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Effective mechanical properties of lattice materials Chopra, Prateek 2011

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Effective Mechanical Properties of Lattice Materials  by Prateek Chopra B.E., University of Delhi, India, 2009  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Mechanical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) November 2011 c Prateek Chopra, 2011  Abstract Lattice materials possess a spatially repeating porous microstructure or unit cell. Their usefulness lies in their multi-functionality in terms of providing high specific stiffness, thermal conductivity, energy absorption and vibration control by attenuating forcing frequencies falling within the band gap region. Analytical expressions have been proposed in the past to predict cell geometry dependent effective material properties by considering a lattice as a network of beams in the high porosity limit. Applying these analytical techniques to complex cell geometries is cumbersome. This precludes the use of analytical methods in conducting a comparative study involving complex lattice topologies. A numerical method based on the method of long wavelengths and Bloch theory is developed here and applied to a chosen set of lattice geometries in order to compare effective material properties of infinite lattices. The proposed method requires implementation of Floquet-bloch transformation in conjunction with a Finite Element (FE) scheme. Elastic boundary layers emerge from surfaces and interfaces in a finite lattice, or an infinite lattice with defects such as cracks. Boundary layers can degrade effective material properties. A semi-analytical formulation is developed and applied to a chosen set of topologies and the topologies with deep boundary layers are identified. The methods developed in this dissertation facilitate rapid design calculation and selection of appropriate core topologies in multifunctional design of sandwich structures employing a lattice core.  ii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xi  Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xii  1  2  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  Why Lattice Materials? . . . . . . . . . . . . . . . . . . . . . . .  1  1.2  Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  6  1.3  Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7  1.4  Scope of the Current Work . . . . . . . . . . . . . . . . . . . . .  11  Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . .  12  2.1  Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  12  2.2  Lattice Materials and Structural Mechanics . . . . . . . . . . . .  13  2.3  Effective Material Properties . . . . . . . . . . . . . . . . . . . .  16  2.4  Finite Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21  2.5  Research Objectives and Outline . . . . . . . . . . . . . . . . . .  25  iii  3  4  Effective Material Properties . . . . . . . . . . . . . . . . . . . . . .  27  3.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  27  3.2  Floquet-bloch Theorem . . . . . . . . . . . . . . . . . . . . . . .  28  3.3  Analysis of Free Wave Motion . . . . . . . . . . . . . . . . . . .  30  3.4  Effective Material Properties . . . . . . . . . . . . . . . . . . . .  35  3.5  Application to a 2-D Orthotropic Lattice . . . . . . . . . . . . . .  37  3.6  Application to an Isotropic Lattice . . . . . . . . . . . . . . . . .  40  3.7  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  41  3.8  Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  42  3.9  Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  47  3.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  56  Elastic Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . .  61  4.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  61  4.2  Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  63  4.3  Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . .  67  4.4  Finite Element Simulations . . . . . . . . . . . . . . . . . . . . .  69  4.4.1  Uni-axial Tension Test . . . . . . . . . . . . . . . . . . .  70  4.5  Configurations with Deep Elastic Boundary Layer . . . . . . . . .  71  4.6  Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  73  4.6.1  Kagome Configurations 1 and 2 . . . . . . . . . . . . . .  75  4.6.2  Diamond Configuration 1 . . . . . . . . . . . . . . . . .  77  4.6.3  Hybrid-A Configurations 1 and 2 . . . . . . . . . . . . .  79  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  84  4.7 5  Conclusions  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  86  5.1  Summary of Present Work and Major Findings . . . . . . . . . .  86  5.2  Suggestions for Future Work . . . . . . . . . . . . . . . . . . . .  87  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  89  iv  List of Tables Table 3.1  Effective properties of Star lattice topology deduced using Euler beam theory . . . . . . . . . . . . . . . . . . . . . . . . . . .  47  Table 3.2  Parent properties of low CTE lattice . . . . . . . . . . . . . . .  47  Table 3.3  Reciprocal and direct basis vectors in Cartesian co-ordinates. .  48  Table 3.4  Bandgap classification of lattice materials for a relative density of 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  52  Table 3.5  Effective properties of different lattice topologies. . . . . . . .  57  Table 4.1  Boundary conditions for pure shear test. . . . . . . . . . . . .  71  Table 4.2  Topologies exhibiting deep boundary layers. . . . . . . . . . .  73  Table 4.3  Comparison of the lattice topologies on the basis of deep elastic boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . .  v  84  List of Figures Figure 1.1  Cellular material permits a taller tree . . . . . . . . . . . . . .  2  Figure 1.2  Material property charts of engineering materials . . . . . . .  3  Figure 1.3  2-D and 3-D lattice materials . . . . . . . . . . . . . . . . . .  5  Figure 1.4  Photographs showing Titanium matrix composite square lattice core sandwich structures at different relative densities . .  5  Figure 1.5  Fabrication process for a tetrahedral lattice material . . . . . .  6  Figure 1.6  A typical stress versus strain curve for a bending dominated lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 1.7  Photograph of a Hex-chiral lattice possessing negative Poisson’s ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 1.8  9  A lattice possessing effective zero coefficient of thermal expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 1.9  8  9  Cylindrical object is cloaked from the acoustic field using a cloaking shell made of lattice material . . . . . . . . . . . . .  10  Figure 2.1  Picture showing a mechanism and a structure . . . . . . . . .  14  Figure 2.2  Picture classifying deformation domination of topologies . . .  17  Figure 2.3  Lattice topologies where cell edges or struts intersect the adjacent unit cell envelopes . . . . . . . . . . . . . . . . . . . . .  18  Figure 2.4  Out of plane bending of a regular hexagonal honeycomb . . .  20  Figure 2.5  Boundary conditions for the finite element simulation conducted by Kueh et al. . . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 2.6  22  Elastic boundary layer emanating from the edge of a Kagome lattice and decaying into the interior . . . . . . . . . . . . . .  vi  23  Figure 2.7  Finite element simulation of a notched Hexagonal, Triangular and Kagome lattice under a uni-axial load . . . . . . . . . . .  Figure 3.1  24  Picture conceptualizing Floquet-bloch theorem using a hypothetical lattice made by tessellating the unit cell along the basis vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  29  Figure 3.3  A reciprocal lattice showing the Brillouin zone . . . . . . . .  34  Figure 3.4  A flowchart showing the wave-based semi-analytical method .  41  Figure 3.5  Topologies of lattice materials with their unit cells and Brillouin zones . . . . . . . . . . . . . . . . . . . . . . . . . . .  43  Figure 3.6  Dispersion curve for Star lattice for a relative density of 0.08 .  50  Figure 3.7  Comparison of effective properties of Star lattice with analytical calculations in Christensen’s paper . . . . . . . . . . . . .  51  Figure 3.8  Dispersion curves of the studied lattice materials . . . . . . .  53  Figure 3.9  Polar plot of effective Youngs modulus non dimensionalised with respect to the Youngs modulus of the parent material (Es ) for the analysed lattice topologies . . . . . . . . . . . . . . .  58  Figure 3.10 Polar plot of effective shear modulus non dimensionalised with respect to the Youngs modulus of the parent material (Es ) for the analysed lattice topologies . . . . . . . . . . . . . . . . .  59  Figure 3.10 Enlarged view of the effective shear modulus polar plot . . . .  60  Figure 4.1  A hypothetical unit cell without corner degrees of freedom is tessellated along the basis vectors to constitute a finite lattice .  Figure 4.2  A lattice presenting the boundary conditions used to simulate Uni-axial tension test and pure shear test . . . . . . . . . . . .  Figure 4.3  64 70  Deep elastic boundary layer configurations comprising of Kagome configurations 1-2, Diamond configuration 1 and Hybrid-A configuration 1 . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 4.4  Deep elastic boundary layer configurations comprising of HybridA configuration 2 . . . . . . . . . . . . . . . . . . . . . . . .  Figure 4.5  72 73  Flowchart showing the procedure to detect elastic boundary layer 74  vii  Figure 4.6  Eigenvectors corresponding to elastic boundary layer modes for Kagome configurations 1 and 2 . . . . . . . . . . . . . . .  Figure 4.7  FE simulation showing deformations resulting from Uni-axial test and pure shear test for Kagome configurations 1 and 2 . .  Figure 4.8  75 76  Modulus of eigenvalues that correspond to elastic boundary layer modes are plotted as a function of relative density for Kagome configurations 1 and 2 . . . . . . . . . . . . . . . .  Figure 4.9  78  Modulus of eigenvalues that correspond to elastic boundary layer modes are plotted as a function of relative density for Diamond configuration 1 . . . . . . . . . . . . . . . . . . . .  79  Figure 4.10 FE simulation showing deformation resulting from Uni-axial test and pure shear test for Diamond configuration 1 . . . . .  80  Figure 4.11 Modulus of eigenvalues that correspond to elastic boundary layer modes are plotted as a function of relative density for Hybrid-A configuration 1 and 2 . . . . . . . . . . . . . . . .  81  Figure 4.12 Modulus of eigenvalues that correspond to elastic boundary layer modes prone to veering are plotted as a function of relative density for Hybrid-A configuration 1 and 2 . . . . . . . .  82  Figure 4.13 FE simulation showing deformations resulting from Uni-axial test and pure shear test for Hybrid-A configuration 1 and 2 . .  viii  83  Glossary Symbols Es ∗ Exx ∗ Eyy G∗xy ∗ , ν∗ νxy yx E∗ G∗ K∗ Ct Cl k1 , k2 ρ¯ L ρ∗ Ω nj v k ε σ r K M D f q ci jkl  Young’s modulus of strut material Effective Young’s modulus in x direction Effective Young’s modulus in y direction Effective shear modulus in x − y direction Effective Poisson’s ratios Isotropic effective Young’s modulus Isotropic effective shear modulus Isotropic in-plane 2-D bulk modulus Magnitude of transverse phase velocity Magnitude of longitudinal phase velocity Components of wavevectors along basis vectors Relative density Strut length Effective density Propagating frequency non-dimensionalized with respect to first pinned-pinned frequency of the strut Unit wave normal Phase velocity Wavevector Strain tensor Stress tensor Position vector Stiffness matrix Mass matrix Dynamic stiffness matrix External force Generalised displacement vector Fourth order constitutive tensor  ix  Technical Terms Veering  Knock down Toughness Stretch dominated  Bending dominated  Tessellated Unit cell Basis vectors  A feature of eigenvalue problems of weakly coupled systems where two eigenvalue branches approach each other as a parameter is varied. The branches do not cross, but veer away, exchanging the eigenvectors in the process. Degradation in stiffness noted due to the presence of deep elastic boundary layer The energy absorbing ability of a material A lattice topology that stores potential energy due to deformation predominantly in tensile deformation of the struts A lattice topology that stores potential energy due to deformation predominantly in bending deformation of the struts To form into a lattice by assembling unit cells along the basis vector directions A repeating unit of a periodic lattice The vectors prescribing the direction along which unit cells should be repeated to constitute a lattice  Abbreviations EMP FE DNA  Effective Material Properties Finite Element Discrete Network Analysis  x  Acknowledgments I would like to thank and acknowledge my supervisor, Dr. A. Srikantha Phani for his support and helpful discussions throughout the project. His influence has helped me shape my thinking and observe the world with a new insightful perspective. I thank my colleagues for healthy discussions and for creating a friendly research atmosphere in the Dynamic and Applied mechanics Laboratory (DAL). This dissertation is dedicated to my loving parents, Rakesh Chopra and Manju Chopra, for being there with me at every step of my life. Financial support from Natural Sciences and Engineering Research Council of Canada (NSERC) is thankfully acknowledged.  xi  Dedication  ❚⑨♦ ▼➉② P⑨❛❼r⑨❡➇♥❼t➁➎  xii  Chapter 1  Introduction 1.1  Why Lattice Materials?  Since the prehistoric times, humans have been advancing materials knowledge to obtain superior properties for structural applications. In ancient times, they would primarily use wood and stones to build houses, bridges and engineering structures, whereas now, they use concrete and metals such as steel. Clearly, the latter are superior from a structural standpoint; however, their use does not permit simultaneous optimization of strength and stiffness for a minimum weight. In nature; porous cellular materials are commonly found as shown in Figure 1.1 for a Balsa tree. The sparse distribution of materials using a porous architecture—regular (or random) arrangement of cells possessing varying sizes and shapes—ensures optimal use of material to withstand imposed stresses. Nature tailors cellular materials for a specific loading condition by removing material from regions susceptible to low stresses. For instance, a tall tree made of a solid material will easily buckle under its own weight. To avoid this and to trap maximum sunlight through well distributed leaves, nature uses cellular materials in its construction. A tree made of a cellular material can grow roughly twice the height of a tree which is made of a solid material using the same quantity of material. Similarly in other contexts, such as human skulls and bones, nature tailors the micro-structure of cellular materials to suit specific functional needs in the most optimized manner with respect to minimum weight and energy. 1  (b)  2h h  (a)  (c)  Figure 1.1: Cellular material permits a taller tree [1]. A tree made of cellular material such as Balsa wood (b) can grow roughly twice the height of a tree (c) which is made of solid material. Both the trees shown are using the same quantity of material.  2  (a) 1000  technical ceramics  Ideal zone  composites  Young’s modulus, E (GPa)  10–1 10–2  Cu alloys  metals lead alloys  concrete PEEK  PS wood Two grai grain  zinc alloys  non-technical ceramics  epoxies  PC  PP  leather  PET PE  aluminium lattice  1/3  E ρ  PTFE  polymers  1/2  E ρ E ρ  CFRP foam  natural materials  aluminium foam  foams  10–3  silicone elastomers  EVA  cork  polyurethane  isoprene  flexible polymer foams  guide lines for minimum mass design  neoprene  butyl rubber  elastomers  10–4  MFA, 2010  (b) 10 000  ceramics  Ideal zone  composites  SiC Al alloys  Si 3N 4 Al 2O 3  Ti alloys steels  metals Ni alloys tungsten alloys  CFRP  1000 polymers and elastomers  Mg alloys GFRP  PC PMMA PMM MMAA woods, woods s,, ll  100  PA A  tungsten carbide  PEEK PE P  copper alloys PET PEE T  CFRP lattice  10  PP P PE P woods,  aluminium lattice  T  strength,σf (MPa)  WC W alloys  polyester p polyes lyes yester esster  PMM PM MA MA PMMA PA  1  steels Ni alloys Ti alloys  glass Mg alloys loys oys yss FRPP GFRP  bamboo wood // grain in  CFRP lattice  Al 2O 3  Al alloys CFRP  100 10  SiC Si 3N 4 B 4C  zinc alloys lead alloys  CFRP foam  1  concrete  aluminium foam  natural materials  foams  0.1  cork  σf ρ  flexible polymer foams  0.01  10  100  silicone butyl rubber elastomers  guide lines for minimum mass design  σf 2/3 1/2 ρ σf ρ  1000 density, ρ (kg m 3)  MFA, 2010  10 000  Figure 1.2: Material property charts of (a) E versus ρ and (b) σ f versus ρ for engineering materials. Lattice materials expand the material property space especially at low densities [1]. 3  With the advancements in sophisticated manufacturing technologies, it is now possible to mimic nature and design micro-architectured materials. Metal foams [2] with random cell arrangement, cellular solids [1, 3] and lattice materials [4] with periodic cell arrangement are being developed for weight sensitive, high performance, multi-functional applications in aerospace and automotive industry. In order to understand the necessity for such materials, a comparison of existing materials is shown in the material property chart in Figure 1.2. In these ‘Ashby charts’ an “ideal” material with high stiffness/strength and low density will naturally be in the top-left corner and a “poor” material will be in the bottom-right corner. By comparing the Young’s modulus and strength of different materials it can be concluded that lattice materials and foams possess superior stiffness to density ratio and strength to density ratio. This is not possible with conventional metals and alloys which have high stiffness and strength but at the cost of high density. In weight sensitive structural applications lattice materials have a clear advantage. Lattice Materials can be two dimensional (2-D) or three dimensional (3-D) as shown in Figure 1.3. For a 2-D Lattice material, the unit cell is repeated in a 2-D plane and is infinitely extruded in the perpendicular plane. For a 3-D Lattice material, the unit cell is tessellated in three directions. Relative density (ρ) is defined as the ratio of the density of lattice material (mass per unit volume) to the density of parent metal. It is frequently used to compare the specific stiffness of different unit cell geometries. Roughly, speaking, relative density is a measure of porosity; low relative densities imply high porosity. Based on the scaling of elastic modulus with relative density, lattice materials can be classified into two categories: stretching dominated (modulus scales linearly with relative density) and bending dominated (modulus does not scale linearly with relative density). Nodal-connectivity (number of bars meeting at a joint) was found to have significant influence through Maxwell’s equations for simply stiff pin-jointed truss structures [5–8]. Stretching dominated lattices are preferred as they are structurally efficient. Foams with random microstructure are bending dominated and they are usually isotropic. Lattice materials on the other hand can be isotropic or anisotropic depending on the symmetries of the repeating unit cell. In addition to metal truss lattices shown in Figure 1.3, composite lattices are also being developed (see Figure 1.4 [9]) which leads to multi-scale hierarchical micro-architectured materials. There are 4  (a) Examples of 2-D lattice materials  (b) 3-D Octet lattice material made from a casting aluminium alloy, LM25  Figure 1.3: Lattice Materials [10, 11].  Figure 1.4: Photographs showing Titanium matrix composite square lattice ¯ [9]. core sandwich structures at different relative densities (ρ) two length scales in Figure 1.4. One at the level of composite strut and the second at the level of unit cell size. Opportunities exist to tailor the material properties via changes at both these two length scales.  5  elongated hexagonal perforated sheet perforation punch roll steel  55°  punch 70°  60°  150°  tetrahedral core  55° die waste  single layer core  mul ti l ayer core  Figure 1.5: Fabrication process for a tetrahedral lattice material [12].  1.2  Fabrication  Several manufacturing techniques exist depending on the length scale of the lattice. For structural sandwich construction, the fabrication process fundamentally involves a template/mould/planar sheet manufacture (see Figure 1.5). Choice of a suitable method for each of these processes depends upon the strut aspect ratio (length to radius) and strut diameter. For instance, a template/mould or planar sheet manufacture can be accomplished by laser micro machining, rapid prototyping, injection moulding, electro deposition or micro contact printing. For a beam aspect ratio less than 5, cost effective injection moulding is suitable. An absolute beam diameter of the order of 50µm would require electro deposition for planar 6  sheet manufacture. The lattice is then manufactured from this template/mould/planar sheet by brazing, punching or investment casting. Figure 1.5 shows a typical process, where a template is manufactured by perforation punching of steel sheets, which are later punched into a tetrahedral core. This tetrahedral core is then brazed with two steel sheets to form a tetrahedral sandwich lattice material.  1.3  Applications  Lattice materials have led to a shift in material perspective from a structural standpoint to a multi-functional standpoint. For instance, apart from providing superior structural properties per unit weight, these micro-architectured materials can offer multifunctional properties besides high specific stiffness and strength. • Utilization of space for sensor integration: Figure 1.7b shows strain gauges  placed inside the hex-chiral lattice material. This allows for integrated structural health monitoring from real time data. The opening space provided by a lattice can also be utilized to fill with sound absorbing materials such as low density polymeric foams.  • Negative Poisson’s ratio: The effective material properties not only depend on the relative density but also on the geometry of the unit cell. For example,  the hex-chiral lattice shown in Fig 1.7a possesses negative Poissons ratio and therefore expands in transverse direction, when pulled along a longitudinal direction. • Effective zero coefficient of thermal expansion: Figure 1.8 shows a bi-metallic material which has an effective zero coefficient of thermal expansion. As this material is heated homogeneously, the relative distance between the nodes remains same. This is achieved by suitably contrasting coefficient of thermal expansion of inner member materials (shown in dark color) with outer member materials. • Acoustic cloaking: A multi-layered material is shown to cloak a cylinder  from acoustic field in Figure 1.9. When insulated with this cloaking material, the cylinder no more alters the acoustic pressure field and hence seems invisible. 7  Figure 1.6: A typical stress versus strain curve for a bending dominated lattice [1]. • High energy absorption capability: Owing to their high area under the stress-  strain curve shown in Figure 1.6, lattice materials can offer high energy absorption capability. A bending dominated lattice is preferred for high energy absorption whereas a stretch dominated lattice is preferred for high stiffness per unit weight.  The above multi functional behaviours are not within the reach of traditional materials, which motivates the development of lattice materials.  8  (a) Hex-chiral Lattice  (b) Sensors are glued with epoxy on the inside wall of a hex-chiral lattice. Sensors can give signal output proportional to mechanical loading on the cell wall.  Figure 1.7: Photograph of a Hex-chiral lattice possessing negative Poisson’s ratio [13].  to e2  e1 (b)  (a)  Figure 1.8: A Lattice possessing effective zero coefficient of thermal expansion. A section of this lattice topology (a) made by tessellating unit cell (b) along the basis vectors e1 and e2 . Note that in (b), thick lines represent a high coefficient of thermal expansion (CTE) material such as Aluminium and thin lines denote low CTE material such as Titanium. The relative distance between the nodes does not change as the material is heated [14]. 9  (a)  (b) Figure 1.9: Cylindrical object is cloaked from the acoustic field using a cloaking shell made of lattice material. Schematic view of a cloaking shell (a). Acoustic pressure map with cloaking shell of 50 layers (left) and 200 layers (right) shown in (b) [15].  10  1.4  Scope of the Current Work  In structural engineering design calculations it is essential to know material moduli such as Young’s modulus and shear modulus. This dissertation studies microarchitecture dependent material properties of different lattice materials. The scope of this study is restricted to static loading and linear material response regime. Specifically, an elastic wave based technique is developed which can predict the material moduli of any planar (or 3-D) lattice topology and its dependence on relative density and direction of loading. In Chapter 2, the literature concerned with structural properties of lattice materials is surveyed. This will be followed by elastic wave technique applied to infinite lattice materials in Chapter 3. The influence of an edge/interface is studied in Chapter 4 using a developed semi-analytical formulation, followed by conclusions and suggestions for future work in Chapter 5.  11  Chapter 2  Literature Review 2.1  Outline  As discussed in Chapter 1, Lattice materials are periodic materials made by tessellating a unit cell along certain basis directions or basis vectors. In a lattice material, these repeating unit cells are typically of the order of a centimetre. The unit cells can be 2-D or 3-D having two or three basis vectors respectively (see Figure 1.3). For example a bee hive or a regular hexagonal honeycomb is a 2-D lattice material formed by tessellating regular hexagons in the two basis directions. Lattice materials have low relative density and are attractive as multifunctional materials, which can simultaneously offer high stiffness per unit weight. This chapter is concerned with the literature outlining various approaches to deduce the Effective Material Properties (EMP) of Lattice materials. Here, the effective material properties represent ci jkl in Equation (2.1). σi j = ci jkl εkl  (2.1)  Equation (2.1) represents the Hooke’s law for an equivalent continuum that corresponds to the lattice medium under consideration. The organization of this chapter is as follows. The literature concerned with the rigidity criteria of lattice materials using pin jointed analysis methods is discussed initially. The rigidity criteria influence the effective material properties of  12  a lattice material. Then, the literature concerning the effective material properties calculation is discussed. Next, the literature related to the phenomena arising due to finiteness of a lattice and its effect on the effective material properties is taken up. Finally, this chapter concludes by bringing out open problems in this area, which would further motivate the work of this thesis.  2.2  Lattice Materials and Structural Mechanics  The structural performance of a lattice material is strongly dependent upon its topology. Based on the topology, the deformation in a lattice material can be classified as stretching dominated or bending dominated. Cell walls deform predominantly by tension or compression for the former and by bending for the latter. The deformation domination affects the effective material properties of a lattice material. Gibson and Ashby [3] demonstrated that the stiffness and strength scale in linear proportion to relative density for stretching dominated lattice topologies. This is given by [3]: Stiffness ∝ ρ¯ , Strength ∝ ρ¯ (Stretch dominated)  (2.2)  Strength and stiffness scale non-linearly for bending dominated lattice topologies. This is given by [3]: Stiffness ∝ ρ¯ 2 , Strength ∝ ρ¯ 1.5 (Bending dominated)  (2.3)  Thus, for a ρ¯ = 0.1, a stretching dominated topology would be 100 times stiffer and 33 times stronger than a bending dominated one. Therefore, the stretching dominated lattice offers higher stiffness per unit weight compared to the bending dominate one. The bending dominated lattice results in a greater deflection per unit load than the stretching dominated lattice, since the bending stiffness of a beam is much lower than its stretching stiffness. Thus, bending dominated lattice is better suited for higher energy absorption. Therefore, it is of much importance to identify the predominant deformation domination in a lattice, which depends on the lattice topology. To identify the dominant deformation domination mechanism, a lattice material 13  Bending dominated for rigid joints  Stretch dominated for rigid joints  Figure 2.1: Picture showing (a) a mechanism and (b) a structure. The joints are pin-jointed. Note that (a) is kinematically indeterminate structure with one mechanism and (b) is a statically and kinematically determinate or simply stiff structure [16]. can be envisioned as a collection of pin jointed struts. At low relative densities, the struts will offer negligible bending moment due to small area moment of inertia. The resulting pin-jointed structure will be stretch dominated if it predominantly constitutes a structure. This can be reasoned by the following argument. Consider a pin jointed arrangement of struts as shown in Figure 2.1. Figure 2.1a is a mechanism and Figure 2.1b is a structure. If a vertical force F is applied, the mechanism will collapse whereas the members of the structure will behave as bars and will deform by tensile deformations thus the structure would be purely stretch dominated. However, if the joints of Figure 2.1a are made rigid, then the 14  nodes would be twisted and the deformation would be resisted by the bending moments originating at the joints. Thus, the struts would bend and Figure 2.1a would be bending dominated. Similarly, if the joints of the structure are made rigid, then for low relative densities, strut thickness t will be small since ρ¯ ∝ t, hence the bending moment arising from the joint would be negligible compared to the axial forces. Hence, the structure at low relative densities would remain stretch dominated even when the joints are rigid. Thus, for a lattice material to be stretch dominated, it should predominantly possess structures in comparison to mechanism when envisioned as pin jointed assembly of struts. Thus, pin jointed techniques could be employed to check for the presence of mechanisms in a lattice. Thus and therefore the predominant deformation domination of a lattice topology could be determined. The algebra pertaining to pin jointed techniques is much simpler and easy to implement compared to the rigid jointed techniques. Therefore, it is useful to study the literature related to pin jointed methods and is examined next. An arrangement of pin jointed struts becomes a structure when it is statically and kinematically determinate (see Figure 2.1). Note that a structure is statically determinate if and only if the equilibrium equations alone can determine the internal member deflections. Kinematic determinacy of a pin jointed frame implies that the position of any joint relative to another joint could be determined purely in terms of individual strut lengths. For a finite pin jointed lattice in space, the necessary conditions for kinematic and static determinacy of a structure, were first given in the form of Maxwell rule by James Clerk Maxwell. This was later extended by Pellegrino (1986) [6] in the form of the Generalised Maxwell rule as a sufficient condition for static and kinematic determinacy of a lattice. The Generalised Maxwell rule, when applied to a unit cell of a lattice, can identify unit cell’s predominant deformation behaviour. However, these criteria are applicable only for a finite lattice. A simple method to determine bending/stretching domination for similarly situated pin jointed infinite lattices was given by Deshpande et al. [8]. By similarly situated lattice, they meant a lattice, which has the same orientation and topology from all nodes. For similarly situated lattices, Nodal connectivity (Z), which is the number of beams meeting at a node, is sufficient to convey the lattice deformation behaviour. Further, Guest and Hutchinson [7] concluded that an in15  finite lattice cannot be both statically and kinematically determinate. The criteria developed by Deshpande et al. were simple but were confined to lattices having similarly situated nodes. Hutchinson et al. [17] developed a formulation to predict the deformation domination of an infinite lattice material using Floquet-bloch theorem. Their formulation was not confined to similarly situated lattices and could be applied to any lattice topology. In particular, their method could identify the presence of periodic and strain producing mechanisms in a pin-jointed lattice. Note that periodic mechanisms do not contribute to macroscopic strains. Hence the topologies that possess only periodic mechanisms are stretch dominated. They observed that Kagome lattice and Triangular lattice are macroscopically stiff as they are incapable of producing macroscopic strain unlike T-T lattice, Square and Hexagonal lattices (See Figure 2.2). Analysis by Hutchinson et al. [17] requires a choice of unit cell, where cell edges or struts do not intersect the adjacent unit cell envelopes. Recently, Elsayed et al. [18] extended their formulation to incorporate choice of unit cells having intersecting envelopes (see Figure 2.3). These ideas can only decipher qualitative information about a lattice by considering a pin jointed analysis. More rigorous techniques are required to quantify the effective material properties, which are taken up in the next section.  2.3  Effective Material Properties  Much of the work in this area gained impetus with Christensen’s review [19], where Christensen surveyed work on 2-D and 3-D low density lattice materials, focused on their effective material properties and presented future applications of these materials. For instance, stiffness and strength anisotropy of these materials can be exploited for carrying directional loads. Moreover, micro-architecturing of unit cell geometries has made it possible to realise materials with negative effective Poisson’s ratio also called Auxetics. Scarpa et al. [20] have conducted experimental investigation on Auxetic honeycomb. They demonstrated high sensitivity of material properties (Young’s modulus, Poisson’s ratio) of the honeycomb to particular ranges of unit cell geometries. Exploitation of their structural capabilities requires the knowledge of their di-  16  Triangular triangular  Square Kagome  Triangular  Hexagonal Strain producing mechanisms  Periodic mechanisms  No mechanisms  Figure 2.2: Picture classifying deformation domination of topologies. Kagome and Triangular lattice topologies are stretch dominated in all orientations. Only periodic mechanisms are found in Kagome whereas hexagonal topology possess no mechanisms. Square and Hexagonal lattice possess periodic and macroscopically strain producing mechanisms. Square and Hexagonal lattice are stretch dominated only in few orientations. Triangular-Triangular (T-T) lattice possess only strain producing mechanism and is bending dominated in all directions. Triangular lattice possess no mechanisms and is stretch dominated in all orientations. Note that periodic mechanisms do not contribute to macroscopic strains [17]. rection dependent effective material properties such as effective Young’s modulus and effective shear modulus in various directions. Effective material properties can act as a a comparative tool for an engineer to compare structural properties of different topologies in order to choose the optimum lattice topology for a specific loading condition. Moreover, computationally expensive brute force F.E simulations could be avoided with the use of effective material properties. In order to deduce, effective material properties, the lattice is envisioned as an effective continuum, therefore, the size of the unit cell should be orders of length smaller than the macroscopic size of the material as a whole to avoid size effects and boundary 17  Lattice Structure y  a2  a1  x Cell Envelope  a2 a1  Figure 2.3: Lattice topologies where cell edges or struts intersect the adjacent unit cell envelopes [18]. layer effects emanating from the edges. Four predominant approaches have been used in the past. They are: 1. Discrete Network Analysis (DNA) 2. Homogenization method 3. Finite Element Approach 4. Wave-based dynamic method (WDM) Gibson and Ashby [1, 3, 21, 22] have derived the effective material properties of two dimensional lattice materials using the DNA approach. Their approach has been tested and validated against extensive experimental data. Gibson and Ashby [3], have established scaling laws for strength and stiffness pertaining to randomised open and closed cell foams. These foams were idealized as a regular two dimensional hexagonal lattice. This is because both have a predominantly 18  bending based deformation mechanism. Gibson and Ashby [3] have shown that stiffness and strength scales proportional to relative density for a stretch dominated lattice whereas it scales as ρ 2 and ρ 1.5 respectively for a bending dominated lattice. Their model is valid for lattices with low relative densities since tensile and shear deformations have been neglected in their analysis. Warren and Kraynik [23] have derived Effective Material Properties by incorporating tensile and bending deformations in their model and have proposed a series expansion for its effective moduli. Their model approximates to Gibson’s model for low relative densities. Masters and Evans [24] further improved this model by incorporating shear along with axial stretching and bending. Wang et al. [25] presented an extensive comparison on different honeycombs with regard to their in-plane material properties and highlighted structurally superior topologies. Their analysis method is similar to the ones used by Gibson et al. Recently, Balawi et al. [26] compared Evans model with experiments. They observed increase in the deviation of Evan’s model with the decrease in relative density. They attributed it to the neglect of curvature of the beam at the nodes, which increases with a decrease in relative density. A model incorporating the effects of curvature has been proposed by Balawi et al. [27]. More recently, Chen [28] looked into the out of plane flexural rigidity of honeycomb and concluded that effective continuum properties cannot be used to predict it (see Figure 2.4). They stated the reason to be the difference in the moments acting on the inclined wall during out of plane bending. These moments are different than the moments acting during the in plane and bending deformations. Chen has proposed a theoretical expression based on torsion and bending of honeycomb wall plates to derive expression for flexural rigidity. In their more recent work, Chen et al. [29] have estimated effective properties for asymmetric honeycomb by incorporating shear, bending and tensile deformation. Though, the above mentioned analytical approach gives a good physical insight, however, its implementation is tedious for a lattice having a complex unit cell. Therefore, this approach cannot serve as an efficient way of comparing various new topologies based on their structural capabilities. Homogenization method results are more rigorous and are often used as a benchmark to verify the results obtained through DNA. Homogenization method was applied by Chen et al. [30] by equating the strain energy of a discrete structure 19  Figure 2.4: Out of plane bending of a regular hexagonal honeycomb [28]. to the strain energy of an equivalent micro-polar continuum. However, their model only considers affine deformation and gives too stiff a moduli, as reported by Fleck and Qui [31]. Kumar et al. [32] have recently used homogenization method by taking terms until second order derivatives in the Taylor series expansion of displacement and rotation field , as suggested by Bazant and Christensen [33]. However, their model cannot be employed for lattices, where a unit cell having one node cannot be realised, such as in Kagome and Hexagonal honeycomb. Gonella et al. [34] and more recently, Lombardo et al. [35] have adopted Kumar et al.’s formulation. They have incorporated higher order terms in the Taylor series expansion and have used their model to predict the wave-bearing characteristics of lattice materials pertaining to the lower branches of the dispersion curves. However, their model owes the same restriction on the choice of unit cell as Kumar et al.’s and hence is limited in its functionality. 20  Direct computation using Finite Element approach requires building a finite element model of a large lattice and applying suitable global boundary conditions to effectively simulate uni-axial tension and pure shear test conditions. By looking at the suitable stress and strain measures from these tests, one can estimate the effective material properties of lattice materials. Wallach and Gibson [36] numerically predicted the effective material properties namely elastic and shear modulus of a 3-D octet lattice using this approach. This approach requires an accurate judgement of boundary conditions to capture the global effects and demands high computational cost as it requires a large lattice sample size. Moreover, anisotropy is a problem since cumbersome finite element computations have to be repeated multiple times to deduce all the parameters of the constitutive tensor (C). Lastly, these effective properties can be deduced by using a wave based dynamic method. This approach has been adopted by Phani et al. [37] using a Timoshenko beam formulation to predict the effective moduli of infinite Triangular, Hexagonal, Square and Kagome lattices. These numerical results agree well with the analytical predictions. Phani et al. have employed this approach to predict the wave bearing characteristics of the lattice materials in the form of dispersion curves. Their approach essentially couples Floquet-bloch formalism to exploit the repeatability of the unit cell and finite element analysis. Since this approach requires only the finite element modelling of a unit cell, it is computationally efficient and could be employed with ease for any complex isotropic lattice topology. However, their model is limited to effective property estimation of isotropic topologies.  2.4  Finite Lattices  Lattice materials, when manufactured, are finite in size. The finiteness of size induces localized effects emerging from the boundaries. These are referred to as boundary layers in this work. Boundary layers spread localized deformations from the boundaries into the interior (see Figure 2.6). Hence, elastic boundary layers degrade effective material properties and the degradation scales with the depth of elastic boundary layer. Boundary layers are a result of the difference between the nodal connectivity of the unit cells at the surface and the unit cells at the core as a result of which the  21  ux = δ/2 0-direction  (a)  y  z  uy , uz , θx , θ y , θ z = 0  x  ux = δ/2 ux , uz , θ y , θ z = 0 90-direction  (b) Figure 2.5: Boundary conditions for the Finite element simulation conducted by Kueh et al. [38]. A 5% and 25% knock down in stiffness was observed for (a) and (b) respectively.  22  u  x2 u  x1 Figure 2.6: Elastic boundary layer emanating from the edge of a Kagome lattice and decaying into the interior. The lattice is subjected to a uniaxial load [39]. stress states at the interior and the core differ. Therefore, boundary layers emerge to smoothly transition from the stress state at the surface to the homogeneous stress state in the core as shown in Figure 2.6. In the area of lattice materials, this phenomena was first observed by Kueh et al. [38] in a Kagome weave made out of a carbon fibre epoxy composite. Its effective moduli differed in different directions and exhibited width dependent effects due to the presence of deep elastic boundary layer. They observed a knock down of 25% and 5% in the Kagome weave shown in Figure 2.5. Elatic boundary layer increases macroscopic toughness. This was observed by Fleck and Qiu [31] in FE simulations conducted on the Kagome, Hexagonal and Triangular lattice topologies. They observed increased macroscopic toughness for Kagome and the presence of a deep elastic boundary layer emerging from the crack tip in a Kagome as shown in Figure 2.7. They reported that the localized elastic zone emerging from the crack tip reduced the stress concentration and enhanced the macroscopic toughness of the Kagome lattice. They showed that elastic boundary 23  (a)  (b)  (c) Figure 2.7: Finite Element simulation of a notched (a) Hexagonal , (b) Triangular and (c) Kagome lattice under a uni-axial load. Note the elastic boundary layer emerging from the crack tips of the Kagome lattice [31].  24  layer also imparted damage tolerance to the Kagome, where below a certain critical crack size, the tensile and shear strength remained unaffected for the Kagome. Phani et al. [39] corroborated their findings and developed a semi-analytical formulation based on a quadratic eigenvalue problem to detect the presence of the elastic boundary layer. These studies have identified the presence of a deep elastic boundary layer as an important parameter in determining the degradation of effective material properties in finite lattice materials. The degradation in the effective moduli scales with the depth of elastic boundary layer and the depth of elastic boundary layer is dependent upon: 1. Lattice topology 2. Relative density: The depth of elastic boundary layer increases with the decrease in relative density. However, these studies are restrictive in the sense that only a particular edge corresponding to a given lattice topology has been examined. In reality, for a finite lattice topology, more than one edge is possible, thus giving rise to a different nodal connectivity at the edge. Thereby, for each of these edges or configurations, elastic boundary layer may or may not be deep. Thus, the effective material property degradation will depend on the choice of edge as noticed. The effective properties of a finite lattice material will be affected by the elastic boundary layer and the size effect. A deep elastic boundary layer can significantly degrade the effective properties of the finite lattice material. Thus, it is of much importance to look at these effects in unison and study the degradation in effective properties due to these effects.  2.5  Research Objectives and Outline  Two objectives for this thesis are identified from the literature review. 1. The first objective is to develop an efficient numerical method to estimate the effective material properties of an infinite anisotropic lattice materials. 2. The second objective is to develop a method which can capture the effects due to the finiteness of lattice materials. 25  Chapter 3 develops a wave-based dynamic method for a 2-D orthotropic infinite lattices. The lattice is envisioned as a network of Timoshenko beams. The method is implemented on a chosen set of 13 topologies. The wave bearing characteristics in the form of dispersion curves and the effective material properties are predicted for this set of topologies. The method is validated against the effective material properties available in the literature. These effective Young’s modulus and the effective shear modulus of the studied topologies are compared using the polar plots at a fixed relative density. Chapter 4 develops a semi-analytical formulation to detect the presence of elastic boundary layers in a lattice topology based on a linear eigenvalue problem. The developed formulation is adopted from the work of Phani et al. [39] based on a quadratic eigenvalue problem. The developed formulation is implemented on the earlier studied set of topologies and 3 topologies are identified. These identified topologies possess deep elastic boundary layer and hence the possibility for significant effective material property degradation. Chapter 5 presents the conclusions pertaining to the current thesis work. The assumptions employed in the methods are summarized and the developed methods are carefully critiqued. Following which, the future areas of research are identified.  26  Chapter 3  Effective Material Properties 3.1  Introduction  Lattice materials possess a spatially repetitive unit cell geometry on the length scales of a few centimeters to a few millimetres [3, 4, 19]. They possess multifunctional properties [13] such as negative Poisson’s ratio [40] and effective zero thermal coefficient of expansion [14]. 2-D lattice materials such as hexagonal honeycombs are frequently used as cores in sandwich panels to provide high structural properties per unit weight. These sandwich panels serve as stiffer weight efficient structural components in industries like missile and aerospace. The structural calculations with regard to these components or sandwich panels using finite element analysis is computationally expensive. The effective material properties of the lattice materials helps to avoid the brute force computationally expensive finite element calculations pertaining to sandwich panels employing lattice materials. The effective structural properties of sandwich panels can be easily calculated using classical sandwich theory, once the effective material properties of the lattice materials employed as core are known. Thus, the effective material properties of lattice materials enables rapid design calculations of structural components involving lattice materials. Moreover, effective material properties aid the designer in choosing an optimum lattice topology for a specific loading condition. Analytical expressions have been proposed in the past to predict effective material properties of lattice materials by considering them as a network of beams [3] 27  are cumbersome to apply to complex geometries. This limitation is overcome in the proposed dynamic wave based formulation using Floquet-bloch theorem [37, 41– 45] in conjunction with FE modelling that can handle any complex unit cell topology with ease. This method enables a rapid calculation of direction dependent effective material properties such as Young’s modulus (E ∗ ), effective shear modulus (G∗ ) and effective Poisson’s ratio (ν ∗ ). Wave bearing characteristics such as bandgaps (frequency zones of no wave propagation) and spatial filtering of waves can also be obtained using the same formalism. The organisation of this chapter is as follows. Floquet-bloch theorem is presented first, followed by the procedure to obtain wave bearing information in the form of dispersion curves. Next, the phase velocities are obtained from the long wavelength assymptotes of dispersion curves and are further used to obtain effective material properties. The developed procedure is implemented for a chosen set of lattice topologies and their wave bearing information and effective material properties are estimated. Finally, the studied topologies are compared with respect to wave bearing information and effective material properties.  3.2  Floquet-bloch Theorem  Floquet-bloch theorem named after Felix Bloch [41] helps to exploit the periodicity of a lattice material. Using Floquet-bloch theorem, the response of the entire lattice can be expressed in terms of the response a single repeating unit cell [37, 42–45]. In order to understand the application of Floquet-bloch theorem in a lattice material, it would be useful to first review the basic concepts in the area of lattice materials. Consider an infinite lattice that is constituted by repeating a unit cell along the basis vectors ei (see Figure 3.1). If a harmonic plane wave solution is admitted in this infinite lattice system , then the displacement q(r,t) as shown in Figure 3.1 of a point with a position vector (r) is of the form q(r,t) = qA ei(ωt−k·r) ,  (3.1)  where qA is the wave amplitude of point A, ω is the frequency in rad/sec and k is the wave vector, which is real for a propagating plane wave. Determination of response of the entire lattice using the above equation requires the estimation of 28  qlt  qt  qrt  ql  qI  qr  qb  qrb  qlb  B  (n1 ,n2)  A  e1  r  y p O  e2  q(p)  q(r)  (a)  x (b)  Figure 3.1: A hypothetical lattice (b) made by tessellating the unit cell (a) along the basis vectors e1 and e2 . The unit cells in the lattice are connected rigidly through the rigid beams that are emanating from the unit cells. The unit cell (a) is partitioned into edge (l, r,t and b), corner (lt, rt, rb and lb) and interior degrees of freedom (I). 1 and 2 represent the basis directions along the basis vectors e1 and e2 respectively. q is a displacement vector of the partitioned degrees of freedom that are represented in the subscript. A reference unit cell having a point A, with position vector p and a displacement vector q(p) is shown with dark thick lines in (b). A point B, similar to point A in an arbitrary unit cell has a position vector r and a displacement vector q(r). This unit cell is identified by the label (n1 , n2 ), which represents n1 unit cell translations along e1 and n2 translations along e2 from the reference unit cell. Using Floquet-bloch theorem, displacements of this arbitrary unit cell (n1 , n2 ) can be represented in terms of displacements of the reference unit cell.  29  wave amplitude (qA ) corresponding to every point in the lattice. Thus, the problem size that is set up to determine parameter (qA ), will scale to the number degrees of freedom of the entire lattice under consideration. However, by employing the Floquet-bloch theorem, the problem size can be reduced to the number degrees of freedom of a single unit cell. The Floquet-bloch theorem is a result of translational symmetry of the lattice. It states that displacement of any arbitrary point (See Figure 3.1) in the unit cell (n1 , n2 ) with the position vector r = p + n1 e1 + n2 e2 is given by q(r, t) = qA (p)e(−ı(k·(r−p)) ,  (3.2)  where qA (p) is the similar point in the reference unit cell with position vector p. Therefore, by virtue of Floquet-bloch theorem, the response of the entire lattice is expressed in terms of the response of a single unit cell. The equations of motion pertaining to a single unit cell in conjunction with Floquet-bloch theorem will reveal the response of the entire infinite lattice. This procedure is demonstrated next.  3.3  Analysis of Free Wave Motion  A unit cell is modelled as a rigid jointed network of Timoshenko beams with each node possessing two translations in the x − y plane and one rotation about the z −  axis normal to the plane of the paper. Timoshenko beam captures the potential energy due to shear, bending and tensile deformations. The equations of motion governing the harmonic dynamic response of the unit cell are given by: Mq¨ + Kq = 0 or Kq − ω 2 Mq = 0, q(t) = qeıωt ,  (3.3)  where M is the mass matrix, K is the stiffness matrix, and q(t) is the generalised displacement vector, which is assumed to be harmonic. A change in unit cell topology will change the corresponding M and K matrices. These matrices are obtained using the Finite Element Method. The response of an infinite lattice system is obtained by applying Floquet-bloch transformation [37, 44] to the equations of motion of a single unit cell. In order 30  8 7 6 5 ωnd  Bandgap  4 3 2 1 0 0  0.1  0.2  kx  0.3  0.4  0  0.1  0.2  k  0.3  0.4  y  Figure 3.2: The dispersion surface for a hexagonal honeycomb. The x − y axis represents the components of wavevectors (k) kx − ky along the x and y directions. Propagating non-dimensional frequency is plotted on the z axis. It is normalized w.r.t the first pinned-pinned frequency of an individual lattice beam. No frequency data point exists in a narrow zone, which is the bandgap for this lattice. to apply the transformation with ease, the degrees of freedom of a single unit cell are partitioned into edge (l, r,t, b), corner (lt, rt, rb, lb, rb) and interior degrees of freedom (I) as presented in Figure 3.1. Note that (l, r, t and b) denote (Le f t, Right, Top and Bottom) degrees of freedom corresponding to the unit cell. Then, by adopting the Floquet-bloch theorem in Equation (3.2), we get qr = e−ık1 ql , qt = e−ık2 qb qrb = e−ık1 qlb , qlt = e−ık2 qlb , qrt = e−ı(k1 +k2 ) qlb  (3.4)  The above equations can be viewed as Equilibrium of forces/Compatibility of  31  displacements. The Equation (3.4) can be neatly expressed using the Transformation matrix (T):   I  0  0  0            T(k1 , k2 ) =           0  I  0  0  0  0  0  I  0  0  0  Ie−ık1  0  0  0  0  I                    0 Ie−ık2 0  0  0  0  Ie−ık1  0  0  0  Ie−ık2  0  0  0  Ie−ı(k1 +k2 )  (3.5)  Then, using T , Equation (3.4) can be better represented in matrix form as:          q=          qi     qb    qt    ql      qr   = T(k1 , k2 )    qlb   qrb    qlt  qrt  qi     qb   = T(k1 , k2 )q, ˜ ql   qlb  (3.6)  where T(k1 , k2 ) is the Floquet-bloch transformation as a function of wavevector components k1 and k2 along the basis vectors e1 and e2 respectively. Consequently, k1 = k · e1 = ε1 + ıδ1 and k2 = k · e2 = ε2 + iδ2 . Wavevector components comprise a  real (ε) and imaginary (δ ) part. They are known as phase and attenuation constants respectively. For an undamped medium, the allowed propagating waves will travel without attenuation and hence wavevector components would be real (δ = 0). Application of Floquet-bloch transformation [37, 44] using the matrix T(k1 , k2 )  32  ˜ to Equation (3.3) yields an eigenvalue problem in Bloch reduced co-ordinates (q): TH KTq˜ − ω 2 TH MTq˜ = 0 or Ax = λ Bx,  (3.7)  where A = TH KT, B = TH MT, x = q˜ and eigenvalue λ = ω 2 . For any given wavevector k with the components k1 and k2 , we can solve the eigenvalue problem to construct the dispersion surface ωn (k1 , k2 ) for each eigenvalue n = 1, 2, . . . N as shown in Figure 3.2. Here, N denotes the number degrees of freedom of the FE model unit cell. The solution of Equation (3.7) is periodic for any lattice medium. Thus dispersion surface exhibits periodicity. Therefore, one need not explore the entire wavevector space but only a predetermined portion of wavevector space can be used to infer the entire propagating frequencies. This zone or portion can be identified by constructing a Brillouin zone. To identify the Brillouin zone, a reciprocal lattice is envisioned first (see Figure 3.3). Physically reciprocal lattice is a wavevector space. It is assembled by tessellating reciprocal lattice along the reciprocal basis vectors e∗1 and e∗2 . These reciprocal basis vectors can be deduced by the relation ei · e∗j =2πδi j , where δi j is the identity matrix. Then, the Brillouin zone is identified on a reciprocal lattice by following these steps:  1. Choose a reference point in the reciprocal lattice. Draw lines connecting the reference point to its adjacent points that surround the chosen point. These lines are displayed in Figure 3.3 by solid lines. 2. Construct the perpendicular bisector of the dashed lines. The smallest closed polygon resulting from the intersection of perpendicular bisectors is the first Brillouin zone. This is displayed in Figure 3.3 by bold lines constituting a square. Owing to the symmetry of the Brillouin zone, it can be further reduced into an irreducible Brillouin zone, shown by a grey triangle O-A-B in Figure 3.3. The wave vectors that are inside the irreducible Brillouine zone results in a unique solution of propagating frequencies. A wave vector chosen outside the irreducible Brillouin zone would result in the frequencies, which would also correspond to a wave vector inside the irreducible Brillouin zone. Furthermore, the frequency extrema of the 33  A  B  O  e2* e1* Figure 3.3: A reciprocal lattice showing the Brillouin zone. Reciprocal lattice is constituted by tessellating the dark circles or points along the recirpocal basis vectors e∗i . The first Brillouin zone is displayed by the square outlined with a bold line. The shaded triangle is the irreducible Brillouine zone. The frequency extrema in the dispersion surface usually occurs at the wavevectors that correspond to the exterior boundary of the triangle O-A-B in a 2-D lattice. dispersion surface occurs at the wave vectors that correspond to the edges of the shaded triangle O-A-B for a 2-D lattice [43]. This is useful when searching for bandgaps (frequency regions of no wave propagation). Furthermore, dispersion surface reveals the phase and group velocities. Physically, the group velocity gives the speed and direction of energy propagation corresponding to a wave whereas the phase velocity gives the speed and direction of wave propagation. The phase and group velocity are deduced by the following relations ω ˆ k |k|  (3.8)  ∂ω ˆ n, |∂ k|  (3.9)  Cp (Phase Velocity) = Cg (Group Velocity) = 34  where kˆ is a unit vector in the wave vector direction and nˆ is a unit vector normal to the iso-frequency contour at the point on the iso-frequency contour in the direction of wave vector. Iso-frequency contour is a 2-D contour plot of propagating wave vectors in all directions that correspond to a fixed propagating frequency. For an isotropic medium, iso-frequency contour is circular and therefore group velocities and phase velocities are co-linear. As a consequence, the energy and the wave are propagated in the same direction for an isotropic medium. This may not be true for an anisotropic medium.  3.4  Effective Material Properties  The effective material properties of a general anisotropic lattice can be obtained from the phase velocity data that pertains to the long wavelength limit (k = 0). In the long wavelength limit, an anisotropic lattice can be treated as an effective anisotropic continuum. The translational equation of motion for the anisotropic continuum without any external traction and body force can be written as: ρ ∗ u¨i = σi j, j ,  (3.10)  where ρ ∗ is the density of the medium, σi j is the stress tensor and ui is the displacement of the particle in the medium. In the linear elastic range, stress and strain for the anisotropic continuum can be related by the Hooke’s in the Voigt matrix or indicial notation respectively: σ = Cε or σi j = ci jkl εkl ,  (3.11)  where εkl is the tensorial strain. Then by inserting Hooke’s law (Equation (3.11)) into translational equation of motion (Equation (3.10)), we get: ρ ∗ u¨i = ci jkl uk,l j  (3.12)  Relationship between the phase velocity data and the independent constants of the matrix(C) can be deduced by inserting the wave solution given by: uk = Apk ei(kr xr −ωt) 35  (3.13)  where: A is a scalar amplitude, pk is a unit displacement vector, ω is the angular frequency of the wave, xr is a space vector, uk is the instantaneous displacement vector of the particle and kr is a wave number; in Equation (3.12), which yields the Kelvin-Christoffel equation [46]: (ci jkl n j nl − ρ ∗ v2 δik )pk = 0 or [Γi j − ρ ∗ v2 δi j ]pk = 0,  (3.14)  where v is the phase velocity, ρ ∗ is the effective density of the anisotropic medium, n j is the unit wave normal. The unit wave normal is perpendicular to the the equiphase surface and corresponds to the direction of wave travel or wave vector k. An equiphase surface is the surface formed by the loci of particles of the medium having the same phase. For a planar wave, equiphase surface is a plane. Equation (3.14) represents an eigenvalue problem. For a given wave normal (n j ), the eigenvalue solution yields three phase velocities since only three translational degrees of freedom per node are allowed in the assumed continuum. Hence, the phase velocity can only have three components. For a continuum having n degrees of freedom of per node, the phase velocities would be n in number. Hence, for all possible wave normals, the solution would represent three closed sheets of phase velocities for the anisotropic continuum and n sheets of phase velocities for the continuum possessing n degrees of freedom per node. For an isotropic medium, these contours would be spherical. Each phase velocity sheet corresponds to a deformation mode. These deformation modes are represented by the longitudinal, shear and transverse phase velocities for a continuum having 3 degrees of freedom per node. The deformation of the particles of the medium pertaining to these phase velocities are as follows: 1. Longitudinal phase velocity: The particles of the medium vibrate in the direction of the wave propagation. 2. Two shear or transverse phase velocities: The particles of the medium vibrate perpendicular to the direction of wave propagation. 36  For the discrete infinite lattice, substituting it as an equivalent continuum having 3 degrees of freedom per node, we are only capturing the first three phase sheets at the long wavelength limit pertaining to zero frequency. By using an advanced continuum, one can capture higher number of phase sheets pertaining to higher frequencies at the long wavelength limit for the discrete infinite lattice. Phase velocity data as a function of wave normals is available from the dispersion curves at the long wavelength limit. Thus, phase velocity data and wave normals can act as an input to the eigenvalue solution in Equation (3.14) to produce a set of algebraic equations. These equations are solved for unknown independent constants of the matrix (C) for an orthotropic medium, which will be shown in the next section. In the proceeding section, we will apply this theory to obtain the effective matrix (C) for a 2-D orthotropic and isotropic lattice.  3.5  Application to a 2-D Orthotropic Lattice  Specializing Equation (3.11) to a 2-D orthotropic continuum in the principal symmetry direction leads to:   σxx      C11 C12  0    εxx          σyy  =  C12 C22 0   εyy  = Cε σxy 0 0 C33 γxy  (3.15)  or   εxx          εyy  =   γxy  ν∗  1 ∗ Exx ∗ νxy − E∗ xx  − Eyx∗  0  0  yy  1 ∗ Eyy  0    σxx      0    σyy  = Sσ , 1 σxy ∗  (3.16)  Gxy  ∗ is the effective Young’s modulus in the x direction, E ∗ is the effective where Exx yy  Young’s modulus in the y direction, G∗xy is the effective shear modulus in the x − y  ∗ and ν ∗ are the effective Poisson’s ratio pertaining to the loading in direction, νyx xy  the x and y directions respectively. Similarly Γi j or ci jkl n j nl in Equation (3.14), is specialized to the 2-D orthotropic  37  case by putting (n3 = 0, C31 = 0, C13 = 0, C13 = 0, C23 = 0 and C32 = 0) to yield:   α11 α12 α13      Γi j =  α12 α22 α23  α13 α23 α33  (3.17)  where α11 = C11 n21 +C33 n22 α22 = C33 n21 +C22 n22 α12 = (C12 +C33 )n1 n2 α33 = 0 α23 = 0 α13 = 0 Substituting Equation (3.17) in Equation (3.14), the solution can be obtained by solving the resulting determinant given by:  ∗ 2  |Γi j − ρ v δi j | =  C11 n21 +C33 n22 − ρ ∗ v2 (C12 +C33 )n1 n2 0  (C12 +C33 )n1 n2  0  C33 n21 +C22 n22 − ρ ∗ v2  0  =0  −ρ ∗ v2  (3.18)  0  Solving the determinant in Equation (3.18), we obtain: 2 2 2 ρ ∗ v2 (C12 n1 n2 − ρ ∗ v4 −C11C33 n41 −C22C33 n42 −C11C22 n21 n22 + 2C12C33 n21 n22 +  +ρ ∗ v2 (C11 n21 +C22 n22 +C33 n21 +C33 n22 )) = 0 (3.19)  38  Equation (3.19) contains four unknowns which can be represented as follows:  C12 = −C33 +  C11 = C22 =  (C11 n21 +C33 n22 − ρ ∗ v2 )(C33 n21 +C22 n22 − ρ ∗ v2 )  n1 n2 2 2 2 2 ∗ 2 ∗ 2 ∗ 2 n1 (C12 n2 + 2C33C12 n2 +C33 ρ v ) + ρ v (−ρ v +C22 n22 +C33 n22 ) −C22C33 n42 n21 (C33 n21 +C22 n22 − ρ ∗ v2 ) 2 n2 + 2C C n2 +C ρ ∗ v2 ) + ρ ∗ v2 (−ρ ∗ v2 +C n2 +C n2 ) −C C n4 n22 (C12 33 12 1 33 11 1 33 1 11 33 1 1 2 2 2 ∗ 2 n2 (C33 n2 +C11 n1 − ρ v ) 2 2 ) −C ρ ∗ v2 ) + ρ ∗ v2 (ρ ∗ v2 −C n2 ) n (n2 (C11C22 −C12 22 11 1 C33 = 2 1 4 2 2 2 4 2 ∗ 2 ∗ −C11 n1 + 2C12 n1 n2 + ρ n1 v −C22 n2 + ρ n2 v2  Solving the above equations for wave normals in the directions of symmetry and the diagonal direction, the constants can be expressed in a simplified manner as: C11 = (Cl/0o )2 ρ ∗  (3.20)  C22 = (Cl/90o )2 ρ ∗  (3.21)  C33 = (Ct/90o or Oo )2 ρ ∗  (3.22)  2 ∗ β = −2Cl/45 o ρ +C33  (3.23)  C12 = + (β +C11 )(β +C22 ) −C33 ,  (3.24)  where Cl/θ and Ct/θ are the longitudinal and shear phase velocities at the long wavelength limit and the subscript θ represent the angle of phase velocity or wave normal with respect to positive x − axis in the counter clock-wise direction. Note  that the value of C12 , which yields a positive definite compliance (S) matrix and an agreeable sign of Poisson’s ratio with regard to its topology should be chosen. Using Equations (3.20)-(3.24), the parameters of the matrix (C) can be represented as a function of phase velocities. C = F(Cl/0o ,Cl/90o ,Ct/90o )  (3.25)  From Equation (3.15)-(3.16), we obtain: C = S−1 39  (3.26)  Using Equations (3.25)-(3.26), we have: ∗ ∗ ∗ ∗ S−1 (Exx , Eyy , νyx , νxy )) = F(Cl/0o ,Cl/90o ,Ct/90o )  (3.27)  Therefore from Equation (3.27), phase velocity or the long wave length assymptotes are obtained as a function of effective material properties. Consequently, the phase velocity or the long wavelength assymptotes for 2-D orthotropic lattice are given by the following relations: Cl/0o =  ∗ E∗ Exx yy ρ∗ ∗ − E ∗ ν ∗2 ) (Eyy xx yx  (3.28)  Cl/90o =  ∗2 Eyy ρ∗ ∗ − E ∗ ν ∗2 ) (Eyy xx yx  (3.29)  Ct/90o /Oo =  G∗xy ρ∗  (3.30)  Repeating the above calculations(3.20)-(3.24) for different relative densities enables one to deduce tensor (C) for different relative densities. Using curve-fitting techniques, C can be expressed as a function of relative density.  3.6  Application to an Isotropic Lattice  For an isotropic lattice, effective shear (G∗ ) and in-plane bulk modulus (K ∗ ) are related to the group velocities at the long wavelength by the relation [37] given by: Cl =  K ∗ + G∗ ,Ct = ρ∗  G∗ , ρ∗  (3.31)  where ρ ∗ is the equivalent density, G∗ is the equivalent shear modulus and K ∗ is the equivalent 2-D in-plane bulk modulus of the isotropic lattice. Using these group velocities effective shear and bulk modulus are calculated. Effective Young’s modulus (E ∗ ) is deduced from the estimated bulk and shear modulus using the  40  Dq=f=0  +  +  Band Gaps  Dispersion Curve Christoffel’s Equations Effective Material Properties  Figure 3.4: A flowchart showing the wave based semi-analytical method. following relation for the in-place isotropic cellular solids [47] given by: E∗ =  3.7  4K ∗ G∗ . K ∗ + G∗  (3.32)  Summary  The developed wave based semi-analytical procedure is explained with the help of a flowchart shown in Figure 3.4. In order to calculate the effective material properties, following procedure is adopted. First, the lattice topology is analysed to identify a repeating unit cell along with the basis vectors e1 and e2 . Then, equations of motion are set-up by using the stiffness (K) and mass matrices (M) obtained through usual FE procedure. Dq = f = 0  41  (3.33)  Since this a wave propagation analysis, hence there is no external force (f = 0). Using the Bloch-Transformation matrix T(k1 , k2 ) of Equation (3.5), equations of ˜ are obtained: motion in Bloch reduced co-ordinates (q) Dq˜ = 0  (3.34)  The re-arranging of Equation (3.34) yields the eigenvalue problem: TH KTq˜ − ω 2 TH MTq˜ = 0 or Ax = λ Bx,  (3.35)  For a fixed value of k, propagating frequencies ω are obtained by the solution of the above eigenvalue problem. The propagating frequencies are obtained by varying k along the edges of irreducible Brillouin zone in the form of dispersion curves to identify bandgaps. Next, the phase velocities or the slopes of the branches of dispersion curve  ω |k|  are obtained at the long wavelength limit (k = 0) and zero  frequency in specific directions. These specific directions correspond to the symmetry and diagonal directions (45o in-plane from the symmetry axis) of the lattice. Hence phase velocities Cl/0o ,Cl/90o and Ct/0o are obtained and are plugged in Equation (3.20)-(3.24) derived from the Christoffel’s equation to obtain the effective material properties C.  3.8  Topologies  The topologies considered for the estimation of effective material properties are as shown in Figure 3.5. The basis vectors and the reciprocal basis vectors for these topologies are presented mathematically in Table 3.3 along with their relative densities. These geometries are chosen owing to their distinct multifunctional benefits. For instance, in Figure 3.5, Topology (a) named Low-CTE can also be designed for effective zero or low thermal co-efficient of expansion by contrasting the thermal co-efficient of inner and outer members and by varying physical parameters like skewness angle (t o ) of (a) in Figure 3.5. The physical properties of the Low-CTE lattice used in this work are mentioned in Table 3.2. Topologies (e),(b),(c),(d) and (g) in Figure 3.5 can provide negative Poissons ratio. Topology (g) in Figure 3.5 can accomodate sensor in its core [13] and hence can provide extra load bearing 42  LowCTE  to  e2*  e2  A O  e1  (a)  e1*  e2  Auxetic  e2* L  θ  A  e1 B  (b)  L  e2*  e2  A  L  θ L  L/3  e1*  O  Hybrid A  (c)  B  Ld  e1  B  O  e1*  (i) Lattice Topologies  Figure 3.5: Topologies of lattice materials under consideration (right), with their unit cells (left) and Brillouin zone (middle). e1 and e2 represent the basis vectors. e∗1 and e∗2 are the reciprocal basis vectors. Perimeter O-A-B marked by guiding arrows represents the points in wave vector space that correspond to frequency extrema. Thick lines in (a) represent a high coefficient of thermal expansion (CTE) such as Aluminum and thin lines denote low CTE material such as Titanium. In (c), (d), (g), (i), (k) and (h) θ = 60o ; in (b) θ = 75o ; in(e) θ = 30o ; in (j) θ = 120o . 43  e2*  Hybrid C  e2  L  θ  A L  *  L  (d)  B  e1  O  e1*  A  e1*  Hybrid D 2L  e2*  e2  B  e1  L  θ  (e)  O  Mixed  e2*  e1  e2  e1*  A B O  L (f ) Hex-Chiral  e2* B  θ (g)  e2  θ  e1* A  O  e1 (ii) Lattice Topologies  Figure 3.5 44  Diamond  e22  e2* A B  e1  (h) Kagome  e2  e2*  e1  θ  e1*  O  A  B O  e1*  L  (i)  L  Hexagonal  e2  θ  e1 L  e2* A  B O  e1*  (j)  Tetragonal  e2*  e2  B  e1 (k)  θ L  A  O  e1* (iii) Lattice Topologies  Figure 3.5 45  e2*  Square  e2  A L  B  e1*  O  e1  (l)  Star - Hex e2  e1*  e2* e1  A B O  (m)  (iv) Lattice Topologies  Figure 3.5 surface. Topology (f), (h), (i), (j), (k), (l) and (m) are the classical topologies whose effective material properties have been derived using the analytical methods in the literature. Some of them are easy to fabricate and their properties have been previously studied in detail by various researchers. These 2-D lattices are envisioned as a network of 2-D rigid jointed beams. A two dimensional Timoshenko beam element B-21 of FE package ABAQUS is used to model the beam elements. The elements permit three degrees of freedom per node i.e., two translations in the x − y plane and one rotation about the axis perpendicular to the plane of the paper.  The effective material properties of the Star topology, shown as (m) in Figure 3.5, is used to validate the present method against the estimated effective property expression deduced by Christensen [19], shown in Table 3.1. With the help of Star topology, Christensen demonstrated that a topology possessing re-entrant angles may not be necessarily Auxetic. In the following section, the effectiveness 46  Table 3.1: Effective properties of Star lattice topology derived by Christensen [19] using Euler beam theory.  Topology  G∗ Es  E∗ Es  ν∗  Isotropic  Stretch Dominated  Star  3 ¯3 16 ρ  9 ¯3 16 ρ  0.5  Yes  No  Table 3.2: Design parameters and properties of low CTE struts and parent material respectively. Note that all struts have a square cross section of area A.  Members  Material  Strut Length (m2 )  Young’s modulus(N/m2 )  ρ (kg/m3 )  Poisson’s ratio  High CTE  Titanium  L1  110 × 109  4500  0.33  Low CTE  Aluminium  L2  69 × 109  2700  0.33  of Floquet-bloch Finite Element scheme is demonstrated for the specific case of Star lattice topology. A typical unit cell of a Star lattice is shown as (m) in Figure 3.5 [19]. The basis vectors e1 and e2 are used to obtain reciprocal basis (e∗1 and e∗2 ) (Table 3.3) to construct the Brillouin zone (See (m) in Figure 3.5). Due to symmetry, the first Brillouin Zone is further reduced to obtain an irreducible Brillouin Zone O-A-B, the boundaries of which define the extrema (usually but not always) for propagating waves. Results for this topology are presented in the next section.  3.9  Results  Eigenvalue analysis is conducted for each wave vector along the irreducible Brillouin zone giving dispersion curves for the star lattice as shown in Figure 3.6. The horizontal axis represents all possible wave directions of propagating waves, given 47  Table 3.3: Reciprocal and direct basis vectors in Cartesian co-ordinate system. Note ei · e j = δi j , where δi j represents the Kronecker delta product ∗ ∗ or the Identity matrix. ρ¯ = ρρ = VV is the relative density of lattice material. Here V ∗ and ρ ∗ are the volume and density of solid material whereas V and ρ are of the lattice material. For the bimetallic√Low CTE lattice, (3)L2  ∗  L2 sin(2t)  ¯ is VVs instead of ρρ∗s ; parameter α = 4 1 − 2 2 + relative density(ρ) √ 2 o −t) (3)L2 + 5L1 L2 sin(60 + 2L2 cos(t)(L2 sin(60o − t) − L2 sin(t) ). The width 4 4 2 of square struts is given by parameter b. (a)  Topology  Direct  Low-CTE  e1 = 2L2 cos(t)ˆi e2 = 2L2 cos(t)( 12 ˆi +  Hex-chiral  Reciprocal  √ 3ˆ 2 j)  e1 = Lˆi + √L3 ˆj e2 = −Lˆi + √L3 ˆj  e∗1 = e∗2 = e∗1 = e∗2 =  ¯ Relative density(ρ)  1 ˆ √1 ˆ 2L2 cos(t) (i − 3 j) 1 √1 ˆ L2 cos(t) ( 3 j)  3b(L1 +L2 ) α  √ 3ˆ 1ˆ i + 2L 2L√j 1ˆ − 2L i + 2L3 ˆj  √ 2 3 Lb  Auxetic  e1 = 2L(cos(t) + 1)ˆi e2 = 2Lsin(t)ˆj  e∗1 = e∗2 =  1 ˆ 2L(cos(t)+1) i 1 ˆ 2Lsin(t) j  b(2+(2cos(t)+1)) 2Lsin(t)(1+cos(t))  HybridA  e1 = L( 38 + 2cos(t))ˆi e2 = 2Lsin(t)ˆj  e∗1 = e∗2 =  3 ˆ 2L(4+3cos(t)) i 1 ˆ 2Lsin(t) j  (29+3(2cos(t)+1))b (4+3cos(t))4sin(t)L  HybridC  e1 = 2L(cos(t) + 1)ˆi e2 = 2Lsin(t)ˆj  e∗1 = e∗2 =  1 ˆ 2L(1+cos(t)) i 1 ˆ 2Lsin(t) j  (3+cos(t))b (1+cos(t))2sin(t)L  HybridD  e1 = 2Lcos(t)ˆi + 2Ljˆ e2 = −2Lcos(t)ˆi + 2Lˆj  1 ˆi + 1 ˆj e∗1 = 4Lcos(t) 4L 1 ˆi + 1 ˆj e∗2 = − 4Lcos(t) 4L  (1+2sin(t))b 4cos(t)sin(t)L  Mixed  e1 = Lˆi + Ljˆ e2 = −Lˆi + Lˆj  1ˆ 1 ˆ e∗1 = 2L i + 2L j 1ˆ 1 ˆ ∗ e2 = − 2L i + 2L j  3.414b L  48  Table 3.3 (b)  Topology  ¯ Relative density(ρ)  Direct  Reciprocal  √ e1 = 3Lˆi + 3Lˆj √ e2 = −3Lˆi + 3Lˆj  1ˆ i + 6L3√ˆj e∗1 = 6L 1ˆ i + 6L3 ˆj e∗2 = − 6L  Diamond  e1 = Lˆi √ e2 = 3Lˆj  1ˆ e∗1 = √ Li e∗2 = 3L3 ˆj  Square  e1 = Lˆi e2 = Lˆj  e∗1 = e∗2 =  1ˆ Li 1ˆ Lj  Kagome  e1 = 2L( 21 ˆi + 23 )ˆj √ e2 = 2L(− 1 ˆi + 3 ˆj)  e∗1 =  1 ˆ √1 ˆ 2L (i + 3 )j 1 ˆ √1 ˆ 2L (−i + 3 j)  e1 = Lˆi  e∗1 = L1 (ˆi − √13 ˆj) e∗ = 1 ( √2 ˆj)  StarHex  √  2  Triangular  e2 = L( 21 ˆi + Hexagonal  √  e∗2 =  2  √ 3ˆ 2 j)  2  √ √ 3L( 21 ˆi + 23 )ˆj √ √ e2 = 3L(− 1 ˆi + 3 )ˆj  e1 =  2  e∗1 = e∗2 =  2  L  √ 2 3b 3L  √ 5 3b 3L  2b L  √ 3b L  √ 2 3( Lb )  3  √1 (ˆi + √1 )ˆj 3L 3 √1 (−ˆi + √1 )ˆj 3L 3  √2b 3L  by the k − space locus along the perimeter O-A-B. The parameter s is a scalar,  which represents the arc length of the wave vector from O to the triangle O-A-B (See (a) in Figure 3.6). The curves in this figure are dispersion curves and each branch can be thought of as a projection of the cross section of dispersion surface cut and pieced together along the O-A-B path. These 2D curves are the extrema of the 3-D dispersion surfaces. The two phase velocities derived by static effective medium properties [19] are superimposed on the dispersion figures in Figure 3.6 as dotted straight lines. 49  k - space locus A B  s  O  Band Gap   Cl  (a)  Ct A  O  ky  k - space  B  O    0.01  (b)  kx Figure 3.6: (a) Dispersion curve for Star lattice for ρ = 8%. The y-axis represents propagating frequencies of free plane waves, non-dimensionalised with respect to first pinned pinned natural frequency of a strut. The xaxis represents the k-space locus or the wave vectors at the boundaries of the irreducible Brillouin zone O-A-B. (b) Iso-frequency contour for the first dispersion surface: circular shape indicates isotropy. Note the group velocities in the long wavelength limit calculated by the effective medium properties and dispersion analysis match well. This validates the present semi-analytical model employing Floquet-bloch theorem. A bandgap at a low frequency can be observed for Star topology as shown in Figure 3.6. Iso-frequency contours (Figure 3.6), generated for first phase constant (dispersion) surface, show that Star lattice is isotropic at lower frequencies. The calculated effective material properties are compared with analytical calculations in Figure 3.7. A close, but not exact, match is observed between finite element calculations using Bloch analysis and analytical results given by Chris50  Floquet-Bloch  (a)  Floquet-Bloch  (b)  Figure 3.7: Comparison of effective properties of Star lattice with analytical calculations in Christensen’s paper [19]. G∗ in (b) and E ∗ in (a) vary non-linearly with relative density. Note the discrepancy at the large relative densities. 51  Table 3.4: Bandgap classification of lattice materials for ρ¯ = 0.2  Low bandgap topologies (Ω < 5)  Medium bandgap topologies (5 < Ω < 15)  High bandgap topologies (Ω > 15)  Low-CTE Mixed  Hybrid-C Hex-chiral Diamond Kagome Hexagonal Triangular  Square Hybrid-D Hybrid-C Hybrid-A Auxetic  tensen [19]. The analytical model of Christensen predicts a much stiffer response, while the FE model suggests a more compliant lattice at higher relative densities. This discrepancy can be resolved by recognising that Christensen’s analytical model hinges upon thin Euler-Bernoulli beam theory. Euler-Bernoulli beam theory does not account for potential energy due to shear which becomes significant at higher relative densities and leads to a stiffer beam in comparison to a Timoshenko beam model used in the current prediction which incorporates shear potential energy. This comparison reveals the limitations of analytical calculations. We note that even the FE model of the unit cell needs to switch from a network of beams to network of periodic inclusions at higher relative densities. However, the procedure presented here remains valid. Using curve fitting technique, a closed form expression is obtained which is presented in Table 3.5. We note that the curve fitted expression agrees reasonably with Christensen’s result [19] (based on thin beam theory) to the leading order term in the relative density. In conclusion, Star lattice is a good candidate for wave filtering at lower frequencies and is isotropic at lower frequencies. The developed semi-analytical method has been applied to the remaining lattice topologies in Figure 3.5. Figure 3.8 displays the dispersion curves of these remaining lattice topologies for the relative density ρ¯ = 0.2. Bandgaps at low frequencies are observed for Low-CTE and Mixed lattice. The bandgaps in these 52  8  45  6  40  Ω4  Ω 35  2  30  0 O  A  BB  k−space  25  O  OO  A  B  k−space  O  (a) Dispersion curve of Low-CTE lattice (b) Dispersion curve of Auxetic lattice mamaterial terial  55  Ω  12  50  Ω  45  B  O  40  O  10 8  A  B B OO k−space  O  B A k−space  O  (c) Dispersion curve of Hybrid-A lattice (d) Dispersion curve of Hybrid-C lattice material material  Figure 3.8: Dispersion curves of the studied lattice materials (see Figure 3.5) for ρ¯ = 0.2. The y-axis represents propagating frequencies of free plane waves, non dimensionalised with respect to the first pinned pinned natural frequency of a strut. The x-axis represents the k-space locus or the wave vectors forming the boundaries of the irreducible Brillouin zone O-A-B (see Figure 3.5). The first grey band represents the first bandgap region.  53  22 20  6  Ω 18  Ω4 2  16  B  O  O  A B B O O k−space  O  A  B  k−space  O  (e) Dispersion curve of Hybrid-D lattice (f) Dispersion curve of Mixed lattice mamaterial terial  B  O  40  8  30  7  Ω 20  Ω6  10  5  0  O  A  B B  k−space  O O  4 O  A  B  k−space  O  (g) Dispersion curve of Hex-chiral lattice (h) Dispersion curve of Diamond lattice material material  Figure 3.8: Dispersion curves of the studied lattice materials (see Figure 3.5) for ρ¯ = 0.2. The y-axis represents propagating frequencies of free plane waves, non dimensionalised with respect to the first pinned pinned natural frequency of a strut. The x-axis represents the k-space locus or the wave vectors forming the boundaries of the irreducible Brillouin zone O-A-B (see Figure 3.5). The first grey band represents the first bandgap region.  54  10  10  Ω  B  O  Ω 5  5 0 O  A  B  k−space  0  O  O  A  B  k−space  O  (i) Dispersion curve of Kagome lattice (j) Dispersion curve of Hexagonal lattice material material  100  20 15  Ω 10  Ω 50  5 B  O  O  A  0 O  B B O O  k−space  A  B  k−space  O  (k) Dispersion curve of Triangular lattice (l) Dispersion curve of Square lattice mamaterial terial  Figure 3.8: Dispersion curves of the studied lattice materials (see Figure 3.5) for ρ¯ = 0.2. The y-axis represents propagating frequencies of free plane waves, non dimensionalised with respect to the first pinned pinned natural frequency of a strut. The x-axis represents the k-space locus or the wave vectors forming the boundaries of the irreducible Brillouin zone O-A-B (see Figure 3.5). The first grey band represents the first bandgap region.  55  two topologies occur below Ω < 5, where Ω is the propagating frequency nondimensionalised with respect to first pinned pinned natural frequency of a strut. These two topologies are identified as low bandgap topologies (Ω < 5). Hybrid-C, Hex-chiral, Diamond, Kagome, Hexagonal and Triangular topologies are identified as medium bandgap topologies (5 < Ω < 15) whereas the remaining are high bandgap topologies (Ω > 15). These results are also tabulated in Table 3.4. Note that the bandgaps are sensitive to relative density and hence the above bandgap classification (low, medium and high) is only valid for ρ¯ = 0.2. Table 3.5 summarizes effective material property results as a function of relative density for lattice topologies in Figure 3.5. For stretch dominated geometries, effective modulus scales in proportion with relative density [3] as can be observed in Table 3.5. Low CTE, Arrow, Mixed, Kagome, Tetragonal and Diamond topologies are stretch dominated and hence are more suitable from the standpoint of providing high stiffness per unit weight. The remaining topologies are bending dominated and therefore may be useful from energy absorption perspective. Low CTE is both isotropic and stretch dominated; hence it can be effectively used in scenarios where high stiffness is required for variable in plane loading direction. By tensor transformation of the constitutive tensor (C) in Equation (3.11), whose parameters are obtained from Table 3.5, polar plot of effective properties are constructed in Figure 3.9-3.10 for a fixed ρ¯ = 0.2. In the polar plot, each contour corresponds to a specific topology and therefore by laying all contours over each other, the studied topologies are compared for effective moduli (E* and G*) as shown in Figure 3.93.10. Clearly in the Figure 3.9-3.10, stretch dominated topologies are seen having a much higher moduli in all directions as compared to the bending dominated topologies. The isotropic topologies can be identified by the circular contours in the polar plot. These topologies are also tabulated in Table 3.5.  3.10  Conclusions  A semi-analytical method to compute effective properties of complex lattice geometries is proposed. The method combines Floquet-bloch theorem with the FE modelling of the unit cell. The usefulness of this method lies in its ability to estimate effective material properties of complex lattice topologies with ease and  56  ∗  Table 3.5: Effective properties of different lattice geometries. ρ¯ = VVs is relative density and Es is the Young’s modulus of parent material. E ∗ ,G∗ and ν∗ are the effective Young’s modulus, Shear modulus and Poisson’s ratio. Note for Low CTE lattice, Titanium and Aluminium are taken respectively as Low and High CTE materials [14] and Es is the young’s modulus of High CTE material.  Topology  G∗xy Es  ∗ Exx Es  ∗ Eyy Es  ∗ νxy  ∗ νyx  Isotropic  Stretch Dominated  Low CTE  0.0940ρ¯  0.2265ρ¯  0.2265ρ¯  0.20  0.20  Yes  Yes  Hex-chiral  0.4283ρ¯ 3  0.5490ρ¯ 3  0.5490ρ¯ 3  −0.42  −0.42  Yes  No  Auxetic  0.0436ρ¯ 3  0.4018ρ¯ 3  2.6987ρ¯ 3  −0.33  −2.61  No  No  HybridA  0.0246ρ¯ 3  0.0988ρ¯  0.5591ρ¯ 3  0  0  No  No  HybridC  0.0373ρ¯ 3  0.4285ρ¯  0.8122ρ¯ 3  0  0  No  No  HybridD  0.0420ρ¯ 3  0.4137ρ¯ 3  0.4972ρ¯  0  0  No  No  Mixed  0.1047ρ¯  0.3693ρ¯  0.3693ρ¯  0.26  0.26  No  Yes  Star  0.1706ρ¯ 3  0.4977ρ¯ 3  0.4977ρ¯ 3  0.48  0.48  Yes  No  Diamond  0.1510ρ¯  0.2000ρ¯  0.3600ρ¯  0.33  0.59  No  Yes  Square  0.0602ρ¯ 3  0.5000ρ¯  0.5000ρ¯  0  0  No  No  Kagome  0.125ρ¯  0.3333ρ¯  0.3333ρ¯  0.33  0.33  Yes  Yes  Triangle  0.125ρ¯  0.3333ρ¯  0.3333ρ¯  0.33  0.33  Yes  Yes  Hexagonal  0.3261ρ¯ 3  1.2259ρ¯ 3  1.2259ρ¯ 3  0.96  0.96  Yes  No  57  0.08 0.06 0.04 0.02 ∗ Eyy 0 Es 0.02 0.04 0.06 0.08 0.08 0.06 0.04 0.02  0 0.02 0.04 0.06 0.08 ∗ Exx  Es Figure 3.9: Polar plot of effective Youngs modulus non dimensionalised with respect to the Youngs modulus of the parent material (Es ) for the studied topologies (see Figure 3.5). Each contour corresponds to a particular lattice topology and provides it’s direction dependent effective modulus information in any given direction. Here, the co-ordinate (0,0) is the origin. The length of a vector drawn from the origin to a specific point on the contour is the magnitude of effective modulus in the direction of this vector. Note that ρ¯ = 0.2 for all lattice topologies. The boxed topologies are stretch dominated. A circular plot indicates isotropy.  58  0.03 0.02 0.01  G∗ yy Es  0  0.01 0.02 0.03 0.03  0.02  0.01  0 ∗ Gxx  0.01  0.02  0.03  Es Figure 3.10: Polar plot of effective shear modulus non dimensionalised with respect to the Youngs modulus of the parent material (Es ) for the studied topologies (see Figure 3.5). Each contour corresponds to a particular lattice topology and provides it’s direction dependent effective modulus information in any given direction. Here, the co-ordinate (0,0) is the origin. The length of a vector drawn from the origin to a specific point on the contour is the magnitude of effective modulus in the direction of this vector. Note that ρ¯ = 0.2 for all lattice topologies. The boxed topologies are stretch dominated. A circular plot indicates isotropy.  59  0.005 0.004 0.003 0.002  G∗ yy Es  0.001 0 0.001 0.002 0.003 0.004 0.004  0.002  0 ∗ Gxx  0.002  0.004  Es Figure 3.10: Enlarged view of the effective shear modulus polar plot. providing additional information about their isotropy and bandgaps. The method is demonstrated for perfect 2-D orthotropic lattice topologies. It could be easily extended to incorporate perfect 3-D orthotropic lattice topologies. Moreover, it is assumed that the lattice topologies are perfect and infinite. In reality, the finiteness and the presence of imperfections can drastically degrade the effective material properties of some lattice topologies. The effect of finiteness on effective material properties of lattice materials is explored in the next chapter.  60  Chapter 4  Elastic Boundary Layers 4.1  Introduction  The effective moduli estimated in Chapter 4 using the Floquet-bloch theorem assumed a lattice of infinite dimension without any boundaries. Lattice materials employed in applications possess edges due to their finite dimensions and therefore their effective material properties differ from that of its infinite counterpart. Owing to the edges in finite lattices, localized deformations emerge from the edges or boundaries and decay normal to the edge into the interior in the form of an elastic boundary layer on the application of small load. Elastic boundary layer spreads the localized deformations from the boundaries into the interior of the lattice medium. Hence, elastic boundary layer degrades the effective moduli and the degradation scales with the depth of elastic boundary layer. The fundamental cause of elastic boundary layer is the difference in the nodal connectivity at the edges and the interior of the cellular materials, as a result of which, the stress states differ in the interior and at the free surface. Therefore, elastic boundary layer emerges to accommodate smooth transition from stress state at the surface to a different stress state in the interior. In the area of lattice materials, this phenomena was first observed by Kueh et al. [38] in a Kagome weave made out of a carbon fibre epoxy composite. Its effective moduli differed in different directions and exhibited width dependent effects due to the presence of deep elastic boundary layer. Later, Fleck and Qiu [31] 61  conducted a Finite Element simulation on the Kagome, Hexagonal and Tetragonal lattice topologies to analyse their macroscopic fracture toughness. They observed increased macroscopic toughness in Kagome due to the presence of deep elastic boundary layer emerging from the crack tip, which is essentially a traction free surface. They reported that an elastic boundary layer resulted in localized deformations around the crack tip causing crack tip blunting. Crack tip blunting reduced the stress concentration at the Kagome crack tip and increased the macroscopic toughness of Kagome lattice topology. Phani et al. [37] corroborated their findings and reported that elastic boundary layer depth scales inversely with relative density for a Kagome lattice and is of the order of a strut length for the Hexagonal and Triangular lattice. Phani et al. [39] developed a semi-analytical formulation based on a quadratic eigenvalue problem to check for the presence of elastic boundary layers. These studies have identified the presence of a deep elastic boundary layer as an important parameter in determining the degradation of effective moduli in finite lattice materials. The degradation in the effective moduli scales with the depth of elastic boundary layer and the depth of elastic boundary layer is dependent on the lattice topology. However, these studies are restrictive in the sense that only a particular edge corresponding to a given lattice topology has been examined. In general, a finite lattice topology may possess multiple types of edge with each type of edge having a different nodal connectivity. Therefore, for each of these edges, elastic boundary layer may or may not bear significant depth. Thus, the effective material property degradation due to elastic boundary layer will depend upon the choice of the edge or its orientation in a finite lattice. In this work, an exhaustive analysis is conducted for all possible types of edges in previously studied topology set introduced in Figure 3.5 of Chapter 3. These configurations are examined for the presence of elastic boundary layer using a dynamic wave formulation [39], which is subsequently specialized for zero frequency; since effective moduli are valid only for a static deformation process. In the formulation, elastic boundary layer is envisioned as an exponentially decaying wave of zero frequency having a wave vector (k) travelling in the +x direction (see Figure 4.1). Thus, its wave vector component in +y direction will be equal to zero (ky = 0). This will yield k2 = 0 since the basis vectors are aligned along the co62  ordinate x−y axis. The edges possessing deep elastic boundary layer are identified. Each edge pertaining to a lattice topology is termed as a specific configuration of that lattice topology in this work. The organization of this chapter is as follows. The semi-analytical dynamic formulation to investigate the elastic boundary layer is presented first. Boundary conditions of pure shear and the Uni-axial Finite Element (FE) simulation are shown next. These FE simulations validate the results of the semi-analytical formulation. The semi-analytical formulation is then applied to all possible configurations of the studied set of topologies (see Figure 3.5) and the configurations exhibiting deep elastic boundary layers are identified. Finally, pure shear and Uni-axial Finite Element (FE) simulation are conducted on the identified configurations to validate the presence of a deep elastic boundary layer.  4.2  Formulation  This formulation as adopted from the work of Phani et al. [39] only differs in formulating a linear eigenvalue problem instead of a quadratic eigenvalue problem. It is applicable to a unit cell without any corner degrees of freedom (see Chapter 3). This implication is not a limitation for the formulation since for any lattice, a unit cell without corner degrees of freedom can always be envisioned. A hypothetical unit cell and the corresponding lattice is shown in Figure 4.1 for the purpose of this formulation. This lattice has one edge defined by x = 0 and is infinite in all the other directions. The Euler-Lagrange equations of motion for this unit cell can be written in the form: Mq¨ + Kq = 0,  (4.1)  where M and K are the assembled mass and stiffness matrices, q and f denote the displacement degrees of freedom and the external nodal force of the unit cell respectively. The unit cell is discretized into a rigid jointed network of Timoshenko beams with each node possessing two translations in the x − y plane and one rota-  tion about the z − axis normal to the plane of the paper. Timoshenko beam captures  the potential energy due to shear, bending and tensile deformations. Assuming harmonic solution of the form q (t) = q e jωt and substituting it in 63  ∞  y  qr  qb  qi  O  ∞  Edge  e2, y  qt  ql (a)  e1, x  x  O  ∞  (b)  Figure 4.1: A hypothetical unit cell in (a) is tessellated along the basis vectors e1 and e2 to constitute a finite lattice in (b). Subscripts [l, r, , b, t and i] denote the [le f t, right, top, bottom and interior] degrees of freedom of the unit cell and q represents the corresponding displacement vector. Note that the lattice in (b) possesses an edge and is infinite in all the other directions. Equation (4.1), we get Kq − ω 2 Mq = 0 or Dq = f,  (4.2)  where D is the dynamic stiffness, q is the harmonic displacement. For the present case of static elastic boundary layer determination, D = K since ω = 0. The degrees of freedom in Equation (4.2) can be partitioned to the form :  64    Dll  Dlr  Dlb  Dlt    Drl Drr Drb Drt   Dbl Dbr Dbb Dbt    Dtl Dtr Dtb Dtt Dil Dir Dib Dit  Dli     Dri     Dbi    Dti   Dii  ql        qr      qb  =   qt   qi  fl     fr   fb  ,  ft  fi  (4.3)  where subscripts (l, r, t, b and i) are the edge and interior degrees of freedom. They are shown in Figure 4.1. For the unit cell in Figure 4.1, satisfying equilibrium and compatibility at the exterior nodes in +y direction using the Floquet-bloch theorem, the following relations are obtained: fr + e−ιk2 fl = 0  (4.4)  fi = 0  (4.5)  qr = e−ιk2 ql ,  (4.6)  where k2 = k · e2 , k is the wave vector and e2 is a basis vector. Here, k2 = 0 since  the static elastic boundary layer is envisioned as an exponentially decaying wave along the +x direction. The wave emerges from the edge of the lattice and decays in amplitude as it propagates into the interior of the lattice medium. Substituting qr in terms of ql in Equation (4.3), and then substituting f in terms of q using Equation (4.3) in Equations (4.4)-(4.5), we obtain the following relations Drl ql + e−ιk2 Drr ql + Drb qb + Drt qt + Dri qi + e−ιk2 [Dll ql + e−ιk2 Dlr ql + ... Dlb qb + Dlt qt + Dli qi ] = 0, (4.7) where −ιk2 qi = −D−1 ql + Dib qb + Dit qt ] ii [Dil ql + Dir e  65  (4.8)  The above equations can be better represented in the matrix form as  Cll Clb Clt  ql       qb  = 0, qt where  Cll = Drl + e  −ιk2  −ιk2  −ι2k2  −ιk2  −ιk2 Drr + e Dll + e Dlr − e Dli D−1 Dir ]... ii [Dil + e −ιk2 −Dri D−1 Dir ] ii [Dil + e −1 Clb = Drb + e−ιk2 Dlb − e−ιk2 Dli D−1 ii Dib − Dri Dii Dib −1 Clt = Drt + e−ιk2 Dlt − e−ιk2 Dli D−1 ii Dit − Dri Dii Dit  (4.9) The equilibrium equations along the +x direction can be written as: fb = Dbl ql + Dbr qr + Dbb qb + Dbt qt + Dbi qi ft = Dtl ql + Dtr qr + Dtb qb + Dtt qt + Dti qi (4.10) From the Equation (4.9), ql can be written as: ql = −C−1 ll [Clb qb + Clt qt ]  (4.11)  Substituting qi from Equation (4.8) and ql from Equation (4.11) in Equation (4.10), we deduce the following equations in matrix form: fb ft  =  Pbb Pbt  qb  Ptb  qt  66  Ptt  ,  (4.12)  where −ιk2 −ιk2 Pbb = Dbb − Dbi D−1 Dbr − Dbi D−1 Dir ]]C−1 ii Dib − [Dbl + e ii [Dil + e ll Clb −ιk2 −ιk2 Pbt = Dbt − Dbi D−1 Dbr − Dbi D−1 Dir ]]C−1 ii Dit − [Dbl + e ii [Dil + e ll Clt  −ιk2 −ιk2 Ptb = Dtt − Dti D−1 Dtr − Dti D−1 Dir ]]C−1 ii Dib − [Dtl + e ii [Dil + e ll Clb −ιk2 −ιk2 Ptt = Dtt − Dti D−1 Dtr − Dti D−1 Dir ]]C−1 ii Dit − [Dtl + e ii [Dil + e ll Clt  (4.13) Equation (4.12) can be re-written in the form: qt ft  =  −P−1 bt Pbb  Ptb − Ptt P−1 bt Pbb  −P−1 bt  qb  −Ptt P−1 bt  −fb  (4.14)  From the Floquet-bloch theorem, the top and bottom degrees of freedom are also connected by the relation: qt ft  = e−ιk1  qb −fb  =λ  qb  (4.15)  −fb  Inserting Equation (4.15) in Equation (4.14), yields the eigenvalue problem [T − λ I]s = 0, where λ = e−ιk1 , s =  qb  −P−1 bt Pbb  and T =  (4.16) −P−1 bt  . −1 −fb Ptb − Ptt P−1 bt Pbb −Ptt Pbt To avoid the problems of ill conditioning, a general eigenvalue problem of  the form As = λ Bs can be formulated by modifying Equation (4.16), where A = Pbb I −Pbt 0 qb ,B= , λ = e−ιk1 and s = . Ptb 0 Ptt I −fb  4.3  Eigenvalue Problem  The solution of the eigenvalue problem (Equation (4.16)), yields 2N solutions of λ , where N is the number of edge degrees of freedom. Out of these 2N solutions, the N solutions correspond to the waves travelling in the +x direction (from the boundary 67  to the interior) are valid. The remaining N solutions are invalid and correspond to the waves travelling in the −x direction (from the interior to the boundary). Hence,  we choose the valid N solutions based on the direction of wave propagation. This  is decided by the eigenvalues (λ ) obtained by solving Equation (4.16). Based on the eigenvalues (λ ), four cases arise. These cases are interpreted by the relation qt = λ qb (Equation (4.6)) • |λ | < 1: This condition implies that the particle amplitudes are decreasing  in the +x direction. For an exponentially decaying wave travelling in the +x direction, this behaviour is expected. Hence, this condition constitutes a valid solution.  • |λ | > 1: The particle amplitude increases in the +x direction. Thus, the wave  travels in the negative x direction and hence we ignore the solution pertaining  to this case. • |λ |  0.99999: In this condition, the wave travels in the +x direction with  very low amplitude attenuation resulting in deep elastic boundary layer.  • |λ | = 1: This case will correspond to the rigid body modes and hence these solutions would be ignored. These will be four in number. Two will pertain to the waves travelling +x direction and the remaining two will pertain to the waves travelling in the −x direction. We obtain the eigenvector (s) in Equation (4.16) corresponding to the valid eigenvalues (λ ). Then, by the back substitution of these eigenvector (s) in Equation (4.15), (4.11), (4.6) and (4.8), the corresponding mode shapes of the unit cell are constructed. Finally, using the Floquet-bloch theorem i.e., by employing Equation (4.11) and Equation (4.15), the corresponding mode shapes of the entire lattice are generated. In order to distinguish between deep, moderate and low elastic boundary layer depths in a given lattice, following convention is chosen. 0.75 < λ < 1 corresponds to deep elastic boundary layer, 0.5 < λ < 0.75 corresponds to moderate depth of elastic boundary layer and 0 < λ < 0.5 points to negligible depth of elastic boundary layer in the present work.  68  4.4  Finite Element Simulations  The above formulation detects elastic boundary layers emerging from a lattice material edge subjected to a uniform macroscopic static loading of Uni-axial or pure shear tests. The finite element simulation is performed using a commercial FE package ABAQUS. A 2-D Timoshenko beam element is chosen from the element library which permits three degrees of freedom (ux , uy , θz ) per node. These are two translations (ux , uy ) in the x − y directions and one rotation (θz ) about the axis per-  pendicular to the paper plane. A large lattice is considered such that the boundary layers emerging from the edges can fully decay and a homogeneous stress state is prevalent in the interior. The stress field resulting from the uniform macroscopic static loading are a summation of a particular solution and a complementary solution. They are explained as follows: • Particular solution: This is referred to as the uniform stress state prevailing in the interior of a lattice medium after the application of a uniform macroscopic load. • Complementary solution: For a given, macroscopic loading condition, the total resulting stress solution is the sum of Particular solution and Complementary solution. Since, particular solution is uniform and the total solution varies near the edges, hence Complementary solution is essentially the transient exponentially decaying stress part. It helps to transition from the stress state on the edges to the uniform stress states referred by particular solution in the interior. Thus, the Complementary solution reduced to zero in the interior and is maximum at the edges. Given a macroscopic loading condition, the complementary solution arising out of it will be the summation of elastic boundary layer eigenvectors in certain proportions. These proportions or contributions of the eigenvectors can be deduced from the boundary conditions prevailing at the edges pertaining to a specific macroscopic loading condition. Note that the eigenvectors are a property of the system and hence do not depend on loading conditions. On the other hand, the contribution factors are dependent on the specific loading condition. Thus, for a given topology possessing a deep elastic boundary layer vector, it is possible that for a certain macroscopic loading condition say Uni-axial test or pure shear, 69  Top Edge  Y  Y Z  Right Edge  Left Edge  Right Edge Left Edge  Top Edge  X  X  Z  Bottom Edge (a)  Bottom Edge (b)  Figure 4.2: A lattice presenting the boundary conditions used to simulate Uni-axial tension test is shown in (a). (b) presents the lattice demonstrating the boundary conditions for a pure shear test. The wedges or triangles correspond to the translational constraints. The arrows denote the prescribed displacement vectors. the contribution factor associated with the deep elastic boundary layer vector may be very small. Thus, the topology might not show the presence of a deep elastic boundary layer in the simulation even though it possess deep elastic boundary layer. Therefore, FE simulation results should not be entirely relied on for the prediction of deep elastic boundary layer for a lattice configuration. Hence, the FE simulations results have to be approached with caution and therefore are only used as a validation tool in this work.  4.4.1  Uni-axial Tension Test  To accurately capture the global effects resulting from the Uni-axial tension pertaining to a continuum in a lattice medium, following displacement boundary conditions are applied. One node corresponding to the bottom edge is pinned (ux = 0, uy = 0) and the translational degree of freedom in the y direction of the remain70  Table 4.1: Boundary conditions assumed to simulate pure shear test.  Node Location  Displacement in +x direction (ux )  Displacement in +y direction (uy )  Rotation (θz )  Left edge nodes Right edge nodes Bottom edge nodes Top edge nodes  free free 0 (Constrained) +1 unit  −1 unit 0 (Constrained) free free  free free free free  ing nodes of the bottom edge is restricted (uy = 0). A unit displacement in the +y direction (uy = 1) is applied to the nodes of the top edge to simulate Uni-axial tension test as shown in Figure 4.2. For the pure-shear test, the translation degree of freedom in the x direction of the bottom edge nodes are restricted (ux = 0) and the top edge nodes are given a unit displacement in +x direction (ux = 1). Similarly, the translation degree of freedom in the y direction for the right edge nodes are restricted (uy = 0) and the left edge nodes are given a unit displacement in the −y direction (uy = −1).These  displacement boundary conditions are summarized in Table 4.1 and are shown in Figure 4.2.  4.5  Configurations with Deep Elastic Boundary Layer  All possible configurations generated from the topologies studied in the previous chapter in Figure 3.5 were analysed. The edges are produced by rotating these topologies and cutting them sideways. The struts emanating from the resulting edges are terminated at the nodes. This procedure ensures that practically feasible edges are produced. The resulting edges which remain parallel to the y axis are configurations assumed to be valid which can be employed in a sandwich configuration effectively for bearing structural loads. A total of 53 configurations resulted from the previously studied topologies. Out of these 53 configurations, 5 configurations showed deep elastic boundary layer. Figures 4.3-4.4 identifies the deep elastic boundary layer configurations. In 71  ∞  e1, x  ∞  e2, y  Edge  e2, y  Edge  ∞  e1, x  ∞ (a)  e1, x  ∞  e2, y  ∞  e1, x  (c)  ∞ (b) ∞  Edge  e2, y  Edge  ∞  ∞  ∞  ∞ (d)  Figure 4.3: Unit cells are tessellated along the basis vectors to constitute finite Kagome configuration 1, Kagome configuration 2, Diamond configuration 1 and Hybrid-A configuration 1 shown in (a), (b), (c) and (d) respectively. e1 and e2 constitute the basis vectors.  72  ∞  Edge  e2, y ∞  ∞  e1, x  Figure 4.4: Unit cells are tessellated along the basis vectors to constitute finite Hybrid-A configuration 2. e1 and e2 constitute the basis vectors. Table 4.2: Deep elastic boundary layer configurations and their parent topologies  Configurations  Parent topology  Clockwise rotation of parent topology  Configuration 1 Configuration 2 Configuration 3 Configuration 4 Configuration 5  Kagome Kagome Diamond Hybrid-A Hybrid-A  0o 0o 30o 90o 90o  the following section, results pertaining to elastic boundary layer are presented for these identified configurations.  4.6  Results  The results pertaining to configurations possessing deep elastic boundary layer are presented in this section. The eigenvectors are plotted for ρ¯ = 0.05 to visually  73  Configuration 4  e2, y  qr  qb  qi ql  qt e1, x  +  +  Deep elastic boundary layer for  =  | |  Figure 4.5: Sequential procedure to detect elastic boundary layer in a lattice configuration. show deep elastic boundary layer modes. Finite element simulations confined to Uni-axial tension and pure shear test are used to verify the presence of deep elastic boundary layer predicted by the semi-analytical formulation. Finally, eigenvalues are plotted as a function of relative density for these deep elastic boundary layer configurations. Next, the results pertaining to Kagome configurations 1 and 2 are presented.  74  (a)  (b)  (c)  (d)  (e)  (f )  Figure 4.6: All eigenvectors corresponding to elastic boundary layer modes for Kagome configuration 1 are shown in (a) eigenvector 1, (b) eigenvector 2 , (c) eigenvector 3 and (d) eigenvector 4 for ρ¯ = 0.05. Eigenvectors corresponding to deep elastic boundary layer modes for Kagome configuration 2 are shown in (e) eigenvector 1 and (f) eigenvector 2 for ρ¯ = 0.05.  4.6.1  Kagome Configurations 1 and 2  Kagome configurations 1 and 2 are derived from the Kagome topology. The procedure to search for elastic boundary layer is summarized using the flowchart in Figure 4.5. For these configurations shown in Figure 4.3, first the equations of motion are set up for the corresponding unit cells. The procedure is then specialized for zero frequency and an eigenvalue problem in λ is set up and solved to search for exponentially decaying waves in the +x direction. Physically, (1 − |λ |) represents  the fractional decay of elastic boundary layer wave amplitude as it traverses a sin75  (a)  (b)  (c)  (d)  Figure 4.7: FE simulation showing deformations resulting from Uni-axial test for Kagome configurations 1 and 2 in (a) and (c) for ρ¯ = 0.05. (b) and (d) show deformations resulting from pure shear test for Kagome configurations 1 and 2 respectively for ρ¯ = 0.05. Boundary layer is deeper in pure shear test as compared to the Uni-axial tension test. gle unit cell (Equation (4.15)). This could be visually observed in the eigenvector corresponding to elastic boundary layer modes. Figure 4.6 presents the corresponding eigenvectors of Kagome configurations 1 and 2 for a relative density of 0.05. Deep elastic boundary layers are observed for Mode I and Mode II (|λ | > 0.75) for these two configurations for ρ¯ = 0.05. The  remaining modes showed negligible depth of elastic boundary layer. In the current work, the eigenvectors are sorted and first mode corresponds to highest eigenvalue and the last mode corresponds to the least eigenvalue. 76  Deep elastic boundary layer are observed in the pure shear and Uni-axial FE simulation in Figure 4.7 for a relative density of 0.05. The elastic boundary layer is deeper in the pure shear FE simulation as compared to the Uni-axial test. These FE simulations verify the results of semi-analytical formulation. In order to understand the elastic boundary layer scaling with relative density, the eigenvalues of these configurations are plotted as a function of relative density in Figure 4.8. Both configurations display the same dependence of eigenvalues on the relative density. The eigenvalues corresponding to deep elastic boundary layer modes (mode I and II) show an approximately linear decrease in magnitude with an increase in relative density. However, the decrease in the deepest elastic boundary layer mode (Mode I) is not appreciable with the increase in relative density (see Table 4.3). Even at a high relative density of 0.2, the eigenvalue corresponding to mode I remains close to 0.75 for these configurations. Hence, these configurations are susceptible to moduli degradation even at large relative densities of the order of 0.2.  4.6.2  Diamond Configuration 1  Diamond configuration 1 is derived from a Diamond topology, Figure 4.9 presents its eigenvectors for a relative density of 0.05. For this configuration, a deep elastic boundary layer is only observed for Mode I (|λ | > 0.75) for this relative density.  The remaining modes show a negligible depth of elastic boundary layer.  FE simulations shown in Figure 4.10 suggests deep elastic boundary layer for the case of pure shear and absence of deep elastic boundary layer for the case of Uni-axial tension. This observation corroborates the need for a semi-analytical formulation to search for the presence of deep elastic boundary layer, since FE simulations may not always demonstrate the presence of a deep elastic boundary layer in a lattice configuration. Moreover, the drop in the depth of the deepest elastic boundary layer with the increase in relative density is appreciable for this configuration (see Figure 4.9). Hence, this confuration is not susceptible to moduli degradation for high relative densities of the order of 0.2.  77  1.2 1  Mode I  0.8  Mode II  |λ | 0.6 0.4 0.2 0  0  0.1  (a)  ρ¯  0.2  1.2 1  Mode I  0.8  Mode II  |λ | 0.6 0.4 0.2 0  (b)  0  0.1  ρ¯  0.2  Figure 4.8: Modulus of eigenvalues that correspond to elastic boundary layer modes are plotted as a function of relative density for Kagome configuration 1 in (a) and Kagome configuration 2 in (b). Both the configurations exhibit the same eigenvalue or boundary layer characteristics. 78  1.2  1  0.8  (b)  Mode I  |λ| 0.6 0.4  0.2  0  0  0.05  0.1  ρ¯  0.15  0.2  0.25  Figure 4.9: Modulus of eigenvalues that correspond to elastic boundary layer modes are plotted as a function of relative density for Diamond configuration 1. The boxed figure represents a schematic of the corresponding eigenvector.  4.6.3  Hybrid-A Configurations 1 and 2  These configurations are derived from the Hybrid-A topology. During the FE simulations for ρ¯ = 0.05 in Figure 4.13, deep elastic boundary layers are only observed in Uni-axial tension test and are absent in pure shear test. This result is unlike the case of Diamond configuration 1, where deep elastic boundary layer is only observed for the pure shear case. In Figure 4.11, the decay in the depth of deepest elastic boundary layer cor-  79  (a)  (b)  Figure 4.10: FE simulation showing deformation resulting from Uni-axial test for Diamond configuration 1 in (a) and for pure shear test in (b) for ρ¯ = 0.05. Boundary layer is absent for the Uni-axial tension case. responding to mode I is not appreciable for these configurations. Hence, these configurations are susceptible to moduli degradation at large relative densities of the order of 0.2. Veering: The eigenvectors for these configurations are shown in Figure 4.12. For a ρ¯ = 0.05, Modes I and II corresponding to Hybrid-A configuration 1 exhibit bending and stretch dominated behaviour respectively. For the Hybrid-A configuration 2, Mode I and Mode II are bending dominated whereas Mode III is stretch dominated for ρ¯ = 0.05. These facts are evident from the visual inspection of the eigenvectors for these configurations. At a relative density of 0.05, mode I shows deep elastic boundary layer and a moderate depth of elastic boundary layer is observed for mode II for these configurations. Mode III of Hybrid-A configuration 2 exhibits moderate depth of elastic boundary layer. These configurations exhibit the phenomena of veering. Modes I-II of configuration Hybrid-A configuration 1 and Modes II-III of Hybrid-A configuration 2 do not intersect and instead veer away from each other. Mode I of Hybrid-A configuration 1 and Mode II of Hybrid-A configuration 2 transitions from a bending dominated mode at ρ¯ = 0.05 to a stretch dominated mode at a ρ¯ = 0.2. Similarly  80  1.2 1 0.8  |λ |  0.6 0.4 0.2 0  (a)  0  0.1  ρ¯  0.2  1.2 1 0.8  |λ | 0.6 0.4 0.2 0  (b)  0  0.1  ρ¯  0.2  Figure 4.11: Modulus of eigenvalues that correspond to elastic boundary layer modes are plotted as a function of relative density for Hybrid-A configuration 1 in (a) and Hybrid-A configuration 2 in (b). The curves do not intersect at ρ¯ = 0.1. Note the phenomenon of veering.  81  1.2 1  Mode I  0.8  |λ | 0.6  Mode II  0.4 0.2 0  0  0.1  ρ¯  0.2  (a) 1.2 1 0.8  Mode II  |λ | 0.6 0.4  Mode III  0.2 0  0  0.1  ρ¯  0.2  (b) Figure 4.12: Modulus of eigenvalues that correspond to elastic boundary layer modes prone to veering are plotted as a function of relative density for Hybrid-A configuration 1 in (a) and Hybrid-A configuration 2 in (b). Along with the eigenvectors corresponding to the eigenvalues are plotted at extreme ends of relative density to highlight the phenomena of veering. Note that the curves do not intersect at ρ¯ = 0.1.  82  (a)  (b)  (c)  (d)  Figure 4.13: FE simulation showing deformations resulting from Uni-axial test for Hybrid-A configuration 1 and 2 in (a) and (c) respectively for ρ¯ = 0.05. (b) and (d) show deformations resulting from pure shear test for Hybrid-A configuration 1 and 2 respectively for ρ¯ = 0.05. Boundary layers are absent in the pure shear case. Mode II of configuration 4 and Mode III of Hybrid-A configuration 2 transitions from a stretch dominated mode at ρ¯ = 0.05 to a bending dominated mode at a ρ¯ = 0.2. Hence, the closely spaced bending and stretching dominated modes are interchanged between each other. Veering is a hallmark phenomenon of linear systems which exhibit closely spaced natural frequencies.  83  Table 4.3: Comparison of the identified configurations on the basis of deep elastic boundary layer  Topology  Kagome configuration 1 Kagome configuration 2 Diamond configuration 1 Hybrid-A configuration 1 Hybrid-A configuration 2  4.7  Percentage decay of deepest eigen mode across a unit cell ρ¯ = 0.05 8% 8% 22% 18% 14%  ρ¯ = 0.1 5% 5% 39% 32% 24%  ρ¯ = 0.15 22% 22% 53% 32% 29%  ρ¯ = 0.2 28% 28% 63% 32% 31%  Conclusions  In this chapter, a semi-analytical formulation has been developed and implemented to identify the presence of a deep elastic boundary layer in a lattice configurations. Five configurations were identified out of the previously studied topologies possessing deep elastic boundary layer modes. A comparison of these configurations on the basis of deep elastic boundary layer is presented in Table 4.3. Excluding Diamond configuration 1, all the other configurations are moderately susceptible to moduli degradation at high relative densities of the order of 0.2. Hence, their effective macroscopic material properties based on an infinite lattice should be approached with caution for high relative densities. Kagome configurations 1 and 2 display an extremely low 8% decay for the deepest Eigen mode at a relative density of 0.05 (see Table 4.3). Therefore, out of the five configurations, Kagome configurations 1 and 2 have highest susceptibility to moduli degradation at low relative densities. The proposed technique in this Chapter is specialised to detect elastic boundary layer in static macroscopic loading conditions only. Suitable adjustments can be employed to enable the formulation to identify elastic boundary layer in dynamic macroscopic dynamic conditions. This formulation can easily be extended to check for elastic boundary layer in 3-D lattice configurations.  84  The formulation developed/adopted hitherto, assumes perfect topologies and does not incorporate the presence of imperfections. The extension of the formulation to incorporate imperfections is not trivial and is a subject of future work.  85  Chapter 5  Conclusions 5.1  Summary of Present Work and Major Findings  1. In the present work, a Wave-based Dynamic Method is developed to predict the Effective Material Properties of infinite lattice materials. The proposed method exploits the periodicity of lattice materials using Floquet-bloch theorem. Consequently, the developed method requires modelling only one unit cell of the periodic lattice, making it computationally efficient. Due to the finite element approach adopted, the current method could be employed for any complex orthotropic lattice topology. Besides deducing effective properties, the method reveals wave propagation information in the form of dispersion curves for fully anisotropic lattices. Dispersion curves can be used to predict band gaps. Band gaps are very useful when lattice materials are to be used for filtering of waves in frequency and spatial domain. Since the lattice is envisioned as a network of Timoshenko beams, the method in the current form is restricted to lattice materials having low relative densities (ρ¯ < 0.2). This is because at higher relative densities, the aspect ratio of struts would decrease and the thick struts would not behave as beams. 2. A semi-analytical formulation based on a linear eigenvalue problem is developed to predict the elastic boundary layers emanating from the edge of a finite lattice. The method is adopted from the formulation developed by Phani  86  et al [39] that is based on a quadratic eigenvalue problem. The developed formulation exploits the periodicity of lattice materials using Floquet-bloch theorem and thus requires modelling only one unit cell. This lends the current method higher computational speed. Moreover, the developed method can be employed to predict boundary layers pertaining to any complex lattice topology which can significantly degrade the predicted effective moduli of lattice materials. The degradation in effective moduli scales with the depth of boundary layer. The boundary layer depth is highly sensitive to relative density and scales inversely with it. The developed formulation was applied to the chosen set of 13 topologies of which 3 topologies were identified to possess deep boundary layers.  5.2  Suggestions for Future Work  1. The present study is simplified by restricting to orthotropic lattice materials. This is because, the effective properties of fully anisotropic lattices cannot be explicitly deduced in terms of the phase velocity data obtained from the dispersion curves, unlike that for the case of orthotropic lattices. The deduction of material properties for the fully anisotropic case will require solving a non-linear inverse problem. Generalising the present method to fully anisotropic remain to be done. 2. Current formulations are accurate for low relative densities since a lattice is envisioned as a collection of Timoshenko beams. Thus, it may be useful to extend these formulations to high relative density regime by envisioning lattices as holes in a shell. Finite element shell elements could be employed for this purpose. 3. The present techniques assume perfect lattices. Imperfections such as cell wall waviness and missing cell walls can influence the predictions of these methods. Since manufactured lattices have imperfections built into them, it is recommended to extend these methods to incorporate imperfections in the future. These future methods can serve as a qualitative tool to probe deeper into imperfections pertaining to lattice materials and their effect on 87  the predicted properties. 4. The methods developed in this work are restricted to the linear elastic range. When selecting lattice materials, it is important for a design engineer to discern the strength parameters of lattice materials. These parameters fall under the non-linear domain. The current techniques employed to predict the strength related parameters require intensive finite element calculations which do not exploit the periodicity of lattice materials. Future work can consider the nonlinear response regime.  88  Bibliography [1] Ashby, M. F., 1983. “The mechanical properties of cellular solids”. Metallurgical and Materials Transactions A, 14, pp. 1755–1769. [2] Ashby, M. 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