1

Mechanical Properties of Hierarchical Honeycomb Structures

A THESIS PRESENTED

BY

Ghanim Alqassim

TO

DEPARTMENT OF MECHANICAL AND INDUSTRIAL

ENGINEERING

IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

NORTHEASTERN UNIVERSITY

BOSTON, MASSACHUSETTS

August 2011

2

Abstract

Cellular solids such as foams are widely used in engineering applications mainly

due to their superior mechanical behavior and lightweight high strength characteristic. On

the other hand, hierarchical cellular structures are known to have enhanced mechanical

properties when compared to regular cellular structures. Therefore, it is important to

understand the mechanical properties and the variation of these properties with the

presence of hierarchy. This investigation builds upon prior works and considers the

mechanical properties of two dimensional hierarchical honeycomb structures using

analytical and numerical methods. However, in contrast to previous research, the

hierarchy in this work is constructed by replacing every three edge vertex of a regular

hexagonal honeycomb with a smaller hexagon. This gives a hierarchy of first order.

Repeating this process builds a fractal appearing second order hierarchical structure. Our

results showed that hierarchical honeycombs of first and second order can be up to 2 and

3.5 times stiffer than regular hexagonal honeycombs with the same relative density.

Another mechanical property considered in this study is the energy absorbance of

hierarchical honeycombs. The in-plane dynamic crushing of hierarchical cellular

structures is yet to be investigated. Most of the previous work performed on the

mechanical behavior of cellular materials, considers an intact structural organization for

the cellular material. Thus, to further explore the energy absorbance of hierarchical

honeycombs, we have studied the response of three dimensional regular hexagonal first

order hierarchical honeycombs under in-plane dynamic crushing. Finite element method

was employed to measure the response of hierarchical cellular structures under impact

3

loading. As it is well established, honeycomb cellular structures behave differently under

dynamic loading, mainly in their deformation modes and stress levels. In addition, for

plastic behavior, the bilinear material properties with two different hardening rates (5%

and 10%) were also considered. Our results demonstrate that there is not much difference

in the energy absorption of the hierarchical structures when compared to regular

honeycombs for elastic perfectly plastic material. However, by applying strain hardening

to the material that makes up the cell walls of the hierarchical honeycomb, the energy

absorbance of the structures significantly increases.

4

Acknowledgment

This dissertation would not have been possible without the guidance and the help

of several individuals who in one way or another contributed or extended their valuable

assistance in the preparation and completion of this study. Firstly, I would like to thank

my thesis advisor, Professor Ashkan Vaziri, for his constant guidance and unlimited

encouragement during this thesis. His support from the initial to the final level enabled

me to develop an understanding of the subject. It started off with him suggesting that I

take a part of the MS/BS program and provided me all the support for it to happen.

Throughout the years I have been taught by him never to give up even when things go

wrong. In addition, he always showed me an inspired way to approach a research

problem and acquire an alternative solution. It has been a pleasure to be in his high

performance materials and structures laboratory team and to accomplish this thesis with

him. I would also like to express my sincere appreciation to Professor Hameed

Metgalchi, the department chairman, for admitting me in the MS/BS program and

believing in me that I would be able to succeed in completing this thesis. I would have

not been able to complete this dissertation without their help.

Besides my thesis advisor, I would like to express my sincere gratitude to

Professor Vaziri’s lab team member, Amin Ajdari for his help and support from the

beginning of the research till the end. The valuable knowledge he shared with me played

a major role in the completion of this thesis. Furthermore, I would also like to thank

Babak Haghpanah for his assistance in some parts of this work. Truly thanks for the

support from everyone in Northeastern University.

5

This acknowledgment would be incomplete if I miss to mention the guidance I

received from my family: my mom, dad, older and younger brother, and sister for their

mental and financial support in pursuing my education in the United States. They

provided me unconditional encouragement and endorsement throughout my whole life. It

is impossible to thank them enough and show my appreciation.

6

Table of Contents Abstract ................................................................................................ 1

Acknowledgment ................................................................................ 4

CHAPTER 1: INTRODUCTION ................................................. 10

1. Introduction to Cellular Structures: .................................................................................11

2. Cellular Structures Applications: ....................................................................................14

3. Introduction to Hierarchical Structures: ..........................................................................15

4. Literature Review:.........................................................................................................19

5. Objectives: ...................................................................................................................23

6. References: ...................................................................................................................24

CHAPTER 2: METHODOLOGY ................................................ 28

1. The Mechanics of Honeycombs: ....................................................................................29

3. In Plane Uniaxial Loading: ............................................................................................30

4. Regular Honeycomb Linear Elastic Deformation:............................................................34

5. Hierarchical Structure Linear Elastic Deformation:..........................................................36

6. Finite Element Method: .................................................................................................41

7. Results and Discussion: .................................................................................................47

8. Conclusion: ..................................................................................................................59

CHAPTER 3: DYNAMIC CRUSHING OF HIERARCHICAL

HONEYCOMBS .............................................................................. 60

1. Introduction: .................................................................................................................61

2. Finite Element Method: .................................................................................................63

3. Effect of Hardening on Plastic Behavior of Hierarchical Cellular Structures:.....................67

4. Results and Discussion: .................................................................................................69

5. Conclusion: ..................................................................................................................81

6. References: ...................................................................................................................82

7

List of Figures

Figure 1- Two dimensional honeycomb .......................................................................... 11

Figure 2- Polygons found in two dimensional cellular materials: (a) equilateral triangle,

(b) isosceles triangle, (c) square, (d) parallelogram, (e) regular hexagon, (f) irregular

hexagon. ............................................................................................................................ 13

Figure 3- Three dimensional polyhedral cells: (a) tetrahedron, (b) triangular prism, (c)

rectangular prism, (d) hexagonal prism, (e) octahedron, (f) rhombic dodecahedron (g)

pentagonal dodecahedron (h) tetrakaidecahedron (i) icosahedrons .................................. 14

Figure 4- Unit cell of an evolved hierarchical first order honeycomb from a regular

honeycomb. ....................................................................................................................... 17

Figure 5- Unit cell of an evolved hierarchical second order honeycomb from a first order

hierarchy............................................................................................................................ 18

Figure 6- A honeycomb with hexagonal cells. The in-plane properties are those relating

koads applied in the X1-X2 plane. Responses to loads applied to the faces normal to X3

are referred to as out of plane.2. In plane deformation: .................................................... 29

Figure 7- Unit cell of an undeformed honeycomb ........................................................... 31

Figure 8- Cell deformation by cell wall bending, giving linear elastic extension or

compression of the honeycomb in the X1 and X2 directions. .......................................... 32

Figure 9- Free body diagrams of first and second order hierarchical honeycombs used in

the analytical estimation.................................................................................................... 33

Figure 10- Two dimensional first order hierarchical honeycomb models: (a) γ1=0.1 (b)

γ1=0.3 (c) γ1=0.5 ............................................................................................................... 42

Figure 11- Y Direction displacement ............................................................................... 43

8

Figure 12- X-direction displacement ............................................................................... 44

Figure 13- Two dimensional second order hierarchical honeycomb models: (a) γ1=0.29

and γ2=0.1, (b) γ1=0.29 and γ2=0.15, (c) γ1=0.335 and γ2=0.1, (d) γ1=0.335 and γ2=0.15.

........................................................................................................................................... 46

Figure 14- Structural response of first order hierarchical honeycombs: (a) γ1=0.1 in the x

direction (b) γ1=0.3 in the x direction (c) γ1=0.1 in the y direction (d) γ1=0.3 in the x

direction. ........................................................................................................................... 49

Figure 15- Normalized effective stiffness of first order hierarchy for different values of

γ1 in the Y direction.......................................................................................................... 50

Figure 16- Normalized effective stiffness of first order hierarchy for different values of

γ1 in the X direction.......................................................................................................... 51

Figure 17- Structural response of second order hierarchical honeycombs: (a) γ1= 0.29,

γ2= 0.1 in the x direction (b) γ1= 0.29, γ2= 0.15 in the x direction (c) γ1= 0.335, γ2= 0.1 in

the y direction (d) γ1= 0.335, γ2= 0.15 in the x direction. ................................................. 54

Figure 18- Normalized effective stiffness of second order hierarchy for different values

of γ2 in the Y direction, with a constant γ1 of 0.29. ......................................................... 55

Figure 19- Normalized effective stiffness of second order hierarchy for different values

of γ2 in the X direction, with a constant γ1 of 0.335. ....................................................... 56

Figure 20- Contour map of the effective stiffness of hierarchical honeycombs with

second order hierarchy for all possible geometries........................................................... 58

Figure 21- Three dimensional first order hierarchical honeycomb models: (a) γ=0.1 (b)

γ=0.3 (c) γ=0.4 (d) γ=0.5 .................................................................................................. 65

Figure 22- Schematic stress-strain curve for materials with different hardening rates. .. 67

9

Figure 23- The effect of hardening on a stress-strain curve............................................. 69

Figure 24- Dynamic crushing of first order hierarchical honeycombs in the Y-direction

with a velocity of v=0.1 (a) regular honeycomb γ=0 (b) γ=0.1 (c) γ=0.3 (d) γ=0.4 (e)

γ=0.5.................................................................................................................................. 72

Figure 25- Energy absorption of first order hierarchical models with different values of γ

and 0% hardening.............................................................................................................. 73

Figure 26- Energy absorption of first order hierarchical models with different values of γ

and 5% hardening.............................................................................................................. 74

Figure 27- Energy absorption of first order hierarchical structures with different

percentages of strain hardening: (a) regular honeycomb γ=0 (b) γ=0.1 (c) γ=0.3 (d) γ=0.4

........................................................................................................................................... 77

Figure 28- Energy absorption of hierarchical models compared with a regular

honeycomb with 0% hardening, and 5% hardening: (a) γ=0.1 with γ=0 (b) γ=0.4 with γ=0

........................................................................................................................................... 79

10

CHAPTER 1:

INTRODUCTION

11

1. Introduction to Cellular Structures:

Cellular structures are made up of solid struts and foams that are interconnected.

This network constructs the faces and edges of individual cells [1]. The simplest type of

cellular structures are honeycombs which are a two dimensional array of polygons

packed together to fill a plane area that looks like the hexagonal cells of a bee (fig. 1).

More common cellular structures come in the form of a polyhedral which is packed in

three dimensions, called foams [3].

Figure 1- Two dimensional honeycomb

The most important feature of a cellular structure is its relative density ρ*/ρs,

where ρ* is the density of the cellular material, and ρs represents the density of the solid

from which the cells are made. Polymeric foams have relative densities that lie in the

range of 5% to 20%. Meanwhile, cork has a relative density of approximately 14%, and

most softwoods have high relative densities (between 15% and 40%) [1] . The thickness

of the cell walls is determined by the value of the relative density. The higher the relative

density, the greater the thickness of the cell walls. This leads the cellular structure to have

12

smaller pore spaces. When the relative density is larger than 30%, the solid is no longer

considered a cellular structure, but instead a solid containing isolated pores. In this work,

we considered cellular structures with relative densities of 6%.

Engineers growing interest in cellular solids comes from the wide range of

mechanical and thermal properties. These properties are measured via the same methods

as those used for fully dense solids. The large extension of properties produces

applications for foams which are not present in fully dense solids. The low density

feature of cellular structures allows the design of light, stiff components for instance

sandwich panels. In addition, cellular structures are known to be reliable thermal

insulators since they have low thermal conductivity. Furthermore, the low stiffness makes

three dimensional cellular structures (foams) perfect for a variety of cushioning

applications, for example, elastomeric foams are the typical material for seats. Cellular

structures also have a lot of energy absorbing applications [4]. This is due to their low

strength and high compressive strains properties.

It is not the cell size that matters, but the cell shape. When the cells are equiaxed

the properties of the cellular structure are isotropic. The unit cells which are packed

together to fill a two dimensional cellular structure are shown in fig. 2 below. These

shapes are available for both isotropic and anisotropic properties. What idealizes the

properties of different two dimensional cellular structures is how they are stacked

together, disregarding the cell shape. Honeycombs that are man-made use the different

types of unit cells in fig. 2. At it is well recognized, a honeycomb with regular hexagonal

cells has six edges surrounding each face.

13

Figure 2- Polygons found in two dimensional cellular materials: (a) equilateral triangle, (b) isosceles triangle, (c) square, (d) parallelogram, (e) regular hexagon, (f) irregular hexagon.

In three dimensions there are more possible cell shapes as seen in fig. 3 below.

The different types include polyhedral, tetrahedron, triangular prism, rectangular prism,

hexagonal prism, and octahedron cells. Idealized unit cell models have demonstrated to

be very useful to engineers in understanding the mechanical behavior of cellular

structures. These behaviors include the effective elastic stiffness and how it depends on

the cells relative density.

14

Figure 3- Three dimensional polyhedral cells: (a) tetrahedron, (b) triangular prism, (c) rectangular prism, (d) hexagonal prism, (e) octahedron, (f) rhombic dodecahedron (g) pentagonal dodecahedron (h) tetrakaidecahedron (i) icosahedrons

2. Cellular Structures Applications:

Cellular structures have many applications which go under four major areas:

packaging, structural use, thermal insulation, and buoyancy. An effective package must

be able to absorb the energy from impacts without affecting the contents with any

damaging stresses. Cellular structures are ideal for this purpose as the strength of a

cellular structure can be adjusted by varying the relative density. Moreover, cellular

structures can undergo large compressive strains at a relatively constant stress which

therefore results in high amounts of energy being absorbed by the structure without

producing large stresses [8]. The use of cellular structures in packaging has another

advantage in that the low relative density makes the package weigh less than different

solids. This lowers the manufacturing, handling, and shipping costs [1].

There are many natural materials that are cellular solids. For instance: wood,

cancellous bone, and coral can handle large cyclic and static loads for a long amount of

15

time. It is well known that the most used structural material is wood. In addition, man-

made cellular structures and honeycombs are used in applications in which they achieve

high energy absorption structures. An example of these structures is sandwich panels

which in today’s world, are made up using glass or carbon-fibre composite skins that are

separated by aluminum or paper-resin honeycombs, providing the panels with extremely

large specific bending stiffness and strength. Other applications include space vehicles,

racing yachts, and portable buildings [6, 9].

The applications of polymeric and glass foams are mainly as thermal insulators.

Products as small disposable coffee cups, and as elaborate as the insulators of booster

rockets of space shuttles. Modern buildings, refrigerated trucks, railway cars, and even

ships all benefit from the low thermal conductivity of cellular structures. In buildings for

example when fire hazards are taken into consideration, glass foams can be used instead.

An advantage that cellular structures have for extremely low temperature research is their

ability to reduce the amount of refrigerant needed to cool the insulation itself. This is due

to their low density. Similarly, this applies at high temperatures in the design of kilt and

furnaces for example, because the lower the mass, the larger the efficiency. The thermal

mass of cellular structures is proportional to its relative density.

3. Introduction to Hierarchical Structures:

There are numerous materials (natural and man-made) that demonstrate

structures in more than one length scale. The concept of structural hierarchy in materials

developed simultaneously in several different fields, in particular structural biology,

polymer science, and fractural science for ceramic and organic aggregates [18]. This is

represented when the structures themselves contain structural elements. The hierarchical

16

cellular structures are known to be large contributors in identifying the bulk mechanical

properties. Many natural hierarchical materials have displayed very high damage

tolerances from impact loading. The main objective of introducing hierarchy to cellular

structures is to further enhance the mechanical behavior of the structures without

compromising the elastic properties of the material. Hierarchical structures are obtained

by adding material where it is most needed to occupy areas of high stress due to impact

loading for instance. This process maximizes the efficiency of the resulting product and

the load bearing component [22].

How the cells are organized or stacked together in a hierarchical structure plays a

huge role in identifying the mechanical properties of the solid. Research has shown that

the hierarchical cell organization of sandwich panels with cores made of composite lattice

structures or foams can result in enhanced mechanical behavior and superior elastic

properties [17-24]. It has also been proven that increasing the levels of hierarchy in

cellular structures produces better performing structures that are lighter weight [16,19]. In

this work, we explored the properties of honeycombs with hierarchical substructure of

first and second order.

As discussed earlier, honeycombs are two dimensional cellular structures that are

used for many applications including energy absorption and thermal insulation. In

addition, honeycombs are used as the core of sandwich panels [1]. The stiffness and

strength of honeycombs is controlled by the bending of the cell walls when exposed to

transverse loading [1-6]. When the honeycomb is subject to uniaxial loading, the

maximum stress takes place at the corners of the cell walls. In other words, the maximum

bending occurs at the vertices of the honeycomb. This means that if we replace the

17

corners of the cell walls by the material in the middle, we can minimize the deformation,

and obtain less bending. Thus, increasing the energy that can be absorbed. In this

investigation, we replaced the corners of the cell walls of a regular hexagonal lattice, with

a new hexagon that is smaller in size. By doing so, we have achieved a first order

hierarchical cellular structure. Results that will be discussed later have shown that the

first order hierarchical honeycombs have enhanced stiffness when compared with regular

non hierarchical honeycombs. Fig. 4 shows the introduction of a first order hierarchical

cellular structure on a regular hexagon. Higher order hierarchical honeycombs can be

achieved by repeating the process of replacing the vertices with even smaller hexagons.

Fig. 5 shows the evolution of a first order hierarchical honeycomb to a second order

hierarchical honeycomb.

Figure 4- Unit cell of an evolved hierarchical first order honeycomb from a regular honeycomb.

18

Figure 5- Unit cell of an evolved hierarchical second order honeycomb from a first order hierarchy.

19

4. Literature Review:

Different models have been developed to study the mechanical properties of

cellular structures in general, and how these properties are further enhanced through the

introduction of hierarchy in regular cellular structures [1,2,4-6]. The failure mechanism

by which the cell walls deform under load is analyzed. The mechanical behavior of

cellular structures has been studied thoroughly and the results can be found in several

different surveys.

Early models for the uniaxial elastic behavior only assumed axial loading for the

cell walls [7-9]. Neglecting transverse loading in cellular structures leads to inconsistent

results obtained from the experiments. In addition, later studies found that bending

deformation of cell walls has more important contribution to the mechanical properties of

the honeycomb cellular structures [1-9].

It is now well known that the in-plane hydrostatic strength of a perfect hexagonal

honeycomb is proportional to the relative density, ρ. Governed by the cell wall stretching,

the deviatory strength is set by how the cell walls bend and in the same time it scales with

the relative density of the cells [1]. This is why the yield surface is elongated along the

hydrostatic axis in biaxial stress space. Klintworth and Stronge developed failure

envelopes for regular honeycombs with respect to different elastic and plastic cell

crushing models [14]. Using the simple beam theory, they managed to describe the in-

plane indentation of a honeycomb by a plane punch. In addition, the biaxial yield surface

of two dimensional honeycombs and the triaxial yield surface of three dimensional open

celled foams were also investigated.

20

Hierarchical structures surrounding us everywhere in nature and can be viewed in

several biological systems and organic materials [16]. The hierarchical organization of

these systems plays a vital role in identifying their properties, while identifying how long

they will survive [22]. Hierarchical structures are used in engineering designs,

architecture, and materials. The mechanical behavior of these structures is generally

governed by the response at different length scales and levels of hierarchy [18].

Increasing levels of structural hierarchy results in lighter weight and better performing

structures [19].

It is well established that introducing hierarchy to different types of cellular

structures provides the structure with improved mechanical behavior [18-26]. Different

types of hierarchical cellular structures have been studied. Taylor and Smith explored the

effects of hierarchy on the in-plane elastic properties of honeycombs [28]. The hierarchy

investigated in the research was the introduction of a square or triangle geometry into the

super and sub-structure cells of the honeycomb. In addition, they also studied the effect

of negative Poisson’s ratio materials with hierarchical honeycombs. Taylor and Smith

have shown through finite element analysis, that with specific designs of functionally

graded unit cells, it is possible to exceed the density specific elastic modulus by values up

to 75%. In addition, they have demonstrated that with negative Poisson ratio materials,

the density modulus can be increased substantially.

Hierarchical cellular structures are not only used for metals, as Burgueno and

Quagliata have shown. Results from their study illustrate that hierarchical cellular designs

can improve the performance of biocomposite beam made of natural fiber polymer

composites [29]. This allows them to compete with other conventional materials for load-

21

bearing applications. In addition, this study also proved that the density of the cellular

structure decreased while maintaining an increase in their performance as load bearing

materials.

The mechanical properties of 2D hierarchical cellular structures made up of

sandwich walls have also been proven by Fan et al., to have enhanced behavior compared

with solid wall cellular materials. The stiffness, Euler buckling strength, plastic collapse

strength, and high damage tolerance, are all mechanical properties that have

demonstrated higher values in sandwich walls hierarchical structures, when compared

with solid walls cellular structures [37].

An interesting study by Zhang et al., where they established that applying

multilayered cell walls to honeybee combs, further enhances their energy absorption

properties [39]. The added layers to the honeybee combs that count as the hierarchical

part of the structure are made of wax. The investigation included four types of honeybee

combs: a two day old (10 combs), a five month old (6 combs), a one year old (10 combs),

and a two year old (6 combs). This research was done to increase the collapse strength of

the honeybee combs due to large increases in temperatures. The study also mentions that

artificial engineering honeycombs (used in all previous works), can only mimic the

macroscopic geometry of natural honeycombs. This means that it does not take into

account the microstructure sophistication of their natural counterparts [39].

In 2007, Kooistra investigated mechanisms for transverse compression and shear

collapse of a first and second order hierarchical corrugated truss structure [30]. The

mechanical properties that were inspected, included elastic buckling and plastic yielding

22

of the larger and smaller struts. Again, as expected, the second order trusses structure

made from alloys, revealed higher compressive and shear collapse strengths than their

equivalent first order parts. This was proven analytically and confirmed experimentally in

the study. In fact, the experimental results demonstrate that the strength of second order

trusses is approximately 10 times higher than that of a first order truss with the same

relative density [30].

Finally, a 3D hierarchical computational model of wood was looked at by Qing

[31]. This model takes into account the structures of wood at different scale levels.

Similar to Zhang and Duang’s work mentioned earlier, the hierarchy in this investigation

took place as multilayered walls attached to the regular wood structure [31, 39].

Equivalent results were expressed where the stiffness in the multi layered wood was

substantially higher than that of the regular wood structure.

23

5. Objectives:

The present investigation builds upon prior works and considers the mechanical

properties of two dimensional hierarchical honeycomb structures using analytical and

numerical methods. The literature search showed a lack in introducing a hierarchy to the

corner of the edge of the cell walls (where the stress is highest). This will enhance the

energy absorption capabilities of the structures. In contrast to the previous works, the

hierarchy in this work is constructed by replacing every three edge vertex of a regular

hexagonal honeycomb with a smaller hexagon. This gives a hierarchy of first order. The

process is then repeated to achieve a second order hierarchical structure that is fractal-

appearing. The stiffness and strength of the structures is controlled by the adjustment of

the ratios for the different hierarchical orders. A finite element method is used through

ABAQUS 6.10 (SIMULIA, Providence, RI), which is used to generate the two

dimensional models of the different hierarchical structures. The structural response of

first and second order hierarchy was simulated to ensure the results from the theoretical

analysis that will be discussed later.

24

6. References:

[1] Gibson, L.J., Ashby, M.F., 1997. Cellular solids: Structure and properties, 2nd ED.

Cambridge, University Press, Cambridge.

[2] Gent, A.N., Thomas, A.G., The deformation of foamed elastic materials. J.Appl.

Polymer. Sci., 15 (1971) 693-703

[3] Ko, W.L., Deformations of foamed elastomers. J. Cell. Plastics, 1 (1965) 45-50

[4] Patel, M.R., Finnie, I., The deformation and fracture of rigid cellular plastics under

multiaxial stress. Lawrence Livermore Labaratory Report UCRL-13420, 1969

[5] Silva, M.J., Hayes, W.C., Gibson, L.J.,The effect of non-periodic microstructure on

the elastic properties of two-dimensional cellular solids. Int. J. Mech. Sci. 37 (1995)

1161-1177

[6] Ashby, M.F., The mechanical properties of cellular solids. Met. Trans., 14 (1983)

1755-1769

[7] Maiti, S.K., Gibson, L.J., Ashby, M.F., Deformation and energy absorption diagrams

for cellular materials. ACTA Metal., 32 (1984) 1963-1975

[8] Maiti, S. K., Ashby, M. F., Gibson, L. J., Fracture toughness of brittle cellular solids.

Scripta Metal., 18 (1984) 213-218

[9] Kurauchi, T., Sato, N., Kamigaito, o., Komatsu, N., Mechanism of high energy

absorption by foamed materials, foamed rigid polyurethane and foamed glass. J. Mat.

Sci., 19 (1984) 871-880

25

[10] Silva, M.J., Gibson, L.J., The effect of non-periodic microstructure and defects on

the compressive strength of two-dimensional cellular solids. Int. J. Mech. Sci., 30 (1997)

549-563

[11] Triantafyllidis, N., Schraad, M.W., Onset of failure in aluminum honeycombs under

general in-plane loading. J. Mech. Phys. Solids., 46 (1998) 1089-1124

[12] Chen, C., Lu, T.J., Fleck, N.A, Effect of imperfections on the yielding of two-

dimensional foams. J. Mech. Phys. Solids., 47 (1999) 2235±2272

[13] Wang, A., Mcdowell, D., Effects of defects on in-plane properties of periodic metal

honeycombs. J. Mech. Sci., 45 (2003) 1799-1813

[14] Klintworth, J., Stronge, W., Elasto-plastic yield limits and deformation

laws for transversely crushed honeycombs. Int. J. Mech. Sci., 30 (1988) 273-292.

[15] Shaw, M. C., Sata, T., The plastic behavior of cellular materials. Int. J. Mech. Sci., 8

(1966) 469-478

[16] Deshpande, V.S., Ashby, M.F., Fleck, N.A., Foam topology bending versus

stretching dominated architectures. ACTA Metal., 49 (2001)1035–1040.

[17] Deshpande, V.S., Fleck, N.A., Ashby, M.F., Effective properties of the octet-truss

lattice material. J. Mech. Phys. Solids., 49 (2001)1747–1769

[18] Fan, H.L., Fang, D.N., Jin, F.N., Mechanical properties of lattice grid composites.

ACTA Mat., 24 (2008) 409–418

[19] Bhat, T., Wang, T.G., Gibson, L.J., Micro-sandwich honeycomb. Sampe. J., 25

(1989) 43–45

[20] Lakes, R., Materials with structural hierarchy. Nature. 361 (1993) 511–516

26

[21] Kooistra, G.W., Deshpande, V.S., Wadley, H.N.G., Hierarchical corrugated core

sandwich panel concepts. J. Appl. Mech., 4 (2007) 259–268

[22] Fleck, N.A., Qiu, X.M., The damage tolerance of elastic–brittle, two-dimensional

isotropic lattices. J. Mech. Phys. Solids., 55 (2007) 562–588

[23] Wicks, N., Hutchinson, J.W., Optimal truss plates. Int. J. Solids. Struct.,

38 (2001) 5165–5183

[24] Yang, M.Y., Huang, J.S., Elastic buckling of regular hexagonal honeycombs

with plateau borders under biaxial compression. Compos. Struct.,

71 (2005) 229–237

[25] Ohno, N., Okumura, D., Noguchi, H., Microscopic symmetric bifurcation condition

of cellular solids based on a homogenization theory of finite deformation. J. Mech. Phys.

Solids., 50(2002) 1125–1253

[26] Fan, H.L., Meng, F.H., Yang, W., Mechanical behaviors and bending effects of

carbon fiber reinforced lattice materials. Arch. Appl. Mech., 75 (2006) 635–647

[27] Grenestedt, J., Influence of wavy imperfections in cell walls on elastic stiffness

of cellular solids. J. Mech. Phys. Solids., 46 (1998) 29–50

[28] Taylor, C., Smith, C., Miller, W., Evans, K.E., The effects of hierarchy on the in-

plane elastic properties of honeycombs. Int. J. Solids. Struct., 48 (2011) 1330-1339

[29] Burgueño, R., Quagliata, M.J., Mohanty, A.K., Mehta, G., Drzal, L.T., Misra, M.,

Hierarchical cellular designs for load bearing biocomposite beams and plates. Mat. Sci.

Eng., A 390 (2005) 178-187

[30] Kooistra, G.W., Deshpande, V., Wadley, H.N.G., Hierarchical corrugated core

sandwich panel concepts. J. App. Mech., 74 (2007) 259-268

27

[31] Qing, H., Mishnaevsky, L., 3D Hierarchical computational model of wood as a

cellular material with fibril reinforced, heterogeneous multiple layers. J. Mech. Mat., 41

(2009) 1034-1049

[32] Bitzer, T., Honeycomb marine applications. J. Plas. Comp. 13 (1994) 355–360

[33] Burlayenko, V.N., Sadowski, T., Analysis of structural performance of sandwich

plates with foam-filled aluminum hexagonal honeycomb core.

Comp. Mat. Sci., 45 (2009) 658–662

[34] Deshpande, V.S., Ashby, M.F., Fleck, N.A., Foam topology bending versus

stretching dominated architectures. ACTA Mat., 49 (2001) 1035–1040

[35] Evans, K.E., Alderson, A., Auxetic materials: functional materials and structures

from lateral thinking. Adv. Mat., 12 (2000) 617-626

[36] Evans, K.E., The design of doubly curved sandwich panels with honeycomb cores.

Comp. Struct., 17 (1991) 95–111

[37] Fan, H.L., Jin, F.N., Fang, D.N., Mechanical properties of hierarchical

cellularmaterials. Part i: Analysis. Comp. Sci. Tech., 68 (2008) 3380–3387

[38] Fratzl, P., Weinkamer, R., Nature’s hierarchical materials. Prog. Mat. Sci., 52 (2007)

1263–1334

[39] Zhang, K., Si, F.W., Duan, H.L., Wang, J., Microstructures and mechanical

properties of silks of silkworm and honeybee. Acta Biomat., 6 (2010) 2165-2171

28

CHAPTER 2:

METHODOLOGY

29

1. The Mechanics of Honeycombs:

For most energy absorption purposes, metallic honeycombs are used. Therefore,

the understanding of their mechanical behavior is very important. Furthermore, finding

out the properties of honeycombs helps in understanding the mechanics of much more

complex three dimensional cellular structures.

To measure the mechanical properties of honeycombs, we need to study the

deformation mechanisms after impact. The most common three dimensional honeycomb

is shown in fig. 6 below. The strengths and stiffness of the honeycomb in the X1-X2 plane

(in plane) are the lowest for the reason that stresses in this plane make the cell walls bend.

Meanwhile, the out of plane strengths and stiffness is highest in the X3 plane since they

require the axial extension or compression of the cell walls.

Figure 6- A honeycomb with hexagonal cells. The in-plane properties are those relating koads applied in the X1-X2 plane. Responses to loads applied to the faces normal to X3 are referred to as out of plane.2. In plane deformation:

30

When a honeycomb is compressed in the X1-X2 direction, the cell walls begin to

bend. As a result, a linear elastic deformation of the cells takes place. When a critical

stress is reached, the individual cells start to collapse. The outcome of the deformation

comes in different forms. For example, if the material was elastic, the cell walls

experience elastic bucking. In addition, plastic hinges start to form in materials with

plastic properties. Brittle fracture can also happen in brittle materials [3, 6-9]. Ultimately,

at high strains the cells collapse effectively causing opposing cells’ broken fragments to

pack together. This is known as the densification of the material [1].

As discussed earlier, the most important feature of a cellular structure is its

relative density. An increase in the relative density of the honeycomb structure increases

the relative thickness of the cell walls. This improves the resistance of the cells to

collapse at a critical stress. Furthermore, at higher densities the densification process

begins at lower crushing strain.

2. In Plane Uniaxial Loading:

In this investigation, we analyze the response of regular hierarchical cellular

structures to loads applied in the X1-X2 direction. A regular honeycomb is one where the

hexagons in the structure are regular. That means that all sides are equal, and all the

angles within the hexagon are 120o. The cell walls must also have the same thickness for

regular honeycombs. In this case, the structure is isotropic. Regular honeycombs have

two independent elastic moduli (a Young’s modulus E and a shear modulus G), and a

single value for the yield stress σ [1].

31

In general, honeycombs do not have equal length cell walls, and the internal

angles are also different. In addition, the cell walls may not have the same thickness. The

in-plane properties of these honeycombs are anisotropic and the structure has four elastic

constants (E1,E2,G12, and which is Poisson’s ratio) and two values for the yield

stress (σy1 and σy2). An undeformed honeycomb cell is shown below in fig. 7. Fig. 8

shows the bending caused by loads in the X1 and X2 directions [1].

Figure 7- Unit cell of an undeformed honeycomb

32

Figure 8- Cell deformation by cell wall bending, giving linear elastic extension or compression of the honeycomb in the X1 and X2 directions.

The equation of the relative density ρ*/ρs can be obtained through simple

geometry analysis:

𝜌∗

𝜌𝑠=

𝑡𝑙 (ℎ𝑙 + 2)

2 cos𝜃(ℎ𝑙 + sin 𝜃) (1)

With θ=30o in regular honeycombs and with h =l, the equation reduces to:

𝜌∗

𝜌𝑠=

2√3

𝑡𝑙

(2)

33

For the theoretical approach, the deformations are assumed to be sufficiently small and

some changes in the geometry are neglected.

The structural organization of the first and second hierarchical order is defined by

the following geometrical parameters: γ1 and γ2. These parameters define the ratio of the

smaller hexagonal edge length, to the original hexagon’s edge length. As demonstrated in

Fig. 9, in the first order hierarchical honeycomb, the edge length is b. For second order

hierarchy, c is the edge length keeping in mind the original hexagon’s cell length is a.

Figure 9- Free body diagrams of first and second order hierarchical honeycombs used in the analytical estimation.

From this, the following relations were acquired:

𝛾1 = 𝑏𝑎 , 𝛾2 = 𝑐

𝑎

In the analysis of hierarchical honeycombs, the cell edge length a of the original

hexagon was assumed to be 1. Therefore the range of values for the 1st order hierarchy is

0 ≤ b ≤ a/2, and thus 0 ≤ γ1 ≤ 0.5. The regular honeycomb structure is attained by setting

γ1 equal to zero. In a second order hierarchical honeycomb, there are two limitations: 0≤

c ≤ b and c ≤ a/2-b. In normalized form, the geometrical constraints are 0 ≤ 𝛾2 ≤ 𝛾1 if

34

𝛾1 ≤ 0.25 and 0 ≤ 𝛾2 ≤ (0.5 - 𝛾1 ) if 0.25 ≤ 𝛾1 ≤ 0.5. From simple geometrical analysis,

the equation to find the relative density of the first order hierarchical honeycomb is:

𝜌∗

𝜌𝑠=𝑡𝑎

(1 + sin 𝜃)

3 ∙ sin 𝜃 ∙ cos𝜃 (1 + 2 ∙ 𝛾1 ) (3)

For the second order hierarchy:

𝜌∗

𝜌𝑠=𝑡𝑎

(1 + sin 𝜃)

3 ∙ sin 𝜃 ∙ cos𝜃 (1 + 2 ∙ 𝛾1 + 6 ∙ 𝛾2) (4)

In regular hexagonal hierarchical cellular structures with θ = 30, the relative density

relations are simplified below:

Regular 1st order hierarchy:

𝜌∗

𝜌𝑠=

2√3

𝑡𝑎

(1 + 2 ∙ 𝛾1) (5)

Regular 2nd order hierarchy:

𝜌∗

𝜌𝑠=

2√3

𝑡𝑎

(1 + 2 ∙ 𝛾1 + 6 ∙ 𝛾2) (6)

4. Regular Honeycomb Linear Elastic Deformation:

When a load is applied in the X1 or X2 direction, the honeycomb deforms in a

linear elastic way, causing the cell walls to bend [1]. While a stress that is parallel to the

X1 direction is applied, the cell walls with length l bend. One cell wall is illustrated in fig.

8 above. The force component C that is parallel to the X2 direction is zero due to

equilibrium. During this research, the cell walls are treated as beams of length l, thickness

t, depth b, and Young’s modulus Es. To find the Moment M that tends to bend the cell

walls we use the following equation:

35

𝑀 = 𝑃𝑙 sin 𝜃

2 (7)

Where the load P is found from the equation below:

𝑃 = 𝜎1(ℎ+ 𝑙 sin 𝜃)𝑏 (8)

From the standard beam theory, the deflection of the cell wall is:

𝛿 = 𝑃𝑙3 sin 𝜃

12𝐸𝑠𝐼 (9)

Where I represents the moment of inertia of a cell wall (beam) .I = bt3 / 12 for a wall of

uniform thickness t. From this, a component 𝛿 sin 𝜃 is parallel to the X1 axis which

results in a strain of:

𝜖1 = 𝛿 sin𝜃𝑙 cos𝜃

=𝜎1(ℎ+ 𝑙 sin 𝜃)𝑏𝑙2 sin2 𝜃

12𝐸𝑠𝐼 cos𝜃 (10)

From Hooke’s law, the Young’s modulus parallel to the X1 direction is 𝐸1∗ = 𝜎1 /𝜖1,

giving:

𝐸1∗

𝐸𝑠= �

𝑡𝑙�3 cos𝜃

(ℎ 𝑙� + sin 𝜃) sin2 𝜃 (11)

In the X2 direction, the forces acting on the cell walls of length l, and depth b, are shown

in figure 8. The force F equals zero through equilibrium and 𝑊 = 𝜎2𝑙𝑏 sin 𝜃 giving:

𝑀 = 𝑊𝑙 cos𝜃

2 (12)

The deflection in the wall in this case is:

36

𝛿 = 𝑊𝑙3 cos 𝜃

12𝐸𝑠𝐼 (13)

Therefore, a component 𝛿 cos𝜃 is parallel to the X2 axis which results in a strain of:

𝜖2 = 𝛿 cos𝜃

ℎ + 𝑙 sin 𝜃=

𝜎2𝑏𝑙4 𝑐𝑜𝑠3 𝜃 12𝐸𝑠𝐼(ℎ + 𝑙 sin 𝜃) (14)

From Hooke’s law, the Young’s modulus parallel to the X2 direction is 𝐸2∗ = 𝜎2 /𝜖2,

giving:

𝐸2∗

𝐸𝑠= �

𝑡𝑙�3 (ℎ 𝑙� + sin 𝜃)

cos3 𝜃 (15)

5. Hierarchical Structure Linear Elastic Deformation:

As for the hierarchical cellular structures, we obtained the equations by exploring

the free body diagrams of the first and second order hierarchy subassembly which are

shown in fig. 9 above. For first order hierarchy, an external load F will be applied at point

3 in the Y-direction keeping in mind that this point is the midpoint of the cell edge. This

causes the cell walls to bend, and produces a moment M1 and M2 at points 1 and 2

respectively. In addition, points 1 and 2 will have reaction forces N1 and N2 in the

positive Y-direction. Using equations of equilibrium in the directions X and Y, we can

express N2 and M2 as functions of N1, M1, and F. As a result, the bending energy U that is

stored in the first order hierarchy can be stated as:

𝑈(𝐹,𝑀1,𝑁1) = ��(𝑀2 2𝐸𝑠𝐼⁄ )𝑑𝑥 (16)

37

Where M represents the bending moment in the beam’s cross section, Es is the elastic

modulus of the material used for the cell walls, and I stands for the moment of inertia of

the beams’ cross sectional area. This comes with the assumption that the cell walls have

rectangular cross sections with a thickness, t. Therefore the moment of inertia is

calculated using:

𝐼 =𝑡3

12 (17)

In the analysis of hierarchical honeycombs, it is also assumed that the displacement and

rotation around point 1 is zero due to symmetry. Therefore, the following equations can

be written:

𝜕𝑈 𝜕𝑁1⁄ = 0 , and 𝜕𝑈 𝜕𝑀1⁄ = 0

From these two relations, the equations for M1 and N1 can be obtained:

𝑁1 = 𝐹 �0.533 +0.15𝛾1

� (18)

𝑀1 = 𝐹𝑎(0.283𝛾1 − 0.017) (19)

However, at point 3 the displacement is not zero:

𝛿 =𝜕𝑈𝜕𝐹

(20)

By substituting this for N1 and M1, we acquire the following:

𝛿 = √3𝐹𝑎3

72𝐸𝐼𝑓(𝛾1) (21)

38

The effective stiffness of the structure is defined as the ratio of the average stress and the

average strain on the cell walls. The average stress depends on the value of F and a, while

the average strain depends on δ. From this E can be calculated from:

𝐸𝐸𝑠

= �𝑡𝑎�3𝑓(𝛾1) (22)

Where:

𝑓(𝛾1) =√3

(0.75− 3.525𝛾1 + 3.6𝛾12 + 2.9𝛾13) (23)

The maximum normalized stiffness for first order honeycomb structures with constant

relative density can be found by rewriting equation (22) as a function of the honeycomb

relative density. The change in the effective stiffness divided by the change in 𝛾1 is zero

(𝜕(𝐸 𝐸𝑠⁄ ) 𝜕𝛾1⁄ = 0). Using this information in equations (5) and (22), we obtain a value

of 𝛾1= 0.32, which results in an effective stiffness of 𝐸 𝐸𝑠⁄ = 2.97𝜌3. To find the

effective stiffness of a regular honeycomb structure, we use the same equations as we did

for first order hierarchy, and setting 𝛾1 = 0. By performing these calculations, a regular

honeycomb structure will only have a stiffness of 𝐸0 𝐸𝑠⁄ = 1.5𝜌3.

Therefore, it is noticed that introducing a first order hierarchy to a regular

honeycomb structure increases the stiffness by approximately twice as much. This is also

proven analytically via a finite element method as will be discussed in a later section.

For the second order hierarchy, a similar method was used to assess the effective

stiffness of the structure. As mentioned earlier, this structure requires two values for γ,

one for the smaller hexagon γ1 (first order), and the other for the smallest hexagon γ2

39

(second order). The free body diagram of the subassembly of this structure is shown in

fig. 9. An external load F, is applied at point 4 in the Y-direction. Here as well we note

that point 4 represents the midpoint of the cell edge. The reaction forces N1, N2, and N3

act on points 1, 2, and 3 respectively. Similarly, the moments M1, M2, and M3 are at the

vertices 1,2, and 3. With second order hierarchy, the same equilibrium equations that are

applied in the X and Y directions for the first order hierarchy are used. In other words,

N3 and M3 can be written as a function of N1, M1, N2, M2, and F. From this, we can

obtain the total energy of the substructure that is being examined. The total energy is the

sum of the bending strain energy of all of the beams in the system. It is expressed as:

𝑈(𝐹,𝑀1,𝑁1,𝑀2,𝑁2) = ��(𝑀2 2𝐸𝑠𝐼⁄ )𝑑𝑥 (24)

In this structure, four boundary conditions apply at points 1 and 2. Due to symmetry, the

displacement of the cells and the rotation is zero. The four boundary conditions are the

following:

𝜕𝑈 𝜕𝑁1⁄ = 0, 𝜕𝑈 𝜕𝑁2⁄ = 0 (displacement)

And

𝜕𝑈 𝜕𝑀1 = 0⁄ , 𝜕𝑈 𝜕𝑀2⁄ = 0 (rotation)

From these conditions, it is possible to solve for the reaction forces N1 and N2, as well as

the moments M1 and M2. Similar to what was applied to first order hierarchy; we can

obtain the effective stiffness through the following:

𝐸𝐸𝑠

= �𝑡𝑎�3𝑓(𝛾1𝛾2) (25)

Where the equation for 𝑓(𝛾1 𝛾2 ) is the following:

𝑓�𝛾1,𝛾2� = �29.62𝛾14− 54.26𝛾1

3𝛾2 + 31.75𝛾12𝛾2

2 − 4.73𝛾1𝛾23 −𝛾2

4�/

40

�−𝛾27 ∙ (2.20)− 𝛾2

6 ∙ �18.13𝛾1 + 1.88�+ 𝛾25 ∙ �159.95𝛾1

2 − 29.38𝛾1 + 3.90�+ �

𝛾24(−270.14𝛾13 + 9.70𝛾12 + 20.50𝛾1 − 0.43) − 𝛾23(195.50𝛾14 − 334.12𝛾13 + 108.06𝛾12 + 2.04𝛾1) +

𝛾22(862.56𝛾15 − 662.32𝛾14 + 123.22𝛾13 + 13.74𝛾12) − 𝛾2 (609.01𝛾16 − 310.43𝛾15 − 12.80𝛾14 +

23.46𝛾13) + �(49.64𝛾17 + 61.73𝛾16 − 60.43𝛾15 + 12.80𝛾14)} .

The maximum normalized stiffness of the second order hierarchy can be obtained

through equation (25) as a function of the relative density. The only difference in this

case is since there is γ2, the change in stiffness and the change in γ2 must also be taken

into account (𝜕(𝐸 𝐸𝑠⁄ ) 𝜕𝛾2⁄ = 0). Using this condition in equations (6) and (25), the

following values of γ1 and γ2 are acquired:

𝛾1 = 0.32, 𝛾2 = 0.135

Plugging these values in the second order stiffness equation (4) we achieve a stiffness of:

𝐸 𝐸𝑠⁄ = 5.26𝜌3

Here, it is perceived that the stiffness of the second order hierarchy is roughly 1.5 times

higher than the first order, and almost 3.5 times the stiffness of the regular honeycomb.

41

6. Finite Element Method:

The structural response of first and second order hierarchy was simulated to

certify the theoretical results explained previously. ABAQUS 6.10 (SIMULIA,

Providence, RI) was used to generate two dimensional models of the different

hierarchical structures. For first order, different models were made to account for

different values of γ1 (the ratio of the smaller hexagon cell edge b, to the original regular

hexagon cell length a). Examples of some of the models are shown below in fig. 10:

(a)

(b)

42

(c) Figure 10- Two dimensional first order hierarchical honeycomb models: (a) γ1=0.1 (b) γ1=0.3 (c) γ1=0.5

The models were designed using the BEAM22 element available in ABAQUS.

This feature in the ABAQUS library enables us to obtain the shear and axial deformation

of the cells, as well as the bending compliance. The significance of those features is

noticed in larger values of t. In this study, aluminum was used as the material of the

structures. As it is well known, the mechanical properties of aluminum are as follows:

An elastic modulus of Es=70 GPa, a yield strength of σys=130 MPa, and a poisson ratio

of ν = 0.3. A rectangular cross section with a unit length in the Z direction was used for

the cell wall beams. As mentioned earlier, the overall relative density of the structure

used in this study was 6%. To ensure this value, the thickness t of the cell walls was

adjusted with different values of γ. Periodic boundary conditions were used to eliminate

the boundary effects on the simulation results. Periodic boundary conditions enable the

borders of the model to maintain the same shape during deformation. In addition, an

analytical rigid shell was coupled to the top and bottom of the structure to ensure that the

displacement applied is equal in all cells. The simulations were examined for

43

displacement in the X-direction, as well as the Y-direction. For displacement in the Y-

direction, the bottom part of the models was fixed from moving in the X and Y direction

and in the same time from rotation about the Z-axis. The top part was fixed in the X

direction and a displacement was applied in the Y direction as displayed no fig. 11 below.

Figure 11- Y Direction displacement

For the displacement in X, the opposite was applied. The model was designed

such that the analytical rigid shell was vertical and held both sides of the structure as

shown below in fig. 12. Another difference would obviously be that the right side would

be fixed in the Y-direction, allowing the cells to deform in the X-direction.

44

Figure 12- X-direction displacement

On the other hand, the models for the second order hierarchical structures are

more complex, since γ2 has to be taken into account as well as γ1. As stated earlier, γ2

represents the ratio of the smallest hexagon relative to the original regular honeycomb.

Fig. 13 demonstrates a few models of second order hierarchical honeycombs with

different values of γ1 and γ2.

45

(a)

(b)

46

(c)

(d) Figure 13- Two dimensional second order hierarchical honeycomb models: (a) γ1=0.29 and γ2=0.1, (b) γ1=0.29 and γ2=0.15, (c) γ1=0.335 and γ2=0.1, (d) γ1=0.335 and γ2=0.15.

To find the thickness of the beams in the above models, equation (6) was used

with the different values of γ1 and γ2 stated above. After adjusting the thickness of the

47

rectangular cross section of the beams, the other options remain the same as what was

applied for the first order hierarchy.

7. Results and Discussion:

To analyze the effective elastic modulus of each structure, the compressive stress-

strain response of the models was utilized. Using Hooke’s law (𝐸 = 𝜎/𝜖), the effective

elastic modulus was calculated from the slope of the stress-strain response. Fig. 14 below

demonstrates a few examples of the deformation of the structures in the x and y direction.

(a)

Von Mises Stress

48

(b)

(c)

Von Mises Stress

Von Mises Stress

49

(d)

Figure 14- Structural response of first order hierarchical honeycombs: (a) γ1=0.1 in the x direction (b) γ1=0.3 in the x direction (c) γ1=0.1 in the y direction (d) γ1=0.3 in the x direction.

The different colors in the cell walls of the structures shown in fig. 14 represent different

levels of stress. The arrow indicates that the stress increases as the color goes from blue

to red. The maximum stress is located at the corners of the cell walls, where the

maximum bending takes place. The graphs shown in fig. 15 and fig. 16 below illustrate

the normalized effective stiffness of first order hierarchical honeycomb structures for

different values of γ1 in the X and Y direction.

Von Mises Stress

50

Figure 15- Normalized effective stiffness of first order hierarchy for different values of γ1 in the Y direction

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5

E 2/E

0

γ1

TheoreticalFEM

51

Figure 16- Normalized effective stiffness of first order hierarchy for different values of γ1 in the X direction

In these graphs, the stiffness is normalized by the effective stiffness of the

counterpart, which is the regular honeycomb, keeping an equal relative density using this

equation:

𝐸0 𝐸𝑠⁄ = 1.5𝜌3 (26)

Solving for E0:

𝐸0 = 1.5𝐸𝑠𝜌3 (27)

Where Es is the elastic modulus of the material used (70 GPA for aluminum), and a

density (ρ) of 6% that was applied in all the different models. By doing so, a single curve

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5

E 1/E

0

γ1

Theoretical

FEM

52

for the normalized effective stiffness of a first order hierarchical honeycomb is obtained

as a function of γ1., The results display good agreement between the theoretical and

numerical approaches as displayed in the graphs above. As previously declared, the

theoretical analysis ignored the axial and shear deformation of the beams. This is why the

results are slightly off and not exactly the same. The results are a good approximation

only for low density honeycombs that have small beam thicknesses. Therefore, the

analytical results match the theoretical values for low density structures only. On the

other hand, with high density beams there would be a noticeable difference in values of

effective stiffness. That’s the reason the results shown in the graph above are a little

dissimilar since the relative density of the beams used for the models in the numerical

approach was 6%. Redoing the simulations for structures with a relative density of 2%

would show a closer match to the theoretical results.

It can be observed from the graphs that in the first order hierarchical structures,

the stiffness in the X and Y direction tends to increase as the value of γ1 rises up to

𝛾1=0.3. The maximum increase in stiffness for both the theoretical and numerical

approaches occurs at 𝛾1=0.3, where a stiffness of approximately twice of that of a regular

hexagon honeycomb is achieved. The effective stiffness then starts to decrease as we

reach values of 𝛾1=0.4 and 0.5.

A similar method was used to analyze the results of second order hierarchical

honeycombs. Fig. 17 displays some examples of the deformation of second order

hierarchies in both the X and Y direction.

53

(a)

(b)

Von Mises Stress

Von Mises Stress

54

(c)

(d)

Figure 17- Structural response of second order hierarchical honeycombs: (a) γ1= 0.29, γ2= 0.1 in the x direction (b) γ1= 0.29, γ2= 0.15 in the x direction (c) γ1= 0.335, γ2= 0.1 in the y direction (d) γ1= 0.335, γ2= 0.15 in the x direction.

Von Mises Stress

Von Mises Stress

55

The process of analyzing the results of the simulations was the same as the first

order hierarchy. The effective elastic stiffness was calculated from the slope of the stress-

strain response of the simulations using Hooke’s law. It was then normalized by the

effective stiffness of the original regular honeycomb. The graphs below in fig. 18 and fig.

19 demonstrate the normalized effective stiffness of second order hierarchical

honeycomb structures for different values of γ1 and γ2.

Figure 18- Normalized effective stiffness of second order hierarchy for different values of γ2 in the Y direction, with a constant γ1 of 0.29.

0.8

1.3

1.8

2.3

2.8

3.3

3.8

0 0.05 0.1 0.15 0.2

E 2/E

0

γ2

Theoretical

FEM

γ1= 0.29

56

Figure 19- Normalized effective stiffness of second order hierarchy for different values of γ2 in the X direction, with a constant γ1 of 0.335.

In the X and Y-direction, the results show that by introducing a second order

hierarchy to regular honeycombs, the effective stiffness of these structures increases

compared to regular hexagonal honeycombs. The maximum stiffness evaluated both

theoretically and through numerical analysis happens where 𝛾1= 0.3, and 𝛾2= 0.135. This

increases the stiffness of regular honeycombs up to 3.5 times.

The theoretical and analytical difference in values of the effective stiffness for

second order hierarchical structures occurs due to the same reasons mentioned earlier for

first order hierarchies.

1.4

1.8

2.2

2.6

0 0.04 0.08 0.12 0.16

E 1/E

0

γ2

Theoretical

FEM

γ1= 0.335

57

The behavior of second order hierarchical structures can be summarized in a

contour map of the effective stiffness for all different values of γ1 and γ2 shown in fig. 20.

Again, the values are normalized with the stiffness of the regular hierarchical structures

with the same relative density of the beams. In the graph, the x-axis represents different

γ1 values ranging from 0 to 0.5. Meanwhile, there are two geometrical constraints for the

values for γ2:

𝛾2 ≤ 𝛾1 ,

0 ≤ 𝛾2 ≤ (0.5 - 𝛾1 )

The plot shows the different combinations of γ1 and γ2 that will produce the maximum

normalized stiffness for second order hierarchy. Moreover, the limitations for the diverse

arrangements of γ values are also shown.

58

Figure 20- Contour map of the effective stiffness of hierarchical honeycombs with second order hierarchy for all possible geometries.

59

8. Conclusion:

The mechanical behavior of two dimensional hierarchical regular hexagonal

honeycomb structures using analytical and numerical methods was investigated.

Hierarchical honeycombs are constructed by replacing every three edge vertex of a

regular hexagonal honeycomb with a smaller hexagon. This is how a first order

hierarchical honeycomb is created. Repeating this process builds a fractal appearing

second order hierarchical structure. The stiffness and strength of the structures were

controlled by the adjustment of the ratios for the various hierarchical models. A finite

element method was used through ABAQUS 6.10 (SIMULIA, Providence, RI), to

generate the two dimensional models of the different hierarchical structures. To analyze

the effective elastic modulus of each structure, the compressive stress-strain response of

the models was used. Our results showed that hierarchical honeycombs of first and

second order can be up to 2 and 3.5 times stiffer than regular hexagonal honeycombs with

the same relative density. This applies for displacement in both the X and Y direction.

60

CHAPTER 3:

DYNAMIC CRUSHING OF HIERARCHICAL HONEYCOMBS

61

1. Introduction:

One of the key applications of cellular structures is in their structural protection

that is due to their superior energy absorption and impact resistance [1-6]. The most

common applications to these characteristics are packaging of fragile components and

various protective products such as helmets and shielding [3]. In addition, the usage of

cellular structures as the core material for metal sandwich panels has proven to show

superior performance over the counterpart solid plates of equal mass under shock

loading. Core topology and relative density have a huge influence on the performance of

the sandwich panels. This is demonstrated as the core crushes at an early stage of

deformation and absorbs a large fraction of the kinetic energy that is conveyed to the

panels as a result of shock loading [4,5, 7-10].

In the quasi-static regime (low impact velocity), the crushing response of most

metal cellular structures shows a typical stress-strain curve that includes three regimes: an

elastic response followed by a plateau regime with almost contact stress and eventually a

densification regime of sharply rising stress. On the other hand, the response of the

metallic cellular structure is affected by complex localized phenomena that include

buckling and micro- inertial resistance [3]. Previous studies have explored these effects,

including the interaction between plastic waves and localized buckling under dynamic

loading [2,6]. In addition, Xue and Hutchinson proved that the micro- inertial resistance

of core webs of a square honeycomb metal core increases its resisting force remarkably at

early stages of dynamic crushing [9]. However at later stages of deformation, the

dynamic effects result in suppression of the buckling of the metal webs. This results in

the formation of buckling shapes with a wavelength much smaller that the core height.

62

These effects lead to a significant increase in the plastic energy dissipation of

honeycombs under dynamic crushing at high strain rates. In a corresponding study in

2007, Vaziri and Xue studied the role of strain rate on the crushing response of folded

plate and truss cores [11]. Their study proved that the initial elevation in the resisting

stress of the metal cores which are due to the micro- inertial resistance mainly depends on

the core relative density. Meanwhile, the strain rate is relatively insensitive to the core

topology.

The energy absorption of hierarchical cellular structures is yet to be investigated.

Thus, to further explore the effect of dynamic crushing on cellular structures, we have

studied the response of three dimensional regular hexagonal first order hierarchical

honeycombs under impact and in-plane dynamic crushing. Most of the previous work

performed on the mechanical behavior of cellular materials, considers an intact structural

organization for the cellular material. The effect of dynamic crushing on general

honeycombs at different kinetic energy levels has been also been explored. This work

compliments previous studies on the dynamic behavior of cellular structures with

uniform cell size and wall thickness, while in the same time providing new insight on the

dynamic behavior of hierarchical cellular structures with strain hardening.

63

2. Finite Element Method:

In addition to the study of the static behavior of honeycombs, the dynamic

crushing of hierarchical honeycombs was also investigated. This enables us to measure

the energy absorption of the hierarchical honeycombs. The influences of honeycomb cells

and impact velocity on the mode of localized deformation are explored. It is well

acknowledged that honeycomb cellular structures behave differently under dynamic

loading, mainly in their deformation modes and stress levels. In this part, the models that

were generated were three dimensional. The designs were made using the same software

(ABAQUS 6.10, SIMULIA). Six different models were made to account for six different

values of γ: γ=0, γ=0.1, γ=0.2, γ=0.3, γ=0.4, and γ=0.5, where γ=0 represents the regular

honeycomb structure. Fig. 21 below displays some examples that correspond to the

different values of γ.

(a)

64

(b)

(c)

65

(d)

Figure 21- Three dimensional first order hierarchical honeycomb models: (a) γ=0.1 (b) γ=0.3 (c) γ=0.4 (d) γ=0.5

The models were designed using the 3D deformable shell extrusion element available in

ABAQUS. Similar to the BEAM element described earlier, the shear and axial

deformation of the cells can be acquired. Moreover, the bending compliance of the cell

walls can be obtained. In this study, the energy absorption was measured, therefore the

plastic dissipation of the cell walls was the key history output to be considered.

The dimensions of the structures were similar to the first order hierarchical

structures described earlier, however since this is a 3D model, a depth of 0.1 was used.

The material used for the structure is aluminum. As it is well established, the mechanical

properties of aluminum are the following: An elastic modulus of Es=70 GPa, a yield

66

strength of σys=130 MPa, and a Poisson’s ratio of ν = 0.3. In this study, the density of the

cells had to be included as well which was calculated to be 2700 using equation (1). This

was applied in the material properties of the model. The overall relative density of the

structures for the dynamic crushing of hierarchical honeycombs was 6%. A homogeneous

shell with a thickness t that is adjusted with different values of γ was used to make sure

all the structures have the same relative densities.

In addition to the 3D deformable shell, a 3D analytical rigid shell was also

modeled. It was then tied to the top and bottom surfaces of the structure. Periodic

boundary conditions were applied to the models to eliminate the side boundary effects on

the simulation results. The benefit of using periodic boundary conditions is that it enables

the borders of the structure to maintain the shape during deformation that takes place due

to the crushing of the cells.

The boundary conditions applied to the structures played a key role in

determining the energy absorption of the cells. The top part of the structures was fixed

from moving in the X and Z directions. It was also fixed from rotating about any axis. In

other words, the top plate was only allowed to move in the Y direction. The bottom plate

was fixed from movement in any direction. It was also prevented from rotation about any

axis. The individual cells that make up the hierarchical honeycomb structure were only

fixed in the z direction. Hence, allowing the cells to deform in the X and Y directions.

The cells were also permitted to rotate about the Z axis. Next, a very low velocity was

used in the negative Y direction to the structure. This enables the cells to deform quasi-

statically.

67

3. Effect of Hardening on Plastic Behavior of Hierarchical Cellular Structures:

The effect of strain hardening in the dynamic crushing of hierarchical honeycomb

structures was studied. For plastic behavior, the bilinear constitute equation with

different hardening rates in the plastic regime was taken into consideration. The

hardening rate represents the tangent modulus in the plastic regime. Fig. 22 below

illustrates a stress-strain curve used in the investigation in order to understand the effect

of strain hardening on the elastic-plastic behavior of the hierarchical honeycomb

structures.

Figure 22- Schematic stress-strain curve for materials with different hardening rates.

The percentages of hardening considered in this work were 5% and 10%. To calculate the

values of the plastic strain to be entered in ABAQUS, the following equations were used:

68

%ℎ𝑎𝑟𝑑𝑒𝑛𝑖𝑛𝑔 =𝐸𝑝𝐸

Where E is the Elastic Modulus that is calculated using Hooke’s law:

𝐸 =𝜎𝑦𝜀𝑦

The values required by the software are σ2 and ε2. For σ2, we assume a very large value of

σ2= 1,000,000 GPA. A smaller value is not used because once the stress on the cells

exceeds the value entered in the material properties in ABAQUS, the software

automatically assumes that the material is linear elastic perfectly plastic again. To

calculate ε2, we used the following:

𝜀2 = 𝜀1 −𝜎2𝐸

Where:

𝜀1 =(𝜎2 − 𝜎𝑦 )

𝐸𝑝+ 𝜀𝑦

The values stated in the equations come from the graph shown in fig. 23 below.

69

Figure 23- The effect of hardening on a stress-strain curve.

The other properties remained the same as the hierarchical structures that were

linear elastic perfectly plastic. The energy absorption of hierarchical structures with strain

hardening was also measured through the plastic dissipation of the cells. Apart from the

energy absorption of the structures, the way the cells deform in the models with strain

hardening applied was also looked at.

4. Results and Discussion:

To evaluate the energy absorption of first order hierarchical honeycomb

structures, the plastic dissipation response of the models was analyzed. At first, we

compared the energy absorption of the models with different γ values.

70

A higher impact velocity was investigated initially with the different first order

hierarchical honeycomb structures to assess the dynamic mode response of the models.

At high crushing velocities, the plastic dissipation strongly depends on its relative

density, and is higher with higher relative density. This is due to dynamic effects and the

non linearity caused by cell walls contact. Since the models investigated in this work only

considered a relative density of 6%, the results showed that the plastic dissipation of the

hierarchical models with different values of γ is relatively equal.

The low crushing velocity of the rigid plate in the Y-direction will cause the

honeycomb structure to deform quasi-statically. This means that for a very low velocity,

as the one applied in this work, the lightly crushed cells form an arch at the top and lower

part of the model. Eventually, the deformed cells form an ‘X’ mode at high strain rates.

Fig. 24 below demonstrates a few examples of the crushed models in the Y direction.

(a)

71

(b)

(c)

(d)

72

(e)

Figure 24- Dynamic crushing of first order hierarchical honeycombs in the Y-direction with a velocity of v=0.1 (a) regular honeycomb γ=0 (b) γ=0.1 (c) γ=0.3 (d) γ=0.4 (e) γ=0.5.

As seen in the previous figures, the deformed cell walls form an arch at the top

and bottom. The localized bands are composed of heavily crushed cells that are

perpendicular to the crushing direction, at the middle position of the models. To compare

the energy absorption of the different γ1 values, a plot of the plastic dissipation versus the

strain was obtained. In these models, the cells were allowed to deform to up to 50%. A

plot of the plastic dissipation of the different models is demonstrated below in fig. 25

with no hardening.

73

Figure 25- Energy absorption of first order hierarchical models with different values of γ and 0% hardening.

Similar to the 2D models, the value of γ must lie in the range between 0 and 0.5. From

the results, we can observe that introducing hierarchy to regular honeycombs does not

necessarily increase the energy absorption of the structure. In this study, no trend is

noticed as the size of the smaller hexagon (the value of γ) increases. In addition, the

energy absorption of the hierarchical models with the variety of γ values is lower than the

regular hexagon. There is one exception however when γ=0.1, the plastic dissipation is at

least twice as much as the other structures.

0

1000000

2000000

3000000

4000000

5000000

6000000

7000000

0 0.1 0.2 0.3 0.4 0.5

Up

ε

γ=0-Hardening =0%

γ=0.1-Hardening =0%

γ=0.2-Hardening =0%

γ=0.3-Hardening =0%

γ=0.4-Hardening =0%

γ=0.5-Hardening =0%

74

For plastic behavior, the bilinear material properties with two different hardening

rates were considered. The effect of a strain hardening of 5% on the plastic dissipation of

the cells is presented in fig. 26 below.

Figure 26- Energy absorption of first order hierarchical models with different values of γ and 5% hardening.

From this graph, we can observe that with the use of linear elastic, plastic with linear

strain hardening of 5%, the energy absorption of the hierarchical models have relatively

higher values when compared with regular honeycombs. Similar to what the no

hardening results showed, the energy absorption of the structure with γ=0.1 is almost

twice as much as the other models.

0

5000000

10000000

15000000

20000000

25000000

0 0.1 0.2 0.3

Up

ε

γ=0-Hardening =5%

γ=0.1-Hardening =5%

γ=0.3-Hardening =5%

γ=0.2-Hardening =5%

75

This significance of using linear elastic, plastic with linear strain hardening is

shown when we compare the energy absorption of the same models, but with different

hardening percentages. For example, there is an extremely large difference between a

first order hierarchical honeycomb structure with γ=0.1 with no hardening, and the same

structure with 5% hardening. In other words, the introduction of hardening to the material

that makes up the cell walls of the hierarchical honeycomb structure, significantly

increases the energy absorbance of the structure. Fig. 27 below shows the substantial

increase in the plastic dissipation of some of the models with different γ values.

(a)

0

10000000

20000000

30000000

0 0.1 0.2 0.3 0.4

Up

ε

γ=0-Hardening =0%

γ=0-Hardening =5%

γ=0-Hardening =10%

76

(b)

(c)

0

5000000

10000000

15000000

20000000

25000000

0 0.1 0.2 0.3

Up

ε

γ=0.1-Hardening =0%

γ=0.1-Hardening =5%

0

5000000

10000000

15000000

20000000

0 0.1 0.2 0.3

Up

ε

γ=0.3-Hardening =0%

γ=0.3-Hardening =5%

77

(d)

Figure 27- Energy absorption of first order hierarchical structures with different percentages of strain hardening: (a) regular honeycomb γ=0 (b) γ=0.1 (c) γ=0.3 (d) γ=0.4

From the plots above we can see that the energy absorption of the structures significantly

rises with the addition of linear strain hardening to the material that makes up the cell

walls. This also applies when the percentage of strain hardening is increased as we notice

the structures can sufficiently absorb higher amounts of energy. For instance, in the first

order hierarchy model with γ=0.4 and a strain hardening of 0%, the plastic dissipation at

30% strain is 1.45 million J. Meanwhile, with a strain hardening of 5%, the plastic

dissipation of the structure increases to 40.4 million J. This means that using a material

with 5% hardening increases the energy absorption of the system by roughly 30 times in

some cases. When the percentage of hardening is raised to 10%, the plastic dissipation of

the same model rises to a value sufficiently greater than the amount of energy a 5%

hardened material of the same structure can absorb.

0

20000000

40000000

60000000

80000000

100000000

0 0.1 0.2 0.3

Up

ε

γ=0.4-Hardening =0%

γ=0.4-Hardening =5%

γ=0.4-Hardening =10%

78

The results also demonstrate that when we compare the energy absorption of the

structures with that of a regular hexagonal honeycomb, we observe that the difference

between the two individual models is extremely larger with hardening included. Fig. 28

below presents a plot of a few examples where we can compare a hierarchical model with

a regular honeycomb at 0% hardening, and 5% hardening.

(a)

0

5000000

10000000

15000000

20000000

0 0.1 0.2 0.3

Up

ε

γ=0-Hardening =0%

γ=0.1-Hardening =0%

γ=0-Hardening =5%

γ=0.1-Hardening =5%

79

(b)

(c)

Figure 28- Energy absorption of hierarchical models compared with a regular honeycomb with 0% hardening, and 5% hardening: (a) γ=0.1 with γ=0 (b) γ=0.4 with γ=0

0

10000000

20000000

30000000

40000000

50000000

0 0.1 0.2 0.3

Up

ε

γ=0-Hardening =0%

γ=0.4-Hardening =0%

γ=0-Hardening =5%

γ=0.4-Hardening =5%

0

2000000

4000000

6000000

8000000

10000000

0 0.1 0.2 0.3

Up

ε

γ=0-Hardening =0%

γ=0.3-Hardening =0%

γ=0-Hardening =5%

γ=0.3-Hardening =5%

80

Furthermore, when we compare the plastic dissipation of the structures with a strain

hardening of 10%, we notice an even higher difference.

81

5. Conclusion:

Finite element method was employed to study the in-plane dynamic crushing of

hierarchical cellular structures. The influences of honeycomb cells and impact velocity on

the mode of localized deformation are investigated. As it is well established, honeycomb

cellular structures behave differently under dynamic loading, mainly in their deformation

modes and stress levels. In this work, we explored three dimensional first order

hierarchical honeycombs under dynamic loading. For plastic behavior, the bilinear

material properties with two different hardening rates were considered in the dynamic

crushing of hierarchical honeycombs.

Our results demonstrate that there is not much difference in the energy absorption

of the hierarchical models when compared to the regular hexagonal honeycomb.

However, there was one exception when γ=0.1, the plastic dissipation is at least twice as

much as the other structures. On the other hand, adding a percentage of strain hardening

to the material that makes up the cell walls of the hierarchical honeycomb, significantly

increases the energy absorbance of the absorbance of the structure.

82

6. References

[1] Tan, P.J., Reid, S.R., Harrigan, J.J., Zou, Z., Li, S., Dynamic compressive strength

properties of aluminum foams. Part II—‘shock’ theory and comparison with

experimental data and numerical models. J. Mech. Phys. Solids., 53 (2005) 2206–2230

[2] Harders, H., Hupfer, K., Rosler, J., Influence of cell wall shape and density on the

mechanical behavior of 2D foam structures. Acta Mat., 53 (2005) 1335–1345

[3] Dharmasena, K.P., Queheillalt, D.T., Wadley, H.N.G., Chen, Y., Dudt, P., Knight,

D.,Wei, Z., Evans, A., Dynamic response of a multilayer prismatic structure to impulsive

loads incident from water. Int. J. Impact Eng., 36 (2009) 632–643

[4] Honig, A., Stronge, W.J., In-plane dynamic crushing of honeycombs. Part I: crush

band initiation and wave trapping. Mech. Sci., 44 (2002) 1665–1696

[5] Honig, A., Stronge, W.J., In-plane dynamic crushing of honeycombs. Part II:

application to impact. Mech. Sci., 44 (2002) 1697–1714

[6] Liu, Y., Zhang, X.C., The influence of cell micro-topology on the in-plane dynamic

crushing of honeycombs. Int. J. Impact Eng., 36 (2009) 98–109

[7] Mori, L.F., Queheillalt, D.T., Wadley, H.N.G., Espinosa, H.D., Deformation and

failure modes of I-core sandwich structures subjected to underwater impulsive loads.

Exp. Mech., 49 (2009) 257–275

[8] Xue, Z., Vaziri, A., Dynamic crushing of square honeycomb structures. J. Mech.

Mater. Struct., 2 (2007) 2025–2048

[9] Xue, Z., Hutchinson, J.W., A comparative study of impulse-resistant metallic

sandwich plates. Int. J. Impact Eng., 30 (2004) 1283–1305

83

[10] Xue, Z., Hutchinson, J.W., Crush dynamics of square honeycomb sandwich cores.

Int. J. Numer. Methods Eng., 65 (2006) 2221–2245

[11] Vaziri, A., Xue, Z., Hutchinson, J.W., Performance and failure of metal sandwich

plates subject to shock loading. J. Mech. Mater. Struct., 2 (2007) 1947–1963