate coefficients were found to be 4277 and 0.6767,

respectively using the Figure 6.13.

y = 0.6767x - 5.658

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 1 2 3 4 5 6 7 8 9

log(V/1.495D)

log(

f crd /f

cr-1

)


Figure 6.13. Graph for evaluating new material strain rate coefficients for Formgrid.

6.9 Dynamic Crush Strength Equation for Double-Flex Honeycombs

The average strain rate for thin-walled structure is given by

hHb

Vave 4

=ε� (6.55)

Substituting the half wavelength H for double flex honeycomb in equation (6.56)

hhHb

Vave 3 21.1**4

=ε� (6.56)


146

Dhb3 25936.0= (6.57)

Substituting the values for the rolling radius, evantually obtained a relation

between the average strain rate in terms of velocity V and the width B of the hexagonal

type of honeycomb.

The average strain rate for double-flex type of honeycomb is given by

D

Vave 612.2

=ε� (6.58)


strength for hexagonal honeycomb is obtained as

fcrDD

Vf

p

m

d

cr *612.2

1/1

�

��

+= (6.59)

The Dm and p material strain rate coefficients are found to be 1926 and 1.64, respectively.

Figure 6.14 shows the graph for evaluating the material strain rate coefficients.

y = 0.8685x - 5.7082

-3

-2.5

-2

-1.5

-1

-0.5

0

0 1 2 3 4 5 6

Log(V/1.84 D)

Log

(fcr

d /fcr

-1)


Figure 6.14. Graph for evaluating new material strain rate coefficients for Double-Flex.

147

6.10 Dynamic Crush Strength Equation for Diamond-Shaped Honeycomb

Using the similar methodology material constant for determining the dynamic

crush strength, dynamic crush strength for a new type of diamond honeycomb was

developed.

The Dm and p material strain rate coefficients for diamond shaped honeycomb were

found out to be 6374 and 2.27, respectively. Figure 6.15 shows the graph for evaluating

the material strain rate coefficients.

y = 0.4405x - 3.8588

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

6.2 6.4 6.6 6.8 7 7.2 7.4 7.6

Log(V/1.5D)

Log

(fcr

d /fcr

-1)


Figure 6.15. Graph for evaluating material strain rate coefficients, Diamond-shaped honeycomb.

148

6.11 Dynamic Crush Strength Equation for Triangular-Shaped Honeycomb

The Dm and p material strain rate coefficients for triangular shaped honeycomb is

found out to be 3275 and 1.14, respectively. Figure 6.16 shows the graph for evaluating

the material strain rate coefficients.

y = 0.8714x - 7.0534

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

6.3 6.4 6.5 6.6 6.7 6.8 6.9 7 7.1 7.2

Log(V/1.122 D)

Log

(fcr

d /fcr

-1)


Figure 6.16. Graph for evaluating material strain rate coefficients-Triangularshaped honeycomb.

6. 12 Validation of Dynamic Crush Strength Obtained from Proposed AnalyticalModels

Validation of the dynamic crush strength obtained from the analytical model was

performed using experimental and validated finite element models. From the Figures 6.21

to 6.26, it can be inferred that the analytical equation that uses material constants Dm and

p from real experimental testing of honeycomb is better than the analytical equation that

uses Cowper-Symond constants for aluminum alloy, namely Dm = 6500 and p = 4. From

149

this it was clear that the approach of using the material constants obtained from the

experimental testing of honeycomb could be used for evaluating dynamic crush strength.

This dynamic crush strength equation could be used to determine crush strength for

metallic honeycomb in a dynamic environment. In order to obtain the material strain-rate

coefficients Dm and p for individual honeycomb, dynamic testing should be performed.

Validated honeycomb models were used to generate the finite element data for range of

impact velocities when the experimental data were not available. Figures 6.17, 6.18, and

6.19 show the validation of the finite element models with experimental data.

6.12.1 Validation of Dynamic Analysis of Honeycombs

Finite element analysis of honeycomb models was performed and validated using

experimental methods. Using the validated honeycomb models, finite element analyses

are performed at certain impact velocities where there is no experimental data available.

0

50

100

150

200

250

300

350

0 100 200 300 400 500 600 700


Cru

shst

reng

th(p

si)



Figure 6.17. Validation of dynamic crush strength of Flexcore.

150

4000

4500

5000

5500

6000

6500

7000

7500

0 100 200 300 400 500 600 700 800


Cru

shS

tren

gth

(psi

)



Figure 6.18. Validation of dynamic crush strength of Half-Hexagonal.

3000

3200

3400

3600

3800

4000

4200

4400

4600

0 100 200 300 400 500 600 700


Cru

shSt

reng

th(p

si)



Figure 6.19. Validation of dynamic crush strength of Hexagonal honeycomb pcf-22.

151

6.12.2 Validation of Dynamic Crush Strength Obtained from Analytical Methods

Table 6.5 shows the dynamic crush strength used for determining the strain rate

coefficients.

TABLE: 6.5

EVALUATION OF MATERIAL RATE SENSITIVE COEFFICIENTS FORHEXAGONAL HONEYCOMB



AverageStrainRate

aveε�

DynamicCrush

Strength

(d

crf )(psi)

Quasi-staticCrush

Strength

( crf )(psi)

Log(d

crf

/ crf -1)

Log(�)

0.125 964 3438.2 1280 830 -0.612 8.140.125 1080 3851.9 1320 830 -0.527 8.250.125 1176 4194.3 1506 830 -0.205 8.340.125 1354 4829.2 1687 830 0.032 8.48

The Dm and p were found out to be 4760 and 0.495, respectively. Figure 6.20 shows the

graph for evaluating the material strain-rate coefficients. Figures 6.21 to 6.26 shows the

validation of dynamic crush strength obtained from new strain rate coefficients.

y = 1.7845x - 14.919

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

8.1 8.15 8.2 8.25 8.3 8.35 8.4 8.45 8.5

log(V/2.242D)

log(

f crd /f

cr-1

)


Figure 6.20. Graph for evaluating material strain-rate coefficients forHexagonal honeycomb.

152

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400


Dyn

amic

crus

hst

reng

th(p

si)


Proposed strain rate coefficients

Cowper-Symond coeffcients [29]

Figure 6.21. Validation of analytical dynamic crush strength of hexagonalhoneycomb -1/8-0.001-5052-8.1pcf.

1300

1400

1500

1600

1100 1120 1140 1160 1180 1200 1220 1240Impact Velocity (inch/sec)

Dyn

amic

Cru

shSt

reng

th(p

si)



Cowper-Symond coefficients [29]

Figure 6.22. Validation of analytical dynamic crush strength of hexagonalhoneycomb -1/8-0.001-5052-8.1pcf.

153

Table 6.6 shows the dynamic crush strength used for determining the strain rate

coefficients.

TABLE: 6.6

EVALUATION OF MATERIAL RATE SENSITIVE COEFFICIENTS FORFORMGRID



Average Strain

Rate

aveε�

Dynamic CrushStrength

(d

crf )(psi)

Quasi-StaticCrush

Strength

( crf )(psi)

Log(d

crf /

crf -1)

Log(�)

0.15 7.5 33.444 170 165 -3.496 3.5090.15 10 44.593 175 165 -2.803 3.7970.15 25 111.48 178 165 -2.540 4.7130.15 498 2220 271 165 -.44 7.7

0 100 200 300 400 500 600Impact Velocity (inch/sec)

Dyn

amic

Cru

shSt

reng

th(p

si)




Figure 6.23. Validation of analytical dynamic crush strength of Formgridhoneycomb, 0.0019-pcf-3.1.

154

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 50 100 150 200 250 300 350 400 450


Dyn

amic

crus

hst

reng

th(p

si)



Cowper-Symonds coefficients [29]

Figure 6.24. Validation of analytical dynamic crush strength of diamond-shapedhoneycomb; strain-rate coefficients obtained from 100, 200, and 300 inches/secvelocities.

700

750

800

850

900

950

1000

250 270 290 310 330 350 370 390 410


Dyn

amic

crus

hst

reng

th(p

si)


Proposed strain rate coefficient


Figure 6.25. Validation of analytical dynamic crush strength of triangular-shaped honeycomb; strain-rate coefficients obtained from 100 and 200inches/sec velocities.

155

400

410

420

430

440

450

460

470

480

490

500

700 750 800 850 900 950 1000 1050 1100

Impact Velocity (Inch/sec)

Dyn

amic

Cru

shS

tren

gth

(psi

)




Figure 6.26. Validation of analytical dynamic crush strength of hexagonalhoneycomb, 1/8-5052-4.5 pcf; strain-rate coefficients obtained from 343, 421,551 and 689 inches/sec velocities.

6.12.3 Methodology of Predicting Dynamic Crush Strength from Low VelocityMaterial Strain Rate Coefficients Dm and p

In order to predict the dynamic crush strength, the experimental data of crush

strength obtained at high velocity was needed. From the experimental data, new material

constants was evaluated. But if the experimental data of crush strength of honeycomb

available is only from low velocity impact, there must be methodology to predict the

dynamic crush strength from low velocity. A new methodology was developed to predict

the dynamic crush strength with low velocity crush strength data. New material strain rate

coefficients were developed for evaluating the dynamic crush strength from low velocity

data. The strain rate coefficients were assumed to be constant for low-high-speed impact

conditions. Figures 6.27 to 6.34 show the validation of dynamic crush strength obtained

from new material strain rate coefficients. Comparisons between dynamic crush strength

156

obtained from new material strain rate coefficients and material coefficients given by

Cowper-Symond are shown clearly. It is noted that the dynamic crush strength obtained

using new material strain-rate coefficients was much closer with the experimental data

than the dynamic crush strength from material constants from Cowper-Symond

coefficients.

0

2000

4000

6000

8000

10000

12000

580 600 620 640 660 680 700 720


Dyn

amic

crus

hst

reng

th(p

si)




Figure 6.27. Analytical dynamic crush strength obtained from low velocitymaterial coefficients for Half-Hexagonal, 1/8-0.006-5052-38-pcf; strain-ratecoefficients obtained from 25 and 50 inches/sec impact velocities.

157

0

200

400

600

800

1000

1200

1400

1600

1800

800 900 1000 1100 1200 1300 1400


Dyn

amic

crus

hst

reng

th(p

si)



Cowper-Symond Coefficients [29]

Figure 6.28. Analytical dynamic crush strength obtained from low velocitymaterial coefficients for hexagonal, 1/8-0.002-5052-8.1pcf; strain- ratecoefficients obtained from 25,100 and 200 inches/sec impact velocities.

265

270

275

280

285

290

295

300

490 500 510 520 530 540 550 560 570 580


Cru

shst

reng

th(p

si)

Finite element analysisProposed strain rate coefficientsCowper-Symond coefficients [29]

Figure 6.29. Analytical dynamic crush strength obtained from low velocities 2.5,7.5, 10, 25 inches/sec FEA data for Formgrid honeycomb.

158

265

270

275

280

285

290

295

300

305

310

450 500 550 600 650 700


Dyn

amic

crus

hst

reng

th(p

si)

Experimental [28]



Figure 6.30. Analytical dynamic crush strength obtained from low velocitymaterial coefficients for Flexcore, F40-0.0019-5052-3.1pcf, strain-ratecoefficients obtained from 7.5 and 25 inches/sec impact velocities.

1300

1350

1400

1450

1500

1550

1600

1650

1700

1750

1800

250 270 290 310 330 350 370 390 410


Dyn

amic

crus

hst

reng

th(p

si)




Figure 6.31. Analytical dynamic crush strength obtained from low velocitymaterial coefficients for diamond-shaped honeycomb; strain-rate coefficientsobtained using impact velocities 100 and 200 inches/sec.

159

0

100

200

300

400

500

600

700

800

0 200 400 600 800 1000 1200


Dyn

amic

crus

hst

reng

th(p

si)



Cowper-Symonds coefficients [29]

Figure 6.32. Analytical dynamic crush strength obtained from low velocitymaterial coefficients for Double Flex; strain-rate coefficients obtained usingimpact velocities 50 and 100 inches/sec.

700

750

800

850

900

950

1000

250 270 290 310 330 350 370 390 410


Dyn

amic

crus

hst

reng

th(p

si)


Proposed strain rate coeffcient

Cowper-Symond coeffcients [29]

Figure 6.33. Analytical dynamic crush strength obtained from low velocitymaterial coefficients for Triangular-shaped honeycomb; strain rate coefficientsobtained from 100 and 200 inches/sec. velocities.

160

410

420

430

440

450

460

470

480

490

400 500 600 700 800 900 1000 1100

Impact Velocity (inch/sec)

Dyn

amic

crus

hst

reng

th(p

si)


Proposed strain rate coeffcients

Cowper-Symonds coeffcients [29]

Figure 6.34. Analytical dynamic crush strength obtained from low velocitymaterial coefficients for hexagonal honeycomb, 1/8-5052-4.5 pcf; strain ratecoefficients obtained from 343 and 421 inches/sec velocities.

6.13 Chapter Summary

This chapter explains how the average strain rate of typical honeycombs, which is

necessary to develop the dynamic crush strength, was determined. New material strain-

rate coefficients were derived from the experimental testing of honeycomb. Dynamic

crush strength equations were developed for typical honeycomb configurations.

Experimental validation was performed on the dynamic crush strength model. The

dynamic crush strength obtained by using the new material strain rate coefficients from

the experimental data was compared with the dynamic crush strength obtained from the

material constant given by Cowper-Symonds coefficients. The new material strain-rate

coefficients for particular honeycomb showed a close match with the experimental data.

The new material strain rate constant for low-speed crush strength can be used to predict

the crush strength at impact.

161

CHAPTER 7

PERFORMANCE ANALYSIS OF HONEYCOMB FOR MAXIMUM ENERGYABSORPTION

7.1 Methodology

The design of energy absorption devices must include a maximum of energy

absorption material. In order to maximize the energy absorption properties, parameters

affecting them should be studied. Performance analysis was done on the honeycombs,

namely, Formgrid and the triangular-shaped honeycomb, to study the responses of

various factors affecting the crush strength of the honeycomb. In order to complete the

performance analysis on the honeycomb models, analytical models were used to generate

the data needed for the surface response analysis. The analytical models were

incorporated into all the geometrical parameters affecting crush strength properties.

7. 2 Performance Analysis Using Response Surfaces

Geometrical parameters, namely, gauge thickness, honeycomb width, and inner

edge angle were studied in Formgrid and Triangular-shaped honeycombs. The analytical

equations for Formgrid and Triangular shaped honeycombs were used to generate the

crush strength data that is the response. Response surface analysis was performed using

Design Expert software [37]. Analytical equations for the honeycomb models follow

The crush strength of the Formgrid honeycomb is given as

3

13

5

03.15D

A

hf

t

ycr

σ= (7.1)

where At is the tributary area of the Formgrid honeycomb

162

)2(sin += θDwAt (7.1)

The crush strength of the triangular honeycomb is given as

θ

σ

sin

32.273/5

3/5

D

h

fy

cr = (7.1)

where

� = inner edge angle

�y = yield strength of honeycomb material

h = gauge thickness

D = width of the honeycomb

w = width of the honeycomb

7. 2.1 Design for Surface Response Analysis

A response surface analysis was performed on the parameters affecting the

maximum crush strength of the metallic honeycomb. The main objective was to perform

the performance analysis of the parameters affecting the crush strength. Various factors

involved in the design of the metallic honeycomb for maximizing the crush strength

properties are as follows:

1. Impact Velocity. Impact velocity obviously affects the crush strength of the

metallic honeycomb. As the velocity of the impact increases, the crush strength deviates

from static to dynamic characteristics.

2. Face Sheet Thickness. Face sheet thickness affects crush strength properties. As

face sheet thickness increases, energy absorbed by the honeycomb core panel also

increases.

163

3. Gauge Thickness. One of the critical factors affecting crush strength properties

is gauge thickness. As gauge thickness increases, crush strength increases.

4. Honeycomb Width. Honeycomb width has an inverse effect on the crush

strength. As width increases, crush strength decreases, due to the loosely packed cells.

Honeycomb width influences the dynamic crush strength through the average strain rate.

5. Inner Edge Angle. Crush strength increases with increases in the inner edge

angle of the cell configuration.

Of these five factors affecting crush strength, three of them gauge thickness;

honeycomb width and inner edge angle, which are involved in the design of the

honeycomb configuration, are considered to be critical. These factors are the geometrical

parameters of the Fromgrid honeycomb. The velocity of impact and the face sheet

thickness have a directly proportional relationship to crush strength. The response in this

performance analysis was that crush strength should be maximized.

7.2.2 Performance Analysis on Formgrid Honeycomb

The design selected for this experiment was a 2k factorial design with a total of

eight observations. Table 7.1 shows details of the design for the Formgrid honeycomb.

Three important factors of the honeycomb configuration are gauge thickness, honeycomb

width, and inner edge angle. Response was the crush strength. The data for the

performance analysis were obtained from the analytical modeling of crush strength of the

honeycomb. The Figure 7.1 shows the half normal plot of the design.

164

TABLE 7.1

23 DESIGN FOR THE CRUSH STRENGTH EXPERIMENT

Std Run Block

Factor 1Gauge

Thickness(inches)

Factor2Honeycomb

Width(inches)

Factor 3InnerEdgeAngle(deg)

CrushStrength

(psi)

1 4 1 0.0019 0.1563 30 169.352 2 1 0.0024 0.1563 30 249.973 5 1 0.0019 0.2200 30 95.744 3 1 0.0024 0.2200 30 141.3275 1 1 0.0019 0.1563 50 202.516 6 1 0.0024 0.1563 50 298.917 7 1 0.0019 0.2200 50 114.498 8 1 0.0024 0.2200 50 146.18

DESIGN-EXPERT Plotcrush strength

A: Gauge ThicknessB: Honey comb widthC: Edge angle

Half Normal plot

Hal

fN

orm

al%

prob

abili

ty

|Effect|

0.00 23.13 46.26 69.39 92.52

0

20

40

60

70

80

85

90

95

97

99

A

B

C

AB

BC

Figure 7.1. Half normal plot.

165

Table 7.2 shows the analysis of interactions among the factors that affect the crush

strength. As can be seen, the BC interaction is not significant, that is, the interaction

between the honeycomb width and the inner edge angle affecting the crush strength is not

significant and is ignored. A reduced regression model is set up by ignoring the

interaction BC. Figure 7.2 shows the half-normal plot for a reduced model and table 7.3

shows the reduced regression model.

TABLE 7.2

ANALYSIS OF VARIANCE TABLE

Source Sum ofsquares

Degreeof

Freedom

MeanSquare

F Value Prob>F

Model 26564.04 5 5312.81 471.19 0.0021 SignificantA 8203.52 1 8203.52 727.57 0.0014 SignificantB 17118.05 1 17118.05 1518.20 0.0007 SignificantC 566.50 1 566.50 50.24 0.0193 Significant

AB 632.26 1 632.26 56.07 0.0174 Significant

BC 43.71 1 43.71 3.88 0.1878Not

SignificantResidual 22.55 2 11.28

Total 26586.59 7

TABLE 7.3

REDUCED REGRESSION MODEL

SourceSum ofsquares

Degreeof

Freedom

MeanSquare

F Value Prob>F

Model 26520.33 4 6630.08 300.18 0.0003A 8203.52 1 8203.52 371.41 0.0003B 17118.05 1 17118.05 775.02 0.0001C 566.50 1 566.50 25.65 0.0149

AB 632.26 1 632.26 28.63 0.0128Residual 66.26 3 22.09

Total 26586.59 7

166



Half Normal plot

Hal

fN

orm

al%

prob

abili

ty

|Effect|

0.00 23.13 46.26 69.39 92.52

0

20

40

60

70

80

85

90

95

97

99

A

B

C

AB

Figure 7.2. Half normal plot for reduced model.

7.2.2.1 Model Validation and Results

Model validation is performed by checking the normal plots, residual, and

outliers. The normal plotting of the residual does not show any abnormality or any

evidence for outliers as shown in Figures 7.3 and 7.5. The residual versus the predicted

plots do not show any funnel shape. Figures 7.3 and 7.4 show the normal plot of residuals

and normal plot of predicted.

167


Studentized Residuals

Nor

mal

%P

roba

bilit

y

Normal Plot of Residuals

-1.53 -0.77 0.00 0.77 1.53

1

5

10

20

30

50

70

80

90

95

99

Figure 7.3. Normal plot of residuals.


Predicted

Stu

dent

ized

Res

idua

ls

Residuals vs. Predicted

-3.00

-1.50

0.00

1.50

3.00

88.75 132.10 175.45 218.80 262.15

Figure 7.4. Residual vs. predicted values.

168


Run Number

Out

lier

T

Outlier T

-3.50

-1.75

0.00

1.75

3.50

1 2 3 4 5 6 7 8

Figure 7.5. Outliers.

Figures 7.6 and 7.7, plots of residual vs. edge angle and residual vs. honeycomb

width, show that edge angle vs. honeycomb width did not shows any inequality in the

variance.

But the residual vs. gauge thickness show inequality in the variance and that the

variance changes were due to the increase in the gauge thickness. An interaction between

A and B, that is, between the gauge thickness and the honeycomb width, was possible.

Figure 7.8 shows the residual vs. honeycomb width.

169


Edge angle

Stu

dent

ized

Res

idua

ls

Residuals vs. Edge angle

-3.00

-1.50

0.00

1.50

3.00

30 33 37 40 43 47 50

Figure 7.6. Residual vs. edge angle.


Gauge Thickness

Stu

dent

ized

Res

idua

ls

Residuals vs. Gauge Thickness

-3.00

-1.50

0.00

1.50

3.00

1.90E-03 2.02E-03 2.15E-03 2.27E-03 2.40E-03

Figure 7.7. Residual vs. Gauge Thickness.

170


Honeycomb width

Stu

dent

ized

Res

idua

ls

Residuals vs. Honeycomb width

-3.00

-1.50

0.00

1.50

3.00

0.16 0.17 0.19 0.20 0.22

Figure 7.8. Residual vs. Honeycomb width.

The response surface plots, as shown in the Figures 7.10 and 7.11 show that when

the edge angle was 30 degrees and the gauge thickness was between 0.0022 and 0.0023

inch, the honeycomb edge ranges between 0.1563 and 0.1722 inch, when the maximum

crush strength 224.478 psi is obtained. When the edge angle is 50 degrees, which is the

maximum range, the gauge thickness is between 0.0023 and 0.0024 inch, the honeycomb

edge is between 0.1563 to 0.165 inch, and the maximum crush strength of 224.478 psi is

obtained. Figure 7.12 shows the three dimensional contour plots of response surfaces.

This leads to the conclusion that if the edge angle is decreased, the gauge

thickness can be lowered, which will obviously reduce the weight of the metallic

honeycomb. This conclusion would be useful for the lightweight design of the energy

absorption systems. This methodology will be useful in the design and selection of

honeycomb panels for particular applications.

171

Figure 7.9. Interaction graph for honeycomb width and gauge thickness.

Figure 7.10. Two dimensional contour plots of response surfaces with edge angle30 degree.

172

Figure 7.11. Two dimensional contour plots of response surfaces with edge angle 50degree.

Figure 7.12. Three dimensional contour plots of response surfaces.

173

7.2.3 Performance Analysis on Triangular-Shaped Honeycomb

Table 7.4 shows the design for crush strength for triangular-shaped honeycomb.

Critical factors namely gauge thickness, honeycomb width and inner edge angle were

selected. The response was crush strength. Figure 7.13 shows the half normal plot of the

design. Analysis of variance is shown in table 7.5.

TABLE 7.4

23 DESIGN FOR THE CRUSH STRENGTH DATA FOR TRIANGULAR SHAPEDHONEYCOMB

Std Run Block

Factor 1Gauge

Thickness(inches)

Factor2Honeycomb

Width(inches)

Factor 3InnerEdgeAngle(deg)

CrushStrength

(psi)

1 4 1 0.0019 0.1563 30 14742 2 1 0.0024 0.1563 30 43523 5 1 0.0019 0.2200 30 16674 3 1 0.0024 0.2200 30 24615 1 1 0.0019 0.1563 50 9626 6 1 0.0024 0.1563 50 28407 7 1 0.0019 0.2200 50 10888 8 1 0.0024 0.2200 50 1606

174



Half Normal plot

Hal

fN

orm

al%

prob

abili

ty

|Effect|

0.00 379.25 758.50 1137.75 1517.00

0

20

40

60

70

80

85

90

95

97

99

A

B

AB

Figure 7.13. Half Normal Plot.

TABLE 7.5

ANALYSIS OF VARIANCE TABLE

Source Sum of SquaresDegrees

ofFreedom

MeanSquare

FValue

Prob>F

Model 7.069E+006 3 2.356E+006 5.22 0.0722Not

SignificantA 4.603E+006 1 4.603E+006 10.19 0.0332 SignificantB 9.842E+005 1 9.842E+005 2.18 0.2140

AB 1.483E+006 1 1.483E+006 3.28 0.1443Residual 1.807E+006 4 4.518E+005

Cor.Total

8.877E+006 7

175

7.2.3.1 Model Validation and Results

Model validation was performed by checking the normal plots, residual, and

outliers. The residual versus predicted plots shows a funnel shape as in Figure 7.15. The

normal plot of the residual did not showed any abnormality or any evidence for outliers

as in Figure 7.16.


Studentized Residuals

Nor

mal

%P

roba

bilit

y

Normal Plot of Residuals

-1.59 -0.80 0.00 0.80 1.59

1

5

10

20

30

50

70

80

90

95

99

Figure 7.14. Normal plot of residuals.

176


Predicted

Stu

dent

ized

Res

idua

ls

Residuals vs. Predicted

-3.00

-1.50

0.00

1.50

3.00

1218.00 1812.50 2407.00 3001.50 3596.00

Figure 7.15. Residual vs. Predicted values.


Run Number

Out

lier

T

Outlier T

-3.50

-1.75

0.00

1.75

3.50

1 2 3 4 5 6 7 8

Figure 7.16. Outliers.

177

Figures 7.17 to 7.19 show the residual plots. The residual versus gauge thickness

show an inequality in the variance and that the variances change due to the increase in

gauge thickness. Similarly residual vs. honeycomb width show variance changes due to a

decrease in honeycomb width. Interaction between A and B, that is, between gauge

thickness and honeycomb width, may be possible.


Edge angle

Stu

dent

ized

Res

idua

ls

Residuals vs. Edge angle

-3.00

-1.50

0.00

1.50

3.00

30 33 37 40 43 47 50

Figure 7.17. Residual vs. Edge angle.

178


Gauge Thickness

Stu

dent

ized

Res

idua

ls

Residuals vs. Gauge Thickness

-3.00

-1.50

0.00

1.50

3.00

1.90E-03 2.02E-03 2.15E-03 2.27E-03 2.40E-03

Figure 7.18. Residual vs. Gauge Thickness.


Honeycomb width

Stu

dent

ized

Res

idua

ls

Residuals vs. Honeycomb width

-3.00

-1.50

0.00

1.50

3.00

0.16 0.17 0.19 0.20 0.22

Figure 7.19. Residual vs. Honeycomb width.

179

Interaction between honeycomb width and guage thickness is shown in Figure

7.20. The surface response plots from Figures 7.21 were useful in selecting the

honeycomb parameters namely, gauge thickness and honeycomb width for maximum

crush strength properties.

Figure 7.20. Interaction graph for honeycomb width and gauge thickness.

Figure 7.21. Two dimensional contour plots of response surfaces with edge angle30 degree.

180

Figure 7.22. Three dimensional contour plots of response surfaces.

7.3 Chapter summary

Performance analysis was performed on the Formgrid and triangular-shaped

honeycombs. Surface response plots were generated and the effects of parameters,

namely, honeycomb gauge thickness, honeycomb width, and inner edge angle on crush

strength were discussed clearly.

181

CHAPTER 8

CONCLUSIONS AND RECOMMENDATIONS

Analytical modeling of the Formgrid, Half-Hexagonal, Flexcore, Double Flex

diamond-shaped, and triangular-shaped honeycombs were developed and validated using

experimental methods. Classifications of honeycomb configurations were made

according to the geometrical parameters and cell connectivity. A parameterized model of

the honeycomb was developed by considering cell connectivity and geometrical

parameters that can accommodate most typical shapes of the honeycomb. The

parameterized honeycomb model was validated using experimental testing and also

applied to new diamond-shaped and triangular-shaped honeycombs. Using this

parameterized honeycomb model the crush strength properties of the newly developed

honeycomb were found. The dynamic crush strength equations were developed for

typical honeycomb configurations using strain-rate equations. New material strain-rate

coefficients were extracted from the experimental testing of honeycombs, in order to

facilitate the designer being able to determine crush strength properties at high strain

rates.

The dynamic crush strength obtained from the newly developed material strain

rate coefficients were compared with the dynamic crush strength obtained from the

Cowper-Symond material constants. The newly developed material strain rate

coefficients proved to be much closer to experimental data. A methodology was

developed to predict the dynamic crush strength at higher strain rates when only low-

velocity crush strength data was available. Finally, performance analysis was performed

182

on the geometrical parameters of the honeycomb, which provided a method for

maximizing the energy absorption with light-weight honeycomb material.

Further research could be carried out for combined out-of-plane impacts of

honeycomb and also composite materials. A generalized model could be developed for

honeycomb filled with foam materials. A rate sensitivity of the sandwich honeycomb

could be developed, and new material strain-rate coefficients can be derived from the

experimental testing of sandwich honeycomb.

183

REFERENCES

184

REFERENCES

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189

APPENDIX

190

APPENDIX

LS-DYNA Key File for Simulation of Honeycomb

*KEYWORD*TITLELS-DYNA USER INPUT

*CONTROL_TERMINATIONENDTIM ENDCYC DTMIN ENDENG ENDMAS

0.0004 0 0.0 0.0 0.0

*CONTROL_BULK_VISCOSITYQ1 Q2 TYPE

1.5 0.060

*HOURGLASS1,4,.05

*CONTROL_HOURGLASS1,0.05

*CONTROL_ENERGY2,2,2

*CONTROL_OUTPUT1,3

*DATABASE_JNTFORCDT BINARY0.00002

*DATABASE_NCFORCDT BINARY0.00002

* DATABASE_RCFORCDT BINARY0.00002

*DATABASE_NODFORDT BINARY0.00002

191

*DATABASE_EXTENT_BINARYNEIPH NEIPS MAXINT STRFLG SIGFLG EPSFLG RLTFLG ENGFLGCMPFLG IEVERP BEAMIP DCOMP SHGE STSSZ N3THDT

1NINTSLD

1

*DATABASE_BINARY_D3PLOTDT/CYCL LCDT BEAM NPLTC0.00002 0 0 0

*PARTHEADINGPart-1PID SECID MID EOSID HGID GRAV ADPOPT TMID

1 1 2 0 1 0 0 0

*PARTHEADINGPart-2PID SECID MID EOSID HGID GRAV ADPOPT TMID

2 2 2 0 1 0 0 0

*PART$HEADINGPart-11PID SECID MID EOSID HGID GRAV ADPOPT TMID

11 3 1 0 0 0 0 0

*SECTION_SHELLSECID ELFORM SHRF NIP PROPT QR/IRID ICOMP SETYP

1 2 0.0 3 0.0 0.0 0 1T1 T2 T3 T4 NLOC MAREA

0.001 0.001 0.001 0.001 0 0.0

*SECTION_SHELLSECID ELFORM SHRF NIP PROPT QR/IRID ICOMP SETYP

2 2 0.0 3 0.0 0.0 0 1T1 T2 T3 T4 NLOC MAREA

0.002 0.002 0.002 0.002 0 0.0

*SECTION_SOLIDSECID ELFORM AET

3 1 0

192

*MAT_RIGIDMAT0002MID RO E PR N COUPLE M ALIAS

1 1.098 1.02E+07 0.33 0.0 0.0 0.0CMO CON1 CON2

1 4 7LCO_OR_A1 A2 A3 V1 V2 V3

*MAT_PLASTIC_KINEMATICMAT0002MID RO E PR SIGY ETAN BETA

2 .000251.0200E+07 0.33 42000.0SRC SRP FS VP

6500 4 1

*INITIAL_VELOCITY_RIGID11,0,0,-809,0,0,0

*DEFINE_CURVELCID SIDR SFA SFO OFFA OFFO DATTYP

115 0 1.0 -1.1557A1 O1

0,0.0070.00002,0.0140.00003,0.0210.00004,0.0280.00005,0.0350.00006,0.0420.00007,0.049

*CONTACT_NODES_TO_SURFACE34,79,2,20.08,,,, 10,,,1

*SET_PART7911

*CONTACT_AUTOMATIC_SINGLE_SURFACESSID MSID SSTYP MSTYP SBOXID MBOXID SPR MPR34,,2FS FD DC VC VDC PENCHK BT DT

0.08 0.08,1

193

*SET_PART341,2

*BOUNDARY_SPC_SET_IDID

1NSID CID DOFX DOFY DOFZ DOFRX DOFRY DOFRZ

1 0 1 1 1 1 1 1

*DATABASE_HISTORY_NODE_SETNSID1 NSID2 NSID3 NSID4 NSID5 NSID6 NSID7 NSID8

2

*DATABASE_NODAL_FORCE_GROUPNSID CID

2 2

*DEFINE_COORDINATE_NODESCID N1 N2 N3 FLAG

2 25358 25266 68759

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