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 spe- cific stiffness, thermal conductivity, energy absorption and vibration control by attenuating forcing frequencies falling within the band gap region. Analytical ex- pressions have been proposed in the past to predict cell geometry dependent ef- fective material properties by considering a lattice as a network of beams in the high porosity limit. Applying these analytical techniques to complex cell geome- tries 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 mate- rial 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 identi- fied. 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 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Why Lattice Materials? . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Scope of the Current Work . . . . . . . . . . . . . . . . . . . . . 11 2 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 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 4 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 4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 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 . . . . . . . . . . . . . . . . . . . . . . . . . . 84 v 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 lat- tice 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Figure 1.7 Photograph of a Hex-chiral lattice possessing negative Pois- son’s ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Figure 1.8 A lattice possessing effective zero coefficient of thermal ex- pansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Figure 1.9 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 adja- cent 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. . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Figure 2.6 Elastic boundary layer emanating from the edge of a Kagome lattice and decaying into the interior . . . . . . . . . . . . . . 23 vi Figure 2.7 Finite element simulation of a notched Hexagonal, Triangular and Kagome lattice under a uni-axial load . . . . . . . . . . . 24 Figure 3.1 Picture conceptualizing Floquet-bloch theorem using a hypo- thetical 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 Bril- louin 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 analyti- cal 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 . 64 Figure 4.2 A lattice presenting the boundary conditions used to simulate Uni-axial tension test and pure shear test . . . . . . . . . . . . 70 Figure 4.3 Deep elastic boundary layer configurations comprising of Kagome configurations 1-2, Diamond configuration 1 and Hybrid-A configuration 1 . . . . . . . . . . . . . . . . . . . . . . . . . 72 Figure 4.4 Deep elastic boundary layer configurations comprising of Hybrid- A configuration 2 . . . . . . . . . . . . . . . . . . . . . . . . 73 Figure 4.5 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 . . . . . . . . . . . . . . . 75 Figure 4.7 FE simulation showing deformations resulting from Uni-axial test and pure shear test for Kagome configurations 1 and 2 . . 76 Figure 4.8 Modulus of eigenvalues that correspond to elastic boundary layer modes are plotted as a function of relative density for Kagome configurations 1 and 2 . . . . . . . . . . . . . . . . 78 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 . . . . . . . . . . . . . . . . . . . . 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 rela- tive 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 . . 83 viii Glossary Symbols Es Young’s modulus of strut material E∗xx Effective Young’s modulus in x direction E∗yy Effective Young’s modulus in y direction G∗xy Effective shear modulus in x− y direction ν∗xy,ν∗yx Effective Poisson’s ratios E∗ Isotropic effective Young’s modulus G∗ Isotropic effective shear modulus K∗ Isotropic in-plane 2-D bulk modulus Ct Magnitude of transverse phase velocity Cl Magnitude of longitudinal phase velocity k1,k2 Components of wavevectors along basis vectors ρ̄ Relative density L Strut length ρ∗ Effective density Ω Propagating frequency non-dimensionalized with respect to first pinned-pinned frequency of the strut n j Unit wave normal v Phase velocity k Wavevector ε Strain tensor σ Stress tensor r Position vector K Stiffness matrix M Mass matrix D Dynamic stiffness matrix f External force q Generalised displacement vector ci jkl Fourth order constitutive tensor ix Technical Terms Veering A feature of eigenvalue problems of weakly cou- pled systems where two eigenvalue branches ap- proach each other as a parameter is varied. The branches do not cross, but veer away, exchanging the eigenvectors in the process. Knock down Degradation in stiffness noted due to the presence of deep elastic boundary layer Toughness The energy absorbing ability of a material Stretch dominated A lattice topology that stores potential energy due to deformation predominantly in tensile deforma- tion of the struts Bending dominated A lattice topology that stores potential energy due to deformation predominantly in bending deforma- tion of the struts Tessellated To form into a lattice by assembling unit cells along the basis vector directions Unit cell A repeating unit of a periodic lattice Basis vectors The vectors prescribing the direction along which unit cells should be repeated to constitute a lattice Abbreviations EMP Effective Material Properties FE Finite Element DNA 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 per- spective. 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 To My Parent 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 simul- taneous 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 opti- mal 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 ma- terials to suit specific functional needs in the most optimized manner with respect to minimum weight and energy. 1 2h h (a) (c) (b) 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 E 1/3 ρ E 1/2 ρ E ρ guide lines for minimum mass design Yo un g’ s m od ul us , E (G Pa ) 10–4 10–3 10–2 10–1 1 10 100 1000 (a) (b) polyester foams polymers metals technical ceramics composites natural materials lead alloys W alloys steels Ti alloys Mg alloys CFRP GFRP Al alloys exible polymer foams Ni alloys Cu alloys zinc alloysPA PEEK PMMA PC PET cork wood butyl rubber silicone elastomers concrete WC Al 2O3 SiC Si 3N4 B 4C epoxiesPS PTFE EVA neoprene isoprene polyurethane leather MFA, 2010 PP PE glass // grain grainT elastomers wood non-technical ceramics bamboo P i rl CFRP lattice CFRP foam aluminium lattice aluminium foam σf 1/2 ρ σf 2/3 ρ σf ρ density, ρ (kg m3) 10 100 1000 10 000 MFA, 2010 guide lines for minimum mass design st re ng th , σ f (M Pa ) 0.01 0.1 1 10 100 1000 10 000 PP PE woods, T foams polymers and elastomers metals ceramics composites natural materials lead alloys tungsten alloys steels Ti alloys Mg alloys CFRP GFRP Al alloys exible polymer foams Ni alloys copper alloys zinc alloys PA PEEK PMMA PC PET cork woods, ll butyl rubber silicone elastomers concrete tungsten carbide Al 2O3SiC Si 3N4 , s CFRP lattice CFRP foam aluminium lattice aluminium foam Ideal zone Ideal zone 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 perfor- mance, multi-functional applications in aerospace and automotive industry. In or- der to understand the necessity for such materials, a comparison of existing mate- rials 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 con- cluded 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 mate- rial, 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 (num- ber 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]. Stretch- ing dominated lattices are preferred as they are structurally efficient. Foams with random microstructure are bending dominated and they are usually isotropic. Lat- tice 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 core sandwich structures at different relative densities (ρ̄) [9]. 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 55° 55°150°60° 70° punch tetrahedral core roll steel single lay al itlumeroc re yer core waste die 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 lat- tice. 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 ra- tio (length to radius) and strut diameter. For instance, a template/mould or planar sheet manufacture can be accomplished by laser micro machining, rapid prototyp- ing, 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/pla- nar 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 stand- point 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 struc- tural 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 ther- mal 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 mate- rial, 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 lat- tice [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 ab- sorption 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 in- side 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]. e1 e2 to (a) (b) Figure 1.8: A Lattice possessing effective zero coefficient of thermal expan- sion. 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 rep- resent 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 mod- uli such as Young’s modulus and shear modulus. This dissertation studies micro- architecture 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 rel- ative density and direction of loading. In Chapter 2, the literature concerned with structural properties of lattice materials is surveyed. This will be followed by elas- tic 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 for- mulation, 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 tessel- lating 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 materi- als 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 jklεkl (2.1) Equation (2.1) represents the Hooke’s law for an equivalent continuum that corre- sponds 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 dis- cussed 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 clas- sified as stretching dominated or bending dominated. Cell walls deform predom- inantly 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 ∝ ρ̄ , Strength ∝ ρ̄ (Stretch dominated) (2.2) Strength and stiffness scale non-linearly for bending dominated lattice topologies. This is given by [3]: Stiffness ∝ ρ̄2 , Strength ∝ ρ̄1.5 (Bending dominated) (2.3) Thus, for a ρ̄ = 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 determi- nate 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 ρ̄ ∝ 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 in- ternal 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 neces- sary 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 suffi- cient 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 pre- dominant 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 in- 15 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 theo- rem. Their formulation was not confined to similarly situated lattices and could be applied to any lattice topology. In particular, their method could identify the pres- ence of periodic and strain producing mechanisms in a pin-jointed lattice. Note that periodic mechanisms do not contribute to macroscopic strains. Hence the topolo- gies that possess only periodic mechanisms are stretch dominated. They observed that Kagome lattice and Triangular lattice are macroscopically stiff as they are in- capable 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, El- sayed 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 con- sidering 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 ma- terials. For instance, stiffness and strength anisotropy of these materials can be ex- ploited for carrying directional loads. Moreover, micro-architecturing of unit cell geometries has made it possible to realise materials with negative effective Pois- son’s ratio also called Auxetics. Scarpa et al. [20] have conducted experimental investigation on Auxetic honeycomb. They demonstrated high sensitivity of mate- rial 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 Triangular Hexagonal Square Kagome No mechanismsPeriodic mechanisms mechanisms producing Strain 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 mecha- nisms. Square and Hexagonal lattice are stretch dominated only in few orientations. Triangular-Triangular (T-T) lattice possess only strain pro- ducing mechanism and is bending dominated in all directions. Triangu- lar lattice possess no mechanisms and is stretch dominated in all orienta- tions. 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 simula- tions could be avoided with the use of effective material properties. In order to deduce, effective material properties, the lattice is envisioned as an effective con- tinuum, 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 Cell Envelope x y 1a2a Lattice Structure 1a 2a 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 regu- lar 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 ρ2 and ρ1.5 respectively for a bending dominated lat- tice. 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 compar- ison 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 in- corporating 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 Fig- ure 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 tak- ing terms until second order derivatives in the Taylor series expansion of displace- ment 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 can- not 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 per- taining 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 look- ing at the suitable stress and strain measures from these tests, one can estimate the effective material properties of lattice materials. Wallach and Gibson [36] numer- ically predicted the effective material properties namely elastic and shear modu- lus 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 dy- namic method. This approach has been adopted by Phani et al. [37] using a Tim- oshenko 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 re- quires 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. How- ever, 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 (a) (b) y z x uy , uz , θx , θy , θz = 0 ux , uz , θy , θz = 0 ux = δ/2 0-direction 90-direction ux = δ/2 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 ob- served for (a) and (b) respectively. 22 uu x1 x2 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 uni- axial 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 effec- tive 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) Tri- angular 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 criti- cal crack size, the tensile and shear strength remained unaffected for the Kagome. Phani et al. [39] corroborated their findings and developed a semi-analytical for- mulation 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 effec- tive 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 de- crease in relative density. However, these studies are restrictive in the sense that only a particular edge cor- responding 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, elas- tic 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 elas- tic 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 signif- icant 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 multi- functional properties [13] such as negative Poisson’s ratio [40] and effective zero thermal coefficient of expansion [14]. 2-D lattice materials such as hexagonal hon- eycombs are frequently used as cores in sandwich panels to provide high struc- tural properties per unit weight. These sandwich panels serve as stiffer weight efficient structural components in industries like missile and aerospace. The struc- tural 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 materi- als. 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 involv- ing 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 ma- terial 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 topol- ogy with ease. This method enables a rapid calculation of direction dependent effective material properties such as Young’s modulus (E∗), effective shear mod- ulus (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 pre- sented 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 effec- tive 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) = qAei(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 O x y A B q(r) r p q(p) qlt qt qrt qr qrbqb qIql e1 e2 (a) (b) (n1 ,n2) qlb 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 con- nected 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 rep- resented 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 displace- ment of any arbitrary point (See Figure 3.1) in the unit cell (n1,n2)with the position vector r = p+n1e1+n2e2 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: Mq̈+Kq = 0 or Kq−ω2Mq = 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 topol- ogy 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 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 1 2 3 4 5 6 7 8 kykx ω nd Bandgap 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−ık1ql, qt = e−ık2qb qrb = e−ık1qlb, qlt = e−ık2qlb, 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 Transforma- tion matrix (T): T(k1,k2) = I 0 0 0 0 I 0 0 0 Ie−ık2 0 0 0 0 I 0 0 0 Ie−ık1 0 0 0 0 I 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 qlb qrb qlt qrt = T(k1,k2) qi qb ql qlb = T(k1,k2)q̃, (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 (q̃): THKTq̃−ω2THMTq̃ = 0 or Ax = λBx, (3.7) where A = THKT, B = THMT, x = q̃ and eigenvalue λ = ω2. 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 eigen- value 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 dis- persion 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 iden- tified 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 wavevec- tor 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=2piδi 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 e1* e2* O A B Figure 3.3: A reciprocal lattice showing the Brillouin zone. Reciprocal lat- tice 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 usu- ally 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. Phys- ically, the group velocity gives the speed and direction of energy propagation cor- responding 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 Cp(Phase Velocity) = ω |k| k̂ (3.8) Cg(Group Velocity) = ∂ω |∂k| n̂, (3.9) 34 where k̂ is a unit vector in the wave vector direction and n̂ 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∗üi = σi j, j, (3.10) where ρ∗ is the density of the medium, σi j is the stress tensor and ui is the displace- ment 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 jklεkl, (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∗üi = ci jkluk,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 = Apkei(krxr−ωt) (3.13) 35 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 jkln jnl−ρ∗v2δik)pk = 0 or [Γi j−ρ∗v2δi 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 vec- tor 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 transla- tional 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 de- grees 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 de- formation 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 di- rection 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 hav- ing 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 dis- persion 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 ma- trix (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 sym- metry direction leads to: σxxσyy σxy = C11 C12 0C12 C22 0 0 0 C33 εxxεyy γxy = Cε (3.15) or εxxεyy γxy = 1 E∗xx − ν∗yxE∗yy 0 − ν∗xyE∗xx 1 E∗yy 0 0 0 1G∗xy σxxσyy σxy = Sσ , (3.16) where E∗xx is the effective Young’s modulus in the x direction, E∗yy is the effective Young’s modulus in the y direction, G∗xy is the effective shear modulus in the x− y direction, ν∗yx and ν∗xy are the effective Poisson’s ratio pertaining to the loading in the x and y directions respectively. Similarly Γi j or ci jkln jnl 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: Γi j = α11 α12 α13α12 α22 α23 α13 α23 α33 (3.17) where α11 =C11n21+C33n 2 2 α22 =C33n21+C22n 2 2 α12 = (C12+C33)n1n2 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: |Γi j−ρ∗v2δi j|= ∣∣∣∣∣∣∣ C11n21+C33n 2 2−ρ∗v2 (C12+C33)n1n2 0 (C12+C33)n1n2 C33n21+C22n 2 2−ρ∗v2 0 0 0 −ρ∗v2 ∣∣∣∣∣∣∣= 0 (3.18) Solving the determinant in Equation (3.18), we obtain: ρ∗v2(C212n 2 1n 2 2−ρ∗v4−C11C33n41−C22C33n42−C11C22n21n22+2C12C33n21n22+ +ρ∗v2(C11n21+C22n 2 2+C33n 2 1+C33n 2 2)) = 0 (3.19) 38 Equation (3.19) contains four unknowns which can be represented as follows: C12 =−C33+ √ (C11n21+C33n 2 2−ρ∗v2)(C33n21+C22n22−ρ∗v2) n1n2 C11 = n21(C 2 12n 2 2+2C33C12n 2 2+C33ρ∗v2)+ρ∗v2(−ρ∗v2+C22n22+C33n22)−C22C33n42 n21(C33n 2 1+C22n 2 2−ρ∗v2) C22 = n22(C 2 12n 2 1+2C33C12n 2 1+C33ρ∗v2)+ρ∗v2(−ρ∗v2+C11n21+C33n21)−C11C33n41 n22(C33n 2 2+C11n 2 1−ρ∗v2) C33 = n22(n 2 1(C11C22−C212)−C22ρ∗v2)+ρ∗v2(ρ∗v2−C11n21) −C11n41+2C12n21n22+ρ∗n21v2−C22n42+ρ∗n22v2 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) 2ρ∗ (3.20) C22 = (Cl/90o) 2ρ∗ (3.21) C33 = (Ct/90o or Oo) 2ρ∗ (3.22) β =−2C2l/45oρ∗+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 repre- sented 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 (3.26) 39 Using Equations (3.25)-(3.26), we have: S−1(E∗xx,E ∗ yy,ν ∗ yx,ν ∗ xy)) = F(Cl/0o ,Cl/90o ,Ct/90o) (3.27) Therefore from Equation (3.27), phase velocity or the long wave length assymp- totes 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∗xxE∗yy (E∗yy−E∗xxν∗2yx ) ρ∗ (3.28) Cl/90o = √ E∗2yy (E∗yy−E∗xxν∗2yx ) ρ∗ (3.29) Ct/90o/Oo = √ G∗xy ρ∗ (3.30) Repeating the above calculations(3.20)-(3.24) for different relative densities en- ables 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∗ ρ∗ ,Ct = √ 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 Dispersion Curve Christoffel’s Equations Effective Material Properties Band Gaps 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∗ = 4K∗G∗ K∗+G∗ . (3.32) 3.7 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 (3.33) 41 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 motion in Bloch reduced co-ordinates (q̃) are obtained: Dq̃ = 0 (3.34) The re-arranging of Equation (3.34) yields the eigenvalue problem: THKTq̃−ω2THMTq̃ = 0 or Ax = λBx, (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 vary- ing 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 sym- metry 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 Equa- tion (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 den- sities. 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 (to) 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 (a) (b) e1 e2 LowCTE e1* e2* to O BA e1 e2 L L θ Auxetic e1* e2* O A B e1* e2* O A B Hybrid A (c) e1 e2 L L Ld L/3 θ (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 * (d) Hybrid C L L L θ e1 e2 O A B e1 e2 θ L 2L e1* e1* O A B (e) e1e2 e1* e2* e2* e2* O A B (f ) Mixed Hybrid D L Hex-Chiral e1e2 θ O A (g) e1*e2 * B θ (ii) Lattice Topologies Figure 3.5 44 L(i) L e1e2 e2* e1* θ O AB (h) Diamond e1 e2 Kagome 2 A e1* e2* B O e1 e2 O A e1* e2* B Tetragonal θ O A e2* Bθ e1 e1* L L (j) Hexagonal e2 (k) (iii) Lattice Topologies Figure 3.5 45 OA e1*e2* B e1 e1* (l) Square e2 O A B L e2* Hex-Star (m) e1e2 (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 pre- viously 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 Fig- ure 3.5, is used to validate the present method against the estimated effective prop- erty 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 Chris- tensen [19] using Euler beam theory. Topology G ∗ Es E∗ Es ν∗ Isotropic Stretch Dominated Star 316 ρ̄ 3 9 16 ρ̄ 3 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 Young’s ρ Poisson’s (m2) modulus(N/m2) (kg/m3) 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 Fig- ure 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 sym- metry, 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 Bril- louin 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 sys- tem. Note ei · e j = δi j, where δi j represents the Kronecker delta product or the Identity matrix. ρ̄ = ρ ∗ ρ = V ∗ V is the relative density of lattice mate- rial. 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, relative density(ρ̄) is V ∗ Vs instead of ρ∗ρs ; parameter α = √ (3)L21 4 − L22sin(2t) 2 +√ (3)L22 4 + 5L1L2sin(60o−t) 4 +2L2cos(t)(L2sin(60 o− t)− L2sin(t)2 ). The width of square struts is given by parameter b. (a) Topology Direct Reciprocal Relative density(ρ̄) Low-CTE e1 = 2L2cos(t)î e∗1 = 1 2L2cos(t) (î− 1√ 3 ĵ) 3b(L1+L2)α e2 = 2L2cos(t)(12 î+ √ 3 2 ĵ) e ∗ 2 = 1 L2cos(t) ( 1√ 3 ĵ) Hex-chiral e1 = Lî+ L√3 ĵ e ∗ 1 = 1 2L î+ √ 3 2L ĵ 2 √ 3 bL e2 =−Lî+ L√3 ĵ e∗2 =− 1 2L î+ √ 3 2L ĵ Auxetic e1 = 2L(cos(t)+1)î e∗1 = 1 2L(cos(t)+1) î b(2+(2cos(t)+1)) 2Lsin(t)(1+cos(t)) e2 = 2Lsin(t)ĵ e∗2 = 1 2Lsin(t) ĵ HybridA e1 = L(83 +2cos(t))î e ∗ 1 = 3 2L(4+3cos(t)) î (29+3(2cos(t)+1))b (4+3cos(t))4sin(t)L e2 = 2Lsin(t)ĵ e∗2 = 1 2Lsin(t) ĵ HybridC e1 = 2L(cos(t)+1)î e∗1 = 1 2L(1+cos(t)) î (3+cos(t))b (1+cos(t))2sin(t)L e2 = 2Lsin(t)ĵ e∗2 = 1 2Lsin(t) ĵ HybridD e1 = 2Lcos(t)î+2Lĵ e∗1 = 1 4Lcos(t) î+ 1 4L ĵ (1+2sin(t))b 4cos(t)sin(t)L e2 =−2Lcos(t)î+2Lĵ e∗2 =− 14Lcos(t) î+ 14L ĵ Mixed e1 = Lî+Lĵ e∗1 = 1 2L î+ 1 2L ĵ 3.414b L e2 =−Lî+Lĵ e∗2 =− 12L î+ 12L ĵ 48 Table 3.3 (b) Topology Direct Reciprocal Relative density(ρ̄) StarHex e1 = 3Lî+ √ 3Lĵ e∗1 = 1 6L î+ √ 3 6L ĵ 2 √ 3b 3L e2 =−3Lî+ √ 3Lĵ e∗2 =− 16L î+ √ 3 6L ĵ Diamond e1 = Lî e∗1 = 1 L î 5 √ 3b 3L e2 = √ 3Lĵ e∗2 = √ 3 3L ĵ Square e1 = Lî e∗1 = 1 L î 2b L e2 = Lĵ e∗2 = 1 L ĵ Kagome e1 = 2L(12 î+ √ 3 2 )ĵ e ∗ 1 = 1 2L(î+ 1√ 3 )ĵ √ 3b L e2 = 2L(−12 î+ √ 3 2 ĵ) e ∗ 2 = 1 2L(−î+ 1√3 ĵ) Triangular e1 = Lî e∗1 = 1 L(î− 1√3 ĵ) 2 √ 3( bL) e2 = L(12 î+ √ 3 2 ĵ) e ∗ 2 = 1 L( 2√ 3 ĵ) Hexagonal e1 = √ 3L(12 î+ √ 3 2 )ĵ e ∗ 1 = 1√ 3L (î+ 1√ 3 )ĵ 2b√ 3L e2 = √ 3L(−12 î+ √ 3 2 )ĵ e ∗ 2 = 1√ 3L (−î+ 1√ 3 )ĵ 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 Gap Band O A B locus space-k xk yk )(a )(b AO B O space-k s 01.0 Cl Ct Figure 3.6: (a) Dispersion curve for Star lattice for ρ = 8%. The y-axis repre- sents propagating frequencies of free plane waves, non-dimensionalised with respect to first pinned pinned natural frequency of a strut. The x- axis 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 cal- culations in Figure 3.7. A close, but not exact, match is observed between finite element calculations using Bloch analysis and analytical results given by Chris- 50 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 rel- ative densities. 51 Table 3.4: Bandgap classification of lattice materials for ρ̄ = 0.2 Low bandgap Medium bandgap High bandgap topologies (Ω< 5) topologies (5 <Ω< 15) topologies (Ω> 15) Low-CTE Hybrid-C Square Mixed Hex-chiral Hybrid-D Diamond Hybrid-C Kagome Hybrid-A Hexagonal Auxetic Triangular tensen [19]. The analytical model of Christensen predicts a much stiffer response, while the FE model suggests a more compliant lattice at higher relative densi- ties. 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 en- ergy. 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 pro- cedure 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 fre- quencies and is isotropic at lower frequencies. The developed semi-analytical method has been applied to the remaining lat- tice topologies in Figure 3.5. Figure 3.8 displays the dispersion curves of these remaining lattice topologies for the relative density ρ̄ = 0.2. Bandgaps at low fre- quencies are observed for Low-CTE and Mixed lattice. The bandgaps in these 52 02 4 6 8 k−space O A B O Ω (a) Dispersion curve of Low-CTE lattice material B O 25 30 35 40 45 k−space O A B O Ω (b) Dispersion curve of Auxetic lattice ma- terial B O 40 45 50 55 k−space Ω O A B O (c) Dispersion curve of Hybrid-A lattice material B O 8 10 12 k−space O A B O Ω (d) Dispersion curve of Hybrid-C lattice material Figure 3.8: Dispersion curves of the studied lattice materials (see Figure 3.5) for ρ̄ = 0.2. The y-axis represents propagating frequencies of free plane waves, non dimensionalised with respect to the first pinned pinned nat- ural 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 B O 16 18 20 22 k−space O A B O Ω (e) Dispersion curve of Hybrid-D lattice material 2 4 6 k−space O A B O Ω B O (f) Dispersion curve of Mixed lattice ma- terial B O 10 20 30 40 O A B O Ω 0 k−space (g) Dispersion curve of Hex-chiral lattice material B O 4 5 6 7 8 k−space O A B O Ω (h) Dispersion curve of Diamond lattice material Figure 3.8: Dispersion curves of the studied lattice materials (see Figure 3.5) for ρ̄ = 0.2. The y-axis represents propagating frequencies of free plane waves, non dimensionalised with respect to the first pinned pinned nat- ural 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 B O 0 5 10 k−space O A B O Ω (i) Dispersion curve of Kagome lattice material 0 5 10 k−space O B Ω A O (j) Dispersion curve of Hexagonal lattice material B O 5 10 15 20 k−space O A B O Ω (k) Dispersion curve of Triangular lattice material B O 0 50 100 O A B O Ω k−space (l) Dispersion curve of Square lattice ma- terial Figure 3.8: Dispersion curves of the studied lattice materials (see Figure 3.5) for ρ̄ = 0.2. The y-axis represents propagating frequencies of free plane waves, non dimensionalised with respect to the first pinned pinned nat- ural 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 non- dimensionalised 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 identi- fied 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 ρ̄ = 0.2. Table 3.5 summarizes effective material property results as a function of rela- tive 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 topolo- gies are stretch dominated and hence are more suitable from the standpoint of pro- viding high stiffness per unit weight. The remaining topologies are bending domi- nated 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 ρ̄ = 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 3.9- 3.10. 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 topolo- gies. 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 ge- ometries 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 es- timate effective material properties of complex lattice topologies with ease and 56 Table 3.5: Effective properties of different lattice geometries. ρ̄ = V ∗ Vs is rel- ative 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 re- spectively as Low and High CTE materials [14] and Es is the young’s modulus of High CTE material. Topology G ∗ xy Es E∗xx Es E∗yy Es ν∗xy ν∗yx Isotropic Stretch Dominated Low CTE 0.0940ρ̄ 0.2265ρ̄ 0.2265ρ̄ 0.20 0.20 Yes Yes Hex-chiral 0.4283ρ̄3 0.5490ρ̄3 0.5490ρ̄3 −0.42 −0.42 Yes No Auxetic 0.0436ρ̄3 0.4018ρ̄3 2.6987ρ̄3 −0.33 −2.61 No No HybridA 0.0246ρ̄3 0.0988ρ̄ 0.5591ρ̄3 0 0 No No HybridC 0.0373ρ̄3 0.4285ρ̄ 0.8122ρ̄3 0 0 No No HybridD 0.0420ρ̄3 0.4137ρ̄3 0.4972ρ̄ 0 0 No No Mixed 0.1047ρ̄ 0.3693ρ̄ 0.3693ρ̄ 0.26 0.26 No Yes Star 0.1706ρ̄3 0.4977ρ̄3 0.4977ρ̄3 0.48 0.48 Yes No Diamond 0.1510ρ̄ 0.2000ρ̄ 0.3600ρ̄ 0.33 0.59 No Yes Square 0.0602ρ̄3 0.5000ρ̄ 0.5000ρ̄ 0 0 No No Kagome 0.125ρ̄ 0.3333ρ̄ 0.3333ρ̄ 0.33 0.33 Yes Yes Triangle 0.125ρ̄ 0.3333ρ̄ 0.3333ρ̄ 0.33 0.33 Yes Yes Hexagonal 0.3261ρ̄3 1.2259ρ̄3 1.2259ρ̄3 0.96 0.96 Yes No 57 0.08 0.06 0.04 0.02 0 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 E∗yy Es E∗xx 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 ρ̄ = 0.2 for all lattice topologies. The boxed topologies are stretch dominated. A circular plot indicates isotropy. 58 0.03 0.02 0.01 0 0.01 0.02 0.03 0.03 0.02 0.01 0 0.01 0.02 0.03 G∗xx Es G∗yy 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 stud- ied topologies (see Figure 3.5). Each contour corresponds to a par- ticular 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 ρ̄ = 0.2 for all lattice topologies. The boxed topologies are stretch dominated. A circular plot indicates isotropy. 59 0.004 0.002 0 0.002 0.004 0.004 0.003 0.002 0.001 0 0.001 0.002 0.003 0.004 0.005 G∗xx Es G∗yy 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 as- sumed a lattice of infinite dimension without any boundaries. Lattice materials employed in applications possess edges due to their finite dimensions and there- fore 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 in- terior 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 ac- commodate 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 ef- fects 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 defor- mations 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 fre- quency; 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 co- 62 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 for- mulation to investigate the elastic boundary layer is presented first. Boundary con- ditions of pure shear and the Uni-axial Finite Element (FE) simulation are shown next. These FE simulations validate the results of the semi-analytical formula- tion. 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: Mq̈+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 re- spectively. 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) = qe jωt and substituting it in 63 ∞∞ Ed ge e1, x e2, y (a) (b) qb qi ql qr qt y xOO ∞ 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−ω2Mq = 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 de- grees of freedom in Equation (4.2) can be partitioned to the form : 64 Dll Dlr Dlb Dlt Dli Drl Drr Drb Drt Dri Dbl Dbr Dbb Dbt Dbi Dtl Dtr Dtb Dtt Dti Dil Dir Dib Dit 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 re- lations are obtained: fr + e−ιk2fl = 0 (4.4) fi = 0 (4.5) qr = e−ιk2ql, (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 Drlql + e−ιk2Drrql +Drbqb+Drtqt +Driqi+ e−ιk2 [Dllql + e−ιk2Dlrql + ... Dlbqb+Dltqt +Dliqi] = 0, (4.7) where qi =−D−1ii [Dilql +Dire−ιk2ql +Dibqb+Ditqt ] (4.8) 65 The above equations can be better represented in the matrix form as [ Cll Clb Clt ] qlqb qt = 0, where Cll = Drl + e−ιk2Drr + e−ιk2Dll + e−ι2k2Dlr− e−ιk2DliD−1ii [Dil + e−ιk2Dir]... −DriD−1ii [Dil + e−ιk2Dir] Clb = Drb+ e−ιk2Dlb− e−ιk2DliD−1ii Dib−DriD−1ii Dib Clt = Drt + e−ιk2Dlt − e−ιk2DliD−1ii Dit −DriD−1ii Dit (4.9) The equilibrium equations along the +x direction can be written as: fb = Dblql +Dbrqr +Dbbqb+Dbtqt +Dbiqi ft = Dtlql +Dtrqr +Dtbqb+Dttqt +Dtiqi (4.10) From the Equation (4.9), ql can be written as: ql =−C−1ll [Clbqb+Cltqt ] (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 Ptb Ptt ][ qb qt ] , (4.12) 66 where Pbb = Dbb−DbiD−1ii Dib− [Dbl + e−ιk2Dbr−DbiD−1ii [Dil + e−ιk2Dir]]C−1ll Clb Pbt = Dbt −DbiD−1ii Dit − [Dbl + e−ιk2Dbr−DbiD−1ii [Dil + e−ιk2Dir]]C−1ll Clt Ptb = Dtt −DtiD−1ii Dib− [Dtl + e−ιk2Dtr−DtiD−1ii [Dil + e−ιk2Dir]]C−1ll Clb Ptt = Dtt −DtiD−1ii Dit − [Dtl + e−ιk2Dtr−DtiD−1ii [Dil + e−ιk2Dir]]C−1ll Clt (4.13) Equation (4.12) can be re-written in the form:[ qt ft ] = [ −P−1bt Pbb −P−1bt Ptb−PttP−1bt Pbb −PttP−1bt ][ qb −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 −fb ] (4.15) Inserting Equation (4.15) in Equation (4.14), yields the eigenvalue problem [T−λ I]s = 0, (4.16) where λ = e−ιk1 , s = [ qb −fb ] and T = [ −P−1bt Pbb −P−1bt Ptb−PttP−1bt Pbb −PttP−1bt ] . To avoid the problems of ill conditioning, a general eigenvalue problem of the form As = λBs can be formulated by modifying Equation (4.16), where A =[ Pbb I Ptb 0 ] , B = [ −Pbt 0 Ptt I ] , λ = e−ιk1 and s = [ qb −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 = λqb (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 eigen- values (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 Equa- tion (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 ma- terial 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 solu- tion. 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 macro- scopic load. • Complementary solution: For a given, macroscopic loading condition, the total resulting stress solution is the sum of Particular solution and Comple- mentary solution. Since, particular solution is uniform and the total solution varies near the edges, hence Complementary solution is essentially the tran- sient 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 in- terior and is maximum at the edges. Given a macroscopic loading condition, the complementary solution arising out of it will be the summation of elas- tic boundary layer eigenvectors in certain proportions. These proportions or contributions of the eigenvectors can be deduced from the boundary con- ditions 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 ZY X Le ft E dg e Ri gh t E dg e Bottom Edge Top Edge Le ft E dg e Ri gh t E dg e Bottom Edge Top Edge Z Y X (a) (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 demon- strating 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 vec- tor 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 en- tirely 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 per- taining to a continuum in a lattice medium, following displacement boundary con- ditions 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 remain- 70 Table 4.1: Boundary conditions assumed to simulate pure shear test. Node Location Displacement in Displacement in Rotation +x direction (ux) +y direction (uy) (θz) Left edge nodes free −1 unit free Right edge nodes free 0 (Constrained) free Bottom edge nodes 0 (Constrained) free free Top edge nodes +1 unit 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 configu- ration effectively for bearing structural loads. A total of 53 configurations resulted from the previously studied topologies. Out of these 53 configurations, 5 configu- rations showed deep elastic boundary layer. Figures 4.3-4.4 identifies the deep elastic boundary layer configurations. In 71 e1, x e2, y ∞ ∞ (a) (b) e1, x e2, y ∞ ∞ Ed ge (c) (d) Ed ge e1, x e2, y ∞ ∞ Ed ge ∞ ∞ Ed ge e1, x e2, y ∞∞ ∞∞ Figure 4.3: Unit cells are tessellated along the basis vectors to constitute fi- nite Kagome configuration 1, Kagome configuration 2, Diamond con- figuration 1 and Hybrid-A configuration 1 shown in (a), (b), (c) and (d) respectively. e1 and e2 constitute the basis vectors. 72 ∞∞ Ed ge e1, x e2, y ∞ Figure 4.4: Unit cells are tessellated along the basis vectors to constitute fi- nite Hybrid-A configuration 2. e1 and e2 constitute the basis vectors. Table 4.2: Deep elastic boundary layer configurations and their parent topolo- gies Configurations Parent topology Clockwise rotation of parent topology Configuration 1 Kagome 0o Configuration 2 Kagome 0o Configuration 3 Diamond 30o Configuration 4 Hybrid-A 90o Configuration 5 Hybrid-A 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 ρ̄ = 0.05 to visually 73 Conguration 4 + + | | = e1, x e2, y qb qi ql qr qt 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) (f )(e) Figure 4.6: All eigenvectors corresponding to elastic boundary layer modes for Kagome configuration 1 are shown in (a) eigenvector 1, (b) eigen- vector 2 , (c) eigenvector 3 and (d) eigenvector 4 for ρ̄ = 0.05. Eigen- vectors corresponding to deep elastic boundary layer modes for Kagome configuration 2 are shown in (e) eigenvector 1 and (f) eigenvector 2 for ρ̄ = 0.05. 4.6.1 Kagome Configurations 1 and 2 Kagome configurations 1 and 2 are derived from the Kagome topology. The pro- cedure 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 mo- tion 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 sin- 75 (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 ρ̄ = 0.05. (b) and (d) show deformations resulting from pure shear test for Kagome configurations 1 and 2 respectively for ρ̄ = 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 ρ̄ = 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 (b) 0 0.1 0.2 0 0.2 0.4 0.6 0.8 1 1.2 ρ̄ |λ | Mode I Mode II (a) 0 0.1 0.2 0 0.2 0.4 0.6 0.8 1 1.2 ρ̄ |λ | Mode I Mode II Figure 4.8: Modulus of eigenvalues that correspond to elastic boundary layer modes are plotted as a function of relative density for Kagome configu- ration 1 in (a) and Kagome configuration 2 in (b). Both the configura- tions exhibit the same eigenvalue or boundary layer characteristics. 78 (b) 0 0.05 0.1 0.15 0.2 0.25 0 0.2 0.4 0.6 0.8 1 1.2 ρ̄ |λ| Mode I Figure 4.9: Modulus of eigenvalues that correspond to elastic boundary layer modes are plotted as a function of relative density for Diamond config- uration 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 simu- lations for ρ̄ = 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 (b)(a) 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 ρ̄ = 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 ρ̄ = 0.05, Modes I and II corresponding to Hybrid-A configuration 1 exhibit bending and stretch dominated behaviour respectively. For the Hybrid-A configu- ration 2, Mode I and Mode II are bending dominated whereas Mode III is stretch dominated for ρ̄ = 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 ob- served 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 config- uration 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 con- figuration 1 and Mode II of Hybrid-A configuration 2 transitions from a bending dominated mode at ρ̄ = 0.05 to a stretch dominated mode at a ρ̄ = 0.2. Similarly 80 0 0.1 0.2 0 0.2 0.4 0.6 0.8 1 1.2 ρ̄ |λ | 0 0.1 0.2 0 0.2 0.4 0.6 0.8 1 1.2 ρ̄ |λ | (a) (b) 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 ρ̄ = 0.1. Note the phenomenon of veering. 81 0 0.1 0.2 0 0.2 0.4 0.6 0.8 1 1.2 ρ̄ |λ | Mode I Mode II 0 0.1 0.2 0 0.2 0.4 0.6 0.8 1 1.2 ρ̄ |λ | (a) (b) Mode III Mode II Figure 4.12: Modulus of eigenvalues that correspond to elastic boundary layer modes prone to veering are plotted as a function of relative den- sity 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 phenom- ena of veering. Note that the curves do not intersect at ρ̄ = 0.1. 82 (b)(a) (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 ρ̄ = 0.05. (b) and (d) show deformations resulting from pure shear test for Hybrid-A configuration 1 and 2 respectively for ρ̄ = 0.05. Bound- ary 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 ρ̄ = 0.05 to a bending dominated mode at a ρ̄ = 0.2. Hence, the closely spaced bending and stretching dominated modes are interchanged between each other. Veering is a hallmark phenomenon of linear sys- tems which exhibit closely spaced natural frequencies. 83 Table 4.3: Comparison of the identified configurations on the basis of deep elastic boundary layer Topology Percentage decay of deepest eigen mode across a unit cell ρ̄ = 0.05 ρ̄ = 0.1 ρ̄ = 0.15 ρ̄ = 0.2 Kagome configuration 1 8% 5% 22% 28% Kagome configuration 2 8% 5% 22% 28% Diamond configuration 1 22% 39% 53% 63% Hybrid-A configuration 1 18% 32% 32% 32% Hybrid-A configuration 2 14% 24% 29% 31% 4.7 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 pos- sessing 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 ap- proached 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 config- urations 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 formula- tion 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 the- orem. 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 prop- erties, the method reveals wave propagation information in the form of dis- persion 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 (ρ̄ < 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 devel- oped to predict the elastic boundary layers emanating from the edge of a fi- nite 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 cur- rent 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 deduc- tion of material properties for the fully anisotropic case will require solv- ing 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. F., Evans, A., Fleck, N. A., Gibson, L. J., Hutchinson, J. W., and Wadley, H. N. G., 2000. Metal foams: a design guide. Elsevier. [3] Gibson, L. J., and Ashby, M. F., 1999. Cellular solids: structure and properties. Cambridge Univ Pr. [4] Fleck, N. A., Deshpande, V. S., and Ashby, M. F., 2010. “Micro-architectured materials: past, present and future”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 466(2121), pp. 2495–2516. [5] Maxwell, J. C., 1864. “On the calculation of the equilibrium and stiffness of frames”. Philosophical Magazine, 27(182), pp. 294–299. [6] Pellegrino, S., and Calladine, C. R., 1986. “Matrix analysis of statically and kinematically indeterminate frameworks”. International Journal of Solids and Structures, 22(4), pp. 409 – 428. [7] Guest, S. D., and Hutchinson, J. W., 2003. “On the determinacy of repetitive structures”. Journal of the Mechanics and Physics of Solids, 51(3), pp. 383 – 391. [8] Deshpande, V. S., Ashby, M. F., and Fleck, N. A., 2001. “Foam topology: bending versus stretching dominated architectures”. Acta Materialia, 49(6), pp. 1035 – 1040. [9] Moongkhamklang, P., Elzey, D. M., and Wadley, H. N. G., 2008. “Titanium matrix composite lattice structures”. Composites Part A: Applied Science and Manufacturing, 39(2), pp. 176 – 187. 89 [10] Evans, A. G., Hutchinson, J. W., Fleck, N. A., Ashby, M. F., and Wadley, H. N. G., 2001. “The topological design of multifunctional cellular metals”. Progress in Materials Science, 46(3-4), pp. 309 – 327. [11] Deshpande, V. S., Fleck, N. A., and Ashby, M. F., 2001. “Effective properties of the octet-truss lattice material”. Journal of the Mechanics and Physics of Solids, 49(8), pp. 1747 – 1769. [12] Wadley, H. N. G., 2006. “Multifunctional periodic cellular metals”. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 364(1838), pp. 31–68. [13] Abramovitch, H., Burgard, M., Edery-Azulay, L., Evans, K. E., Hoffmeister, M., Miller, W., Scarpa, F., Smith, C. W., and Tee, K. F., 2010. “Smart tetrachiral and hexachiral honeycomb: Sensing and impact detection”. Composites Science and Technology, 70(7), pp. 1072 – 1079. Special issue on Chiral Smart Honeycombs. [14] Steeves, C. A., dos Santos e Lucato, S. L., He, M., Antinucci, E., Hutchinson, J. W., and Evans, A. G., 2007. “Concepts for structurally robust materials that combine low thermal expansion with high stiffness”. Journal of the Mechanics and Physics of Solids, 55(9), pp. 1803 – 1822. [15] Torrent, D., and Sánchez-Dehesa, J., 2008. “Acoustic cloaking in two dimensions: a feasible approach”. New Journal of Physics, 10, p. 063015. [16] Deshpande, V. S., Ashby, M. F., and Fleck, N. A., 2001. “Foam topology: bending versus stretching dominated architectures”. Acta Materialia, 49(6), pp. 1035 – 1040. [17] Hutchinson, R. G., and Fleck, N. A., 2006. “The structural performance of the periodic truss”. Journal of the Mechanics and Physics of Solids, 54(4), pp. 756 – 782. [18] Elsayed, M. S. A., and Pasini, D., 2010. “Analysis of the elastostatic specific stiffness of 2d stretching-dominated lattice materials”. Mechanics of Materials, 42(7), pp. 709 – 725. [19] Christensen, R. M., 2000. “Mechanics of cellular and other low-density materials”. International Journal of Solids and Structures, 37(1-2), pp. 93 – 104. 90 [20] Scarpa, F., Panayiotou, P., and Tomlinson, G., 2000. “Numerical and experimental uniaxial loading on in-plane auxetic honeycombs”. The Journal of Strain Analysis for Engineering Design, 35(5), pp. 383–388. [21] Gibson, L. J., and Ashby, M. F., 1982. “The mechanics of three-dimensional cellular materials”. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 382(1782), pp. pp. 43–59. [22] Gibson, L. J., Ashby, M. F., Schajer, G. S., and Robertson, C. I., 1982. “The mechanics of two-dimensional cellular materials”. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 382(1782), pp. pp. 25–42. [23] Warren, W. E., and Kraynik, A. M., 1987. “Foam mechanics: the linear elastic response of two-dimensional spatially periodic cellular materials”. Mechanics of Materials, 6(1), pp. 27 – 37. [24] Masters, I. G., and Evans, K. E., 1996. “Models for the elastic deformation of honeycombs”. Composite Structures, 35(4), pp. 403 – 422. [25] Wang, A. J., and McDowell, D. L., 2004. “In-plane stiffness and yield strength of periodic metal honeycombs”. Journal of Engineering Materials and Technology, 126(2), pp. 137–156. [26] Balawi, S., and Abot, J. L., 2008. “A refined model for the effective in-plane elastic moduli of hexagonal honeycombs”. Composite Structures, 84(2), pp. 147 – 158. [27] Balawi, S., and Abot, J. L., 2008. “The effect of honeycomb relative density on its effective in-plane elastic moduli: An experimental study”. Composite Structures, 84(4), pp. 293 – 299. [28] Chen, D. H., 2010. “Bending deformation of honeycomb consisting of regular hexagonal cells”. Composite Structures, In Press, Corrected Proof, pp. –. [29] Chen, D. H., and Yang, L., 2010. “Analysis of equivalent elastic modulus of asymmetrical honeycomb”. Composite Structures, In Press, Corrected Proof, pp. –. [30] Chen, J. Y., Huang, Y., and Ortiz, M., 1998. “Fracture analysis of cellular materials: A strain gradient model”. Journal of the Mechanics and Physics of Solids, 46(5), pp. 789 – 828. 91 [31] Fleck, N. A., and Qiu, X., 2007. “The damage tolerance of elastic-brittle, two-dimensional isotropic lattices”. Journal of the Mechanics and Physics of Solids, 55(3), pp. 562 – 588. Cited by 23. [32] Kumar, R. S., and McDowell, D. L., 2004. “Generalized continuum modeling of 2-d periodic cellular solids”. International Journal of Solids and Structures, 41(26), pp. 7399 – 7422. [33] Bazant, Z. P., and Christensen, M., 1972. “Analogy between micropolar continuum and grid frameworks under initial stress”. International Journal of Solids and Structures, 8(3), pp. 327 – 346. [34] Gonella, S., and Ruzzene, M., 2008. “Homogenization and equivalent in-plane properties of two-dimensional periodic lattices”. International Journal of Solids and Structures, 45(10), pp. 2897–2915. [35] Lombardo, M., and Askes, H., 2011. “Higher-order gradient continuum modelling of periodic lattice materials”. Computational Materials Science. [36] Wallach, J. C., and Gibson, L. J., 2001. “Mechanical behavior of a three-dimensional truss material”. International Journal of Solids and Structures, 38(40-41), pp. 7181 – 7196. [37] Phani, A. S., Woodhouse, J., and Fleck, N. A., 2006. “Wave propagation in two-dimensional periodic lattices”. The Journal of the Acoustical Society of America, 119(4), pp. 1995–2005. [38] Kueh, A., Soykasap, O., and Pellegrino, S., 2005. “Thermo-mechanical behaviour of single-ply triaxial weave carbon fibre reinforced plastic”. In Spacecraft Structures, Materials and Mechanical Testing 2005, Vol. 581, p. 105. [39] Phani, A. S., and Fleck, N. A., 2008. “Elastic boundary layers in two-dimensional isotropic lattices”. Journal of Applied Mechanics, 75, p. 021020. [40] Lakes, R., 1987. “Foam structures with a negative Poisson’s ratio”. Science, 235, pp. 1038–1040. [41] Bloch, F., 1929. “ber die quantenmechanik der elektronen in kristallgittern”. Zeitschrift fr Physik, 52(7), pp. 555–600. [42] Kittel, C., and McEuen, P., 1976. Introduction to solid state physics, Vol. 7. Wiley New York. 92 [43] Brillouin, L., 1953. Wave propagation in periodic structures. Dover Pubns. [44] Langley, R. S., Bardell, N. S., and Ruivo, H. M., 1997. “The response of two-dimensional periodic structures to harmonic point loading: a theoretical and experimental study of a beam grillage”. Journal of Sound and Vibration, 207(4), pp. 521–535. [45] Mead, D. J., 1996. “Wave propagation in continuous periodic structures: research contributions from Southampton, 1964-1995”. Journal of Sound and Vibration, 190(3), pp. 495–524. [46] Musgrave, M. J. P., 1970. Crystal acoustics. Holden-Day San Francisco. [47] Torquato, S., Gibiansky, L. V., Silva, M. J., and Gibson, L. J., 1998. “Effective mechanical and transport properties of cellular solids”. International Journal of Mechanical Sciences, 40(1), pp. 71 – 82. 93
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Effective mechanical properties of lattice materials
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Effective mechanical properties of lattice materials Chopra, Prateek 2011
pdf
Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.
Page Metadata
Item Metadata
Title | Effective mechanical properties of lattice materials |
Creator |
Chopra, Prateek |
Publisher | University of British Columbia |
Date Issued | 2011 |
Description | 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. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-12-01 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0072404 |
URI | http://hdl.handle.net/2429/39436 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2012-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
Download
- Media
- 24-ubc_2012_spring_chopra_prateek.pdf [ 9.42MB ]
- Metadata
- JSON: 24-1.0072404.json
- JSON-LD: 24-1.0072404-ld.json
- RDF/XML (Pretty): 24-1.0072404-rdf.xml
- RDF/JSON: 24-1.0072404-rdf.json
- Turtle: 24-1.0072404-turtle.txt
- N-Triples: 24-1.0072404-rdf-ntriples.txt
- Original Record: 24-1.0072404-source.json
- Full Text
- 24-1.0072404-fulltext.txt
- Citation
- 24-1.0072404.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
data-media="{[{embed.selectedMedia}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.24.1-0072404/manifest