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 1
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 2
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. 3
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. 4
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. 5
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 6
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 X 1 -X 2 plane. Responses to loads applied to the faces normal to X 3 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) γ 1 =0.1 (b) γ 1 =0.3 (c) γ 1 =0.5... 42 Figure 11- Y Direction displacement... 43 7
Figure 12- X-direction displacement... 44 Figure 13- Two dimensional second order hierarchical honeycomb models: (a) γ 1 =0.29 and γ 2 =0.1, (b) γ 1 =0.29 and γ 2 =0.15, (c) γ 1 =0.335 and γ 2 =0.1, (d) γ 1 =0.335 and γ 2 =0.15.... 46 Figure 14- Structural response of first order hierarchical honeycombs: (a) γ 1 =0.1 in the x direction (b) γ 1 =0.3 in the x direction (c) γ 1 =0.1 in the y direction (d) γ 1 =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) γ 1 = 0.29, γ 2 = 0.1 in the x direction (b) γ 1 = 0.29, γ 2 = 0.15 in the x direction (c) γ 1 = 0.335, γ 2 = 0.1 in the y direction (d) γ 1 = 0.335, γ 2 = 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 8
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 9
CHAPTER 1: INTRODUCTION 10
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 dime nsional honeycomb The most important feature of a cellular structure is its relative density ρ*/ρ s, where ρ* is the density of the cellular material, and ρ s 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 11
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. 12
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. 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 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 14
time. It is well known that the most used structural material is wood. In addition, manmade 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 Hie rarchical 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 15
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 16
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. 17
Figure 5- Unit cell of an evolved hierarchical second order honeycomb from a first order hierarchy. 18
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 inplane 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. 19
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- 20
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 21
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. 22
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 fractalappearing. 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. 23
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 24
[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 twodimensional 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 25
[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 inplane 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 26
[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 27
CHAPTER 2: METHODOLOGY 28
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 X 1 -X 2 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 X 3 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 X 1 -X 2 plane. Responses to loads applied to the faces normal to X 3 are referred to as out of plane.2. In plane deformation: 29
When a honeycomb is compressed in the X 1 -X 2 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 X 1 -X 2 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 120 o. 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]. 30
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 (E 1,E 2,G 12, and which is Poisson s ratio) and two values for the yield stress (σ y1 and σ y2 ). An undeformed honeycomb cell is shown below in fig. 7. Fig. 8 shows the bending caused by loads in the X 1 and X 2 directions [1]. 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. The equation of the relative density ρ*/ρ s can be obtained through simple geometry analysis: ρ ρ s = t l (h l + 2) 2 cos θ( h (1) l + sin θ) With θ=30 o in regular honeycombs and with h =l, the equation reduces to: ρ = 2 t ρ s 3 l (2) 32
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: γ 1 and γ 2. 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: γ 1 = b a, γ 2 = 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 1 st order hierarchy is 0 b a/2, and thus 0 γ 1 0.5. The regular honeycomb structure is attained by setting γ 1 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 γ 2 γ 1 if 33
γ 1 0.25 and 0 γ 2 (0.5 - γ 1 ) if 0.25 γ 1 0.5. From simple geometrical analysis, the equation to find the relative density of the first order hierarchical honeycomb is: ρ ρ s = t a For the second order hierarchy: ρ ρ s = t a (1 + sin θ) 3 sin θ cos θ (1 + 2 γ 1 ) (3) (1 + sin θ) 3 sin θ cos θ (1 + 2 γ 1 + 6 γ 2 ) (4) In regular hexagonal hierarchical cellular structures with θ = 30, the relative density relations are simplified below: Regular 1 st order hierarchy: ρ ρ s = 2 t 3 a (1 + 2 γ 1 ) (5) Regular 2 nd order hierarchy: ρ ρ s = 2 t 3 a (1 + 2 γ 1 + 6 γ 2 ) (6) 4. Regular Honeycomb Linear Elastic Deformation: When a load is applied in the X 1 or X 2 direction, the honeycomb deforms in a linear elastic way, causing the cell walls to bend [1]. While a stress that is parallel to the X 1 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 X 2 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 E s. To find the Moment M that tends to bend the cell walls we use the following equation: 34
M = Pl sin θ 2 (7) Where the load P is found from the equation below: P = σ 1 (h + l sin θ)b (8) From the standard beam theory, the deflection of the cell wall is: δ = Pl3 sin θ 12E s I (9) Where I represents the moment of inertia of a cell wall (beam).i = bt 3 / 12 for a wall of uniform thickness t. From this, a component δ sin θ is parallel to the X 1 axis which results in a strain of: ε 1 = δ sin θ l cos θ = σ 1 (h + l sin θ)bl2 sin 2 θ 12E s I cos θ (10) From Hooke s law, the Young s modulus parallel to the X1 direction is E 1 = σ 1 /ε 1, giving: E 1 E s = t l 3 cos θ ( h l + sin θ) sin 2 θ (11) In the X 2 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 = σ 2 lb sin θ giving: M = Wl cos θ 2 (12) The deflection in the wall in this case is: 35
δ = Wl3 cos θ 12E s I (13) Therefore, a component δ cos θ is parallel to the X 2 axis which results in a strain of: ε 2 = δ cos θ h + l sin θ = σ 2 bl4 cos 3 θ 12E s I(h + l sin θ) (14) From Hooke s law, the Young s modulus parallel to the X 2 direction is E 2 = σ 2 /ε 2, giving: E 2 = t 3 E s l ( h l + sin θ) cos 3 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 M 1 and M 2 at points 1 and 2 respectively. In addition, points 1 and 2 will have reaction forces N 1 and N 2 in the positive Y-direction. Using equations of equilibrium in the directions X and Y, we can express N 2 and M 2 as functions of N 1, M 1, and F. As a result, the bending energy U that is stored in the first order hierarchy can be stated as: U(F, M 1, N 1 ) = (M 2 2E s I)dx (16) 36
Where M represents the bending moment in the beam s cross section, E s 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 N 1 = 0, and U M 1 = 0 From these two relations, the equations for M 1 and N 1 can be obtained: N 1 = F 0.533 + 0.15 γ 1 (18) M 1 = Fa(0.283γ 1 0.017) (19) However, at point 3 the displacement is not zero: δ = U F (20) By substituting this for N 1 and M 1, we acquire the following: δ = 3Fa 3 72EIf(γ 1 ) (21) 37
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: E E s = t a 3 f(γ 1 ) (22) Where: f(γ 1 ) = 3 (0.75 3.525γ 1 + 3.6γ 12 + 2.9γ 13 ) (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 γ 1 is zero ( (E E s ) γ 1 = 0). Using this information in equations (5) and (22), we obtain a value of γ 1 = 0.32, which results in an effective stiffness of E E s = 2.97ρ 3. To find the effective stiffness of a regular honeycomb structure, we use the same equations as we did for first order hierarchy, and setting γ 1 = 0. By performing these calculations, a regular honeycomb structure will only have a stiffness of E 0 E s = 1.5ρ 3. 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 γ 1 (first order), and the other for the smallest hexagon γ 2 38
(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 N 1, N 2, and N 3 act on points 1, 2, and 3 respectively. Similarly, the moments M 1, M 2, and M 3 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, N 3 and M 3 can be written as a function of N 1, M 1, N 2, M 2, 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, M 1,N 1,M 2, N 2 ) = (M 2 2E s I)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: And U N 1 = 0, U N 2 = 0 (displacement) U M 1 = 0, U M 2 = 0 (rotation) From these conditions, it is possible to solve for the reaction forces N 1 and N 2, as well as the moments M 1 and M 2. Similar to what was applied to first order hierarchy; we can obtain the effective stiffness through the following: Where the equation for f(γ 1 γ 2 ) is the following: f γ 1,γ 2 = 29.62γ 1 4 54.26γ 13 γ 2 + 31.75γ 12 γ 2 2 4.73γ 1 γ 2 3 γ 2 4 / E E s = t a 3 f(γ 1 γ 2 ) (25) 39
γ 2 7 (2.20) γ 2 6 18.13γ 1 + 1.88 + γ 2 5 159.95γ 1 2 29.38γ 1 + 3.90 + γ 24 ( 270.14γ 1 3 + 9.70γ 1 2 + 20.50γ 1 0.43) γ 23 (195.50γ 1 4 334.12γ 1 3 + 108.06γ 1 2 + 2.04γ 1 ) + γ 22 (862.56γ 1 5 662.32γ 1 4 + 123.22γ 1 3 + 13.74γ 12 ) γ 2 (609.01γ 1 6 310.43γ 1 5 12.80γ 1 4 + 23.46γ 1 3) + (49.64γ 1 7 + 61.73γ 1 6 60.43γ 1 5 + 12.80γ 14 )}. 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 γ 2, the change in stiffness and the change in γ 2 must also be taken into account ( (E E s ) γ 2 = 0). Using this condition in equations (6) and (25), the following values of γ 1 and γ 2 are acquired: γ 1 = 0.32, γ 2 = 0.135 Plugging these values in the second order stiffness equation (4) we achieve a stiffness of: E E s = 5.26ρ 3 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. 40
6. Finite Ele ment 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 γ 1 (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) 41
(c) Figure 10- Two dimensional first order hierarchical honeycomb models: (a) γ 1 =0.1 (b) γ 1 =0.3 (c) γ 1 =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 E s =70 GPa, a yield strength of σ ys =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 42
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 displace ment 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. 43
Figure 12- X-direction displacement On the other hand, the models for the second order hierarchical structures are more complex, since γ 2 has to be taken into account as well as γ 1. As stated earlier, γ 2 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 γ 1 and γ 2. 44
(a) (b) 45
(c) (d) Figure 13- Two dimensional second order hierarchical honeycomb models: (a) γ 1 =0.29 and γ 2 =0.1, (b) γ 1 =0.29 and γ 2 =0.15, (c) γ 1 =0.335 and γ 2 =0.1, (d) γ 1 =0.335 and γ 2 =0.15. To find the thickness of the beams in the above models, equation (6) was used with the different values of γ 1 and γ 2 stated above. After adjusting the thickness of the 46
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 stressstrain 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. Von Mises Stress (a) 47
Von Mises Stress (b) Von Mises Stress (c) 48
Von Mises Stress (d) Figure 14- Structural response of first order hierarchical honeycombs: (a) γ 1 =0.1 in the x direction (b) γ 1 =0.3 in the x direction (c) γ 1 =0.1 in the y direction (d) γ 1 =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 γ 1 in the X and Y direction. 49
2 1.5 Theoretical FEM E 2 /E 0 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 γ 1 Figure 15- Normalized effective stiffness of first order hierarchy for different values of γ1 in the Y direction 50
2.5 2 Theoretical FEM 1.5 E 1 /E 0 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 γ 1 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: E 0 = 1.5ρ 3 (26) E s Solving for E 0 : E 0 = 1.5E s ρ 3 (27) Where E s 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 51
for the normalized effective stiffness of a first order hierarchical honeycomb is obtained as a function of γ 1., 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 γ 1 rises up to γ 1 =0.3. The maximum increase in stiffness for both the theoretical and numerical approaches occurs at γ 1 =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 γ 1 =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. 52
Von Mises Stress (a) Von Mises Stress (b) 53
Von Mises Stress (c) Von Mises Stress (d) Figure 17- Structural response of second order hierarchical honeycombs: (a) γ 1 = 0.29, γ 2 = 0.1 in the x direction (b) γ 1 = 0.29, γ 2 = 0.15 in the x direction (c) γ 1 = 0.335, γ 2 = 0.1 in the y direction (d) γ 1 = 0.335, γ 2 = 0.15 in the x direction. 54
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 stressstrain 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 γ 1 and γ 2. 3.8 3.3 Theoretical FEM 2.8 E 2 /E 0 2.3 1.8 1.3 γ 1 = 0.29 0.8 0 0.05 0.1 0.15 0.2 γ 2 Figure 18- Normalized effective stiffness of second orde r hie rarchy for different values of γ2 in the Y direction, with a constant γ1 of 0.29. 55
2.6 Theoretical FEM 2.2 E 1 /E 0 1.8 γ 1 = 0.335 1.4 0 0.04 0.08 0.12 0.16 Figure 19- Normalized effective stiffness of second orde r hie rarchy 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 γ 1 = 0.3, and γ 2 = 0.135. This γ 2 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. 56
The behavior of second order hierarchical structures can be summarized in a contour map of the effective stiffness for all different values of γ 1 and γ 2 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 γ 1 values ranging from 0 to 0.5. Meanwhile, there are two geometrical constraints for the values for γ 2 : γ 2 γ 1, 0 γ 2 (0.5 - γ 1 ) The plot shows the different combinations of γ 1 and γ 2 that will produce the maximum normalized stiffness for second order hierarchy. Moreover, the limitations for the diverse arrangements of γ values are also shown. 57
Figure 20- Contour map of the effective stiffness of hierarchical honeycombs with second order hierarchy for all possible geometries. 58